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[
"AN UPPER BOUND FOR THE REGULARITY OF BINOMIAL EDGE IDEALS OF TREES",
"AN UPPER BOUND FOR THE REGULARITY OF BINOMIAL EDGE IDEALS OF TREES"
]
| [
"A V Jayanthan ",
"N Narayanan ",
"B V Raghavendra Rao "
]
| []
| []
| In this article we obtain an improved upper bound for the regularity of binomial edge ideals of trees. | 10.1142/s0219498819501706 | [
"https://arxiv.org/pdf/1808.06374v1.pdf"
]
| 119,165,912 | 1808.06374 | 5d9a0589f9b72fb77c7d4b01b1df4244d34700f1 |
AN UPPER BOUND FOR THE REGULARITY OF BINOMIAL EDGE IDEALS OF TREES
20 Aug 2018
A V Jayanthan
N Narayanan
B V Raghavendra Rao
AN UPPER BOUND FOR THE REGULARITY OF BINOMIAL EDGE IDEALS OF TREES
20 Aug 2018arXiv:1808.06374v1 [math.AC]
In this article we obtain an improved upper bound for the regularity of binomial edge ideals of trees.
obtain an improved upper bound for reg(S/J T ). The upper bound obtained is better than the presently known bound, n − 1, for most of the trees.
Upper bound for regularity of trees
Let G be a block graph. If two distinct blocks in G share a vertex, then it is a cut vertex. A block is said to be an end-block if it contains at most one cut vertex. We define the block degree bd(v) of a cut vertex to be the number of blocks incident to v. A spine of a block graph G is defined to be a maximum length path P in G where every edge of P is a block in G. Note that it is possible that the spine is a single vertex. For v ∈ V (G), let lbd(v) denote the number of large blocks, i.e., blocks of size at least three, incident at v.
One of the terminology that we need is that of gluing of two graphs at a vertex. Let G be a graph. For a subset W of V (G), let G[W ] denote the induced subgraph of G on the vertex set W . For a cut vertex v in G, let G 1 , . . . , G k denote the components of G \ {v}. Let
G ′ i = G[V (G i ) ∪ {v}].
Then we say that G is obtained by gluing G 1 , . . . , G k at v, [12]. A vertex v in a graph G is said to be a free vertex if it is part of exactly one maximal clique. Let G be a block graph and v be a vertex which is not a free vertex of a graph G. Let G ′ denote the graph obtained by adding edges between all the vertices of N G (v), G ′′ denote the graph G \ {v} and H denote the graph G ′ \ {v}. Then there is an exact sequence, [3,1]:
0 −→ S J G −→ S J G ′ ⊕ S J G ′′ −→ S J H −→ 0(1)
If G is obtained by identifying a vertex each of k cliques of size at least three, then by [6], reg(S/J G ) = k. Now, we consider block graphs having non-trivial spine.
Theorem 1. Let G be a connected block graph in which every block of size at least three is an end-block. Let P be a spine of G of length ℓ(G) ≥ 1,
e 2 (G) = |{{a, b} ∈ E(G) \ E(P ) | bd(a) ≤ 2 and bd(b) ≤ 2}|, C G = {v ∈ V (G) \ V (P ) | bd G (v) ≥ 3}
and b(G) be the number of large end-blocks that intersect the spine P . Then,
reg(S/J G ) ≤ e 2 (G) + ℓ(G) + b(G) + v∈C G max{lbd(v), 2}.
Proof. If there is no cut vertex in G, then G is an edge and hence the assertion holds, since e 2 (G) = 0, b(G) = 0 and C G = ∅.
Assume that G has at least one cut vertex. Let d(x, P ) denote the distance of the vertex x from the spine P and d(G) =
x is a cut vertex in G d(x, P ). We apply induction on d(G). If d(G) = 0, then G is a graph with a spine P and some cliques attached to P . Therefore, the assertion follows from [6,Theorem 4.5].
Let d(G) > 0. Let v be a cut vertex in G such that d(v, P ) is maximum.
Case I: If bd(v) = 2, then there exists a graph G 1 containing v as a free vertex and a clique C such that G is obtained by gluing G 1 and C at v. Then e 2 (G 1 ) = e 2 (G) − 1, ℓ(G) = ℓ(G 1 ), C G = C G 1 and b(G) = b(G 1 ). Moreover, d(G 1 ) < d(G). By induction,
reg(S/J G 1 ) ≤ e 2 (G 1 ) + ℓ(G 1 ) + b(G 1 ) + v∈C G 1 max{lbd(v), 2}.
By [6, Theorem 3.1], reg(S/J G ) = reg(S/J G 1 ) + 1. Hence the assertion follows.
Case II: Assume that bd(v) ≥ 3. Then v ∈ C G . Since v is not a free vertex, it follows from the exact sequence (1) that
reg S J G ≤ max reg S J G ′ ⊕ S J G ′′ , reg S J H + 1 .
Since H is an induced subgraph of G ′ , reg(S/J H ) ≤ reg(S/J G ′ ). Therefore, we get
reg S J G ≤ max reg S J G ′′ , reg S J G ′ + 1 .
We show that both the entries on the right hand side of the above inequality satisfies the bound given in the assertion. Note that v is not a cut vertex in G ′ and if v = y ∈ V (G) is a cut vertex of G ′ , then it is a cut vertex of G as well. Therefore, d(
G ′ ) = d(G) − d(v, P ) < d(G). We also have C G ′ = C G \ {v}.
It can be seen that e 2 (G ′ ) ≤ e 2 (G) and ℓ(G) = ℓ(G ′ ). By induction hypothesis,
reg(S/J G ′ ) ≤ e 2 (G ′ ) + ℓ(G ′ ) + b(G ′ ) + x∈C G ′ [max{lbd(x), 2}].
If d(v, P ) = 1, then b(G ′ ) = b(G) + 1 and for every u ∈ C G ′ , lbd G ′ (u) = lbd G (u). Therefore,
x∈C G [max{lbd G (x), 2}] = x∈C G ′ [max{lbd G ′ (x), 2}] + max{lbd G (v), 2}. Hence reg(S/J G ′ ) ≤ e 2 (G) + ℓ(G) + b(G) + 1 + x∈C G ′ [max{lbd G (x), 2}] ≤ e 2 (G) + ℓ(G) + b(G) + x∈C G [max{lbd G (x), 2}] − 1.
If d(v, P ) > 1, then b(G ′ ) = b(G). Further, there is a vertex u v ∈ C G ′ which is the unique cut vertex neighbor of v. Morever, we have lbd G ′ (u v ) = lbd G (u v ) + 1 and for every u ∈ C G ′ \ {u v }, lbd G ′ (u) = lbd G (u). Therefore,
x∈C G ′ [max{lbd G (x), 2}] = x∈C G ′ \{uv} [max{lbd G ′ (x), 2}] + max{lbd G ′ (u v ), 2} = x∈C G ′ \{uv} [max{lbd G (x), 2}] + max{lbd G (u v ) + 1, 2} ≤ x∈C G ′ \{uv} [max{lbd G (x), 2}] + max{lbd G (u v ), 2} + 1 ≤ x∈C G ′ \{uv} [max{lbd G (x), 2}] + max{lbd G (u v ), 2} + max{lbd G (v), 2} − 1 = x∈C G [max{lbd G (x), 2}] − 1. Therefore, reg(S/J G ′ ) ≤ e 2 (G) + ℓ(G) + b(G) + x∈C G [max{lbd G (x), 2}] − 1.
Now we consider the graph G ′′ = G \ {v}. Let lbd G (v) = r. Then G ′′ is the disjoint union of G 1 which is the connected component of G ′′ containing P and C 1 , . . . , C r maximal cliques on at least 2 vertices and possibly some isolated vertices. Hence reg(S/J G ′′ ) = reg(S/J G 1 )+r.
For all x ∈ V (G 1 ), lbd G 1 (x) = lbd G (x) and d(G 1 ) = d(G) − d(v, P ) < d(G). Therefore, by induction hypothesis reg(S/J G 1 ) ≤ e 2 (G ′′ ) + ℓ(G 1 ) + b(G 1 ) + x∈C G 1 [max{lbd G 1 (x), 2}].
Now, there are two possibilities, namely e 2 (G ′′ ) = e 2 (G) + 1 or e 2 (G ′′ ) = e 2 (G).
If e 2 (G ′′ ) = e 2 (G) + 1, then the unique cut vertex neighbor u v of v has block degree 2 in G 1 . Therefore
C G 1 = C G \ {v, u v } so that reg(S/J G 1 ) ≤ e 2 (G) + 1 + ℓ(G 1 ) + b(G 1 ) + x∈C G 1 max{lbd G 1 (x), 2} ≤ e 2 (G) + ℓ(G 1 ) + b(G 1 ) + x∈C G 1 [max{lbd G 1 (x), 2}] + max{lbd G 1 (u v ), 2} ≤ e 2 (G) + ℓ(G 1 ) + b(G 1 ) + x∈C G\{v} [max{lbd G (x), 2}].
Note also that r = lbd G (v). Hence reg(S/J G ′′ ) = reg(S/J G 1 ) + r ≤ e 2 (G) + ℓ(G) + b(G) + x∈C G max{lbd G (x), 2}.
For the case when e 2 (G ′′ ) = e 2 (G), we have C G 1 = C G\{v} . Now as argued in the previous case, one can conclude that reg(S/J G ′′ ) ≤ e 2 (G) + ℓ(G) + b(G) + x∈C G max{lbd G (x), 2}.
Acknowledgement: We thank the Science and Engineering Research Board (SERB) of Government of India for partially funding this work through the Extra Mural Project Grant No. EMR/2016/001883. We also thank the National Board for Higher Mathematics for partially funding this work through the project No. 02011/23/2017/R&D II/4501. We also thank the anonymous reviewer for the valuable comments.
Following the notation of Theorem 1, b(T ) = 0 and lbd T (x) = 0 for each x ∈ V (T ) so that max{lbd T (x), 2} = 2. Now the assertion follows directly from Theorem 1. Proof, Proof. Following the notation of Theorem 1, b(T ) = 0 and lbd T (x) = 0 for each x ∈ V (T ) so that max{lbd T (x), 2} = 2. Now the assertion follows directly from Theorem 1.
We have computed the regularity of this graph using Macaulay 2 and have found that the graph attains the regularity upper bound. We also note that the upper bound we obtained in Theorem 1 coincides with the lower bound for the regularity of Flower graph F h,k (v). 9Following the notation in Theorem 1, we can see that e 2 (G) = 1, ℓ(G) = 4, b(G) = 0 and |C G | = 2. Therefore, we get reg. proved in Corollary 3.5 of [9Let G be the graph given on the right side. Following the notation in Theorem 1, we can see that e 2 (G) = 1, ℓ(G) = 4, b(G) = 0 and |C G | = 2. Therefore, we get reg(S/J G ) ≤ 9. We have computed the regularity of this graph using Macaulay 2 and have found that the graph attains the regu- larity upper bound. We also note that the upper bound we obtained in Theorem 1 coincides with the lower bound for the regularity of Flower graph F h,k (v) proved in Corollary 3.5 of [9].
Following the notation in Theorem 1, we get e 2 (G) = 0 and b(G) = h. If k ≤ 2, then C(G) = ∅ and if k > 2, then C(G) consists of all the certer vertices of k − 2 copies of K 1,3 outside a fixed spine. Therefore, it follows from Theorem 1 that reg(S/J G ) ≤ 2k + h. Following the notation in the article. Proof, Let G = F H,K (v), 9)) = k + 1 and cdeg(v) = h + k. This proves the assertion. It may also be noted that the upper bound is not attained by all block graphs. For example, in the case of the graph considered in [9, Example 3Proof. Let G = F h,k (v). Following the notation in Theorem 1, we get e 2 (G) = 0 and b(G) = h. If k ≤ 2, then C(G) = ∅ and if k > 2, then C(G) consists of all the certer vertices of k − 2 copies of K 1,3 outside a fixed spine. Therefore, it follows from Theorem 1 that reg(S/J G ) ≤ 2k + h. Following the notation in the article [9], it can be seen that i(F (v)) = k + 1 and cdeg(v) = h + k. This proves the assertion. It may also be noted that the upper bound is not attained by all block graphs. For example, in the case of the graph considered in [9, Example 3
Binomial edge ideals of bipartite graphs. D Bolognini, A Macchia, F Strazzanti, European Journal of Combinatorics. 70D. Bolognini, A. Macchia, and F. Strazzanti. Binomial edge ideals of bipartite graphs. European Journal of Combinatorics, 70:1 -25, 2018.
On the binomial edge ideals of block graphs. An. Ştiinţ. Univ. F Chaudhry, A Dokuyucu, R Irfan, Ovidius" Constanţa Ser. Mat. 242F. Chaudhry, A. Dokuyucu, and R. Irfan. On the binomial edge ideals of block graphs. An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat., 24(2):149-158, 2016.
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Regularity of Binomial Edge Ideals of Certain Block Graphs. A V Jayanthan, N Narayanan, B V Rao, prints:1601.01086A. V. Jayanthan, N. Narayanan, and B. V. Raghavendra Rao. Regularity of Binomial Edge Ideals of Certain Block Graphs. ArXiv e-prints:1601.01086, Jan. 2016.
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The Castelnuovo-Mumford regularity of binomial edge ideals. D Kiani, S , Saeedi Madani, J. Combin. Theory Ser. A. 139D. Kiani and S. Saeedi Madani. The Castelnuovo-Mumford regularity of binomial edge ideals. J. Combin. Theory Ser. A, 139:80-86, 2016.
Krull dimension and regularity of binomial edge ideals of block graphs. C Mascia, G Rinaldo, ArXiv e-printsC. Mascia and G. Rinaldo. Krull dimension and regularity of binomial edge ideals of block graphs. ArXiv e-prints, Mar. 2018.
Regularity bounds for binomial edge ideals. K Matsuda, S Murai, J. Commut. Algebra. 51K. Matsuda and S. Murai. Regularity bounds for binomial edge ideals. J. Commut. Algebra, 5(1):141- 149, 2013.
Graphs and ideals generated by some 2-minors. M Ohtani, Comm. Algebra. 393M. Ohtani. Graphs and ideals generated by some 2-minors. Comm. Algebra, 39(3):905-917, 2011.
Construction of Cohen-Macaulay binomial edge ideals. A Rauf, G Rinaldo, Comm. Algebra. 421A. Rauf and G. Rinaldo. Construction of Cohen-Macaulay binomial edge ideals. Comm. Algebra, 42(1):238-252, 2014.
Binomial edge ideals of graphs. S , Saeedi Madani, D Kiani, Electron. J. Combin. 192Department of Mathematics, Indian Institute of Technology MadrasE-mail address: [email protected]. Saeedi Madani and D. Kiani. Binomial edge ideals of graphs. Electron. J. Combin., 19(2):Paper 44, 6, 2012. Department of Mathematics, Indian Institute of Technology Madras, Chennai, INDIA - 600036. E-mail address: [email protected]
| []
|
[
"An IoT-Based Framework for Remote Fall Monitoring",
"An IoT-Based Framework for Remote Fall Monitoring"
]
| [
"Ayman Al-Kababji \nDepartment of Electrical Engineering\nQatar University\nDohaQatar\n",
"Abbes Amira \nInstitute of Artificial Intelligence\nDe Montfort University\nLeicesterUnited Kingdom\n",
"Faycal Bensaali \nDepartment of Electrical Engineering\nQatar University\nDohaQatar\n",
"Abdulah Jarouf \nDepartment of Electrical Engineering\nQatar University\nDohaQatar\n",
"Lisan Shidqi \nDepartment of Electrical Engineering\nQatar University\nDohaQatar\n",
"Hamza Djelouat \nFaculty of Information Technology and Electrical Engineering\nUniversity of Oulu\nOuluFinland\n"
]
| [
"Department of Electrical Engineering\nQatar University\nDohaQatar",
"Institute of Artificial Intelligence\nDe Montfort University\nLeicesterUnited Kingdom",
"Department of Electrical Engineering\nQatar University\nDohaQatar",
"Department of Electrical Engineering\nQatar University\nDohaQatar",
"Department of Electrical Engineering\nQatar University\nDohaQatar",
"Faculty of Information Technology and Electrical Engineering\nUniversity of Oulu\nOuluFinland"
]
| []
| Fall detection is a serious healthcare issue that needs to be solved. Falling without quick medical intervention would lower elderly's chances of survival, especially if living alone. Hence, the need is there for developing fall detection algorithms with high accuracy. This paper presents a novel IoT-based system for fall detection that includes a sensing device transmitting data to a mobile application through a cloud-connected gateway device. Then, the focus is shifted to the algorithmic aspect where multiple features are extracted from 3-axis accelerometer data taken from existing datasets. The results emphasize on the significance of Continuous Wavelet Transform (CWT) as an influential feature for determining falls. CWT, Signal Energy (SE), Signal Magnitude Area (SMA), and Signal Vector Magnitude (SVM) features have shown promising classification results using K-Nearest Neighbors (KNN) and E-Nearest Neighbors (ENN). For all performance metrics (accuracy, recall, precision, specificity, and F 1 Score), the achieved results are higher than 95% for a dataset of small size, while more than 98.47% score is achieved in the aforementioned criteria over the UniMiB-SHAR dataset by the same algorithms, where the classification time for a single test record is extremely efficient and is real-time.outcome of high life expectancy around the world where previously mentioned systems present a solid solution for, particularly that there is an unavoidable shortage in healthcare personnel.Contrary to what has been mentioned earlier, healthcare systems have evolved significantly in the last century, where a lot of new technological advancements and clinical tests are implemented and discovered. However, because these systems have succeeded in raising people's life expectancy, another challenge arose as a result, which is the total increase in population, especially the elderly portion. From this point-of-view, healthcare systems have to re-adjust the provided services following a "smart" and automated methodology in order to accommodate the new needs that are currently arising. | 10.1016/j.bspc.2021.102532 | [
"https://arxiv.org/pdf/2105.09461v1.pdf"
]
| 233,565,271 | 2105.09461 | 3a3194a6ec5375c3db593a3b4b4ee86edbe3e4ab |
An IoT-Based Framework for Remote Fall Monitoring
Ayman Al-Kababji
Department of Electrical Engineering
Qatar University
DohaQatar
Abbes Amira
Institute of Artificial Intelligence
De Montfort University
LeicesterUnited Kingdom
Faycal Bensaali
Department of Electrical Engineering
Qatar University
DohaQatar
Abdulah Jarouf
Department of Electrical Engineering
Qatar University
DohaQatar
Lisan Shidqi
Department of Electrical Engineering
Qatar University
DohaQatar
Hamza Djelouat
Faculty of Information Technology and Electrical Engineering
University of Oulu
OuluFinland
An IoT-Based Framework for Remote Fall Monitoring
2010 MSC: 00-01, 99-00Wearable sensing device3-axis accelerometerFeature extraction algorithm selectionCWTMobile application
Fall detection is a serious healthcare issue that needs to be solved. Falling without quick medical intervention would lower elderly's chances of survival, especially if living alone. Hence, the need is there for developing fall detection algorithms with high accuracy. This paper presents a novel IoT-based system for fall detection that includes a sensing device transmitting data to a mobile application through a cloud-connected gateway device. Then, the focus is shifted to the algorithmic aspect where multiple features are extracted from 3-axis accelerometer data taken from existing datasets. The results emphasize on the significance of Continuous Wavelet Transform (CWT) as an influential feature for determining falls. CWT, Signal Energy (SE), Signal Magnitude Area (SMA), and Signal Vector Magnitude (SVM) features have shown promising classification results using K-Nearest Neighbors (KNN) and E-Nearest Neighbors (ENN). For all performance metrics (accuracy, recall, precision, specificity, and F 1 Score), the achieved results are higher than 95% for a dataset of small size, while more than 98.47% score is achieved in the aforementioned criteria over the UniMiB-SHAR dataset by the same algorithms, where the classification time for a single test record is extremely efficient and is real-time.outcome of high life expectancy around the world where previously mentioned systems present a solid solution for, particularly that there is an unavoidable shortage in healthcare personnel.Contrary to what has been mentioned earlier, healthcare systems have evolved significantly in the last century, where a lot of new technological advancements and clinical tests are implemented and discovered. However, because these systems have succeeded in raising people's life expectancy, another challenge arose as a result, which is the total increase in population, especially the elderly portion. From this point-of-view, healthcare systems have to re-adjust the provided services following a "smart" and automated methodology in order to accommodate the new needs that are currently arising.
Introduction
Enabling technology use in health sector brings many advantages that can save lives. The introduction of such systems for autonomous alarming and monitoring is significant in providing continuous supervision on patients' and elderlies' status. According to the United Nations (UN), people aged 65 and above constituted around 9.1% of the total population in 2019 (∼700 millions) [1]. This percentage is expected to reach 15.9% from the expected population (9.735 billions) in 2050 [1]. Not only that, but for the first time in human history, in 2019, people aged 65 or more have exceeded the number of children below five [1]. This is a life-changing in [9]. This demonstrates that a complete automated monitoring system can be implemented on a device that the majority of, if not all, people have.
However, coming back to reality, the complexity of utilized algorithms, the continuous computations, and the classical battery limitations, render it to be impractical. Therefore, in an effort to build a complete feasible system, this paper introduces a novel fall detection system incorporating a sensing device, a gateway transceiver and a mobile application. Firstly, it investigates different systems mentioned in previous literature related to fall • Using the ENN classifier and with the specific optimal features' concatenation, especially that it is rarely utilized in the context of fall detection as the main classifier.
• Investigating optimal number of neighbors for both KNN and ENN, highlighting how ENN is less sensitive to variations within the number of neighbors exhibiting more robustness in its performance
• Finding the optimal features vector over a dataset of small size, which is then calculated over the UniMiB-SHAR dataset, showing the resilience of our hand-crafted engineered features and the extremely high achieved results +98.5% for all performance criteria.
The paper's sections are organized as follows. Section 2 shows fall detection-related work describing the concept and the sensors in use, and highlighting the pros and cons of each system. Section 3 describes our proposed system for fall detection. Section 4 demonstrates the methodology to examine different feature extraction algorithms with three different classifiers: KNN, ENN, and BDT. Section 5 shows the results of different features combinations, different number of neighbors for KNN and ENN, and individual classifier's performance. Moreover, a comparison is also drawn with available commercial products and other state-of-the-art studies. Finally, Section 6 concludes the paper revisiting the most important highlights of this paper.
Related Work
Looking into previous fall detection-related work in the last few years, many systems with a variety of monitoring ideas have been proposed and discussed. The main categories are based on the sensing device in use that include externally placed cameras, floor sensors, radars, WSD, smartphones, and more interestingly, through Wi-Fi Channel State Information (CSI).
Cameras
Camera-based systems use camera(s) and image processing techniques to detect elderlies' movements and analyze falls. It involves tracking the significant parts of the elderlies' body. Omnidirectional-Camera as in [10] is used for fall detection purposes with input data as the RGB components of each frame. However, recently the usage of depth cameras seems to be the trend, where an Infrared (IR) emitter works together with a camera.
Authors in [11][12][13][14][15] utilize such cameras due to their ability in operating well under weak light conditions; since these cameras do not depend on the visible light spectrum. Hence, more reliable information can be attained. In [11], the aim is to detect the 3D trajectory of the head joint, while in [12], the goal is to monitor the individual's vertical state in each image frame using Microsoft Kinect depth imaging sensor. Similarly, in [13], Microsoft Kinect camera is utilized on 70 video samples, to detect falls using two different methods, Support Vector Machine-Machine Learning (SVM ML) and thresholding. However, in [14], the authors claim that trying to locate the human key joints is difficult especially due to the occluded body parts and sudden posture changes.
They proposed their own skeleton-free fall detection system by predicting individuals' correct postures. Lastly, in [15], authors use an IR depth camera for detecting the elderly's silhouette, from which they extract two types of features, kinematic and mel-cepstrum features. Following, they feed the extracted features to multiple classifiers such as SVM ML, Artificial Neural Network (ANN) and Naïve Bayes classifiers for performance comparison.
Camera-based systems have a major concern; which is the privacy of the patient, since cameras are constantly recording private locations, hence, it might deter certain elderlies from opting into the system.
Floor Sensors
Floor sensors-based systems collect data using nearly unnoticeable sensors as in [16][17][18]. In [16], the acoustic signals (i.e. vibration and sound), propagating through room's floor, are acquired by accelerometer and microphone sensors stationed at the room's corner. In a more recent publication, the authors in [17] use the INRIA-Nancy Smart Tiles prototype (around 100 tiles) where a 3-axis accelerometer is situated in the middle of the tile accompanied by four pressure sensors distributed at the corners. However, in [18], a different type of input is acquired based on a conductive film under the floor powered to generate electric fields. A good analogy in explaining the mechanism behind this conductive film would be the way that touch screens work. The presence of human and resultant falls can be detected where a change in the film's impedance will be calculated [18].
This type of systems, especially the one mentioned in [18], is promising in nursing homes where elderlies have limited mobility. However, they need a pre-installation phase, before furnishing the rooms, where sensors should be equipped under the rooms' floor.
Radars
Radar-based fall detection systems use radar sensors and require a base station for signals' reception. Such systems, presented in [19][20][21][22], take advantage of the Doppler effect that signals exhibit when received/reflected by moving objects causing them to change in frequency. In [19] and [20], the utilized feature generating highest accuracy is the Global Alignment (GA) Kernel belonging to the Least Squares Support Vector Machine (LS-SVM) classifier. Whereas in [21], the extracted feature from the radar signal is the Discrete Wavelet Transform (DWT) and it is fed to KNN classifier with K = 1. On the other hand, autoencoders (a form of ANN) are used in [22] to learn the features from the input radar signals, and the Softmax regression classifier to predict human's movement.
Softmax regression is the generalized case of logistic regression, and is used for multi-class classification scenarios.
Wi-Fi CSI
Wi-Fi CSI-based systems, as stated in [23], aim to detect falls by observing indoor environment's effects on the Wi-Fi's radio signal through time delay, amplitude attenuation and phase shift. Thus, information estimation about the channel's properties can be done. It utilizes the Wi-Fi devices as transmitters and have processing units as receivers such as computers. The CSI information is then extracted from the received packets [23]. Authors in [23][24][25] have all utilized this type of systems for detecting falls. Main difference between WiFall [23] and RT-Fall [24] is that the WiFall system depends only on the amplitude changes in the CSI, while RT-Fall investigates both amplitude and phase difference information produced by multiple antennas. In [25], the proposed FallSense system uses Dynamic Template Matching (DTM) that relies initially on less training data that keeps updating during system's usage. Moreover, the authors claim that their system is less complex since they use DTM with small training set instead of SVM and Hidden Markov Model (HMM) as in [23] and [24].
It should be noted that Wi-Fi-based systems and radar-based systems use Radio Frequency (RF) similarly, however, Wi-Fi-based systems investigates human's movements effects on CSI, while radar-based systems observe their effects on the signal's frequency (Doppler effect). Since both systems relatively yield similar results, it is perhaps more convenient to use the Wi-Fi since it is pre-existing in homes whereas radar-based systems require extra installation.
Camera, floor, radar and Wi-Fi-based systems have two serious challenges in common. They would be useless if the elderly had a walk outside and fell. Meaning that for them to function correctly, the elderlies must stay indoor all the time, which imposes restricted mobility on elderlies since the sensors are static. In addition, since these systems rely on having a single individual or elderly indoor, pets and other people movement would generate false alarms [26].
WSD & Smartphones
The most common type of fall detection systems found in literature were the ones depending on WSD. It is safe to say that most researchers are opting to use the triaxial accelerometer as part of their system disregarding where the wearable sensor is placed as in [27][28][29][30][31][32][33][34][35]. However, Ozcan et al. took a completely different approach where the device in use is a camera placed at the waist for entropy distance measurement [36]. Comparison is not straightforward for fall detection systems of this type as there are many differences to account for. For instance, some researchers use only the 3-axis accelerometer sensor placed on the chest [29,30], on the waist [31], and on the thigh [35]. However, authors in [28] and [34] use a sensing device generating 3-axis accelerometer, gyroscope and magnetometer data, which was placed on the waist and the wrist, respectively. Hence, giving them more information about the orientation, velocity and displacement of the elderly. For authors in [32], they utilize the 3-axis accelerometer and barometric pressure sensors, on the neck for sit-to-stand detection, while Wang et al. use the same type of sensor and placement for power-efficient signal features extraction [33]. In [32] Lee et al. talk about the usage of 3-axis accelerometer and gyroscope sensors on the elderly's waist for detecting near-fall scenarios as well as falls [27].
More interestingly, since smartphones nowadays possess multiple sensors including 3-axis accelerometer and gyroscope, researchers utilized them for fall detection as both a sensing and a processing device. The idea formulated in early 2010s as smartphones processing capabilities started emerging at the time. A system that utilizes a smartphone is initially introduced in [37] where fall detection is implemented through a simple thresholding-based learning algorithm. In [38], a more advanced learning algorithm such as Finite-State Machine (FSM) cascaded with SVM is used on the smartphone. Kau et al, interestingly argues that the power consumption resulting from running fall detection application is around 9%, and it is the same consumption of a gaming application. Authors in [39], use ANN on the smartphone for fall detection and, as expected, it drains a lot of power as shown in their results. FallDroid, developed by [40], uses a thresholding-based algorithm followed by multiple kernel SVM learning algorithm. Implementing such systems, as in [38][39][40], with high complexity on smartphones is mainly due to the technological enhancements mobile phones have undergone recently.
Both WSD and smartphones fall detection systems are challenged by the limited amount of energy batteries can provide. Hence, it is not possible to deploy these systems on the same device 24/7. Moreover, both systems are susceptible to be forgotten by elderlies where forgetfulness is a common trait in that age category.
It is true that systems based on WSD can create uncomfortableness as well to the wearer, however, it is the most reliable source of data since they are coming from the elderlies themselves. In other words, existence of other people or pets near the elderly would not trigger a false positive as in the case of camera, floor, radar and Wi-Fi fall detection systems. Furthermore, they can be used to implement a system where the elderly does not have to be imprisoned in a house, which reduces mobility restrictions other systems impose. Table 1 summarizes the reviewed technologies, highlighting their advantages and disadvantages.
In this paper, extracting different features, from two pre-existing datasets [7,29], containing 3-axis accelerometer data is investigated. Furthermore, examining different feature combinations and focusing on CWT as a significant feature for fall detection where the WSD systems did not opt to use, except the one mentioned in [32].
Then, the optimal feature combination is fed to three different classifiers (KNN, ENN and BDT) and a majority Voting Machine (VM). KNN and ENN number of neighbors are varied where the optimal neighbor's number is chosen based on specific performance criteria. Then, a comparison between individual classifiers is presented to show which one would be the best choice if only one is to be deployed. Finally, a comparison with commercial products and state-of-the-art studies is conducted, highlighting the competitiveness of the designed classifiers.
Overall System
In this section, a description of the overall system, from both its hardware and software aspects, is presented elaborating on the role of each device.
Hardware Implementation
As illustrated in Figure 1, several components and communication protocols are involved. A WSD (Shim-mer3ECG) transmits acquired 3-axis accelerometer and electrocardiogram (ECG) data to a gateway device
(ODROID-XU4) through Bluetooth.
On ODROID-XU4, as shown in Figure 2, feature extraction algorithms are applied to the received data where the most significant features are looked for. Then, they are fed to chosen classifiers where they are trained to distinguish between Activities of Daily Living (ADL) and fall events. If a fall is detected, ODROID-XU4 will send an alarm to caregivers' smartphones through a well-structured cloud database channel. Hence, ensuring a quick response from caregivers to aid the elderly in danger.
The Shimmer3ECG sensor is used mainly due to its small size, its power efficiency, programmability, and its capability in streaming ECG and accelerometer data in real-time simultaneously [29]. On the other hand, having an octa-core Exynos5422 big.LITTLE processor, a 2GB LPDDR3 RAM, and an advanced Mali GPU are the main reasons behind choosing ODROID-XU4 [41]. In addition, a Wi-Fi dongle can be connected to the ODROID-XU4 to transmit data to the cloud.
Software Implementation
The proposed communication structure in Figure 3 highlights interlinked entities within the cloud-hosted database platform, and illustrates the role of both the ODROID-XU4 and the user's mobile application for data visualization [9]. As illustrated by Figure 3, the elderly-to-caregiver communication system involves the link from the Shimmer3ECG, the ODROID-XU4 as the gateway and the cloud-hosted Firebase Realtime database up to the Vitals Monitoring mobile application.
RN42 Class 2 Bluetooth
Shimmer3ECG
Mobile Phone The Firebase platform provides a back-end service that allows for the real-time storage of large datasets within a database that synchronizes all clients with internet access.
ODROID-XU4 Firebase Realtime Database
The database is meant to act as a temporary buffer for the uploaded signals data, which should be accessed remotely by another client device with authenticated access, in this case, a mobile application that displays the stream of accelerometer and/or ECG data. The uploaded data are updated continuously with new values during streaming, while the mobile application "listens" to data changes on the cloud and handles them appropriately.
The interactions of the different entities with the cloud-hosted database are illustrated in Figure 4 (a) for data visualization, and Figure 4 (b) for alerting caregivers.
ODROID XU4
Prepare data to be sent
Mobile App
Visualize incoming data Realtime Database Moreover, they can receive and visualize the streamed data from the elderly's side, or even call the elderly for a quick check-up. In the Elderly View, he/she can visualize the data that is streamed from the Shimmer3ECG connected to him/her, as well as ask for help from the caregiver in time of need.
Lastly, all users' data are available on the cloud database for user authentication. Furthermore, previous fall records for each elderly, containing ECG and 3-axis accelerometer data stored at falling time, are saved under the Elderlies Records path, illustrated in Figure 3. As shown in Figure 6, only caregivers have access to these records for further analysis.
Methodology
In this section, the pursued methodology to reach the presented results in the following section are discussed.
Firstly, the used dataset is tabulated showing the total number of fall and ADL records. Secondly, the features that are extracted from the previously mentioned dataset are discussed. Thirdly, the machine learning classifiers in use are explained and the criteria for classifiers' performance check are also demonstrated. These criteria are important as they assist in validating the classifier prediction performance.
Utilized Training Datasets
Supervised machine learning algorithms depend heavily on having a well-structured and properly labeled dataset. Building such dataset requires significant amount of time, thus, a part of the dataset generated by authors in [29] is used as it employed the previous version of the used WSD in this paper. The content used from the dataset is described in Table 2. Moreover, after the verification of optimal features and classification algorithms over the Gibson et al. dataset [29], they are trained and tested on a more comprehensive dataset named "University of Milano Bicocca Smartphone-based Human Activity Recognition" (UniMiB-SHAR) [7], where it has a total of 11,771 records (ADL: 7,579 and Falls: 4,192) collected from a Samsung Galaxy Nexus I9250 equipped with a Bosh BMA220 acceleration sensor. Further details about that dataset and its distribution are presented in Table 3. the UniMiB-SHAR dataset [7].
For our case, we consider the two classes classification issue, a fall or a non-fall (ADL) event. All soft and strong fall events were grouped in a single category under "fall" and the same is applied to ADL for the Gibson et al. dataset. Similar approach is applied over the UniMiB-SHAR dataset, where the classification of fall and non-fall events is named "AF-2" in their work [7].
Feature Extraction Algorithms
Depending on the problem in hand and the signal in measurement, features will significantly vary in what they represent and the technique to extract them.
Triaxial accelerometer data are to be dealt with and fall events are to be detected here. Hence, the following features, existing in time, frequency, and time-frequency domains are chosen as it is foreseen that they would contribute the most.
CWT
CWT is an excellent decomposition tool where a non-static wave's changing properties will be captured in small wavelets localized in time [42]. These wavelets are shifted and scaled versions of the original mother wavelet. CWT outdoes the famous Short-Time Fourier Transform (STFT) in providing varying time windows for different frequencies. By allocating smaller time windows for high frequencies and large time windows for lower frequencies, this increases the resolution in the time-frequency domain as frequencies become higher [43].
Equation (1) shows the general equation for CWT, which gives features in the time-frequency domain:
CW T x (a, b) = 1 |a| ∞ −∞ x (t) ψ t − b a dt(1)
where: a is the scaling factor b is the translational factor
x(t) is the processed signal ψ(t) is a mother wavelet function (filter)
Triaxial accelerometer data representing events (whether ADL or fall events) have unique distinguishable traits from one another. For instance, the signal of a person jumping three times would have a repetitive behavior, while if he/she was to fall, the signal would have a high peak and a semi-constant signal afterwards (for one second at least). Hence, using CWT, as a feature to extract these traits would enhance the performance of the used classifiers.
Signal Vector Magnitude (SVM)
SVM is a time domain feature that calculates the total acceleration magnitude generated by the existing triaxial acceleration data [44]. It shows how significant the change in acceleration in a moment of time, disregarding on which axis that change was, through measuring the magnitude using equation (2):
SV M i = Ax 2 i + Ay 2 i + Az 2 i(2)
where: SV M i is the i th acceleration vector magnitude
Total |SVM|
Total |SVM| feature is based on the previously mentioned SVM feature. The difference appears in summing the absolute value of all calculated SVMs as shown in equation (3):
T otal |SV M | = M i=1 |SV M i |(3)
where: M is the number of samples (the discrete equivalent of time interval)
By doing so, this feature becomes independent from time, since all the |SV M i |s are summed into one single feature.
Signal Magnitude Area (SMA)
SMA is another feature used to capture the observed amount of change from the acquired triaxial accelerometer data. Equation (4) calculates this feature [40]:
SM A = M i=1 |Ax i | + |Ay i | + |Az i |(4)
Triaxial Accelerometer Data Range
The range of each accelerometer axis in time domain is the difference between the maximum and minimum values found in the tested record. They are calculated through equation (5):
Range x = max(Ax) − min(Ax) Range y = max(Ay) − min(Ay) Range z = max(Az) − min(Az)(5)
where: Ax, Ay & Az are the x, y & z accelerometer vectors respectively This feature holds information of how big the difference in each axis, but it does not necessarily mean that the record is a fall event if the range is large (see Figure 7).
Signal Energy (SE)
SE is a frequency domain feature that is calculated using equation (6):
E x = N i=1 a 2 x,i , E y = N i=1 a 2 y,i , E z = N i=1 a 2 z,i(6)
where: a x,i , a y,i & a z,i are the i th Fast Fourier Transform (FFT) coefficients of the x, y & z axes, respectively E x , E y & E z are the energy features for the x, y & z axes, respectively N is the total number of FFT coefficients per axis
The frequency coefficients are calculated for the 3-axis accelerometer data using FFT. Then, the resulting coefficients in each axis will be squared and summed to find the energy exerted by each axis [45]. Table 4 summarizes the extracted features in MATLAB and in which domain each feature resides. It is worth noting that the aforementioned feature extraction algorithms are used in the literature, some more than the others, such as the SVM where it is almost used in every fall detection system that utilizes acceleration data.
However, to the best of our knowledge, CWT features are not investigated deeply in the manner we portray, highlighting the effect of varying parameters such as the scale and the utilized wavelet function. Moreover, an investigation is carried out to find the best concatenation of such features, evaluating their overall performance and highlighting the best combination that yields the most promising results.
Machine Learning Classification Algorithms
The extracted features, as shown in Table 4, are fed to the following implemented supervised machine learning classifiers: KNN, ENN, and BDT. Then, the outputs of each classifier are inserted into VM that yields an output based on the majority of the aforementioned classifiers.
KNN
KNN algorithm is an instance-based classification algorithm that uses the whole dataset to classify events based on the provided features. It calculates the distance between features of the new record with the existing training dataset features. After that, it uses the K nearest neighbors in predicting which class does the new record belong to.
ENN
ENN has the same basis of operation as KNN; however, it differs from KNN in that it has a two way of communication. Meaning that it checks the nearest neighbors to the test record, and at the same time, takes into consideration the neighbors that see the test record as one of their nearest neighbors [46]. Because of that,
BDT
BDT is a model-based classification algorithm. It builds a tree structure where in each node a decision must be made between two choices (binary). It breaks the dataset down into smaller subsets, at the same time, the associated decision tree keeps developing until it reaches a leaf (decision).
VM
VM checks what each classifier yielded, and based on that, makes the decision to detect an ADL or a fall event. Thus, it is not a classifier by itself.
Both KNN and ENN are convenient to be used here as the datasets are relatively small in size. For problems with much bigger datasets, it is not recommended to use such algorithms. They calculate the distance between the new incoming record's features and the features of all the records, making them computationally expensive for huge datasets.
To test whether these classifiers are accomplishing satisfying results, calculating the following quantities, shown by their equations (7-11) is a must:
Accuracy (Ac) = T P + T N T P + T N + F P + F N * 100% (7)
Recall (Re) = T P T P + F N * 100% (8) P recision (P r) = T P T P + F P * 100% (9) Recall measures the percentage of falls that were "correctly" detected from the fall test set. Precision on the other hand weighs how many of the detected falls were actual true falls. Moreover, F 1 Score is also important especially if recall and precision were not showing promising results and a choice is to be made between different models/classifiers. However, if both were high enough, F 1 Score will also have high percentage. Lastly, specificity shows how many ADL are "correctly" predicted from the overall ADL test set. It is similar to recall but for ADL events instead of fall events. The pseudo code describing the methodology of generating the results in the following section is outlined by Algorithm 1.
F 1 Score (F 1 ) = 2 * P recision * Recall P recision + Recall(
Results Analysis
The results are based on randomizing the dataset and splitting it into 70% for training and 30% for testing [47] by convention when the number of records in the dataset is low. Furthermore, the data are folded five times and for each iteration, accuracy, specificity, sensitivity, recall, and F 1 Score are calculated. The results shown in this paper are the average of these five folds.
The number of extracted features from the triaxial accelerometer data for both datasets from [29] (2 seconds records) and UniMiB-SHAR [7] (3 seconds records), via the extraction algorithms, is presented in Table 5. Feeding individual features into previously mentioned classifiers generates good accuracy, however, concatenating multiple features generate better results as shown in Table 6. reducing the scale to be less than 100 shows reduction in classifiers' performance.
The accuracy values depicted in Table 6 are ordered in an ascending order based on the classifiers' overall performance. The last two rows show most promising results. Either one would be an excellent candidate to show next results, thus, the features in the last row are chosen. Feature normalization was examined for features in the last row but KNN and ENN performance has worsen. For BDT performance was relatively the same.
Thus, feature normalization was not applied to the dataset. Choosing the correct number of neighbors considered for both KNN and ENN is a hyperparameter that significantly affects the aforementioned performance criteria and this can be clearly observed in Figure 8 and Figure 9. The results in Figure 8 and Figure 9 indicate that the optimal number of neighbors is K = 5 and E = 5 for KNN and ENN, respectively. In addition, F 1 Score is calculated each iteration and the one presented in these figures is the mean of these five folds. To compare with the individual classifiers in [29], KNN and ENN are achieving better results due to the deployed features concatenations where CWT is concatenated with SE, SMA and SVM. Although the number of features is higher than the one reported in [29], the classification time is still adequate and is considered to be real-time as shown in Table 7. Note that the recorded classification durations in Table 7 are acquired on a "Lenovo ideapad 500-15ISK" laptop with 8GB single-channel RAM, 1TB SSD, and an Intel Core i7-6500 dual-core processor with a clock speed of [2.50 -2.60] GHz on MATLAB R2020a.
97
After analyzing the best feature vector combinations on the Gibson et al. [29] dataset, and to further validate our results, the larger dataset UniMiB-SHAR [7] has been used. The performance comparison on both datasets is shown in Table 7, highlighting the classifiers' performance, with more variance on train/test ratio on the UniMiB-SHAR dataset due to its large size. It is worth noting that the reported results are the average over the results obtained from each fold. Moreover, the VM classification time is the summation of all three classifiers along the time it takes to compute the majority vote.
If one of the classifiers is to be chosen and others to be discarded, a check on the overall performance for each classifier must be done. between the test record and the other training records need to be computed to predict the group it belongs to.
The huge number of provided records within the UniMiB-SHAR dataset allows for enhancing the classifiers' performance, as the classifiers have "seen" more variant records for ADL and falling scenarios. Thus, more tests on the performance of the aforementioned classifiers are created. It is also worth mentioning that ENN has a pre-processing phase that is heavily dependent on the number of records, and of high computational complexity O(M 2 log(M )) to build the weighted KNN maps [46], where M is the number of records. It can be discarded as a training phase that will not be accounted into the classification time but it is mentioned here as following:
i
Comparison with the State-of-the-Art
In Table 8, some of the commercial products available on the market are mentioned, where they offer an automatic fall detection feature. Similar products to the ones mentioned in Table 8, but not limited to, are
LifeFone, Bay Alarm Medical, and GreatCall [48]. It is worth noting that these systems do have automatic fall detection scheme that either is an add-on feature mentioned explicitly, or embedded within the monthly fee that they collect. Moreover, the majority of these products also offer a 24/7 call service for the elderlies to talk with, and they come with a push button to request help in times of distress. MyNotifi is different than the other products in three ways: i) Although they provide automatic fall detection and communication with relatives, they do not have 24/7 call service, which can be thought of to be autonomous; ii) The system is much cheaper than others, as it is only a single-time payment while the others are monthly-based as long as the elderly is using the service; and iii) most importantly, they give details on the used algorithms, ANN, and obtained system accuracy, 96.2%. Ours would be a single payment system, where if Shimmer3ECG is utilized as the sensory device, the system would cost around $604, and it would cost $59 if a smartphone is used as the sensor ($59 for the ODROID-XU4), assuming that every individual already has a smartphone. The reason we utilized Shimmer3ECG in the first place is to include ECG data, which can work as a complementary source of information to the accelerometer data. It would prove its vitalness if no fall has occurred but the elderly is in a critical condition due to a health related-issue. However, if ECG information is not needed in a certain installment, it would be possible to replace it with the elderly's smartphone to collect the accelerometer data, similar to the UniMiB-SHAR dataset.
The reason behind mentioning these systems is to emphasize that fall detection issue is an actual and serious concern that many companies are tackling, however, can be quite expensive, especially for financially-incapable senior individuals.
Moreover, in Table 9, a comparison with the state-of-the-art is made, specifically in terms of the results that the algorithm portrays, and the ML algorithm it uses over the UniMiB-SHAR dataset. Firstly, the authors who published the dataset set a high bar for other researchers to exceed by obtaining an accuracy of 98.71% for the AF-2 classification task [7]. Shahiduzzaman et al. [54] the state-of-the-art in the fall and non-fall classification task (AF-2), highlighting the novelty of our work. It is worth noting that WSD or smartphone-based systems are advantageous from multiple aspects when compared to other available systems. For instance, from privacy point-of-view, they are more private when compared to camera-based systems, where the elderlies would feel their privacy being violated. In comparison to other systems such as floor sensors, radars, WiFi-CSI, privacy can be considered maintained, however, the elderly can still be located, by an eavesdropper, within the household since the acquired data are spatially correlated.
From a financial and logistics point-of-view, WSD-based systems are easier to install as all other systems require sensors that are either expensive, e.g. multiple cameras in each room, or restrictive, e.g. cameras, floor sensors, radars, and WiFi-CSI-based systems that require elderlies to be inside a household. WSD and smartphone-based systems are much less restrictive when it comes to mobility, as elderlies can carry them wherever they go. This allows them to leave the house more frequently, and have more physical activity in more refreshing environments instead of being imprisoned 24/7 within their households. From an accuracy point-of-view, all other systems are more prone to predicting false positives as they are more susceptible to noise coming from within the monitored environment. For instance, a relative or a pet can induce artifacts to the environment where the classifier might consider a mixture of these signals as a fall, simply because the environment is noisy, putting the elderly's life in significant risk. On contrast, a fall might not be detected (false negative) due to the noise within the environment, which can exhaust caregivers or hospitals if happens frequently. This is not the case with WSD and smartphone-based systems as they collect the information related to the elderly only, disregarding potential sources of noise within the environment. However, two main disadvantages that come to the surface when using WSD and smartphone-based systems are: 1) they can easily become uncomfortable; and 2) their reliance on battery charge. Lastly, within WSD-based systems, some of them use other sensors, along with 3-axis accelerometer, due to their availability such as gyroscope and magnetometer as in [28]. Although the achieved results following our method are already superior, adding more sensory data can be extremely useful, which could further perfect the system performance and make it more robust against noise or bias coming from a single sensor.
Conclusion
To conclude, in order to provide elderlies with 24/7 healthcare service to support them in the event of a fall, multiple feature extraction and classification algorithms for the presented fall detection system were evaluated.
The most promising classifiers were KNN and ENN, but ENN outperformed KNN in both the performance criteria and its processing time on both examined datasets. Moreover, both performed extremely well on the UniMiB-SHAR dataset, showing state-of-the-art results. BDT's results are also good but further inspection on how to improve them is a must, as detecting falls should have extremely low error rate. VM also showed promising results but its output was dictated by the results of both KNN and ENN since both are showing relatively similar behavior. The usage of F 1 Score for classifiers' performance comparison was also shown where the models with the highest F 1 Score showed the most promising results. From the presented results, it is evident that the extracted features, centered around the CWT, along with the selected classifiers are performing extremely well in detecting falls (Re = +99%). Challenges that can face WSD-based systems is regarding their uncomfortability, and the limited battery charge. For the former issue, manufacturers are tackling it by making WSDs that can be worn comfortably on the wrist as a watch, or around the neck as a pendant (in the case of a smartphone the issue is almost non-existing, as the smartphone can be placed inside the pocket). The latter issue is much more significant as the elderly is susceptible to falling when the device is charging. A potential solution would be to design the sensors to have long battery lives, and use multiple ones, so that when one is charging, the other can be used, preventing the elderly from being exposed to undetected falls.
detection, focusing then on Wearable Sensing Device (WSD) systems showing that Continuous Wavelet Transform (CWT) is seldom used in such systems. Then, it shows the significance of extracting different features, emphasizing on CWT as a feature with high influence on fall detection. In addition, multiple classifiers, such as K-Nearest Neighbors (KNN), Extended-Nearest Neighbors (ENN), and Binary Decision Tree (BDT) are investigated. Finally, results on two different datasets are demonstrated, using only 3-axis accelerometer and in real-time, where a comparison is drawn with available products in the market in terms of their prices, and with ones found in literature in terms of achieved performance criteria. Even though some of the investigated feature extraction algorithms and classifiers have been extensively utilized before in this domain, our main contributions can be summarized as follows: • Designing and concatenating CWT features, which are seldom utilized and optimized in literature, along with the Signal Energy (SE), Signal Vector Magnitude (SVM), and Signal Magnitude Area (SMA) to generate the best results. The exact values for the scale and the different wavelet functions for the CWT are identified, which help in generating the best results.
Figure 1 :Figure 2 :Figure 3 :
123Overall Received Communication system
Figure 4 :Figure 5 :
45Communication protocols: a) data streaming request; b) alerting caregivers request A separate Python program would be running by the ODROID-XU4 acting as a client device, standing by for when data streaming is required. To communicate with the cloud database, an appropriate helper library (Pyrebase) in Python is utilized where it provides necessary Application Programming Interface (API) to communicate with the Firebase cloud. The Python program would receive the data pool to be uploaded and stores them temporarily. Using the Pyrebase library, it establishes the proper channel connection with the cloud database through internet, where the program then sends the data. As a result, synchronization happens when the data are put into the cloud database.The developed mobile application (Vitals Monitoring) is intended to run on Android devices as a proof of concept. Its role revolves mainly on providing data visualization for both elderlies and caregivers and facilitating an alarming feature for caregivers in case of an emergency from any elderly. There are two separate portals of the Vitals Monitoring application, categorized as the Elderly View and the Caregiver View as shown inFigure 5. Application usage cycle Caregiver View provides a list of all registered elderlies, and caregivers can see any of the elderlies' details.
Figure 6 :
6Previous falling records
Ax i ,
iAy i & Az i are the i th acceleration element from x, y & z accelerometer vectors respectively It is worth mentioning that the existence of the time parameter, through the order of samples, plays an important role in classifying an event whether it is a fall or not.
Figure 7 :
7(a) Strong fall event vs. (b) ADL (Jumping three times) event
Positive (T P ): A fall occurs, and the system properly detects it False Positive (F P ): The system detects a fall although it did not occur True Negative (T N ): An ADL is performed, and the system does not detect a fall False Negative (F N ): A fall occurs but the system does not detect it Accuracy measures how good the classifier's predictions are with respect to the whole test set. It would be convenient if the dataset was structured to have classes of equal number of records. For instance, collecting 200 falling events and 200 ADL events would be an ideal case where accuracy could indicate if the classifier is performing well. Calculating other performance criteria such as recall and precision will surely help in assessing the classifier's performance, especially if an equal number of records for each class is not available.
Figure 9 :
9Testing ENN with different No. of neighbors that as well. The same can be seen for ENN when E ∈ {3, 5, 7} fromFigure 9. Using low number of neighbors for KNN and ENN is equivalent to overfitting on the training data. Increasing number of neighbors would reduce this overfitting making the classification algorithm generalize better when new records arrive. However, if the number of neighbors increases significantly, it would be equivalent to the underfitting scenario, meaning that the classifier is not trained well. Consequently, error on new incoming records will also be larger. Hence, this shows the need for tuning the number of neighbors to find the optimal one resulting in least classification error on the test set. The performance of ENN worsens in a slower pace than that of KNN, possibly showing the significance of ENN's "two-way communication".
) 0.020 seconds for a training set of 159 records; ii) 284.5 seconds for a training set of 8,239 (70% of 11,771) classification time for both classifiers significantly increases with the number of used records, while BDT remains relatively low because it is being trained based on the number of features within a single record, which is 408 for the case of Gibson et al. dataset and 608 in UniMiB-SHAR dataset's case.
Table 1 :
1Literature review summaryTechnology
Studies
Advantages
Disadvantages
Cameras
[10-15]
Minimal human intervention
Privacy, multiple cameras & indoor
Floor Sensors
[16-18]
Minimal human intervention &
more private than cameras
Frequent false alarms [14] & indoor
Radars & Wi-Fi
CSI
[19-22] &
[23-25]
More private than floor sensors
& cost-effective
High-sensitivity to noise [21] & indoor
WSD & Smart-
phones
[27-36]
Immunity to noise & minimal
number of sensors, mobility
Depends on Elderlies' interaction, un-
comfortable, needs recharging
Table 2 :
2Gibson et al. Dataset content[29] Both datasets contain ADL of different types covering some of the most common daily activities elderlies do.For the fall events, they recorded falls in all directions and with different scenarios to maximize the possibility of detecting these events regardless of their cause or direction. Sampling frequency for both datasets was set to beType
Event
Records
ADL
Jumping (3 times)
6
Jumping (1 time)
6
Lie down from sitting position
6
Lie down from sitting position quickly
6
Running
15
Sitting on chair
7
Sitting on chair quickly
6
Standing up
6
Standing up quickly
6
Walking
22
Walking quickly
6
Total ADL Events
92
Fall
Soft front fall
19
Soft back fall
19
Soft left fall
19
Soft right fall
19
Strong front fall
15
Strong back fall
15
Strong left fall
15
Strong right fall
15
Total Fall Events
136
Dataset size = 228
50 Hz and the period of each measured record (event) is 2 seconds for the Gibson et al. [29] and 3 seconds for
Table 3 :
3UniMiB-SHAR Dataset content[7] Type
Event
Records
ADL
Standing up from sitting
153
Standing up from lying
216
Walking
1,738
Running
1,985
Going upstairs
921
Jumping
746
Going downstairs
1,324
Lying down from standing
296
Sitting down
200
Total ADL Events
7,579
Fall
Falling forward
529
Falling right
511
Falling backward
526
Falling left
534
Hitting obstacle
661
Falling with protection strategies
484
Falling backward sitting on chair
434
Syncope
513
Total Fall Events
4,192
Dataset size = 11,771
Table 4 :
4Feature extraction algorithms summaryExtraction Method
Brief Explanation
Domain
CWT
Finds the CWT equivalent for a given signal
Time-Frequency
SVM
Resultant magnitude vector of the 3-axial data
Time
Total |SVM|
Sum all absolute values of SVMs
Time
SMA
Area under the acceleration curve
Time
Accelerometer
Data Range
Subtract the minimum value from the maximum value
per axis
Time
SE
Calculate FFT and sum the square of each axis data
separately
Frequency
the algorithm builds weighted KNN map in a pre-processing "training" step that is done only once and is then
utilized in the testing phase.
Extract features (CWT, SVM, Total |SVM|, SMA, range, SE);while K ≤ 17 & E ≤ 17 doAlgorithm 1: Classification pseudo code
Result: Accuracy, Recall, Precision, F 1 Score & Specificity
folds := 5;
Import dataset;
Randomize dataset;
while i ≤ f olds do
1. Divide dataset to 70% training and 30% testing;
2. Train classifiers (KNN, ENN, BDT);
3. Predict testing data labels & evaluate performance metrics;
end
end
Calculate the average of performance criteria;
5. Results & Discussion of Different Feature Extraction Combinations & Different Classifiers
Table 5 :
5Number of extracted featuresNo. of Extracted Features
CWT SVM Total |SVM|
SMA Data Range
SE Total
Gibson et al. [29]
303
101
1
1
3
3
412
UniMiB-SHAR [7]
453
151
1
1
3
3
612
Table 6 :
6Testing different features combinations accuraciesFeature Combination
Table 7
7shows all the performance results. The time needed for extracting the utilized features (CWT, SVM, SMA, and SE) is in the [7.5ms -9ms] range, which is highly dependent over the records' time length t and the used sampling frequency f s . In other words, it is dependent on the total number of samples within each record (N = t×f s ). Nonetheless, the portrayed computational time values are very efficientfrom the real-time factor perspective. If f s is 50Hz for example, as it is the case with both utilized datasets,
the feature extraction time for a single record is faster than acquiring a single triaxial acceleration sample from
the record (T s = 20ms). To highlight the features vector's length, a total of 408 features for the Gibson et al.
dataset are generated, while 608 features are generated for the UniMiB-SHAR dataset (refer back to Table 4).
From observing Table 7, ENN is outperforming KNN on both datasets in the performance criteria especially
on the smaller dataset size, due to its ability to significantly extract meaningful information from the available
records. Moreover, it consumes less time during classification noting that the ENN has a pre-processing (training)
phase that is done only once, while the KNN does not have any. From time-efficiency point-of-view, BDT is
significantly outperforming both ENN and KNN as it builds a model that does not expand with higher number of
records, but gets finely-tuned. Contrastingly, both ENN and KNN are model-free in the sense that the distance
Table 7 :
7Classifiers' performance criteria comparisonPerformance Criteria
KNN
ENN
BDT
VM
Gibson et al. dataset [29], 5 Folds, (70/30) Train/Test Ratio
Accuracy (%)
96.23
97.68
94.20
97.68
Recall (%)
96.72
98.63
95.14
98.63
Precision (%)
97.16
97.67
95.19
97.67
F 1 Score (%)
96.94
98.13
95.15
98.13
Specificity (%)
95.45
96.25
92.59
96.25
Average Classification Time for a Single
Record (ms)
5.78
0.36
0.31
6.50
UniMiB-SHAR Dataset [7], 5 Folds, (70/30) Train/Test Ratio
Accuracy (%)
98.94
99.07
96.79
99.14
Recall (%)
98.84
98.98
95.53
99.09
Precision (%)
98.19
98.42
95.45
98.52
F 1 Score (%)
98.51
98.70
95.49
98.80
Specificity (%)
98.99
99.12
97.49
99.17
Average Classification Time for a Single
Record (ms)
43.56
33.81
0.52
77.96
UniMiB-SHAR Dataset [7], 10 Folds, (90/10) Train/Test Ratio
Accuracy (%)
99.09
99.15
96.70
99.19
Recall (%)
99.01
99.05
95.54
99.12
Precision (%)
98.47
98.58
95.21
98.63
F 1 Score (%)
98.73
98.81
95.37
98.87
Specificity (%)
99.14
99.21
97.34
99.23
Average Classification Time for a Single
Record (ms)
54.53
43.31
0.54
98.45
used the UniMiB-SHAR dataset to create longer streams of records and were combined with camera-imagery input into an SVM ML algorithm. On the other hand, Ivascu et al. [55] and Casilari et al. [56] both used an ANN-based model achieving an accuracy of 96.73% and 91.09%, respectively. Delgado-Escaño et al. [57] utilized KNN ML algorithm where the results exceeded the ones mentioned earlier. Ours is simulated over the UniMiB-SHAR dataset where the results portray ENN as the best classifier over the 90/10 train/test ratio averaged over the 10-fold cross-validation, taken from Table 7. As can be observed from the table, the results shown by ENN, along with the engineered features, are surpassing
Table 8 :
8Comparison with systems available in marketsBrand
Fall Detection
Placement
Total Price
Mini Guardian [49]
Yes ($10 monthly)
Pendant/Pocket (Other
models exist)
+$750 (yearly)
On The Go [50]
Yes ($10 monthly)
Pendant/Pocket
+$430 (yearly)
HomeSafe
with
Au-
toAlert/ GoSafe 2 [51]
Yes
Pendant/Pocket
+$540 (yearly)
MobileHelp Solo [52]
Yes
Pendant/Pocket
∼$395 (yearly)
MyNotifi [53]
Yes (96.2% Accuracy)
Hip/Wrist
$200
Ours (Sensor:
Shim-
mer3ECG Unit)
Yes
Hip
$604
Ours (Sensor: Smart-
phone)
Yes
Pocket
$59
Table 9 :
9Comparison with the state-of-the-art over UniMiB-SHAR dataset (AF-2 Task) Ac (%) Re (%) Pr (%) F 1 (%) Sp (%)Study
Algorithm
Metrics
Data Type
Micucci et al. [7]
SVM ML
98.71
-
-
-
-
Accelerometer
Shahiduzzaman et
al. [54]
SVM ML
96.67
96.67
98.38
97.52
-
Accelerometer +
Camera
Ivascu et al. [55]
DNN
96.73
-
-
-
-
Accelerometer
Casilari et al. [56]
CNN
91.09
71.71
-
-
97.53
Accelerometer
Delgado-Escaño et
al. [57]
KNN
97.08
-
-
-
-
Accelerometer
Ours
ENN
99.15 99.05 98.58 98.81 99.21
Accelerometer
The statements made herein are solely the responsibility of the authors.
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. S Pope, Best Medical Alerts with Fall Detection -ConsumerAffairs. 7S. Pope, 7 Best Medical Alerts with Fall Detection -ConsumerAffairs (2021).
. Medical Guardian, Order Now -Mini Guardian. Medical Guardian, Order Now -Mini Guardian (2021).
On the Go 2-Day Medical Alert System -Medical Alert. Medical Alert. Medical Alert, On the Go 2-Day Medical Alert System -Medical Alert (2021).
Philips Lifeline, HomeSafe with AutoAlert -Philips Lifeline. Philips Lifeline, HomeSafe with AutoAlert -Philips Lifeline (2021).
. Mobilehelp Mobilehelp, Solo, MobileHelp, MobileHelp Solo (2021).
. Mynotifi, Product Accuracy -MyNotifi Medical Alert Devices & Accessories. MyNotifi, Product Accuracy -MyNotifi Medical Alert Devices & Accessories (2021).
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A cross-dataset deep learning-based classifier for people fall detection and identification. R Delgado-Escaño, F M Castro, J R Cózar, M J Marín-Jiménez, N Guil, E Casilari, 10.1016/j.cmpb.2019.105265Computer Methods and Programs in Biomedicine. 184105265R. Delgado-Escaño, F. M. Castro, J. R. Cózar, M. J. Marín-Jiménez, N. Guil, E. Casilari, A cross-dataset deep learning-based classifier for people fall detection and identification, Computer Methods and Programs in Biomedicine 184 (2020) 105265. doi:https://doi.org/10.1016/j.cmpb.2019.105265. URL http://www.sciencedirect.com/science/article/pii/S0169260719311770
| []
|
[
"Cubic sublattices",
"Cubic sublattices"
]
| [
"Márton Horváth [email protected] \nDepartment of Geometry\nInstitute of Mathematics\nBudapest University of Technology and Economics\nHungary\n"
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"Department of Geometry\nInstitute of Mathematics\nBudapest University of Technology and Economics\nHungary"
]
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| A sublattice of the three-dimensional integer lattice Z 3 is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector v ∈ Z 3 whose squared length is divisible by d 2 , there exists a cubic sublattice containing v with edge length d. This improves one of the main result of a paper [3] of Goswick et al., where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices. | null | [
"https://arxiv.org/pdf/2203.01901v1.pdf"
]
| 247,222,921 | 2203.01901 | a94383dc8e28e19a9b51063b7c9a1d62dfe9a4b9 |
Cubic sublattices
3 Mar 2022
Márton Horváth [email protected]
Department of Geometry
Institute of Mathematics
Budapest University of Technology and Economics
Hungary
Cubic sublattices
3 Mar 2022
A sublattice of the three-dimensional integer lattice Z 3 is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector v ∈ Z 3 whose squared length is divisible by d 2 , there exists a cubic sublattice containing v with edge length d. This improves one of the main result of a paper [3] of Goswick et al., where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.
Introduction
Constructing lattice squares in the integer lattice Z 2 is easy as we can take a vector and its image under a 90 degree rotation. The analogous problem in dimension three is to find lattice cubes in Z 3 . Although there are axis-parallel lattice cubes, still many other constructions exist, which are much more complicated to construct as there is no canonical rotation in dimension three.
One early result on lattice cubes is due to A. Sárközy [7], who described some constructions of lattice cubes and determined the number of certain lattice cubes in 1961. This topic is still researched, E. J. Ionascu studied lattice cubes in dimensions 2, 3, 4 and determined their Ehrhart polynomial in his recently published paper [4].
If the edge length of a lattice cube is d, then its volume d 3 is the determinant of the edge vectors, so it is an integer, while the squared edge length d 2 is also an integer. This means that the edge length d is an integer. The following statement is an easy consequence of the description of the Pythagorean quadruples (see [1] or [8]). If the length of a vector v ∈ Z 3 is an integer, then v can be extended to a lattice cube. An algorithmic proof is found in [6]. Similar questions are discussed in dimension four using Hurwitz integral quaternions by E. W. Kiss and P. Kutas [5].
A lattice cube can be easily extended to a cubic sublattice, which is a sublattice having such basis whose elements are pairwise orthogonal and of equal lengths. In this situation, it is naturally to ask which cubic sublattices contain a given vector v ∈ Z 3 not necessarily as a basis vector. If the edge length of the cubic sublattice is d, then d 2 must divide the squared length of v. Our goal is to show that this condition is sufficient in the following sense.
Theorem 1. For a vector v ∈ Z 3 whose squared length is divisible by d 2 for an integer d, there exists a cubic sublattice containing v with edge length d. If v is primitive, then this cubic sublattice is unique.
For example, the squared length of the vector v = (5, 5, 2) is 54, which is divisible by 9, so there exists a cubic sublattice containing v with edge length 3, see Figure 1. For the uniqueness, it is enough to assume that the greatest positive divisor k of v and d are coprime. Indeed, we show that the primitive vector u = v/k is also contained in the cubic sublattice Γ given by the theorem for v when d and k are coprime. In the group Z 3 /Γ, the order of uΓ divides k and the order d 3 of the group by Lagrange's theorem, hence the order of uΓ is 1.
For the greatest possible value of d, i.e., in the case when v 2 /d 2 is square-free, Theorem 1 was proved in [3] by L. M. Goswick, E. W. Kiss, G. Moussong and N. Simányi using the decomposition theory of Hurwitz integral quaternions. Our proof builds solely on the structure of the three-dimensional Euclidean space and some basic facts about lattice geometry.
Finally, we give a number-theoretic corollary by considering the squared length of the vector v. When a number d 2 m (where d and m are integers) is a sum of three squares, then m is also a sum of three squares. This is not surprising because Legendre's three-square theorem states that a natural number is a sum of three squares if and only if it is not of the form 4 n (8k + 7). As a primitive vector remains primitive vector in the cubic sublattice, we can formulate a similar corollary: If an integer d 2 m is a sum of three coprime squares, then m is also a sum of three coprime squares. We will discuss the converse of this claim after Theorem 3 at the end of Section 4.
In Section 2, we introduce some definitions and propositions from lattice geometry. Section 3 is devoted to the proof of Theorem 1. Then we characterize the cubic sublattices of Z 3 and we prove a kind of reverse theorem in Section 4. Finally, we investigate cubic sublattices as a partial ordered set in Section 5.
Preliminaries
In this section, we summarize some basic definitions and statements without proof. We refer to [2] for more details on lattice geometry. After that we prove some easy propositions which will be used later.
Let Λ be an n-dimensional lattice in a real vector space. We say that a subgroup K < Λ is a sublattice if K is also an n-dimensional lattice. This is equivalent to the finiteness of the index of K in Λ. If some linearly independent vectors in Λ form a basis in the intersection of Λ and the linear subspace generated by them, then these vectors can be extended to a basis of Λ. The parallelepiped generated by the basis of the lattice is called fundamental parallelepiped. When a scalar product or just a volume form is given on the vector space, the volume of the fundamental parallelepiped does not depend on the choice of the basis of the lattice. For a sublattice K ⊆ Λ, the ratio of the volumes of the fundamental parallelepipeds is equal to the index of K in Λ as a subgroup.
We say that a vector v ∈ Λ is divisible by a positive integer k if there exists a vector u ∈ Λ such that v = ku. A vector v ∈ Λ is called primitive if there is not any positive integer k = 1 which divides v. For a non-zero vector v ∈ Λ, there exists a unique primitive vector u and a unique positive integer k such that v = ku. In this case, we say that k is the greatest divisor of v. The vectors that are divisible by a positive integer k form a sublattice in Λ, it will be denoted by kΛ. The index of kΛ in Λ is k n .
Fixing a basis {e 1 , . . . , e n } of Λ, we can identify Λ with Z n by using the coordinates (v 1 , . . . , v n ) of a vector v ∈ Λ with respect to this basis. A vector is divisible by k if and only if its coordinates so are. A vector is primitive if and only if its coordinates are coprime. The standard embedding Z n ⊆ R n and the Euclidean vector space structure of R n define the dot product on Λ. In particular, the perpendicularity of two vectors of Λ and the length v of a vector v ∈ Λ are defined. In this case, the volume of the fundamental parallelepiped is the determinant of the basis vectors.
From now, we consider the case n = 3. The advantage of this dimension is the applicability of the cross product. We say that a sublattice Γ of the standard lattice Z 3 is a cubic sublattice if there exists a basis of Γ whose elements are pairwise orthogonal and of equal lengths. Such basis is called cubic basis, and their common length d is the edge length. When we have a cubic sublattice Γ ⊂ Z 3 , we can identify Γ with Z 3 , so we can measure the vectors of Γ with respect to this identifying.
For a non-zero vector v ∈ Z 3 , the set of the orthogonal vectors to v will be denoted by v ⊥ .
Proposition 1. For a non-zero vector
v = (v 1 , v 2 , v 3 ) ∈ Z 3 , the set v ⊥ is a two-dimensional lattice.
Proof. Obviously, v ⊥ is a discrete subgroup. Its dimension is at most 2, we show that it is exactly 2. We can assume that v 3 = 0, so none of the linear combinations of e 1 , e 2 is parallel to v. In this case, the linearity of the cross product yields that the vectors e 1 × v, e 2 × v ∈ v ⊥ are linearly independent.
Proposition 2. For a primitive vector v ∈ Z 3 , the area of the fundamental parallelogram in v ⊥ is equal to the length of v.
Proof. The area of the fundamental parallelogram equals the length of the cross productṽ of the generating vectors. As v is primitive,ṽ is a multiple of v. Every pair of the vectors e i × v (i = 1, 2, 3) generates a sublattice of v ⊥ . The cross product of the generators is
(e i × v) × (e j × v) = v · (e i × e j ) v = ±v k v,
where i, j, k ∈ {1, 2, 3} are distinct indices. These vectors are multiples ofṽ. As v is primitive, the coefficients v 1 , v 2 , v 3 are coprime, henceṽ = ±v.
Fixing a primitive vector v, let the vectors a, b ∈ Z 3 be equivalent if their difference is a multiple of v. Then the cross product with v is a well-defined map Φ v from the equivalence classes to v ⊥ .
Proposition 3. The map Φ v is a bijection.
Proof. If the cross products with v are equal for two vectors of Z 3 , then their difference is parallel to v. This means that Φ v is injective.
For the surjectivity, consider a vector ku ∈ v ⊥ , where u is primitive and k is an integer. The vector v is primitive in u ⊥ , so there exists a vector w such that {v, w} is a basis of u ⊥ . Then the cross product w × v is ±u, so ±kw × v = ku, hence Φ v is surjective.
The proof of Theorem 1
It is enough to prove the theorem for primitive vector v. Indeed, if v = ku for a primitive vector u ∈ Z 3 , and d 2 divides the squared length v 2 = k 2 u 2 , then there exists a decomposition d = d 1 d 2 such that d 1 divides k and d 2 2 divides u 2 . Applying the theorem for d 2 and u, we get a cubic sublattice Γ. Then the cubic sublattice d 1 Γ has edge length d 1 d 2 = d and contains d 1 u, therefore also v.
First we prove the uniqueness part of the theorem in case v is primitive. Suppose that we have a cubic sublattice Γ containing v with edge length d. The first lemma is the key observation.
Lemma 1. If a, b ∈ Γ, then a × b is divisible by d, and a × b/d ∈ Γ.
Proof. Computing the cross product of a and b with respect to the Euclidean structure of Γ ∼ = Z 3 ⊂ R 3 , we get a × b/d, which implies the statement.
Lemma 1 yields that a × v is divisible by d for a ∈ Γ, so consider the following subset of v ⊥ M (v, d) = {a ∈ v ⊥ | a × v is divisible by d}.
We have that M (v, d) contains the intersection v ⊥ ∩ Γ. Later we will see that these sets are coincide. By the linearity of the cross product, M (v, d) is a subgroup. The next lemma shows that it is a sublattice.
Lemma 2. The index of M (v, d) in v ⊥ is d. Proof. Let i denote the index of M (v, d) in v ⊥ . Firstly we prove that i is divisible by d. As the vector v = (v 1 , v 2 , v 3 ) is primitive, there exist integers t 1 , t 2 , t 3 such that t 1 v 1 + t 2 v 2 + t 3 v 3 = 1. Consider the vector t = (t 1 , t 2 , t 3 ), and put u = t × v ∈ v ⊥ . Then u × v = (t × v) × v = (t · v)v − (v · v)t = v − ℓ 2 t = v 1 − ℓ 2 t 1 , v 2 − ℓ 2 t 2 , v 3 − ℓ 2 t 3 .
This vector is not divisible by any non-unit divisor of d, otherwise such a divisor would divide
ℓ 2 and v 1 , v 2 , v 3 , which would contradict the primitiveness of v. Thus, the vectors u × v, 2u × v, . . . , (d − 1)u × v are not divisible by d, so the vectors u, 2u, . . . , (d − 1)u are not contained in M (v, d). The vector du belongs to M (v, d) by definition. In the group u generated by u ∈ v ⊥ , the subgroup u ∩ M (v, d) has index d, hence i is a multiple of d by Noether's isomorphism theorem.
Now we prove that i divides d. Consider the vectors
r 1 = de 1 × v = (0, −dv 3 , dv 2 ),s 1 = (r 1 × v)/d = (e 1 × v) × v = −v 2 2 − v 2 3 , v 1 v 2 , v 1 v 3 .
We have that r 1 ∈ M (v, d) and r 1 ⊥ s 1 . As r 1 is perpendicular to v, we have
s 1 × v = 1 d (r 1 × v) × v = − ℓ 2 d r 1 ,
which means that s 1 ∈ M (v, d). The area of the rectangle spanned by r 1 and s 1 is r 1 2 ℓ/d = dℓ(v 2 2 +v 2 3 ). By Proposition 2, the area of the fundamental parallelogram in v ⊥ is ℓ. The index of the sublattice generated by r 1 and s 1 in v ⊥ is the quotient of the areas of the fundamental parallelograms, which is equal to
d(v 2 2 + v 2 3 ). This shows that i divides d(v 2 2 + v 2 3 ). We get similarly that i also divides d(v 2 1 + v 2 3 ) and d(v 2 1 + v 2 2 )
, which yields that i divides the greatest common divisor
gcd d(v 2 2 + v 2 3 ), d(v 2 1 + v 2 3 ), d(v 2 1 + v 2 2 ) = d gcd v 2 2 + v 2 3 , v 2 1 + v 2 3 , v 2 1 + v 2 2 = d gcd v 2 2 + v 2 3 , v 2 1 + v 2 3 , v 2 1 + v 2 2 , v 2 2 − v 2 3 , v 2 1 − v 2 3 , v 2 1 − v 2 2 = d gcd 2v 2 1 , 2v 2 2 , 2v 2 3 , v 2 2 + v 2 3 , v 2 1 + v 2 3 , v 2 1 + v 2 2 .
Using
gcd(v 1 , v 2 , v 3 ) = 1, we obtain gcd(d(v 2 2 + v 2 3 ), d(v 2 1 + v 2 3 ), d(v 2 1 + v 2 2 )) = d if v 1 , v 2 , v 3
are not all odd. This implies the statement in this case. If v 1 , v 2 , v 3 are all odd, then gcd(d
(v 2 2 + v 2 3 ), d(v 2 1 + v 2 3 ), d(v 2 1 + v 22 )
) = 2d, so we have that i divides 2d. In this case, , d), and its index is d 2 . Therefore i divides gcd(2d, d 2 ) = d. d). This motivates the definition of the set
ℓ 2 = v 2 1 + v 2 2 + v 2 3 is odd, so is d. The sublattice dv ⊥ is contained in M (v
Lemma 1 implies for a vector
a ∈ Γ that a × v is divisible by d and a × v/d ∈ Γ. Since a × v/d ∈ v ⊥ as well, we obtain a × v/d ∈ M (v,Γ(v, d) = {a ∈ Z 3 | a × v/d ∈ M (v, d)} = {a ∈ Z 3 | a × v is divisible by d, and (a × v) × v is divisible by d 2 }.
We have Γ ⊆ Γ(v, d) and v ∈ Γ(v, d). As the cross product is linear, Γ(v, d) is a subgroup. The following lemma shows that Γ(v, d) is a sublattice.
Lemma 3. The index of Γ(v, d) in Z 3 is d 3 . Proof. The sublattice Γ(v, d) contains such vectors a ∈ Z 3 that a × v ∈ dM (v, d). By Proposition 3, the index of Γ(v, d) in Z 3 is equal to the index of dM (v, d) in v ⊥ . As M (v, d) has index d in v ⊥ , the index of the sublattice dM (v, d) in v ⊥ is d 3 .
Since the index of Γ is d 3 as well, if there exists an appropriate cubic sublattice, then it is Γ(v, d), which proves the uniqueness part of Theorem 1. Now we prove that Γ(v, d) is indeed a cubic sublattice. The first step is to show that the dot products of the elements of Γ(v, d) are divisible by d 2 , but we have to start with weaker lemmas.
Lemma 4. For a vector a ∈ Γ(v, d), the dot product a · v is an integer and it is divisible by d 2 .
Proof. For a ∈ Γ(v, d), we have that d 2 divides
(a × v) × v = (a · v)v − (v · v)a, which means that a · v d 2 v − v · v d 2 a ∈ Z 3 .
Since v · v is divisible by d 2 , we earn a · v d 2 v ∈ Z 3 . This gives the statement as v is primitive.
Lemma 5. For vectors a, b ∈ Γ(v, d), the vector a × b d is contained in Γ(v, d).
Proof. Our goal is to prove that M (v, d) contains the vector
a×b d × v d = a · v d 2 b − b · v d 2 a,
where the coefficients a · v d 2 and b · v d 2 are integers by Lemma 4. The above vector is in
M (v, d) if a · v d 2 b × v d − b · v d 2 a × v d ∈ Z 3 , which is satisfied, because b × v d , a × v d ∈ Z 3 as a, b ∈ Γ(v, d).
Lemma 6. For vectors a, b ∈ Γ(v, d), the dot product a · b is divisible by d 2 .
Proof. We have a × v d ∈ Γ(v, d) by Lemma 5. Applying Lemma 5 again, we get that the vector
(a × v) × b = (a · b)v − (v · b)a
is divisible by d 2 . Lemma 4 implies that d 2 divides the coefficient v · b. Therefore the vector (a · b)v is also divisible by d 2 , which gives the statement.
The second step is to find the cubic basis in Γ(v, d).
Lemma 7.
There exists a vector a ∈ Γ(v, d) of length d.
Proof. Suppose the contrary. Lemma 6 implies that the squared length of every vector of Γ(v, d) is divisible by d 2 . This and the indirect assumption yield that the length of every non-zero vector of Γ(v, d) is at least √ 2d. Thus, the balls centered at the elements of Γ(v, d) with radius √ 2d/2 are disjoint. The parts of the intersection of a fundamental parallelepiped and these balls form an entire ball. Its volume is
4π 3 √ 2d 2 3 > √ 2d 3 ,
while the volume of the fundamental parallelepiped is d 3 . This is a contradiction. Proof. We prove by contradiction in a way similar to the proof of Lemma 7. We suppose that the length of every non-zero vector in Λ is at least √ 2d. In this case, the disks around the elements of Λ with radius √ 2d/2 in the plane perpendicular to a are disjoint. The area of the intersection of a fundamental parallelogram and these disks is
√ 2d 2 2 π = π 2 d 2 > d 2 ,
which is the area of the fundamental parallelogram, this is a contradiction.
Characterization of cubic sublattices
Although the following two lemmas are well-known, we prove them for the sake of completeness.
Lemma 9. Let K be a sublattice of a two-dimensional lattice Λ and let k be the greatest common divisor of the vectors of K. Then there exists a vector in K whose greatest divisor is k.
Proof. Let the generators of K be v 1 = k 1 u 1 and v 2 = k 2 u 2 , where u 1 , u 2 are primitive, and k 1 , k 2 are positive integers. We can assume that k = gcd(k 1 , k 2 ) is equal to 1. Unfortunately, the vector v 1 + v 2 is not necessarily primitive, see Figure 2, so we have to be more tricky. As u 1 is primitive, there exists such a vector w that u 1 and w generate Λ. We can write u 2 = au 1 + bw, where a, b are coprime integers. Let c be the product of those prime numbers which divide b, but do not divide k 1 k 2 (set c = 1 if there is not any such prime number). We show that the vector v = cv 1 + v 2 ∈ K is a primitive vector. Suppose the contrary, i.e., there exists a prime number p which divides v = cv 1 + v 2 = ck 1 u 1 + k 2 (au 1 + bw) = (ck 1 + ak 2 )u 1 + bk 2 w.
Since u 1 , w are the generators of Λ, p divides ck 1 + ak 2 and bk 2 . As p is prime, p divides b or k 2 . If p divides k 2 , then p also divides ck 1 , which contradicts the definition of c or gcd(k 1 , k 2 ) = 1. If p does not divide k 2 , then p divides b. By the definition of c, p divides ck 1 , hence p divides ak 2 and consequently also a. Thus, p divides u 2 = au 1 + bw, which is a contradiction. The next lemma is the three-dimensional version of Lemma 9.
Lemma 10. If the greatest common divisor of the vectors of a sublattice K of a three-dimensional lattice Λ is k, then K contains a vector whose greatest divisor is k.
Proof. Denote the generators of K by v 1 , v 2 , v 3 and its greatest divisors by k 1 , k 2 , k 3 , respectively. We have that k = gcd(k 1 , k 2 , k 3 ). Let M be the sublattice generated by v 1 and v 2 . By Lemma 9, there is a vector v in M whose greatest divisor is gcd(k 1 , k 2 ). If we apply Lemma 9 again to the sublattice generated by v and v 3 , we get a vector in K, whose greatest divisor is gcd(gcd(k 1 , k 2 ), k 3 ) = k.
We remark that similar statement holds for arbitrary dimensional lattices, and one can prove it by induction on the dimension. Now we can characterize the cubic sublattices of Z 3 . We show that every cubic sublattice can be obtained from our construction. Proof. We have that k is the greatest common divisor of the vectors of Γ and d is the quotient of the edge length of Γ and k. Consider the cubic sublattice Γ ′ such that kΓ ′ = Γ. By Lemma 10, there exists a primitive vector v ∈ Γ ′ . Theorem 1 implies Γ ′ = Γ(v, d), which gives the statement.
When a cubic sublattice is constructed from a primitive vector, this vector is also a primitive vector in the cubic sublattice. Our next goal is to show that every primitive vector can be such a vector. It requires two lemmas.
Lemma 11. If a cubic sublattice Γ ⊆ Z 3 has edge length d, then it contains the vectors divisible by d 2 , i.e., the inclusion d 2 Z 3 ⊆ Γ is fulfilled.
Proof. When Γ = Γ(v, d) for some primitive vector v, we have d 2 a ∈ Γ(v, d) for any a ∈ Z 3 by the construction. In the general case, Γ = kΓ(v, d/k) for some integer k by Theorem 2. For an arbitrary a ∈ Z 3 , we obtain
d 2 /k 2 a ∈ Γ(v, d/k), hence d 2 a ∈ Γ.
The second lemma is a claim from number theory.
Lemma 12. For an odd prime number p, there exists a primitive vector in Z 3 whose squared length is divisible by p 2 .
Proof. It is enough to construct a vector whose squared length is divisible by p 2 and which is not divisible by p, as we can divide it by its greatest divisor.
If −1 is a quadratic residue (when p = 4k + 1 for a positive integer k), then there exists a positive integer x such that x 2 = ap − 1 for some integer a. As p is odd, there exists integer b such that −2b ≡ a modulo p. Then (0, x, bp + 1) is a suitable vector.
When −1 is not a quadratic residue (if p = 4k + 3), we search for x, y ∈ Z p such that x 2 + y 2 = −1. If there were no such elements, then each of the pairs (1, p − 2), (2, p − 3), . . . , ((p − 1)/2, (p − 1)/2) would contain only one quadratic residue (in particular the last one would not contain any), which would contradict the well-known fact that there exist (p − 1)/2 quadratic residues modulo p. Denoting the corresponding integers by x, y as well, we obtain that x 2 + y 2 = ap − 1 for some integer a. As in the previous case, we have an integer b such that −2b ≡ a modulo p, thus, the vector (x, y, bp + 1) is a suitable vector.
Lemma 12 does not hold for p = 2. Indeed, the square numbers are congruent 0 or 1 modulo 4, so if 4 divides the sum of the square of the coordinates, then all of them are even, hence the vector is divisible by 2.
If we construct a cubic sublattice in a cubic sublattice Λ instead of Z 3 , we will use the notion Γ Λ (v, d) for the cubic sublattice of Λ with edge length d which contains the primitive vector v ∈ Λ.
Theorem 3. For an arbitrary primitive vector v = (v 1 , v 2 , v 3 ) ∈ Z 3 and an odd d, there exists a primitive vector u ∈ Z 3 for which we can choose a cubic basis of the cubic sublattice Γ(u, d) so that the coordinates of
u are (v 1 , v 2 , v 3 ).
Proof. First we show that it is enough to prove the theorem for odd prime number d. Applying the theorem for a vector v = (v 1 , v 2 , v 3 ) ∈ Z 3 and d 1 yields a primitive vector u ∈ Z 3 . If we apply again the theorem for u and for d 2 , we get a primitive vector t ∈ Z 3 . The sublattice Γ Γ(t,d2) (u, d 1 ) is a cubic sublattice with edge length d 1 d 2 , where the coordinates of the vector t are (v 1 , v 2 , v 3 ). The uniqueness part of Theorem 1 implies that this sublattice is Γ(t, d 1 d 2 ). In the following, we suppose that d is an odd prime number.
As v is primitive, d does not divide at least one of its coordinates, let it be the first coordinate v 1 . Lemma 12 provides a primitive vector w = (w 1 , w 2 , w 3 ) whose squared length is divisible by d 2 . Reordering the coordinates, we can achieve that d does not divide the first coordinate w 1 . Setw = (−w 1 , w 2 , w 3 ). Now one of the dot products v · w or v ·w is not divisible by d, otherwise d would divide their difference 2v 1 w 1 , which would contradict our assumptions on v 1 and w 1 as d is odd. Assume that v · w is not divisible by d.
Finally, construct the cubic sublattice Γ(w, d). By Lemma 11, it contains the vector d 2 v. Consider this vector as a vector of the cubic sublattice Γ(w, d), and denote it by u. This vector is primitive since dv is not contained in Γ(w, d) by Lemma 4. The sublattice d 2 Z 3 is a cubic sublattice in Γ(w, d) with edge length d and it contains u. Thus, the cubic sublattice Γ Γ(w,d) (u, d) is d 2 Z 3 . Choosing the basis (d 2 e 1 , d 2 e 2 , d 2 e 3 ) of d 2 Z 3 , we have that the coordinates of u are (v 1 , v 2 , v 3 ).
Theorem 3 allows us to formulate the converse of the number-theoretic corollary mentioned in the Introduction. If an integer m is a sum of three coprime squares, then for an odd d, the number d 2 m is also a sum of three coprime squares. The assumption on the oddness of d is necessary, as a number divisible by 4 cannot be a sum of three coprime squares by the investigation of the remainders modulo 4.
Cubic sublattices as partially ordered set
The inclusion gives a partial order over the cubic sublattices of Z 3 . This makes the set of the cubic sublattices to a partially ordered set. We decide whether this set is a lattice in the algebraic sense.
Similarly to the cubic sublattices, we can investigate the square sublattices of Z 2 . A square sublattice can be characterized by the invariance of the rotation of 90 • . Therefore the intersection and the union of square sublattices are also square sublattices. These are the supremum and the infimum of the given square sublattices, so the partially ordered set of the square sublattices of Z 2 forms a lattice in algebraic sense.
We show that the partially ordered set of the cubic sublattices of Z 3 is not a lattice in algebraic sense. The squared length of v = (1, 2, 2) is 9 = 3 2 , so there exists the cubic sublattice Γ(v, 3) ≤ Z 3 . We have the inclusions 9Z 3 ≤ 3Z 3 ≤ Z 3 and 3Γ(v, 3) ≤ Γ(v, 3) ≤ Z 3 , and from the last one, 3Γ(v, 3) ≤ 3Z 3 , see Figure 3. Lemma 11 gives 9Z 3 ≤ Γ(v, 3). In all of the mentioned inclusions, the (relatively) edge lengths of the cubic sublattices are 3, so there is not any cubic sublattice between the inclusions. This means that there is no supremum of 3Γ(v, 3) and 9Z 3 . And there is no infimum of Γ(v, 3) and 3Z 3 . Figure 3: This example shows that the partially ordered set of the cubic sublattices of Z 3 does not form a lattice in algebraic sense.
3Γ(v, 3) 9Z 3 Γ(v, 3) 3Z 3 Z 3
However, the partially ordered set of such cubic sublattices that contain a given primitive vector v ∈ Z 3 is a lattice in algebraic sense. If the squared length of v is kd 2 , where k is square-free, then this lattice is isomorphic to the lattice of the divisors of d.
Figure 1 :
1The vector v = (5, 5, 2) is contained in the cubic sublattice with edge length 3 generated by vectors (−1, 2, 2), (2, −1, 2), (2, 2, −1). The figure shows the domain [−1, 6] × [−1, 6] × [−2, 4]. There might be several suitable cubic sublattices if v is not primitive. For instance, the vector v = (5, 0, 0) is contained in the cubic sublattices with edge length d = 5 generated by bases {(5,
Fix
a vector a ∈ Γ(v, d) of length d. By Lemma 6, for a vector b ∈ Γ(v, d), the dot product a · b is divisible by d 2 , so the length of the projection of b to a is a multiple of d. Therefore the elements of Γ(v, d) lie in a ⊥ and its translates by the multiples of a. Consider the sublattice Λ = a ⊥ ∩ Γ(v, d). The area of the fundamental parallelogram in Λ is d 2 as we can get the basis of the fundamental parallelepiped of Γ(v, d) by adding a to the basis of the fundamental parallelogram of Λ.Lemma 8. There exists a vector b ∈ Λ of length d.
Fix
a vector b ∈ Λ of length d. The vector c = a×b d is in Γ(v, d) by Lemma 5. Since the lengths of a and b are d and they are perpendicular, the length of c is also d. The vectors a, b, c are pairwise perpendicular, so the volume of the parallelepiped generated by them is d 3 . As the index of Γ(v, d) in Z 3 is also d 3 , the vectors a, b, c generate the sublattice Γ(v, d). This means that Γ(v, d) is indeed a cubic sublattice with edge length d.
Figure 2 :
2The vector v 1 + v 2 is divisible by 2.
Theorem 2 .
2For an arbitrary cubic sublattice Γ ⊆ Z 3 , there exist unique positive integers k and d such that Γ = kΓ(v, d) for a primitive vector v ∈ Z 3 .
AcknowledgmentThe author is immensely grateful for Gábor Moussong, who recommended this topic and proposed to search an elementary proof for Theorem 1.
R D Carmichael, Diophantine Analysis. Mathematical Monographs. John Wiley & SonsR. D. Carmichael. Diophantine Analysis. Mathematical Monographs. John Wiley & Sons, 1915.
An Introduction to the Geometry of Numbers. J W S Cassels, Classics in Mathematics. Springer-VerlagCorrected reprint of the 1971 editionJ. W. S. Cassels. An Introduction to the Geometry of Numbers. Classics in Mathematics. Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition.
Sums of squares and orthogonal integral vectors. L M Goswick, E W Kiss, G Moussong, N Simányi, J. Number Theory. 1321L. M. Goswick, E. W. Kiss, G. Moussong, and N. Simányi. Sums of squares and orthogonal integral vectors. J. Number Theory, 132(1):37-53, 2012.
Ehrhart polynomial for lattice squares, cubes and hypercubes. E J Ionascu, Rev. Roumaine Math. Pures Appl. 641E. J. Ionascu. Ehrhart polynomial for lattice squares, cubes and hypercubes. Rev. Roumaine Math. Pures Appl., 64(1):57-80, 2019.
Cubes of integral vectors in dimension four. E W Kiss, P Kutas, Studia Sci. Math. Hungar. 494E. W. Kiss and P. Kutas. Cubes of integral vectors in dimension four. Studia Sci. Math. Hungar., 49(4):525-537, 2012.
Lattice cubes. R Parris, College Math. J. 422R. Parris. Lattice cubes. College Math. J., 42(2):118-125, 2011.
On lattice cubes in the three-space. A Sárközy, Mat. Lapok. 12In HungarianA. Sárközy. On lattice cubes in the three-space. Mat. Lapok, 12:232-245, 1961. In Hungarian.
The Diophantine equation x 2 + y 2 + z 2 = m 2. R Spira, Amer. Math. Monthly. 695R. Spira. The Diophantine equation x 2 + y 2 + z 2 = m 2 . Amer. Math. Monthly, 69(5):360-365, 1962.
| []
|
[
"The circumstellar disc in the Bok globule CB 26 Multi-wavelength observations and modelling of the dust disc and envelope",
"The circumstellar disc in the Bok globule CB 26 Multi-wavelength observations and modelling of the dust disc and envelope"
]
| [
"Jürgen Sauter \nInstitut für Theoretische Physik und Astrophysik\nChristian-Albrechts-Universität zu Kiel\nLeibnizstr. 1524098KielGermany\n\nMax-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany\n",
"Sebastian Wolf \nInstitut für Theoretische Physik und Astrophysik\nChristian-Albrechts-Universität zu Kiel\nLeibnizstr. 1524098KielGermany\n",
"Ralf Launhardt \nMax-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany\n",
"Deborah L Padgett \nCalifornia Institute of Technology\n1200 E California Blvd, Mail code 220-691125PasadenaCAUSA\n",
"Karl R Stapelfeldt \nJPL\n4800 Oak Grove Drive, Mail Stop 183-90091109PasadenaCAUSA\n",
"Christophe Pinte \nSchool of Physics\nUniversity of Exeter\nStocker RoadEX4 4QLExeterUK\n\nLaboratoire d'Astrophysique de Grenoble\nCNRS\nUJF\n\nUMR 5571\nB.P. 53F-38041Grenoble Cedex 9France\n",
"Gaspard Duchêne \nLaboratoire d'Astrophysique de Grenoble\nCNRS\nUJF\n\nUMR 5571\nB.P. 53F-38041Grenoble Cedex 9France\n\nAstronomy Department\nUniversity of California Berkeley\n601, 94720-3411Campbell Hall, BerkeleyCAUSA\n",
"François Ménard \nLaboratoire d'Astrophysique de Grenoble\nCNRS\nUJF\n\nUMR 5571\nB.P. 53F-38041Grenoble Cedex 9France\n",
"Caer-Eve Mccabe \nCalifornia Institute of Technology\n1200 E California Blvd, Mail code 220-691125PasadenaCAUSA\n",
"Klaus Pontoppidan \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n150-21, 91125PasadenaMS, CAUSA\n",
"Michael Dunham \nDepartment of Astronomy\nUniversity of Texas at Austin\n1 University Station C140078712AustinTXUSA\n",
"Tyler L Bourke \nHarvard-Smithonian Center for Astrophysics\n60 Garden StMA02138CambridgeUSA\n",
"Jo-Hsin Chen \nDepartment of Astronomy\nUniversity of Texas at Austin\n1 University Station C140078712AustinTXUSA\n"
]
| [
"Institut für Theoretische Physik und Astrophysik\nChristian-Albrechts-Universität zu Kiel\nLeibnizstr. 1524098KielGermany",
"Max-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany",
"Institut für Theoretische Physik und Astrophysik\nChristian-Albrechts-Universität zu Kiel\nLeibnizstr. 1524098KielGermany",
"Max-Planck-Institut für Astronomie\nKönigstuhl 1769117HeidelbergGermany",
"California Institute of Technology\n1200 E California Blvd, Mail code 220-691125PasadenaCAUSA",
"JPL\n4800 Oak Grove Drive, Mail Stop 183-90091109PasadenaCAUSA",
"School of Physics\nUniversity of Exeter\nStocker RoadEX4 4QLExeterUK",
"Laboratoire d'Astrophysique de Grenoble\nCNRS\nUJF",
"UMR 5571\nB.P. 53F-38041Grenoble Cedex 9France",
"Laboratoire d'Astrophysique de Grenoble\nCNRS\nUJF",
"UMR 5571\nB.P. 53F-38041Grenoble Cedex 9France",
"Astronomy Department\nUniversity of California Berkeley\n601, 94720-3411Campbell Hall, BerkeleyCAUSA",
"Laboratoire d'Astrophysique de Grenoble\nCNRS\nUJF",
"UMR 5571\nB.P. 53F-38041Grenoble Cedex 9France",
"California Institute of Technology\n1200 E California Blvd, Mail code 220-691125PasadenaCAUSA",
"Division of Geological and Planetary Sciences\nCalifornia Institute of Technology\n150-21, 91125PasadenaMS, CAUSA",
"Department of Astronomy\nUniversity of Texas at Austin\n1 University Station C140078712AustinTXUSA",
"Harvard-Smithonian Center for Astrophysics\n60 Garden StMA02138CambridgeUSA",
"Department of Astronomy\nUniversity of Texas at Austin\n1 University Station C140078712AustinTXUSA"
]
| []
| Context. Circumstellar discs are expected to be the nursery of planets. Grain growth within such discs is the first step in the planet formation process. The Bok globule CB 26 harbours such a young disc. Aims. We present a detailed model of the edge-on circumstellar disc and its envelope in the Bok globule CB 26. Methods. The model is based on HST near-infrared maps in the I, J, H, and K bands, OVRO and SMA radio maps at 1.1 mm, 1.3 mm and 2.7 mm, and the spectral energy distribution (SED) from 0.9 µm to 3 mm. New photometric and spectroscopic data from the Spitzer Space Telescope and the Caltech Submilimeter Observatory have been obtained and are part of our analysis. Using the self-consistent radiative transfer code MC3D, the model we construct is able to discriminate parameter sets and dust properties of both its parts, namely envelope and disc. Results. We find that the disc has an inner hole with a radius of 45 ± 5 AU. Based on a dust model including silicate and graphite the maximum grain size needed to reproduce the spectral millimetre index is 2.5 µm. Features seen in the near-infrared images, dominated by scattered light, can be described as a result of a rotating envelope. Conclusions. Successful employment of ISM dust in both the disc and envelope hint that grain growth may not yet play a significant role for the appearance of this system. A larger inner hole gives rise to the assumption that CB 26 is a circumbinary disc. | 10.1051/0004-6361/200912397 | [
"https://arxiv.org/pdf/0907.1074v1.pdf"
]
| 1,424,339 | 0907.1074 | 51e1024ddbc07d974a2f70a71a69270a398dfd2a |
The circumstellar disc in the Bok globule CB 26 Multi-wavelength observations and modelling of the dust disc and envelope
6 Jul 2009 July 6, 2009
Jürgen Sauter
Institut für Theoretische Physik und Astrophysik
Christian-Albrechts-Universität zu Kiel
Leibnizstr. 1524098KielGermany
Max-Planck-Institut für Astronomie
Königstuhl 1769117HeidelbergGermany
Sebastian Wolf
Institut für Theoretische Physik und Astrophysik
Christian-Albrechts-Universität zu Kiel
Leibnizstr. 1524098KielGermany
Ralf Launhardt
Max-Planck-Institut für Astronomie
Königstuhl 1769117HeidelbergGermany
Deborah L Padgett
California Institute of Technology
1200 E California Blvd, Mail code 220-691125PasadenaCAUSA
Karl R Stapelfeldt
JPL
4800 Oak Grove Drive, Mail Stop 183-90091109PasadenaCAUSA
Christophe Pinte
School of Physics
University of Exeter
Stocker RoadEX4 4QLExeterUK
Laboratoire d'Astrophysique de Grenoble
CNRS
UJF
UMR 5571
B.P. 53F-38041Grenoble Cedex 9France
Gaspard Duchêne
Laboratoire d'Astrophysique de Grenoble
CNRS
UJF
UMR 5571
B.P. 53F-38041Grenoble Cedex 9France
Astronomy Department
University of California Berkeley
601, 94720-3411Campbell Hall, BerkeleyCAUSA
François Ménard
Laboratoire d'Astrophysique de Grenoble
CNRS
UJF
UMR 5571
B.P. 53F-38041Grenoble Cedex 9France
Caer-Eve Mccabe
California Institute of Technology
1200 E California Blvd, Mail code 220-691125PasadenaCAUSA
Klaus Pontoppidan
Division of Geological and Planetary Sciences
California Institute of Technology
150-21, 91125PasadenaMS, CAUSA
Michael Dunham
Department of Astronomy
University of Texas at Austin
1 University Station C140078712AustinTXUSA
Tyler L Bourke
Harvard-Smithonian Center for Astrophysics
60 Garden StMA02138CambridgeUSA
Jo-Hsin Chen
Department of Astronomy
University of Texas at Austin
1 University Station C140078712AustinTXUSA
The circumstellar disc in the Bok globule CB 26 Multi-wavelength observations and modelling of the dust disc and envelope
6 Jul 2009 July 6, 2009Submitted: April 27, 2009 / Accepted July 6, 2009Astronomy & Astrophysics manuscript no. cb26circumstellar matter -planetary systems: proto-planetary discs -radiative transfer -stars: formation, individual: CB 26
Context. Circumstellar discs are expected to be the nursery of planets. Grain growth within such discs is the first step in the planet formation process. The Bok globule CB 26 harbours such a young disc. Aims. We present a detailed model of the edge-on circumstellar disc and its envelope in the Bok globule CB 26. Methods. The model is based on HST near-infrared maps in the I, J, H, and K bands, OVRO and SMA radio maps at 1.1 mm, 1.3 mm and 2.7 mm, and the spectral energy distribution (SED) from 0.9 µm to 3 mm. New photometric and spectroscopic data from the Spitzer Space Telescope and the Caltech Submilimeter Observatory have been obtained and are part of our analysis. Using the self-consistent radiative transfer code MC3D, the model we construct is able to discriminate parameter sets and dust properties of both its parts, namely envelope and disc. Results. We find that the disc has an inner hole with a radius of 45 ± 5 AU. Based on a dust model including silicate and graphite the maximum grain size needed to reproduce the spectral millimetre index is 2.5 µm. Features seen in the near-infrared images, dominated by scattered light, can be described as a result of a rotating envelope. Conclusions. Successful employment of ISM dust in both the disc and envelope hint that grain growth may not yet play a significant role for the appearance of this system. A larger inner hole gives rise to the assumption that CB 26 is a circumbinary disc.
Introduction
CB 26 is a small cometary-shaped Bok globule located about 10 • north of the Taurus/Auriga dark cloud at a distance of 140 pc (Launhardt & Sargent 2001). An IRAS point source (IRAS 04559+5200) at its southwest rim suggests an embedded Class I young stellar object (YSO) (Stecklum et al. 2004) source. Launhardt & Henning (1997) found an unresolved 1.3mm continuum source associated with the IRAS source.
Interferometric observations by Launhardt & Sargent (2001) showed that the major fraction of thermal dust emission at millimetre wavelengths has its origin in a young circumstellar disc with a diameter of about 400 AU and a mass of about 0.1 M ⊙ . This disc is seen almost edge-on and the central star is not visible directly. However, the spectral energy distribution suggests a Class I YSO with L ≥ 0.5 L ⊙ (Stecklum et al. 2004). From the 13 CO line emission and the Keplerian rotation curve, Launhardt & Sargent (2001) derive a central stellar mass of M * = 0.5 ± 0.1 M ⊙ . Furthermore, Launhardt et al. (2008) detected a jet-like molecular outflow emanating perpendicular to the plane of the disc. This outflow seems to be co-rotating with the disc.
We present new observations and a model for this source that accounts for spatially resolved data sets over more than 3 orders of magnitude as well as for the unresolved SED of the object.
Our modelling is based on spatially resolved maps of CB 26 in the millimetre regime from the Sub-millimetre Array (SMA) and the Owens Valley Radio Observatory (OVRO), high resolution images in the I,J,H, and K bands obtained with the the Near Infrared Camera and Multi-Object Spectrometer (NICMOS) at the Hubble Space Telescope (HST) and the Advanced Camera for Surveys (ACS), which is as well an instrument of the HST. The photometric data for the SED upon our model is based are provided by the Multiband Imaging Photometer for Spitzer (MIPS) and the Infrared Array Camera (IRAC) aboard the Spitzer Space Telescope (SST) and millimetre photometry. We also obtained a spectrum with the Infra-Red Spectrograph (IRS) aboard the SST.
In this paper, we use all available NIR to mm continuum data on CB26 to develop a self-consistent model of the source. This model consists of two parts: First, an optically thick dust disc which accounts for the dark lane seen in images obtained with the HST of the object and the significantly elongated intensity profiles in the mm range. The second component is an optically thin envelope that reproduces the scattered light nebulosity.
While the millimetre observations are sensitive only to radiation being emitted from dust in the dense region within the disc, the near-infrared images are dominated by scattered stellar light from dust in the circumstellar envelope and the disc's upper optically thin layers, often referred to as the "disc surface". These observations trace different physical processes in different regions of the circumstellar environment, but they are both strongly related to the dust properties in the system.
Thus, we are in the position not only to model observations on the common basis of one set of parameters, but we are also able to investigate whether the dust properties are different in the disc and the envelope. This has been suggested by investigations of the dust evolution in circumstellar discs where dust grain growth alters the dust grain properties in the circumstellar disc quite considerably whilst it is of less importance in the low-density envelope. Evidence for grain growth has been found, for instance, for the circumstellar discs IM Lupi , GG Tau (Duchêne et al. 2004;Pinte et al. 2007), HH 30 (Watson & Stapelfeldt 2004), IRAS 04302+2247 (Wolf et al. 2003), and VV Serpens (Alonso-Albi et al. 2008).
In order to compare our model with the available observations, we use the self-consistent radiative transfer code MC3D (Wolf et al. 1999;Wolf 2003b) in a parameter space study on the free parameters of the model. We aim at finding the best-fit model, which we define to be the model that reproduces certain predefined features among the observational data (such as width of the dark lane; see below for further details) best.
Observations and data reduction
In the case of the disc in the Bok Globule CB 26, we are in the fortunate situation of having a large variety of observational data at hand. This data includes not only the spectral energy distribution (SED) from 0.9 µm to 2.7 mm but also resolved maps in the near-infrared and in the millimetre regime. We will now briefly discuss those observations and the data reduction in this section.
HST Imaging
The NICMOS and ACS data were taken by the GEODE 1 team. An overview of the complete program, its objects, 1 Group for Edge-On Disc Exploration and its objectives can be found in Padgett et al. (2009, in preparation).
ACS
CB 26 was observed with the Advanced Camera for Surveys Wide Field Channel on 2005 September 09. Two 1250sec exposures were made in the F814W filter (λ = 0.80 µm, ∆λ = 0.15 µm), corresponding to Johnson I band. With a pixel scale of 0.05 ′′ , the ACS provides a field of view of 201 ′′ × 100 ′′ . The images were reduced, combined to reject cosmic rays, and corrected for geometric distortion by the STScI pipeline. Residual hot pixels and cosmic rays were manually removed by replacing the affected pixels by local median values.
NICMOS
CB 26 was observed using NICMOS on 2005 September 15 with the F110W (λ = 1.12 µm, ∆λ = 0.16 µm ), F160W (λ = 1.60 µm, ∆λ = 0.12 µm ) and F205W (λ = 2.06 µm, ∆λ = 0.18 µm) filters on the NIC2 array. With a pixel scale of 0.075 ′′ , the NIC2 array provides a field of view of 19.2 ′′ on a side. All data were taken in a MULTIACCUM step=128 sequence, with total exposure times of 512secs for the F110W data and 384secs for the F160W and F205W data. The observations were dithered back and forth by 0.75 ′′ for bad pixel removal.
All three of the NICMOS data sets were taken less than 1900 secs after emergence from the South Atlantic Anomaly (SAA) and therefore suffer from significant cosmic ray persistence. The data were re-reduced using the most recent calibration files available. The standard STScI CALNICA pipeline was used as a basis for the re-reduction, in addition to which, we employed both the biaseq and pedsub IRAF routines to remove the pedestal effects visible in each of the quadrants. The biaseq task removes the timedependent variations in the bias level for each quadrant, and the pedsub task then removes the fixed bias offset. After the data have had the pedestal effect removed, we used the IDL SAACLEAN program to model and iteratively remove the persistent cosmic rays in the image. After emerging from the SAA, NICMOS powers back up and takes 2 dark frames. These dark frames provide the model for the cosmic ray persistence pattern; SAACLEAN takes this model and iteratively removes it from the data until the noise in the background reaches a global minimum. Further details on this procedure can be found in Bergeron & Dickinson (2003) (ISR 2003-010). Following the SAACLEAN procedure, the data are run through a second round of pedestal subtraction, have any remaining bad pixels removed, and are then run through the CALNICB pipeline procedure to generate the final mosaics.
Spitzer photometry and spectroscopy
Mid-and far-infrared photometry for CB 26 were retrieved from the Spitzer Archive. IRAC observations were carried out under GTO program 94 (PI: Charles Lawrence). The data were taken 2004 February 11, AOR key 4916224, using the high dynamic range mode and 5 dithered exposures of 30sec each. Aperture photometry was measured from the Spitzer pipeline "post-BCD" mosaics (version 11.0.2), using a standard 10 pixel radius aperture for the source and background annulus with 12-20 pixel radius. MIPS observations were made under GTO program 53 (PI: George Rieke) on 2005 March 08, AOR key 12020480. A medium scan map covering 0.2 • × 0.5 • was performed, providing multiple dithered 4sec exposures and total integration times of 168, 84, and 16 sec at 24, 70, and 160 µm, respectively. Spitzer post-BCD pipeline mosaics (version 14.4.0) were used for aperture photometry. Photometry was measured in apertures whose radii and background annuli were 7 ′′ /7 ′′ − 13 ′′ (24µm), 35 ′′ /39 ′′ − 65 ′′ (70µm), and 48 ′′ /64 ′′ − 128 ′′ (160 µm) with aperture correction factors as given by Engelbracht et al. (2007); Gordon et al. (2007); Stansberry et al. (2007).
We carried out our own Spitzer low-resolution spectroscopy of CB 26 on 2006 October 18 under program 30765, AOR key 18964992. The ramp duration and number of observing cycles used were 6×14 sec, 4×14 sec, 8×30 sec, and 8×30 sec for the SL2, SL1, LL2, and LL1 modules of the IRS (respectively). The spectra were processed beginning with the intermediate droopres products from pipeline version 15.3.0. The 2D images were co-added and the nod pairs subtracted to remove stray light and other additive artifacts, as well as rogue pixels. 1D spectra of each nodding beam were then extracted using a fixed aperture of 5 pixels. Following Bouwman et al. (2008) the spectra were flux calibrated using a spectral response function determined by comparing standard star observations with template spectra. These response functions are also used with the "C2D" Legacy data set (e.g. Kessler-Silacci et al. 2006). It was not necessary to apply any scaling to match the short-low and long-low modules, indicating that the source is unresolved at the IRS wavelengths.
Millimetre and Sub-millimetre measurements
SMA
Observations at 270 GHz (1.1 mm) with the Sub-Millimetre Array (SMA, Ho et al. 2004) were made in December 2006, in two configurations providing baselines in the range 12 -62 kλ. Typical system temperatures were 350-500 K. The quasar 3C279 was used for bandpass calibration, and the quasars B0355+508 and 3C111 for gain calibration. Uranus was used for absolute flux calibration, which is accurate to 20-30%. The data were calibrated using the IDL MIR package (Qi 2005) and imaged using MIRIRAD Sault et al. (1995). The cleaned and restored 1.1 mm continuum map was constructed with robust uv-weighting using line-free channels in both sidebands. Here we only use the continuum map. The observations together with the molecular line data are described in more detail in a forthcoming paper (Launhardt et al. in prep.).
OVRO
CB 26 was also observed with the Owens Valley Radio Observatory (OVRO) between January 2000 and December 2001. Four configurations of the six 10.4 m antennas provided baselines in the range 6 -180 kλ at 2.7 mm (110 GHz) and 12 -400 kλ at 1.3 mm (232 GHz). Average SSB system temperatures of the SIS receivers were 300 -400 K at 110 GHz and 300 -600 K at 232 GHz. The raw data were calibrated and edited using the MMA software package (Scoville et al. 1993). Mapping and data analysis used the MIRIAD toolbox. Observing parameters are described in detail in Launhardt & Sargent (2001). The data presented here include additional observations conducted in 2001. All maps are generated with robust uv-weighting, cleared, and restored with a clean beam. Effective synthesised beam sizes of all interferometric millimetre continuum maps used here are summarised in Table 3.1.
CSO
Submillimeter observations of CB26 at 350 µm were obtained with the Submillimeter High Angular Resolution Camera II (SHARC-II) at the Caltech Submillimeter Observatory (CSO) on 2007 October 21. SHARC-II is a 12 × 32 element bolometer array giving a 2.59 ′ × 0.97 ′ field of view (Dowell et al. 2003). The beam-size at 350 µm is 8.5 ′′ .
We used the Lissajous observing mode to map a region approximately 1 ′ × 0.5 ′ , centred at the position of the source. We obtained two scans, each 10 minutes long, for a total integration time of 20 minutes in good weather (τ 225GHz ∼ 0.06). During both scans the Dish Surface Optimisation System (DSOS) 2 was used to correct the dish surface for gravitational deformations as the dish moves in elevation.
The raw scans were reduced with version 1.61 of the Comprehensive Reduction Utility for SHARC-II (CRUSH), a publicly available, 3 Java-based software package. CRUSH iteratively solves a series of models that attempt to reproduce the observations, taking into account both instrumental and atmospheric effects (Kovács 2006;Kovács et al. 2006;Beelen et al. 2006). Pointing corrections to each scan were applied in reduction based on a publicly available 4 model fit to all available pointing data. Pixels at the edges of the map with a total integration time less than 25% of the maximum were removed to compensate for the increased noise in these pixels. We then used Starlink's stats package to assess the rms noise of the map, calculated using all pixels in the off-source regions. The final map has a 1σ rms noise of 70 mJy beam −1 .
Photometry was measured in a 20 ′′ aperture centred at the peak position of the source. Calibration was performed according to the method used by Shirley et al. (2000); Wu et al. (2007). This method is based on the requirement that a point source should have the same flux density in all apertures with diameters greater than the beam FWHM (8.5 ′′ for these observations). To briefly summarise, a flux conversion factor (FCF) is calculated for a 20 ′′ aperture by dividing the total flux density of a calibration source in Jy by the calculated flux density in the native instrument units of µV in a 20 ′′ aperture. Flux densities of science targets are then derived by multiplying the 20 ′′ aperture flux density (in the instrument units) of the source by the FCF. We measure a 350 µm flux density for CB 26 of 2.8 ± 0.6 Jy.
Results from Observations -Basis for modelling
The focus of this section are the results by the previously described observations. The presented data set forms the basis of our modelling.
Images
The maps from the HST in I, J, H, and K bands are shown in Fig. 1. All all four images have been rotated in order to align the major axis of the dark lane with the horizontal axis. The emission seen on these maps is pure scattered light from the central star. The dark shadowy line that intersects the bipolar structure is present in all images. Especially the dependency of its width on the wavelength is clearly visible. Since one expects the circumstellar disc to be optically thick at these wavelengths, the dark lane is interpreted as the disc's shadow in the encompassing envelope structure. The bipolar nebula also shows a complex morphology far above and especially below the disc. For further discussion we refer the interested reader to Padgett et al. (2009, in preparation).
The interferometric millimetre continuum maps at 1.1 mm, 1.3 mm, and 2.7 mm, together with the dirty beam maps (see also Table 3.1), are shown in Fig. 2. For the ease of modelling, these images have been rotated 30 • in order to align the major axis of the elongated structure with the horizontal axis. While the source is unresolved vertically in all images, it is well-resolved along its horizontal axis, especially in the highest-resolution map at 1.3 mm. The 1.3 mm map also recovers some extended emission from the envelope that might be related to a disc wind (see Fig. 2. Inverse, reconstructed images from millimetre interferometric observations (left column) and corresponding dirty beam maps (right column), linear colour scale. All images have been rotated by −30 • in order to align the major axis of the brightness distribution with the horizontal axis. For the image scale a distance to the object of 140 pc is assumed (100 AU = 0.7 ′′ ). The contour lines are drawn at (from inside out) 90%, 50%, and 25% of the image maximum flux value. In the dirty beam images, the 50% contour, marking the FWHM of the Gaussian clean beam, is marked bold. Launhardt & Sargent 2001). For all images shown, radio and NIR, North is the same direction. Fig. 3 shows an overlay of a NIR colour-composite image and the 1.3 mm dust continuum emission. The spatial colocation of the millimetre dust emission and the dark lane in the scattered light images confirms the hypothesis of an edge-on optically thick disc as explanation for the observed features. However, the HST pointing is only good to about 1 arcsec. Due to the high extinction in the cloud core and the small NICMOS field of view, there are no reference stars in the image that could be used to correct for this pointing uncertainty.
Spectral energy distribution
The results of photometric measurements are presented in Table 3.2 and Fig.4. The spectrum we obtained from the IRS very well complements the photometric points as seen in Fig. 4. As is evident in this Figure, the 350 µm SHARC-II flux density is lower than expected based on comparison to the complete SED. This discrepancy can be explained by the fact that the Lissajous observing mode is insensitive to extended emission, as noted by Wu et al. (2007). This issue will be explored in more detail by M. Dunham et al. (2009, in preparation), but preliminary results suggest that flux densities measured from this particular observing mode may underestimate the true flux density by up to a factor of 2, even for relatively compact objects.
Model & Modelling
In this section we provide the reader with an introduction to the concepts and techniques we use.
The Model
First, we discuss the various components of our model. That is, the disc and envelope dust density distribution. The employment of both these parts in the model is readily suggested by the data.
The Disc
The main part of our model for CB 26 is a parametrised disc. The disc is seen as the radially extended luminous structure in the millimetre maps. The dark lane in the nearinfrared maps is another indication for a disc as it can be understood as the disc's shadow on the surrounding envelope.
We employ a parametric approach to such a disc which can be written as
ρ disc (r) = ρ 0 R * r cyl α exp − 1 2 z h 2(1)
where z is the usual cylindrical coordinate with z = 0 corresponding to the disc midplane and r cyl is the radial distance from this z-axis (see e.g. Wolf et al. 2003;Stapelfeldt et al. 1998). In our model, the parameter ρ 0 is determined by the mass of the entire disc. R * is the stellar radius and h, the vertical scale height, is a function of r cyl
h(r cyl ) = h 0 r cyl R * β .(2)
Here the quantities α, β, and h 0 in Equation (2) are geometrical parameters. These parameters allow us to adjust the disc structure and shape in order to fit the data. This modelling strategy has already been successfully applied to various other edge-on seen discs, such as the Butterfly-Star IRAS 04302+2247 (Wolf et al. 2003), HK Tau (Stapelfeldt et al. 1998), IM Lupi , and HV Tau .
Integrating Equation (1) along the z axis yields the surface density
Σ(r) = Σ 0 R * r cyl p .(3)
Comparison with Equation (1) yields the following relation between the exponent of the surface density power law and the geometrical parameters of the ansatz we use (2001) The radial size of the disc r out is another parameter and is mainly determined by the size of the elongated structure in the millimetre maps. As Hughes et al. (2008) argues, the millimetre continuum does not trace the outermost region of the disc. However, in the case of CB 26, the extent of the dark lane is also consistent with the outer radius estimate we get.
p = −β + α.(4)
The Envelope
In order to reproduce the pattern of scattered light we see in the observations in the I, J, H, and K bands, we add an envelope-like dust distribution to the model. The density within this envelope is understood to be orders of magnitude lower than that of the disc. The HST optical and NIR data show while the envelope has a high enough density to produce scattered light, it is orders of magnitude lower than the density of the circumstellar disc which is optically thick enough to obscure the central star completely. Further evidence for the low density in the envelope are the millimetre maps where the envelope is cannot be seen.
For the model of the envelope structure we follow the ideas of Ulrich (1976). We thus implement a model for a rotating envelope resulting from in-falling matter the same way as done by Whitney et al. (2003); Eisner et al. (2005):
ρ env =Ṁ 8π √ GM * r 3 1 + µ µ 0 − 1 2 1 2 µ µ 0 + r cf r µ 2 0 −1 . (5)
HereṀ is the dust in-fall rate, M * the stellar mass, r cf the centrifugal radius and µ = cos θ. The initial in-fall path of dust particles is given by µ 0 as r → ∞. As this is the only occurrence ofṀ and M * , the factorṀ (8π √ GM * r 3 ) −1 in Equation 5 is merely a coupling constant that scales the mass of the envelope just as ρ 0 does in case of Equation 1.
Hence, we are usingṀ (8π √ GM * r 3 ) −1 =ρ as a fitting parameter. To avoid introducing a constant likeρ with no direct physical interpretation into the discussion later on, we shall in this work refer to M * andṀ separately.
As a criterion to decide whether the disc dust distribution or the envelope dust distribution at a given point should be considered, we compare the two densities and choose the larger value:
ρ(r) = ρ disc (r) : ρ disc (r) ≥ ρ env (r) ρ env (r) : ρ disc (r) < ρ env (r) .(6)
In this manner, we embed the disc into the envelope and guarantee a smooth transition from the disc to the envelope without the need to alter the density structure of the optically thick, millimetre-glowing part of the object. For the radius of the complete model space we take twice the outer radius of the disc. Since we do employ a maximum size for the outer disc radius r out , the remaining space from the disc's edge to the end of the model space is readily filled by the envelope. With a computational domain going out to 2r out , we are able to model the scattered light from the envelope.
The Dust
Since gas is optically thin 5 in the wavelength regime we deal with, we limit ourselves to radiative transfer through the dust. For the mass relation of dust and gas we assume the standard value of MGas MDust = 100 which is in agreement with the findings of Glauser et al. (2008) in another disc surrounding a low-mass T Tauri star. Therefore, it is the dust whose density structure is described by Equations (1) and (5) in the disc and the envelope, respectively. The dust grain properties in our model can be divided in three groups: The shape of the dust grains, their chemical composition, and their size distribution.
Grain shape We assume the dust grains to be homogeneous spheres. Real dust grains, of course, are expected to feature a much more complex and fractal structure. As discussed by Voshchinnikov (2002), chemical composition, size and shape of dust grains cannot be determined separately, but only as a combination. We therefore limit our model to the less complex but also less ambiguous approach of spherical, non-aligned and non-orientated dust grains.
Grain chemistry For the chemical composition of the dust grains we employ a model that incorporates both silicate and graphite material. This grain model has already been used to model the "Butterfly star" by Wolf et al. (2003). For the optical data we use the complex refractive indices of "smoothed astronomical silicate" and graphite as published by Weingartner & Draine (2001). Since the longest wavelength considered in our modelling is 2.7 mm, we extrapolate the refractive indices to that wavelength. This is readily done since for this wavelength regime both the real and the imaginary part of the refractive index show asymptotic behaviour. For graphite we adopt the common " 1 3 − 2 3 " approximation. That means, if Q ext is the extinction efficiency factor, then
Q ext,graph = 1 3 Q ext (ǫ ) + 2 3 Q ext (ǫ ⊥ ),(7)
where ǫ and ǫ ⊥ are the graphite dielectric tensor's components for the electric field parallel and orthogonal to the crystallographic axis, respectively. As has been shown by Draine & Malhotra (1993), this graphite model is sufficient for extinction curve modelling. Applying an abundance ratio from silicate to graphite of 1 × 10 −27 cm 3 H −1 : 1.69 × 10 −27 cm 3 H −1 , we get relative abundances of 62.5% for astronomical silicate and 37.5% graphite ( 1 3 ǫ and 2 3 ǫ ⊥ ).
Grain sizes For the grain size distribution we assume a power law of the form n(a) da ∼ a −3.5 da with a min < a < a max .
Here, a is the dust grain radius and n(a) the number of dust grains with a specific radius. For a min = 5 nm and a max = 250 nm this distribution becomes the commonly known MRN distribution of the interstellar medium by Mathis et al. (1977). We choose those values as the starting point of the present study.
To model the different grain sizes and chemical populations, one has to consider an arbitrary number of separate dust grain sizes within a given interval [a min : a max ]. But the observables derived from radiative transfer considering each grain species separately, are close to the observables resulting from radiative transfer (RT) simulations based on weighted mean dust grain parameters of the dust grain ensemble (Wolf 2003a). Thus, we use weighted mean values for the efficiencies factors, cross sections, albedo, and scattering matrix elements. For each dust grain ensemble, 1000 logarithmically equidistantly distributed grain sizes within the interval [a min : a max ] have been taken into account for each chemical component in the averaging process.
There are arguments that the grain size can not be governed by a power law as in Equation (8). Furthermore, in the complex environment of a circumstellar disc, we expect dust settling and grain growth to make the grain size distribution quite dependent on the location within the disc.
Unfortunately, a consideration of grain sizes that takes into account the effects of e.g. dust settling requires more than just one parameter as Equation (8) does. Given the data we have, we are not able to disentangle these parameters in our study. Therefore, we assume that the power law distribution (8) to be valid in the whole disc and envelope structure and only use the maximum grain size as a parameter in our modelling efforts.
It will turn out that even with these simplifying assumptions we are able to model observational data of the system, although different a max in the disc and envelope have been found -but for other objects -in the past (e.g. Wolf et al. 2003).
We assume an average grain mass density of ρ grain = 2.5 g cm −3 . This density does not have any influence on the optical properties of the dust, as they are governed by the chemical composition and the particle size of the grains. The average grain mass density merely controls, together with the disc mass and particle size, the number of dust grains in the disc.
Heating Sources
There are two sources of energy for the disc that need our attention. The disc can be heated by stellar radiation and/or accretion of in-falling matter. Accretion Heating Our model involves a parameter with the dimensionality of an "accretion rate":Ṁ in Equation (5) as part of the description of the envelope structure. However, this quantity is really an in-fall rate within the envelope and may not necessarily at the same time describe mass accretion onto the star itself. But it is in general the latter mass flow that, as e.g. in FU Orionis like objects, accounts for significant contributions to the system's luminosity. For a T Tauri like system as CB 26 we thus neglect accretion as a major source of energy. As our study shows,Ṁ is rather small, around ∼ 10 −8 M ⊙ yr −1 . This is small compared to FU Orionis objects but average for T Tauri Stars and strongly supports our ansatz.
Besides matter in-fall in the envelope, accretion in the disc might be an important source of energy. As shown by Wolf et al. (2003), the accretion luminosity from within the disc is about two of magnitude smaller than the stellar contribution. As the model setup here is similar, we neglect accretion heating as a significant source of energy.
Stellar heating
The discussion above leaves the star as the only primary source of energy in our model. Its radiation heats the dust which then in turn itself re-emits at longer wavelengths. In this sense the disc in our model is passive. That is, we neglect accretion or turbulent processes within the disc as a possible other primary energy source.
We do not observe the star directly. This has mainly two reasons:
1. In the far infrared and at longer wavelengths, where the disc becomes less opaque with increasing wavelength, the contribution to the spectral energy distribution of the dust is orders of magnitude larger than of the star. 2. In the optical, near, and mid-infrared bands, the disc becomes opaque and the star is shielded from our direct view.
Therefore, we have to assume the stellar parameters, that is temperature and luminosity, since we are not able to derive them directly from observations. Observations only hint at a luminosity being L ≥ 0.5 L ⊙ . As a starting point we choose an "average" T Tauri star as described by Gullbring et al. (1998). This star has a radius of r = 2 R ⊙ and a luminosity of L * = 0.92L ⊙ . Assuming the star to be a black body radiator, this yields an effective surface temperature of T eff = 4000 K . Both of these parameters have been kept fixed in our parameter study to avoid degeneracies between parameters of the model. Except for the total flux, this choice has no impact on the near-infrared images.
As Natta (1993) showed, under certain conditions stellar light scattered back to the disc can have significant implications for the thermal structure of the disc. Here, the outer envelope regions 400 AU ≤ r env ≤ 1000 AU is shown to be of importance. An important assumption in the argumentation is the scattering phase function to be independent of the scattering angle. However, in our modelling framework the scattering phase function is highly asymmetrical and favours forward scattering by orders of magnitude. Thus, the amount of radiation scattered back to the disc from the envelope outside our model space, i. e. at distance larger then 400 AU can be neglected.
Means of modelling
In this section we discuss how we proceed with the aforesaid model. We discuss the free parameters of the model, their range, and the sampling of the resulting parameter space. Finally, we review the constraints imposed upon our model by the various observation and give a criterion for the bestfit model.
Radiative Transfer
For our continuum radiative transfer simulations we made use of the program MC3D (Wolf et al. 1999;Wolf 2003b). It is based on the Monte-Carlo method and solves the continuum radiative transfer problem self-consistently. It estimates the dust temperature distribution taking into account any heating sources, in our case the central star's radiation. It makes use of the temperature correction technique as described by Bjorkman & Wood (2001), the absorption concept as introduced by Lucy (1999) and the enforced scattering scheme as proposed by Cashwell & Everett (1959). The optical properties of the dust grains (scattering, extinction and absorption cross sections, scattering phase function) and their interaction with the radiation field is calculated using Mie theory. Multiple and anisotropic scattering is considered. The phase function is highly asymmetrical (e. g. at the peak of stellar emission at λ = 0.7 µm one has cos(θ scatter ) = 0.86), strongly favouring forward-scattering.
In order to derive a spatially resolved dust temperature distribution, the model space has to be subdivided into volume elements inside which a constant temperature is assumed. Both the symmetry of the density distribution and the density gradient distribution have to be taken into account. For the present study, we use a spherical model space, centred on the illuminating star and an equidistant subdivision of the model in the θ-direction, whilst a loga-rithmic radial scale is chosen in order to resolve the temperature gradient at the very dense inner region of the disc. The required spatial resolution at the disc inner radius rim of our model ranges from 10 −4 AU up to 10 −1 AU and every grid cell outwards is 1% larger than its next inner neighbour.
The radiative transfer is simulated at 101 wavelengths. The first 100 wavelengths are logarithmically distributed in the wavelength range [λ min , λ max−1 ] = [50 nm, 2.0 mm]. The largest wavelength used is λ max = 2.7 mm.
With MC3D we compute observables from the model. These observables are then compared to the observed data in the quest for the best-fit model. Namely, the quantities we derive with MC3D from the model are 1. Images in the NIR, that is in the I, J, H, and K Band, 2. Images in the millimetre regime at 1.1 mm, 1.3 mm and 2.7 mm, 3. 101 points for the SED accordingly to the above wavelength distribution.
Constraints from observations
Facing the broad variety of available observational data, one has to point out what the main features are that we want to reproduce with our model. This also determines the criteria for the best-fit model. The case is simple for the spectral energy distribution. There we aim at reproducing the complete spectrum over three orders of magnitudes from the optical bands down to the millimetre regime. We can divide the maps of the disc in two major groups: the maps in the millimetre regime, and the maps in the near-infrared. Both groups trace different physical processes and different spatial regions of the object.
Millimetre maps For resolved images, the issue is not as simple as for the SED. Our model is rotational symmetric and thus does not provide for any related asymmetry as seen in observations. The morphology of the millimetre maps has its origin in the dust that is heated by the star and re-emits light at those wavelengths. Although the images at 1.1 mm, 1.3 mm and 2.7 mm are rather simply structured they impose two major features that constrain our models. These are 1. the peak flux and 2. the spatial brightness distribution.
Since in all three maps the beam size is larger or comparable to the vertical extent of the disc, we can not constrain the flux distribution on the z-axis, perpendicular to the disc mid plane. Any feature there is smoothed out by the beam. Therefore, we focus on reproducing the flux distribution along the midplane of the disc. All images are fitted in the image plane.
Maps in the near-infrared The four images in the nearinfrared show more structures and details than the millimetre maps. Besides the disc appearing as dark lane in the near-infrared also a complex, wavelength-dependent morphology of the surrounding envelope is seen.
Considering that the circumstellar disc CB 26 is located at the edge of a Bok globule, one realises that the environment of the disc can account for the majority of the sub-structure seen on these maps, yet the Bok Globule itself is not part of our model. Hence, we have to restrict ourselves to the following two points that we want to reproduce with our model:
1. the dependence of the width of the dark dust lane on wavelength and 2. the relative peak height of the brightness distribution above and below the dust lane.
We restrict our modelling efforts to a simple envelope structure and the above two points since it is hard to distinguish whether the appearance of the object in the observations is due to envelope or environmental structure. We also put no emphasis on the vertical width of the upper and lower lobe nor the exact morphology.
Quality of the fit
For each comparison between model and observation on the aforesaid points, we get an individual χ 2 i . The total χ 2 of one model is then just the sum over all the individual χ 2 s:
χ 2 total = 1 n χ 2 SED + mm−maps χ 2 i + NIR−maps χ 2 j (9)
Here, n = 8 as we have one χ 2 from the SED, three from the millimetre maps and four from the scattered light images. Based on χ 2 total we get from Equation (9), we give our modelling errors as the range where we can alter the parameter values without changing χ 2 total more than 10%. This value is rather arbitrary as there is no mathematical reasoning behind it. Yet, it has proved within our study to reflect quite well the adjustability of the model. Allowing for a larger variation of χ 2 total than 10% gives generally worse results.
Parameter space study
Based on the model outlaid in the previous sections, we are left with ten adjustable parameters to reproduce the characteristics as described in the previous section. If not stated otherwise, we choose the range of a parameter for our study based on modelling of other yet similar objects (see e.g. Wolf et al. 2003). Then we first sample that range of an individual parameter in four coarse steps, select the two best values and go with the same procedure to the next parameter. Secondly, we take the results as an indicator how to refine the stepping in which smaller range. This process is being iterated to reach our final results. As the starting values in our parameter space study we chose the values obtain for circumstellar disc IRAS 04302+2247 which has at first glance a similar appearance as the disc in CB 26.
In detail the parameters we have are 1. The exponents α and β which describe the radial density profile of the disc (see Equation (1) for details). From D' Alessio et al. (1999), we choose for the flaring parameter β = 1.25 and get then a corresponding value for α = 2.25 from the relation α = 3(β− 1 2 ), which is a result of accretion disc physics (see e.g. Shakura & Syunyaev (1973)). Those values are taken as a starting point and we then look for agreement between observations and modelling at and beyond those values in steps of 0.1 and 0.2, respectively. 2. The scale height h 0 of the disc at a given radius. In the following we will fix the scale height at a radial distance from the star of r cyl = 100 AU and consider the cases h 0 = 5 AU, 10 AU, 15 AU, 20 AU, and 25 AU. 3. From the density Equation (5) for the envelope we have the centrifugal radius r cf and 4. the coupling constantρ which is a function of the mass accretion rateṀ and the mass of the central star M * . For the centrifugal radius we probe values in the range 100 AU ≤ r cf ≤ 800 AU. As forρ, we chose for the accretion rate values from the inter-valṀ ∈ [10 −5 M ⊙ yr −1 ; 10 −10 M ⊙ yr −1 ] and calculateρ under the consideration of the stellar mass by Launhardt et al. (2008). In this paper a dynamical mass of M * = 0.5 ± 0.1 M ⊙ is derived. 5. The inner and outer disc radius. The inner radius was initially set to 0.1 AU, which is approximately the dust sublimation radius. However, we do not get good agreement especially with the 1.3 mm map unless we choose values of ∼ 45 AU. For the outer radius, we chose a value of 200 AU but also considered configurations with 150 AU and 250 AU. 6. The disc mass. We consider 7 different disc dust masses in the range from 1.0 × 10 −5 M ⊙ up to 4.5 × 10 −3 M ⊙ . 7. The disc's inclination is such that the system is seen almost edge-on. We restrict the range for θ to be part of our parameter space to values between 60 • and 90 • with a stepping of 1 • between 80 • and 90 • . 8. The maximum grain size a max . Grain growth is a major issue for protoplanetary discs as it is the first step towards the formation of planets from the dust in the interstellar medium. We allow for a maximum grain size of 1 mm.
For a summary of those ten adjustable parameters of our model and the range we covered in the study see Table 3. A sketch showing all components of our model is presented in Fig. 5. Table 3. Overview of parameter ranges and best-fit values. For the definition of the uncertainty see section 3.2.3. The first group of parameters contains purely geometric parameters, the second group physical parameters and the last group the inclination of the disc as seen by the observer as an observational parameter.
Results
The values of the parameters of our best-fit model can be found in Table 3. Our geometrical parameters α and β of the disc density structure yield with the Equation (4) a surface density power-law exponent of p = −0.8. In Fig. 6 the density and temperature distribution of the best-fit model is shown while in Fig. 7 radial profiles of the density and temperature distribution in the midplane and 20 AU above are shown. One can see, that the high density in the midplane on the inner rim provides enough opacity such that material behind it in the midplane is not being heated up directly by the stellar radiation. The highest dust temperature at the inner disc rim amounts to ≈ 90K and is reached ∼ 20 AU above the midplane. However, due to the high density in the midplane, the stellar radiation does not penetrate deep into the midplane which results in the steep temperature gradient. In the less dense upper (and lower) layers of the disc, the stellar radiation also heats more distant parts of the disc resulting in a less steep temperature gradient.
An average dust temperature ofT dust = 16 K is obtained from the temperature distribution by weighting it with the mass distribution. This goes nicely with Fig. 6 if one bears in mind that the bulk of the dust is located in the midplane and well shielded against stellar radiation by inner parts of the disc. High temperatures are only reached in a very narrow region at the inner disc rim and in the very low density regions of the disc and thus contributing little to the mass averaged dust temperature.
Since our dust grain model only uses refractive indices, an effective dust grain opacity can be calculated by rearranging
M dust = S ν D 2 κ ν B ν (T dust )(10)
and assuming gas-to-dust ratio of 100. Here, κ ν is the wavelength depended mass absorption coefficient, S ν is the observed flux, D the distance of the object, and B ν (T dust ) the Planck-function at a certain temperature. The calculation yields for our model a dust opacity of κ 1.3 mm = 0.26 cm 2 g −1 . This value is very close to the ISM dust opacity given by Draine & Lee (1984, 1987. Compared to opacities for coagulated dust grains and ice-coated grains Ossenkopf & Henning (1994); Beckwith et al. (1990), our model yields an effective dust grain opacity at the lower end of the range of commonly employed opacities. Fig. 4 shows the spectral energy distribution of the bestfit model in comparison with the spectral data. We achieve quite a good match to the observational data except for the optical wavelengths. Since the circumstellar disc is embedded in the Bok globule CB 26 we need to care about the dust outside our model space as well. Thus, we add a screen that mimics the effect of foreground extinction between the object and the observer. For this screen, we assume the extinction properties of interstellar dust grains. Such a screen is described in detail by Cardelli et al. (1989). Using A V as a parameter with a minimum value of 2, we find in our study that a screen with a visual extinction of A V = 10 can easily account for the missing flux in the optical. The result is shown in Fig. 4 as the dashed curve, whereas the dasheddotted line corresponds to the best-fit model without the screen.
It needs to be stressed, that this screen only applies extinction law to all observable quantities. It is not subject to any radiative transfer or thermal re-emission. We estimated the possible contribution to re-emitted radiation of a such a screen with A V = 10 composed of ISM grains at the same distance as CB 26 with a temperature of 16 K. We found the screen to be clearly optically thin (e.g. τ 1.3 mm = 8 × 10 −5 ) and has only enough mass to have about ≈ 1% of the observed flux in the millimetre regime.
Furthermore, the spectral energy distribution of the model shows that the contribution of the envelope is quite important for shorter wavelengths. In this regime, the main contribution to the spectral energy distribution comes from the envelope whilst in the radio regime the flux comes completely from the dust in the disc that glows at those wavelengths. Fig. 8 illustrates this.
Discussion
The following paragraphs are now dedicated for a more detailed discussion of some results of our model.
Grain size and growth
It needs to be pointed out, that we found a model capable of explaining all major elements of the observations without the need to increase the maximum grain size in our parameter study significantly. The maximum grain size of our best fit model is a max = 2.5 µm. While this is a factor ten larger than the smallest maximum grain size considered in our parameter space, the value found is only marginally larger Fig. 6. Upper plot: Contours of dust density distribution on a plane perpendicular to the midplane normalised to the peak density of 0.34 × 10 −8 g cm −3 . The contour levels are at 10 −7 , 10 −3 ,10 −2 , 5 × 10 −2 , and 2 × 10 −1 . Lower plot: Contours of the temperature distribution on the same plane as above. Contour levels are at 20K, 40K, 60K, 70K, and 80K. The maximum temperature is 90K. then upper grain sizes given for the ISM in the literature. In fact, this is only true for the dust in the the disc component of our model. The maximum grain size in the envelope is the same as in found in the ISM, a max = 0.25 µm. If the maximum grain size in the envelope were bigger, then the short wavelength part of the SED could not be reproduced.
It is certainly smaller by several orders of magnitude than the maximum grain size found in other disc models such as in the work of Pinte et al. (2008). There, a maximum grain size of a few millimetres has been found. In contrast, models in our parameter space featuring values of a max ≈ 1 mm fail to fit the SED in the millimetre regime as the slope of the model SED is not steep enough. Also, Fig. 7. Radial profiles of density (upper plot ) and temperature (lower plot ) in the midplane along r (solid line) and along z/r = 0.1 (dashed line). The latter profile is chosen in order to include the scale height h 0 at r = 100 AU. The maximum density is normalised to 1 (corresponding to 0.34 × 10 −8 g cm −3 ). Fig. 8. Contributions to the spectral energy distribution from disc (dashed line) and envelope (solid line). The transition from envelope to disc as the major source of radiation (re-emission and scattering) is at λ = 217 µm.
we did not succeed to reproduce the default value for the maximum grain size of a max = 250 nm of ISM. In particular, the model would be off by a factor of ten for the SED data point at 1.3 mm.
One needs to discuss why the slight change for a max from 250 nm to 2.5 µm allows for a fit in the millimetre part of the SED -at wavelengths three orders of magnitude larger than the largest grains in the model. Intuitively, one would expect the millimetre part of the SED to remain unaltered by a change of grain size at that level. However, this expectation is based on the assumption that the absorption efficiency of the grains C abs in the millimetre regime is also insensitive to a change in the grain size at the micrometre level. Yet, this only holds true for only two of the three dust species in the model. For the astronomical silicate and the graphite component with an alignment of the crystals' optical axis perpendicular to the propagation direction of the electromagnetic field C abs has the same slope in the millimetre regime for grain size distributions with a maximum grain size of 250 nm and 2.5 µm. But for the third dust species, namely the graphite component with the crystals' optical axis aligned with the electromagnetic field, this is different. Here, the slope of C abs is significantly larger for a max = 250 nm than for a max = 2.5 µm. This effect is large enough to dominate the sensitivity of the SED to changes in the maximum grain size even at the level discussed despite the fact that the dust species responsible for this behaviour has only a 12.5% share of the total dust. This is due to the fact that graphite is a far more effective absorber than silicate.
A look on the millimetre spectral index of the data yields α mm = 3.1 ± .27. The corresponding millimetre opacity slope is β mm = 1.1 ± .27, if the millimetre emission is assumed to be optically thin. A β mm = 1 and smaller is understood to indicate dust grain particles larger than in the interstellar medium to be present. A value of β mm = 2 is expected if only ISM grains were present in the disc. The latter is true only for grains whose absorption efficiency C abs behaves like silicate. The value of β mm we obtain from data and model of CB 26 close to what is expected for large grains despite having still only micrometre sized grains in the model is due to the unorthodox behaviour of the parallel graphite component on the one hand side and on the other hand side to the non-vanishing optical depth in the millimetre regime (e.g. τ 1.3 mm ∼ 0.6). Draine (2006) considered also the behaviour of the millimetre opacity slope β mm for dust mixtures of graphite and silicate. There, no dependency of the opacity index as in our work was found. However, in this analysis not crystalline graphite was used but the optical properties of amorphous carbonaceous solids instead.
We summarise that we do not need mm sized grains to model the circumstellar disc CB 26 but grains with a maximum grain size still close to what is found in the ISM. This is in contrast to the modelling of the Butterfly star (see Wolf et al. 2003) as well as for the circumstellar disc HH 30 (see Wood et al. 2002) where in both cases the authors found it necessary, to have their largest grains at least four times larger than the largest grain of the interstellar matter. Of course, this result is based directly on the choice of the grain model we made. For another model, especially one without graphite, larger grains might be needed to fit the observed SED. Yet, due to the poorly constrained dust composition of circumstellar discs -in particular in the disc interior -this degeneracy between dust model and grain size must remain. As the dust model used in this work is also used in the context of other studies of circumstellar discs such as the Butterfly star (Wolf et al. 2003), it is a reasonable choice as it keeps the models of similar objects comparable as they are built on common assumptions.
Inner hole
The most unexpected result of our modelling is the inner disc radius for the model. In Fig. 9 the solid line shows the flux profile along the disc midplane at 1.3 mm as seen by OVRO whilst the thin solid line is the PSF of the observation. At the centre of the disc, the profile shows a plateau in the brightness distribution. There are two explanations in stock for this dip. Fig. 9. Horizontal cut through the spatial brightness distribution at 1.3 mm. The thick solid line represents the OVRO observation, the dash-dotted lines gives the addition/subtraction of one σ, the dotted line indicates the 2σlevels, the dashed line corresponds to our best-fit model, the thin solid is the dirty beam of the observation.
First, the minimum could indicate that emitting dust in that region is present but not visible. If the optical depth in the midplane is sufficiently high, the flux contribution from the inner parts of the disc compared to the contribution of the optically thin parts on the disc's surface would be smaller.
A high optical depth can easily be reached in regions of high dust densities. For a given total disc mass, a disc with small inner radius yields higher densities than the model with the large inner radius. Hence, the smaller the inner radius in our model, the more matter we find to be close to the star and thus reaching higher optical depths in the inner disc regions. However, even for the smallest inner radius of our parameter space, 0.1 AU we did not reach an optical depth that obscures enough flux from the disc centre.
This behaviour would also be more obvious when compared with maps at shorter wavelengths since the optical depth increases with decreasing wavelength. But in the images at 1.1 mm and 2.7 mm we do not observe a dip at the centre of the disc. Unfortunately, the available images do not help us to conclude whether the absence of the dip is really an indicator for a big void in the disc. This is because the point spread function at those two images is far too large to resolve the feature (see Table 3.1). Wolf et al. (2008) reasoned this way in the case of IRAS 04302+2247 where they found a similar dip in the brightness distribution at λ = 894 µm but not at 1.3 mm.
The second possible cause for the spatial brightness distribution in the 1.3 mm map is the actual lack of dust in the inner region of the disc. Whilst we started with an inner radius in the order of magnitude of a few tens of the stellar radius, it was not possible to match the plateau structure of the 1.3 mm map. On this account, we allow the inner disc radius to be as large as 50 AU. Unfortunately, within our model and parameter framework we are not at liberty to increase the total mass, and hence the density at the disc centre because the flux at 2.7 mm sets already an upper limit for the total disc dust mass. This is because at this wavelength any model is optically thin and thus we see the total matter of the disc.
Another way to think about the issue is to consider the optical thickness of the disc at 1.3 mm. We have run one simulation of the disc with exactly the same parameter values as for our best-fit model except for the inner radius. For the comparison model we chose r in = 0.1 AU. For both runs we computed the optical depth along the line of sight from the observer through the disc. As a result, for the large inner radius we have τ 1.3 mm = 0.6 and for the small inner radius τ 1.3 mm = 1.9. In the first case we deal with an optically thin system. As for the second, this is not so clearly said. An optical depth of 1.9 means that the initial flux is reduced by a factor of e −1.9 = 0.15. A much larger value of τ along the line of sight would be required to hide all emitting dust in the central region and produce the observed plateau structure. In our study, we were not able to fit the millimetre profile with a small inner radius.
We are therefore forced to conclude, that in the millimetre regime we do see the entire disc. Hence, the plateau structure rises from the wide spatial separation of the inner rim from the star, whereas a disc with a small inner radius would have a central peak in the brightness distribution.
Based on this line of arguments, our model provides predictive power for high resolution images at wavelengths longer than 1.3 mm. Since we conclude that the plateau structure in the brightness profile is due to lack of dust, it should also be visible at longer wavelengths. There, the dust becomes ever optical thinner and thus provides us with literal insight inside the disc. Now, if the lack of flux in the disc centre would be due to an effect of optical depth, it should vanish with longer wavelengths and the brightness profile should have one central peak instead of a plateau-structure. Unfortunately, the image at 2.7 mm has a point spread function far too large to allow us to disentangle between the two possibilities. Future observations of the disc CB 26 with a high spatial resolution will provide a perfect opportunity to confirm this prediction. Fig. 10 shows what we expect the disc to look like at different wavelengths according to our model with the inner void. With the Atacama Large Millimeter Array (ALMA) it will be quite easy to confirm our findings.
It needs to be pointed out, that the spectral energy distribution from CB 26 itself does not hint on the large inner hole. Its general shape is similar to other edge-on seen disc's. This is because the inner regions of the disc, while they completely dominate the disc emission in the 1-20 micron regime, is completely hidden from view in the edge-on configuration. Scattered starlight from the outer disc and/or the envelope accounts for the short wavelength bump of the SED, irrespective of the amount of emission from the disc itself. Only the quality of the OVRO map at 1.3 mm provides us with the possibility (and the need) to conclude about the inner clearing.
A possible explanation for a large void with 90 AU in diameter as we found it here might be, that the disc in the Bok globule CB 26 is actually a circumbinary disc. Binarity is also a possible explanation for the rotating molecular outflow described by Launhardt et al. (2008). However, a detailed dynamical study is not the scope of this paper.
Of course, another idea might be that the dust in the inner region has already bean processed to planetesimals, or at least to bodies that are large enough to decouple from the disc an its dynamics and do not contribute to the mm-flux. However, a first indicator for a low age of the system is the fact that it is still deeply embedded in its parental cloud. Another indication to the young age of the system is the absence of dust grains larger than found in the interstellar medium. Hence, we do not expect that we may be dealing with a so called "transitional disc" 6 and a planet population in the centre that has already cleaned up those formerly dusty regions.
Disc mass
Within our model a rather high disc mass is needed to account for the observed flux in the millimetre regime. With a total of about 3 × 10 −3 M ⊙ and dust-to-gas ratio of
M Gas M Dust ∼ 100,(11)
we end up with a total disc mass of ∼ 0.3 M ⊙ which is close to the star's mass. Hence, we need to reconsider the mass we computed for the disc. The derived disc mass essentially builds upon the assumptions we had to made about the dust grain chemistry and shape.
6.3.1. The dust grain structure, temperature and density
The radiative transfer in our modelling is performed under the assumption of spherical dust grains to avoid equivocalities. A possibility to have a lower estimate of the dust mass is to dismiss this assumption. This very much complicates the radiative transfer calculations since we now need to take into account all the possible fractal grain shapes as well as the spatial orientation of every grain. So far, we do not have the abilities to do such a simulation. But we can make a gedankenexperiment on parts of the results.
Of course, one expects strong changes in the scattering behaviour of dust grains. But besides that, one also can think of a plenitude of dust grains with almost the same absorption cross section as spherical dust grains but with much less mass. Voshchinnikov et al. (2007) discusses very fluffy particles with a porosity up to 90%. We furthermore want to point out, that the grain density ρ grain is not one of the fitting parameters. The disc's mass is proportional to this density and the number of grains. We only can constrain within our study the disc structure, the grain size and their number, but not the density of one grain. For our investigation of the system we used ρ grain = 2.5 g cm −3 , but it might well be less than this value. In turn, this will alter our estimate for the disc mass by the same factor.
The snow line
For the estimation of the total disc mass, gas and dust, one makes use of the canonical relation (11). This relation assumes, that there is some gas and dust of the disc inside the snow line and some outside. The snow line indicates the largest radius for which the temperature in the disc is high enough to keep water from freezing onto the dust grains. This line is usually set at a radius where the disc temperature drops to 170 K. As reported above, the maximum temperature we reach within our disc is about 90 K, and the average is only 16 K. This means, that our entire disc is outside the snow line. Thus we need to adjust the dust to gas ratio (11) to about 50 as published by Kokubo & Ida (2002).
However, the very same dust model we made use of in this work was also build upon in other modelling projects (eg. Wolf et al. 2003). In order to allow comparison of our model for CB 26 with those models, we also give considerations about the disc stability when we keep the dust model.
Stability, Binarity & Dynamics
A criterion for a disc to become unstable was first shown by Toomre (1964). In order to allow self-gravity of the disc to take over, it requires to satisfy
Q = c s κ πGΣ ≃ 1(12)
where c s is the local sound speed, κ the angular frequency of the disc, G the gravitational constant and Σ the surface density. For values of Q smaller then unity, the disc is assumed to be gravitationally unstable whereas for Q larger unity the disc is supposed to be stable against gravitational collapse. As Fig. 11 clearly shows, in our model Q < 1 throughout the entire disc by a factor of two to five. Yet, the Toomre criterion only can be consistently employed for a system that is formed by a central star and a surrounding disc. As discussed above, the inner hole we find in CB 26 can be an indication for a binary. An example for such a system would be the young binary system GG Tau. Here, a dusty ring around the central stars has been observed by Guilloteau et al. (1999) with a total ring mass of 0.13 M ⊙ , which is about a factor of two small than the total mass of CB 26. Yet, having two stars and the disc, the discussion of stability of such a system is quite delicate and not the topic of the present paper. However, if one accepts to hold on to a single central star, there are still effects that we did not take care of in our model but might be important for the system's stability. Following the discussion in the paper of Gammie (2001), discs violating the Toomre criterion are not stable but nevertheless might be in a steady gravoturbulent state. The paper investigates gravitationally unstable thin Keplerian discs and concludes, that the actual outcome of the instability depends on the cooling time τ c .
However, the parametric ansatz we use does not account for any dynamical interaction within the dust. Therefore, it would be interesting to couple our radiative transfer code MC3D with hydrodynamic simulations of circumstellar discs in a future study.
The width of the dark lane & envelope structure
Another aim of our modelling efforts was to mimic the chromaticity of dust lane, which is narrowing with increasing wavelength. Fig. 12 shows cuts from north to south through the centre of the NIR maps from modelling and observations. Our model of the spherical envelope is quite successful in reproducing the overall flux at each wavelength. Also, wavelength-dependent width of the dark lane is correctly reproduced. Fig. 13 shows an overlay of the HST/NICMOS H-band image with the contours given by the best-fit model. The Fig. clearly shows, that the dark lane is well reproduced. Since our envelope model, Equation (5), is axial symmetric, we cannot expect the asymmetries of the lower lobe to be modelled as well. Thus, the model yields the seen discrepancy to the observation.
A second effect of the aforesaid disregard is that the lower lobe flux, which in the image is concentrated on the left hand side, is in the model distributed among the complete lower lobe. Since both fluxes are equal in magnitude we end up with a smaller maximum in the model which also can be seen in Fig. 13. Fig. 12. Vertical cuts at the horizontal centre through the near-infrared scattered light images. The solid line is from observational data, the dashed line from our best-fit model. For an illustration where those cuts are obtained in the respective maps, see Fig. 13. The plots are normalised to 1. In case of the I-Band, this corresponds to 1.3 × 10 −7 Jy/beam, 5×10 −6 Jy/beam in the J-Band, 2×10 −5 Jy/beam in the H-Band, and 2.2×10 −4 Jy/beam ('beam' refers to the FWHM area of the PSF). Fig. 13. Inverse H-Band image from HST/NICMOS. The contour lines are from our best-fit model. From the outer to the inner lines are they at 27%, 41%, 68%,of the peak height. The second dark lane is not reproduced by our modelling as we did not aim at envelope asymmetries. The PSF for those images is rather small ∼ 7 AU FWHM. The vertical line illustrates how the cuts were taken for Fig. 12. 6.4.1. Alternative model for the envelope Besides the rotating, in-falling envelope model as described in Equation (5), we also tested a spherical symmetric density distribution as a model for the envelope:
ρ env (r) = ρ 0,env r γ cyl(13)
This brings two parameters into the model, ρ 0,env and γ. With this parameters we are able to model the dependency of the dust lane wideness on wavelength as well as the overall SED. Yet, this approach has two major drawbacks: 1. The spatial flux distribution as seen in the I, J, H, and K images cannot be reproduced. In fact, we obtain a much more concentrated flux distribution above and below the dark lane than what we see in Fig. 3. Fig. 14 illustrates this. 2. In the SED appears a strong silicate emission feature between 8 µm ≤ λ ≤ 10 µm. We are not able to have the feature disappearing as required by observations except for the inclination approaching values 75 • > θ. But this clearly contradicts the edge-on nature of the system. Therefore, we discarded Equation (13) as a model for the envelope.
Inclination of the disc
Within our study, we considered two ways to get a hand on the disc's inclination. The first one is to infer it from the millimetre observations and the second one from the maps in the near-infrared. In Fig. 15 the spatial brightness distribution along a horizontal cut in the 1.3 mm map from the best-fit model at different inclinations is shown. For inclination values in the range of θ = [80 • ; 90 • ] the profile does not change considerably. For inclinations smaller than 80 • the profile gets lower. This is because for an observer the warm inner rim of the disc looks at exactly 90 • like a line and becomes more and more a circle as one goes from edge-on to face-on orientation. As long as the complete rim fits into the size of the point spread function, one cannot distinguish between the different inclinations. But as soon as the rim gets less ellipsoidal and is not longer within the scope of one beam, the flux in the centre decreases. This is what can be seen in Fig. 15. Thus, from the comparison of models at different inclinations with the observations, we can infer an value for theta in the aforesaid range of θ = [90 • ; 80 • ]. Naturally, this way of reasoning is only valid for optically thin systems. As outlined earlier, this is the case for CB 26.
The second possibility to conclude about the disc's inclination is to compare the relative peak heights from vertical cuts through the spatial brightness distributions in the near-infrared. Assuming a symmetric system one expects that both peaks exhibit the same height if the inclination is exactly edge-on. From an analysis of the actual peak heights (see Fig. 12) in the HST/NICMOS images we deduce an inclination of θ = 85 • ± 5 • which is in good agreement with the numbers we obtained from the millimetre maps.
Comparison with similar objects
In general our model has disc and envelope parameters that are comparable with those of the circumstellar environment of other young stellar objects. Yet, there are not many edgeon seen circumstellar discs in the sky for which the same richness of observational data is available for modelling as it is for CB 26. Hence, the number of comparable multiwavelength studies on edge-on discs is equally limited. Two objects, HH 30 and the "Butterfly-star", share a large number of features with the disc in CB 26. All three are seen circumstellar discs, seen almost edge-on. Wolf et al. (2003) has compiled a similar data set for a model for IRAS 04302+2247, the so called "Butterfly-star". The data set includes millimetre maps and high-resolution near-infrared images obtained with HST/NICMOS. Utilising the same techniques as in this study, the authors' main result is that the dust properties must be different in the circumstellar disc and in the envelope. Whilst a grain size distribution with grain radii up to 100 µm is required to reproduce the millimetre observations of the disc, the envelope is dominated by smaller grains similar to those of the interstellar medium. This is quite in contrast to our model, where we find grains with almost ISM-like properties are needed in both, disc and envelope. However, the millimetre maps of the model do not suggest any presence of a large inner void as the spatial resolution is not good enough. In a later investigation (Wolf et al. 2008) on the "Butterfly-star" with the Sub-Millimetre Array the authors as well discover a "dip" in the horizontal brightness distribution. According to the model however, in this case the effect is to be understood as an effect of the optical depth and not as an inner void.
HH 30 was identified as a circumstellar disc with a large inner void present by HH 30 by Guilloteau et al. (2008). The authors found a inner hole of radius r in = 40±5 AU and an outer disc radius of r out = 128 ± 3 AU. At this numbers HH 30 and CB 26 are quite comparable. The separation of two central stars is postulated to be about 15 AU, so this might also be good guess for CB 26. Yet, in contrast to our model for CB 26, no envelope structure is needed to explain the appearance of HH 30 in the near-infrared bands. This suggests that CB 26 is less dynamically evolved than HH 30.
Both systems drive a molecular outflow, but CB 26 shows clear signatures of outflow rotation (Launhardt et al. 2008), while HH 30 does not (Pety et al. 2006). The reason for this difference remains unclear, though the early dynamically state of CB 26 as compared to HH 30 may provide the key. Yet another difference between HH 30 and our object is that the presence of cm size grains is needed to model the value for β mm (Wood et al. 2002).
As a summary, we see that, although all three objects, CB 26, HH 30, and IRAS 04302+2247 feature the same structure, a detailed investigation as presented in this work shows, that they are systems at different evolutionary states.
Errors & caveats
We conclude this section by discussing simplifications and resulting possible sources of errors within the framework of our model which have not been mentioned so far.
A matter that has not been touched so far is the uniqueness of or model. Despite the simplifications we applied the volume of the parameter space is still to vast to be completely scanned in a single study. We therefore employed the above described step-by-step technique for the parameters to find our best-fit model.
For instance, we started to model the millimetre maps as for those the envelope structure is not a dominant contribution. Thus, we obtain information about the disc total mass as well as its inner and outer radius and the apparent absence of large dust grains. The exploration of the near infrared images then showed the requirement for a rotational envelope structure rather then a simple power law distribution. By treating the parameters in this independent and serial manner, the order in which they are fitted might become an important issue. For instance, the study could first have been focusing on the near infrared images and wideness of the dust lane. As we know from our experience from modelling other objects this wideness usually requires small grains at least in the envelope. Modelling the SED in the millimetre regime as a second step would still have lead us to the overall usage of ISM grains.
This raises the question, if our model is really a global minimum in the parameter space or just a local one and if there exists in that space a model, that reproduces the observations on CB 26 even better. We exclude this possibility. The model we obtained exhibits some unexpected features. Therefore, in the framework of our model set we thoroughly explored the range of those parameters upon which these features sensitively depend. For example, we did all the combinations for values of the inner radius and the scale height and disc mass. If adjust accordingly, all three can produce the plateau observed in the 1.3 mm image. However, only the larger inner radius prevailed. A small scale height might squeeze the dust tight enough for high optical depth, but clearly contradicts the wavelength dependence of the dark lane in the NIR images. The same holds for the maximum grain size. Whereas here we do not even have the possibility to mimic ISM behaviour by means of disc geometry.
As a remaining issue, we need to think about parameters not varied at all in our modelling effort. As explained in sections above, model assumptions such as spherical grains versus fractal grains can hold the key to the mass problem of the disc. However, this would not alter the model we have at hand, especially this does not provide a hint to the "real" global χ 2 -minimum if one thinks the model is trapped in a local minimum.
This might also be an issue for the stellar parameters. Varying the luminosity and effective temperature of the embedded T Tauri star in general changes the total energy throughput in the radiative transfer and the location of the peak of the stellar spectrum in the wavelength space. Deviations from our assumed "typical" T Tauri star of course will affect the numbers of our best-fit model. For instance, since the total mass critically depends on the flux in the millimetre regime and this flux in turn on the total energy provided by the central star. Also, the screen introduced to mimic interstellar extinction is affected by the choice of surface temperature. However, no choice of stellar parameters is able to affect the main conclusions of our model. These are the presence of ISM grains in the disc and the inner void.
Despite the discussed caveats, the fact we actually found a good model means that we do not need to call for complex physics, such as grain growth or dust settling. The data do not require this.
Summary
For a large span of wavelengths, we have compiled a high quality data set for the circumstellar disc in the Bok globule CB 26. We obtained images in the near infrared and in the millimetre regime as well as photometric data and spectra. Together with literature values, we have constructed a detailed model that allows interpretation of observations with one single set of parameters. The conclusions we obtained are multifarious:
1. In order to account for the brightness distribution in the 1.3 mm map we needed to include an inner hole with a radius of 45 AU. Observations to come, especially in the sub-millimetre regime are suitable to confirm our interpretation of a low optical depth along the line of site at 1.3 mm. 2. Our model very nicely reproduces the prominent chromaticity of the dark lane as seen in the near infrared images. 3. Based on the chosen dust composition (astronomical silicate, graphite) and the resulting opacity structure of the best-fit disc model, we find that the millimetre SED indicates that the grains feature the same grain size distribution and almost the same upper limit to the grain size as the interstellar medium. 4. The disc is massive with a total dust and gas mass of 0.3 M ⊙ under the assumption of spherical grains and ρ grain = 2.5 g cm −3 , compared to a mass of the central T Tauri star with 0.5 M ⊙ and therefore possibly, but not inevitably, unstable. We discussed that due to possible fractal structure and other effects the real disc mass may be small enough to have a stable system.
Fig. 1 . 2 5
12Inverse HST images obtained with ACS and NICMOS in the I, J, H and K band. The colour scale is ∼ S All four images have been rotated by −30 • in order to align the major axis of the dark lane with the horizontal axis. For the image scale a distance to the object of 140 pc is assumed (100 AU = 0.7 ′′ ). The image orientation is PA 330 • up, and PA 60 • to the left.
Fig. 4 .
4Spectral energy distribution. Data with error-bars from Table 3.2. The IRS spectrum is the solid line. The dashed-dotted line corresponds to the best-fit model. The dashed line is the best-fit model with the dust screen.
Fig. 5 .
5Sketch showing all components of our model and their spatial arrangement. All sizes are not to scale.
Fig. 10 .
10Predicted appearance of the inner region of the disc.
Fig. 11 .
11Contour lines of the Toomre parameter Q (see Equation 12). The lines, from right to left, are at levels of Q = 0.1, 0.3, 0.5, 0.7.
Fig. 14 .
14J Band image from a simulation with a power-law envelope structure (left) and from rotating envelope (right). See alsoFig. 1for the observation.
Fig. 15 .
15Dependency of the brightness distribution along a central cut on the inclination. All the curves for θ = 90 • , 85 • , 80 • overlay.
Table 1 .
1Overview of the beam sizes in the millimetre-mapsInstrument λ[mm] PSF (FWHM) [ ′′ ] Orientation
SMA
1.1
1.00 × 0.84
−59.3 •
OVRO
1.3
0.61 × 0.36
−53.8 •
OVRO
2.7
1.21 × 0.87
−58.3 •
Table 2 .
2Photometricdata points for CB 26. References: (1) Stecklum et al. (2004); (2) Launhardt et al., in prep.; (3)
Launhardt & Sargent
See www.cso.caltech.edu/dsos/DSOS MLeong.html 3 See www.submm.caltech.edu/~sharc/crush/index.htm 4 See www.submm.caltech.edu/~sharc/analysis/pmodel/
This implies that we neglect line emission and absorption by the gas. We do not aim at modelling those as almost throughout the entire disc, the dust is by far the dominant coolant of the disc. Hence, thermal equilibrium obtained in radiative transfer calculations based on dust only will provide a reliable description of the disc's thermal structure.
Jürgen Sauter et al.: The circumstellar disc in the Bok globule CB 26
Those discs are considered to be predecessors of evolved debris-type discs.
Acknowledgements. The authors thank all members of the GEODEteam for their help in this project. J. Sauter thanks Owen Matthews, Jens Rodmann, and Arjan Bik for enlightening discussions. This work is supported by the DFG through the research group 759 "The Formation of Planets: The Critical First Growth Phase". F. Menard thanks financial support from Programme national de Physique Stellaire (PNPS) of CNRS/INSU, France and from Agence Nationale pour la Recherche of France under contract ANR-07-BLAN-0221. This work has been supported by NASA funding from the Space Telescope Science Institute, HST general observer program 10603; and by NASA funding from the Jet Propulsion Laboratory, under Spitzer general observer program 30765. C. Pinte acknowledges the funding from the European Commission's Seventh Framework Program as a Marie Curie Intra-European Fellow (PIEF-GA-2008-220891). The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
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| []
|
[
"Detection of quantum phase boundary at finite temperatures in integrable spin models",
"Detection of quantum phase boundary at finite temperatures in integrable spin models"
]
| [
"Protyush Nandi \nDepartment of Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700009KolkataIndia\n",
"Sirshendu Bhattacharyya \nDepartment of Physics\nRaja Rammohun Roy Mahavidyalaya712406Radhanagar, HooghlyIndia\n",
"Subinay Dasgupta \nDepartment of Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700009KolkataIndia\n"
]
| [
"Department of Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700009KolkataIndia",
"Department of Physics\nRaja Rammohun Roy Mahavidyalaya712406Radhanagar, HooghlyIndia",
"Department of Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700009KolkataIndia"
]
| []
| Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. Here we identify a physical quantity that shows non-analytic behaviour at finite temperatures, when an interaction parameter is quenched across the line of quantum phase transition. This quantity under consideration is the long time limit of a form of quantum fidelity. Our treatment is analytic for XY chain and 2D Kitaev model and is numerical for a 3D Hamiltonian applicable to Weyl semimetals. arXiv:2111.11126v3 [cond-mat.stat-mech] 24 May 2022 | 10.1103/physrevlett.128.247201 | [
"https://arxiv.org/pdf/2111.11126v3.pdf"
]
| 248,366,740 | 2111.11126 | 4e5c55ccde5959b70063a5780db6565439691c39 |
Detection of quantum phase boundary at finite temperatures in integrable spin models
Protyush Nandi
Department of Physics
University of Calcutta
92 Acharya Prafulla Chandra Road700009KolkataIndia
Sirshendu Bhattacharyya
Department of Physics
Raja Rammohun Roy Mahavidyalaya712406Radhanagar, HooghlyIndia
Subinay Dasgupta
Department of Physics
University of Calcutta
92 Acharya Prafulla Chandra Road700009KolkataIndia
Detection of quantum phase boundary at finite temperatures in integrable spin models
Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. Here we identify a physical quantity that shows non-analytic behaviour at finite temperatures, when an interaction parameter is quenched across the line of quantum phase transition. This quantity under consideration is the long time limit of a form of quantum fidelity. Our treatment is analytic for XY chain and 2D Kitaev model and is numerical for a 3D Hamiltonian applicable to Weyl semimetals. arXiv:2111.11126v3 [cond-mat.stat-mech] 24 May 2022
The dynamics of quantum many-body system at non-zero temperatures has always been an intriguing area of study, primarily because of the interplay between the quantum and the thermal fluctuations [1][2][3]. The dominance of thermal fluctuation with increasing temperature makes the perception of quantum noise limited to low temperatures only [4][5][6][7]. The question is, whether a quantum phase transition (QPT), exclusively driven by quantum fluctuations at zero temperature, has any impact on the behaviour of the system at non-zero temperature and whether some physical quantity measured at finite temperature bears the signature of the QPT occurring at zero temperature [8]. Over the past decades, this issue has been addressed through the studies of quantum fidelity. At zero temperature, fidelity generally vanishes in the thermodynamic limit at a quantum critical point as on two sides of this point the ground state wave functions are structurally different (Anderson's orthogonality catastrophe) [9][10][11][12][13][14]. At finite temperatures, generalized forms of fidelity have been studied in different systems [15][16][17][18][19][20][21], and some of them do detect the QPT through non-analytic signature in their logarithms at low temperatures [15,16]. An important work in this direction is by Li, Zhang and Lin [22] who have calculated quantum coherence for XXZ chain using transfer matrix renormalisation group technique and could detect the presence of QPT at finite values of temperature. Recently, Hou et. al. [23] have studied at finite temperatures, a form of rate function for Loschmidt amplitude and detected the presence of QPT. However, all these works were limited to 1D systems and were not applicable to very high temperatures.
The objective of this paper is to look for a quantity that has a robust non-analytic behaviour at zero as well as finite temperature while moving across the quantum phase boundary through the quench of a parameter. At a temperature T , we perform a sudden quench of the Hamiltonian from H to H at time t = 0 and define quantum fidelity as
F t ≡ Tr [ρ t · ρ 0 ] Tr [ρ t ] Tr [ρ 0 ](1)
where ρ 0 is the density matrix at t = 0 and ρ t is the same after the system has evolved for time t under the Hamiltonian H . At zero temperature, this expression reduces to the usual expression for the probability, | ψ(0)|ψ(t) | 2 (where |ψ(t) is the normalized wave function at time t) called the Loschmidt echo and the logarithm of it shows singularities as a function of time, indicating a dynamical quantum phase transition [24]. However, at finite temperatures, there is no such singularity (See however [23]). Since logarithm of F t is proportional to the system size, we may define a measurable quantity called rate function as,
r(t, β) ≡ − lim N →∞ 1 N log F t(2)
The quantity which turns out to be useful is the long-time average of the rate function, defined as
r a ≡ lim τ →∞ 1 τ τ 0 r(t, β) dt(3)
We shall consider two integrable quantum spin models, namely the XY chain and the Kitaev model on a honeycomb lattice. Each of these Hamiltonians show a QCP at T = 0. We shall show analytically that the quantity r a shows a non-analytic behaviour at the QCP at any finite temperature just as at T = 0. Of course, there does not exist an actual quantum phase transition at T > 0 but our detector bears a signature of the zero-temperature QCP even at T > 0. One important strength of our detector is that for a d-dimensional lattice, the calculation of the relevant quantity boils down to the evaluation of a d-dimensional integral. This enables our method to be applicable to higher dimensional systems. In fact, we shall also show (numerically) that for a three-dimensional Hamiltonian applicable to Weyl semimetals, the quantity r a shows a non-analyticity at the phase boundary, although only for low temperatures. We shall now discuss the case of 2D Kitaev honeycomb lattice is defined as ,
H = i,j J α s α i s α j(4)
where i, j run over all the nearest-neighbouring pairs on the honeycomb lattice, α is 1 or 2 or 3 depending on the location of the sites, as shown in Fig. 1, and s α = σ α where σ are the Pauli spin matrices. This model contains three interaction parameters J α . It can be shown that in the vortex-free sector, this Hamiltonian can be written as a sum of commuting Hamiltonians [25][26][27]
H = q H q , H q = a q σ 3 + b q σ 1(5)
where each component of q = (q x , q y ) spans over a square lattice in the range (−π, π) and the coefficients are given by [28,29] a q = J 3 − J 1 cos q x − J 2 cos q y , b q = −J 1 sin q x + J 2 sin q y
The ground state shows a gapless phase in the region satisfying triangular inequality |J i − J j | ≤ J k ≤ J i + J j where i, j, k are cyclic permutations of 1, 2, 3. The gapless and gapped phases are separated by a phase transition line. The two phases are topologically different [25][26][27] and can be detected by studying Loschmidt echo [30]. We set J 1 = J 2 = 1 so that the critical line occur at J 3 = 2. We shall prove analytically that for a quench J 3 = J 0 → J 3 = J , the second derivative (with respect to J ) of the rate function r a diverges algebraically with an exponent 1/2 at the phase boundary J = 2. We shall consider only the vortex-free sector and comment on this aspect later. The decomposition of the Hamiltonian in the Eq (5) enables one to express the density matrix ρ = exp(−βH)/Tr[exp(−βH)] in terms of exp (−βH q ) where β is the inverse temperature scaled by Boltzmann constant. To calculate the exponential of H q , we write,
H q = λ q G q(7)
and exploiting the fact G 2 q is unit matrix, obtain the quantum fidelity of Eq.(1) and from there the rate function, as
r(t, β) = log 2 − 1 4π 2 q A q d q(8)
with
A q = log 1 + tanh 2 βλ q 1 − 2 sin 2 λ q t sin 2 (φ q )(9)
where prime refers to the post-quench Hamiltonian and φ q = θ q − θ q , where θ q is defined by
cos θ q = a q /λ q , sin θ q = b q /λ q(10)
and θ q by similar expressions with primed quantities. As mentioned earlier, at zero temperature, the tanh term is 1, and A q shows singularities as a function of time, but at finite temperature no such singularity occurs since the argument of the logarithm never vanishes. The long-time average of the rate function, as defined in Eq.
(3) can be calculated from Eq. (8) using standard results [31].
r a = 3 log 2 − 1 4π 2 q d q log(1 + α q ) − 1 2π 2 q d q log 1 + 1 − γ q sin 2 (θ q − θ q )(11)
where α q = tanh 2 βλ q , γ q = 2α q /(1 + α q ). A few subtle issues need to be discussed. (1) The angle θ (θ ) is undefined where λ (λ ) is zero in the q plane, but this fact will not spoil the integration in Eq. (8) because we may exclude small regions R and R around λ = 0 and λ = 0 respectively, from the integral and evaluate it in the limit R → 0, R → 0. (2) In Eq.
(3), the quantity λ τ → ∞ as τ → ∞ since the point λ = 0 is excluded from the integration. (3) The long time limit of fidelity has also been studied in Ref. [32], but our procedure of calculating the long-time limit is different from theirs. They have first calculated the longtime limit of fidelity and then taken the logarithm to get the rate function while we have taken the long-time limit of the rate function itself, since the experimentally measurable quantity is the rate function [33] and not the fidelity F t . (4) When ρ t in Eq. (1) is replaced by the equilibrium density matrix for the post-quench Hamiltonian, namely, ρ 0 (H ) = exp (−βH ) one gets a measure
∂ra / ∂J (b) J 0 1 2 −0.4 −0.2 0.2 0 ∂ 2 ra / ∂J 2 (c) J 1 2 −20 −10 0 0 −30 − ∂ 2 ra / ∂J 2 0.3/√x 2 -J 0.001 0.01 0.0001 1 10 100 (d) FIG. 2.
Rate function ra and its derivatives computed numerically from Eqs. (11,12,13) with J0 = 1 and inverse temperature β = 2. Non-analyticity appears only at the phase boundary J = 2. Fig. (d) shows the scaling of the second derivative as J approaches the value 2 from below. Also, in (d) the violet squares are obtained by integrating over the entire region −π < (qx, qy) < π while the green crosses are obtained for −0.1π < (qx, qy) < 0.1π. This shows that only the region near the origin is important for the nature of divergence.
study how the rate function r a depends on J when J approaches 2 from being within the gapless phase. Thus, the first and second derivatives of the rate function is obtained as (omitting the suffix q)
∂r a ∂J = − 1 4π 2 q d q λ BC (12) ∂ 2 r a ∂J 2 = − 1 4π 2 q d q λ 2 − 1 + 1 2D B 2 C 2 −2αB sin θ sin(3θ − 2θ)](13)with B = (2 − γ)/(D + D 2 ), C = α sin θ sin(2θ − 2θ) and D = √ [1 − γ sin 2 (θ − θ)].
Numerical integration shows that when J approaches the phase boundary from below, there appears a nonanalyticity at any finite temperature -the rate function shows a kink, the first derivative remains continuous but undergoes a change of slope, while the second derivative shows power-law divergence with exponent 1/2 (Fig. 2). The expressions for the first and second derivatives contain 1/λ and 1/λ 2 respectively in the integrand. Indeed, whenever J lies within the gapless phase, the region of integration includes a point q = q c where λ vanishes. However, the presence of this point leads to a non-analyticity only when J is on the phase boundary (see the Supplemental Material).
We now set out to study analytically the behaviour of the rate function as a function of the post-quench parameter J. Instead of using Eqs. (12, 13), we shall rather start with the expression of r a as in Eq. (11). Using the power series expansion of log 1 + √ 1 − x for any x in the range 0 < x < 1, we obtain (14) with c 1 = 1, c 2 = 3 8 , c 3 = 5 24 etc. We now express the integrand as
r a = log 2 − 1 4π 2 q d q log(1 + α) + 1 8π 2 n=1,2,··· c n d q γ n sin 2n (θ − θ )γ n sin 2n (θ−θ ) = tanh βλ λ 2n 2(J − J 0 ) 2 1 + tanh 2 βλ n b λ 2n(15)
and observe that any non-analytic behaviour of this function may arise, if at all, only from a small region around the point where λ = 0. The location of this point is given by q = (q c , q c ) with q c = cos −1 (J/2) for J ≤ 2. Around this point λ ≈ |J 0 − J| and when J is close to 2 from below, we obtain
r a = log 2 − 1 4π 2 q d q log(1 + α) + 1 8π 2 n c n d q (J − J 0 )b λ 2n(16)
where c n = c n [2 tanh 2 β(J 0 − 2)/(1 + tanh 2 β(J 0 − 2))] n . Indeed, this equality will not work away from the phase boundary. Numerical results also support this equality (Fig. 2). Hence, we only need to calculate the integral
I n ≡ (J −J 0 ) 2n π qx,qy=−π dq x dq y b λ 2n
, n = 1, 2, · · · (17) As we are interested only in the behaviour of rate function when the post-quench parameter J approaches the value 2 within the gapless phase, we introduce a parameter by J = 2 − 2 , = √ 2 − J and express I n as a power series:
I n = a 0 + a 1 + a 2 2 + a 3 3 + · · ·(18)
It can be shown that (see the Supplemental Material) upto leading order in , for any value of n, a 1 = 0 and a 3 = 0. In view of Eq. (16), this proves that at any temperature,
∂ 2 r a ∂J 2 ∼ 1 √ 2 − J(19)
We mention here that a quantity related to fidelity has been previously observed to show logarithmic divergence in zero temperature [36]. We also mention some related is observed when it is studied as a function of J 0 (which is not surprising, since the right-hand side of Eq. (15) does not show any non-analytic behaviour at λ = 0).
(iii) If we approach the phase boundary keeping J > 2, indeed we observe a non-analyticity but the nature of singularity is different from the one for J < 2. (iv) As we have remained within the vortex-free sector of the Kitaev model, a question arises as to whether the excitation of vortices at finite temperatures destroys the singularity in the rate function r a . In this connection, we note that, the expression of our rate function Eq. (16) has the form of a series, each term of which is an integral with a pre-factor. The pre-factor involves temperature but not the post-quench value of the parameter, while the integral involves pre-and post-quench value of the parameter but not temperature. Also, it has been shown [37] that the vortex excitation in 2D Kitaev model being adiabatic with temperature, does not induce any phase transition.Hence, the pre-factor will not show any nonanalytic behaviour as a function of temperature, and it is expected that the rate function will not also show any singularity at a finite temperature due to the presence of vortex excitations. However, for 3D Kitaev model the situation is different since the excitation of vortices triggers a thermal phase transition here. This particular model therefore opens up a future direction of study [38].
It is interesting to consider the case of zero temperature. The rate function in Eq. (11) is now,
r a = 2 log 2 − 1 2π 2 q d q log 1 + | cos(θ q − θ q )|(20)
This rate function shows a kink at the phase boundary only, irrespective of whether the pre-quench parameter J 0 is in the gapless phase or in the gapped phase (Fig. 3). It has been shown [30] that the Loschmidt probability as a function of time shows peaks wrongly for quenches within the gapless phase. Hence, we conclude that after taking the long time limit the rate function r a correctly shows a kink only at the phase boundary.
We shall now turn to XY Hamiltonian defined by,
H XY = − 1 2 (1+h) N i=1 s 1 i s 1 i+1 − 1 2 (1−h) N i=1 s 2 i s 2 i+1 −Γ N i=1 s 3 i (21)
where h is the anisotropy parameter, Γ is the transverse field, and s α = σ α , the Pauli spin matrices. It can be shown by using Jordan-Wigner transformation that this Hamiltonian can be written as a sum of commuting Hamiltonians [3,39]
H = q H q , H q = a q σ 3 + b q σ 1(22)
where q spans over (0, π) and the coefficients are given by a q = Γ + cos q, b q = h sin q. The line −1 < Γ < 1, h = 0 separates two ordered phases and the lines Γ = ±1 separate the ordered and the disordered phases. Using the expressions for a q and b q , one can define θ q from Eq. (10) and obtain the rate function and its derivatives from Eqs. (11,12,13) noting that q should now be integrated from 0 to π. Numerical integration shows that (see the Supplemental Material for figures) (i) for a quench from h = h 0 → h = h (with any Γ in the range −1 < Γ < 1) the first derivative (with respect to h ) of the longtime rate function r a shows a discontinuity at the QCP h = 0 and (ii) for a quench from Γ = Γ 0 → Γ = Γ at any h, the first derivative (with respect to Γ ) of r a shows a discontinuity at the QCP Γ = ±1. Using the approximation in Eq. (16), one can calculate the amount of discontinuity at infinitely large temperature (see the Supplemental Material).
(∂r a /∂Γ ) Γ =1+ − (∂r a /∂Γ ) Γ =0− = β 2 (1 − Γ 2 0 )/h (23) (∂r a /∂h ) h =0+ − (∂r a /∂h ) h =0− = 2β 2 h 2 0 (1 − Γ 2 ) (24)
We shall now turn to 3D Hamiltonians. The Hamiltonian for Weyl semimetals with broken time reversal symmetry can be written as [40,41],
H q = a q σ 3 + b q σ 1 + c q σ 2(25)
where a q = J 3 − cos q x − cos q y − cos q z , b q = sin q x , c q = sin q y , and q runs over a simple cubic lattice in the range (−π, π). The ground state of this Hamiltonian shows a gapless phase for J 3 < 3 and a gapped phase for J 3 > 3. We consider a quench J 3 = J 0 → J 3 = J and note that one can arrive at Eqs. (8,9) in a straightforward manner, with φ q being the angle between the unit vectors (b q /λ q , c q /λ q , a q /λ q ) and (b q /λ q , c q /λ q , a q /λ q ). The evaluation of the rate function r a now reduces to the computation of an integral in the q space and we observe numerically that the first derivative (with respect to J) of r a shows a change of slope at the QCP J = 3 both for J 0 < 3 and > 3 (see the Supplementary Material for figures). It is important to mention that, unlike the previous two cases, this singularity is visible only at low temperatures. When the time reversal symmetry is not broken, c q = 0 in Eq. (25) and one has the topological nodal line semimetals [42]. In this case also, one observes a change of slope at J = 3 of the curve ∂r a /∂J vs J.
To conclude, we explore the finite temperature behaviour of three integrable quantum spin models and observe a non-analyticity in the mixed state fidelity at the phase boundary. The rate function can be written (Eq. (16)) as a series, each term of which is an integral independent of temperature with a pre-factor involving temperature. It is the integral from which the non-analytic behaviour originates(at all temperatures in 2D and at low temperatures in 3D). The fact that the quantity in question is insensitive to thermal fluctuations, makes it a potential candidate to be studied experimentally as a good detector of quantum phase transition. It would be interesting to explore how our rate function behaves for other integrable and non-integrable Hamiltonians. Work in this line is under progress. The quench under consideration is from (J 1 , J 2 , J 3 ) = (1, 1, J 0 ) to (1, 1, J) and we have from Eq. (13)
r = ∂ 2 r a /∂J 2 = π qx,qy=−π Fdq x dq y with F = 1 4π 2 λ 2 1 + 1 2D B 2 C 2 + 2αB sin θ sin(3θ − 2θ) (S1)
According to Eq. (10)
λ cos θ = J 0 − cos q x − cos q y , λ sin θ = − sin q x + sin q y , λ cos θ = J − cos q x − cos q y λ sin θ = − sin q x + sin q y (S2)
We first observe that if q is replaced by − q, the quantities θ, θ becomes −θ, −θ . Also, such reversal of sign of θ, θ keeps F invariant. Thus, it is sufficient to integrate only over the region −π < q x ≤ π, 0 ≤ q y < π (shaded region in Fig. S1a) and write
∂ 2 r/∂J 2 = 2 π qx=−π π qy=0
Fdq x dq y (S3)
For any J consider a circle of radius R around the point where λ = 0 (We shall call this point as node). The location of this node is given by q xc = q yc = cos −1 (J/2). The quantity F diverges at the node and decreases as one moves away from it. When J lies well inside the gapless region, the shaded integration region covers such circles for R = 0 to a fairly large value. However, when J is close to the phase boundary, the shaded region cannot cover the whole circle unless the radius R is too small (Fig. S1). It is seen numerically (but is difficult to verify analytically), that the values of F plotted over a full circle sums up to zero, although being asymmetric about zero. Hence after integration, r vanishes when J is well-inside the gapless region. However, when J is close to the phase boundary, the integration is over part of a circle and the cancellation of F values does not occur. Hence, r does not vanish. This effect is aggravated by the fact that F attains numerically larger values over a wider region when J is close to the phase boundary, as can be easily seen from Fig. S1.
EVALUATION OF THE INTEGRAL In
To evaluate the integral I n as defined in Eq. (17), let us transform to the variables u = (q x + q y )/2, v = (q y − q x )/2 and set J 0 = 1. This gives, π ω qy (a) 3 However, when J 0 3 is close to the phase boundary, the integration region contains truncates circles for small r (Fig. 1). It is seen numerically (but is di cult to verify analytically), that the values of F plotted over a full circle is asymmetric about zero but sums up to zero. Hence after integration, r 00 vanishes when J 0 3 is well-inside the gapless region. However, when J 0 3 is close to the phase boundary, the integration is over part of a circle and the cancellation of F values do not occur. Hence, r 00 does not vanish. This e↵ect is aggravated by the fact that F attains numerically larger values (WHY? DUE TO WHICH TERM?) when J 0 3 is close to the phase boundary. qx π -π -π π ω qy -π 2 6 2 02 = 2r 2 sin 2 (1 + F ) 2 + (✏ cos + 1 2 r) 2 ,
2 = 02 + (1 ✏ 2 ) 2 cos(✓ ✓ 0 ) = aa 0 + bb 0 0 = 02 a 0 (1 ✏ 2 ) 0 sin(✓ ✓ 0 ) = b 0 (1
The region of integration becomes (see Fig. 2)
Z ⇡ ⇡ dq x Z ⇡ ⇡ dq y = Z 0 dr Z ⇡ ⇡ d + Z dr Z 0 0 d
where 0 = cos 1 (✏/r), and is some number larger than ✏ and smaller than 1.
Retaining dominant terms in each expression, (2) Why the curve for r 00 is not smooth for large beta ?
b 0 = p 2r sin , a 0 = r( p 2✏ cos + 1 2 r), 02 = 2r 2 sin 2 + (✏ cos + 1 2 r) 2 , b = p
This seems to be a nasty and tricky problem and we have to try some reliable package for numerical integration using Matlab or Mathematica. As ! 1, ↵ ! 1 and p (C 2 D 2 ) ! 2| cos |. Therefore, the integrand of r 00 diverges both for 0 = 0 and cos = 0. It is found that cos = 0 vanishes at many places on a closed line (Fig. 2). We could not achieve desired numerical precision by increasing the precision of numerical integration. This problem does not occur for the first derivative r 0 , since there the cos term in the denominator gets cancelled by a sin 2 term in the numerator, and one is left with sign(cos ). Thus, the integrand of r 0 reverses sign but remains finite at the zeros of cos . One must note that when is not large, C D = 1 + ↵ cos 2 and p (C 2 D 2 ) does not vanish anywhere.
Also near = 0 point, ↵ = tanh( ) is not close to 1 unless is very high (say 1000). When = 5, ↵ is not close to 1 over an appreciable region, and hence zeros of cos will not matter. Profile of F, the integrand of ∂ 2 ra/∂J 2 at β = 2. We consider J = 1.90 and 1.98 for which the node is at qxc = qyc = 0.32 and 0.14 respectively. (a) Circle around the nodes for two J values. The points on the circle have polar angle ω measured with respect to the qx axis. Note that the region of integration runs over positive values of qy. For J somewhat smaller than 2, the circle lies entirely within the qy > 0 region while for J close to 2, part of the circle goes outside this region. (b) Variation of F over the two circles of radius R = 0.2. For J = 1.90, the circle is in the qy > 0 region and the positive and negative values nearly cancel out, while for J = 1.98 such cancellation does not occur since part of the circumference is excluded. Furthermore, the magnitude of F is larger in the latter case.
is to express the integrand as For u < q c , the poles inside the unit circle are z 1 = 0 and z 2 = J/µ, while for u > q c , the poles inside are z 1 = 0 and z 3 = µ/J. Denoting the residue of F (n) at z i as R (n) i for i = 1, 2, 3,
I n = 8π(J − 1) 2n (4J) n 3 i=1 π/2 0 R (n) i du(S7)
We define = √ 2 − J so that q c = cos −1 (1 − 2 /2) = + 1 24 3 + · · · and express I n as a power series:
I n = a 0 + a 1 + a 2 2 + a 3 3 + · · · (S8) so that
∂ 2 I n ∂J 2 = − a 1 4 · 1 (2 − J) 3/2 + 3a 3 4 · 1 √ 2 − J + · · ·
It is important to note that the divergence behaviour is controlled by the coefficients a 1 and a 3 only. At high temperatures, only the first term in the summation in Eq. (14) is dominant and the residues of the integrand
FIG. 3 .
3results of interest: (i) If the pre-quench value of J 0 is chosen to be in the gapped phase, the divergence with respect to variation of J remains unchanged (see the Supplementary Material). (ii) No singularity in rate function Rate function ra and its derivatives computed from Eq.(20) at zero temperature with the pre-quench parameter J0 = 1 (in gapless phase) and J0 = 3 (in gapped phase).Non-analyticity appears only at the phase boundary J = 2.
the invariance of the integral under u → π ± u, v → π ± v. Recall that θ and θ are defined by Eq. (10), and the expressions for a and b are now a = 1 − 2 cos u cos v, b = 2 cos u sin v = b , a = J − 2 cos u cos v. The crucial step
FIG. 1: (a) Circle (of radius r = 0.01) around the nodes for J 0 3 = 1.5 and 1.95. T measured with respect to the qx axis. Note that the region of integration runs ove over the two circles.
Circle (of radius r = 0.01) around the nodes for J 0 3 = 1.5 and 1.95. The points on the circle have polar angle ! measured with respect to the qx axis. Note that the region of integration runs over positive values of qy. (b) Variation of F over the two circles.
FIG. S1. Profile of F, the integrand of ∂ 2 ra/∂J 2 at β = 2. We consider J = 1.90 and 1.98 for which the node is at qxc = qyc = 0.32 and 0.14 respectively. (a) Circle around the nodes for two J values. The points on the circle have polar angle ω measured with respect to the qx axis. Note that the region of integration runs over positive values of qy. For J somewhat smaller than 2, the circle lies entirely within the qy > 0 region while for J close to 2, part of the circle goes outside this region. (b) Variation of F over the two circles of radius R = 0.2. For J = 1.90, the circle is in the qy > 0 region and the positive and negative values nearly cancel out, while for J = 1.98 such cancellation does not occur since part of the circumference is excluded. Furthermore, the magnitude of F is larger in the latter case.
= exp(iv) and µ = 2 cos u. This givesI n = − 4i(J − 1)
model in detail and then go over to XY chain and the Hamiltonian for semimetals.The Hamiltonian of the Kitaev spin-1/2 model on a
2
J1 , J2 , J3 = J0
J1 , J2 , J3 = J
FIG. 1. Kitaev model on honeycomb lattice. The continuous,
dashed and dotted lines correspond to xx, yy, and zz inter-
actions respectively. We study a quench, where J1 and J2 are
kept unchanged and J3 is changed instantaneously from J0 to
J. For most of the results here, we shall keep J0 = 1.
of fidelity[34,35] different from ours, since the Hamiltonian being integrable, the t → ∞ limit of ρ t is not the same as ρ 0 (H ).In this work, we shall study a particular type of quench where the interaction parameters J 1 and J 2 of the Kitaev Hamiltonian Eq. (4) is kept fixed at 1, so that the gapless phase exists for 0 ≤ J 3 ≤ 2 and the gapped phase, for J 3 > 2. We quench the parameter J 3 from J 0 to J and
Protyush Nandi, 1, * Sirshendu Bhattacharyya, 2, † and Subinay Dasgupta 1, ‡
Acknowledgement: PN acknowledges UGC for financial support (Ref. No. 191620072523).(z − z 2 ) n F (n) = G n z and observe thatwhere T contains the other terms that will necessarily involve G k or G k /z as a factor with k ≥ 1.Obviously, (G/z) z=z2 = R2 . Also,Hence, upto leading order terms in ,Using Eqs. (S13) and (S12) in Eq. (S11) and then performing the integration over u in Eq.(21)we can conclude that the second integral in Eq. (S7) contributes an 3 term but no term linear in . Similar conclusion will hold for the residue at z = z 3 too. This implies that a 1 = 0 and a 3 = 0, not only for n = 1, but for any n. In view of Eqs.(16), this completes the proof that at any temperature. First derivative of rate function computed numerically from Eqs.(11,12)for quench of the anisotropy parameter at β = 0.1, Γ = 0.5 in (a) and for quench of the transverse field at β = 0.1, h = 1 in (b), (c). For (b) Γ0 is within the ordered state, while for (c) Γ0 is within the disordered state. In both cases the quantity ∂ra/∂Γ is discontinuous at the critical point Γ = 1.RESULTS FOR KITAEV MODEL ON HONEYCOMBFIG. S3. Rate function ra and its two derivatives computed numerically from Eqs.(11,12)for quench of the parameter J3 at β = 2, J1 = J2 = 1 with the pre-quench value J0 = 3. A non-analyticity appears at J = 2 in the same way as inFig. 2.S7. Rate function ra and its two derivatives computed numerically from Eqs.(11,12)for quench of the parameter J at β = 5 in the case of 3D Hamiltonian with time reversal symmetry. For (a), (b) and (c), the pre-quench value is within the gapless phase, while for (d), (e) and (f) it is within the gapped phase. In both cases the quantity ∂ 2 ra/∂J 2 diverges at the phase boundary J0 = 2.
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| []
|
[
"Black Hole Spindown by Light Bosons",
"Black Hole Spindown by Light Bosons"
]
| [
"Andrei Gruzinov \nPhysics Department\nCCPP\nNew York University\n4 Washington Place10003New YorkNY\n"
]
| [
"Physics Department\nCCPP\nNew York University\n4 Washington Place10003New YorkNY"
]
| []
| The saturation mechanism for the fastest-growing instability of massive scalar field in Kerr metric is identified, assuming saturation by cubic or quartic nonlinearities of the field potential. The resulting spindown rate of the black hole is calculated. The (rather involved) saturation scenario is confirmed by numerical simulations. | null | [
"https://arxiv.org/pdf/1604.06422v1.pdf"
]
| 119,107,480 | 1604.06422 | ce1401609e92e57d27694a9e3346d8937e9d1274 |
Black Hole Spindown by Light Bosons
Andrei Gruzinov
Physics Department
CCPP
New York University
4 Washington Place10003New YorkNY
Black Hole Spindown by Light Bosons
arXiv:1604.06422v1 [astro-ph.HE] 19 Apr 2016
The saturation mechanism for the fastest-growing instability of massive scalar field in Kerr metric is identified, assuming saturation by cubic or quartic nonlinearities of the field potential. The resulting spindown rate of the black hole is calculated. The (rather involved) saturation scenario is confirmed by numerical simulations.
I. INTRODUCTION
We will assume that bosons with small masses exist and calculate how they spin down black holes. In practice, our results can be used to rule out bosons with certain masses and self-interactions, whenever a spinning black hole is observed. If, on the other hand, an otherwise unexplainable lack of spin of black holes in a certain mass interval is discovered, it can be interpreted as indirect evidence for a light boson [1,2].
II. SUPERRADIANCE
As explained by Zeldovich [3], rotating absorbing body (say a black hole) amplifies waves of frequency ω and angular momentum projection onto the spin axis m, provided mΩ > ω, where Ω is the angular velocity of the body. This leads to the so-called superradiant instability of massive boson fields in Kerr metric [4,5].
In Planck units, for the Kerr hole of mass M and dimensionless spin parameter a (a < 1), the fastest growing mode of the Klein-Gordon field φ of mass µ ≪ M −1 is the 2p state, with the growth rate
γ 1 = 1 24 aM 8 µ 9 .(1)
If the number of e-foldings in the black hole lifetime t H is large (γ 1 t H 100 amplifies the field from zero-point oscillations to the Planck scale values φ ∼ M P ), the exponential growth has to be terminated by the gravitational or φ radiation. The φ radiation dominates if the nonlinear scale of the φ field f ≪ M P . With this assumption, we can study the saturation of the instability by the nonlinearities of the φ field using the unperturbed Kerr metric and neglecting gravitational radiation.
III. NONLINEAR SATURATION OF THE INSTABILITY
The following (rather involved) saturation scenario has originally been seen in numerical simulations. The numerical results are presented in §IV. Here we The crucial feature of our scenario is the simultaneous presence and interaction of the 2p and 3d bound states. As relevant modes are approximately nonrelativistic, they can be analyzed in the Schrodinger equation approximation. Both cubic and quartic self-interactions of the φ field then give the same |Ψ| 2 Ψ nonlinearity in the Schrodinger equation.
• (A)
Let N 1 , N 2 be the number of particles in the 2p(m = 1) and 3d(m = 2) states. The dimensionless nonrelativistic energies (frequencies) of the modes are ω 1 = −1/8 and
ω 2 = −1/18. Let Ψ m ∝ N 1/2 m e −iωmt+imφ(2)
be the corresponding wave functions.
In the beginning, more precisely, after a few e-foldings, N 2 ≪ N 1 . The leading self-interaction is then |Ψ 1 | 2 Ψ 1 . This nonlinearity only reshapes the potential well, without affecting the fate of the instability.
The next to leading order interaction is Ψ 2 1 Ψ * 2 . This interaction excites a forced l = 0 oscillation of the wave function. The frequency of this oscillation is negative, 2ω 1 − ω 2 < 0, so the oscillation is trapped. As l = 0, the forced oscillation has a finite amplitude at small r and is therefore damped by the black hole.
The damping reduces the number of particles N , without reducing the angular momentum L. Schematically (with arbitrary normalizations):
N = 2γ 1 N 1 − N 2 1 N 2 (3) L = 2γ 1 N 1 ,(4)
where γ 1 is the Detweiler's growth rate (1). As the forced oscillation amplitude is small, the number of particles and the angular momentum are approximately equal to
N = N 1 + N 2 ,(5)L = N 1 + 2N 2 .(6)
It follows thatṄ
1 = 2γ 1 N 1 − 2N 2 1 N 2 ,(7)N 2 = N 2 1 N 2 .(8)
We see that in the presence of N 1 , the mode N 2 becomes linearly unstable with the growth rate ∝ N 2 1 . Formally, the system asymptotes to N 2 ≫ N 1 . So, even though we start with N 2 ≪ N 1 , the nonlinear interactions will try to reverse the situation. However, as N 2 approaches N 1 , we must also include the interaction Ψ 2 2 Ψ * 1 . This forces the m = 3 oscillation, which, importantly, is unbound:
2ω 2 − ω 1 > 0 (9)
The m = 3 oscillation carries particles and angular momentum to infinity. Again schematically, treating eqs.(5, 6) as the definitions of N and L, we havė
N = 2γ 1 N 1 − N 2 1 N 2 − N 1 N 2 2 (10) L = 2γ 1 N 1 − 3N 1 N 2 2 ,(11)givingṄ 1 = 2γ 1 N 1 − 2N 2 1 N 2 + N 1 N 2 2 ,(12)N 2 = N 2 1 N 2 − 2N 1 N 2 2 .(13)
Equations (12, 13) have a single equilibrium point N 1 = 2N 2 = (8γ 1 /3) 1/2 , which is stable for all γ 1 . At equilibrium, particles and angular momentum are extracted from the black hole. Then some particles are returned to the black hole and some particles, carrying all the extracted angular momentum, are radiated to infinity.
B. Schrodinger equation
To proceed, we need Schrodinger equation describing nonrelativistic nonlinear superradiance. Consider Klein-Gordon equation with cubic and quartic nonlinearities in the field potential:
V = 1 2 µ 2 φ 2 + λ 3 3! µ 2 f φ 3 + λ 4 4! µ 2 f 2 φ 4 ,(14)
where f is the nonlinear scale of the field φ. The coefficients λ 3 , λ 4 are dimensionless. For example, λ 3 = 0, λ 4 = −1 give the first expansion terms of the axion potential.
For M µ ≪ 1, the growing mode and all other modes participating in the nonlinear saturation of the instability are nonrelativistic. The nonlinear Klein-Gordon equation then reduces to the nonlinear Schrodinger equation. Namely, define Ψ by
φ = µ −1/2 Ψe −iµt + µ −1/2 Ψ * e iµt .(15)
Then, with Newtonian gravitational field, we have
iΨ = − 1 2µ ∇ 2 Ψ − M µ r Ψ − λ|Ψ| 2 Ψ.(16)
Here
λ ≡ λ 2 3 /3 − λ 4 /4 f 2 = ± 1 4f 2 ,(17)
with the last equality redefining the nonlinear scale f (we do not consider the degenerate case λ 2 3 /3 = λ 4 /4). In the axion case, λ > 0, meaning that the self-interaction is attractive.
As it is, eq.(16) does not have any growing or decaying modes, because the near-horizon effects disappear in the nonrelativistic approximation. But, as will become clear from what follows, all we need for our calculation is: (i) correct growth rates for the 2p mode with m = 1, (ii) correct decay rates for the l = 0 modes, (iii) negligibly small growth/decay rates for l > 1. This can be achieved by adding a rotating absorber to eq.(16):
iΨ = − 1 2µ ∇ 2 Ψ − M µ r Ψ − λ|Ψ| 2 Ψ + σθ(b − r)( Ω µ ∂ φ − i)Ψ. (18)
Here σ is the decay rate, b is the radius of the absorber, Ω is the angular velocity. So far these parameters are arbitrary. However, only with
b = 5 3 1/2 M, Ω = a 2r + , σ = 6 3 5 3/2 r + M 2 ,(19)
the first-order perturbation theory growth rates of eq.(18),
γ nl = 2 2l+2 (n + l)! n 2l+4 (n − l − 1)!(2l + 1)! 2 (2l + 3) σ mΩ µ − 1 (M bµ 2 ) 2l+3 ,(20)
coincide with those given by Detweiler (for all l = 0 and for 2p in the limit of small µ; note that our n is the standard principle quantum number, with n > l).
C. Estimates
To convert the schematics of §III A into estimates, we note that the Schrodinger equation (18) gives the length scales of the modes
r ∼ (M µ) −1 µ −1 ,(21)
frequencies
ω ∼ µ −1 r −2 ∼ (M µ) 2 µ,(22)
amplitudes, in terms of the particle numbers N ,
Ψ ∼ N 1/2 r −3/2 ,(23)
and nonlinearly induced amplitudes
Ψ ind ∼ ω −1 λΨ 3 ∼ µ f 2 (M µ) 5/2 µ 3/2 N 3/2 .(24)
The induced m = 0 mode is damped by the absorber (the black hole) at the ratė
N 0 ∼ −σb 3 Ψ 2 ind ,(25)givingṄ 0 = −γ 0 N 2 1 N 2 ,(26)γ 0 =γ 0 r + M (M µ) 7 µ f 4 µ,(27)
whereγ 0 is a dimensionless coefficient to be calculated in the next section. The induced m = 3 mode is radiated at the ratė
N 3 ∼ −vr 2 Ψ 2 ind ,(28)
where the velocity is
v ∼ (µ −1 ω) 1/2 ∼ M µ,(29)givingṄ 3 = −γ 3 N 1 N 2 2 ,(30)γ 3 =γ 3 (M µ) 4 µ f 4 µ,(31)
whereγ 3 is a dimensionless coefficient to be calculated in the next section. We thus havė
N = 2γ 1 N 1 − γ 0 N 2 1 N 2 − γ 3 N 1 N 2 2 (32) L = 2γ 1 N 1 − 3γ 3 N 1 N 2 2 ,(33)givingṄ 1 = 2γ 1 N 1 − 2γ 0 N 2 1 N 2 + γ 3 N 1 N 2 2 ,(34)N 2 = γ 0 N 2 1 N 2 − 2γ 3 N 1 N 2 2 .(35)
The saturated particle numbers are
N 1 = 8γ 1 γ 3 3γ 2 0 1/2 , N 2 = 2γ 1 3γ 3 1/2 .(36)
The resulting torque on the black hole is K = 2γ 1 N 1 , or
K = 1 36γ 3 1/2 γ 0 a 3/2 M r + (M µ) 7 f 2 µ 2 µ (37)
D. Final Results and Calculations
We start by listing the final results. Then we outline the calculations.
Final Results
We will show that γ 0 ≈ 3.4 × 10 −6 ,γ 3 ≈ 4.5 × 10 −8 .
(38)
Then the torque on the black hole is
K ≈ 1.7a 3/2 M r + (M µ) 7 f 2 µ 2 µ(39)
We must also make sure that the instability saturates at φ < f , otherwise our calculation would be meaningless. And, indeed, we get for the maximal values of the m = 1 and the m = 2 modes
φ 1max f ≈ 0.4a 1/4 M r + 1/2 (M µ),(40)φ 2max f ≈ a 1/4 (M µ) 5/2 .(41)
Calculations
With M = µ = 1, the wave functions of the 2p(m = 1) and 3d(m = 2) modes are (42) where
Ψ 1 = N 1/2 1 R 21 Y 11 e −iω1t , Ψ 2 = N 1/2 2 R 32 Y 22 e −iω2tω m = − 1 2(m + 1) 2 ,(43)R 21 = 1 2 √ 6 re −r/2 , R 32 = 4 81 √ 30 r 2 e −r/3 .(44)
The Ψ 2 1 Ψ * 2 forced oscillation: We first calculate the product of spherical harmonics:
Y 2 11 Y * 22 = 1 2π 15 2 1/2 1 35 Y 40 + 2 7 √ 5 Y 20 + 1 5 Y 00 .
(45) Only the l = 0 forced oscillation will penetrate to the absorber and damp the particle number. The l = 0 forced oscillation is
Ψ ind = Ψ(r)Y 00 e −iω ind t ,(46)ω ind = 2ω 1 − ω 2 = − 7 36 .(47)
The (first-order perturbation theory) damping rate iṡ
N 0 = − 2 3 σb 3 |Ψ(0)| 2 ,(48)
and Ψ(r) satisfies
Ψ ′′ + 2 r Ψ ′ + 2 r Ψ − 7 18 Ψ = Ar 4 e − 4 3 r ,(49)A = − 1 4π · 5 · 3 5 λN 1 N 1/2 2 .(50)
Numerical integration of
F ′′ + 2 r F ′ + 2 r F − 7 18 F = r 4 e − 4 3 r ,(51)
gives (for the solution which asymptotes to zero at infinity) |F (0)| = C 0 , where
C 0 ≈ 56.(52)
We have
|Ψ(0)| = C 0 4π · 5 · 3 5 λN 1 N 1/2 2 ,(53)
and, finally,Ṅ
0 = −γ 0 N 2 1 N 2 ,γ 0 ≈ 3.4 × 10 −6 .(54)
The Ψ 2 2 Ψ * 1 forced oscillation: The product of spherical harmonics:
Y 2 22 Y * 11 = 1 2π 5 42 1/2 2Y 33 + 1 √ 11 Y 53 .(55)
Both the l = 3 and the l = 5 forced oscillations radiate the particle number and angular momentum to infinity. The l forced oscillation is
Ψ ind = Ψ(r)Y l3 e −iω ind t ,(56)ω ind = 2ω 2 − ω 1 = 1 72 .(57)
The radiation rate iṡ
N 3 = −k l=3,5 (r|Ψ(r)|) 2 , r → ∞,(58)k = √ 2ω ind = 1 6 .(59)
Ψ(r) satisfies
Ψ ′′ + 2 r Ψ ′ − l(l + 1) r 2 Ψ + 2 r Ψ + k 2 Ψ = A l r 5 e − 7 6 r ,(60)A 3 = − 4 π · 3 10 √ 35 λN 1/2 1 N 2 ,(61)A 5 = − 2 π · 3 10 √ 385 λN 1/2 1 N 2 .(62)
Numerical integration of
F ′′ + 2 r F ′ − l(l + 1) r 2 F + 2 r F + k 2 F = r 5 e − 7 6 r ,(63)
gives (for the solution which asymptotes to an outgoing mode at infinity, for large r) r|F (r)| = C l , where
C 3 ≈ 570, C 5 ≈ 80.(64)
We have, for large r,
r|Ψ 3 (r)| = 4C 3 π · 3 10 √ 35 λN 1/2 1 N 2 ,(65)r|Ψ 5 (r)| = 2C 5 π · 3 10 √ 385 λN 1/2 1 N 2 ,(66)
and, finally,Ṅ
3 = −γ 3 N 1 N 2 2 ,γ 3 ≈ 4.5 × 10 −8 .(67)
FIG. 1: Upper curves: (arbitrarily normalized) |φm(r)|r 1/2 vs. r at the end of the simulation for m = 0, 1, 2, 3 harmonics (black, yellow, magenta, blue). Lower curves: log 1000 |φm|max as a function of (arbitrarily normalized) time for the entire simulation (same color coding for m = 0, 1, 2, 3, and green for m = 4, 5, 6.
IV. NUMERICAL SIMULATIONS
The figures show simulation results for the "heavy Zeldovich cylinder" -two dimensional self-interacting scalar field in the presence of a "gravitating" rotating absorber, described bÿ
φ = ∇ 2 φ − µ 2 (1 − 2r g r )(φ − φ 3 6 )− σθ(r g − r)(∂ t + Ω∂ θ )φ,(68)
with r g = 1, Ω = 0.5, µ = 0.4, σ = 3.
The calculation is done by decomposing
φ(t, r, θ) = mmax m=−mmax φ m (t, r)e imθ , φ −m = φ * m ,(70)
and solving the resulting set of m max + 1 coupled (1+1)d equations. As unbound modes do get excited, we put a "damping buffer" into the last quarter of the simulated r domain. So long as m max ≥ 3 the number of modes does not matter. This is the way it should be theoretically, and this what the figures show -other modes are not efficiently excited.
We have also simulated the cases of cubic potential and repulsive quartic potential. As expected ( §III A), we got similar results, but with one interesting exception. For quartic repulsion, with not too small µ, like what's listed in eq.(69), the saturation scenario is entirely different. Repulsion reduces the mode-mode interaction so much that the m = 2 mode never grows to a significant level. Instead the most unstable m = 1 mode grows asymptotically to the marginally bound state:
φ ′′ 1 + 1 r φ ′ 1 − 1 r 2 φ 1 − µ 2 (1 − 2r g r )(φ 1 + φ 3 1 2 ) = −µ 2 φ 1 . (71)
As this state has an infinite number of particles (φ 1 ∝ r −1/2 at large r), the growth of the number of particles lasts forever, while the maximal value of φ saturates. We see that numerics does confirm the theoretical saturation scenario. There is no "bose-nova" . Rather a steady state is formed, with boson radiation from the black hole [6].
V. CONCLUSION
A physically clear, even if somewhat intricate, scenario for the nonlinear saturation of superradiance is proposed: (i) black hole (BH) pumps 2p, (ii) 2p pumps 3d, (iii) 2p and 3d pump unbound f. In more details:
• BH pumps particles, N , and angular momentum, L, into 2p • interaction (2p) 2 (3d) pumps bound s • s pumps N but not L back into BH, meaning that 2p pumps 3d
• interaction (3d) 2 (2p) pumps unbound f
The resulting torque on the black hole, for M µ ≪ 1, is
K ∼ a 3/2 (M µ) 7 f 2 µ 2 µ.(72)
I thank Sergei Dubovsky for many discussions. I thank Mina Arvanitaki and Sergei Dubovsky for explaining to me the potential uses of these results.
give a qualitative picture• (B) derive Schrodinger equation for nonrelativistic
nonlinear superradiance
• (C) estimate and then
• (D) calculate the mode amplitudes at saturation
and the resulting black hole spindown rate.
A. Qualitative Picture
FIG. 2: Same as fig.1, but for repulsive self-interaction. FIG. 2: Same as fig.1, but for repulsive self-interaction.
. A Arvanitaki, S Dimopoulos, S Dubovsky, N Kaloper, J March-Russell, Phys. Rev. D. 81123530A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, J. March-Russell, Phys. Rev. D, 81, 123530 (2010)
. A Arvanitaki, M Baryakhtar, X Huang, Phys. Rev. D. 9184011A. Arvanitaki, M. Baryakhtar,X. Huang, Phys. Rev. D, 91, 084011 (2015)
. . B Ya, Zeldovich, JETP Letters. 14180Ya. B. Zeldovich, JETP Letters, 14, 180 (1971)
. S Detweiler, Phys. Rev. D. 222323S. Detweiler, Phys. Rev. D, 22, 2323 (1980)
. S R Dolan, Phys. Rev. D. 7684001S. R. Dolan, Phys. Rev. D, 76, 084001 (2007)
. H Yoshino, H Kodama, C Q Gra, 32214001H. Yoshino, H. Kodama, C.Q. Gra., 32, 214001 (2015)
| []
|
[
"Explainable time series tweaking via irreversible and reversible temporal transformations",
"Explainable time series tweaking via irreversible and reversible temporal transformations"
]
| [
"Isak Karlsson [email protected] \nDepartment of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden\n",
"Jonathan Rebane [email protected] \nDepartment of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden\n",
"Panagiotis Papapetrou [email protected] \nDepartment of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden\n",
"Aristides Gionis [email protected] \nAalto University\nEspooFinland\n"
]
| [
"Department of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden",
"Department of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden",
"Department of Computer and Systems Sciences\nDepartment of Computer Science\nStockholm University\nStockholmSweden",
"Aalto University\nEspooFinland"
]
| []
| Time series classification has received great attention over the past decade with a wide range of methods focusing on predictive performance by exploiting various types of temporal features. Nonetheless, little emphasis has been placed on interpretability and explainability. In this paper, we formulate the novel problem of explainable time series tweaking, where, given a time series and an opaque classifier that provides a particular classification decision for the time series, we want to find the minimum number of changes to be performed to the given time series so that the classifier changes its decision to another class. We show that the problem is NP-hard, and focus on two instantiations of the problem, which we refer to as reversible and irreversible time series tweaking. The classifier under investigation is the random shapelet forest classifier. Moreover, we propose two algorithmic solutions for the two problems along with simple optimizations, as well as a baseline solution using the nearest neighbor classifier. An extensive experimental evaluation on a variety of real datasets demonstrates the usefulness and effectiveness of our problem formulation and solutions. | 10.1109/icdm.2018.00036 | [
"https://arxiv.org/pdf/1809.05183v1.pdf"
]
| 52,282,388 | 1809.05183 | a2c8c4baa4f1b7a0697b6d59a34fa7796d0a0a80 |
Explainable time series tweaking via irreversible and reversible temporal transformations
Isak Karlsson [email protected]
Department of Computer and Systems Sciences
Department of Computer Science
Stockholm University
StockholmSweden
Jonathan Rebane [email protected]
Department of Computer and Systems Sciences
Department of Computer Science
Stockholm University
StockholmSweden
Panagiotis Papapetrou [email protected]
Department of Computer and Systems Sciences
Department of Computer Science
Stockholm University
StockholmSweden
Aristides Gionis [email protected]
Aalto University
EspooFinland
Explainable time series tweaking via irreversible and reversible temporal transformations
Index Terms-time series classificationinterpretabilityex- plainabilitytime series tweaking
Time series classification has received great attention over the past decade with a wide range of methods focusing on predictive performance by exploiting various types of temporal features. Nonetheless, little emphasis has been placed on interpretability and explainability. In this paper, we formulate the novel problem of explainable time series tweaking, where, given a time series and an opaque classifier that provides a particular classification decision for the time series, we want to find the minimum number of changes to be performed to the given time series so that the classifier changes its decision to another class. We show that the problem is NP-hard, and focus on two instantiations of the problem, which we refer to as reversible and irreversible time series tweaking. The classifier under investigation is the random shapelet forest classifier. Moreover, we propose two algorithmic solutions for the two problems along with simple optimizations, as well as a baseline solution using the nearest neighbor classifier. An extensive experimental evaluation on a variety of real datasets demonstrates the usefulness and effectiveness of our problem formulation and solutions.
I. INTRODUCTION
Time series classification has been the center of attention in the time series community for more than a decade. The problem typically refers to the task of inferring a model from a collection of labeled time series, which can be used to predict the class label of a new time series. 1 Example applications of time series classification include historical document or projectile point classification [1], classification of electrocardiograms (ECGs) [2], or anomaly detection in streaming data [3].
Several time series classification models have been proposed in the literature, including distance-based classifiers (see Ding et al. [4] for a thorough review), shapelet-based classifiers [1], [1] along with optimizations for shapelet selection or generation [5]- [7], and ensemble-based classifiers [8]. Recently, the random shapelet forest (RSF) [9] has been proposed for classifying univariate and multi-variate time series. The main idea is to build a set of decision trees, where each feature corresponds to a shapelet. The decision condition on an internal node is the presence or absence of a shapelet in a test time series example.
Despite its competitive performance in terms of classification accuracy on a large collection of time series datasets, RSF 1 To appear in International Conference on Data Mining, 2018 is an opaque classification model. It is, hence, not feasible to come up with any reasoning behind the predictions that could possibly be helpful to domain experts and practitioners. Interpretability studies within the time series domain have been largely dominated by the explanatory power provided by shapelets, which are class-discriminatory subsequences extracted from training examples [1], [10], [11]. However, a clear gap has been present within the time series domain regarding explainability, which this study has sought to address.
Consider the task of binary time series classification, where a times series may belong to either the positive ('+') or negative class ('-'). Our main objective in this paper is to study the following simple problem: given a time series T and an opaque classification model (e.g., an RSF) expressed by function f (·), such that f (T ) = '-', we want to identify the minimum number of changes that need to be applied to T in order to switch the classifier's decision to the positive class. That is, we want to define a transformation of T to T , such that f (T ) = '+'. We call this problem explainable time series tweaking. By solving this problem, practitioners will not only be able to understand the reasoning behind decisions produced by opaque time series classification models, but will also be able to take action to change a given time series instance from an undesired state (e.g., sick) to a desired state (e.g., healthy).
A. Examples
We present two motivating examples for the problem of explainable time series tweaking. Example I: Abnormal vs. normal heartbeats. Consider an electrocardiogram (ECG) recording, such as the one shown in Figure 1. The original signal (blue curve), denoted as T , corresponds to a patient suffering from a potential myocardial disease. An explainable time series tweaking algorithm would suggest a transformation of the original time series to T (yellow curve), such that the classifier considers it normal. Example II: Gun-draw vs. finger-point. Consider the problem of distinguishing between two motion trajectories, one corresponding to a gun-draw and the other to a finger-point. In Figure 2 we can see the trajectory of a regular finger pointing motion (blue time series), denoted as T . The objective of explainable time series tweaking would be to suggest a transformation of T to T (yellow curve), such that the classifier considers it a gun-point motion instead.
B. List of contributions and organization
The main contributions of this paper are summarized as follows:
• we formulate the novel problem of interpretable time series tweaking, and focus on two instantiations of the problem using the random shapelet forest classifier; • we show that the problem is NP-hard by a transformation from the Hitting Set problem; • we propose two methods for solving the problem for the random shapelet classifier, which are based on shapelet feature tweaking, along with optimization techniques; • we provide an extensive experimental evaluation of the two proposed methods and compare them with a baseline Nearest Neighbour approach in terms of three metrics: cost, compactness and speed of transformation. The remainder of this paper is organized as follows: in Section II we discuss the related work in the area of time series classification with emphasis on interpretability, while in Section III we provide the formal problem formulation. In Section IV we describe the two proposed methods, along with optimization strategies and theoretical properties, while in Section V we present our experimental evaluation and results. Finally, in Section VI we conclude the paper and provide directions for future work.
II. RELATED WORK
The majority of time series classification methods typically rely on instance-based classification techniques, For example, the k-Nearest Neighbor (k-NN) classifier, employs various similarity (or distance) measures, of which the most common and simplest is the Euclidean norm. To improve accuracy, elastic distance measures have been proposed, such as dynamic time warping (DTW) or longest common subsequence [12] and variants, e.g., cDTW [13], EDR [14], ERP [15], which are robust to misalignment and time warps. By regularization using, e.g., a band [16], the search performance and generalization behavior of k-NN can be greatly improved [4]. For a more complete overview of instance based univariate time series classifiers, the reader is referred to, e.g., Ding et al. [4].
A growing body of research is related to the domain of interpretable models, in which investigators have sought to provide greater clarity to decisions made by machine learning classifiers [17]- [19]. Such a need for interpretability often stems from a stakeholder desire to trust a model in order to find it useful; a trust which can be built both through the transparency of the model itself and post-hoc interpretability such as from local explanations [20]. As mentioned, a variety of studies in the time series domain highlight shapelets as the main vehicle for providing interpretability [1], [10], [11] with at least one study providing an alternative Symbolic Aggregate approximation (SAX) combined with a vector space model approach [21].
Moreover, instance-based classifiers are supplemented by feature-based classifiers that typically use class-discriminant features, called shapelets [1], which correspond to time series subsequences with high utility, measured by different discriminative measures, such as information gain [22]. For shapeletbased classifiers, the idea is to consider all subsequences of the training data recursively in a divide-and-conquer manner, while assessing the quality of the shapelets using a scoring function to estimate their discriminative power, constructing an interpretable shapelet tree classifier [1].
Shapelet transformation is one instance of a more general concept of feature generation, which has been thoroughly investigated for time series classification. For example, the generated features can range from statistical features [23], [24] to interval-based features [25] or other interpretable features, such as correlation or entropy [26]. A typical grouping of features produced by these transformations includes: correlationbased, auto-correlation-based, and shape-based, each denoting similarity in time, change, and shape, respectively. For example, a time series forest based on interval features, such as averages, standard deviations and slope has been proposed by Deng et al. [24] and a transformation based on time series bagof-words Baydogan et al. [27]. Moreover, in order to achieve performance improvements, Hilles et al. [5] introduce a heuristic approach for providing an estimation of the shapelet length. The described optimization algorithm repeatedly selects the ten best shapelets in a subset of ten randomly selected time series, searching for subsequences of all possible lengths.
Regarding multivariate time series classification methods, a shapelet forest approach has been introduced by Patri et al. for heterogeneous sensor data [28]. The algorithm employs the Fast Shapelet selection approach for extracting the most informative shapelets per dimension. In a similar manner, a shapelet tree is built from each time series dimension [29] using several additional techniques for providing search speedups. Moreover, various voting approaches are evaluated for providing the final classification label, demonstrating that one shapelet tree per dimension outperforms shapelets defined over multiple dimensions [29]. More recently, the generalized random shapelet forest has been proposed for univariate and multivariate time series classification, by expanding the idea of random shapelet trees and randomly selecting shapelet features per dimension [9]. While this approach can achieve competitive performance against existing classifiers in terms of classification accuracy it is a black-box classifier with limited interpretability and explainability of the predictions.
Complementary to interpretability, a number of studies have focused on actionable knowledge extraction, where the focus is placed on identifying a transparent series of input feature changes intended to transform particular model predictions to a desired output with low cost. Many actionability studies exist with a business and marketing orientation, investigating actions necessary to alter customer behaviour for mostly treebased models [30], [31]. In addition, several studies place particular focus on actionability which can be performed in an efficient and optimal manner [32], [33]. For example, Cui et al. specified an algorithm to extract a knowledgeable action plan for additive tree ensemble models under a specified minimum cost for a given example [34]. Similarly, an actionability study by Tolomei et al. investigated actionable feature tweaking in regards to converting true negative instances into true positives; employing an algorithm which alters feature values of an example to the point that a global tree ensemble prediction is switched under particular global cost tolerance conditions [35].
Despite the expansion of explainability, this is an unexplored prospect within the time series domain. In this paper, we study the problem of altering the prediction of examples, through the alteration of examples themselves, such that the prediction of a tree ensemble is changed with minimal cost. Moreover, we achieve such class alterations in an effective and efficient manner, proving and addressing the NP hard nature of the problem in accord with several optimization strategies. We then examine the real-world relevancy of this approach in regards to both medical and biomechanical time series datasets.
III. PROBLEM FORMULATION
In this section, we present our notation, and formally define the problem of explainable time series tweaking. In this paper, we only consider uni-variate time series, but the proposed framework and methods can be easily generalized to the multi-variate case. For the remainder of this paper, we will refer to uni-variate time series simply as time series.
A local, continuous segment of a time series is called a time series subsequence. Definition 2: (Time series subsequence or shapelet) Given a time series T , a time series subsequence or shapelet [1] of T is a sequence of contiguous elements of T , denoted as T [s, ] = {T s , . . . , T s+ −1 }, where s is the starting position and is its length.
Time series classification mainly relies on the chosen distance or similarity measure used to discriminate between instance pairs. The main task is to employ a distance function d(·) that compares two time series of equal length, and then given a time series subsequence (corresponding to a candidate discriminant shapelet) identify the closest subsequence match in the target time series. Depending on the application domain and the nature of the time series, various distance measures can be used.
Definition 3: (Time series subsequence distance) Given two time series S and T of lengths and m, respectively, such that ≤ m, the time series subsequence distance between S and T , is the minimum distance between S and any subsequence of T of length , i.e.:
d s (S, T ) = m− +1 min s=1 {d(S, T [s, ])} .(1)
A typical instantiation of d(·), given two time series T and T of equal length , is the Euclidean distance, i.e.:
d(T , T ) = d E (T , T ) = i=1 (T i − T i ) 2 .(2)
Definition 4: (Time series classification function) Given a time series T and a finite set of class labels C, a classification function is a mapping f from the set of all possible time series to the set C, such that:
f (T ) =ŷ ∈ C .
Note thatŷ denotes the predicted class for T , and f can be any type of time series classification function.
In this paper we study the problem of explainable time series tweaking, which is formulated below.
Problem 1: (Explainable time series tweaking) Given a time series T , a desired class y , and a classifier f , such that f (T ) =ŷ, withŷ = y, we want to find a transformation function τ , such that T is transformed to T = τ (T ), with f (T ) = y , and c(T , T ) is minimized, where c(T , T ) defines the cost of the transformation. We call a transformation that changes the class successful and the transformation that minimizes the cost the most successful transformation.
Any distance or similarity measure can be employed as a cost function. In this paper, we use the Euclidean distance, and consider two instantiations of Problem 1, where f is the random shapelet forest (RSF) classifier [9]. IV. EXPLAINABLE TIME SERIES TWEAKING In this section, we first formulate the problem of explainable time series tweaking, then describe the shapelet transformation function, which is the building block of our solution, followed by the two algorithms to tackle the problem. In addition, we present simple optimization strategies for both algorithms. Finally we prove that the problem we study is NP-hard.
A. Problem formulation
In short, an RSF, denoted as R = {F 1 , . . . , F |R| }, is a shapelet tree ensemble of size |R|, where each F j denotes a shapelet tree, constructed using a random sample of shapelet features [1]. Each shapelet tree F j ∈ R comprises a set of t decision paths {P
(y1)j 1 , . . . , P (yt)j t }, where y i is the decision class of path i. Let p j ik denote the k th non-leaf node in the i th path P (yi)j i , such that P (yi)j i = {p j i1 , . . . , p j iu } → y i , where u is the length of path P (yi)j i , i.e., |P (yi)j i | = u and each p j ik is described by a tuple S j k , θ j k , δ j ik
defining a condition over shapelet S j k using a distance threshold θ j k ∈ R and a comparison operator {≤, >}, such that δ j ik equals −1 or 1 if the comparison operator is ≤ or >, respectively.
Definition 5: (Condition test) Given a non-leaf node p j ik = S j k , θ j k , δ j ik of shapelet tree F j and a time series T , a condition test of path i on non-leaf node k is defined as:
φ(T , p j ik ) = true, if (d s (S j k , T ) − θ j k )δ j ik ≤ 0 false, otherwise.(3)
More concretely, φ(·) returns true if T fulfills the k-th condition of the i-th path of the j-th tree.
To clarify the notation consider the simple tree in Figure 3, with two internal nodes and three terminal nodes. This tree can be converted into a set F 1 of 3 distinct paths:
P ('+')1 1 = { S 1 1 , θ 1 1 , 1 } P ('-')1 2 = { S 1 1 , θ 1 1 , −1 , S 1 2 , θ 1 2 , 1 } P ('+')1 3 = { S 1 1 , θ 1 1 , −1 , S 1 2 , θ 1 2 , −1 }.
Finally, observe that each non-leaf node performs a binary split depending on whether the time series subsequence distance between S (i,k) and T is within a distance range θ. The decision label of F j for T is denoted as
y j = f (T , F j ), while the decision label of R for T is defined asŷ = f (T , R) = M (y 1 , . . . , y |R| ), where M (·)
is the majority function. For more details on the actual structure and implementation of RSF the reader may refer to [9].
The final step is to define a suitable transformation function τ (·) for explainable time series tweaking. Given a time series example T and an RSF classifier R, we define the transformation function τ (·) used at each conversion step while traversing a decision path in each tree of the ensemble. Recall that our goal is to suggest the transformation of T , such that the transformation cost is minimized and the classifier changes its classification decision. The smallest cost corresponds to the transformation that imposes the lowest Euclidean distance between the original and transformed time series.
We study two versions of τ (·), hence defining the following two subproblems, reversible time series tweaking and irreversible time series tweaking. Problem 2: (Reversible time series tweaking) Given a time series T , a desired class y , and a RSF classifier R,
such that f (T , R) =ŷ, withŷ = y , we want to transform T to T = τ (T ), such that f (T , R) = y , the Euclidean distance d E (T , T ) is minimized, and τ (T ) defines a sequence of transformations T → T 1 → T 2 → . . . → T , where each subsequent transformation T i can override any earlier transformation T j , with j ≤ i.
Problem 3: (Irreversible time series tweaking) Given a time series T , a desired class y and a RSF classifier R, such that f (T , R) =ŷ, withŷ = y , we want to transform T to T = τ (T ), such that f (T , R) = y , the Euclidean distance d E (T , T ) is minimized, and τ (T ) defines a sequence of transformations T → T 1 → T 2 → . . . → T , where each subsequent transformation T i cannot override any earlier transformation T j , with j ≤ i.
Note that Problem 2 is a more general version of Problem 3 as the first one allows any change applied to the time series to be overridden by a later change, while the second one "locks" the time series segments that have already been changed, hence not allowing for any change to be reversed. By restricting overriding transformations in Problem 3, the Euclidean distance between the current and transformed time series is guaranteed to be monotonically increasing as more transformations are applied; hence allowing for early abandoning a transformation if the cumulative cost is above the currently best successful transformation. In contrast, reversible time series tweaking does not guarantee that the Euclidean cost is monotonically increasing, hence, it does not allow for early abandoning of the transformation. Despite this, we will show in Section IV-C that a simple optimization can achieve substantial speedups for Problem 2.
B. Time series tweaking
Given a non-leaf node p j ik containing a shapelet S j k , and a threshold θ j k , we define two types of time series tweaks: • increase distance: if S j k exists in the current version of the time series (i.e., d s (S j k , T ) ≤ θ j k ) and the current k th condition demands that S j k does not (i.e., demanding that d s (S j k , T ) > θ j k ), we want to increase the distance of all matches falling below θ j k to > θ j k ; • decrease distance: if S j k does not exist in the current version of T (i.e., d s (S j k , T ) > θ j k ) and the current k th condition demands that S j k does (i.e., it demands that d s (S j k , T ) ≤ θ j k ), we want to decrease the distance of its best match to ≤ θ j k . As depicted in Figure 4, these time series tweaks can be achieved by considering any shapelet S as an m-dimensional point, and by defining an m-sphere with point S j k as its center and radius θ j k . Intuitively, if d E (S, S j k ) ≤ θ j k , then S falls inside the circle, and hence the resulting time series corresponds to the point on the circle that intersects the line connecting the two points. Given a desired distance threshold (radius) θ j k , the transformed time series that has exactly the desired distance threshold is given by:
τ S (S, p j ik , ) = S j k + S j k − S S j k − S 2 (θ j k + ( δ j ik ))(4)
where ∈ R, > 0 is a parameter that control if the transformed time series distance fall inside the m-sphere ( < 0), outside the m-sphere ( > 0) or exactly at the circumference ( = 0). Note that in Equation 4, we use δ j ik to control the direction of the move, i.e., for condition k with a ≤ test is negated and for conditions with a > test is not.
In summary, transforming a time series T predicted asŷ to a time series T predicted as y for as single decision tree is a matter of changing the time series such that all conditions of the decision path, resulting in a transformation with the lowest cost, is successfully. Next, we will present two greedy algorithms for giving approximate solutions to Problem 2 and Problem 3 using forests of randomized shapelet trees.
C. Greedy algorithm I: reversible tweaking
Given an ensemble R of shapelets trees, where each tree F j is converted to a set of decision paths {P (y1) ij , . . . , P (yt) tj }, a desired class label y and a transformation strength parameter , which controls the amount of transformation applied, Algorithm 1 enumerate and apply the changes recommended by each condition k for each path i of all trees in the forest that is labeled with the desired label y .
In Algorithm 1 transformations are applied one condition, from a path with the desired class, at a time. Consequently, the first step, on Line 5, is to check if the current condition test k is fulfilled for the time series T , which we want to transform to y . The check investigates if we need to apply any of the two tweaks in order for the current condition to hold. In the case where the condition does not hold, i.e., if 16 if c(T, T ) < c min ∧ f (T, R) = y then 17 T ← T 18 c min ← d(T , T ) 19 return T φ(·, ·) return false, we check, on Line 7, if there is a need to increase or decrease the distance to fulfill the k th condition. In the first case i.e., when the closest distance is larger than the threshold, but it needs to be smaller, the transformation is simple: we find the shapelet (starting at idx and ending at idx + |S j k |) with the closest distance and apply Equation 4 to tweak the shapelet such that it distance is slightly smaller than θ j k and subsequently replaces the shapelet in T[idx : idx + |S j k |] with the new subsequence, S . In the second case, i.e., when the closest distance is smaller than the threshold but the distance needs to larger, the transformation is slightly more convoluted since there might exist many position where the distance is smaller than θ j k . In the presented algorithm, we find and transform each lowest distance position incrementally, on Line 8, until there exists no subsequence in the transformed time series with a distance smaller than θ j k . After all conditions k, . . . , u of the i th path has been applied, the algorithm computes the cost of transforming T to T , i.e., c(T , T ), and if this cost is lower than the best so far and the classification according to f (T, R) has changed to y , we record the current score as the lowest and keep track of the best transformation. This procedure is repeated for all paths, until the path with the lowest cost is returned. Optimization via prediction ordering. Since the cost of prediction of the random shapelet forest is more costly than the cost of transforming a time series, one possible optimization is to compute all transformations, T 1 , . . . , T I for a particular time series T and order the transformed time series in increasing order according the transformation cost, c(T , T i ), where i ∈ {1, . . . , K}. By ordering the prediction, the first transformation for which f (T , R) = y is true, is by definition the transformation with lowest cost that also changes the class label. Although, this might seem a simple optimization, the pruning power and runtime reduction is significant in practice, as seen in Section V.
1 T ← T .copy 2 c min ← ∞ 3 for j ← 1 to |R|, k ← 1 to |F j | do 4 for i ← 1 to u do 5 if y k = y ∧ φ(T , p j ik ) is false then 6 T ← T .copy 7 if d s (S j k , T ) ≤ θ j k then 8 while d s (S j k , T ) ≤ θ j k do 9 idx ← start index of subsequence with lowest distance, d s (S j k , T) 10 S ← τ S (T[idx : idx + |S j k |], p j ik , ) 11 Assign S to T[idx : idx + |S j k |] 12 else 13 idx ← start index of subsequence with lowest distance, d s (S j k , T) 14 S ← τ S (T[idx : idx + |S j k |], p j ik , ) 15 Assign S to T [idx : idx + |S j k |]
D. Greedy algorithm II: irreversible tweaking
The irreversible tweaking algorithm (τ IRT ), introduces a "locking" data structure that stores the start and end positions (i.e., idx and idx +|S · · | in Algorithm 1) of transformed regions of the time series T . As such, we modify Algorithm 1 to store these locked regions after transformations have been applied, at Line 11 and Line 15. We also ensure that the subsequence with the lowest distance does not overlap with a region has been previously locked, by introducing an additional check on Line 8 and after Line 13. Note that by introducing the irreversible criterion, we are not guaranteed that the changes introduced by the algorithm changes the prediction of even the current tree j. However, as we show in Section V this does not significantly affect the transformation cost, but the irreversible criterion allows the algorithm to produce more compact transformations. Optimization via early abandoning. For the reversible tweaking problem, early abandoning of transformation is not possible; however, if we specifically "lock" regions, as in the τ IRT algorithm, of the time series that have already been transformed by an earlier condition p · ik, the cost is guaranteed to be monotonically increasing as we progress with further transformations. As such, as soon as a transformation is successfully, i.e., f (T , R) = y , on Line 7, conditions for which the partial cost is greater than or equal to c(T , T ) can be safely ignored, by introducing a partial cost indicator and check if the the current cost is increased above this value after each transformation, i.e., after Line 15. Using this simple technique, we can eliminate both predictions and transformations. An example of the both shapelet transformation algorithms are shown in Figure 5 (top/middle), where a time series representing different insects flying through a audio recording device are transformed from being predicted as class 1 (blue) and transformed to be predicted as 6 (yellow). We can note that both tweaking algorithms increases the amplitude around time t = 35 and reduces the amplitude around t = 100 and t = 175, all changes that seems to correspond well with the intuition provided by the average time series for each class ( Figure 5 (bottom)).
E. NP-hardness
Let us consider a very simple model where time series are sequences of binary values, and tree classifier test whether certain elements of the time series have a certain value. Let T be the time series, and let R = {F 1 , . . . , F m } be the set of all decision trees in the ensemble.
Theorem 1: Given a time series T and an ensemble R, the problem of making the smallest number of changes in the time series so as to change the ensemble prediction is NP-hard.
Proof 1: We consider the decision version of the problem, where a number k is given and we ask whether there exists a solution that requires at most k changes in the time series.
We reduce a variant of the Hitting Set problem to the problem of explainable time-series tweaking (Problem 1). An instance of the Hitting Set problem is the following: We are given a ground set U of n elements, subsets S 1 , . . . , S m ⊆ U , and an integer k. We ask whether there is a set H ⊆ U of cardinality |H| at most k, so that H ∩ S j = ∅, for all j = 1, . . . , m, that is, whether there are at most k elements in U that "hit" all the sets S 1 , . . . , S m . Here we consider the variant where we ask whether there are at most k elements in U that hit at least half of the sets S 1 , . . . , S m . This is also an NP-hard problem, as it is equivalent to Maximum Cover.
Given an instance of this variant of Hitting Set problem, we create an instance to the explainable time-series tweaking problem as follows. We first create a time series of length n, where all its entries are 0s, that is, T [i] = 0, for all i = 1, . . . , n. Then we create an ensemble R = {F 1 , . . . , F m }, so that there is a tree F j for each subset S j . In particular, the tree F j is constructed as follows. If i ∈ S j , then the tree F j contains a node of the form "if T [i] = 1 then T is classified to class 1, otherwise pointer to another node ". The tree F j is organized in an left-unbalanced manner, so that if none of these rules are satisfied, they will all be checked. The last (leftmost) leaf has the form "if T [i] = 1 then T is classified to class 1, otherwise to class 0". It follows that the tree F j classifies the time-series to 1 if and only if the series has a value equal to 1 in at least one position that corresponds to an element of the input set S j .
We see that T is initially classified to class 0. We ask whether it is possible to change at most k positions in T so that it is classified to class 1. One can easily see that the answer to this question is affirmative, if and only if there exists a solution to the instance of the Hitting Set variant that is given as input. Thus, we conclude that the explainable time series tweaking problem is NP-hard.
Note that we prove NP-hardness for a very special case of our problem. As a result, the most general case of our problem, where we have real-valued time series, complex shapelets, and arbitrary decision trees in the ensemble is also NP-hard.
V. EXPERIMENTAL EVALUATION
A. Experimental setup
We evaluate the proposed algorithms on datasets from the UCR Time Series repository [36]. The datasets represent a wide range of different classification tasks varying from motion classification, e.g., Gun Point to sensor reading classification, e.g., ECG200. In the paper, we have selected all binary classification tasks to empirically evaluate the proposed time series tweaking algorithms. Hence, our task is to convert time series classified asŷ = 1 to y = −1 and to convert time series classified asŷ = −1 to time series classified as y = 1 by RSF; as such, the results presented in Table I are the average of both transformations. Although, we limit the empirical evaluation to binary datasets, we note that the proposed algorithms can be used for multiclass problems. In the experiments, we set aside 20% of the data for transformation and testing and use the remaining 80% for training the model.
Note that the τ NN is guaranteed to find the transformation among the time series in the training set that minimizes the transformation cost as long as the transformation cost is the same as the 1-NN distance measure.
2) Parameters: The random shapelet algorithm requires several hyper-parameters to be set, namely the number of shapelets to sample at each node, the number of trees in the forest, and the minimum and maximum shapelet size. Since the purpose of this work is not to evaluate the effectiveness of the shapelet forest algorithm, the hyper-parameters are set to their default values, which amounts to 100 random shapelets at each node and shapelets of all possible sizes. To have a viable number of paths to use for transformation, we let the learning algorithm grow 100 trees. Moreover, we set the transformation strength for both the reversible and irreversible tweaking algorithms to = 1, which corresponds to relatively small changes.
B. Performance metrics
We compare the two algorithms and the baseline as the average cost over the test set, which we define as:
c µ (τ, y ) = 1 n n i=1 c(T i , τ (T i , y ))
where n is the number of time series in the test set not classified as y . We report the average of c µ (·, y ) with y ∈ {'+', '-'}. Moreover, we examine which fraction of the original time series must be altered under both the tweaking algorithms and the 1-NN approach. Given T , its transformation T , and a threshold e ∈ R, assuming that |T | = |T | we define the compactness of a transformation of T to T as
compact(T , T ) = 1 |T | |T | i=1 diff (T i , T i ) , where diff (T i , T i ) = 1, if |T i − T i | ≤ e 0, otherwise.
Note that a compactness of 1 means that the entire time series is changed, whereas a compactness of 0 indicates that the transformed and original time series are identical. We report the average compactness, defined as:
compact µ (τ, y ) = 1 n n i=1 compact(T i , τ (T i , y ))
where n is the number of time series in the test set not classified as y . We report the average of compact µ (·, y ) with y ∈ {'+', '-'}. Finally, we examine the fraction of correct predictions, i.e., the accuracy, produced by our classifiers as a means of judging the trustworthiness of the classification approaches, and consequently the trustworthiness of the transformations.
C. Results
Tables I and II show a comprehensive comparison in terms of the running time and the solution quality measured by the cost, transformation fraction, and runtime per transformation. In Table I we observe, in regards to cost and compactness, that both the reversible tweaking τ RT and irreversible tweaking τ IRT approaches greatly outperform the nearest neighbor τ NN approach, with τ RT demonstrating the best average cost by a small degree compared to τ IRT , and τ IRT showing the best level of compactness by a small degree compared to τ RT . In terms of accuracy, RSF on average provides more trustworthy predictions compared to τ NN and thus the explainable tweaking produced by RSF would, not only result in less cost and more compactness, but potentially be considered more trustworthy by domain experts. In Table II we present a runtime comparison of τ IRT against τ RT with and without pruning. We observe that τ RT with pruning provides the best runtime performance on average, which can be explained by the fact that the relative cost of an ensemble prediction is, on average, more costly than a transformation. As such, the superior performance of optimized τ RT can be attributed to its ability to prune more predictions than τ IRT . In fact, for datasets with costly transformations (e.g., PhalangesOutlinesCorrect), the τ IRT algorithm, which is able to prune 90% of the transformations, outperform τ RT . As a result, one should prefer τ IRT when transformations are complex and the compactness of transformations are deemed important.
D. Use-case examples
In this section, we provide two use-case examples of the proposed time series tweaking framework by revisiting the motivating examples from Section I. 1) Electrocardiograms: Revisiting the problem of heartbeat classification (Example I), we demonstrate a use-case example from the ECG200 dataset, which contains measurements of cardiac electrical activity as recorded from electrodes at various locations on the body; each time series contains the measurements recorded by one electrode. The binary classification objective is to distinguish between Normal and Abnormal heartbeats.
In Figure 6, we observe that the original time series T (blue curve) exhibits a low-amplitude QRS complex, which may suggest a pericardial effusion or infiltrative myocardial disease [37], and is hence classified as Abnormal by the RSF classifier. Our explainable time series tweaking algorithm τ RT suggests a transformation of the original time series to T (yellow curve), such that the low-amplitude QRS complex is changed, by increasing the amplitude of the S-wave. This is illustrated by the yellow curve. Since τ RT is the best performing transformation of the two proposed ones in terms of cost for this dataset, we apply it to T resulting in the classifier to label T as Normal. Moreover, we observe that the baseline competitor τ NN suggests a much costlier transformation (dotted curve).
2) Gun-draw vs. finger-point: Revisiting the problem of motion recognition (Example II), we demonstrate a use-case example from the Gun Point dataset, which contains motion trajectories of an actor making a motion with his or her hand. The objective is to distinguish whether that motion corresponds to a gun-draw or to finger-pointing.
In Figure 7 we observe the trajectory of a pointing motion (blue curve), for three consecutive motion recording, classified as Finger-point by the RSF classifier. By observing the bottom recording (yellow curve), we see the transformations needed to change the decision of RSF to Gun-point, using τ RT (which has again achieved the lowest transformation cost, according to II: Summary of the runtime of the two algorithms including the pruning power of the proposed optimization protocols. While τ RT pruning does not prune any transformations, the τ IRT pruning algorithm does. Hence, for τ IRT the fraction of early abandoned transformations is the same as the fraction of pruned predictions. Fig. 6: Abnormal vs. Normal heartbeat identification: the original time series is depicted in blue. We observe that a classifier f labels the three segments of the input time series T as Abnormal (top). By applying τ RT , we can transform these heartbeats to the normal class (bottom). We also show the transformations using τ NN (bottom). Table I). Following our experimental findings, we also observe that the baseline competitor τ NN suggests a much costlier transformation (dotted curve).
VI. CONCLUSIONS
In this study we have sought to exploit and expand upon the interpretability afforded by shapelets in the time series domain as a means of permitting explainability. We showed that the proposed problem formulation is NP-hard and provided two instantiations of the problem using the random shapelet forest classifier. Experiments were performed to examine our approach in-depth and enable a comparison to a nearest neighbor solution in terms of Euclidean distance cost, compactness of transformations, and time needed for altering time series examples. We have demonstrated that the two proposed solutions outperform the baseline nearest neighbor solution in terms of cost and compactness, both of which are important factors in permitting actions pertaining to time series that are actually feasible in the sense that alterations can be realistically performed in a given domain. Future work includes the investigation of alternative distance measures, such as dynamic time warping, as well as expanding our approach to permit transformations exploiting trade-offs between cost and trustworthiness of classifier predictions. Reproducibility. Source code and data is available at: http://github.com/isakkarlsson/tsexplain.
Fig. 1 :
1Abnormal vs. Normal heartbeat identification. The original time series is depicted in blue. We observe that a classifier f , classifies the input time series T as Abnormal (blue curve). By applying time series tweaking, we change the classifier's decision to the normal class (yellow curve).
Fig. 2 :
2Gun-draw identification. The original time series is depicted in blue. We observe that a classifier f classifies the input time series T as class Finger-point. When transforming T to T by changing two small segments (indicated in yellow) converts it to class Gun-draw.
Definition 1 :
1(Time series) A time series T = {T 1 , . . . , T m } is an ordered set of real values, sampled at equal time intervals, where each T i ∈ R.
Fig. 3 :
3A simple decision tree example of two internal nodes and three leaf nodes.
Fig. 4 :
4Example of moving the point T to the closest point on the circle representing the distance threshold θ, where the distance between d(S, T ) = θ. Algorithm 1: Reversible time series tweaking algorithm (τ RT ) input : A shapelet forest R, a time series T and a desired class y and transformation strength output: A transformed time series T
Fig. 5 :
5(top/middle) Example of transforming time series (blue) classified as 1 by the classifier transformed to time series (yellow) labeled as 6 by the classifier, using both the reversible and irreversible tweaking algorithms. (bottom) Average time series belonging to each of the classes, used for comparison.
1 )
1Baseline: The two proposed time series tweaking algorithms are compared to a baseline defined as the 1-nearest neighbour (1-NN) under the Euclidean distance, among the time series labeled as the target transformation label, which we call the training set; i.e., τ NN (T , y ) = arg min {T |(ŷ,T )∈D,ŷ=y } d E (T , T ).
fFig. 7 :
7(T )=Gun-draw f (T )=Gun-draw f (T )=Gun-draw τ RT (T ) τ N N (T ) Gun-draw vs. Finger-point identification. The original time series is depicted in blue. We observe that RSF classifies the three segments of the input time series T as Finger-point (top). By applying τ RT , we can transform these Finger-point motion trajectories to Gun-draw trajectories (bottom). We also show the transformations using τ NN (bottom).
TABLE I :
ISummary of the results for the evaluation metrics. The best performing method for each metric is highlighted.Cost
Compactness
Accuracy
Dataset
τ RT
τ IRT
τ NN
τ RT
τ IRT
τ NN
RSF NN(1)
BeetleFly
7.3810 7.3810
26.6223
0.5737 0.5737
1.0000
0.8750
0.7500
BirdChicken
4.5071
4.5098 15.6695
0.5048
0.5169 1.0000
1.0000
0.6250
Coffee
1.1447
1.1846
1.9178
0.3824
0.1809
1.0000
1.0000
1.0000
Computers
2.2197
2.5132 22.4809
0.4123
0.4044
1.0000
0.7000
0.4900
DistalPhalanxOutlineCorrect
0.9314
1.1150
1.1704
0.5917
0.4466
0.9999
0.7886
0.7143
Earthquakes
2.2725
3.1455 30.0943
0.7449
0.7577 1.0000
0.7826
0.6630
ECG200
1.8730
1.9080
4.1428
0.7976
0.7686
1.0000
0.8750
0.9500
ECGFiveDays
1.9722
2.0158
4.2143
0.5215
0.4913
1.0000
1.0000
0.9944
GunPoint
1.9787
1.9942
3.6975
0.4712
0.4460
0.9998
1.0000
0.9250
Ham
2.1744
2.2187
7.8253
0.6791
0.6621
0.9999
0.8605
0.7907
Herring
1.2492
1.2488
3.5817
0.4563
0.4060
0.9999
0.5000
0.3846
ItalyPowerDemand
1.1791
1.2645
1.3088
0.7262
0.6397
0.9998
0.9726
0.9589
Lightning2
3.2741
3.9266 18.9703
0.7470
0.7071
1.0000
0.6667
0.6667
MiddlePhalanxOutlineCorrect
0.6685
0.9877
0.6791
0.6182
0.4493
0.9999
0.8258
0.7753
MoteStrain
2.4413
2.5313
6.0249
0.5602
0.4834
1.0000
0.9685
0.9213
PhalangesOutlinesCorrect
0.6979
0.9568
0.7574
0.6186
0.5116
0.9998
0.8421
0.7782
ProximalPhalanxOutlineCorrect 0.5895
1.0056
0.5326
0.6552
0.4121
0.9997
0.8315
0.8090
SonyAIBORobotSurface1
1.7384
1.7260
4.7213
0.4429
0.4394
1.0000
0.9919
1.0000
SonyAIBORobotSurface2
1.8601
1.8566
5.6126
0.4133
0.3584
1.0000
0.9796
0.9949
Strawberry
1.2082
1.3628
1.2802
0.6644
0.5464
0.9999
0.9695
0.9797
ToeSegmentation1
3.1200
3.1436 14.7768
0.3871
0.3718
1.0000
0.9259
0.7407
ToeSegmentation2
5.4407
5.8238 17.8733
0.6173
0.5705
1.0000
0.9697
0.7879
TwoLeadECG
0.9112
1.0671
1.3517
0.4966
0.4028
0.9999
1.0000
0.9957
Wafer
3.0135
3.1419
8.6207
0.7152
0.6676
0.9999
0.9958
0.9979
Wine
0.5052
0.9301
0.1708
0.7529
0.3452
0.9996
1.0000
1.0000
WormsTwoClass
5.7723
7.2023 28.7383
0.4416
0.4219
1.0000
0.8269
0.7308
Avg.
2.3132 2.5329
8.9552
0.5733 0.4942 0.9999
0.8924
0.8240
TABLE
ACKNOWLEDGMENTSThis work was partly supported by grants provided by the Stockholm County Council and partly by the project Temporal Data Mining for Detective Adverse Events in Healthcare, ref.no. VR-2016-03372. The work was also supported by three Academy of Finland projects (286211, 313927, and 317085), and the EC H2020 RIA project "SoBigData" (654024). Finally, we acknowledge the UCR Time Series Classification Archive[36]for the datasets used in this paper.
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| COVID-19, caused by SARS-CoV-2, has resulted in a global pandemic recently. With no approved vaccination or treatment, governments around the world have issued guidance to their citizens to remain at home in efforts to control the spread of the disease. The goal of controlling the spread of the virus is to prevent strain on hospitals. In this paper, we focus on how non-invasive methods are being used to detect COVID-19 and assist healthcare workers in caring for COVID-19 patients. Early detection of COVID-19 can allow for early isolation to prevent further spread. This study outlines the advantages and disadvantages and a breakdown of the methods applied in the current state-of-the-art approaches. In addition, the paper highlights some future research directions, which need to be explored further to produce innovative technologies to control this pandemic. | 10.3390/s20195665 | null | 220,919,689 | 2007.16063 | 5a71c17589c69839b698335bf686aeada20c2545 |
A Review of the State of the Art in Non-Contact Sensing for COVID-19
William Taylor
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
Qammer H Abbasi [email protected].
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
Kia Dashtipour [email protected].
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
Shuja Ansari [email protected].
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
Syed Aziz Shah [email protected]
Centre for Intelligent Healthcare
Coventry University
CV1 5RWCoventryUK
Arslan Khalid
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
Muhammad Ali Imran [email protected].
James Watt School of Engineering
University of Glasgow
G12 8QQGlasgowUK
( A K
Muhammad Ac Imran@glasgow
M A I Uk
A Review of the State of the Art in Non-Contact Sensing for COVID-19
10.3390/s20195665Received: 5 August 2020; Accepted: 29 September 2020; Published: 3 October 2020sensors Review * Correspondence:COVID-19population healthSars-Cov-2AIMLdisease diagnosticssensing
COVID-19, caused by SARS-CoV-2, has resulted in a global pandemic recently. With no approved vaccination or treatment, governments around the world have issued guidance to their citizens to remain at home in efforts to control the spread of the disease. The goal of controlling the spread of the virus is to prevent strain on hospitals. In this paper, we focus on how non-invasive methods are being used to detect COVID-19 and assist healthcare workers in caring for COVID-19 patients. Early detection of COVID-19 can allow for early isolation to prevent further spread. This study outlines the advantages and disadvantages and a breakdown of the methods applied in the current state-of-the-art approaches. In addition, the paper highlights some future research directions, which need to be explored further to produce innovative technologies to control this pandemic.
Introduction
Since late 2019, countries around the world have been experiencing a global pandemic through the surfacing and spread of the potentially fatal COVID-19 (COronaVIrusDisease 2019) caused by SARS-CoV-2 (Severe Acute Respiratory Syndrome CoronaVirus 2) virus [1]. COVID-19 causes victims to develop a fever and display respiratory difficulties causing coughing or shortness of breath [2][3][4]. Data collected from victims of the virus shows that most deaths occurred in patients with underlying health issues with elderly people being at a higher risk of death [5]. The first confirmed case of the virus is considered to be in Wuhan, China in December 2019 with some of the early cases thought to be traced to seafood markets trading live animal species such as bats and snakes [6][7][8][9]. The virus has been discovered to likely be related to bats. It is suspected that the virus may have been transmitted to humans through bats which were being sold as food items [10,11]. The exact cause of the virus is still unknown, and it has also been suggested that the virus could originate from pangolins, which are natural hosts of corona viruses [12]. Pangolin is unlikely to be linked to the outbreak as the corona viruses found on the animal differ to COVID-19 [13]. However, it is possible the pangolin could have served as an intermediate host. As a result, these markets were shut down in China [14]. The virus rapidly spread throughout China and eventually spread throughout the world. The virus was officially declared a global pandemic by the World Health Organisation (WHO) on 30th January 2020 [15,16]. Although new discoveries are being made at the time of writing this paper, the virus has been found to be highly contagious and this has led to its rapid spread throughout the world [17]. The virus is spread primarily through respiratory droplets from an infected person [18]. These droplets can be dispensed by an infected person when coughing or sneezing. The droplets can then infect others directly via the eyes, mouth or nose when they are within a one or two meters radius of an infected person [19]. Some examples can show where 2 m is not enough distance such as with tobacco smoke traveling over 9 m from a lung source [20]. This uncertainly has led to the recommendation of using facial masks as a protective measure. There is debate on the effectiveness of masks, but it is recommended by the WHO to use masks if in contact with COVID-19 patients [21,22]. The droplets can also be passed to others indirectly due to their long-term presence on surfaces [23]. Another leading factor in the rapid spread is that those infected with COVID-19 can be contagious during the early stages of infection while they are showing no symptoms [24]. This leads to people believing they are not sick while unknowingly spreading the virus. One of the main challenges of the COVID-19 pandemic is the how the spread of the virus can be controlled. The rapid spread of COVID-19 has highlighted how the world's population interacts when faced with a pandemic [25]. Governments around the world have outlined guidelines to their citizens to adhere to lockdown rules. Currently, the best strategy to control the spread of COVID-19 is to ensure social distancing until a vaccine or an effective treatment can be produced [26,27]. The National Health Service (NHS) of the United Kingdom is expecting an increased demand for their services as more COVID-19 patients are admitted and staff sick leave increases as staff members contract the disease [28]. Technology is being rapidly introduced in healthcare applications to develop systems that can ease the demand of the health service [29][30][31]. Any assistance via healthcare technology will free up valuable clinical resources to focus on other areas of care. In this paper, we look at the state-of-the-art non-contact sensing techniques and how these technologies can be used to assist in the care and detection of people suffering from COVID-19 and how these methods can help to reduce the spread of the disease, primarily the spreading of the disease from patients to healthcare workers such as doctors, nurses, and career staff.
Search Strategy
The following search terms and variation of search terms were used in Google Scholar, MDPI, Science Direct and IEEE databases: radar breathing detection tachypnea, RGB-thermal breathing detection, Terahertz COVID-19, ultrasound non-contact lungs, ultrasound imaging, CT Scanning COVID-19, X-ray COVID-19, Camera COVID-19 Detection, Radar COVID-19 diagnosis, Thermography COVID-19, Terahertz COVID-19 detection, thermography non-contact, COVID-19 symptoms, Ultrasound Non-contact.
Non-Contact Sensing to Detect COVID-19 Symptoms
Non-contact sensing is the ability to detect information without direct contact with a subject. In terms of healthcare, non-contact can be used monitor the human body without devices physically touching the body. Non-contact techniques are considered highly valuable in dealing with a highly infectious disease such as COVID-19, as contact may contribute to the spread of disease. This is because healthcare workers will not need to make physical contact with patients to enable the monitoring of the patient. Using wearable devices can cause risks to healthcare workers as they will need to have physical contact with patients to attach the device. Despite precautions being undertaken such as wearing gloves and face masks, there will be lower risk if contact with patients can be successfully removed completely. Healthcare sensing technologies aim to collect information from a person which can be processed by Artificial Intelligence (AI) to provide decision support or directly analyzed by a clinician to diagnose a disease or monitor existing conditions. The use of AI can help to relieve pressure on hospital staff while they work hard to manage resources during the global pandemic. Non-contact remote sensing technology can sense such healthcare markers without introducing anything to the body (e.g., wearable devices). Wearable devices can be uncomfortable for some which will entice users to remove the device and results in misplacement or damage [32]. The non-contact techniques can assist in the detection of COVID-19 and the care of patients suffering from COVID-19. This will allow for quick diagnosis and allow for healthcare professionals to make clearer judgements on the treatment of the patient and allow for quarantine action to be undertaken. Vital-sign monitoring can provide great assistance in the fight against COVID-19 for several reasons. These reasons include detection of irregular breathing patterns, which is a major symptom of COVID-19, but it can also monitor the health conditions of patients suffering with COVID-19. Although COVID-19 affects the respiratory system [33,34], it has also been shown to take effect on the cardiovascular system [16]. These non-contact methods can also monitor heartbeats and therefore provide a monitoring system of the patient cardiovascular system. It can be concluded that non-contact sensing that monitors these vital signs can be used to aid in the detection and treatment of COVID-19. Examples of non-contact techniques described in this paper include computed tomography (CT) scans, X-rays, Camera Technology, Ultrasound Technology, Radar Technology, Radio Frequency (RF) signal sensing Thermography and Terahertz. Table 1 details the advantages and disadvantages of each technique. These methods can be used with AI to help give diagnosis. Currently testing for COVID-19 is done by doing a swab test. The results of these tests are currently returned the next day, but may be delayed by up to 72 h [35]. The paper will provide a review of the state-of-the-art literature that is using these non-contact methods to be able assist patients suffering with COVID-19. Table 2 provides a summary table of the current literature contained within this review paper.
CT Scanning
An example of a non-invasive technique to detect COVID-19 is using computed tomography (CT) scans [47]. This process involves taking several X-ray images of a person's chest to create a 3D image of the lungs. The images can be reviewed by professionals to look for abnormalities in the lungs. The professionals are trained to review the images and they can tell from the captured image what is normal tissue of the lungs and which part of the lungs look to be infected. Infection can lead to inflammation of the tissue which will be present in the CT images. This method has been used to look for pneumonia which is an infection of the lungs which can affect the lungs similarly to how COVID-19 has an effect on the lungs of a patient. The activity of COVID-19 in the lungs is more prominent in the later stages of infection; however, ultimately, research has shown that CT scans showed a sensitivity of 86-98% [48]. This technique is non-contact as nothing is directly introduced into the body of the patient. However if a patient has been found to be infected with COVID-19 then the surface of the CT scanning machine is likely to contain droplets of the infection dispensed by the patient. This will therefore need to be cleaned effectively to prevent the spread of the virus to another patient who will be tested using the CT scanner apparatus. It can be noted that cleaning of surfaces can be considered safer for healthcare workers than physical contact with a patient. This is because droplets that are present on surfaces are likely to be static, whereas infected patients will dispense these droplets from their bodies during breathing and possibly through coughing, which is a symptom of COVID-19. CT scans can achieve high precision with high image resolution, however the technology used to perform CT scans is expensive. CT Scanners are paid for out of hospital budgets and are part of the dedicated equipment used to assist hospital staff in patient diagnosis. Their cost is proportionate to the level of accuracy they can provide within the healthcare industry. The equipment is not portable, and it requires skilled professionals for image analysis. The CT scanning machine is a massive piece of equipment. The machine is big enough to scan the entire length of an adult laying down. This also ensures the machine is of a high weight which will further remove the portability of the device. Another disadvantage of CT scanning is that the patient is exposed to radiation [49]. The radiation levels in CT scans have been found to result in an estimated cancer mortality risk of 0.08% within a 45-year-old adult [50]. Recently, AI has been used on CT images for diagnosis of COVID-19 [51]. Again, AI can allow for support for the skilled professionals analyzing the CT images produced by the CT scanners. If AI can assist with the detection and predictions of any disease in the lungs, this can help to ease the workload of the CT scan professionals. The advantages of this can allow for greater care of patients and more opportunity to ensure the appropriate safety prosecutions are being taken to prevent the spread of COVID-19 to the hospital staff or other patients who could potentially be classed as at high risk of COVID-19.
Fei Shan et al. [45] developed a deep-learning model which was able to detect COVID-19 and the level of infection within the lungs. Their model adopted a human-in-the-loop (HITL) strategy. Human-in-the-loop is when specialists are used to label a small amount of training data. Then an initial model is trained. Then this initial model is used to classify new data. The specialist then corrects any incorrect labels and the data set can be used to train further models. This task can be iterated numerous times to reduce the tedious task of labeling large amounts of data. The experiment used 249 confirmed cases of COVID-19 for training. The experiment achieved a high result of 91.6% accuracy. The experiments of this paper used 3 iterations. The first iteration made classifications on the validation data using 36 labeled images as a data set with an accuracy score of 85.1%. The labels are then corrected and added to the second iteration. The second iteration used 114 images for training and achieved an accuracy result of 91.0%. The labels are then corrected and passed to the third iteration. The third iteration is used on all 249 training images and achieved an accuracy result of 91.6%. The improved accuracy greatly reduces the human involvement and time devoted to labeling the full data. Figure 1 displays a flow chart of the process of human-in-the-loop. Li, Lin, et al. [37] used a COVNet, a custom deep-learning neural network to predict COVID-19 in CT images. The complete data set used included 400 COVID-19 CT images, 1396 Pneumonia CT images and 1173 non-infected CT images. The model takes CT images as input and extracts features of COVID-19 and pneumonia evidence found in the CT images. The features are then combined and the neural network can be applied to make predictions on whether the CT images contain COVID-19 or pneumonia features or if the CT images are of that of a non-infected person. Results found that the model was able to predict COVID-19 in patients with 90% sensitivity. The model proved to not only be able to detect infected and non-infected lungs but was also able to differentiate between COVID-19 and pneumonia with pneumonia having a sensitivity of 87%. Once the model was trained it was able to classify new samples within 4.51 s [37]. Figure 2 shows the process followed in this research. . Support Vector Machine (SVM) algorithm was then used to classify the extracted features of each of the methods. Support Vector Machine was used on the features using 2-fold, 5-fold and 10-fold cross-validation. Cross-fold validation is the process of using each fold to work as both training and testing data for the model to make predictions. Each fold will take a turn as being the testing data while the others are used as training. This is repeated for however many folds there are so that each fold serves as the testing data at least once. Then the results are compiled and each sample will have predictions made on it as it served as the testing data through each fold. The best accuracy result achieved out of the various methods of experimentation was 99.64%. This result was achieved using Discrete Wavelet Transform feature extraction method with 10-fold cross-validation using the 48 × 48 patch dimension CT images. A flow chart of the methodology followed in this research is shown in Figure 3. The above papers have shown through experimentation that CT scanning can display the signs of COVID-19 within a person's lungs. The research has also shown how AI can be used to make predictions of CT images and provide assistance in the determination of whether COVID-19 is present in the lungs or not. The studies have also shown that AI can determine the level of infection present in COVID-19 patients. The AI has also been able to differentiate between pneumonia and COVID-19 infections which is a positive as COVID-19 and pneumonia is similar in the way that both diseases attack the lungs. Table 3 provides a breakdown of the above research papers on using CT scans for non-contact COVID-19 diagnosis and care of patients.
X-Ray Imaging
X-ray images can provide an analysis of the health of the lungs and are used frequently to diagnose pneumonia [52]. The same strategy is used with X-ray images of the lungs to display the visual indicators of COVID-19 [53,54]. This is due to the similarities between COVID-19 and pneumonia as diseases that take an effect on the respiratory system. Similar to CT scans, X-ray equipment is also expensive and requires professionals to analyze the X-ray image.
The paper entitled "Automatic detection of coronavirus disease (COVID-19) using x-ray images and deep convolutional neural networks" used X-ray images taken of COVID-19-infected lungs and patients with lungs that were non-infected with COVID-19 to create a data set of x-ray images which was then used to predict COVID-19 automatically in patients. The X-ray images are passed into a ResNet-50 Convolutional Neural Network (CNN) which successfully obtained results of 98% accuracy in the differentiating between COVID-19 infected X-ray images and the non-infected x-ray images [38].
The paper of Zhang, Jianpeng, et al. [44] used deep-learning techniques on a data set of X-ray images of 70 patients confirmed to have COVID-19. Additional images of patients with pneumonia are added from a public chest X-ray image data set. The model is used to identify differences in X-ray images between patients infected with COVID-19 and patients suffering from pneumonia. The proposed deep-learning model was able to achieve a sensitivity of 90% detecting COVID-19 and a specificity of 87.84% in detecting non-COVID-19 cases.
Ozturk, Tulin, et al. [39] also conducted experiments using deep learning to classify X-ray images of patients with real-time classification of COVID-19. The experiments made use of a custom deep-learning model named DarkNet to perform binary and multi-class classifications. The binary classification is the process of deep learning, making predictions based on two choices. In the case of this experiment, the binary classification seeks to distinguish between COVID-19 and no findings of disease. Multi-class classification is when AI is tasked with making classifications on more than two possible classifications. This differs from binary classifications as the model must make decisions on which class data belongs to rather than just making distinctions between data. The multi-class classification distinguishes between no findings of disease and or if disease is found, and then whether the disease is pneumonia or COVID-19. The experiments used a publicly available data set of COVID-19 X-ray images and another publicly available data set for non-infected and pneumonia X-ray images. The complete data set included 127 COVID-19 X-ray images and 500 pneumonia X-ray images and 500 non-infected X-ray images. The deep-learning process made use of the developed DarkNet neural network. The complete X-ray image data set was divided between 80% training data and 20% testing data. The deep learning was run for 100 epochs using 5-fold cross-validation. Each epoch is an iteration of when the data is passed through the neural network. The neural network will learn about the data being passed through. Repeating epochs can allow for the model to fine-tune its biases and weights on what it believes data should be classified as. Then the model can improve the accuracy as it learns what works and does not until it can provide the best results obtained. The results produced an accuracy score of 98% for binary classification and an accuracy score of 87.02% for multi-class classification. It is expected that the result will fall as the number of classifications increase as the AI will need to recognize more features to distinguish between classes rather than differentiate between two data patterns. The complete process followed in this work is detailed in Figure 4.
Camera Technology
Camera technology can be used to provide non-contact sensing by observing the chest movements of an individual [55]. This can be achieved by capturing video footage of movements of the chest or, in the case of depth cameras, they are able to calculate depth by using two sensors with a known range [56]. The information captured using camera technology can be used provide assistance in the detection of COVID-19 as one of the symptoms of the disease includes an increase in the breathing rate of patients.
The paper "Combining Visible Light and Infrared Imaging for Efficient Detection of Respiratory Infections such as COVID-19 on Portable Device" used RGB-thermal camera footage for the detection of COVID-19. The footage was used with machine-learning binary classification to detect normal and abnormal breathing from people wearing protective masks. This research is relevant as masks are now commonly worn by people around the world as a preventive measure against COVID-19. The research collected real-world data and applied deep learning to achieve a high result of 83.7% accuracy which is the highest result found in the literature in regards to breathing detection using RGB-thermal imaging with deep-learning models. This research can provide a scanning method which can be used to control the spread of the virus and work with protective masks, thus reducing spread of COVID-19 [41].
Wang, Yunlu, et al. [36] used Microsoft Kinect cameras to take depth images of volunteers breathing. A total of 20 volunteers were asked to sit on a chair and simulate 6 different breathing patterns. The breathing patterns were eupnea, bradypnea, tachypnea, biots, Cheyne -Stokes and central apnea. Each of these patterns display a different breathing rate in the individuals. Patients of COVID-19 display the rapid breathing pattern of tachypnea. During data collection, a spirometer was used to ensure the breathing pattern was being simulated correctly by the volunteers. The depth images taken using the camera were used in a deep-learning neural network model to classify the abnormal breathing patterns of tachypnea associated with COVID-19. The deep-learning model used was the BI-AT-GRU algorithm. Gated Recurrent Unit (GRU) is a simplified version of the Long-Term Short Memory (LTSM) algorithm. The BI-AT-GRU algorithm results achieved a high accuracy score of 94.5%. This research shows how depth images can be used to identify the tachypnea breathing patterns observed in COVID-19 patients in real time. The process map for this research is shown in Figure 5. The primary disadvantage of using this method is the cost of thermal and depth cameras and the camera operators. Although the price of these cameras is falling gradually, it remains substantially high [57]. The cost of the equipment is of course less expensive than methods such as CT and X-ray scanning, but still more expensive than other methods discussed further in this paper. The research done with cameras has shown that the devices can be used with AI in the detection of COVID-19 and without contact with the body. This allows for more techniques to be implemented where diagnosis of COVID-19 can be achieved in a safe manner without increasing the risk of spreading the disease.
Ultrasound Technology
Ultrasound technology can be applied to detect respiratory failure of the lungs. An ultrasound machine is a device that uses high-frequency sound waves to image body movements [58]. The sound waves bounce off different parts of the body which create echoes that are detected by the probe and used to create a moving image. Lung ultrasounds have seen great development in recent years [59]. The use of ultrasound technology can be used in the detection of COVID-19 in a non-contact method where the risk of healthcare professionals becoming infected from patients can be decreased [60,61]. Ultrasound technology becomes contactless by using an ultrasound transmitter and receiver. Respiratory movement can then take place between the transmitter and receiver and creates a Doppler affect. This can then be used to create a contactless breathing monitor [62][63][64]. Ultrasound technology can be performed using smartphones for the signal and processing of ultrasound images in a portable setting [65]. The disadvantage of ultrasound technology is that patients must prepare themselves before an ultrasound can effectively create an image of the body [66]. This preparation can include not eating for a few hours before.
The work of Born, Jannis, et al. [46] shows that ultrasound technology can be used in deep-learning models to distinguish the differences in COVID-19, pneumonia, and no infection within the lungs. The research collects a data set of lung ultrasound images which contain video recordings of lung ultrasound scans. The data set includes a total of 64 video recordings with 39 of the recordings of COVID-19 patients, 14 videos of pneumonia patients and 11 videos of non-infected patients. The paper has developed a deep-learning convolutional neural network named POCOVID-Net. The deep-learning algorithm was able to achieve an accuracy score of 89%. These ultrasound devices can diagnose 4 to 5 patients per hour. Figure 6 shows a simplified flow graph of the experiment undertaken in this paper.
Radar Technology
Radar technology can be used to monitor the respiratory system within a home environment and provide a quick response if abnormalities are found, which suggests COVID-19 being present. Radar systems use frequency-modulated continuous wave (FMCW) to observe the Doppler effect when a person moves [67][68][69][70]. This can be used to monitor the fine movements associated with breathing. This is achieved by using the images captured by the radar systems then applying AI to classify the images. AI models can be used to give real-time classification on new images [71][72][73]. Research done shows that radar technology can achieve 94% accuracy for the detection of breathing rates and 80% accuracy for heart-rate detection [34,74,75]. The Israeli military force has made use of radar systems for monitoring the vital signs of COVID-19 patients. The goal of using this method is to prevent medical staff from becoming infected while caring for patients [40,76]. Tachypnea is a symptom of COVID-19 and can be detected in a patient by using radar sensing technology [63,68,77]. Using radar technology to monitor vital signs can provide non-interference monitoring; however the disadvantage of radar systems is that it has high power requirements and the technology comes at a high cost [78].
Radio Frequency Signals
The use of radio frequency (RF) signal sensing can detect the vital signs of individuals by sensing the minute movements of the chest made while breathing as the heart beats ( [73,[79][80][81][82]). This technique can be used for monitoring the vital signs of patients independent of their activities [83]. The RF signals detect the movement by observing the Channel State Information (CSI), which can show amplitudes of the RF signals while movement occurs between a RF transmitter and receiver [84,85]. The Emerald system has been developed to monitor COVID-19 patients using RF signals. The system uses RF signals to detect the breathing rate of COVID-19 patients and then uses AI to infer the breathing rate of the patient. This allows for doctors treating the patients to be able to monitor the patient from a safe distance. This method prevents the risk of infection to staff and provides the patient comfort as they do not need to wear monitoring devices [43]. RF signals have been used in previous research to detect breathing rates. RF signals can be used to detect abnormal breathing patterns such as tachypnea [86], which is a symptom of COVID-19 [36]. Systems have been developed to allow for real-time monitoring of breathing patterns using RF signals [87]. RF signals can be vulnerable to other movements within the room. The other movements create noise in the Channel State Information which can then in turn cause false readings [88,89].
Thermography
Thermography is a widely used non-contact technique within the medical community [90,91]. It has been used for mass screening of people in other pandemics such as H1N1 and Ebola so it can be applied in this current pandemic of COVID-19 [92]. Thermography works by using infrared radiation to calculate the temperature of the human body [93]. Abnormal body temperatures are a well-known indication of infection [94]. Symptoms of COVID-19 have been found to include high temperatures over the normal body temperature of 36-37 degrees Celsius [95,96]. Thermography can also be used to monitor the respiratory systems of patients and provide detection of breathing patterns such as bradypnea or tachypnea using AI [97]. Thermography has been recommended as an early detection strategy for COVID-19 among large amounts of people in places such as in airports [98]. Deep learning has been applied to thermal images where classifications on new images can be made in under a second [99,100].
Terahertz
Terahertz sensing technology is the process of directing terahertz beams to a person's body to detect the motion of the chest created by a heart beating or lungs inhaling or exhaling breath [101,102]. Terahertz sensing is a non-contact method which can achieve superior penetration depth [103]. This can be helpful when penetrating a patient's clothes. These terahertz systems can be produced in a similar fashion to how the radar imaging takes place, except with using terahertz waves and observing the Doppler effect of the Terahertz wave while a patient performs the breathing issue [104]. Terahertz waves refer to electromagnetic frequencies around 0.1-10 Terahertz(THZ) [103,105]. The use of terahertz can detect disease such as COVID-19 [106]. This will work similarly to the radar system with AI being used to make classifications on the images showing the Doppler effect of terahertz waves. Deep learning can be applied to these images and give fast classifications of new models once an AI model has been fully trained. Terahertz radiation is considered the first choice in radiation exploitation due to the non-harmful properties to living cells [107]. A terahertz spectroscopy is an example of a powerful tool in medical research and diagnosis used for analysis of human breath samples and it offers a low cost [108].
Comparison to Contact Methods
The methods discussed in this paper have looked at non-contact techniques for diagnosing COVID-19. Due to the nature of the disease, it has been widely acknowledged that reducing contact between people is the best action to reduce the spread. Therefore non-contact technologies for diagnosis are the preferred method. Wearable devices can also be used for monitoring vital signs [109,110]. This monitoring of vital signs can therefore be used to detect any displays of COVID-19 symptoms. Popular devices such as AppleWatch, FitBit and Oura ring are highly available and provide monitoring of the heart rate [111]. The Oura ring has been found to show changes in body temperature associated with COVID-19 and has led to several studies being conducted into the use of Oura rings in early detection of COVID-19 [112,113]. These technologies are known as personal health trackers and in terms of COVID-19 detection, these devices will be better for self-diagnosis. If these devices can inform users that they are displaying COVID-19 symptoms then the user can take action. Non-contact methods will serve healthcare workers better as they can provide assistance to patients while still reducing contact with the patient and thus reducing risk of infection.
Future Directions
This section will detail some of the future directions which may be suitable for expanding on the research presented in this paper. The research has highlighted how the detection of COVID-19 is possible using various techniques. This section will now discuss how this research can be taken further to work within real-life scenarios.
•
One of the biggest challenges with CT scanning to diagnose COVID-19 is the lack of portability. This means that although the method is non-contact, its use still requires individuals to travel to a location where the machine is available. As the CT images can provide high resolution, the AI can be used for the detection of COVID-19. Therefore, future directions of this method should look to creating highly accurate models that can eventually lead to the automation of COVID-19 detection. This can allow for faster diagnosis, which can allow for more patients to be tested and increase availability of staff operating and analyzing CT scans. • X-rays, similarly to CT scans, are not portable. Like CT scans, professionals are required to operate these machines and analyze the X-ray images. The research presented in this paper has shown that AI can be used to make predictions if COVID-19 is present in the lungs. This can be useful similarly to CT scans where AI can be applied to make the predictions and speed up the process. The more data collected, the more advanced the model will become. Perhaps initially the predictions will need to be confirmed by humans but eventually the checks can become less frequent. Since the research above has displayed an ability of AI to distinguish between not just COVID-19 and non-infected but also pneumonia at high accuracy, then the AI has proved to be capable of accurate classifications.
•
Thermal and depth cameras can detect the irregular breathing patterns that are associated with COVID-19 symptoms. The issue here is that even though the camera can detect the irregular breathing pattern, it is unable to categorically define COVID-19 as the cause for individuals displaying the irregular breathing patterns. In a real-life situation, the camera method may be better suited to monitoring vulnerable people who are considered high risk from COVID-19. Then once the monitoring system has identified the irregular breathing patterns, an alarm can be raised with a career or family member. Then, appropriate action can be taken for greater accuracy such as diagnosis with CT scanning or X-ray scanning.
•
Ultrasound technology can take moving images of the lungs and detect COVID-19. This can also be made portable by using mobile devices. AI can be applied to recognize if COVID-19 or pneumonia is present in the lungs. This research can be further applied to develop applications on a mobile device that can capture an ultrasound of the lungs then compare it to an AI model to predict if COVID-19 is present. Although not all phones may not have the necessary hardware to achieve this, the non-contact method can allow for others to be able to use the devices for diagnosis at a safe distance.
•
Radar technology can identify the breathing patterns of individuals. Much like camera technology, the identification of breathing patterns can raise cause of concern but it cannot isolate COVID-19 as the sole cause. Radar technology can again be used to monitor individuals but due to the high costs it is more likely to be used as a monitoring system within a hospital and not a home environment.
•
Any future directions should consider the use of RF signals to detect the breathing patterns which give indication of COVID-19 symptoms. The RF systems can be implemented inexpensively using existing WiFi technology present within many homes. This allows for the monitoring of individuals without the costs incurred in implementing radar or camera technologies highlighted in this paper. • Thermography has shown in previous research to be able to detect body temperatures of large amounts of people in previous pandemics. Therefore, it can be implemented in mass screening in the current COVID-19 pandemic. With the use of thermography being able to detect respiratory issues, it is clear that these systems can also be implemented for COVID-19 detection.
•
Terahertz can provide deeper penetration and detect smaller movements such as the chest movements while breathing. This can therefore be used in early detection of COVID-19. The earlier the disease is detected, the sooner isolation can begin and ensure that further spread is reduced.
Conclusions
The works listed in this paper have shown that COVID-19 can be detected using contactless techniques. Techniques such as CT scans and X-ray imaging provide high accuracy and high image resolution, but the cost of the equipment is high and not portable. Thermal and depth camera technology has been used to detect breathing patterns, which is associated with COVID-19 symptoms. However, these cameras are expensive and need to be operated by a professional. Radar technology is also able to detect breathing patterns but carries disadvantages of high operating expenses and capital expenditures. RF signals provide low cost and high accuracy as compared with other non-invasive technologies. The technologies can work on AI which can allow for skilled professionals to be available to assist in other areas of healthcare during the pandemic. The non-contact methods also protect healthcare workers from contracting the disease. The future direction of non-contact detection should look at the use of RF systems as the cost is cheap and it is easier to implement within a home environment in comparison to other methods. This gives the advantage of allowing the users to remain within isolation.
Figure 1 .
1Flow chart of work for detection of COVID-19 from CT scan (Reproduced from[45]).
Figure 2 .
2Flow chart of work for detection of COVID-19 from CT scan (Reproduced from[37]).The research of Barstugan, Mucahid et al.[42] used machine learning on a data set of 150 CT images. The data set contains 53 infected CT images. Patches of the images are taken. Patches in image processing is the process of taking images and dividing them into containers of different sizes of pixels. Different sized patches are used to create 4 different samples of patches. The patch sizes are 16 × 16, 32 × 32, 48 × 48 and 64 × 64. The images were labeled as infected CT images and non-infected CT images in regard to COVID-19. The research used different methods of feature extraction on the images. These methods include Grey-Level Co-occurrence Matrix (GLCM), Local Directional Patterns (LDP), Grey-Level Run Length Matrix (GLRLM), Grey-Level Size Zone Matrix (GLSZM) and Discrete Wavelet Transform (DWT)
Figure 3 .
3Flow chart of work for detection of COVID-19 from CT scan (Reproduced from[37]).
Figure 4 .
4Flow chart of work for detection of COVID-19 from X-ray images (Reproduced from[39]).
Figure 5 .
5Flow chart of work for detection of COVID-19 from Depth Camera Image (Reproduced from[36]).
Figure 6 .
6Flow chart of work for detection of COVID-19 from Ultrasound Technology (Reproduced from[46]).
Author
Contributions: Conceptualization, W.T., Q.H.A., S.A., S.A.S., A.K., M.A.I.; formal analysis, W.T., S.A.S., K.D., A.K.; investigation,W.T., S.A.S., K.D.; resources, writing, review and editing, W.T., Q.H.A., S.A., S.A.S., A.K., M.A.I.; funding acquisition, Q.H.A., M.A.I.; All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding.
Table 1 .
1Summary of Non-Invasive Techniques.Method
Accuracy
Cost
Time for Measurement
Time for
Results
Harm to
Body
Skills of
Operators
Possibility
of AI
CT
High
High
Moderate
Fast
Low
High
Yes
X-Ray
High
High
Moderate
Fast
Low
High
Yes
Camera
High
Medium
Real Time
Real Time None
Medium
Yes
Ultrasound High
Medium/High Moderate
Medium
Low
High
Yes
Radar
High
High
Real Time
Real Time None
Medium
Yes
RF
High
Low
Real Time
Real Time None
Low
Yes
IR Thermo High
Medium
Fast
Fast
None
High
Yes
THz
High
Medium
Fast
Fast
None
High
Yes
Table 2 .
2Summary of Current Literature.Title of Paper
Citation Year
Key Themes
Authority
Abnormal respiratory patterns
classifier may contribute to
large-scale screening of people
infected with COVID-19 in
an accurate and unobtrusive
manner
[36]
2020
The paper details that COVID-19
patients display tachypnea
(Rapid breathing). The paper
looks at taking depth images to
identify the breathing patterns of
volunteers using deep learning
Peer reviewed paper.
24 citations on Google
Scholar.
Artificial
intelligence
distinguishes COVID-19 from
community acquired pneumonia
on chest CT
[37]
2020
CT scan images are used in
a COVNet neural network to
distinguish between COVID-19,
Pneumonia and Non-infected
scan images.
Peer reviewed paper.
157
citations
on
Google Scholar.
Automatic
detection
of
coronavirus disease (COVID-19)
using x-ray images and deep
convolutional neural networks
[38]
2020
X-ray scan images are used
in a ResNet-50 Convolutional
Neural Network (CNN) to
distinguish between COVID-19
and non-infected scan images.
Peer reviewed paper.
102
citations
on
Google Scholar.
Automated
detection
of
COVID-19 cases using deep
neural networks with X-ray
images
[39]
2020
X-ray images are processed
using the DarkNet neural
network
to
test
binary
classification
between
COVID
and
Non-infected
and multi-class classification
between COVID, Pneumonia
and Non-infected.
Peer reviewed paper.
22 citations on Google
Scholar.
Can Radar Remote Life Sensing
Technology Help to Combat
COVID-19?
[40]
2020
Radar systems have been used
to monitor the vital signs of
patients in a contact less manner
to protect healthcare workers
Paper uploaded on
researchgate.net.
Combining Visible Light and
Infrared Imaging for Efficient
Detection
of
Respiratory
Infections such as COVID-19 on
Portable Device
[41]
2020
RGB-Terminal camera footage
used in a BiGRU neural network
model between healthy and ill.
Peer reviewed paper.
Coronavirus
(COVID-19)
classification using CT images
by machine-learning methods
[42]
2020
CT scan images are used
to experiment with various
methods of feature extraction
and deep learning algorithms to
achieve the best results
Peer reviewed paper.
157
citations
on
Google Scholar. 157
citations on Google
Scholar.
CSAIL device lets doctors
monitor COVID-19 patients
from a distance
[43]
2020
Radio Frequencies have been
used to monitor the vital signs of
patients in a contactless manner
to protect healthcare workers
Article found on MIT
Computer Science &
Artificial Intelligence
Laboratory website.
Covid-19
screening
on
chest x-ray images using
deep-learning-based anomaly
detection
[44]
2020
X-ray images are used with deep
learning to identify if samples
are COVID-19 or Pneumonia
Peer reviewed paper.
32 citations on Google
Scholar.
Lung infection quantification of
COVID-19 in CT images with
deep learning
[45]
2020
CT scan images are used in deep
learning to identify COVID-19.
Human-in-the-loop technique is
used to focus on increasing
accuracy
Peer reviewed paper.
52 citations on Google
Scholar.
POCOVID-Net:
automatic
detection of COVID-19 from a
new lung ultrasound imaging
data set (POCUS)
[46]
2020
Lung Ultrasound videos of
COVID-19, Pneumonia and
non-infected patients used deep
learning for classification.
Peer reviewed paper.
2 citations on Google
Scholar.
Table 3 .
3Summary of CT Scanning works.Citation Training Data
Algorithms
Results
[45]
249 CT images of COVID-19
showing different levels of infection.
Custom
Convolutional
neural network (CNN) called
"VB-Net"
91.6% Accuracy
[37]
400 COVID-19 CT images, 1396
Pneumonia CT images and 1173
non-infected CT images
Custom
Convolutional
neural network (CNN) called
"COVNet"
90%
sensitivity
of
COVID-19 samples.
[42]
150 CT images including 53
COVID-19 cases.
Support Vector Machine
99.64% Accuracy
Acknowledgments: William Taylor's studentship is funded by CENSIS UK through Scottish funding council in collaboration with British Telecom. This work is supported in parts by EPSRC DTG EP/N509668/1 Eng, EP/T021020/1 and EP/T021063/1.Conflicts of Interest:The authors declare no conflict of interest.
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| []
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[
"Entropic Mechanics: towards a stochastic descripton of quantum mechanics",
"Entropic Mechanics: towards a stochastic descripton of quantum mechanics"
]
| [
"Vitaly Vanchurin [email protected] \nDepartment of Physics\nDuluth Institute for Advanced Study\nUniversity of Minnesota\n55812, 55804Duluth, DuluthMinnesota, Minnesota\n"
]
| [
"Department of Physics\nDuluth Institute for Advanced Study\nUniversity of Minnesota\n55812, 55804Duluth, DuluthMinnesota, Minnesota"
]
| []
| We consider a stochastic process which is (a) described by a continuous-time Markov chain on only short time-scales and (b) constrained to conserve a number of hidden quantities on long time-scales. We assume that the transition matrix of the Markov chain is given and the conserved quantities are known to exist, but not explicitly given. To study the stochastic dynamics we propose to use the principle of stationary entropy production. Then the problem can be transformed into a variational problem for a suitably defined "action" and with time-dependent Lagrange multipliers. We show that the stochastic dynamics can be described by a Schrödinger equation, with Lagrange multipliers playing the role of phases, whenever (a) the transition matrix is symmetric or the detailed balance condition is satisfied, (b) the system is not too far from the equilibrium and (c) the number of the conserved quantities is large. | 10.1007/s10701-019-00315-6 | [
"https://arxiv.org/pdf/1901.07369v2.pdf"
]
| 84,845,425 | 1901.07369 | 35341abf380c705bf48d16e9e1b6ef30f8a30166 |
Entropic Mechanics: towards a stochastic descripton of quantum mechanics
20 Mar 2019
Vitaly Vanchurin [email protected]
Department of Physics
Duluth Institute for Advanced Study
University of Minnesota
55812, 55804Duluth, DuluthMinnesota, Minnesota
Entropic Mechanics: towards a stochastic descripton of quantum mechanics
20 Mar 2019Contents 1 Introduction 1
We consider a stochastic process which is (a) described by a continuous-time Markov chain on only short time-scales and (b) constrained to conserve a number of hidden quantities on long time-scales. We assume that the transition matrix of the Markov chain is given and the conserved quantities are known to exist, but not explicitly given. To study the stochastic dynamics we propose to use the principle of stationary entropy production. Then the problem can be transformed into a variational problem for a suitably defined "action" and with time-dependent Lagrange multipliers. We show that the stochastic dynamics can be described by a Schrödinger equation, with Lagrange multipliers playing the role of phases, whenever (a) the transition matrix is symmetric or the detailed balance condition is satisfied, (b) the system is not too far from the equilibrium and (c) the number of the conserved quantities is large.
Introduction
From the early days of quantum mechanics physicists tried to come up with a classical or statistical model which would explain the bizarre prediction of quantum mechanics. The literature on the subject is rather vast and we are not going to discuss it here. The interested reader is referred to recent books on the subject [1][2][3] (and references therein) where some new and original proposals are also presented (e.g. trace dynamics [1], entropic dynamics [2], cellular automata interpretation [3]). Perhaps it is worth mentioning that any attempts to derive quantum mechanics from classical or statistical mechanics should be taken with great caution due to severe experimental constraints on local hidden variables theories [4,5] imposed by Bell's inequalities [6]. We are not going to discuss Bell's inequalities here either just because we are not yet at the stage of matching experimental results. See, however Ref. [7], for a recently proposed duality between a quantum system of spinors and a classical system of scalars. The coupling between scalars is non-trivial (e.g. a model on 2-sphere configuration space), but the locality structure of the dual systems is preserved and so the duality can potentially be used to study the Bell's inequalities in context of local hidden variables theories.
In this paper, our main goal is to identify the conditions under which a statistical system would evolve according to rules of quantum mechanics. More precisely, we want to construct a stochastic process (which need not be Markovian) whose dynamics would be (if not exactly, but approximately) described by a Schrödinger equation. The stochastic process will be assumed to obey Markovian dynamics on the shortest time-scales (e.g. Planck time) and at the same time the dynamics will be constrained to conserve a large number of conserved quantities. As a result, the overall process may no longer be Markovian, but is a martingale in a sense that expectation values of the conserved quantities in the initial state remain unchanged throughout evolution. Whether such a process would be a generic consequence of coarse-graining (or of lumping of states) is an important question which deserves a separate study. We will provide an example of a physical system for which these two conditions (i.e. Markovian on short time-scales and martingale on long time-scales) are satisfied, but for the most part of the paper we shall assume that such a process exists and the main problem will be to determine the most probable path that a statistical state of such a system would take.
To tackle the problem we propose to use the principle of stationary entropy production. We are not going to prove the principle but it is, once again, something which deserves a separate analysis. Note, however, the apparent similarities between the stationary action principle (also known as least action principle) in classical mechanics and the stationary entropy production principle which we propose to use in context of statistical systems, or what we shall call entropic mechanics. In classical mechanics, action is extremized along classical paths and in entropic mechanics, entropy production is what must be extremized along statistical paths. In fact, our framework of entropic mechanics is very similar in spirit to both classical mechanics and also entropic dynamics [2], but the underlying principles (i.e. stationary entropy production principle vs. stationary action principle or maximum entropy principle) are not the same. Of course, it is possible that all these principles are intimately related to each other.
The paper is organized as follows. In the next section we define a stochastic process which is described by a Markov process on only short time-scales and highly constrained on longer time-scales. To study the process we propose the stationary entropy production principle which is introduced in Sec. 3. In Sec. 4 we apply the condition of detailed balance and derive an expression for entropy production near equilibrium. In Sec. 5 we introduce an auxiliary wave function and in Sec. 6 we derive an equation which governs its dynamics in the limit of a large number of constraints. In Sec. 7 we discuss the main results of the paper.
The Stochastic Process
Consider a continuous-time Markov chain described by a master equation
dp m dt = n ∆ mn p n , (2.1)
where p n is the probability to find the system in state n and ∆ mn is the transition rate from state n to state m. We assume that the total number of states, N , is finite and summation over all states is implied unless stated otherwise (as, for example, in equation (2.4)). From conservation of probabilities it follows that 0 = m dp m dt = m,n ∆ mn p n , (2.2) but since this must be true for any vector p, including p n = δ nl , we get
m ∆ ml = 0. (2.3) or ∆ ll = − m =l ∆ ml (2.4)
for any l. The master equation (2.1) can be integrated to obtain a time-dependent solution p m (t). At very large times the solution will be dominated by an eigenvector of ∆ mn with eigenvalue zero. For irreducible Markov processes (an assumption we are going to make) the eigenvector is unique due to Perron-Frobenius theorem. Now consider another stochastic process which is described well by the master equation (2.1) only on short times-scales (e.g. Planck time), but for longer time-scales all that we know is that there are certain conserved quantities due to some (perhaps hidden microscopic) symmetries. We do not know what these conserved quantities are, but we know they exist. It may be useful to keep in mind a concrete (but toy-) model of a process of this type. For instance, per unit time (e.g. Planck time) the process might be such that the state first changes according to (2.1) and then it is projected back to the surface described by K < N constraints. Note that the constraint surface is N − K dimensional and the equientropic surface (surface on which entropy is constant) is N − 1 dimensional and so they would generically intersect along N − K − 1 dimensional surface. On Fig. 1 we provide an illustration of such a process with N = 2, K = 1 and N − 1 − K = 0 and so there is a unique choice for the path. Vertical lines represent the short time-scales Markovian evolution, the dashed blue lines represent equi-entropic surfaces, the thick solid green line represents a single constraint surface and the zigzagged line with alternating red and blue segments represents respectively the Markovian dynamics and projections to the constraint surface. If the projection is only a small correction to the path then the short-time dynamics would be approximated by (2.1), but the constraints would be satisfied on long time-scales. In general N − 1 − K > 0 and so an additional principle must be imposed to single out a unique path. We will discuss one such principle in the next section.
Perhaps a more physically-relevant example of the stochastic process can be described starting with a gas of molecules (see Fig. 2) and then introducing pairwise interactions so that the molecules form long polymer-like chains (see Fig. 3). We are not interested in following individual molecules exactly, but only in a coarse-grained dynamics of a probability distribution of molecules. For the gas of molecules there is really only very few constraint (conservation of energy and normalization condition), but with molecules forming long chains (which we assume, for simplicity, cannot break) the overall distribution becomes highly con- strained. The individual molecules can only experience an approximately Brownian motion (which is Markovian) on short time-scales as they must also respect the motion of their immediate neighbors on the chain. And the constraints imposed by the neighbors is what introduces a large number of conserved quantities on long time-scales. The situation is very similar to a fluid of strings where, in addition to the conservations of energy and momentum, an anti-symmetric tensor, which describes tangent vectors of strings, must also be conserved [8][9][10].
In what follows, we are not going to consider a particular process, but a class of processes for which the short-time dynamics is described by (2.1) and the long-time dynamics is constrained to conserve a large number of conserved quantities. The conserved quantities will be denoted by Θ for α = 1, ..., K. Note that the normalization condition can also be imposed as a conserved quantity with, for example, Θ As in the case of a pure Markov process, we are interested in finding p n (t) for any t, but now the problem does not have a unique solution and some principle must be postulated in order to single out a unique (time-dependent) distribution p n (t).
The Principle of Stationary Entropy Production
The principle of maximum entropy [11] states that when a number of different probability distributions are consistent with the same set of constraints, the most reasonable choice corresponds to a distribution which has the largest Shannon entropy
S ≡ − n p n log p n . (3.1)
Then the problem can be solved using the method of Lagrange multipliers. If we define a "Lagrangian"
L(λ 1 , ...., λ K ; p 1 , ..., p N ) ≡ − n p n log p n + α λ α n p n Θ (α) n − θ α , (3.2)
then at a (local) maxima of L the partial derivatives with respect to p n 's and λ α 's must vanish
0 = ∂L ∂p n = α λ α Θ (α) n − log p n − 1 (3.3) 0 = ∂L ∂λ α = n Θ (α) n p n − θ α (3.4)
and, thus, the maximum entropy distribution would be given by
p n = exp −1 + α λ α Θ (α) n (3.5)
with Lagrange multipliers λ α determined from the constraints (3.4). Using this principle a number of fundamental results of statistical mechanics can be derived from information theory, but it remains unclear what role the principle may play in context of quantum mechanics. (See, however, Ref. [2] where the principle was used to study a possible emergence of quantum mechanics from information theory.). In our problem we are not interested in time-independent equilibrium states (and not even in time-independent non-equilibrium steady states), but in how the state evolves in time. In other words, we must obtain a solution for the entire path p m (t) from some initial time t = 0 to some final time t = T and thus we are forced to use another principle. In this paper we shall consider the following principle:
Principle of Stationary Entropy Production: The path taken by the system is the one for which the entropy production is stationary.
The principle can be thought of as a generalization of both, the maximum entropy principle [11] and the minimum entropy production principle [12,13] (which is often used to study steady states in non-equilibrium thermodynamics.) The main difference, however, is that instead of specifying only a single state (e.g. an equilibrium or a steady state) the principle of stationary entropy production is supposed to describes the entire path p m (t).
According to the second law of thermodynamics, the entropy must grow and all that the principle says is that this growth has to be extremized (either as slow as possible or as fast as possible). In other words what we want to extremize is the entropy production or the total entropy change,
∆S ≡ S(T ) − S(0) = T 0 dt dS(t) dt = − T 0 dt m dp m (t) dt (log p m (t) + 1) = − T 0 dt m,n p n (t)∆ mn log p m (t),(3.6)
subject to whatever constraints. Note that we have explicitly assumed that the entropy production is due entirely to our Markovian dynamics on the short time-scales and contributions coming from restricting trajectories to remain on the constraint surface are negligible (see Fig 1). More generally, this assumption may be relaxed if the overall entropy production can still be expressed using equation (3.6) with perhaps a different matrix used on place of ∆ mn . Given (3.6) we want to apply the principle of stationary entropy production to find a path which corresponds to either the least or the greatest amount of produced entropy. To accomplish this task, we define not a "Lagrangian", but an "action" that is extremized along the paths of stationary entropy production,
S ≡ T 0 dt dS dt + α λ α (t) n p n (t)Θ (α) n − θ α = T 0 dt − m,n p n (t)∆ mn log p m (t) + α λ α (t) n p n (t)Θ (α) n − θ α . (3.7)
To guarantee that the constraints (2.5) are imposed at all times the Lagrange multipliers λ α (t) are now time-dependent.
Entropy Production
The entropy production term (3.6) can be re-expressed as where we used (2.4). If the detailed balance condition is satisfied, i.e.
∆ mn π n = ∆ nm π m (4.2)
where π m is the equilibrium state, then the system can be transformed into a system with symmetric transition matrix after appropriate splitting of states into substates. To simplify the analysis we will assume that the transition matrix is symmetric
∆ mn = ∆ nm ,(4.3)
but our analysis and conclusions will be equally valid for systems with detailed balance given that the state space was appropriately redefined. Using (4.3) the entropy production (4.1) becomes dS dt = 1 2 m,n p n ∆ mn (log p n − log p m ) + 1 2 m,n p n ∆ nm (log p n − log p m )
= 1 2 m,n p n ∆ mn (log p n − log p m ) + 1 2 m,n p m ∆ mn (log p m − log p n ) = 1 2 m,n (p n − p m )∆ mn (log p n − log p m ) . (4.4)
Since log(p) is a monotonically increasing function, (p n − p m )(log p n − log p m ) ≥ 0, and offdiagonal elements of ∆ mn are non-negative, the entropy production must also be non-negative dS dt ≥ 0. (4.5) This means that the second law of thermodynamics is satisfied as it should.
For a symmetric transition matrix (4.3) the conditions (2.3) becomes
m ∆ ml = m ∆ lm = 0 (4.6)
and, therefore, the uniform distribution π m = 1 N , is also an equilibrium distribution
dπ l dt = m ∆ lm π m = m ∆ lm 1 N = 0. (4.7)
Then near equilibrium we should be able to expand log p n around p m ,
log(p n ) = log(p m + (p n − p m )) = log(p m ) + (p n − p m ) p m − (p n − p m ) 2 2p 2 m + O((p n − p m ) 3 ) (4.8)
which can be substituted back to the entropy production (4.4),
dS dt = 1 2 m,n ∆ mn (p n − p m ) 2 p m − (p n − p m ) 3 2p 2 m + O((p n − p m ) 4 ) . (4.9)
Note, however, that
√ p n = p m + (p n − p m ) = √ p m + (p n − p m ) 2 √ p m − (p n − p m ) 2 8 √ p m 3 + O((p n − p m ) 3 ) (4.10)
and so
4 ( √ p m − √ p n ) 2 = (p n − p m ) 2 p m − (p n − p m ) 3 2p 2 m + O((p n − p m ) 4 ). (4.11)
Therefore, up to the fourth (!) order in p n − p m , the entropy production is simply
dS dt = 2 m,n ∆ mn ( √ p m − √ p n ) 2 = 2 m,n ∆ mn p m − 4 m,n ∆ mn √ p m p n + 2 m,n ∆ mn p n = −4 m,n √ p m ∆ mn √ p n . (4.12)
where we used (4.6).
Auxiliary Wave Function
The action (3.7) with entropy production approximated by (4.12) can be rewritten as
S = T 0 dt m,n p m (t) −4∆ mn + δ mn α λ α (t)Θ (α) n p n (t) − α λ α (t)θ α . (5.1)
To understand the behavior of the system it is useful to introduce a new set of Lagrange multipliers defined by
Λ α (t) ≡ t 0 λ α (τ )dτ + Λ α (0),(5.2)
and then the action (5.1) takes the following from
S = T 0 dt m,n p m (t) −4∆ mn + δ mn α Θ (α) n dΛ α (t) dt p n (t) + α (Λ α (0) − Λ α (T )) θ α . (5.3)
Moreover, the symmetric transition matrix can be decomposed as
∆ mn = p,q Q T mp D pq Q qn , (5.4)
where D is a real diagonal matrix, Q is an orthogonal matrix,
p Q T mp Q pn = δ mn ,(5.5)
and Q T is the transpose of Q, Q T mn = Q nm .
p m (t)Q T mp −4D pq + δ pq α Θ (α) q dΛ α (t) dt Q qn p n (t) + α (Λ α (0) − Λ α (T )) θ α . (5.7)
(Note that Λ α (0), Λ α (T ) and λ α (t) = dΛα(t) dt are not all independent (see Eq. (5.2)) and so one can only vary Λ α (t) for either t ∈ [0, T ), i.e. solving for a stationary path for a given initial state, or t ∈ (0, T ], i.e. solving for a stationary path for a given final state.)
It is now convenient to define an auxiliary wave-function ψ m ≡ n e iϕm Q mn √ p n (5.8) with phases given by
ϕ m (t) ≡ 1 4 α Θ (α) m Λ α (t) = α Θ (α) m t 0 λ α (τ )dτ + Λ α (0) (5.9)
and then
S = T 0 dt −4 m,n ψ * m D mn ψ n − dϕ n dt ψ * m δ mn ψ n + α (Λ α (0) − Λ α (T )) θ α = T 0 dt −4 m,n ψ * m D mn ψ n + iψ * m δ mn dψ n dt − i m,n,l √ p m Q T ml Q ln d √ p n dt + α (Λ α (0) − Λ α (T )) θ α = T 0 dt −4 m,n ψ * m D mn ψ n + iψ * m δ mn dψ n dt − i m √ p m d √ p m dt + α (Λ α (0) − Λ α (T )) θ α = T 0 dt −4 m,n ψ * m D mn ψ n + iψ * m δ mn dψ n dt − 1 2 i m dp m dt + α (Λ α (0) − Λ α (T )) θ α = T 0 dt −4 m,n ψ * m D mn ψ n + iψ * m δ mn dψ n dt + α (Λ α (0) − Λ α (T )) θ α . (5.10)
Schrödinger Equation
The action (5.10) can be expressed using the bra-ket notations
S = T 0 dt −4 ψ|∆|ψ + i ψ| d dt |ψ + α (Λ α (0) − Λ α (T )) θ α (6.1) where |ψ ≡ N n=1 ψ n |n (6.2) ψ| ≡ N n=1 n|ψ * n (6.3) ∆ ≡ N n=1
D nn |n n| (6.4) and |n are eigenvectors of the transition operator∆ with eigenvalues D nn . Although (6.1) was derived in a particular basis, it is invariant under U (N ) transformations and so the corresponding equations should also be valid in any basis. By setting variations of (6.1) with respect to ψ n 's to zero (and ignoring the boundary terms) we arrive at our main result:
i d dt |ψ = −∆|ψ . (6.5)
Note, that in the original problem the path of the stationary entropy production should have been determined by setting variations of (6.1) with respect to probabilities p n and Lagrange multipliers λ α (or equivalently Λ α ) to zero. However, to obtain (6.5) the action (6.1) was varied with respect to ψ n (or p n and ϕ n ). Strictly speaking, these two procedures are not equivalent, is only true if Θ (α) n is an invertible square matrix, or if there are K = N linearly independent constraints. In our case K < N and so (6.5) is at most an approximation which can only be a valid approximation if the number of constraints is sufficiently large K N .
δS δψ n = 0 ⇔ δS δp n = δS δϕ n = 0 ? ⇔ δS δp n = δS δΛ α = 0 ⇔ δS δp n = δS δλ α = 0.
It is important to emphasize that in the above description the auxiliary wave function ψ n was only a bookkeeping device which keeps track of the information about a system with a large number of hidden symmetries/constraints. These constraints were first described explicitly by Θ (α) n and θ α , but in the Schrödinger-like equation (6.5) they only appear implicitly in the form of phases ϕ n defined as linear combinations of the Lagrange multipliers Λ α . This suggests that the state vector |ψ should not be considered as an ontic state, i.e. representing the real state of a quantum system, but rather as an epistemic state, i.e. representing the state of knowledge about the state of a statistical system with hidden constraints.
In conclusion, let us mention an interesting connection between equations (2.1) and (6.5). It is well known that one can go from (2.1) to (6.5) by performing the so-called Wick rotation, i.e. by replacing t with it in (2.1) we arrive at (6.5). (The fact that one equation is written for p n 's and the other one for ψ n 's is of no importance for this argument.) But now we also have a physical interpretation of what the Wick rotation actually means. It takes us from a statistical system with no hidden constraints (e.g. Markov process described by (2.1)) to a statistical system with many hidden constraints (e.g. non-Markov process described by (6.5)).
Discussion
In this paper we considered a stochastic process whose dynamics is described by a Markov chain on only short time-scales and, at the same time, constrained to conserved a number of quantities on long time-scales. To study the system we applied the principle of stationary entropy production and used the method of Lagrange multipliers to derive the action (6.1) and the corresponding equation (6.5). Despite of the apparent similarities between (6.5) and the usual Schrödinger equation, the considered statistical system is not equivalent to quantum mechanics. First of all, to derive (4.4) the detailed balance condition had to be assumed and thus our results would have to be modified for more general systems. Secondly, the entropy production was approximated by (4.12) which is only valid up to the fourth order in p n − p m , i.e. not too far from an equilibrium. Thirdly, equation (6.5) is only accurate when the number of linearly independent constraints is sufficiently large. Therefore, we must conclude that from our statistical system we do not get quantum mechanics per se, but something which can in certain limits reduce to quantum mechanics.
If our analysis is correct (and it is a big if) then the framework of entropic mechanics presented here may be considered as more general than quantum mechanics. Since it is based entirely on only the principle of stationary entropy production nothing stops us from applying this principle to other (than considered here) stochastic processes. In fact one such process was analyzed in Ref. [14] where it was argued that the entropy production could gives rise to an emergent dynamics of the metric. By taking a phenomenological approach (in Sec. 6 of Ref. [14]) the dynamics was described approximately by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry the Onsager tensor is expected to be symmetric. Then it was shown that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein-Hilbert term, i.e. general relativity.
We would like to stress that our results (here and in Ref. [14]) suggest that both quantum mechanics and general relativity may not be exact, but only approximate near-equilibrium limits of some statistical systems for which conservations/symmetries play a crucial role. As such we expect that both theories should break down further away from the equilibrium where some of the symmetries are expected to be broken. It would be interesting to see if one can attribute the breaking of these symmetries to other mysteries such as dark matter, dark energy or cosmic inflation. On the other hand, if quantum mechanics and general relativity are only limits of some other theory, then there is no need to look for a theory of quantum gravity -it simply does not exist.
Figure 1 .
1Probability space of a stochastic process with Markovian dynamics (red vertical segments) and projections to the constrained surface (blue segments). Equi-entropic surfaces are plotted with blue dashed lines and the constrained surface with green solid line.
Figure 2 .
2Gas of molecules.
Figure 3 .
3Gas of molecules with constraints.
p n ∆ mn log p m + p m ∆ mm log p p n ∆ mn (log p n − log p m ) = m,n p n ∆ mn (log p n − log p m ) .
Acknowledgments. The author wishes to acknowledge the hospitality of the Pacific Science Institute where this work began, the University of Niš where the key results were obtained and the Duluth Institute for Advance Study where much of the work in completing the paper was carried out. The work was supported in part by the Foundational Questions Institute (FQXi).
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Principle of minimum entropy production. M J Klein, P H E Meijer, Phys. Rev. 96Klein, M. J. and Meijer, P. H. E. 1954. "Principle of minimum entropy production." Phys. Rev. 96: 250-255
Covariant Information Theory and Emergent Gravity. V Vanchurin, Int. J. Mod. Phys. A. 33341845019V. Vanchurin, "Covariant Information Theory and Emergent Gravity," Int. J. Mod. Phys. A 33, no. 34, 1845019 (2018)
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"Nuclear skin emergence in Skyrme deformed Hartree-Fock calculations",
"Nuclear skin emergence in Skyrme deformed Hartree-Fock calculations"
]
| [
"P Sarriguren \nInstituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain\n",
"M K Gaidarov \nInstituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain\n\nInstitute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria\n",
"E Moya De Guerra \nInstituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain\n\nDepartamento de Fisica Atomica\nMolecular y Nuclear\nFacultad de Ciencias Fisicas\nUniversidad Complutense de Madrid\nE-28040MadridSpain\n",
"A N Antonov \nInstitute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria\n"
]
| [
"Instituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain",
"Instituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain",
"Institute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria",
"Instituto de Estructura de la Materia\nCSIC\nSerrano 123E-28006MadridSpain",
"Departamento de Fisica Atomica\nMolecular y Nuclear\nFacultad de Ciencias Fisicas\nUniversidad Complutense de Madrid\nE-28040MadridSpain",
"Institute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria"
]
| []
| A study of the charge and matter densities and the corresponding rms radii for even-even isotopes of Ni, Kr, and Sn has been performed in the framework of deformed self-consistent mean field Skyrme HF+BCS method. The resulting charge radii and neutron skin thicknesses of these nuclei are compared with available experimental data, as well as with other theoretical predictions. The formation of a neutron skin, which manifests itself in an excess of neutrons at distances greater than the radius of the proton distribution, is analyzed in terms of various definitions. Formation of a proton skin is shown to be unlikely. The effects of deformation on the neutron skins in even-even deformed nuclei far from the stability line are discussed. | 10.1103/physrevc.76.044322 | [
"https://arxiv.org/pdf/0710.0542v1.pdf"
]
| 26,907,749 | 0710.0542 | a73f6b775d0cc4bc88a2152ddfa0a40fe2a25052 |
Nuclear skin emergence in Skyrme deformed Hartree-Fock calculations
2 Oct 2007
P Sarriguren
Instituto de Estructura de la Materia
CSIC
Serrano 123E-28006MadridSpain
M K Gaidarov
Instituto de Estructura de la Materia
CSIC
Serrano 123E-28006MadridSpain
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences
1784SofiaBulgaria
E Moya De Guerra
Instituto de Estructura de la Materia
CSIC
Serrano 123E-28006MadridSpain
Departamento de Fisica Atomica
Molecular y Nuclear
Facultad de Ciencias Fisicas
Universidad Complutense de Madrid
E-28040MadridSpain
A N Antonov
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences
1784SofiaBulgaria
Nuclear skin emergence in Skyrme deformed Hartree-Fock calculations
2 Oct 2007
A study of the charge and matter densities and the corresponding rms radii for even-even isotopes of Ni, Kr, and Sn has been performed in the framework of deformed self-consistent mean field Skyrme HF+BCS method. The resulting charge radii and neutron skin thicknesses of these nuclei are compared with available experimental data, as well as with other theoretical predictions. The formation of a neutron skin, which manifests itself in an excess of neutrons at distances greater than the radius of the proton distribution, is analyzed in terms of various definitions. Formation of a proton skin is shown to be unlikely. The effects of deformation on the neutron skins in even-even deformed nuclei far from the stability line are discussed.
I. INTRODUCTION
The detailed study of the properties of unstable nuclei far from the stability line has been one of the main goals of nuclear physics in the last years. Recently, the development of radioactive ion beam (RIB) facilities in GSI (Germany) and in RIKEN (Japan) has opened a new field for such study, making possible the production of a variety of exotic nuclei which may have large neutron or proton excess.
So far studies have largely dealt with the light nuclei which became accessible by the use of RIB produced in fragmentation reactions. Prior experiments have revealed a halo phenomenon (e.g. in Refs. [1,2]) whose occurrence is due to both the small separation energy of the last few nucleons and their occupation on the orbits with low angular momentum. As well as halos, heavier systems may lead to formation of pronounced neutron skins [3] due to availability of far more neutrons than protons in these nuclei.
Immediate determination of the neutron skin thickness usually involves the precise measurement of the root mean square (rms) radii of both charge and mass distributions. Electron-nucleus scattering has proven to be an excellent tool for the study of nuclear structure. In particular, it has accumulated much reliable information on charge density distributions of stable nuclei. Therefore, it is believed that the new facilities in GSI [4,5] and RIKEN [6] will provide a good opportunity to study the charge density, and consequently the proton density distribution, of unstable nuclei by elastic electron scattering. Unfortunately, a measurement of the neutron density distributions to a precision and detail comparable to that of the proton one is hardly possible. The nuclear matter distribution in 6 He and 8 He has been determined recently at GSI by using small angle proton scattering in inverse kinematics at relativistic energy [7], and data has also been collected for 11 Li. It turned out that to get information on the neutron skin thickness one needs data obtained with probes having different sensitivities to the proton and neutron distributions. The methods for extracting the neutron skin thickness mostly include hadron scattering [8,9], antiprotonic atoms [10], parity violating electron scattering [11,12,13], as well as giant dipole resonance method [14] and spin-dipole resonance method [15,16].
On the theoretical side, calculations of nuclear charge and matter radii of exotic nuclei are usually made in the framework of mean-field approaches, namely Hartree-Fock (HF) method (see for example [17,18]) or Hartree-Fock-Bogoliubov (HFB) method including pairing correlations [19,20,21,22,23,24]. The latter predicts well the monotonic increase of the neutron skin thickness for different chains of isotopes up to the drip line [19]. Recently, the self-consistent relativistic mean-field (RMF) model has been widely applied to both stable and unstable nuclei (e.g., Refs. [25,26,27,28]). Also the relativistic Hartree-Bogoliubov approach has been employed to study the nuclear skin thickness in neutron/proton-rich sodium isotopes [29]. Many calculations show that the RMF model can reproduce with good precision a number of ground-state nuclear properties including the charge radii [30]. The charge rms radii were successfully described very recently in Ref. [24], where a generator coordinate method (GCM) on top of Gogny HFB calculations was explored.
Theoretical identification of skin and/or halo structure in neutron-rich weakly bound nuclei, however, is still a matter of discussion. In Ref. [31] a definition of the neutron skin and its appearance were presented in terms of spherical HF calculations. The proposed criteria which deal with proton and neutron densities allowed one to predict neutron skins in nuclei far from the β stability line. It has been also shown in [31] that the formation of proton skin appears to be rather difficult. The Helm model [32,33] has been applied in Ref. [20] to analyze neutron and proton skins, as well as halos, of even-even Ni, Sn, and Pb isotopes in terms of form factors. In Ref.
[20] three different definitions were proposed for neutronproton radii differences. Among them, the one based on the Helm model was chosen as a measure of the skin. The latter was shown to have a smooth gradual dependence on the neutron excess and to be almost unaffected by shell effects. The Helm model was used very recently also by Bertulani [34] to investigate electron scattering from light unstable nuclei.
Hitherto, the different definitions for the skin thickness mentioned above have been explored within different nuclear structure models. We would like to emphasize that a comparison of skins extracted by using various definitions is not very meaningful unless the same nuclear model is used and this has not been done in the past. Such an analysis of neutron skins within a given microscopic nuclear structure model could be very useful also in respect to demonstrate their expected spreading when different definitions of the nuclear skin are used.
Another interesting question is to explore how the neutron skin emerges in the presence of deformation. The latter is defined by the non-spherical components of the proton and neutron density distributions. In particular, studies of deformed exotic nuclei and skins can be found in Refs. [35,36]. It is desirable to study the evolution of shape and skin formation, not only because deformation influences the nuclear rms radii, but also because of the possible skin anisotropies that may take place.
In the present study, the properties of even-even Ni (A=48-78), Kr (A=70-100), and Sn (A=100-136) isotopes are described using the deformed self-consistent mean-field Skyrme HF+BCS method. We have used three parametrizations of the Skyrme force, namely SG2, Sk3 and SLy4, which were able to give an appropriate description of bulk properties of spherical and deformed nuclei in the past. As in our previous paper [22], we choose some medium and heavy Ni, Kr, and Sn isotopes because many of these sets, which lie in the nuclear chart between the proton and neutron drip lines can be formed as radioactive ions to perform scattering experiments. The main goal of this study is to clarify theoretically the emergence of the neutron and proton skins in neutron-rich and neutron-deficient isotopes, respectively, by testing different definitions for the skin thickness in the framework of the deformed Skyrme HF+BCS model. Alternatively to one of the criteria for the neutron skin proposed in Ref. [31] we consider another one which treats proton and neutron densities in a similar way. We extend the analysis of nuclear sizes presented in Ref. [22] by performing a more systematic study of a larger set of exotic nuclei and calculating also neutron skin thicknesses. The calculated charge rms radii are compared with the laser or muonic atoms spectroscopy measurements of isotope shifts performed on Sn [37,38,39,40], Ni [41,42], and Kr [43] isotopes. The neutron skin thicknesses obtained in this paper are compared with the available experimental data extracted from methods mentioned above for eveneven Sn isotopes with masses from 112 to 124. We also study whether the emergence of a skin is influenced by the nuclear shape, an issue that has not been sufficiently studied so far. The question of skin formation in nuclei having a non-spherical shape is discussed in detail on the example of Kr isotopes, assuming axial symmetry.
The paper is organized in the following way. Section II contains the formalism of the deformed Skyrme HF+BCS method that provides the model density distributions, form factors, and nuclear radii. The numerical results and discussions are presented in Sec. III. Finally, we draw the main conclusions of this study in Sec. IV.
II. DEFORMED SKYRME HF+BCS FORMALISM
The results discussed in the next sections have been obtained from self-consistent deformed Hartree-Fock calculations with density dependent Skyrme interactions [44] and pairing correlations. Pairing between like nucleons has been included by solving the BCS equations at each iteration either with a fixed pairing gap parameter (determined from the odd-even experimental mass differences) or with a fixed pairing strength parameter.
We consider in this paper the Skyrme force SLy4 [45]. We also show in some instances the results obtained from other parametrizations, namely Sk3 [46] and SG2 [47] because they are among the most extensively used Skyrme forces and are considered as standard references.
Assuming time reversal, the single-particle Hartree-Fock solutions for axially symmetric deformed nuclei are characterized by the eigenvalue Ω i of the third component of the total angular momentum on the symmetry axis and by the parity π i . The state i can be written as
Φ i R, σ, q = χ qi (q) Φ + i (r, z)e iΛ − ϕ χ + (σ) + Φ − i (r, z)e iΛ + ϕ χ − (σ) ,(1)
where χ qi (q), χ ± (σ) are isospin and spin functions, Λ ± = Ω i ± 1/2 ≥ 0. r, z, ϕ are the cylindrical coordinates of R. The wave functions Φ i are expanded into the eigenfunctions, φ α , of an axially symmetric deformed harmonic-oscillator potential in cylindrical coordinates. We use 12 major shells in this expansion,
Φ i R, σ, q = χ qi (q) α C i α φ α R, σ ,(2)
with α = {n r , n z , Λ, Σ} and
φ α R, σ = ψ Λ nr (r)ψ nz (z) e iΛϕ √ 2π χ Σ (σ) ,(3)
in terms of Hermite and Laguerre polynomials
ψ nz (z) = 1 √ π2 nz n z ! β 1/2 z e −ξ 2 /2 H nz (ξ) ,(4)ψ Λ nr (r) = n r (n r + Λ)! β ⊥ √ 2 η Λ/2 e −η/2 L Λ nr (η) ,(5)
with
β z = (mω z / ) 1/2 , β ⊥ = (mω ⊥ / ) 1/2 , ξ = zβ z , η = r 2 β 2 ⊥ .(6)
A. Density distributions and root mean square radii
The spin-independent proton and neutron densities are given by
ρ( R) = ρ(r, z) = i 2v 2 i ρ i (r, z) ,(7)
in terms of the occupation probabilities v 2 i resulting from the BCS equations and the single-particle densities ρ i
ρ i ( R) = ρ i (r, z) = |Φ + i (r, z)| 2 + |Φ − i (r, z)| 2 ,(8)with Φ ± i (r, z) = 1 √ 2π × α δ Σ,±1/2 δ Λ,Λ ∓ C i α ψ Λ nr (r) ψ nz (z) . (9)
The multipole decomposition of the density can be written as [44,48] ρ(r, z) = λ ρ λ (R)P λ (cos θ)
= ρ 0 (R) + ρ 2 (R) P 2 (cos θ) + . . . ,(10)
with multipole components λ
ρ λ (R) = 2λ + 1 2 +1 −1 P λ (cos θ) × ρ(R cos θ, R sin θ)d(cos θ) ,(11)
and normalization given by
ρ( R)d R = X; 4π R 2 dRρ 0 (R) = X ,(12)
with X = Z, N for protons and neutrons, respectively. The mean square radii for protons and neutrons are defined as
< r 2 p,n >= R 2 ρ p,n ( R)d R ρ p,n ( R)d R ,(13)
and the rms radii for protons and neutrons are simply given by r p,n =< r 2 p,n > 1/2 .
The mean square radius of the charge distribution in a nucleus can be expressed as
< r 2 ch > = < r 2 p > + < r 2 ch > p +(N/Z) < r 2 ch > n + r 2 CM + r 2 SO ,(15)
where < r 2 p > is the mean square radius of the point proton distribution in the nucleus (13), < r 2 ch > p and < r 2 ch > n are the mean square charge radii of the charge distributions in a proton and a neutron, respectively. r 2 CM is a small correction due to the center of mass motion, which is evaluated assuming harmonic-oscillator wave functions. The last term r 2 SO is a tiny spin-orbit contribution to the charge density. Correspondingly, we define the charge rms radius
r c =< r 2 ch > 1/2 .(16)
B. Form factors and diffraction parameters
Besides the mean square radii, additional characteristics of the density distributions can be deduced from the Fourier transforms of these densities. The form factors are defined as
F p,n ( q) = ρ p,n ( R)e i q· R d R ρ p,n ( R)d R .(17)
In the Plane Wave Born Approximation (PWBA) the elastic electron scattering cross sections are related to the Fourier transform of the charge density
F ch ( q) = 1 Z ρ ch ( R)e i q· R d R ,(18)
where q is the momentum transfer by the virtual photon in the scattering process. For each density multipole λ, one defines a Cλ form factor
F Cλ (q) = 4π X ∞ 0 R 2 dRρ λ (R)j λ (qR) .(19)
In particular,
F C0 (q) = 4π X ∞ 0 R 2 dRρ 0 (R)j 0 (qR)(20)
has the limit at q → 0
F C0 (q) → 4π X R 2 dRρ 0 (R) = 1 .(21)
Elastic and inelastic electron scattering have been extensively used to extract the various multipoles of the charge density, which show up in different transitions. In particular, in even-even deformed nuclei, F C0 (and correspondingly ρ 0 ) show up in the elastic cross section, while F C2 (and hence ρ 2 ) show up in the inelastic cross section for the transition 0 + → 2 + between the band-head and first excited rotational state [49,50].
In the next sections we will study the neutron skin thickness. We will use first the difference between the neutron and proton rms radii to characterize the different spatial extensions of neutron and proton densities. But as already noticed [20], the rms radii (second moments of the densities) provide a very limited description of the nucleon density distributions. A more effective tool to analyze skins [20,34] is the Helm model [32,33]. This is a model that allows one to extract from the form factor in a simple way the two main characteristics of the density, a diffraction radius and a surface thickness. In this model one describes the density by convoluting a hard sphere (hs) density having diffraction radius R d with a gaussian of variance σ,
ρ Helm (r; R d , σ) = ρ hs (r; R d ) * ρ G (r; σ) ,(22)
where
ρ hs (r, R d ) = 3X 4πR 3 d Θ(R d − r),(23)
and
ρ G (r; σ) = (2πσ 2 ) −3/2 e (−r 2 /2σ 2 ) .(24)
The corresponding Helm form factor is
F Helm (q) = F hs (q; R d )F G (q; σ) = 3 qR d j 1 (qR d )e −σ 2 q 2 /2 .(25)
Now, the most prominent feature of the density distribution, namely its extension, can be related to the first zero in the form factor, this is the diffraction radius
R d = 4.49341/q 1 ,(26)
where q 1 is the first zero of the form factor. The nuclear surface width σ can be related to the height of the second maximum of the form factor located at q max :
σ 2 = 2 q 2 max ln 3j 1 (q max R d ) R d q max F (q max ) .(27)
The variance σ is related to the surface thickness t (defined as the distance over which the density decreases from 90% to 10% of the central value) by t = 2.54 σ. Moreover, the surface thickness t is also related to the diffuseness a in the two-parameter Fermi distribution, by t = 4a ln 3 = 4.39 a.
Taking into account that the second moment of a convoluted distribution is given by the sum of the second moments of the two single distributions, one gets the Helm rms radius
R Helm rms = 3 5 (R 2 d + 5σ 2 ) .(28)
Taking out the factors 3/5, which relate the rms radii to the radii of the equivalent uniform hard spheres, we define R hs = 5/3 < r 2 > 1/2 (29) and
R Helm = 5/3 R Helm rms = R 2 d + 5σ 2 .(30)
From these definitions we construct the following neutron-proton radius differences that will be used in the next sections
∆R d = R d (n) − R d (p) ,(31)∆R hs = R hs (n) − R hs (p) = 5/3 < r 2 n > 1/2 − < r 2 p > 1/2 ,(32)∆R Helm = R Helm (n) − R Helm (p) .(33)
III. RESULTS AND DISCUSSION
A. Root mean square radii and density distributions
We start by showing our results for the rms radii of the charge distributions [Eq. (16)]. We compare them to the available experimental information obtained from various methods including laser and muonic atoms spectroscopy [37,38,39,40,41,42,43]. We also compare our results with different theoretical calculations. They include RMF calculations with NL3 parametrization and pairing correlations in BCS approach (RMF in Fig. 1) [26], nonrelativistic calculations performed within HFB approach deduced under triaxial symmetry from D1S Gogny effective interaction (HFB in Fig. 1), as well as calculations performed within a configuration mixing approach in the space spanned by the constrained HFB states. The latter are done within the GCM under Gaussian overlap approximation for the complete quadrupole collective space (GCM in Fig. 1) [24].
Beginning with Sn isotopes for which more data and calculations are available, we show on the right panel of Fig. 1 our results for the squared charge radii differences in Sn isotopes obtained from three different Skyrme forces, SLy4, SG2 and Sk3. We compare them to experiment, taking the radius of 120 Sn as the reference [40].
On the left panel we compare our SLy4 results for the charge radii with the other theoretical approaches mentioned above. The general purpose of Fig. 1 is firstly to show that different Skyrme forces do not differ much in their predictions of charge rms radii and secondly, to show that our results with SLy4 are comparable to other theoretical predictions including approaches that go beyond the mean-field approximation, as well as relativistic approaches. Then, by comparing our results with experiment and with other theoretical results, we have evaluated the quality of our calculations. We conclude that our method reproduces the experimental data with a similar accuracy to other microscopic calculations that, as explained above, may be more sophisticated but may also be more time consuming. This agreement provides a good starting point to make predictions for other quantities such as neutron-proton radii differences, where the experimental information is scarce and it is not as accurate as in the case of charge radii.
We complete this comparison of charge radii in Fig. 2. On the left panel we show our results for Ni isotopes and compare them with experiment [41,42] and with results from RMF calculations [26]. On the right panel we show the same comparison for Kr isotopes. Data are taken from [43]. In the Ni isotopes, we can see that the lower values of the rms radii occur around the double magic nucleus N = Z = 28, and around the semi-magic N = 50 in Kr isotopes. It is also worth mentioning that the bump shown around A = 76 in the RMF calculations of Kr isotopes has its origin in the change of the ground-state nuclear shape from oblate to prolate. In our case we obtain a smooth line because we only consider oblate shapes in this figure, as they correspond to the equilibrium shapes in most cases.
Once we have confirmed that the agreement between our calculations with the experimental r c radii is satisfactory, we have guarantees that meaningful results will be obtained for the neutron and proton mean square radii (14) by using the same formalism with the same forces. Figure 3 contains our results with the SLy4 force for those radii in the three isotopic chains. They are compared with the predictions from RMF [26]. We see that the tendency in the radii as a function of the mass number A is quite similar in both approaches, but in general the proton rms radii with Skyrme are systematically larger than the results from RMF. The situation is the opposite with respect to the neutron rms radii. At the same time the latter increase more slowly when calculated with SLy4. As a result we will get systematically differences between the neutron and proton rms radii, which are larger in the case of RMF as compared to the case of Skyrme forces. This is clearly seen in Fig. 4 where we plot the differences between the rms of neutrons and protons ∆r np = r n − r p . On the left panel we show our results for Sn isotopes and compare them to RMF results and to experimental data taken from (p, p) scattering [8,9], antiprotonic atoms [10], giant dipole resonance method [14], and spin dipole resonance method [15,16]. As we can see in Fig. 4 the experimental data are located between the predictions of both theoretical approaches and in general, there is agreement with experiment within the error bars. On the right panels we see the predictions for ∆r np in the cases of Ni and Kr isotopes, where there are no data.
The RMF results for the difference ∆r np systematically overestimate the Skyrme HF results, as it can be seen from Fig. 4. The reason for this is related to the difference in the nuclear symmetry energy and, consequently, to the different neutron equation of state (EOS) which has been extensively studied in recent years [51,52,53,54]. It was shown that there exists a linear correlation between the derivative of the neutron EOS (or the pressure of neutron matter) and the neutron skin thickness in heavy nuclei (defined as ∆r np = r n − r p ) in both Skyrme HF [55,56] and RMF [56,57] models. We note that also a relation between ∆r np and both volume and surface symmetry energy parameters was established recently by Danielewicz [58] and Steiner et al. [59] which provides a consistent description of nuclei with neutron excess. Typel and Brown [57] demonstrated that the relativistic models produce larger neutron radii compared with the nonrelativistic ones, reflecting the fact that the saturation density of asymmetric matter is lower in the EOS when phenomenological nucleon interaction in the RMF theory is used [60]. The results shown for neutron radii in Fig. 3 and correspondingly for neutron thicknesses in Fig. 4 support the above general conclusion.
In the next figures we show the proton and neutron density distributions ρ 0 (R) (10) of some selected isotopes in the three chains considered. We have chosen two extreme neutron-deficient and neutron-rich isotopes and one stable isotope between them. Figure 5 shows the neutron (solid) and proton (dashed) densities in the 100,120,136 Sn isotopes. From left to right we see the evolution of these densities as we increase the number of neutrons. In the case of 100 Sn (N =Z=50) we see that the two densities are practically the same except for Coulomb effects that make the protons to be more extended and, therefore, this has to be compensated with a small depression in the interior. The effect of adding more and more neutrons is to populate and extend the neutron densities. This makes also the proton distribution to follow the neutron one, increasing its spatial extension. The cost of this radius enlargement in the case of protons is a depression in the nuclear interior to preserve the normalization to the constant number of protons Z = 50. Then, it can be seen graphically the emergence of a region at the surface where the protons have practically disappeared while the neutrons still survive. We will quantify later this region in terms of the neutron skin thickness definitions. Figures 6 and 7 show the same information as in Fig. 5 but for 50,64,78 Ni and 70,84,98 Kr isotopes, respectively. The behavior of these densities corroborates the comments made on the case of Sn isotopes.
As we mentioned in the last chapter, we will also characterize the skin thickness in terms of diffraction parameters R d and σ deduced from the form factors. [26], HFB [24] and GCM [24]. Experimental data are from [37,38,39,40]; Right panel: Theoretical (with different Skyrme forces) and experimental isotope shifts δ r 2 c of tin isotopes relative to 120 Sn. 48 [26]. Experimental data are from [43].
(entering in Eq. (27)) needed to extract R d and the surface width σ, change with the neutron number. Thus, we see that the q 1 values diminish with increasing neutron number and therefore R d increases accordingly for both protons and neutrons. The values of q max are also reduced when A increases but the values of the form factor at these q max are rather similar. Consistently, the parameters σ extracted from (27) are fairly similar.
B. Neutron skin thickness
The thickness of a neutron skin in nuclei may be defined in different ways. One of these possibilities is to define it as the difference between the root mean square radius of neutrons and that of protons, as we have plotted in Fig. 4. Similarly, it can be defined as the difference between the neutron and proton radii of the equivalent uniform spheres [Eq. (32)]. Alternatively, it can be defined as the difference between the neutron and proton diffraction radii (31) or Helm radii (33). All of these quantities have already been discussed and used in the past as possible ways to quantify the skin thickness (see for example [20]), arriving to the conclusion that the radii difference defined in (32) contains contribution from halo effects and the radii difference defined in (33) is a better measure of the skin. Nevertheless, qualitatively the difference between the two definitions becomes only apparent when dealing with very neutron-rich isotopes, which are presently beyond the experimentally observed isotopes and out of the scope of this paper.
On the other hand, the skin thickness can be also defined in terms of some criteria that the neutron and proton densities must fulfill. In Ref. [31] the neutron skin thickness is defined as the difference between two radii, R 1 and R 2 . R 1 is the radius at which the ratio of the neutron density to the proton density is equal to some given value (4 in [31]). R 2 is the radius at which the neutron density becomes smaller than some percentage of the density at the center of the nucleus (1 % in [31]). When this difference, ∆R = R 2 − R 1 , is larger than some established value (in [31] this value is 1 fm, which is comparable to the range of the nuclear force), a neutron skin with skin thickness ∆R is said to occur. [8,9], antiproton atoms (full stars) [10], giant dipole resonance method (full circles) [14] and spin dipole resonance method (full and open squares) [15,16] are also shown. The factors used to define the skin thickness in the above criteria could have been differently chosen in rather arbitrary manners. Therefore, the absolute sizes of the skin thickness do not have a very precise meaning. Nevertheless, these values are useful to judge how the nucleon skins develop as the number of nucleons change. Indeed, we have also considered the case where the first criterion for the inner radius R 1 of the neutron skin is changed. We use instead of the above criterion for R 1 , the radius at which the proton density becomes smaller than 1% of the latter at the center, which is similar to the criterion used to define the outer radius R 2 , but in this case for proton density instead of the neutron density. When we use the conditions in Ref. [31], we call it criterion (a). When we use the alternative condition for R 1 , we call it criterion (b).
We show in Fig. 9 the results obtained for the neutron skin thickness in Sn isotopes according to the different definitions discussed above. The left panel contains the results for definitions involving directly the difference between neutron and proton radii, either the equivalent hard spheres radii ∆R hs [Eq. (32)] corresponding to the rms radii, the diffraction radii ∆R d [Eq. (31)], and the Helm radii ∆R Helm [Eq. (33)]. The skin thickness predicted by the difference of the very simple diffraction radii is in general smaller than the thickness predicted by the other two more involved options that are very similar in this range of masses. The right panel contains the neutron skin thickness defined according to the criteria on the density distributions (a) (solid line) and (b) (dashed line). They only differ in the way in which the starting radius of the skin R 1 is chosen. One can see that we obtain larger neutron skin thicknesses when using criterion (b) in the lighter isotopes, but this is reversed for heavier isotopes and we get larger thickness when using criterion (a). This fact is confirmed also by the values of the radii R 1 and R 2 and their differences ∆R listed in Table I for the heaviest three isotopes in each chain considered. In general, the formation of a skin when using (a) starts at distances smaller than those in case (b) or comparable with them, which leads to larger absolute size of the neutron skin produced by criterion (a). It is in this region of heavier isotopes where we can properly talk about a neutron skin formation. In this region, criterion (b) somehow establishes a lower limit for the skin thickness. The latter can be arbitrarily enlarged by relaxing the ρ n /ρ p condition to values lower than 4. Similar comments apply also to the next figures, Fig. 10 for Ni isotopes and Fig. 11 for Kr isotopes. Nuclei R (a) 1 We would like to emphasize that although different definitions of the neutron skin thickness produce different absolute values for it, the relative skin thicknesses corresponding to the evolution as the number of neutrons increase indicates the formation of such a skin that can be expected to start at A > 132 in Sn, A > 74 in Ni, and A > 96 in Kr isotopes, as it is observed in Figs. 9-11.
R (b) 1 R2 ∆R (a) ∆R (b)
Finally, we also consider the most neutron-deficient region of Ni isotopes in a search for the formation of a proton skin. We have already seen in the left panel in Fig. 10 that the neutron skin thickness defined in terms of differences between neutron and proton radii become negative at some point, indicating that the proton distribution extends beyond the neutron one. This can be further explored by reversing the definitions of R 1 and R 2 interchanging the role of protons and neutrons. We show the results in the inset of the right panel in Fig. 10, where we have applied the criterion (b) with protons and neutrons interchanged. We find no proton skin when applying criterion (a). One can see that a small skin starts developing in these isotopes but we cannot push it further because 48 Ni is already at the proton drip line. The results are then not conclusive enough to assess the existence of a proton skin in these isotopes. This possibility could be explored in the future in the most proton-rich nuclei approaching the proton drip lines of lighter nuclei with Z > N .
C. Neutron skin and deformation
When the nucleus is deformed, the thickness of the neutron skin might depend on the direction. It is an interesting and natural question to ask whether the deformed densities give rise to a different skin size in the different directions. It is also interesting to know whether the emergence of the skin may be influenced by the nuclear shape. We study in this work such a dependence on the example of Kr isotopes, which are examples of well deformed nuclei characterized by a large variety of competing nuclear shapes [61]. Constraint HF+BCS calculations [61,62] show also the possibility of shape coexistence in these nuclei. The results which we obtain for the binding energy of the three previously selected Kr isotopes as a function of the quadrupole parameter β = π/5Q p /(Zr 2 p ) (Q p being the proton quadrupole moment) are presented in Fig. 12. In this figure the distance between two ticks in the vertical axis is always 1 MeV but the origin is different for each curve. As we can see, both prolate and oblate shapes produce minima very close in energy. Then, we have chosen the neutron rich isotope 98 Kr to study the sensitivity of the neutron skin thickness to the various directions in the two shapes.
GCM calculations built on the constrained HF+BCS states may be carried out in order to describe more properly some ground-state properties in deformed nuclei. In the case of 98 Kr the potential energy curve (Fig. 12) shows pronounced minima at oblate and prolate shapes, which are separated by an energy barrier of about 6 MeV. Thus, one expects the ground state of 98 Kr to be basically described by a linear combination of these two configurations.
We first study the intrinsic density distributions ρ( R) in various selected directions. For that purpose we show in Figs. 13 and 14 the densities of 98 Kr for oblate and prolate shapes, respectively. We can see the spatial distributions for neutrons (solid) and protons (dotted) in three different directions: z-direction (r = 0), r-direction (z = 0), and r = z direction. We can observe that the profiles of the densities as well as the spatial extensions change with the direction. Clearly, the densities are more extended in the z-direction in the case of prolate shapes. The opposite is true in the case of oblate shapes. The case r = z gives always intermediate densities. We have added in the three directions a couple of full dots, indicating the radii R 1 and R 2 that defines the skin thickness according to the above mentioned criterion (a).
The dependence of the intrinsic density on the different directions can be also seen in Fig. 15, where we plot as an example the proton densities in the three directions mentioned above for oblate (left) and prolate (right) shapes in the same plane. We see more clearly how the extension of the density in the z-direction (labeled r = 0) is the largest for the prolate shape and the shortest for the oblate shape. We also plot for comparison the monopole component ρ 0 (R) (10) that lies between the two extreme cases and it is close to the density in the r = z direction.
It is also worth looking at the points in the (r, z) plane that define the ellipses where the criteria for R 1 and R 2 are met. Figure 16 shows these points for protons (thin lines) and neutrons (thick lines) and for the two shapes, prolate (solid) and oblate (dashed). We can see that the size of the skin changes little with the directions perpen- FIG. 14: Same as in Fig. 13, but for prolate shape of 98 Kr. dicular to the surface, but shows a tendency to increase on the shorter axis. It is interesting to note that the skin size of the spherical component ρ 0 (R) is an intermediate value. The overall skin thickness is also similar in the oblate and prolate equilibrium shapes. From this example we may conclude that the skin thickness does not depend much on the oblate or prolate character of the deformation. This is in line with the conclusions reached in Ref. [35] on the example of Dy isotopes, where it was shown that the neutron skin is nearly independent of the size of deformation (spherical, deformed or superdeformed). Figure 17 shows the monopole, ρ 0 (R), and quadrupole, ρ 2 (R), components of the intrinsic density ρ( R) (10) for protons (dashed lines) and neutrons (solid lines) and for the oblate and prolate shapes in 98 Kr. We can see that ρ 2 (R) is peaked at the surface positively in the case of the prolate deformation and negatively in the case of the oblate one. This makes the total density in the zdirection to be incremented with respect to the ρ 0 density in the prolate case and to be decreased in the oblate one. The opposite is true with respect to the direction perpendicular to the symmetry axis z. We can also see that the skin thickness derived from the ρ 0 components is quite similar to the thickness derived from the quadrupole components ρ 2 . This explains the approximately constant skin thickness observed in the different directions in Fig. 16.
IV. CONCLUSIONS
In this work we perform a theoretical analysis of nuclear skins, exploring various definitions. For this purpose we examine three chains of Ni, Kr, and Sn isotopes which might be of particular interest in the future experiments in GSI and RIKEN. The densities of these nuclei are calculated within a deformed HF+BCS approach with Skyrme-type density-dependent effective interactions [61]. We have shown that this model gives a very reasonable description of the charge rms radii of the Sn, Ni, and Kr isotopes and of the differences between neutron and proton rms radii for several Sn isotopes. This is confirmed by the good agreement with the available experimental data, as well as with other theoretical predictions. Three Skyrme parametrizations have been involved in the calculations: SG2, Sk3 and SLy4. Most of the results shown in the paper are obtained with SLy4 force, but the other Skyrme interactions produce similar results.
For the first time the various definitions which have been previously proposed to determine the neutron skin thickness, involving both matter radii and tails of nuclear densities, have been compared within a deformed Skyrme HF+BCS model. We find that all definitions of the neutron skin predict to a different extent the existence of a skin in nuclei far from the stability line. Particularly, a pronounced neutron skin can be attributed to heavier isotopes of the three chains considered, namely with A > 132 for Sn, A > 74 for Ni, and A > 96 for Kr isotopes. We also find that for a given isotopic chain the increase of the skin with the neutron number in the neutron-rich nuclei exhibits a rather constant slope, which is different depending on the definition of nuclear skin. More significant neutron skin is obtained when analyzing its formation by means of definition from Ref. [31] (called criterion (a)) or using an alternative one (called criterion (b)). In this case we get an absolute size of the skin larger than 0.4 fm and almost reaching 1 fm for the heaviest isotopes (in the case of criterion (a)). At the same time, the neutron skin determined by the difference between neutron and proton radii using diffraction parameters defined in the Helm model shows a more smooth gradual increase with the neutron excess and it is in size of around 0.3-0.4 fm. We would like to note that our results for Sn isotopes are consistent with the results of calculations from Ref. [20] with SLy4 parametrization. In both calculations the analysis of neutron skin formation is based on the nuclear form factors, which are well suited for such study since the diffraction parameters are mainly sensitive to the nuclear densities in the surface region.
We also show on the example of the neutron-deficient Ni isotopes the possibility to find a proton skin in a similar way to the neutron skin. Although the analysis, which was performed in our paper for this case, uses an alternative criterion to that applied in [31], it indicates a situation close to proton skin formation in Ni isotopes very close to the proton drip line. However, the search for the existence of proton skin could be explored in the most proton-rich nuclei approaching the proton drip lines of lighter nuclei, where Z > N .
In the present work the effects of deformation on the skin formation are studied in Kr isotopes which are well deformed nuclei. Taking as an example 98 Kr isotope, we find that the profiles of the proton and neutron densities, as well as the spatial extensions change with the direction in both oblate and prolate shapes. At the same time, the neutron skin thickness remains almost equal along the different directions perpendicular to the surface. Same type of calculations have been also performed on the example of 100 Kr, exhibiting a similar potential en-ergy curve. In this case, the conclusion concerning neutron skin thickness on the different directions remains unchanged. We find a very weak dependence of the neutron skin formation on the character of deformation. This is a useful information, worth to be known before complete GCM calculations are performed because it indicates that no drastic changes in the neutron skin thickness are expected when such more sophisticated calculations are performed.
The results obtained in the present paper demonstrate the ability of our microscopic theoretical method to predict the nuclear skin in exotic nuclei. They also illustrate the range of the skin sizes to be expected depending on the adopted skin definition. More definite conclusions on the emergence of nuclear skin will be drawn when direct measurements of proton and neutron form factors, and thus the corresponding proton and neutron densities, for these nuclei will become available at the upcoming experimental facilities.
FIG. 1 :
1Figure 8contains these form factors(17) for protons and neutrons of the three Sn isotopes shown inFig. 5. We can see how the diffraction zeroes at q 1 (26) and the location and magnitude of the second maximum, q max and F (q max ) Left panel: Charge rms radii rc of tin isotopes. The SLy4 result is compared with the results from RMF calculations
FIG. 5 :FIG. 6 :
56HF+BCS proton and neutron densities ρ0(R) of 100 Sn, 120 Sn, and 136 Sn calculated with SLy4 force. Same as inFig. 5, but for 50 Ni, 64 Ni, and 78 Ni.
FIG. 7 :
7Same as in Fig. 5, but for 70 Kr, 84 Kr, and 98 Kr.
FIG. 8 :
8Proton (left panel) and neutron (right panel) form factors for the 100 Sn, 120 Sn, and 136 Sn isotopes are calculated in the PWBA.
R2 and their differences (skin thicknesses) ∆R (a) = R2 − R
FIG. 9 :
9Neutron skin thicknesses for tin isotopes. Left panel: ∆R d [Eq. (31)], ∆R hs [Eq. (32)], and ∆R Helm [Eq. (33)]; Right panel: corresponding to criterion (a) (solid line) and criterion (b) (dotted line).
FIG. 10 :FIG. 11 :
1011Same as inFig. 9, but for Ni isotopes. A formation of proton skin thickness with the criterion (b) is also shown. Same as inFig. 9, but for Kr isotopes.
FIG. 12 :
12Binding energies E calculated with the SLy4 force as a function of the quadrupole parameter β for the even-even 70 Kr, 84 Kr, and 98 Kr isotopes.
FIG. 13 :
13Neutron (solid line) and proton (dotted line) density distributions ρ( R) in different directions for oblate shape of 98 Kr. The full dots shown on the (r, z) plane correspond to radii R1 and R2 according to criterion (a).
FIG. 17 :
17.16: Radii R1 and R2 according to criterion (a) for neutrons (thick lines) and protons (thin lines) in 98 Kr nucleus (shown in rz plane) corresponding to its oblate (solid lines) and prolate (dashed lines) shape. Monopole ρ0(R) (thin lines) and quadrupole ρ2(R) (thick lines) neutron (solid lines) and proton (dashed lines) density distributions of 98 Kr with oblate and prolate shape.
FIG. 2: Left panel: Charge rms radii rc of Ni isotopes. The SLy4 results are compared with the results from RMF calculations[26]. Experimental data are from[41,42]; Right panel: Charge rms radii rc of Kr isotopes. The SLy4 results are compared with the results from RMF calculations52
56
60
64
68
72
76
80
3.66
3.72
3.78
3.84
3.90
3.96
4.02
72
76
80
84
88
92
96
100
4.05
4.10
4.15
4.20
4.25
4.30
4.35
4.40
4.45
A
r
c
[fm]
r
c
[fm]
A
exp
SLy4
RMF
Ni
Kr
exp
SLy4
RMF
Proton rp and neutron rn rms radii of Sn, Ni and Kr isotopes calculated by using SLy4 force. The results from RMF calculations[26] are also given.102
108
114
120
126
132
4.2
4.4
4.6
4.8
5.0
48
52
56
60
64
68
72
76
80
3.2
3.4
3.6
3.8
4.0
4.2
4.4
68
72
76
80
84
88
92
96
100
3.8
4.0
4.2
4.4
4.6
4.8
r
n,p
[fm]
A
r
n,p
[fm]
r
n
r
n,p
[fm]
A
Sn
SLy4
RMF
r
p
r
p
r
n
SLy4
RMF
Ni
r
p
r
n
A
Kr
SLy4
RMF
FIG. 3:
Difference between neutron and proton rms radii ∆rnp of Sn, Ni, and Kr isotopes calculated with SLy4 force. The RMF calculation results are from Ref.[26]. The experimental data for Sn isotopes measured in (p, p) reaction (open stars)108
112
116
120
124
128
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
48 52 56 60 64 68 72 76
-0.2
0
0.2
0.4
72
76
80
84
88
92
96 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
A
A
r
np
[fm]
r
np
[fm]
r
np
[fm]
A
Sn
SLy4
RMF
Ni
SLy4
RMF
SLy4
RMF
Kr
FIG. 4:
TABLE I :
IRadii R
FIG. 15: Proton density distributions ρp(R) corresponding to different directions for oblate (left panel) and prolate (right panel) shape of 98 Kr. The monopole component ρ0(R) is also shown.0
2
4
6
8
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
2
4
6
8
10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
p
(R) [fm
-3
]
R [fm]
p
(R) [fm
-3
]
R [fm]
0
(R)
r=0
r=z
z=0
98
Kr (oblate)
98
Kr (prolate)
0
(R)
r=0
r=z
z=0
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| []
|
[
"Finite sample forecasting with estimated temporally aggregated linear processes",
"Finite sample forecasting with estimated temporally aggregated linear processes"
]
| [
"Lyudmila Grigoryeva ",
"Juan-Pablo Ortega "
]
| []
| []
| We propose a finite sample based predictor for estimated linear one dimensional time series models and compute the associated total forecasting error. The expression for the error that we present takes into account the estimation error. Unlike existing solutions in the literature, our formulas require neither assumptions on the second order stationarity of the sample nor Monte Carlo simulations for their evaluation. This result is used to prove the pertinence of a new hybrid scheme that we put forward for the forecast of linear temporal aggregates. This novel strategy consists of carrying out the parameter estimation based on disaggregated data and the prediction based on the corresponding aggregated model and data. We show that in some instances this scheme has a better performance than the "all-disaggregated" approach presented as optimal in the literature. | null | [
"https://arxiv.org/pdf/1209.4188v1.pdf"
]
| 88,512,739 | 1209.4188 | eebcd0032850fc81359683c3bd17f815fa056f95 |
Finite sample forecasting with estimated temporally aggregated linear processes
Lyudmila Grigoryeva
Juan-Pablo Ortega
Finite sample forecasting with estimated temporally aggregated linear processes
linear modelsARMAtemporal aggregationforecastingfinite sample forecastingflow temporal aggregationstock temporal aggregationmultistep forecasting
We propose a finite sample based predictor for estimated linear one dimensional time series models and compute the associated total forecasting error. The expression for the error that we present takes into account the estimation error. Unlike existing solutions in the literature, our formulas require neither assumptions on the second order stationarity of the sample nor Monte Carlo simulations for their evaluation. This result is used to prove the pertinence of a new hybrid scheme that we put forward for the forecast of linear temporal aggregates. This novel strategy consists of carrying out the parameter estimation based on disaggregated data and the prediction based on the corresponding aggregated model and data. We show that in some instances this scheme has a better performance than the "all-disaggregated" approach presented as optimal in the literature.
Introduction
The success of parametric time series models as a tool of choice in many research fields is due in part to their good performance when it comes to empirical forecasting based on historical samples. Once a data generating process (DGP) has been selected and estimated for the forecasting problem at hand, there is a variety of well studied forecasting procedures and algorithms available in the literature. The most widespread loss function used in the construction of predictors is the mean square forecasting error (MSFE); see the monographs [Bro06,BD02,Ham94,L05] and references therein for detailed presentations of the available MSFE minimization-based techniques. This is the approach to prediction that we follow in this work; the reader is referred to [Gra69] or Section 4.2 in [GN86] for forecasting techniques based on other optimality criteria.
The stochastic nature of the time series models that we consider implies that the forecasts produced with them, carry in their wake an error that cannot be minimized even if the parameters of the model are known with total precision; we refer to this as the characteristic error of the model. Additionally, all that is known in most applications is a historical sample of the variable that needs to be forecasted, out of which a model needs to be selected and estimated. There are well-known techniques to implement this, which are also stochastic in nature and that increase the total error committed when computing a forecast; we talk in that case of model selection error and estimation error. All these errors that one incurs in at the time of carrying out a forecasting task are of different nature and much effort has been dedicated in the literature in order to quantify them in the case of linear multivariate VARMA processes.
Most results obtained in this direction have to do with the combination of the estimation and the characteristic errors; this compound error is always studied assuming independence between the realizations of the model that are used for estimation and the ones used for prediction; we refer the reader to [Bai79,Rei80,Yam80,Yam81,Duf85,BS86,L87,SH88]. Explicit expressions for these errors in the VARMA context are available in the monograph [L05]. Indeed, if we assume that the sample out of which we want to forecast is a realization of the unique stationary solution of a VAR model, this error can be written down [L05, page 97] using the time-independent autocovariance of the process; the situation in the VARMA context is more complicated and the expression provided [L05, page 490] requires Monte Carlo simulations for its estimation.
The knowledge regarding the error associated to model selection is much more rudimentary and research in this subject seems to be in a more primitive state. A good description of the state of the art can be found in [L06, page 318] as well as in [L86b, page 89], and references therein. We do not consider this source of forecasting error in our work and hence in the sequel we will use the denomination total error to refer to the combination of the characteristic with the estimation errors.
In this paper we concentrate on one dimensional linear processes, a subclass of which is the ARMA family. The first contribution in this paper is the formulation of a MSFE based predictor that takes as ingredients a finite sample and the coefficients of a linear model estimated on it, as well as the computation of the corresponding total error. The main improvements that we provide with respect to preexisting work on this question are:
• We make no hypothesis on the second order stationarity of the sample at hand; in other words, we do not assume that the sample is a realization of the stationary solution of the recursions that define the model. Such a hypothesis is extremely difficult to test in small and finite sample contexts and it is hence of much interest to be able to avoid it.
• The expression for the total forecasting error is completely explicit and does not require the use of Monte Carlo simulations.
The interplay between the characteristic error, the estimation errors, and the forecasting horizon is highly nonlinear and can produce surprising phenomena. For example, as it is well known, the characteristic error is an increasing function of the horizon, that is, the further into the future we forecast, the more error we are likely to commit. When we take into account the estimation error, the total error may decrease with the forecast horizon! We study this finite sample related phenomenon with the total error formula introduced in Theorem 3.3 and illustrate it with an example in Section 5.1. The characterization of the total forecasting error that we described serves as a basis for the second main theme of this paper, namely, the interplay between multistep forecasting, the prediction of temporal aggregates, and the use of temporal aggregation estimation based techniques to lower the total forecasting error. In this part of the paper we work strictly in the ARMA context. The temporal aggregation of ARMA processes is a venerable topic that is by now well understood [AW72, Tia72, Bre73, TW76, Wei79, SW86, Wei06, SV08] and has been extensively studied and exploited in the context of forecasting [Abr82, L86b, L86a, L87, L89a, L89b, RS95, L06, L09, L10] mainly by H. Lütkepohl. A recurrent question in this setup consists of determining the most efficient way to compute multistep forecasts or, more generally, predictions of linear temporal aggregates of a given time series. More specifically, given a sample and an underlying model, we can imagine at least two ways to construct a h time steps ahead forecast, or in general the one that is a linear combination of the h steps ahead values for the time series. First, we can simply compute the h time steps ahead forecasts of the time series out of the original disaggregated sample and to determine the needed aggregate prediction out of them; another possibility would be to temporally aggregate the sample and the time series model in such a way that the required forecast becomes a one time step ahead forecast for the new aggregated sample and model. If we do not take into consideration estimation errors and we only consider the characteristic error, there is a general result that states that the forecast based on high frequency disaggregate data has an associated error that is smaller or equal than the one associated to the aggregate sample and model (we will recall it in Proposition 4.2). In the VARMA context, H. Lütkepohl [L86b, L87, L09] has characterized the situations in which there is no loss of forecasting efficiency when working with temporally aggregated ingredients.
When estimation errors are taken into account, the inequality that we just described becomes strict [L86b, L87], that is, forecasts based on models estimated using the dissagregated high-frequency samples perform always better than those based on models estimated using aggregated data. This is so even in the situations described in [L86b, L87, L09] for which the characteristic errors associated to the use of the aggregated and the disaggregated models are identical; this is intuitively very reasonable due to the smaller sample size associated to the aggregated situation, which automatically causes an increase in the estimation error.
In Section 4.3 we propose a forecasting scheme that is a hybrid between the two strategies that we just described. We first use the high frequency data for estimating a model. Then, we temporally aggregate the data and the model and finally forecasting is carried out based on these two aggregated ingredients. We will show that this scheme presents two major advantages:
• The model parameters are estimated using all the information available with the bigger sample size provided by the disaggregated data. Moreover, these parameters can be updated as new high frequency data becomes available.
• In some situations, the total error committed using this hybrid forecasting scheme is smaller than the one associated to the forecast based on the disaggregated data and model and hence our strategy becomes optimal. Examples in this direction for both stock and flow temporal aggregates are presented in Section 5. The increase in performance obtained with our method comes from minimizing the estimation error; given that the contribution of this error to the total one for univariate time series models is usually small for sizeable samples, the differences in forecasting performance that we will observe in practice are moderate. As we will show in a forthcoming work, this is likely to be different in the multivariate setup where in many cases, the estimation error is the main source of error.
To our knowledge, this forecasting scheme has not been previously investigated in the literature and the improvement stated in the last point seems to be the first substantial application of temporal aggregation techniques in the enhancing of forecasting efficiency.
Finite sample forecasting of linear processes
In this section we introduce notations and definitions used throughout the paper and describe the framework in which we work. Additionally, since we are interested in finite sample based forecasting, we spell out in detail the predictors as well as the information sets on which our constructions are based.
Linear processes
Let ε = {ε t } ∞ t=−∞ be a set of independent and identically distributed random variables with mean zero and variance σ 2 . We will write in short
ε = {ε t } ∼ IID 0, σ 2 .
Finite sample forecasting with estimated temporally aggregated linear processes
We say that X = {X t } ∞ t=−∞ is a linear causal process whenever it can be represented as
X t = ∞ i=0 ψ i ε t−i , for all t ∈ Z, (2.1) where {ψ i } ∞ i=0
is a set of real constants such that ∞ i=0 |ψ i | < ∞. Expression (2.1) can also be rewritten as
X = Ψ (L) ε,
where L is the backward shift operator and Ψ (z) is the power series
Ψ (z) = ∞ i=0 ψ i z i . The process X defined in (2.1) is called invertible if there exist constants {π j } ∞ j=0 such that ∞ j=0 |π j | < ∞ and ε t = ∞ j=0 π j X t−j , for all t ∈ Z, (2.2) or equivalently, ε = Π (L) X, where Π (z) is the power series Π (z) = ∞ j=0 π j z j . Ψ (L)
and Π (L) can also be referred to as causal linear filter and invertible linear filter, respectively.
Finite sample forecasting of causal and invertible ARMA processes
Consider the causal and invertible ARMA(p, q) specification determined by the equivalent relations
Φ (L) X t = Θ (L) ε t , X t = ∞ i=0 ψ i ε t−i , ε t = ∞ j=0 π j X t−j .
(2.
3)
The innovations process ε = {ε t } can be either independent and identically distributed IID(0, σ 2 ) or white noise WN(0, σ 2 ). In this subsection we focus on how to forecast out of a finite sample ξ T = {x 1 , ..., x T } that satisfies the relations (2.3) and that has been generated out of a presample {x 1−p , ..., x 0 } and a preinnovations set {ε 1−q , ..., ε 0 }. A standard way to solve this problem [Bro06,BD02] consists of assuming that ξ T is a realization of the unique stationary process X that satisfies the ARMA relations (2.3) and to use its corresponding time independent autocovariance functions to formulate a linear system of equations whose solution provides the linear projection X T +h of the random variable X T +h onto {x 1 , ..., x T } using the L 2 norm; this projection X T +h minimizes the mean square error. We recall that writing the unique stationary solution of (2.3) usually requires knowledge about the infinite past history of the process. For example, for an AR(1) model of the form X t − φX t−1 = ε t , the unique stationary solution is given by
X t = ∞ i=0 φ i ε t−i .
Given that we are concentrating in the finite sample context, we prefer for this reason to avoid the stationarity hypothesis and the use of the corresponding autocovariance functions and to exploit in the forecast only the information that is strictly available, that is:
(i) The model specification (2.3): we assume that the model parameters are known with certainty and we neglect estimation errors.
(ii) The sample ξ T = {x 1 , ..., x T }.
(iii) The presample {x 1−p , ..., x 0 } and preinnovations {ε 1−q , ..., ε 0 } that have been used in the sample generation.
We now define the preset I as
I := {x 1−p , ..., x 0 } ∪ {ε 1−q , .
.., ε 0 } , when p, q = 0 {x 1−p , ..., x 0 } , when q = 0 {ε 1−q , ..., ε 0 } , when p = 0.
Let r = max {p, q} and define the enlarged preset I * as
I * := {x 1−r , ..., x 0 } ∪ {ε 1−r , ..., ε 0 } ,
where:
• if p > q: r = p and ε t := t+p−1 j=0
π j X t−j , 1 − p ≤ t < 1 − q; • if q > p: r = q and x t = t+q−1 i=0 ψ i ε t−i , 1 − q ≤ t < 1 − p;
• if q = p: I = I * .
The enlarged preset I * is defined as a function of the elements in I. Consequently, the sigma-algebras σ(I) and σ(I * ) generated by I and I * , respectively, coincide, that is, σ(I) = σ(I * ). The following result is basically known (see for example [Ham94,L05]) but we include it in order to be explicit and self-contained about the forecasting scheme that we are using in the rest of the paper and also to spell out the peculiarities of the finite sample setup in which we are working. We include a brief proof in the appendix.
Proposition 2.1 In the conditions that we just described:
(i) The information sets (sigma algebras) σ T := σ (I, ε 1 , ..., ε T ) and σ ξ T := σ (I, x 1 , ..., x T ) generated by the preset and the past histories of the innovations T := {ε 1 , . . . , ε T } and the sample
ξ T := {x 1 , . . . , x T } coincide, that is, σ (ξ T ) = σ ( T ) . (2.4) (ii)
If the innovations process is IID (respectively WN) then the optimal multistep forecast X T +h (respectively optimal linear forecast) based on σ (ξ T ) that minimizes the mean square forecasting error (MSFE) E X T +h − X T +h 2 is:
X T +h = T +h−1+r i=h ψ i ε T +h−i = T −1+r i=0 ψ i+h ε T −i = T −1+r i=0 T −i−1+r j=0 ψ i+h π j X T −i−j . (2.5) (iii)
The MSFE associated to this forecast is
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 .
(2.6) (iv) For ARMA models, the forecasts constructed in (2.5) for different horizons with respect to the same information set F T := σ (ξ T ) = σ ( T ), satisfy the following recursive formula:
X T +h = φ 1 X T +h−1 + ... + φ p X T +h−p + θ h ε T + ... + θ q ε T +h−q , q ≥ h, φ 1 X T +h−1 + ... + φ p X T +h−p , q < h. (2.7)
Remark 2.2 Testing the stationarity of small or in general finite samples is a difficult task in practice. We emphasize that the prediction in Proposition 2.1 does not require any stationarity hypothesis. Moreover, we underline that the forecast (2.5) does not coincide in general neither with the standard linear forecast for second order stationary series that uses the corresponding time independent autocovariance function (see for example [Bro06], page 63), nor with the usual finite sample approximation to the optimal forecast (see [Ham94], page 85). The main difference with the latter consists of the fact that the innovations associated to the presample are not assumed to be equal to zero but they are reconstructed out of it so that there is no loss of information.
In the examples 2.3 and 2.4 below we show how our forecast allows us to construct a predictor that:
(i) is different from the one obtained assuming stationarity;
(ii) has a better performance in terms of characteristic forecasting error.
These statements do not generalize to arbitrary ARMA models; for example, for pure AR models, the predictor that we propose and those cited above coincide.
Example 2.3 Finite sample forecasting for the MA(1) process. We consider the MA(1) model X t = ε t + θε t−1 (2.8) and the trivial sample consisting of just one value x 1 at time t = 1; this sample is generated by the preset I = {ε 0 } and the innovation ε 1 . In this case, the enlarged preset I * = {x 0 , ε 0 } with x 0 = ε 0 . Moreover, we have
• ψ 0 = 1, ψ 1 = θ and ψ i = 0, for any integer i > 1,
• π 0 = 1, π j = (−1) j θ j , for any integer j ≥ 1.
Consequently by (2.5), the forecast X 2 based on the information set F 1 = σ ({I, x 1 }), is given by
X 2 = θε 1 = θ (x 1 − θx 0 ) = θx 1 − θ 2 ε 0 ,
and has the associated error MSFE( X 2 ) = σ 2 .
On the other hand, the forecast that assumes that x 1 is a realization of the unique stationary solution of (2.8) and that uses the corresponding autocovariance function [Bro06, page 63] is given by
X S 2 := θ 1 + θ 2 x 1 ,
and has the associated error
MSFE( X S 2 ) = σ 2 1 + θ 2 − σ 2 θ 2 1 + θ 2 . We note that MSFE( X S 2 ) = σ 2 1 + θ 2 − θ 2 1 + θ 2 > σ 2 = MSFE( X 2 ),
which shows that the forecast that we propose has a better performance than the one based on the stationarity hypothesis.
The better performance of the forecast that we propose in the preceding example can be in part due to the fact that we are using for X 2 additional information on the preinnovations that is not taken advantage of at the time of writing X S 2 . In the following example we consider an ARMA(1,1) model and we see that the statements of Remark 2.2 also hold, even though in this case, unlike in the MA(1) situation, the information sets on which the two forecasts considered are based are identical.
Example 2.4 Finite sample forecasting for the ARMA(1,1) process. Consider the model
X t − φX t−1 = ε t + θε t−1 .
Then,
• π 0 = 1, π j = (−1) j (φ + θ) θ j−1 , for any integer j ≥ 1,
• ψ 0 = 1, ψ i = (φ + θ) φ i−1 , for any integer i ≥ 1.
We consider the trivial sample x 1 generated by the preset I = {x 0 , ε 0 } = I * . Using Proposition 2.1, we have that the one-step ahead forecast X 2 based on the information set F 1 = σ ({I, x 1 }), is given by
X 2 = (φ + θ) x 1 − θ (φ + θ) x 0 , with MSFE( X 2 ) = σ 2 .
On the other hand, the forecast based on the stationarity hypothesis using the same information set, yields
X S 2 := θ 2 + φθ + 1 (θ + φ) (φθ + 1) (θ 2 + θφ + 1) 2 − θ 2 x 1 − (θ + φ) (θφ + 1) θ (θ 2 + θφ + 1) 2 − θ 2 x 0 , and MSFE( X S 2 ) = θ 2 + φθ + 1 θ 4 + θ 3 φ + θφ + 1 (θ 2 + θφ + 1) 2 − θ 2 σ 2 .
It is easy to check that the statement MSFE( X S 2 ) > MSFE( X 2 ) is equivalent to θ 4 (θ + φ) 2 > 0, which is always satisfied and shows that the forecast that we propose has a better performance than the one based on the stationarity hypothesis.
Forecasting with estimated linear processes
In Proposition 2.1 we studied forecasting when the parameters of the model are known with total precision. In this section we explore a more general situation in which the parameters are estimated out of a sample. Our goal is to quantify the joint mean square forecasting error that comes both from the stochastic nature of the model (characteristic error) and the estimation error; we will refer to this aggregation of errors as the total error. This problem has been extensively studied in the references cited in the introduction always using the following two main constituents:
• Estimation and the forecasting are carried out using independent realizations of the time series model.
• The model parameter estimator is assumed to be asymptotically normal (for example, the maximum likelihood estimator); this hypothesis is combined with the use of the so called Delta Method [Ser80] in order to come up with precise expressions for the total error.
The most detailed formulas for the total error in the VARMA context can be found in [L05] where the Delta Method is applied to the forecast considered as a smooth function of the model parameters. If we assume that the sample out of which we want to forecast is a realization of the unique stationary solution of a VAR model, an explicit expression for this error can be written down by following this approach [L05, page 97] that involves the time-independent autocovariance of the process. In the VARMA setup, the situation is more complicated [L05, page 490] and the resulting formula requires the use of a Monte Carlo estimation.
In subsection 3.1 we start by obtaining a formula for the total error using a different approach at the time of invoking the Delta Method; our strategy uses this method at a more primitive level by considering the parameters of the linear representation of the process seen as a function of the ARMA coefficients. We show that discarding higher order terms on 1/ √ T , where T is the sample size used for estimation, the resulting formula for the total error can be approximated by a completely explicit expression that involves only the model parameters and the covariance matrix associated to the asymptotically normal estimator of the ARMA coefficients.
In subsection 3.2 we rederive the total error formula by H. Lütkepohl [L05, page 490] and show that it can be rewritten as explicitly as ours without using any stationarity hypothesis or Monte Carlo simulations. Moreover, we show that this formula coincides with the approximated one obtained in subsection 3.1 by discarding higher order terms on 1/ √ T .
The total error of finite sample based forecasting
Consider the causal and invertible ARMA(p, q) process {X t } determined by the equivalent relations
Φ (L) X t = Θ (L) ε t , X t = ∞ i=0 ψ i ε t−i , ε t = ∞ j=0 π j X t−j , {ε t } ∼ IID(0, σ 2 ), (3.1)
and denote Ψ := {ψ 0 , ψ 1 , . . .}, Π := {π 0 , π 1 , . . .}. In Proposition 2.1 we studied forecasting for the process (3.1) when the parameters Ψ or Π of the model are known with total precision; in this section we suppose that these parameters are estimated by using a sample independent from the one that will be used for forecasting. A more preferable assumption would have been that the parameters Ψ are estimated based on the same sample that we intend to use for prediction, but exploiting only data up to the forecasting origin; Samaranayake [SH88] and Basu et al [BS86] have shown that many results obtained in the presence of the independence hypothesis remain valid under this more reasonable assumption. Under the independence hypothesis, the model coefficients Ψ or Π become random variables independent from the process X and the innovations ε. Moreover, we assume that these random variables are asymptotically normal, as for example in the case of maximum likelihood estimation of the ARMA coefficients.
For the sake of completeness, we start by recalling the Delta Method, that will be used profusely in the following pages. A proof and related asymptotic statements can be found in [Ser80].
Lemma 3.1 (Delta Method) Let β be an asymptotically normal estimator for the vector parameter β ∈ R n , that is, there exists a covariance matrix Σ such that
√ T β − β dist − −−− → T →∞ N (0, Σ),
where T is the sample size. Let f : R n → R m be a vector valued continuously differentiable function and let J f be its Jacobian matrix, that is,
((J f )(β)) ij := (∂f i /∂β j )(β). If J f (β) = 0, then √ T f ( β) − f (β) dist − −−− → T →∞ N (0, J f ΣJ f ).
The next ingredient needed in the formulation of the main result of this section is the covariance matrix Σ Ξ P associated to the asymptotic normal character of the estimator Ξ P for the parameters Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ), for some integer P . This is spelled out in the following lemma whose proof is a straightforward combination of the Delta Method with the results in Section 8.8 of [Bro06].
Lemma 3.2 Let {X t } be a causal and invertible ARMA(p,q) process like in (3.1). Let Φ := (φ 1 , . . . , φ p ) , Θ := (θ 1 , . . . , θ q ) , and β := Φ , Θ be the ARMA parameter vectors and let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) be a collection of length 2P of the parameters that provide the linear causal and invertible representations of that model. Then:
(i) The maximum likelihood estimator β of β is asymptotically normal √ T β − β dist − − → N (0, Σ β ), with Σ β = σ 2 E [U t U t ] E [U t V t ] E [V t U t ] E [V t V t ] −1 , where U t := (U t , . . . , U t+1−p ) , V t := (V t , . . . , V t+1−q ) , and {U t } and {V t } are the autoregressive processes determined by Φ(L)U t = ε t and Θ(L)V t = ε t .
(ii) Consider the elements in Ξ P as a function of β, that is, Ξ P (β) := (ψ 1 (β), . . . , ψ P (β), π 1 (β), . . . , π P (β)).
Then, by the Delta Method we have that
√ T Ξ P − Ξ P d −−→ N (0, Σ Ξ P ), where Σ Ξ P := J Ξ P Σ β J Ξ P (3.2) and (J Ξ P ) ij = ∂(Ξ P ) i ∂β j , i = 1, . . . , 2P , j = 1, . . . , p + q.
Details on how to algorithmically compute the Jacobian J Ξ P are provided in Appendix 7.2.
The next theorem is the main result in this section. Its proof can be found in the appendix.
Theorem 3.3 Let ξ T = {x 1 , . .
. , x T } be a sample obtained as a realization of the causal and invertible ARMA(p,q) model in (3.1) using a preset I. In order to forecast out of this sample, we first estimate the parameters of the model Ψ = ψ 0 , ψ 1 , . . . , Π = { π 0 , π 1 , . . . } based on another sample ξ T that we assume to be independent of ξ T , using the maximum likelihood estimator β := Φ , Θ of the ARMA parameters. If we use the forecasting scheme introduced in Proposition 2.1, then:
(i) The optimal multistep forecast X T +h for X T +h based on the information set F T generated by the sample ξ T and using the coefficients estimated on the independent sample ξ T is
X T +h = T +h−1+r i=h ψ iεT +h−i , (3.3)
where r = max{p, q} andε t := t+r−1 j=0
π j x t−j .
(ii) The mean square forecasting error (MSFE) associated to this forecast is
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + σ 2 P i=h ψ i 2 − 2 P i=h P −i j=0 P −i−j k=0 ψ i+j+k ψ k E ψ i π j + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ i+j+k,i +j +k , (3.4) where P = T + h − 1 + r.
The first summand will be referred to as the characteristic forecasting error and the second one as the estimation based forecasting error. Notice that the characteristic error coincides with (2.6) and amounts to the forecasting error committed when the model parameters are known with the total precision.
(iii) Let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) with P = T + h − 1 + r. Using the notation introduced in Lemma 3.2 and discarding higher order terms in 1/ √ T , the MSFE in (3.4) can be approximated by
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + σ 2 1 T P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k , (3.5)
where Σ Ξ P is the covariance matrix in (3.2).
On Lütkepohl's formula for the total forecasting error
As we already pointed out, H. Lütkepohl [L05, pages 97 and 490] has proposed formulas for VARMA models similar to the ones presented in Theorem 3.3 based on a different application of the Delta Method.
In this section, we rederive Lütkepohl's result in the ARMA context and show that it is identical to the approximated formula (3.5) presented in part (iii) of Theorem 3.3. In passing, this conveys that Lütkepohl's result can be explicitly formulated and computed using neither stationarity hypotheses nor Monte Carlo simulations.
The key idea behind Lütkepohl's formula is applying the Delta Method by thinking of the forecast X T +h in question as a differentiable function X T +h (β) of the model parameters β := (Φ , Θ ) . In order to develop further this idea, consider first the information sets F T := σ(ξ T ) and F T := σ(ξ T ) generated by two independent samples ξ T and ξ T of the same size. The sample ξ T is used for forecasting and hence F T determines the forecast X T +h (β) once the model parameters β have been specified. The sample ξ T is in turn used for model estimation and hence F T determines β. Consequently, the random variable X T +h ( β) is fully determined by F T and F T . In this setup, a straightforward application of the statement in Lemma 3.1 shows that
√ T X T +h β − X T +h (β) | F T dist − −−− → T →∞ N 0, ∂ X T +h ∂β Σ β ∂ X T +h ∂β , (3.6)
which, as presented in the next result is enough to compute the total forecasting error.
Theorem 3.4 In the same setup as in Theorem 3.3, the total error associated to the forecast in (3.3) can be approximated by
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + 1 T E ∂ X T +h ∂β Σ β ∂ X T +h ∂β . (3.7)
We refer to this expression as Lütkepohl's formula for the total forecasting error. Moreover:
(i) Lutkepohl's formula coincides with the approximate expression for the total error stated in (3.5). In particular, the second summand in Lütkepohl's formula, which describes the contribution to the total error given by the estimation error, can be expressed as:
Ω(h) :=E ∂ X T +h ∂β Σ β ∂ X T +h ∂β = σ 2 P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k .
(3.8)
(ii) If we assume that the samples used for forecasting are second order stationary realizations of the model (3.1) and γ : Z → R is the corresponding time invariant autocovariance function, then the estimation error can be expressed as:
1 T Ω(h) = 1 T P i=h P −i j=0 P i =h P −i j =0 π j π j (Σ Ξ P ) i,i + 2π j ψ i (Σ Ξ P ) i,j +P + ψ i ψ i (Σ Ξ P ) j+P,j +P γ(i+j −i −j ).
(3.9)
Finite sample forecasting of temporally aggregated linear processes
The goal of this section is proposing a forecasting scheme for temporal aggregates based on using high frequency data for estimation purposes and the corresponding temporally aggregated model and data for the forecasting task. We show, using the formulas introduced in the previous section, that in some occasions this strategy can yield forecasts of superior quality than those based exclusively on high frequency data that are presented in the literature as the best performing option [L86b, L87, L09]. We start by recalling general statements about temporal aggregation that we need in the sequel. We then proceed by using various extensions of the results in Section 3 regarding the computation of total forecasting errors with estimated series in order to compare the performances of the schemes that we just indicated.
Temporal aggregation of time series
The linear temporal aggregation of time series requires the use of the elements provided in the following definition.
Definition 4.1 Given K ∈ N, X a time series, and w = (w 1 , ..., w K ) ∈ R K , define the K-period projection p K of X as
p K (X) := X (K) j j∈Z ∈ j∈Z R K , where X (K) j := X (j−i)K+1 , .
.., X jK ∈ R K , and the corresponding temporally aggregated time series Y as
Y := I w • p K (X) , (4.1) where I w : j∈Z R K −→ j∈Z R, j∈Z v j −→ j∈Z < w, v j > .
The integer K is called the temporal aggregation period and the vector w the temporal aggregation vector. Notice that the aggregated time series Y is indexed using the time scale τ = mK, with m ∈ Z and its components are given by the X-aggregates X w t+K defined by
X w t+K := w 1 X t+1 + ... + w K X t+K = Y τ . (4.2)
Definition 4.1 can be reformulated in terms of the backward shift operator L as:
Y = Π K • K−1 i=0 w K−i L i (X) , where Π K : j∈Z R −→ j∈Z R, (X j ) j∈Z −→ (X Kj+1 ) j∈Z , (4.3)
and the indices of the components (Z j ) j∈Z of Z :=
K−1 i=0 w K−i L i (X) are uniquely determined by the choice Z 1 := w 1 X 1 + ... + w K X K .
There are four important particular cases covered by Definition 4.1, namely:
(i) Stock aggregation (also called systematic sampling, skip-sampling, point-in-time sampling): it is obtained out of (4.1) or (4.2) by setting w = (0, 0, ..., 0, 1) .
(ii) Flow aggregation: w = (1, 1, ..., 1) .
(iii) Averaging: w = (1/K, 1/K, ..., 1/K) .
(iv) Weighted averaging:
w = 1 K (ξ 1 , ..., ξ K ) such that ξ 1 + ... + ξ K = 1.
Multistep approach to the forecasting of linear temporal aggregates
Let X be a time series and w = (w 1 , ..., w K ) an K-period aggregation vector. Given a finite time realization ξ T = {x 1 , ..., x T } of X such that T = M K with M ∈ N, we aim at forecasting the aggregate w 1 X T +1 + ... + w K X T +K . There are two obvious ways to carry this out; first, we can produce a multistep forecast X T +1 , ..., X T +K for X out of which we can obtain the forecast of the aggregate by setting X w T +K := w 1 X T +1 + ... + w K X T +K . Second, we can temporally aggregate X using (4.1) into the time series Y given by
Y = I w • p K (X)
and produce a one-step forecast for Y . The following result recalls a well known comparison [AW72, L84, L86b, L89a] of the forecasting performances of the two schemes that we just described using the mean square characteristic error as an optimality criterion. In that setup, given an information set encoded as a σ-algebra, the optimal forecast is given by the conditional expectation with respect to it [Ham94, page 72]. Given a time series X, we will denote in what follows by σ X T the information set generated by a realization ξ T = {x 1 , . . . , x T } of length T of X and the preset I used to produce it; more specifically
σ X T := σ(I ∪ {x 1 , . . . , x T }).
Proposition 4.2 Let X be a time series and w = (w 1 , ..., w K ) a K-period aggregation vector. Let Y = I w • p K (X) be the corresponding temporally aggregated time series. Let T = M K with M, T ∈ N and consider F T = σ X T , F M = σ Y M the information sets associated to two histories of X and Y of length T and M , respectively, related to each other by temporal aggregation. Then:
MSFE E X w T +K |F T ≤ MSFE (E [Y M +1 |F M ]) . (4.4)
Remark 4.3 The inequality (4.4) has been studied in detail in the VARMA context by H. Lütkepohl [L86b, L87, L09] who has fully characterized the situations in which the two predictors are identical and hence have exactly the same performance. This condition is stated and exploited in Section 5, where we illustrate with specific examples the performance of the forecasting scheme that we present in the following pages.
In the next two results we spell out the characteristic and the total errors associated to a multistep approach to the forecast of linear aggregates. The characteristic error is given in Proposition 4.4 and the total error is provided in Theorem 4.7 under the same independence hypothesis between the samples used for estimation and forecasting that were already invoked in Theorem 3.3.
Proposition 4.4 Let X be a time series model as in (3.1), r = max{p, q}, K a temporal aggregation period, w = (w 1 , ..., w K ) an aggregation vector, and F T := σ X T the information set generated by
a realization ξ T = {x 1 , . . . , x T } of length T of X. Let X w T +K be the forecast of X-aggregate X w T +K := K i=1
w i X T +i based on F T using the forecasting scheme in Proposition 2.1. Then: (i) The forecast X w T +K is given by:
X w T +K = K i=1 w i T +i−1+r j=i ψ j ε T +i−j . (4.5) (ii)
The corresponding mean square forecasting characteristic error is:
MSFE X w T +K = E X w T +K − X w T +K 2 = σ 2 K i=1 w 2 i i−1 l=0 ψ 2 l + 2 K−1 i=1 K j=i+1 w i w j i−1 l=0 ψ l ψ j−i+l .
(4.6)
Example 4.5 Forecast of stock temporal aggregates. It is a particular case of the statement in Proposition 4.4 obtained by taking w = (0, ..., 0, 1) . In this case
X w T +K = X T +K = T +K−1+r j=K ψ j ε T +K−j .
This shows that the forecast of the stock temporal aggregate coincides with the K-multistep forecast of the original time series. Consequently, it is easy to see by using (4.6) and (2.6) that MSFE X w T +K = MSFE X T +K . Example 4.6 Forecast of flow temporal aggregates. It is a particular case of the statement in Proposition 4.4 obtained by taking w = (1, ..., 1) . In this case
X w T +K = K i=1 T +i−1+r j=i ψ j ε T +i−j .
Consequently,
MSFE X w T +K = σ 2 K−1 j=0 (K − j) ψ 2 j + 2 K−1 i=1 K j=i+1 i−1 l=0 ψ l ψ j−i+l = σ 2 K−1 j=0 (K − j) ψ 2 j + 2 K−1 i=1 K−1 j=i (K − j) ψ j−i ψ j .
Theorem 4.7 (Multistep forecasting of linear temporal aggregates) Consider a sample ξ T = {x 1 , . . . , x T } obtained as the realization of a model of the type (3.1) using the preset I. In order to forecast out of this sample, we first estimate the parameters of the model Ψ = ψ 0 , ψ 1 , . . . and Π = { π 0 , π 1 , . . . } based on another sample ξ T of the same size that we assume to be independent of ξ T . Let w = (w 1 , . . . , w K ) be an aggregation vector and let X w T +K be the forecast of the aggregate X w T +K := K h=1 w h X T +h based on F T := σ (I ∪ ξ T ) using Proposition 4.4 and the estimated parameters Ψ, Π. Then:
(i) The optimal forecast X w T +K for X w T +K given the samples ξ T and ξ T is
X w T +K = K h=1 w h T +h−1+r j=h ψ jεT +h−j , (4.7)
where r = max{p, q} andε t = t+r−1 j=0
π j x t−j .
(ii) The mean square forecasting error associated to this forecast is
MSFE X w T +K = σ 2 < w, (A + B + C) w >, (4.8)
where A, B, C are the matrices with components given by
A hh = P (h) i=0 P (h ) i =0 ψ i ψ i δ h−i,h −i , (4.9) B hh = −2 P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ k E ψ i π j δ h−l,h −i−j−k , (4.10) C hh = P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ h−i−j−k,h −i −j −k , (4.11) with P (h) = T + h − 1 + r, P (h ) = T + h − 1 + r. Notice that A hh = A char hh + A res hh , where A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , A res hh := P (h) i=h P (h ) i =h ψ i ψ i δ h−i,h −i ,
and σ 2 < w, A char w > is the characteristic forecasting error in part (ii) of Proposition 4.4.
(iii) Let Ξ P := ψ 1 , . . . , ψ P (K) , π 1 , . . . , π P (K) , with P (K) = T + K − 1 + r. Using the notation introduced in Lemmas 3.1 and 3.2 and discarding higher order terms in 1/ √ T , the MSFE in (4.8) can be approximated by
MSFE X w T +K = σ 2 < w, A char + D + F + G w >, (4.12) where A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , D hh = 1 T P (h) i=h P (h ) i =h (Σ Ξ P ) i,i δ h−i,h −i , F hh = 2 T P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k (Σ Ξ P ) i,P (K)+j δ h−i,h −i −j −k , G hh = 1 T P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P (K),j +P (K) δ h−i−j−k,h −i −j −k .
Remark 4.8 In order to compute the total error in (4.12) it is necessary to determine the covariance matrix Σ Ξ P . By Lemma 3.2, it can be obtained out of the covariance Σ β matrix associated to the estimator of the ARMA parameters combined with the Jacobian J Ξ P . Details on how to algorithmically compute this Jacobian are provided in Appendix 7.2.
Remark 4.9 Notice that all the matrices involved in the statement of Theorem 4.7 are symmetric except for B and F .
A hybrid forecasting scheme using aggregated time series models
In the previous subsection we presented a forecasting method for linear temporal aggregates based exclusively on the use of high frequency data and models. The performance of this approach has been compared in the literature [L86b, L87] with the scheme that consists of using models estimated using the aggregated low frequency data; as it could be expected due to the resulting smaller sample size, this method yields a performance that is strictly inferior to the one based on working in the pure high frequency setup.
In this section, we introduce and compute the performance of a hybrid recipe that consists of estimating first the model using the high frequency data so that we can take advantage of larger sample sizes and of the possibility of updating the model as new high frequency data become available. This model and the data used to estimate it are subsequently aggregated and used for forecasting. We will refer to this approach as the hybrid forecasting scheme. The main goal in the following pages is writing down explicitly the total MSFE associated to this forecasting strategy so that we can compare it using Theorem 4.7 with the one obtained with the method based exclusively on the use of high frequency data and models. In the next section we use the resulting formulas in order to prove that there are situations in which the hybrid forecasting scheme provides optimal performance for various kinds of temporal aggregation.
The main tool at the time of computing the MSFE associated to the hybrid scheme is again the use of the Delta Method [Ser80] in order to establish the asymptotic normality of the estimation scheme resulting from the combination of high frequency data with the subsequent model temporal aggregation.
In order to make this more explicit, consider a time series model X determined by the parameters β X for which an asymptotically normal estimator β X is available, that is, there exists a covariance matrix Σ βX such that
√ T β X − β X dist − −−− → T →∞ N (0, Σ βX )
with T being the sample size on which the estimation is based. Now, let K ∈ N be an aggregation period, w ∈ R K an aggregation vector, and Y := I w • p K (X) the linear temporally aggregated process corresponding to X and w.
Proposition 4.10 In the setup that we just described, suppose that the temporally aggregated process Y is also a parametric time series model and that the parameters β Y that define it can be expressed as a C 1 function β Y (β X ) of the parameters β X that determine X. Using the estimator β X , we can construct an estimator β Y for β Y based on disaggregated X samples by setting
β Y := β Y β X . Then √ T β Y − β Y dist − −−− → T →∞ N (0, Σ βY ) , (4.13)
where T is the disaggregated sample size and
Σ βY = J βY Σ βX J T βY , (4.14) with (J βY ) ij = ∂(β Y ) i ∂(β X ) j
the Jacobian matrix corresponding to the function β Y (β X ).
Once the model temporal aggregation function and its Jacobian have been determined, this proposition can be used to formulate an analog of Theorem 4.7 for the hybrid forecasting scheme by mimicking the proof of Theorem 3.3; the only necessary modification consists of replacing the asymptotic covariance matrix Σ βX of the estimator for the disaggregated model by that of the aggregated model Σ βY obtained using Proposition 4.10.
We make this statement explicit in the following theorem and then describe how to compute the model aggregation function β Y (β X ) and its Jacobian J βY in order to make it fully functional. The construction of these objects is carried out in the ARMA context where the model aggregation question has already been fully studied. Even though all necessary details will be provided later on in the section, all we need to know at this stage in order to state the theorem is that the linear temporal aggregation of an ARMA(p,q) model is another ARMA(p, q * ) model where
q * := K (p + 1) + q − p − K * K , (4.15)
K is the temporal aggregation period, K * is the index of the first nonzero entry of the aggregation vector, and the symbol · denotes the integer part of its argument. We emphasize that if the innovations that drive the disaggregated model are independent with variance σ 2 , this is not necessarily the case for the resulting aggregated model, whose innovations may be only uncorrelated with a different variance σ 2 * , making it into a so called weak ARMA model. We forecast the value of the temporal aggregate Y M +1 = X w T +K out of the sample η M by first estimating the parameters β X of the model X using another disaggregated sample ξ T of the same size, that we assume to be independent of ξ T . Let β Y (β X ) be the function that relates the ARMA parameter values of the disaggregated and the aggregated model and let J βY be its Jacobian. Consider the ARMA(p, q * ) model, with q * as in (4.15), determined by the parameters β Y := β Y ( β X ). Then:
(i) The optimal forecast Y M +1 of the temporal aggregate Y M +1 = X w T +K based on the information set F M := σ (I K ∪ η M ) using Proposition 4.4 and the estimated parameters β Y is given by
Y M +1 = T +r * j=1 ψ jεT +h−j ,(4.
16)
where I K is the preset obtained out of the temporal aggregation of I, r * = max{p, q * }, and ε t := t+r * −1 j=0 π j y t−j , with Ψ = ψ 0 , ψ 1 , . . . and Π = { π 0 , π 1 , . . . } the parameters corresponding to the causal and invertible representations of the temporally aggregated ARMA model with parameters β Y .
(ii) Let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) with P = T + r * . Discarding higher order terms in 1/ √ T , the MSFE corresponding to the forecast (4.16) can be approximated by
MSFE Y M +1 = σ 2 * + σ 2 * 1 T P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=1 P −i j=0 P −i−j k=0 P i =1 P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k , (4.17)
where σ 2 * is the variance of the innovations of the aggregated ARMA(p, q * ) model Y and Σ Ξ P is the covariance matrix given by
Σ Ξ P = J Ξ P Σ βY J Ξ P = J Ξ P J βY Σ βX J T βY J Ξ P ,
with J βY the Jacobian matrix corresponding to the function β Y (β X ) and J Ξ P the Jacobian of
Ξ P (β) := (ψ 1 (β Y ), . . . , ψ P (β Y ), π 1 (β Y ), . . . , π P (β Y )) .
As we announced above, we conclude this section by describing in detail the parameters aggregation function β Y (β X ) and its Jacobian J βY , so that all the ingredients necessary to apply formula (4.17) are available to the reader. In order to provide explicit expressions regarding these two elements, we provide a brief review containing strictly the results on the temporal aggregation of ARMA processes that are necessary for our discussion; for more ample discussions about this topic we refer the reader to [AW72, Tia72, Bre73, TW76, Wei79, SW86, Wei06, SV08] and references therein.
The temporal aggregation function β Y (β X ). Consider the ARMA(p,q) model Φ (L) X = Θ (L) ε, where Φ (L) = 1 − p i=1 φ i L i and Θ (L) = 1 + q i=1 θ i L i .
In order to simplify the discussion and to avoid hidden periodicity phenomena, we place ourselves in a generic situation in which the autoregressive and moving average polynomials of the model that we want to temporally aggregate have no common roots and all roots are different (see [Wei06] for the general case). Consider now a K-period aggregation vector w = (w 1 , ..., w K ) . Our first goal is to find polynomials T (z) and Φ * (z) that satisfy
T (L) Φ (L) = Φ * L K • Π K • K i=1 w i L i ,(4.18)
with Π K : j∈Z R −→ j∈Z R as in (4.3). The intuition behind (4.18) is that for any time series X, its temporal aggregation
Y = Π K • K i=1 w i L i (X) satisfies T (L) Φ (L) X = Φ * L K Y. (4.19)
In other words, the polynomial T (L), that we will call the temporal aggregation polynomial transforms the AR polynomial for X into an AR polynomial for Y in the aggregated time scale units. Let
T (L) = t 0 + t 1 L + ... + t n L n and Φ * L K = 1 − φ * 1 L K − ... − φ * c L Kc be the unknown polynomials. Equation (4.18) can be written in matrix form as [Bre73]:
t 0 0 0 . . . 0 t 1 t 0 0 . . . 0 t 2 t 1 t 0 . . . 0 . . . . . . . . . . . . . . . t p t p−1 t p−2 . . . t 0 . . . . . . . . . . . . . . . t n t n−1 t n−2 . . . t n−p 0 t n t n−1 . . . t n−p−1 0 0 t n . . . t n−p−2 . . . . . . . . . . . . . . . 0 0 0 . . . t n A 1 −φ 1 . . . −φ p Z = w −φ * 1 w . . . . . . . . . . . . . . . . . . −φ * c w D (4.20)
where w = (w K , w K−1 , ..., w 1 ) is the reflection of w. We start by determining the unknown values n and c using the two following dimensional restrictions:
• Since A is a matrix of size (n + p + 1) × (p + 1), Z and D are vectors of size p + 1 and cK + K, respectively, and we have AZ = D then, necessarily n + p + 1 = cK + K.
(4.21)
• The system AZ = D contains n + p + 1 equations that need to coincide with the number of unknowns, that is, the n + 1 + c coefficients (t 0 , t 1 , . . . , t n ) and (φ * 1 , . . . , φ * c ) of the polynomials T (z) and Φ * (z), respectively. Consequently, n + p + 1 = n + c + 1.
(4.22)
The conditions (4.21) and (4.22) yield c = p, n = pK + K − p − 1 = (p + 1) (K − 1) .
(4.23)
The first condition in (4.23) shows that the autoregressive order does not change under temporal aggregation.
Let now K * ≤ K be the index of the first nonzero component in the vector w. This implies that w is a vector of the form w = (w K , w K−1 , ..., w K * , 0, ..., 0) with w K , w K−1 , . . . , w K * = 0. Since (4.20) is a matrix representation of the polynomial equality in (4.18), we hence have that deg (T (L) Φ (L)) = deg (D (L)), where D (L) is the polynomial associated to the vector in the right hand side of (4.20). It is clear that deg (D (L)) = K (p + 1) − K * ; as deg (T (L) Φ (L)) = n + p, the degree n of T (L) is therefore n = K (p + 1) − p − K * .
(4.24)
Solving the polynomial equalities (4.20), we have found polynomials T (L) and Φ * L K such that the temporally aggregated time series Y satisfies
Φ * L K Y = T (L) Φ (L) X (4.25)
Our goal now is showing the existence of a polynomial Θ * L K and a white noise {ε * } ∼ WN 0, σ 2 * such that T (L) Θ (L) ε lK = Θ * L K ε * lK for any l ∈ Z.
(4.26)
This equality, together with (4.25) shows that the temporally aggregated process Y out of the ARMA process X is a weak ARMA process, as it satisfies the relation
Φ * (B) Y = Θ * (B) ε * , {ε * } ∼ WN 0, σ 2 * , (4.27) where B = L K , Φ * (B)
is a polynomial of degree p, and Θ * (B) a polynomial of degree n + q K whose coefficients will be determined in the following paragraphs. We recall that the symbol · denotes the integer part of its argument. Indeed, by (4.24) we have that deg (T (L) Θ (L)) = K (p + 1)−p−K * +q = n + q. Additionally, by (4.25) The coefficients of the polynomial Θ * (B) are obtained by equating the autocovariance functions of the processes on both sides of (4.30) at lags 0, K, 2K, ..., q * K, which provide q * + 1 nonlinear equations that determine uniquely the q * + 1 unknowns corresponding to the coefficients θ * 1 , . . . , θ * q * of the polynomial Θ * (B) and the variance σ 2 * of the white noise of the aggregated model. In order to explicitly write down the equations that we just described, let us denote C (L) := T (L) Θ (L) as in (4.30) and set C (L) = n+q i=0 c i L i . Let now γ and Γ be the autocovariance functions of the MA (n + q) and MA (q * ) processes Consequently, the coefficients of the polynomial Θ * (B) and the variance σ 2 * are uniquely determined by the q * + 1 equations The equations (4.34) can be written down in matrix form, which is convenient later on at the time of spelling out the Jacobian of the aggregation function. Indeed, we can write:
T (L) Θ (L) ε = T (L) Φ (L) X = Φ * L K Y.V t = C (L) ε t and U τ = Θ * (B) ε * τ , respectively,(4.γ (i) = σ 2 ε T (L) Θ (L) S i T (L) Θ (L) = σ 2 ε C (L) S i C (L) , (4.35) Γ (i) = σ 2 * Θ * (B) S i Θ * (B) ,(4.36)
where the bars over the polynomials in the previous expressions denote the corresponding coefficient vectors, that is, given a polynomial q (x) = n i=1 a i x i , then q (x) = (a 0 , a 1 , . . . , a n ) . Additionally S i is the lower ith-shift matrix, that is, In conclusion, if we denote β X = (Φ, Θ) and β Y = (Φ * , Θ * ), the construction that we just examined shows that
(S i ) jl = δ j−l,i .β Y (β X ) = (Φ * (Φ) , Θ * (Φ, Θ)) . (4.38)
The function Φ * (Φ) is given by the solution of the polynomial equalities (4.20) and Θ * (Φ, Θ) by the coefficients (t 0 , t 1 , . . . , t n ) determined by (4.20) and the solutions of (4.37).
Example 4.12 Stock temporal aggregation of an ARMA(p,q) model. In this case, w = (0, . . . , 0, 1) and hence K * = K, n = p (K − 1), and q * = p (K − 1) + q K .
Example 4.13 Flow temporal aggregation of an ARMA(p,q) model. In this case, w = (1, . . . , 1) and
hence K * = 1, n = (p + 1) (K − 1), q * = (p + 1) (K − 1) + q K .
The Jacobian J βY of the temporal aggregation function β Y (β X ). The goal of the following paragraphs is the computation of the Jacobian J βY of the function β Y (β X ) = (Φ * (Φ) , Θ * (Φ, Θ)) in (4.38). We first compute the Jacobian of the function Φ * (Φ) by taking derivatives with respect to the components of the vector Φ on both sides of the equations (4.20) that determine Φ * (Φ). This results in the following p matrix equations that will be needed later on in the computation of the remaining blocks of the Jacobian. Given that there is no Θ dependence on the function Φ * (Φ), the (1, 2)-block of the Jacobian is a zero matrix of size p × q * . The remaining two blocks are computed by using the function Θ * (Φ, Θ) uniquely determined by the equations (4.37). Its derivatives are obtained out of a new set of equations resulting from the differentiation of both sides of this relation, namely,
∂t 0 ∂φ i 0 . . . 0 ∂t 1 ∂φ i ∂t 0 ∂φ i . . . 0 ∂t 2 ∂φ i ∂t 1 ∂φ i . . . 0 . . . . . . . . . . . . ∂t p ∂φ i ∂t p−1 ∂φ i . . . ∂t 0 ∂φ i . . . . . . . . . . . . ∂t n ∂φ i ∂t n−1 ∂φ i . . . ∂t n−p ∂φ i 0 ∂t n ∂φ i . . . ∂t n−p−1 ∂φ i 0 0 . . . ∂t n−2 ∂φ i . . . . . . . . . . . . 0 0 . . . ∂t n ∂φ i 1 −φ 1 −φ 2 . . . −φ p + t 0 0 . . . 0 t 1 t 0 . . . 0 t 2 t 1 . . . 0 . . . . . . . . . . . . t p t p−1 . . . t 0 . . . . . . . . . . . . t n t n−1 . . . t n−p 0 t n . . . t n−p−1 0 0 . . . t n−p−2 . . . . . . . . . . . . 0 0 . . . t n 0 − ∂φ 1 ∂φ i − ∂φ 2 ∂φ i . . . − ∂φ p ∂φ i = 0 −w ∂φ * 1 ∂φ i −w ∂φ * 2 ∂φ i . . . −w ∂φ * p ∂φ i ,σ 2 ε ∂T (L) ∂(β X ) i Θ (L) + T (L) ∂Θ (L) ∂(β X ) i S jK T (L) Θ (L) + T (L) Θ (L) S jK ∂T (L) ∂(β X ) i Θ (L) + T (L) ∂Θ (L) ∂(β X ) i = ∂σ 2 * ∂(β X ) i Θ * (B) S j Θ * (B) + σ 2 * ∂Θ * (B) ∂(β X ) i S jK Θ * (B) + Θ * (B) S j ∂Θ * (B) ∂(β X ) i ,
(4.41) j = 0, 1, . . . , q * , i = 1, . . . , p + q.
We recall that the entries of the vector ∂T (L) ∂φ i correspond to the values previously obtained in (4.40)
and that ∂T (L) ∂θ i = 0. Expression (4.41) provides (q * + 1) (p + q) equations that allow us to find the values of the (q * + 1) (p + q) unknowns
∂ Θ * (B) j ∂φ r , ∂ Θ * (B) j ∂θ s , ∂σ 2 * ∂φ r , ∂σ 2 * ∂θ s , j = 1, .
. . , q * , r = 1, . . . , p, s = 1, . . . , q.
(4.42)
5 Comparison of forecasting efficiencies. Examples.
In the previous section we proposed a new hybrid scheme for the forecasting of temporal aggregates coming from ARMA processes. We recall that this strategy consists of first using high frequency disaggregated data for estimating a model; then we temporally aggregate both the data and the model, and finally we forecast using these two ingredients. As we announced in the introduction, there are situations in which our strategy is optimal with respect to the total error, that is, the predictor constructed following this procedure performs better than the one based exclusively on high frequency data and the underlying disaggregated model. In this section we give a few examples of specific models for which our scheme provides optimal efficiency of prediction. Before we proceed, we introduce abbreviations for the various predictors that we will be working with:
(i) Temporally aggregated multistep predictor (TMS predictor): this is the denomination that we use for the forecast of the aggregate that is constructed out of the disaggregated data and the underlying disaggregated model estimated on them.
(ii) Temporally aggregated predictor (TA predictor): this is the forecast based on use of the temporally aggregated sample and a model estimated on it.
(iii) Hybrid predictor (H predictor): this is the predictor introduced in Section 4.3 whose performance is spelled out in Theorem 4.11. In this scheme, a first model is estimated on the disaggregated high frequency data sample, then the data and the model are temporally aggregated with an aggregation period that coincides with the forecasting horizon; finally, both the temporally aggregated model and the sample are used to produce a one-step ahead forecast that amounts to a prediction of the aggregate we are interested in.
(iv) Optimal hybrid predictor (OH predictor): this predictor is constructed by taking the multistep implementation of the H predictor that yields the smallest total error. More explicitly, suppose that the aggregate that we want to forecast involves K time steps; let {K 1 , . . . , K r } be the positive divisors of K and {C 1 , . . . , C r } the corresponding quotients, that is, K = K i C i for each i ∈ {1, . . . , r}. There are aggregation schemes (stock and flow for example) for which a K-temporal aggregate can be obtained as the aggregation of C i K i -temporal aggregates, for all i ∈ {1, . . . , r}. The total error associated to the forecasting of these aggregates using a multistep version of the H predictor obviously depends on the factor K i used. The OH predictor is the one associated to the factor K i that minimizes the total error.
As we already mentioned, the forecasting performance of the TMS predictor is always superior or equal than that of the TA predictor when we take into account only the characteristic error, and it is strictly superior when the total error is considered. In view of these results and given that the H and the OH predictors carry out the forecasting with temporally aggregated data, they are going to produce worse characteristic errors than their TMS counterpart; hence, the only way in which the H and OH predictors can be competitive in terms of total error performance is by sufficiently lowering the estimation error. In order to check that they indeed do so, we will place ourselves in situations that are particularly advantageous in this respect and will choose models for which the TMS and the TA predictors have identical characteristic errors and hence it is only the estimation error that makes a difference as to the total error. The linear models for which this coincidence of characteristic errors takes place have been identified in the works of H. Lütkepohl [L86b, L87, L09] via the following statement.
Theorem 5.1 (Lütkepohl) Let X t = ∞ i=0 ψ i ε t−i be a linear causal process and let w = (w 1 , . . . , w K ) ∈ R K be a K-period aggregation vector. Then the TMS and TA predictors for the K-temporal aggregate determined by w have identical associated characteristic errors if and only if the following identity holds:
K−1 i=0 w K−i L i Ψ (L) = ∞ j=0 K−1 i=0 w K−i ψ jK−i L jK K−1 j=0 j i=0 w K−i ψ j−i L j .
(5.1)
The equality (5.1) is satisfied for both stock (w = (0, . . . , 0, 1) ) and flow aggregation (w = (1, . . . , 1) ) if {X t } is a purely seasonal process with period K, that is,
X t = ∞ i=0 ψ iK ε t−iK . (5.2)
Given a specific model we want to compare the performances of the H and the TMS predictors for a variety of forecasting horizons. Given that condition (5.1) is different for each aggregation period K and cannot be solved simultaneously for several of them, we will content ourselves either with approximate solutions that are likely to produce very close H and TMS characteristic errors for several periods K or with exact solutions that provide exactly equal errors for only a prescribed aggregation period. The following points describe how we have constructed examples following the lines that we just indicated:
• We first choose the orders p and q of the disaggregated ARMA(p,q) model that we want to use as the basis for the example.
• We fix an aggregation period K and a number n of parameters ψ i for which the equation (5.1) will be solved. The choice of p and q imposes a minimal number n min = q − p + 1.
• We determine a vector Ψ * = (ψ 0 , ψ 1 , ..., ψ n−1 ) that consists of the n first components of the set Ψ = {ψ 0 , ψ 1 , ...} that satisfies condition (5.1). We emphasize that in general this condition does not determine uniquely the vector Ψ * and that arbitrary choices need to be made. The vector Ψ * is a truncation at order n − 1 of the MA representation of the ARMA process that we want to construct.
• We conclude the construction of the ARMA(p,q) model that we are after by designing either an AR(p) polynomial Φ consistent with causality or a MA(q) polynomial Θ consistent with invertibility. Then:
-In the first case, the required model is given by
Φ(L)X = Θ * (L)ε, with Θ * = Ψ * · Φ. (5.3)
-In the second case, the required model is given by
Φ * (L)X = Θ(L)ε, with Φ * = (Ψ * ) −1 · Θ. (5.4)
In both cases, the MA and AR polynomials that are obtained in this way have to be checked regarding invertibility and causality, respectively. Additionally, the finite truncation of Ψ is likely to give rise to common roots between the AR and MA polynomials in (5.3) or in (5.4) which may make necessary a slight perturbation of the coefficients in order to be avoided.
• We emphasize that the resulting ARMA model satisfies (5.1) only approximately and hence the characteristic errors of the two predictors will be not identical but just close to each other for the specific aggregation period K used. For pure MA models no truncation is necessary and hence exact equality can be achieved.
Stock aggregation examples
In the particular case of stock temporal aggregation, condition (5.1) is written as:
Ψ (L) = ∞ j=0 ψ jK L jK K−1 j=0 ψ j L j . (5.5)
We now consider the truncated vector Ψ * with n components, that is, Ψ * = (ψ 0 , . . . , ψ n−1 ) . Then, the truncated version of (5.5) is:
n−1 j=0 ψ j L j = (n−K+1)/K j=0 ψ jK L jK K−1 j=0 ψ j L j , (5.6)
where the symbol · denotes the integer part of its argument. We now provide a few examples of models whose specification is obtained following the approach proposed in the previous subsection and the relation (5.6).
Example 5.2 MA(10) model.
Let p = 0, q = 10, n = n min = 11 and let K = 2. Equation (5.6) becomes
10 j=0 ψ j L j = 5 j=0 ψ 2j L 2j 1 j=0 ψ j L j ,
which imposes the following relations:
ψ 0 = 1, ψ 1 ψ 2i = ψ 2i+1 , i = 0, . . . , 5; ψ i = 0 for i ≥ n.
This system of nonlinear equations has many solutions. We choose one of them by setting ψ j = 0, for j = 1, . . . , 9, and ψ 10 = 0.3. This way we can trivially determine a MA(10) model which satisfies exactly the relation (5.5) by taking θ j = 0 for j = 1, . . . , 9 and θ 10 = 0.3. Figure 1 shows the values of the characteristic errors for different values of the forecasting horizon for the TMS predictor, the H predictor, and the OH predictor. For the horizon h = 2, the values of the characteristic errors of all the predictors coincide, which is a consequence of the fact that the model has been constructed using the relation (5.5) with K = 2. Moreover, it is easy to see by looking at (5.2), that the particular choice of MA coefficients that we have adopted ensures that the resulting model is seasonal for the periods 2, 5, and 10; this guarantees that (5.5) is also satisfied for the corresponding values of K and hence there is coincidence for the characteristic errors at those horizons too.
The total errors for a sample size of T = 50 are then computed using the formulas presented in sections 3 and 4. The corresponding results are also plotted in Figure 1 for the different forecasting schemes. This plot shows that for several forecasting horizons both the H and the OH predictors perform better than the TMS predictor.
A quick inspection of this plot reveals another interesting phenomenon consisting on the decrease of the total error associated to the three predictors as the forecasting horizon increases; this feature is due to the decrease of the estimation error using these forecasting schemes as the horizon becomes longer. The characteristic errors for the H and OH predictors do not increase monotonically with the forecasting horizon either; however, in this case, this is due to the fact that for each value of the forecasting horizon, these predictors are constructed using a different model since the aggregation period changes and hence so does the aggregated model used for forecasting.
In conclusion, in this particular example, both the H and the OH predictors exhibit a better forecasting performance than the TMS predictor and, additionally, the results regarding the OH predictor help in making a decision on what is the best possible aggregation period to work with in order to minimize the associated total forecasting error. Figure 2 shows the errors with respect to the forecasting horizon for all the predictors as in the previous example. The H and the OH predictors perform better than the TMS predictor for h = 3, 6, 9, 10. Additionally the OH predictor performs better than the H predictor for h = 4. Taking into account the initial choice of K = 3 when constructing the example, it becomes clear why the characteristic errors associated to the H and the OH predictors are very close to those associated to the TMS predictor for horizons h that are multiples of 3. Figure 3 shows that the H and the OH predictors have equal associated total errors and exhibit a better forecasting efficiency than the TMS predictor for all forecasting horizons except at h = 2. The initial choice of K = 4 at the model construction stage results in the fact that for h = 4 the values of the characteristic errors associated to the H and the OH predictors are very close to the one committed by the TMS predictor.
Flow aggregation examples
In the particular case of flow temporal aggregation, condition (5.1) can be written as:
Ψ (L) K−1 i=0 L i = ∞ j=0 L jK K−1 i=0 ψ jK−i K−1 j=0 L j j i=0 ψ j−i . (5.7)
We now consider the truncation Ψ * of Ψ with n components, that is, Ψ * = (ψ 0 , . . . , ψ n−1 ). Then, the truncated version of (5.7) can be expressed as:
K−1 i=0 L i n−1 j=0 ψ j L j = (n−K+1)/K j=0 L jK K−1 i=0 ψ jK−i K−1 j=0 L j j i=0 ψ j−i , (5.8)
where symbol · denotes the integer part of its argument. We now provide a few examples of models whose specification is obtained following the approach described in the beginning of the section and the relation (5.8).
Example 5.5 MA(10) model.
Let p = 0, q = 10, n = n min = 11, and let K = 2. Then the expression (5.8) reads
1 i=0 L i 10 j=0 ψ j L j = 5 j=0 L 2j 1 i=0 ψ 2j−i 1 j=0 L j j i=0 ψ j−i ,
and consequently ψ 0 = 1, (1 + ψ 1 )(ψ i + ψ i+1 ) = ψ i+1 + ψ i+2 , for i = 1, . . . , 9, and ψ i = 0 for i ≥ n.
(5.9)
A solution for these equations is given by the choice ψ j = 0, for j = 1, . . . , 9, and ψ 10 = 0.3. Since the order of the AR polynomial is zero, the method that we proposed determines uniquely in this case the MA(10) polynomial that we are after with Θ = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0.3) . The evolution of the forecasting errors versus the forecasting horizon in plotted in the Figure 4. Both the H and the OH predictors perform better than the TMS predictor. For h = 4 the OH predictor has the smallest total error among the three predictors. Figure 5 shows the corresponding characteristic and total errors for the three predictors. The H and the OH predictors exhibit better performance than the TMS predictor for horizons h = 2, 4, 5, 6, 7. Remark 5.7 The model that we just presented can be used to illustrate the fact that the construction method that we presented in this sections is not the unique source of examples in which the H and the OH predictors perform better than the TMS scheme. Indeed, as it can be seen in Figure 6, the very same ARMA(3,10) model prescription used in the context of stock aggregation also shows this feature even though it has not been obtained by finding a solution of the equation (5.5).
Remark 5.8 Notice that when the forecasting horizon h equals one all predictors coincide because there is no temporal aggregation and hence they obviously have the same errors associated.
Conclusions
In this work we have carried out a detailed study of the total error committed when forecasting with one dimensional linear models by minimizing the mean square error. We have introduced a new hybrid scheme for the forecasting of linear temporal aggregates that in some situations shows optimal performance in comparison with other prediction strategies proposed in the literature. We work in a finite sample context. More specifically, the forecasting is based on the information set generated by a sample and a model whose parameters have been estimated on it and we avoid the use of second order stationarity hypotheses or the use of time independent autocovariance functions.
In this setup, we provide explicit expressions for the forecasting error that incorporate both the error incurred in due to the stochastic nature of the model (we call it characteristic error) as well as the one associated to the sample based estimation of the model parameters (estimation error). In order to derive these expressions we use certain independence and asymptotic normality hypotheses that are customary in the literature; our main contribution consists of providing expressions for the total error that do not require neither stationarity on the samples used nor Monte Carlo simulations to be evaluated. We subsequently use these formulas to evaluate the performance of a new forecasting strategy that we propose for the prediction of linear temporal aggregates; we call it hybrid scheme. This approach consists of using high frequency data for estimation purposes and the corresponding temporally aggregated data and model for forecasting. This scheme uses all the information available at the time of estimation by using the bigger sample size provided by the disaggregated data, and allows these parameters to be updated as new high frequency data become available. More importantly, as we illustrate with various examples, in some situations the total error committed using this scheme is smaller than the one associated to the forecast based on the disaggregated data and model; in those cases our strategy becomes optimal. As the increase in performance obtained with our method comes from minimizing the estimation error, we are persuaded that this approach to forecasting may prove very relevant in the multivariate setup where in many cases the estimation error is the main source of error. (ii) Suppose first that the innovations {ε t } are IID(0, σ 2 ). Then the forecast X T +h that minimizes the mean square forecasting error E X T +h − X T +h 2 is given by the conditional expectation (see for example [Ham94], page 72):
X T +h = E X T +h |σ ξ T = E X T +h |σ T = T +h−1+r i=0 ψ i E ε T +h−i |σ T = T +h−1+r i=h ψ i ε T +h−i = T −1+r i=0 ψ i+h ε T −i = T −1+r i=0 T −i−1+r j=0 ψ i+h π j X T −i−j ,
as required.
When {ε t } is WN(0, σ 2 ) our goal is finding the linear combination
T −1+r j=0 a j X T −j that minimizes E X T +h − T −1+r i=0 a i X T −i 2 .
Given that by (7.1), the elements X T −i can be written as a linear combination of the elements in T −i , this task is equivalent to finding the vector b = (b 0 , ..., b T −1+r ) that minimizes the function
S (b 0 , ..., b T −1+r ) = E X T +h − T −1+r i=0 b i ε T −i 2 = E T +h−1+r i=0 ψ i ε T +h−i − T −1+r i=0 b i ε T −i 2 = E h−1 i=0 ψ i ε T +h−i + T −1+r i=0 (ψ i+h − b i ) ε T −i 2 = σ 2 h−1 i=0 ψ 2 i + T −1+r i=0 (ψ i+h − b i ) 2 .
Hence, in order to minimize the function S (b 0 , ..., b T −1+r ) we compute the partial derivatives ∂S/∂b i and we set them to zero, which shows that the optimal values are attained when b i = ψ i+h . Consequently, the optimal linear forecast is given by X T +h = T −1+r i=0 ψ i+h ε T −i , as required.
(iii) We first compute X T +h − X T +h . By (2.5) and (7.1) we have
X T +h − X T +h = h−1 i=0 ψ i ε T +h−i . Therefore MSFE X T +h = E h−1 i=0 ψ i ε T +h−i 2 = σ 2 h−1 i=0 ψ 2 i .
(iv) Given the model Φ (L) X = Θ (L) ε, we have
X T +h − φ 1 X T +h−1 + ... + φ p X T +h−p = ε T +h + θ 1 ε T +h−1 + ... + θ q ε T +h−q .
We first recall that by (2.4) we have that σ ξ T = σ T =: F T . We now project both sides of this equality onto the information set F T by thinking of this σ-algebra as σ ξ T for the left hand side projection and as σ T for the right hand side. We obtain:
X T +h − φ 1 X T +h−1 + ... + φ p X T +h−p = E [ε T +h + θ 1 ε T +h−1 + ... + θ q ε T +h−q |F T )] = θ h ε T + ... + θ q ε T +h−q , q ≥ h 0, otherwise.
In the presence of white noise innovations, the conditional expectation in the previous equality should be replaced by a linear projection.
7.2 Computation of the Jacobian J Ξ P In this section we provide a simple algorithmic construction for the computation of the Jacobian J Ξ P when the elements in Ξ P are considered as a function of β, that is, Ξ P (β) := (ψ 1 (β), . . . , ψ P (β), π 1 (β), . . . , π P (β)).
We will separately compute the components ∂ψ i ∂β k and ∂π i ∂β k , i = 1, . . . , P , k = 1, . . . , p + q.
The causality and invertibility hypotheses on the ARMA process we are working with, guarantee that for any P ≥ max {p, q} there exist polynomials Ψ P (z), Π P (z) of order P uniquely determined by the relations:
Φ(z)Ψ P (z) = Θ(z), (7.2) Φ(z) = Π P (z)Θ(z), (7.3)
which are equivalent to
Ψ P (z) = Φ −1 (z)Θ(z), Π P (z) = Φ(z)Θ −1 (z).
These polynomial relations determine the functions Ψ P (Φ, Θ), Π P (Φ, Θ) needed in the computation of the Jacobian. We now rewrite (7.2) and (7.3) as Φ(z)Ψ P (Φ, Θ)(z) = Θ(z), (7.4) Φ(z) = Π P (Φ, Θ)(z)Θ(z).
(7.5)
If we take derivatives with respect to θ j and φ i , j ∈ {1, . . . , q}, i ∈ {1, . . . , p} on both sides of (7.4), we obtain a set of p + q polynomial equations:
Φ(z) ∂Ψ P (φ, θ) ∂θ j (z) = z j , j ∈ {1, . . . , q} , z i Ψ P (φ, θ) + Φ(z) ∂Ψ P (φ, θ) ∂φ i = 0, i ∈ {1, . . . , p} ,
that determine uniquely the corresponding entries of the Jacobian due to the invertibility of Φ(z). At the same time, taking derivatives on both the right and left hand sides of (7.5) with respect to θ j and φ i , we obtain another set of p + q polynomial equations ∂Φ ∂θ j (z) = ∂Π P (φ, θ) ∂θ j (z) + Π P z j , j ∈ {1, . . . , q} ,
z i = ∂Π P (φ, θ) ∂φ i , i ∈ {1, . . . , p} ,
that determine uniquely the corresponding entries of the Jacobian due to the invertibility of Θ(z).
Proof of Theorem 3.3
(i) It is a straightforward consequence of the independence hypothesis between the samples ξ T and the ξ T , and part (ii) of Proposition 2.1.
(ii) By (3.1) and part (i) of Proposition 2.1 we have that
MSFE X T +h = E X t+h − X t+h 2 = E P i=0 ψ i ε T +h−i − P i=h ψ iεT +h−i 2 .
In order to compute this error notice thatε T +h−i can be rewritten in terms of the original innovations asε
T +h−i = P −i j=0 π j x T +h−i−j = P −i j=0 P −i−j k=0 π j ψ k ε T +h−i−j−k . (7.6)
Hence,
MSFE X T +h = E P i=0 ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E P i=0 ψ i ε T +h−i 2 − 2 P l=0 P i=h P −i j=0 P −i−j k=0 ψ l ψ i π j ψ k ε T +h−l ε T +h−i−j−k + P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = σ 2 P i=h ψ i 2 − 2 P l=0 P i=h P −i j=0 P −i−j k=0 ψ l ψ k E ψ i π j δ l,i+j+k + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ i+j+k,i +j +k + σ 2 h−1 i=0 ψ i 2 .
(iii) By part (i) the forecast X T +h is given by
X T +h = P i=h ψ iεT +h−i . (7.7)
According to the statement (3.2), both ψ i and π j can be asymptotically written as
ψ i = ψ i + r i √ T , π j = π j + t j √ T ,
with r i and t j as Gaussian random variables of mean 0 and variances (Σ Ξ ) i,i and (Σ Ξ ) j+P,j+P , respectively. Consequently by (7.7) and (7.6)
X T +h = P i=h ψ iεT +h−i = P i=h P −i j=0 P −i−j k=0 ψ i + r i √ T π j + t j √ T ψ k ε T +h−i−j−k = P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k + ψ i t j ψ k √ T + r i π j ψ k √ T + r i t j ψ k T ε T +h−i−j−k .
We now recall that
P −i j=0 P −i−j k=0 π j ψ k ε T +h−i−j−k = ε T +h−i ,
and we eliminate in this expression the term that decays as 1/T; we hence approximate X T +h as
X T +h P i=h ψ i + r i √ T ε T +h−i + P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k = P i=h ψ i ε T +h−i + P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k .
Using this approximation we compute now the MSFE:
MSFE X T +h = E X t+h − X t+h 2 = E P i=0 ψ i ε T +h−i − P i=h ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E h−1 i=0 ψ i ε T +h−i + P i=h ψ i − ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = σ 2 h−1 i=0 ψ i 2 + σ 2 1 T P i=h (Σ Ξ ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ ) j+P,j +P δ i+j+k,i +j +k .
Proof of Theorem 3.4
The mean square error associated to the forecast X T +h ( β) carried out using estimated parameters β is given by:
MSFE X T +h ( β) = E X T +h − X T +h ( β) 2 = E X T +h − X T +h (β) + X T +h (β) − X T +h ( β) 2 = E X T +h − X T +h (β) 2 + 2E X T +h − X T +h (β) X T +h (β) − X T +h ( β) + E X T +h (β) − X T +h ( β) 2 .
(7.8)
We now recall that
X T +h (β) = P i=h ψ i ε T +h−i , with P = T + h − 1 + r,
and notice that
E X T +h − X T +h (β) X T +h (β) − X T +h ( β) = E h−1 j=0 ψ j ε T +h−j P i=h ψ i ε T +h−i − ψ iεT +h−i = E h−1 j=0 ψ j ε T +h−j P i=h ψ i ε T +h−i − P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k = 0,
since the first term in the product involves the innovations {ε T +1 , . . . , ε T +h } and the second one {ε 1−r , . . . , ε T }; these two sets are disjoint and hence independent. Consequently by (2.6) and (7.8) we have
MSFE X T +h ( β) = σ 2 h−1 i=0 ψ 2 i + E X T +h (β) − X T +h ( β) 2 .
The second summand of this expression can be asymptotically evaluated using (3.6). Indeed, ∂ψ k ∂β j π l + ψ k ∂π l ∂β j ∂ψ m ∂β i π n + ψ m ∂π n ∂β i (Σ β ) ij E [X T +h−k−l X T +h−m−n ] . ((Σ Ξ P ) k,m π l π n + (Σ Ξ P ) k,n+p π l ψ m +(Σ Ξ P ) m,l+p ψ k π n + (Σ Ξ P ) l+p,n+p ψ k ψ m ) ψ u ψ v δ k+l+u,m+n+v .
E X T +h (β) − X T +h ( β) 2 = E E X T +h (β) − X T +h ( β) 2 |F T = 1 T E [Ω(h)] = 1 T E p+q i,j=1 J i J j (Σ β ) ij ,(7.
(7.12)
The required identity (3.8) follows directly from (7.12) by noticing that π n ψ v δ k,m+n+v = δ k,m .
Proof of Proposition 4.4
First, we notice that by (4.2) we have F M ⊂ F T .
The same relation guarantees that X w T +k = Y M +1 . Hence the result is a consequence of the following general fact:
Lemma 7.1 Let z be a random variable in the probability space (Ω, P, F). Let F * be a sub-sigma algebra of F , that is, F * ⊂ F. Then
E (z − E [z|F * ]) 2 ≥ E (z − E [
Proof of Proposition 4.4
Part (i) is a straightforward consequence of (2.5). Regarding (ii), we first have that
X w T +K − X w T +K = K i=1 w i i−1 j=0 ψ j ε T +i−j .
Consequently,
MSFE X w T +K = E K i=1 K j=1 w i w j i−1 l=0 j−1 m=0 ψ l ψ m ε T +i−l ε T +j−m = σ 2 K i=1 w 2 i i−1 l=0 ψ 2 l + 2 K−1 i=1 K j=i+1 w i w j i−1 l=0 ψ l ψ j−i+l .
7.7 Proof of Theorem 4.7
(i) It is a straightforward consequence of part (i) in Theorem 3.3.
(ii) By (4.7) we have that
MSFE X w T +K = E X w t+K − X w t+K 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i − P (h) i=h ψ iεT +h−i 2 ,
where P (h) = T + h − 1 + r. We now notice that
ε T +h−i = P (h)−i j=0 P (h)−i−j k=0
π j ψ k ε T +h−i−j−k .
(7.14)
Hence,
MSFE X w T +K = E K h=1 w h P (h) i=0 ψ i ε T +h−i − K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i 2 − 2 K h=1 K h =1 w h w h P (h ) l=0 P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ i π j ψ k ε T +h −l ε T +h−i−j−k + K h=1 w h P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E K h=1 K h =1 w h w h P (h) i=0 P (h ) i =0 ψ i ψ i ε T +h−i ε T +h −i 2 (7.15) − 2σ 2 K h=1 K h =1 w h w h P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ k E ψ i π j δ h−l,h −i−j−k + E K h=1 K h =1 w h w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0
ψ k ψ k ψ i π j ψ i π j ε T +h−i−j−k ε T +h −i −j −k = σ 2 < w, (A + B + C)w >, (7.16)
where A, B, C are the matrices with components given by
A hh = P (h) i=0 P (h ) i =0 ψ i ψ i δ h−i,h −i , (7.17) B hh = −2 P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0
ψ l ψ k E ψ i π j δ h−l,h −i−j−k , (7.18)
C hh = P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ h−i−j−k,h −i −j −k ,(7.19)
and P (h) = T + h − 1 + r, P (h ) = T + h − 1 + r.
(iii) We recall that by (4.7), the forecast X w T +K is given by
X w T +K = K h =1 w h P (h) i=h ψ iεT +h−i .
According to (3.2), both ψ i and π j can be asymptotically written as
ψ i = ψ i + r i √ T and π j = π j + t j √ T ,
with r i and t j Gaussian random variables of mean 0 and variances (Σ Ξ P ) i,i and (Σ Ξ P ) j+P (K),j+P (K) , respectively. Consequently,
X w T +K = K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i + r i √ T π j + t j √ T ψ k ε T +h−i−j−k = K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i π j ψ k + ψ i t j ψ k √ T + r i π j ψ k √ T + r i t j ψ k T ε T +h−i−j−k .
(7.20)
We now recall that
P (h)−i j=0 P (h)−i−j k=0 π j ψ k ε T +h−i−j−k = ε T +h−i ,
and we eliminate in (7.20) the term that decays as 1/T. We hence approximate X w T +K as
X w T +K K h=1 w h P (h) i=h ψ i + r i √ T ε T +h−i + P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k = K h=1 w h P (h) i=h ψ i ε T +h−i + P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k .
(7.21)
Using this approximation we now compute the MSFE:
MSFE X w T +K = E X w T +K − X w T +K 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i − P (h) i=h ψ i ε T +h−i − P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E K h=1 w h h−1 i=0 ψ i ε T +h−i + P (h) i=h ψ i − ψ i ε T +h−i − P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E K h=1 K h =1 w h w h h−1 i=0 h −1 i =0 ψ i ψ i ε T +h−i ε T +h −i + E K h=1 K h =1 w h w h P (h) i=h P (h ) i =h ψ i − ψ i ψ i − ψ i ε T +h−i ε T +h −i + E K h=1 K h =1 w h w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k t j t j T ε T +h−i−j−k ε T +h −i −j −k + 2E K h=1 K h =1 w h w h P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ i − ψ i ( π j − π j ) ε T +h−i ε T +h −i −j −k .
Using Lemma 3.2, this expression can be asymptotically approximated by:
MSFE X w T +K = σ 2 < w, A char + D + F + G w >,
where A char , D, F , G are matrices whose components are given by
A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , D hh = 1 T P (h) i=h P (h ) i =h (Σ Ξ P ) i,i δ h−i,h −i , F hh = 2 T P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k (Σ Ξ P ) i,P (K)+j δ h−i,h −i −j −k , G hh = 1 T P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P (K),j +P (K) δ h−i−j−k,h −i −j −k .
Theorem 4.11 (Hybrid forecasting of linear temporal aggregates) Let ξ T = {x 1 , . . . , x T } be a sample obtained as a realization of a causal and invertible ARMA(p,q) model X as in (3.1). Let w = (w 1 , . . . , w K ) be a temporal aggregation vector such that T = M K, for some M ∈ N, and let Y = I w • p K (X) be the temporal aggregation of the model X and η M := {y 1 , . . . , y M } the temporal aggregated sample obtained out of ξ T .
right hand side of this expression only involves time steps that are integer multiples of K, the relation (4.30) only imposes requirements on the left hand side at those time steps. Moreover, it is easy to see thatE [(T (L) Θ (L) ε lK ) (T (L) Θ (L) ε lK+jK )] = 0, (4.29) for any Kj > K (p + 1) + q − p − K * . This implies that the process is {T (L) Θ (L) ε lK } l∈Z is (K (p + 1) + q − p − K * )-correlated,which guarantees in turn by [Bro06, Section 3.2] the existence of a weak MA(q * ) representation T (L) Θ (L) ε lK = Θ * (B) ε * lK , l ∈ Z, (4.30) where deg (Θ * (B)) = K (p + 1) + q − p − K * K := q * . (4.31)
l c l+jK = σ 2 * q * −j l=0θ * l θ * l+j , j = 0, 1, . . . , q * .(4.34)
For any given vectorv = (v 1 , . . . , v n ) , S i v = (0, . . . , 0 i , v 1 , .. . , v n−i ) . With this notation, the equations (4.34) can be rewritten asσ 2 ε T (L) Θ (L) S jK T (L) Θ (L) = σ 2 * Θ * (B) S j Θ * (B) , j = 0, 1, . . . , q * .(4.37)
= 1, ..., p. These equations uniquely determine the (1, 1)-block ∂Φ * ∂Φ = ∂φ * i ∂φ j i,j of the Jacobian J βY , as well as the derivatives d ij := ∂t i ∂φ j (4.40)
Figure 1 :
1Characteristic and total errors associated to the forecast of the temporal stock aggregate of the MA(10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.3 ARMA(3,11) model.Let p = 3, q = 11, n = 10 and let K = 3. In this case, the relation (5.ψ j ψ 3i = ψ 3i+1 , i = 0, . . . , 2, j = 1, 2, ψ i = 0 for i ≥ n − 1.We choose a solution for these relations of the form Ψ * = (1, −0.9, 0.8, 0, 0, 0, −0.7, 0.63, −0.56, 0) . We now introduce an AR(3) polynomial of the form Φ = (−0.9, 0.8, −0.4) . We then determine the MA(11) part of the model by using(5.3), which yields the coefficients Θ =(−1.8, 2.41, −1.84, 1, −0.32, −0.7, 1.26, −1.687, 1.288, −0.7, 0.224). In order to avoid the common roots between the AR and the MA polynomials that are obtained when the coefficients of the MA part are derived in this manner, we slightly perturb the values of some of the components of the vector Θ that we now set to be Θ = (−1.8, 2.4102, −1.8403, 1, −0.32, −0.7, 1.26, −1.687, 1.288, −0.7, 0.224) .
Figure 2 :
2Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(3,11) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.4 ARMA(1,4) model.Let p = 1, q = 4, n = 5 and let K = 4. In this setup, relation (5.L j , and consequently ψ 0 = 1 and ψ 4 = 0, necessarily, while the values of the coefficients ψ 1 , ψ 2 , and ψ 3 are not subjected to any constraint. We hence set Ψ *= (1, 0.3, −0.3, 0.3, 0). We now introduce the AR(1) polynomial determined by the coefficient Φ = 0.8. We then determine the MA(4) part of the model by using (5.3) which yields Θ = (−0.5, −0.54, 0.54, −0.24) . Again in order to avoid common roots between the AR and the MA polynomials, we perturb the polynomial Θ by setting: Θ = (−0.5, −0.5403, 0.54, −0.24) .
Figure 3 :
3Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(1,4) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
Figure 4 :
4Characteristic and total errors associated to the forecast of the temporal flow aggregate of the MA(10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.6 ARMA(3,10) model. In the previous example we chose ψ 10 = 0. Let us now use another solution of the system (5.9) in order to obtain another model with target orders p = 3 and q = 10. If we set ψ 10 = 0, then a possible solution is Ψ * = (1, −0.5, 0.45, −0.475, 0.3, −0.3875, 0.1, −0.2438, 0, 0, 0) . Let now Φ = (0.21, 0.207, 0.0162) be a causal AR(3) polynomial which determines via (5.3) the MA(10) polynomial Θ = (−0.71, 0.348, −0.4822, 0.3147, −0.3595, 0.1270, −0.1894, 0.0368, 0.0488, 0.0039) . In order to avoid common roots for the AR and MA polynomials, we perturb the MA coefficients and set Θ = (−0.71, 0.3481, −0.4823, 0.3148, −0.3595, 0.1270, −0.1894, 0.0368, 0.0488, 0.0039) .
Figure 5 :
5Characteristic and total errors associated to the forecast of the temporal flow aggregate of the ARMA(3,10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
Figure 6 :
6Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(3,10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
It is a straightforward consequence of the causality and invertibility hypotheses on the ARMA model that we are dealing with. Indeed, we can writeε t = t−1+r j=0 π j X t−j and X t = t−1+r i=0 ψ i ε t−i ,(7.1) which proves (2.4).
presence of the stationarity hypothesis in part (ii) of the theorem we have thatE [X T +h−k−l X T +h−m−n ] = γ(k + l − m − n)and hence (3.9) follows. Otherwise, since we have in general thatE [X t X s ] ψ k ψ m (Σ β ) ij ψ u ψ v δ k+l+u,m+n+v
l ψ u δ k+l+u,m+n+v ) = δ k,m+n+v , l ψ u δ k+l+u,m+n+v ) π n ψ v =
Proof of Lemma 7.1E (z − E [z|F * ]) 2 = E (z − E [z|F * ] − E [z|F] + E [z|F]) 2 = E (z − E [z|F]) + E (E [z|F] − E [z|F * ]) 2 + 2E [(z − E [z|F]) (E [z|F] − E [z|F * ])] . that E (z − E [z|F * ]) 2 ≥ 0,the inequality (7.13) follows if we show that E [(z − E [z|F]) (E [z|F] − E [z|F * ])] = 0. E [(z − E [z|F]) (E [z|F] − E [z|F * ]) |F] = E zE [z|F] − zE [z|F * ] − E [z|F] 2 + E [z|F] E [z|F * ] |F = E [z|F] 2 − E [z|F] E [z|F * ] − E [z|F] 2 + E [z|F] E [z|F * ] = 0.z|F])
2 .
(7.13)
2
Given Indeed,
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Forecasting Economic Time Series. Clive W J Granger, P Newbold, Academic PressSan Diego, CAsecond editionClive. W. J. Granger and P Newbold. Forecasting Economic Time Series. Academic Press, San Diego, CA, second edition, 1986.
Prediction with a Generalized Cost of Error Function. Clive W J Granger, Clive. W. J. Granger. Prediction with a Generalized Cost of Error Function, 1969.
Time series analysis. D James, Hamilton, Princeton University PressPrinceton, NJJames D Hamilton. Time series analysis. Princeton University Press, Princeton, NJ, 1994.
Predictors for temporally and contemporaneously aggregated stationary processes. Helmut Lütkepohl, Fourth International Symposium on Forecasting. LondonHelmut Lütkepohl. Predictors for temporally and contemporaneously aggregated stationary processes. In Fourth International Symposium on Forecasting, London, 1984.
Comparison of predictors for aggregated time series. Helmut Lütkepohl, International Journal of Forecasting. 2Helmut Lütkepohl. Comparison of predictors for aggregated time series. International Journal of Forecasting, 2:461-475, 1986.
Forecasting temporally aggregated vector ARMA processes. Helmut Lütkepohl, Journal of Forecasting. 5Helmut Lütkepohl. Forecasting temporally aggregated vector ARMA processes. Journal of Forecasting, 5:85-95, 1986.
Forecasting Aggregated Vector ARMA Processes. Helmut Lütkepohl, Springer-VerlagBerlinHelmut Lütkepohl. Forecasting Aggregated Vector ARMA Processes. Springer-Verlag, Berlin, 1987.
Prediction of temporally aggregated systems involving both stock and flow variables. Helmut Lütkepohl, Statistical Papers. 2Helmut Lütkepohl. Prediction of temporally aggregated systems involving both stock and flow variables. Statistical Papers, 2:279-293, 1989.
Prediction of temporally aggregated systems involving both stock and flow variables. Helmut Lütkepohl, Statistical Papers. 30Helmut Lütkepohl. Prediction of temporally aggregated systems involving both stock and flow variables. Statistical Papers, 30:279-293, 1989.
New introduction to multiple time series analysis. Helmut Lütkepohl, Springer-VerlagBerlinHelmut Lütkepohl. New introduction to multiple time series analysis. Springer-Verlag, Berlin, 2005.
Forecasting with VARMA models. Helmut Lütkepohl, Handbook of Economic Forecasting. Graham Elliott, Clive Granger, and Allan TimmermannBerlinElsevier1Helmut Lütkepohl. Forecasting with VARMA models. In Graham Elliott, Clive Granger, and Allan Timmermann, editors, Handbook of Economic Forecasting, volume 1, pages 287-325. Elsevier, Berlin, 2006.
Forecasting aggregated time series variables: A survey. EUI Working Papers. Helmut Lütkepohl, Helmut Lütkepohl. Forecasting aggregated time series variables: A survey. EUI Working Papers, 2009.
Forecasting nonlinear aggregates and aggregates with time-varying weights. EUI Working Papers. Helmut Lütkepohl, Helmut Lütkepohl. Forecasting nonlinear aggregates and aggregates with time-varying weights. EUI Working Papers, 2010.
Asymptotic properties of prediction errors for multivariate autoregressive model using estimated parameters. G , Journal of the Royal Statistical Society. Series B. 42G. Reinsel. Asymptotic properties of prediction errors for multivariate autoregressive model using estimated parameters. Journal of the Royal Statistical Society. Series B, 42:328-333, 1980.
Temporal aggregation and economic time series. J Robert, John J Rossana, Seater, Journal of Business and Economic Statistics Economic Statistics. 134Robert J. Rossana and John J. Seater. Temporal aggregation and economic time series. Journal of Business and Economic Statistics Economic Statistics, 13(4):441-451, 1995.
Approximation theorems of mathematical statistics. Robert J Serfling, John Wiley \& SonsRobert J. Serfling. Approximation theorems of mathematical statistics. John Wiley \& Sons, 1980.
Properties of predictors for multivariate autoregressive models with estimated parameters. V A Samaranayake, David P Hasza, Journal of Time Series Analysis. 94V. A. Samaranayake and David P. Hasza. Properties of predictors for multivariate autoregressive models with estimated parameters. Journal of Time Series Analysis, 9(4):361-383, 1988.
Temporal aggregation of univariate and multivariate time series models: a survey. Andrea Silvestrini, David Veredas, Journal of Economic Surveys. 223Andrea Silvestrini and David Veredas. Temporal aggregation of univariate and multivariate time series models: a survey. Journal of Economic Surveys, 22(3):458-497, July 2008.
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Asymptotic behaviour of temporal aggregates of time series. G C Tiao, Biometrika. 593G. C. Tiao. Asymptotic behaviour of temporal aggregates of time series. Biometrika, 59(3):525- 531, December 1972.
Effect of temporal aggregation on the dynamic relationship of two time series variables. G C Tiao, W S William, Wei, Biometrika. 633G. C. Tiao and William W. S. Wei. Effect of temporal aggregation on the dynamic relationship of two time series variables. Biometrika, 63(3):513-523, December 1976.
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Predictions of multivariate autoregressive-moving average models. Taku Yamamoto, Biometrika. 65Taku Yamamoto. Predictions of multivariate autoregressive-moving average models. Biometrika, 65:485-492, 1981.
| []
|
[
"Electromagnetic Calorimeter for MPD Spectrometer at NICA Collider on behalf of MPD collaboration",
"Electromagnetic Calorimeter for MPD Spectrometer at NICA Collider on behalf of MPD collaboration"
]
| [
"A Yu Semenov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"S Bazylev \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"E Belyaeva \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"M Bhattacharjee \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nGauhati University\n781014GuwahatiAssamIndia\n",
"B Dabrowska \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nPlovdiv University \"Paisii Hilendarski\"\nTzar Assen 24PlovdivBulgaria\n",
"D Egorov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"V Golovatyuk \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"Yu Krechetov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"A Shutov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"V Shutov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"S Sukhovarov \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"A Terletskiy \nJoint Institute for Nuclear Research\n141980DubnaRussia\n",
"I Tyapkin \nJoint Institute for Nuclear Research\n141980DubnaRussia\n"
]
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"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Gauhati University\n781014GuwahatiAssamIndia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Plovdiv University \"Paisii Hilendarski\"\nTzar Assen 24PlovdivBulgaria",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
"Joint Institute for Nuclear Research\n141980DubnaRussia",
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| []
| A: The Multi-Purpose Detector (MPD) is designed to study a hot and dense baryonic matter formed in heavy-ion collisions at √ s N N =4-11 GeV at the NICA accelerator complex (Dubna, Russia). Large-sized electromagnetic calorimeter (ECal) of the MPD spectrometer will provide precise spatial and energy measurements for photons and electrons in the central pseudorapidity region of |η|<1.2. The Shashlyk-type sampling structure of the ECal is optimized for the photons energy range from about 40 MeV to 2-3 GeV. Fine segmentation and projective geometry of the calorimeter allow to deal with high multiplicity of secondary particles from Au-Au reactions. In this talk, we report on a design, a construction status and expected parameters of the ECal.K : Heavy-ion detectors, Instrumentation and methods for heavy-ion reactions and fission studies, Calorimeters | 10.1088/1748-0221/15/05/c05017 | [
"https://arxiv.org/pdf/2002.07709v1.pdf"
]
| 211,146,599 | 2002.07709 | 130513003a89b1baee1d977d219374b9436208c3 |
Electromagnetic Calorimeter for MPD Spectrometer at NICA Collider on behalf of MPD collaboration
17 Feb 2020
A Yu Semenov
Joint Institute for Nuclear Research
141980DubnaRussia
S Bazylev
Joint Institute for Nuclear Research
141980DubnaRussia
E Belyaeva
Joint Institute for Nuclear Research
141980DubnaRussia
M Bhattacharjee
Joint Institute for Nuclear Research
141980DubnaRussia
Gauhati University
781014GuwahatiAssamIndia
B Dabrowska
Joint Institute for Nuclear Research
141980DubnaRussia
Plovdiv University "Paisii Hilendarski"
Tzar Assen 24PlovdivBulgaria
D Egorov
Joint Institute for Nuclear Research
141980DubnaRussia
V Golovatyuk
Joint Institute for Nuclear Research
141980DubnaRussia
Yu Krechetov
Joint Institute for Nuclear Research
141980DubnaRussia
A Shutov
Joint Institute for Nuclear Research
141980DubnaRussia
V Shutov
Joint Institute for Nuclear Research
141980DubnaRussia
S Sukhovarov
Joint Institute for Nuclear Research
141980DubnaRussia
A Terletskiy
Joint Institute for Nuclear Research
141980DubnaRussia
I Tyapkin
Joint Institute for Nuclear Research
141980DubnaRussia
Electromagnetic Calorimeter for MPD Spectrometer at NICA Collider on behalf of MPD collaboration
17 Feb 2020P JINST C H E F C (CHEF ) N -, F , J 1Corresponding author.
A: The Multi-Purpose Detector (MPD) is designed to study a hot and dense baryonic matter formed in heavy-ion collisions at √ s N N =4-11 GeV at the NICA accelerator complex (Dubna, Russia). Large-sized electromagnetic calorimeter (ECal) of the MPD spectrometer will provide precise spatial and energy measurements for photons and electrons in the central pseudorapidity region of |η|<1.2. The Shashlyk-type sampling structure of the ECal is optimized for the photons energy range from about 40 MeV to 2-3 GeV. Fine segmentation and projective geometry of the calorimeter allow to deal with high multiplicity of secondary particles from Au-Au reactions. In this talk, we report on a design, a construction status and expected parameters of the ECal.K : Heavy-ion detectors, Instrumentation and methods for heavy-ion reactions and fission studies, Calorimeters
Introduction
The main goal of the Multi-Purpose Detector (MPD) at Nuclotron-based Ion Collider fAcility (NICA) is a study of signals from hot baryonic matter with maximal density that might be produced in collisions of heavy ions at √ s N N =4-11 GeV [1,2]. The experimental program with heavy ions at NICA includes event-by-event measurements of observables that are expected to be sensitive to high-density effects and phase transitions: particle yields, particle yields ratios, fluctuations, and correlations. The MPD spectrometer ( Fig. 1) has 2-π acceptance in azimuth, low material budget, and is able to handle event rate up to 6 kHz. Solenoid magnet with 0.5 T magnetic field and the barrel Time-Projection-Chamber (TPC) tracking system provide the measurements of momenta of charged particles with sufficient accuracy (about 2% at p t =300 MeV/c), and vertex finding. dE/dx measurements with TPC together with Time-of-Flight measurements allow π/K separation up to 1.5 GeV/c, and K/p separation up to 3 GeV/c. Electromagnetic barrel calorimeter (ECal) as an important part of MPD provides an access to the electromagnetic probes such as direct photons and lepton pairs, decays of neutral mesons, and significantly improves electron/hadron separation.
Electromagnetic Calorimeter Modules
Large-sized (about 6-meters-long and 4.5-meters in diameter) ECal covers the central pseudorapidity region of |η|<1.2 ( Fig. 1), and is optimized for precise spatial and energy measurements for photons and electrons in the energy range from about 40 MeV to 2-3 GeV [3]. To deal with a high multiplicity of secondary particles from central heavy-ions collisions, ECal has a fine segmentation and consists of 38,400 cells (towers). Taking all requirements (high energy resolution, large enough distance to the vertex, small Moliere radius, ability to work in the high magnetic field, high time resolution, and a reasonable price) into consideration, a "shashlyk"-type electromagnetic calorimeter was selected [4]. Each tower has a sandwich structure of 210 polystyrene scintillator [5, 6] and 210 lead plates with 16 Wave Length Shifting (WLS) fibers Kuraray Y-11 (200) [7] that penetrate the plates to collect the scintillation light and transport it to the photodetector; the far (from the photodetector) end of fiber is painted with white light-reflecting paint to increase an amount of the collected light. The thickness of each scintillator plate is 1.5 mm, and the thickness of lead plate is 0.3 mm.
Monte-Carlo simulations suggest that such a proportion of scintillators and lead coverters provides the sampling fraction of about 34-39% (depending on energy), and results into relatively small statistical term and a good energy resolution in the energy range below 1 GeV. A dark side of this calorimeter design is that the limited space inside the MPD magnet leads to the limited ECal thickness just above 11 X 0 and the correspondent visible energy leak from the backend of the calorimeter; though the leak does not exceed 10-12% in the ECal energy range. Each ECal module consists of 16 towers that are glued together. The geometry of the each module depends on the module Z-coordinate (beam direction) location in respect to the beams interception point (Fig. 2). The advantage the calorimeter with the projective geometry (where the towers are inclined along the beam axis to keep the tower axis to be consistent with the direct view to the beams intersection region) is a reduction of dead zones, increase of the detector efficiency, improvement of a linearity and an energy resolution of the calorimeter measurements in conditions of high multiplicity of secondary particles from the collisions of heavy ions.
In total, ECal will contain 2,400 modules of 8 different types. The production of the ECal modules is divided between Russian (25%) and Chinese (75%) facilities. Production of the modules in Russia is started in 2019, and the production in China is expected to be started in 2020.
ECal Power Structure
From geometrical point of view, ECal is divided in 25 sectors or 50 half-sectors; each half-sector contains 6×8=48 modules of 8 different types. These modules are located in the half-sector container (basket) made of fiberglass material. Rigidity of the container is enough to provide deformation less than 0.5 mm under full half-sector load of about 1.5 tons. Original MPD construction plan was to build ECal as a self-supporting structure. After sharing the modules production between Russian and Chinese institutions and corresponding differences in the modules delivery schedule, the decision was made to locate the calorimeter half-sectors into carbon-composite power MPD frame which is strong enough to hold the total weight of ECal and other MPD detectors (about 100 tons) with a maximal deformation of 2-3 mm, that makes possible to install and extract any ECal half-sector without dismounting whole MPD spectrometer.
In addition to the modules, half-sector includes ECal readout electronics. To keep ability to extract and reinstall calorimeter electronics (for service and repair), special electronics installation system was developed. Electronics support is provided by 3-m-long boxes; each box serves 2×8=16 modules. Outside each box, 16 front-end boards with 16 photodetectors (6×6 mm 2 Hamamatsu S13360-6025PE MAPD [8,9]), preamplifiers and slow-control electronics (that controls the temperature and makes corresponding correction of about 45 mV/deg. on photodetectors supply voltage) are located, while the JINR-designed 64-channel 14-bit 62.5 MS/s Pipelined ADC64ECAL boards [10] are housed inside heat-isolated box to minimize influence on photodetectors. The hit production is estimated as about 150 W per half-sector (or about 7.5 kW for whole ECal), and the water-cooling system is used to evacuate the heat from the boxes.
ECal Module Tests
Tests of the prototype modules were performed with electron beams at DESY (Hamburg, Germany) and Lebedev Physics Institute of Russian Academy of Science (Troitsk, Russia). For electron beam energies above 1 GeV, a visible deviation from linearity for the ECal response was observed (see left panel in Fig. 5). It was found that this deviation is connected mostly with the signal saturation because of the limited number of pixels in the MAPD; the correction on this effect restores the linearity. The signal time was produced from an analysis of ADC waveform front, and the time resolution for each tower is presented in Fig. 5 (right panel). The blue star in this plot is belong to the time measurement with cosmic muons that travels through ECal modules in a transverse direction, and this result is consistent with measurements on electron beam. The green line on the plot presents results of a special time measurement with high-precision and high-frequency ADCs, and allows to estimate the contribution of "standard" ECal electronics into measured time resolution. An energy resolution of the single ECal module was measured recently with relatively-lowenergy electron beam in Troitsk. The obtained data (shown in blue in Fig. 6) are in a good agreement with results of Monte-Carlo simulations for a single module (shown in red in Fig. 6). The same Monte-Carlo simulation made with whole calorimeter (dash-dotted black line in Fig. 6) allows to
Conclusions
Large-sized barrel electromagnetic calorimeter for MPD spectrometer is under construction in Joint Institute for Nuclear Research (Dubna, Russia). The "shashlyk"-type calorimeter is optimized to deal with high multiplicity of secondary particles from heavy-ion collisions at NICA accelerator complex. Production of the calorimeter modules is shared between Russian and Chinese institutions. The prototype modules tests with electron beams in GSI and Troitsk demonstrate that the measured calorimeter parameters are in a good agreement with expectations.
Figure 1 .
1MPD setup. The ECal location is shown in blue.
Figure 2 .
2Top: evolution of ECal modules shape along the beam direction from the center to the edge of MPD. Bottom left: photo of the module from the MPD center (type 0 shown in pink in the top panel). Bottom right: photo of the module from the edge of MPD (type 7 shown in medium-blue in the top panel).
Figure 3 .
3The power frame of ECal. On the right panel, the half-sector containers (baskets) are shown in blue, and Time-of-Flight modules are shown in green.
Figure 4 .
4Left panel shows the container for ECal half-sector (basket). Right panel shows the installation of boxes with electronics into the baskets.
Figure 5 .
5Results of the linearity (left panel) and the time resolution (right panel) measurements of prototype ECal modules with the electron beam at GSI.
Figure 6 .
6Results of the energy resolution measurements with the electron beam in Troitsk. make a preliminary estimation of the expected energy resolution of ECal:
Acknowledgments
. V D Kekelidze, Phys. Part. Nucl. 49457V.D. Kekelidze, Phys. Part. Nucl. 49 (2018) 457.
The Multi-Purpose Detector (MPD) of the collider experiment. V Golovatyuk, V Kekelidze, V Kolesnikov, The European Physical Journal A. 528212V. Golovatyuk, V. Kekelidze, V. Kolesnikov, et al., The Multi-Purpose Detector (MPD) of the collider experiment, The European Physical Journal A 52 (8) (2016) 212.
. G S Atoian, Nucl. Instrum. Meth. A. 584291G.S. Atoian et al., Nucl. Instrum. Meth. A 584 (2008) 291.
| []
|
[
"Mathematics Subject Classification: Primary 11R11; Secondary 11N80",
"Mathematics Subject Classification: Primary 11R11; Secondary 11N80"
]
| [
"Colin Defant [email protected] \nDepartment of Mathematics\nUniversity of Florida United States\n\n"
]
| [
"Department of Mathematics\nUniversity of Florida United States\n"
]
| []
| Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call n-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of 2-powerfully perfect numbers in the rings O Q( √ −1) , O Q( √ −2) , and O Q( √ −7), the three imaginary quadratic rings with unique factorization in which 2 is not a prime. | null | [
"https://arxiv.org/pdf/1412.3072v2.pdf"
]
| 118,671,093 | 1412.3072 | 014e5130c18384771c93c38ad4f57799e8f1ec60 |
Mathematics Subject Classification: Primary 11R11; Secondary 11N80
2010
Colin Defant [email protected]
Department of Mathematics
University of Florida United States
Mathematics Subject Classification: Primary 11R11; Secondary 11N80
2010Abundancy indexquadratic ringsolitary numberperfect num- ber
Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call n-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of 2-powerfully perfect numbers in the rings O Q( √ −1) , O Q( √ −2) , and O Q( √ −7), the three imaginary quadratic rings with unique factorization in which 2 is not a prime.
Introduction
Throughout this paper, we will let N denote the set of positive integers, and we will let N 0 denote the set of nonnegative integers.
The arithmetic functions σ k are defined, for every integer k, by σ k (n) = c|n c>0 c k . For each integer k = 0, σ k is multiplicative and satisfies σ k (p α ) = p k(α+1) − 1 p k − 1 for all (integer) primes p and positive integers α. The abundancy index of a positive integer n is defined by I(n) = σ 1 (n) n . A positive integer n is said to be t-perfect if I(n) = t for a positive integer t ≥ 2, and 2-perfect numbers are called perfect numbers.
For any square-free integer d, let O Q( √ d) be the quadratic integer ring given by
O Q( √ d) = Z[ 1+ √ d 2 ], if d ≡ 1 (mod 4); Z[ √ d], if d ≡ 2, 3 (mod 4).
Throughout the remainder of this paper, we will work in the rings O Q(
√ d)
for different specific or arbitrary values of d. We will use the symbol "|" to mean "divides" in the ring O Q( √ d) in which we are working. Whenever we are working in a ring other than Z, we will make sure to emphasize when we wish to state that one integer divides another in Z. For example, if we are working in Z[i], the ring of Gaussian integers, we might say that 1 + i|1 + 3i and that 2|6 in Z. We will also refer to primes in O Q( √ d) as "primes," whereas we will refer to (positive) primes in Z as "integer primes." For an integer prime p and a nonzero integer n, we will let υ p (n) denote the largest integer k such that p k |n in Z. For a prime π and a nonzero number x ∈ O Q( √ d) , we will let ρ π (x) denote the largest integer k such that π k |x. Furthermore, we will henceforth focus exclusively on values of d for which O Q( √ d) is a unique factorization domain and d < 0. In other words, d ∈ K, where we will define K to be the set {−163, −67, −43, −19, −11, −7, −3, −2, −1}. The set K is known to be the complete set of negative values of d for which O Q( √ d) is a unique factorization domain [4].
For an element a+b √ d ∈ O Q( √ d) with a, b ∈ Q, we define the conjugate by a + b √ d = a−b √ d.
The norm and absolute value of an element z are defined, respectively, by N(z) = zz and |z| = N(z). We assume familiarity with the properties of these object, which are treated in Keith Conrad's online notes [1]. For x, y ∈ O Q( √ d) , we say that x and y are associated, denoted
x ∼ y, if and only if x = uy for some unit u in the ring O Q( √ d) . Furthermore, we will make repeated use of the following well-known facts.
Fact 1.1. Let d ∈ K.
If p is an integer prime, then exactly one of the following is true.
• p is also a prime in O Q( √ d) . In this case, we say that p is inert in O Q( √ d) .
• p ∼ π 2 and π ∼ π for some prime π ∈ O Q( √ d) . In this case, we say p ramifies (or p is ramified) in O Q( √ d) .
• p = ππ and π ∼ π for some prime π ∈ O Q( √ d) . In this case, we say p
splits (or p is split) in O Q( √ d) . Fact 1.2. Let d ∈ K. If π ∈ O Q( √ d)
is a prime, then exactly one of the following is true.
• π ∼ q and N(π) = q 2 for some inert integer prime q.
• π ∼ π and N(π) = p for some ramified integer prime p.
• π ∼ π and N(π) = N(π) = p for some split integer prime p.
Fact 1.3. If d ∈ K, q is an integer prime that is inert in O Q( √ d) , and x ∈ O Q( √ d) \{0}, then υ q (N(x)) is even and ρ q (x) = 1 2 υ q (N(x)√ −2) , splits in O Q( √ −7) , and is inert in O Q( √ d) for all d ∈ K\{−1, −2, −7}. Fact 1.5. Let O * Q( √ d) be the set of units in the ring O Q( √ d) . Then O * Q( √ −1) = {±1, ±i}, O * Q( √ −3) = ±1, ± 1 + √ −3 2 , ± 1 − √ −3 2 , and O * Q( √ d) = {±1} whenever d ∈ K\{−1, −3}.
For a nonzero complex number z, let arg(z) denote the argument, or angle, of z. We convene to write arg(z) ∈ [0, 2π) for all z ∈ C. For each d ∈ K, we define the set A(d) by Remark 1.1. We note that, for each x in the summation in the above definition, we may cavalierly replace x with one of its associates. This is because associated numbers have the same absolute value. In other words, the only reason for the criterion x ∈ A(d) in the summation that appears in Definition 1.1 is to forbid us from counting associated divisors as distinct terms in the summation, but we may choose to use any of the associated divisors as long as we only choose one. This should not be confused with how we count conjugate divisors (we treat 2 + i and 2 − i as distinct divisors
A(d) = {z ∈ O Q( √ d) \{0} : 0 ≤ arg(z) < π 2 }, if d = −1; {z ∈ O Q( √ d) \{0} : 0 ≤ arg(z) < π 3 }, if d = −3; {z ∈ O Q(of 5 in Z[i] because 2 + i ∼ 2 − i). Remark 1.2. We mention that the function δ n is different in each ring O Q( √ d) .
Perhaps it would be more precise to write δ n (z, d), but we will omit the latter component for convenience. We note that we will also use this convention with functions such as I n (which we will define soon).
We will say that a function f :
O Q( √ d) \{0} → R is multiplicative if f (xy) =
f (x)f (y) whenever x and y are relatively prime (have no nonunit common divisors). The author has shown that, for any integer n, δ n is multiplicative [2].
I n : O Q( √ d) \{0} → [1, ∞) by I n (z) = δ n (z) |z| n . For a positive integer t ≥ 2, we say that a number z ∈ O Q( √ d) \{0} is n-powerfully t-perfect in O Q( √ d) if I n (z) = t, and, if t = 2, we simply say that z is n-powerfully perfect in O Q( √ d) .
As an example, we will let
d = −1 so that O Q( √ d) = Z[i]. Let us compute I 2 (9 + 3i). We have 9 + 3i = 3(1 + i)(2 − i), so δ 2 (9 + 3i) = N(1) + N(3) + N(1 + i) + N(2 − i) + N(3(1 + i)) + N(3(2 − i)) + N((1 + i)(2 − i)) + N(3(1 + i)(2 − i
)) = 1 + 9 + 2 + 5 + 18 + 45 + 10 + 90 = 180. (b) I n is multiplicative.
Then I 2 (9 + 3i) = 180 N(3(1 + i)(2 − i)) = 2, so 9 + 3i is 2-powerfully perfect in O Q( √ −1) . Theorem 1.1. Let n ∈ N, d ∈ K, and z 1 , z 2 , π ∈ O Q( √ d) \{0} with π a(c) I n (z 1 ) = δ −n (z 1 ). (d) If z 1 |z 2 , then I n (z 1 ) ≤ I n (z 2 ), with equality if and only if z 1 ∼ z 2 .
We refer the reader to [2] for a proof of Theorem 1.1. The author has already investigated 1-powerfully t-perfect numbers in imaginary quadratic rings with unique factorization, and he has shown that, for any integers n ≥ 3 and t ≥ 2, no n-powerfully t-perfect numbers exist in these rings [3]. Hence, the remainder of this paper will focus on the interesting topic of 2-powerfully t-perfect numbers.
Investigating 2-powerfully t-perfect Numbers
Trying to find 2-powerfully t-perfect numbers is quite a pleasant activity.
One reason for this is that 2 is the only positive integer n for which there exist n-powerfully t-perfect numbers that are not associated to integers [3]. For example, in O Q( √ −1) , 3 + 9i is 2-powerfully perfect, and 30 + 30i is 2powerfully 3-perfect. We will also utilize the helpful that, for any d ∈ K and
z ∈ O Q( √ d) \{0}, we have N(z), δ 2 (z) ∈ N.
In this section, we will focus on the rings O Q(
√ −1) , O Q( √ −2) , and O Q( √ −7) , which are the only rings O Q( √ d) with d ∈ K in which 2 is not inert. Theorem 2.1. Let us work in a ring O Q( √ d) with d ∈ {−1, −2}. Then 2 ramifies in O Q( √ d) , so we may write 2 ∼ ξ 2 for some prime ξ satisfying ξ ∼ ξ and N(ξ) = 2. Suppose z is 2-powerfully perfect in O Q( √ d) and ξ|z. Then we may write z = ξ γ x, where γ ∈ N, x ∈ O Q( √ d) , ξ ∤ x, and 2 γ+1 − 1 is a Mersenne prime that is inert in O Q( √ d) . Furthermore, there exists an odd positive integer m such that δ 2 (x) = 2 γ+1 m and N(x) = (2 γ+1 − 1)m.
Proof. We know the first part of the theorem, which is stated simply to introduce notation. All that we need to prove is the final sentence of the theorem, as well as the fact that 2 γ+1 − 1 is a Mersenne prime that is inert
in O Q( √ d) . As z is 2-powerfully perfect in O Q( √ d) , we have δ 2 (z) = 2N(z) = 2N(ξ γ )N(x) = 2 γ+1 N(x).
However, we also have
δ 2 (z) = δ 2 (ξ γ )δ 2 (x) = γ j=0 N(ξ j ) δ 2 (x) = γ j=0 2 j δ 2 (x) = (2 γ+1 − 1)δ 2 (x).
Therefore, 2 γ+1 N(x) = (2 γ+1 − 1)δ 2 (x). As 2 γ+1 − 1 is odd, we find that 2 γ+1 |δ 2 (x) in Z. We may then write δ 2 (x) = 2 γ+1 m for some positive integer m. Substituting this new expression for δ 2 (x) into the equation 2 γ+1 N(x) = (2 γ+1 − 1)δ 2 (x), we find N(x) = (2 γ+1 − 1)m. This tells us that m is odd
because ξ ∤ x (implying that 2 ∤ N(x) in Z). Suppose that 2 γ+1 − 1 is not a prime in O Q( √ d) so that we may write 2 γ+1 − 1 = y 1 y 2 , where y 1 , y 2 ∈ O Q( √ d) satisfy 1 < N(y 1 ) ≤ N(y 2 ) < N(2 γ+1 − 1) = (2 γ+1 − 1) 2 . Then, because N(y 1 )N(y 2 ) = N(2 γ+1 − 1) = (2 γ+1 − 1) 2 , we see that N(y 1 ) ≤ 2 γ+1 − 1.
Now, let π 0 be a prime that divides y 1 . Then π 0 |N(x), which implies that either π 0 |x or π 0 |x. If π 0 |x, write π = π 0 . Otherwise, write π = π 0 . Then N(π) ≤ N(y 1 ) ≤ 2 γ+1 − 1, and x π is a nonunit proper divisor of x. This implies that
δ 2 (x) ≥ 1 + N x π + N(x) = 1 + N(x) N(π) + N(x) = 1 + (2 γ+1 − 1)m N(π) + (2 γ+1 − 1)m ≥ 1 + (2 γ+1 − 1)m 2 γ+1 − 1 + (2 γ+1 − 1)m = 1 + 2 γ+1 m.
However, this contradicts the fact that δ 2 (x) = 2 γ+1 m, so we conclude that 2 γ+1 − 1 is a prime in O Q( √ d) . Furthermore, because 2 γ+1 − 1 is an integer, we conclude that 2 γ+1 − 1 is an inert integer prime that is also a Mersenne prime.
Theorem 2.2. Let z, m, γ, and x be as in Theorem 2.1. Write q = 2 γ+1 − 1 and m = q k v, where k ∈ N 0 , v ∈ N, and q ∤ v in Z. Then k is odd, v ≥ q + 2, and
m ≥ q k+1 + (q + 3) k−1 2 j=0 q 2j ≥ q 2 + q + 3.
Proof. First, note that q is inert and υ q (N(x)) = k + 1. Therefore, Fact 1.3 implies that k is odd and ρ q (x) = k + 1 2 . Next, assume that v = 1.
Then m = q k , so x ∼ q k+1 2 . This implies that δ 2 (x) = k+1 2 j=0 q 2j ≡ 1 (mod q).
However, this contradicts Theorem 2.1, which tells us, under the assumption m = q k , that δ 2 (x) = 2 γ+1 m = (q + 1)m = (q + 1)q k ≡ 0 (mod q). Therefore, v > 1. Now, write y = x q (k+1)/2 . Then, using Theorem 2.1,
N(y) = N(x) N(q k+1 2 ) = qm q k+1 = q k+1 v q k+1 = v.
Because ρ q (x) = k + 1 2 , we see that y and q k+1 are relatively prime. Therefore,
δ 2 (x) = δ 2 (y)δ 2 (q k+1 2 ) = δ 2 (y) k+1 2 j=0 q 2j ≥ (v + 1) k+1 2 j=0 q 2j .
Theorem 2.1 states that δ 2 (x) = 2 γ+1 m = (q + 1)m, so we have
(q + 1)m ≥ (v + 1) k+1 2 j=0 q 2j = q k+1 v + q k+1 + (v + 1) k−1 2 j=0 q 2j = qm + q k+1 + (v + 1) k−1 2 j=0 q 2j .
We can simplify this last inequality to get
m ≥ q k+1 + (v + 1) k−1 2 j=0 q 2j .(1)Therefore, v = m q k ≥ q + (v + 1) k−1 2 j=0 q 2j−k > q.
As v and q are both odd and v > q, we conclude that v ≥ q + 2. Substituting this into (1), we have
m ≥ q k+1 + (q + 3) k−1 2 j=0 q 2j ≥ q 2 + q + 3,
which completes the proof.
It is interesting to note that, in the case z = 3 + 9i in O Q( √ −1) , the inequalities in Theorem 2.2 are, in fact, equalities. That is, q = 3, v = q + 2 = 5, and m = q 2 + q + 3 = 15. It seems likely, in light of the inequalities in Theorem 2.2, that the value of k in Theorem 2.2 should have to be 1.
2 = εε, where ε = 1+ √ −7 2 . Suppose z is 2-powerfully perfect in O Q( √ −7) and 2|N(z) in Z. Then either z = ε γ x or z = ε γ x, where γ ∈ N, x ∈ O Q( √ −7) , 2 ∤ N(x) in Z, and 2 γ+1 − 1 is a Mersenne prime that is inert in O Q( √ −7)
. Furthermore, there exists an odd positive integer m such that δ 2 (x) = 2 γ+1 m and N(x) = (2 γ+1 − 1)m.
Proof. We know that we may write
z = ε γ 1 ε γ 2 x, where γ 1 , γ 2 ∈ N 0 , x ∈ O Q( √ −7)
, and 2 ∤ N(x) in Z. Furthermore, we know from the fact that 2|N(z) in Z that γ 1 and γ 2 are not both zero. We must prove that either γ 1 = 0 or γ 2 = 0. Then, after setting γ = γ 1 + γ 2 , we need to prove the final sentence of the theorem and the fact that 2 γ+1 − 1 is a Mersenne prime that is inert in O Q( √ −7) .
As z is 2-powerfully perfect in O Q( √ −7) , we have
δ 2 (z) = 2N(z) = 2N(ε γ 1 )N(ε γ 2 )N(x) = 2 γ 1 +γ 2 +1 N(x).
However, we also have
δ 2 (z) = δ 2 (ε γ 1 )δ 2 (ε γ 2 )δ 2 (x) = γ 1 j=0 N(ε j ) γ 2 j=0 N(ε j ) δ 2 (x) = γ 1 j=0 2 j γ 2 j=0 2 j δ 2 (x) = (2 γ 1 +1 − 1)(2 γ 2 +1 − 1)δ 2 (x).
Therefore, 2 γ 1 +γ 2 +1 N(x) = (2 γ 1 +1 −1)(2 γ 2 +1 −1)δ 2 (x). As (2 γ 1 +1 −1)(2 γ 2 +1 −1) is odd, we find that 2 γ 1 +γ 2 +1 |δ 2 (x) in Z. We may then write δ 2 (x) = 2 γ 1 +γ 2 +1 m for some positive integer m. Substituting this new expression for δ 2 (x) into the equation 2 γ 1 +γ 2 +1 N(x) = (2 γ 1 +1 − 1)(2 γ 2 +1 − 1)δ 2 (x), we find N(x) = (2 γ 1 +1 − 1)(2 γ 2 +1 − 1)m. This tells us that m is odd because 2 ∤ N(x) in
Z. Now, 2 γ 1 +γ 2 +1 m = δ 2 (x) ≥ 1 + N(x) = 1 + (2 γ 1 +1 − 1)(2 γ 2 +1 − 1)m, so 2 γ 1 +γ 2 +1 > (2 γ 1 +1 − 1)(2 γ 2 +1 − 1) = 2 · 2 γ 1 +γ 2 +1 − 2 γ 1 +1 − 2 γ 2 +1 + 1.
Simplifying this inequality, we have 2 γ 1 +1 +2 γ 2 +1 > 2 γ 1 +γ 2 +1 +1, which is impossible unless γ 1 = 0 or γ 2 = 0. Therefore, either z = ε γ 1 x or z = ε γ 2 x. Either way, if we write γ = γ 1 + γ 2 , then we have δ 2 (x) = 2 γ+1 m and N(x) = (2 γ+1 − 1)m. Suppose that 2 γ+1 − 1 is not a prime in O Q( √ −7) so that we may write 2 γ+1 − 1 = y 1 y 2 , where y 1 , y 2 ∈ O Q( √ −7) satisfy 1 < N(y 1 ) ≤ N(y 2 ) < N(2 γ+1 − 1) = (2 γ+1 − 1) 2 . Then, because N(y 1 )N(y 2 ) = N(2 γ+1 − 1) = (2 γ+1 − 1) 2 , we see that N(y 1 ) ≤ 2 γ+1 −1. Now, let π 0 be a prime that divides y 1 . Then π 0 |N(x), which implies that either π 0 |x or π 0 |x. If π 0 |x, write π = π 0 . Otherwise, write π = π 0 . Then N(π) ≤ N(y 1 ) ≤ 2 γ+1 − 1, and x π is a nonunit proper divisor of x. This implies that
δ 2 (x) ≥ 1 + N x π + N(x) = 1 + N(x) N(π) + N(x) = 1 + (2 γ+1 − 1)m N(π) + (2 γ+1 − 1)m ≥ 1 + (2 γ+1 − 1)m 2 γ+1 − 1 + (2 γ+1 − 1)m = 1 + 2 γ+1 m.
However, this contradicts the fact that δ 2 (x) = 2 γ+1 m, so we conclude that 2 γ+1 − 1 is a prime in O Q( √ −7) . Furthermore, because 2 γ+1 − 1 is an integer, we conclude that 2 γ+1 − 1 is an inert integer prime that is also a Mersenne prime.
Theorem 2.4. Let z, m, γ, and x be as in Theorem 2.3. Write q = 2 γ+1 − 1 and m = q k v, where k ∈ N 0 , v ∈ N, and q ∤ v in Z. Then k is odd, v ≥ q + 2, γ ≡ 1 (mod 3), q ≡ 3 (mod 7), and
m ≥ q k+1 + (q + 3) k−1 2 j=0 q 2j ≥ q 2 + q + 3.
Proof. Fact 1.4 tells us that an integer prime is inert in
O Q( √ −7)
if and only if that integer prime is congruent to 3, 5, or 6 modulo 7. Also, it is easy to see that powers of 2 cannot be congruent to 6 or 7 modulo 7. Therefore, as q is a Mersenne prime that is inert in O Q( √ −7) , we must have q ≡ 3 (mod 7). This implies that 2 γ+1 ≡ 4 (mod 7), so γ ≡ 1 (mod 3). The proof of the rest of the theorem is identical to the proof of Theorem 2.2, except all references to Theorem 2.1 should be replaced with references to Theorem 2.3. These numbers are somewhat analogous to perfect numbers in Z. The analogues of odd perfect numbers are then 2-powerfully perfect numbers with odd norms. We now briefly explore some of the properties that such numbers would need to exhibit.
Theorem 2.5. Let us work in a ring O Q( √ d) with d ∈ K. Suppose z ∈ O Q( √ d) \{0}
is such that I 2 (z) = 2 and N(z) is odd (suppose such a z exists). Then we may write z ∼ π k x 2 , where π, x ∈ O Q( √ d) \{0}, π is prime, and k ∈ N. Furthermore, k ≡ N(π) ≡ 1 (mod 4).
Proof. First, let π 0 be a prime whose norm is odd, and let α be a positive integer. As δ 2 (π α 0 ) = α j=0 N(π j 0 ) = α j=0 N(π 0 ) j and N(π 0 ) is odd, we see that α and δ 2 (π α 0 ) have opposite parities. Now, from I 2 (z) = 2, we have δ 2 (z) = 2N(z). Because N(z) is odd, we find that δ 2 (z) ≡ 2 (mod 4). Write z = r j=1 π α j j , where, for all distinct j, l ∈ {1, 2, . . . , r}, π j is prime, α j is a positive integer, and π j ∼ π l . Then δ 2 (z) = r j=1 δ 2 (π α j j ). Because δ 2 (z) ≡ 2 (mod 4), we find that there must be exactly one value of j ∈ {1, 2, . . . , r} such that δ 2 (π α j j ) is even. This means that there is exactly one value of j ∈ {1, 2, . . . , r} such that α j is odd. Therefore, z ∼ π k x 2 , where π, x ∈ O Q( √ d) , π is prime, and k is an odd positive integer. Furthermore, δ 2 (π k ) ≡ 2 (mod 4).
If N(π) = q 2 , where q is an inert integer prime, then
δ 2 (π k ) = k l=0 N(π l ) = k l=0 q 2l ≡ k l=0 1 ≡ k + 1 (mod 4).
Therefore, in this case, we have k ≡ 1 (mod 4). Also, because N(π) = q 2 and q is odd, we know that N(π) ≡ 1 (mod 4).
On the other hand, if N(π) = p is an integer prime, then
δ 2 (π k ) = k l=0 N(π l ) = k l=0 p l ≡ 2 (mod 4),
which implies that p ≡ k ≡ 1 (mod 4).
Theorem 2.6. Let us work in a ring O Q( √ d) with d ∈ {−1, −2}. Let z ∈ O Q( √ d) \{0}
be such that I 2 (z) = 2 and N(z) is odd (suppose such a z exists). Then z has at least five nonassociated prime divisors.
Proof. Suppose z has four or fewer nonassociated prime divisors. Then we may write z ∼ π α 1 1 π α 2 2 π α 3 3 π α 4 4 , where, for all distinct j, l ∈ {1, 2, 3, 4}, π j is prime, α j is a nonnegative integer, and π j ∼ π l .
First, let us deal with the case d = −1. In the ring O Q( √ −1) , the five primes (up to units) that have the smallest odd norms are 2 + i, 1 + 2i, 3, 3 + 2i, and 2 + 3i, which have norms 5, 5, 9, 13, and 13, respectively. Therefore, I 2 (z) = I 2 (π α 1 1 π α 2 2 π α 3 3 π α 4 4 )
= α 1 j=0 1 N(π 1 ) j α 2 j=0 1 N(π 2 ) j α 3 j=0 1 N(π 3 ) j α 4 j=0 1 N(π 4 ) j < ∞ j=0 1 N(π 1 ) j ∞ j=0 1 N(π 2 ) j ∞ j=0 1 N(π 3 ) j ∞ j=0 1 N(π 4 ) j ≤ ∞ j=0 1 5 j ∞ j=0 1 5 j ∞ j=0 1 9 j ∞ j=0 1 13 j = 5 4 · 5 4 · 9 8 · 13 12 < 2,
which is a contradiction.
I 2 (z) ≥ I 2 ((1 + √ −2) 2 )I 2 ((1 − √ −2) 2 ) = 1 + 1 3 + 1 9 2 > 2,
which is a contradiction. This implies that 1 + √ −2 and 1 − √ −2 cannot both divide z. Now, the six primes (up to units) that have the smallest odd norms are 1 + √ −2, 1 − √ −2, 3 + √ −2, 3 − √ −2, 3 + 2 √ −2, and 3 − 2 √ −2, which have norms 3, 3, 11, 11, 17, and 17, respectively. Because 1 + √ −2 and 1 − √ −2 cannot both divide z, we have I 2 (z) = I 2 (π α 1 1 π α 2 2 π α 3 3 π α 4 4 ) \{0} be such that I 2 (z) = 2 and N(z) is odd (suppose such a z exists). Then z has at least eleven nonassociated prime divisors.
= α 1 j=0 1 N(π 1 ) j α 2 j=0 1 N(π 2 ) j α 3 j=0 1 N(π 3 ) j α 4 j=0 1 N(π 4 ) j < ∞ j=0 1 N(π 1 ) j ∞ j=0 1 N(π 2 ) j ∞ j=0 1 N(π 3 ) j
Proof. Suppose z has ten or fewer nonassociated prime divisors. Then we may write z ∼ 10 m=1 π αm m , where, for all distinct m, l ∈ {1, 2, . . . , 10}, π m is prime, α m is a nonnegative integer, and π m ∼ π l . In O Q( √ −7) , the eleven primes (up to units) that have the smallest odd norms are √ −7, 3, 2 + √ −7, 2 − √ −7, 4 + √ −7, 4 − √ −7, 5, 1 + 2 √ −7, 1 − 2 √ −7, 3 + 2 √ −7, and 3 − 2 √ −7, which have norms 7, 9, 11, 11, 23, 23, 25, 29, 29, 37, and 37, respectively. Therefore, I 2 (z) = 1 N(π m ) j which would guarantee that some specific multiple of an n 1 -powerfully t 1perfect number is n 2 -powerfully t 2 -perfect (for some n 1 , n 2 , t 1 , t 2 ∈ N with t 1 , t 2 ≥ 2)?
\{0} : 0 ≤ arg(z) < π}, otherwise. Thus, every nonzero element of O Q( √ d) can be written uniquely as a unit times a product of primes in A(d). Also, every z ∈ O Q( √ d) \{0} is associated to a unique element of A(d). The author has defined analogues of the arithmetic functions σ k in quadratic rings O Q( √ d) with d ∈ K [2], and we will state the important definitions and properties for the sake of completeness.
Definition 1 . 1 .
11Let d ∈ K, and let n ∈ Z. Define the function δ n : O Q(
Definition 1 . 2 .
12For each positive integer n, define the function
prime. Then, if we are working in the ring O Q( √ d) , the following statements are true. (a) The range of I n is a subset of the interval [1, ∞), and I n (z 1 ) = 1 if and only if z 1 is a unit in O Q( √ d) . If n is even, then I n (z 1 ) ∈ Q.
Theorem 2 . 3 .
23Let us work in the ring O Q( √ −7) so that 2 splits as
some properties of 2-powerfully perfect numbers with even norms.
. 7 .
7Let us work in the ring O Q(
Conjecture 3. 1 .
1The value of k in Theorem 2.2 must be 1. Similarly, if there is a 2-powerfully perfect number in O Q( √ −7) , then the value of k in Theorem 2.4 must be 1.
). Fact 1.4. Let p be an odd integer prime. Then p ramifies in O Q( √ d) if and only if p|d in Z. If p ∤ d in Z, then p splits in O Q( √ d) if and only if d is a quadratic residue modulo p. Note that this implies that p is inert in O Q(√
d)
if and only if p ∤ d in Z and d is a quadratic nonresidue modulo p. Also, the
integer prime 2 ramifies in O Q(
√
−1) and O Q(
Second, let us deal with the case d = −2. In the ring O Q( must both appear with even exponents in the prime factorization of z. In particular, (1 + Therefore, by part (d) of Theorem 2.2,√
−2) , the integer
prime 3 splits as 3 = (1+
√
−2)(1−
√
−2). Suppose 1+
√
−2|z and 1−
√
−2|z.
Then, because N(1 +
√
−2) = N(1 −
√
−2) = 3 ≡ 1 (mod 4), Theorem 2.5
implies that 1 +
√
−2 and 1 −
√
−2 √
−2) 2 (1 −
√
−2) 2 |z.
This work was supported by National Science Foundation grant no. 1262930.
AcknowledgmentsThe author would like to thank Professor Pete Johnson for inviting him to the 2014 REU Program in Algebra and Discrete Mathematics at Auburn University.. This is simply because, under these assumptions, we find thatFurther Ideas and a ConjectureWe admit that we directed almost all of our attention toward 2-powerfully perfect numbers, rather than the more general 2-powerfully t-perfect numbers. Hence, the subject of 2-powerfully t-perfect numbers awaits exploration. We also concentrated so heavily on the rings O Q(
the concluding paragraph of Section 2, we might ask if there are other relationships between different types of n-powerfully t-perfect numbers. More specifically. in a given ring O Q( √ d) , are there certain criteriathe concluding paragraph of Section 2, we might ask if there are other relationships between different types of n-powerfully t-perfect num- bers. More specifically, in a given ring O Q( √ d) , are there certain criteria
Factoring in Quadratic Fields. Keith Conrad, Conrad, Keith. Factoring in Quadratic Fields. Available at http://www.math.uconn.edu/∼kconrad/blurbs/ugradnumthy /quadraticundergrad.pdf.
An extension of the abundancy index to certain quadratic rings. Colin Defant, SubmittedDefant, Colin. An extension of the abundancy index to certain quadratic rings, Submitted (2014).
Multiperfect numbers in certain quadratic rings. Colin Defant, SubmittedDefant, Colin. Multiperfect numbers in certain quadratic rings, Submit- ted (2014).
A complete determination of the complex quadratic fields of class-number one. H M Stark, Michigan Math. J. 14Stark, H. M. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 14 1967 1-27.
| []
|
[
"Role of Bound Magnon in Magnetic Domain Wall Motion",
"Role of Bound Magnon in Magnetic Domain Wall Motion"
]
| [
"Tae-Suk Kim \nInstitute for Materials Research\nTohoku Univ\n980-8577SendaiJapan\n\nDepartment of Physics\nPohang University of Science and Technology\n790-784PohangKorea\n",
"J Ieda \nInstitute for Materials Research\nTohoku Univ\n980-8577SendaiJapan\n\nCREST\nScience and Technology Agency (JST)\n102-0075TokyoJapan, Japan\n",
"S Maekawa \nInstitute for Materials Research\nTohoku Univ\n980-8577SendaiJapan\n\nCREST\nScience and Technology Agency (JST)\n102-0075TokyoJapan, Japan\n"
]
| [
"Institute for Materials Research\nTohoku Univ\n980-8577SendaiJapan",
"Department of Physics\nPohang University of Science and Technology\n790-784PohangKorea",
"Institute for Materials Research\nTohoku Univ\n980-8577SendaiJapan",
"CREST\nScience and Technology Agency (JST)\n102-0075TokyoJapan, Japan",
"Institute for Materials Research\nTohoku Univ\n980-8577SendaiJapan",
"CREST\nScience and Technology Agency (JST)\n102-0075TokyoJapan, Japan"
]
| []
| We report on a quantum description of the domain wall (DW) motion under a spin current. A bound magnon, which is the zero mode of DW, is found to play a dominant role in DW dynamics. The bound magnon acquires its inertia by the hard axis anisotropy and is a free particle even under the spin current. The full transfer of spin angular momentum from the spin current to DW via the bound magnon leads to the DW motion with the adiabatic velocity, decoupling of spin waves from DW, and no Doppler shift in spin waves. | null | [
"https://arxiv.org/pdf/0901.3066v1.pdf"
]
| 118,400,290 | 0901.3066 | 026869b853bfeb81404e30e4a211f962194e996d |
Role of Bound Magnon in Magnetic Domain Wall Motion
20 Jan 2009
Tae-Suk Kim
Institute for Materials Research
Tohoku Univ
980-8577SendaiJapan
Department of Physics
Pohang University of Science and Technology
790-784PohangKorea
J Ieda
Institute for Materials Research
Tohoku Univ
980-8577SendaiJapan
CREST
Science and Technology Agency (JST)
102-0075TokyoJapan, Japan
S Maekawa
Institute for Materials Research
Tohoku Univ
980-8577SendaiJapan
CREST
Science and Technology Agency (JST)
102-0075TokyoJapan, Japan
Role of Bound Magnon in Magnetic Domain Wall Motion
20 Jan 2009(Received January 20, 2009)
We report on a quantum description of the domain wall (DW) motion under a spin current. A bound magnon, which is the zero mode of DW, is found to play a dominant role in DW dynamics. The bound magnon acquires its inertia by the hard axis anisotropy and is a free particle even under the spin current. The full transfer of spin angular momentum from the spin current to DW via the bound magnon leads to the DW motion with the adiabatic velocity, decoupling of spin waves from DW, and no Doppler shift in spin waves.
Magnetic DW motion [1,2] has attracted much interest to both experiments and theories, owing to its potential device applications such as DW logic [3] and DW memory [4]. DW motion under magnetic fields is now well documented and understood [1] in terms of classical Landau-Lifshitz-Gilbert (LLG) equations. Recently many experimental groups [5,6,7,8,9,10] have observed the magnetic DW motion under spin currents. However, theoretical description [11,12,13,14,15] is very controversial over several issues. Even for a perfect ferromagnetic (FM) nanowire, the hard axis anisotropy was claimed [12] to induce the intrinsic pinning so that a DW does not move until a finite spin current is applied. But this intrinsic pinning is not consistent with translational symmetry of DW [14]. In the generalized LLG equation approach to the DW motion, the spin current gives rise to the so-called nonadiabatic [13,14,15] as well as adiabatic torques. The relative magnitude of nonadiabatic and Gilbert damping torques remains an unresolved problem.
In this paper we study the DW motion under spin currents, using the one dimensional s-d model Hamiltonian and the full quantum mechanical description of the DW motion. In the LLG equation approach, macroscopic magnetization is treated as a classical vector with fixed magnitude and two Euler angles. In this work we treat both electron and local spin systems quantum mechanically in order to address a microscopic mechanism for spin transfer from a spin current to a DW. We find that a bound magnon (with zero energy) in the DW plays a dominant role in absorbing angular momentum from the spin current and thereby in the DW motion. This bound magnon is the zero mode of magnetic soliton (DW) [16]. The mass of the bound magnon is derived quantum mechanically for the first time and is in agreement with the classical DW or Döring mass [17]. Our study shows that the bound magnon is a free particle even under a spin current and a DW can start to move under any finite spin currents for a perfect FM nanowire. The bound magnon-mediated spin transfer mechanism is compatible with the balance of the nonadiabatic and Gilbert damping torques. The role of the bound magnon is very similar to that of zero phonon mode in solitons of charge density wave state [18].
We consider the one-dimensional s-d model to study the DW motion which is driven by spin currents. Our model system consists of three parts:
H = H e +H S +H eS H e = −t iα c † iα c i+1α + H.c. − µ iα c † iα c iα ,(1)H S = −J i S i · S i+1 − A i ( S i ·ẑ) 2 + K i ( S i ·ŷ) 2 ,(2)H eS = −J H i S ci · S i .(3)
Conduction s electrons are described by H e and are spin polarized by the Hund coupling H eS to the ordered local spins. H S describes a system of local spins with an easy z-axis (A > 0) along the wire direction and a hard yaxis (K > 0). Ferromagnetically (J > 0) coupled local spins are assumed to have a transverse DW. DWs can be induced in the FM wire by ingenious experimental techniques. The mutually orthogonal unit vectors,x,ŷ andẑ, define the laboratory frame.
Representing the local spins in terms of two Euler angle fields, θ and φ, the DW structure can be derived by the energy minimization. The ordered local spins lie on the easy x-z plane (φ w = 0, π) and are rotated away from the easy axis by the angle θ w (z) = 2 cot −1 e −(z−q)/∆ . q is the DW position, ∆ = Ja 2 /2A is the DW width and a is the lattice spacing between two neighboring spins. The rotation angle θ w at each spin defines the quantization axis along which local spins are aligned.
Local spins may well fluctuate away from the ordered DW state. Small fluctuating spin fields can be represented by small fluctuating Euler angle fields. But instead, we adopt in this work the small fluctuating transverse spin fields, S ix and S iy , which are a more natural description of spin fluctuations. Using the local coordinate frames defined by the local spin quantization axis, the spin fields can be represented in terms of local transverse spin fields,
x · S i = S ix cos θ w (z i ) + S iz sin θ w (z i ),(4)y · S i = S iy ,(5)z · S i = S iz cos θ w (z i ) − S ix sin θ w (z i ).(6)
Here
S iz = S(S + 1) − S 2 ix − S 2 iy .
Expanding the transverse spin fluctuations away from the quantization axis, we find the magnon Hamiltonian H mag = H 0 + H 1 , keeping only up to quadratic terms in transverse spin components.
H 0 = i a=x,y J 2 (S ia − S i+1a ) 2 + A(cos 2 θ i − sin 2 θ i )S 2 ia ,(7)
and
H 1 = K i S 2 iy , where θ i = θ w (z i ).
The transverse spins in the continuum limit (S ix − S i+1x ≃ −a∂S x /∂z) satisfy the following equations of motion,
∂ t S x = H dw S y + 2KS S y ,(8)∂ t S y = −H dw S x .(9)
Here H dw = −JSa 2 ∂ 2 z + 2AS(cos 2 θ w − sin 2 θ w ) and acts as a Hamiltonian for normal modes of magnons.
Winter [19] found two types of normal modes for the transverse DW: the magnon bound to the DW and the extended spin waves. The static Schrödinger equation of H dw accommodates one bound state with energy ǫ = 0 and extended states of wave number k with eigen energy ǫ k = JSa 2 k 2 + 2AS [19].
ψ B (z) = a 2∆ sech z − q ∆ ,(10)ψ k (z) = a L ik∆ − tanh z−q ∆ √ 1 + k 2 ∆ 2 e ik(z−q) .(11)
ψ B (z) is the normal mode with the energy eigenvalue ǫ = 0. This normal mode is bound within the potential well formed in the DW. Due to localization in the DW, this magnon mode may be called as the bound magnon.
The wave function ψ B of the bound magnon is related to the DW structure function θ w by its spatial derivative ∂ z θ w , which implies a translation of the DW. This bound magnon is none other than the zero mode [16] of a DW, and tends to restore the translation symmetry of the FM nanowire. On the other hand, extended spin waves ψ k (z) of wave number k (L: length of FM wire) are plane waves with reduced amplitude in the DW, are characterized by the excitation energy gap and deforms the DW. Local spin fields can be represented with the Holstein-
Primakoff magnons: S i+ = 2S − b † i b i b i ≃ √ 2Sb i , S i− = b † i 2S − b † i b i ≃ √ 2Sb † i and S iz = S − b † i b i .
With identification of the normal modes, we can define the corresponding magnon operators:
b k = i ψ * k (z i )b i (spin wave operators) and b w = i ψ * B (z i )b i (a bound
magnon operator). Since ψ B and ψ k 's form a complete set of orthonormal wave functions for H dw , b w and b k 's exhaust all possible normal modes of H 0 and the inverse relation can be readily written down,
b i = k ψ k (z i )b k + ψ B (z i )b w .(12)
In terms of normal modes b w and b k 's, H 0 is already diagonalized:
H 0 = k ǫ k b † k b k .
Due to its zero energy, the bound magnon does not show up formally in H 0 .
Including the hard axis anisotropy (HAA), the magnon Hamiltonian can be diagonalized as H mag = H B + H sw . The bound magnon still remains as a normal mode with energy E B = 0.
H B = − 1 2 KS b w − b † w 2 ,(13)S x S y ∝ ψ B (z) 1 0 .(14)
Note that the bound magnon has no spin component along the hard axis, S y = 0, but instead, its spin lies on the easy plane. S x has the zero mode, which means that spins can rotate freely on the easy plane so that the DW can be shifted freely along the FM wire direction.
No zero mode in S y simply reflects no free rotation of spins away from the easy plane. On the other hand, the spin waves have spin excitations along two transverse directions with the increased excitation energy gap E k = ǫ k (ǫ k + 2KS) under the HAA.
H sw = k E k a † k a k ,(15)S x S y ∝ ψ k (z) u x (k) u y (k) .(16)
Here a k is the spin wave boson operator under the HAA and a linear combination of b k and b † −k , and u x/y corresponds to the amplitudes of transverse spins.
For the ordered local spins, the spin texture can be described by the magnetization unit vectors, m i 's, where m i = m(z i ) and m =ẑ cos θ +x sin θ cos φ +ŷ sin θ sin φ.
Under unitary transformation u i = exp − i 2 θ iφi · σ , which rotates the quantization axis of conduction electrons at site i from the z axis into m i (c iα → d iα = u * iβα c iβ ), i.e., u † i σ · m i u i = σ z , the Hund coupling H eS is diagonalized and results in spin polarized conduction bands. The kinetic term in H e introduces the currentspin coupling H cS or the Berry phase term [11],
H cS = v s S i (1 − cos θ i )∂ z φ i .(17)
Here v s = − aIs 2Se and I s is the spin current flowing in the system under electric field. I s is computed from thermal average ofÎ si = et
i α α[d † i+1α d iα −d † iα d i+1α ]
, which measures the spin polarized electric current from i to i+1. H cS can be written in a compact form as
H cS = v s P,(18)P = S i (1 − cos θ i )∂ z φ i .(19)
P is the DW linear momentum [21] or the generator of DW translation as will be shown below. Here angles are field variables. We now prove that P is the generator of DW translation or the linear momentum for DW. For this purpose we consider the DW spin texture |Ψ(
{z i }) >= i |S i ; m i >= i U i ( m i )|S i ;ẑ >, where U i ( m j ) = exp −iθ jφj · S i
rotates the orientation of S i from the z axis into m j . The DW state shifted to right by a lattice constant a can be written as |Ψ({z i − a}) >=
i |S i ; m i−1 >= i U i ( m i−1 )U † i ( m i )|Ψ({z i }) >. Writ- ing |Ψ({z i −a}) >= exp −i a P |Ψ({z i }) >, we can iden- tify P as P = i S i · [ m i (1 − cos θ i )∂ z φ i − m i × ∂ z m i ] . (20)
This quantum definition of P can also be obtained from Eq. (19) by allowing small fluctuating angle or spin fields as in the normal mode expansion. Angles or m i represent the DW solution. The first term is c-number ( S i · m i = S iz ≃ S), while the second is the quantum correction P and
P = − i ∂ z θ i S iy ,(21)
for the transverse DW in our case.
Under the finite spin current, θ w (z) now becomes dynamical and thus, the DW position q is time-dependent. Two coupled Eqs. (8) and (9), under the current-spin coupling (18), are modified by two effects: dynamic θ w and the spin current. In the rotating frame about the hard axis or the y axis, the equation of motion for an operator A is i ∂ t A = [A, H eff ], where the effective Hamiltonian H eff = H 0 + H 1 + H cS − θ S y has an additional contribution from rotating angle θ.
∂ t S x = H dw S y + 2KS S y − S(v s ∂ z θ + ∂ t θ), (22) ∂ t S y = −H dw S x .(23)
In general, the spin waves are coupled to the DW. Normal modes under spin current are decoupled from DW only when dq dt = v s , i.e., the DW moves with the adiabatic velocity v s . If the DW absorbs with full efficiency the spin angular momentum transferred from the spin current, there will be no Doppler shift [11,12,13,20] in the spin wave energy spectrum. If not, spin angular momentum from the spin current will be transferred to exciting spin waves.
The DW dynamics is determined by the bound magnon Hamiltonian, H B and H cS . Since ∂ z θ ∝ ψ B is finite only near the DW, the main contribution to P comes from spins in the DW and the number of contributing spins is roughly ∆/a. Furthermore, owing to H dw sin θ w = 0, we have the identity [P, H mag ] = 0 such that P is the constant of motion. This is a simple mathematical manifestation of translational symmetry for a DW in an infinite FM nanowire. P can be represented in terms of the bound magnon as
P = −i S a∆ (b † w − b w ).(24)
H B can be interpreted as the kinetic Hamiltonian of the bound magnon by noting that H B can be written in terms of P as
H B = K∆a 2 2 P 2 = P 2 2M dw ,(25)
where the bound magnon mass is defined as M dw ≡ 2 K∆a . The DW or Döring mass defined in the classical approach [17] is none other than the mass of the bound magnon or the zero mode in a ferromagnetic DW.
The effect of the hard axis anisotropy K is threefold. For extended spin waves, their energy gap is enhanced such that they become much harder to excite. The hard axis anisotropy confines the bound magnon to have spin components only on the easy plane, but no component along the hard axis. The bound magnon acquires its inertia due to the hard axis anisotropy.
Dropping c-number from P, the current-spin coupling Eq. (18) becomes
H cS = v s P,(26)
which is the same for both φ w = 0, π. Note that P (φ w = π) = −P (φ w = 0). Under spin currents, the system retains a translational symmetry. The action of H cS on the DW can be most easily understood in terms of the Schrödinger equation,
i ∂ t |Ψ({z i }, t) > = (H B + H cS )|Ψ({z i }, t) > . (27)
Denoting the DW state as |Ψ(
{z i }) > when v s = 0, we find that |Ψ({z i }, t) >= exp (−iv s tP/ ) |Ψ({z i }) >= |Ψ({z i − v s t}) >.
The DW motion with velocity v s is induced by the spin current. The intrinsic pinning was claimed [12] to be induced by HAA of a perfect FM nanowire. Spin currents rotate the local spins away from the easy plane and the HAA field acts as a blockade [12] to the DW motion and generates the intrinsic pinning. The quantum approach clearly shows the absence of the intrinsic pinning and is consistent with the translation symmetry of DWs under spin currents. The DW absorbs the spin angular momentum from the spin current via the bound magnon, and thereby avoids the tilting of local spins away from the easy plane.
There is (no) translation symmetry for DW in FM wires under spin currents (magnetic fields). The energy damping torque like the Gilbert type is prerequisite for the steady domain wall motion under magnetic fields. Energy dissipation via damping uses up the Zeeman energy and sets the DW in motion. On the other hand, the DW under spin currents absorbs spin angular momentum via the bound magnon from conduction electrons and can move even without damping.
Our theory is based on the perfect FM nanowires without spin damping. With the fully efficient absorption of spin angular momentum, the spin current sets the DW in motion with the adiabatic velocity v s , the spin waves are decoupled from the DW motion, and no Doppler effect is expected in the spin wave energy spectrum. Note that the spin wave energy shift under spin currents was observed [22] in FM nanowires with uniform magnetization. According to the phenomenological LLG equation [13,14,15], the DW velocity is modified from the adiabatic value v s by the so-called α [23] and β [13] damping torques. In this case more careful study [24] is required for elucidation of the Doppler effect in the spin wave energy spectrum.
In summary we studied the domain wall motion under spin current, based on the s-d model Hamiltonian. We found that the bound magnon plays an important role in the domain wall dynamics. Since the bound magnon is localized to the domain wall and has zero excitation energy, a spin current, without energy cost, transfers spin angular momentum to the domain wall via the bound magnon. Furthermore the hard axis anisotropy confines the bound magnon on the easy plane so that local spins can absorb spin angular momentum from the spin current and rotate about the hard axis without tilting away from easy plane. The bound magnon acquires its inertia due to the hard axis anisotropy and remains a free particle even under spin currents. With the full transfer of spin angular momentum from the spin current to the DW, spin waves are decoupled from the DW motion and no Doppler shift is expected for the spin waves. The bound magnon-mediated spin transfer mechanism leads to the free motion of bound magnon or domain wall under spin currents.
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| []
|
[
"Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization *",
"Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization *"
]
| [
"Yiming Wang \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n",
"Zhidan Li \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n",
"Qiang Han \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n\nBeijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices\nRenmin University of China\n100872BeijingChina\n"
]
| [
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices\nRenmin University of China\n100872BeijingChina"
]
| []
| The fermionic and bosonic zero modes of the 1D interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su-Schrieffer-Heeger model, in which we show that these two types of zero modes can coexist in certain region of its phase diagram.The Kitaev chain has stimulated intense research interest in the community of condensed matter physics since it was first proposed in the pioneer work of Kitaev. [1] As a model of one-dimensional (1D) topologicalwave superconductor, the Kitaev chain hosts two unpaired Majorana zero modes (MZMs) nonlocally distributed at the two ends of the chain. [1-4] These MZMs are exotic quasiparticles which are their own antiparticles and topological superconductors possessing well-separated MZMs are potential platforms to realize fault tolerant topological quantum computation due to their non-Abelian statistics and immunity to local perturbations. [5-7] Signatures of observing the MZMs have been reported by several experimental groups [8-11] based on theoretical proposals of realizing effective -wave pairing in the spin-orbit coupled semiconductor nanowires in proximity to -wave superconductors. [12-15]To gain deeper insight into the MZMs beyond the single-particle picture, the interacting Kitaev chain has been studied theoretically.[16][17][18][19][20][21]The variation of the zero-energy peak in the local density of states was examined numerically [17] as a reflection of the effect of interaction on the MZMs. Futhermore, the interacting Kitaev chain at the symmetric point was shown to be exactly solvable and exact solutions of all the eigenstates were given.[21] By investigating the two degenerate ground states, it was pointed out that there are fermionic or * | 10.1088/1674-1056/27/6/067101 | [
"https://arxiv.org/pdf/2102.05247v1.pdf"
]
| 125,245,679 | 2102.05247 | 06d4dc0a6710d69878a48430bee3290309999780 |
Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization *
February 11, 2021 10 Feb 2021
Yiming Wang
Department of Physics
Renmin University of China
100872BeijingChina
Zhidan Li
Department of Physics
Renmin University of China
100872BeijingChina
Qiang Han
Department of Physics
Renmin University of China
100872BeijingChina
Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices
Renmin University of China
100872BeijingChina
Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization *
February 11, 2021 10 Feb 2021Majorana zero modesbosonic zero modesinteracting Kitaev chain
The fermionic and bosonic zero modes of the 1D interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su-Schrieffer-Heeger model, in which we show that these two types of zero modes can coexist in certain region of its phase diagram.The Kitaev chain has stimulated intense research interest in the community of condensed matter physics since it was first proposed in the pioneer work of Kitaev. [1] As a model of one-dimensional (1D) topologicalwave superconductor, the Kitaev chain hosts two unpaired Majorana zero modes (MZMs) nonlocally distributed at the two ends of the chain. [1-4] These MZMs are exotic quasiparticles which are their own antiparticles and topological superconductors possessing well-separated MZMs are potential platforms to realize fault tolerant topological quantum computation due to their non-Abelian statistics and immunity to local perturbations. [5-7] Signatures of observing the MZMs have been reported by several experimental groups [8-11] based on theoretical proposals of realizing effective -wave pairing in the spin-orbit coupled semiconductor nanowires in proximity to -wave superconductors. [12-15]To gain deeper insight into the MZMs beyond the single-particle picture, the interacting Kitaev chain has been studied theoretically.[16][17][18][19][20][21]The variation of the zero-energy peak in the local density of states was examined numerically [17] as a reflection of the effect of interaction on the MZMs. Futhermore, the interacting Kitaev chain at the symmetric point was shown to be exactly solvable and exact solutions of all the eigenstates were given.[21] By investigating the two degenerate ground states, it was pointed out that there are fermionic or *
bosonic zero modes in the topologically nontrivial or trivial phase. [21] However, the exact structures of the zero modes have not been given explicitly for the interacting Kitaev chain even at the symmetric point, although the MZMs in the presence of interaction are expected to be adiabatically connected to the noninteracting ones. [1,18] In this paper we will present the analytic expressions of the MZMs as well as the bosonic zero modes (BZMs) in the interacting (dimerized) Kitaev chain at the symmetric point. The main purpose of this paper is to give the many-body generalization of MZM. With the help of the explicit forms of the zero modes in the interacting (dimerized) Kitaev chain, the topological phase diagram are also given.
Before giving the exact results, we first discuss some general restrictions on the boundary zero modes in the fermion system. [3,18,19] The many-body zero modeˆof the interacting Kitaev chain is a Hermitian operator which commutes with the system Hamiltonian, [22] [ˆ,ˆ] = 0,ˆ † =ˆ.
(1)
The MZM (BZM) is of the fermionic (bosonic) type and therefore anticommutes (commutes) with the parity operator of the system,
{(−)^,ˆF} = 0, [(−)^,ˆB] = 0,(2)
whereˆis the operator of total fermion number. As a physical zero mode,ˆacts on a normalizable state | ⟩ to get another normalizable stateˆ| ⟩, which demands thatˆis unitary and thus satisfies the normalization condition,ˆ2
F( ) = 1.(3)
In addition, as a boundary mode, the components of the MZM or BZM is exponentially small with the distance from the boundary.
Next we address the zero modes in the interacting Kitaev chain with open boundary condition, of which the Hamiltonian is written as,ˆ=ˆK +ˆI.
The system Hamiltonian is composed of a single-particle partˆK and a fermion-fermion interaction partˆI.
K denotes the tight-binding Hamiltonian of 1D Kitaev chain, which is written aŝ
K = − −1 ∑︁ =1 ( † +1 + ∆ † † +1 + ℎ. .) − ∑︁ † ,(5)
with the nearest-neighbor hopping integral, ∆ the -wave superconducting pairing potential, and the chemical potential. denotes the length of the chain. and † are the fermion annihilation and creation operators, respectively. I is expressed asˆI
= −1 ∑︁ =1 (︂ † − 1 2 )︂ (︂ † +1 +1 − 1 2 )︂ ,(6)
where is the interaction between nearest-neighbor fermions.
In the following, we focus on the symmetric case where = ∆ and = 0, and the thermodynamic limit → ∞ is taken. Without loss of generality we set > 0. For later convenience, we rewrite Eq. (4) in the Majorana basis and into a dimensionless form,
ℎ =ˆ/ =ĥ 0 +ĥ 1 ,(7)
2 where = /4 is a dimensionless parameter describing the relative strength of the interaction. 7) as a small quantity and expandˆF into power series of as follows,
ℎ 0 = −1 ∑︁ =1 +1 ,(8)ℎ 1 = − −1 ∑︁ =1 +1 +1 .(9)F = F ∑︁ =0 ,(10)
where F is the normalization factor and the coefficients 's are operators to be determined. Substituting
Eq. (10) into [ĥ,ˆF] = 0 and comparing the coefficient of , we find that satisfies the following recurrence relation,
[ĥ 0 , ] + [ĥ 1 , −1 ] = 0.(11)
Starting from the leading term 0 = 1 , one can obtain successively and the general expression is,
= 2 +1 ∏︁ =1 2 −1 2 .(12)
's are many-body Majorana fermion operators which satisfy † = , { , ′ } = 2 , ′ . Substituting Eq. (12) into Eq. (10), we have the analytic expression of the zero mode,
F = F ∑︁ =0 2 +1 ∏︁ =1 2 −1 2 ,(13)
which starts from the left-most Majorana fermion 1 and decays exponentially with the distance on condition that | | < 1. Similarly, one can obtain the right-boundary zero mode starting from ,
F = F ∑︁ =0 −2 ∏︁ =1 −2 +2 −2 +1 .(14)
F andˆF satisfy Eqs.
B = B ∑︁ =0 − ,(15)
where B is the normalization factor. Substituting Eq. (15) into [ĥ,ˆB] = 0 and comparing the coefficient of − , we find that satisfies the following recurrence relation,
[ĥ 0 , ] + [ĥ 1 , +1 ] = 0.(16)
Starting from the first term 0 = 1 1 , one can obtain successively and the general expression is,
= 2 +1 2 +1 ∏︁ =1 2 −1 2 .(17)
Similar to , 's are also many-body Majorana fermion operators which satisfy † = , { , ′ } = 2 , ′ .
Substituting Eq. (17) into Eq. (15), we have the expression for the left-boundary zero mode,
B = B ∑︁ =0 − 2 +1 2 +1 ∏︁ =1 2 −1 2 .(18)
Likewise, we obtain the right-boundary zero mode starting from ,
B = B ∑︁ =0 − −2 −2 ∏︁ =1 −2 +1 −2 +2 .(19)
Eqs. (18) and (19) We next discuss the coexistence of both the MZMs and the BMZs in a hybrid system which comprises of the interacting Kitaev chain model and the Su-Schrieffer-Heeger (SSH) model [23] as shown in Fig.2, which is a dimerized generalization of the interacting Kitaev chain studied above. The noninteracting version of this model has been studied extensively in the literature. [24][25][26][27][28][29][30][31] This so-called interacting Kitaev-SSH chain consists of unit cells each of which hosts two inequivalent lattice site, and . The model Hamiltonian is written as,
= K-SSH + I ,(20)
where
K-SSH = − −1 ∑︁ =1 ( 1 † + 2 † +1 + ℎ. .), − −1 ∑︁ =1 (∆ 1 † † + ∆ 2 † † +1 + ℎ. .) − ∑︁ =1 ( 1 † + 2 † )(21)
and
I = 1 ∑︁ =1 (︂ † − 1 2 )︂ (︂ † − 1 2 )︂ , + 2 −1 ∑︁ =1 (︂ † − 1 2 )︂ (︂ † +1 +1 − 1 2 )︂ .(22)
Here is the index of unit cell. , are associated with the sublattice while , with . In the Majorana representation,
we have
K-SSH = ∑︁ ( 1 + 2 +1 ),(23)
and
I = − ∑︁ ( 1 + 2 +1 +1 ).(24)
Applying the forgoing method of perturbation expansion, we obtain two types of zero modes. The left-boundary MZM is written asˆF
= F ∑︁ =0 (︂ 1 4 2 )︂ +1 ∏︁ =1 ,(25)
which emerges under the condition that | 1 | < 4| 2 |, and the left-boundary BZM iŝ
B = B ∑︁ =0 (︂ 4 1 2 )︂ +1 +1 ∏︁ =1 .(26)
whose existence is guaranteed by the condition that 4| 1 | < | 2 |. The corresponding right-boundary zero modes can be obtained similarly (not shown here). One can readily check that the when 1 = 2 = , 1 = 2 = , i.e.
the dimerization is absent, the former equations (13) and (18) are recovered from Eqs. (25) and (26), respectively.
From the explicit forms of the zero modes, we derive the phase diagram of the interacting Kitaev-SSH chain as illustrated in Fig. 3. We find four phases:
(i) | 1 | > 4| 2 |, 4| 1 | > | 2 |,
where no boundary zero modes exist;
(ii) | 1 | < 4| 2 |, 4| 1 | > | 2 |,
where there is one MZM at each end of the chain and no BZM; (iii) | 1 | > 4| 2 |,
4| 1 | < | 2 |,
where there is one BZM at each end of the chain and no MZM; (iv)
| 1 | < 4| 2 |, 4| 1 | < | 2 |,
where the MZM and the BZM coexist at each end of the chain. In this coexistence region, considering that {ˆF,ˆB} = 0, we can construct another MZM bŷ
′ F =ˆFˆB,(27)
which fulfills Eqs. (1), (2), and (3) and anticommutates withˆF. Therefore in the coexistence region, there are two MZMs at each end of the chain as illustrated in Fig. 3. Furthermore one can construct a complex fermion by pairing these two MZMs at the same end,ˆ=ˆF +ˆ′ F , and forms a many-body SSH-like [24] zero mode.
In summary, the many-body structures of the MZMs and BZMs in the symmetric interacting (dimerized)
Kitaev chain have been investigated in this paper. From the explicit forms of the zero modes, the many-body
MZMs of the interacting model are adiabatically connected to the one-body MZMs of the noninteracting one.
For the symmetric interacting Kitaev chain without dimerization, the MZMs and BZMs are found in different regions of the phase diagram. The dimerized generalization of the model has four phases depending on the numbers of zero modes and we find that both types of zero modes can coexist with each other in certain region of its phase diagram.
and = − ( − † ) are Majorana fermion operators which satisfy † = , † = , and the anti-commutation relations { , } = 2 , , { , } = 2 , and { , } = 0. When = 0, it was shown by Kitaev [1] that there is an exact MZM at each end of the chain, namely [ĥ 0 , 1 ] = [ĥ 0 , ] = 0, indicating that the MZMs are composed of one-body Majorana fermions. In the presence of interaction, manybody contributions are involved and higher order terms occur in the expression ofˆF. For small , we can treat ℎ 1 in Eq. (
( 1 )
1,(2) and the anti-commutation relation {ˆF,ˆF} = 0. To guaranteeˆ( condition Eq. (3) must be satisfied. From Eq. (3) we obtain 2 < 1, or equivalently | | < 4 .The normalization factor is F = √ 1 − 2 . Therefore the many-body MZM emerges in the topological region | | < 4 .[21] In addition, according to Eqs.(13) and(14)the many-body MZMs are adiabatically connected to the one-body MZMs 1 and as → 0. In the topologically trivial region 2 > 1, on the other hand, the many-body MZMs do not exist. On the contrary we find BZMs localized at ends of the chain. Note that when = 0, i.e. → ∞, there are two bosonic boundary zero modes 1 1 and satisfying [ĥ 1 , 1 1 ] = [ĥ 1 , ] = 0. For large but finite , one can treatĥ 0 in Eq. (7) as perturbation and expandˆB as power series of −1 ,
indicate thatˆB andˆB are BZMs and in additon they satisfy the commutation relation [ˆB,ˆB] = 0. Furthermore from Eq. (3),ˆ( ) Bis normalizable under the condition 2 > 1, which shows that the two BZMs exist only in the topologically trivial region | | > 4 . The corresponding normalization factor isB = √ 1 − −2 .
Figure 1 :Figure 2 :
12(Color online) Phase diagram of the interacting Kitaev chain at the symmetric point with = ∆ and = 0, where the topologically nontrivial (trivial) phase in the region | | < 4 (| | > 4 ) hosts one MZM (BZM) at each end of the chain. The corresponding conditions for the emergence of different types of zero modes can be employed to depict the phase diagram of the symmetric interacting Kitaev chain as shown in Fig. 1 which is identical with that obtained by exact diagonalization of the system Hamiltonian. [21] Illustration of the interacting Kitaev-SSH chain model. Dashed squares denote the unit cells and green solid (empty) circles denote sites on sublattice C (D). See the main text for more details.
( † ) and ( † ) denote the annihilation (creation) operators of fermion on the site of the sublattice ( ), respectively. The intra(inter)-unit-cell hopping integral, pairing potential, and fermion-fermion interaction are denoted by 1(2) , ∆ 1(2) , and 1(2) , respectively. Hereafter, we focus on the symmetric point with 1 = ∆ 1 , 2 = ∆ 2 and 1 = 2 = 0.
Figure 3 :
3Phase diagram of the symmetric interacting Kitaev-SSH chain model with 1 = ∆ 1 , 2 = ∆ 2 and 1 = 2 = 0. The Majorana fermion operators are defined as = + † , = − ( − † ), = + † , = − ( − † ).
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| []
|
[
"ON THE HAMILTONIAN STRUCTURE OF HIROTA-KIMURA DISCRETIZATION OF THE EULER TOP",
"ON THE HAMILTONIAN STRUCTURE OF HIROTA-KIMURA DISCRETIZATION OF THE EULER TOP"
]
| [
"Matteo Petrera \nZentrum Mathematik\nTechnische Universität München\nBoltzmannstr. 3D-85747Garching bei MünchenGermany\n",
"Yuri B Suris \nZentrum Mathematik\nTechnische Universität München\nBoltzmannstr. 3D-85747Garching bei MünchenGermany\n"
]
| [
"Zentrum Mathematik\nTechnische Universität München\nBoltzmannstr. 3D-85747Garching bei MünchenGermany",
"Zentrum Mathematik\nTechnische Universität München\nBoltzmannstr. 3D-85747Garching bei MünchenGermany"
]
| []
| This paper deals with a remarkable integrable discretization of the so(3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi-Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville-Arnold sense. † [email protected]. ⋄ [email protected]. | 10.1002/mana.200711162 | [
"https://arxiv.org/pdf/0707.4382v1.pdf"
]
| 17,781,610 | 0707.4382 | 3e0d2d946f3cf8f77bb26b26b3b3747ede8a179a |
ON THE HAMILTONIAN STRUCTURE OF HIROTA-KIMURA DISCRETIZATION OF THE EULER TOP
30 Jul 2007
Matteo Petrera
Zentrum Mathematik
Technische Universität München
Boltzmannstr. 3D-85747Garching bei MünchenGermany
Yuri B Suris
Zentrum Mathematik
Technische Universität München
Boltzmannstr. 3D-85747Garching bei MünchenGermany
ON THE HAMILTONIAN STRUCTURE OF HIROTA-KIMURA DISCRETIZATION OF THE EULER TOP
30 Jul 2007arXiv:0707.4382v1 [math-ph]
This paper deals with a remarkable integrable discretization of the so(3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi-Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville-Arnold sense. † [email protected]. ⋄ [email protected].
Introduction
This paper deals with a remarkable integrable discretization for one of the basic integrable systems, the three-dimensional Euler top, which describes the motion of the free rigid body with a fixed point. Equations of motion of the Euler top in the body frame readẋ 1 = α 1 x 2 x 3 ,ẋ 2 = α 2 x 3 x 1 ,ẋ 3 = α 3 x 1 x 2 .
(1) where x = (x 1 , x 2 , x 3 ) ∈ R 3 , and the real coefficients α i are parameters of the system. We will denote the vector of parameters by α = (α 1 , α 2 , α 3 ) ∈ R 3 . Throughout this paper we will use an abbreviated notation, according to which (ijk) stands for any cyclic permutation of (123). Thus, system (1) takes with this notation the forṁ
x i = α i x j x k .(2)
The coordinates x i stand either for the angular velocities Ω i , in which case the coefficients α i are given by
α i = I j − I k I i ,(3)
or otherwise for the angular momenta M i , in which case the coefficients α i are given by
α i = 1 I k − 1 I j .(4)
Here I i are the principal moments of inertia of the body. The relation between the two formulations is given by M i = I i Ω i . Integrability features of the Euler top include [1,10,11]: a bi-Hamiltonian structure, i.e. the existence of two compatible invariant Poisson structures on the phase space; two independent integrals of motion, which are in involution with respect to any of the invariant Poisson brackets; a Lax representation; explicit solutions in terms of elliptic functions. For the reader's convenience, some of these features are briefly exposed in Sect. 2. The general problem of integrable discretization of integrable systems is dealt with in the monograph [11]. One finds there also a detailed exposition of an integrable discretization of the Euler top, due to Veselov and Moser [8,12]. The basic feature of this discretization is that it comes from a discrete Lagrangian formulation on the Lie group SO(3). Upon a reduction to so(3) * , it produces a correspondence, i.e. a multi-valued map, each branch of which is Poisson with respect to the Lie-Poisson bracket on so(3) * , like the original phase flow. Moreover, it shares the integrals of motion and the Lax representation with the original continuous time flow. This Lax representation is related to matrix factorizations.
A class of discretizations of the Euler top sharing the integrals of motion with the continuous system has been introduced and studied in [2]. These discretizations are characterized by the equations of motioñ
x i − x i = γα i (x j + x j )(x k + x k ).(5)
Here and below tilde denotes the shift t → t + ǫ in the discrete time ǫZ, where ǫ is a (small) time step. In other words, in Eq. (5) (and in similar situation throughout the paper) we consider x i as functions on ǫZ, and we write x i for x i (nǫ) andx i for x i (nǫ + ǫ), n ∈ Z. In Eq. (5), it is assumed that γ ∼ ǫ/4 is some real-valued function on the phase space. Then the map (5) approximates, for small ǫ, the time ǫ shift along the trajectories of the continuous flow (1). In [2] the functions γ have been characterized for which the map (
x 1 , x 2 , x 3 ) → (x 1 ,x 2 ,x 3 ) defined by Eq. (5)
shares the invariant Poisson structure with the continuous system. In particular, the function γ for the Veselov-Moser discretization has been determined. A further integrable discretization of the Euler top belonging to the family (5) was proposed in [4]. Interestingly, the simplest choice γ = ǫ/4 leads to a a map which does not preserve the original Poisson structure. Discretizations (5) share the Lax matrix with the continuous time Euler top. They are implicit, since these formulas represent a system of algebraic (nonlinear) equations for (x 1 ,x 2 ,x 3 ) which does not possess a simple closed-form solution.
The present paper deals with the following beautiful explicit discretization of equations of motion (2), introduced by Hirota and Kimura [5]:
x i − x i = δ i (x j x k + x jxk ).(6)
Here one can take
δ i = ǫα i 2 ,(7)
we will adopt this choice for the vector of parameters δ = (δ 1 , δ 2 , δ 3 ) ∈ R 3 throughout the paper. This discretization is explicit, since the algebraic equations (6) are linear with respect to (x 1 ,x 2 ,x 3 ), and thus they can be solved in a closed form (see Sect. 3 for further details). Hirota and Kimura presented some of the integrability attributes for their discretization: two independent integrals of motion and a solution in terms of elliptic functions. Other attributes, like the Hamiltonian formulation and the Lax representation, has not been mentioned by them. The main goal of the present paper is to fill the first of these two gaps by providing a bi-Hamiltonian structure for the Hirota-Kimura discretization, and thereby to prove its integrability in the standard Liouville-Arnold sense.
We found it worthwhile to give also a simplified and streamlined presentation of the results found in [5]. Indeed, the discretization of the Lagrange top given by Kimura and Hirota later in [7], as well as some preliminary results by Ratiu [9], indicate that the map (6) might be just a tip of an iceberg, a huge collection of discretizations of integrable systems of classical mechanics. We plan to develop this topic in a series of upcoming publications.
It is an established fact that many of the most important integrable systems can be found in the classical literature on differential geometry. Usually this refers to solitonic partial differential equations, like the sine-Gordon equation, but it turns out to be true also for the integrable map (6): a 1951 paper in "Mathematische Nachrichten" by H. Jonas is devoted to a birational map (x, y, z) → (x,ỹ,z) given by
x +x + yz + zỹ = 0, y +ỹ + zx + xz = 0, z +z + xỹ + yx = 0,(8)
which differs only unessentially from (6). The map (8) has an origin in the spherical geometry, (x, y, z) and (x,ỹ,z) being the cosines of the side lengths of two spherical triangles with complementary angles. Jonas' results include integrals of the map (8) and its solution in terms of elliptic functions. Thus, [6] seems to be one of the earliest precursors of the theory of integrable maps.
Euler top
The aim of this Section is to recall some of the main features of the integrable continuoustime Hamiltonian flow (2).
Proposition 1. Let β = (β 1 , β 2 , β 3 ) ∈ R 3
be a constant vector. A quadratic function
H (β) = 1 2 (β 1 x 2 1 + β 2 x 2 2 + β 3 x 2 3 )(9)
is an integral of motion for (2) if and only if β ⊥ α, i.e. if β 1 α 1 + β 2 α 2 + β 3 α 3 = 0.
Proof: An easy computation based on Eq.
(2) shows that
d dt H (β) = (β 1 α 1 + β 2 α 2 + β 3 α 3 )x 1 x 2 x 3 .
Since the orthogonal complement of the vector α is two-dimensional, there are two independent integrals of motion. It is sometimes convenient to use a special basis of the orthogonal complement just mentioned, consisting of vectors with one vanishing component.
Corollary 1. The three quadratic functions
G i = 1 2 (α j x 2 k − α k x 2 j )(10)
are integrals of motion for (2). Of course, only two of them are (linearly) independent since
α 1 G 1 + α 2 G 2 + α 3 G 3 = 0.
Notice that any function H (β) is a linear combination of the G i 's:
α i H (β) = β j G k − β k G j .
In the angular velocities formulation, a basis of the orthogonal complement α ⊥ can be chosen consisting of β (1) = (I 1 , I 2 , I 3 ) and β (2) = (I 2 1 , I 2 2 , I 2 3 ). In the angular momenta formulation, a basis of α ⊥ consists of β (1) = (1/I 1 , 1/I 2 , 1/I 3 ) and β (2) = (1, 1, 1).
Proposition 2. Let β ⊥ α, and let γ = (γ 1 , γ 2 , γ 3 ) ∈ R 3 satisfy α i = β j γ k − β k γ j ,(11)
so that γ ⊥ α. Then the system (2) is Hamiltonian with the Hamilton function H (β) with respect to the Poisson bracket
{x i , x j } (γ) = γ k x k .(12)
Proof: A direct verification:
{x i , H (β) } (γ) = β j x j {x i , x j } (γ) + β k x k {x i , x k } (γ) = = (β j γ k − β k γ j )x j x k = α i x j x k .
Propositions 1 and 2 show the bi-Hamiltonian property of the Euler top. Referring to the angular velocities the system has two Hamiltonian formulations:
H = 1 2 (I 1 Ω 2 1 + I 2 Ω 2 2 + I 3 Ω 2 3 ) with {Ω i , Ω j } = I k I i I j Ω k , and H = 1 2 (I 2 1 Ω 2 1 + I 2 2 Ω 2 2 + I 2 3 Ω 2 3 ) with {Ω i , Ω j } = 1 I i I j Ω k .
Referring to the angular momenta, the system also has two Hamiltonian formulations:
H = 1 2 M 2 1 I 1 + M 2 2 I 2 + M 2 3 I 3 with {M i , M j } = M k , and H = 1 2 (M 2 1 + M 2 2 + M 2 3 ) with {M i , M j } = 1 I k M k .
Hirota-Kimura discretization of the Euler top
We now turn to the study of the map (6). Though the vector of parameters δ is arbitrary, we will think of it as related to α as in Eq. (7).
3.1. Integrals of motion. An explicit form of this map can be easily obtained. Considering Eq. (6) as a system of linear equations for the updated variablesx i , one finds immediately its solution:
x 1 x 2 x 3 = 1 −δ 1 x 3 −δ 1 x 2 −δ 2 x 3 1 −δ 2 x 1 −δ 3 x 2 −δ 3 x 1 1 −1 x 1 x 2 x 3 .
Note also that, considering Eq. (6) as a system of linear equations forx i , one finds the alternative formula
x 1 x 2 x 3 = 1 δ 1x3 δ 1x2 δ 2x3 1 δ 2x1 δ 3x2 δ 3x1 1 x 1 x 2 x 3 .
We will use the notation
A(x, δ) = 1 −δ 1 x 3 −δ 1 x 2 −δ 2 x 3 1 −δ 2 x 1 −δ 3 x 2 −δ 3 x 1 1 ,
so that the equations of the map can be written as
x = A −1 (x, δ)x = A(x, −δ)x.
Proposition 3. The quantities
F i = 1 − δ k δ i x 2 j 1 − δ i δ j x 2 k ,(13)
are integrals of motion for the map (6). Of course, there are only two independent integrals since F 1 F 2 F 3 = 1.
Proof: EquationF i = F i can be re-written as
(1 − δ k δ ix 2 j )(1 − δ i δ j x 2 k ) = (1 − δ i δ jx 2 k )(1 − δ k δ i x 2 j ),
which is equivalent to
δ j (x 2 k − x 2 k ) − δ k (x 2 j − x 2 j ) = δ i δ j δ k (x 2 j x 2 k − x 2 jx 2 k ),
that is, to
δ j (x k + x k )(x k − x k ) − δ k (x j + x j )(x j − x j ) = δ i δ j δ k (x j x k + x jxk )(x k x j − x kxj ).
Using the equations of motion (6) on both sides of the latter formula, we arrive at
(x k + x k )(x i x j + x ixj ) − (x j + x j )(x k x i + x kxi ) = (x i − x i )(x k x j − x kxj ),
which is an algebraic identity.
The relation between F i 's and the integrals of the continuous time Euler top is straightforward:
F i = 1 + ǫ 2 α i 4 G i + O(ǫ 4 ).
Corollary 2. Let β ⊥ δ. Then the following three functions are integrals of motion for the map (6):
H (β) i = H (β) 1 − δ j δ k x 2 i ,
where the common numerator H (β) is an integral of the continuous time Euler top given in Eq. (9).
Proof: We show that H (β) i can be expressed in terms of the F i 's given in Eq. (13):
δ i H (β) i = −(β j δ j + β k δ k )x 2 i + β j δ i x 2 j + β k δ i x 2 k 1 − δ j δ k x 2 i = β j (δ i x 2 j − δ j x 2 i ) + β k (δ i x 2 k − δ k x 2 i ) 1 − δ j δ k x 2 i = β j δ k 1 − 1 − δ k δ i x 2 j 1 − δ j δ k x 2 i + β k δ j 1 − 1 − δ 1 δ j x 2 k 1 − δ j δ k x 2 i = β j δ k 1 − 1 F k + β k δ j (1 − F j ).
3.2. Invariant volume form. Next, we establish the existence of an invariant measure for the map (6). Let us first give the following useful Lemma.
Lemma 1.
For the map (6) the following holds:
x i − δ ixjxk 1 − δ j δ kx 2 i = x i + δ i x j x k 1 − δ j δ k x 2 i ,(14)
and, as a corollary,
(x i − δ ixjxk ) 2 (1 − δ i δ kx 2 j )(1 − δ i δ jx 2 k ) = (x i + δ i x j x k ) 2 (1 − δ i δ k x 2 j )(1 − δ i δ j x 2 k )
.
(15)
Proof: We prove, for instance, Eq. (14). It is equivalent to
(x i − δ ixjxk )(1 − δ j δ k x 2 i ) = (x i + δ i x j x k )(1 − δ j δ kx 2 i ), or tox i − x i − δ ixjxk − δ i x j x k = −δ j δ k x ixi (x i − x i ) − δ i δ j δ k (x 2
ix jxk +x 2 i x j x k ). Upon using equations of motion (6) on both sides of the latter formula, we find that it is equivalent to
(x j − x j )(x k − x k ) = δ j δ k (x ixj +x i x j )(x ixk +x i x k )
, which is a direct consequence of Eq. (6). Now we are in the position to prove the following claim.
det ∂x ∂x = φ(x) φ(x) ,
where φ(x) is any of the functions
φ(x) = (1 − δ i δ j x 2 k )(1 − δ j δ k x 2 i ), (16) φ(x) = (1 − δ i δ j x 2 k ) 2 .(17)
(The ratio of any two different functions φ(x) is an integral of motion for (6) due to Proposition 3). Equivalently, the three-form
Ω = 1 φ(x) dx 1 ∧ dx 2 ∧ dx 3(18)
is invariant under the map (6).
Proof: First of all, we derive the following formula for the Jacobian of the map (6):
det ∂x ∂x = det A(x, −δ) det A(x, δ) .(19)
Indeed, differentiating Eq. (6) with respect to x 1 , x 2 , x 3 , one obtains the columns of the matrix equation
1 −δ 1 x 3 −δ 1 x 2 −δ 2 x 3 1 −δ 2 x 1 −δ 3 x 2 −δ 3 x 1 1 ∂x ∂x = 1 δ 1x3 δ 1x2 δ 2x3 1 δ 2x1 δ 3x2 δ 3x1 1 .
Computing determinants leads to Eq. (19), which can be written in length as Define ω to be the dual n-vector field to Ω such that ω Ω = 1. Here the symbol denotes the contraction between multi-vector fields and forms. If I 1 , . . . , I n−2 are integrals of f with dI 1 ∧ · · · ∧ dI n−2 = 0, then the bi-vector field σ = ω dI 1 · · · dI n−2 is an invariant Poisson structure for f . If J 1 , . . . , J n−2 is another set of independent integrals and τ = ω dJ 1 · · · dJ n−2 is the corresponding Poisson structure, then σ and τ are compatible, i.e., for any constants a, b, the bi-vector field aσ + bτ is a Poisson structure, again.
det ∂x ∂x = 1 − δ j δ kx 2 i − δ i δ kx 2 j − δ i δ jx 2 k + 2δ i δ j δ kxixjxk 1 − δ j δ k x 2 i − δ i δ k x 2 j − δ i δ j x 2 k − 2δ i δ j δ k x i x j x k = (1 − δ i δ kx 2 j )(1 − δ i δ jx 2 k ) − δ j δ k (x i − δ ixjxk ) 2 (1 − δ i δ k x 2 j )(1 − δ i δ j x 2 k ) − δ j δ k (x i + δ i x j x k ) 2 .
In particular, for n = 3, if a three-form (18) is invariant under f , so that the dual tri-vector field is given by
ω = φ(x) ∂ ∂x 1 ∧ ∂ ∂x 2 ∧ ∂ ∂x 3 ,
then for any integral I of f the bi-vector field
σ = ω dI = φ(x) ∂I ∂x 3 ∂ ∂x 1 ∧ ∂ ∂x 2 + ∂I ∂x 1 ∂ ∂x 2 ∧ ∂ ∂x 3 + ∂I ∂x 2 ∂ ∂x 3 ∧ ∂ ∂x 1(20)
is an invariant Poisson structure for f , as well as any linear combination of such bi-vector fields. The Poisson brackets of coordinate functions are given by
{x i , x j } = φ(x) ∂I ∂x k .(21)
Applying this result to the integrals log F 1 , log F 2 , log F 3 , with the three volume densities (16), we arrive at the following statement. (6):
{x i , x j } = C i δ j x k (1 − δ k δ i x 2 j ) − C j δ i x k (1 − δ j δ k x 2 i ),(22)
where C 1 , C 2 , C 3 are arbitrary constants.
Notice that the Poisson brackets (22) yield three compatible polynomial Poisson structures. Indeed, setting C 2 = C 3 = 0 and C 1 = 1, we get
{x 1 , x 2 } 1 = δ 2 x 3 (1 − δ 3 δ 1 x 2 2 ), {x 2 , x 3 } 1 = 0, {x 3 , x 1 } 1 = −δ 3 x 2 (1 − δ 1 δ 2 x 2 3 ),(23)
setting C 1 = C 3 = 0 and C 2 = 1, we get
{x 1 , x 2 } 2 = −δ 1 x 3 (1 − δ 2 δ 3 x 2 1 ), {x 2 , x 3 } 2 = δ 3 x 1 (1 − δ 1 δ 2 x 2 3 ), {x 3 , x 1 } 2 = 0,(24)
while setting C 1 = C 2 = 0 and C 3 = 0, we get
{x 1 , x 2 } 3 = 0, {x 2 , x 3 } 3 = −δ 2 x 1 (1 − δ 3 δ 1 x 2 2 ), {x 3 , x 1 } 3 = δ 1 x 2 (1 − δ 2 δ 3 x 2 1 ). (25)
It is easy to verify that the brackets (23), (24), (25) admit as Casimir functions the integrals F 1 , F 2 , F 3 , respectively.
In the continuous limit ǫ → 0 these three brackets correspond to the invariant linear brackets {·, ·} (γ) of the Euler top, given in Proposition 2, with γ = (0, −α 3 , α 2 ), γ = (α 3 , 0, −α 1 ), and γ = (−α 2 , α 1 , 0), respectively. Clearly, these three linear brackets are linearly dependent. On the contrary, the three polynomial brackets (23), (24), (25) are linearly independent, if one considers linear combinations with scalar coefficients. However, they become linearly dependent, if one considers more general linear combinations. Indeed, the volume density φ in Eq. (20) can be multiplied by an arbitrary integral without violating the Poisson property. Thus, in formulating the compatibility property of such Poisson tensors it is natural to consider their linear combinations with coefficients being integrals of motion rather than just numbers. In particular, the linear combination of the brackets (23), (24), (25) with the coefficients
(C i , C j , C k ) = δ i F j , δ j F i , δ k
vanishes, so that there are only two independent brackets among them.
Explicit solutions.
Explicit solutions were given in [5], but it has not been explained there how to determine the parameters of the elliptic functions involved in their formulas, using the initial conditions. We would like to fill in this gap here. We use the following addition formulas for the Jacobi elliptic functions:
cn(ξ + η) − cn(ξ − η) = − 2 sn ξ dn ξ sn η dn η 1 − k 2 sn 2 ξ sn 2 η ,
sn(ξ + η) − sn(ξ − η) = 2 cn ξ dn ξ sn η 1 − k 2 sn 2 ξ sn 2 η , dn(ξ + η) − dn(ξ − η) = − 2k 2 sn ξ cn ξ sn η cn η 1 − k 2 sn 2 ξ sn 2 η , and the related formulas sn(ξ + η)dn(ξ − η) + sn(ξ − η)dn(ξ + η) = 2 sn ξ dn ξ cn η 1 − k 2 sn 2 ξ sn 2 η , cn(ξ + η)dn(ξ − η) + cn(ξ − η)dn(ξ + η) = 2 cn ξ dn ξ cn η dn η 1 − k 2 sn 2 ξ sn 2 η , sn(ξ + η)cn(ξ − η) + sn(ξ − η)cn(ξ + η) = 2 sn ξ cn ξ dn η 1 − k 2 sn 2 ξ sn 2 η .
Assume that the coefficients δ i are given by formulas (7) with α i coming from Eqs. (3) or (4) with I 1 < I 2 < I 3 , so that
δ 1 < 0, δ 2 > 0, δ 3 < 0.
Then the above addition formulas suggest to look for the solution in one of two forms:
x 1 = A 1 cn(νn + ϕ 0 ), x 2 = A 2 sn(νn + ϕ 0 ), x 3 = A 3 dn(νn + ϕ 0 ),(26)
or
x 1 = A 1 dn(νn + ϕ 0 ), x 2 = A 2 sn(νn + ϕ 0 ), x 3 = A 3 cn(νn + ϕ 0 ),(27)
with ν being a parameter to be determined and ϕ 0 an arbitrary phase. Both possibilities (26) and (27) are realized (in different regions of the phase space). Consider first the possibility (26). It is easy to see that equations of motion (6) are satisfied by functions (26), if and only if the following conditions hold [5]:
A 1 = −δ 1 A 2 A 3 cn(ν/2) sn(ν/2)dn(ν/2) ,(28)A 2 = δ 2 A 1 A 3 cn(ν/2)dn(ν/2) sn(ν/2) ,(29)A 3 = −δ 3 A 1 A 2 dn(ν/2) k 2 sn(ν/2)cn(ν/2) .(30)
The amplitudes A i should be determined from the values of the integrals of motion. Substitute the ansatz (26) into the integrals (13), then a direct computation based on the relations cn 2 ξ = 1 − sn 2 ξ and dn 2 ξ = 1 − k 2 sn 2 ξ leads to
A 2 1 = 1 − F 3 δ 2 δ 3 , A 2 2 = 1 − F −1 3 δ 1 δ 3 , A 2 3 = 1 − F −1 1 δ 1 δ 2 , and k 2 = 1 − F −1 3 1 − F 1 .
Thus, this ansatz holds, if and only if F 1 < F −1
3 < 1, that is, if F 2 > 1.
With the values just found, relations (28)-(30) lead to sn 2 (ν/2) = 1 − F 1 .
Turning to the possibility (27) (omitted in [5]), we find that equations of motion (6) are satisfied by functions (27), if and only if the following conditions hold:
A 1 = −δ 1 A 2 A 3
dn(ν/2) k 2 sn(ν/2)cn(ν/2) , (31)
A 2 = δ 2 A 1 A 3 cn(ν/2)dn(ν/2) sn(ν/2) ,(32)
A 3 = −δ 3 A 1 A 2 cn(ν/2) sn(ν/2)dn(ν/2) .
Substituting the ansatz (27) into the integrals (13), we find:
A 2 1 = 1 − F 3 δ 2 δ 3 , A 2 2 = 1 − F 1 δ 1 δ 3 , A 2 3 = 1 − F −1 1 δ 1 δ 2 ,
and
k 2 = 1 − F 1 1 − F −1 3 .
Theferore, this ansatz holds, if and only if F −1 3 < F 1 < 1, that is, if F 2 < 1, and then relations (31)-(33) lead to sn 2 (ν/2) = 1 − F −1 3 . Thus, in both cases all parameters of the solution are expressed in terms of the initial data (more precisely, in terms of the integrals of motion).
Concluding remarks
In this paper, we studied a remarkable birational map of R 3 , which serves as an integrable discretization of the Euler top, on one hand, and plays a role in the spherical geometry, on the other. Along with a streamlined presentation of results obtained previously in [6] and in [5], namely the conserved quantities and the solution in terms of elliptic functions, we found an invariant volume form and a family of compatible invariant Poisson tensors for this map. Thus, it becomes a well-established representative of integrable maps, with a standard definition of integrability in the Liouville-Arnold sense. One more standard attribute of integrable systems remains to be found for this map, namely the Lax representation. This would provide a key to understanding the nature of analogous discretizations proposed in [7], [9], which seem to belong to the most mysterious objects in the universe of integrable maps.
Proposition 4 .
4There holds:
Now the claim of Proposition with φ as in Eq. (16), say, follows from Eq. (15). 3.3. Invariant Poisson structure. In the construction of an invariant Poisson structure for the map (6) we shall use the following results from [3] (Proposition 15 and Corollary 16 there). Let f : M → M be a smooth mapping of an n-dimensional manifold M, and let Ω be a volume form invariant under f , i.e., f * Ω = Ω.
Aknowledgments M.P. has been supported by the European Community through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652).
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Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures. G B Byrnes, F A Haggar, G R W Quispel, Physica A. 272G.B. Byrnes, F.A. Haggar, G.R.W. Quispel, Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures, Physica A, 1999, 272, 99-129.
Yu N Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3). 12supplYu.N. Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3), Jour. Nonlinear. Math. Phys., 2005, 12, suppl. 2, 77-94.
Discretization of the Euler top. R Hirota, K Kimura, Jour. Phys. Soc. Japan. 69R. Hirota, K. Kimura, Discretization of the Euler top, Jour. Phys. Soc. Japan, 2000, 69, 627-630.
Deutung einer birationalen Raumtransformation im Bereiche der sphärischen Trigonometrie. H Jonas, Math. Nachrichten. 62H. Jonas, Deutung einer birationalen Raumtransformation im Bereiche der sphärischen Trigonome- trie, Math. Nachrichten, 1951/2, 6, 303-314.
Discretization of the Lagrange top. K Kimura, R Hirota, Jour. Phys. Soc. Japan. 69K. Kimura, R. Hirota, Discretization of the Lagrange top, Jour. Phys. Soc. Japan, 2000, 69, 3193- 3199.
Discrete versions of some classical integrable systems and factorization of matrix polynomials. J Moser, A P Veselov, Commun. Math. Phys. 139J. Moser, A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Commun. Math. Phys., 1991, 139, 217-243.
Nonabelian semidirect product orbits and their relation to integrable systems. T Ratiu, Talk at the International Meeting "Geometric Integration" at Mathematisches Forschungsinstitut Oberwolfach. T. Ratiu, Nonabelian semidirect product orbits and their relation to integrable systems, Talk at the International Meeting "Geometric Integration" at Mathematisches Forschungsinstitut Oberwolfach, March 2006.
Group theoretical methods in the theory of finitedimensional integrable systems. A G Reyman, M A Semenov-Tian-Shansky, Dynamical systems VII. SpringerA.G. Reyman, M.A. Semenov-Tian-Shansky, Group theoretical methods in the theory of finite- dimensional integrable systems, in Dynamical systems VII, Springer, 1994.
The problem of integrable discretization: Hamiltonian approach. Yu B Suris, Progress in Mathematics. BaselBirkhäuser Verlag219Yu.B. Suris, The problem of integrable discretization: Hamiltonian approach, Progress in Mathe- matics, 219, Birkhäuser Verlag, Basel, 2003.
Integrable discrete time systems and difference operators. A P Veselov, Funkt. Anal. Prilozh. 22Funct. Anal. Appl.A.P. Veselov, Integrable discrete time systems and difference operators, Funkt. Anal. Prilozh., 1988, 22, 2, 1-13 (English translation: Funct. Anal. Appl., 1988, 22, 83-93).
| []
|
[
"Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs",
"Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs"
]
| [
"Maria Chiara Angelini \nDipartimento di Fisica\nUniversità \"La Sapienza\"\nP.le A. Moro 500185RomeItaly\n",
"Federico Ricci-Tersenghi \nDipartimento di Fisica\nUniversità \"La Sapienza\"\nP.le A. Moro 500185RomeItaly\n\nINFN\nSezione di Roma1\n\nCNR-Nanotec\nRome unit, P.le A. Moro 500185RomeItaly\n"
]
| [
"Dipartimento di Fisica\nUniversità \"La Sapienza\"\nP.le A. Moro 500185RomeItaly",
"Dipartimento di Fisica\nUniversità \"La Sapienza\"\nP.le A. Moro 500185RomeItaly",
"INFN\nSezione di Roma1",
"CNR-Nanotec\nRome unit, P.le A. Moro 500185RomeItaly"
]
| []
| The effectiveness of stochastic algorithms based on Monte Carlo dynamics in solving hard optimization problems is mostly unknown. Beyond the basic statement that at a dynamical phase transition the ergodicity breaks and a Monte Carlo dynamics cannot sample correctly the probability distribution in times linear in the system size, there are almost no predictions nor intuitions on the behavior of this class of stochastic dynamics. The situation is particularly intricate because, when using a Monte Carlo based algorithm as an optimization algorithm, one is usually interested in the out of equilibrium behavior which is very hard to analyse. Here we focus on the use of Parallel Tempering in the search for the largest independent set in a sparse random graph, showing that it can find solutions well beyond the dynamical threshold. Comparison with state-of-the-art message passing algorithms reveals that parallel tempering is definitely the algorithm performing best, although a theory explaining its behavior is still lacking. | 10.1103/physreve.100.013302 | [
"https://arxiv.org/pdf/1904.02231v1.pdf"
]
| 102,353,977 | 1904.02231 | 01c0da5659e57c7459ab112fc14ee91647f19abb |
Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs
3 Apr 2019
Maria Chiara Angelini
Dipartimento di Fisica
Università "La Sapienza"
P.le A. Moro 500185RomeItaly
Federico Ricci-Tersenghi
Dipartimento di Fisica
Università "La Sapienza"
P.le A. Moro 500185RomeItaly
INFN
Sezione di Roma1
CNR-Nanotec
Rome unit, P.le A. Moro 500185RomeItaly
Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs
3 Apr 2019(Dated: April 5, 2019)arXiv:1904.02231v1 [cond-mat.dis-nn]
The effectiveness of stochastic algorithms based on Monte Carlo dynamics in solving hard optimization problems is mostly unknown. Beyond the basic statement that at a dynamical phase transition the ergodicity breaks and a Monte Carlo dynamics cannot sample correctly the probability distribution in times linear in the system size, there are almost no predictions nor intuitions on the behavior of this class of stochastic dynamics. The situation is particularly intricate because, when using a Monte Carlo based algorithm as an optimization algorithm, one is usually interested in the out of equilibrium behavior which is very hard to analyse. Here we focus on the use of Parallel Tempering in the search for the largest independent set in a sparse random graph, showing that it can find solutions well beyond the dynamical threshold. Comparison with state-of-the-art message passing algorithms reveals that parallel tempering is definitely the algorithm performing best, although a theory explaining its behavior is still lacking.
I. INTRODUCTION
Discrete optimization problems defined on graphs are widespread among many scientific disciplines and commonly found in real-world applications. Depending on the properties of the underlying graph, these optimization problems may become so hard to solve that all known algorithms find only very suboptimal solutions, while the optimal ones remain unreachable to algorithms running in polynomial time.
A common benchmark to test the effectiveness of search algorithms is represented by optimization problems defined on random graphs (typical case analysis). In this case, the hardness of the optimization problem can be usually controlled by varying continuously a model parameter (e.g. the random graph mean degree or the solution size), and different algorithms can be quantitatively compared on the basis of how close to optimality they can go.
Unfortunately, in optimization problems that, in the worst case analysis, are NP-hard and also hard to approximate, a large algorithmic gap is often present in the typical case analysis, i.e. all known algorithms stop working at an algorithmic threshold which is bounded far away from the optimal (information theoretical) threshold. Computing the ultimate algorithmic threshold in these hard problems and understanding whether and why such an algorithmic threshold remains below the optimal one are fundamental open questions. The present work takes a step towards the answer of these questions, by studying the problem of finding a large Independent Set (IS) in a Random Regular Graph (RRG).
Given a graph G = (V, E), an IS is a subset of vertices S ⊂ V such that no vertices in S are adjacent, that is (ij) / ∈ E, ∀ i, j ∈ S. Finding the largest IS in a graph is a fundamental problem (NP-hard in the worst case), tightly related to minimum vertex cover and maximum clique [1]. In physics, the problem is known under the name of hard-core model [2], because vertices in S can be seen as particles that have a hard-core interaction and cannot be adjacent. The largest IS thus corresponds to the densest packing configuration in the hard-core model.
We call ρ the relative size of the IS, that is |S| = ρ|V | = ρN . On RRG of constant degree d it has been proved that, in the large N limit, IS with ρ < ρ max ∼ 2 log d/d do exist with high probability for d large enough [3,4]. However algorithms running in polynomial time cannot find IS with ρ > ρ alg ∼ log d/d for d large enough [5]. And actually this algorithmic threshold ρ alg can be achieved with very simple algorithms [6]. The algorithmic gap, that is the strict inequality ρ alg < ρ max , has been proven for a class of local algorithms in the large d limit [7]. In this case, the origin of the algorithmic failure is due to the ergodicity breaking taking place at ρ alg : this is a common phenomenon in optimization problems [8,9], also called clustering or shattering of the solution space.
One expects the ergodicity breaking taking place at ρ alg to affect also other types of algorithms. In particular, the sampling of the optimal solutions through numerical methods based on Monte Carlo Markov Chain should become much slower when ergodicity is broken, due to the need of overcoming large barriers. However if one is just interested in finding a single optimal or very close to optimal solution maybe Monte Carlo methods may work better than expected. This is a question never investigated in detail (to the best of our knowledge) and its answer is one of the main motivations for the present work.
We are going to analyze the performances of different algorithms, dedicating particular attention to those based on Monte Carlo Markov Chains, and we will try to relate such performances to the relevant phase transitions taking place in the space of IS in the limit of large RRG. Indeed, studying the thermodynamics of the problem via the cavity method the authors of Ref. [10] showed how the space of IS changes while increasing ρ: for d < 16 it undergoes a continuous phase transition from a Replica Symmetric (RS) phase to a phase described by a Full Replica Symmetry Breaking (FRSB) solution; while for d ≥ 16 the space of IS undergoes a random first-order transition (RFOT) and it can be described by a solution with one step of Replica Symmetry Breaking (1RSB).
Let us briefly review the important phase transitions in the RFOT case, each one corresponding to a drastic change in the structure of the set of ISs. At small densities ρ, the ISs form a single large cluster (two ISs are considered adjacent if they differ in o(N ) vertices) and can be well described by an RS solution that assumes the existence of a single state. Increasing the density, one first finds a dynamical threshold ρ d above which the space of ISs is divided into an exponential number in N of distinct clusters. This is the ergodicity breaking phase transition that affects local search algorithms and Monte Carlo methods for sampling. At the condensation threshold ρ c > ρ d the number of clusters becomes sub-exponential, and beyond the maximum density ρ max there are no more ISs. This last threshold is the equivalent to the sat/unsat threshold in constraint satisfaction problems (CSP).
Beside the above thermodynamic transitions, another property has been conjectured to be important for understanding the origin of the algorithmic complexity in CSP: the concept of frozen clusters [9,11,12]. A cluster of solutions is said to be frozen if it contains frozen variables that take the same value in all the solutions of that cluster. The rigidity threshold ρ r is defined such that for ρ > ρ r typical clusters are frozen, while above the freezing transition ρ f all clusters are frozen. In CSP many smart algorithms can find solutions in the clustered phase, but even the most performing ones do not find frozen solutions [13]. For this reason, the freezing threshold is conjectured to be the ultimate algorithmic threshold. Unfortunately, its analytic computation is a very difficult task, which has been achieved only in random hypergraph bi-coloring at present [14].
We will analyze different kinds of algorithms running in polynomial times. We avoid using algorithms that are known to find the largest IS in time typically growing exponentially in the graph size since these are impractical. Three main classes of polynomial algorithms will be considered: greedy algorithms, Monte Carlo methods and message passing algorithms. Greedy algorithms are very popular [15][16][17], because extremely fast and often provide a reasonably large IS.
We will mainly focus on Monte Carlo based algorithms that have been much less studied. Indeed the common belief is that a slow enough Simulated Annealing (SA) is able to reach densities not larger than the bottom of the equilibrium states at ρ d [18]. Above ρ d ergodicity is broken and Monte Carlo methods should not be able to sample correctly the equilibrium properties of the model. However, it could always be possible that there are states accessible to the out of equilibrium dynamics that terminate at densities ρ > ρ d and thus an out-of-equilibrium process can find very large IS with ρ > ρ d .
Recently it has been proposed to enhance the weight of deep, large states in an efficient way coupling some replicas of the system, for example in an SA algorithm [19]. The Replicated SA (RSA) has been seen to enhance the performances of learning in some models of neural networks. Here we apply RSA to the problem of finding the largest IS problem, discovering indeed that this algorithm is able to find solutions when the standard SA is not able to, well beyond ρ d . However, this seems to be true only if the transition is strongly discontinuous (RFOT). In case the transition is weakly discontinuous (or continuous) RSA and SA show similar performances.
Finally, we will analyze the behavior of Parallel Tempering (PT). Although PT has been invented to sample at equilibrium the very rough energy landscape of disordered systems and posterior distributions [20,21], it can be used in the out-of-equilibrium regime to try to reach some of the lowest energy configurations [22]. Recently the PT has been applied to the planted IS problem, allowing to find the planted configuration in the supposedly hard regime (i.e. when the planted IS is very small) in a time that seems to scale polynomially with the system size [23]. In the random case, we show here that PT is able to find solutions above the algorithmic threshold of the SA and of all the other analyzed algorithms, included the Belief-Propagation with Reinforcement, that is usually the best-performing message passing algorithm in other optimization problems, able to go beyond the rigidity transition [24]. We will measure the scaling of the convergence time for PT, showing that indeed it stays polynomial for ρ > ρ d .
II. PROBLEM DEFINITION AND DESCRIPTION OF ANALYZED ALGORITHMS
In this section, we report the details of the problem, and of the algorithms whose performances are analyzed in the rest of the paper.
The optimization problem we try to solve is to find the largest IS in a given RRG. Being K the size of an IS, we call ρ = K/N its density. Finding the largest IS problem is clearly a zero-temperature problem since it imposes strong constraints on any pair of nearest neighbor vertices not to be in the IS. As usual in a statistical mechanics approach, we can add a temperature parameter T = 1/β and relax the strong constraints into soft ones. The probability measure can be written as
P (n) ∝ exp µ N i=1 n i + β (ij)∈E n i n j(1)
where n i ∈ {0, 1}. In the T → 0 limit, vertices with n i = 1 form the IS, and the largest IS can in principle be achieved by sending µ → ∞ afterwards.
In practice, we are going to approach such a limit (T → 0 and µ → ∞) in two different ways. In the first way, we fix the IS size K, such that the first term in the measure in Eq. (1) is constant and can be ignored, and we study the problem in temperature. In the second way we fix T = 0, making constraints hard, that is we rewrite the measure as follows
P (n) ∝ exp µ N i=1 n i (ij)∈E (1 − n i n j )(2)
and we study the problem increasing µ.
We will use many different algorithms, described in the following list. Each algorithm will show its own algorithmic threshold ρ alg above which that algorithm is not able to find IS.
• Greedy algorithms (GA) Greedy algorithms are linear time algorithms where variables are set just once during the process of finding an IS. They differ according to the rule which is used to select the next vertex to include in the growing IS. Schematically they work as follows:
start with all n i = 0; -at each step choose a vertex v from the graph and add it to the IS, i.e. set n v = 1; -the vertex is chosen uniformly at random in the 'random vertex' version (RV GA) and such as to have the smallest degree in the 'minimum degree' version (MD GA);
all the neighbors of the chosen vertex are removed from the graph.
The random vertex version has been designed by Karp and Sipser [25] and produces with high probability an IS of size N log(d + 1)/d both at finite and large d. The minimum degree version has been introduced in Ref. [26] and gives better results, at least for finite d, while it has the same scaling at large d values.
The computational time of the greedy algorithm scales as O(dN ), that is linear in the graph size.
• Monte Carlo in temperature (βMC)
We fix the size K of the IS we would like to find and the temperature T = 1/β to be used in the Monte Carlo algorithm. The algorithm will sample configurations with exactly K variables set to n = 1, that is i n i = K; each of these configurations can be equivalently described in terms of the subset of vertices containing a particle I ≡ {i ∈ V : n i = 1}. To each configuration we associate the energy E(n) = (ij)∈E n i n j counting how many pairs of nearest neighbours are filled (n = 1). A configuration of zero energy is an IS of size K.
We start by choosing I as a random subset of K vertices of V . At each step of the algorithm we propose to move a randomly chosen particle to a randomly chosen empty vertex; the particle and the empty vertex do not need to be nearest neighbors, so the algorithm is not standard diffusion. Calling n the current configuration and n ′ the proposed configuration, we follow standard Metropolis rule for accepting the proposed configuration, that is we accept the change with probability 1 if E(n ′ ) ≤ E(n), and with probability exp[β(E(n ′ ) − E(n))] otherwise. As done conventionally, we define a Monte Carlo Sweeps (MCS) the attempt to move a randomly chosen particle, repeated K times. We stop the algorithm when a configuration n IS with E(n IS ) = 0 is found, that corresponds to an IS.
• Parallel Tempering in temperature (βPT)
We consider N β replicas, each one with exactly K variables set to n = 1 as in the βMC method discussed above. Each replica undergoes a standard Metropolis evolution at inverse temperature
β i = β max − i · ∆β, i ∈ [0, N β − 1]
. Every 5 steps of βMC a temperature swapping step is attempted for each pair of configurations at nearby temperatures β i and β i+1 ; the temperature swap is accepted with probability
p = min 1, e (βi−βi+1)(Ei−Ei+1) ,(3)
where E i is the current value of the energy of the i-th replica. The algorithm is stopped when a replica (usually the one with the lowest temperature) reaches a zero energy configuration.
• Simulated Annealing in chemical potential (µSA)
Working directly at zero temperature, i.e. sampling the measure in Eq. (2), we run a Simulated Annealing scheme in the following way. We start from the empty configuration n i = 0 ∀i that certainly satisfy all the constraints and from a null chemical potential µ = 0. At each step of the SA algorithm we increase the chemical potential by ∆µ and we do a Monte Carlo sweep, that corresponds to the attempt to update each of the N variables n i following the usual Metropolis rule: in practice if n i = 0, we set n i = 1 only if all the nearest neighbors are empty, and if n i = 1 we set n i = 0 with probability exp(−µ). We stop the SA algorithm at a value µ max where we observe the IS density ρ = i n i /N not increasing any more on any reasonable timescale.
The algorithm, at fixed parameter ∆µ, is linear in the size N .
• Replicated Simulated Annealing in chemical potential (µRSA)
In Ref. [19] a replicated version of the SA is proposed to sample with higher probability states with larger entropy.
To define the Replicated SA, we introduce R replicas of the variables on the same RRG, and a coupling between the different replicas according to the following measure:
P (n 1 , . . . , n R ) ∝ exp µ R a=1 N i=1 n a i + γ a<b N i=1 n a i n b i R a=1 (ij)∈E (1 − n a i n a j )(4)
We then run the SA algorithm on this replicated system, fixing the value of γ and incrementing the value of µ as in the µSA. At variance to numerical experiments in Ref. [19], where γ is incremented during the annealing, we prefer to keep γ fixed as we have seen that varying γ does not improve the final result.
• Parallel Tempering in chemical potential (µPT)
We consider N µ replicas of the system, each replica being at a different chemical potential:
µ i = µ max − i · ∆µ, i ∈ [0, N µ −1].
For each replica, we run 5 Metropolis Monte Carlo sweeps at the corresponding chemical potential and then we try to swap configurations between close by values of the chemical potential with probability
p = min 1, e (µi−µi+1)(−Ki+Ki+1) ,(5)
where K i is the actual number of variables set to 1 in the i-th replica. We stop the simulation if a replica (usually the one of index 0) reaches the IS size K we aim at.
• Belief Propagation with Reinforcement (BPR)
The Belief Propagation equations for the present problem were already derived in Ref. [10]:
π i→j = e µ k∈∂i\j (1 − π k→i ) 1 + e µ k∈∂i\j (1 − π k→i ) ,(6)
where π i→j is the probability to have n i = 1 in a modified graph where edge (ij) has been removed.
These equations for µ < µ c converge to a homogeneous paramagnetic fixed point (FP). To turn the BP equations into a solver, one can add a reinforcement term, initially introduced in Ref. [27], with two parameters γ, dt that tune respectively the strength and the speed of update of the reinforcement term. Practically the eqs. for the update of the messages becomes:
π t+1 i→j = e µ [θ i (t)] 1−γt k∈∂i\j (1 − π t k→i ) 1 + e µ [θ i (t)] 1−γt k∈∂i\j (1 − π t k→i ) ,(7)
with θ i (t) = k∈∂i (1 − π t−1 k→i ) and γ t = γ ⌊t dt⌋ . The FP reached when reinforcement is present is a completely magnetized one, that is the marginal probabilities for the values of n i are such that P [n i = 1] ∈ {0, 1}, and thus each variable is surely in the IS or surely outside of it. Thus the FP reached by BPR does correspond to an IS. [10] and the algorithmic thresholds found in this work for many different algorithms searching for the largest IS in a RRG of degree d = 20 and d = 100
The attentive reader probably notices that the above list is not including all possible Monte Carlo schemes: for example (Replicated) Simulated Annealing in temperature and simple Monte Carlo in chemical potential are missing. For this reason, we spend now a few words discussing our choice of the analyzed algorithms and explaining how the present work is organized.
The first algorithms in the list are greedy algorithms. They are clearly suboptimal and have been run just to give an idea of the IS size which is very easy to find in linear time. This information will also be useful to set the parameters of more refined algorithms as PT. The algorithmic thresholds for the GA, as for all the other analyzed algorithms, are reported in Table I.
We then analyze in Sec. III the algorithms at fixed density, βMC and βPT. In these algorithms, the size of the IS one is looking for is fixed to K and what is changed is the inverse temperature parameter β, that in turn varies the number of links within the set I representing the putative IS. Naturally, in the limit β → ∞, no more links inside I are allowed and we obtain a true IS. We start the discussion about stochastic algorithms with the analysis of βMC because this is an adaptation to the IS problem of commonly used local search algorithms, e.g. WALKSAT or ASAT [28,29], which have been applied with success to problems like random K-SAT or random graph coloring: the main difference being that βMC respects the detailed balance condition, while WALKSAT or ASAT do not. We then study the βPT algorithm because this is the most common way to improve Monte Carlo sampling methods in glassy systems. This algorithm seems to scale superlinearly, but still polynomially, with the problem size N as shown in detail in Sec. III A.
Then in Sec. V we move to analyse stochastic algorithms that work directly at zero temperature, µSA, µRSA and µPT, where links inside I are not allowed and the tuning parameter is the chemical potential µ. We do not study the βSA because the extrapolation of the algorithmic threshold in that case is a long and difficult task [30]: one should find the threshold for any given µ and then extrapolate int the µ → ∞ limit. The extrapolation of the algorithmic threshold is instead direct for the µSA algorithm, and for this reason, we prefer to study this version of SA. We will see that the µPT have an algorithmic threshold similar to the βPT one, thus showing that the performances of PT are rather robust.
Finally in Sec. VI we compare the results obtained via the stochastic algorithms with the outcome of BPR, which is a powerful message passing algorithm, widely used to solve problems defined on random graphs.
We will mainly analyze the problem at d = 20, where the transition is still near to the continuous one, and d = 100 where the transition is distinctly 1RSB. In Table I, the values for the ρ d , ρ c and ρ max , together with the thresholds for the maximum density reached by the analyzed algorithms are reported.
III. MAXIMUM DENSITY REACHED BY FIXED-DENSITY ALGORITHMS
In this section we look at the performances of the fixed-density algorithms, namely βMC and βPT. For these algorithms, if we measure the running time in Monte Carlo Sweeps (MCS) a linear dependence on N is hidden in the single MCS (that takes a time proportional to N ) and we can limit ourselves to measure the number of MCS needed to reach the wanted solution in order to understand the computational complexity of this class of algorithms. In Fig. 1 we show the number of MCS needed by βMC and βPT to converge to an IS of a given density ρ. Also, the results for µPT are shown for comparison.
For what concerns βMC, the optimal value of β maximizing the probability of reaching an IS, i.e. a zero energy configuration, is likely to depend on N . Consequently, the convergence time will depend on N , since we expect the Monte Carlo dynamics to slow down when the temperature is decreased. Nevertheless, we are not going to make this detailed study, because, as shown in Fig. 1, standard Monte Carlo run at a single temperature is easily outperformed by Parallel Tempering.
The time to find an IS of a given density ρ is clearly diverging approaching the algorithmic threshold ρ alg . In order to estimate the algorithmic threshold we need to perform an extrapolation. The best data interpolation is obtained via a power law divergence where C, ν and ρ alg are the fitting parameters (specific to each different algorithm). The best fitting curves are shown with full lines in Fig. 1. The extrapolated algorithmic threshold are reported in Table I, while the best fitting values for the ν exponent can be found in Table II. Data in Fig. 1 are for size N = 5 · 10 4 , that is large enough that finite size effects are not present in the estimation of ρ alg . The dependence of C and ν on the size will be discussed in Sec. III A. We notice that both versions of PT (in temperature and chemical potential) have very similar algorithmic thresholds. This may suggest that at that density value there is some unavoidable hardness that affects both versions of PT. Our PT scheduling is not particularly optimized on purpose, because we believe that if an unavoidable algorithmic barrier arises at a certain density value, this should affect any version of Monte Carlo based algorithms. The only parameter that we decide to fix in an (almost) optimal way is β min , i.e. the lowest value for the inverse temperature: indeed a too low β min requires a larger running time without any performances improvement (too many replicas at high temperature are useless), while a too large β min does not allow the configurations to decorrelate fast enough. We find that a very good choice for β min is the inverse temperature such that the actual density of the larger IS among the K variables with n = 1 is almost the maximum IS density reached by the best greedy algorithm. This means that the replica at β min can easily travel in the whole configurational space and this is enough for the PT algorithm to work properly.
τ = C (ρ alg − ρ) ν ,(8)
A. Scaling with N for the βPT
We have seen that the PT algorithm is able to find solutions in a region of ρ where other algorithms fail. The next important question to answer is how the number of PT iterations needs to be scaled with N in order to find an IS of density ρ. The issue is particularly relevant above ρ d and approaching ρ alg where the convergence time diverges. To analyze the scaling with N , we implement an optimized choice of the temperatures in the PT algorithm, whose derivation is in Appendix . The optimized temperatures scheduling requires a number of replicas in a range β ∈ [0, β max ] that scales as √ N . However, the replicas in the range β ∈ [0, β min ] are useless and can be safely ignored without altering PT performances. In practice we end up with N β ∼ 40 in the worst case studied (d = 100, N = 10 5 and ρ = 0.0646).
To study the size dependence of the convergence time, we run all our βPT simulations with the temperature set defined in Eq. (A.4) with r = r opt , between β min and β max . In Fig. 2
(N ) = a(ρ) · N b(ρ) ,(9)
where the main ρ dependence is in the prefactor a(ρ) that diverges at ρ alg as in Eq. (8). However there is also a slight dependence on ρ in the exponent b. We plot b as a function of ρ in the right panel of Fig. 2, together with a fit of the type b(ρ) = c 1 + c 2 · log(ρ alg − ρ), that interpolates nicely the data. We notice that this behaviour is the one to make Eqs. (8) and (9) compatible, since they are particular cases of the more general expression
log(τ (ρ, N )) = log(c) − ν ′ log(ρ alg − ρ) + c 1 log(N ) + c 2 log(N ) log(ρ alg − ρ) .(10)
For a fixed value of N we recover Eq. (8) with C = cN c1 and ν = ν ′ − c 2 log(N ). From the data shown in Fig. 2 it is evident that the exponent b is positive even in the "easy" region and it seems to go to zero only for ρ ≃ 0. This means that using PT to find IS always requires a running time growing more than linearly in N . We think this is due to the fact that PT is a sophisticated algorithm developed to find solutions when the energy landscape is complex. For ρ < ρ d , when there is just a single state, PT is thus suboptimal (maybe with a different choice of the parameters it could become a linear algorithm in this region, this kind of optimization is, however, out of our scope: we introduced PT to reach solutions in the hard region).
The time divergence as a power law approaching a given density, as in Eq. (8), is reminiscent of what happens in a first order phase transition, thus suggesting that at ρ alg an extensive barrier develops that makes impossible to reach states with ρ > ρ alg in polynomial time. The weak dependence on N , instead, suggests that some long range correlations may develop in the states in which the dynamics fall for ρ < ρ alg (this is discussed in the next section).
IV. LOOKING AT THE FREEZING
As already mentioned, it has been conjectured for other optimization problems that the threshold for the appearance of hardness in polynomial time algorithms corresponds to the freezing threshold, that is the lowest density such that all clusters are frozen. We want to check this conjecture in the present problem.
For practical purposes let us define a cluster as the set of solutions (i.e. valid ISs) that are "connected" via paths where each step is the flip of just two variables. A cluster of solutions is frozen if it contains frozen variables, that is if there is at least one variable fixed to a given value in all the configurations of the cluster. Above the rigidity threshold almost all the dominant clusters are frozen (but clusters with larger internal entropy might be not frozen). The freezing threshold corresponds to the density at which each cluster of solutions is frozen.
In this section we study the escape time, t esc , which is the time needed by an algorithm that moves only between solutions to go away from the initial configuration. More precisely, we first find a solution with a given algorithm, then we apply the βMC algorithm at β = ∞ (that is a kind of diffusive dynamics at fixed zero energy and fixed size of the IS) and we measure the time needed to "free" each variable from its starting value, that is to find that variable in a value different from the starting one. Looking at Fig. 3, the first important observation is that all the analyzed algorithms show the same t esc at a fixed density of the IS. This means that they all find the same kind of solutions (when they can find one).
The escape time diverges as a power law at a threshold density ρ r (see the fits in Fig. 3). From the data we estimate ρ r (d = 20) = 0.1890(6) and ρ r (d = 100) = 0.0639 (2). The observation that the same threshold holds for different kind of algorithms suggests us to conjecture that ρ r does actually correspond to the rigidity threshold, that is the density where the typical clusters become frozen and the escape time from it thus diverges.
The values of ρ r are compatible with the thresholds for the βMC algorithm, while βPT and µPT can find solutions of densities greater than ρ r . At this point, it is natural to check whether the solutions found by the PT algorithms at densities larger than ρ r are frozen or not. To answer this question we find a solution at density ρ > ρ r with the βPT algorithm, then we run the βMC algorithm at β = ∞ (the diffusive algorithm) and we look at the persistence, that is the fraction of variables that have not changed during the diffusive dynamics.
The results are shown in Fig. 4 for a single sample: the fraction of frozen variables seems to decrease in an extremely slow way, mostly logarithmically in time with evident jumps (corresponding to avalanches of variables that are set free altogether). It is worth noticing that the slowness of the diffusive dynamics around the initial solution found by PT is only due to entropic effects, given than the diffusive dynamics keeps the energy constant.
In Fig. 4 we also notice some interesting finite size effects. For the largest sizes, the diffusive dynamics eventually makes every variable unfrozen, although the escape time is some orders of magnitude larger than the time needed by PT to reach that particular solution (suggesting that PT follows a smart path that is not affected by entropic barriers!). For smaller sizes, frozen variables persist longer and eventually we observe that the diffusive dynamics is not able to leave the cluster: the fraction of frozen variables becomes constant in time. This is a strong evidence that the IS found by PT for small enough N belong to frozen clusters (a similar phenomenon has been observed also in other models when solved for example via the Reinforcement algorithm [31]).
The above observations support the following scenario: the PT algorithm is able to find IS beyond the rigidity threshold ρ r and in this rigid phase, for small enough sizes, there is a non zero probability that PT finds a solution in a rare frozen cluster. However, for large N , the solutions found by PT seems to be all unfrozen and thus we deduce that the PT algorithmic threshold is bounded above by the freezing threshold. We are strongly tempted to conjecture that the two thresholds, ρ alg for PT and ρ f , do actually coincide, but we do not have firm arguments in support.
We have also checked that the solutions found by the PT algorithms above ρ d are not equilibrium solutions. To do this, we find a solution at ρ > ρ d with the βPT or µPT algorithms. We then initialize BP on that solution and we check whether BP converges to a fixed point close to the solution found by PT. If it is so, this means that the PT solution lays inside one of the states (and replica symmetry holds within a state) that form the 1RSB structure that characterises the equilibrium measure for densities slightly above ρ d . However, we find that BP does not converge (neither to the paramagnetic fixed point nor to a fixed point close to the PT solution). This lack of convergence suggests that the solution found by PT is probably inside a state that is not replica symmetric, but probably FRSB, as found in other models [18]. Indeed it is well known that states reached by the out-of-equilibrium dynamics may be FRSB even when equilibrium states are 1RSB [32,33].
V. ZERO TEMPERATURE ALGORITHMS
In this section, we analyze a different kind of algorithms, the ones running directly at zero temperature. This means that links inside I are not allowed, i.e. the algorithm always works with a valid IS. For this class of algorithms, the varying parameter is the chemical potential µ that changes the average density of the IS. The limit µ → ∞ should correspond to the largest possible IS.
First of all, we run µSA. It is a common belief that a slow enough SA should reach the bottom of the equilibrium states at ρ d . The algorithmic thresholds, computed as the average over 100 samples of the maximum density ρ reached when µ → ∞ in a SA with ∆µ = 10 −7 and N = 5 · 10 4 , are reported in Table I. As one can notice, for d = 20 the inequalities ρ alg > ρ c > ρ d hold, implying that the states that dominate the measure at ρ d can be followed deeply beyond ρ c . This is compatible with the fact that at d = 20 the transition is still near to a continuous FRSB one and thus the ergodicity breaking is less pronounced. For d = 100 instead ρ alg < ρ c , consistently with the fact that the transition is distinctly 1RSB and ergodicity breaking takes place in a much more marked way.
We then pass to analyze the µRSA algorithm. We take inspiration from Ref. [19], where a replicated version of the SA is proposed to sample with higher probability states with larger entropy. In the context of ISs one can identify a state in the following way: starting from a maximal IS, that is an IS that cannot be increased any further by just adding vertices to the IS itself, and considering this maximal IS as the "bottom of a valley" in a usual energy landscape, one can build a state by the set of the ISs which are a subset of the maximal one (the construction has to be refined when one finds ISs which are a subset of more than one maximal IS, but we do not need such a detailed description for the present argument). According to this construction it is likely that states corresponding to the largest IS are also those of largest entropy. So the use of an algorithm that favours states according to their entropy is likely to be beneficial also in the search for the largest IS.
We run µRSA with parameters R = 3, γ = 1. In Fig. 5 its performances are compared with those of µSA in the case d = 100 (their algorithmic thresholds can be found in Table I). It is remarkable that the improvement of RSA with respect to SA is practically null for d = 20 and very tiny for d = 100. While for d = 20 one may claim that the improvement is absent because the model has a very weakly discontinuous phase transition (the range where the phase transition is continuous is very close by), for d = 100 the 1RSB scenario holds clearly, but we do not see any improvement by reweighting states according to their internal entropy. This observation raises some doubts about what RSA is actually doing and why is not working as expected.
Moreover, given that the performances of RSA are clearly worst than those of PT (see their algorithmic thresholds in Table I), we arrive at the conclusion that there are more and less efficient ways to couple replicas.
We move now to the analysis of the µPT algorithm. We use N µ = 21 replicas evenly spaced by ∆µ = 0.2 in the range µ ∈ [2, 6] for d = 20 and N µ = 31 replicas evenly spaced by ∆µ = 0.15 in the range µ ∈ [2, 6.5] for d = 100.
We have already anticipated in Sec. III that the behavior of µPT is equivalent to the one of βPT and in particular the algorithmic thresholds of the two algorithms are compatible. In Fig. 6 we show that also the scaling of their running times with N is similar, as the time needed to reach a solution of a given density scales as τ = aN b . In Table III we make the comparison between the exponent b of the two algorithms at the same values of d and ρ. These data confirm that the PT algorithm is a very robust one.
VI. COMPARISON WITH ADVANCED MESSAGE PASSING ALGORITHMS
We have seen that Monte Carlo based algorithms easily outperform greedy algorithms and can reach densities well above the dynamical threshold ρ d , passing also the rigidity threshold ρ r and for d = 20 even beyond the condensation threshold ρ c , thus approaching closely the maximum density ρ max . This looks like a great result, but in order to put it under the right light, we need a comparison with a some other algorithm that is expected to work efficiently on this kind of optimization problems. Since the problem is defined on a random graph we expect message passing algorithms to be particularly well suited. For this reason, we have run also BPR on this problem.
In Fig. 7 we show the average density of IS found by the BPR algorithm, as a function of the chemical potential µ, for different values of the BPR parameters. Let us just mention that below a certain chemical potential µ L , the solutions found by the BPR algorithm are always n i = 0, ∀i. The value of µ L is the one that generates, using the RS solution of the model from [10], a density ρ L that roughly corresponds to the threshold density for the random vertex GA (for both d = 20 and d = 100 we have µ L = 2.15 (5)). We have run the BPR algorithm in a broad range of chemical potentials and for different choices of the BPR parameters. The best results have been obtained with the choice γ = 0.999 and dt = 10. The maximum density reached can be deduced from the data shown in Fig. 7 and it is clearly lower than the thresholds for the PT algorithms. Our best estimates are reported in Table I. We notice that the threshold density for the BPR algorithm is very similar to the one of the RSA algorithm and this is expected from Ref. [19].
VII. CONCLUSIONS
We have done a comparative study of algorithms to find the largest IS in a RRG of degree d = 20 and d = 100. Our aim was to understand the actual performances of different kind of algorithms (greedy, message passing and especially Monte Carlo based), and to connect their algorithmic thresholds with thermodynamical phase transitions. For both values of d the set of IS undergoes a RFOT varying the IS density ρ, however for d = 20 the transition is weakly discontinuous because of the vicinity to the range where the transition is continuous (d < 16); for d = 100 the transition is markedly discontinuous as in the large degree limit.
While Table I summarizes thermodynamical and algorithmic thresholds, we list below the most relevant conclusions that we achieved:
• Only greedy algorithms get stuck below the dynamical threshold, while all the other algorithms easily pass beyond ρ d ; the relevance of the dynamical threshold for smart optimization algorithms seems very limited.
• Also the condensation threshold at ρ c seems to play no role at all in describing the performances of the best optimization algorithms.
• The simplest version of Monte Carlo algorithms seems to work roughly until the rigidity threshold at ρ r , defined as the density where the time to diffuse away from a typical IS diverges.
• More sophisticated Monte Carlo schemes (SA and PT) find IS beyond ρ r , but without frozen variables, thus showing the ability of finding IS in atypical unfrozen states.
• Replicated SA does not show any sensible improvement over standard SA for this problem, especially for d = 20.
• Belief Propagation with Reinforcement has an algorithmic threshold similar to Replicated SA.
• Parallel Tempering is by far the best algorithm for solving this problem and can find IS of a very large density that no other algorithm can find.
• Different versions of PT (in temperature and chemical potential) show almost the same algorithmic threshold, and this strongly suggests an universal behavior linked to an underlying phase transition. We conjecture the PT algorithmic threshold to coincide with the freezing threshold, i.e. PT is able to find an unfrozen IS as long as there is one.
• Running times of PT are super-linear, but still polynomial in N . Algorithmic thresholds for super-linear algorithms are likely to be larger than those for linear algorithms, but a theory for the formers is completely lacking.
Our results clearly show the need for a theory for advanced Monte Carlo algorithms, like Parallel Tempering, which is at present lacking. Only by understanding analytically this class of algorithms we can hope to approach the ultimate algorithmic threshold for a broad class of hard optimization problems.
implying p swap = erfc(r/2). The optimal value for r can be obtained by maximizing the mean squared distance traveled by a random walker performing jumps of size r with probability erfc(r/2), that is r opt = argmax erfc r 2 r 2 ≃ 1.68376 , (A. 5) leading to an optimal swapping rate equal to erfc(r opt /2) ≃ 0.23381 (this is the well-known 0.23 rule [34]). With the set of temperatures defined in Eq. (A.4) the optimized PT would require O( √ N ) replicas. However, we empirically observe that the time of convergence of the algorithm does not change if replicas in the range β ∈ [0, β min ] are removed. We find empirically that the largest possible value for β min roughly corresponds to the inverse temperature at which the equilibrium magnetization coincide with the maximum IS density reached by the greedy algorithm, ρ GA . This is very reasonable, indeed for ρ < ρ GA we do not expect any relevant barrier to be present and so the PT replicas at β min can easily travel the entire configurational space.
For β max we choose the lowest inverse temperature at which the condition E(β) − N |e ′ (β)| < 0 is satisfied, implying that a typical spontaneous fluctuation can lead the algorithm to find a configuration of zero energy.
We observe that the optimized version of βPT finds solutions up to a ρ max (d = 20) = 0.1941 (5), compatible with the non-optimized version, but with a smaller exponent b = 2.4 (3). For d = 100 the optimized βPT algorithm reaches ρ max (d = 100) = 0.06572(9) with b = 3.2(1).
Things are different for the µPT algorithm. For this algorithm, the RS magnetization is the one written in Eq. (6) of Ref. [10]. However, in the hard region, the real magnetization it is quite different and so we can not use the RS result to optimize the µPT algorithm.
FIG. 1 :
1Convergence time for βMC (with parameter β = 11), βPT (with parameters βmax = 11, ∆β = 0.4, N β = 20) and µPT algorithm (with parameters µmax = 6, ∆µ = 0.2, Nµ = 20) for N = 5 · 10 4 and d = 20 (left) or d = 100 (right). The vertical lines show the theoretical thresholds for comparison. : Fitting the divergence of the convergence time shown in Fig. 1 via the power law τ = C(ρ alg − ρ) −ν , the best fitting values for ν are the ones shown in this table.
FIG. 2 :
2Left: MCS to find a solution for d = 100 for different values of ρ as a function of the size N of the graph for the optimized βPT. Errors are smaller than points. The fits are of the kind τ (N ) = aN b . Right: Dependence of the exponent b on the distance from the algorithmic threshold ρ alg − ρ. The fit is of the type b = c1 + c2 · log(ρ alg − ρ). The right border of the plot corresponds to ρ = 0. of densities (similar behaviour is observed for d = 20). The running times grow as a power law in N τ
FIG. 3 :
3Escape time from the solution reached by different algorithms (d = 20, N = 5 · 10 4 ).
FIG. 4 :
4Starting from an IS of density ρ found via the βPT algorithm, we measure the fraction of variables that have not changed their value during a pure diffusive dynamics (βMC algorithm with β = ∞). Results are for d = 20 (left), d = 100 (right) and a single sample of the size indicated in the legend.
FIG. 5 :FIG. 6 :
56Comparison between µSA and µRSA for 50 samples of size N = 5 · 10 4 and d = 100 (parameters are ∆µ = 10 −7 , Number of iterations to find a solution for d = 20 and ρ = 0.19, 0.192 as a function of the problem size N for the optimized βPT (left) and the µPT (right) algorithms. The behaviour of the two algorithms is very similar.
TABLE I :
IRelevant physical thresholds ρ d , ρc and ρmax reported from Ref.
we show for d = 100 the results in a wide range100
1000
10000
100000
1e+06
1e+07
1000
10000
100000
iteration steps
N
ρ=0.05
ρ=0.056
ρ=0.06
ρ=0.062
ρ=0.064
ρ=0.0646
ρ=0.0649
ρ=0.0651
TABLE III :
IIIThe convergence time in Figs. 6 diverges as τ (N ) = a · N b . In the table the comparison between values of b for βPT and µPT FIG. 7: Average density of the ISs found by BP+Reinforcement as a function of the chemical potential µ at different values of N and of the BPR parameters.0.1918
0.192
0.1922
0.1924
0.1926
0.1928
0.193
0.1932
0.1934
3
3.2
3.4
3.6
3.8
4
4.2
4.4
<ρ>
µ
N=1e4 γ=0.999 dt=10
N=1e4 γ=0.9995 dt=3
N=5e4 γ=0.999 dt=10
N=5e4 γ=0.9995 dt=3
N=1e5 γ=0.999 dt=10
N=1e5 γ=0.9995 dt=3
0.0632
0.0634
0.0636
0.0638
0.064
0.0642
0.0644
0.0646
0.0648
0.065
0.0652
5.6
5.8
6
6.2
6.4
6.6
6.8
7
<ρ>
µ
N=1e4 γ=0.999 dt=10
N=1e4 γ=0.9995 dt=3
N=5e4 γ=0.999 dt=10
N=5e4 γ=0.9995 dt=3
N=1e5 γ=0.999 dt=10
N=1e5 γ=0.9995 dt=3
AcknowledgmentsThis research has been supported by the European Research Council under the European Unions Horizon 2020 research and innovation programme (grant No. 694925 -Lotglassy, G. Parisi).Appendix: Optimizing the choice of temperatures in the PT algorithmsHere we explain how we have implemented an optimized choice for the temperatures in the Parallel Tempering algorithm. We assume that the energy is close to its equilibrium value that can be computed via the RS solution. For the βMC algorithm at a fixed density ρ, the RS mean energy can be computed noticing that the RS marginals are equal for each site and assume the value p RS (σ) = ρ δ σ,1 + (1 − ρ)δ σ,0 , thus getting.(A.1)In the large N limit we can assume that the extensive energy at inverse temperature β is a Gaussian variables with mean E(β) = N e(β) and variance σ 2 (β) = −N e ′ (β). This Gaussianity assumption (which is rather well satisfied, but in the vicinity of the ground state) allows us to compute the probability of swapping two replicas at inverse temperatures β 1 and β 2 ,In the limit ∆β = β 2 − β 1 ≪ 1 we can approximate E(β 2 ) − E(β 1 ) ≃ N e ′ (β)∆β with β = (β 1 + β 2 )/2 and σ(β 1 ) ≃ σ(β 2 ) ≃ σ(β) = −N e ′ (β), thus gettingThe best way to allow replicas to wander fast between temperatures is to fix a constant p swap between any pair of successive temperatures and this can be achieved with the choice β n+1 = β n + r N |e ′ (β n )| (A.4)
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| []
|
[
"Measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions",
"Measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions"
]
| [
"Yoshihide Nakamiya \nGraduate School of Science\nHiroshima University\nKagamiyama\n\nHigashi-Hiroshima\n739-8526HiroshimaJapan\n",
"Kensuke Homma \nGraduate School of Science\nHiroshima University\nKagamiyama\n\nHigashi-Hiroshima\n739-8526HiroshimaJapan\n"
]
| [
"Graduate School of Science\nHiroshima University\nKagamiyama",
"Higashi-Hiroshima\n739-8526HiroshimaJapan",
"Graduate School of Science\nHiroshima University\nKagamiyama",
"Higashi-Hiroshima\n739-8526HiroshimaJapan"
]
| []
| We discuss measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions, equivalently at different pion multiplicities per heavy-ion collision dN π 0 +π ± /dy = 1000 and 2700 intended for the most central Au+Au collisions at √ sNN = 200GeV (RHIC) and the most central Pb+Pb collisions at √ sNN = 5.5 TeV (LHC), by evaluating the signal-to-background ratios and the statistical significance for the idealized detection system in the numerical simulation. The simulation study provides a guideline to be applicable to a concrete detector design by focusing on only the key experimental issues relevant to the measurement of di-electrons. The results suggest that there are realizable parameter ranges to measure light vector mesons via di-electrons with the reasonable significance level, even in the highest multiplicity case. | 10.1093/ptep/ptt088 | [
"https://arxiv.org/pdf/1309.5732v2.pdf"
]
| 118,727,796 | 1309.5732 | ce34c882dbbfa4574fbd1423c4e1099cd8a0bc46 |
Measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions
24 Sep 2013
Yoshihide Nakamiya
Graduate School of Science
Hiroshima University
Kagamiyama
Higashi-Hiroshima
739-8526HiroshimaJapan
Kensuke Homma
Graduate School of Science
Hiroshima University
Kagamiyama
Higashi-Hiroshima
739-8526HiroshimaJapan
Measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions
24 Sep 2013arXiv:1309.5732v2 [nucl-ex]
We discuss measurability of φ, ω and ρ mesons via di-electron decays in high-temperature states produced in heavy-ion collisions, equivalently at different pion multiplicities per heavy-ion collision dN π 0 +π ± /dy = 1000 and 2700 intended for the most central Au+Au collisions at √ sNN = 200GeV (RHIC) and the most central Pb+Pb collisions at √ sNN = 5.5 TeV (LHC), by evaluating the signal-to-background ratios and the statistical significance for the idealized detection system in the numerical simulation. The simulation study provides a guideline to be applicable to a concrete detector design by focusing on only the key experimental issues relevant to the measurement of di-electrons. The results suggest that there are realizable parameter ranges to measure light vector mesons via di-electrons with the reasonable significance level, even in the highest multiplicity case.
I. INTRODUCTION
High energy heavy-ion collisions provide a unique opportunity to liberate quarks and gluons from nucleons and cause the phase transition from nuclear to quarkgluon matter. Heavy-ion collisions have advantage of studying the QCD phase diagram especially in hightemperature and low-baryon density domains [1][2][3][4][5].
The mass modification [6][7][8][9][10] of light vector mesons such as φ, ω and ρ is an important signature of the QCD phase transition, because their masses are sensitive to chiral condensate qq . Chiral condensate is one of the most prominent order parameters characterizing the QCD phase structure. The behavior of the chiral condensates near the critical temperature is studied by several kinds of lattice QCD calculations [11][12][13][14][15][16][17]. The light vector mesons can be candles to study the properties of quark-gluon matter produced in heavy-ion collisions. The mass modification inside quark-gluon matter is potentially visible because their lifetimes are supposed to be comparable to duration of the thermal equilibrium state, unless the interactions with hadronic matter in the later stage dominate. In addition, electron-positron pairs, which are referred to as "di-electrons", decaying from light vector mesons are considered to be a clear probe, since charged leptons carry the original information in the early stage without perturbation by hadronic matter via strong coupling in the relatively later stage of the system evolution.
In low-temperature and high-baryon density domains, the symptoms of the mass modification are intensively discussed [18][19][20][21]. In high-temperature and low-baryon density domains, the enhancement of the di-electron yield is observed in the mass range below 0.7 GeV/c 2 in central Au+Au collisions at √ s N N = 200 GeV [1,22], and it indicates a deviation from the di-electron spectrum in p+p collisions [1,[22][23][24]. Therefore, quantitative studies of * Electronic address: [email protected] the mass spectra including the light vector mesons grow in importance to understand the phenomena in the high temperature domain. Furthermore, the measurement of the light vector mesons in higher-temperature states, at the LHC energy, would give a deeper insight into these phenomena.
The detection of di-electrons is, however, challenging from the experimental point of view, because a heavyion collision event produces a huge number of particles. The experiments at RHIC and LHC energies report that O (100-1000) particles are produced at midrapidity in a heavy-ion collision [25][26][27]. The produced particles are dominated by π ± and π 0 mesons. The production cross sections of pions are approximately hundred times larger than that of φ meson, and ten times larger than that of ω/ρ meson [81]. In addition, the branching ratio (BR) to a di-electron pair is on the order of 10 −4 for φ meson and 10 −5 for ω/ρ meson. Therefore, the measurement of the signal di-electrons requires state-of-the-art technology to identify electrons and positrons among a large amount of background hadrons by combining information from Ring Imaging Cherenkov counters [28], dE/dx [29,30], Time of Flight [31], most notably, the Hadron-Blind Detector [32] and so forth.
In addition to the small yield of di-electrons from the light vector mesons, the difficulty of the measurement is severely caused by background contaminations from processes other than the light vector meson decays. Dominant sources of such backgrounds are listed as follows, 1. Dalitz decays π 0 → γe + e − and η → γe + e − , 2. pair creations by decay photons from π 0 and η meson, 3. semi-leptonic decays from hadrons, 4. charged hadron contaminations by electron misidentification.
The key issues are in the increase of combinatorial pairs from different sources. The amount of these pairs has approximately quadratic dependence on multiplicity because combination between all possible electrons and positrons must be taken, while the number of true dielectron pairs from the light vector mesons approximately linearly scales with multiplicity [82]. Detector upgrades are planned in several experiments at RHIC and LHC. For instance, the ALICE experiment at LHC plans to upgrade the internal tracking system with the relatively low detector materials and enhance low-p T tracking capability [33]. These upgrades are expected to suppress the backgrounds from the photonconversion and Dalitz decay processes, and improve the Time-of-Flight measurement for the hadron identification. Moreover, the tracking system will improve the secondary vertex resolution. It allows to separate the backgrounds from the semi-leptonic decay of open charms. The key issues from (1) to (4) above are actually relevant to the items of the upgrade plan.
The measurement of light vector mesons is still important but challenging. The detector system applied to high-energy heavy ion collisions is, in general, multipurpose. Thus it is not necessarily optimized only for the measurement of light vector mesons. In this situation, unless the key issues above are simultaneously considered in advance of the other irrelevant design issues, for instance, only installing a state-of-the-art particle identifier might not be sufficient. In order to prove the performance of an upgraded detector system, a very detailed detector simulation is required. It is usually a time-consuming process to conclude whether a reasonable signal-to-background ratio is guaranteed with the upgraded complex detector system or not. Thus the applicability of such a detailed simulation is rather limited to a specific detector system. If one knows the key issues, however, considering only these issues in the idealized case would provide a more general guideline applicable to any complex detector system.
In this paper, therefore, we do not focus on details of individual detector designs. Instead, we investigate the necessary conditions which must be minimally satisfied in advance in order to gain reasonable signal-to-background ratios with idealized detector systems for given multiplicities, collision species and energies in heavy-ion collisions.
For this purpose, we first give the estimates on the production yields of light vector mesons and the other hadrons producing di-electrons in the final state. They are applied to p+p and A+A collisions at the RHIC and LHC energy regime, in concrete, dN π 0 +π ± /dy = 3, 6, 1000 and 2700 in p+p 200 GeV, p+p 7 TeV, Au+Au 200 GeV and Pb+Pb 5.5 TeV, respectively. We then implement the key physical background processes in the numerical simulation including the Dalitz decay and the photon conversion as well as the semi-leptonic decay of hadrons under the key experimental conditions: the amount of the detector material, the charged pion contamination, the geometrical acceptance, the electron tagging efficiency, the measurable momentum cutoff and the smearing effect by momentum resolution. These experimental conditions are parameterized and implemented into the simulation. The flowchart of the simulation is explained in Section II. Starting from a set of the idealized baseline parameters, we explore the expected signalto-background ratios and the statistical significance as a function of the individual experimental parameters for given pion multiplicities in high-temperature states. The results of the simulation study are provided in Section III. The non-trivial residual effects, though some of them depend on a specific detection system, are discussed in Section IV. In Section V, we conclude that we can find reasonable parameter ranges to measure the light vector mesons via di-electrons even at dN π 0 +π ± /dy = 2700.
II. NUMERICAL SIMULATION FOR IDEALIZED DETECTION SYSTEMS
The numerical simulation is developed to estimate the signal-to-background ratios and the statistical significance of light vector mesons via di-electron decays in heavy-ion collisions. Instead of directly simulating the multi-particle production with detailed dynamics in heavy-ion collisions, high multiplicity states are first represented by the form of the total pion multiplicity dN π 0 +π ± /dy. The production of the relevant particles other than pions is determined based on the individual production cross sections relative to that of pions as a function of the transverse momentum p T . They are evaluated by the measured data points, or the extrapolation via the proper scaling for missing data points. In addition, the key experimental parameters for the di-electron measurement are set as the inputs. As the idealized baseline parameters on the experimental conditions, we choose the following set of parameters:
1. photon conversion probability P cnv : 1 %, 2. rejection factor of charged pions R π ± , which is defined by the inverse of the probability that charged pions are identified as electrons: 500,
(0.01 · p T ) 2 + (0.0056) 2 GeV/c.
Some of experimental parameters are correlated in fact, but these correlations are neglected in this simulation. Figure 1 shows the flowchart of the numerical simulation. The step (1) sets the input parameters above. In the step (2), primary particles are generated with the weights of the invariant p T spectra. The p T spectra are provided by the experimental data and the proper scaling for missing data. The details of the input p T spectra for each are explained later. Rapidity y of a particle is uniformly generated in |y| ≤ 0.5 [25,35]. Primary particles branch into subsequent decay processes according to their branching ratios. The branching ratios are summarized in Table II of C. φ, ω and ρ mesons decay into di-electrons through the two-body decay process in the step (3). The phase space of di-electrons from the light vector mesons is determined by the Gounaris-Sakurai model [36]. π 0 and η mesons branch into 2γ or the Dalitz decay process (γe + e − ) in the step (3) or (4). The decaying γ's are subsequently converted into di-electrons with the given photon-conversion probability in the step (5). Kinematics of di-electron in the photonconversion process, that is, energy and scattering angle, are simulated by the well-established GEANT algorithm [37][38][39]. All photon-conversion points are fixed to the primary vertex points [83]. The phase space of Dalitz decaying di-electrons is determined by the Kroll-Wada formula [40,41]. The detailed formula is expressed in D.
Charged kaons decay into electrons through the threebody decay process in the step (6). In the step (7), electrons and positrons from open charms are directly generated to be consistent with the input p T spectra of single electrons [84] as shown in Fig.2. They are randomly generated in azimuth with the branching ratio of 9.5 % [46]. The correlation of a di-electron pair originating from the open charm production is neglected in this simulation. The effect on the correlation is discussed in Section IV. Charged pions are identified as electrons with the given probability corresponding to the rejection factor of charge pions in the step (8). At the final stage of the simulation, final-state electrons are filtered by the geometrical acceptance, the electron tagging efficiency and p T threshold in the step (9). The algorithms of decaying processes mentioned above are developed based on the EXODUS simulator [47]. The dN π 0 +π ± /dy and the invariant p T spectra are applied to the simulation taking the given collision species and energies into account. The input dN π 0 +π ± /dy is estimated by the measured dN ch /dy [25][26][27]. dN π 0 +π ± /dy = 3, 6 and 1000 are set for the simulation of p+p 200 GeV, p+p 7 TeV and central Au+Au 200 GeV collisions, respectively. For the simulation of central Pb+Pb 5.5 TeV collisions, the dN π 0 +π ± /dy is estimated by the extrapolation of the scaling curve as a function of collision energy [27]. The extrapolated dN π 0 +π ± /dy corresponds to 2700.
The input p T spectra for the p+p 200 GeV simulation are determined by the measured data points with the fits in the panel (a) of Fig.2. The Tsallis function, whose properties are explained in A, is used as the fitting function.
The invariant p T spectra in central Au+Au collisions are well known to be suppressed and scaled by the number of participant nucleons N part in the high p T region. The scaling parameter N part is calculated by the Monte Carlo simulation based on the Glauber model [48,49]. The N part scaling helps to extend the p T spectra in p+p collisions to those in heavy-ion collisions, even if there is lack of data points. Therefore we can determine the input p T spectra in the wide range by combining the existing data points with the scaling properties. The panel (b) of Fig.2 shows the data points and the scaling curves in Au+Au 200 GeV at the centrality class of 0-10 % [85]. The dotted curves are obtained by the N part scaling. The spectrum shape is assumed to be the same as that of p+p 200 GeV. These curves are reasonably consistent with the data points. The solid curves on the data points of φ mesons, charged kaons and single electrons from heavy flavor decays are obtained by directly fitting with the Tsallis function. For charged kaons, the Tsallis parameter q is fixed because of the missing data in the high p T region.
The panel (c) of Fig.2 shows the invariant p T spectra in p+p 7 TeV. The p T spectra of π 0 , η, φ and single electrons are determined by the data points and the fits. The dotted curves show the p T spectra of the other hadrons. The spectrum shape is estimated by the m T scaling based on the π 0 data points, where m T = p 2 T + m 2 0 and m 0 is the rest mass of a particle. Their absolute production cross sections are estimated by the inclusive ratios between pions and the other hadrons in p+p 200 GeV. The invariant p T spectra in p+p 7 TeV are commonly used for the simulations of p+p 7 TeV and Pb+Pb 5.5 TeV, since the relative production cross sections between pions and the other hadrons are expected to be common for both collision systems, as long as the particle production between 7 TeV and 5.5 TeV has little dependence on the collision energy. Table I in B summarizes these production cross sections and inclusive yields over all p T ranges in p+p 200 GeV, p+p 7 TeV and Au+Au 200 GeV at the centrality class of 0-10 %. [50] and (π + + π − )/2 [51,52] are obtained by the simultaneous fitting. The curves on the data points of (K 54] are also obtained by the simultaneous fitting. The star symbols show η → γγ [55] and η → π 0 π + π − [56]. The open diamonds show ρ → π + π − [57]. The triangles show ω → e + e − , π 0 π + π − and π 0 γ [53]. The squares show φ → e + e − and K + K − [53]. The asterisks show single electrons from heavy flavor decays [46]. (b) The invariant pT spectra in Au +Au 200 GeV at the centrality class of 0-10%. The dotted curves are scaled by the Npart and assumed to be the same spectrum shape as that of p+p 200 GeV. The scaling curves are consistent with the data points of pions [58][59][60] , η [61] and ω [62], respectively. The solid curves are the fitting results to the data points of K ± [60], φ [63] and single electrons from heavy flavor decays [64]. For K ± , the Tsallis parameter q is fixed since there is no data point in the high pT region. (c) The differential cross section in p+p 7 TeV. The data points of π 0 /η [65], φ [66] and single electrons from heavy flavor decays [67] are shown in this figure. Tsallis fitting curves are depicted as the solid curves. The dotted curves of ρ, ω and K ± are obtained by assuming the same spectrum shape of π 0 and normalizing the individual production ratios with respect to pions in p+p 200 GeV. The invariant mass spectrum of photon-conversion pairs obeys dynamics of the pair-creation process in materials. The invariant mass spectrum of Dalitz decaying pairs has a character whose leading edge is the summation of masses of decay products and the distribution continues up to their parent masses. The mass spectrum of cc → e + e − is reconstructed by randomly pairing dielectrons in azimuth. The inclusive invariant mass spectra are shown in Fig.5. The component of signal pairs, combinatorial background pairs and all background pairs is superimposed in the figures. The fraction of individual components depends on the given multiplicities, collision species and energies. The peaks of the light vector mesons are clearly seen at the multiplicities in p+p collisions, but hardly seen above dN π 0 +π ± /dy = 1000, though the statistical significance is not necessarily small. The quantitative evaluations of the signal-to-background ratios and the statistical significance are discussed in the next section.
+ + K − )/2 [51, 52], K 0 s → π 0 π 0 [53] and K 0 s → π + π − [
III. SIGNAL-TO-BACKGROUND RATIOS AND THE STATISTICAL SIGNIFICANCE
The feasibility to measure φ/ω/ρ → e + e − is evaluated by the signal-to-background ratios and the statistical significance in the signal mass region. The signal mass region for each meson is defined as the invariant mass range of M φ,ω,ρ ± 3 × Γ 2 φ,ω,ρ + σ 2 φ,ω,ρ , where M φ,ω,ρ is the mass center and Γ φ,ω,ρ is the decay width. M φ,ω,ρ and Γ φ,ω,ρ are cited from the particle data group [68]. The mass resolutions σ φ,ω,ρ are calculated by the single particle simulation [86] and result in 7.6, 5.7 and 5.6 MeV/c 2 for φ, ω and ρ mesons, respectively. Figure 6 and 7 show the signal-to-background ratios S/B as a function of the experimental parameters in central Au+Au collisions at √ s N N = 200 GeV (dN π 0 +π ± /dy = 1000) and central Pb+Pb collisions at √ s N N = 5.5 TeV (dN π 0 +π ± /dy = 2700), respectively. Only one parameter is changed by fixing the other parameters at the baseline values: P cnv = 1 %, R π ± = 500, ǫ acc = 100 %, ǫ tag = 100 %, p th T = 0.1 GeV/c and
σ ref pT = (0.01 · p T ) 2 + (0.0056) 2 GeV/c.
The top-left figure shows the S/B as a function of photon-conversion probability P cnv . The minimum amount of detector materials typically corresponds to P cnv = 1-2 %, because photon conversions from the beam pipe and the first layer of the innermost detector are unavoidable in any detector system, even though electron trajectories coming from the off-axis point are rejected by tracking algorithm.
dN π 0 +π ± /dy with Pcnv = 1 %, R π ± = 500, ǫacc = 100 %, ǫtag = 100 %, σ ref p T = (0.01 · pT ) 2 + (0.0056) 2 GeV/c and without pT cutoff.
Thus the tendency below P cnv = 10 % is important for the detector system with typical amount of the materials.
The dependence on the rejection factor of charged pions R π ± is shown in the top-right plot of Fig.6 and 7. Typical devices for the electron identification have the rejection factor of a few hundreds in the stand-alone operation [23,28,69,70], although it varies by the principle of detection. Therefore, the information in the range of R π ± = 100-1000 are useful. The S/B can be changed by a factor of 3-5 for φ/ω meson in this range.
The bottom-left figure shows the S/B as a function of the azimuthal acceptance ǫ acc . The S/B depends on decay kinematics of the signal particles and the backgrounds. Therefore the geometrical configuration in azimuthal coverage as well as the absolute acceptance in azimuth should be taken into account. Two types of geometrical configurations are considered in this simulation. Type I simply covers the azimuthal range of 0 ≤ φ ≤ φ 1 . Type II covers two separated domains which are symmetrically arranged in azimuth with respect to the collision point, that is, the coverage is set to 0 ≤ φ ≤ φ1 2 and π ≤ φ ≤ π + φ1 2 . Both of them have the same total acceptance in azimuth with different geometry. The difference between the two geometrical configurations increases in the case of the imperfect coverage. If ǫ acc is 40 %, for instance, the S/B differs by a factor of 3-4 for φ/ω meson depending on the detector geometry.
The statistical significance S/ √ S + B depends on the square root of the number of events, in other words, de-pends on available luminosity in experiments. Depending on the available statistics in the specific collision centrality and the detector conditions, we can evaluate whether a detection system is able to measure the light vector mesons with a reasonable statistical significance or not.
IV. RESIDUAL EFFECTS
The signal-to-background ratios and the statistical significance of the light vector mesons are evaluated with the idealized detection system so far. In this section, we discuss the non-trivial aspects originating from the real data analysis and the correlations in the open charm production. As the other issues beyond the scope of the numerical simulation, we mention the track reconstruction algorithm bias, the correlation between the electron identification and the rejection of charged hadrons, and the fiducial effect on the acceptance to charged particles in the magnetic field. The studies of this section are performed by simulating 5 M events in Au+Au 200 GeV and Pb+Pb 5.5 TeV with the baseline parameters set: P cnv = 1 %, R π ± = 500, ǫ acc = 100 %, ǫ tag = 100 %, p th T = 0.1 GeV/c and σ ref pT = (0.01 · p T ) 2 + (0.0056) 2 GeV/c. The numerical simulation so far is performed under the assumption that we know the exact number of signals and backgrounds. In the real data analysis, however, the source of any electron cannot be identified. Therefore all electrons and positrons are combined into pairs and reconstructed into the invariant mass. The mass distribution of pairs from one source ("true pairs") is extracted by subtracting that of pairs from different source ("combinatorial pairs") statistically. The mass shape of combinatorial pairs is estimated by mixing an electron in an event and a positron in another event ("event mixing") [71,72]. The mass distribution of event-mixing pairs is normalized by 2 N ++ N −− /N mix +− , where N ++ , N −− and N mix +− are the numbers of positron-positron pairs, electron-electron pairs and event-mixing electronpositron pairs, respectively. This normalization factor is valid as long as electrons and positrons are produced as pairs and they have the same acceptance [1]. The mass distribution of true pairs includes the light vector mesons and the other sources. The contributions from the light vector mesons and the background sources are separately estimated by the fits based on the linear combination between the Breit-Wigner function convoluted with the Gauss function and an empirical function. We apply a series of procedures used in the real data analysis to the simulated data and evaluate how much the signal-to-background ratios change by applying these procedures. The top plots of Fig.14 show the comparison of the invariant mass spectra between the combinatorial pairs and the event-mixing pairs. The comparison is performed in central Au+Au collisions at √ s N N = 200 GeV (dN π 0 +π ± /dy = 1000) and central Pb+Pb collisions at √ s N N = 5.5 TeV (dN π 0 +π ± /dy = 2700), respectively. The ratio between the number of combinatorial pairs and that of event-mixing ones is close to unity within a few % of statistical fluctuations below the mass of 1.0 GeV/c 2 in both collision systems as shown in the middle plots of Fig.14. Therefore the event-mixing pairs in the real data analysis can provide the reliable baseline representing the combinatorial pairs which is known only at the simula-tion study [87]. The bottom plots of Fig.14 show the invariant mass distribution after subtracting the eventmixing distribution. The linear combination between the Breit-Wigner function convoluted with the Gauss function and a first-order polynomial function is used as the fitting function. In the mass range of φ meson, we use
dN e + e − dM e + e − = A F φ (M ′ ) G gauss (M e + e − − M ′ ) dM ′ +H bg (M e + e − ) .(1)
In the mass range of ω/ρ meson, we use
dN e + e − dM e + e − = A {RF ω (M ′ ) + (1 − R) F ρ (M ′ )} G gauss (M e + e − − M ′ ) dM ′ + H bg (M e + e − ) , R = N ω BR (ω → e + e − ) N ω BR (ω → e + e − ) + N ρ BR (ρ → e + e − ) ,(2)
where N ω and N ρ are the inclusive yields of ω and ρ meson, respectively. The absolute values of the inclusive yields are fixed to the measured values listed in Table I
F φ,ω,ρ (M ′ ) = Γ φ,ω,ρ /2π (M ′ − M φ,ω,ρ ) 2 + (Γ φ,ω,ρ /2) 2 ,(3)G gauss (M e + e − − M ′ ) = 1 √ 2πσ e −(M e + e − −M ′ ) 2 /2σ 2 ,(4)H bg (M e + e − ) = BM e + e − + C,(5)
where the mass center M φ,ω,ρ and the width Γ φ,ω,ρ of the light vector mesons are fixed to their intrinsic values [68], whereas the mass resolution σ is a free parameter. A, B and C in the equations are normalization factors. The fitting ranges are from 0.9 to 1.2 GeV/c 2 for φ meson and from 0.6 to 0.9 GeV/c 2 for ω/ρ meson. The number of the light vector mesons is counted by the integration of the convolution function over the signal mass region. The definition of the signal mass region is mentioned in Section III. The signal-to-background ratios estimated by fitting are 8.4 × 10 −2 , 2.0 × 10 −2 and 4.1 × 10 −4 for φ, ω and ρ mesons, respectively, in central Au+Au collisions at √ s N N = 200 GeV (dN π 0 +π ± /dy = 1000) with the baseline parameter set. The differences of the signal-to-background ratios are 4.9 % (φ), 7.4 % (ω) and 8.8 % (ρ) compared to those by the simple counting of the simulated true pairs. In central Pb+Pb collisions at √ s N N = 5.5 TeV (dN π 0 +π ± /dy = 2700), The signal-tobackground ratios are 1.7 × 10 −2 , 6.7 × 10 −3 and 1.7 × 10 −4 for φ, ω and ρ mesons, respectively. They correspond to 3.2 % (φ), 12.1 % (ω) and 14.3 % (ρ) differences with respect to the case of the simple counting of the simulated true pairs. The differences depend on the experimental parameters. It is unlikely for them to exceed 50 % at a realistic range of the experimental parameters [88].
Electrons and positrons from open charms are randomly generated and combined into pairs in this simulation. These pairs are, in fact, azimuthally correlated at mid-rapidity, because they originate from the jets due to the large mass of charm quarks. We assume the back-toback e + e − correlation in azimuth as the extreme case of the open charm production. Realistic correlations would exist between the random pairing case and the back-toback correlated case. The top plots in Fig.15 show the invariant mass spectra reconstructed from all true pairs, combinatorial pairs and cc → e + e − , respectively, in central Au+Au collisions at √ s N N = 200 GeV (dN π 0 +π ± /dy = 1000) on the left and in central Pb+Pb collisions at √ s N N = 5.5 TeV (dN π 0 +π ± /dy = 2700) on the right. The distributions of the random di-electron pairs and the back-to-back correlated ones in azimuth are superimposed in the same plot. The middle plots show the ratio of the number of cc → e + e − as a function of the invariant mass. The denominator is the number of di-electrons with random pairing and the numerator is the number of di-electrons with the back-to-back correlation. This ratio varies by a factor of 1.5 to 3 around the mass range of the light vector mesons. The ratio between the number of combinatorial pairs in the random pairing case and in the back-to-back correlated case is consistent within only a few % in both collision systems as shown in the bottom figures. Therefore the correlation of the cc production has little influence on the signal-to-background ratios of the light vector mesons.
In addition to above issues, the other residual effects, which are beyond the scope of this study, are listed below.
• Track reconstruction algorithms bias: Track reconstruction algorithms can bias momentum measurement of charged particles. For example, the algorithm based on the combinatorial Hough transform technique [73][74][75] reconstructs higher momentum than true one, especially for a charged particle producing from the off-axis point. The photonconversion electrons at the off-axis point contribute to the background shape in the relatively higher mass region. In addition, especially under a high multiplicity environment, fake tracks are reconstructed by chance depending on the algorithms. These tracks can contribute as the additional backgrounds.
• The correlation between the tagging efficiency of electrons and the rejection factor of charged hadrons: The correlation between the electron tagging efficiency and the rejection factor of charged hadrons depends on the method of particle identification. For instance, if particles are identified by dE/dx, the correlation has a trade-off relation. Another example is the degradation by the situation where a number of particles simultaneously pass through the detector. If a hadron and an electron enter the same area of the electron identification device, either or both of them can be wrongly identified. In more general case, complicated correlations may appear, since particles are identified with a combination of multiple devices.
• The fiducial effect in the magnetic field: This simulation considers the detector acceptance under the assumption that di-electron kinematics is completely reconstructed. In real experiments, charged particles are bent in the magnetic field and entered into the imperfect coverage of the detectors. The fiducial effect becomes apparent at the edge of the acceptance. Therefore the inefficiency of the electron detection should be taken into account as a function of the magnetic field, the detector positions from the collision point and the detector configurations.
Nevertheless, our simulation study would provide a useful guideline to evaluate the effect of the non-residual factors on measurability of the light vector mesons.
V. CONCLUSION
This paper provides a guideline to evaluate the minimum requirements in order for a given idealized detection system to achieve reasonable statistical significance for the measurement of light vector mesons via di-electron decays in different temperature states, dN π 0 +π ± /dy = 1000 and 2700 intended for the most central Au+Au collisions at √ s N N = 200 GeV and the most central Pb+Pb collisions at √ s N N = 5.5 TeV, respectively. The simulation codes used for this study are openly available [76].
The results suggest that parameter ranges for the measurement of φ and ω mesons are selectable in designing a detector system depending on the number of events at the highest centrality class, even if the residual effects caused by the procedure of the real data analysis and the correlations in the open charm production are considered. The statistical significance of ρ mesons is less than that of φ and ω mesons due to its broad mass shape and the limited mass resolution, even if sufficiently high luminosity is prospected. However, the mass spectrum of ρ mesons in the vacuum can be indirectly evaluated as long as φ and ω mesons spectra are accurately determined. This would provide the baseline to understand the properties of the low-mass di-electron continuum. The production cross sections and the inclusive yields over all pT ranges at midrapidity for different collision systems. The used data points to calculate the production cross sections and the inclusive yields are cited from the publications listed in the second column for each collision system. The production cross section of single electrons is obtained by the Tsallis fit to the measured data points and converted into the cc cross section with the branching ratio of 9.5 % [46]. The production cross sections of the other particles are obtained by fitting to the measured data points with the Tsallis function, or assuming the proper scaling for missing data points. The details are explained in Section II. The errors of the production cross sections and the inclusive yields are expected to be from 10 to 30 % depending on particles. These errors are neglected in the simulation. Masses and branching ratios are cited from the particle data group [68]. The branching ratio of c → e is assumed to be 9.5 % [46].
Appendix D: Dalitz decays of pseudo-scalar mesons
Pseudo-scalar mesons such as π 0 and η mesons mainly decay into two photons. The Dalitz decay corresponds to the case where photons become off-shell and subsequently decay into di-electrons. The relation between 2 γ decay process (P → γγ) and the Dalitz decay process (P → γe + e − ) is described by Kroll-Wada formula [40,41]
F P Q 2 = 1 − Q 2 Λ 2 P −1 ,(D1)
where M e + e − is the invariant mass of di-electrons, m e is the rest mass of electron and m P is the rest mass of a parent meson. F P Q 2 is the electro-magnetic transition form factor. Q 2 is equivalent to the square of the virtual photon mass (i.e. Q = M e + e − ). The measurements of the form factor by the experiments [79,80] show Λ P ≃ M ρ , where M ρ is the rest mass of ρ meson. The Kroll-Wada formula determines the branching ratio and the phase space of Dalitz decaying di-electron.
FIG. 1 :
1The flowchart of the numerical simulation.
The main targets of this simulation are central Au+Au collisions at √ s N N = 200 GeV and central Pb+Pb collisions at √ s N N = 5.5 TeV. The results of p+p 200 GeV and p+p 7 TeV simulations are used as the references.
Figure 3
3shows the simulated results of the p T spectra for the final-state electrons from individual sources with the baseline parameter set in p+p collisions at √ s = 200 GeV, p+p collisions at √ s = 7 TeV, central Au+Au colli-online) (a) The differential cross sections of different particles in p+p 200 GeV. Tsallis fitting curves are depicted as the solid curves on the data points. The solid curve on π 0 → γγ
sions at √ s N N = 200 GeV and central Pb+Pb collisions at √ s N N = 5.5 TeV, separately. The parents of electrons are all indicated with different symbols specified inside the plot.
Figure 4
4shows the invariant mass distributions of dielectron pairs with dN π 0 +π ± /dy = 3, 6, 1000, and 2700, respectively. The components from individual di-electron sources are indicated with different types of curves in the figure. The mass shapes of φ, ω and ρ characterize their short lifetimes and show Breit-Wigner resonance peaks.
FIG. 3 :
3(color online) The transverse momentum spectra of final-state electrons and misidentified charged pions from individual sources for the given
Figure 8 -
813 show the statistical significance as a function of the experimental parameters in central Au+Au collisions at √ s N N = 200 GeV (dN π 0 +π ± /dy = 1000) and in central Pb+Pb collisions at √ s N N = 5.5 TeV (dN π 0 +π ± /dy = 2700) for φ, ω and ρ meson, separately. The data points and the empirical curves are shown as fulfilled symbols and the solid curves in the figures. The number of simulated events for central Au+Au collisions and central Pb+Pb collisions corresponds to 5M and 1M events, respectively. The other dotted curves show the scaled curves with the square root of the expected number of events with the highest centrality selection. The two horizontal lines indicate S/ √ S + B = 3 and 5. The S/B is independent of the electron tagging efficiency ǫ tag , whereas the S/ √ S + B scales with the square root of the statistics. Therefore we added the dependence on the ǫ tag to the bottom figure for the discussion on the statistical significance.
FIG. 4 :
4(color online) The invariant mass spectra of dielectrons from individual sources for the given dN π 0 +π ± /dy with Pcnv = 1 %, R π ± = 500, ǫacc = 100 %, ǫtag = 100 %, p th T = 0.1 GeV/c and σ ref p T = (0.01 · pT ) 2 + (0.0056) 2 GeV/c. The mass spectra from individual origins are shown with different curves specified inside the plot. The curves of the combinatorial pairs are reconstructed by all combinations between electrons and positrons but only true combinations are excluded.
FIG. 5 :
5(color online) The inclusive mass spectra compared to the components of signal pairs, all background pairs and combinatorial background pairs for the given dN π 0 +π ± /dy with Pcnv = 1 %, R π ± = 500, ǫacc = 100 %, ǫtag = 100 %, p th T = 0.1 GeV/c and σ ref p T = (0.01 · pT ) 2 + (0.0056) 2 GeV/c. The simulated number of events for each collision system is shown inside the plot.
FIG. 6 :FIG. 7 :FIG
67(color online) The signal-to-background ratio S/B of φ, ω and ρ meson as a function of the experimental parameters Pcnv, R π ± and ǫacc in central Au+Au collisions at √ sNN = 200 GeV (dN π 0 +π ± /dy = 1000). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. Type I on the bottom figure shows the azimuthal coverage of 0 ≤ φ ≤ φ1. Type II shows two separated coverages of 0 ≤ φ ≤ φ 1 2 and π ≤ φ ≤ π + φ 1 2 . (color online) The signal-to-background ratio S/B of φ, ω and ρ meson as a function of the experimental parameters Pcnv, R π ± and ǫacc in central Pb+Pb collisions at √ sNN = 5.5 TeV (dN π 0 +π ± /dy = 2700). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. Type I of the bottom figure shows the azimuthal coverage of 0 ≤ φ ≤ φ1. Type II shows two separated coverages of 0 ≤ φ ≤ φ 1 2 and π ≤ φ ≤ π + φ 1 2 . . 8: (color online) The statistical significance S/ √ S + B of φ mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Au+Au collisions at √ sNN = 200 GeV (dN π 0 +π ± /dy = 1000). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
FIG
. 9: (color online) The statistical significance S/ √ S + B of ω mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Au+Au collisions at √ sNN = 200 GeV (dN π 0 +π ± /dy = 1000). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
GeV (dN FIG. 10: (color online) The statistical significance S/ √ S + B of ρ mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Au+Au collisions at √ sNN = 200 GeV (dN π 0 +π ± /dy = 1000). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
FIG. 11: (color online) The statistical significance S/ √ S + B of φ mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Pb+Pb collisions at √ sNN = 5.5 TeV (dN π 0 +π ± /dy = 2700). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
FIG. 12: (color online) The statistical significance S/ √ S + B of ω mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Pb+Pb collisions at √ sNN = 5.5 TeV (dN π 0 +π ± /dy = 2700). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
FIG. 13: (color online) The statistical significance S/ √ S + B of ρ mesons as a function of the experimental parameters Pcnv, R π ± , ǫacc and ǫtag in central Pb+Pb collisions at √ sNN = 5.5 TeV (dN π 0 +π ± /dy = 2700). Only one parameter is changed by fixing the other parameters at the baseline values for each plot. The results of the simulation are shown as the symbols and the empirical curves are superimposed on the data points as the solid curves. The other dotted curves are the scaled curves with the square root of the expected number of events found in the highest centrality class. Two horizontal lines indicate S/ √ S + B = 3 and 5.
of B. BR (ω → e + e − ) and BR (ρ → e + e − ) are the branching ratios to a di-electron for ω and ρ meson, respectively. F φ,ω,ρ (M ′ ) in Eq.(1)and(2)indicate the Breit-Wigner function describing the intrinsic mass spectra of the light vector mesons and G gauss (M e + e − − M ′ ) shows the Gauss function expressing the smearing effect caused by the transverse momentum resolution. The residual backgrounds are assumed to follow a first-order polynomial function H bg (M e + e − ). These functions are expressed as
FIG. 15: (color online) The comparison of the invariant mass spectra for different correlations of di-electrons from charm quarks. The panel (a) and (b) on the top figures show the invariant mass spectra in cases of random di-electron pairs and back-to-back correlated di-electron ones in azimuth, respectively. The middle figures show the ratio between the number of cc → e + e − in the random pairing case and in the back-to-back correlated one. The bottom figures show the ratio between the number of combinatorial pairs in the random pairing case and in the back-to-back correlated one.
TABLE I :
I
Appendix C: Masses, decay products and branching ratiosParticle Mass (GeV/c 2 ) Decay products Branching ratioφ
1.01946
e + e −
2.954 × 10 −4
ω
0.78265
e + e −
7.28 × 10 −5
ρ
0.77549
e + e −
4.72 × 10 −5
π 0
0.13498
γγ
0.98823
γe + e −
0.01174
η
0.54785
γγ
0.03931
γe + e −
7.0 × 10 −3
π +
0.13957
N/A
N/A
π −
0.13957
N/A
N/A
K +
0.49368
e + π 0 νe
0.0507
K −
0.49368
e − π 0ν
e
0.0507
cc
N/A
e + e −
0.095
TABLE II :
IIMasses, decay products and branching ratios of the light vector mesons and the other background particles.
AcknowledgmentsWe thank Prof. T. Sugitate for his suggestions on this paper and supports especially for establishing the computing environment. We thank Prof. K. Shigaki and all the colleagues at Hiroshima university for many discussions. The numerical simulation study has been performed on the parallel processing system at the Data Analysis Laboratory for High-Energy Physics at Hiroshima University.The Tsallis function[77]is widely used for explaining the properties of particle productions. At midrapidity, the total energy of each particle is approximately represented by the transverse mass m T = p 2 T + m 2 0 , where m 0 is the rest mass of a particle. The Tsallis function is formulated as a function of m T as follows,where n = −1/ (1 − q), and dσ/dy is the production cross section over all p T ranges at midrapidity. Equation (A1) simultaneously represents the power-law behavior at high p T and the exponential behavior at low p T by the two parameters q and T .In the limit of m 0 → 0, Eq.(A1) becomesEquation (A2) is similar to the pQCD expression by Hagedron[78]. The parameter n is directly related to the exponent k of the pure power-law shape of pQCD calculations. The relation between n and k is expressed asIn the limit of m 0 → 0 and q → 1 (i.e. n → −∞), Eq. (A1) becomeswhere A is a normalization factor. Equation (A4) is equivalent to the Boltzmann distribution. The slope parameter T characterizes the thermal production of particles. [81] The production ratios between pions and light vector mesons depend on the collision species and energies. The exact values of these ratios are calculated one-by-one for the given collision species and energies. Refer toTable Iin B.[82] The yields and kinematics of produced particles vary by the collision species and energies. They are fully taken into account in the simulation. The details are explained in Section II.[83] The contribution to the di-electron background shape depends on where the photon conversion takes place, in other words, depends on the arrangement of the detector materials. They should be considered in association with the track reconstruction algorithms. The biases from photon-conversion electrons at the off-axis point are discussed in Section IV.[84] The pT spectrum of single electrons originates from not only charm quarks but also beauty quarks. The contributions from them are calculated by the fixedorder-plus-next-to-leading-log perturbative QCD calculation (FONLL)[42]and its calculation is compared to the measurements[43][44][45]. The results suggest that N b→e / (Nc→e + N b→e ) is smaller than 0.2 at pT ≤ 2.0 GeV/c in p+p 200 GeV and N b→e /Nc→e are smaller than 0.3 at pT ≤ 2.0 GeV/c in p+p 7 TeV, where N b→e and Nc→e is the number of electrons from beauty quarks and charm quarks, respectively. These ratios tend to drop rapidly as the pT reduces. Therefore we neglect the contributions from beauty quarks in the calculation of particle production and decay kinematics in this simulation.[85] We use the published data points of π ± and K ± at the centrality class of 0-5 % and those of ω at the centrality class of 0-20 %. The mismatch of the centrality class is corrected by the weights with the Npart. The corrected data points, which are equivalent to the data at the centrality class of 0-10 %, are used for the comparison to the scaling curves in the panel (b) ofFig.2.[86] The mass resolutions are calculated as follows. φ, ω and ρ mesons are singly generated under the condition that their mass widths are set at zero, respectively. The mass distributions fluctuate around individual mass centers due to only the transverse momentum resolution σ ref p T . The mass resolutions σ φ,ω,ρ are estimated by the fits with the Gauss function.[87] We note that the normalization factor of 2 √ N++N−−/N mix +− overestimates the combinatorial backgrounds by 0.05-0.3 %. For instance, if a detection system has large amount of materials (i.e. Pcnv is high) or has a poor capability of hadron rejection (i.e. R π ± is low), this estimation excessively subtracts the backgrounds.[88] The differences of the signal-to-background ratios between the estimation with the fit and the counting of the simulated true pairs originate from the background shape mainly depending on the subtraction of the combinatorial background. The differences are studied at the experimental parameter range of 1 < Pcnv < 5 % or 100 < R π ± < 500 and result in ∼ 20 % for the three mesons in both collision systems.
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| []
|
[
"Stability and upper bounds for statistical estimation of unbalanced transport potentials",
"Stability and upper bounds for statistical estimation of unbalanced transport potentials"
]
| [
"Adrien Vacher [email protected] \nINRIA Paris\n2 rue Simone Iff75012ParisFrance\n",
"François-Xavier Vialard [email protected] \nINRIA Paris\n2 rue Simone Iff75012ParisFrance\n",
"\nLaboratoire Informatique Gaspard Monge\nUniv Gustave Eiffel\nCNRS\nLIGM\nF-77454Marne-la-ValléeFrance\n"
]
| [
"INRIA Paris\n2 rue Simone Iff75012ParisFrance",
"INRIA Paris\n2 rue Simone Iff75012ParisFrance",
"Laboratoire Informatique Gaspard Monge\nUniv Gustave Eiffel\nCNRS\nLIGM\nF-77454Marne-la-ValléeFrance"
]
| []
| In this note, we derive upper-bounds on the statistical estimation rates of unbalanced optimal transport (UOT) maps for the quadratic cost. Our work relies on the stability of the semi-dual formulation of optimal transport (OT) extended to the unbalanced case. Depending on the considered variant of UOT, our stability result interpolates between the OT (balanced) case where the semi-dual is only locally strongly convex with respect the Sobolev semi-normḢ 1 and the case where it is locally strongly convex with respect to the H 1 norm. When the optimal potential belongs to a certain class C with sufficiently low metric-entropy, local strong convexity enables us to recover super-parametric rates, faster than 1/ √ n. | null | [
"https://arxiv.org/pdf/2203.09143v1.pdf"
]
| 247,519,190 | 2203.09143 | 8a680eab58623733159e8aa3eee179a478db2cda |
Stability and upper bounds for statistical estimation of unbalanced transport potentials
Adrien Vacher [email protected]
INRIA Paris
2 rue Simone Iff75012ParisFrance
François-Xavier Vialard [email protected]
INRIA Paris
2 rue Simone Iff75012ParisFrance
Laboratoire Informatique Gaspard Monge
Univ Gustave Eiffel
CNRS
LIGM
F-77454Marne-la-ValléeFrance
Stability and upper bounds for statistical estimation of unbalanced transport potentials
In this note, we derive upper-bounds on the statistical estimation rates of unbalanced optimal transport (UOT) maps for the quadratic cost. Our work relies on the stability of the semi-dual formulation of optimal transport (OT) extended to the unbalanced case. Depending on the considered variant of UOT, our stability result interpolates between the OT (balanced) case where the semi-dual is only locally strongly convex with respect the Sobolev semi-normḢ 1 and the case where it is locally strongly convex with respect to the H 1 norm. When the optimal potential belongs to a certain class C with sufficiently low metric-entropy, local strong convexity enables us to recover super-parametric rates, faster than 1/ √ n.
Introduction
In its original formulation, OT is a tool to compare probability distributions: it seeks a map that optimally transports one distribution µ to an other distribution ν with respect to some fixed cost c and it returns the associated transport cost. This problem was later relaxed into a linear program by Kantorovitch and its primal formulation consists into seeking a coupling instead of a map with minimal cost and whose marginals are constrained to be µ and ν; under suitable assumptions on the measures and the cost, this relaxation is tight (Brenier, 1991). Quite recently, OT was extended to arbitrary positive measures (Chizat, 2017), with possibly different masses, thus the name Unbalanced Optimal Transport (UOT). On the primal problem, the hard marginal constraints are relaxed by soft entropic penalties. From an applied point of view, the mass constraint relaxation is indeed a desirable feature: it allows not only displacement of mass but also local growth or shrinkage (Séjourné et al., 2019). In image processing for instance, it can remove or at least decreases blurred areas in favor of sharper contrasts Feydy et al. (2017). From a statistical point of view, UOT may appear as a more robust version of OT as it is able to cut down the outliers. At the heart of classical OT, rather than the mere OT distance value, the main object of interest is the transport map: in generative imaging, we use the transport map to generate images from noise , for point cloud approximation, the particle flow is driven by the maps (Mérigot et al., 2021) and in Domain Adaptation, the source distribution is transported on the target using an OT map (Courty et al., 2017). Novel applications include predicting the evolutions of cells from measurements Schiebinger et al. (2019); Yang et al. (2020). Notably in the case of a quadratic cost, Brenier showed that these maps are linked with the dual formulation of Kantorovitch relaxation: the map from µ to ν corresponds to the the gradients of the first variable of the dual problem that we shall refer to as a potential. This potential is the solution of a linear program, yet with an infinite dimensional constraint. Hence, even in the case where the measures µ and ν are known analytically, there is in general no closed form to compute the OT potentials. In particular, recent methods instead rely on n-samples empirical counterparts of µ and ν to statistically estimate the cost and the potentials (Genevay et al., 2016;Seguy et al., 2018). Such methods are thus concerned with the statistical estimation of optimal transport quantities, such as the cost or the potentials, see Zemel (2019, 2020) for an overview of this rapidly growing field. If the distributions µ, ν are only assumed to have a density w.r.t. the Lebesgue measure, the error achieved by the plug-in estimator, which is simply the OT between the empirical samples, scales in O(n − 2 d ) (Chizat et al., 2020). In particular, without further assumptions the OT problem is said to suffer the curse of dimension. However, in the seminal work of Hütter and Rigollet (2021), the authors showed that if the original potential is α-smooth, then its gradient could be at best estimated in n − 2(α+1) 2α+d with respect to the squared L 2 distance. They also provided an estimator that actually achieved this rate of estimation, hence providing a minimax rate under smoothness assumptions. Yet, we emphasize the fact that their estimator as such in infeasible as it requires in particular to project on the space of convex, k-times differentiable functions. This result has triggered follow-ups on computationally feasible and efficient estimators of the optimal transport maps, for instance Muzellec et al. (2021) leveraging the underlying smoothness of the optimal maps and Pooladian and Niles-Weed (2021); Pooladian et al. (2022);Deb et al. (2021) using entropic regularization.
In this note, we propose to explore the results of statistical estimation of transport potentials to the unbalanced setting and derive upper-bounding rates. In particular, we restrict ourselves to probability measures instead of positive measures, yet it does not affect the relevance of using UOT instead of classical OT as it allows to handle outliers. Instead of making explicit smoothness assumptions on the ground truth, we assume that it belongs to a certain class C and derive rates of estimation depending on the complexity of C, namely its metric entropy. In particular, our statistical analysis relies on an unbiased estimation, where our search space for the empirical candidate is the same set C, that is assumed to contain the ground truth. Using the recent regularity results on Unbalanced Optimal transport (Gallouët et al., 2021), we shall in particular cover the case where the smoothness assumption is not directly made on the potential z 0 but instead on the measures µ, ν. As in the balanced case, we rely on the semi-dual formulation of UOT for which we derive stability results; interestingly, thanks to the extra convexity brought by the entropic relaxation of the marginal in some cases, we do not need to assume smoothness of the potentials to derive those stability estimates. In the case where the metric entropy of C slowly diverges, the strong convexity enables us to use localization arguments and derive super-parametric rates. In particular, we obtain two different regimes that depend on the metric entropy of C; under smoothness assumptions, our rates closely match those of Hütter and Rigollet (2021) in the highly smooth case.
Assumptions and notations In this paper X, Y are compact subsets of R d , µ and ν are positive measures over X and Y respectively with their nindependent samples empirical counterpartsμ,ν when µ and ν are probability measures. We shall denote by supp(µ), supp(ν) the support of µ and ν respectively. We shall denote by ·, · the pairing between radon measures and continuous functions and by q the quadratic function q(x) = 1 2 x 2 . The notation · L p Z for p ∈ [1, +∞] shall refer to the L p norm over functions defined on Borel sets Z as
f L p Z = ( Z |f (x)| p dx) 1 p .
Conversely, for a probability measure β, we shall denote for p ∈ [1, +∞], g L p (β) = ( x |g(x)| p dβ(x)) 1 p .
Unbalanced Optimal Transport
In this section, we present unbalanced optimal transport via primal and dual formulations. The latter is used to show stability estimates, generalizing standard strong convexity estimates for the semi-dual in optimal transport. Unbalanced optimal transport (UOT) is a generalization of standard optimal transport which relaxes the marginal constraints using a convex divergence between positive measures. The primal formulation of UOT uses Csizár divergences which are defined as follows.
Definition 1 (Csizár divergences). Let F : R + → R + ∪ {+∞} be a convex lower semicontinuous function such that F (1) = 0. Its recession constant is
F ∞ = lim ∞ F (r)
r . Let µ, ν be non negative Radon measures on a convex domain Ω in R d . The Csiszàr divergence associated with F is
D F (µ, ν) = Ω F dµ(x) dν(x) dν(x) + F ∞ Ω dµ ⊥ ,(1)
where µ ⊥ is the orthogonal part of the Lebesgue decomposition of µ with respect to ν.
The Kullback-Leibler divergence is obtained for F (x) = x log(x) − x + 1. The primal formulation of UOT is defined by, for ρ 0 , ρ 1 ∈ M + (Ω),
UOT(ρ 0 , ρ 1 ) = inf γ∈M+(X×Y ) D F0 (γ 0 , ρ 0 ) + D F1 (γ 1 , ρ 1 ) + X×Y c(x, y) dγ(x, y) ,
where F 0 , F 1 are two possibly different entropy functions. Note that standard OT is recovered for the entropy function
F (x) = ι {1} (x) the convex indicator function of {1}.
The optimization problem associated with UOT is convex and its dual formulation reads, denoting F * the Legendre transform of F ,
sup z0,z1∈C b (X),C b (Y ) − X F * 0 (−z 0 (x)) dρ 0 (x) − Y F * 1 (−z 1 (y)) dρ 1 (y)(2)
under the constraint z 0 (x) + z 1 (y) ≤ c(x, y) .
The following proposition shows that at optimality, z 0 is a standard optimal transport potential between modified versions of (µ, ν) for the cost c(x, y).
Proposition 1 (see Lemma 3 in Gallouët et al. (2021)). Assume that ϕ * is differentiable on its domain. At optimality of (2), the pair
(z 0 , z 1 ) reads z 1 = z c 0 := inf x c(x, ·) − z 0 (x) and z 0 is an optimal transport potential betweenμ = ∂F * 0 (−z 0 )µ andν = ∂F * 1 (−z 1 )ν.
In the rest of the paper, we shall from now on focus assume that the cost is quadratic c(x, y) = q(x − y). Furthermore, to avoid heavy notations, we shall assume F 0 = F 1 := ϕ. However, similar results can be obtained when the entropy functions are different.
It is possible to optimize Formula (2) with respect to the second variable to obtain the so-called semi-dual formulation of UOT. Indeed, the optimal z 1 is the c-conjugate of z 0 . Using this argument on z c 0 , one can further assume that z 0 is the c-conjugate of a function z 1 , which says thatz 0 = q − z 0 is convex. In this case, the semi-dual UOT problem can be read as
− inf z∈C b (X),z convex ϕ * (z − q), µ + ϕ * (z * − q), ν ,(4)
where we injected (q − z) c = q − z * . Thus, let us introduce Definition 2 (Semi-Dual UOT). Given nonnegative measures µ, ν, the UOT semi-dual is defined by
J µ,ν (z) = ϕ * (z − q), µ + ϕ * (z * − q), ν .(5)
When confusion is possible, we shall denote J µ,ν by J. This semi-dual objective J remains convex and even gains in convexity with respect to the original objective. This phenomenon is well-known in standard OT and we show how it extends in the unbalanced setting. The important difference with standard OT is that when ϕ * is strongly convex, the stability is expressed in an H 1 norm instead of the L 2 norm of the gradient.
Proposition 2 (Stability estimate). The semi-dual functional J is convex. Assume that ϕ * is differentiable and that ν is absolutely continuous with repsect to the Lebesgue measure. Let z a λ-strongly convex potential and the optimal potential z 0 . Then, it holds
J(z) − J(z 0 ) ≥ 1 2λ Eν[ ∇(z * 0 − z * ) 2 ] + C z * E ν [(z * 0 − z * ) 2 ] + C z E µ [(z 0 − z) 2 ] ,(6)
with nonnegative constants C z and C z * depends on ϕ * and z. If z and z * and uniformly bounded on the support of µ and ν respectively and if ϕ * is strongly convex on every compact, then C z and C z * are uniformly lower bounded by a constant C > 0.
Proof. We start by applying the convexity inequality
ϕ * (y) − ϕ * (x) ≥ ϕ * (x)(y − x) + 1 2 m xy |x − y| 2 ,(7)
where m xy ≥ 0. We apply it in the difference of dual values and we get
J(z) − J(z 0 ) ≥ z − z 0 , (ϕ * ) (z 0 − q)µ + z * − z * 0 , (ϕ * ) (z * 0 − q)ν + C z (z 0 − z) 2 , µ + C z * (z * 0 − z * ) 2 , ν , where C z = inf x∈supp(µ) m z(x)−q(x),z0(x)−q(x) C z * = inf y∈supp(ν) m z * (y)−q(y),z * 0 (y)−q(y) .(8)
Now, recall that z 0 is an optimal potential for the transport of measureμ := ϕ * (z 0 −q)µ onto the measureν :
= ϕ * (z * 0 −q)ν. DenotingJ(z) = z,μ + z * ,ν , Equation (9) reads J(z) − J(z 0 ) ≥J(z) −J(z 0 ) + C z (z 0 − z) 2 , µ + C z * (z * 0 − z * ) 2 , ν .(9)
We now apply the stability of optimal transport guaranteed by the absolute continuity of ν which gives the lower boundJ
(z)−J(z 0 ) ≥ 1 2λ Eν[ ∇(z * −z * 0 ) 2 ].
Note that this upper-bound encompasses the balanced case and that the entropy can bring extra convexity. In the generic case, we shall write
J(z) − J(z 0 ) ≥ d λ H • (z, z 0 ) where the pseudo distance d λ H • is defined as d λ H • (f, g) 2 := 1 2λ Eν[ ∇(f * − g * ) 2 ] + C (E ν [(f * − g * ) 2 ] + E µ [(f − g) 2 ]) ,
where C ≥ 0. When C is strictly positive, J is (locally, at the optimum) strongly convex with respect to an H 1 norm while in the balanced setting, there it is formulated with the semi-normḢ 1 . Furthermore, when C > 0, J(z) − J(z 0 ) not only controls z * 0 − z * in an L 2 sense but also z 0 − z in an L 2 sense. This is a notable improvement with respect to balanced OT: in this case, in order to upper-bound z − z 0 with J(z) − J(z 0 ), a smoothness assumption must be made on z 0 ; in the UOT case, if the entropy is sufficiently convex, we can obtain an upper-bound without this extra smoothness assumption. In the next section, we use this property to apply a localization technique and obtain fast rates without requiring smoothness of the functions in C.
Estimation of UOT maps
In this section, we restrict ourselves to the case where µ, ν are probability measures which we only access through their n-samples stochastic counterpartsμ,ν. Even though this setting is much more restrictive than the original UOT setting where µ and ν can be arbitrary positive radon measures, it remains nonetheless a relevant setting in ML applications for instance. Indeed, the relaxation of the hard marginal OT constraint by a divergence allow to better handle outliers as shown experimentally by Mukherjee et al. (2021).
In this stochastic setting, a natural way to estimate UOT map is to solve the empirical semi-dual over a given search space C.
Definition 3 (Stochastic Semi-Dual Unbalanced OT). Let C be a set of real- valued function, we define UOT C UOT C = − inf z∈CĴ (z) ,(10)
whereĴ = Jμ ,ν . Conversely,we define the empirical potential
z C = arg min z∈CĴ (z) .(11)
When no confusion is possible, we shall simply denote itẑ.
If the true unbalanced potential z 0 belongs to C, we can prove that the empirical potentialẑ converges toward z 0 with respect to d λ H • at a rate that will depend on the complexity of C.
Generic case
We show that under suitable assumptions, the solutions of empirical unbalanced semi-dual OT converges toward the ground truth in the d λ H • sense.
Assumption 1. The measures µ, ν have support included in B R , where B r is the euclidean ball of R d centered in 0 and of radius r.
Assumption 2. The measures µ, ν have densities with respect to the Lebesgue measure on B R .
Assumption 3. There existsz 0 ∈ C such thatz 0 coincides with z 0 on supp(µ) and withz * 0 coincides with z * 0 on supp(ν).
Assumption 4. The functions in C are uniformly bounded by M (r) over B r , uniformly lower bounded by l and are λ-strongly convex.
The goal of Assumption 2 is to ensure the existence of the unbalanced transport map between µ, ν. The goal of Assumption 3 is to ensure the absence of bias in the model. We believe that under a finer analysis, Assumption 1 could be replaced with sub-gaussian measures. We show in the following Lemma that Assumption 4 ensures that the conjugate of the functions in C are both bounded and smooth on every ball. Proof. For z ∈ C, we have that z * is 1 λ -smooth. In particular, for
x ∈ B r ∇z * (x) = ∇z * (x) − ∇z * (0) + ∇z * (0) (12) ≤ ∇z * (x) − ∇z * (0) + ∇z * (0) (13) ≤ r λ + ∇z * (0) .(14)
Now recall that ∇z * (0) = arg min x∈R d z(x). Since z is λ-strongly convex, we have the following inequality
z(0) ≥ z(x * ) + λ 2 x * 2 ,(15)
where x * = arg min x∈R d z(x). Using that z(0) ≤ M (0) and −z ≤ −l, we recover
x * ≤ 2(M (0) − l) λ .(16)
The bound on z * L ∞ Br follows the definition of the Fenchel-Legendre transform
z * (x) = x ∇z * (x) − z(∇z * (x)) .(17)
Finally, combined with the previous Lemma, Assumption 4 also ensures that the conjugate on C has a Lipschitz behavior with respect to the sup norm.
Lemma 2. Let z 1 , z 2 be λ-strongly convex functions such that z 1 , z 2 are lowerbounded by l and bounded by M (r) on B r . We have z
* 1 − z * 2 L ∞ B R ≤ z 1 − z 2 L ∞ B G(R)
, where G(r) := r λ + 2(M (0)−l) λ as in Lemma 1.
Proof. Let x ∈ B R . By definition of the Fenchel transform, we have for all
y ∈ R d z * 1 (x) ≥ x y − z 1 (y) ,(18)
with equality when y = ∇z * 1 (x). Hence, we have for all y
z * 1 (x) − z * 2 (x) ≥ x y − z 1 (y) + z 2 (∇z * 2 (x)) − x ∇z * 2 (x) .(19)
In particular, for y = ∇z * 2 (x), we obtain
z * 1 (x) − z * 2 (x) ≥ z 2 (∇z * 2 (x)) − z 1 (∇z * 2 (x)) ,(20)
and applying Lemma 1 yields z *
1 (x) − z * 2 (x) ≥ − z 1 − z 2 L ∞ B G(R)
. Conversely, flipping the role of z 1 , z 2 , we obtain
z * 2 (x) − z * 1 (x) ≥ z 1 (∇z * 1 (x)) − z 2 (∇z * 1 (x)) ,(21)which yields |z * 1 (x) − z * 2 (x)| ≤ z 1 − z 2 L ∞ B G (R) .
We have now all the ingredients to derive our result.
Proposition 3. Denotingẑ C the solution of problem (10), we have under Assumptions 1-4, if the unbalanced optimal transport potential z 0 between µ and ν belongs to C, then we have for all
δ ≤ M L E[d λ H • (ẑ C , z 0 ) 2 ] δ + 1 √ n M L δ 4 n(C, L ∞ B R , Lu)du
where n(C, · , u) is the logarithm of the covering number, also called the metric entropy, of C with respect to the · (semi)-norm at scale u, M = (M, R, λ, l, ϕ), R = (M, R, λ, l), L = (M, R, λ, l, ϕ) and hides a factor 64.
Proof. We start by applying the strong convexity inequality of the semi-dual and the optimality conditions
d λ H • (ẑ, z 0 ) 2 ≤ J(ẑ) − J(z 0 ) (22) = J(ẑ) −Ĵ(ẑ) +Ĵ(ẑ) −Ĵ(z 0 ) +Ĵ(z 0 ) − J(z 0 ) .(23)
Using Assumption 3, the termĴ(ẑ) −Ĵ(z 0 ) is negative hence we have
d λ H • (ẑ, z 0 ) 2 ≤ J(ẑ) −Ĵ(ẑ) +Ĵ(z 0 ) − J(z 0 ) (24) ≤ sup z∈C φ * (z − q), µ −μ (25) + sup z∈C * φ * (z − q), ν −ν(26)+Ĵ(z 0 ) − J(z 0 ) ,(27)
where we denoted C * = {z * , z ∈ C}.
Bound on term (25) Denoting C 0 = {φ * (g − q), g ∈ C}, we apply Luxburg and Bousquet (2004, Theorem 16) to bound our empirical process
W := sup z∈C φ * (z − q), µ −μ ,
and we obtain for all δ > 0
E[W ] ≤ 2δ + 4 √ 2 √ n ∞ δ 4 n(C 0 , L 2 (μ), u) du .(28)
Noting that g L 2 (μ) ≤ g L ∞ (µ) almost surely, we recover the upper bound
E[W ] ≤ 2δ + 4 √ 2 √ n ∞ δ 4 n(C 0 , L ∞ (µ), u) du .(29)
Since the functions in C are uniformly bounded by M (R) on B R and that µ is supported on B R , we have ∀(g 1 , g 2 ) ∈ C 2 ,
φ * (g 1 − q) − φ * (g 2 − q) L ∞ (µ) ≤ L 1 φ * g 1 − g 2 L ∞ (µ) ,(30)
where L 1 φ * is defined as
L 1 φ * := sup x∈[−M1,M1] |∂φ * (x)| ,(31)
and M 1 = 2M (R) + R 2 . In particular, we get the new upper-bound for all
δ 4 ≤ 2M (R) L 1 φ * E[W ] ≤ 2δ + 4 √ 2 √ n 2M (R) L 1 φ * δ 4 n(C, L ∞ (µ), L 1 φ * u) du ≤ 2δ + 4 √ 2 √ n 2M (R) L 1 φ * δ 4 n(C, L ∞ B R , L 1 φ * u) du .
Bound on term (26) Lemma 1 ensures that the functions in C * are uniformly bounded on every ball B r by some constant M (r). In particular, we can proceed as in the last paragraph and obtain
E[W * ] ≤ 2δ + 4 √ 2 √ n 2M (R) L 2 φ * δ 4 n(C * , L ∞ B R , L 2 φ * u) du ,
where W * := sup z∈C * z, ν −ν and L 2 φ * is defined as
L 2 φ * := sup x∈[−M2,M2] |∂φ * (x)| ,(32)
with M 2 = 2M (R) + R 2 . Using Lemma 2 that states
z * 1 − z * 2 L ∞ B R ≤ z 1 − z 2 L ∞ B G(R) ,(33)
for some constant G(R), we can control the covering number of C * with respect to the L ∞ B R and we have the upper-bound for δ 4 ≤ 2M (R)
L 2 φ * E[W * ] ≤ 2δ + 4 √ 2 √ n 2M (R) L 2 φ * δ 4 n(C, L ∞ B G(R) , L 2 φ * u) du .
Final upper bound Since the term (27) is zero in average, we obtain our final bound
d λ H • (ẑ, z 0 ) 2 ≤ 4δ + 8 √ 2 √ n M L δ 4 n(C, L ∞ B R , Lu)du , where M = 2 max(M (R), M (R)) and L = max(L 1 φ * , L 2 φ * )
Leveraging the recent regularity results on UOT derived in Gallouët et al. (2021), we can deduce from Proposition 3 an upper-bound for the statistical estimation of UOT potentials.
Corollary 1. Assume that µ and ν have compact and convex support with densities (f, g) bounded away from zero and infinity and assume that ϕis strictly convex with infinite slope at 0. If (f, g) are k-times continuously differentiable with k ∈ N then, denoting z 0 an optimal unbalanced OT potential, there exists C such that the empirical potentialẑ C verifies
E[d λ H • (ẑ C , z 0 ) 2 ] n − k+2 d if k + 2 < d/2 , E[d λ H • (ẑ C , z 0 ) 2 ] log(n) √ n if k + 2 = d/2 , E[d λ H • (ẑ C , z 0 ) 2 ] 1 √ n if k + 2 > d/2 .(34)
Proof. Using the Corollary 9 of Gallouët et al. (2021), we can ensure that z 0 , z * 0 are (k + 2)-times continuously differentiable over the support of µ and ν respectively. Recalling that for all x ∈ supp(ν)
∇ 2 z 0 (x) = [∇ 2 z * 0 (∇z 0 (x))] −1 ,(35)
and using the fact that ∇z 0 is a diffeomorphism between the support µ and ν, we recover that z 0 is λ-strongly convex over supp(µ) where we defined
1 λ := sup y∈supp(ν) ∇ 2 z * 0 (y) .(36)
Now, recall that in order to apply our previous result, we need to globally bound the strong-convexity constant as well as controlling the sup norm over every ball. To achieve this, we can extend these potentials to the whole domain. Proposition 1.5 in Azagra and Mudarra (2019) provides a (k + 2)-times continuously differentiable convex extensiong 0 of z 0 − λq on the whole domain R d . Defining z 0 =g 0 + λq, we have thatz 0 coincides with z 0 on supp(µ). Using again the diffeomorphism property of ∇z 0 between supp(µ) and supp(ν), we have thatz * 0 coincides with z * 0 on supp(ν). Now let us define
C = {z | z L ∞ Br ≤ z 0 L ∞ Br , ∇ k+2 z L ∞ Br ≤ ∇ k+2z 0 L ∞ Br , z ≥ l, z is λ-strongly convex} ,
where l is the minimum ofz 0 . The set C indeed meets Assumption 4 and Assumption 3 hence we can apply Prop. 3 which yields
E[d λ H • (ẑ C , z 0 ) 2 ] δ + 1 √ n M L δ 4 n(C, L ∞ B R , Lu)du .(37)
Finally, using van der Vaart and Wellner (1996, Theorem 2.7), we have n(C,
L ∞ B R , Lu) u − d k+2 . If k+2 d < 1/2, take δ = n − k+2 d . For this choice of δ, 1 √ n M L δ 4 n(C, L ∞ B R , Lu)du 1 √ n (n − k+2 d ) 1− d 2(k+2)(38)1 √ n n − 2(k+2)−d 2d (39) = n − k+2 d .(40)
If k+2 d = 1/2, take δ = 1 √ n . For this choice of δ, the integral is of order log(n) which yields the upper-bound
E[d λ H • (ẑ C , z 0 ) 2 ] log(n) √ n .(41)
Finally, if k+2 d > 1/2, taking δ = 0 yields
E[d λ H • (ẑ C , z 0 ) 2 ] 1 √ n .(42)
We see that we have two distinct regimes: a regime where the extra smoothness directly improves the rate of estimation and a highly smooth regime where the rate saturates at 1/ √ n. In particular, we do not recover the asymptotic rate (with respect to the smoothness) 1/n which in known to be minimax in the case of balanced OT (see Hütter and Rigollet (2021)). We show in the next paragraph that this 1/ √ n rate can be improved under suitable assumptions on the metric entropy of C.
Low metric entropy case
When the metric entropy of C slowly diverges, we can obtain faster rates than 1/ √ n. The central argument is the localization: thanks to the strong convexity of the semi-dual, we can localize the empirical potentialẑ in a certain neighborhood of the ground truth z 0 . The next assumption allows to apply the stability result of the semi-dual with a L 2 control onẑ − z 0 andẑ * − z * 0 ; From the statistical point of view, it allows to employ localization arguments.
Assumption 5. The conjugate of the entropy ϕ * is strongly convex on every compact.
Hence, instead of controlling the global empirical processes
W = sup z∈C z, µ −μ ,(43)
and
W * = sup z∈C z * , ν −ν ,(44)
we simply must control the localized empirical processes
W (τ ) := sup z∈C∩B • (z0,τ ) φ * (z − q) − φ * (z 0 − q), µ −μ ,(45)
and
W * (τ ) := sup z∈C∩B • (z0,τ ) φ * (z * − q) − φ * (z * 0 − q), ν −ν ,(46)
where τ is a suitable radius and B • (z 0 , τ ) is the ball centered in z 0 of radius τ with respect to the d λ H • pseudo-distance. In the case where the metric entropy of C grows too fast, the localized processes (45) and (46) behave like the global processes (43) and (44) and we cannot apply localization. However, when the metric entropy of C is sufficiently low, the following lemma shows that W (τ ) and W * (τ ) are upper bounded with high probability by τ 1−α/2 √ n with 2 > α > 0.
Lemma 3. Under Assumptions 4-5, if we assume that there exists (P µ , P ν ) and α < 2 such that for every u ∈ R ≥0 , n(C, L 2 (µ), u) ≤ P µ u −α and n(C, L 2 (ν), u) ≤ P ν u −α , it holds with probability at least 1 − e −t
W (τ ) ≤ 8 √ 2Pµ (1− α 2 ) n(L 1 φ * ) α (Kτ ) 1−α/2 + Kτ 2t n + 2M (R)L 1 φ * n W * (τ ) ≤ 8 √ 2Pν (1− α 2 ) n(L 2 φ * ) α (K τ ) 1−α/2 + K τ 2t n + 2M (R)L 2 φ * n ,(47)
where L 1 φ * , L 2 φ * are defined in Equations (31) and (32) respectively and measure local lipschitz behaviors of ϕ * , M (R) is defined in Assumption 2 and is a uniform bound over B R of the potentials in C, M (R) is defined in Lemma 1 and is a uniform bound over B R of the conjugate of the potentials in C, and K = K(R, M, φ * ), K = K (R, M, φ * , λ, l) are the embedding constants of (C, L 2 (µ)) and (C * , L 2 (ν)) in H • .
Proof. The proof relies on the Lipschitz behavior of the Legendre transform that preserves the metric entropy of C and on the Bousquet concentration inequality. We start by analyzing the term W (τ ).
Term W (τ ) Let us denote C 0 = {φ * (z − q) − φ * (z 0 − q), z ∈ C ∩ B • (z 0 , τ )}. For g ∈ C 0 of the form g = φ * (z − q) − φ * (z 0 − q) with z ∈ C ∩ B • (z 0 , τ ), we have the pointwise bound for all x ∈ B R , |g(x)| ≤ L 1 φ * |z(x) − z 0 (x)| ,(48)
where L 1 φ * := sup x∈[−M1,M1] |∂φ * (x)| with M 1 = 2M (R) + R 2 as in the previous proof. This implies g L 2 (µ) ≤ L 1 φ * z − z 0 L 2 (µ) . Since we assumed φ * strongly convex on every compact, there exists K = K(R, M, φ * ) > 0 such that z − z 0 L 2 (µ) ≤ Kd λ H • (z, z 0 ) and in particular, all g ∈ C 0 verifies g L 2 (µ) ≤ Kτ . Hence, applying Luxburg and Bousquet (2004, Theorem 16), we obtain for all
δ 4 ≤ Kτ E[W (τ )] ≤ 2δ + 4 √ 2 √ n Kτ δ 4 n(C 0 , L 2 (µ), u) du .(49)
Again, taking (g 1 , g 2 ) ∈ C 2 0 of the form
g 1 = φ * (z 1 − q) − φ * (z 0 − q) and g 2 = φ * (z 2 − q) − φ * (z 0 − q) with (z 1 , z 2 ) ∈ (C ∩ B • (z 0 , τ )) 2 , we have g 1 − g 2 L 2 (µ) ≤ L 1 φ * z 1 − z 2 L 2 (µ) ,(50)
and in particular, we recover the upper-bound
E[W (τ )] ≤ 2δ + 4 √ 2 √ n Kτ δ 4 n(C, L 2 (µ), L 1 φ * u) du.(51)
Now, we assumed that for all u ∈ R + we had the upper-bound, n(C, L 2 (µ), u) ≤ P µ u −α with α < 2, we obtain taking δ = 0 our final upper bound
E[W (τ )] ≤ 4 2P µ (1 − α 2 ) n(L 1 φ * ) α (Kτ ) 1−α/2 .(52)
There remains to bound the process W (τ ) with high probability. We use for this the Bousquet concentration inequality.
Lemma 4 (Bousquet, see Theorem 26 in Hütter and Rigollet (2021)). Let F be a class of functions such that for every f ∈ F, f 2 L 2 (µ) ≤ σ 2 and f L ∞ (µ) ≤ M , then for all t > 0, we have with probability at least
1 − e −t sup f ∈F √ n| f, µ −μ | ≤ 2E[sup f ∈F √ n| f, µ −μ |] + σ √ 2t + M √ n t .(53)
Applying this result to W (τ ) yields that with probability at least 1 − e −t ,
W (τ ) ≤ 8 2P µ (1 − α 2 ) n(L 1 φ * ) α (Kτ ) 1−α/2 + Kτ 2t n + 2tM (R)L 1 φ * n ,(54)
where we used the pointwise upper-bound (48) and where M (R) is the constant such that ∀z ∈ C, z L ∞
B R ≤ M (R).
where κ and κ are given in Lemma 3 defined as
κ = 8 √ 2 (1− α 2 ) √ PµK 1−α/2 (L 1 ϕ * ) α 2 + √ Pν (K ) 1−α/2 (L 2 ϕ * ) α 2 κ = 2(M (R)L 1 ϕ * + M (R)L 2 ϕ * ) .(59)
.
Let A n = {τ ∈ A, τ ≥ 1 √ n }. For τ ∈ A n , we have τ 2 4 ≤ κ τ 1−α/2 √ n + (K + K )τ 2t n + tκ τ √ n .(60)
Assuming that t ≥ 1, we have two cases
Case 1 If τ ≤ 1, we have τ 2 4 ≤ tητ 1−α/2 √ n ,(61)
where η = (κ + κ + √ 2(K + K )) and we recover τ ≤ (4ηt)
1 1+α/2 n 1 2+α . Case 2 If τ ≥ 1, we have τ 2 4 ≤ tητ √ n i.e. τ ≤ 4tη √ n .
In any case, for t ≥ 1, we have with probability at least 1 − e −t sup(A) ≤ (4η t)
1 1+α/2 + (4η t) n 1 2+α ,(62)
where we defined η = max(η, 1). Now, by definition of A, we have for all > 0, d λ H • (ẑ, z 0 ) ≤ sup(A) + . Taking → 0 gives that with probability at least 1 − e −t , for t ≥ 1
d λ H • (ẑ, z 0 ) ≤ (4η t) 1 1+α/2 + (4η t) n 1 2+α (63) ≤ 8η t n 1 2+α .(64)
And in particular, d λ H • (ẑ, z 0 ) 2 ≤ 64(η ) 2 t 2 n 1 1+α/2 with probability at least 1 − e −t for t ≥ 1. We denote X the random variable d λ H • (ẑ, z 0 ) 2 . Since X is nonnegative almost surely, we can apply Fubini's formula The integrand in the first term is upper-bounded by 1 and the integrand on the second term is upper bounded by te −t . Hence we obtain
E[X] = ∞ 0 P (X > u) du .(65)E[d λ H • (ẑ, z 0 ) 2 ] ≤ 128(η ) 2 n 1 1+α/2 (1 + ∞ 1 te −t dt) = 128(1 + 2e −1 )(η ) 2 n 1 1+α/2 .
An immediate consequence of this result is that we can improve the rate derived in Corollary 1 and recover the asymptotic behavior in 1/n when the smoothness grows.
Corollary 2. Assume that µ and ν have compact and convex support with densities (f, g) bounded away from zero and infinity and assume that ϕ * is strongly convex on every compact. If (f, g) are k-times continuously differentiable with k + 2 > d/2 then, denoting z 0 an optimal unbalanced OT potential, there exists C such that the empirical potentialẑ C verifies
E[d λ H • (ẑ C , z 0 ) 2 ] n − 1 1+ d 2(k+2) .(66)
Proof. As in the proof of Corollary 1, the original potential z 0 is λ-strongly convex and (k + 2)-times continuously differentiable over supp(µ). It can be extended toz 0 that is also λ-strongly convex and (k + 2)-times continuously differentiable but over the whole domain R d . The functionz 0 is such that it coincides with z 0 on supp(µ) and its conjugatez 0 coincides with z 0 on supp(ν). We define the set
C = {z | z L ∞ Br ≤ z 0 L ∞ Br , ∇ k+2 z L ∞ Br ≤ ∇ k+2z 0 L ∞ Br , z ≥ l, z is λ-strongly convex} ,
where l is the minimum ofz 0 . Theorem 2.7 of van der Vaart and Wellner (1996) ensures that the metric entropy at scale u of C with respect to L 2 (µ) and L 2 (ν) is upper bounded by u − k+2 d . Hence we can apply Prop. 4 and obtain
E[d λ H • (ẑ C , z 0 ) 2 ] n − 1 1+ d 2(k+2) .(67)
Hence, using Corollary 1 and Corollary 2, we obtain a rate of n − α+2 d when α + 2 < d/2 and n − 1 1+ d 2(α+2) when α + 2 > d/2; note that we continuously transition from one rate to another when α + 2 = d/2 where we recover (up to log factor) the parametric rate 1/ √ n. Even though the fast rates do not encompass the balanced case as they require ϕ * to be at least locally strongly convex, we still compare our rates to the ones of Hütter and Rigollet (2021). Under the same assumptions, they propose estimators that achieve a rate in n − α+1 α+d/2 . As shown in Fig. 1 for the case d = 100, their rate is faster for any α > 0 yet when we transition in the highly smooth regime α + 2 > d/2, our rate closely matches theirs. This discrepancy is due to the fact that we have no bias in our model i.e. we assumed z 0 ∈ C. On their side, Hütter and Rigollet (2021) fixed C to be a finite wavelet basis that does not necessarily contain z 0 . In particular, they improve the bias variance trade-off on two levels: in the low smooth regime, they can benefit the acceleration given by the localization as they choose a small class of functions and for the same reason, in the highly smooth regime, they better leverage the localization.
Lemma 1 .
1For all z that are λ-strongly convex and such that z ≥ l, z L ∞ Br ≤ M (r), we have ∇z * Br ≤ M (r) := rG(r) + M (G(r)).
Let us make the change of variable u = 64(η )
Figure 1 :
1Comparison of our rates against the rates of Hutter and Rigollet (2021): on the left for d = 12 and on the right for d = 100.
Term W * (τ ) We can apply the same reasoning as previously. Indeed, as shown in Lemma 1, there exists a constant M (R) such that for all z ∈ C, z * L ∞ B R ≤ M (R). In particular, since the potentials z * are bounded, we can also leverage the local strong convexity of φ * that yields a constant K = K (R, M, φ * , λ, l) > 0 such that for every z ∈ C, (z−z 0 ) * L 2 (ν) ≤ K d λ H • (z, z 0 ). Hence we recover that with probability at least 1 − e −t ,Then, the informal reasoning is as follows:Proposition 4. Under Assumptions 1-5, if we assume that there exists (P µ , P ν ) and α < 2 such that for every u ∈ R ≥0 , n(C, L 2 (µ), u) ≤ P µ u −α and n(C, L 2 (ν), u) ≤ P ν u −α then ∀n ≥ 1,where hides constants that do not depend on n.Proof. For τ > 0, define s = τ τ +d λ H • (ẑ,z0) andẑ s = (1 − s)z 0 + sẑ. By local strong convexity of J, we haveLet us decompose the right hand side as J(ẑ s ) −Ĵ(ẑ s ) − (J(z 0 ) −Ĵ(z 0 )) + J(ẑ s ) −Ĵ(z 0 ). By convexity ofĴ, the last term can be upper-bounded by sĴ(ẑ) + (1 − s)Ĵ(z 0 ) −Ĵ(z 0 ) = s(Ĵ(ẑ) −Ĵ(z 0 )). Sinceẑ is the minimizer of the empirical semi-dual, we have in particular that s(Ĵ(ẑ) −Ĵ(z 0 )) ≤ 0 which givesLet us now consider A = {τ, d λ H • (ẑ, z 0 ) ≥ τ }. We wish to recover an upperbound on A. Remark that A = {τ, d λ H • (ẑ s , z 0 ) ≥ τ 2 }. In particular, every τ ∈ A verifies with probability at least 1 − e −t τ 2 4 ≤ κ τ 1−α/2 √ n + (K + K )τ 2t n + tκ n ,
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Statistical aspects of wasserstein distances. M Victor, Yoav Panaretos, Zemel, 10.1146/annurev-statistics-030718-104938Annual Review of Statistics and Its Application. 61Victor M. Panaretos and Yoav Zemel. Statistical aspects of wasserstein dis- tances. Annual Review of Statistics and Its Application, 6(1):405-431, Mar 2019. ISSN 2326-831X. doi: 10.1146/annurev-statistics-030718-104938. URL http://dx.doi.org/10.1146/annurev-statistics-030718-104938.
Entropic estimation of optimal transport maps. Aram-Alexandre Pooladian, Jonathan Niles-Weed, Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps, 2021.
Aram-Alexandre Pooladian, Marco Cuturi, Jonathan Niles-Weed, arXiv:2202.08919Debiaser Beware: Pitfalls of Centering Regularized Transport Maps. arXiv e-prints, art. Aram-Alexandre Pooladian, Marco Cuturi, and Jonathan Niles-Weed. Debiaser Beware: Pitfalls of Centering Regularized Transport Maps. arXiv e-prints, art. arXiv:2202.08919, February 2022.
Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, Cell. Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subrama- nian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies de- velopmental trajectories in reprogramming. Cell, 2019.
Large-scale optimal transport and mapping estimation. Vivien, Seguy, B Bharath, Remi Damodaran, Nicolas Flamary, Antoine Courty, Mathieu Rolet, Blondel, In ICLR. Vivien. Seguy, Bharath B. Damodaran, Remi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel. Large-scale optimal transport and mapping estimation. In ICLR, 2018.
Sinkhorn divergences for unbalanced optimal transport. Thibault Séjourné, Jean Feydy, François-Xavier Vialard, Alain Trouvé, Gabriel Peyré, arXiv:1910.12958arXiv preprintThibault Séjourné, Jean Feydy, François-Xavier Vialard, Alain Trouvé, and Gabriel Peyré. Sinkhorn divergences for unbalanced optimal transport. arXiv preprint arXiv:1910.12958, 2019.
Weak convergence and empirical processes. Jon A Aad W Van Der Vaart, Wellner, SpringerAad W van der Vaart and Jon A Wellner. Weak convergence and empirical processes. Springer, 1996.
Predicting cell lineages using autoencoders and optimal transport. Karren D Yang, Karthik Damodaran, Saradha Venkatachalapathy, Ali C Soylemezoglu, G V Shivashankar, Caroline Uhler, PLoS Computational Biology. 16Karren D. Yang, Karthik Damodaran, Saradha Venkatachalapathy, Ali C Soyle- mezoglu, G. V. Shivashankar, and Caroline Uhler. Predicting cell lineages using autoencoders and optimal transport. PLoS Computational Biology, 16, 2020.
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"Right submodules of finite rank for von Neumann dynamical systems",
"Right submodules of finite rank for von Neumann dynamical systems"
]
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"Paul Jolissaint "
]
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| Let (M, τ, σ, Γ) be a (finite) von Neumann dynamical system and let N be a unital von Neumann subalgebra of M which is Γ-invariant. If V ⊂ L 2 (M ) is a right N -submodule whose projection p V has finite trace in M, e N and is Γ-invariant, then we prove that, for every ε > 0, one can find a Γ-invariant submodule W ⊂ V which has finite rank and such that T r(p V − p W ) < ε. In particular, this answers Question 4.2 of [1].Mathematics Subject Classification: 46L10. | null | [
"https://arxiv.org/pdf/1009.0684v3.pdf"
]
| 119,632,359 | 1009.0684 | 38570ed322bb2fe5b18710b35b6623d76a431a6d |
Right submodules of finite rank for von Neumann dynamical systems
10 Jan 2011 January 11, 2011
Paul Jolissaint
Right submodules of finite rank for von Neumann dynamical systems
10 Jan 2011 January 11, 2011arXiv:1009.0684v3 [math.OA]Finite rank submodules, von Neumann dynamical systems
Let (M, τ, σ, Γ) be a (finite) von Neumann dynamical system and let N be a unital von Neumann subalgebra of M which is Γ-invariant. If V ⊂ L 2 (M ) is a right N -submodule whose projection p V has finite trace in M, e N and is Γ-invariant, then we prove that, for every ε > 0, one can find a Γ-invariant submodule W ⊂ V which has finite rank and such that T r(p V − p W ) < ε. In particular, this answers Question 4.2 of [1].Mathematics Subject Classification: 46L10.
Introduction
Let (M, τ, σ, Γ) be a von Neumann dynamical system, i.e. M is a finite von Neumann algebra endowed with
• a normal, finite, faithful, normalized trace τ ,
• an action σ of a countable group Γ such that every σ g is τ -preserving for every g ∈ Γ.
In order to prove an extention to von Neumann dynamical systems of multiple recurrence Szemerédi's Theorem, the authors of [1] need the following lemma ( [1], Lemma 4.1, stated here somewhat vaguely):
Let (M, τ, α) be a finite von Neumann endowed with a τ -preserving action of Z : n → α n . If N is an α-invariant von Neumann subalgebra of the center of M and if V ⊂ L 2 (M) is an invariant right-N-submodule with finite trace, then one can find an invariant N-submodule W ⊂ V of finite rank r, arbitrarily close to V in an appropriate sense, and the associated action of α on W is described by a suitable unitary u ∈ U(M r (N)) (all relevant definitions are recalled below).
The proof rests on the decomposition of L 2 (M) in a direct integral based on N, and that is why N is assumed to lie in the center of M. Guessing that the conclusions of the lemma should hold under more general hypotheses, the authors of [1] ask if it is indeed the case in Question 4.2 of their article.
Thus, the aim of the present notes is to provide more general statements of the above lemma. As a matter of fact, we present two variants. In the first one (Proposition 1), we assume that N is at the opposite of being central, namely that N is a type II 1 factor, and we get a neater version than in the general case which is the subject of Proposition 2. The first result and its proof are contained in Section 2, and the second one is treated in the last section. We recall below the main definitions and results that will be needed in the forthcoming sections.
Standard notations in the framework of finite von Neumann algebras will be used; they are borrowed from A. Sinclair's and R. Smith's monograph [5]. Let (M, τ ) be as above, and let L 2 (M) = L 2 (M, τ ) be the associated Hilbert space given by GNS construction. It is an M-bimodule, and M embeds into L 2 (M) as a dense subspace. We denote by ξ the image of 1 ∈ M in L 2 (M), so that τ (x) = xξ, ξ for every x ∈ M. Let 1 ∈ N ⊂ M be a von Neumann subalgebra of M. We denote by E N the τ -preserving conditional expectation of M onto N, and by e N the extention of E N to L 2 (M); M, e N = JN ′ J denotes the associated basic construction algebra.
Recall also that L 1 (M) is the completion of M with respect to the norm
x 1 := τ (|x|) = sup{|τ (xa)| : a ∈ M, a ≤ 1}.
It is an M-bimodule because it is easy to check that axb 1 ≤ a x 1 b for all a, x, b ∈ M. Furthermore, the mapping (x, y) → xy defined from M × M to M extends to a mapping from L 2 (M) × L 2 (M) to L 1 (M) because xy 1 ≤ x 2 y 2 by Cauchy-Schwarz inequality. In particular, setting y = 1, we get the inequality x 1 ≤ x 2 which implies that L 2 (M) ⊂ L 1 (M). Similarly, the conditional expectation E N extends to a contraction from L 1 (M) to L 1 (N).
All elements of L 1 (M) can (and will) be interpreted as densely defined linear operators affiliated with M acting on L 2 (M); full details are contained for instance in Appendix B of [5]. As x * p = x p for all x ∈ M, p = 1, 2, conjugation x → x * extends to an antilinear, isometric bijection of L p (M) onto itself. It is usually denoted by J in the case p = 2.
We denote by T r the faithful, normal, semifinite trace on M, e N characterized by T r(xe N y) = τ (xy) ∀x, y ∈ M, and we will also make use of the so-called pull-down map Φ defined first on the dense subalgebra
Me N M := { finite a i e N b i : a i , b i ∈ M ∀i} by Φ(xe N y) = xy ∀x, y ∈ M.
It extends to an M-bimodule * -map from L 1 ( M, e N , T r) to L 1 (M) and it has the following properties (see Section 4.5 in [5], in particular Theorem 4.5.3):
(i) τ (Φ(x)) = T r(x) for all x ∈ L 1 ( M, e N , T r); (ii) xe N = Φ(xe N )e N for all x ∈ L 1 ( M, e N , T r); (iii) Φ(x) 1 ≤ x 1,T r for all x ∈ L 1 ( M, e N , T r); (iv) Φ maps Me N onto M with Φ(x) 2 = x 2,T r for all x ∈ Me N ;
(v) the equations Φ(xe N y) = xJy * Jξ and Φ(ye N x) = Jx * Jyξ hold and both are in
L 2 (M)
for all all x ∈ M and y ∈ M, e N .
Moreover, we need the following extention of claim (iv) above: as T r(e N ) = 1, one has e N ∈ L 2 ( M, e N , T r) as well, hence Xe N ∈ L 2 ( M, e N , T r) for every X ∈ M, e N , and Φ extends to M, e N e N isometrically, i.e. one has Xe N = Φ(Xe N )e N with
Xe N 2,T r = Φ(Xe N ) 2
for every X ∈ M, e N . Indeed, it suffices to prove the above equality for X = i a i e N b i where the sum is finite and a i , b i ∈ M for every i. One has
Xe N = i a i e N b i e N = i a i E N (b i )e N and Φ(Xe N ) = i a i E N (b i ) hence Xe N 2 2,T r = T r(e N X * Xe N ) = T r( i,j e N b * i e N a * i a j e N b j e N ) = i,j τ (E N (b * i )E N (a * i a j )E N (b j )) = i,j τ (E N (b * i )a * i a j E N (b j )) = Φ(Xe N ) 2 2 .
Let N be a Γ-invariant, unital von Neumann subalgebra of M. We denote by u σ (g) the unitary operator on L 2 (M) defined by u σ (g)xξ = σ g (x)ξ for every x ∈ M. Invariance implies that E N σ g = σ g E N , and that u σ (g) and e N commute for all g ∈ Γ. We extend σ g to an automorphism of M, e N as follows: σ g (X) = u σ (g)Xu σ (g −1 ) for all X ∈ M, e N ; it is characterized by σ g (ae N b) = σ g (a)e N σ g (b) for all a, b ∈ M and g ∈ Γ.
Moreover, there is a bijective correspondence between the set of projections of M, e N and the right N-submodules of L 2 (M) given by
p → V = pL 2 (M). The reciprocal map is denoted by V → p V . Note that V is u σ (g)-invariant if and only if σ g (p V ) = p V . Let V be a right N-submodule of L 2 (M); following [1], we say that V is of finite lifted trace if T r(p V ) < ∞. Furthermore, V is said to be of finite rank if there exist finitely many vectors ξ 1 , . . . , ξ r ∈ V such that V = r i=1 ξ i N . Following [4], we say that ξ 1 , . . . , ξ r is an orthonormal basis of V over N if E N (ξ * i ξ j ) = δ i,j f i where f i is a non zero projection for every i = 1, . . . , r.
If it is the case, it follows immediately that, if η ∈ V can be expressed as f 1 a 1 , . . . , a r = f r a r ∈ N, then such a decomposition is unique: if η = j ξ j b j with b j = f j b j ∈ N for all j, then a j = b j for all j = 1, . . . , r.
η = r j=1 ξ j a j with a 1 =
We will also freely use comparison theory of projections in factors and its relationships with (semi)finite traces as it appears for instance in Part III of [2] or in Chapter V of [6]: we just recall that, if M is a von Neumann algebra, if e, f ∈ M are projections, then e f if there exists a partial isometry u ∈ M such that u * u = e and uu * ≤ f . We denote by Ip(M) the set of all partial isometries of M.
Finally, if σ : Γ → Aut(M) is an action of Γ on M, a map u : Γ → Ip(M) is a σ-cocycle if it satisfies u(gh) = u(g)σ g (u(h)) for all g, h ∈ Γ.
The case of subfactors
Here is our first result. Its proof is strongly inspired by that of Proposition 1.3 in [3].
Proposition 1 Suppose that (M, τ, σ, Γ) is a von Neumann dynamical system and that N is a Γ-invariant type II 1 subfactor of M. Let p be a projection in M, e N such that 0 < t := T r(p) < ∞ and which is Γ-invariant. Then V := pL 2 (M) has finite rank r = t if t is an integer, or r = n + 1 where n is the integer part of t otherwise, and there exists a family (ξ j ) 1≤j≤r ⊂ L 2 (M) such that:
(a) E N (ξ * j ξ k ) = 0 for j = k; (b) E N (ξ * j ξ j ) = 1 for 1 ≤ j ≤ n, and E N (ξ * n+1 ξ n+1 ) := f n+1 is a projection of N with trace equal to T r(p) − n;
(c) each ξ j e N is a partial isometry in M, e N and j ξ j e N ξ * j = p;
(d) every η ∈ V has a unique decomposition η = j ξ j E N (ξ * j η), thus in particular, the family (ξ j ) 1≤j≤r is an orthonormal basis of V over N;
(e) there exists a σ ⊗ i r -cocycle u : Γ → Ip(M r (N)), g → u(g) = (u i,j (g)) 1≤i,j≤r , such that
u σ (g)ξ j = i ξ i u i,j (g) ∀j, g.
Proof. Let us assume that T r(p) is not an integer, hence that 0 < T r(p) − n < 1. Since M, e N is a factor of type II, there exist projections g 1 , . . . , g n , g n+1 ∈ M, e N such that (i) g j g k = 0 if j = k;
(ii) j g j = p; (iii) T r(g j ) = T r(e N ) = 1 for all j ≤ n, and T r(g n+1 ) = T r(p) − n.
Hence, for every j ≤ n, g j is a projection in M, e N which is equivalent to e N (since they have the same trace, and since JN ′ J = M, e N is a factor), and g n+1 e N . Thus, there exist partial isometries v j ∈ M, e N , j = 1, . . . , n + 1, such that v j v * j = g j for all j = 1, . . . , n + 1, v * j v j = e N for all j ≤ n, and v * n+1 v n+1 ≤ e N . Since v j = v j e N for all j, the use of the pull-down map Φ implies that v j = ξ j e N with ξ j = Φ(v j ) = Φ(v j e N ) ∈ L 2 (M), and this proves immediately statement (c).
For j = k, one has proves statement (a). Similarly, the fact that v * j v j = e N for j ≤ n shows that E N (ξ * j ξ j ) = 1 for j ≤ n. If j = n + 1, then v * n+1 v n+1 = E N (ξ * n+1 ξ n+1 )e N is a subprojection of e N , hence E N (ξ * n+1 ξ n+1 ) := f n+1 is a projection, too. Thus claim (b) holds true.
0 = v * j v k = e N v * j v k e N = E N (ξ * j ξ k )e N hence E N (ξ * j ξ k ) = 0, whichIf x ∈ M, one has p(xξ) = p(xe N ξ) = j ξ j e N ξ * j xe N ξ = j ξ j E N (ξ * j xξ)
thus, by density of M in L 2 (M), this proves statement (d).
Finally, put u i,j (g) = E N (ξ * i u σ (g)ξ j ) ∈ L 1 (N) for all i, j, so that u σ (g)ξ j = i ξ i u i,j (g) ∀j, g.
Then we claim that k u k,i (g) * u k,j (g) = δ i,j σ g E N (ξ * i ξ j ); this will prove that all u i,j (g)'s are bounded operators (by taking i = j), and that u(g) ∈ Ip(M r (M)). One has, indeed,
k u k,i (g) * u k,j (g)e N = k e N u k,i (g) * u k,j (g)e N = k e N (u σ (g)ξ i ) * ξ k e N ξ * k u σ (g)ξ j e N = e N (u σ (g)ξ i ) * pu σ (g)ξ j e N = δ i,j σ g (E N (ξ * i ξ j ))e N
since u σ (g) commutes with p and pξ i = ξ i . This shows that u k,i (g) is bounded, hence belongs to N and that
u(g) * u(g) = 1 0 . . . 0 0 1 . . . 0 . . . . . . . . . . . . 0 0 . . . σ g (f r ) ,
proving that u(g) is a partial isometry for all g. As obviously f r u r,r (g)σ g (f r ) = u r,r (g) for all g, the decomposition of each u σ (g)ξ j is unique, we see that u(gh) = u(g)σ g ⊗ i r (u(h)) for all g, h ∈ Γ as follows: in the proof of Proposition 4.5), but they satisfy u i,j ≤ 1 for all i, j, and this suffices to get the conclusion in the proof of Proposition 4.5 in [1].
u σ (gh)ξ j = u σ (g)(u σ (h)ξ j ) = u σ (g) i ξ i u i,j (h) = i u σ (g)(ξ i u i,j (h)) = i u σ (g)ξ i σ g (u i,j (h)) = k ξ k i ξ i u k,i (g)σ g (u i,j (h)) = k ξ k u k,j (gh).
The general case
Here is our last result.
Proposition 2 Let (M, τ, σ, Γ) be a von Neumann dynamical system, let N be a unital, Γinvariant von Neumann subalgebra of M, let p be a projection in M, e N such that 0 < T r(p) < ∞ and σ g (p) = p for all g ∈ Γ. For every ε > 0, there exists a projection q ≤ p in M, e N such that σ g (q) = q for all g, T r(p − q) < ε and there exists a finite family (ξ j ) 1≤j≤r ⊂ L 2 (M) with the following properties: (c) each ξ j e N is a partial isometry and j ξ j e N ξ * j = q;
(a) E N (ξ * j ξ k ) = 0 for all j = k; (b) E N (ξ * j ξ j ) =: f j is
(d) every η ∈ qL 2 (M) has a decomposition η = j ξ j E N (ξ * j η), thus in particular, W := qL 2 (M) is a right N-submodule of finite rank and the family (ξ j ) 1≤j≤r is an orthonormal basis of W over N;
(e) there exists a σ ⊗ i r -cocycle u : Γ → Ip(M r (N)), g → u(g) = (u i,j (g)) 1≤i,j≤r , such that
u σ (g)ξ j = i ξ i u i,j (g) ∀j, g.
Proof. Assume first that p e N . Then there exists a partial isometry v ∈ M, e N such that vv * = p and v * v ≤ e N . Thus v = ve N = ξ 1 e N where ξ 1 = Φ(v) = Φ(ve N ) ∈ L 2 (M). One checks as in the proof of Proposition 1 that v * v = E N (ξ * 1 ξ 1 ) = f is a projection in N and that pη = ξ 1 E N (ξ * 1 η) for every η ∈ L 2 (M), hence pL 2 (M) = ξ 1 N. Moreover, if we set u(g) = E N (ξ * 1 u σ (g)ξ 1 ), then it is easy to see that u(g) * u(g)e N = σ g (f )e N , hence that u(g) ∈ N is a partial isometry and one can check as in the proof of the preceeding proposition that u is a cocycle.
Next, if p e N , by the Comparability Theorem (see Theorem V.1.8 of [6], for instance), there exists a central projection z ∈ M, e N such that e N z pz and p(1 − z) e N (1 − z). By our present assumption, z = 0 and e N z = 0, since the central support of e N is 1. Then we define the following set Z, ordered by inclusion: it is the set of all families of projections (g i ) i∈I ⊂ M, e N such that g i g j = 0 for all i = j, 0 = g i e N , σ g (g i ) = g i for all i ∈ I and all g ∈ Γ and finally i∈I g i ≤ p.
Let us see that Z = ∅: as e N z pz, there exists a partial isometry u = 0 such that u * u = e N z ≤ e N and uu * ≤ pz ≤ p.
Then e := g∈Γ σ g (uu * ) is a Γ-invariant non zero projection, e ≤ p and, by Lemma V.2.2 of [6], e e N . Hence, by Zorn's Lemma, let (g i ) i∈I be a maximal element of Z. We claim that i∈I g i = p. Indeed, if it is not the case, put f = p − i g i = 0. Then σ g (f ) = f for all g, f ≤ p, and, as the central support of e N is 1, there exists a non zero projection f ′ ≤ f such that f ′ e N (Lemma V.1.7 of [6]). As above, the projection f ′′ = g∈Γ σ g (f ′ ) is non zero, Γ-invariant, and the family (g i ) i∈I ∪ {f ′′ } contradicts the maximality of (g i ) i∈I .
As i T r(g i ) = T r(p) < ∞, if ε > 0 is given, there exists {1, . . . , r} ⊂ I such that T r(p − q) < ε and σ g (q) = q for all g, where we have set q = r i=1 g i . Now, as in the proof of Proposition 1, there exist isometries v 1 , . . . , v r ∈ M, e N such that v i v * i = g i and v * i v i ≤ e N for all i. One can find vectors ξ 1 , . . . , ξ r ∈ L 2 (M) such that v i = ξ i e N for all i, and it is easy to see that statements (a), (b), (c) and (d) hold true. Furthermore, if we set u i,j (g) = E N (ξ * i u σ (g)ξ j ) so that u σ (g)ξ j = i ξ i u i,j (g), one can prove as in the previous section that k u k,i (g) * u k,j (g)e N = δ i,j σ g (f i )e N for all i, j, proving that u(g) = (u i,j (g)) ∈ M r (N) is a partial isometry. The cocycle identity is proved exactly as in Proposition 1.
Remark.
Contrary to what is stated in Lemma 4.1 of [1], the entries u i,j of the unitary operator u are not unitary operators themselves (neither are the entries u (−n) i,j
a non zero projection of N for all j = 1, . . . , r;
Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems. T Austin, T Eisner, T Tao, ArXiv:math.OA/09125093T. Austin, T. Eisner, and T. Tao. Nonconventional ergodic averages and multiple recur- rence for von Neumann dynamical systems. ArXiv:math.OA/0912.5093 v5, 2010.
Von Neumann Algebras. J Dixmier, North-Holland Mathematical Library, AmsterdamJ. Dixmier. Von Neumann Algebras. North-Holland Mathematical Library, Amsterdam, 1981.
Entropy and index for subfactors. M Pimsner, S Popa, Ann. Scient. ENS. 19M. Pimsner and S. Popa. Entropy and index for subfactors. Ann. Scient. ENS, 19:57-106, 1986.
Classification of Subfactors and Their Endomorphisms. S Popa, CBMS Lecture Notes. 86Amer. Math. SocS. Popa. Classification of Subfactors and Their Endomorphisms. CBMS Lecture Notes, 86, Amer. Math. Soc., 1995.
A Sinclair, R Smith, Finite von Neumann Algebras and Masas. CambridgeCambridge University PressA. Sinclair and R. Smith. Finite von Neumann Algebras and Masas. Cambridge University Press, Cambridge, 2008.
Theory of Operator Algebras I. M Takesaki, Springer-Verlag11New YorkUniversité de Neuchâtel, Institut de MathémathiquesM. Takesaki. Theory of Operator Algebras I. Springer-Verlag, New York, 1979. Université de Neuchâtel, Institut de Mathémathiques, Emile-Argand 11
Case postale 158 CH-2009 Neuchâtel. Switzerland [email protected] postale 158 CH-2009 Neuchâtel, Switzerland [email protected]
| []
|
[
"Octonacci photonic crystals with negative refraction index materials",
"Octonacci photonic crystals with negative refraction index materials"
]
| [
"E R Brandão \nDepartamento de Física Te orica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil\n",
"M.SVasconcelos \nEscola de Ciências e Tecnologia\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil\n",
"D H A L Anselmo \nDepartamento de Física Te orica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil\n"
]
| [
"Departamento de Física Te orica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil",
"Escola de Ciências e Tecnologia\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil",
"Departamento de Física Te orica e Experimental\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalRNBrazil"
]
| []
| a b s t r a c tWe investigate the optical transmission spectra for s-polarized (TE) and p-polarized (TM) waves in onedimensional photonic quasicrystals on a quasiperiodic multilayer structure made up by alternate layers of SiO 2 and metamaterials, organized by following the Octonacci sequence. Maxwell's equations and the transfer-matrix technique are used to derive the transmission spectra for the propagation of normally and obliquely incident optical fields. We assume Drude-Lorentz-type dispersive response for the dielectric permittivity and magnetic permeability of the metamaterials. For normally incident waves, we observe that the spectra does not have self-similar behavior or mirror symmetry and it also features the absence of optical band gap. Also for normally incident waves, we show regions of full transmittance when the incident angle q C ¼ 0 in a particular frequency range. | 10.1016/j.optmat.2016.11.013 | [
"https://repositorio.ufrn.br/jspui/bitstream/123456789/29136/4/OctonacciPhotonic_Anselmo_2016.pdf"
]
| 100,207,931 | 1611.05303 | 8cb45e18cb48dea47e1153e16ddabd9cbbf23a9d |
Octonacci photonic crystals with negative refraction index materials
Available online 23 November 2016
E R Brandão
Departamento de Física Te orica e Experimental
Universidade Federal do Rio Grande do Norte
59072-970NatalRNBrazil
M.SVasconcelos
Escola de Ciências e Tecnologia
Universidade Federal do Rio Grande do Norte
59072-970NatalRNBrazil
D H A L Anselmo
Departamento de Física Te orica e Experimental
Universidade Federal do Rio Grande do Norte
59072-970NatalRNBrazil
Octonacci photonic crystals with negative refraction index materials
Available online 23 November 201610.1016/j.optmat.2016.11.013Article history: Received 19 October 2016 Accepted 10 November 2016a r t i c l e i n f oPhotonic crystals Metamaterials Transmission Multilayers
a b s t r a c tWe investigate the optical transmission spectra for s-polarized (TE) and p-polarized (TM) waves in onedimensional photonic quasicrystals on a quasiperiodic multilayer structure made up by alternate layers of SiO 2 and metamaterials, organized by following the Octonacci sequence. Maxwell's equations and the transfer-matrix technique are used to derive the transmission spectra for the propagation of normally and obliquely incident optical fields. We assume Drude-Lorentz-type dispersive response for the dielectric permittivity and magnetic permeability of the metamaterials. For normally incident waves, we observe that the spectra does not have self-similar behavior or mirror symmetry and it also features the absence of optical band gap. Also for normally incident waves, we show regions of full transmittance when the incident angle q C ¼ 0 in a particular frequency range.
Introduction
Recently, the idea of complex materials in which both the permittivity and the permeability possess negative real values at certain frequencies has received considerable attention. In 1967, Veselago theoretically investigated plane-wave propagation in a material whose permittivity and permeability were assumed to be simultaneously negative [1]. His theoretical study showed that for a monochromatic uniform plane wave in such a medium the direction of the Poynting vector is antiparallel to the direction of the phase velocity, contrary to the case of plane-wave propagation in conventional simple media. In recent years, Smith, Schultz, and their group constructed such a composite medium for the microwave regime and demonstrated experimentally the presence of anomalous refraction in this medium [2,3]. For metamaterials with negative permittivity and permeability, several names and terminologies have been suggested, such as "left-handed" media [1e7]; media with negative refractive index (NRI) [1e4,6]; "backwardwave media" (BW media) [8]; and "double-negative" (DNG) metamaterials [9], to name a few. Many research groups all over the world are now studying various aspects and applications of these materials which have been proposed.
Also, since the discovery of quasicrystals by Shechtman et al. in 1984 [10], many efforts have been conducted to understand the physical properties of these aperiodic materials, which possess a long range order without having a translational symmetry. Hence, quasicrystals are regarded to have a degree of order intermediate between crystals and disordered systems. Quasicrystalline systems have been extensively studied, not only with respect to their structure, which shows uncommon rotational symmetries [10e12], and their electronic states, which were found to show a Cantor-set spectrum in one dimension [13e16], but also phonons and magnetic properties of these materials have been investigated [17e21]. Although the term quasicrystal is more appropriate when applied to natural compounds or artificial alloys, in one dimension (1D) there is no difference between this and the quasiperiodic structures formed by the incommensurate arrangement of periodic unit cells. An appealing motivation for studying such structures is that they exhibit a highly fragmented energy spectrum displaying a selfsimilar pattern.
Further, the photonic properties of those quasiperiodic structures are of special interest because the complex symmetries in quasicrystals make them suitable for the application in several optical devices such as single-mode light-emitting diodes, polarization switching and microelectronic devices that are based on photons rather than on electrons, which potentially can be the electromagnetic analogue to semiconductors [22e26]. The theoretical study of photonic properties of one-dimensional systems is based on the transfer matrix method and the concept of aperiodic mathematical sequences, as the Fibonacci sequence, the Thue-Morse sequence, and Cantor sequences [23,27e29]. Such onedimensional systems can be relatively easily produced in reality and a comparison of the theoretical and the experimental results shows a good agreement [30,31]. For the theoretical study of quasiperiodic systems one often applies the concept of aperiodic mathematical sequences/tilings. Especially, the photonic properties of one-dimensional systems have been extensively analyzed with this approach, where transfer matrix methods can be applied.
There are examples based upon the Fibonacci, Thue-Morse and Cantor sequences [23,27,28,32e34], and also systems with negative refractive indices have been studied [35e38, 40,41].
It is the aim of this work to study the propagation of light waves in multilayer photonic structures composed of SiO 2 /metamaterial (labeled A and B on this work) layers stacked alternately following the Octonacci sequence, which describes the arrangement of spacing of the Ammann quasilattice (8-grid), namely, the octagonal Ammann-Becker tiling [42]. The quasiperiodic structure follows the Octonacci sequence, and the multilayer photonic structure can be grown by juxtaposing the two building blocks A and B, where the nth-stage of the superlattice S N is given iteratively by the rule S N ¼ S NÀ1 S NÀ2 S NÀ1 , for N ! 3, with S 1 ¼ A and S 2 ¼ B. The number of the building blocks increases according to the Pell number, P N ¼ 2P NÀ1 þP NÀ2 (with P 1 ¼ 1 e P 2 ¼ 1). This structure can also be grown by following a recurrence rule, namely: A/ B, B/ BAB. Recently, we have studied the transmission spectra in onedimensional photonic quasicrystals, made up of SiO 2 (A) and TiO 2 (B) materials, organized following the Octonacci sequence [43]. In that work we report, for normally incident waves and for a same generation, that the transmission spectra for transverse electric (TE) and transverse magnetic (TM) waves presents a perfect scaling property where a self-similar behavior is obtained, as an evidence that these spectra are fractals. Also we show regions where the omnidirectional band gaps emerges for specific generations of Octonacci photonic structure, except for TM waves. On the other hand, M.-R. Wu et al. [44], have studied theoretically the photonic bandgap structure for a polaritonic photonic crystal containing lithium tantalate (LiTaO 3 ) in the NRI region. They have concluded that in NRI region we have a multi band gap structure and that the gap falling in the anomalous dispersion region can be treated as a zero-index gap which is further shown to be omnidirectional. However in both works, the authors do not explore the NRI region in order to study the transmission spectra. It is one of our aims to fill this gap, by studying the optical transmission in this region in order to search perfect, or almost perfect, transmission peaks.
Specifically, in this paper we want to investigate the behavior of the light when it pass through an Octonacci photonic layered system, considering the central wavelength l 0 ¼ 700 nm [45]. We also intend to investigate the influence of the oblique incidence at the system, by searching for frequency regions where the band gaps are independent from polarization and the incident angle q C .
The plan of this work is as follows. In Sec. 2, we present the method of calculation employed here, which is based on the transfer-matrix approach, together with a discussion of the transfer matrices for the quasiperiodic structure presented here. In Sec. 3 we present our results and discuss them. This is followed by a brief conclusion in Sec. 4.
Theoretical model
In the present work we make use of a theoretical model based on a transfer-matrix treatment (for a review see Refs. [38,40]).
Consider a s-polarized (TE wave) light of frequency u, normally incident from a transparent medium C at an arbitrary angle q C with respect to the normal direction of the layered system (see Fig. 1). The layered system is formed from an array of slabs of different materials (A or B). The reflectance and the transmittance coefficients are simply given by
R ¼ M 21 M 11 2 and T ¼ 1 M 11 2 ;(1)
where M ij (i,j ¼ 1,2) are the elements of the optical transfer-matrix M N , which links the coefficients of the electromagnetic fields in the region z < 0 to the coefficients of the electromagnetic fields in the region z > L, L being the size of the quasi-periodic structure. Let us consider first, to illustrate our method, the optical transfer-matrix calculation for the quasiperiodic multilayer which is characterized by having two dielectric media A and B with thicknesses d A and d B , and refractive indexes n A and n B , respectively, organized in accordance to the Octonacci sequence. It is surrounded by the transparent medium C with refractive index n C (see Fig. 1). The transmission of an obliquely incident light wave across the
interfaces a / b (i.e., C / A, A / B, …, B / C) is represented by the matrix M ab ¼ 1 2 1 þ k za . k zb 1 À k za . k zb 1 À k za . k zb 1 þ k za . k zb ! : (2) with k za ¼ h ðn a u=cÞ 2 À k 2 x i 1=2(3)
and
k x ¼ n c ðu=cÞsinðq C Þ:(4)
The propagation of the light wave within one of the layers g
(g ¼ A or B)
is characterized by the propagation matrices
M g ¼ exp À Àik g d g Á 0 0 exp À ik g d g Á ! ;(5)
We assume that, in each layer, the electrical field is given by
S 4 ¼ [BjAjBjBjBjAjB], with P 4 ¼ 7. L is the size of the whole superlattice structure. E ! ðNÞ j ¼ 0; E ðNÞ yj ; 0 (6) ðNÞ yj ¼ h A ðNÞ 1j exp À Àik zj z Á þ A ðNÞ 2j exp À ik zj z Á i  expðik x x À iutÞ(7)
where A ðNÞ 1j and A ðNÞ 2j (j ¼ A or B; N ¼ 0,1,2,/) are the amplitudes. Application of Maxwell's electromagnetic boundary conditions at the interface C/A, yields
" A ð0Þ 1C A ð0Þ 2C # ¼ M CA " A ð1Þ 1A A ð1Þ 2A # : (8) At the interface A/B, we find " A ð1Þ 1A A ð1Þ 2A # ¼ M A M AB " A ð3Þ 1B A ð3Þ 2B # :(9)
Successive application of the matrices M along the finite structure gives
" A ð0Þ 1C A ð0Þ 2C # ¼ T CABAB…BC " A ðNÞ 1C A ðNÞ 2C # ;(10)
where
We now intend to investigate the optical transfer matrices T, given by (11) for the periodic case, for structures that exhibit deterministic disorders, i.e. Octonacci structures. Therefore, from Equation (11) we can observe that the transfer matrix T N is formed by a product of matrices M ab and M g . The ordering of these matrices in the product depends upon the type of quasiperiodic array and the generation number N of the quasiperiodic sequence which is the same index used in the amplitudes of the electromagnetic field. The transfer matrices of all quasiperiodic systems considered here can be straightforwardly determined by induction method [38]. Therefore, by following the Octonacci sequence, it is easy to show that these matrices are given by
T N ¼ ðT NÀ1 Þ 2 T NÀ2 for N even;(12)T N ¼ ðT NÀ1 M BA Þ 2 T NÀ2 for N odd;(13)
whose initial conditions are
T 1 ¼ M A , and T 2 ¼ M A M AB M B .
Numerical results
We have chosen medium A as silicon dioxide (SiO 2 ), whose refraction index is n A ¼ 1.45, while medium B is a metamaterial, considering n B as frequency-dependent. Also, we assume the individual layers as quarter-wave layers, for which the quasiperiodicity is expected to be more effective [46], with the central wavelength l 0 ¼ 700 nm. These conditions yield the physical
thickness d J ¼ (175/n J ) nm, J ¼ A or B, such that n A d A ¼ n B d B , which
gives the very reversed phase shift in the two materials.
We consider the case where all realized artificial negative refractive index metamaterials have an electric permittivity ε and a magnetic permeability m that are frequency dispersive. Neglecting any damping term, the corresponding dielectric permittivity ε(u) and magnetic permeability m(u) are respectively given by Ref. [47].
εðuÞ ¼ 1 À u 2 p . u 2 ;(14)mðuÞ ¼ 1 À Fu 2 . u 2 À u 2 0 ;(15)
where the plasma frequency u p , the resonance frequency u 0 , and the fraction F are given by u 0 /2p ¼ 4 GHz, u p /2p ¼ 10 GHz, and F ¼ 0.56 (see Ref. [2]). Fig. 2 shows the plot of the permittivity, permeability, and real and imaginary part of the refractive index versus the reduced frequency U ¼ u/u p . In the present work, investigations have been carried out in yellow and green regions. As frequency increases, four important regions of investigation have been observed. In the white region, ε B < 0 and m B > 0. In the yellow region, ε B , m B and refractive index n B are negative. In the green region, ε B , m B and refractive index n B are positive. Bragg peaks They are also observed. The transmission spectrum for the periodic sequence analyzed depending on both the reduced frequency u/u p and incidence angle q ¼ q C is shown in Fig. 5 for TE and TM polarizations. In Fig. 5a (TE waves) we observe that with the increasing of the incidence angle, the complete band gaps keep the same width in the frequency region u/u p where n B has considerable negative values. For all angles q > 30 it arises a Bragg pseudo reflector (Bragg mirror), where the refraction index n B tends to zero for u close to 6 GHz (see Fig. 2).
On Fig. 5b (TM waves) the complete band gaps are no longer seen, and we observe only the Bragg pseudo reflector near the frequencies u/u p ¼ 0.4 and u/u p ¼ 0.6 when q > 9 + .
We now proceed to analyze the optical transmission spectrum, for the Octonacci quasiperiodic sequence (OQS), as a function of the reduced frequency u/u p , considering normal incidence for the three regions of Fig. 2, separately. For the frequency range 0.0 < u/ u p < 0.4, in the periodic sequence, where n B > 0, a complete reflection occurs. The same is observed for the Octonacci sequence, irrespective of the generation considered. In Fig. 6 we show the transmission spectra for the fifth, sixth, seventh and eighth Octonacci generations (with P 5 ¼ 17, P 6 ¼ 41, P 7 ¼ 99 and P 8 ¼ 239 layers, respectively), for s polarization (TE waves) and normal incidence where n B < 0, which depends on frequency, on the frequency range 0.4 < u/u p < 0.6. One can see that the spectra are not self-similar.
Instead of a symmetric distribution, it has three Bragg gaps or complete band gaps, which are characteristic of photonic crystals and conventional Bragg reflectors [39], when compared to quasiperiodic structures found by Medeiros and colleagues [40]. Notice that the complete band gaps are centered around 4.09 GHz, 4.24 GHz and 4.44 GHz, as shown in the orange (color online) emphasized areas. Also, the light transmission spectra present several Bragg peaks, whose number increases with the generation number N. These Bragg peaks where T ¼ 1 denote a completely transparent system (of course we are considering a lossless case!). On these special peaks we have the famous superlenses, a phenomenon studied by Pendry and Ziolkowski et al. [4,9]. Notice that this intriguing phenomenon would occur in the interval where the refraction index is negative.
In Fig. 7 we show the transmission coefficients for the OQS as a function of the reduced frequency u/u p and of the incidence angle q ¼ q C (C is the vacuum where the light beam comes from), in four situations: fifth (Fig. 8a), sixth (Fig. 8b), seventh (Fig. 8c), and eighth generation (Fig. 8d), respectively, in the frequency range 0.4 < u/ u p < 0.6. One can see narrow regions with a well defined photonic band gap. These band gaps are independent of the incidence angle, and are located around the same region of the reduced frequency Bragg pseudo reflector arises. In Fig. 8 we present the same analysis as Fig. 7, but now for the TM polarization. In this case there are no band gaps. Also, for all angles q > 9 a Bragg pseudo reflector is observed. Fig. 6. Transmission spectra T as functions of the reduced frequency u/u p for the Octonacci multilayers system (TE polarization) in the frequency range 0.4 < u/u p < 0.6, where n B is negative. We have so far analyzed the first two regions of Fig. 2, from a total of four. Now the make an analysis in the third region, where there must be absorption. However, in the present work, in the layer B (metamaterial), where the refraction index depends on the frequency, is a dispersive medium, so it causes some losses. Because of this, the absorption that could be observed is minimal. In the region where ε < 0, m > 0 and n B > 0 (imaginary), in the frequency range 0.6 < u/u p < 1.0, the light transmission spectrum with normal incidence for the fifth, sixth, seventh and eighth generations of the OQS is shown in Fig. 9a, b, 9c) and 9d, respectively. As one can see, the spectra features a complete transmission, or complete reflection (complete band gap in the frequency range 0.678 u/u p ¼ 0.754) for infinite systems. These features also are noticeable for others frequency range, as 0.1 < u/u p < 10.0, of the OQS (see Fig. 10).
Conclusions
In summary, we have shown the transmission spectra of light waves propagating through periodic and quasiperiodic Octonacci multilayers. We analysed the four frequency regions, namely, n B > 0 (imaginary), n B < 0 (real), n B > 0 (imaginary) e n B > 0 (real), respectively, as shown in Fig. 2. Specifically, one of the superlattice components has a negative refraction index, so it is a dispersive medium. In this case, we observe an abrupt behavior of the negative refraction index (NRI) near the frequency 4 GHz, and a smooth profile up to the 6 GHz frequency. In the periodic case, the transmission spectra, for normal incidence, presents complete band gaps for some frequency regions and also features several Bragg peaks.
When the incidence angle is oblique, for a varying q C , the band gaps became independent of q C , for TE polarization, a behavior not observed in the TM polarization. For the quasiperiodic structure, we consider a more realistic situation where the refraction index is frequency-dependent (layer B). The spectra are also analysed through the four frequency regions, in both normal and oblique incidence. In the frequency region 0.0 < u/u p < 0.4 the spectra presents a complete reflection for all generations of the OQS. Next, in the frequency range 0.4 < u/u p < 0.6, where the refraction index is negative, which means the B layer is a metamaterial, the spectra for normal incidence give rise to a rich profile of transmission of Bragg peaks, without self-similarity or mirror symmetry, but perfect transmission peaks (T ¼ 1). When q C is allowed to vary, we observe that the band gaps are independent from incidence angle, for TE polarization. For TM polarization the complete band gaps are no longer found. For 0.6 < u/u p < 1.0, where the absorption is small, we have, for normal incidence, a transmission spectre where one can see complete band gaps around 7.16 GHz (0.678 u/u p ¼ 0.754) for such infinite systems, becoming much more visible for higher generations. On the other hand, for oblique incidence, a complete reflection is more noticeable for TE polarization. However, for TM polarization we found that the spectra is totally reflected (complete reflection), irrespective of the incidence angle. Finally, for the frequency range 1.0 < u/u p < 10.0, the spectra exhibit several Bragg peaks and regions with complete band gaps. These intriguing properties could be used to make resonant optical cavities where a system of Octonacci multilayers (e.g. S4) featuring complete transmission, is built between two optical mirrors. Also we call attention to the possibility of design the so called superlenses studied theoretically by Pendry et al. [4,9], for very particular frequencies. We hope that our work could inspire experimentalists to probe the results presented here.
Fig. 1 .
1Schematic representation showing the geometry of the Octonacci quasiperiodic multilayer system considered in this work. More precisely, for sequence
Fig. 2 .Fig. 3 .
23Permittivity (red line), permeability (green line), real part of the refractive index (blue line) and imaginary part of the refractive index (orange line) of layer A versus the reduced frequency U. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Light transmission spectra for the case of incidence normal in a periodic photonic structure. (A) The transmittance T as a function of reduced frequency u/u p for frequency range 0.0 < u/u p < 0.4. (B) For the group of often reduced 0.4 < u/u p < 0.6. (C) For the group of often reduced 0.6 < u/u p < 1.0. (D) and to the track reduced frequency 1.0 < u/ u p < 10.0. T CABAB/BC ¼ M CA M A M AB M B …M B M BC :
Fig. 3
3shows the transmission spectrum of electromagnetic radiation, as a function of reduced frequency u/u p , for s polarization (TE wave) and a normally incident wave, for the periodic case at intervals of 20 unit cell repeats in four main regions frequencies shown inFig. 2in the range of omega frequency 0.0<u/u p < 10.0. Polarization p (TM wave) is not taken into account here, because in both polarizations S and p is obtained the same results standards. InFig. 3 to where refractive index n B is positive and imaginary in the frequency range 0.0 < u/u p < 0.4, shows a complete reflection (Bragg reflector). In Fig. 3b where n B < 0 in 0.4 < u/u p < 0.6, shows complete band gaps in the tracks frequency 0.4003 u/ u p ¼ 0.4004, 0.4005 u/u p ¼ 0.4007, 0.4009 u/u p ¼ 0.4010, 0.402 u/u p ¼ 0.403 and 0.408 u/u p ¼ 0.414, which can be observed with more details in Fig. 4 also We observe the Bragg peaks. Fig. 3 c, where n B is positive and imaginary in 0.6 < u/ u p < 1.0, It shows only some Bragg peaks. In the end, Fig. 3 d where n B > 0 in 1.0 < u/u p < 10.0 shows the complete band gaps in frequency bands 2.84 u/u p ¼ 3.71 and 6.13 u/u p ¼ 6.76, and some
Fig. 4 .
4Light transmission spectra of Fig. 3 b in the frequency range 0.4 u/u p ¼ 0.42.
Fig. 5 .
5Transmittance T in color map (see color scale at left), as a function of the reduced frequency u/u p and incidence angle q for the periodic sequence for light polarizations. (a) s polarization (TE waves) (b) p polarization (TM waves).
( 0 .
0408 u/u p ¼ 0.410, 0.421 u/u p ¼ 0.424, 0.438 u/u p ¼ 0.442) and they have approximately the same width. Again, we can assume the arising of these band gaps are due to the long range order of the arrangement of the layers in the OQS. For all angles q > 20 + a
Fig. 7 .
7Transmittance T in color map (see color scale at left), as function of reduced frequency u/u p for the frequency range 0.4 < u/u p < 0.6 and TE polarization. Here, medium B has a frequency dependent negative refraction index, whose dielectric function has a photonic character: (a) fifth generation; (b) sixth generation; (c) seventh generation; (d) eighth generation of the Octonacci quasiperiodic sequence.
Fig. 8 .
8The same ofFig. 7, but for the frequency range 0.4 < u/u p < 0.6 and TM polarization. Here, medium B has a frequency dependent negative refraction index, whose dielectric function has a photonic character: (a) fifth generation; (b) sixth generation; (c) seventh generation; (d) eighth generation of the Octonacci quasiperiodic sequence.
Fig. 9 .
9Transmission spectra T as function of reduced frequency u/u p for the Octonacci multilayers system (TE polarization) in the frequency range 0.6 < u/u p < 1.0, where n B is positive (complex).
Fig. 10 .
10Transmission spectra T as function of reduced frequency u/u p for the Octonacci multilayers system (TE polarization) in the frequency range 1.0 < u/u p < 10.0, onde n B is positive (real).
E.R. Brandão et al. / Optical Materials 62 (2016) 584e592
AcknowledgmentsThe authors would like to thank CAPES and CNPq (Brazilian Science Funding Agencies) for the financial support.
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| []
|
[
"ALGEBRAIC S-INTEGERS OF FIXED DEGREE AND BOUNDED HEIGHT",
"ALGEBRAIC S-INTEGERS OF FIXED DEGREE AND BOUNDED HEIGHT"
]
| [
"Fabrizio Barroero "
]
| []
| []
| Let k be a number field and S a finite set of places of k containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of S-integers of k. Moreover, we give an asymptotic formula for the number of S-integers of bounded height and fixed degree over k, where S is the set of places of k lying above the ones in S. | 10.4064/aa167-1-4 | [
"https://arxiv.org/pdf/1309.7849v3.pdf"
]
| 119,586,959 | 1309.7849 | f8549a52ca8ea73a6e39633b7561e4c0b5533a0d |
ALGEBRAIC S-INTEGERS OF FIXED DEGREE AND BOUNDED HEIGHT
28 Mar 2014
Fabrizio Barroero
ALGEBRAIC S-INTEGERS OF FIXED DEGREE AND BOUNDED HEIGHT
28 Mar 2014arXiv:1309.7849v2 [math.NT]
Let k be a number field and S a finite set of places of k containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of S-integers of k. Moreover, we give an asymptotic formula for the number of S-integers of bounded height and fixed degree over k, where S is the set of places of k lying above the ones in S.
Introduction
In this article we give asymptotic estimates for the cardinality of certain subsets of Q n of bounded height. Here and in the rest of the article, by height we mean the multiplicative absolute Weil height H on the affine space Q n , whose definition will be recalled in Section 2.
Let k be a number field of degree m over Q and let n and e be positive integers. We fix an algebraic closure k of k and set k(n, e) = α ∈ k n : [k(α) : k] = e ,
where k(α) is the field obtained by adjoining all the coordinates of α to k. By Northcott's Theorem [12], subsets of k(n, e) of uniformly bounded height are finite. Therefore, for any subset A of k(n, e) and H > 0, we may introduce the following counting function Various results about this counting function appeared in the literature. One of the earliest is a result of Schanuel [13], who gave an asymptotic formula for N (k(n, 1), H). Schmidt was the first to consider the case e > 1. In [14], he found upper and lower bounds for N (k(n, e), H) while in [15], he gave asymptotics for N (Q(n, 2), H). Shortly afterwards, Gao [8] found the asymptotics for N (Q(n, e), H), provided n > e. Later Masser and Vaaler [11] established an asymptotic estimate for N (k(1, e), H). Finally, Widmer [17] proved an asymptotic formula for N (k(n, e), H), provided n > 5e/2 + 5 + 2/me. However, for general n and e even the correct order of magnitude for N (k(n, e), H) remains unknown.
In this article we are interested in counting algebraic S-integers. Let S be a finite set of places of k containing the archimedean ones. As usual O S indicates the ring of S-integers of k. Let S be the set of places of k that lie above the places in S and let O S be the ring of S-integers of k. Alternatively, we could think of O S as the ring of those algebraic numbers having minimal polynomial over k that is monic and has coefficients in O S .
Given n and e positive integers, we set O S (n, e) = k(n, e) ∩ O n S = α ∈ O n S : [k(α) : k] = e . Let S ∞ be the set of archimedean places of k. If we choose S = S ∞ , then O S = O k is the ring of algebraic integers of k and we use the notation O k (n, e) with the obvious meaning. Besides the trivial cases O Q (n, 1) = Z n , the first asymptotic result can probably be found in Lang's book [9]. Lang states, without proof, where m = [k : Q], q is the rank of the unit group of O k , and γ k is an unspecified positive constant, depending on k. More recently, Widmer [16] established the following asymptotic formula N (O k (n, e), H) = t i=0 D i H men (log H men ) i + O(H men−1 (log H) t ), (1.1) provided e = 1 or n > e + C e,m , for some explicit C e,m ≤ 7. Here t = e(q + 1) − 1, and the constants D i = D i (k, n, e) are explicitly given. Our Theorem 1.1 generalizes Widmer's result in the case e = 1 to asymptotics for N (O S (n, 1), H). However, we do not obtain a multiterm expansion as in (1.1).
Chern and Vaaler, in [6], proved an asymptotic formula for the number of monic polynomials in Z[X] of given degree and bounded Mahler measure. Theorem 6 of [6] immediately implies the following estimate for some explicit constant C e . This was extended by the author in [1], where an asymptotic estimate is given for N (O k (1, e), H). Our Theorem 1.2 generalizes this result and gives an asymptotic estimate for N (O S (1, e), H) for any finite set of places S containing the archimedean ones.
We write S fin for the set of non-archimedean places of S. Suppose that S fin = {v 1 , . . . , v L } and that v l corresponds to the prime ideal p l of O k . We indicate by N(A) the norm from k to Q of the fractional ideal A and by N(S) the L-tuple (N(p 1 ), . . . , N(p L )). Let r and s be, respectively, the number of real and pairs of conjugate complex embeddings of k. Moreover, we indicate by ∆ k the discriminant of k. Let n be a positive integer, we set
(1.2) B (n) k,S = n r+s−1 2 sn m |S|−1 (|S| − 1)! |∆ k | n L l=1 1 log N(p l ) 1 − 1 N(p l ) n ,
and
C R,n = 2 n−M M j=1 2j 2j + 1 n−2j n M M ! ,
with M = ⌊ n−1 2 ⌋ (as usual ⌊x⌋ indicates the floor of x ∈ R), and C C,n = π n n n (n!) 2 .
In this article, as usual, empty products are understood to be 1. For non-negative real functions f (X), g(X), h(X) and X 0 ∈ R, we write f (X) = g(X)
+ O(h(X)) as X ≥ X 0 tends to infinity, if there is C 0 such that |f (X) − g(X)| ≤ C 0 h(X) for all X ≥ X 0 .
Theorem 1.1. Let n be a positive integer and let k be a number field of degree m over Q. Moreover, let S be a finite set of places of k containing the archimedean ones. Then, as H ≥ 2 tends to infinity,
N (O S (n, 1), H) = (2 r π s ) n B (n) k,S H mn (log H) |S|−1 + O H mn (log H) |S|−2 , if |S| > 1, O H mn−1 , if |S| = 1.
The implicit constant in the error term depends on m, n and N(S).
Theorem 1.2. Let e be a positive integer and let k be a number field of degree m over Q. Moreover, let S be a finite set of places of k containing the archimedean ones. Then, as H ≥ 2 tends to infinity,
N (O S (1, e), H) = e |S| C r R,e C s C,e B (e) k,S H me 2 (log H) |S|−1 + O H me 2 (log H) |S|−2 , if |S| > 1, O H e(me−1) L , if |S| = 1,
where L = log H if (m, e) = (1, 2) and 1 otherwise. The implicit constant in the error term depends on m, e and N(S).
As mentioned before, if S = S ∞ , then Theorem 1.1 reduces to (1.1), although with a larger error term, and Theorem 1.2 to the result in [1]. However, for the case S ∞ = S the results appear to be new.
As in [1], our proof relies on a work of the author and Widmer [2] about counting lattice points in definable sets in o-minimal structures. Our approach is similar to the one in [1], but in the case S = S ∞ the result is more straightforward, because the embedding of O k in R m is a lattice. On the other hand, if S S ∞ , the embedding of O S is dense in R m , and a more elaborate proof is needed.
Let us apply our theorems in a few simple examples. Fix a prime number p. One can see, as an easy exercise and as a special case of both theorems, that the number of elements of Z 1
p of height at most H is 2 log p 1 − 1 p H log H + O(H).
Now, let d be a square-free positive integer with d ≡ 3 mod 4. Consider k = Q( √ d) and set S to consist of the place corresponding to the prime ideal (2, 1 + √ d), in addition to the two archimedean places. Then
N (O S (n, 1), H) = 2n(2 n − 1) d n 2 log 2 H 2n (log H) 2 + O H 2n log H .
Now consider k = Q again and suppose the non-archimedean places in S are associated to the primes 2 and 3. Then
N (O S (1, 2), H) = 32 3 log 2 log 3 H 4 (log H) 2 + O H 4 log H .
In [11], Masser
Preliminaries
Let k be a number field of degree m over Q and let M k be the set of places of k. For v ∈ M k , we indicate by k v the completion of k with respect to v. We write Q v for the completion of Q with respect to the unique place of Q that lies below v. Moreover, we set d v = [k v : Q v ] to be the local degree of k at v.
Any v ∈ M k corresponds either to a non-zero prime ideal p v of O k or to an embedding of k into C. In the first case v is called a finite or non-archimedean place and we write v ∤ ∞. In the second case v is called an infinite or archimedean place and we write v | ∞. We set, for v ∤ ∞,
|α| v = N(p v ) − ord pv (α) dv ,
for every α ∈ k \ {0}, where ord pv (α) is the power of p v in the factorization of the principal fractional ideal αO k . Furthermore, |0| v = 0. If v | ∞ corresponds to σ v : k ֒→ C, we set
|α| v = |σ v (α)|,
for every α ∈ k, where | · | is the usual absolute value on C. The absolute multiplicative Weil height H : k n → [1, ∞) is defined by
(2.1) H(α 1 , . . . , α n ) = v∈M k max{1, |α 1 | v , . . . , |α n | v } dv m .
Note that for α ∈ k \ {0}, |α| v = 1 for finitely many v. Therefore, the product above is actually finite. Moreover, this definition is independent of the field containing the coordinates, and therefore the height is defined on Q n . For properties of the Weil height we refer to the first chapter of [4]. We conclude this section introducing semialgebraic sets and stating the Tarski-Seidenberg principle.
Definition 2.1. Let N and M i , for i = 1, . . . , N , be positive integers. A semialgebraic subset of R n is a set of the form
N i=1 M i j=1 {x ∈ R n : f i,j (x) * i,j 0},
where f i,j ∈ R[X 1 , . . . , X n ] and the * i,j are either < or =.
Let A ⊆ R n be a semialgebraic set, a function f : A → R n ′ is called semialgebraic if its graph Γ(f ) is a semialgebraic set of R n+n ′ .
If we identify C with R 2 , then the definitions of semialgebraic set and function are extended to subsets of C n and to functions of complex variables in a natural way. We will need the following theorem which is usually known as the Tarski-Seidenberg principle.
Theorem 2.1 ([3], Theorem 1.5). Let A ∈ R n+1 be a semialgebraic set, then π(A) ∈ R n is semialgebraic, where π : R n+1 → R n is the projection map on the first n coordinates.
A generalization
In this section we formulate a theorem which will be used later to derive Theorems 1.1 and 1.2.
In the following definition we consider functions whose domain is R n+1 or C n+1 . We use the notation z to indicate a vector with entries in a generic field, while x will be a vector with real coordinates. We are often going to identify a function f : Let r and s be non-negative integers, not both zero. A system N of r real and s complex semialgebraic distance functions (of dimension n) is called (r, s)-system (of dimension n).
C n → R with f : R 2n → R, where, if x = (x 1 , . . . , x 2n ) ∈ R 2n , f (x) = f (x 1 + ix 2 , . . . , x 2n−1 + ix 2n ).
Let us fix a number field k with [k : Q] = m. Let r and s be, respectively, the number of real and pairs of conjugate complex embeddings of k. These induce r + s archimedean places of k, with respective completions R or C. Given an (r, s)-system N of dimension n, we can associate to every archimedean place v a semialgebraic distance function N v on k n+1 v . We will mostly use the alternative notation N 1 , . . . , N r for the r real distance functions and N r+1 , . . . , N r+s for the s complex ones and we put d i = 1, for i = 1, . . . , r, and d i = 2 for i = r + 1, . . . , r + s. For the non-archimedean places we set
N v (z) = max {|z 0 | v , . . . , |z n | v } , for z = (z 0 , . . . , z n ) ∈ k n+1 v
. Now we can define, for α ∈ k n+1 , a height function associated to N ,
H N (α) m = v∈M k N v (σ v (α)) dv , where σ v is the embedding of k into k v corresponding to v, extended componentwise to k n+1 . Now, let O N S (H) be the set of a ∈ O n S with H N (1, a) ≤ H.
We are interested in obtaining an estimate for |O N S (H)| as H → ∞. In view of the application of such estimate we impose conditions on the functions N i . Namely, for i = 1, . . . , r +s, we set N i (z) = N i (1, z) and suppose that N i (z) ≥ 1 for every z ∈ R n or C n and that the sets
(3.1) Z i (T ) = z : N i (z) ≤ T have volume p i (T ) for every T ≥ 1, where p i (X) ∈ R[X]
is a polynomial of degree d i n and leading coefficient C i .
Theorem 3.1. Let N be an (r, s)-system of dimension n satisfying the above hypothesis. Moreover, suppose S is a finite set of places of k as fixed in Section 1. Then, for every
H 0 > 1 there exists a positive C 0 = C 0 (N , N(S), H 0 ), such that for every H ≥ H 0 |O N S (H)| − C N ,k,S H mn (log H) |S|−1 ≤ C 0 H mn (log H) |S|−2 , if |S| > 1, C 0 H mn−1 , if |S| = 1, where (3.2) C N ,k,S = n r+s−1 2 sn m |S|−1 (|S| − 1)! |∆ k | n r+s i=1 C i L l=1 1 log N(p l ) 1 − 1 N(p l ) n .
Proof of Theorems 1.1 and 1.2
In this section we apply Theorem 3.1 to prove Theorems 1.1 and 1.2. Let us start with the first one. We choose our system N to consist of the max norm
N v (z) = |z| ∞ = max {|z 0 |, . . . , |z n |} ,
for every archimedean place v of k. These N v clearly satisfy the definition of semialgebraic distance function. The sets Z i (T ) defined in (3.1) have volume (2T ) n for i = 1, . . . , r and π n T 2n for i = r + 1, . . . , r + s, for every T ≥ 1. Therefore, the hypotheses of Theorem 3.1 are satisfied.
Note that, for every a ∈ k n ,
H N (1, a) = v N v (1, σ v (a)) dv m = v max {1, |a 1 | v , . . . , |a n | v } dv m = H(a).
Therefore H N is the usual absolute Weil height defined in (2.1). The claim of Theorem 1.1 follows applying Theorem 3.1 with H 0 = 2. Now let us prove Theorem 1.2. We choose N to consist of the Mahler measure function:
N i (z 0 , . . . , z n ) = M (z 0 X n + z 1 X n−1 + · · · + z n ) = M (z 0 , . . . , z n ),
for every i = 1, . . . , r +s. Let us recall its definition. If f = z 0 X d +z 1 X d−1 +· · ·+z d is a non-zero polynomial of degree d with complex coefficients and roots α 1 , . . . , α d , the Mahler measure of f is defined to be:
(4.1) M (f ) = |z 0 | d h=1 max {1, |α h |} .
Moreover, we set M (0) = 0. Mahler ([10], Lemma 1) proved that M is continuous as a function of the coefficients and it is easy to see that it satisfies conditions i. and ii. of Definition 3.1. We now prove that it is a semialgebraic function. Proof. We start by proving the claim for the complex Mahler measure. We need to prove that, for every positive integer n, the function
M n : R 2(n+1) → [0, ∞) (x 0 , . . . , x 2n+1 ) → M ((x 0 + ix 1 )X n + · · · + (x 2n + ix 2n+1 ))
is semialgebraic, i.e., its graph
Γ(M n ) = (x 0 , . . . , x 2n+1 , t) ∈ R 2(n+1)+1 : M (x 0 , . . . , x 2n+1 ) = t is a semialgebraic set.
We prove this by induction on n. For n = 1,
Γ(M 1 ) = (x 0 , x 1 , x 2 , x 3 , t) ∈ R 5 : max x 2 0 + x 2 1 , x 2 2 + x 2 3 = t 2 , t ≥ 0 is clearly semialgebraic. Now suppose n > 1. Let Γ(M n ) = A ∪ B, where A = (x 0 , . . . , x 2n+1 , t) ∈ Γ(M n ) : x 2 0 + x 2 1 = 0 , and B = {(x 0 , . . . , x 2n+1 , t) ∈ Γ(M n ) : x 0 = x 1 = 0} . By the inductive hypothesis, B is a semialgebraic set since B = {(0, 0)} × Γ(M n−1 ). Now let A ′ be the set of points (x 0 , . . . , x 2n+1 , t, α 1 , β 1 , . . . , α n , β n ) ∈ R 2(n+1)+1+2n ,
such that x 2 0 +x 2 1 = 0, α h +iβ h , for h = 1, . . . , n, are the roots of (x 0 +ix 1 )X n +· · ·+(x 2n +ix 2n+1 ) and
(4.2) |x 0 + ix 1 | n h=1 max {1, |α h + iβ h |} = t.
This set A ′ is defined by the symmetric functions that link the coefficients of a polynomial with its roots and by (4.2). It is therefore semialgebraic. Since A is the projection of A ′ on the first 2(n + 1) + 1 coordinates, it is also semialgebraic by the Tarski-Seidenberg principle (Theorem 2.1). We have the claim for the complex Mahler measure. For the real one it is sufficient to note that its graph is nothing but the projection that forgets the coordinates
x 1 , x 3 , . . . , x 2n−1 , x 2n+1 of Γ(M n ) ∩ {(x 0 , . . . , x 2n+1 , t) : x 2j+1 = 0 for j = 0, . . . , n}.
Since M satisfies the three conditions of Definition 3.1, it is a semialgebraic distance function. Moreover, in [6], Chern and Vaaler calculated the volume of the sets of the form (3.1) for the real and the complex monic Mahler measure. By (1.16) and (1.17) of [6], for every T ≥ 1 the volumes of the sets {(z 1 , . . . , z n ) ∈ R n : M (1, z 1 , . . . , z n ) ≤ T }, and {(z 1 , . . . , z n ) ∈ C n : M (1, z 1 , . . . , z n ) ≤ T } are, respectively, polynomials p R (T ) and p C (T ) of degree n and 2n and leading coefficients
C R,n = 2 n−M M j=1 2j 2j + 1 n−2j n M M ! , 1 with M = ⌊ n−1 2 ⌋, and C C,n = π n n n (n!) 2 .
We just showed that N satisfies the hypothesis of Theorem 3.1 and we have that for every H 0 > 1 there exists a positive C 0 = C 0 (m, n, N(S), H 0 ), such that for every H ≥ H 0 ,
(4.3) O N S (H) − C r R,n C s C,n B (n) k,S H mn (log H) |S|−1 ≤ C 0 H mn (log H) |S|−2 , if |S| > 1, C 0 H mn−1 , if |S| = 1, where B (n)
k,S is the constant defined in (1.2). Let us reformulate these considerations in terms of polynomials. We proceed in a similar way as done in Section 2 of [1]. For any positive integer n we fix the system N n of dimension n to consist of Mahler measure distance functions and we define
M k : k[X] → [0, ∞) a 0 X n + a 1 X n−1 + · · · + a n → H Nn (a 0 , a 1 , . . . , a n ).
Therefore we can write for every H ≥ 1.
M k (a 0 , . . . , a n ) = r+s i=1 M (σ i (a 0 )X n + · · · + σ i (a n )) d i m v∤∞ max {|a 0 | v , . . . , |a n | v } dv m . Let M k,S (n, H) be the set of of monic polynomials f ∈ O S [X] of degree n with M k (f ) ≤ H.
Note that, for every α ∈ k,
(4.5) M k (X − α) = v∈M k max {1, |α| v } dv m = H(α).M k,S (n, H) − C r R,n C s C,n B (n) k,S H mn (log H) |S|−1 ≤ D 0 H mn (log H) |S|−2 , if |S| > 1, D 0 H mn−1 L, if |S| = 1,
where L = log H if (m, n) = (1, 2) and 1 otherwise.
Proof. For n = 1, there is nothing to prove. Suppose n > 1. We show that, up to a constant, the number of all monic reducible f ∈ O S [X] of degree n with M k (f ) ≤ H is not larger than the right hand side of (4.3), except for the case |S| = 1 and (m, n) = (1, 2). Consider all f = gh ∈ M k,S (n, H) with g, h ∈ O S [X] monic of degree a and b respectively, with 0 < a ≤ b < n and a + b = n. We have 1 ≤ M k (g), M k (h) ≤ H because g and h are monic. Thus, there exists a positive integer d such that 2 d−1 ≤ M k (g) < 2 d . Note that d must satisfy
(4.6) 1 ≤ d ≤ log H log 2 + 1 ≤ 2 log H + 1.
Since M k is multiplicative,
M k (h) = M k (f ) M k (g) ≤ 2 1−d H.
Using The last step of the proof links such irreducible polynomials with their roots and M k with the height of these roots. Recall that S is the set of places of k that lie above the places in S. Proof. If an algebraic number β ∈ O S has degree e over k, then it is clearly a root of a monic irreducible polynomial f ∈ O S [X] of degree e, and vice-versa. We claim that
H(β) e = M k (f ).
The function M k is independent of the choice of k since it is possible to define an absolute M Q over Q[X] that, restricted to any k[X], coincides with M k . To see this one can simply imitate the proof of the fact that the Weil height is independent of the field containing the coordinates (see [4], Lemma 1.5.2).
Suppose f = (X − α 1 ) · · · (X − α e ). By (4.5) we have
M Q(α i ) (X − α i ) = H(α i ),
and the α i have the same height because they are conjugate (see [4], Proposition 1.5.17). Finally, by the multiplicativity of M k we can see that
M k (f ) = M Q (f ) = e i=1 M Q (X − α i ) = H(α j ) e ,
for any α j root of f .
Counting lattice points
We start this section introducing the counting theorem that will be used to prove Theorem 3.1. The principle dates back to Davenport [7] and was developed by several authors. In a previous work [2], the author and Widmer formulated a counting theorem that relies on Davenport's Theorem and uses o-minimal structures. We do not need Theorem 1.3 of [2] in its full generality as we count lattice points in semialgebraic sets.
For a semialgebraic set Z ⊆ R n+n ′ , we call Z t = {x ∈ R n : (x, t) ∈ Z} the fiber of Z lying above t ∈ R n ′ and Z a semialgebraic family. It is clear that the fibers Z t are semialgebraic subsets of R n . Let Λ be a lattice of R n and let λ i = λ i (Λ), for i = 1, . . . , n, be the successive minima of Λ with respect to the unit ball B 0 (1), i.e., λ i = inf{λ : B 0 (λ) ∩ Λ contains i linearly independent vectors}.
The following theorem is a special case of Theorem 1.3 of [2].
Theorem 5.1. Let Z ⊂ R n+n ′ be a semialgebraic family and suppose the fibers Z t are bounded. Then there exists a constant c Z ∈ R, depending only on the family, such that
|Z t ∩ Λ| − Vol(Z t ) det Λ ≤ n−1 j=0 c Z V j (Z t ) λ 1 · · · λ j ,
where V j (Z t ) is the sum of the j-dimensional volumes of the orthogonal projections of Z t on every j-dimensional coordinate subspace of R n and V 0 (Z t ) = 1.
Let us introduce the family we want to apply Theorem 5.1 to. We fix an (r, s)-system N of dimension n consisting of r real and s complex semialgebraic distance functions. Recall that we defined N i (z) = N i (1, z). Moreover, we see the complex N i as functions from R 2n , i.e.,
N i (x 1 , x 2 , . . . , x 2n−1 , x 2n ) = N i (z 1 , . . . , z n ),
for (x 1 , x 2 , . . . , x 2n−1 , x 2n ) = (ℜ(z 1 ), ℑ(z 1 ), . . . , ℜ(z n ), ℑ(z n )).
Recall that d i = 1, for i = 1, . . . , r, and d i = 2, for i = r + 1, . . . , r + s, and m = r + 2s. Let
(5.1) Z = (x 1 , . . . , x r+s , t) ∈ R n(r+2s)+1 : r+s i=1 N i (x i ) d i ≤ t , where x i ∈ R d i n .
We need to show that Z is a semialgebraic family and that the fibers Z t are bounded for every t ∈ R.
Lemma 5.2. The set Z defined in (5.1) is semialgebraic.
Proof. First note that, since the N i are semialgebraic functions, also the N i are semialgebraic. Indeed, one can get Γ N i by intersecting Γ(N i ) with an appropriate affine subspace. Let us define the following sets:
S (i) = (x 1 , . . . , x r+s , t, t 1 , . . . , t r+s ) ∈ R mn × R 1+r+s : N i (x i ) = t i ,
for i = 1, . . . , r + s, and
A = (x 1 , . . . , x r+s , t, t 1 , . . . , t r+s ) ∈ R mn × R 1+r+s : r+s i=1 t d i i ≤ t .
All these sets are clearly semialgebraic. Let π be the projection map of R mn+1+r+s to the first mn + 1 coordinates. By the Tarski
i=1 t d i i ≤ t, i.e., r+s i=1 N i (x i ) d i ≤ t.
Therefore B = Z, and we proved the claim.
Since the N i are bounded distance functions, there exist positive real constants δ i such that
δ i |z| ∞ ≤ N i (z),
for every z in R n+1 or C n+1 (see [5], Lemma 2, p. 108). We define γ i = max{δ i : δ i |z| ∞ ≤ N i (z)} and N ′ i (z) = γ i |z| ∞ . As before, we use the notation
N ′ i (z) for N ′ i (1, z). Let N ′ be the (r, s)-system consisting of N ′ i (z) = γ i |z| ∞ for every i = 1, . . . , r + s. Each (x 1 , . . . , x r+s , t) such that r+s i=1 N i (x i ) d i ≤ t satisfies r+s i=1 N ′ i (x i ) d i ≤ t. Therefore, if Z ′ = (x 1 , . . . , x r+s , t) ∈ R mn+1 : r+s i=1 N ′ i (x i ) d i ≤ t ,
we have Z ⊆ Z ′ . For every x ∈ R d i n we have, by definition, N ′ i (x) ≥ γ i and therefore, for every (x 1 , . . . ,
x r+s ) ∈ Z ′ t , N ′ i (x i ) d i ≤ t j =i γ d j j
holds. This implies
|x i | d i ∞ ≤ t j γ d j j ,
for every i = 1, . . . , r + s. We have just showed that the fibers Z ′ t , and therefore Z t , are bounded.
From now on we use the notation Z(T ) for Z T . Recall that V j (Z(T )) is the sum of the jdimensional volumes of the orthogonal projections of Z(T ) on every j-dimensional coordinate subspace of R n and V 0 (Z(T )) = 1.
Since Z ⊆ Z ′ , we have V j (Z(T )) ≤ V j (Z ′ (T )). By Theorem 5.1 there exists a constant c Z , depending only on Z, such that
(5.2) |Z(T ) ∩ Λ| − Vol(Z(T )) det Λ ≤ mn−1 j=0 c Z V j (Z ′ (T )) λ 1 · · · λ j ,
for every T ∈ R.
We have to calculate Vol(Z(T )) and we need upper bounds for V j (Z ′ (T )).
Recall we supposed that, for every i = 1, . . . , r + s, N i (x) ≥ 1 and the volume of the set Z i (T ) defined in (3.1) is p i (T ) for every T ≥ 1, where p i is a polynomial of degree d i n and leading coefficient C i . Lemma 5.3. Let q = r + s − 1. Under the hypotheses above we have that, for every T ≥ 1,
Vol (Z(T )) = Q T 1 2 , log T , where Q(X, Y ) ∈ R[X, Y ], deg X Q = 2n, deg Y Q = q and the coefficient of X 2n Y q is n q q! q+1 i=1 C i .
Proof. This is a special case of Lemma 5.2 of [1].
The V j (Z ′ (T )) were already computed in [1].
Lemma 5.4. For each j = 1, . . . , mn − 1, there exists a polynomial P j (X, Y ) in R[X, Y ], with deg X P j ≤ 2n, deg Y P j ≤ q, and the coefficient of X 2n Y q is 0, such that, for every T ≥ 1, we have V j (Z ′ (T )) = P j T 1 2 , log T .
Proof. See [1], Lemma 5.4.
For an integer u, we will use the notation
X (u) = X u , for u > 0, 1, for u ≤ 0,
in order to avoid possible appearances of 0 0 , for instance in the following proposition, where we must consider (log T ) q for T ≥ 1 and q can be 0.
Proposition 5.5. Let N be a (r, s)-system of dimension n that satisfies the above hypotheses on the volumes of the sets Z i (T ) and Λ a lattice. There exist two positive real constants E and E ′ , depending only on N , such that, for every T ≥ 1,
|Z(T ) ∩ Λ| − n q q+1 i=1 C i q! det Λ T n (log T ) (q) ≤ D(Λ) ET n (log T ) (q−1) + E ′ , if q ≥ 1, D(Λ)ET n− 1 m , if q = 0, where D(Λ) = 1 det Λ + mn−1 j=0 1 λ 1 ...λ j . Moreover, if T < 1, then Z(T ) = ∅.
Proof. For T < 1, Z(T ) = ∅ since we supposed N i (x) ≥ 1 for every x. Suppose T ≥ 1.
We start with the case q = 0. In this case, our system N consists only of one function N 1 that can be either real (d 1 = m = 1) or complex (d 1 = m = 2). In any case, the volume of the set Z(T ) ⊆ R mn equals p 1 T 1 m for every T ≥ 1, where p 1 has degree mn and leading coefficient C 1 .
Fix a j, 1 ≤ j ≤ mn − 1. Any projection of Z ′ (T ) to a j-dimensional coordinate subspace has volume at most F j T j m , for some positive real constant F j . Therefore, there exists an E ′′ such that V j (Z ′ (T )) ≤ E ′′ T n− 1 m , for every T ≥ 1, and by (5.2) we have the claim if q = 0.
Suppose q > 0. By (5.2), Lemma 5.3 and Lemma 5.4, we have the following inequality, for every T ≥ 1,
|Z(T ) ∩ Λ| − n q q+1 i=1 C i q! det Λ T n (log T ) (q) ≤ D(Λ)P T 1 2 , log T ,
for some polynomial P (X, Y ) ∈ R[X, Y ] with deg X P ≤ 2n, deg Y P ≤ q, whose coefficients depend on N and the coefficient of X 2n Y q is 0. Since P satisfies such conditions, there exists a positive E such that P T
1 2 , log T ≤ ET n (log T ) (q−1) ,
for every T ≥ 3. For T ∈ [1,3], the function of T given by P T 1 2 , log T is bounded, say by E ′ .
Then P T 1 2 , log T ≤ ET n (log T ) (q−1) + E ′ ,
for every T ≥ 1. Clearly, E and E ′ depend only on the coefficients of P and therefore only on N .
6. Proof of Theorem 3.1
In this section we prove Theorem 3.1.
Recall that we fixed a number field k of degree m over Q. Let σ 1 , . . . , σ r be the real embeddings of k and σ r+1 , . . . , σ r+2s be the complex ones, indexed in such a way that σ i = σ i+s , for every i = r + 1, . . . , r + s. For a = (a 1 , . . . , a n ) ∈ k n , we set σ i (a) = (σ i (a 1 ), . . . , σ i (a n )) ∈ R n for i = 1, . . . , r and σ i (a) = (ℜ(σ i (a 1 )), ℑ(σ i (a 1 )), . . . , ℜ(σ i (a n )), ℑ(σ i (a n ))) ∈ R 2n for i = r + 1, . . . , r + s.
Let A be a non-zero fractional ideal of k. The image of A via the embedding σ : a ֒→ (σ 1 (a), . . . , σ r+s (a)) is a lattice in R m . If we set Λ A = τ (A n ), where τ (a) = (σ 1 (a), . . . , σ r+s (a)), for a ∈ k n , then Λ A is a lattice in R mn . Recall that N(A) indicates the norm of A and ∆ k the discriminant of k. 1 m . Proof. In [11] this Lemma is stated for integral ideals ( [11], Lemma 5). The same arguments work also for non-zero fractional ideals.
To prove Theorem 3.1 we need an estimate for the cardinality of O N S (H), i.e., the set of points a ∈ O n S such that H N (1, a) ≤ H. Recall that we set d i = 1, for i = 1, . . . , r, and d i = 2, for i = r + 1, . . . , r + s. As in Section 1, we call S fin the set of non-archimedean places in S.
First suppose S fin = ∅, then O S = O k and |S| = q + 1 = r + s. Note that, if a is a vector with integer coordinates, its non-archimedean absolute values are smaller than or equal to 1. Then From now, to avoid confusion between Cartesian powers and powers of an ideal with respect to the operation of ideal multiplication, we indicate the latter by A ⋆(d) for a non-zero fractional ideal A and an integer d. Now, suppose S fin = {v 1 , . . . , v L }, with L > 0. In this case we cannot apply Proposition 5.5 to τ (O n S ) directly because it is dense in R mn . Recall that v l corresponds to the prime ideal p l of O k . Let I S be the set of non-zero integral ideals A in O k which are products of the prime ideals we fixed, i.e., A = p
H N (1, a) = v∈M k N v (1, σ v (a)) dv m = r+s i=1 N i (σ i (a))⋆(g 1 ) 1 . . . p ⋆(g L ) L
for some non-negative integers g 1 , . . . , g L . An a ∈ k n is in O n S if and only if there exists an ideal A ∈ I S such that a u ∈ A ⋆(−1) for every u = 1, . . . , n, i.e., τ (a) = (σ 1 (a), . . . , σ r+s (a)) ∈ Λ A ⋆(−1) which is a lattice in R mn . We will therefore apply Proposition 5.5 to lattices of this form and then combine the obtained estimates.
We set
V k,N = n q 2 sn q! |∆ k | n q+1 i=1 C i .
For a non-zero integral ideal A and T > 0, by Z(A, T ) we indicate the set of a ∈ k n such that τ (a) ∈ Λ A ⋆(−1) ∩ Z(T m ).
Lemma 6.2. There exist two positive constants F and F ′ , depending only on N such that, for T ≥ 1 and every non-zero integral ideal A, we have
|Z(A, T )| − V k,N N(A) n T mn (log T m ) (q) ≤ N(A) n F T mn (log T m ) (q−1) + F ′ , if q ≥ 1, N(A) n F T mn−1 , if q = 0. Moreover, if T < 1, Z(A, T ) = ∅.
Proof. Note that, by Lemma 6.1, the first successive minimum of Λ A ⋆(−1) is greater than or equal to N(A) − 1 m . Since N(A) is a positive integer, we have
j i=1 λ i ≥ N(A) − j m ≥ N(A) − mn−1 m = N(A) −n+ 1 m ≥ N(A) −n ,
for every j = 1, . . . , mn − 1. Moreover, |∆ k | ≥ 1. The claim follows from Proposition 5.5 and Lemma 6.1, after noting that
D (Λ A ⋆(−1) ) ≤ mnN(A) n + 2 sn N(A) n |∆ k | n ≤ N(A) n (mn + 2 sn ) .
We fix a T ≥ 1. For a non-zero integral ideal A, let Z * (A, T ) be the subset of Z(A, T ) consisting of the points a such that, for every B strictly dividing A, there is a u ∈ {1, . . . , n} such that a u ∈ B ⋆(−1) . In other words, a corresponds to a lattice point of Λ A ⋆(−1) that is not contained in any sublattice of the form Λ B ⋆(−1) where B is a strict divisor of A. We have with H N (1, a) ≤ H. Lemma 6.3. For every H ≥ 1 we have Let I S (T ) be the set of ideals in I S with norm not exceeding T and recall that the norm is multiplicative. Combining (6.2) with (6.1), we have that
(6.1) |Z * (A, T )| − V k,N B|A µ k (B)N AB ⋆(−1) n T mn (log T m ) (q) ≤ B|A |µ k (B)|N AB ⋆(−1) n F T mn (log T m ) (q−1) + F ′ , if q ≥ 1, F B|A |µ k (B)|N AB ⋆(−1) n T mn−1 , if q = 0, and Z * (A, T ) = ∅ if T < 1. Recall that O N S (H) is the set of points a ∈ O n S(6.2) O N S (H) = A∈I S , N(A) −1 H m ≥1 Z * A, N(A) − 1 m H . Proof. Let A = p ⋆(g 1 ) 1 . . . p ⋆(g L ) L and recall d v l = [k v l : Q v l ] isO N S (H) − V k,N A∈I S (H m ) B|A µ k (B) N(B) n H mn log H m N(A) (q)
is smaller than or equal to
A∈I S (H m ) B|A |µ k (B)| N(B) n F H mn log H m N(A) (q−1) + F ′ N(A) n if q ≥ 1 andΨ (1) (A) log H m N(A) (q) ≤ A∈I S (H m ) Ψ (2) (A) F H mn log H m N(A) (q−1) + F ′ N(A) n , if q ≥ 1, F H mn−1 A∈I S (H m ) Ψ (2) (A)N(A) 1 m , if q = 0.
Let K be a non-negative integer, we set
L (h) S (H, K) = A∈I S (H m ) Ψ (h) (A) log H m N(A) (K)
, for h = 1, 2. Recall that we defined N(S) = (N(p 1 ), . . . , N(p L )). In the next lemma we allow S fin to be empty as the base step for the induction.
L (h) S (H, K) − L l=1 F (h) l K+L i=K+1 1 i (log H m ) (K+L) ≤ U K,N(S) (log H m + 1) (K+L−1) , for every H ≥ 1, where F (h) l = Ψ (h) (p l ) log N (p l ) .
Proof. We proceed by induction on the cardinality of S fin . Clearly, we can define L We have
L (h) S (H, K) = B∈R A(B) g L =0 Ψ (h) Bp ⋆(g L ) L log H m N(B) − g L log N (p L ) (K) = B∈R A(B) g L =1 Ψ (h) Bp ⋆(g L ) L K i=0 (−1) i K i (log N (p L )) i g i L log H m N(B) (K−i) + L (h) S ′ (H, K).
Using the definitions of Ψ (h) , it is easy to see that 1/2 ≤ Ψ (h) (p l ) ≤ 3/2 for every l and, if g L ≥ 1,
(6.4) Ψ (h) Bp ⋆(g L ) L = Ψ (h) (Bp L ) = Ψ (h) (B)Ψ (h) (p L ) > 0.
Therefore,
L (h) S (H, K) = Ψ (h) (p L ) K i=0 (−1) i K i (log N (p L )) i B∈R Ψ (h) (B) log H m N(B) (K−i) A(B) g L =1 g i L +L (h)
S ′ (H, K). where Q i is a polynomial of degree i (except Q 0 = 0) whose coefficients depend only on i. Then
A(B) g L =1 g i L − 1 i + 1 log H m N(B) −1 log N (p L ) i+1 ≤ Q ′ i log H m N(B) ,
where Q ′ i is a polynomial of degree at most i, whose coefficients depend on i and N (p L ). Finally, after noting that K i=0 (−1) i K i 1 i + 1 = 1 K + 1 , by (6.5), we can derive the following inequality:
L (h) S (H, K) − F (h) L K + 1 B∈R Ψ (h) (B) log H m N(B) (K+1) ≤ L (h) S ′ (H, K) + B∈R Ψ (h) (B)Q log H m N(B) ,
where Q is a polynomial of degree at most K whose coefficient depend only on K and N (p L ). Therefore, we have Note that, L ≤ |S| − 1 and if q ≥ 1, then L ≤ |S| − 2. Moreover,
L (h) S (H, K) − F (h) L K + 1 L (h) S ′ (H, K + 1) ≤ K i=0 b i L (h) S ′ (H, i),F (1) l = Ψ (1) (p l ) log N (p l ) = 1 log N (p l ) 1 − 1 N (p l ) n .
We apply Lemmas 6.4 and 6.5 and we can conclude that there exists a positive G = G(N , N(S)) such that |O n S (H)| − C N ,k,S H mn (log H) |S|−1 ≤ GH mn (log H + 1) |S|−2 , for every H ≥ 1, where C N ,k,S was defined in (3.2). Now, for every H 0 > 1, there exists a positive C 0 , clearly depending on N , N(S) and H 0 such that GH mn (log H + 1) |S|−2 ≤ C 0 H mn (log H) |S|−2 , and we have the claim of Theorem 3.1
N
(A, H) = | {α ∈ A : H(α) ≤ H} |.
N
(O Q (1, e), H) = C e H e 2 + O H e 2 −1 ,
rational number depending only on e, r and s, as already pointed out in[1] for the case S = S ∞ . As Masser and Vaaler did, one can ask again whether lim H→∞ N O S (n,
Definition 3 . 1 .
31Let n be a positive integer. A semialgebraic distance function (of dimension n) is a continuous function N from R n+1 or C n+1 to the interval [0, ∞) satisfying the following conditions:i. N (z) = 0 if and only if z is the zero vector; ii. N (wz) = |w|N (z) for any scalar w in R or in C; iii. N is a semialgebraic function.
Lemma 4. 1 .
1The Mahler measure M , as a function of the coefficients of a polynomial, is a semialgebraic function.
Clearly O N S (H) = |M k,S (n, H)| and (4.3) is an estimate for such cardinality. Fixing m, n, |S| and an |S|-tuple of prime powers, and letting k vary among all number fields of degree m, and S among the sets of places of the chosen number field with the predescribed set of norms of the non-archimedean places, the constants C r R,n , C s C,n and B (n) k,S are bounded and therefore there exists a constant G (n) m,N(S) , depending on n, m and N(S), such that |M k,S (n, H)| ≤ G (n) m,N(S) H mn (log H + 1) |S|−1 , (4.4)
It is clear from the definition of Mahler measure (4.1) thatM (f g) = M (f )M (g),and therefore, by Lemma 1.6.3 of[4], one can see thatM k (f g) = M k (f )M k (g), for every f, g ∈ k[X].Now we want to restrict to monic f irreducible over k. Let M k,S (n, H) be the set of monic irreducible polynomials f ∈ O S [X] of degree n with M k (f ) ≤ H, i.e., the polynomials in M k,S (n, H) that are irreducible over k.
Corollary 4. 2 .
2For every H 0 > 1 there exists a positive D 0 , depending on n, m, N(S) and H 0 , such that for every H ≥ H 0 we have
(4.4) and noting that 2 d ≤ 2H, we can say that there are at mostG H + 2) |S|−1 possibilities for h. Therefore, we have at most H (n) m,N(S) H mb 2 md(a−b) (log H + 2) 2(|S|−1) (4.7) possibilities for gh with M k (gh) ≤ H and 2 d−1 ≤ M k (g) < 2 d , where H (n) m,N(S) is a real constant depending on n, m and N(S). If a = b = n 2 , then (4.7) is H (n) m,N(S) H m n 2 (log H + 2) 2(|S|−1) . Summing over all d, 1 ≤ d ≤ ⌊2 log H⌋ + 1 (recall (4.6)), gives an extra factor 2 log H + 1. Therefore, when a = b, there are at most H f = gh, with M k (f ) ≤ H. If |S| > 1 or (m, n) = (1, 2), this has smaller order than the right hand side of (4.3), since mn > 2 implies mn 2 < mn − 1. In the case |S| = 1 and (m, n) = (1, 2), we get H (n) m,N(S) H (2 log H + 2) and we need an additional logarithm factor. In the case a < b, summing 2 md(a−b) over all d, 1 ≤ d ≤ ⌊2 log H⌋ + 1 =: D, recalling b ≤ n − 1, if a < b there are at most H (n) m,N(S) H m(n−1) (log H + 2) 2(|S|−1)possibilities for f = gh, with M k (f ) ≤ H. This is again not larger than the right hand side of (4.3).
Lemma 4 . 3 .
43An algebraic number β ∈ O S has degree e over k and H(β) ≤ H if and only if it is a root of a monic irreducible polynomial f ∈ O S [X] of degree e with M k (f ) ≤ H e .
This implies that |N (O S (1, e), H)| = e M k,S (e, H e ) because there are e different β ∈ O S with the same minimal polynomial over k. We have that, for every H 0 > 1, there exists a positive E 0 = E 0 (m, e, N(S), H 0 ) such that, for every H ≥ H 0 ,N (O S (1, e), H) − e |S| C r R,e C s C,e B (e) k,S H me 2 (log H) |S|−1 ≤ E 0 H me 2 (log H) |S|−2 , if |S| > 1, E 0 H e(me−1) L, if |S| = 1,where L = log H if (m, e) = (1, 2) and 1 otherwise. We obtain Theorem 1.2 by choosing H 0 = 2.
S
(i) ∩ A is semialgebraic. A point (x 1 , . . . , x r+s , t) belongs to B, if and only if there are t 1 , . . . , t r+s such that N i (x i ) = t i for every i and r+s
Lemma 6 . 1 .
61We have det Λ A = 2 −s N(A) |∆ k | n , and the first successive minimum of Λ A with respect to the Euclidean distance is λ 1 ≥ N(A)
a ∈ O n k . Therefore, the number of a ∈ O n k such thatH N (1, a) ≤ H is the number of lattice points of Λ O k = τ (O n k ) in Z(H m ). By Lemma 6.1, det Λ O k = 2 −s |∆ k | n and λ 1 ≥ 1. Thus, D(Λ O k ) ≤ mn + 2 sn . Moreover, for every H 0 > 1 there exists a C 0 = C 0 (N , H 0 ) such that, if q ≥ 1, (mn + 2 sn ) EH mn (log H m ) (q−1) + E ′ ≤ C 0 H mn (log H) (q−1) ,for every H ≥ H 0 and, in case q = 0, (mn + 2 sn )E ≤ C 0 . The claim of Theorem 3.1 follows applying Proposition 5.5.
|Z(A, T )| = B|A |Z * (B, T )|. If µ k is the Möbius function for the non-zero ideals of O k , the Möbius inversion formula implies that |Z * (A, T )| = B|A µ k (B) Z AB ⋆(−1) , T .Lemma 6.2 gives us an estimate for |Z * (A, T )|, for every T ≥ 1,
the local degree of k at v l . Every point a ∈ Z * (A, T ) is such that max u∈{1,...,n} |a u | dv l v l = N (p l ) g l , for every l = 1, . . . , L, and max u∈{1,...,n} |a u | v ≤ 1 for all v ∈ S. This means that every a ∈ Z * (A, T ) satisfies v∤∞ max u {1, |a u | v } dv = N(Therefore, a ∈ O N S (H) if and only if there exists an A ∈ I S such that a ∈ Z * A, N(A) − 1 m H . Since such an A is unique and recalling that, if T < 1, then Z * (A, T ) is empty, we obtain the claim.
F
A∈I S (H m ) B|A |µ k (B)| N(B) n N(A) 1 m H mn−1 if q = 0, for every H ≥ 1. Now, let Ψ (1) (A) = B|A µ k (B) N(B) n and Ψ (2) (A) = B|A |µ k (B)| N(B) n . Therefore (6.3) O N S (H) − V k,N H mn A∈I S (H m )
Lemma 6. 4 .
4For every non-negative integer K, there exists a positive constant U K,N(S) , depending only on K and N(S), such that for h = 1, 2
S
′ (H, K) and I S ′ for S ′ = S \ {v L }. If S fin is empty, i.e., L = 0, then I S (H m ) = {O k } and L (h) S (H, K) = (log H m ) (K) , for every H ≥ 1. Now suppose S fin has cardinality L > 0. The sum over all A ∈ I S (H m ) can be viewed as two sums: the first over all B ∈ I S ′ (H m ), and the second over all non-negative integers g L , with N p ⋆(g L ) L ≤ H m N(B) −1 . For typographical convenience we set A(B) = log H m N(B) −1 log N (p L ) , and R = I S ′ (H m ).
's formula, for every i = 0, . . . , K, we have H m N(B) −1 log N (p L ) i+1 = Q i log H m N(B) −1 log N (p L ) ,
S
where the b i are real coefficients again depending on K and N (p L ). Now, by the inductive hypothesis, there exist U K+1,N(S ′ ) andU ′ i,N(S ′ ) , for i = 0, . . . , K, log H m ) (K+L) ≤ U K+1,N(S ′ ) (log H m + 1) ′ (H, i) ≤ U ′ i,N(S ′ ) (log H m + 1) (i+L−1) ,for every i = 0, . . . , K. The claim follows easily.Lemma 6.5. There exists a real constant V m,N(S) , depending only on m and N(S), such that A∈I S (H m ) Ψ (2) (A)N(A) 1 m ≤ V m,N(S) H (log H + 1) (L−1) , for every H ≥ 1.
Proof.S
We proceed by induction on the cardinality of S fin as before. If S fin is empty, thenA∈I S (H m ) Ψ (2) (A)N(A)1 m = 1 and the claim holds. Now suppose S fin = {v 1 , . . . , v L }, with L > 0, and again p 1 , . . . , p L are the prime associated to the places in S fin . Let S ′ = S \ {v L } and again A(B) = log H m N(B) −1 log N(p L ) .Note that Ψ (2) (p L ) are ready prove Theorem 3.1. We already dealt with the case S fin = ∅. Suppose S fin = ∅. By (6.3) we haveO N S (H) − V k,N H mn L (H, q − 1) + F ′ H mn L (2) S (H, 0), if q ≥ 1, F H mn−1 A∈I S (H m ) Ψ (2)
and Vaaler observed that the limit for H → ∞ ofN k(1, e), H
1
e
N (k(e, 1), H)
is a rational number. Moreover, they asked if this can be extended to some sort of reciprocity
law, i.e., whether
lim
H→∞
N k(n, e), H
1
e
N k(e, n), H
1
n
∈ Q.
Analogously we notice that
lim
H→∞
N O S (1, e), H
There is a misprint in (1.16) of [6], 2 −N should read 2 −M .
AcknowledgementsThe author would like to thank Jeffrey Vaaler for many useful discussions and the hospitality at the Department of Mathematics at UT Austin and Martin Widmer for his encouragement and his advice that significantly improved this article.
Counting algebraic integers of fixed degree and bounded height. F Barroero, 10.1007/s00605-013-0599-6to appear in Monatshefte für MathematikF. Barroero, Counting algebraic integers of fixed degree and bounded height, to appear in Monatshefte für Mathematik, DOI: 10.1007/s00605-013-0599-6.
Counting lattice points and o-minimal structures, to appear in Int. F Barroero, M Widmer, 10.1093/imrn/rnt102Math. Res. Not. IMRN. F. Barroero and M. Widmer, Counting lattice points and o-minimal structures, to appear in Int. Math. Res. Not. IMRN, DOI: 10.1093/imrn/rnt102.
Semianalytic and subanalytic sets. E Bierstone, P D Milman, Inst. HautesÉtudes Sci. Publ. Math. 67E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. HautesÉtudes Sci. Publ. Math. (1988), no. 67, 5-42.
Heights in Diophantine Geometry. E Bombieri, W Gubler, New Mathematical Monographs. 4Cambridge University PressE. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006.
An Introduction to the Geometry of Numbers, Grundlehren der mathematischen Wissenschaften. J W S Cassels, Springer-VerlagJ.W.S. Cassels, An Introduction to the Geometry of Numbers, Grundlehren der mathematischen Wis- senschaften, Springer-Verlag, 1971.
The distribution of values of Mahler's measure. S Chern, J D Vaaler, J. reine angew. Math. 540S. Chern and J. D. Vaaler, The distribution of values of Mahler's measure, J. reine angew. Math. 540 (2001), 1-47.
On a principle of Lipschitz. H Davenport, J. London Math. Soc. 26H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179-183.
On Northcott's Theorem. X Gao, University of ColoradoPh.D. ThesisX. Gao, On Northcott's Theorem, Ph.D. Thesis, University of Colorado (1995).
Fundamentals of Diophantine Geometry. S Lang, Springer-VerlagNew YorkS. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983.
On the zeros of the derivative of a polynomial. K Mahler, Proc. Roy. Soc. Ser. A. 264K. Mahler, On the zeros of the derivative of a polynomial, Proc. Roy. Soc. Ser. A 264 (1961), 145-154.
Counting algebraic numbers with large height. D Masser, J D Vaaler, Trans. Amer. Math. Soc. II1D. Masser and J. D. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc. 359 (2007), no. 1, 427-445.
An inequality in the theory of arithmetic on algebraic varieties. D G Northcott, Proc. Cambridge Philos. Soc. 45D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502-509.
S H Schanuel, Heights in number fields. 107S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433-449.
Northcott's theorem on heights. I. A general estimate. W M Schmidt, 169-181. 15Acta Arith. LXX. 1151-2Monatsh. Math.W. M. Schmidt, Northcott's theorem on heights. I. A general estimate, Monatsh. Math. 115 (1993), no. 1-2, 169-181. 15. , Northcott's theorem on heights II. The quadratic case, Acta Arith. LXX.4 (1995), 343-375.
Integral points of fixed degree and bounded height, submitted. 17. , Counting points of fixed degree and bounded height. M Widmer, Acta Arith. 1402M. Widmer, Integral points of fixed degree and bounded height, submitted. 17. , Counting points of fixed degree and bounded height, Acta Arith. 140 (2009), no. 2, 145-168.
. Scuola Normale Superiore, [email protected] dei Cavalieri. 7Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail address: [email protected]
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| Here we present a detailed study of the broadband noise in the power density spectra of the black hole X-ray binary MAXI J1820+070 during the hard state of its 2018 outburst, using the Hard X-ray Modulation Telescope (Insight-HXMT) observations. The broadband noise shows two main humps, which might separately correspond to variability from a variable disk and two Comptonization regions. We fitted the two humps with multiple Lorentzian functions and studied the energy-dependent properties of each component up to 100-150 keV and their evolution with spectral changes. The lowest frequency component is considered as the sub-harmonic of QPO component and shows different energy dependence compared with other broadband noise components. We found that although the fractional rms of all the broadband noise components mainly decrease with energy, their rms spectra are different in shape. Above ∼ 20-30 keV, the characteristic frequencies of these components increase sharply with energy, meaning that the high-energy component is more variable on short timescales. Our results suggest that the hot inner flow in MAXI J1820+070 is likely to be inhomogeneous. We propose a geometry with a truncated accretion disk, two Comptonization regions. | null | [
"https://arxiv.org/pdf/2204.00739v3.pdf"
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| 247,939,648 | 2204.00739 | a01cf19e1ea80946537e5e7bbf166d73034ca603 |
The accretion flow geometry of MAXI J1820+070 through broadband noise research with Insight-HXMT
Z -X Y
L Z
Q -C B
Y H
H -X L
P J W
L T
J L Q
S Z
S N Z
X M
L M S
S M J
M Y G
Q Z L
J Z Y
D K Z
T M L
B Y W
X Q R
R C M
Y X Z
Y C X
B Y M
Y F D
Y C F
Y X X
Institute of High Energy Physics
Key Laboratory for Particle Astrophysics
Chinese Academy of Sciences
19B Yuquan Road100049BeijingPeople's Republic of China
University of Chinese Academy of Sciences
Chinese Academy of Sciences
100049BeijingPeople's Republic of China
Institut für Astronomie und Astrophysik
Kepler Center for Astro and Particle Physics
Eberhard Karls Universität
Sand 1, GermanyãĂŊ 4 Purple Mountain Observatory72076Tübingen
Chinese Academy of Sciences
210034NanjingPeople's Republic of China
Kapteyn Astronomical Institute
University of Groningen
Postbus 8009700 AVGroningenThe Netherlands
The accretion flow geometry of MAXI J1820+070 through broadband noise research with Insight-HXMT
(Received January 1, 2018; Revised January 7, 2018; Accepted April 11, 2022) Submitted to ApJD A 11, 2022 Typeset using L A T E X twocolumn style in AASTeX62Accretion disksCoronaLMXBInsight-HXMT
Here we present a detailed study of the broadband noise in the power density spectra of the black hole X-ray binary MAXI J1820+070 during the hard state of its 2018 outburst, using the Hard X-ray Modulation Telescope (Insight-HXMT) observations. The broadband noise shows two main humps, which might separately correspond to variability from a variable disk and two Comptonization regions. We fitted the two humps with multiple Lorentzian functions and studied the energy-dependent properties of each component up to 100-150 keV and their evolution with spectral changes. The lowest frequency component is considered as the sub-harmonic of QPO component and shows different energy dependence compared with other broadband noise components. We found that although the fractional rms of all the broadband noise components mainly decrease with energy, their rms spectra are different in shape. Above ∼ 20-30 keV, the characteristic frequencies of these components increase sharply with energy, meaning that the high-energy component is more variable on short timescales. Our results suggest that the hot inner flow in MAXI J1820+070 is likely to be inhomogeneous. We propose a geometry with a truncated accretion disk, two Comptonization regions.
INTRODUCTION
Black hole X-ray binaries (BHXBs) in a complete outburst usually show a counter-clockwise 'q-shaped' evolution pattern in the hardness-intensity diagram (HID). The different branches of HID correspond to different spectral and temporal states (see Remillard & McClintock 2006;Belloni & Motta 2016, for reviews). At the beginning of an outburst, the source is observed in a hard state (HS), where its energy spectrum is dominated by a hard power-law component with a and width), we divide them into QPO (Q > 2) and broadband noise (Q < 2) components in LHS and IMS (Belloni et al. 2002). According to different central frequencies and quality factors etc., QPO can be divided into A, B and C types. In low/hard state and hard immediate state, type-C QPO is believed to come from Lense-Thirring precession based on truncated disk/hot inner flow model (Schnittman et al. 2006;Ingram et al. 2009). For other broadband continuum components which represent fast aperiodic variability, several models have been presented to explain this noise component, including shot noise model, coronal flare model and fluctuation propagation model (Terrell 1972;Nolan et al. 1981;Belloni & Hasinger 1990;Mineshige et al. 1994;Stern & Svensson 1996). Considering that shot noise model predicts a stationary power spectrum and cannot produce a linear rms-flux relation for different timescale, it is not accepted to explain the broadband noise (Li et al. 2012). Furthermore, Uttley (2004) showed that rms-flux relation in the accreting millisecond pulsar SAX J1808.4-3658 is coupled with the 401 Hz pulsation. This relation put strict constrain on the origin of rms-flux relation from magnetic caps of the neutron star, which means that the linear relation does not favor the coronal flare model for the X-ray variability. Up to now, fluctuation propagation model is widely accepted because it naturally explains the rms-flux relation for different timescale which is common in X-ray binaries (Uttley & McHardy 2001;Negoro & Mineshige 2002;Uttley 2004;Gleissner et al. 2004;Uttley et al. 2005;Li et al. 2010Li et al. , 2012. In the fluctuation propagation model (Lyubarskii 1997;Ingram & Done 2012;Ingram 2016;Mushtukov et al. 2019), the broadband noise components are believed to break down at local viscous frequency visc ∝ 1/ 2 in power spectrum with Lorentzian shape 1/(1+ ( / visc ( )) 2 ) in PDS (Rapisarda et al. 2014(Rapisarda et al. , 2016bIngram 2016;Rapisarda et al. 2017a,b;Turner & Reynolds 2021). Perturbation occurs at each radius of the accretion flow, but the fluctuation from the outer region will modulate the inner region because the inward motion of accretion flow. This is the reason why we call it the fluctuation propagation model. Rapisarda et al. (2014) applied for the first time PROPFLUC on the BHB MAXI J1543-564 fitting the single-hump power spectrum in a single energy band. After that, Rapisarda et al. (2016a) applied PROPFLUC on BHB MAXI J1659-152 using for the first time the hypothesis of fluctuations stirred up and propagating from the disc to hot flow. They fitted simultaneously the power spectra in two energy bands and cross-spectra between two bands. Rapisarda et al. (2017b) further updated PROPFLUC by introducing the hypothesis of extra variability in the hot flow, damping and different propagation speeds of the fluctuations. Rapisarda et al. (2017a) modelled the power spectra, time lags and coherence in hard and soft states of Cyg X-1. Mushtukov et al. (2018) considered more realistically both forward and backward propagation for the first time and find propagating fluctuations also produce soft lags at high frequency as the reflection process does by numerical simulations. Mahmoud & Done (2018a,b) built a spectral-timing model to explain the energy dependence of power spectra and phase lag spectra with two Comptonization zones basing on fluctuation propagating model. Stiele & Yu (2015) showed a noise component with a characteristic frequency above 1 Hz in the hard energy band (4-8 keV) and the same component at a lower frequency in the soft band (1-2 keV) in a large BHXB sample. The dependence was interpreted as a hint that the soft photons originating in the outer region of the Comptonizing corona whereas hard band locates inner region. But due to the detector energy band limit, only the energy band below 10 keV was implemented (Stiele & Yu 2015). Apart from the energy dependence of characteristic frequency, in LHS, the noise is also slightly stronger at lower energy. The fractional rms of noise as a function of energy is flat or decreases by a few percent from 2 to 20 keV (Belloni et al. 2011). In previous studies, some authors (Yu & Zhang 2013;Stiele & Yu 2014) also investigated the energy dependence of power spectra but is limited to narrow energy range. It is necessary to provide more information about the high energy dependence of broadband noise. As a result, we present a wider range band energy band dependence in more detail with the help of Insight-HXMT LE, ME and HE detectors for the first time.
MAXI J1820+070 is a new X-ray transient discovered on 2018 March 11 by MAXI/GSC (Matsuoka et al. 2009;Kawamuro et al. 2018). Optical follow-up observations identified a optical counterpart coinciding with ASASSN-18ey (Denisenko 2018). Torres et al. (2019) derived a mass function ( ) = 5.18 ± 0.15 M , dynamically confirming the black hole nature of the source. Torres et al. (2020) estimated the mass of the black hole to be 5.73 < (M ) < 8.34 under 95% confidence level limits. A precise distance of 2.96 ± 0.33 kpc was obtained from radio parallax (Atri et al. 2020), with a jet inclination angle = 63 ± 3°. The similar distance result = 2.81 +0.70 −0.39 kpc was obtained by Gaia EDR3 parallax measurement (Bailer-Jones et al. 2021). By fitting the temperature and radius of the donor, Mikolajewska et al. (2022) also constrained the distance at ≈ 3kpc. MAXI J1820+070 is likely to harbor a slowly spinning black hole. Guan et al. (2021) constrained the spin of the black hole to be * = 0.2 +0.2 −0.3 by fitting the Insight-HXMT broadband spectra. A similar low spin result * = 0.14±0.09 was obtained by Zhao et al. (2021).
During the outburst, MAXI J1820+070 stayed in the HS for almost 4 months from 2018 March to early 2018 July. The unchanging shape of the Fe line profile (Buisson et al. 2019), together with the shortening thermal reverberation lags, suggest that the HS evolution is driven by the changes of the corona, rather than the disk (Kara et al. 2019). However, several recent results are inconsistent with this picture and support a truncated disk geometry Dziełak et al. 2021). Type-C QPOs were detected in Optical and X-ray wavelengths (Yu et al. 2018;Zampieri et al. 2018;Fiori et al. 2018;Stiele & Kong 2020;Ma et al. 2021;Thomas et al. 2021). Starting from 2018 July 4 (MJD 58303), the source underwent a rapid hard-to-soft transition. During the transition, an extremely powerful superluminal ejection was detected close in time to the appearance of the type-B QPO (Homan et al. 2020). After the transition, the source moved to the SS and stayed there for over 2 months before the final soft-to-hard transition (see Stiele & Kong 2020 for the details of the outburst evolution). Dziełak et al. (2021) studied the properties of the broadband noise in the PDS of MAXI J1820+070 with NICER data. They found that the broadband noise can be fitted with four Lorentzians and the spectra of these variability components are quite different in shape. At least two Comptonization regions with different temperatures and optical depths are required to fit both the variability spectra and the time-averaged spectra. Kawamura et al. (2022) proposed a model based on the fluctuations propagation by considering that the hot inner flow is spectrally inhomogeneous, and the viscous time-scale is discontinuous between the disk and the hot flow. This model reconstructs the shape of the broadband noise below 10 keV in MAXI J1820+070. The large effective area of Insight-HXMT at high energies enables us to perform detailed analysis on fast X-ray variability at energy bands above 100 keV (e.g., Huang et al. 2018;Liu et al. 2020Liu et al. , 2021Huang et al. 2021). In Ma et al. (2021), the authors re-ported the discovery of low-frequency QPOs above 200 keV in MAXI J1820+070 for the first time. In this work, we present a qualitative study of the evolution of the broadband noise and its energy dependence using Insight-HXMT observations of MAXI J1820+070. For the first time, we extend the study of the broadband noise up to 100-150 keV. In Section 2, we describe the observation and data reduction. The analysis and main results are presented in Section 3 and discussed in Section 4. In Section 5, we summarize our results.
OBSERVATIONS AND DATA REDUCTION
Insight-HXMT is China's first X-ray astronomy satellite, launched on 2017 June 15 . It carries three slat-collimated instruments: the High Energy X-ray telescope (HE: 20-250 keV, Liu et al. 2020), the Medium Energy Xray telescope (ME: 5-30 keV, Cao et al. 2020), and the Low Energy X-ray telescope (LE: 1-15 keV, Chen et al. 2020).
Four days after the discovery of MAXI J1820+070 , Insight-HXMT started monitoring MAXI J1820+070 at a high cadence. During its initial full outburst between 2018 March and October, Insight-HXMT accumulated a total exposure of 2560 ks. The LE (1-10 keV), ME (10-30 keV) and HE (30-150 keV) light curves of this outburst have been shown in Ma et al. (2021). In Figure 1, we show the HID of this outburst. For our analysis, we only selected observations with the LE exposure time longer than 2500 s and the LE counts rate larger than 440 counts s −1 , where their PDS show a significant broadband noise and there are enough photons to do detailed timing analysis. Table 1 lists the log of the observations used in this work. The data are extracted from all three instruments using the Insight-HXMT Data Analysis software (HXMTDAS) v2.04
, and filtered with the following criteria: (1) pointing offset angle less than 0.04°; (2) Earth elevation angle larger than 10°;
(3) the vaule of the geomagnetic cutoff rigidity larger than 8 GV; (4) at least 300 s before and after the South Atlantic Anomaly passage. To avoid possible contamination from the bright Earth and nearby sources, we only use data from the small field of view (Chen et al. 2018).
ANALYSIS AND RESULTS
To study the fast X-ray variability, we produce PDS from different energy bands for each observation we used. We use 128-s long intervals and 1/256-s time resolution, corre-The data analysis software is available from http://hxmten.ihep.ac.cn/software.jhtml. sponding to a Nyquist frequency of 128 Hz. The PDS is then applied to Miyamoto normalization, namely normalized to fractional rms (Miyamoto et al. 1991). In Figure 2, we show representative PDS for LE (1-10 keV), ME (10-30 keV) and HE (30-150 keV), respectively. The PDS of ME and HE are separately multiplied by a factor of 1.4 and 2.1 to keep the QPO aligned between energy bands. The PDS we show here are extracted from two relatively long observations (ObsIDs P0114661003 and P0114661004) with a similar PDS shape. It is apparent from this figure that, although the shape of the noise component below the QPO frequency has not changed too much, the shape of the noise component above the QPO frequency changes significantly between energies. In order to further study the properties of the QPO and the broadband noise, we fit the PDS with a multiple-Lorentzian model (Belloni et al. 2002). In Figure 3, we show a representative PDS of the HE (30-150 keV) band with its best fit. The QPO and its second harmonic are fitted with one Lorentzian each ( 1 10 1
Hardness (3-10 keV/1-3 keV) and 2 ). The broadband noise shows a two-humped shape, and can be well-fitted with four Lorentzians, i.e., a very lowfrequency noise ( 1 ), a low-frequency noise ( 2 ), and two high-frequency noise components ( 3 and 4 ). After the fitting process, we calculate the characteristic frequency and the fractional rms amplitude for each component. The characteristic frequency, max , is defined as max = √︃ 2 0 + ( /2) 2 , where 0 is the centroid frequency and is the full width at half maximum (FWHM) of the Lorentzian function (Belloni et al. 1997).
In order to study the energy-dependent properties of the different components, we extract PDS from 12 energy bands: LE (1-3 keV, 3-7 keV), ME (7-10 keV, 10-20 keV, 20-30 keV), HE (30-40 keV, 40-50 keV, 50-60 keV, 60-70 keV, 70-80 keV, 80-90 keV, 90-150 keV). In this paper, we only show the results combined from ObsIDs P0114661003 and P0114661004. The energy-dependent properties of the other observations are similar.
3.1. Properties of 1 Figure 4 shows the energy dependence of the fractional rms and characteristic frequency of 1 . The characteristic frequency of 1 remains more or less constant in the 1-150 keV energy band. This is similar to what was found in Ma et al. (2021) for the QPO and its second harmonic. The evolution of the fractional rms of 1 with energy is more complicated. Below ∼30 keV, the fractional rms shows a slight decreasing Power*Frequency L 1 L 2 L 3 L 4 Q 1 Q 2 Figure 3. A representative PDS of HE (30-150 keV) plotted along with the best multi-Lorentzian fit (red). The PDS are calculated using the data of ObsIDs P0114661003 and P0114661004, and fitted with a multiple-Lorentzian model. 1 and 2 represent the QPO and its second harmonic, respectively. 1 , 2 , 3 and 4 represent the four broadband noise components on different timescales. trend with energy; while above ∼30 keV, the fractional rms increases monotonously with energy.
In Figure 5 we show the characteristic frequency of the QPO as a function of the characteristic frequency of 1 . It can be seen that the data points follow the correlation QPO = 2 1 , rather than the WK correlation found by Wijnands & van der Klis (1999) and Bu et al. (2017) in other BHXRBs. In LHS, the correlation between characteristic frequencies of QPO and low frequency broadband noise can be fitted by a power-law function the so called WK correlation. This suggests that 1 is more like a sub-harmonic of the QPO rather than broadband noise. We have tried to fit the low-frequency part of the PDS (below the QPO frequency) with two Lorentzian functions. However, adding another Lorentzian does not improve the fits a lot and this extra component is not statistically needed. Note that, the quality factor for 1 is relatively low (< 1) compared with QPO 1 and harmonic 2 .
In order to compare the fractional rms relation between 1 and 1 , we show the rms(E) of QPO and the ratio of rms between 1 and 1 in Figure 6. The ratio keeps almost constant at 1.2-1.3 when energy is lower than 30 keV. However, when energy is higher than 30 keV, the ratio starts to decline sharply from 1.3 to 0.87.
3.2. Properties of 2 , 3 and 4
Evolution with spectral hardness
Hereafter we mainly focus on the noise components above the QPO frequency. Figure 7 shows the evolution of the fractional rms of 2 , 3 and 4 with hardness ratio. From top to bottom, the rms are calculated in the LE (1-10 keV), ME (10-30 keV) and HE (30-150) keV bands, respectively. In all panels, the hardness ratio is defined as the ratio of the count rate between the 1-3 keV and the 3-10 keV bands. In the LE and ME bands, the fractional rms of all the three components generally decrease as the spectrum softens. However, in the HE band, the evolution of the fractional rms is more complicated. In the hardness range ∼0.54-0.60, the fractional rms of 2 and 3 increase with hardness; while the fractional rms of 4 remains almost constant. In the hardness range ∼0.60-0.62, the fractional rms of 2 and 3 instead tend to decrease; whereas the fractional rms of 4 starts increasing sharply. In the hardness range > 0.62, we do not have a good monitoring coverage. Overall, it seems that the fractional rms of all the three components show an increasing trend with hardness. Figure 8 shows the corresponding evolution of the characteristic frequencies of the three components with spectral hardness. It can be seen that the characteristic frequencies of all the three components generally increase as the spectra softens. At hardness > ∼0.62, the increasing trend is seems to be flatter than that at hardness < ∼0.62.
It is worth noting that 3 is not always present in all cases. In the PDS of LE, 3 is only seen in the observation (ObsID P0114661002) where the spectrum is the hardest. In the PDS of ME, it only appears when the hardness ratio is larger than ∼0.61. While in the HE band, we can see this component in all observations. Except for the evolution trend with hardness ratio for 2 , 3 and 4 , we also plot the evolution of the fractional rms, characteristic frequency for 1 component in Figures 7 and 8. As we can see from Figure 7, the fractional rms of 1 shows no obvious evolutionary trend with hardness ratio which is totally different from other broadband noise components. In contrary, from Figure 8, the evident decreasing trend of 1 implicates the physical relation between radiation region and spectral evolution. Figure 9 shows the energy dependence of the fractional rms and characteristic frequency of 2 (left), 3 (middle) and 4 (right). As we can see from Figure 9 right panel, fractional rms decreases from 23% to 15% with increasing energy while characteristic frequency always increases with energy. In the left panel ( 2 ), the fractional rms shows the same trend but the characteristic frequency first keeps almost constant below 20-30 keV, whereas above 20-30 keV, the frequency increases rapidly with energy. This case is also true in middle panel ( 3 ), although with a large uncertainty below 20 keV. It is worth noting that the characteristic frequency of 4 above 90 keV is almost five times the frequency in 1-10 keV (from ∼ 2 Hz to ∼ 10 Hz). Similar results in LE band can be found in Kawamura et al. (2022). They used NICER data to study the relationship between asymmetric Lorentzian function P1, P2 and energy which actually reflects the evolution of characteristic frequency with energy for 2 , 3 and 4 components (P1 corresponds 2 ; P2 corresponds 3 and 4 ) . In order to fit the PDS phenomenally, Kawamura et al. (2022) used two asymmetric Lorentzian functions. Nevertheless, considering the Lorentzian function that fluctuation propagation predicts, we decide to use three standard Lorentzian functions to fit PDS.
Energy
In order to compare the relation between fractional rms of 2 , 3 and 4 , Figure 10 shows the energy dependence of fractional rms ratio for three different Lorentzian components. As we can see from Figure 10, with increasing energy, when
The ObsID we choose and Kawamura et al. (2022) choose are no more than one day apart, so we can easily match P1, P2 with 2 , 3 and 4 . P1 corresponds to 2 , P2 corresponds to 3 and 4 . energy is below 20-30 keV, the ratio between 3 and 2 changes slightly and is less than 1. When energy is higher than 20-30 keV, the rms ratio starts to increase robustly to 1.8. As for 4 and 3 , unlike 2 , there is a totally different trend. The rms ratio between 4 and 3 decreases from 1.6 to 1.1 with increasing energy. Although the downward trend is opposite to the upward trend for 3 / 2 , but the rms ratio is still larger than 1 which means 4 is more variable than 3 at high energy band. Figure 11 shows a representative power spectrum and corresponding phase lag spectrum. At higher frequency above QPO frequency, hard lag feature is prominent. In logarithmic coordinates, the shape of hard lag is similar to a normal distribution with large normalization plus another normal function with small normalization at low frequency (see Figure 11). We will only focus on the high frequency (higher than QPO frequency) part corresponding to the broadband noise components. Then we use two log-normal functions to fit the hard lag part to get the peak value frequency of two log-normal functions (hereafter called g1 and g2 ). Meanwhile, we can approximately see that g1 L3 and g2 L2 . Then, we produce the phase lag spectrum for different energy band, with reference to the 1-10 keV band (Nowak et al. 1999;Al-tamirano & Méndez 2015;Zhang et al. 2017). The phase lag spectrum are similar between different energy bands as Figure 11 shows and has a similar shape as the phase lag spectra given by Ma et al. (2021). As Figure 12 clearly shows, as the photon energy increases, the phase lag peak value for two normal functions also increases with energy band.
Phase lag spectra
)UHTXHQF\+] /( L 1 L 2 L 3 L 4 )UHTXHQF\+] 0( L 1 L 2 L 3 L 4 +DUGQHVV )UHTXHQF\+] +( L 1 L 2 L 3 L 4
DISCUSSION
In this paper, we have investigated the evolution of the broadband noise in the PDS of MAXI J1820+070 using observations from Insight-HXMT. We uncover the possible subharmonic component, 1 , and find the different energy dependence of fractional rms, characteristic frequency. We extend the study of the energy-dependent broadband noise up to 100-150 keV for the first time. It is found that the fractional rms of 2 , 3 and 4 generally increase with hardness in LE/ME/HE band, whereas characteristic frequency decreases with hardness. As for energy dependence, the rms of all three components decreases with energy. In particular, the characteristic frequencies of 2 and 3 remain unchanged below 20-30 keV then increase to 150 keV. The characteristic frequency of 4 always increases with energy from 1-150 keV.
Properties of 1
As shown in Section 3.1, we suggest that the peak of 1 observed at half the QPO frequency. This implicates that the 1 component is the sub-harmonic of QPO with low Q value.
In the jet precessing model Ma et al. (2021) proposed, the QPO signal is believed to originate from the precession of a small-scale jet to explain the LFQPO's high energy, soft lag and the maximum value, and energy-related behaviours for frequency, fractional rms and phase lag. As a result, considering the tight relation in Figure 5, we suggest that the same origin from small-scale jet for 1 . However, we cannot provide a physical model to interpret QPO and harmonics components at the same time (Rao et al. 2010;Ratti et al. 2012). Considering the poor understanding for harmonics and our study mainly focusing on the broadband noise component, we will mainly discuss the 2 , 3 and 4 .
Evolution of the fractional rms and characteristic frequency with hardness
As shown in Figure 8, the characteristic frequencies of all the high-frequency components ( 2 , 3 and 4 ) increase with spectral softening. This is consistent with the evolution trend commonly found in other BHXBs (Psaltis et al. 1999;Bult & van der Klis 2015;Zhang & Yu 2015). The trend of the characteristic frequency to increase with spectral softening can be explained naturally under the truncated disk/corona geometry (Esin et al. 1997;Done et al. 2007). In the truncated disk model, the outer part of the accretion flow forms a geometrically thin, optically thick accretion disk truncated at a larger radius. The inner part of the flow is a hot, geometrically thick and optically thin configuration. The reduction in hardness ratio generally means a higher accretion rate, thus corona and disk should also move closer to black hole (Belloni et al. 2005;Belloni 2010). In the fluctuation propagating model Lyubarskii (1997); Kotov et al. (2001) proposed, characteristic frequency is related to the outer radius of hot flow, and as the accretion rate increases, the characteristic frequency increases while outer radius decreases. Hence the viscous frequency of each component also increases with decreasing hardness.
In view of fractional rms, with increasing hardness, the fractional rms also increases in 2 , 3 and 4 . Based on truncated disk/corona model, corona shows more variability than standard accretion disk (Sobolewska & Życki 2006;Axelsson et al. 2013). Consequently, when hardness ratio increases, more corona components contribute a higher fractional rms value.
In summary, the different components show similar evolutionary trends for fractional rms, characteristic frequency with hardness ratio in LE/ME/HE energy bands. The evolutionary trend can be explained under the truncated disk/corona model (Esin et al. 1997;Done et al. 2007). Actually, there are also still debates on the truncation of accretion disk in MAXI J1820+070. Buisson et al. (2019) discovered a steady inner accretion disc measured by relativistic reflection. Kara et al. (2019) found that the reverberation time lags between the continuum-emitting corona and the irradiated accretion disk are much shorter than previously seen in truncated accretion disk, and the timescale of the reverberation lags shortens by an order of magnitude over a period of weeks, whereas the shape of the Fe K emission line remains remarkably constant. The similar results are also obtained from spectral analysis. Meanwhile, there are some other studies that support the truncated accretion disk argument from either spectral analysis or timing analysis Marino et al. 2021;Zdziarski et al. 2021a,b). De found that the frequency of thermal reverberation lags increases steadily, and, on the other hand, the temperature of the quasi-thermal component grows as the source softens, which can be explained in terms of a decrease in the disc inner radius. Moreover, De Marco et al. (2021) measured that the values of lag amplitude are a factor of 3 longer than those reported in Kara et al. (2019). The longer lags might not be easily reconciled with the conclusion of a disc extending close to the ISCO. Zdziarski et al. (2021a,b) confirm the opticallythick disk at least > 10 from joint spectral analysis. To sum up, so far, all arguments in favor of the non-truncated disk model can be reasonably explained under the truncated disk model. Except for the methods mentioned above, in the present paper, we confirm the truncated accretion disk model in MAXI J1820+070 from the evolutionary trend of broadband noise components with a new perspective by means of the correspondence relation between break frequency and radiation region radius. Quantitatively, we can calculate the radiation region at different energy band of 2 which represents the variable emission from the outermost region (see Figure 13). We take a parameter set for standard -disk:
=0.1, BH = 10M , scale height / = 0.1 to calculate the viscous frequency at certain radius (Kato et al. 2008). As Figure 13 shows, the characteristic frequency of 2 component shows energy-dependence: the emitting region spans from ∼ 34 to ∼ 27 corresponding to 1-150 keV photon energy. Like Dziełak et al. (2021) and Kawamura et al. (2022) indicated, with frequency-resolved spectral analysis, the 2 component is supposed to come from variable disc emission. However, making use of the ME and HE data from Insight-HXMT, we actually detect the emission from high energy emission > 100 keV from the 2 component which cannot be attributed to a simple standard accretion disc. Considering the change of radius plotted in Figure 13, even though we can attribute the high energy emission to the propagation of fluctuation from disc to hot flow, then we should also expect a constant characteristic frequency for 2 at high energy band according to fluctuation propagation. The characteristic frequency of 2 remains unchanged below 20-30 keV as Kawamura et al. (2022) found the constant peak frequency for P1. However, when the photon energy is greater than 20-30 keV, the radiation radius of 2 starts to decrease to 27 . This phenomenon cannot be easily interpreted by fluctuation from disk propagating to hot flow. Therefore, we speculate that 2 may be originated from a warm extended variable disk region. We therefore should consider more complicated accretion flow geometry where a standard accretion disc transits to a hot ADAF geometry.
Energy dependence of broadband noise
First, the fractional rms of 2 , 3 and 4 all generally show decreasing trend with energy. This phenomenon also exists for another black hole transient MAXI J1348-630 for broadband noise component in LHS ). This phenomenon that fractional rms decreases with energy can be interpreted as variable input soft photon flux connected with Comptonization process (Gierliński & Zdziarski 2005). Generally speaking, this can be connected with an accretion disk located outer corona. Because of Magnetorotational Instability (MRI), the magnetic field will excite the fluctuation of accretion rate in accretion disk (Hawley & Balbus 1991;Balbus & Hawley 1998;Balbus 2005;Beckwith et al. 2008;Dexter & Fragile 2011). The fluctuation of mass accretion rate will cause variable emission. Then this mechanism will cause variable input soft photons flux. The variable seed photon is up-scattered in corona to higher energy radiation. Besides, by comparing the rms ratio between 2 , 3 and 4 , we can find that with energy increasing, more and more variable high energy photon comes from 3 than 2 . In other words, the radiation region that 3 corresponds to should be hotter than 2 . As for 4 and 3 , the decreasing trend can be interpreted by different variable seed photon flux in Comptonization process. If we consider a radially-extended corona, 3 comes from outer region whereas 4 comes from inner region, the inner region contributes more radiation than outer region at 1-150 keV. In other words, the rms ratio between 4 and 3 should always be higher than 1. Meanwhile, we should note that the inner region receives more variable soft photons than outer region (inner region not only receives soft photons from disk but also from the outer region, the flux from the outer region should be more variable than the flux from standard accretion disk, this phenomenon has been found in Dziełak et al. (2021), most of the disc photons upscattered in the outer Comptonization region (Zone II) are used as seed photons for the inner Comptonization region), therefore, the decreasing slope of fractional rms with energy for 4 should be greater than 3 which means a downward trend for 4 / 3 rms ratio with energy.
Then, the energy dependence of characteristic frequency is comparatively complicated. For 2 and 3 components, the characteristic frequency keeps almost constant below 20-30 keV, then it increases with energy up to 90-150 keV. For 4 component, the characteristic frequency always increases with energy from 1-150 keV. When energy is below 20-30 keV, the constant characteristic frequency for 2 and 3 may reflect the relatively uniform radiation area in the outer region. However, when energy is above 30 keV, the increasing characteristic frequency of 2 and 3 may reflect the increasing optical depth in hot flow from outer to inner region (This will be discussed in Section 4.4).
In summary, the energy dependence of fractional rms, characteristic frequency indicates that a complicated stratified accretion flow consisting of multiple coronae is need.
Phase lag spectra
As shown in Figure 11, at frequencies above QPO, the value of phase lag is positive and frequency-dependent. The positive lag means that hard emission lags the soft one. According to fluctuation propagating model, hard photons coming from the inner region will lag behind soft photons coming from the outer region (Kotov et al. 2001). More interestingly, there are two humps in Figure 11. From Figure 11, we can approximately see that g1 L3 and g2 L2 . This correspondence is very similar to the simulation results Rapisarda et al. (2017a) gave. According to the PROPFLUC model developed by Ingram & Done (2012), if we consider a disk+corona geometry, then we will also have two humps in phase lag spectra as two visible humps in PDS (Rapisarda et al. 2014(Rapisarda et al. , 2016b(Rapisarda et al. ,a, 2017a.
In addition, as Figure 12 shows, the energy dependence of phase lag also reflects that harder photons come from the inner region and cause a greater delay (Kotov et al. 2001).
In summary, by combining power spectra and phase lag spectra, we speculate that 2 originates from the outer region whereas 3 and 4 originates from the inner region. Meanwhile, we confirm the applicability of fluctuation propagation model.
Implication for accretion structure
In this paper, we investigate the accretion flow qualitatively at LE, ME, HE band and the quantitative fitting will be done in next paper preparing. Dziełak et al. (2021) found significant spectral differences among Lorentzians for MAXI J1820+070 using the NICER 0.3-10 keV data. The model they presented comprises of an outer Comptonization region fueled by thermal photons from the cool disc, and an inner Comptonization region fueled by a fractional of the upscattered photons from the outer Comptonization region. Similarly, Kawamura et al. (2022) also model the spectra and timing variability of MAXI J1820+070 to assume geometry consisting of variable disc and two hot flow regions. As for this paper, we also have found the two humps structure in PDS as papers above mentioned but one more QPO component in LE, ME and HE band. As Kawamura et al. (2022) said, the dip in PDS is assumed to be caused by drop of viscous timescale between accretion disk and hot flow (Rapisarda et al. 2017b;Kawamura et al. 2022). The different viscous timescale between the disc and hot flow are physically natural because the scale height / of the accretion flow is expected to be different between these regions ). Thus, we see two evident humps in PDS (separately 2 and 3 , 4 ).
In addition to the common evolutionary trend for 2 , 3 and 4 , 3 shows more sophisticated evolution trend. 3 first appears in all three energy band in ObsID P0114661002, but as hardness decreases, 3 disappears in LE band first, then disappears in ME band after nine ObsIDs (∼ 18 days) too. This may implicate the change of emitting spectrum where 3 originates: when hardness ratio is less than 0.61, the region represented by 3 emits a relatively hard spectrum so that 3 only appears in HE PDS. The distinct evolution trend of 3 highlights the strong contrast with 2 and 4 . proportional to radius. In view of the toy model we discussed, combining with the relation between characteristic frequency and photon energy, we speculate that the region farther from black hole in the outer corona has a relatively uniform distribution in parameters such as density and temperature to cause the constant frequency below 20-30 keV for 3 . Then when photon energy is greater than 20-30 keV, with increasing optical depth, the emitted spectrum becomes harder at a smaller radius to cause increasing characteristic frequency. This explanation also applies to 4 only with more seed photon from the outer corona. Then we use the energy-dependence of characteristic frequency to make some simple quantitative estimation. We consider that the outer radius of hot flow out equals to the inner radius of accretion disk disk ∼ 27 (see Figure 13). It is difficult to connect the characteristic frequency with regions of the hot flow because of the poor understanding of hot flow now. Nevertheless, because the break frequency of broadband noise component is proportional to −3/2 in hot flow, then the inner radius of the outer corona is in1 ∼ 20 , the inner radius of the inner corona is in2 ∼ 7 . In calculation, we assume that the 90-150 keV radiation comes from the innermost region of each corona and 1-3 keV radiation comes from the outer region. This result is basically consistent with that in Kawamura et al. (2022) and in 10 in De Marco et al. (2021). Meanwhile, as Figure 12 shows, we found that g1 3 and g2 2 . This phenomenon can be interpreted under fluctuation propagation model. This shows that the region where 2 and 3 originate really locates at outer region in accretion flow to cause two humps in phase lag spectra. As for the 4 component, we should similarly see a hump in phase lag spectra whereas that was not the case. We consider that it is because the inner region emits mostly in the hard band and locates close to ISCO (Rapisarda et al. 2017b).
All discussions above are based on the two coronae model. In fact, we note that there are still some results that are not easy to interpret. For instance, in Figure 7, 8, we find that the disappearance of 3 in LE, ME energy band as spectra evolves. In the two coronae model, the emission region where 3 represents locates between outer accretion disk and inner corona. It is relatively difficult to explain why 3 emits so hard spectra when hardness ratio is less than ∼ 0.61. Besides, from Figure 9, when photon energy is less than 20-30 keV, the characteristic frequency of 3 and 3 shows no evident change. It seems to be contrary to our usual understanding that harder photon comes from inner region. As a result, as Rapisarda et al. (2017b); Mushtukov et al. (2018) and other paper discuss (Marino et al. 2021), we may consider more sophisticated accretion flow geometry (such as bending wave, viscous diffusion, outward fluctuation propagation effect, hot jet-emitting disk etc.) and multi-wavelength observations to explain all the results. To sum up, MAXI J1820+070 is an ideal laboratory to study inhomogeneous stratified corona.
Except the model we discuss in this paper, similar to the model that QPO comes from precession of jet (Ma et al. 2021), we can also attribute broadband noise to jet's contribution (Markoff et al. 2005;Nowak et al. 2011). It should be note that the results in Ma et al. (2021) reveal the relationship between the jet precession and the LFQPO in the high energy band, jet precession model does not depend on whether the accretion disk is truncated. Malzac (2013) has proposed the internal shock model to consider the effect of fluctuation in accretion flow on jet ejecta. Especially for Wang et al. (2020), by studying the relation between hard time lag in the high-frequency range, the high-frequency time lags are significantly correlated to the photon index derived from the fit to the quasi-simultaneous NICER spectrum. They suggested that this result is qualitatively consistent with a model in which the high-frequency time lags are produced by Comptonization in a jet. As Wang et al. (2020) shows, the evolution of the high-frequency lags is highly correlated to that of the photon index of hard spectral component by integrating the continuum broadband noise. Different from the method used in Wang et al. (2020), we investigate the broadband noise components through Lorentzian function fitting method and mainly focus on the energy-dependence of each component in PDS. Overall, the results in Wang et al. (2020), Ma et al. (2021) and our work suggest that there are two hard emission regions during the studied period of the outburst in MAXI J1820: one is a hot flow liked corona and another one is a jet. As a result, this model needs further more investigation for connection between hot flow and jet base.
CONCLUSION
In summary, radial-stratified hot flow with truncated accretion disk is needed to explain our results based on fluctuation propagating model. We should combine timing analysis and spectral fitting especially in HE energy band to improve our understanding of the inhomogeneous corona in the future.
We are grateful the anonymous referee's helpful comments and suggestions. This work has made use of the data from the Insight-HXMT mission, a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS), and data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), a service of the Astrophysics Science Division at NASA/GSFC. This work is supported by the National Key RD Program of China (2021YFA0718500) and the National Natural Science Foundation of China (NSFC) under grants U1838201, U1838202, 11733009, 11673023, U1938102, U2038104, U2031205, the CAS Pioneer Hundred Talent Program (grant No. Y8291130K2) and the Scientific and Technological innovation project of IHEP (grant No. Y7515570U1).
Softwares: XSPEC (Arnaud 1996), Astropy (Astropy Collaboration et al. 2013), Numpy (van der Walt et al. 2011), Matplotlib (Hunter 2007), Stingray (Huppenkothen et al. 2019a,b).
DATA AVAILABILITY
The raw data underlying this article are available at http://hxmten.ihep.ac.cn/.
Figure 1 .
1Insight-HXMT hardness-intensity diagram (HID). Red points mark the observations used in this work to study the evolution of the different PDS components with hardness. Magenta points mark the observations we used to study the energy-dependent properties of these components.
Figure 2 .
2Representative PDS for LE (1-10 keV), ME (10-30 keV) and HE (30-150 keV), respectively. The PDS are calculated using the data of ObsIDs P0114661003 and P0114661004. The PDS of the two observations have a similar shape and consistent fractional rms of different energy bands. The PDS of ME and HE are separately multiplied by a factor of 1.4 and 2.1 to keep the QPO aligned between energy bands.
Figure 4 .Figure 5 .
45Energy dependence of the fractional rms and characteristic frequency of 1 . Green, blue and red points represent the LE, ME and HE data, respectively. The characteristic frequency does not show significant changes with energy. Therefore, we fixed the centroid frequency and FWHM of 1 when calculating the energy dependence of the fractional rms. The data are extracted from ObsID P0114661003 and P0114661004. Characteristic frequency of the QPO as a function of the characteristic frequency of 1 . The dotted line represents the correlation QPO = 2 1 . The characteristic frequencies are measured by fitting the PDS of HE band.
Figure 6 .
6The fractional rms of QPO (top panel) and the ratio of fractional rms between QPO and 1 (bottom panel) versus energy band. The data are extracted from ObsID P0114661003 and P0114661004.
Figure 7 .
7Evolution of the fractional rms of 1 , 2 , 3 and 4 with hardness. From top to bottom, the fractional rms are calculated in the LE (1-10 keV), ME (10-30 keV) and HE (30-150) keV bands, respectively.
Figure 8 .
8Evolution of the characteristic frequencies of 1 , 2 , 3 and 4 with hardness.
Figure 9 .Figure 10 .
910Energy dependence of the fractional rms and characteristic frequency for 2 (left), 3 (middle) and 4 (right). Green, blue and red points represent the LE, ME and HE data, respectively. The ratio of fractional rms between 2 and 3 (green points), 3 and 4 (red points) versus energy band. The data points are extracted from ObsID P0114661003 and P0114661004.
Figure 11 .
11PDS for HE 50-70 keV and phase lag spectrum. The phase lag spectra was calculated for the LE 1-10 keV relative to the HE 50-70 keV. Insight-HXMT ObsIDs P0114661003 and P0114661004 are used.
Figure 12 .Figure 13 .
1213Phase lag spectrum for Insight-HXMT ObsIDs P0114661003 and P0114661004. Reference energy band is 1-10 keV. From bottom to top is 10-20 keV (red), 20-30 keV (yellow), 30-50 keV (green), 50-70 keV (cyan), 70-100 keV (blue), 100-150 keV (purple) separately.Then we investigate the hot accretion flow to explain other results especially the energy-dependence of 3 and 4 . Similar to the argumentKawamura et al. (2022) presented, we attribute 3 to the outer corona while 4 comes from the inner corona. The fluctuation propagation model predicts that the characteristic frequency of broadband noise is inversely Truncation radius of accretion disk from the black hole as a function of photon energy for 2 component. The standard −disk model was applied. A parameter set was used: =0.1, M = 10M , scale height / = 0.1.
Table 1 .
1The log of the Insight-HXMT observations used in this work. The hardness is the ratio of the count rate between the 1.0-3.0 keV and the 3.0-10.0 keV bands.ObsID
Start time (UTC)
Exposure (s) Hardness QPO
P0114661002 2018/03/16 10:01:24
32717
0.67(1)
None
P0114661003 2018/03/22 10:46:58
23245
0.62(1)
C
P0114661004 2018/03/24 07:19:14
28893
0.61(1)
C
P0114661005 2018/03/20 00:00:10
6343
0.62(1)
C
P0114661006 2018/03/27 08:29:37
3325
0.61(1)
C
P0114661008 2018/03/29 20:56:55
4649
0.61(1)
C
P0114661009 2018/03/30 16:02:31
3728
0.61(1)
C
P0114661010 2018/03/31 20:41:05
5356
0.61(1)
C
P0114661011 2018/04/01 20:33:24
26322
0.61(1)
C
P0114661012 2018/04/03 23:29:38
8122
0.61(1)
C
Table 1 continued
Table 1 (continued)
1ObsID
Start time (UTC)
Exposure (s) Hardness QPO
P0114661013 2018/04/05 20:04:33
8490
0.61(1)
C
P0114661014 2018/04/06 15:11:06
12809
0.60(1)
C
P0114661015 2018/04/08 13:22:00
8151
0.60(1)
C
P0114661016 2018/04/09 10:04:01
5744
0.60(1)
C
P0114661017 2018/04/10 14:43:27
8337
0.60(1)
C
P0114661018 2018/04/11 20:58:02
7313
0.60(1)
C
P0114661019 2018/04/12 16:03:53
9654
0.60(1)
C
P0114661020 2018/04/14 11:01:36
7600
0.60(1)
C
P0114661021 2018/04/15 01:20:44
2992
0.59(1)
C
P0114661024 2018/04/18 15:14:50
7265
0.59(1)
C
P0114661025 2018/04/19 15:06:26
7360
0.59(1)
C
P0114661026 2018/04/20 14:58:02
6492
0.59(1)
C
P0114661027 2018/04/23 19:19:13
2812
0.58(1)
C
P0114661028 2018/04/25 14:16:17
4639
0.58(1)
C
P0114661029 2018/04/27 09:13:33
4215
0.58(1)
C
P0114661031 2018/04/22 19:27:34
4025
0.58(1)
C
P0114661032 2018/04/28 13:51:51
4029
0.57(1)
C
P0114661035 2018/05/02 10:09:17
3617
0.56(1)
C
P0114661038 2018/05/05 09:46:03
2817
0.56(1)
C
P0114661040 2018/05/07 11:05:59
2719
0.55(1)
C
P0114661041 2018/05/08 10:58:08
5513
0.54(1)
C
P0114661042 2018/05/09 14:01:08
2632
0.54(1)
C
P0114661043 2018/05/10 13:53:07
2633
0.54(1)
C
P0114661044 2018/05/12 04:04:08
3950
0.54(1)
C
P0114661045 2018/05/13 00:45:01
2729
0.54(1)
C
P0114661048 2018/05/16 13:03:38
3388
0.54(1)
C
P0114661052 2018/05/20 15:41:03
15578
0.53(1)
C
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| []
|
[
"D 2 -Tree: A New Overlay with Deterministic Bounds",
"D 2 -Tree: A New Overlay with Deterministic Bounds"
]
| [
"Gerth Stølting Brodal \nDepartment of Computer Science\nMADALGO\nUniversity of Aarhus\nDenmark\n",
"Spyros Sioutas \nDepartment of Informatics\nIonian University\nCorfu Greece\n",
"Kostas Tsichlas [email protected] \nDepartment of Informatics\nAristotle University of Thessaloniki\nGreece\n",
"Christos Zaroliagis \nDept. of Computer Eng. & Informatics\nInstitute of Theoretical Informatics\nUniversity of Patras\nGreece\n\nKarlsruhe Institute of Technology\nGermany\n"
]
| [
"Department of Computer Science\nMADALGO\nUniversity of Aarhus\nDenmark",
"Department of Informatics\nIonian University\nCorfu Greece",
"Department of Informatics\nAristotle University of Thessaloniki\nGreece",
"Dept. of Computer Eng. & Informatics\nInstitute of Theoretical Informatics\nUniversity of Patras\nGreece",
"Karlsruhe Institute of Technology\nGermany"
]
| []
| We present a new overlay, called the Deterministic Decentralized tree (D 2 -tree). The D 2 -tree compares favourably to other overlays for the following reasons: (a) it provides matching and better complexities, which are deterministic for the supported operations; (b) the management of nodes (peers) and elements are completely decoupled from each other; and (c) an efficient deterministic load-balancing mechanism is presented for the uniform distribution of elements into nodes, while at the same time probabilistic optimal bounds are provided for the congestion of operations at the nodes. The load-balancing scheme of elements into nodes is deterministic and general enough to be applied to other hierarchical tree-based overlays. This load-balancing mechanism is based on an innovative lazy weight-balancing mechanism, which is interesting in its own right. | 10.1007/s00453-014-9878-4 | [
"https://arxiv.org/pdf/1009.3134v2.pdf"
]
| 7,047,710 | 1009.3134 | 0b51559d299483af305f370c946e2f52741a4e0e |
D 2 -Tree: A New Overlay with Deterministic Bounds
8 Mar 2012 March 9, 2012
Gerth Stølting Brodal
Department of Computer Science
MADALGO
University of Aarhus
Denmark
Spyros Sioutas
Department of Informatics
Ionian University
Corfu Greece
Kostas Tsichlas [email protected]
Department of Informatics
Aristotle University of Thessaloniki
Greece
Christos Zaroliagis
Dept. of Computer Eng. & Informatics
Institute of Theoretical Informatics
University of Patras
Greece
Karlsruhe Institute of Technology
Germany
D 2 -Tree: A New Overlay with Deterministic Bounds
8 Mar 2012 March 9, 2012Overlayindexing schemedecentralized systemdistributed data structureload-balancing
We present a new overlay, called the Deterministic Decentralized tree (D 2 -tree). The D 2 -tree compares favourably to other overlays for the following reasons: (a) it provides matching and better complexities, which are deterministic for the supported operations; (b) the management of nodes (peers) and elements are completely decoupled from each other; and (c) an efficient deterministic load-balancing mechanism is presented for the uniform distribution of elements into nodes, while at the same time probabilistic optimal bounds are provided for the congestion of operations at the nodes. The load-balancing scheme of elements into nodes is deterministic and general enough to be applied to other hierarchical tree-based overlays. This load-balancing mechanism is based on an innovative lazy weight-balancing mechanism, which is interesting in its own right.
Introduction
Decentralized systems and in particular Peer-to-Peer (P2P) networks have become very popular of late and are widely used for sharing resources and store very large data sets. Data are stored at the nodes (or peers) and the most crucial operations are data search (identify the node that stores the requested information) and updates (insertions/deletions of data). Searching and updating is typically done by building a logical overlay network that facilitates the assignment and indexing of data at the nodes. Sometimes, we distinguish between the overlay structure per se and the indexing scheme used to access the data.
Following the typical modeling, a decentralized communication network is represented by a graph. Its nodes correspond to the network nodes, while its edges correspond to communication links. We assume constant size messages between nodes through links and asynchronous communication. It is assumed that the network provides an upper bound on the time needed for a node to send a message and receive an acknowledgment. The complexity of an operation is measured in terms of the number of messages issued during its execution. Throughout the paper, when we refer to cost we shall mean number of messages (internal computations at nodes are considered insignificant). The overlay is another graph defined over the communication network. The nodes of the overlay correspond to nodes of the original network, while its edges (links) may not correspond to existing communication links, but to communication paths.
With respect to its structure, the overlay supports the operations Join (of a new node v; v communicates with an existing node u in order to be inserted into the overlay), and Departure (of an existing node u; u leaves the overlay announcing its intent to other nodes of the overlay).
The overlay is used to implement an indexing scheme for the stored data. Such a scheme supports the operations search for an element, insert a new element, delete an existing element, and range query for elements in a specific range.
In terms of efficiency, an overlay network should address the following issues:
-Fast queries and updates: updates and queries must be executed in a minimal number of communication rounds and using a minimal number of messages. -Ordered data: keeping the data in order facilitates the implementation of various enumeration queries when compared to a simple dictionary that can only answer membership queries, including those arising in DNA databases, location-based services, and prefix searches for file names or data titles. Indeed, the ever-wider use of P2P infrastructures has found applications that require support for range queries (e.g., [6]). -Size of nodes (peers): the size of a node is the routing information (links and related data) maintained by this node and it is not related to the number of data elements stored in it. Keeping the size of a node small allows for more efficient update operations, but in general reduces the efficiency of access operations while aggravating fault tolerance. -Fault Tolerance: the structure should be able to discover and heal failures at nodes or links. -Congestion: it refers to the distribution of the load of search (access) operations per node, aiming at distributing this load equally across all nodes. The congestion is an expected quantity defined as the maximum, among all nodes, of the fraction of the expected number of search operations at a node, due to a random sequence of search operations on the structure, divided by the total number of search operations. -Load Balancing: it refers to the distribution of data elements on the nodes.
The goal of load balancing is to distribute equally the n elements stored in the N nodes of the network (typically N ≪ n). That is, if there are N nodes and n data elements, ideally each node should carry approximately k elements, where ⌊n/N ⌋ ≤ k ≤ ⌊n/N ⌋ + 1.
There has been considerable recent work in devising effective distributed search and update techniques. Existing structured P2P systems can be clas-sified into two broad categories: distributed hash table (DHT)-based systems and tree-based systems. Examples of the former, which constitute the majority, include Chord [13], CAN [18], Pastry [17], Symphony [14], and Tapestry [23]. DHT-based systems support exact match queries well and use (successfully) probabilistic methods to distribute the workload among nodes equally. DHT-based systems work with little synchrony and high churn (the collective effect created by independent burstly arrivals and departures of nodes), a fundamental characteristic of the Internet. Since hashing destroys the ordering on keys, DHT-based systems typically do not possess the functionality to support straightforwardly range queries, or more complex queries based on data ordering (e.g., nearest-neighbor and string prefix queries). Some efforts towards addressing range queries have been made in [9,19], getting however approximate answers and also making exact searching highly inefficient. The most recent effort towards range queries is reported in [22].
Tree-based systems are based on hierarchical structures. They support range queries more naturally and efficiently as well as a wider range of operations, since they maintain the ordering of data. On the other hand, they lack the simplicity of DHT-based systems, and they do not always guarantee data locality and load balancing in the whole system. Important examples of such systems include Family Trees [21], BATON [11], and Skip List-based schemes [16] like Skip Graphs (SG) [4,7], NoN SG [15], SkipNet (SN), Deterministic SN [10], Bucket SG [3], Skip Webs [1], Rainbow Skip Graphs (RSG) [8] and Strong RSG [8] that use randomized techniques to create and maintain the hierarchical structure.
In this work, we focus on tree-based overlay networks that support directly range and more complex queries. Let N be the number of nodes present in the network and let n denote the size of data (N ≪ n). Let M be the size of available memory at each node, Q(n, N ) be the cost of a single query, U (n, N ) be the cost of an update, C(n, N ) be the congestion per node (measuring the load) incurred by search operations, and let L(n, N ) be the cost for load balancing the overlay with respect to (w.r.t.) element updates. Regarding congestion, each node issues one operation, while the destination node of the operation is assumed to be selected uniformly at random among all nodes of the network. Congestion depends on the distribution of elements into nodes as well as on the topology of the overlay. It provides hints as to how well the structure avoids the existence of hotspots (i.e., nodes which are accessed multiple times during a sequence of operations -the root of a tree is usually a hotspot in decentralized tree structures).
A comparison of the aforementioned tree-based overlays is given in Table 1. We would like to emphasize that w.r.t. load balancing, there are solutions in the literature either as part of the overlay (e.g., [11]) or as a separate technique (e.g., [3,7]). These solutions are either heuristics, or provide expected bounds under certain assumptions, or amortized bounds but at the expense of increasing the memory size per node. In particular, in BATON [11], a decentralized overlay is provided with load balancing based on data migration. However, their O(log n) amortized bound is valid only subject to a probabilistic assumption about the number of nodes taking part in the data migration process, and thus it is in fact an amortized expected bound. In the case of Bucket Skip Graphs [3], elements are structured in buckets attached to nodes. Although it is a solution which can be applied to a large set of P2P structures, it has two drawbacks: (i) a list of free nodes is required, and (ii) a global control for the size of the buckets is imperative. The latter is very crucial and is tackled by heuristics with no analysis whatsoever. The solution proposed in this paper can be used to tackle the problem of bucket size control efficiently in an amortized sense. A deterministic solution for load-balancing comes from [7], in which a O(log N ) amortized bound w.r.t. the elements transferred is provided. Their solution, stemming from a centralized parallel database framework, is a node migration process in which a lightweight node is selected, its load is moved to an adjacent node and then it shares the load of the heavyweight node. This process was initially developed for a parallel database in which there is central control. The original process was translated to a decentralized framework by applying a second overlay on the nodes where the order is defined w.r.t. the load of the nodes. In particular, they maintain two skip graphs on the nodes, one w.r.t. the order of elements and one w.r.t. the load of the nodes (in fact the second one can be replaced by a decentralized min-heap [20]). Apart from this deficit, one more problem with this method is that it assumes that node migration is possible and each time an update takes place the structure of the overlay is changed. This incurs an additive cost equal to the cost update of the structure. Additionally, in structures that strive for deterministic bounds (like BATON) this is not possible since such structures are quite strict and do not allow the placement of a node anywhere in the structure.
Methods N M Q(n, N ) U(n, N ) C(n, N ) L(n, N ) SG [4,7] ≤ n O(log N ) O(log N ) w.h.p. O(log N ) w.h.p. O( log N N ) O(log N ) NoN SG [15] n O(log 2 n) O( log n log log n ) O(log 2 n) O( log 2 n n ) - Determ. SN [10] n O(log n) O(log n) O(log 2 n) O( n 0,32 n ) - BATON [11] ≤ n O(log N ) O(log N ) O(log N ) - O(log n) Family Trees [21] n O(1) O(log n) O(log n) O( log n n ) - Bucket SG [3] ≤ n O( n N + log N ) O(log N ) O(log N ) O( 1 N + log N n ) No Bounds Skip Webs [1] n O(log n) O( log n log log n ) O( log n log log n ) O( log n n ) - Rainbow SG [8] n O(1) O(log n) w.h.p. O(log n) w.h.p. O( log n n ) - Strong RSG [8] n O(1) O(log n) O(log n) O( n ǫ n ) - D 2 -tree ≤ n O(1) O(log N ) O(log N ) O( log N N ) O(log N )
The basic characteristic of a decentralized overlay is that the balancing information is local. Locality is a must in a decentralized structure since there are no means to acquire global information. For example, internal memory heightbalanced trees have local balancing information and thus lend themselves nicely to P2P environments but they have problems with congestion w.r.t. updates. In particular, in a sequence of n operations the root can be accessed O( √ n)
times. However, weight balanced trees avoid this bottleneck having very good congestion w.r.t. updates but they need a lazy mechanism as the one described in this paper to update the weight information.
Our Contribution. In this paper we present a new tree-based overlay, called the Deterministic Decentralized tree or D 2 -tree. The D 2 -tree (see also Table 1):
-uses O(1) space per node; -achieves a deterministic O(log N ) query bound; -achieves a deterministic (amortized) O(log N ) update bound for elements as well as for node joins and departures; -achieves optimal congestion; -exhibits a deterministic (amortized) O(log N ) bound for load-balancing; -supports ordered data queries optimally, and tolerates node failures.
The D 2 -tree is an overlay consisting of two levels. The upper level is a perfect binary tree. The leaves of this tree are representatives of the buckets that constitute the lower level of the D 2 -tree. Each bucket is a set of nodes and these nodes are structured as a doubly linked list. Each bucket contains O(log N ) nodes. Since N changes, the size of buckets is dynamically maintained by the overlay.
In the D 2 -tree, we separate the index from the overlay structure using the load-balancing mechanism. The number of elements per node is dynamic w.r.t. node joins and departures and it is controlled by the load-balancing mechanism. Moreover, the number of nodes of the perfect binary tree is not connected by any means to the number of elements stored in the structure. The overlay structure supports the operations of node join and node departure, while at the same time it tackles failures of nodes whenever these are discovered.
Our load-balancing technique distributes almost equally the elements among nodes by making use of weights. Weights are used to define a metric of loadbalance, which shows how uneven is the load between nodes. When the load is uneven, then a data migration process is initiated to equally distribute elements.
Our load-balancing technique is quite general and can be applied to any hierarchical decentralized overlay (e.g., BATON, Skip Graphs) with the following specifications:
-The overlay structure must be a tree with height O(log N ) and with each node having O(1) children. -Nodes at level i having the same father have approximately (within constant factors) the same weight, which is Ω(i 4 ). -Updates are performed at the leaves. Alternatively, if each node has access to a leaf in O(1) messages then this is enough, since the update is simply forwarded to this leaf.
The rest of the paper is organized as follows. Section 2 presents some definitions and notation used throughout the paper. We discuss the load balancing technique in Section 3, and present the D 2 -tree in Section 4. We conclude in Section 5. A preliminary version of this work appeared as [5].
Definitions and Notation
In this section, we give some definitions regarding tree structures that will be used throughout the paper.
Let T be a tree. Based on T ancestor-descendant relationships are defined in a natural way. There is a node that has no ancestor (the root ) and there are nodes with no descendants (the leaves). All nodes which are not leaves are called internal. The subgraph induced by the descendants of node v (including v) in T is the subtree of v. The height of node v is the length (in number of edges) of the longest path from v to one of its leaves. The depth or level of node v is the length of the path from v to the root. Two nodes are called brothers when they have the same father and they are consecutive in his child list.
The weight w(v) of a node v is equal to the number of elements stored in its subtree. The number of elements residing in a node v is denoted by e(v).
We define the size of v, denoted by |v|, as the number of nodes of the subtree
of v (including v) in T . The density d(v) of node v is defined as d(v) = w(v)
|v| and represents the mean number of elements per node in the subtree of v.
Let v be a node at height h, let p be a child of v and let q be the right brother of p; both p and q are at height h − 1.
The criticality c(p, q) of the two brother nodes p and q is defined as c(p, q) = d(p) d(q) and represents their difference in densities. Let T ′ be a perfect binary tree. The node criticality nc v of a node v ∈ T ′ at level ℓ with left and right children w and z at level ℓ + 1, respectively, is defined as nc v = |w| |v| . The node criticality represents the difference in size between a node (v) and its left child (w).
Deterministic Load Balancing
The main idea of our load-balancing mechanism is as follows. It distributes almost equally the elements among nodes by making use of weights, which are used to define a metric showing how uneven is the load between nodes. When the load is uneven, then a data migration process is initiated to equally distribute the elements.
We describe the load-balancing mechanism in two steps. First, we provide a mechanism that allows for efficient and local update of weight information in a tree when elements are added or removed at the leaves. This is necessary to avoid hotspots. Then, we describe the load-balancing scheme in a tree overlay. In the following, we assume that the overlay structure is a tree T .
A Technique for Amortized Constant Weight Updating
We provide a technique that lazily updates the weights on the nodes of a tree. When an element is added/removed to/from a leaf u in T , the weights on the path from u to the root must be updated. If the height of T is H, then the cost of the weight updating is O(H). Assume that node v lies at height h and its children are v 1 , v 2 , . . . , v s at height h − 1. We relax the weight of a node and its recomputation. We define the virtual weight b(v) of v as the weight stored in node v. In particular, for node v the following invariants are maintained
Invariant 1 b(v) > e(v) + (1 − ǫ h ) ( s i=1 b(v i )) Invariant 2 b(v) < e(v) + (1 + ǫ ′ h ) ( s i=1 b(v i ))
where ǫ h and ǫ ′ h are appropriate constants. These invariants imply that the weight information is approximate, at most by a multiplicative constant.
Assume that an update takes place at leaf u. Apparently, only the weight of its ancestors need to be updated by ±1 and no other node is affected. We traverse the path from u to the root until we find a node z for which Invariants 1 and 2 hold. Let v be its child for which either Invariant 1 or 2 does not hold on this path. We recompute all weights on the path from u to v. In particular, for each node z on this path, we update its weight information by taking the sum of the weights written in its children plus the number of elements that z carries.
The constants ǫ h and ǫ ′ h are chosen such that for all nodes the virtual weight will be within a constant factor c > 0 of the real weight, i.e.,
1 c · w(v) < b(v) < c · w(v) First we prove the lower bound on v. At height h: b(v) > (1 − ǫ h ) s j=1 b(v j ) + e(v)
By recursing and lower bounding to get clean bounds we get
b(v) > w(v) h j=2 (1 − ǫ j ) Choosing 5 ǫ j = 1 j 2 , we get h j=2 1 − 1 j 2 = h j=2 (j − 1) × h j=2 (j + 1) h j=2 j 2 = (h − 1)! h j=3 j (h!) 2 = h + 1 2h > 1 2
Similarly, for the upper bound we get
b(v) < w(v) h j=1 (1 + ǫ ′ j )
. Choosing ǫ j = 1 j 2 and taking into account that 1 + 1
j 2 < 1 1− 1 j 2 , we have h j=2 1 + 1 j 2 < h j=2 1 1 − 1 j 2 = 1 h j=2 1 − 1 j 2 < 1 1 2 = 2 As a result, by choosing ǫ h = ǫ ′ h = 1 h 2 we get that: 1 2 · w(v) < b(v) < 2 · w(v)(1)
The following lemma states how frequently the weight information in each node changes.
Lemma 1. The minimum number of updates in the subtree of
v, causing a weight update at v, is Θ(ǫ h w(v)).
Proof. The weight update of node v is a result of the violation of either of Invariants 1 or 2. After the update, it holds that
b(v) = s i=1 b(v i ) + e(v). Node v has its weight updated again when b(v) < (1 − ǫ h ) ( s i=1 b(v i ) + e(v)) (or b(v) > (1 + ǫ h ) ( s i=1 b(v i ) + e(v)
) symmetrically). This will happen only when the weight of the subtree of v changes by ǫ h (
s i=1 b(v i ) + e(v)
). This change is a lower bound on the number of operations performed in this subtree, no matter when they have been performed. Taking into account (1), we get the lemma. ⊓ ⊔
The following theorem states that the weight updating mechanism is efficient in an amortized sense.
Theorem 1. The amortized cost of the weight update algorithm is O(1).
Proof. Lemma 1 states that if we make ǫ h w(v) update operations then the maximum number of weight changes at node v is 1. As a result, the amortized cost per update operation at height h is 1 ǫ h b(v) . In the following, given that v (i) is the node on the path at height i and by the assumption that b(v (i) ) = Ω(i 4 ) we get that the amortized cost is:
H i=0 1 ǫ i b(v (i) ) = H i=0 i 2 b(v (i) ) = H i=0 O i 2 Ω(i 4 ) = O(1) ⊓ ⊔
Updates and Load Balancing
We now investigate how load balancing is realized on the balanced tree structure T . For clarity of exposition, we assume that T is a binary tree. The following discussion can be easily generalized for trees with O(1) maximum degree, simply by looking between brother nodes. First, bear in mind that this mechanism does not tamper with the structure of T . An update operation (either insertion or deletion of an element) is initiated at node v. Node v issues a search for the involved element and the appropriate node u is returned. Then, the update request is forwarded from v to u. Node u executes the update operation and signals v for the status of the update. The load balancing mechanism redistributes the elements among nodes when the load between nodes is not distributed equally enough.
Assume that node v at height h has child p and its right brother q at height h − 1. Recall that |v| denotes the size of v (number of nodes in the subtree
of v, including v) in the overlay structure, d(v) = w(v)
|v| denotes the density of v (representing the mean number of elements per node in the subtree of v), and that c(p, q) = d(p) d(q) denotes the criticality of the two brother nodes p and q (representing their difference in densities). The following invariant guarantees that there will not be large differences between densities.
Invariant 3 For two brothers p and q, it holds that
1 c ≤ c(p, q) ≤ c, 1 < c ≤ 2.
For example, choosing c = 2 we get that the density of any node can be at most twice or half of that of its brother. In the more general case where the number of children of node v is O(1), we get that no child of v has more density than a constant factor w.r.t. the other children of v.
When an update takes place at leaf u, weights are updated by using the mechanism described in Section 3.1. In this way, we guarantee that no hotspot exists w.r.t. weight updating as implied by Lemma 1. Then, starting from u, the highest ancestor w is located that is unbalanced w.r.t. his brother z, meaning that Invariant 3 is violated. Finally, the elements in the subtree of their father v are redistributed uniformly so that the density of the brothers becomes equal; this procedure is henceforth called redistribution of node v. Assume that the redistribution phase has a cost of O(f (w(v))), for some increasing function f : N → N. The following theorem provides amortized bounds for the redistribution. Proof. If a node v with weight w(v) has the elements in its subtree redistributed, then this node will go through this process again after O(w(v)) updates of elements in its subtree. In particular, when v is redistributed the criticality c(p, q) of its children p, q is 1. To move the criticality out of bounds again at least w(p) 2 or w(q) 2 elements must be inserted or deleted from p or q respectively. By the assumption that the number of nodes in the subtree of p is approximately equal (within constant factors) to that of q, we deduce that O(w(v)) elements must be inserted or deleted from v. Since the cost of the redistribution of v is O(f (w(v))),
the amortized cost for node v is O f (w(v)) w(v)
. This is true for all nodes on the path from a leaf to the root, and thus the amortized cost is O H f (w(root)) w(root)
. ⊓ ⊔
The D 2 -tree
In this section we design and analyze the D 2 -tree overlay. We first describe the overlay structure, then move to the description of the index, and finally discuss efficiency issues regarding congestion and fault-tolerance.
The D 2 -tree Structure
The D 2 -tree is a binary tree, where each node maintains an additional set of links to other nodes apart from the standard links which form the tree. Each node v in the tree maintains the following links:
1. Links to its father (if there is one) and its children. 2. Links to its adjacent nodes based on an inorder traversal of the tree. 3. Links to nodes at the same level as v. These links facilitate an exponential search on the nodes of the same level. Assume that node v lies at level ℓ.
In a binary tree, the maximum number of nodes at level ℓ is equal to 2 ℓ .
Node v maintains at most 2ℓ links: ℓ links to nodes to the right and ℓ links to nodes to the left. The links are distributed in exponential steps, that is Each node of the bucket points to the node which is a leaf of the PBT and is called the representative of the bucket. Additionally, it maintains its routing table w.r.t. the nodes of all buckets. When a node z makes a join request to v, then this node is forwarded to its adjacent leaf u w.r.t. the inorder traversal. Then, node z is added to the doubly linked list representing the bucket of u by manipulating a constant number of links. The routing table of z is updated by using Lemma 2(ii). When a node v leaves the network, then it is replaced by its right adjacent node u (if there is no right adjacent node then we choose the left one) which in turn is replaced by its first node z in its bucket (Figure 1). Link and data information are copied from v to u and from u to z. When a node v is discovered to be unreachable, its adjacent node u is first located. This is accomplished by traversing the path to the rightmost or leftmost leaf starting from the left or right child respectively. Node u fills the gap of v and the first child z in the bucket of u fills the gap left by u. The contents of u are not moved to another node except from the navigation data (routing tables and other links) which are moved to node z that takes its place. Node u has its routing tables recomputed.
The join and departure of nodes may cause the size of the buckets to be uneven, which in the long run renders the structure unbalanced (imagine a bucket holding almost all nodes). To control the size of the buckets we employ a weightbased approach 6 . Each node v of the PBT maintains its size |v|, which is equal to the number of nodes in the buckets of its subtree. The size control is accomplished by using the method introduced in Section 3.1, in order to avoid the existence of hotspots.
Recall that the node criticality nc v of a node v at level ℓ with left and right children w and z at level ℓ+1, respectively, is defined as nc v = |w| |v| . The following invariant bounds the criticality of nodes.
Invariant 4
The node criticality of all nodes is in the range 1 4 , 3 4 .
Invariant 4 implies that the number of nodes in buckets in the left subtree of a
node v is at least half and at most twice the corresponding number of its right subtree (this definition can be easily generalized when v has a O(1) number of children). When an update takes place at bucket x, then we locate the highest ancestor v of x whose node criticality is out of bounds, w.r.t. Invariant 4, and we redistribute the nodes in its subtree.
The redistribution is carried out as follows. A traversal of all buckets of the subtree of v at level ℓ is performed in order to determine the exact value of |v|.
Then, the number of nodes per bucket should be |v| 2 ℓ + 1. The redistribution of nodes in the subtree of v starts from the rightmost bucket and it is performed in an inorder fashion so that elements in the nodes are not affected. The transfer of nodes is accomplished by maintaining a link (called dest henceforth) for the position in which nodes should be put or taken from. In addition, this pointer plays the role of a token indicating which node implements the redistribution process. The transfer process involving bucket b is implemented by its representative that maintains the pointer dest.
Assume that bucket b has q extra nodes which must be transferred to other buckets. Pointer dest points to a bucket b ′ in which these extra nodes should be put. All these nodes are put in b ′ as well as in adjacent nodes if necessary. Note that during this procedure internal nodes of PBT are also updated since dest implements an inorder traversal following the respective pointers. When bucket b has the correct size, the link dest is transferred to the representative of the next bucket and the same procedure applies again. In each visited bucket there are nodes which have been transferred and are in their correct position and there are nodes which are to be transferred. The distinction between these nodes is quite easy by the total number of nodes in the bucket as well as by the keys they contain. The case where q nodes must be transferred to bucket b from bucket b ′ is completely symmetric. The cost for the redistribution for node v is f (|v|) = O(|v|).
The redistribution guarantees that if there are z nodes in total in the y buckets of the subtree of v, then after the redistribution each bucket maintains either ⌊z/y⌋ or ⌊z/y⌋+1 nodes. However, the following discussion still holds (with minor changes) even if the redistribution phase guarantees that the minimum and maximum size of the buckets is within constant factors. The cost for the redistribution we propose for node v is f (|v|) = O(|v|).
We guarantee that each bucket contains O(log N ) nodes, throughout joins or departures of nodes, by employing two operations on the PBT, the contraction and the extension. When a redistribution takes place at the root of the PBT, we also check whether any of these two operations can be applied to the PBT. The extension operation adds one more level of nodes at the PBT from existing nodes in the buckets, thus increasing its height by one. The contraction operation removes one level of nodes from the PBT and puts them into the buckets, thus decreasing its height by one. In order to decide whether the PBT needs extension or contraction we compare the size of the buckets B after the redistribution with the height of the PBT. Note that after redistribution, the sizes of all buckets may differ by at most 1. If the size is larger than the height of the PBT by at least 1 then an extension takes place. If the size of the bucket is smaller than the height of the PBT by at least 1 then a contraction takes place (see Figure 2). The height of the PBT can be deduced by the size of the routing table in the nodes of the last level of the PBT. These two operations involve a reconstruction of the overlay which rarely happens as shown in the following lemma.
Lemma 3. If a redisribution operation is performed at a node with size s, then this node will be redistributed again after Ω(s) joins or departures have been performed in its subtree.
Proof. Assume that node v with size s is redistributed. Then, nc v = 0.5, meaning that the number of nodes in the buckets for both subtrees are equal. The bound of 0.5 on criticality after redistribution is not strict in the sense that any bound in the interval [ 1 4 + ζ, 3 4 − ζ], where ζ > 0, suffices. The same holds for their subtrees recursively. Node v will be redistributed again only when the criticality of one of its children gets out of bounds. Since it was 0.5 at least s/4 joins or departures of nodes must be performed in order to redistribute v. This is a worst-case sequence of operations that trigger a redistribution at v. Assuming a uniform distribution of updates, a much larger bound can be obtained. ⊓ ⊔ Lemma 3 states that the expensive operations of extension and contraction take place when the number of nodes has at least doubled or halved. Assuming that the redistribution of v has O(f (|v|)) cost, it follows by Lemma 3 that the amortized cost for join/departure of a node v at height h is O f (|v|) |v| . Since the PBT has height H, we establish the following. O(1) Space per Node. The routing tables require O(log N ) space for each node. To make the space consumption constant, one could apply on the overlay the schemes described in [8,21]. However, on the one hand the complexities will not be deterministic while on the other hand even in the case of Strong Rainbow Skip Graphs [8] with deterministic bounds our congestion for searching is much better than theirs. To achieve constant space we distribute the routing tables to many nodes doing the same also for nodes in the buckets. A set of nodes with constant degree is grouped together and a routing table is distributed on all these nodes, such that each node uses constant space. Thus, a node can recreate approximately its routing table by accessing nodes inside the same group. We call each such group a hypernode.
A hypernode at level ℓ consists of at most ℓ nodes, numbered from left to right 1, 2, . . .. This number is the rank of the node within the hypernode. A node v with rank i maintains two links to the nodes that are approximately 2 i positions to the right and to the left. In particular, node v either points to a node z in the same hypernode whose distance is 2 i or to a node z ′ whose rank is i and lies in a different hypernode than that of v which contains a node whose distance is 2 i from v. The concatenation of all such links constitutes the routing table for the hypernode. Additionally, each node with rank i maintains two links to nodes with ranks i − 1 and i + 1, if there are such nodes. Finally, each node with rank i in the hypernode maintains a link to the node with the largest rank. The following lemma translates Lemma 2(ii) in the setting of hypernodes. Proof. Direct implication of the distances between successive links in the routing tables as well as of the increasing ranks in the hypernodes.
⊓ ⊔ Using Lemma 5 we can update the links of a node v by simply looking at the links of its siblings u and w and update the links of v by pointing to the adjacent nodes of the nodes pointed to by u and w. Hypernodes are static in the overlay and only in the case of contraction we destroy the hypernodes of the last level while in the case of extension we create new hypernodes for the new level. A faulty node inside a hypernode will not disconnect it since by accessing the parents we can find its siblings and reconstruct the missing routing information.
The Index Structure of the D 2 -tree
The overlay provides the infrastructure for the index to efficiently support various operations. The overlay is used as a node-oriented tree. The range of all values stored in the overlay is partitioned into subranges each one of which is assigned to a node of the overlay. An internal node v with range [x v , x ′ v ] may have a left child u and a right child w with ranges [
x u , x ′ u ] and [x w , x ′ w ] respectively such that x u < x ′ u < x v < x ′ v < x w < x ′ w . Thus, if an element x ∈ [x v , x ′ v ]
then it must be stored at node v. Ranges are dynamic in the sense that they depend on the values maintained by the node.
In the following, we discuss the search and update operations supported by the index. Our arguments refer to the case where nodes use O(log N ) space but they can be trivially changed to hold in the case they use O(1) space. In the few cases where these arguments do not transfer trivially we make further explanations.
Search and Range Queries. The search for an element α in the overlay may be initiated from any node v at level ℓ. Let z be the node with range of values containing α. Assume O(log N ) space per node and assume that w.l.o.g. x ′ v < α. Then, by using the routing tables we search at level ℓ for a node u with right sibling w (if there is such sibling) such that x ′ u < α and x w > α unless α is in the range of u and the search terminates. This step has O(ℓ) cost, since we simulate a binary search. If the search continues, then node z will either be an ancestor of u or in the subtree rooted at u. If u is a leaf, then we move upwards (or in its corresponding bucket) until we find node z in O(log N ) steps. If u is an internal node, by following the respective link we move to the left adjacent node y of u which is certainly a leaf (inorder traversal). If x ′ y > α then an ordinary top down search from node u will suffice to find z in O(log N ) steps (or in its bucket). Otherwise, node z is certainly an ancestor of u and thus we can move upwards from u until we find it in O(log N ) steps. The following lemma establishes the complexity of the search operation. Proof. The case of O(log N ) space per node was analyzed in the paragraph preceding the statement of the Lemma.
In the case of O(1) space per node, assume that node v belongs in the hypernode V at level ℓ. The only change concerns the discovery of node u at level ℓ. By following the respective link, node p ∈ V with highest rank is reached. Then, by following the backward links we make the search on the level ℓ. In particular, assume that during our search in hypernode V we find that node u is somewhere between the nodes pointed to by nodes with rank i and i + 1 in V . Assume that the node pointed by node with rank i is in hypernode V ′ . This means that we have narrowed down the search in a subproblem consisting of 2 i+1 − 2 i = 2 i nodes after having made ℓ − i steps due to the backward exponential search. The procedure is applied again from node with rank i in V ′ until we find node u. This node in V ′ can be located in O(1) steps, since the node with rank i of hypernode V maintains a link to node with rank i in hypernode V ′ . Thus, the number of steps is at most ℓ. The vertical search on a path from a node towards the root or a leaf is exactly the same as before.
⊓ ⊔
A range query [a, b] reports all elements x such that x ∈ [a, b]. A range query [a, b] initiated at node v, invokes a search operation for element a. Node u that contains a returns to v all elements in this range. If all elements of u are reported then the range query is forwarded to the right adjacent node (inorder traversal) and continues until an element larger than b is reached for the first time.
Updates and Load Balancing. Assume that an update operation is initiated at node v involving element α. By invoking a search operation we locate node u with range containing element α. Finally, the update operation is performed on u. The main issue is how to balance the load to all nodes of the overlay as much equally as possible. To do that we employ the machinery developed in Section 3. Assume that w is the node for which the redistribution must be applied. It remains to determine how the redistribution will be realized. An implementation of this redistribution follows.
First we make a scan of all nodes in the subtree of w by forwarding a message which simply counts the number of nodes and the number of elements in the subtree. Finally, this message ends up in the leftmost leaf of the subtree of w. Thus, w now knows exactly how many elements should be distributed in each node in order to have a uniform load. Then, a data migration procedure is initiated.
The idea is to migrate the elements to their final destination nodes in a simple step and in an inorder traversal fashion which is facilitated by adjacency links. The link dest facilitates the transfer of elements between nodes and at the same time functions as a token which designates the node that implements at the moment the data migration. Starting from the rightmost node of the rightmost bucket in the subtree of w, it checks whether the number of elements is less or more than the ideal load. If they are less, then by using the dest link the necessary number of elements is transferred from the designated node to the node containing dest. If they are more, the necessary number of elements are moved to the node designated by dest. If during this procedure the node designated by dest fills up (meaning it reaches the desired load) or empties (meaning we transferred a lot of elements) then dest is moved to the next node w.r.t. the inorder traversal. When the node containing dest has reached its ideal load then dest is moved to the next node w.r.t. the inorder traversal and the procedure continues. This procedure requires a linear number of messages w.r.t. the number of elements in the subtree of node w.
The cost for the redistribution of a node v is O(|v| log N ) for the case of O(log N ) space per node or O(|v|) for the case of O(1) space per node. This is because, during the transfer of elements the routing tables must be reconstructed. The following lemma states that the load balancing is efficient in an amortized sense when the structure is subject to insertions and deletions of elements. Proof. This is a direct implication of Theorem 2 and the space used by the nodes.
⊓ ⊔
One final comment is that the redistribution of elements may be affected by the redistribution of nodes in the weight-balanced overlay. In order to avoid such a phenomenon, the redistribution of nodes in the subtree of node v in the overlay is preceded by a redistribution of elements.
Other Efficiency Issues and the Main Result
We are now ready to tackle the congestion and the fault-tolerance of the D 2 -tree overlay, and to present the main results of this work.
Congestion. We assume that a sequence of searches s 1 , s 2 , . . . , s N is initiated from each of the N nodes of the overlay. We assume that search s i is looking for an element residing in a node z i (target node for s i ). The target nodes z 1 , z 2 , . . . , z N are chosen independently and uniformly at random from all nodes of the overlay. There are two phases in the search. The first is the horizontal search phase, which makes use of the routing tables, and the second is the vertical search phase on a path from a node either towards the root or towards a leaf.
To establish a bound on the congestion, we need to provide bounds on the horizontal and vertical searches. These bounds are provided by Lemmata 9 and 10 below. Before proving these lemmata, we need the following result. Proof. Since the destinations are chosen uniformly at random, the destination nodes at level ℓ for searches starting from this level depend on the weight of each node plus the weight of the nodes on the path to the root which is almost equal. The weight of each node at level ℓ is approximately equal for all nodes. Thus, it is expected that O(1) searches will have as a destination any node at level ℓ. ⊓ ⊔
The following lemma bounds the congestion due to the horizontal search. Proof. Level ℓ contains O(2 ℓ ) nodes. We number the nodes from left to right by 0, 1, . . .. A path from a node j to a node k is the sequence of nodes that we access when we search from node j to find node k at level ℓ by using the routing tables. Let X i,j be the random indicator variable defined as follows:
X i,j =
1 if node i is in the path that starts from node j 0 otherwise X i,j is a random variable since node j can choose its target among all nodes at level ℓ uniformly at random as implied by Lemma 8. The following quantity bounds the expected number of paths passing through an arbitrary node i when all searches from nodes at level ℓ are accounted for.
E O(2 ℓ ) j=0 X i,j = O(2 ℓ ) j=0 E [X i,j ]
Since X i,j is a random indicator variable it follows that
E [X i,j ] = Pr {X i,j = 1}
This probability is equal to the number of paths going through i divided by the total number of paths starting from j and ending at all nodes of level ℓ.
Pr {X i,j = 1} = # of paths passing through i from j Total number of paths starting at j
The total number of paths starting from j to all nodes of level ℓ is equal to the number of target nodes which is O(2 ℓ ). Note that we only count the number of search paths as defined by the search procedure between two nodes and not all possible paths. It is a little trickier to compute the number of paths going through node i. The crucial observation is that the binary representations of the nodes, in their left to right numbering at level ℓ, provide a way to count the number of paths passing through a particular node. Let the binary representation of node i be i ℓ−1 . . . i 1 i 0 , where i ℓ−1 is the most significant bit. Then, if there is a link of length 2 ℓ−1 between node i and node j it holds that i ℓ−2 . . . i 1 i 0 = j ℓ−2 . . . j 1 j 0 . The following observation holds. Proof. This is an implication of the construction of the routing tables as well as from the fact that during searching the sequence of links that are followed are of monotonically decreasing length by powers of 2.
⊓ ⊔ Thus, we have to compute:
1 O(2 ℓ ) O(2 ℓ ) j=0
(# of paths passing through i from j)
The number of paths that go through i starting from j with destination any node at level ℓ can be deduced by using Observation 1 and the properties of the binary representations. In particular, if the m less significant bits of numbers i and j are equal and i m = j m , then at most 2 m paths go through i by Observation 1. The number of different nodes j that go through i in this case is 2 ℓ−m since those are the possible numbers that have the m least significant bits the same as i. Thus, the previous sum can be expressed by summing over all possible m:
1 O(2 ℓ ) ℓ−1 m=0 2 ℓ−m 2 m = ℓ2 ℓ O(2 ℓ ) = O(ℓ)
and the lemma follows.
⊓ ⊔
The following lemma bounds the congestion due to the vertical search.
Lemma 10. The vertical phase of the search starting at level ℓ contributes to congestion O(1) in expectation at each node in its subtree or on the path to the root.
Proof. By Lemma 8, only an expected O(1) number of searches will stop at any node due to the horizontal search phase. Assume a node u at level ℓ. This node has ℓ − 1 ancestors and 2 H−ℓ descendants. Thus, in total at most O(2 H−ℓ + ℓ) searches in expectation can affect node u. We start by investigating how ancestors affect node u. The ancestor at level ℓ − 1 can choose between two children, the one of which is u, as well as from its path of ancestors. Thus, the probability of choosing u is O
2 H−ℓ 2 H−ℓ+1 +ℓ−2 .
In general, the probability of node z at level
ℓ ′ < ℓ going through u is O 2 H−ℓ 2 H−ℓ ′ +ℓ ′ −1 .
Thus, the expected number of searches going through u due to its ancestors is
ℓ ℓ ′ =1 O 2 H−ℓ 2 H−ℓ ′ + ℓ ′ − 1 = O 2 H−ℓ ℓ ℓ ′ =1 1 2 H−ℓ ′ = O(1)(2)
Now we move to the descendants of u. The probability that the leaves of the subtree of u go through u during a search is O ℓ 2 H−ℓ n . This is because the probability of choosing any node as a destination node of the search operation is 1 n , the number of leaves is O(2 H−ℓ ) and there are ℓ nodes in total from u to the root. Similarly, for the i-th level, i > ℓ, the probability of going through u is O ℓ 2 H−ℓ−i n . Thus, in total we get that the expected number of searches going through u from its descendants is (2) and (3) we get the lemma.
ℓ−1 i=0 O ℓ 2 H−ℓ−i n = O ℓ2 H−ℓ+1 n = O ℓ2 H−ℓ+1 2 H−1 = O ℓ 2 ℓ−2 = O(1) (3) By
⊓ ⊔
The following theorem establishes the congestion bound. Proof. By Lemmata 9 and 10, we deduce that O(log N ) searches in expectation will go through each node of the tree. Since the tree has N nodes, the theorem is established. ⊓ ⊔
The following theorem extends Theorem 3 by using O(1) space per node. Proof. This proof is very similar to the proof of Theorem 3 and we simply sketch it. Lemma 8 still holds. Searching is again divided into two phases. The vertical search phase is identical to the one in Lemma 10 and hence this lemma still holds. However, horizontal search has slightly changed and Observation 1 is not valid anymore. First, the search always starts from the highest rank node in a hypernode V which results in O(ℓ) accesses from the searches that start from all nodes of V . From this point and on, the horizontal search is similar to the one of Lemma 9. The proof that the congestion remains optimal is a result of the following similar argument to Lemma 9. The probability that a node i will be part of the search path which starts at node j is large for very few nodes j among the O(2 ℓ ) such possible nodes. Most of the nodes have a very small probability of using node i, since node i can be accessed after O(ℓ) steps. This follows directly from Lemma 8. Using this fact and the fact that highest rank nodes have at least O(ℓ) accesses we are driven to the conclusion that the expected bound on the number of accesses to nodes of level ℓ due to the horizontal search is O(1) and the theorem follows.
⊓ ⊔ Fault Tolerance. If a node v discovers (during the execution of an operation) that node u is unreachable, then it contacts a sibling of u through the routing tables of the siblings of v (by making use of Lemma 2(ii)). This sibling of u is able by Lemma 2(ii) (or Lemma 5) to reconstruct all links of node u and a node departure for u is initiated, which resolves this failure. Searches and updates in the D 2 -tree do not tend to favour any node, and in particular nodes near the root. This is a direct consequence of the way the search operation is implemented by first moving horizontally at the same level as the node that initiated the search and then by moving vertically (see Theorem 4). As a result, near to root nodes are not crucial and their failure will not cause more problems than the failure of any node. However, a single node can be easily disconnected from the overlay simply when all nodes with which it is connected fail. This means that 4 failures (two adjacent nodes and two children) are enough to disconnect the root (recall that the routing table of the root is empty). For the O(1) space per node solution, a O(1) number of failures is enough to disconnect any node. For the O(log N ) space per node solution, a node at level ℓ can be disconnected after O(ℓ) failures in the worst-case.
When routing tables have O(log N ) size, to disconnect a group of k nodes at least k failures must happen. The most easily disconnected nodes are those which are near the root since their routing tables are small in size. Thus, they can be disconnected by simply letting their respective adjacent nodes (which are leaves) fail which provides the bound. When routing tables have O(1) size, fault tolerance is naturally deteriorated. When the representative of a bucket fails then the leftmost node among the nodes of the bucket replaces it, initiating a departure operation.
Main Result. We are now ready for the main result of this work.
Discussion and Future Work
Our load-balancing scheme (Section 3) can be applied straightforwardly to BA-TON [11]. BATON is a balanced tree-like overlay that satisfies the specifications set in the Introduction. The same goes also for Skip Graphs [4] with the exception that the specifications hold probabilistically and thus the bounds are also probabilistic. Additionally, it provides a mechanism to control the bucket size of [3].
We provide a technique that lazily updates the weights on the nodes of a tree (Section 3.1). This technique is interesting by itself and can be straightforwardly applied to weighted balanced trees [2] in the Pointer Machine model of computation for single processor internal memory machines. In this manner, the update of balancing information is supported in O(1) amortized time, an improvement over the currently best known bound of O(log n).
Future work includes the extension of the load-balancing mechanism to accommodate weighted elements (weights representing preference). Additionally, the load balancing mechanism provides amortized complexities which results in the existence of very few indeed but very costly rebalancing operations (imagine the root being redistributed). To fully tackle the existence of churn, one needs to come up with worst-case complexities for the load balancing mechanism. Note that churn is the collective effect created by independent burstly arrivals and departures of nodes.
With respect to the overlay, future work includes tackling multidimensional data, integrating the network topology with the overlay topology as well as taking into account locality of reference.
It is also an open problem the application of the proposed balancing scheme to the BATON * [12] structure (the latest version of BATON), where the overlay structure is a tree with height O(log m N ) with each node having O(m) children.
Finally, the mechanisms we provide require extensive experimental verification.
Theorem 2 .
2The load balancing has an amortized cost of O H f (n) n .
the first link points to a node (if there is one) 2 0 positions to the left (right), the second 2 1 positions to the left (right), and the i-th link 2 i−1 positions to the left (right). These links constitute the routing table of v. The next lemma captures some important properties of the routing tables w.r.t. their construction. It follows immediately from the aforementioned link structure and the fixed distances between successive links in the routing tables. Lemma 2. (i) If a node v contains a link to node u in its routing table, then the parent of v also contains a link to the parent of u, unless u and v have the same father. (ii) If a node v contains a link to node u in its routing table, then the left (right) sibling of v also contains a link to the left (right) sibling of u, unless there are no such nodes. (iii) Every non-leaf node has two adjacent nodes in the inorder traversal, which are leaves. A Weight-Balanced Overlay. The overlay consists of two levels. The upper level of the overlay is a Perfect Binary Tree (PBT). The leaves of the tree are representatives of buckets that constitute the lower level of the overlay. Each bucket is a set of O(log N ) nodes and it is structured as a doubly linked list.
Fig. 1 .
1To the left (right) the join of z (leave of v) is depicted. The dotted labeled arrows represent the movement of the nodes denoted by the label.
Fig. 2 .
2In the middle, the structure of the weight balanced overlay is depicted. To the left (right) is the result of the application of an extension (contraction) operation.
Lemma 4 .
4The amortized cost of join/departure of a node v is O H f (N ) N .
Lemma 5 .
5If node v contains a link to node u, then the left (right) sibling of v also contains a link to the left (right) sibling of u, unless ∄ such nodes.
Lemma 6 .
6The search for an element α in a D 2 -tree of N nodes is carried out in O(log N ) steps.
Lemma 7 .
7The load rebalancing operation of the index has an amortized cost of O(log N ).
Lemma 8 .
8The number of searches that stop at a node v at level ℓ during the horizontal phase of the search is O(1) in expectation.
Lemma 9 .
9The horizontal phase of the search at level ℓ contributes to congestion O(ℓ) in expectation at each node of this level.
Observation 1
1Node i will be accessed by a link of length at most 2 m in a search path starting from j if i m−1 . . . i 1 i 0 = j m−1 . . . j 1 j 0 .
Theorem 3 .
3The (expected) congestion due to the search operations is O log N N in a D 2 -tree with N nodes, when each node uses O(log N ) space.
Theorem 4 .
4The (expected) congestion due to the search operations is O( log N N ) in a D 2 -tree with N nodes, where each node uses O(1) space.
Theorem 5 .
5A D 2 -tree overlay with N nodes and n data elements residing on them achieves: (i) O(1) space per node; (ii) deterministic O(log N ) searching cost; (iii) deterministic amortized O(log N ) update cost both for element update and for node joins and departures; (iv) optimal congestion of O log N N expected cost; (v) deterministic amortized O(log n) bound for load-balancing. The D 2 -tree overlay supports ordered data queries optimally, and tolerates node failures. Proof. Space usage is O(1) by construction. The search cost follows from Lemma 6. Node join and departures are O(log N ) amortized by Lemma 4 and the fact that f (n) = O(N ). The congestion bound comes from Theorem 4. Finally, the loadbalancing bound comes from Lemma 7.⊓ ⊔
Table 1 .
1A comparison between previous methods and the D 2 -tree. By O we represent expected bounds, by O we represent amortized bounds, and by O expected amortized bounds. All other bounds are worst-case. Typically, N ≪ n.
We have chosen this ǫj for simplicity. In fact for any η > 0, choosing ǫj = 1 j 1+η is sufficient.
The alternative of following a height-based approach, resulting in a height (instead of weight) balanced overlay, would render update operations inefficient.
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| []
|
[
"Study of dust-induced beam losses in the cryogenic arcs of the CERN Large Hadron Collider",
"Study of dust-induced beam losses in the cryogenic arcs of the CERN Large Hadron Collider"
]
| [
"A Lechner \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n",
"P Bélanger \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n\nTRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada\n",
"I Efthymiopoulos \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n",
"L Grob \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n",
"B Lindstrom \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n\nDepartment of Physics and Astronomy\nUppsala University\nBox 51675120UppsalaSweden\n",
"R Schmidt \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n",
"D Wollmann \nEuropean Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland\n"
]
| [
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"TRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"Department of Physics and Astronomy\nUppsala University\nBox 51675120UppsalaSweden",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland",
"European Organization for Nuclear Research (CERN)\nEspl. des Particules 11211GenevaSwitzerland"
]
| []
| The interaction of dust particles with the LHC proton beams accounts for a major fraction of irregular beam loss events observed in LHC physics operation. The events cease after a few beam revolutions because of the expulsion of dust particles from the beam once they become ionized in the transverse beam tails. Despite the transient nature of these events, the resulting beam losses can trigger beam aborts or provoke quenches of superconducting magnets. In this paper, we study the characteristics of beam-dust particle interactions in the cryogenic arcs by reconstructing key observables like nuclear collision rates, loss durations and integral losses per event. The study is based on events recorded during 6.5 TeV operation with stored beam intensities of up to ∼ 3 × 10 14 protons per beam. We show that inelastic collision rates can reach almost 10 12 collisions per second, resulting in a loss of up to ∼ 1.6 × 10 8 protons per event. We demonstrate that the experimental distributions and their dependence on beam parameters can be described quantitatively by a previously developed simulation model if dust particles are assumed to be attracted by the beam. The latter finding is consistent with recent time profile studies and yields further evidence that dust particles carry a negative charge when entering the beam. We also develop different hypotheses regarding the absence of higher-loss events in the measurements, although such events are theoretically not excluded by the simulation model. The results provide grounds for predicting dust-induced beam losses in presence of higher-intensity beams in future runs of the High-Luminosity LHC. | 10.1103/physrevaccelbeams.25.041001 | [
"https://arxiv.org/pdf/2112.11068v1.pdf"
]
| 245,353,457 | 2112.11068 | 0652e8d43a4cd64fff356425214c129af603a1ab |
Study of dust-induced beam losses in the cryogenic arcs of the CERN Large Hadron Collider
A Lechner
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
P Bélanger
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
TRIUMF
4004 Wesbrook MallV6T 2A3VancouverBCCanada
I Efthymiopoulos
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
L Grob
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
B Lindstrom
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
Department of Physics and Astronomy
Uppsala University
Box 51675120UppsalaSweden
R Schmidt
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
D Wollmann
European Organization for Nuclear Research (CERN)
Espl. des Particules 11211GenevaSwitzerland
Study of dust-induced beam losses in the cryogenic arcs of the CERN Large Hadron Collider
The interaction of dust particles with the LHC proton beams accounts for a major fraction of irregular beam loss events observed in LHC physics operation. The events cease after a few beam revolutions because of the expulsion of dust particles from the beam once they become ionized in the transverse beam tails. Despite the transient nature of these events, the resulting beam losses can trigger beam aborts or provoke quenches of superconducting magnets. In this paper, we study the characteristics of beam-dust particle interactions in the cryogenic arcs by reconstructing key observables like nuclear collision rates, loss durations and integral losses per event. The study is based on events recorded during 6.5 TeV operation with stored beam intensities of up to ∼ 3 × 10 14 protons per beam. We show that inelastic collision rates can reach almost 10 12 collisions per second, resulting in a loss of up to ∼ 1.6 × 10 8 protons per event. We demonstrate that the experimental distributions and their dependence on beam parameters can be described quantitatively by a previously developed simulation model if dust particles are assumed to be attracted by the beam. The latter finding is consistent with recent time profile studies and yields further evidence that dust particles carry a negative charge when entering the beam. We also develop different hypotheses regarding the absence of higher-loss events in the measurements, although such events are theoretically not excluded by the simulation model. The results provide grounds for predicting dust-induced beam losses in presence of higher-intensity beams in future runs of the High-Luminosity LHC.
I. INTRODUCTION
The Large Hadron Collider (LHC) [1] at CERN is the first machine with positively charged hadron beams where interactions with micrometer-sized dust particles caused perceivable disruptions of beam operation [2][3][4][5][6][7][8][9][10]. The trapping of ionized dust particles in the beam, accompanied by a drop of beam lifetime, is a well-known phenomenon in electron storage rings [11][12][13][14][15][16][17]. Dust or macroparticle-related beam losses were, however, not expected to be perturbing for a proton collider since dust particles become rapidly ionized and repelled from the beams. Simulations and experimental observations in the LHC indeed support the hypothesis that dust grains are expelled before reaching the beam core [18][19][20][21][22]. The events typically last less than a millisecond, i.e. less than twelve beam revolutions, with many events being as fast as one or two revolutions [6]. Despite the short loss duration, it was already apparent in LHC run I (2009-2013) that dust particles can still generate sufficient beam losses to provoke beam aborts by the Beam Loss Monitor (BLM) system [2,3,[5][6][7]. Beam aborts already occurred at stored intensities as low as 10 12 protons [6], which is two orders of magnitude below the LHC design intensity. After increasing the operation energy from 3.5 TeV and 4 TeV in run I to 6.5 TeV in run II (2015-2018), the first dust-induced quenches of superconducting magnets were observed [8][9][10]. All quenches concerned bending dipoles in the arcs or dispersion suppressors. The quench events typically required a recovery time between 8 and 12 hours before regular cryogenic conditions could be restored. In total, about a hundred BLM abort triggers and eight quenches attributed to dust particles were observed in the first two LHC runs, resulting in the loss of several hundred hours of beam time. No other beam loss mechanism caused more magnet quenches than the interaction of the beam with dust particles.
Dust-induced loss events occur all around the LHC circumference, including the room temperature insertion regions and the cryogenic arcs and dispersion suppressors [2][3][4][5][6][7][8][9][10]. Besides the events causing beam dumps or quenches, a copious amount of smaller events has been registered by the BLMs every operational year. These events appear as a localized transient loss spike on BLMs, but do not have any detrimental effect on operation and luminosity production. The beam intensity loss is smaller than the resolution of the LHC beam current monitors (∼ 10 9 charges) and cannot be measured directly. The secondary particle showers can nonetheless be detected by the BLMs even for beam losses as small as 10 4 -10 5 protons. Although these events are harmless, they are still carefully monitored as they provide insight about the correlation with beam parameters and the long-term evolution of event rates.
Dust particles were not the only cause of beam-induced aborts and quenches in run II. Performance limitations arose also from a macroscopic obstacle in one of the dispersion suppressor dipoles [23,24] and from solid micrometer-sized aggregates of residual air molecules TABLE I. Proton beam parameters (beam energy E, bunch spacing ∆t b , number of bunches per beam N b , bunch intensity I b , and normalized transverse emittance εn) in previous and future runs of the LHC, and in the HL-LHC era. The values correspond to the start of physics production, referred to as the stable beams period. Besides the standard beam production scheme, an alternative low-emittance scheme is used, called Batch Compression, Merging and Splitting (BCMS) scheme. For run I and run II, maximum performance values are given, corresponding to standard 50 ns beams in 2012 [29] and to 25 ns BCMS beams in 2017/18 [30], respectively. For run III, optimistic and pessimistic values are given for the transverse emittance (BCMS beams), whereas the bunch intensity is the maximum intensity expected in run III [31]. The last column shows nominal HL-LHC beam parameters for standard 25 ns beams [28]. [24][25][26][27]. The latter were leftovers from an accidental air inflow in a certain arc cell. While the occurrence of these localized loss events was limited in time, dustinduced quenches remain a concern for future runs of the LHC, in particular in the High-Luminosity (HL)-LHC era [28]. A large increase of event rates can potentially be expected after long shutdowns, as was the case when restarting the LHC for its second run in 2015 [24]. Such an increase can have a detrimental impact on the operational performance in the first year after a shutdown. Increasing the operation energy from 6.5 TeV in run II to 6.8 TeV in run III (2022-2024) and further to 7 TeV in the HL-LHC era, will in addition increase the susceptibility for magnet quenches since the temperature rise, which superconducting coils can tolerate, will decrease. Besides the reduced quench margin, the beam parameters will become more challenging in future runs, with the stored beam intensity increasing from 3.2 × 10 14 protons in run II to possibly 5 × 10 14 protons in run III and further to 6.1 × 10 14 in the HL-LHC era (see Table I).
Dust-induced loss events have been put under scrutiny since their first occurrence in run I, by studying event rates and empirical correlations with beam parameters [2-6, 8, 9], by analysing dust samples from the vacuum chamber of magnets [7,32], and by modeling the motion of dust particles in the beam [18-20, 33, 34]. In combination with bunch-by-bunch beam diagnostics, individual dust particle trajectories could be reconstructed [21,22]. The simulation studies and measurements also indicated that the radius of dust particles is smaller than 100 µm [19,20] and that the dust grains likely carry a negative charge when entering the beam [22,35]. Despite these findings, a better understanding of these events is needed for quantifying the performance impact on the LHC, the HL-LHC and other future high-energy proton colliders like the FCC-hh [36]. In particular, the physical mechanism governing the release of dust particles into the beam still lacks a theoretical explanation, which is fundamental for predicting the likelihood of events. It is also essential to reliably quantify dust-induced beam losses as a function of beam parameters and dust properties. The induced beam losses have been estimated previously by means of simulations [33,34], but the predictions still lack a systematic experimental verification.
In this paper, we study the characteristics of beamdust particle interactions through a comprehensive experimental analysis of inelastic nuclear collisions between beam protons and dust grains. Inelastic collisions are the main mechanism for dust-induced beam losses in superconducting magnets. The characteristic features of loss profiles, like peak collision rates and integral losses, are of practical importance since they directly relate to the risk of magnet quenches. They also reveal more about the nature of these events and the properties of dust particles. Owing to a better coverage of arc dipoles with beam loss monitors in run II, observables related to beam-dust collisions could be systematically reconstructed for a large ensemble of events. Based on these data, we probe the ability of the previously developed simulation model [18-20, 22, 33, 34] to reproduce different experimental distributions by constraining the properties of dust particles. We further probe the ability of the model to reproduce the dependence of proton losses on beam parameters. This enables more accurate predictions for future runs with higher-intensity beams.
Although dust particle events are observed in all regions of the LHC, the studies presented in this paper focus exclusively on the cryogenic arc sectors (see Fig. 1). The arcs are believed to be the regions where dustinduced losses might have the highest impact in future runs because of the higher risk of magnet quenches. The arc sectors offer ideal conditions for a systematic study of dust events. They are less exposed to other types of beam losses than the insertion regions or dispersion suppressors, which facilitates the identification of such transient loss events. Another advantage is the cumulative arc length of more than 19 km and the cell-by-cell periodicity of the BLM layout, which help in enhancing the statistical significance of observations. The paper is organized as follows. Section II describes the methods for reconstructing the number of nuclear beam-dust particle collisions from BLM measurements in the arc sectors. Section III explores the characteristics of dust-induced losses at 6.5 TeV by comparing reconstructed distributions (loss rates, loss durations and integral losses) with predictions from dust particle dynamics simulations. By constraining the dust properties, we attempt to find the best agreement between simulations and measurements. Based on these results, Sec. IV analyses the dependence of observables on beam param- eters, while Sec. V provides predictions of dust-induced losses for beam intensities in the HL-LHC era. Section VI summarizes the studies and provides some concluding remarks.
II. RECONSTRUCTION OF DUST-INDUCED BEAM LOSSES IN THE LHC ARCS
Dust-induced loss events can only be studied indirectly by analysing beam losses through the shower-induced energy deposition in BLMs. In this section, we discuss the methods for reconstructing the number of inelastic encounters from spatial BLM signal patterns near the collision vertex. The event reconstruction is based on FLUKA [37][38][39] Monte Carlo simulations. Generalpurpose transport codes like FLUKA can provide an estimate of macroscopic observables like monitor signals by describing the propagation of showers in complex geometries based on microscopic interaction models. The predictive ability of the FLUKA code for BLM response studies in the LHC radiation environment has been demonstrated in Ref. [40]. In the following, we use this simulation technique for reproducing dust-induced signal patterns recorded during 6.5 TeV operation in run II.
A. Detection of dust events in LHC operation
The eight LHC arcs are composed of superconducting twin-bore magnets, which host the two counter-rotating proton beams in physically separated apertures [1]. Each of the arcs consists of 23 lattice cells. A cell is made up of two 53.45 m long half-cells (see Fig. 1), which are composed of three bending dipoles, one quadrupole and corrector magnets. The mechanical clearance for the beams is defined by racetrack-shaped beam screens inside the vacuum chambers. The beam screens protect the cold magnets from different heat sources such as electron clouds and synchrotron photons. The beam screens are perforated at the top and bottom surfaces and are maintained at a higher temperature (5-20 K) than the vacuum chambers and magnets (1.9 K) [41]. They are made of stainless steel with a layer of copper on the inner side. The beam screens of neighboring magnet cryostats are connected by means of radiofrequency bridges, which provide a passage for the beam-image current [42]. The bridges consist of gold-plated copper-beryllium fingers, which can slide along a copper tube. Dust samples collected from the vacuum components of an arc dipole in run II showed that dust contamination is present on all components, including beam screens, vacuum chambers and the plug-in modules containing the radiofrequency bridges [32].
When a dust grain enters the LHC proton beam, it gets ionized by the traversing beam particles and is rapidly ejected due to the repelling force exerted by the electric field of the beam. While a dust particle travels in the beam, a small fraction of the impacting protons will be subject to an inelastic nuclear collision and will be lost from the beam. The energetic collision products, mainly π ± , protons, neutrons, kaons, as well as photons from decaying π 0 , impact on the machine aperture and induce hadronic and electromagnetic showers in the surrounding beam screens, vacuum chambers and magnets. Most of the secondary particles are lost within the same or the neighboring lattice half-cell, i.e. nearby the primary collision vertex. An exception are diffractive protons, which can travel longer distances in the collider rings. Contrary to the inelastic collision products, beam protons undergoing an elastic nuclear collision either stay in the transverse beam acceptance or they are intercepted by collimators in the cleaning or experimental insertions if the deflection angle is large enough [43].
The LHC is equipped with almost 4000 ionization chambers which constantly monitor beam losses around the two rings [44,45]. The chambers are filled with nitrogen gas and have a sensitive volume of ∼1500 cm 3 . About three quarters of these monitors are located in the arc sectors. The arc BLMs are mounted on the outside of magnet cryostats (see Fig. 2) and detect electromagnetic and hadronic particle showers induced by beam losses in magnets and other equipment. The BLM system records the deposited dose in twelve sliding time windows of different length (from ∆t = 40 µs to 83.9 s). These running b) Quadrupole: sums enable a customized monitoring of beam losses with different time characteristics, from very fast losses up to steady-state losses. Dust events represent some of the fastest beam losses in the LHC. They are recorded in real time during operation by a dedicated software application developed in run I [6]. A beam loss event is classified as a dust particle event if the dose measured in the 640 µs sliding time window of at least two BLMs exceeds a certain trigger threshold. The two BLMs must be located within 40 m in order to exclude false triggers on uncorrelated signals. In addition, the detection algorithm verifies that dose values recorded in shorter time windows (40-320 µs) do not exhibit an unphysical correlation. This should minimize false triggers because of spurious noise spikes. The noise suppression parameters were optimized in run I [6] and the final settings were retained in run II. The trigger threshold for the 640 µs time window was adapted a few times throughout the years, and hence the data must be post-processed to remove any bias.
Each of the arc half-cells is equipped with six BLMs. In run I, all six BLMs were installed in the vicinity of quadrupoles, as illustrated in Fig. 2. Quadrupoles were expected to be the bottleneck for beam losses in the arcs due to local restrictions of the effective aperture (maxima in the β and dispersion functions), aperture discontinu-ities (beam position monitors) and possible imperfections (magnet misalignment). Because of the absence of BLMs along the bending dipoles, which account for more than 85% of the arc length, this configuration provided only a limited resolution for detecting and localizing dustinduced loss events. As dust particles posed the major source of transient beam losses in the arcs in run I, hundreds of arc BLMs were relocated from quadrupoles to dipoles in the shutdown between run I and run II (2013)(2014)(2015) to improve the detection of such loss events [46]. The run II BLM layout is illustrated at the bottom of The number of inelastic nuclear collisions of beam protons in pointlike obstacles like dust particles can be estimated empirically from BLM signals using particle shower simulations [40]. This requires finding the longitudinal position of dust particles since the BLM response per collision depends on the distance to the loss location due to the shower attenuation by nearby magnets. The spatial pattern of BLM signals along an arc half-cell, i.e. the measured dose as a function of the longitudinal BLM position, exhibits a relatively high sensitivity to the longitudinal loss location. The collision vertex can therefore be determined empirically by finding the best match between measured and simulated patterns. This method already proved to be successful for localizing the aforementioned obstacle in the dispersion suppressor [24] as well as the location of accidental air inflow in one of the arc cells [27]. The same approach was also used in run I for estimating the location of dust events at 3.5 TeV and 4 TeV in a specific arc cell equipped with additional BLMs [40].
Since dust events appear at any longitudional position in the arcs, BLM patterns were calculated for different loss locations inside a representative arc half-cell consisting of three bending dipoles and a quadrupole. The FLUKA geometry model is illustrated in Fig. 3. The proton beam energy was 6.5 TeV. The dose is given per proton-nucleus collision. The assumed loss location is indicated by a cross on the s-axis. The beam direction is from the left to the right. The statistical error of the simulations is less than 3% for the highest signal, but can be as large as few 100% for the smallest signals. The data points are connected by lines to guide the eye.
Neighboring magnets in adjacent half-cells were included as well. The BLM signals were calculated as described in Ref. [40], by recording the energy deposition in the sensitive gas volume between the electrodes of the BLM model. The dust particles were assumed to be static and pointlike [27,40]. The loss locations were spaced by 0.5-2 meters in order to achieve a good resolution in the vertex identification. Because of the significant computational requirements, losses in the two upstream dipoles were only studied for the anti-clockwise rotating beam (beam 2) and the obtained signal patterns were then mapped to the other beam (beam 1). This approach is justified since the dipole BLMs, located on top of the dipole interconnects, are equally exposed to losses from both beams. The situation is different for beam-dust particle collisions in or nearby quadrupoles since the relative BLM positions are not fully identical for the two counter-rotating beams. In addition, the layout around quadrupoles differs for the two beams because of the asymmetric sequence of corrector magnets in the quadrupole assemblies [1]. Losses in the third dipole and the quadrupole were hence studied separately for the two beams as the different geometry can affect the shower leakage to BLMs.
In all studies, it was assumed that the clockwise rotating beam circulates in the inner aperture of the twinaperture magnets, while the other beam circulates in the outer aperture. This assumption holds only for half of the arc sectors since the beams change from the inner to the outer aperture and vice versa due to the beam crossing in the four detectors (see Fig. 1). It was further assumed that the field of the quadrupole in the considered half-cell is defocusing in the horizontal plane. These assumptions may slightly increase the uncertainty of the pattern reconstruction, but it is not expected to impact the general conclusions.
The composition of dust particles can possibly vary from event to event. The analysis of dust samples collected in run II showed that macroparticles of different chemical position are present in the vacuum system [32]. Figure 4 presents simulated BLM signal patterns for dust particles composed of carbon and copper, respectively. The BLM signals are given per proton-nucleus collision. The different dipoles (MBs) and quadrupoles (MQs) are illustrated by gray boxes. The results show that BLM signals up to 40 m from the loss location agree within 30% percent for the two compositions (the highest signal agrees within a few percent). Larger discrepancies can be observed for more distant BLMs, which can at least partially be explained by the higher statistical uncertainty of the simulation results because of the much smaller dose values. The good agreement of the BLM patterns around the loss location suggests that the number of inelastic collisions can be reconstructed from the measurements with reasonable accuracy even without having an exact knowledge of the dust particle composition. In the shower simulations presented in the following, dust particles were assumed to be made of carbon.
C. Matching of simulated and measured BLM dose patterns
In order to estimate the number of inelastic hadronic proton-dust particle collisions N i for a measured loss event, the following expression was minimized as a function of N i and the discrete loss location s j :
Q(N i , s j ) = k [D tot,k − N i d k (s j )] 2 N i d k (s j ) ,(1)
where D tot,k is the time integral of the dose rateḊ k (t) measured in BLM k during the loss event,
D tot,k = Ḋ k (t )dt(2)
and d k (s j ) is the corresponding simulated dose per proton-nucleus collision assuming that the collisions occur at s j . The time-integrated dose was obtained from the ∆t = 2.56 ms long sliding window of BLMs. This time window is sufficiently long to contain the full loss event. The BLM signals were corrected for the noise floor. For each case, six BLMs were used to find the best match between simulated and measured patterns (typically 1-2 BLMs upstream of the loss location and 4-5 BLMs downstream). The statistical error of the simulated signals was usually between 1-10%, except for some BLMs upstream of the loss location and for BLMs >30-40 m downstream of the loss location. Figure 5 shows a selection of simulated BLM patterns which were matched to measured loss patterns using Eq. (1). All measurements were recorded at 6.5 TeV in run II. The x-axis represents the s-coordinate of the curvilinear coordinate system of the concerned beam; the In four out of the six cases a dipole quench occurred (labels of dipoles which quenched due to showers are in bold red). The simulated patterns were scaled according to Eq. (1) to obtain the best match with the measurements. The scaling factors, which represent estimates of the inelastic proton-dust particle collisions, are indicated in the labels. The data points are connected by lines to guide the eye. The estimated s-location of the dust particles is identified by 'x' labels.
origin is arbitrarily set to coincide with the center of the quadrupole upstream of the loss location. The measurements shown in the figure exhibited some of the highest BLM signals of all dust events in run II. In four out of the six cases, the losses induced a dipole quench, while no quench was observed in the other two events. The simulated patterns generally show a good agreement with the measurements. It is estimated that in most cases analysed in the following section, the number of inelastic collisions per event can be determined with an uncertainty less than a factor of two despite the unknown dust particle composition and the model approximations.
III. CHARACTERISTICS OF BEAM-DUST PARTICLE INTERACTIONS
The characteristics of dust-induced BLM signals have been studied since run I [2,6,[19][20][21][22]47]. Different observations were made, which provided some insight into the nature of these events. It was found that the distribution of BLM signals in run I was proportional to 1/D 2 , where D is the time-integrated BLM dose in the BLM with the highest signal near the collision vertex [2,6,47]. This observation provided an approximate indication about the size distribution of dust particles in the cold arc sectors, with the caveat that BLM signals depend on the loss position and the dust particle trajectory in the beam. The latter can vary from event to event even for similar-sized dust particles. Hence BLM signals do not uniquely reflect the dust particle size for a given event. The signal distribution nonetheless demonstrates the abundance of smaller dust particulates in the arc vacuum system, which is also confirmed by the dust samples collected in run II [32]. It was also observed that the time profiles of events can typically be described by a skewed Gaussian distribution [4,6,22] (see Fig. 6). Some of the profiles had a shorter rise time, whereas others exhibited a faster fall time. The latter can be explained by the rapid repulsion of dust particulates once they get ionized in the beam, but the opposite observation (faster rise times) still lacks a theoretical understanding [6,22].
The trajectory of a dust particle and hence the induced nuclear collisions depend on various parameters like the beam intensity, the transverse beam size, the dust particle radius, the dust particle composition and density, as well as the initial dust particle position. The trajectory is also strongly influenced by the initial charge carried by the dust particle when it enters the beam. A numerical simulation model has been developed for studying the motion of dust particles in the LHC beams as a function of these parameters [33,34]. The model assumes that the dust particle is initally located on the beam screen. Different physics improvements have been incorporated over time, as described in Refs. [18][19][20] and more recently in Ref. [22]. The simulations suggest that the dust particles are expelled before reaching the beam core [19,33,34]. This was confirmed in recent experimental and numerical studies in run II, where a maximum penetration depth of ∼3σ from the beam center was found for particular events recorded at 5.5 TeV and 6.5 TeV, respectively [21,22]. The model also explained other features of dust events, like the asymmetry of time profiles (in case of faster fall times) [19,20,22] and the decrease of the loss duration as a function of beam intensity [4]. The model calculations further made it possible to better understand dust particle properties. It could be shown that measured BLM signals in run I can be reproduced by assuming dust particle radii between 5 and 100 µm [19,20], which was consistent with the dust samples collected in run II [32]. In a more recent article, based on BLM data from run II, it was demonstrated that the rising slope of certain loss profiles can only be explained if dust particles are initially attracted by the beam [22]. As proposed in this article, this can be explained if dust particles possess a negative charge.
In this section, we extend previous studies by performing an absolute comparison between simulations and beam loss observables. The latter were reconstructed from BLM measurements at 6.5 TeV in run II. By constraining the material properties of dust particulates, we probe the ability of the simulation model to reproduce the experimental distributions and their dependency on beam parameters. This is an important prerequisite for assessing the predictive ability of model calculations. We also discuss possible implications for future operation with higher-intensity.
n . i (t) (1/s) Time t (ms) FIG. 6.
Rate of inelastic nuclear collisions between 6.5 TeV protons and a dust particle entering the LHC beam. The two events were recorded in the LHC arcs in run II. In both cases, the losses lead to a magnet quench. The loss rate was reconstructed from BLM measurements using particle shower simulations. The time resolution of the measurements is 40 µs.
A. Observables under study
Dust-induced loss events can be characterized by the inelastic nuclear collision rateṅ i (t) between beam protons and the dust grain. Figure 6 illustrates two typical collision rate profiles for events which lead to a quench at 6.5 TeV. Time profiles, as shown in the figure, are only recorded for a subset of events. In absence of such profiles, we can nonetheless describe dust particle events by different observables, which can be reconstructed from the maximum dose recorded in the different sliding time windows of BLMs. The maximum dose per window is logged by the monitoring application and is therefore available for all events. The observables under study include the integral number of inelastic proton-dust particle collisions per event (as already introduced in the previous section),
N i = ṅ i (t)dt,(3)
the maximum collision rate
R i = max[ṅ i (t)],(4)
and the loss duration τ . We define the latter as the full width at half maximum (FWHM) ofṅ i (t) profiles,
τ = t 2 − t 1 ,(5)
where t 1 < t max , t 2 > t max anḋ
n i (t 1 ) =ṅ i (t 2 ) = R i 2 .(6)
t max refers to the time where the collision rate is maximum. These quantities can provide more insight into the nature of dust particle events. As can be seen in Fig. 6, the two profiles exhibit different features, one being twice as long as the other, but featuring a lower peak loss rate. The integral number of inelastic collisions was similar in both cases. Evidently, the considered observables N i , R i and τ are not independent from each other, but we still consider it instructive to study all three together. While N i can be obtained from Eq. (1), the reconstruction of R i and τ entails a few additional approximations, as detailed below.
The peak collision rate was derived from the following expression,
R i = N i D tot D 40µs max ∆t ,(7)
where D 40µs max is the maximum dose recorded in the shortest sliding time window of BLMs (∆t = 40µs), D 40µs max = max tn+1 tnḊ (t )dt : t n = n × 40 µs , (8) and D tot is the time-integrated dose in the same BLM where D 40µs max was measured. Eq. (7) provides only an approximation of the real peak collision rate because of the signal delay introduced by the readout cables and the delayed charge collection in the ionization chambers. The collection time is around 300 ns for electrons, but is around 80 µs for ions [44,45]. In general one can assume that about 40% of the signal is registered within 40 µs, and about 70% within 80 µs. Because of this delay, the actual peak dose rate and hence the peak collision rate might be underestimated. This underestimation is expected to be more pronounced for very fast events, lasting only one to two turns (80-160 µs), which account for about one third of the considered events. Even without the delayed signal collection, the measurement of ultrafast events would be intrinsically limited by the 40 µs time resolution of the BLMs.
The loss duration τ was reconstructed by assuming that theṅ i (t) profiles are of Gaussian shape, neglecting any possible asymmetry. For consistency, this approximation was applied to all events, even if time profiles were available. The integral of the Gaussian profiles was assumed to be N i . The width τ of the Gaussian profiles was determined by calculating the number of standard deviations m σ contained in an 80 µs time window centered around the mean (see bottom graph in Fig. 7); m σ
N 80µs i = N i D 80µs max D tot ,(9)
where D tot is the time-integrated dose and D 80µs max is the maximum dose in the 80 µs sliding time window of the BLM, which is updated every 40 µs, Although the approach could have also been based on a different window length, 80 µs was found to be the best compromise between faster and slower events. The calculated τ values can differ from the actual width of loss rate profiles because of the neglected skewness, which is unknown for events where no time profile was recorded. The approximation is nonetheless suitable for identifying trends. For very fast events, with a duration τ < 10 −4 seconds, the obtained values become unreliable because of the limited time resolution.
D 80µs max = max
B. Distribution of events as a function of Ni, Ri and τ
About 21000 dust particle events were detected at 6.5 TeV in the LHC arcs in run II. The smallest of these events gave rise to about 10 4 inelastic proton-nucleus collisions, while the largest event resulted in more than 10 8 collisions. The shower simulations indicate that events, which generated less than ∼ 5 × 10 5 collisions within a time interval of ∆t = 640 µs, were not uniformly detected along arc cells even with the improved BLM layout in run II. These events were only registered if the loss location was nearby a BLM. In the following, we therefore discard all events with less than 5 × 10 5 collisions (∼15500 out of the 21000) since they represent an incomplete sample of the true event population. The detection limit of 5 × 10 5 collisions within ∆t = 640 µs depends on the adopted dose threshold in the monitoring software. Although the dose threshold was temporarily raised in 2015 and 2016, it is estimated that >98% of all events with more than 5 × 10 5 collisions in 640 µs were recorded in run II. The considered 5500 events therefore represent an almost complete sample of the true number of events. Figure 8 presents the obtained distributions of inelastic collisions per event N i , peak collision rates R i , and loss durations τ . The different R i and τ histograms represent subsets of events, where a minimum number of collisions N i was exceeded. The results show that events with a higher peak collision rate R i also yield a higher number of integral collisions N i . Considering the delayed signal registration in BLMs, the actual R i values can be higher. In case of a fast event, which lasts 40 µs, R i is underestimated by a factor of 2.5 since only 40% of the dose is registered within such a time interval. For faster events, the systematic error can be more significant due to the intrinsic time resolution of BLMs. This concerns however only a fraction of events. In a majority of the considered cases, the underestimation of R i is estimated to be less than a factor of 2.5 since the events last longer than 40 µs.
The large majority of events shown in Fig. 8 did not cause any disruption of LHC operation, i.e. neither a beam-induced quench nor a BLM abort. Events, which resulted in a quench, are represented by separate histograms (dashed lines). Besides the six dipole quenches in the arcs at 6.5 TeV, the figure also includes one dustinduced quench in the energy ramp (at 6.39 TeV) and one quench in the dispersion suppressor (at 6.5 TeV). In both cases, the quenched magnet was also a bending dipole. In all events where a quench occurred, the estimated number of inelastic proton-dust particle collisions was higher than ∼ 6×10 7 . This corresponds to a fraction of ∼ 2 × 10 −7 of the maximum beam intensity in run II, or to a fraction of 5 × 10 −4 of the intensity of a single nominal bunch (1.2 × 10 11 protons). The occurrence of a quench depends not only on the number of lost protons, but also on the longitudinal loss location and the resulting energy deposition density in magnet coils. The quench level depends also on the loss duration and on the local temperature margin in the volume heated by the showers [48]. This explains why not all events with > 6 × 10 7 collisions resulted in a quench. The maximum number of collisions observed in a single event was ∼ 1.6 × 10 8 .
In case of a magnet quench, the beam abort does not shorten the loss event since the delay until a quench is detected is generally much longer [O(10 ms)] than the duration of dust events. On the other hand, BLM abort triggers can be fast enough to induce the extraction of the beams while the circulating protons still interact with the dust grain. In such cases, the number of beam-dust particle collisions could have been higher if the event would have been unperturbed. The histograms in Fig. 8 include 17 events, where the BLM abort thresholds were exceeded (14 events without quench and three events where a quench developed despite the BLM abort trigger). We estimate that in four out of these 17 cases, N i could have possibly exceeded 1.6 × 10 8 collisions if the beams would not have been extracted. In another four to six cases, we can neither exclude that the abort trigger shortened the events, but the number of collisions would have likely stayed below 1.6 × 10 8 collisions. In the remaining seven to nine cases, the time profiles of the events suggest that the losses had already ceased at the moment of beam extraction [8].
C. Constraints on the properties of dust particles
The reconstructed N i , R i and τ distributions can be used to constrain some of the properties of dust particles. A similar approach was adopted in the study of event rise times [22], where evidence of a negative pre-charge of dust particles was found. The amount of negative pre-charge, together with the composition and mass of a dust grain, are the key quantities, which govern the induced beam losses. Using the latest simulation model as described in Ref. [22], a random sample of loss events was generated for comparison with the distributions presented in the previous section. For simplicity, the dust particles were assumed to be of spherical shape. The volumes V were sampled according to a 1/V 2 distribution. As discussed in Ref. [6], this describes well the measured distribution of dust particle sizes in the magnet test hall. It was also hypothesized in [6] that such a distribution can explain the 1/D 2 BLM dose distribution although there is no unique relationship between V and D. The minimum and maximum radii adopted in the volume sampling were 5 µm and 100 µm, respectively. An analysis of dust samples showed that an abundance of particulates with r < 5 µm is present in the vacuum chamber [32], but these are estimated to be irrelevant for the comparison with the measurement sample. Such small dust grains are still expected to cause copious smaller events. In many cases, the beam losses would, however, be too small to be detected by the BLMs.
Other assumptions about the dust particle properties were similar to the ones used in Ref. [22], which are briefly summarized in the following. The dust particle composition was randomly selected from four chemical elements (C, Al, Si, Cu) since the actual composition may vary from event to event. This accounts for the observation that dust particles of different chemical composition are present in the vacuum system, which can possibly be explained by different fabrications methods and by different materials used for vacuum system components [32]. The dust samples collected in run II showed also the presence of other chemical elements [32], but the four selected materials are considered to be a representative subset of the actual dust constituents. For simplicity, no mixtures of multiple elements were considered and each element was sampled with equal probability. The dust particles were assumed to carry a negative charge when entering the beam. The charge-to-mass ratio |Q/m| was sampled from a log-uniform distribution between 10 −7 C/kg and 10 −1 C/kg, albeit different subintervals were studied as discussed below. The dust particles were assumed to be initially attached to the top surface of the beam screen. The horizontal offset of the initial position with respect to the beam was assumed to be uniformly distributed up to two millimeters from the center, which is sufficient to cover the horizontal extent of the beam. Figure 9 presents scatter plots of events sampled within the defined parameter space. The two axes show the maximum collision rate R i and integral number of collisions per event N i , respectively. Only events which generated more than 5 × 10 5 collisions within 640 µs are shown. The color coding indicates the distribution of loss durations, charge-to-mass ratios and radii. The loss duration was calculated in the same way as for the experimental data. For this purpose, the simulated collision rate profiles were discretized in time, using 80 µs intervals. The time window with the highest fraction of N i was used for the calculation of τ . The figures illustrate that for a given number of collisions N i , the peak collision rate and hence the loss duration can vary by more than one order of magnitude. The highest peak collision rate and therefore the shortest loss duration for a given N i can be attributed to smaller dust particles with a high |Q/m| ratio. The figure also shows that the simulation results include events which exceed the maximum N i and R i values observed in the experimental distribution (see horizontal and vertical lines in Fig. 9). The dust properties hence need to be further constrained. Figure 10 compares measured and simulated distributions of N i , R i and τ , for events with ≥ 5 × 10 5 collisions within 640 µs. The distribution of beam intensities was 9. Scatter plots of randomly sampled dust events based upon the simulation model described in Ref. [22]. The graphs show the number of inelastic proton-dust particle collisions per event versus the maximum collision rate. The color coding indicates the loss duration (left), the charge-to-mass ratio (center) and the radius (right). Only events with ≥ 5 × 10 5 collisions within 640 µs are shown. See text for more details about the simulation parameters. The horizontal and vertical lines indicate the maximum Ni and Ri values observed in the experimental distributions in Fig. 8. modeled to be the same as in the measurement sample to remove any bias due to the intensity dependence of the considered quantities. About two third of the events occurred at a stored beam intensity between 1 × 10 14 and 2.5 × 10 14 protons. The figure includes different simulation sub-samples from Fig. 9, with different constraints on |Q/m| and r. Each simulation sample was normalized such that the total number of events was the same as in the measurements. The first sample (simulation "A") excludes high charge-to-mass ratios (|Q/m| > 10 −2 C/kg), whereas the second sample (simulation "B") excludes lower charge-to-mass ratios (|Q/m| < 10 −3 C/kg). In both cases, the upper radius was 100 µm. In the third sample (simulation "C"), the range of charge-to-mass ratios was the same as in "B", but the mass of dust particles was restricted to less than 0.4 µg. This corresponds to a maximum radius of 35 µm for spherical carbon grains and 22 µm for spherical copper grains. As discussed in the following, sample "C" reproduces best the features of the measured distributions.
The comparison between simulated and measured R i and τ distributions suggests that |Q/m| must have been higher than 10 −2 C/kg in at least a fraction of the events, otherwise the most probable peak loss rate (loss duration) is underestimated (overestimated). The most probable R i and τ values can be best reproduced if |Q/m| ranges from 10 −3 C/kg to 10 −1 C/kg, as in the samples "B" and "C". This finding is in good agreement with the charge-to-mass ratio inferred from the rise time of loss events [22], where |Q/m| was found to be higher than 5 × 10 −3 C/kg.
The results also show that the assumed distribution of dust volumes (1/V 2 ) reproduces well the measured distribution of N i , but overestimates the actual number of higher-loss events (N i > 3 × 10 7 ) if r max = 100 µm (as in sample "A" and "B"). In particular, the simulation predicts events with N i > 1.6 × 10 8 collisions, which are absent in the measurement sample, apart from possibly four cases where the BLM abort trigger shortened the event. A similar observation can be made for the distribution of maximum collision rates, which extends to higher values in the simulations than in the measurements. The absence of larger events in the measurements was already observed in BLM signal distributions in run I [6]. This observation was solely attributed to the shortening of events due BLM abort triggers, which, according to our analysis, does not apply to the run II data. It is also unlikely that the absence of large events in the run II measurements is due to the limited sample size. Assuming that dust volumes are distributed as 1/V 2 and that r max =100 µm, then one would have expected with 95% confidence to observe ≥12 events with N i > 1.6 × 10 8 in run II. A more likely explanation is that the maximum radius of dust grains which interact with the beam is smaller than 100 µm, or that the propensity of reaching a high charge-to-mass ratio |Q/m| diminishes for larger dust grains. As shown in Fig. 10, the measured distribution can be well reproduced by sample "C", where the mass of dust particles was limited to 0.4 µg. Such dust grains can still generate a higher number of losses N i than observed in the measurements, but the likelihood of such events diminishes significantly. The remaining discrepancies between simulated and measured τ and R i distributions can be explained by the limited time resolution of the measurements and by delayed charge collection in BLMs.
The reason that dust particles heavier than 0.4 µg would not interact with the beam could be twofold. One possibility could be that more massive dust particles are less susceptible to being detached from the cold beam screen in presence of the beams. An alternative explanation could be that the assumed 1/V 2 distribution does not correctly describe the population of larger dust grains in the vacuum system. If the population of dust grains with larger radii is sufficiently smaller than a 1/V 2 distribution, then the absence of higher-loss events could be of statistical nature. The dust samples extracted from a dipole in run II showed that dust grains with masses larger than 0.4 µg are present in the vacuum system [32]. However, the analysis also indicated that the size distribution can vary depending on the location where the dust sample was taken. We can therefore not entirely exclude that the global size distribution deviates from the assumed 1/V 2 dependence. This might in particular apply to dust grains adhering to the top surface of the beam screen.
The measured distributions can also be reproduced in a similar way as in Fig. 10 by constraining the absolute dust charge |Q| instead of the mass. This assumption implies that larger dust grains cannot reach the same charge-to-mass ratio |Q/m| as smaller ones. Considering a maximum |Q| of 2 × 10 −11 C, similar curves would be obtained as if the mass m is limited to 0.4 µg. Based on these findings, it cannot be established with certainty if the population of larger dust particles is smaller than expected, if larger dust particles are not released into the beam even if they possess a high charge-to-mass ratio, or if the maximum charge dust particles can acquire is limited. The latter would reduce their ability to induce higher losses.
IV. DEPENDENCE OF PROTON LOSSES ON BEAM PARAMETERS
The correlation between dust-induced beam losses and beam parameters has been studied both experimentally (based on data from LHC run I operation) [3,49] and through simulations [20,34]. However, no direct comparison of simulations and measurements was carried out so far, mainly due to a lack of a common observable. The experimental studies primarily relied on BLM signals, which could not be compared directly with the number of inelastic collisions. In this section, we use the experimental and simulated distributions from the previous section to study the dependence of observables on the beam intensity and transverse emittance. Since the reconstructed distributions contain the full event popula-tion above a given N i threshold, an absolute comparison can be performed. As dust properties, we assume the same as in simulation "C" (see Fig. 10), i.e. |Q/m| ranging from 10 −3 C/kg to 10 −1 C/kg and m being smaller than 0.4 µg.
A. Correlation between Ni, Ri and τ and the beam intensity Figure 11 shows scatter plots of N i , R i and τ as a function of the beam intensity I. Each point represents a reconstructed loss event from run II. The measurements are the same as in Fig. 10, i.e. only events with more than 5 × 10 5 collisions within 640 µs were considered. The data points exhibit a large spread at all beam intensities. Relatively high losses and loss rates can occur already with low-intensity beams. In particular, three of the eight dust-induced quenches in run II occurred at I = 9.4 × 10 12 , 3.2×10 13 and 5.6×10 13 protons, which corresponds to 3%, 11% and 18% of the maximal intensity achieved in run II, respectively.
The solid and dashed lines represent the average and median values of N i , R i and τ for beam 1 (blue) and beam 2 (red), respectively. The intensity dependence is very similar for both beams. While the average and median peak collision rates, R i and R i , show a gradual increase as a function of I, the opposite trend can be observed for the average and median loss duration, τ and τ . The behaviour is compatible with the rather weak dependence of N i and N i on I. The decrease of τ as a function of I has already been observed in previous empirical studies, which were based on data from 3.5 TeV operation in run I [3,49]; the same trend has also been predicted by the theoretical model [34]. In the present data set, the average duration τ is found to be around 500 µs for low beam intensities and decreases to ∼200 µs for intensities higher than 1.5 × 10 14 protons.
The simulation results, represented by the yellow lines, reproduce well the absolute intensity dependence of the different observables, in particular the increase of the maximum collision rate and the decrease of the loss duration. The simulation results suggest that the average number of collisions is rather independent of I, while a very slight increase can be observed in the measurements. The absolute agreement between simulations and measurements is nevertheless satisfactory.
Previous simulation studies [18][19][20]34] suggested that R i would decline with increasing I if the gravitational force would be the sole force acting on the dust particle before entering the beam, i.e. if dust particles were not pre-charged. This decline of R i with I can be explained by the smaller degree of ionization needed to reach the point of repulsion in presence of higher-intensity beams, which then results in a reduced inelastic collision rate at the turning point of the dust particle trajectory. Observing the opposite trend in the present data provides another strong indication that dust particles are initially 11. Number of inelastic proton-nucleus collisions (top), maximum collision rate (center) and loss duration (bottom) of dust particle events as a function of the beam intensity. Every dot represents a loss event reconstructed from BLM measurements in run II. All considered events occurred at 6.5 TeV, except one event at 6.39 TeV. Only events with more than 5×10 5 collisions within 640 µs were considered. The crosses indicate events, which resulted in a magnet quench. The average and median values are indicated by the solid and dashed lines, respectively (beam 1 in blue and beam 2 in red). The yellow lines represent the simulation results (simulation "C" in Fig. 10).
Measurement (beam 1) − average (N − i , R − i , τ − ) Measurement (beam 1) − median (Ñ i , R i , τ) Measurement (beam 2) − average (N − i , R − i , τ − ) Measurement (beam 2) − median (Ñ i , R i , τ) Simulation − average (N − i , R − i , τ − ) Simulation − median (Ñ i , R i , τ) DS 6.39 TeV FIG.
attracted by the beam. The increasing attraction negatively pre-charged dust particles experience in case of higher-intensity beams outweighs the aforementioned effect, i.e. the dust particles can penetrate deeper into the beam and hence the peak collision rate increases as a function of I.
The results also suggest that the charge-to-mass ratio of dust particles interacting with the beam is similar at all beam intensities. The physical mechanism, which causes dust particles to acquire a negative charge, is beyond the scope of this study and is investigated in another paper [35]. The results nonetheless suggest that, if the charging mechanism depends on the presence of the proton beams, the amount of charge picked up by the dust particle does not or only moderately depend on the number of circulating protons.
B. Correlation between Ni, Ri and τ and the transverse beam emittance Figure 12 shows scatter plots of N i , R i and τ as a function of the normalized transverse emittance ε n . Like in the previous section, the average and median values of the measured distributions are represented by solid and dashed lines, respectively. Only events which occurred at beam intensities I ≥ 1.5 × 10 14 protons were considered due to the weaker dependence on I. This shall reduce any intensity-related bias when studying the dependence on the transverse emittance. The emittance at the time of each the dust event was approximated by ε n = ε n,i + k × ∆t, (11) where ε n,i is the averaged convoluted emittance at the start of stable proton-proton collisions for physics data taking, k is the emittance growth rate and ∆t is the time which elapsed between the start of stable collisions and the dust particle event. The initial emittance ε n,i was reconstructed considering the luminosity measurements in the ATLAS and CMS experiments and represents the convoluted emittance in the horizontal and vertical plane, averaged over all bunches. Experimental studies showed that the emittance growth in 2018 was about k = 0.07 − 0.08 (0.04 − 0.06) µm/h in the horizontal (vertical) plane [50]. In this study, we consider k = 0.05 µm/h as the average convoluted emittance growth for all years, accepting that this represents only a rough estimate of the actual emittance at the moment of dust particle events. Describing the emittance growth by a single constant value is considered sufficient for identifying trends. Since Eq. (11) relies on luminosity measurements, only dust events during stable beam collisions were included in this analysis, and only for the years 2016-2018 of run II. The yellow symbols in Fig. 12 represent the simulation results. The emittances were sampled from discrete values (ε n = 2, 3, 4, 5 µm). The corresponding beam size was calculated by using β-and dispersion functions at
Measurement (beam 1) − average (N − i , R − i , τ − ) Measurement (beam 1) − median (Ñ i , R i , τ) Measurement (beam 2) − average (N − i , R − i , τ − ) Measurement (beam 2) − median (Ñ i , R i , τ) Simulation − average (N − i , R − i , τ − ) Simulation − median (Ñ i , R i , τ)
FIG. 12. Number of inelastic proton-nucleus collisions (top), maximum collision rate (center) and loss duration (bottom) of dust particle events as a function of the normalized transverse beam emittance. The measurements are a subset of the data shown in Fig. 11. The average and median values of the measurements are indicated by the solid and dashed lines, respectively (beam 1 in blue and beam 2 in red). Only events from 2016-2018 at beam intensities I ≥ 1.5×10 14 protons were considered. Simulations (yellow symbols) were performed for discrete emittance values only (average values are given by circles, median values are given by crosses).
randomly selected dust particle positions inside a standard arc cell [18]. The dust particles were assumed to be uniformly distributed along the cell. The minimum and maximum β-functions were 30 m and 180 m, respectively. In general, the simulations and measurements are in good agreement, although some discrepancies are visible. For example, the simulation predicts a slight increase of τ and τ with ε n , while no clear trend can be identified in the measurements. The latter can at least partially be attributed to fluctuations in the experimental data resulting from the limited sample size. The results nevertheless indicate that the average and median N i , R i and τ values vary at most by a few ten percent in the considered emittance interval between 2 µm rad and 5 µm rad.
V. PROTON LOSSES AT HIGHER BEAM INTENSITIES
The proton beam intensity in the HL-LHC era will be twice as high as in run II, whereas beam emittances as small as the nominal HL-LHC emittance (2.5 µm rad) have already been achieved in run II (see Table I). A possible worsening of dust-induced losses can therefore be mainly expected due to the higher number of circulating protons. Based on the dust properties found in Sec. III, we can derive predictions about the distribution of N i , R i and τ at higher intensities. As in the previous sections, we consider it instructive to study the distribution of events exceeding a minimum number of collisions N i . The dust dynamics simulations predict that the number of events above a given N i threshold grows with increasing I. This is illustrated in Fig. 13, which shows the relative increase of events for different N i thresholds. The results were arbitrarily normalized to the number of events, which result in more than 5 × 10 5 collisions (∆t = 640 µs) at a beam intensity of I = 3 × 10 14 protons. As indicated by the blue line in Fig. III, the number of events with N i above 5×10 5 collisions increases by about 17% at HL-LHC beam intensities (∼ 6 × 10 14 protons) compared to the highest beam intensity in run II (∼3 × 10 14 protons). Figure 14 presents the distribution of N i , R i and τ for three different beam intensity intervals, up to the maximum intensity expected in HL-LHC. The stored beam intensity declines throughout physics fills due to the proton burn-off in the experiments and due to beam losses in the collimation system. Dust events may therefore exhibit different characteristics depending on the time when they occur in a fill. The intensity intervals in the figure do not represent specific operational scenarios, but provide a general comparison of event characteristics for different intensity regimes. For each interval, an equal number of events was generated. Two different N i thresholds were adopted for the R i and τ distributions, N i = 10 6 inelastic collisions (solid histograms) and N i = 10 7 inelastic collisions (dashed histograms). With increasing intensity, the distribution of peak collision rates shifts towards higher values while the τ distribution shifts towards smaller values. This behaviour could already be observed in the average and median values of the distri- butions shown in Sec. IV. Comparing the distributions for the two upper intensity intervals, the characteristics of dust particle events are not expected to get significantly worse in the HL-LHC era. The minimum energy deposition density for inducing a quench decreases for shorter heating times, but this decrease is small for τ < 10 −4 s [48,51]. The shift of the τ distribution is therefore not expected to considerably increase the risk of quenches. As shown in Sec. III, the minimum number of inelastic proton-dust collisions for inducing a quench at 6.5 TeV was 6 × 10 7 . This loss threshold will decrease in 7 TeV operation because of the reduced quench margin and the higher energy density deposited in coils. Electro-thermal model calculations suggest that the quench level of arc dipoles will decrease by about 20% to 30% for fast beam losses between 10 −4 s and 10 −3 s [51]. On the other hand, FLUKA simulations show that the average energy density per proton lost increases by about 14% because of the higher particle energy and the narrower angular distribution of secondary collision products. We therefore estimate that about 4×10 7 inelastic collisions can lead to a quench at 7 TeV. The simulation results in Fig. 14 show that, independently of the considered I interval, the number of events with N i ≥ 4×10 7 collisions is about a factor of two higher than the number of events with N i ≥ 6×10 7 collisions. This suggests that an increase of the beam energy from 6.5 TeV to 7 TeV will presumably have a larger impact on the likelihood of dust-induced quenches than the increase of the beam intensity.
Apart from the higher fraction of events which can induce a quench, the impact of dust events in future runs will strongly depend on the frequency of events. The physical mechanism, which governs the rate at which dust particles are released into the beam, still lacks a theoretical understanding. Much of what is known about the occurrence of these events derives from experimental observations. In particular, it was observed that the rate of dust events gradually decreases during operational years, while the situation could deteriorate after winter shutdowns [6]. The largest increase of the event rate occurred after the two year-long shutdown between run I and run II. The increase of the event rate after future shutdowns will therefore be decisive for the number of dust-induced quenches.
VI. CONCLUSIONS
The interaction of the LHC proton beams with dust particles was the dominant source of beam-induced magnet quenches in the second physics run of the LHC (6.5 TeV). Besides these detrimental events, thousands of harmless beam-dust particle encounters have been observed in the cryogenic arcs every year. In this paper, we studied the characteristics of dust events by reconstructing the number of inelastic nuclear collisions between the beam and dust grains. The paper demonstrated that the experimental distribution of peak collision rates and loss durations can be consistently reproduced by dust dynamics simulations if dust particles are negatively precharged and if the charge-to-mass ratio |Q/m| ranges from 10 −3 C/kg to 10 −1 C/kg. This finding is in good agreement with recent studies of the rise time of loss profiles [22].
The assumed range of charge-to-mass ratios also describes well the beam intensity-dependence of observables. In particular, the observed increase of the average peak collision rate with the number of circulating protons can only be explained if dust particles carry sufficient negative charge when entering the beam. The opposite trend would be expected if dust particles were neutral or only weakly charged (|Q/m| < 10 −5 C/kg). Within the resolution achieved in this study, we could not find any evidence that the charge acquired by dust particles depends significantly on the number of circulating protons, i.e. the experimental data could be well reproduced by assuming the same |Q/m| distribution at all beam intensities, up to the maximum intensity achieved in run II (∼ 3 × 10 14 protons).
The paper also illustrated that dust events with more than 1.6×10 8 inelastic collisions were absent in the run II measurements, although events with higher losses were theoretically not excluded by the simulation model. The absence of these events can have several reasons. It could indicate that the mass of dust particles interacting with the beam is limited ( 0.4 µg), either because more massive dust grains are less likely to detach from the cold aperture or because the population of larger dust particles is overestimated by the assumed 1/V 2 volume distribution. In the latter case, the absence of higher-loss events could be of statistical nature. An alternative hypothesis is that dust particles can only acquire a limited negative charge (|Q| 2 × 10 −11 C), which would constrain the attainable charge-to-mass ratio for larger dust grains. This in turn diminishes their ability to penetrate deeper into the beam and hence their ability to induce higher losses. A better understanding of the charging mechanism and the detachment of the dust grains from the cold beam screen is needed to conclusively explain the absence of higher-loss events. Independently of the underlaying mechanism, the absence of such loss events was one of the main reason why dust particles did not have a higher impact on the operational performance in run II.
Dust-induced loss events remain a concern for future LHC runs. This applies in particular to the HL-LHC era, when the operational energy will be raised to 7 TeV and the stored beam intensity will increase by a factor of two compared to previous years. We showed that already 6 × 10 7 inelastic proton-dust particle collisions could lead to a dipole quench at 6.5 TeV. The loss threshold for inducing a quench will decrease to 4×10 7 inelastic collisions at nominal energy (7 TeV) because of the reduced quench margin and the higher energy density induced by the particle showers in superconducting coils. Hence, smaller dust particles will have the ability to provoke a quench. The data from run II suggests that the number of detrimental events could increase by about a factor of two to three compared to 6.5 TeV, even if the total rate of events remains unchanged.
The simulations also showed that the number of dust events exceeding a certain particle loss threshold increases with beam intensity. This increase is, however, expected to be less than 20% at the HL-LHC design intensity (∼ 6 × 10 14 protons) compared to the maximum beam intensity in run II. Hence, the increase in beam energy is expected to be the most important parameter change compared to previous runs. The dust dynamics simulations also showed that the time profiles of dust events will become shorter at higher intensities, while the peak collision rate will increase.
Dust-induced quenches may also pose a challenge for the operational efficiency of future high-energy hadron colliders like the FCC-hh, which are being designed to operate with unprecedented stored beam energies. The results presented in this paper, which provide a first quantitative analysis of dust-induced beam losses in a high-intensity hadron storage ring, can serve as a basis for estimating the impact of dust events in such future colliders. The studies demonstrated that the nature of beam-dust particle interactions can be well reproduced by modeling the ionization of dust particles and the resulting repelling force exerted on the dust grain. A similar simulation approach can therefore be used to predict dust-induced beam losses at higher beam energies and intensities. Such studies can provide critical information about the tolerable dust contamination in the vacuum system of energy-frontier hadron machines.
FIG. 2 .
2Illustration of BLM positions in a standard LHC arc half-cell. The two layout plots at the bottom show a top view of the different BLM positions in run I and run II, respectively. Gray boxes represent magnets (bending dipoles: MB, quadrupoles: MQ), yellow boxes represent monitors. Monitors around the quadrupole are located on the horizontal plane (figure top right), while monitors between dipoles (run II only) are located on top of the dipole interconnects (figure top left). The blue and red arrows indicate the beam direction of beam 1 and beam 2, respectively.
Fig. 2 .
2The BLMs were installed on top of dipole-dipole interconnects, as shown in the upper figure. B. Methods for reconstructing the number of collisions
FIG. 3 .
3Geometry model of a LHC arc cell used in the particle shower simulations. The cryostat and a dipole are cut open to show the interior. The beam loss monitors (yellow cylinders) are mounted on the outside of the cryostat.
FIG. 4 .
4Comparison of simulated BLM dose patterns for beam collisions with dust particles of different composition.
FIG. 5 .
5Comparison of measured and simulated BLM dose patterns along LHC arc cells for dust-induced loss events in run II physics operation at 6.5 TeV.
FIG. 7 .
7Measured time profile of a dust event recorded in LHC operation (top) and Gaussian profile used for estimating the loss duration τ (bottom). The standard deviation of the Gaussian distribution was calculated from the relative number of collisions N 80µs i /Ni contained in the central 80 µs time window. The relative number of collisions in this interval was estimated from the measurements (ratio of red area and total area in top graph). depends on the number of collisions N 80µs i in this time interval. We assume that N 80µs i corresponds to the maximum number of collisions measured in the 80 µs sliding time window of BLMs:
(t )dt : t n = n × 40 µs . ( 10 )Figure 7
107illustrates the time profile of a typical dust particle event measured in run II (top graph) together with the derived Gaussian distribution (bottom graph). The red area in the top graph indicates the number of collisions as given by Eq.(9), which was in turn used to calculate the width τ of the Gaussian. Numerically, the results for τ would be the same if the total area of the Gaussian inFig. 7would be D tot and if the area within the central 80 µs window would be D 80µs max . For clarity, the formulas and figures were still expressed in terms of collisions rather than in terms of dose.
FIG. 8 .
8Reconstructed experimental distributions of dust events as a function of collisions per event (top), maximum collision rate (center), and loss duration (bottom). The figure considers about N = 5500 loss events, which were recorded in the LHC arcs during 6.5 TeV proton operation in run II. In addition, one dust-induced quench at 6.39 TeV and one quench in the dispersion suppressor are included (at 6.5 TeV). The two bottom figures show different subsets of events exceeding a minimum number of collisions indicated by the vertical lines in the top figure. Events leading to a quench are represented by the dashed histograms.
FIG. 9. Scatter plots of randomly sampled dust events based upon the simulation model described in Ref. [22]. The graphs show the number of inelastic proton-dust particle collisions per event versus the maximum collision rate. The color coding indicates the loss duration (left), the charge-to-mass ratio (center) and the radius (right). Only events with ≥ 5 × 10 5 collisions within 640 µs are shown. See text for more details about the simulation parameters. The horizontal and vertical lines indicate the maximum Ni and Ri values observed in the experimental distributions in Fig. 8.
FIG. 10 .
10Measured and simulated distributions of dust-induced loss events as a function of collisions per event (top), maximum collision rate (center), and loss duration (bottom). The measurements are the same as inFig. 9, including about N = 5500 events. The simulation results correspond to different intervals of |Q/m| and different maximum radii (see top of the figure).
emittance ε n (µm rad)
FIG. 13 .
13Simulation predictions of the relative number of dust events exceeding a given number of collisions Ni. The absolute number of events was assumed to be independent of the intensity I.
FIG. 14 .
14Simulated distributions of Ni, Ri and τ for different beam intensity intervals; N indicates the number of events. The different Ri and τ distributions correspond to different minimum Ni thresholds as indicated in the figures (Ni ≥ 10 6 : solid histograms, Ni ≥ 10 7 : dashed histograms).
and arcs (gray). Each of the eight arcs consists of 23 cells. A cell is made up of two 53.45 m long half-cells (see top), which are composed of three bending dipoles (MBs), one quadrupole (MQ) and corrector magnets (not shown). The two counter-rotating beams are represented by the blue line (beam 1) and the red line (beam 2), respectively. The beams are crossing each other in four interaction points.0
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FIG. 1. Illustration of the LHC layout, showing the different
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ACKNOWLEDGEMENTSWe would like to thank the LHC operations team, in particular M. Albert, and the LHC Beam Instrumentation group, in particular C. Zamantzas, for their support concerning the monitoring software.
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T Baer, M J Barnes, E Carlier, F Cerutti, B Dehning, L Ducimetière, A Ferrari, N Garrel, A Gérardin, B Goddard, E B Holzer, S Jackson, J M Jimenez, V Kain, A Lechner, V Mertens, M Misiowiec, R Ballester, E Del Busto, L Drosdal, A Nordt, J Uythoven, B Velghe, V Vlachoudis, J Wenninger, C Zamantzas, F Zimmermann, N. Fuster Martinez, 10.5170/CERN-2012-006.294UFOs in the LHC after LS1, in Proceedings of the Chamonix 2012 Workshop on LHC Performance. T. Baer, M. J. Barnes, E. Carlier, F. Cerutti, B. Dehn- ing, L. Ducimetière, A. Ferrari, N. Garrel, A. Gérardin, B. Goddard, E. B. Holzer, S. Jackson, J. M. Jimenez, V. Kain, A. Lechner, V. Mertens, M. Misiowiec, R. Morón Ballester, E. Nebot del Busto, L. Norder- haug Drosdal, A. Nordt, J. Uythoven, B. Velghe, V. Vla- choudis, J. Wenninger, C. Zamantzas, F. Zimmermann, and N. Fuster Martinez, UFOs in the LHC after LS1, in Proceedings of the Chamonix 2012 Workshop on LHC Performance (2012) pp. 294-298.
Testing beam-induced quench levels of lhc superconducting magnets. B Auchmann, T Baer, M Bednarek, G Bellodi, C Bracco, R Bruce, F Cerutti, V Chetvertkova, B Dehning, P P Granieri, W Hofle, E B Holzer, A Lechner, E Del Busto, A Priebe, S Redaelli, B Salvachua, M Sapinski, R Schmidt, N Shetty, E Skordis, M Solfaroli, J Steckert, D Valuch, A Verweij, J Wenninger, D Wollmann, M Zerlauth, 10.1103/PhysRevSTAB.18.061002Phys. Rev. ST Accel. Beams. 1861002B. Auchmann, T. Baer, M. Bednarek, G. Bellodi, C. Bracco, R. Bruce, F. Cerutti, V. Chetvertkova, B. Dehning, P. P. Granieri, W. Hofle, E. B. Holzer, A. Lechner, E. Nebot Del Busto, A. Priebe, S. Redaelli, B. Salvachua, M. Sapinski, R. Schmidt, N. Shetty, E. Sko- rdis, M. Solfaroli, J. Steckert, D. Valuch, A. Verweij, J. Wenninger, D. Wollmann, and M. Zerlauth, Testing beam-induced quench levels of lhc superconducting mag- nets, Phys. Rev. ST Accel. Beams 18, 061002 (2015).
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[]
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"Julio Andrade "
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| Let F be a finite field of odd cardinality q, A = F [T ] the polynomial ring over F , k = F (T ) the rational function field over F and H the set of square-free monic polynomials in A of degree odd. If D ∈ H, we denote by O D the integral closure of A in k( √ D). In this note we give a simple proof for the average value of the size of the groups K 2 (O D ) as D varies over the ensemble H and q is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over finite fields. | null | [
"https://arxiv.org/pdf/1504.06205v1.pdf"
]
| 119,614,706 | 1504.06205 | 2ea09662f7146be1ef0b581798e3e540089e68f8 |
23 Apr 2015
Julio Andrade 23 Apr 2015arXiv:1504.06205v1 [math.NT] A SIMPLE PROOF OF THE MEAN VALUE OF |K 2 (O)| IN FUNCTION FIELDS UNE DÉMONSTRATION SIMPLE DE LA VALEUR MOYENNE DE |K 2 (O)| EN CORPS DE FONCTIONS
Let F be a finite field of odd cardinality q, A = F [T ] the polynomial ring over F , k = F (T ) the rational function field over F and H the set of square-free monic polynomials in A of degree odd. If D ∈ H, we denote by O D the integral closure of A in k( √ D). In this note we give a simple proof for the average value of the size of the groups K 2 (O D ) as D varies over the ensemble H and q is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over finite fields.
Introduction
In [10], the author established average value results for the size of the algebraic K groups K 2 (O) over function fields. His proof crucially depends on the mean value of quadratic Dirichlet L-functions over function fields which in turn was first obtained with the help of functions defined on the metaplectic two-fold cover of GL(2, k ∞ ), where k ∞ is the completion of k = F q (T ) at the prime at infinity.
In this paper we provide a simple proof for the mean value of the size of the groups K 2 (O) over the rational function field F q (T ). Our proof is simpler in the sense that we avoid the Eisenstein series construction involved in the proof given by Hoffstein and Rosen [6], and we do this by computing the mean value of the required quadratic Dirichlet L-function in F q (T ) through character sum estimates.
We start by setting the notation. Let F = F q be a finite field with q elements (q odd), A = F q [T ], and k = F q (T ). For f ∈ A we define |f | = q deg(f ) if f = 0 and |0| = 0. If D ∈ A is square-free then O D is the integral closure of A in the quadratic function field K D = k( √ D). The zeta function associated to A is defined by Making use of (1.2), together with the results of Quillen [9] and Tate [12], Rosen [10] was able to relate the number L(2, χ D ) to the size of the group K 2 (O D ). We will use such relationship to prove our main result.
(1.1) ζ A (s) = f ∈A monic |f | −s = P ∈A monic irreducible (1 − |P | −s ) −1 . A straightforward calculation shows that ζ A (s) = (1 − q 1−s ) −1 . If D ∈ A is square-free we set χ D (f ) = (D/f ) where (D/f )
2. The algebraic K groups K 2 (O D ) and a theorem of Rosen Let D ∈ A be a monic and square-free polynomial. For ease of discussion we only consider the case where degree of D is odd since the case with degree of D even is similar and there are no important differences. Let F = F q and K/F be a function field in one variable with a finite constant field F q . The primes in K are denoted by v, and O v is the valuation ring at v. We denote by P v the maximal ideal of O v and by F v the residue class field at v. The tame symbol ( * , * ) v is a mapping from
K * × K * to F * v defined by (2.1) (a, b) v = (−1) v(a)v(b) a v(b) /b v(a) modulo P v .
Let a ∈ K * such that a = 0, 1 so the group K 2 (K) is defined to be K * ⊗ K * modulo the subgroup generated by the elements a⊗(1−a). Moore (see [12] for more details) proved that the following sequence is exact
(2.2) (0) −→ ker(λ) −→ K 2 (K) λ −→ v F * v µ −→ F * −→ (0), where λ : K 2 (K) → v F * v is the sum of the tame symbol maps, and µ : v F * v → F * is the map given by µ(. . . , a v , . . .) = v a mv/m v where m v = N P v − 1 and m = |F * | = q − 1.
By making use of the above discussion with Tate's proof [12] of the Birch-Tate conjecture concerning the size of ker(λ), i.e.,
(2.3) | ker(λ)| = (q − 1)(q 2 − 1)ζ K (−1), where ζ K (s) = v (1 − N P −s v ) −1 ,
the product being over all the primes v of the function field K, Rosen [10, Proposition 2] established that
(2.4) #K 2 (O D ) = q (3/2)deg(D) q −3/2 L(2, χ D ).
With this in hand Rosen [10, Proposition 2(a)] proves the following Theorem 2.1 (Rosen). Let m be a square-free polynomial of degree M , with M odd, and ε > 0 given. Then
(2.5) (q − 1) −1 (q M − q M−1 ) −1 m∈A m square−free |K 2 (O m )| = ζ A (2)ζ A (4)c(2)q −3/2 q 3(M/2) + O(q M(1+ε) ), where (2.6) c(2) = P 1 − |P | −2 − |P | −5 + |P | −6 ,
the product is taken over all monic irreducible polynomials in A.
The Main Result
Without further postponements we present below the main result of this note.
Theorem 3.1. Let H = {D ∈ A monic, square-free and deg(D) = 2g + 1} and ε > 0. Then
(3.1) 1 #H D∈H #K 2 (O D ) = q 3 2 (2g+1) q −3/2 ζ A (4)P (4) + O(q (2g+1)(1+ε) ), where (3.2) P (s) = P ∈A monic irreducible 1 − 1 (|P | + 1)|P | s .
A theorem similar to this was previously studied by Rosen. There are essentially two differences between Rosen's result (Theorem 2.1) and the main result in this paper. First is that the average value of |K 2 (O D )|, as presented by Rosen, is an average taken over all square-free D and in our result we only consider the monic and square-free D, i.e., positive and fundamental discriminants over function fields. Comparing equations (2.5) and (3.1) we observe that the constants multiplying the main term are close but not equal. And this is due to the fact that in our result we are summing over monic and square-free while in Rosen's result he is summing over all square-free, and in this sense Rosen's result is more general. This phenomenon is not new and it has appeared when you compare the main theorem of [2] with [6,Theorem 5.2] and it also appears in number fields when you compare the first moment of quadratic Dirichlet L-functions at the central point as given by [8, Theorem 1] and [5, Theorem (1)], in both comparisons we see again that the constants multiplying the leading term in the mean values are different. The second, and most important difference, is the argument used to prove such result. In Rosen, he needs to invoke a mean value of L(s, χ D ) that was previously proved by himself and Hoffstein [6] through the use of the theory of Eisenstein series and the metaplectic two-fold cover of GL(2, k ∞ ), whereas our method is solely based on estimating characters sums and in the use of the Riemann hypothesis for curves over finite fields.
Preparatory Results
In this section we present a few auxiliary results that will be used in the proof of the main theorem of this note. Lemma 4.1. Let f ∈ A be a fixed monic polynomial. Then for all ε > 0 we have that
(4.1) D∈H gcd(D,f )=1 1 = |D| ζ A (2) P monic irrducible P |f |P | |P | + 1 + O |D| 1 2 |f | ε . Proof. See [2, Proposition 5.2].
We also need the following lemma.
Lemma 4.2. We have
(4.2) f monic deg(f )=n P |f (1 + |P | −1 ) −1 = q n d monic deg(d)≤n µ(d) |d| P |d (|P | + 1) −1 .
Proof. See [2,Lemma 5.7].
The last result that we need before to proceed to the proof of our main theorem is given below and it has appeared in a different form in [2,3,4] and its proof, as appears here, was first given in [1].
Lemma 4.3. If f ∈ A is not a perfect square then (4.3) D∈H f = D f ≪ |D| 1/2 |f | 1/4 . Proof. we write D∈H D f = 2α+β=2g+1 deg(B)=β deg(A)=α µ(A) A 2 B f = 0≤α≤g deg(A)=α µ(A) A 2 f deg(B)=2g+1−2α B f ≤ 0≤α≤g deg(A)=α deg(B)=2g+1−2α B f . (4.4) If f = then deg(B)=2g+1−2α B f
is a character sum to a non-principal character modulo f . So using [7, Proposition 2.1] (which is the Pólya-Vinogradov inequality for F q [T ]) we have that
(4.5) deg(B)=2g+1−2α B f ≪ |f | 1/2 .
Further we can estimate trivially the non-principal character sum by
(4.6) deg(B)=2g+1−2α B f ≪ |D| |A| 2 = q 2g+1−2α .
Thus, if f = , we obtain that
D∈H D f ≪ 0≤α≤g deg(A)=α min |f | 1/2 , |D| |A| 2 ≪ |D| 1 2 |f | 1 4 , (4.7)
upon using the first bound (4.5) for α ≤ g − deg(f ) 4 and the second bound (4.6) for larger α. And this concludes the proof of the lemma.
Proof of the Main Theorem
From now on we are assuming that all the sums are being taken over monic polynomials and the products are over monic and irreducible polynomials P in F q [T ].
By [11,Proposition 4.3] we have
D∈H L(2, χ D ) = D∈H deg(f )≤2g χ D (f )|f | −2 = D∈H deg(f )≤2g f = χ D (f )|f | −2 + D∈H deg(f )≤2g f = χ D (f )|f | −2 . (5.1)
For the sum above, where f is not a square of a polynomial, we use Lemma 4.3, which depends on the Riemann hypothesis for curves over finite fields, to obtain that
(5.2) D∈H deg(f )≤2g f = χ D (f )|f | −2 ≪ q g .
For the sum with f a square of a polynomial in (5.1) we need some extra manipulations. First we use Lemma 4.1 so we can write
D∈H deg(f )≤2g f = χ D (f )|f | −2 = |D| ζ A (2) g m=0 1 q 4m deg(l)=m P |l |P | |P | + 1 + O |D| 1/2 2g n=0 q nǫ−n . (5.3) From Lemma 4.2 we have D∈H deg(f )≤2g f = χ D (f )|f | −2 = |D| ζ A (2) deg(d)≤g µ(d) |d| P |d 1 |P | + 1 deg(d)≤m≤g q −3m + O q −g (q 2gǫ+ǫ − q 2g+1 ) q ǫ − q . (5.4)
After some arithmetic manipulations and summing the geometric series we can rewrite (5.4) as
D∈H deg(f )≤2g f = χ D (f )|f | −2 = ζ A (4) |D| ζ A (2) d monic − deg(d)>g µ(d) |d| 4 P |d 1 |P | + 1 − q −3g q 3 − 1 |D| ζ A (2) d monic − deg(d)>g µ(d) |d| P |d 1 |P | + 1 + O q −g (q 2gǫ+ǫ − q 2g+1 ) q ǫ − q . (5.5)
The sums over deg(d) > g in (5.5)
1 |d| 2 = n>g q −n ≪ q −g ,(5.7)
and therefore does not contribute to the main term.
By expressing the sums over all monic d in (5.5) as Euler products we derive that (5.8)
D∈H deg(f )≤2g f = χ D (f )|f | −2 = ζ A (4) |D| ζ A (2) P (4) + O q −g (q 2gǫ+ǫ − q 2g+1 ) q ǫ − q ,
where P (s) is given as in the statement of Theorem 3.1. Combining (5.2) and (5.8) we get that (5.9) D∈H L(2, χ D ) = |D| ζ A (2) ζ A (4)P (4) + O(q g ) + O q g + q g(2ǫ−1)+ǫ q ǫ − q .
We invoke [11,Proposition 2.3], which shows that #H = |D|/ζ A (2), together with equation (2.4) and a few arithmetic maneuvers to complete the proof of the main theorem in this letter.
is the Kronecker symbol in A, and we form the quadratic Dirichlet L-function L(s, χ D ) = f χ D (f )|f | −s . Lastly, the zeta function of the ring O D is defined by ζ OD (s) = a N a −s where a runs through the nonzero ideals of O D = A[ √ D] and N a denotes the norm of a, i.e., the number of elements in O D /a. Similar to number fields [11, Proposition 17.7], one has the relation (1.2) ζ OD (s) = ζ A (s)L(s, χ D ).
are respectively bounded by O(q −4g ) and O(q −g ) as can be seen from below, P |d(|P | + 1) −1 ≪ deg(d)>gdeg(d)>g
µ(d)
|d| 4
1
|d| 4
P |d
1
|P |
≪
deg(d)>g
1
|d| 5
=
n>g
q −4n ≪ q −4g
(5.6)
and
deg(d)>g
µ(d)
|d|
P |d
(|P | + 1) −1 ≪
deg(d)>g
1
|d|
P |d
1
|P |
≪
deg(d)>g
Acknowledgment. This research was supported by EPSRC grant EP/K021132X/1. The author is thankful to the comments of an anonymous referee which helped to give more clarity to the presentation of this note. The author also wishes to thank Professor Alain Connes for the several discussions related to the problem treated in this paper.
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Conjectures for the Integral Moments and Ratios of L-functions over function fields. J C Andrade, J P Keating, J. Number Theory. 142J.C. Andrade and J.P. Keating, Conjectures for the Integral Moments and Ratios of L-functions over function fields, J. Number Theory, 142, 102-148 (2014).
Statistics of the zeros of zeta functions in families of hyperelliptic curves over a nite field. D Faifman, Z Rudnick, Compos. Math. 146D. Faifman and Z. Rudnick, Statistics of the zeros of zeta functions in families of hyperelliptic curves over a nite field, Compos. Math., 146, 81-101 (2010).
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| []
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[
"Protection of the surface states in topological insulators: Berry phase perspective",
"Protection of the surface states in topological insulators: Berry phase perspective"
]
| [
"Ken-Ichiro Imura \nDepartment of Quantum Matter\nHiroshima University\nAdSM, Higashi-Hiroshima739-8530Japan\n",
"Yositake Takane \nDepartment of Quantum Matter\nHiroshima University\nAdSM, Higashi-Hiroshima739-8530Japan\n"
]
| [
"Department of Quantum Matter\nHiroshima University\nAdSM, Higashi-Hiroshima739-8530Japan",
"Department of Quantum Matter\nHiroshima University\nAdSM, Higashi-Hiroshima739-8530Japan"
]
| []
| The metallic surface state of a topological insulator (TI) is not only topologically protected, but exhibits a remarkable property of inducing an effective vector potential on curved surfaces. For an electron in the surface state of a spherical or a cylindrical TI (TI nanoparticle or nanowire) a pseudo-magnetic monopole or a fictitious solenoid is effectively induced, encoding the geometry of the system. Here, by taking an example of a hyperbolic surface we demonstrate that as a consequence of this property stemming from its active spin degree of freedom, the surface state is by itself topologically protected. Neither being a metal nor an insulator, the topological insulator has now been recognized as a basic form of solid that exhibits both gapped bulk and gapless surface states[1][2][3]. Such a classification is well-defined in the continuum limit, while the situation is less trivial in the case of lattice models often employed as a concrete implementation of topological insulators, there remaining a question, "where actually is the surface?" A lattice model is sparse, and in a somewhat extreme point of view, existing only on sites and links, so that its surface is not restricted to the macroscopic boundary of the system, but could be also chosen e.g., such that it is partly extended to a rectangular-prism-shaped region (RPSR) penetrating into the bulk as depictedFig. 1. Or one can also think of an atomic scale closed surface isolated in the bulk[4]. However, in reality the protected surface state appears only on its macroscopic surface, exhibiting no symptom of penetrating into the bulk even in the case of sparse lattice systems.Why is the surface state thus noninvasive into the bulk? What prevents it from penetrating into the sparsely filled interior of the lattice models? In this Communication we demonstrate that the existence of a Berry phase π, or a spin connection associated with what is often called spin-to-surface locking [5-10], plays a central role in this issue. Though existence of a protected surface state is a defining property of the topological insulator, topological protection does not exclude the possibility of finite-size gap opening. As we have demonstrated previously[10,11], Dirac electrons on the surface of a topological insulator encodes information on the geometry of the sample in the form of spin connection that appears in the effective surface Dirac Hamiltonian. On a cylindrical surface a fictitious solenoid threading the cylinder is effectively induced [10], while in the case of a spherical system, an effective magnetic monopole[11,12]is induced, determining the gapped electronic spectrum on the surface.A Dirac electron on the surface of a topological insulator, especially, its spin state is susceptible of two types of constraints, and "locked" both in the momentum and FIG. 1: (Color online) Which is the genuine surface?real spaces. Spin-to-momentum locking is a direct consequence of the strong spin-orbit coupling in this system. Here, we focus on another phenomenon that manifests on a curved surface, often represented by the term, "spin-tosurface locking". Through the bulk-boundary correspondence, the entangled nature of the spin and the configuration spaces encoded in the bulk Hamiltonian is transcribed to the surface Dirac equation. The helical surface state thus inherits a geometrical constraint imposed on its spin state, and an electron in this state is susceptible of a specific type of Berry phase, or the spin connection, inducing an effective monopole or a flux tube. In the somewhat special case of cylindrical geometry, the constraint on spin manifests as spin-to-surface locking, i.e., the spin of the surface state is constrained onto the tangential plane of the curved surface[5][6][7][8][9][10]. Appearance of the spin connection in the surface Dirac equation is more universal, unrestricted to the case of specific geometry.There is an inverse effect in this specific property of the surface state. Along a flux tube of strength π (half unit flux quantum) piercing a TI sample a pair of gapless helical modes bound to the tube is induced. These 1D helical channels are shown to be perfectly conducting[13], and topologically protected as well[9,14,15]. In the presence of a surface at which the flux tube is terminated, how are these 1D channels connected to the 2D helical surface states? InFig. 2we demonstrate that the noninvasive surface state becomes gradually invasive into the bulk with the aid of the flux tube. When the | 10.1103/physrevb.87.205409 | [
"https://arxiv.org/pdf/1211.2088v2.pdf"
]
| 119,295,288 | 1211.2088 | 6fe6c4055e75333470000752bce23ef9d1802ded |
Protection of the surface states in topological insulators: Berry phase perspective
28 Feb 2013
Ken-Ichiro Imura
Department of Quantum Matter
Hiroshima University
AdSM, Higashi-Hiroshima739-8530Japan
Yositake Takane
Department of Quantum Matter
Hiroshima University
AdSM, Higashi-Hiroshima739-8530Japan
Protection of the surface states in topological insulators: Berry phase perspective
28 Feb 2013(Dated: May 5, 2014)
The metallic surface state of a topological insulator (TI) is not only topologically protected, but exhibits a remarkable property of inducing an effective vector potential on curved surfaces. For an electron in the surface state of a spherical or a cylindrical TI (TI nanoparticle or nanowire) a pseudo-magnetic monopole or a fictitious solenoid is effectively induced, encoding the geometry of the system. Here, by taking an example of a hyperbolic surface we demonstrate that as a consequence of this property stemming from its active spin degree of freedom, the surface state is by itself topologically protected. Neither being a metal nor an insulator, the topological insulator has now been recognized as a basic form of solid that exhibits both gapped bulk and gapless surface states[1][2][3]. Such a classification is well-defined in the continuum limit, while the situation is less trivial in the case of lattice models often employed as a concrete implementation of topological insulators, there remaining a question, "where actually is the surface?" A lattice model is sparse, and in a somewhat extreme point of view, existing only on sites and links, so that its surface is not restricted to the macroscopic boundary of the system, but could be also chosen e.g., such that it is partly extended to a rectangular-prism-shaped region (RPSR) penetrating into the bulk as depictedFig. 1. Or one can also think of an atomic scale closed surface isolated in the bulk[4]. However, in reality the protected surface state appears only on its macroscopic surface, exhibiting no symptom of penetrating into the bulk even in the case of sparse lattice systems.Why is the surface state thus noninvasive into the bulk? What prevents it from penetrating into the sparsely filled interior of the lattice models? In this Communication we demonstrate that the existence of a Berry phase π, or a spin connection associated with what is often called spin-to-surface locking [5-10], plays a central role in this issue. Though existence of a protected surface state is a defining property of the topological insulator, topological protection does not exclude the possibility of finite-size gap opening. As we have demonstrated previously[10,11], Dirac electrons on the surface of a topological insulator encodes information on the geometry of the sample in the form of spin connection that appears in the effective surface Dirac Hamiltonian. On a cylindrical surface a fictitious solenoid threading the cylinder is effectively induced [10], while in the case of a spherical system, an effective magnetic monopole[11,12]is induced, determining the gapped electronic spectrum on the surface.A Dirac electron on the surface of a topological insulator, especially, its spin state is susceptible of two types of constraints, and "locked" both in the momentum and FIG. 1: (Color online) Which is the genuine surface?real spaces. Spin-to-momentum locking is a direct consequence of the strong spin-orbit coupling in this system. Here, we focus on another phenomenon that manifests on a curved surface, often represented by the term, "spin-tosurface locking". Through the bulk-boundary correspondence, the entangled nature of the spin and the configuration spaces encoded in the bulk Hamiltonian is transcribed to the surface Dirac equation. The helical surface state thus inherits a geometrical constraint imposed on its spin state, and an electron in this state is susceptible of a specific type of Berry phase, or the spin connection, inducing an effective monopole or a flux tube. In the somewhat special case of cylindrical geometry, the constraint on spin manifests as spin-to-surface locking, i.e., the spin of the surface state is constrained onto the tangential plane of the curved surface[5][6][7][8][9][10]. Appearance of the spin connection in the surface Dirac equation is more universal, unrestricted to the case of specific geometry.There is an inverse effect in this specific property of the surface state. Along a flux tube of strength π (half unit flux quantum) piercing a TI sample a pair of gapless helical modes bound to the tube is induced. These 1D helical channels are shown to be perfectly conducting[13], and topologically protected as well[9,14,15]. In the presence of a surface at which the flux tube is terminated, how are these 1D channels connected to the 2D helical surface states? InFig. 2we demonstrate that the noninvasive surface state becomes gradually invasive into the bulk with the aid of the flux tube. When the
The metallic surface state of a topological insulator (TI) is not only topologically protected, but exhibits a remarkable property of inducing an effective vector potential on curved surfaces. For an electron in the surface state of a spherical or a cylindrical TI (TI nanoparticle or nanowire) a pseudo-magnetic monopole or a fictitious solenoid is effectively induced, encoding the geometry of the system. Here, by taking an example of a hyperbolic surface we demonstrate that as a consequence of this property stemming from its active spin degree of freedom, the surface state is by itself topologically protected. Neither being a metal nor an insulator, the topological insulator has now been recognized as a basic form of solid that exhibits both gapped bulk and gapless surface states [1][2][3]. Such a classification is well-defined in the continuum limit, while the situation is less trivial in the case of lattice models often employed as a concrete implementation of topological insulators, there remaining a question, "where actually is the surface?" A lattice model is sparse, and in a somewhat extreme point of view, existing only on sites and links, so that its surface is not restricted to the macroscopic boundary of the system, but could be also chosen e.g., such that it is partly extended to a rectangular-prism-shaped region (RPSR) penetrating into the bulk as depicted Fig. 1. Or one can also think of an atomic scale closed surface isolated in the bulk [4]. However, in reality the protected surface state appears only on its macroscopic surface, exhibiting no symptom of penetrating into the bulk even in the case of sparse lattice systems.
Why is the surface state thus noninvasive into the bulk? What prevents it from penetrating into the sparsely filled interior of the lattice models? In this Communication we demonstrate that the existence of a Berry phase π, or a spin connection associated with what is often called spin-to-surface locking [5][6][7][8][9][10], plays a central role in this issue. Though existence of a protected surface state is a defining property of the topological insulator, topological protection does not exclude the possibility of finite-size gap opening. As we have demonstrated previously [10,11], Dirac electrons on the surface of a topological insulator encodes information on the geometry of the sample in the form of spin connection that appears in the effective surface Dirac Hamiltonian. On a cylindrical surface a fictitious solenoid threading the cylinder is effectively induced [10], while in the case of a spherical system, an effective magnetic monopole [11,12] is induced, determining the gapped electronic spectrum on the surface.
A Dirac electron on the surface of a topological insulator, especially, its spin state is susceptible of two types of constraints, and "locked" both in the momentum and real spaces. Spin-to-momentum locking is a direct consequence of the strong spin-orbit coupling in this system. Here, we focus on another phenomenon that manifests on a curved surface, often represented by the term, "spin-tosurface locking". Through the bulk-boundary correspondence, the entangled nature of the spin and the configuration spaces encoded in the bulk Hamiltonian is transcribed to the surface Dirac equation. The helical surface state thus inherits a geometrical constraint imposed on its spin state, and an electron in this state is susceptible of a specific type of Berry phase, or the spin connection, inducing an effective monopole or a flux tube. In the somewhat special case of cylindrical geometry, the constraint on spin manifests as spin-to-surface locking, i.e., the spin of the surface state is constrained onto the tangential plane of the curved surface [5][6][7][8][9][10]. Appearance of the spin connection in the surface Dirac equation is more universal, unrestricted to the case of specific geometry.
There is an inverse effect in this specific property of the surface state. Along a flux tube of strength π (half unit flux quantum) piercing a TI sample a pair of gapless helical modes bound to the tube is induced. These 1D helical channels are shown to be perfectly conducting [13], and topologically protected as well [9,14,15]. In the presence of a surface at which the flux tube is terminated, how are these 1D channels connected to the 2D helical surface states? In Fig. 2 we demonstrate that the noninvasive surface state becomes gradually invasive into the bulk with the aid of the flux tube. When the total amount of the flux is not precisely π, penetration of the surface state into the bulk is exponentially suppressed. When the flux is exactly π, the surface state can penetrate into the bulk as deeply as the system's configuration allows it. In a sense the π-flux drags the surface state into the bulk, making it invasive.
This Communication is intended to reveal the nature of this noninvasive metallic state that appears on topological insulator surfaces by demonstrating that the surface state is by itself topologically protected. This provides with a scenario alternative to the standard bulk-boundary correspondence picture that attributes the same protection to (the non-trivial value of) a bulk topological invariant. We start by simulating the behavior of the surface wave function along a flux tube. Then, as a complementary to this, we analytically establish the correspondence between the bulk and the surface descriptions. This is achieved in the second half of the paper, by employing a configuration in which the surface state can partly penetrate into the bulk. To ease analytic treatments the surface is designed to shape a smooth hyperbolic form, which may look like a "drain" [see Fig. 3, panel (a)]. Mathematically, this is the locus of a hyperbola depicted in Fig. 3 (b) when it revolves around the z-axis. In the limit of sharply edged hole (R → 0) this reproduces the situation described by the tight-binding model employed in the first part for numerical simulations.
Let us briefly describe the model employed in the tightbinding simulation. The model is based on the following 3D Wilson-Dirac type effective Hamiltonian in the bulk [16,17],
H bulk = m(p)τ z + A(p x σ x + p y σ y + p z σ z )τ x , (1)
where m(p) = m 0 + m 2 p 2 are Einstein and Newtonian mass terms encoding a band inversion due to strong spin-orbit coupling. Note that two types of Pauli matrices σ and τ represent physically real and orbital spins. It is then implemented on a cubic lattice with nearest-neighbor hopping terms. Periodic boundary conditions are applied in the x-and y-directions (no surfaces on the corresponding sides), while the model is restricted in the z-direction to 0 ≤ z ≤ N z − 1. We consider a system of N x × N y × N z and introduce a pair of flux tubes piercing RPSRs respectively, in the z and −z-directions at (x, y) = Nx 2 − 1 2 ,
Ny 4 − 1 2 and at (x, y) = Nx 2 − 1 2 , 3Ny 4 − 1 2 .
The actual simulation is done in a system of size, (N x , N y , N z ) = (10,20,20), in which a moderate strength of potential disorder is also included [18]. Both the 2D surface and 1D helical modes are shown to be robust against disorder.
Depicted in Fig. 2 is the evolution of the profile of the lowest energy surface wave functions when a magnetic flux of different strength Φ is introduced. As the flux approaches π, the surface state tends to penetrate into the bulk along a RPSR (compare different panels of Fig. 2 in which only a half of the system is shown). When the flux is null, the RPSR is empty. Yet, one can still hypothesize an electronic motion bound to it. But then, its energy levitates because of the spin Berry phase π; recall half-odd integral quantization of the orbital angular momentum. Here, since the circumference of the RPSR is atomically small (= 4a 0 with a 0 being the lattice constant), the corresponding energy scale of finite-size quantization is huge. Clearly, he is no longer compatible with the gapless (zero-energy) surface state. The gapless surface state, in turn, does not penetrate into the bulk along the RPSR. As the flux is introduced, this Berry phase π is either partly or completely cancelled depending on the amount inserted. Then, at least a small portion of 1D state along the flux tube starts to merge with the gapless surface state. From the viewpoint of the surface state, a portion of the wave function is dragged into the RPSR (the wave function gets also accumulated around the RPSR). This effect should be compared with the asymptotic behavior of the analytic formula.
We have seen so far through numerical simulations how the surface state loses its noninvasive character when a flux tube is inserted piercing plaquettes of the bulk crystalline structure. We have seen that when the strength of the flux is precisely π, it becomes completely invasive. These imply, in turn, that the noninvasiveness of the surface state stems from the Berry phase π, which is in a sense omnipresent. Penetration of the surface state into any hypothetical RPSR of the lattice is banned by the existence of this Berry phase π.
To reinforce the above argument we formulate this analytically in the remainder of the paper by solving a corresponding electronic state on the hyperbolic surface as depicted in Fig. 3. To find the surface Dirac equation on this curved surface it is convenient to introduce a set of curvilinear coordinates (ξ, θ, φ) [19], defined in terms of the hyperbolic surface:
x 2 0 + y 2 0 − a z 0 = R 2 ; its cross section in the xz-plane is shown in Fig. 3. The original cartesian coordinates are expressed as x = r cos φ, y = r sin φ, z = ξ cos θ + R √ tan θ, where is an auxiliary parameter dependent on ξ and θ. The derivatives are represented by
r = r(ξ, θ) = ξ sin θ + a + R √ cot θ(2)∇ = e ξ ∂ ξ − 1 η(θ) − ξ e θ ∂ θ + 1 r(ξ, θ) e φ ∂ φ ,(3)
where the unit vectors e ξ , e θ , e φ are those of the standard 3D polar (spherical) coordinates [21]. η(θ) represents geometrically the radius of curvature of the hyperbolic curve at r 0 = (x 0 , y 0 , z 0 ):
η(θ) = |∂ θ r 0 | 2 = R 2 1 √ sin 3 θ cos 3 θ .(4)
The subsequent analyses are based on the complex amplitudes of the surface wave function at the point (ξ, θ, φ), which is vanishingly small when ξ significantly exceeds the penetration depth. If this is much smaller than R, only the range of ξ ≪ R is physically relevant. In this regime we focus on hereafter apparent singularities in the expressions of Eq. (3) cause no mathematical difficulty. With the aid of these new coordinates we deduce the surface Dirac Hamiltonian on the hyperbolic surface from the bulk effective theory. In the standard procedure [10,11] this is done by restricting the space of state vectors |ψ associated with the bulk Hamiltonian H bulk to a set of surface states, i.e., those states that are localized in the vicinity of the hyperbolic surface. Any surface solution |ψ of H bulk can be written as a linear combination of two basis solutions,
|± = 1 √ c(θ) e −κ1ξ − e −κ2ξ |± ,
i.e., |ψ = ψ + |+ + ψ − |− , where ψ ± are (scalar) functions of θ and φ. With an appropriate choice of κ 1,2 and |± , |± can be made indeed (two degenerate) zeroenergy eigenstates of H bulk at the "Dirac point". The ξ-dependence of the wave function is determined such that it vanishes on the hyperbolic surface. The spinor part of the wave function |± can be chosen as
|+ = 1 √ 2 cos(θ/2) e iφ sin(θ/2) ⊗ 1 i , |− = 1 √ 2 sin(θ/2) −e iφ cos(θ/2) ⊗ 1 −i .(5)
Notice that here we have chosen this single-valued [in contrast to the standard SU(2) spinor] with respect to φ → φ + 2π. Though this is a confusing point of this formulation, whether the basis is double or single valued is simply a matter of choice [10]. The θ-dependent normalization constant c(θ) in |± is defined as
c(θ) = ∞ 0 dξ r(ξ, θ)(η(θ) − ξ) e −κ1ξ − e −κ2ξ 2 ,(6)
in which r(η − ξ) is a measure of the integral associated with the volume integral element r(η − ξ)dξdθdφ.
The same measure appears also in the evaluation of the matrix elements, ±|H bulk |± (see below). The surface Dirac Hamiltonian H surf is obtained in the spirit of k · p-theory [10,11,19]. Or, in the language of degenerate perturbation theory this can be regarded as a secular equation for the coefficients, ψ ± (θ, φ); they are solutions of the Dirac equation, H surf ψ = Eψ with ψ = (ψ + , ψ − ) T . We find the coefficient matrix H surf by evaluating the matrix elements ±|H bulk |± as
H surf = +|H bulk |+ −|H bulk |+ +|H bulk |− −|H bulk |− = 0 D − D + 0 ,(7)where D ± = ±A θ ∂ θ ± ∂ θ A θ 2 + A φ −i∂ φ + 1 2 ,(8)
and [22]
A θ = r r(η − ξ) A + r η−ξ r m 2 ≡ r r(η − ξ) Ã θ , A φ = η − ξ r(η − ξ) A − η−ξ r sin θ η − ξ m 2 .(9)
Notice that in Eq. (8) the φ-derivative in D ± is shifted by 1/2, which is nothing but the "Berry phase" of amount π. Since we have chosen the spinor part of wave function single-valued, the orbital angular momentum L z , defined as ψ(θ, φ) = e iLzφ Z(θ), takes formally an integral value, L z = 0, ±1, ±2, · · · . But due to the Berry phase π the physical angular momentumL z = L z + 1/2 becomes halfodd integral [10,20]. In Eqs. (9) the ξ-average f of a function f (ξ) is defined in terms of a ξ-integral similar to Eqs. (6), i.e.,
f = ∞ 0 dξ f (ξ) e −κ1ξ − e −κ2ξ 2 ∞ 0 dξ (e −κ1ξ − e −κ2ξ ) 2 .(10)
The effective "Dirac theory" on the hyperbolic surface is prescribed by Eqs. (7), (8) and (9). We now attempt to construct zero energy solutions of this effective model. To ease physical interpretation of the results it is useful to modify one of the coordinates by using, instead a (dimensionless) angle θ, a linear coordinate ζ(θ) having the dimension of length such that
ζ(θ) = θ π/4 dθ ′ r(ξ, θ ′ )(η(θ ′ ) − ξ) r(ξ, θ ′ ) .(11)
Notice that (η(θ) − ξ )dθ is a line integral element associated with the locus of the point r 0 = (x 0 , y 0 , z 0 ) along a hyperbola at fixed φ. Thus, at a large distance r ≫ R on the surface (xy-plane) from the z-axis (θ ≪ π/4), −ζ(θ) can be identified as the radial component r of the standard 2D polar coordinates (r, φ), while in the opposite limit of θ ≫ π/4, ζ(θ) can be identified as z, the depth into the hole. Since dζ/dθ = r(η − ξ) / r , the off diagonals in H surf [see Eq. (7)] becomes in the (ζ, φ)-basis,
D ± = ±Ã θ ∂ ζ ± ∂ ζÃθ 2 + A φ −i∂ φ + 1 2 .(12)
How does the wave function penetrate (or not penetrate) into the hyperbolic hole? What happens to the Berry phase π on the surface sufficiently away from the hole? Answers to these questions are encoded in the explicit form of H surf . Let us focus on the zero energy solutions for comparison with the result of numerical simulations. There are two of such solutions, either with spin up or down, ψ (±) E=0 = e iL±φ Z ± (ζ)e ± , where e + = (1, 0) T , e + = (0, 1) T , which satisfy, respectively, D ± ψ (±) E=0 = 0. This can be readily solved as
Z ± (ζ) = 1 Ã θ (ζ) exp ∓L ± ζ 0 dζ ′ A φ (ζ ′ ) A θ (ζ ′ ) ,(13)
whereL ± = L ± + 1/2 [23]. In the asymptotic limit ζ → ∞, A φ /Ã θ in the exponent can be readily approximated as
A φ A θ ≃ 1 r 1 − 1 r m2
A . Deep inside the hyperbolic hole, ζ ≃ z, and also, r ≃ a + ξ and 1 r ≃ 1 a+ξ become constant, therefore Eqs. (13) decay exponentially under the convergence conditions:L + ≥ 1/2 for Z + (ζ) andL − ≤ −1/2 for Z − (ζ) [24]. In this regime, the wave function decays exponentially as it penetrates deeper into the hyperbolic hole, in other words, it actually barely penetrates the bulk (noninvasiveness).
How about the opposite limit, i.e., on the surface as ζ → −∞? In this limit the profile of the wave functions can be directly compared with those of the 2D Dirac equation solved in terms of the Bessel functions J n (|E|r/A) with the use of the polar coordinates (r, φ). And also, we expect that the Berry phase π becomes ineffective on the surface, which seems a priori contradictory to Eqs. (8) and (12). A clue to resolve this discrepancy is in the normalization of the wave function. On the 2D surface, the wave function ψ 2D (r, φ) should be normalized in terms of the surface integral element, rdrdφ, while in the normalization of ψ(ζ, φ) this measure r is not taken into account. Indeed, what should be interpreted as the 2D surface wave function is
ψ 2D (ζ, φ) = ψ(ζ,φ) √ r(ζ)
. Here, the corresponding effective "2D Hamiltonian" H 2D for ψ 2D is deduced from H surf by the replacement, D ± → D ± = D ± ±Ã θ 2 ∂ ζ log r , which can be rewritten as D ± = ±Ã θ ∂ ζ ± ∂ ζÃθ 2 + A φ L ± , by noticingà θ ≃ A and A φ ≃ −A/ζ in the limit of ζ → −∞, where L + = L + , L − = L − + 1. The 1 √ r factor in the normalization of ψ 2D compensates the effects of Berry phase π. Thus, as expected, the Berry phase π is shown to be ineffective on the flat surface away from the hyperbolic hole. Since in the present limit, A φ /à θ ≃ 1/ r and r(ζ) ≃ −ζ, the ζ ′ -integral in the exponent of Eqs. (13) diverges logarithmically, implying Z±(ζ) √ r ∝ |ζ| ±L± . These solutions are bounded only when L + ≤ 0 for Z + (ζ), and L − ≥ 0 for Z − (ζ). This implies, combined with the convergence conditions for the opposite asymptotics, that the zero energy solution is possible only when L + = 0 for Z + (ζ), and when L − = −1 for Z − (ζ). In these two cases Z ± (ζ) becomes constant, consistently with the fact only the zeroth order Bessel function J 0 (|E|r/A) is compatible with the zero energy condition E = 0.
Let us finally remark how the introduction of a flux tube piercing the hyperbolic hole modifies the above argument. In the extreme case of Φ = π, the Aharonov-Bohm flux Φ and the Berry phase π (the shift of 1/2) in Eqs. (8) and (12) cancel out each other. As a result, the bare angular momentum L ± appears in the exponent of the zero energy solutions (13). This modifies the asymptotic condition deeply inside the hyperbolic hole (in the limit of ζ → ∞) to L + ≥ 0 and L − ≤ 0. In the opposite limit (on the surface away from the hole) the two solutions behave asymptotically as Z±(ζ) √ r ∝ |ζ| ±L± , i.e., formally as before, but L ± now replaced with L ± = L ± ∓ 1/2. The two solutions are legitimate only when L + ≤ 0 and L − ≥ 0. The only possible choice of L ± compatible with these two asymptotic conditions is L + = L − = 0. This signifies that the wave function deeply inside the hyperbolic hole stays constant in contrast to the previous case (exponential decay). The π-flux tube transforms the surface state invasive, penetrating the bulk to attain the opposing surface. The asymptotic behaviors on the surface are modified accordingly, reproducing those of the Bessel function J −1/2 (|E|r/A) in the limit of E → 0.
The surface state of a topological insulator is always cited as being a manifestation of the topological nontriviality of the bulk (bulk-boundary correspondence), while the exotic nature of the surface state itself was apt to be ignored. Here, in this Communication we have revealed that the surface state is by itself topologically pro-tected. The proposed scenario makes this point explicit, providing with a viewpoint alternative to the standard bulk-boundary correspondence picture.
KI acknowledges Tomi Ohtsuki for stimulating discussions. The authors are supported by KAKENHI; K.I. by the "Topological Quantum Phenomena" (No. 23103511), and Y.T. by a Grant-in-Aid for Scientific Research (C) (No. 24540375).
PACS numbers: 73.20.-r, 73.22.-f
FIG. 1 :
1(Color online) Which is the genuine surface?
FIG. 2 :
2(Color online) Penetration of the surface wave function along a flux tube of strength Φ.
FIG. 3 :
3(Color online) (a) Image of the "drain". (b) Cross section of the hyperbolic surface on the xz-plane (only the x > 0 part is shown).
Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan (Dated: May 5, 2014)
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The factor 1/ Ã θ arising from the second term of Eq. (12) represents accumulation of the wave function around θ ∼ π/4 due to velocity renormalization. 10, 19The factor 1/ Ã θ arising from the second term of Eq. (12) represents accumulation of the wave function around θ ∼ π/4 due to velocity renormalization [10, 19].
Recall thatL± are quantized to be half-odd integral values. Recall thatL± are quantized to be half-odd integral val- ues.
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"DIFFRACTIVE DIS CROSS SECTIONS AND PARTON DISTRIBUTIONS",
"DIFFRACTIVE DIS CROSS SECTIONS AND PARTON DISTRIBUTIONS"
]
| [
"F.-P Schilling [email protected] \n(H1 COLLABORATION) Physics Department\nCERN\nCH-1211Geneva 23Switzerland\n"
]
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"(H1 COLLABORATION) Physics Department\nCERN\nCH-1211Geneva 23Switzerland"
]
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| Highlights are presented mainly from two recent measurements of the diffractive Deep Inelastic Scattering cross section at HERA. In the first, the process ep → eXp is studied by tagging the leading final state proton. In the second, events of this type are selected by requiring a large rapidity gap devoid of hadronic activity in the proton direction. The two measurements are compared in detail and the kinematic dependences are interpreted within the framework of a factorisable diffractive exchange. Diffractive parton distributions are determined from a next-toleading order QCD analysis of the large rapidity gap data, which can be applied to the prediction of diffractive processes, also at the TEVATRON and the LHC. | 10.1142/9789812790873_0113 | [
"https://arxiv.org/pdf/hep-ex/0608064v1.pdf"
]
| 15,639,551 | hep-ex/0608064 | 3be41ae5403b9ef7eecd419ef311bf11c31337ca |
DIFFRACTIVE DIS CROSS SECTIONS AND PARTON DISTRIBUTIONS
arXiv:hep-ex/0608064v1 31 Aug 2006 July 17, 2018
F.-P Schilling [email protected]
(H1 COLLABORATION) Physics Department
CERN
CH-1211Geneva 23Switzerland
DIFFRACTIVE DIS CROSS SECTIONS AND PARTON DISTRIBUTIONS
arXiv:hep-ex/0608064v1 31 Aug 2006 July 17, 20182:18 WSPC/Trim Size: 10in x 7in for Proceedings proc
Highlights are presented mainly from two recent measurements of the diffractive Deep Inelastic Scattering cross section at HERA. In the first, the process ep → eXp is studied by tagging the leading final state proton. In the second, events of this type are selected by requiring a large rapidity gap devoid of hadronic activity in the proton direction. The two measurements are compared in detail and the kinematic dependences are interpreted within the framework of a factorisable diffractive exchange. Diffractive parton distributions are determined from a next-toleading order QCD analysis of the large rapidity gap data, which can be applied to the prediction of diffractive processes, also at the TEVATRON and the LHC.
Introduction
This report a summarises recent results on measurements of the diffractive deepinelastic scattering (DIS) cross section obtained with the H1 detector at the HERA ep collider, in particular from two recent publications 1,2 . The measurements cover an unprecedented kinematic range of photon virtualities 3.5 < Q 2 < 1600 GeV 2 with unprecedented precision (5% statistical, 5% systematic and 6% normalisation errors in the best-measured region).
In the first paper 1 , the Forward Proton Spectrometer (FPS) is used to detect and measure the four-momentum of the outgoing proton in the process ep → eXp. This selection method has the advantages that the proton unambiguously scatters elastically and that the squared four-momentum transfer at the proton vertex t can be reconstructed. However, the available statistics are limited by the FPS acceptance. A high statistics sample of diffractive DIS events is selected on the basis of a large rapidity gap (LRG) in the outgoing proton direction, as described in the second paper 2 . The measured process is ep → eXY where Y corresponds to any baryonic state with mass M Y < 1.6 GeV.
a Talk presented at ICHEP 2006, Moscow Together, the FPS and LRG data provide a means of studying inclusive diffraction as a function of all relevant variables. In addition to t and the usual DIS variables x and Q 2 , measurements are made as a function of the fractional proton longitudinal momentum loss x I P and of β = x/x I P , which corresponds to the fraction of the exchanged longitudinal momentum carried by the quark coupling to the virtual photon.
The data exhibit a remarkable consistency with proton vertex factorisation 3 , where the dependences on x I P , t and M Y describing the proton vertex are completely independent of β and Q 2 , which describe the hard interaction with the photon. The dependences on x I P and t can then be expressed in terms of an effective pomeron flux of colourless exchange, whilst the β and Q 2 dependences can be interpreted in terms of diffractive parton distributions (DPDFs), which describe the partonic structure of that exchange 4 .
Only a short summary of a few highlights is possible here. Much more detail, including the multi-differential cross section measurements themselves, can be found in 1,2 . The first charged current diffractive measurement, as well as ratios of the diffractive and inclusive cross sections, are also presented Fig. 1. The ratio of the cross section for M Y < 1.6 GeV and |t| < 1 GeV 2 (LRG data) to that for Y = p and |t| < 1 GeV 2 (FPS data) as a function of Q 2 , β and x I P , averaged over the other variables. A 13% normalisation uncertainty is not shown. in 2 , but not covered here. New H1 diffractive DIS measurements with increased statistical precision, but in a limited kinematic range, were also presented 5 .
Comparison between Data Sets
Since the LRG and FPS data sets are statistically independent and have very different systematics, the two measurements constitute a powerful mutual cross-check. Compatibility between them is established in detail by performing t-integrated measurements by both techniques with identical binning and forming the ratio of the two measurements for each (Q 2 , β, x I P ) point 1 .
The dependences of this ratio on each kinematic variable individually is shown in Fig. 1 after taking statistically weighted averages over the other two variables. Within the uncertainties of typically 10% per data point, there is no significant dependence on β, Q 2 or x I P . The ratio of overall normalisations, LRG / FPS, is σ(M Y < 1.6 GeV)/σ(Y = p) = 1.23 ± 0.03 (stat.) ± 0.16 (syst.), consistent with predictions for the proton-elastic cross section and the proton dissociation cross section with M Y < 1.6 GeV 1 . The FPS data are also consistent with the corresponding measurement obtained with the ZEUS Leading Proton Spectrometer 6 .
Dependences on x I P and t
The t dependences of diffractive cross sections are commonly parameterised with an exponential, dσ/dt ∼ e Bt . The values of B resulting from such fits to the FPS data are shown as a function of x I P in Fig. 2. At low x I P , the data are compatible with a constant slope parameter, B ≃ 6 GeV −2 . In a Regge approach with a single linear exchanged pomeron trajectory, α I P (t) = α I P (0) + α ′ I P t, the slope parameter decreases with increasing x I P according to B = B 0 − 2α ′ I P ln x I P . The low x I P data thus favour a small value of α ′ I P ≃ 0.06 GeV −2 , though α ′ I P ≃ 0.25, as obtained from soft hadronic interactions, cannot be excluded.
The x I P dependences of both measurements are interpreted in terms of effective pomeron intercepts. The two results are consistent, the more precise value of α I P (0) = 1.118 ± 0.008 (exp.) +0.029 −0.010 (model) coming from the LRG data. The dominant error arises from the strong positive correlation between α I P (0) and α ′ I P , such that α I P (0) increases to around 1.15 if α ′ I P is set to 0.25 GeV −2 rather than 0.06 GeV −2 . The extracted α I P (0) is slightly higher than the 'soft pomeron' value of α I P (0) ≃ 1.08, obtained from long distance hadronic interactions. The values of both α I P (0) and α ′ I P describing diffractive DIS are compatible with the results obtained for soft exclusive photoproduction of ρ 0 mesons 7 . This similarity supports the picture of diffractive DIS as probing the structure of a 'soft' pomeron. 'Hard' perturbative two gluon exchange contributions are likely to be small, as is also suggested by the lack of a signal for exclusive dijet production 8 .
Further analysis in which either the slope B or the intercept α I P (0) is allowed to vary with β or Q 2 shows no significant dependences (Fig. 3), confirming the validity of proton vertex factorisation for the present data. This contrasts with the Q 2 dependent effective pomeron intercept extracted in a Regge approach to inclusive low x proton structure function data, as studied in detail via the ratio of diffractive to inclusive cross sections in 2 .
Dependences on β and Q 2 : Diffractive Parton Densities
In 2 , the cross section is presented differentially in β, Q 2 and x I P . After dividing out the x I P dependence using a flux factor with parameters obtained as described in section 3, the results from different x I P values are compatible, as expected where proton vertex factorisation holds. The β and Q 2 dependences (Fig. 4) of the data are interpreted in a next-to leading order (NLO) DGLAP QCD fit 2 in order to extract DPDFs. For the first time, experimental and theoretical uncertainties are evaluated for these partons. The results are shown in Fig. 5. The singlet quark density is very closely related to the measured diffractive cross section and is thus well constrained, with a typical error of 5%. According to the DGLAP evolution equations, the logarithmic Q 2 derivative (shown in Fig. 6 for x I P = 0.01) contains contributions due to the splittings g → qq and q → qg, convoluted with the diffractive gluon and quark densities, respectively. The derivative is determined almost entirely by the diffractive gluon density up to β ≃ 0.3. The large positive ln Q 2 derivatives in this region can thus be attributed to a large gluonic component in the DPDFs. For β > 0.3, the contribution to the Q 2 evolution from quark splittings q → qg becomes increasingly important and the derivatives become less sensitive to the gluon density. The gluon density is thus known to around 15% at low β, with an uncertainty that grows quickly for β > 0.3.
These DPDFs provide important input to final state measurements such as those involving jets and charm quarks 9 , which may also provide important additional constraints 10 on the gluon at high β. In- tegrated over β, the gluon density carries around 70% of the total momentum. A similar fraction of the total proton momentum is carried by the inclusive gluon density in the low x region where valence quark effects are small. This similarity of the ratio of quarks to gluons in the DPDFs and the inclusive proton parton densities is reflected 2 in a ratio of the two cross sections which, to good approximation, is flat as a function of Q 2 at fixed x and x I P .
The DPDFs may also be used in calculations of diffractive cross sections at the TEVATRON as well as the LHC.
Fig. 2 .
2Measurements of the slope parameter B by H1 and ZEUS and a parameterisation of the H1 data as used to describe the pomeron flux factor.
Fig. 3 .
3Effective pomeron intercept α I P (0) as extracted from the LRG data, showing no significant variation with Q 2 or β.
Fig. 4 .
4Q 2 dependence of the diffractive DIS cross section at x I P = 0.01 for different values of β.
Fig. 5 .
5DPDF Fit A (exp. error) (exp.+theor. error) Quark singlet and gluon distributions from the NLO QCD fit 'H1 2006 DPDF Fit A', as a function of the momentum fraction z carried by the relevant parton.
Fig. 6 .
6Logarithmic Q 2 derivatives of the diffractive cross section at fixed x I P = 0.01 as a function of β.
. hep-ex/0606003Eur. Phys. J. C. H1 Coll., DESY06-048, hep-ex/0606003, acc. by Eur. Phys. J. C.
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. J Collins, Phys. Rev. D. 573051Erratum-ibid. D61, 019902 (2000)J. Collins, Phys. Rev. D 57, 3051 (1998) [Erratum-ibid. D61, 019902 (2000)].
. Zeus Coll, Eur. Phys. J. C. 3843ZEUS Coll., Eur. Phys. J. C 38, 43 (2004).
. K Krueger, these proceedingsK. Krueger, these proceedings.
. H1 Coll, Eur. Phys. J. C. 2029H1 Coll., Eur. Phys. J. C 20, 29 (2001)
. M Kapishin, these proceedingsM. Kapishin, these proceedings.
. H1 Coll, H1 Coll., H1prelim-06-016.
| []
|
[
"Construction and application of algebraic dual polynomial representations for finite element methods",
"Construction and application of algebraic dual polynomial representations for finite element methods"
]
| [
"V Jain \nFaculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands\n",
"Y Zhang \nFaculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands\n",
"A Palha \nDepartment of Mechanical Engineering\nEindhoven University of Technology\nP.O. Box 5135600 MBEindhovenThe Netherlands\n",
"M Gerritsma \nFaculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands\n"
]
| [
"Faculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands",
"Faculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands",
"Department of Mechanical Engineering\nEindhoven University of Technology\nP.O. Box 5135600 MBEindhovenThe Netherlands",
"Faculty of Aerospace Engineering\nDelft University of Technology\nP.O. Box 50582600 GBDelftThe Netherlands"
]
| []
| Given a polynomial basis Ψ i which spans the polynomial vector space P, this paper addresses the construction and use of the algebraic dual space P and its canonical basis. Differentiation of dual variables will be discussed. The method will be applied to a Dirichlet and Neumann problem presented in [1] and it is shown that the finite dimensional approximations satisfy φ h = div q h on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions. | null | [
"https://arxiv.org/pdf/1712.09472v1.pdf"
]
| 54,877,836 | 1712.09472 | e88e35ff0577afa290317c605202c424374aeb00 |
Construction and application of algebraic dual polynomial representations for finite element methods
V Jain
Faculty of Aerospace Engineering
Delft University of Technology
P.O. Box 50582600 GBDelftThe Netherlands
Y Zhang
Faculty of Aerospace Engineering
Delft University of Technology
P.O. Box 50582600 GBDelftThe Netherlands
A Palha
Department of Mechanical Engineering
Eindhoven University of Technology
P.O. Box 5135600 MBEindhovenThe Netherlands
M Gerritsma
Faculty of Aerospace Engineering
Delft University of Technology
P.O. Box 50582600 GBDelftThe Netherlands
Construction and application of algebraic dual polynomial representations for finite element methods
Finite element methodSpectral elementsAlgebraic dual polynomialsRiesz Representation Theorem
Given a polynomial basis Ψ i which spans the polynomial vector space P, this paper addresses the construction and use of the algebraic dual space P and its canonical basis. Differentiation of dual variables will be discussed. The method will be applied to a Dirichlet and Neumann problem presented in [1] and it is shown that the finite dimensional approximations satisfy φ h = div q h on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions.
Introduction
With every linear vector space V we have the algebraic dual V = L(V, R), see [2, §2.10] or [3, §2.10]. If e i forms a basis for V, then one can construct a canonical basis e * i , which satisfies e * i , e j := e * i (e j ) = δ i j . The finite dimensional polynomial spaces we use in finite element methods also form a linear vector space and therefore the existence of algebraic dual polynomial space directly follows from functional analysis, or more precisely in the finite dimensional case, from linear algebra, [4, §3.F].
Let P be a finite dimensional function space with basis Ψ i (x), then we will give a construction of the dual space P and its canonical dual basis Ψ j (x).
As applications we will demonstrate that the Dirichlet-Neumann problems discussed in [1] can be represented in a finite dimensional setting which preserves the relation between the solutions of both problems, see Section 2.
The construction of dual polynomial spaces in the one dimensional case is presented in Section 3. In this section it is also shown how nodal sampling and edge sampling from polynomial spaces extend to Sobolev spaces.
In Section 4 the construction of dual spaces is presented in two dimensions. In Section 5 the Dirichlet-Neumann problem is discretized where we use a representation in primal degrees of freedom for the Neumann problem and a representation in terms of dual degrees of freedom for the Dirichlet problem. It will be shown that φ h = div q h continues to hold point-wise in these finite-dimensional approximations. As a result, we prove that φ h H 1 = q h H(div) just as in the continuous setting. These results are illustrated with a computational example.
In Section 6 a dual polynomial representation is used for the mixed formulation of the Poisson equation. It will be shown that a primal-dual formulation results in a very sparse matrix where two of the sub-matrices consist of only incidence matrices. This observation is relevant for incompressible flow equation where we encounter a similar divgrad pair. These techniques may also be valuable in electromagnetism to represent the involution constraint div B = 0 in a way that is very sparse and independent of the shape and size of the mesh.
Finally, in Section 7 conclusions are drawn and future work is discussed.
Duality relations
In the proof of Lemma 2.2 of [1] use is made of an equivalence between a Dirichlet and a Neumann problems. We start with the two problems given by: Givenφ ∈ H 1/2 (∂K) 1. The Dirichlet problem: Find φ ∈ H 1 (K) such that
φ =φ on ∂K −div (grad φ) + φ = 0 in K .(1)
2. The Neumann problem: Find q ∈ H(div; K) such that
div q =φ on ∂K −grad (div q) + q = 0 in K .(2)
If q solves the Neumann problem (2), then φ solves the Dirichlet problem, (1), if and only if φ = div q. Furthermore, it follows that, [1] φ H 1/2 (∂K) = φ H 1 (K) = q H(div;K) .
The purpose of this paper is to find suitable, finite dimensional function spacesĜ h ⊂ H 1/2 (∂K), G h ⊂ H 1 (K) and D h ⊂ H(div; K), such that for anyφ h ∈Ĝ h we have 1. The Dirichlet problem: Find φ h ∈ G h such that
φ h =φ h on ∂K −div (grad φ h ) + φ h = 0 in K .(3)
2. The Neumann problem: Find q h ∈ D h such that
div q h =φ h on ∂K −grad (div q h ) + q h = 0 in K ,(4)
such that the solutions φ h and q h will satisfy φ h = div q h identically in the element K. Furthermore, we wish to prove that in this case
φ h H 1/2 (∂K) = φ h H 1 (K) = q h H(div;K) .
Construction of dual finite elements
For the definition of the finite element spaces, we will use the definition in terms of the triplet (K, P, N) by Ciarlet, [5], see also Ern and Guermond, [6, §1.2] and Brenner and Scott, [7, §3.1].
Definition 1.
A finite element consists of the triplet (K, P, N) with i K is a compact, connected, Lipschitz subset of R d with non-empty interior; ii P is a (finite dimensional) linear vector space with domain K. Usually, P is a polynomial vector space; iii N is a set of linear functionals {N i }, i = 1, . . . , ndo f , acting on elements of P, such that the linear map
p ∈ P → (N 1 (p), . . . , N ndo f (p)) ∈ R ndo f (p) ,(5)
is bijective.
N i (Ψ j ) = δ i j , 1 ≤ i, j ≤ ndo f . Example 1. Consider the interval K = [−1, 1] ⊂ R. Let ξ i ∈ K, i = 0, .
. . , N, be the roots of the polynomial (1 − ξ 2 )L N (ξ), where L N (ξ) is the Legendre polynomial of degree N and L N (ξ) its derivative. These nodes are referred to as the Gauss-Lobatto-Legendre (GLL) points, [8]. Let P be the space of polynomials of degree N defined on the interval K. For any p ∈ P define the degrees of freedom by
N 0 i (p) := p(ξ i ) , i = 0, . . . , N .(6)
Because polynomials are continuous, (6) is well-defined. The superscript '0' indicates that we sample in points. The basis which satisfies the Kronecker-delta property from Proposition 1 is given by the Lagrange polynomials through the GLL-points
h i (ξ) = (ξ 2 − 1)L N (ξ) N(N + 1)L N (ξ i )(ξ − ξ i ) .
This example also corresponds to [
N 1 i (p) = ξ i ξ i−1 p(ξ) , i = 1, . . . , N .(7)
For polynomials the integral in (7) is well-defined. The superscript '1' in N 1 i expresses the fact that the degree of freedom is associated to line segments [ξ i−1 , ξ i ]. The basis functions, e j (ξ), which satisfy the Kronecker-delta property from Proposition 1 need to satisfy
N 1 i (e j ) = ξ i ξ i−1 e j (ξ) dξ = δ i j .
Lemma 1. The basis functions e j (ξ) on the GLL-grid defined in Example 1 are given by
e j (ξ) = − j−1 k=0 dh k dξ (ξ) , j = 1, . . . , N ,(8)
where h k (ξ) are the Lagrange polynomials defined in Example 1.
Proof.
ξ i ξ i−1 e j (ξ) dξ = − j−1 k=0 ξ i ξ i−1 dh k (ξ) = − j−1 k=0 h k (ξ i ) − h k (ξ i−1 ) = δ i j ,(9)
where we repeatedly use the Kronecker-delta property of the Lagrange polynomials. If the Lagrange polynomials h k (ξ) are polynomials of degree N, then dh k (ξ)/dξ is a polynomial of degree (N − 1). This proves that the polynomials defined by (8) form a basis for the linear functional defined in Example 2.
Corollary 1. From (8) it follows that dh j dξ = e j (ξ) − e j+1 (ξ) .
So if p ∈ P is expanded in terms of Lagrange polynomials as
p(ξ) = N i=0 N i (p)h i (ξ) ,
then its derivative is given by
dp dξ (ξ) = N i=0 N 0 i (p) dh i dξ = N i=0 N 0 i (p) e i (ξ) − e i+1 (ξ) = N i=1 N 0 i (p) − N 0 i−1 (p) e i (ξ) ,(10)
where we used that e 0 (ξ) = e N+1 (ξ) = 0.
Let E 1,0 be the N × (N + 1) matrix
E 1,0 = −1 1 −1 1 . . . . . . −1 1 −1 1 ,
then we can write (10) as dp dξ
(ξ) = N i=1 N j=0 E 1,0 i, j N 0 j (p) e i (ξ) .
Taking the derivative of a nodal expansion changes the nodal degrees of freedom discussed in Example 1 to integral degrees of freedom discussed in Example 2. The matrix E 1,0 is called the incidence matrix, which converts nodal degrees of freedom to integral degrees of freedom. Differentiation is a map d/dξ : P → Q.
Construction of dual basis
Consider the finite element constructed in Example 1. Any element p ∈ P can be represented as
p(ξ) = N i=0 N 0 i (p)h i (ξ) ,
where N 0 i (p) are the nodal degrees of freedom and h i (ξ) are the associated basis functions. To simplify the notation, we will write this as
p(x) = Ψ 0 (ξ)N 0 (p) ,(11)
where
Ψ 0 (ξ) = (h 0 (ξ) h 1 (ξ) . . . h N−1 (ξ) h N (ξ)) and N 0 (p) = N 0 1 (p) N 0 2 (p) . . . N 0 N−1 (p) N 0 N (p) .
Let p, q ∈ P be both represented as in (11), then the L 2 -inner product is given by
(p, q) L 2 (K) := K pq dK = N 0 (p) T M (0) N 0 (q) .(12)
Here M (0) denotes the mass matrix associated with the nodal basis functions
M (0) = K Ψ 0 T Ψ 0 dK .
Definition 2.
Let N 0 (p) be the degrees of freedom for p ∈ P, then the dual degrees of freedom, N 1 i (p) for p ∈ P are defined by N 0 (q) T N 1 (p) := N 0 (q) T M (0) N 0 (p) , ∀q ∈ P . The dual degrees of freedom N 1 (p) are linear functionals acting on the primal degrees of freedom N 0 (q). Linearity of the functional follows from the linearity of the L 2 inner-product.
Corollary 2. The dual basis functionsh j (ξ) need to satisfy the Kronecker-delta property
N 1 i (h j ) = δ i j .
This will be the case when the dual basis functions are given bỹ
Ψ 0 (ξ) = Ψ 0 (ξ) M (0) −1 . Proof. p(ξ) = Ψ 0 (x)N(p) = Ψ 0 (x)M (0) −1 M (0) N(p) =Ψ 0 (ξ) N(p) .(14)
Remark 2. Note that in (14) an element p ∈ P can be represented in P and in P . This is due to the fact that L 2 (K) is the pivot space in this duality relation, see also [3,Ex.6.7.2]. Proof.
M (0) := K Ψ 0 (ξ) T Ψ 0 (ξ) dK = M (0) −1 K Ψ 0 (x) T Ψ 0 (x) dK · M (0) −1 = M (0) −1 .
In Figure 1 the Lagrange polynomials through the GLL-points and the associated dual polynomials are presented for N = 4.
Analogous to the construction of the dual nodal polynomials, we can also construct the dual polynomials to the edge functions. Let an element p ∈ Q be represented as In the simplified notation this can be written as
p(ξ) = N i=1 N 1 i (p)e i (ξ) .p(ξ) = Ψ 1 (ξ)N 1 (p) , with Ψ 1 (ξ) = e 1 (ξ) e 2 (ξ) . . . e N−1 (ξ) e N (ξ) and N 1 (p) = N 1 1 (p) N 1 2 (p) . . . N 1 N−1 (p) N 1 N (p) .
Similarly, we can write the L 2 -inner product for two functions expanded in this way as
(p, q) L 2 (K) = N 1 (p) T M (1) N 1 (q) ,
with M (1) the mass matrix associated with the edge polynomials
M (1) = K Ψ 1 (ξ) T Ψ 1 (ξ) dK .
Definition 3. Let N 1 (p) be the degrees of freedom for p ∈ P, then the associated dual degrees of freedom N 0 (p) are defined as
N 1 (q) T N 0 (p) := N 1 (q) T M (1) N 1 (p) = ∀q ∈ P .(15)
Following Corollary 2, the dual edge functions are then given by
Ψ 0 (ξ) := Ψ 1 (ξ)M (1) −1 .
In Figure 2 the edge polynomials e i (ξ) and their dual polynomials are shown for N = 3. From the Definitions 2 and 3 we see that the dual degrees of freedom act as linear functionals on the primal degrees of freedom. These two definitions essentially are a particular form of the Riesz Representation Theorem, [7, §2.4] or [2, §3.8]. We have, in particular that
N 0 (p) T M (0) N 0 (p) = N 1 (p) T M (1) N 1 (p) and N 1 (p) T M (1) N 1 (p) = N 0 (p) T M (0) N 0 (p)
The mass matrices M (0) and M (1) which map the primal degrees of freedom to the dual degrees of freedom are called the Riesz maps, [3, corr]. A direct consequence is that
N k (p) 2 M (k) = N k (p) T M (k) N k (p) = N 1−k (p) T M (1−k) N 1−k (p) = N 1−k (p) 2 M (1−k) , k = 0, 1 ,(16)
which just states that the Riesz map preserves the norm. One can compare this construction with covariant and contravariant representation of vectors. Let v = v i e i ∈ V be the contravariant representation and α = α i e i ∈ V a covariant representation, then for every α ∈ V there exists a v ∈ V such that α, w = (v, w), for all w ∈ V. This is the Riesz representation theorem. Compare this with Definitions 2 and 3. In components the connection between α and v is written as
α i = g i j v j , where g i j = (e i , e j ) is the metric tensor. If we compare this with N 1−k (p) = M k N k (p)
for all p ∈ P and k = 0, 1, we see that the mass matrix plays the role of the metric tensor g i j . Note also that in this case we have that e i , e j = δ i j , which states that e i is a canonical dual basis of e j . A similar relation holds for the primal and dual polynomials.
Lemma 2.
Let Ψ(ξ) k and Ψ 1−k (ξ) be the primal and the dual bases as defined above, then we have
K Ψ 1−k (ξ) T Ψ k (ξ) dK = I , k = 0, 1 .
where I is the (N + 1) × (N + 1) identity matrix for k = 0 and the N × N identity matrix for k = 1.
Proof. Using the definition of the dual basis
Ψ 1−k (ξ) = M (k) −1 Ψ k (ξ) gives K Ψ 1−k (ξ) T Ψ k (ξ) dK = M (k) −1 K Ψ k (ξ) T Ψ k (ξ) dK = M (k) −1 M (k) = I .
In Remark 1 it was stated that nodal sampling of a function is only possible in the space P of continuous functions. In a Sobolev space the element consist of equivalence classes of functions and in this case nodal sampling is not defined.
(ξ) = 1 if ξ ξ i 0 if ξ = ξ i for i = 0, . . . , N , ξ ∈ [−1, 1] .
As elements of L 2 ([−1, 1]) the functions f and g are the same, but N 0 ( f ) N 0 (g). For a well-posed degree of freedom, we require that the operation should be independent of the representation we take from an equivalence class.
Lemma 3. Let p ∈ P, then the nodal degrees of freedom are given by
N 0 (p) = K Ψ 1 (ξ) T p(ξ) dK .
Proof. Every p ∈ P can be written as p(ξ) = Ψ(ξ)N 0 (p), therefore
K Ψ 1 (ξ) T p(ξ) dK = K Ψ 1 (ξ) T Ψ(ξ)N 0 (p) dK = N 0 (p) ,
where in the last equality we used Lemma 2.
Example 3 demonstrated that nodal sampling of a f ∈ L 2 ([−1, 1]) is not well-defined. But Lemma 3 allows us to extend nodal sampling to square integrable functions.
N 0 ( f ) := K Ψ 1 (ξ) T f (ξ) dK .
Corollary 4. Using now the fact that Ψ 1 (ξ) = M (0) −1 Ψ 0 (ξ) this 'nodal sampling' can be written as
N 0 ( f ) = M (0) −1 K Ψ 0 (ξ) T f (ξ) dK ,
which is just the L 2 -projection of f onto the basis functions. Analogous we have
N 1 ( f ) := K Ψ 0 (ξ) f (ξ) dK , N 0 ( f ) := K Ψ 1 (ξ) f (ξ) dK and N 1 ( f ) := K Ψ 0 (ξ) f (ξ) dK .
Differentiation of dual variables
Using (10), we can define the derivative of the dual variables. Let q be expanded in Lagrange polynomials and and φ in edge polynomials
q(ξ) = N i=0 N 0 i (q)h i (ξ) and φ(ξ) = N j=1 N 1 j (φ)e j (ξ) .
Then, using (10), we have
K dq dξ φ dK = N 0 (q) T E 1,0 T M (1) N 1 (φ) = N 0 (q) T E 1,0 T N 0 (φ) . The identity K dq dξ φ dK = q(1)φ(1) − q(−1)φ(−1) − K q dφ dξ dK , gives K q dφ dξ dK = −N 0 (q) T E 1,0 T N 0 (φ) + N 0 N (q) N 0 N+1 (φ) − N 0 0 (q) N 0 0 (φ) , ∀q ∈ P ,(17)
where N 0 N+1 (φ) and N 0 0 (φ) are the nodal values of φ(ξ) at the end points ξ = −1 and ξ = 1, respectively. Since (17) needs to hold for all q ∈ P, we define the degrees of freedom of derivative of the dual representation of φ as
N 1 dφ dξ := −E 1,0 T N 0 (φ) + N 0 N+1 (φ) − N 0 0 (φ) .(18)
With these degrees of freedom we can expand the derivative of φ as
dφ dξ = Ψ 1 (ξ) N 1 dφ dξ = Ψ 1 (ξ) −E 1,0 T N 0 (φ) + N 0 N+1 (φ) − N 0 0 (φ) .(19)
Remark 3. Note that while φ(ξ) is a polynomial of degree (N − 1), its derivative, as defined by (19), is a polynomial of degree N.
Two-dimensional dual spaces
In order to address more challenging problems, it is important to consider in more detail the case K = [−1, 1] 2 ⊂ R 2 . For d := dim K = 2, we now have three different function spaces H(curl; K), H(div; K), and L 2 (K). These function spaces constitute a de Rham complex
H(curl; K) ∇× −→ H(div; K) ∇· −→ L 2 (K) .(20)
In order to preserve this structure at the discrete level, we will introduce three different finite elements such that the associated discrete functional spaces, C h (K) ⊂ H(curl; K), D h (K) ⊂ H(div; K), and S h (K) ⊂ L 2 (K), also constitute a De Rham complex
C h (K) ∇× −→ D h (K) ∇· −→ S h (K) .(21)
The basis which satisfies the Kronecker-delta property from Proposition 1 is given by the Lagrange (or nodal) polynomials, (0) k , k = 0, . . . , (N + 1) 2 − 1, through the two-dimensional GLL-nodes x i(N+1)+ j = (ξ i , η j ), i, j = 0, . . . , N, such that
(0) i(N+1)+ j (ξ, η) := h i (ξ)h j (η), i, j = 0, . . . , N ,(23)
where h i are the 1D nodal interpolants introduced in Example 1. A visual representation of these basis functions for N = 2 is presented in Figure 3a.
The dual finite element
The construction of the dual basis functions follows the ideas presented in Section 3.1. Here we outline the direct application to the 2D case of constructing the dual basis of the space C h . The degrees of freedom of the dual element are given by
N 2 (p) := M (0) N 0 (p) .(24)
Since the dual basis functions˜ (2) j need to satisfy the Kronecker-delta property, we have
N 2 i (˜ (2) j ) = δ i j ,(25)
by Corollary 2, we have that the dual basis functions can be expressed in terms of the primal basis functions as
Ψ 2 (ξ, η) :=˜ (2) 0 . . .˜ (2) (N+1) 2 −1 = (0) 0 . . . (0) (N+1) 2 −1 M (0) −1 = Ψ 0 (ξ, η)M (0) −1 ,(26)
with
M (0) i j := K (0) i (ξ, η) (0) j (ξ, η) dK , i, j = 0, . . . , (N + 1) 2 − 1 .(27)
A visual representation of these basis functions for N = 2 is presented in Figure 3b. Consider now the polynomial tensor product spaces Q 1 ξ := P N ⊗ P N−1 and Q 1 η := P N−1 ⊗ P N . We introduce for any polynomial vector field p ∈ Q 1 ξ × Q 1 η the degrees of freedom as
N 1 iN+ j ( p) := (ξ i ,η j ) (ξ i ,η j−1 )
p · e ξ dξ, i = 0, . . . , N and j = 1, . . . , N ,
N 1 (i−1)(N+1)+ j+1+N(N+1) ( p) := (ξ i ,η j ) (ξ i−1 ,η j )
p · e η dη, i = 1, . . . , N and j = 0, . . . , N ,
where e ξ , and e η are the unit vectors in the ξand η−directions, respectively. In a polynomial vector space these integrals are well-defined. It is possible to show, see [9][10][11][12], that the basis functions that satisfy the Kronecker-delta property from Proposition 1 are the edge polynomials, (1) k , k = 1, . . . , 2N(N + 1), defined as
where h i are the 1D nodal interpolants introduced in Example 1, and e j are the 1D edge interpolants introduced in Example 2. A visual representation of these basis functions for N = 2 is presented in Figure 4a.
Application of (10) shows that R(curl; C h (K)) ⊂ D h (K), which is a necessary requirement for C h (K) and D h (K) to form a finite dimensional De Rham sequence, (21).
Dual finite element
The construction of the dual basis functions of the space D h (K) is done in the same manner as for the dual basis functions of the space C h (K). In this case, the dual basis functions can be expressed in terms of the primal basis functions as
Ψ 1 (ξ, η) := ˜ (1) 1 . . .˜ (1) N(N+1) 0 . . . 0 0 . . . 0˜ (1) N(N+1)+1 . . .˜ (1) 2N(N+1) = (1) 1 . . . (1) N(N+1) 0 . . . 0 0 . . . 0 (1) N(N+1)+1 . . . (1) 2N(N+1) M (1) −1 =: Ψ 1 (ξ, η)M (1) −1 ,(30)
with
M (1) i j := K (1) i (ξ, η) · 1) j (ξ, η) dK , i, j = 1, . . . , 2N(N + 1) .(31)
The degrees of freedom of the dual element are given by
N 1 ( p) := M (1) N 1 ( p) .(32)
An important point to remark is that in this case, given the orthogonality between the basis functions, we have that M (1) is block diagonal
M (1) = M (1,ξ) 0 0 M (1,η) ,(33)with M (1,ξ) i j := K (1) i (ξ, η) 1) j (ξ, η) dK , i, j = 1, . . . , N(N + 1) , M (1,η) i j := K (1) i (ξ, η) 1) j (ξ, η) dK , i, j = N(N + 1) + 1, . . . , 2N(N + 1) .(34)
A visual representation of these basis functions for N = 2 is presented in Figure 4b. The degrees of freedom for this finite element can be introduced for any polynomial p ∈ Q 2 as
N 2 (p) := η j+1 η j ξ i+1 ξ i p dξdη , i, j = 1, . . . , N .(35)
These integrals are well-defined in a polynomial space. It is possible to demonstrate, see [9][10][11][12], that the basis functions that satisfy the Kronecker-delta property from Proposition 1 are the surface polynomials, (2) k , k = 1, . . . , N 2 , defined as (2)
(i−1)N+ j (ξ, η) := e i (ξ) e j (η), i, j = 1, . . . , N ,(36)
where, as before e j are the 1D edge interpolants introduced in Example 2. A visual representation of these basis functions for N = 2 is presented in Figure 5a. Application of (10) shows that R(div; D h (K)) ⊆ S h (K), which is required for the spaces D h (K) and S h (K) to be part of the finite dimensional De Rham sequence, (21).
An element from q h ∈ D h (K) can be represented in the basis functions of D h (K) as
q h (ξ, η) = N i=0 N j=1 u i, j h i (ξ)e j (η) N i=1 N j=0 u i, j e i (ξ)h j (η) .
If we take the divergence of this vector field and use (10) repeatedly, we have
div q h (ξ, η) = N i=1 N j=1 u i, j − u i−1, j + v i, j − v i, j−1 e i (ξ)e j (η) .(37)
So, we see that the divergence modifies the degrees of freedom (the expansion coefficients) and changes the basis functions from basis functions in D h (K) to basis functions for S h (K). We can write this as
div q h (ξ, η) = Ψ 2 (ξ, η)E 2,1 N 1 (q h ) ,(38)
where the incidence matrix E 2,1 is a sparse matrix which only contains the non-zero entries −1 and 1 as can be seen from (37).
Dual finite element
The dual basis functions of the space S h (K) follow the same steps as performed for the spaces C h (K) and D h (K). The degrees of freedom for the dual element are given by
N 0 (p) := M (2) N 2 (p) .(39)
The associated dual basis functions are expressed in terms of the primal basis functions as
Ψ 0 (ξ, η) := ˜ (0) 1 . . .˜ (0) N 2 = (2) 1 . . . (2) N 2 M (2) −1 =: Ψ 2 (ξ, η)M (2) −1 ,(40)
with
M (2) i j := K (2) i (ξ, η) (2) j (ξ, η) dK , i, j = 1, . . . , N 2 .(41)
A visual representation of these basis functions for N = 2 is presented in Figure 5b.
Discrete Dirichlet-Neumann problems
The Neumann problem
Consider K = [−1, 1] 2 ⊂ R d , with d = 2.
Then the variational formulation of the Neumann problem, (4), is given by:
Forφ ∈Ĝ h find q h ∈ D h such that divq h , div q K + q h , q h K = ∂K (q h · n)φ dΓ , ∀q h ∈ D h .(42)
Let q h be represented as
q h (ξ, η) = Ψ(ξ, η) 1 N 1 (q) ,
Then, using (37), the divergence is given by
div q h = Ψ 2 (ξ, η)E 2,1 N 1 (q h ) ,
If we use this in the variational formulation (42), we obtain
divq h , div q K + q h , q h K = N 1 (q h ) T E 2,1 T M (2) E 2,1 N 1 (q h ) + N 1 (q h ) T M (1) N 1 (q h ) .(43)
The dual degrees of freedom of the prescribed boundary conditionφ are obtained from Corollary 4. For the right boundary, for instance, where ξ = 1, we have
N 0 b (φ) = 1 −1 Ψ 1 (η)φ(1, η) dη ,
where the subscript 'b' indicates that these integrals are boundary integrals. Similarly, we evaluate the integrals over the other boundaries. Collecting all boundary terms and combining with (43) gives
E 2,1 T M (2) E 2,1 N 1 (q h ) + M (1) N 1 (q h ) = ± N 0 b (φ) ,(44)
where we have a plus sign on the right and top boundary and a minus sign on the left and bottom boundary. Note that the vector N 0 b (φ) only contributes to those N 1 (q h ) which are located at the boundary of the domain.
The Dirichlet problem
Consider now the Dirichlet problem given by (3)
Forφ h ∈Ĝ h find φ h ∈ G h gradφ h , grad φ h K + φ , φ K = ∂Kφ ∂φ ∂n dΓ .(45)
We discretize φ h based on the dual degrees of freedom N 0 (φ h ). Then the degrees of freedom of the gradient is given analogous to (18) by
N 1 (grad φ h ) = −E 2,1 T N 0 (φ h ) ± N 0 b (φ) , where N 0 b (φ)
are the degrees of freedom of the prescribed boundary condition. We have plus sign on the right and the top boundary and the minus sign on the left and bottom boundary based on the direction of the outward unit normal n. Then we know that the gradient of φ h is given by
grad φ h (ξ, η) = Ψ 1 (ξ, η) −E 2,1 T N 0 (φ h ) ± N 0 b (φ) .(46)
The gradient of the test functionsφ h is discretized similarly, but then the values on the boundary are set to zero, therefore
gradφ h (ξ, η) = − Ψ 1 (ξ, η)E 2,1 T N 0 (φ h ) .
If we use this in the variational formulation (45) we have
gradφ h , grad φ h K + φ h , φ h K = N 0 (φ h ) T E 2,1 M (1) E 2,1 T N 0 (φ h ) ∓ N 0 b (φ) + N 0 (φ h ) T M (0) N 0 (φ h ) = 0 .
Note that the boundary conditions are strongly imposed in terms of the dual variables. So the discrete formulation is given by
E 2,1 M (1) E 2,1 T N 0 (φ h ) + M (0) N 0 (φ h ) = ±E 2,1 M (1) N 0 b (φ) .(47)
Relation between Dirichlet and Neumann problem
What we need to check now is that the solutions of (44) and (47) are related by φ h = div q h . This discrete relation translates into
N 0 (φ h ) = M (2) E 2,1 N 1 (q h ) .(48)
In order to establish this relation, we fill in (48) in (47) to obtain
E 2,1 M (1) E 2,1 T M (2) E 2,1 N 1 (q h ) + M (0) M (2) E 2,1 N 1 (q h ) = ±E 2,1 M (1) N 0 b (φ) .(49)
Form (44) we have that
E 2,1 T M (2) E 2,1 N 1 (q h ) = −M (1) N 1 (q h ) ± N 0 b (φ) . If we use this in (49), we have −E 2,1 M (1) M (1) N 1 (q h ) ± E 2,1 M (1) N 0 b (φ) + M (0) M (2) E 2,1 N 1 (q h ) = ±E 2,1 M (1) N 0 b (φ) .(50)
Then we use the fact that M (1) M (1) = I and M (0) M (2) = I to get
−E 2,1 N 1 (q h ) ± E 2,1 M (1) N 0 b (φ) + E 2,1 N 1 (q h ) = ±E 2,1 M (1) N 0 b (φ) ,(51)
which proves the relation between the Dirichlet and the Neumann problem. It remains to show that φ h H 1 (K) = q h H(div;K) . Using (46) we have that
φ h 2 H 1 (K) = N 0 (φ h ) T M (0) N 0 (φ h ) + ± N 0 b (φ h ) T − N 0 (φ h ) T E 2,1 M (1) ± N 0 b (φ h ) − E 2,1 T N 0 (φ h ) .(52)
Since we have just established that N 0 (φ h ) = M (2) E 2,1 N 1 (q h ), we can insert this in (52)
φ h 2 H 1 (K) = N 1 (q h ) T E 2,1 T M (2) M (0) M (2) E 2,1 N 1 (q h ) + ± N 0 b (φ h ) T − N 1 (q h ) T E 2,1 T M (2) E 2,1 M (1) ± N 0 b (φ h ) − E 2,1 T M (2) E 2,1 N 1 (q h ) (44) = N 1 (q h ) T E 2,1 T M (2) E 2,1 N 1 (q h ) + N 1 (q h ) T M (1) M (1) M (1) N 1 (q h ) = N 1 (q h ) T M (1) N 1 (q h ) + N 1 (q h ) T E 2,1 T M (2) E 2,1 N 1 (q h ) (53) = q h 2 H(div;K) ,(54)
where we used again that M
Test case
In this section the Dirichlet and Neumann problems are discretized on one spectral element and the solutions φ h and div q h are compared. For this test case we use a 'standard' orthogonal spectral element, shown in the left plot of Figure 6 and a deformed spectral element shown on the right in Figure 6.
The deformed mesh coordinates (x, y) are obtained by mapping the orthogonal coordinates (ξ, η) with the mapping
x = 1 2 + 1 2 (ξ + c sin(πξ) sin(πη)) y = 1 2 + 1 2 (η + c sin(πξ) sin(πη)) ,(55)
where c is the deformation coefficient. The boundary conditionφ prescribed along the boundary is given bŷ In Figure 7a the numerical solution div q h is shown on the orthogonal mesh, c = 0 for N = 8. The solution φ h on the same mesh is graphically indistinguishable from Figure 7a, therefore in Figure 7b the difference between div q h and φ h is shown. The difference between both solutions is of the order of machine accuracy.
φ = 0 for x = 0 and y = 0 sin(πy) for x = 1 − log (1 − 3x(1 − x)) for y = 1 .(56)
In Figure 8a div q h is plotted for the deformed grid with c = 0.3. On the deformed mesh the solution is less accurate than on the orthogonal mesh, but φ h computed on the same mesh again shows results which are graphically identical to Figure 8a. The differences between div q h and φ h are shown in Figure 8b. (a) Solution div q h of (4) (b) The difference between div q h of (4) and the solution φ h from (3) Figure 7: Comparison between φ h obtained from (3) and div q h calculated using (4) for N = 8 on an orthogonal mesh with c = 0 (a) Solution div q h of (4) (b) The difference between div q h of (4) and the solution φ h from (3) In order to corroborate that the norms φ h H 1 (K) and q h H(div;K) are identical according to (54), Table 1 lists these norms on three different grids, the orthogonal grid, c = 0, slightly deformed mesh, c = 0.15 and the highly deformed mesh, c = 0.3. This table shows that on all meshes and for all polynomial degrees we have φ h H 1 (K) = q h H(div;K) . All three meshes show convergence to a limiting value φ
Mixed formulation of the Poisson equation
The second application of dual polynomial representations concerns a constrained minimization problem which will lead to the mixed formulation of the Poisson problem. Let K = [0, 1] 2 , then for φ ∈ L 2 (K) and q ∈ H(div; K) we define the functional L(φ, q; f ) := for a prescribed function f ∈ L 2 (K). The optimality conditions for this functional are given by
K 1 2 |q| 2 dK + K φ (div q − f ) dK ,(57)c = 0 c = 0.15 c = 0.3 N φ h H 1 q h H(div) φ h H 1 q h H(div) φ h H 1 q h H(div (q, q) K + (divq, φ) K = ∂K (q, n)φ dΓ ∀q ∈ H(div; K) (φ, div q) K = (φ, f ) K ∀φ ∈ L 2 (K) .(58)
We will consider two different discretizations for this problem. For the first approximation we choose (q h , φ h ) ∈ D h (K) × S h (K), while in the second case we we approximate the solution as (
q h , φ h ) ∈ D h (K) × S h (K). For (q h , φ h ) ∈ D h (K) × S h (K)
the discrete system is given by
M (1) E 2,1 T M (2) M (2) E 2,1 0 N 1 (q h ) N 2 (φ h ) = N 0 n (φ) N 2 ( f ) ,(59)
where the degrees of freedom of f are given by
N 2 ( f ) := K Ψ(x) T f (x) dK ,
analogous to Lemma 3. The incidence matrix E 2,1 is a very sparse topological matrix which only contains entries −1, 1 and 0, which does not depend on mesh size, the shape of the mesh (orthogonal mesh or highly curved grid) and independent of the polynomial degree of the approximation, see [10][11][12]. All metric properties are contained in the mass matrices M (1) and M (2) . For high order methods, these matrices are full matrices which destroy the sparsity of the incidence matrix with which they are multiplied. We will refer to this formulation as the primal-primal formulation, because both q h and φ h are expanded in primal basis functions. If the mesh is deformed, all sub-matrices in (59) will change and need to be recomputed. Alternatively, we may approximate φ h ∈ S h (K). In this case the discrete system is given by
M (1) E 2,1 T E 2,1 0 N 1 (q h ) N 0 (φ h ) = N 0 n (φ) N 2 ( f ) .(60)
We see that if we discretize the Lagrange multiplier φ h as dual polynomials, the constraint matrix for the Lagrange multiplier is very sparse and no longer depends on the mesh size, mesh shape or polynomial order. The difference in sparsity pattern is shown in Figure 9. This formulation will be referred to as the primal-dual formulation.
In Figure 9 we see the sparsity structure of the matrix for the orthogonal mesh domain for N = 3. The non-zero elements in dual grid approach -144, are much less than that in the primal grid approach -504. Remark 4. We can immediately convert (59) to (60). The mass matrix M (2) in the constraint can be eliminated directly, while the mass matrix M (2) in the Lagrange multiplier can be contracted with the degrees of freedom N 2 (φ h ) to give M (2) N 2 (φ h ), but these new unknowns are just the dual degrees of freedom N 0 (φ h ) according to (39). We solve (59) and (60) on two different mesh configurations c = 0.0 and c = 0.3 as shown in Figure 6. In the top-left plot of Figure 10 we show the L 2 -error in the constraint (div q h − f h ) for polynomial degrees from 5 to 50. The constraint in the primal-primal and the primal-dual formulation is satisfied up to machine precision over the entire range of polynomial degrees for the orthogonal mesh and the highly curved mesh. In the top right plot of Figure 10 we see the convergence in L 2 -error of the fluxes on the orthogonal mesh and the curved mesh. The results from the primal-primal formulation and the primal-dual formulation coincide with each other. Both methods converge exponentially towards the exact solution, but the convergence is slower in the case of curved mesh. Once machines precision is reached convergence stalls as can be seen for the grid c = 0.
In the plot at the bottom of Figure 10 we see the convergence in L 2 -error of φ h . Again, the results from the primalprimal formulation and the primal-dual formulation overlap. We see exponential convergence on both meshes, but the convergence is slower in the case of curved mesh. In terms of accuracy Figure 10 shows that the primal-primal formulation and primal-dual formulation are comparable.
In Table 2, we list the condition number of the system of the two formulations, for polynomial degree N = 5, 10, ..., 50. For both the orthogonal and the curved domain we observe that the condition number for the primal-primal formulation is much higher than that of the primal-dual formulation. Moreover, the increase in condition number is more rapid for the single grid system than the dual grid system. These results are presented in Figure 11.
Conclusions
In this paper a dual polynomial basis is constructed. Duality pairing between a primal and a dual representation reduces to a vector inner product between the primal and dual degrees of freedom. The dual polynomials have been used to to show the equivalence of a Dirichlet-Neumann pair of equations, taken from [1], at the discrete level. This equivalence is proven and illustrated by a test case. The second example, where the use of a dual representation is beneficial, concerns the mixed formulation of the Poisson problem. When a primal-dual formulation is used, two sub-matrices in the mixed formulation become very sparse, even though very high order methods are used. These two submatrices do not changed when the mesh is deformed.
In future work we will expand these ideas to a hybrid multi-element case. In this paper only one of the Dirichlet-Neumann problems is discussed. In future work we will also address the Dirichlet-Neumann problems in H(curl) as discussed in [1].
Figure 1 :
1The nodal Lagrange polynomial basis functions and the associated dual polynomials.
Corollary 3 .
3The mass matrix M (0) is the inverse of the mass matrix M (0) .
Figure 2 :
2The edge polynomial basis functions and the associated dual polynomials.
Example 3 .
3Let ξ i be the Gauss-Lobatto-Legendre points, which were defined in Example 1. Consider the functions f (ξ) = 1 and g
Definition 4 .
4For f ∈ L 2 ([−1, 1]) we define the nodal degrees of freedom by
Figure 3 :
3Visualization of primal, (0) i(N+1)+ j (ξ, η), and dual,˜ (0) i(N+1)+ j (ξ, η), basis functions of the spaces C h (K) and C h (K) for polynomial degree N = 2.
4. 1 .
1The function space C h (K) 4.1.1. Primal finite element Let ξ i , η i ∈ [−1, 1], i = 0, . . . , N, be Gauss-Lobatto-Legendre (GLL) points, and P N denote the space of polynomials of degree N defined on the interval [−1, 1], see Example 1. Consider now the polynomial tensor product space C h (K) := P N ⊗ P N . Given the set x of 2D nodes x k defined as x := {x i(N+1)+ j = (ξ i , η j ) | i, j = 0, . . . , N}, we can introduce for any p ∈ C h (K) the degrees of freedom as N 0 k (p) := p(x k ), k = 0, . . . , (N + 1) 2 − 1 .
Let ξ i , η i ∈ [−1, 1], i = 0, . . . , N, be Gauss-Lobatto-Legendre (GLL) points, and P N denote the space of polynomials of degree N defined on the interval [−1, 1], see Example 1.
( 1 )
1iN+ j (ξ, η) := h i (ξ) e j (η) e ξ , i = 0, . . . , N and j = 1, . . . , N N+1)+ j+1+N(N+1) (ξ, η) := e i (ξ) h j (η) e η , i = 1, . . . , N and j = 0, . . . , N
Figure 4 :
4Visualization of the primal basis functions,(1) iN+ j (ξ, η) (top left), (1) (i−1)(N+1)+ j+1+N(N+1) (ξ, η) (bottom left), and dual basis functions,(1)iN+ j (ξ, η) (top right), and˜ (1) (i−1)(N+1)+ j+1+N(N+1) (ξ, η) (bottom right),for the spaces D h (K) and D h (K) with N = 2. 4.3. The function space S h (K) 4.3.1. Primal finite element Once again, let ξ i , η i ∈ [−1, 1], i = 0, . . . , N, be Gauss-Lobatto-Legendre points, and P N represent the space of polynomials of degree N on the interval [−1, 1]. Consider now the polynomial tensor product space Q 2 := P N−1 ⊗P N−1 .
Figure 5 :
5Visualization of primal basis functions,(2) iN+ j (ξ, η), and dual basis functions,˜ (0) iN+ j (ξ, η), for the spaces S h (K) and S h (K) with N = 2.
( 0 )
0M (2) = I and M (1) M (1) = I and the fact that the degrees of freedom of q h satisfy equation (44).
Figure 6 :
6Meshes generated by the transformation (55) for c = 0 and c = 0.3.
Figure 8 :
8Comparison between φ h obtained from (3) and div q h calculated using (4) for N = 8 on curvilinear mesh with c = 0.3
Figure 9 :
9Sparsity structure for orthogonal mesh, c = 0.0, for N = 3, on: i) Left: Primal-dual formulation, ii) Right: Primal-primal formulation We compare both formulations with a manufactured solution φ ex = sin(2πx) sin(2πy) which gives for f f = −div(grad φ ex ) .
2 :
2Condition number for the single grid method and the dual grid method, on mesh geometries c = 0.0 and c = 0.3 for polynomial degree N = 5, 10, ..., 50.
Figure 10 :
10Top left: L 2 -error in constraint (div q h − f h ). Top right: L 2 -error in flux q h . Bottom: L 2 -error in potential φ h .
Figure 11 :
11Condition numbers for the meshes c = 0 and c = 0.3 as a function of the polynomial degree
The linear functional {N i } are called the local degrees of freedom. The following Proposition taken from[6] defines the basis functions: Proposition 1. There exists a basis {Ψ 1 , . . . , Ψ ndo f } in P such that
Remark 1. Note that the degrees of freedom are linear functionals on P. The nodal sampling of functions in P is essentially the Dirac delta distribution which is well defined when the vector P consists of continuous functions, see[3, Example 2.10.2]. Extension of this functional to Sobolev space in general will not be possible.Example 2. Let K and ξ i be defined as in Example 1. Let Q be the space of polynomial degree (N − 1). The degrees of freedom, N i , will be defined in this case by6, Prop.1.34] for d = 1.
on the domain K = [−1, 1] 2 ⊂ R d , with d = 2.The variational formulation for this problem is given by:
Table 1 :
1Norms φ h
H 1 (K) and q h
H(div;K) on three different meshes as a function of the polynomial degree N.
Primal-Primal, c=0.0 Primal-Dual, c=0.0 Primal-Primal, c=0.3 Primal-Dual, c=0.3
N=5
825.0252
12.4568
3.6714e+03
85.1068
10
9.8191e+03
26.2636
4.4249e+04
299.379
15
4.6040e+04
66.8577
2.1424e+05
1.0578e+03
20
1.4036e+05
138.4703
6.1383e+05
1.9300e+03
25
3.6434e+05
250.7597
1.6331e+06
3.9074e+03
30
8.0614e+05
413.7121
3.4321e+06
6.9518e+03
35
1.582e+06
637.1410
6.3343e+06
1.0209e+04
40
2.8439e+06
931.1191
1.0912e+07
1.5291e+04
45
4.7756e+06
1.3054e+03
1.7483e+07
2.2209e+04
50
7.5999e+06
1.7702e+03
2.6566e+07
2.9444e+04
Table
Breaking spaces and forms for the DPG method and applications including Maxwell equations. C Carstensen, L Demkowicz, J Gopalakrishnan, CAMWA. 72C. Carstensen, L. Demkowicz, J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, CAMWA 72 (2016) 494-522.
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| []
|
[
"Optical Lattice Modulation Spectroscopy for Spin-orbit Coupled Bosons",
"Optical Lattice Modulation Spectroscopy for Spin-orbit Coupled Bosons"
]
| [
"Sangita De Sarkar \nTheoretical Physics Department\nIndian Association for the Cultivation of Science\n700032Jadavpur, KolkataIndia\n",
"Rajdeep Sensarma \nDepartment of Theoretical Physics\nTata Institute of Fundamental Research\n400005MumbaiIndia\n",
"K Sengupta \nTheoretical Physics Department\nIndian Association for the Cultivation of Science\n700032Jadavpur, KolkataIndia\n"
]
| [
"Theoretical Physics Department\nIndian Association for the Cultivation of Science\n700032Jadavpur, KolkataIndia",
"Department of Theoretical Physics\nTata Institute of Fundamental Research\n400005MumbaiIndia",
"Theoretical Physics Department\nIndian Association for the Cultivation of Science\n700032Jadavpur, KolkataIndia"
]
| []
| Interacting bosons with two "spin" states in a lattice show novel superfluid-insulator phase transitions in the presence of spin-orbit coupling. Depending on the parameter regime, bosons in the superfluid phase can condense to either a zero momentum state or to one or multiple states with finite momentum, leading to an unconventional superfluid phase. We study the response of such a system to modulation of the optical lattice potential. We show that the change in momentum distribution after lattice modulation shows distinct patterns in the Mott and the superfluid phase and these patterns can be used to detect these phases and the quantum phase transition between them. Further, the momentum resolved optical modulation spectroscopy can identify both the gapless (Goldstone) gapped amplitude (Higgs) mode of the superfluid phase and clearly distinguish between the superfluid phases with a zero momentum condensate and a twisted superfluid phase by looking at the location of these modes in the Brillouin zone. We discuss experiments which can test our theory. arXiv:1505.06372v2 [cond-mat.str-el] | 10.1103/physrevb.92.174529 | [
"https://arxiv.org/pdf/1505.06372v2.pdf"
]
| 119,230,446 | 1505.06372 | 88d07e92fbf10232460d64896219510e79e67c52 |
Optical Lattice Modulation Spectroscopy for Spin-orbit Coupled Bosons
Sangita De Sarkar
Theoretical Physics Department
Indian Association for the Cultivation of Science
700032Jadavpur, KolkataIndia
Rajdeep Sensarma
Department of Theoretical Physics
Tata Institute of Fundamental Research
400005MumbaiIndia
K Sengupta
Theoretical Physics Department
Indian Association for the Cultivation of Science
700032Jadavpur, KolkataIndia
Optical Lattice Modulation Spectroscopy for Spin-orbit Coupled Bosons
(Dated: October 13, 2015)
Interacting bosons with two "spin" states in a lattice show novel superfluid-insulator phase transitions in the presence of spin-orbit coupling. Depending on the parameter regime, bosons in the superfluid phase can condense to either a zero momentum state or to one or multiple states with finite momentum, leading to an unconventional superfluid phase. We study the response of such a system to modulation of the optical lattice potential. We show that the change in momentum distribution after lattice modulation shows distinct patterns in the Mott and the superfluid phase and these patterns can be used to detect these phases and the quantum phase transition between them. Further, the momentum resolved optical modulation spectroscopy can identify both the gapless (Goldstone) gapped amplitude (Higgs) mode of the superfluid phase and clearly distinguish between the superfluid phases with a zero momentum condensate and a twisted superfluid phase by looking at the location of these modes in the Brillouin zone. We discuss experiments which can test our theory. arXiv:1505.06372v2 [cond-mat.str-el]
I. INTRODUCTION
Ultracold atoms have emerged in recent years as a valuable platform to study strongly interacting many-particle Hamiltonians relevant to condensed matter systems, nuclear matter and many other different fields of physics 1,2 . The unprecedented control over the Hamiltonian parameters and easy access to strongly interacting regimes have opened up the possibilities of systematically studying interacting many body models which have been used as paradigms to describe a multitude of phenomena from the realms of condensed matter physics, nuclear physics, astrophysics and high energy physics. In the arena of condensed matter physics, many body lattice Hamiltonians like the Bose-and the Fermi-Hubbard models, which are used as paradigms to study strongly interacting bosons and fermions, have been experimentally implemented and studied 3,4 . These provided a wealth of information which is relevant to the phenomena of superfluidinsulator transition 5 and high temperature superconductors 6 respectively. In addition, various spin-models have also been realized using ultracold atoms 7,8 , which will be useful in study of frustrated magnetic systems and spin liquids 9 .
Gauge fields and their interaction with matter are at the heart of our understanding of the physical phenomena around us. While the electromagnetic interactions governing most of condensed matter physics is described by a U(1) Abelian gauge theory, more complicated non-Abelian versions govern the weak and strong interactions. In condensed matter systems, the presence of externally imposed gauge configurations, e.g. a fixed electric or magnetic field, can lead to interesting and qualitative change in properties of the system, e.g. in presence of magnetic fields, vortex lattices can form and melt in a superconductor 10 , the ground state of the system can have non-trivial topology and associated quantized conductance in quantum Hall effect etc 11 . Non-Abelian gauge fields, which can take the form of spin-orbit coupling 12 is an essential ingredient in the realization of topological insulators 13 and topological superconductors 14 and plays a crucial role in understanding novel phenomena like anomalous quantum Hall effect (AQHE) in systems with strong spin orbit coupling.
Ultracold atoms have been dressed by laser fields 15 , so that in the lowest manifold of dressed states, the effective Hamiltonian is that of the bosons/fermions, whose spin and orbital degrees of freedom are coupled. Alternate proposals 16,17 of realizing effective spin-orbit coupling terms are present in the literature. Recently time-varying magnetic field gradients have been used to realize spin-orbit coupling 18 . The implementation of specific types of spin-orbit coupling can lead to interesting phases of matter like topological insulators and topological superconductors. The ability to tune both the effective spin-orbit coupling and the interaction strength in these systems, which is very hard to achieve in material based systems, has opened up the possibility of studying novel Mott insulators and superfluids with Bose Einstein condensation into states with finite momenta in these systems [19][20][21][22] . Compared to the wide array of experimental techniques available to probe material systems, cold atom systems suffer from a paucity of experimental probes. The main tool of obtaining time of flight absorption images, which translates to observation of momentum distribution in lattice systems, is a rather blunt instrument to differentiate between the myriad phases of matter which can occur in these systems. Further, the simple time of flight measurement does not provide any dynamic (energy dependent) information about the system, which is crucial in understanding its low temperature properties. A few spectroscopic techniques like rf spectroscopy 23 , Bragg spectroscopy 24 and lattice modulation spectroscopy 25,26 are available to obtain energy resolved information about these systems. The latter method constitutes an ultracold atom counterpart of standard angle-resolved photoemission spectroscopy and provides energy and momentum resolved information regarding the single particle spectral function of the bosons. This method has been proposed for single species bosons in the strong-coupling regime 25 ; however to the best of our knowledge, it has never been applied to spinor bosonic systems with spin-orbit coupling. Here, we will focus on the lattice modulation spectroscopy of such systems.
In this paper, we will consider two component bosons 27,28 in a 2D square optical lattice, which are interacting with a local Hubbard type interaction. We will consider "spin"-orbit coupling [29][30][31][32] in these systems, implemented either by Raman dressing of the atoms or modulation of magnetic field gradi-ents. We consider the response of this system to optical lattice modulation spectroscopy and demonstrate the following. First, we show that optical lattice modulation spectroscopy can resolve the Mott and the superfluid phases both by looking at the absence/presence of gapless Goldstone modes and by identifying the unique pattern of appearance and disappearance of excitation contours in the Brillouin zone. Second, we demonstrate that the momentum resolved nature of the optical modulation spectroscopy can be used to clearly distinguish superfluids with condensate at zero momentum from those with condensate at finite momentum. Third, we provide a general theory of extracting the spectral function of spin-orbit coupled bosons in the strong coupling regime and near their superfluidinsulator phase transition by computing their response to the lattice modulation. Finally, we use the results of this theory to make definite predictions about the excitation spectrum of these bosons both in the Mott and the superfluid phases which can tested in realistic experiments and demonstrate that lattice modulation spectroscopy can reliably identify and characterize both the gapless (Goldstone) and the gapped amplitude (Higgs) modes of spin-orbit coupled superfluid bosons; we note that such an analysis of the properties of excitation of the SF and Mott phases of these systems is beyond the scope of standard time-of-flight measurement which can also distinguish between the Mott and the SF phases.
The plan of the rest of the paper is as follows. In Sec. II, we discuss the general theory of optical modulation spectroscopy for spin-orbit coupled bosons. This is followed by Secs. III and IV where we inspect the detailed response of the system to optical modulations in the Mott and the superfluid phases respectively. Finally, we discuss possible experiments that can be carried out to validate out theory, sum up our main results, and conclude in Sec. V.
II. LATTICE MODULATION SPECTROSCOPY WITH NON-ABELIAN GAUGE FIELDS
In a system with multiple species of bosons, Raman lasers can be used to generate non-abelian gauge fields which couple the different species of bosons. In most experimental situations, the different boson species are actually different hyperfine state of the same bosonic species allowing one to treat them as a system of multi-component bosons. In fact, if one can trap two bosonic states it can be considered as an effective pseudo-spin 1/2 system, which is however made of bosonic atoms. A particularly interesting non-abelian gauge field configuration is the Rashba spin-orbit coupling in the two component bosonic system. There are various proposals to implement the Rashba spin-orbit coupling, although currently experiments have focussed on the more easily implementable configuration of equal Rashba and Dresselhouse coupling, which leads to spins coupling to momenta in a particular direction. Interacting bosons with Rashba spin-orbit coupling shows interesting chiral Mott and superfluid phases as various parameters are tuned in the Hamiltonian 32,33 .
The Hamiltonian for two-species bosons in a square optical lattice in the presence of Rashba spin-orbit coupling term can be written as 27,28,34
H 0 = iσ [(−Ωσ z − µ)n iσ + U 2 n iσ (n iσ − 1)] + ζU n iσ n iσ −J ij σ b † iσ b jσ + iγ ij Ψ † iẑ · σ × d ij Ψ j(1)
Here b iσ annihilates a boson of spin σ =↑, ↓ on the i th site,
n iσ = b † iσ b iσ is the number of σ bosons, U (ζU )
is the intra-(inter-)species interaction strength between the bosons, and J denotes the nearest neighbor hopping amplitude. Here µ and Ω are the species independent and the species dependent chemical potentials; the latter acts as an effective Zeeman magnetic field for these bosons. The last term represents the lattice analogue of the Rashba spin-orbit coupling generated by the Raman lasers 34,35 , with a coupling constant γ. Here, d ij is unit vector along the x − y plane between the neighboring sites i and j, σ is the vector of Pauli matrices, and Ψ i = (b i↑ , b i↓ ) is the two-component boson field. The non-interacting part of the Hamiltonian is given by
H K = k Ψ † k [Λ(k) − Ωσ z − µ]Ψ k Λ(k) = k 1 + γ k σ + + γ * k σ −(2)
where k ≡ k = (k x , k y ) is the 2D quasi-momentum, k = −2J(cos k x + cos k y ) and γ k = −2γi(sin k x − i sin k y ). This can be diagonalized to obtain chiral bands touching each other at the zone center for Ω = 0, while a finite Ω opens up a gap between the two bands. The ratio γ/J controls the bare dispersion including the location of the band-minima. As γ/J becomes larger the location of the minimum of the lowest band shifts away from the zone center, [0, 0]. The spin-orbit coupled bosons undergo a Mott insulator to superfluid quantum phase transition as a function of J/U and γ/U 20,29 . In the strongly interacting limit, each site has exactly n 0 σ bosons of spin σ, with the value of n 0 σ decided by µ and Ω. The system is thus a Mott insulator with no number or spin fluctuations. As J/U or γ/U increases, the system undergoes a phase transition to a state with delocalized bosons. This is the superfluid state with phase coherent Bose condensate and gapless Goldstone excitations due to broken U (1) symmetry. If the transition takes place when γ/J is large, the minimum of the effective dispersion occurs at [±k 0 , ±k 0 ], and the bosons condense into these finite momentum states, leading to a twisted superfluid phase 20 . At low values of γ/J one recovers the standard superfluid with a condensate at zero momentum. Thus this system shows two remarkable qualitative changes : (a) a superfluid-Mott insulator transition as a function of J/U and γ/U and (b) a change from a standard superfluid to a twisted superfluid as a function of γ/J. As we will show, lattice modulation spectroscopy can distinguish both these phenomena and hence provide us with a wealth of information about this system. The optical lattice modulation spectroscopy protocol that we propose consists of the following steps: (a) The optical lattice potential is weakly modulated with an a.c. field on top of the static field that forms the lattice in the original system, with the Raman fields and the trapping potential turned on. This leads to a modulation of the hopping parameter and the spin-orbit coupling. (b) The modulation is turned off after some time, making sure that the system is still in perturbative regime. At the same time, the Raman lasers, the optical lattice lasers and the trapping potential is also turned off and the system undergoes ballistic expansion, from which the (spinresolved) momentum distribution of the system right after the modulation can be measured. (c) The change in the momentum distribution (from the unperturbed/ unmodulated system) will provide us with information about the spectrum and spectral weight of one particle excitations in these systems 25 .
In experiments, the lattice Hamiltonian Eq. 1 is implemented by putting a system of bosons, characterized by mass m, a continuum spin-orbit coupling γ c and an effective magnetic field h generated by the Raman lasers, under a periodic optical potential V 0 [cos 2 (x/a) + cos 2 (y/a)], where a = λ op /(2π) is a transverse confinement scale used to con- struct the 2D square lattice, and λ op is the wavelength of the laser forming the optical lattice. The interaction between these bosons in continuum are characterized by a intra-species scattering length a s and an inter-species scattering length ζa s . In the limit of a deep lattice, the continuum parameters are related to the lattice parameters by
U = 4 V 0 E R a s /a, J = V 0 E R e − 1 4 √ V0/E R γ = γ c √ 2mV 0 e − 1 4 √ V0/E R ,(3)
where E R = 2 /2mλ 2 op is the recoil energy. It is evident that J and γ both depend exponentially on the lattice depth V 0 , while U has only polynomial dependence on lattice height. Thus, an ac modulation put on the optical lattice depth V 0 (t) = V 0 + δV cos ωt will lead to simultaneous modulation of the hopping parameter, the spin orbit coupling as well as the interaction parameter U . Using U as an overall scale for the problem, the perturbation Hamiltonian, to linear order in variations of the optical lattice potential, can be written as
H 1 = δU H 0 − U δ J U ij σ b † iσ b jσ +iU δ γ U ij Ψ † iẑ · σ × d ij Ψ j(4)
where the first term, proportional to the variation of U, commutes with the unperturbed Hamiltonian and hence does not create excitations. This term can thus be neglected as far as modulation spectroscopy is concerned. In this case, the perturbation consists of modulating the hopping and spin-orbit terms with amplitudes 36
λ = δJ/J = δγ/γ = 1 8 (δV / V 0 E R )(5)
The perturbation Hamiltonian can then be written as
H 1 (t) = λ cos ωt k Ψ † kΛ (k)Ψ k .(6)
In linear response regime, the momentum distribution right after the modulation is turned off oscillates with the frequency of the perturbation 25 δn σ,k (t) = δn (1) (σ, k, ω) cos ωt + δn (2) (σ, k, ω) sin ωt (7) The out-of phase response contains the information about excitation spectrum of the unperturbed system. In the next section, we will define the response function Π σ (k, iω), where iω is the Matsubara frequency, and relate this to the single particle Green's function of the bosons and hence to the excitation spectra. The imaginary part of Π σ (k, ω + i0 + ) then measures the amplitude of the out of phase modulation of the spin dependent momentum distribution due to the perturbation. The structure of the response is qualitatively different in the Mott and superfluid phase owing to the broken U (1) symmetry in the superfluid phase and associated anomalous propagators. Hence we will treat the response in the Mott and superfluid phase separately.
III. RESPONSE IN THE MOTT PHASE
In the strongly interacting limit, the system is in an incompressible Mott insulating state with gapped excitation spectrum. In the atomic limit, (γ = J = 0), the ground state has a fixed number of bosons of each spin, n 0 σ at all the sites. In general, n 0 σ is determined by the values of µ and Ω. We will restrict our analysis to the case where n 0 ↑ = 1 and n 0 ↓ = 0, but the analysis can be easily extended to arbitrary integer values of n 0 σ . The single particle Green's function in the Mott phase is a 2 × 2 matrix in the spin space, in terms of which the response function is given by
Π σ (k, iω n ) = λ β ω l [Ĝ(k, iω l )Λ(k)Ĝ(k, iω l + iω n )] σσ +(ω n → −ω n )(8)
The boson Green's function can be worked out in a strong coupling expansion around the localized atomic limit 20,29,37 .
In the Mott phase, it is given by
G −1 (k, iω n ) = F 1 (iω n ) − k −γ k −γ * k F 2 (iω n ) − k F 1 (iω n ) = iω n + E 0 − 2U + 2U 2 /(iω n + E 0 + U ), F 2 (iω n ) = iω n − E 1(9)
where E 0 = µ + Ω and
E 1 = Ω + ζU − µ.
It is evident that this Green's function is not diagonal in the basis of non-interacting bands, as the atomic limit local propagator is different for the 2 spin species. This can be traced to the fact that in the atomic limit (J = 0, γ = 0) the Ω term lifts the degeneracy between the spin states and as a result one obtains a polarized Mott state which is n 0 = 1 for the ↑ spins and n 0 = 0 (vacuum) for the ↓ spins. The Green's function can, however, be diagonalized to obtain where the diagonal components are given by
G D (k, iω n ) = 1 ζ−(k,iωn) 0 0 1 ζ+(k,iωn)(10)ζ ± (k, iω n ) = F + (iω n ) − k ± (F 2 − (iω n ) + |γ k | 2 ) 1/2 ,(11)
and
F ± (iω n ) = (1/2)[F 1 (iω n ) ± F 2 (iω n )].
The basis transform, which diagonalizes the Green's function is given by
M (k, iω n ) = u(k, iω n ) v(k, iω n ) −v * (k, iω n ) u * (k, iω n ) |u(k, iω n )| 2 = γ 2 k /N (k, iω n ) v(k, iω n ) = γ −1 k u * (k, iω n )[F 1 (iω n ) − k − ζ + (k, iω n )], N (k, iω n ) = [F 1 (iω n ) − k − ζ + (k, iω n )] 2 + γ 2 k . (12) Note that |u(k, iω n )| 2 + |v(k, iω n )| 2 = 1.
It is evident from the above expressions that even in the transformed basis, the diagonal Green's function (Eq. 10) will have a complicated frequency dependence. However, we are only interested in the out of phase response of the system to the external perturbation, which depends solely on the imaginary part of the Green's function, analytically continued to the real frequency domain (iω n → ω + i0 + ). It can be easily shown that the retarded Green's function only has simple poles, and so, for the purpose of calculating the out of phase response, the complicated expression of Eq. 10 can be replaced by the simpler form
G D (k, iω n ) z − pk iωn−E − pk 0 0 z + pk iωn−E + pk + z + hk iωn−E + hk (13)
where ω = E + p(h)k are the two zeroes of ζ + (k, ω) and correspond to the particle and hole excitations of the + band, while ω = E − pk is the zero of ζ − (k, ω). The residues z are given by the ω derivative of the components of the Green's function, Eq. 10, evaluated at the corresponding poles. The three poles have a relatively simple explanation in terms of the par-ticle and hole excitations of a Mott insulator. In absence of the spin-orbit coupling (γ = 0), the ↑ spins form a n 0 = 1 Mott insulator and has the corresponding particle and hole excitations. This corresponds to the limiting form of E + p(h)k as γ → 0. The ↓ spins , however, form a n 0 = 0 Mott insulator, and hence has only particle excitations. This is the limiting form of E − pk in the limit γ → 0. In presence of spin-orbit coupling, the spin states get mixed, but this three pole structure of the Green's function persists. We note that the Green's function in Eq. 13 has the same imaginary part as the actual Green's function (Eq. 10) and hence there is no approximation made in the above replacement, as far as computation of the out of phase response of the system is concerned.
The response of the system can now be calculated in terms of the diagonal Green's function as
Π σ (k, iω n ) = λ β iω l M (k, iω l )G D (k, iω l )M −1 (k, iω l )Λ(k)M (k, iω l + iω n )G D (k, iω l + iω n )M −1 (k, iω l + iω n ) σσ +ω n → −ω n ,(14)
whereΛ(k) is given by Eq. 2. Note that the transformation matrices are themselves function of the Matsubara frequencies iω l and iω n . However, it can be easily seen that the transformation matrix elements, analytically continued to the real frequency domain (iω l → ω + i0 + ), have no imaginary part, and hence the real frequency response will be dominated by the singularities of the diagonal Green's function only. Analytically continuing to real frequencies and working out the Matsubara sums (and noting that matrix elements of M are real in this limit), we get
n (2) (σ, k, ω) = 2λ pq ∞ −∞ dω π α σ pq (k, ω , ω + ω )G p D (k, ω )G q D (k, ω + ω )[n B (ω ) − n B (ω + ω)](15)
where p, q = ±, G p D indicates the imaginary part of the Green's function component, and the matrix element
α σ pq (k, ω 1 , ω 2 ) = mn M σp (k, ω 1 )M −1 pm (k, ω 1 )Λ mn (k)M nq (k, ω 2 )M −1 qσ (k, ω 2 )(16)
We now restrict ourselves to the response at T = 0. From the simple pole structure of the Green's function the response is then obtained as
n (2) (σ, k, ω) = 2πλ α σ ++ (k, E + pk , E + hk )z + pk z + hk δ(ω − E + pk + E + hk ) + α σ +− (k, E − pk , E + hk )z + hk z − pk δ(ω − E − pk + E + hk ) +α σ ++ (k, E + hk , E + pk )z + pk z + hk δ(ω − E + hk + E + pk ) + α σ +− (k, E + hk , E − pk )z + hk z − pk δ(ω − E + hk + E − pk )(17)
It is clear from Eq. 17 that the response shows up on two contours specified by ω = E + pk − E + hk and ω = E − pk − E + hk , the intra and interband particle-hole excitations. In the Mott phase, both these excitations are gapped with the lowest Mott gap corresponding to the intra-band excitations. As the frequency of modulation is swept, there is no response till the frequency crosses the Mott gap. Beyond this point, two distinctive set of phenomena can be seen depending on the ratio, γ/J. For small γ/J the minimum of the intraband excitations is at the zone center, [0, 0]. This is seen in Fig. 1(a), where the intraband excitation in the Brillouin zone is plotted as a color plot for the following set of parameters: J = 0.02U , γ = 0.01U , Ω = 0.01U , µ = 0.2U and ζ = 0.4, where the minimum of the spectrum (the Mott gap) is 0.24U at the zone center. The dispersion increases as one moves away from the zone center and the highest excitation energy occur at [π, π with ω = 0.57U . The corresponding inter-band excitations are shown in Fig. 1(b). They follow a similar pattern as the intra-band excitations with a minimum energy of 0.73U and a maximum energy of 1.22U .
The optical modulation response in this case is plotted for four different frequencies in this case in Fig. 2. As frequency increases beyond the gap, 0.24U , a contour of excitations is seen, which become larger and moves towards the edge of the Brillouin zone, before disappearing at ω = 0.57U . This contour, corresponding to intra-band excitations, are shown in Fig 2(a) and (b). As the frequency is further increased beyond ω = 0.73U , the contours corresponding to the inter-band excitations appear around the zone center and move outwards, before disappearing at ω = 1.22U , as shown in Fig 2(c) and (d). We would like to note here that the full dispersion of the excitations can be tracked from the optical lattice modulation spectroscopy. To show how this is done, in Fig. 4 we plot the lattice modulation response in the Mott phase as a function of frequency along various cuts in the Brillouin zone, going radially outwards at different angles θ with the k x axis. Fig. ref-fig8 (a)-(c) correspond to the system parameters J = 0.02U , γ = 0.01U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4 with the band minimum at the zone center, which is clearly seen from the plots. Further, the modulation spectroscopy also provide information about the spectral weight of the various excitations. In this case, it is evident the interband excitations carry much less spectral weight than the intraband excitations.
The phenomenology changes dramatically if the spin orbit coupling γ is larger than hopping J. Fig. 1(c) and (d) shows a plot of the intra and interband particle hole excitations in the Mott phase for a system with parameter values J = 0.01U , γ = 0.04U , Ω = 0.01U , µ = 0.2U and ζ = 0.4. In this case the spin orbit coupling is larger than the hopping and consequently the intraband excitations have a minimum at [±k 0 , ±k 0 ] along the zone diagonal. The location of the minima corresponding [k 0 , k 0 ] is shown in Fig. 1(c); the corresponding figures for other minima can be obtained by a reflection of the contours in the first quadrant (0 ≤ k x , k y ≤ π) about the k x and k y axes and the origin. The interband excitations however have a minimum at the zone center, as seen in Fig. 1(d). Once again, the system will not show any response as long as the modulation frequency is below the Mott gap of 0.28U . Once the Mott gap is crossed, excitation contours around [±k 0 , ±k 0 ] appear in the response, as seen (for [k 0 , k 0 ]) in Fig 3(a). These contours spread out and disappear at ω = 0.45U , as seen in Fig 3(b). Another contour, corresponding to interband excitations and centered around k = 0 appears as the modulation frequency is swept beyond 0.87U . This spreads out and finally disappears at ω = 1.1U , the upper limit of the interband excitation energy, as seen in Fig. 3(c) and (d). To get a better idea of the dispersion, the lattice modulation response for a system in Mott phase with a band minimum shifted to finite wavevectors is plotted in Fig. 4 (d) -(f), where the parameters of the system are J = 0.01U , γ = 0.04U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case, the band minimum is at (k 0 , k 0 ) with k 0 = 1.131a −1 . It is clearly seen that the intraband excitations have a shifted band minimum, while the interband excitations continue to have a minima at the zone center. In this case, it is clear from comparison with corresponding response in Fig. 2, that although the intraband excitations have larger spectral weight than the in-terband excitations, the interband excitations carry relatively larger spectral weight than the case where the minimum was at the zone center.
The lattice modulation spectroscopy can identify the Mott phase from the existence of a gap in the spectrum. However the momentum resolved nature of the spectroscopy provides much more detailed information about the single particle spectral function of the system. It provides information about the spectrum and one should be clearly able to see the minimum of the excitation spectrum shift from the zone center as the parameters are varied. The modulation spectroscopy also provides information about the spectral weight of the excitations of the system, which is of immediate relevance for figuring out both near equilibrium and far from equilibrium response of the system to different stimuli.
IV. RESPONSE IN THE SUPERFLUID PHASE
The spin-orbit coupled bosons undergo a Mott insulatorsuperfluid quantum phase transition as either the hopping or the spin-orbit coupling is increased as both terms help to delocalize the bosons. If the transition occurs at a large value of J/γ, the superfluid phase has a BEC at the zone center k = [0, 0] with associated anomalous propagators and Goldstone modes. On the other hand, if the transition takes place at a large value of γ/J, the system exhibits the twisted superfluid phase, with a BEC at a finite momentum. In general there can be four such momentum values given by [±k 0 , ±k 0 ] which are degenerate minima of the spectrum. This in principle allows the possibility of formation of a square superlattice (i.e. a supersolid) with the incommensurate momentum k 0 providing the inverse lattice constant. However, in a cold atom system with the presence of a trapping potential which breaks translational symmetry, the more likely effect is the formation of domains 39 , each of which corresponds to choosing one of the four values of allowed k 0 . Assuming domains of size much larger than k −1 0 , the momentum distribution signal will be an incoherent weighted sum of the signal from a single domain with a fixed condensation wavevector. In this paper, we shall assume that condensation occurs at only one possible momentum which we choose to be (k 0 , k 0 ) and work out the signal from a single domain. The important features, like the low energy features around(k 0 , k 0 ) will occur in different parts of the Brillouin zone for different domains and thus will not be washed out by formation of domains. Although the detailed signal requires knowledge of domain distribution, a first approximation can be obtained from our calculation by symmetry, together with an idea of the density of each type of domains. The lattice modulation spectroscopy is uniquely suited to distinguish between gapless superfluid phase from the Mott phase, and owing to its momentum resolved nature, it can also distinguish between the normal and twisted superfluid phases.
Since this method gives detailed information about the spectral function, useful quantities like the speed of sound can be calculated from the spectrum, while the relative intensity of the contours will provide information about transfer of spectral weight among the different kinds of excitations as the en- ergy is varied.
In the superfluid phase the bosons occupying the lower energy band form a BEC at the appropriate momentum. In this case the Green's functions expand to a 4 × 4 matrix to accommodate the anomalous propagators. It is easier to work in the ± basis, earlier described in the Mott phase, since the structure of the Green's functions are simplest in this basis.
Working with a 4 component vector [φ − (k, iω), φ − * (2k 0 − k, −iω), φ + (k, iω), φ + * (2k 0 − k, −iω)]
, the inverse Green's function is given by 20,25,37
G −1 (k, iω n ) ζ − (k, iω n ) − r 0 0 0 0 ζ − (2k 0 − k, −iω n ) − r 0 0 0 0 ζ + (k, iω n ) − r −r 0 0 −r ζ + (2k 0 − k, −iω n ) − r (18)
where ζ ± (k, iω) is defined by Eq. 11 and r = ζ + (k 0 , k 0 , ω = 0) incorporates the effects of the presence of the condensate. The lower 2 × 2 block simply looks like the Green's func-tion of a Bose gas in the Bogoliubov approximation, with the complicated function ζ + (k, ω), incorporating correlations due to proximity to a Mott insulator, replacing a simple free par- ticle piece. In the diagonal upper block, the presence of the condensate leads to a Hartree correction due to inter-band interactions.
The unitary transform which converts the original "spin" basis, i.e.
[φ ↑ (k, iω), φ * ↑ (2k 0 − k, −iω), φ ↓ (k, iω), φ * ↓ (2k 0 − k, −iω)]
, to the ± basis is given bỹ
M (k, iω n ) M * 11 (k, iω) 0 M * 21 (k, iω) 0 0 M 11 (2k 0 − k, −iω) 0 M 21 (2k 0 − k, −iω) M * 12 (k, iω) 0 M * 22 (k, iω) 0 0 M 12 (2k 0 − k, −iω) 0 M 22 (2k 0 − k, −iω) ,(19)
where the matrix elements M ij are given by Eq. 12. The final Green's function has the form
G(k, iω n ) = G − (k, iω n ) 0 0 0 0 G − (2k 0 − k, −iω n ) 0 0 0 0 G + (k, iω n ; k 0 ) F (k, iω n ; k 0 ) 0 0 F (2k 0 − k, −iω n ; k 0 ) G + (2k 0 − k, −iω n ; k 0 ) .(20)
The contours : two open contours due to two phase modes, and two closed contours (outward from the zone center) due to two amplitude modes, a phase and an amplitude mode (d) closed contour due to a phase and an amplitude mode and open contours due to two amplitude modes, (e) a single contour: two amplitude modes (f) a single contour: a phase mode and an interband transition at finite momentum (g) 2 contours (outward from zone center): an amplitude mode and an interband transition, a phase and an interband transition (h) a single contour: an amplitude mode and an interband transition. Note that the spectral weight of the interband transitions are much smaller than the weights of other transitions.
G − (k, iω) = g − pk iω − E − pk , G + (k, iω; k 0 ) = g +p 1k iω − E + 1k + g +h 1k iω + E + 1k + g +p 2k iω − E + 2k + g +h 2k iω + E + 2k (21) F (k, iω; k 0 ) = f + 1k iω − E + 1k − f + 1k iω + E + 1k + f + 2k iω − E + 2k − f + 2k iω + E + 2k(22)
Here the particle excitation in the − branch, E − pk is the solution of ζ − (k, E − pk ) = r and the four excitation poles ±E + 1(2)k are obtained from the solutions of
[ζ + (k, ω) − 2r][ζ + (2k 0 − k, −ω) − 2r] − r 2 = 0. (23)
The quasiparticle residues are obtained from the frequency derivatives of the Green's functions at the corresponding poles. Note that in the + band, the particle and hole excitations are now mixed due to formation of a condensate and that the energies E + 1(2)k , g +p(h) 1(2)k , and f + 1(2)k depends on k 0 through Eq. 23; this is in contrast to the corresponding quantities in the − band which has no particle-hole mixing. This necessitates the use of additional k 0 argument in the definition of G + and F in Eq. 20; however, we refrain from putting such additional label of k 0 in E + 1(2)k , g +p(h) 1(2)k , and f + 1(2)k for notational brevity.
The response function for the lattice modulation spectroscopy is then given by where a = 1 for σ =↑ and a = 2 for σ =↓, and the perturbation matrix Λ is given by
Π σ (k, iω n ) = λ β iω l M (k, iω l )G(k, iω l )M −1 (k, iω l )Λ (k)M (k, iω l + iω n )G(k, iω l + iω n )M −1 (k, iω l + iω n ) 2a−1,2a−1 +ω n → −ω n(24)Λ (k) = 1 2 k 0 γ k 0 0 k 0 γ * −k γ * k 0 k 0 0 γ −k 0 k (25)
Working out the Matsubara sums, and analytically continuing to real frequencies,
n (2) (σ, k, ω) = 2λ pq ∞ −∞ dω π α σ pq (k, ω , ω + ω )G p (k, ω ; k 0 )G q (k, ω + ω ; k 0 )[n B (ω ) − n B (ω + ω)] +β σ (k, ω , ω + ω )F (k, ω ; k 0 )F (k, ω − ω ; k 0 )[n B (ω ) − n B (ω − ω )(26)
where p, q = ±, G p and F indicate the imaginary part of the Green's function component, and it is understood that
G − (k, ω; k 0 ) ≡ G − (k, ω).
Here the matrix elements
α σ pq (k, ω 1 , ω 2 ) = mnM 2σ−1p (k, ω 1 )M −1 pm (k, ω 1 )Λ mn (k)M nq (k, ω 2 )M −1 q2σ−1 (k, ω 2 ) β σ (k, ω 1 , ω 2 ) = mnM 2σ−1,3 (k, ω 1 )M −1 2m,4 (k, ω 1 )Λ 2n,2m (k)M 2n,4 (k, ω 2 )M −1 2σ−1,3 (k, ω 2 )(27)
We now restrict ourselves to the response at T = 0 for ω > 0. The ω < 0 response can then be obtained from the fact that the imaginary part of the response function is an odd function of ω. From the simple pole structure of the Green's function the response is then obtained as
n (2) (σ, k, ω) = 2πλ ρ σ 1 (k)δ(ω − E − pk − E + 1k ) + ρ σ 2 (k)δ(ω − E − pk − E + 2k ) ρ σ 3 (k)δ(ω − 2E + 1k ) + ρ σ 4 (k)δ(ω − 2E + 2k ) + ρ σ 5 (k)δ(ω − E + 1k − E + 2k )(28)
where the weight of the different contours are given by
ρ σ 1 (k) = 2g − pk g +h 1k α σ 12 (k, E − pk , −E + 1k ) ρ σ 2 (k) = 2g − pk g +h 2k α σ 12 (k, E − pk , −E + 2k ) ρ σ 3 (k) = 2g +p 1k g +h 1k α σ 22 (k, E + 1k , −E + 1k ) − (f + 1k ) 2 [β σ (k, E + 1k , E + 1k ) + β σ (k, −E + 1k , −E + 1k )](29)ρ σ 4 (k) = 2g +p 2k g +h 2k α σ 22 (k, E + 2k , −E + 2k ) − (f + 2k ) 2 [β σ (k, E + 2k , E + 2k ) + β σ (k, −E + 2k , −E + 2k )] ρ σ 5 (k) = 2g +p 1k g +h 2k α σ 22 (k, E + 1k , −E + 2k ) + 2g +p 2k g +h 2k α σ 22 (k, −E + 1k , E + 2k ) − 2f + 1k f + 2k [β σ (k, E + 1k , E + 2k ) + β σ (k, −E + 2k , −E + 1k )]
It is clearly seen that the system would show response on the contours which correspond to energies 2E + 1k , 2E + 2k , E + 1k + E + 2k , E − pk + E + 1k and E − pk + E + 2k . Here E + 2k is the Goldstone phase mode which goes gapless, E + 1k is the gapped amplitude or Higgs mode and E − pk is the particle excitation to the upper band, which also has an excitation gap. The location of these contours in the Brillouin zone are dramatically different depending on whether the BEC is formed at the zone center (large J/γ) or at a finite momentum (small J/γ).
The dispersions of 2E + 1k , 2E + 2k , E + 1k + E + 2k , E − pk + E + 1k and E − pk + E + 2k are plotted in Fig. 5 for a system with following set of parameters: J = 0.03U , γ = 0.01U , Ω = 0.01U , µ = 0.35U and ζ = 0.4. In this case, the system is in a superfluid with a BEC at the zone center. Fig. 5 (a) shows 2E + 2k , which ranges from 0 at the zone center (the gapless point) to 0.58U . All the other plots in Fig. 5 show similar features except the fact that these excitations are gapped. In Fig. 5 The optical modulation response for the above system is shown in Fig 6. Fig. 6(a) shows the response at ω = 0.2U , where a single contour of excitations corresponding to exciting two phase modes, ω = 2E + 2k is seen. Fig. 6(b) shows the response at ω = 0.4U , where two contours of excitations are seen, the outer one corresponding to exciting two phase modes, ω = 2E + 2k , and the inner one corresponding to exciting a phase and an amplitude mode ω = E + 2k + E + 1k . Fig. 6(c) shows the response at ω = 0.55U , where three contours of excitations are seen, the outermost one corresponding to exciting two phase modes, ω = 2E + 2k , the middle one corresponding to exciting a phase and an amplitude mode ω = E + 2k + E + 1k , and the innermost contour corresponding to exciting a phase mode and a band transition to the − band, ω = E + 2k +E − pk . As the frequency is increased to ω = 0.6U , the two phase modes disappear and two contours corresponding to ω = E + 2k + E + 1k
and ω = E + 2k + E − pk remain, as seen in Fig. 6(d). As the frequency is further increased to ω = 0.8U , an additional contour (the innermost) corresponding to ω = 2E + 1k appears in Fig. 6(e) along with the other excitations seen in Fig. 6(d). At ω = 0.85U , the contours in Fig 6(f) corresponds to ω = 2E + 1k and ω = E + 2k + E − pk , while at ω = U , Fig 6(g) has an additional innermost contour of ω = E + 1k + E − pk . Finally, in Fig 6(h), there are two contours corresponding to ω = 2E + 1k and ω = E + 1k + E − pk at ω = 1.15U , while at ω = 1.4U (Fig. 6(i) ), only the contour corresponding to ω = E + 1k + E − pk remains.The presence the gapless phase mode as well as four other gapped modes is clearly seen in Fig. 9 (a)-(c), where the lattice modulation response is plotted as a function of frequency along different radial cuts in the Brillouin zone making angles θ = π/2 , θ = π/4 and θ = π/6 with the k x axis respectively. The speed of sound in the system can be obtained from the slope of linearly dispersing phase modes seen in the figure. The gapless mode and the unique pattern of obtaining up to three contours at certain frequencies distinguishes the superfluid phase from the Mott phase and can be used to detect the superfluid-insulator quantum phase transition in this system. One can also obtain detailed information about the spectrum and relative spectral weights of the various modes from the lattice modulation response. We now consider the changes that appear in the optical modulation spectroscopy when the spin-orbit coupling is greater than the hopping amplitude and a BEC is formed at a finite momentum. To see this, in Fig. 7, we plot the dispersions of 2E + 1k , 2E + 2k , E + 1k + E + 2k , E − pk + E + 1k and E − pk + E + 2k for a system with following set of parameters: J = 0.01U , γ = 0.06U , Ω = 0.01U , µ = 0.2U and ζ = 0.4. In this case, the system is in a superfluid with a BEC at the momen-
tum [1.34a −1 , 1.34a −1 ]
where a is the lattice constant. Fig. 7 (a) shows 2E + 2k , which ranges from 0 at BEC wavevector (the gapless point) to 0.4U . All the other plots in Fig. 7 show similar features except the fact that these excitations are gapped. In Fig. 7 (b), E + 1k + E + 2k ranges from 0.1U to 0.42U . 2E + 1k ranges from 0.21U to 0.51U as shown in Fig. 7 (c). The excitations involving band transitions, E − pk + E + 2k ranges from 0.86U to 1.06U [ Fig. 7 (d)], and E − pk +E + 1k [ Fig. 7 (e)] ranges from 0.9U to 1.13U .
The optical modulation response for the above system is shown in Fig 8. Fig. 8(a) shows the response at ω = 0.02U , where a single contour of excitations corresponding to exciting two phase modes, ω = 2E + 2k is seen around the BEC wavevector [k 0 , k 0 ]. Fig. 8(b) shows the response at ω = 0.15U , where two contours of excitations are seen, the outer one corresponding to exciting two phase modes, ω = 2E + 2k , and the inner one corresponding to exciting a phase and an amplitude mode ω = E + 2k + E + 1k . Fig. 8(c) shows the response at ω = 0.25U , where three contours of excitations are seen, the outermost one corresponding to exciting two phase modes, ω = 2E + 2k , the middle one corresponding to exciting a phase and an amplitude mode ω = E + 2k + E + 1k , and the innermost contour corresponding to exciting two amplitude modes, ω = 2E + 1k . As the frequency is increased to ω = 0.42U , the two phase modes disappear and two contours corresponding to ω = E + 2k + E + 1k and ω = 2E + 2k remain, as seen in Fig. 8(d). As the frequency is further increased to ω = 0.5U , only the contour corresponding to ω = 2E + 1k appears in Fig. 8(e). As modulation frequency is increased further, the system shows no response, till the frequency reaches ω = 0.86U . Beyond this point, a contour of excitations corresponding to ω = E + 2k +E − pk appears around the BEC wavevector, as seen in Fig. 8(f) for ω = 0.87U . Beyond ω = 0.9U , a second contour, ω = E + 1k + E − pk appears in Fig. 8(g). Finally Fig. 8(h) shows the response at ω = 1.1U , where a single contour corresponding to ω = E + 1k + E − pk is present. The distinct pattern of the spectrum is better visualized in Fig. 9 (d)-(f) where the lattice modulation response is plotted as a function of frequency along different radial cuts in the Brillouin zone making angles θ = π/2 , θ = π/4 and θ = π/6 with the k x axis respectively. For the θ = π/2 and θ = π/6 cut, which does not pass through the condensate wavevector, all the spectra are gapped, while the θ = π/4 cut, which passes through the condensate location shows the gapless Goldstone mode. This clear qualitative distinction along various cuts can lead to a precise location of the condensate wavevector. In addition information about spectral weights can also be obtained from the lattice modulation signal.
V. DISCUSSION
In this work, we have studied a system of two species of interacting bosons in an optical lattice to modulation of the lattice potential. The spin states of the bosons are coupled through a spin-orbit coupling, which is implemented either by Raman dressing or by time-dependent magnetic field gradients. This system shows a superfluid-Mott insulator quantum phase transition as a function of increasing interaction strength. In addition, as a function of the relative strength of the spin-orbit coupling to the hopping amplitude, the system in the superfluid phase shows a transition from an ordinary superfluid phase with a BEC at the zone center to a twisted superfluid phase with a BEC at a finite momentum. We have provided a technique for differentiating between the different phases of such a system based on the response of such a system on to a modulating optical lattice.
In addition to finding the location of the precursor peaks in the MI phase and condensate position in the SF phase which can also be obtained by other standard experimental techniques such as time-of-flight measurements. However, lattice modulation spectroscopy provides several additional information. First, one can use this technique to map out the single-particle excitation spectrum of the bosons. In the Mott phase, it provides the effective mass of the dispersion around the minimum gap. In the superfluid phase, this technique not only shows the presence of gapless excitations, a quantitative estimate of the speed of sound v s (the slope of the gapless mode) and the mass of the Higgs (gapped) mode can be extracted from the experimental data. Further, since the matrix elements of the lattice modulation operator can be explicitly calculated within our technique, the spectral weight of the different modes can also be extracted from the modulation spectroscopy response. In this context, we would like to note that the expression of the response in terms of the spectral function is more general than the particular approximation used to calculate it in this paper. For example, in general the Higgs mode will develop a width due to decay to two phase modes. This can also be computed from the modulation spectroscopy response. Finally, we would like to point out that there is a technical advantage of our calculation over its counterpart for Bragg spectroscopy. This advantage stems from the fact that the optical modulation response, being a zero momentum transfer process, does not receive large contribution from the vertex correction terms. Thus, even if the poles in the single particle Greens function are broadened due to self energy corrections, the optical modulation response function will pick out the frequency convolution of the one particle spectral function at the same momentum. Thus this specific spectroscopic method provides direct access to single particle excitations in the system.
The experimental verification of our theory involves use of standard spectroscopy experiment techniques 23,24 . The specific experiment that we propose involves modulating J and γ by a laser creating an additional optical lattice with modulation frequency. After this modulation, one turns off the trap and the lattice and measures the position distribution of the outgoing bosons as done in any standard time-of-flight measurement. The position distribution of these bosons under standard experimental conditions 1 reproduces their momentum distribution n mod (k, ω) inside the trap (in the presence of the modulating lattice). We suggest a comparison of n mod (k, ω) to the momentum distribution n(k) of the bosons without the modulating lattice potential to obtain δn(k, ω) = n mod (k, ω) − n(k). We expect δn(k, ω) to provide the necessary information about the boson spectral function and carry the signature of the Mott and the superfluid states as described in Secs. III and IV. We note that realization of these experiments requires that the thermal smearing of the contours would be small enough to distinguish between the different phases. In the Mott phases this requires k B T 0.2U , where k B is the Boltzman constant and k B T * = 0.2U is the melting temperature of the MI phase 38 . This is readily achieved in standard experiments where U ∼ 2−5 KHz ∼ 200−500 nK. In the SF phase, the precise nature of the low-energy contours for the Goldstone modes may be difficult to discern since it would require a small T . However, the amplitude modes which occurs at finite energy scale would still be observable in a straightforward manner within current experimental resolution. An estimate of the maximal allowed thermal smearing for the SF phase comes from the criteria T T c where T c is the critical temperature of the SF phase which can be estimated to be k B T c z 0 Max[J, γ] 0.1U 20 − 50 nK. We note that T ∼ 1 nK have been achieved in standard ultracold atom experiments 38 .
In conclusion, we have shown, via explicit computation of δn(k, ω) in the strong coupling regime, that the response of a spin-orbit coupled Bose system to a modulated optical lattice can differentiate between (a) Mott and superfluid phases and (b) systems with finite momentum BEC and zero momentum BEC. The momentum resolved nature of the optical modulation spectroscopy, which provides information about the one particle spectral function of the system, resolves the superfluid phase from the insulator phase by presence/absence of gapless Goldstone modes in the two phases. Further, the pattern of excitation contours appearing in the Brillouin zone as the frequency is tuned is distinct in the two phases and further helps to distinguish the phases. The momentum resolution can also resolve states with condensates at finite momentum from states with condensate at the zone center by looking at the location of the low energy excitations, which are always centered around the momentum where the BEC forms. We have suggested concrete experiments which can verify our theory.
FIG. 1 .
1Density plots showing (a) intraband and (b) interband particle-hole excitation spectrum across the Brillouin zone in the Mott phase. The parameters of the system are J = 0.02U , γ = 0.01U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. The intraband excitations have energies between 0.24U to 0.57U , while the interband excitations have energies between 0.73U and 1.2U . (c) Intraband and (d) Interband particle-hole excitations in the Mott phase for J = 0.01U , γ = 0.04U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case minimum of intraband excitations are shifted from the zone center to (k0, k0) with k0 = 1.131a −1 . The figure shows the contours in the interval 0 ≤ kx, ky ≤ π; the contours in the other quadrants can be obtained by the reflection of the present figure around kx and ky axes and the origin. The intraband excitations have energies between 0.28U to 0.5U , while the interband excitations have energies between 0.87U and 1.1U . Note that the color scheme has same absolute value in all the panels to show which excitations overlap with each other in energy.
FIG. 2 .
2Lattice Modulation response in the Mott phase for different modulation frequencies. The parameters of the system are J = 0.02U , γ = 0.01U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. As the frequency of modulation crosses the Mott gap, a contour of intraband excitations are seen to disperse across the Brillouin zone ( a and b). Then, as the frequency crosses the interband threshold, another contour of interband excitations disperse across the Brillouin zone (c and d). Note that the spectral weight of the interband transitions are much smaller than the weights of other transitions.
FIG. 3 .
3Lattice Modulation response in the Mott phase for different modulation frequencies. The parameters of the system are J = 0.01U , γ = 0.04U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case minimum of intraband excitations are shifted from the zone center to (±k0, ±k0) with k0 = 1.131a −1 . The figure shows the contours in the interval 0 ≤ kx, ky ≤ π; the contours in the other quadrants can be obtained by the reflection of the present figure around kx and ky axes and the origin. As the frequency of modulation crosses the Mott gap, a contour of intraband excitations are seen to disperse across the Brillouin zone starting from a circle around [k0, k0] ( a and b). Then, as the frequency crosses the interband threshold, another contour of interband excitations disperse across the Brillouin zone starting from the zone center (c and d).
FIG. 4 .
4Lattice modulation response in the Mott phase with the variation in frequency for various cuts in the Brillouin zone going radially outwards at an angle θ with the kx axis. In (a) -(c) the parameters of the system are J = 0.02U , γ = 0.01U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4 with the band minimum at the zone center. (a) θ = π/2, (b) θ = π/4 and (c) θ = π/6. In (d) -(f) the parameters of the system are J = 0.01U , γ = 0.04U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case minimum of intraband excitations are shifted from the zone center to (k0, k0) with k0 = 1.131a −1 . (d) θ = π/2, (e) θ = π/4 and (f) θ = π/6.
FIG. 5 .
5Excitation spectrum in the superfluid phase. The parameters of the system are J = 0.03U , γ = 0.01U , µ = 0.35U , Ω = 0.01U and ζ = 0.4. In this case the BEC is formed at the zone center. The excitations are (a) two phase modes (b) one phase and one amplitude mode (c) one phase mode and one interband transition (d) two amplitude modes and (e) an amplitude mode and an interband transition. Note that the color scheme has same absolute value in all panels to show which excitations overlap with each other in energy.
FIG. 6 .
6Lattice modulation response in the superfluid phase. The parameters of the system are J = 0.03U , γ = 0.01U , µ = 0.35U ,Ω = 0.01U and ζ = 0.4. In this case the BEC is formed at the zone center. (a) A single contour corresponding to the two phase modes (b) 2 contours (outward from zone center): a phase and an amplitude mode, two phase modes (c) 3 contours (outward from zone center): a phase mode and an interband transition, a phase and an amplitude mode, two phase modes (d) 2 contours (outward from zone center): a phase mode and an interband transition, a phase and an amplitude mode (e) 3 contours (outward from zone center): two amplitude modes, a phase mode and an interband transition, a phase and an amplitude mode (f) 2 contours (outward from zone center): two amplitude modes, a phase mode and an interband transition (g) 3 contours (outward from zone center): an amplitude mode and an interband transition, two amplitude modes, a phase and an interband transition (h) 2 contours (outward from zone center): an amplitude mode and an interband transition, two amplitude modes (i) a single contour: an amplitude mode and an interband transition. Note that the spectral weight of the interband transitions are much smaller than the weights of other transitions.
FIG. 7 .
7Excitation spectrum in the superfluid phase. The parameters of the system are J = 0.01U , γ = 0.06U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case the BEC is formed at a finite momentum [±k0, ±k0], with k0 = 1.34a −1 , away from the zone center. The excitations are (a) two phase modes (b) one phase and one amplitude mode (c) one phase mode and one interband transition (d) two amplitude modes and (e) an amplitude mode and an interband transition. Note that the color scheme has same absolute value in all panels to show which excitations overlap with each other in energy.
FIG. 8 .
8full expression for the Green's functions are complicated, but, as in the case of the Mott phase, the only singularities of the Green's functions are simple poles. So, as far as the imaginary part of the Green's function is concerned, they are faithfully reproduced by Lattice modulation response in the superfluid phase. The parameters of the system are J = 0.01U , γ = 0.06U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case the BEC is formed at a finite momentum [k0, k0] (k0 = 1.34a −1 ) away from the zone center. A single contour corresponding to the two phase modes (b) 2 contours (outward from zone center): a phase and an amplitude mode, two phase modes (c) 4
FIG. 9 .
9Lattice modulation response in the superfluid phase with the variation in frequency for various cuts in the Brillouin zone going radially outwards at an angle θ with the kx axis. In (a) -(c) the parameters of the system are J = 0.03U , γ = 0.01U , µ = 0.35U ,Ω = 0.01U and ζ = 0.4 with the condensate at the zone center. (a) θ = π/2, (b) θ = π/4 and (c) θ = π/6. All the five branches including the gapless linear Goldstone mode near k = 0 is clearly seen. In (d) -(f) the parameters of the system are J = 0.01U , γ = 0.06U , µ = 0.2U ,Ω = 0.01U and ζ = 0.4. In this case the condensate is located at (k0, k0) with k0 = 1.131a −1 . (d) θ = π/2, (e) θ = π/4 and (f) θ = π/6. Only the cut along[1,1] direction passes through the condensate location and shows the gapless Goldstone mode, while the spectrum is gapped along the other cuts.
(b), E + 1k + E + 2k ranges from 0.37U to 0.83U . Similarly E − pk + E + 2k [Fig. 5 (c)] ranges from 0.51U to 1.2U , 2E + 1k [Fig. 5 (d)] ranges from 0.74U to 1.1U , and E − pk + E + 1k [Fig. 5 (e)] ranges from 0.88U to 1.5U .
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We neglect the fact that the Raman dressed "spin states" are slightly different from the original spin states. We neglect the fact that the Raman dressed "spin states" are slightly different from the original spin states.
the coefficients δγ/γ and δV /V would be slightly different. In principle one can also modulate only the spin-orbit term by modulating the Raman lasers. These cases will change the matrix elements in our calculations a little bit. In reality, when the deep lattice assumption is relaxed. but the main conclusions remain unaffectedIn reality, when the deep lattice assumption is relaxed, the co- efficients δγ/γ and δV /V would be slightly different. In princi- ple one can also modulate only the spin-orbit term by modulating the Raman lasers. These cases will change the matrix elements in our calculations a little bit, but the main conclusions remain unaffected.
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| []
|
[
"V us determination from kaon decays",
"V us determination from kaon decays"
]
| [
"P Massarotti \nUniversita' degli studi di Napoli \"Federico II\"\nINFN Napoli\nNaplesItaly\n"
]
| [
"Universita' degli studi di Napoli \"Federico II\"\nINFN Napoli\nNaplesItaly"
]
| [
"Heavy Quarks and Leptons"
]
| This review presents the analysis of leptonic and semileptonic kaon decays data done by the FlaviaNet Kaon Working group, as described in[1]. Data include all recent results by BNL-E865, KLOE, KTeV, ISTRA+, and NA48. Experimental results are critically reviewed and combined, taking into account theoretical (both analytical and numerical) constraints on the semileptonic kaon form factors. We report on a very accurate determination of Vus as well as on many other tests of the SM which can be performed with leptonic and semileptonic kaon decays. * WWW access at www.lnf.infn.it/wg/vus/; for sake of completeness and brevity for all references we refer to the Note written by the FlaviaNet Kaon WG [1]. | null | [
"https://arxiv.org/pdf/1002.1654v1.pdf"
]
| 118,472,098 | 1002.1654 | f941f07d2c29ca0bef74498dedaf51f699b5eef7 |
V us determination from kaon decays
2008
P Massarotti
Universita' degli studi di Napoli "Federico II"
INFN Napoli
NaplesItaly
V us determination from kaon decays
Heavy Quarks and Leptons
Melbourne20081
This review presents the analysis of leptonic and semileptonic kaon decays data done by the FlaviaNet Kaon Working group, as described in[1]. Data include all recent results by BNL-E865, KLOE, KTeV, ISTRA+, and NA48. Experimental results are critically reviewed and combined, taking into account theoretical (both analytical and numerical) constraints on the semileptonic kaon form factors. We report on a very accurate determination of Vus as well as on many other tests of the SM which can be performed with leptonic and semileptonic kaon decays. * WWW access at www.lnf.infn.it/wg/vus/; for sake of completeness and brevity for all references we refer to the Note written by the FlaviaNet Kaon WG [1].
I. INTRODUCTION
In the Standard Model, SM, transition rates of semileptonic processes such as d i → u j ℓν, with d i (u j ) being a generic down (up) quark, can be computed with high accuracy in terms of the Fermi coupling G F and the elements V ji of the Cabibbo-Kobayashi Maskawa (CKM) matrix. Measurements of the transition rates provide therefore precise determinations of the fundamental SM couplings A detailed analysis of semileptonic decays offers also the possibility to set stringent constraints on new physics scenarios. While within the SM all d i → u j ℓν transitions are ruled by the same CKM coupling V ji (satisfying the unitarity condition k |V ik | 2 = 1) and G F is the same coupling appearing in the muon decay, this is not necessarily true beyond the SM. Setting bounds on the violations of CKM unitarity, violations of lepton universality, and deviations from the V − A structure, allows us to put significant constraints on various new-physics scenarios (or eventually find evidences of new physics). In the case of leptonic and semileptonic kaon decays these tests are particularly significant given the large amount of data recently collected by several experiments: BNL-E865, KLOE, KTeV, ISTRA+, and NA48. The analysis of these data provides precise determination of fundamental SM couplings, sets stringent SM test almost free from hadronic uncertainties, and finally can discriminate between new physics scenarios. The high statistical precision of measurements and the detailed information on kinematical distributions have provided substantial progress on the theory side, in particular the theoretical error on hadronic form factors has been reduced at the 1% level. The paper is organized as follows. First in Sec. II we present fits to world data on the leading branching ratios and lifetimes, for K L , K S , and K ± mesons. Sec. III summarizes the status of the knowledge of form factor slopes from K ℓ3 decays. The physics re- sults obtained are described in Sec. IV, in particular the measurement of |V us f + (0)|.
II. EXPERIMENTAL DATA: BRS AND LIFETIME
Numerous measurements of the principal kaon BRs, or of various ratios of these BRs, have been published recently. For the purposes of evaluating |V us f + (0)|, these data can be used in a PDG-like fit to the BRs and lifetime. A detailed description to the fit procedure and the references of all experimental input used can be found in Ref. [1]. For K L the results are given in table I, while table II gives the results for K ± . For the K S , the fit is dominated by the KLOE measurements of BR(K S → πeν) and of BR(π + π − )/BR(π 0 π 0 ). These, together with the constraint that the K S BRs must add to unity, and the assumption of universal lepton couplings, completely determine the K S leading BRs. In particular, BR(K S → πeν) = 7.046(91) × 10 −4 . For τ KS we use 8.958 ×10 −11 s, where this is the non-CP T constrained fit value from the PDG, [2].
The fit takes into account the correlation between these values, as well as their dependence on the K ± lifetime. The world average value for τ ± is nominally quite precise; the 2006 PDG quotes τ ± = The hadronic K → π matrix element of the vector current is described by two form factors (FFs), f + (t) and f 0 (t), defined by π − (k) |sγ µ u|K 0 (p) =
(p + k) µ f + (t) + (p − k) µ f − (t) and f − (t) = m 2 K −m 2 π t f 0 (t) − f + (t) where t = (p − k) 2 .
By construction, f 0 (0) = f + (0). In order to compute the phase space integrals we need experimental or theoretical inputs to determinate the t-dependence of FF. In principle, Chiral Perturbation Theory (ChPT) and Lattice QCD are useful tools to set theoretical constraints. However, in practice the t-dependence of the FFs at present is better determined by measurements and by combining measurements and dispersion relations. Many approaches have been used, and all have been described in detail in [1]. For K e3 decays, recent measurements of the quadratic slope parameters of the vector form factor (λ ′ + , λ ′′ + ) are available from KTeV, KLOE, ISTRA+, and NA48. The same collaborations recentely measured also the slope parameters (λ ′ + , λ ′′ + , λ 0 ) for K µ3 decays Here we list only the averages of quadratic fit results for K e3 and K µ3 slopes (III) used to determine |V us |f + (0). It is important to stress that the significance of the quadratic term in the vector form factor is strong for both K e3 (4.2σ) and K µ3 (3.6σ) fit to all data.
χ 2 /ndf 54/13 (7 × 10 −7 ) λ ′ + × 10 3 24.9 ± 1.1 (S = 1.4) λ ′′ + × 10 3 1.6 ± 0.5 (S = 1.3) λ0 × 10 3 13.4 ± 1.2 (S = 1.9) ρ(λ ′ + , λ ′′ + ) −0.94 ρ(λ ′ + , λ0) +0.33 ρ(λ ′′ + , λ0) −0.44 I(K 0 e3 ) 0.15457(29) I(K ± e3 ) 0.15892(30) I(K 0 µ3 ) 0.10212(31) I(K ± µ3 ) 0.10507(32) ρ(Ie3, Iµ3) +0.63 mode |Vus|f+(0) % err BR τ ∆ Int KL → πeν
IV. PHYSICS RESULTS
A. Determination of |Vus|f+(0) and |Vus|/|V ud |fK /fπ
The value of |V us |f + (0) has been determined from the decay rate of kaon semileptonic decays:
Γ(K ℓ3(γ) ) = G 2 µ M 5 K 192π 3 C K S ew |V us | 2 f + (0) 2 ×(1)I ℓ K (λ +,0 ) 1 + δ K SU(2) + δ Kℓ em 2
using the world average values reported in previous sections for lifetimes, branching ratios and phase space integrals and the radiative and SU (2) breaking corrections discussed in [1].
The results are shown in figure 1 and given in Table IV, for K L → πeν, K L → πµν, K S → πeν, K ± → πeν, K ± → πµν, and for the combination. The average, |V us |f + (0) = 0.21664(48), has an uncertainty of about of 0.2%. The results from the five modes are in good agreement, the fit probability is 58%. In particular, comparing the values of |V us |f + (0) obtained from K 0 ℓ3 and K ± ℓ3 we obtain a value of the SU(2) breaking correction δ K SU(2)exp. = 2.9(4)% in agreement with the CHPT calculation δ K SU(2) = 2.36(22)%. Moreover, recent analyzes on the so-called violations of Dashen's theorem in the kaon electromagnetic mass splitting point to δ K SU (2) values of about 3%. The test of Lepton Flavor Universality (LFU) between K e3 and K µ3 modes constraints a possible anomalous lepton-flavor dependence in the leading weak vector current. It can therefore be compared to similar tests in τ decays, but is different from the LFU tests in the helicity-suppressed modes π l2 and K l2 . The results on the parameter r µe = R Exp Kµ3/Ke3 /R SM Kµ3/Ke3 is r µe = 1.0043 ± 0.0052, in excellent agreement with lepton universality. With a precision of 0.5% the test in K l3 decays has now reached the sensitivity of other determinations: r µe (τ ) = 1.0005 ± 0.0041 and r µe (π) = 1.0042 ± 0.0033 [2] An independent determination of V us is obtained from K ℓ2 decays. The most important mode is K + → µ + ν, which has been recently updated by KLOE reaching a relative uncertainty of about 0.3%. Hadronic uncertainties are minimized considering the ratio Γ(K + → µ + ν)/Γ(π + → µ + ν):
0.215 0.2175 f + (0) V us K Le3 K Lm3 K Se3 K ± e3 K ± m3Γ(K ± ℓ2(γ) ) Γ(π ± ℓ2(γ) ) = V us V ud 2 f 2 K m K f 2 π m π 1 − m 2 ℓ /m 2 K 1 − m 2 ℓ /m 2 π 2 ×(1 + δ em )
Using the world average values of BR(K ± → µ ± ν) and of τ ± given in Section II and the value of Γ(π ± → µ ± ν) = 38.408(7) µs −1 from [2] we obtain: |V us |/|V ud |f K /f π = 0.2760 ± 0.0006.
B. Theoretical estimates of f+(0) and fK /fπ
The main obstacle in transforming these highly precise determinations of |V us |f + (0) and |V us |/|V ud |f K /f π into a determination of |V us | at the precision of 0.1%, are the theoretical uncertainties on the hadronic parameters f + (0) and f K /f π . By construction, f + (0) is defined in the absence of isospin-breaking effects of both electromagnetic and quark-mass origin. More explicitly f + (0) is defined by the K 0 → π + matrix element of the vector current in the limit m u = m d and α em → 0, keeping kaon and pion masses to their physical values. This hadronic quantity cannot be computed in perturbative QCD, but it is highly constrained by SU (3) and chiral symmetry. In the chiral limit and, more generally, in the SU Several analytical approaches to determine f 4 have been attempted over the years, essentially confirming the original estimate by Leutwyler and Roos. The benefit of these new results, obtained using more sophisticated techniques, lies in the fact that a better control over the systematic uncertainties of the calculation has been obtained. However, the size of the error is still around or above 1%, which is not comparable to the 0.2% accuracy which has been reached for |V us |f + (0). Recent progress in lattice QCD gives us more optimism in the reduction of the error on f + (0) below the 1% level. Most of the currently available lattice QCD results have been obtained with relatively heavy pions and the chiral extrapolation represents the dominant source of uncertainty. There is a general trend of lattice QCD results to be slightly lower than analytical approaches. An important step in the reduction of the error associated to the chiral extrapolation has been recently made by the UKQCD-RBC collaboration. Their preliminary result f + (0) = 0.964(5) is obtained from the unquenched study with N F = 2 + 1 flavors, with an action that has good chiral properties on the lattice even at finite lattice spacing (domainwall quarks). They also reached pions masses (≥ 330 MeV) much lighter than that used in previous studies of f + (0). The overall error is estimated to be 0.5%, which is very encouraging. In contrast to the semileptonic vector form factor, the pseudoscalar decay constants are not protected by the Ademollo-Gatto theorem and receive corrections lin- ear in the quark masses. Expanding f K /f π in power of quark masses, in analogy to f + (0), f K /f π = 1+r 2 +. . . one finds that the O(p 4 ) contribution r 2 is already affected by local contributions and cannot be unambiguously predicted in ChPT. As a result, in the determination of f K /f π lattice QCD has essentially no competition from purely analytical approaches. The present overall accuracy is about 1%. The novelty are the new lattice results with N F = 2 + 1 dynamical quarks and pions as light as 280 MeV, obtained by using the so-called staggered quarks. These analyzes cover a broad range of lattice spacings (i.e. a=0.06 and 0.15 fm) and are performed on sufficiently large physical volumes (m π L ≥ 5.0). It should be stressed, however, that the sensitivity of f K /f π to lighter pions is larger than in the computation of f + (0) and that chiral extrapolations are far more demanding in this case. In the following analysis we will use as reference value the MILC-HPQCD result f K /f π = 1.189(7).
C. Test of CKM unitarity
To determine |V us | and |V ud | we use the value |V us |f + (0) = 0.2166(5), the result |V us |/|V ud |f K /f π = 0.2760(6), f + (0) = 0.964(5), and f K /f π = 1.189(7). From the above we find: |V us | = 0.2246 ± 0.0012 from K ℓ3 only, and |V us |/|V ud | = 0.2321 ± 0.0015 from K ℓ2 only. These determinations can be used in a fit together with the the recent evaluation of V ud from 0 + → 0 + nuclear beta decays: |V ud |=0.97418±0.00026. This global fit gives V ud = 0.97417(26) and V us = 0.2253(9), with χ 2 /ndf = 0.65/1 (42%). This result does not make use of CKM unitarity. If the unitarity constraint is included, the fit gives V us = 0.2255(7) and χ 2 /ndf = 0.80/2 (67%). Both results are illustrated in Figure 2. The test of CKM unitarity can be also interpreted as a test of universality of the lepton and quark gauge couplings. Using the results of the fit (without imposing unitarity) we obtain: G CKM ≡ G µ |V ud | 2 + |V us | 2 + |V ub | 2 1/2 = (1.1662 ± 0.0004) × 10 −5 GeV −2 , in perfect agreement with the value obtained from the measurement of the muon lifetime: G µ = (1.166371 ± 0.000007) × 10 −5 GeV −2 . The current accuracy of the lepton-quark universality sets important constraints on model building beyond the SM. For example, the presence of a Z ′ would affect the relation between G CKM and G µ . In case of a Z ′ from SO(10) grand unification theories we obtain m Z ′ > 700 GeV at 95% CL, to be compared with the m Z ′ > 720 GeV bound set through the direct collider searches [2]. In a similar way, the unitarity constraint also provides useful bounds in various supersymmetrybreaking scenarios.
D. K ℓ2 sensitivity to new physics
A particularly interesting test is the comparison of the |V us | value extracted from the helicity-suppressed K ℓ2 decays with respect to the value extracted from the helicity-allowed K ℓ3 modes. To reduce theoretical uncertainties from f K and electromagnetic corrections in K ℓ2 , we exploit the ratio BR(K ℓ2 )/BR(π ℓ2 ) and we study the quantity
R l23 = V us (K ℓ2 ) V us (K ℓ3 ) V ud (0 + → 0 + ) V ud (π ℓ2 ) .
Within the SM, R l23 = 1, while deviation from 1 can be induced by non-vanishing scalar-or right-handed currents. Notice that in R l23 the hadronic uncertainties enter through (f K /f π )/f + (0). In the case of effect of scalar currents due to a charged Higgs, the unitarity relation between |V ud | extracted from 0 + → 0 + nuclear beta decays and |V us | extracted from K ℓ3 remains valid as soon as form factors are experimentally determined. This constrain together with the experimental information of log C MSSM can be used in the global fit to improve the accuracy of the determination of R l23 , which in this scenario turns to be R l23 | exp scalar = 1.004 ± 0.007. Here (f K /f π )/f + (0) has been fixed from lattice. This ratio is the key quantity to be improved in order to reduce present uncertainty on R l23 . This measurement of R l23 can be used to set bounds on the charged Higgs mass and tan β. Figure 3 shows the excluded region at 95% CL in the M Htan β plane. The measurement of BR(B → τ ν) can be also used to set a similar bound in the M H -tan β plane. While B → τ ν can exclude quite an extensive region of this plane, there is an uncovered region in the exclusion corresponding to a destructive interference between the charged-Higgs and the SM amplitude. This region is fully covered by the K → µν result.
E. A test of lattice calculation
The vector and scalar form factors f +,0 (t) are analytic functions in the complex t-plane, except for a cut along the positive real axis, starting at the first physical threshold t th = (m K + m π ) 2 , where they develop discontinuities. They are real for t < t th . Cauchy's theorem implies that f +,0 (t) can be written as a dispersive integral along the physical cut where all possible on-shell intermediate states contribute to its imaginary part. A number of subtractions is needed to make the integral convergent. Particularly appealing is an improved dispersion relation recently proposed where two subtractions are performed at t = 0 (where by definition,f 0 (0) ≡ 1) and at the so-called Callan-Treiman point t CT ≡ (m 2 K − m 2 π ). Since the Callan-Treiman relation fixes the value of scalar form factor at t CT to the ratio (f K /f π )/f + (0), the dispersive parametrization for the scalar form factor allows to transform the available measurements of the scalar form factor into a precise information on (f K /f π )/f + (0), completely independent of the lat-tice estimates. Figure 4 shows the values for f + (0) determined from the scalar form factor slope measurements obtained using a dispersive parametrization and the Callan-Treiman relation, and f K /f π = 1.189(7). from result on the FF slope using the dispersive parameterization The value of f + (0) = 0.964(5) from UKQCD/RBC is also shown. The NA48 result is difficult to accommodate. Here one can see that this result is also not consistent with the theoretical estimates of f + (0). In particular, it violates the Fubini-Furlan bound f + (0) < 1. For this reason, the NA48 result will be excluded when using the Callan-Treiman constraint.
FIG. 1 :
1Display of |Vus|f+(0) for all channels.
u = m d = m s ) the conservation of the vector current implies f + (0)=1. Expanding around the chiral limit in powers of light quark masses we can write f + (0) = 1 + f 2 + f 4 + . . . where f 2 and f 4 are the NLO and NNLO corrections in ChPT. The Ademollo-Gatto theorem implies that (f + (0) − 1) is at least of second order in the breaking of SU (3) This in turn implies that f 2 is free from the uncertainties of the O(p 4 ) counter-terms in ChPT, and it can be computed with high accuracy: f 2 = −0.023. The difficulties in estimating f + (0) begin with f 4 or at O(p 6 ) in the chiral expansion.
FIG. 3 :
3Excluded region in the charged Higgs mass-tan β plane. The region excluded by B → τ ν is also indicated.
FIG. 4 :
4Values for f+(0) determined from the scalar form factor slope using the Callan-Treiman relation and fK /fπ = 1.189(7).
TABLE I :
IResults of fit to KL BRs and lifetime. S is the scale factor applied to the error in order to obtain chi 2 = ndf.Parameter
Value
S
BR(Ke3)
0.4056(7)
1.1
BR(Kµ3)
0.2705(7)
1.1
BR(3π 0 )
0.1951(9)
1.2
BR(π + π − π 0 )
0.1254(6)
1.1
BR(π + π − )
1.997(7) ×10 −3 1.1
BR(2π 0 )
8.64(4) ×10 −4 1.3
BR(γγ)
5.47(4) ×10 −4 1.1
τL
51.17(20) ns 1.1
TABLE II :
IIResults of fit to K ± BRs and lifetime.Parameter
Value
S
BR(Kµ2)
63.57(11)% 1.1
BR(ππ 0 )
20.64(8)% 1.1
BR(πππ)
5.595(31)% 1.0
BR(Ke3)
5.078(26)% 1.2
BR(Kµ3)
3.365(27)% 1.7
BR(ππ 0 π 0 ) 1.750(26)% 1.1
τ±
12.384(19) ns 1.7
12.385(25) ns. However, the error is scaled by 2.1;
the confidence level for the average is 0.17%. The two
new measurements from KLOE [3] agree with the
PDG average, and give a smaller scale factor to the
τ ± value.
III. EXPERIMENTAL DATA: K ℓ3 FORM
FACTORS
TABLE III :
IIIAverages of quadratic fit results for Ke3 and Kµ3 slopes.KL and K −
Measurements
16
TABLE IV :
IVSummary of |Vus|f+(0) determination from
all channels.
FIG. 2: Results of fits to |V ud |, |Vux|, and |Vus|/|V ud |.0.225
0.230
0.970
0.975
V ud
V us
V ud (0
+
® 0
+ )
V us (K l3 )
fit with unitarity
fit
V us
/Vud (K m2
)
un ita rit y
lavi
net Kaon WG
f + (0) = 0.9644(49)
f K /f p = 1.189(7)
AcknowledgmentsThis document is adapted from the instructions provided to the authors of the proceedings papers at FPCP 06, Vancouver, Canada [4], and from eConf templates [5].
arXiv:0801.1817[hep-ph]11Precision test of the Standard Model with leptonic and semileptonic kaon decays. Precision test of the Standard Model with leptonic and semileptonic kaon decays arXiv:0801.1817[hep-ph] 11
W.-M Yao, Particle Data GroupG3320061 and 2007 web updates. W.-M. Yao. et al. (Particle Data Group), J. Phys. G3320061 and 2007 web updates.
. F Ambrosino, KLOE CollaborationJ. High Energy Phys. F. Ambrosino, et al. (KLOE Collaboration), J. High Energy Phys.012008073.
| []
|
[
"On World Religion Adherence Distribution Evolution",
"On World Religion Adherence Distribution Evolution"
]
| [
"M Ausloos \nGRAPES\nB5 Sart-Tilman\nB-4000LiègeEuroland\n",
"F Petroni \nGRAPES\nUniversité de Liège\nB5 Sart-Tilman\nB-4000LiègeBelgium\n\nUniversitá dell'Aquila\nI-67010L'AquilaItaly\n"
]
| [
"GRAPES\nB5 Sart-Tilman\nB-4000LiègeEuroland",
"GRAPES\nUniversité de Liège\nB5 Sart-Tilman\nB-4000LiègeBelgium",
"Universitá dell'Aquila\nI-67010L'AquilaItaly"
]
| []
| Religious adherence can be considered as a degree of freedom, in a statistical physics sense, for a human agent belonging to a population. The distribution, performance and life time of religions can thus be studied having in mind heterogeneous interacting agent modeling in mind. We present a comprehensive analysis of 58 so called religion (to be better defined in the main text) evolutions, as measured through their number of adherents between 1900 and 2000, -data taken from the World Christian Encyclopedia: 40 are considered to be "presently growing" cases, including 11 turn overs in the XX century; 18 are "presently decaying", among which 12 are found to have had a recent maximum, in the XIX or the XX century. The Avrami-Kolmogorov differential equation which usually describes solid state transformations, like crystal growth, is used in each case in order to obtain the preferential attachment parameter introduced previously [1]. It is often found close to unity, indicating a smooth evolution. However large values suggest the occurrence of extreme cases which we conjecture are controlled by so called external fields. A few cases indicate the likeliness of a detachment process. We discuss different growing and decaying religions, and illustrate various fits. Some cases seem to indicate the lack of reliability of the data. Others, departure from Avrami law. We point out two difficulties in the analysis : (i) the "precise" original time of apparition of a religion, (ii) the time of its maximum, both informations being necessary for integrating reliably any evolution equation. Moreover the Avrami evolution equation might be surely improved, in particular, and somewhat obviously, for the decaying religion cases. | 10.1007/978-4-431-53853-0_15 | [
"https://arxiv.org/pdf/0801.1010v1.pdf"
]
| 119,297,643 | 0801.1010 | 9ca89a7583b92d5cf89ab772ecd76ca3a4e53f31 |
On World Religion Adherence Distribution Evolution
M Ausloos
GRAPES
B5 Sart-Tilman
B-4000LiègeEuroland
F Petroni
GRAPES
Université de Liège
B5 Sart-Tilman
B-4000LiègeBelgium
Universitá dell'Aquila
I-67010L'AquilaItaly
On World Religion Adherence Distribution Evolution
dynamicsopinion formationreligionsociophysics PACS numbers:
Religious adherence can be considered as a degree of freedom, in a statistical physics sense, for a human agent belonging to a population. The distribution, performance and life time of religions can thus be studied having in mind heterogeneous interacting agent modeling in mind. We present a comprehensive analysis of 58 so called religion (to be better defined in the main text) evolutions, as measured through their number of adherents between 1900 and 2000, -data taken from the World Christian Encyclopedia: 40 are considered to be "presently growing" cases, including 11 turn overs in the XX century; 18 are "presently decaying", among which 12 are found to have had a recent maximum, in the XIX or the XX century. The Avrami-Kolmogorov differential equation which usually describes solid state transformations, like crystal growth, is used in each case in order to obtain the preferential attachment parameter introduced previously [1]. It is often found close to unity, indicating a smooth evolution. However large values suggest the occurrence of extreme cases which we conjecture are controlled by so called external fields. A few cases indicate the likeliness of a detachment process. We discuss different growing and decaying religions, and illustrate various fits. Some cases seem to indicate the lack of reliability of the data. Others, departure from Avrami law. We point out two difficulties in the analysis : (i) the "precise" original time of apparition of a religion, (ii) the time of its maximum, both informations being necessary for integrating reliably any evolution equation. Moreover the Avrami evolution equation might be surely improved, in particular, and somewhat obviously, for the decaying religion cases.
I. INTRODUCTION
Religion like sex, age, wealth, political affiliation, language, ... can be considered to characterize a group or an individual status. Whence religion or sex, age, wealth, political affiliation, language distributions can be studied as a function of time, space, auto-correlated, or correlated with any other variable or "parameter" characterizing a population, going toward socio-economic studies pertaining to attitudes, behaviors, opinion formations [2], ... Several interesting considerations well known in statistical physics can be found in most sociological systems : the role of nucleation, growth, aging, death, criticality, self-organization, epidemic spreading, and subsequent avalanches. If some geometric-like transition or some thermodynamical-like transition exists then fluctuations should be seen.
Recently the dynamics of world's languages, especially on their disappearing due to competition with other languages [3] has been of interest [4] in such a respect. One of our aims has been recently to approach a set of similar questions on religions, through a statistical physics point of view, attempting to quantify religion dynamics as seen from individual adherence distribution functions [1]. We emphasize that we are not interested here in any religion's origin, history or in finding any hierarchy, but rather in the statistical physics-like aspects of a complex non-equilibrium biological agent based system [5,6].
History is full of examples of individuals or entire groups of people changing their religion, -for various reasons: following the "leader", e.g. Constantinus, Clovis, ... or "external pressure" , leading to martyrdom, or "conversely" like at inquisition time, or following a fatwah, ... or "internal pressure" (Khazars, ...) or so called adaptation under proselytism action, e.g. sun worshipping Incas in presence of catholic missionaries, zoroastrian Persians in presence of muslim arabs, ... "Competition" through interactions or under "external field conditions" exist in many cases. In so doing the number of adherents can much evolve due to such various conditions [7]. However notice that external field conditions can be rather more drastic in the religious domain than in language history [8]. See also Appendix A for some discussion outlining a few aspects, i.e. "differences" between languages and religions, from a physics point of view, perspective or input into modeling such sociological features.
We consider as the fundamentally relevant variable the number of adherents of each religion [9] Only this number is treated as the physics object. Thus a religion is hereby considered as a (socially based) variable, like a language or wealth, to be so studied like any other organizational parameter for defining a thermodynamic-like state. We recognize that a religious state [10] is more individualistic than a linguistic state. Thus, in some sense one can better define the religious adherence of an agent than the linguistic one. Indeed one can hardly be multi-religious but one can be a polyglot. Of course one can switch more easily, i.e. through "conversion" from one religious denomination to another than in language cases. Thus the observation time of a religious state needs careful attention in surveys.
From another point of view, time and time scales, one can notice that a religion can seem to appear rather instantaneously, often as a so called sect, at the beginning, and its number of adherent can grow steadily (see the recent Mormon or Rastafarianism case) or not; a religion can also "rather quickly"' disappear (see the Antoinists in some coal mine regions of Western Europe), -in both cases for quite "interesting" reasons or causes, actually outside the realm of this paper. Thus the time life, aging, of a religion can be studied through the number of adherents, surely for modern times, -with some caution.
In so doing several pertinent questions can be raised, e.g. from a "macroscopic" point of view : (i) how many religions exist at a given time? (ii) how are they spatially distributed ? ... From a "microscopic" view point: (iii) How many adherents belong to one religion? (iv) Does the number of adherents increase or not, and how? -and maybe why? (!), (v) Last but not least is there some modelization, ... some agent based model possible?
We recognize that the definition of a religion or an adherent (or adept) might not be accepted univocally, but the same can be said about languages; we recognize that there are various denominations which can impair data gathering and subsequent analysis; like many, we admit to put on the same footing religions, philosophies, sects and rituals. Idem, we do not distinguish between adherents or adepts; there are also agnostics, atheists or "not concerned". In fact, a similar set of considerations exists when discussing languages and dialects, slangs, etc. There are e.g. three definitions of a language [11]. Similarly one could "weight" the level of adherence to a religion, one could try as for languages to define a religion through its rituals, and quantity of practitioners. Many other indicators are possible (see Appendix B). To consider such variants would lead us too far away from the main stream of the present research and is left for further investigations when possible.
Thus to address some of these issues, we have followed classical scientific steps as in physics investigations [1]. We have accepted as such and subsequently analyzed "empirical" data on the number of adherents of religions. We have discussed in [1] two different freely available data sets. The exactness of both data sets from an experimental (laboratory or naturally based) physics point of view is debatable. Some discussion will be rejuvenated in Sec. II. Yet, it has been found in [1] that empirical laws can be deduced for the number of adherent, I.e. the probability distribution function (pdf). Two quite different statistical models were proposed, both reproducing well the data, with the same precision, one being a preferential attachment model [12], like for heterogeneous interacting agents on evolving networks, e.g. it is more likely that one has the religion of one's mother or neighbor..... (leading to a log-normal distribution), another based on a "time of failure" argument (leading to a Weibull distribution function).
Moreover, a population growth-death equation has been conjectured to be a plausible modeling of the evolution dynamics in a continuous time framework, i.e. the time evolution of several "main" religions, from a microscopic interpretation is plausible along the lines of the growth Avrami-Kolmogorov equation describing solid state formation in a continuous time framework, which solution is usually written as
F (t) = 1 − exp[−Kt n ](1)
where F (t) is the volume fraction being transformed from one phase to another; K and n are adjustable parameters (Fig. 1). For n = 1, this equation reproduces the loading of a capacitance in series with a resistance R, for which the differential equation for the voltage V (t) across the capacitance C reads
d dt V (t) = E − V RC(2)
in terms of the emf E, and for which one remembers that one interprets RC as a relaxation time τ . It is also the behavior of the Verhulst logistic map above the inflection point; indicating that this Avrami equation is of interest for so called late stage growth, i.e.F For n = 1, Eq.(1) can correspond to complex non linear electronic circuits containing an infinity of elements, or also to an infinite combination of springs and damping elements [13] in mechanics.
(t) = 1 1 + exp[−Kt](3)
A priori in analogy with crystal growth studies [14,15], we have considered that a microscopic-like, continuous time differential equation can be written for the evolution of the number of adherents, in terms of the percentage with respect to the world population, of the world main religions, as for competing phase entities in Avrami sense
d dt g(t) = γt −h [1 − g(t)].(4)
It can be solved easily giving the evolution equation for the fraction g(t) of religion adherents
g(t) = 1 − ηe − γ 1−h t 1−h(5)
where, adapting to our case this Eq. (4), η is related to the initial condition, γ is a (positive for growth process) rate (or scaling) parameter to be determined, see discussion in Sect. III, and h is a parameter to be deduced in each case, measuring the attachment-growth (or death) process in this continuous time approximation. The relaxation time τ n , since n ≡ 1 − h, of this stretched exponential growth is
τ n = ( γ 1 − h ) −1 1−h(6)
which is markedly rate (K) dependent. For further consideration, let us explicitly write the "very (infinitely) slow" growth case h=1, i.e.,
d dt g(t) = γt −1 [1 − g(t)],(7)
whence
g(t) = 1 − βt −γ ,(8)
where β, being positive (negative) for a growth (decay) case, is set by initial conditions; for h = 1, there is no "relaxation time", but a scaling time τ 1 = β 1 γ , or β = τ γ 1 . The h-cases which can be illustrated through an Avrami equation are shown in arbitrary time units in Fig.1 for various h values, for η =1 and γ = 1 − h. They are compared to the (generalized, i.e. n = 1) logistic map.
What should be emphasized is the fact that religions have appeared at some time t 0 which is somewhat unknown, or to say the least, loosely defined, due to a lack of historical facts but also due to the inherent process of the creation of a religion. Yet this initial time is a parameter much more important than in crystal growth, but less well defined. Therefore we rewrite the Avrami equation as
g(t) = 1 − e − " t−t 0 t 1 " 1−h ,(9)
thereby allowing also for a time scaling through t 1 related to some growth (or death) rate process. Notice or so that the maximum in such theoretical laws occurs at zero or +/-infinity, -a time information on which there is not much data in the case of religions. If (t − t 0 )/t 1 is much smaller than 1, Eq. (9) can be expanded in Taylor series, taking only the first order, and gives
g(t) = α + t t 1 1−h (10)
where we have chosen t starting from 0 (instead of 1900, as in our data, being this completely arbitrary and purely conventional) and α represent the initial condition, i.e. the value of the number of adherents for t = 0.
A few examples of religions for which the number of adherents is increasing (e.g., Islam), decaying (e.g., Ethnoreligions) or rather stable (e.g., Christianity and Buddhism) is already shown in Fig. 4 of [1]. In such cases we have found that h -1.8, 6.9, 1.5 and 1.4, respectively in the time range of interest (1900-2050). However in [1] the main denominations were "loosely grouped". To be more specific: Christians in [1] were the results of grouping together 12 denominations; similarly for Muslims we grouped 15 denominations.
Here we present a more complete and somewhat more detailed analysis of the values of h and its meaning for 58 "time series", where 58 means "55"+"1"+"2" religions. More precisely there are 56 data sets for specific religions, in the WCE and WCT references [16,17], most of them being in the main denomination bracket, i.e. in the upper part of the pdf as obtained from the surveys taken between 1900 and 2000. The "1" refers to some data containing 3000 religions which are put together, as "other religions" in the WCT tables. The "2" refers to the set of data on atheists and nonreligious persons, as mentioned in Table 1-2 of ref. [17]. Thereafter for conciseness, we will also identify/call those three sets as "religions".
Emphasis will be on distinguishing between growing and decaying cases, discussing our "theoretical" fit, comparing to the forecasting in ref. [17], for 2025 and later, and observing diverse anomalies, thus raising questions to be further investigated.
The remainder of the paper is organized as follows: in Section II the data bank is briefly discussed, -and criticized, though accepted for further research and subsequent analysis along the theoretical and methodological tools used here which we adapt to the considered time series set. The results are largely presented and discussed in Section III under the form of Tables and graphs for various groups of religions, grouping according to the apparent behavior. Some concluding remarks are done in Section IV.
II. DATA BANK. THEORETICAL AND METHODOLOGICAL FRAMEWORK
The data [18] analyzed here were taken from the World Christian Trends [17]. It is fair to say that this is a remarkable compilation work. Their tables give information on the number of adherents of the world's main religions and their main denominations : 55 specific ("large") religious groups + atheists + nonreligious, plus a set called other religionists made of 3000 religions which however contains, Yezidis and Mandeans which we consider also, so that we examine 53+2= 55 (truly recognized) religions. From this data set we have also information on changes during one century of the number of adherents of each religion from 1900 till 2000 (information in the data set are given for the following years 1900, 1970, 1990, 1995 and 2000) -with a forecast for 2025 and 2050. Let us point out that it is not understood (or barely understandable) how such a forecast is made in the data bank.
A critical view of this data has to follow: we have already [1] noticed a break at 10 7 , in the pdf, indicating in our view an overestimation of adepts/adherents in the most prominent religions, or a lack of distinctions between denominations, for these, -as can be easily understood either in terms of propaganda or politics, or because of the difficulty of surveying such cases precisely. Yet one paradoxical surprise stems in the apparent precision of the data. E.g., in several cases in which the religion adherent numbers are reported, the data in [17] seems to be precise up to the last digit i.e., in mid-2000 , there are 1057328093 and 38977 roman catholics and mandeans respectively. In strong contrast there are 7000000 and 1650000 wahhabites and black muslims respectively, numbers which are quite well rounded. Thus a mere reading of the numbers warns about the difficulty of fully trusting the data. Nevertheless the analysis is pursued bearing this caveat here below.
III. RESULTS
Results of the h-fit to Avrami equation of the WCT surveys [17] are summarized in Tables I, II and III: the 58 "denominations" of interest are given. The parameters are obtained by a least-square best fit of the data (not considering the WCT forecast) to the equations mentioned in each table caption for the various cases. The ranking in the tables is according to the fit parameter h or A.
The parameter h values and their meaning deserve some short explanation and discussion here. According to the standard growth (Avrami) process h should be positive and less than 1, since n ≡ 1 − h; if it is greater than 1, this is indicating the possibility for detachment. We consider that if |h| is outside the (0, 1) interval, we have to imagine that the nucleation growth process is heterogeneous and/or conjecture that it is due to external field influences. Moreover notice that when h is greater than 1, the Avrami equation solution decays, ... from a maximum at the time t 0 . However it is hardly difficult to know when a religion has attained its maximum number of adherents. Thus the time scale or the initial appearance time of a religion are questionable points. Another point is obvious from Fig.1. The theoretical expressions do not allow a fit in the vicinity of a maximum of minimum. We should expect deviations, if such a case occurs, whence other empirical functions to be of interest.
A. Intermediary comments
In order to read the figures, let us point out the way we have here chosen for their display. It seems somewhat obvious, from a mathematical or physics point of view that one should consider (i) strictly increasing or decreasing cases, (ii) cases of growth after a minimum, or (iii) of decay after a maximum. This hints also to consider the curvature as a relevant indicator as for other (financial) time series [21]. Therefore we have grossly ranked the figures and data according to whether the religion number of adherents seems to be increasing (Table I), with h ≤ 0, starting from the lowest value and increasing, in a power law fit. Next we display the "decreasing" cases along an Avrami law, again ranking in order of increasing h, corresponding to fits with parameters given in (Table II).
Sometimes it is readily observed from the WCT tables that there are "presently growing" religions but for which a minimum is observed during the XX-th century, or a few are decaying after some maximum. For such "religions" the number of adherents can be in a first approximation fitted with a second order polynomial y = A + Bx + Cx 2 for which the parameters are given in Table III).
B. Growing Religions
In this subsection, we show cases of small or large size religions which are strictly increasing (Figs. 2-6). The illustrations are sufficiently readable and understandable that we do not convey much more hereby than in the figure captions. We distinguish the case of rather good fits (Fig. 4) , or not; we emphasize that we either overshoot or underestimate the WTE forecast for 2025 and thereafter. We suggest to the reader to compare the figures with the h values in the Table, and observe that the y-scales are evidently quite different from figure to figure depending on the rank of the religion. We observe (Fig.5) that there are three cases where the growth appears to be linear and in all cases underestimate the WTE forecast. We emphasize that growth does not mean lack of saturation, as it should be appreciated: this is illustrated by two cases in Fig. 6.
C. Decaying religions
Turning to apparently decaying religions (Figs. 7-8) , the same classification can be made: either there are strongly decaying cases or smooth decays, with remarkable fits, even though there is no a priori knowledge of the time when the number of adherents is maximum. The empirical forecast is in almost all cases in rather good agreement with the WTE one.
D. Cases with presently observed extremum
Sometimes there are "presently decaying" ("growing") religions for which a maximum (minimum) is observed during the present centuries. For such "religions" the number of adherents can be fitted with a second order polynomial y = A + Bx + Cx 2 ; the relevant parameters for these cases are given in Table III). Notice that these are rather large size religions. In the Fig. 9 cases we overshoot the WTE forecast, -not in Fig. 10. In Fig. 11, the maximum occurs during the XIX-th century, and apparently there is a strong prognosis for the collapse of these religions. In Fig. 12, we underestimate the WTE forecast, which is thus more optimistic than the XX-th century data evolution indicates. In some (3) cases (Fig. 13) the collapse seems rather obvious, though the tail of the evolution law has some (sometimes large) error bar, not shown for data readability. Observe that the "other religions", containing 3000 or so smaller denominations present also a parabolic convex shape. One can debate whether the parabola makes physical sense of course. It might happen that such religions will disappear, but it might be through a tail evolution, like presumably the cases of Fig. 14. More data should be of interest in such cases.
IV. DISCUSSION AND CONCLUSIONS
We consider that the religious practice is more likely more diverse than WCE and WTE surveys indicate. yet claiming the interest of the data we suggest to let religious adherence to be a degree of freedom of a population, and take it through statistical physics considerations for our enlightment. Therefore we have analyzed 58 cases of growing and decaying (so called) religions observing several groups through the analytical behaviors. We indicate that with an Avrami equation the fit can be quite often good, in particular for the growing cases. Physically speaking that gives some support to the conjecture of religions grow like crystals [22]. However we cannot expect that the Avrami equation holds true for ever; the system should saturate at some point, except if only a few religions are excessively predominating, and not "allowing" (in a thermodynamic sense) the probability of existence of others.
The same is true for the parabolic fit, which either indicates a quite quickly forthcoming disappearance of a religion or allows for infinite growth. We recognize that these are approximations.
It seems that we often overestimate/underestimate the WTE theoretical trend in the decaying cases and in several growing cases, though we sometimes agree in the latter cases. Again we claim that the WTE trends can be quite arbitrary, supposedly predicting a linear evolution from the last three data points in the surveys. It might be interesting to use other types of statistical analysis to conclude whether the forecasts so much differ from one another. ... As Table I; our empirical law underestimates the WTE forecast well perform detailed analyses taking into account error bars in the original data. E.g., the case of Zoroastrians (in Fig. 3) indicates an anomalous point corresponding to 1975, while other cases seem to indicate major (but unknown) error bars (like on Figs. 5) on the data from the surveys. To resolve such questions is outside the scope of this report.
Turning to the data displayed on different figures, a "high" growth is seen for Hanfites, Shafiites and Malikites which are all Sunnists. Maybe we should not need to add a comment based on "political considerations" here, but we may consider that the meaning of h makes sense again. In fact this is emphasized when considering two of the highest growth rates, i.e. as found for Charsimatics and Independents, though a strict late growth stage theory might be debated upon. One case where one case trust the data points is likely that of the black muslims (Fig. 3 ) since they are hardly existed before 1900, whence for which an Avrami equation would hold. It would be very interesting to check soon the number of adherents in such a case.
Finally observe that the "non religion" adherent data finds a remarkable position as the fourth growing "denomination". Observe the maximum in the number of "adherents" in such a case near 1970, rendering the theory (or the FIG. 14: Four cases of equivalent size, but rather small, religions having a relatively complex behavior, apparently decaying during the XX-th century, but with debatable forecasting for the XIX-th century data !) to be debated upon.
In conclusion, here above we have shown that we can attempt to make a statistical physics like analysis of the number of adherents in religions, going beyond our first paper [1] on the subject. However the data seems sometimes barely reliable.
Nevertheless one can, expecting better surveys, at a more limited scale, suggest further lines of research. One could suggest agent based models like for languages, including the role of external fields. One could try to have a Langevin equation connexion to Avrami equation; of course we need to define a hamiltonian H and a current : that implies interactions thus competitions between entities; what we do not see here yet. However the hamiltonian can be obtained following standard ideas, like turning over the pdf into its log and defining some temperature. Religions seem to be an interesting field of study for statistical mechanics! Through this Appendix A we wish to outline what we consider are a few aspects, i.e. "differences" , between languages and religions, from a physics point of view, perspective or input into modeling their sociological features; see Table IV, as a summary of to be considerations of interest.
We insist that in physics one should study the response of the system to intrinsic or extrinsic fields. We may describe the population of agents through a free energy, Hamiltonian formalism or Langevin equation indicates that all terms, ordered along the increasing size of the cluster, should be included
VI. APPENDIX B. INDICATORS OF RELIGION STATUS
The time dependence of the number of adherents can be considered to be a very restrictive way to "measure" the evolution of a religion. One could also "weight" the level of adherence to a religion. For example, one could try as for languages to define a religion through its quantity of practitioners, rituals, .... Many other indicators are possible. One can measure diverse quantities related to the religious efffect. As in physics one can search for the relation between causes and effects, the response to internal or/and external fields.
As there are several definitions of a language [11], similarly one could also define what a religion "is" in different ways [23].
First let us list a few definitions of religions form the conventional literature :
FIG. 1 :
1(left) Logistic map or Verhulst law (Eq.(2)) and (right) theoretical behavior of the solution of an Avrami Equation as (Eq.(5) in reduced units for various typical h values
FIG. 2 :
2Six illustrative cases of actually increasing religions, ... with h ≤ 0. Observe the overshooting in the forecast with respect to WCT in all cases
FIG. 3 :
3Six illustrative cases of actually increasing (small size) religions ... with h ≤ 0; our empirical law does not confirm the WTE forecast in the next years, but overshoots the WTE value
FIG. 4 :FIG. 5 :
45Six illustrative cases of actually increasing (small size) religions ... ; our empirical law confirms the WCT forecast Three small size religions, with increasing number of adherents; decreasing h from left to right, with h close to 0; see
FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 :FIG. 10 :
678910Two illustrative cases of actually increasing (indicated) religions, with saturating like forecast. Observe the rather good fits, parameters inTable I, h positive and ≤ 1, and even a rather good confirmation of the WCT forecast Two sharply decaying religions, with very small t1 and h ≥ Four smoothly decaying religions with h much larger than 1; our forecast being similar to that predicted in WTE Seven large size religions indicating a turn over with a minimum in XX-th century; theoretical forecasting with respect to WTE is debatable though our fit slightly overshoots the WTE data Two large size religions indicating a turn over with a minimum in XX-th century; our theoretical forecasting slightly underestimates the WTE data
FIG. 11 :FIG. 12 :FIG. 13 :
111213Two cases of religions having a markedly predicted collapse after having had a maximum in the XIX-Nine religions having had a maximum during the XX-th century; the parabolic forecast undershoots the WTE expectation Case of so called 3000 other religions for which a decreasing behavior is observed; notice the marked underestimate of our forecast with respect to WTE, -predicting an increase in this XXI-
TABLE I :
IValues of the parameters h, α, and t1, used for fitting the data for "increasing religions" with a power law formula; see Eq. (10); religions are hereby ranked based on the size of the attachment parameter h which can be negative or positive but ≤ 1Religion
h
α
t1
Shaivites
-5.32
0.032
239
Hanbalites
-4.66 0.000305
527
Hanafites
-3.84
0.0629
211
Zoroastrians
-3.64 3.29e-005
530
Kharijites
-2.88 0.000196 1.15e+003
Afro-Caribbean religionists
-2.75 -5.06e-007 1.8e+003
Black Muslims
-2.36 -8.06e-006 1.11e+003
Pentecostals/Charismatics
-2.19 -0.00186
208
Independents
-1.61
0.00427
288
Shafiites
-1.49
0.024
528
Afro-American spiritists
-1.32 6.87e-005 4.99e+003
Ithna-Asharis
-1.26
0.0137
812
Afro-Brazilian cultists
-1.21 5.52e-005 2.61e+003
Zaydis
-1.10 0.000741 3.45e+003
Alawites
-1.09 0.000154 7.56e+003
Ismailis
-1.04
0.00142 1.87e+003
Yezidis
-1.01 1.84e-005 2.23e+004
High Spiritists
-0.83 2.33e-005 5.69e+003
Sikhs
-0.792
0.00182 3.13e+003
Ahmadis
-0.789 4.32e-005 4.17e+003
Baha is
-0.368 4.87e-006 1.38e+004
Druzes
-0.366 4.38e-005 8.85e+004
Neo-Hindus
-0.212 6.19e-005 1.28e+004
Marginal Christians
-0.206 0.000569
1e+004
Mandeans
-0.0667
5e-006 3.17e+007
Malikites
0.0566
0.0167 6.38e+003
Other sectarian Muslims
0.0929 0.000311 2.62e+006
crypto-Christians
0.230
0.0022 1.8e+004
Reform Hindus
0.384 0.000154 1.78e+007
TABLE II :
IIValues of the parameters h, t0, and t1 used for fitting the data on "ecreasing religions" with Eq. (9); h is in this
case ≥ 1
Religion
h
t0
t1
Chinese folk-religionists 1.07 -3.36e-007 3.49e-015
Orthodox
1.14
-0.821 1.06e-008
Theravada
1.81
-242
3.27
Mahayana
2.04
-321
16.2
Karaites
2.09
-99.3 0.00226
Lamaists
2.77
-614
29.7
TABLE III :
IIIValues of the parameter used for fitting data on 12 "decreasing" and 11 "increasing" religions with the polynomial equation Cx 2 + Bx + A; for a warning on the six "central"(in the table) religions, see textReligion
C
B
A
Nonreligious
-2.61e-005
0.103
-102
Atheists
-1.31e-005
0.0514
-50.3
unaffiliated Christians
-4.38e-006
0.017
-16.5
Roman Catholics
-4.2e-006
0.0165
-16
New-Religionists (Neoreligionists)
-3.88e-006
0.0153
-15
Shamanists
-4.87e-007
0.00185
-1.74
Confucianists
-2.11e-007 0.000831
-0.815
Wahhabites
-5.53e-008 0.000215
-0.208
Taoists
-4.4e-008 0.000174
-0.171
Other religionists (in 3000 religions) -3.81e-008
0.00015
-0.148
Ashkenazis
-2.46e-008 4.26e-005
0.0149
Oriental Jews
-3.23e-009 1.22e-005 -0.0112
Samaritans
7.14e-012 -3.01e-008 3.18e-005
Sefardis
6.52e-010 -2.8e-006 0.00315
Jains
8.97e-009 -3.61e-005
0.0371
Shintoists
2.05e-007 -0.000836
0.853
Saktists
2.12e-007 -0.000824
0.806
Protestants
7.66e-007 -0.00306
3.11
Anglicans
9.75e-007 -0.00386
3.83
Vaishnavites
1.25e-006 -0.00487
4.82
Sufis
2.34e-006 -0.00921
9.11
Animists
2.77e-006
-0.0111
11.2
Evangelicals
5.95e-006
-0.0233
22.8
TABLE IV :
IVComparison : similarities and differences between languages and religions seen from a statistical physics point of view V. APPENDIX A. LANGUAGES VS. RELIGIONSLanguages
Religions
more than 6000
more than 3000
agents
multilingual
frequent
polyreligious
rare
variety
huge:
dialects, slangs
huge:
denominations, sects
time scales
nucleation
slow
nucleation
fast
growth
slow
fast
through avalanches
decay
fast
slow
semantics
grammar
vocabulary
images
rituals
applied fields
rare
many, strong
AcknowledgmentsThe work by FP has been supported by European Commission Project E2C2 FP6-2003-NEST-Path-012975 Extreme Events: Causes and Consequences. Critical comments by A. Scharnhorst have to be mentioned.
Cambridge) Encyclopedia (1990): "...no single definition will suffice to encompass the varied sets of traditions, practices, and ideas which constitute different religions. & Barns, Noble, Barns & Noble (Cambridge) Encyclopedia (1990): "...no single definition will suffice to encompass the varied sets of traditions, practices, and ideas which constitute different religions."
Human recognition of superhuman controlling power and especially of a personal God entitled to obedience. The Concise Oxford Dictionary. The Concise Oxford Dictionary (1990): "Human recognition of superhuman controlling power and especially of a personal God entitled to obedience.
New World Dictionary (Third College Edition): "any specific system of belief and worship. ' Webster, Webster's New World Dictionary (Third College Edition): "any specific system of belief and worship, often involving a code of ethics and a philosophy.
a cause, principle, or system of beliefs held to with ardor and faith. Merriam-Webster's Online, Dictionary, In fact, we can admit thatMerriam-Webster's Online Dictionary: "a cause, principle, or system of beliefs held to with ardor and faith." In fact, we can admit that
!) accepting that those not included in the above are "non religious", in which one can distinguish between atheists, agnostics, non-interested ones, etc. , while we can contain an adherent of whatever "denomination" into a definition like 1. An adept is an individual identified as having attained a specific level of knowledge, skill. Religion is any specific system of belief about deity, often involving rituals. or aptitude in doctrines relevant to a particular (author or) organizationReligion is any specific system of belief about deity, often involving rituals, a code of ethics, a philosophy of life, and a worldview. (!) accepting that those not included in the above are "non religious", in which one can distinguish between atheists, agnostics, non-interested ones, etc. , while we can contain an adherent of whatever "denomination" into a definition like 1. An adept is an individual identified as having attained a specific level of knowledge, skill, or aptitude in doctrines relevant to a particular (author or) organization.
Next one may imagine a Potts vector or ferrroelectric type of (Hamiltonian) models for describing an ensemble of religious agent evolution or state. Quantitative and qualitative dynamical evolutions of agents and groups ("denominations") can also find some basis in many competition and organization physics models. Moreover, one should consider religions from another ensemble of point of views also called sometimes indicators) 1. Number of groups, sects, 2. analog of an up or down spin, is rather a vector for which each element can be a quantity of value like considered in sociology, i.e. a "quality. Number of churches, parishes, 3. Number of chapels, sites, 4. Number of priests, (clergyIt is indeed clear that a religious adherent instead of being an analog of an up or down spin, is rather a vector for which each element can be a quantity of value like considered in sociology, i.e. a "quality". Next one may imagine a Potts vector or ferrroelectric type of (Hamiltonian) models for describing an ensemble of religious agent evolution or state. Quantitative and qualitative dynamical evolutions of agents and groups ("denominations") can also find some basis in many competition and organization physics models. Moreover, one should consider religions from another ensemble of point of views also called sometimes indicators) 1. Number of groups, sects, 2. Number of churches, parishes, 3. Number of chapels, sites, 4. Number of priests, (clergy)
Number of believers, sex, age, wealth, language, 6. Intensity of participations, in rituals. in practicing principles, 7. Wealth and financing, 8. Type of hierarchyNumber of believers, sex, age, wealth, language, 6. Intensity of participations, in rituals, in practicing principles, 7. Wealth and financing, 8. Type of hierarchy, ...
No need to say that physicists are not the first ones to reflect on variability in religion distribution or adherence level. We may find already such considerations in books and papers by specialists of the history or sociology of religions. 23No need to say that physicists are not the first ones to reflect on variability in religion distribution or adherence level. We may find already such considerations in books and papers by specialists of the history or sociology of religions [23].
M Ausloos, F Petroni, Statistical dynamics of religions and adherents. 774M. Ausloos and F. Petroni, Statistical dynamics of religions and adherents, Europhys. Lett. 77 (2007) 38002 (4pp)
Phase transitions in social impact models of opinion formation. J Holyst, K Kacperski, F Schweitzer, Physica A. 285199J. Holyst, K. Kacperski, and F. Schweitzer, Phase transitions in social impact models of opinion formation, Physica A 285 (2000) 199.
Modelling the dynamics of language death. D M Abrams, S H Strogatz, Nature. 424900D.M. Abrams and S.H.Strogatz, Modelling the dynamics of language death, Nature 424 (2003) 900.
Theoretical model for the evolution of the linguistic diversity. V M De Oliveira, M A F Gomes, I R Tsang, Physica A. 361361V.M. de Oliveira, M.A.F. Gomes, and I.R. Tsang, Theoretical model for the evolution of the linguistic diversity, Physica A 361 (2006) 361.
K Kaneko, I Tsuda, Complex Systems: Chaos and Beyond. A constructive Approach with Applications in Life Sciences. BerlinSpringerK. Kaneko and I. Tsuda, Complex Systems: Chaos and Beyond. A constructive Approach with Applications in Life Sciences, Springer, Berlin (1996).
Exploring complexity. G Nicolis, I Prigogine, W.H. FreemanNYG. Nicolis and I. Prigogine, Exploring complexity, W.H. Freeman, NY (1989).
The medieval inquisition: scale-free networks and the suppression of heresy. P Ormerod, A P Roach, Physica A. 339P. Ormerod and A.P. Roach, The medieval inquisition: scale-free networks and the suppression of heresy, Physica A 339 (2004) 645-652
Matines Brugeoises" , when flemish "peasants" killed the french nobles, recognized as such, because they could not pronounce correctly "schild and vriend. However one can recall the case of. Another case is that of the khmer rouges killing vietnamese educated intellectuals in CambodiaHowever one can recall the case of "Matines Brugeoises" , when flemish "peasants" killed the french nobles, recognized as such, because they could not pronounce correctly "schild and vriend". Another case is that of the khmer rouges killing vietnamese educated intellectuals in Cambodia.
is truly a member of a religious denomination or church. However this caveat pertains to usual problems encountered in sociological investigations. It is sometimes hard to know or to be sure whether an adherent, a disciple,It is sometimes hard to know or to be sure whether an adherent, a disciple, ... is truly a member of a religious denomination or church. However this caveat pertains to usual problems encountered in sociological investigations.
as a sort of religion, from our point of view. , Admitting, admitting indifference, atheism, agnosticism, ... as a sort of religion, from our point of view.
J M Klinkenberg, it Des langues romanes. Duculot, Louvain-la-NeuveJ.M. Klinkenberg, it Des langues romanes, Duculot, Louvain-la-Neuve (1994).
Emergence of scaling in random networks. A L Barabási, R Albert, Science. 286509A.L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286, (1990) 509.
private communication after and during an invited talk at FENS07. R Kutner, Wroclaw, PolandR. Kutner, private communication after and during an invited talk at FENS07, Wroclaw, Poland
Simulations of the kinetic growth of Y Ba2Cu3O 7−d grains. R Cloots, N Vandewalle, M Ausloos, J. Cryst. Growth. 166816R. Cloots, N. Vandewalle, and M. Ausloos, Simulations of the kinetic growth of Y Ba2Cu3O 7−d grains. J. Cryst. Growth 166 (1996) 816.
Stretched exponential kinetics of the pressure induced hydration of model lipid membranes. A possible scenario. A Gadomski, J. Phys. II France. 61537A. Gadomski, Stretched exponential kinetics of the pressure induced hydration of model lipid membranes. A possible scenario. J. Phys. II France 6 (1996) 1537.
D Barrett, G Kurian, T Johnson, World Christian Encyclopedia. New YorkOxford University Press2nd editionD. Barrett, G. Kurian, and T. Johnson, World Christian Encyclopedia (2nd edition). New York: Oxford University Press (2001)
World Christian Trends. D Barrett, T Johnson, William Carey LibraryD. Barrett and T. Johnson, World Christian Trends. William Carey Library (2001)
Data Source Information: The sources used in the WCT database were so numerous and diverse that we only mention here few of them, for a more exhaustive discussion the readers are referred to the WCE. The major physical collections of data built up may be summarized here: around 5000 statistical questionnaires returned by churches and national collaborators over the period 1982-2006; field surveys and interviews on the spot in over 200 countries conducted by the authors, who over the years 1965-2006 visited virtually every country in the world; the collection of 600 directories of denominations, Christian councils, confessions and topics; a collection of 4500 printed contemporary descriptions of the churches, describing denominations, movements, countries and confessions; officially-published reports of 500 government-organized national censuses of population each including the question on religion, in over 120 countries, covering most decades over the period 1900-2005; bibliographical listings from searches (including computerized enquiries on key-words) in a number of major libraries including those of the British Library. London), Library of Congress; Washington), Propaganda (Rome), Missionary Research Library; New Yorkand a score of universitiesData Source Information: The sources used in the WCT database were so numerous and diverse that we only mention here few of them, for a more exhaustive discussion the readers are referred to the WCE. The major physical collections of data built up may be summarized here: around 5000 statistical questionnaires returned by churches and national collaborators over the period 1982-2006; field surveys and interviews on the spot in over 200 countries conducted by the authors, who over the years 1965-2006 visited virtually every country in the world; the collection of 600 directories of denominations, Christian councils, confessions and topics; a collection of 4500 printed contemporary descriptions of the churches, describing denominations, movements, countries and confessions; officially-published reports of 500 government-organized national censuses of population each including the question on religion, in over 120 countries, covering most decades over the period 1900-2005; bibliographical listings from searches (including computerized enquiries on key-words) in a number of major libraries including those of the British Library (London), Library of Congress (Washington), Propaganda (Rome), Missionary Research Library (New York), and a score of universities.
An assessment of the human attachment as a mechanism for human sexual network formation. J H Jones, M S Handcock, Proc. R. Soc. Lond. B. 270J. H. Jones and M. S. Handcock, An assessment of the human attachment as a mechanism for human sexual network formation, Proc. R. Soc. Lond. B 270 (2003) 1123-1128.
In one step from KCl to YBa2Cu3O7 crystal growth : understanding of dendritic morphology in crystals. M Ausloos, N Vandewalle, R Cloots, Phil. Mag. Lett. 73M. Ausloos, N. Vandewalle, and R. Cloots, In one step from KCl to YBa2Cu3O7 crystal growth : understanding of dendritic morphology in crystals, Phil. Mag. Lett. 73 (1996) 101-105.
Low order variability diagrams for short range correlation evidence in financial data: BGL-USD exchange rate, Dow-Jones Industrial Average, Gold ounce price. K Ivanova, M Ausloos, Physica A. 265K. Ivanova and M. Ausloos, Low order variability diagrams for short range correlation evidence in financial data: BGL-USD exchange rate, Dow-Jones Industrial Average, Gold ounce price, Physica A 265 (1999) 279-286.
Breaking the Spell: Religion as a Natural Phenomenon. D Dennett, Penguin Group. D. Dennett, Breaking the Spell: Religion as a Natural Phenomenon, Penguin Group, (2006).
| []
|
[
"Manifold-based isogeometric analysis basis functions with prescribed sharp features",
"Manifold-based isogeometric analysis basis functions with prescribed sharp features"
]
| [
"Qiaoling Zhang \nDepartment of Engineering\nUniversity of Cambridge\nTrumpington StreetCB2 1PZCambridgeU.K\n",
"Fehmi Cirak [email protected] \nDepartment of Engineering\nUniversity of Cambridge\nTrumpington StreetCB2 1PZCambridgeU.K\n"
]
| [
"Department of Engineering\nUniversity of Cambridge\nTrumpington StreetCB2 1PZCambridgeU.K",
"Department of Engineering\nUniversity of Cambridge\nTrumpington StreetCB2 1PZCambridgeU.K"
]
| []
| We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was first demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to the construction in Majeed et al. no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish the local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. The convergence and the utility of the new manifold-based basis functions are demonstrated with linear and nonlinear Kirchhoff-Love shell finite elements by means of beam, plate and shell examples. | 10.1016/j.cma.2019.112659 | [
"https://arxiv.org/pdf/1904.03258v2.pdf"
]
| 102,352,177 | 1904.03258 | 35809ab911e3d619cc9e95b0c5963445e69a6447 |
Manifold-based isogeometric analysis basis functions with prescribed sharp features
April 9, 2019 5 Apr 2019
Qiaoling Zhang
Department of Engineering
University of Cambridge
Trumpington StreetCB2 1PZCambridgeU.K
Fehmi Cirak [email protected]
Department of Engineering
University of Cambridge
Trumpington StreetCB2 1PZCambridgeU.K
Manifold-based isogeometric analysis basis functions with prescribed sharp features
April 9, 2019 5 Apr 2019isogeometric analysismanifoldssmooth basis functionspartition of unity methodsharp features * Corresponding author
We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was first demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to the construction in Majeed et al. no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish the local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. The convergence and the utility of the new manifold-based basis functions are demonstrated with linear and nonlinear Kirchhoff-Love shell finite elements by means of beam, plate and shell examples.
Introduction
Smooth approximation schemes for unstructured meshes are crucial for isogeometric design and analysis of parts with arbitrary topology. Until recently isogeometric analysis was dominated by NURBS basis functions, which is the prevailing technology in present CAD systems. In order to represent the bounding surface of parts with arbitrary topology CAD systems resort to trimmed NURBS. Trimming involves the computation of the intersection between spline surfaces with other surfaces or curves. The respective nonlinear root-finding problem looks deceptively simple but is extremely hard to robustly solve and leads to non-watertight geometries [1][2][3]. In the analysis context, the nonwatertight geometries obtained from trimming pose unique challenges. Without turning to trimming, unstructured meshes with extraordinary vertices, i.e. vertices with different than four attached patches inside the domain, are required to represent parts with arbitrary topology. In computer-aided design numerous techniques have been developed to deal with extraordinary vertices, including geometrically G k and parametrically C k continuous constructions and subdivision surfaces. The application and further development of these techniques is currently a very active area of research in isogeometric analysis, see e.g. [4][5][6][7][8][9][10][11][12][13][14][15]. Unfortunately, only very few techniques from computer-aided design seem to give optimal finite element convergence rates without further modifications. Hence, the search for easy to implement and optimally convergent schemes, especially with C k≥2 and G k≥2 , is still open. In a complementary line of research, there has been progress in isogeometric analysis of shells based on trimmed surfaces and weak enforcement of mechanical continuity conditions across patch boundaries, see e.g. [16,17].
Manifold techniques for mesh-based construction of C k continuous surfaces were first introduced in Grimm et al. [18]. In contrast to most other smooth surface construction techniques, which essentially rely on glueing of surface patches along their edges, in manifold techniques a surface is created by blending of overlapping surface patches. The surface patches are first defined over unconnected planar chart domains in R 2 and subsequently mapped to R 3 . On the collection of the unconnected planar chart domains, the partition of unity method is used to construct the smooth surface patches. To that end, on each planar chart domain a local polynomial approximant and a partition of unity, or blending, function is needed. In addition, transition functions are required in order to be able to navigate between the different chart domains. In manifold techniques the planar chart domains, the transition functions, the local polynomial approximants and the blending functions can all be relatively freely chosen, which makes them extremely versatile. Based on Grimm et al.'s seminal work a number of complementary mesh-based manifold approaches have been proposed in geometric modelling [18][19][20][21][22]. Most of these approaches differ in the choice of the planar chart domains. For instance, in [18] each vertex, edge and face of the mesh have a corresponding planar chart domain with a suitably chosen geometry. In [20] and [21] only the vertices have a corresponding planar chart domain in the form of a conformally mapped star-shaped polygonal disk or a circular disk, respectively. In [19] the chart domains are chosen similar to the characteristic map in subdivision surfaces [23]. Each of the mentioned choices for the chart domains implies a corresponding choice for the transition functions. The manifold-based surface techniques were first introduced to isogeometric analysis in Majeed et al [24]. Following the construction proposed in Ying et al. [20] a new set of manifold-based basis functions were derived which can yield high convergence rates in finite element analysis. From a finite element viewpoint, the manifold-based basis functions resemble spline basis functions in the sense that each basis function has a local support and has one corresponding vertex.
In this paper, we derive new manifold-based basis functions for the isogeometric design and analysis of smooth surfaces with boundaries and with C 0 continuous sharp features, like creases and corners, by extending Majeed et al. [24]. The crease edges are tagged as such on the control mesh by the user and during finite element analysis different mechanical continuity conditions can be imposed across the crease edges and along the boundary edges, see Figure 1. In mesh-based construction of surfaces using manifold techniques creases have previously been considered in [25] and boundaries in [26]. The essential idea can be reduced to the choice of special local polynomials in the partition of unity approximation. That is, the local polynomials on each chart domain have to consist out of several polynomial pieces that are C 0 continuously connected across the crease edges. As in [24], in our present construction each vertex and its attached elements have a corresponding star-shaped planar chart domain consisting of images of conformally mapped unit squares placed around a centre vertex. There is one chart domain per vertex and each quadrilateral element is present on four different chart domains. The choice of the special C 0 continuous local polynomials can be simplified by slightly modifying the geometry of the chart domains. The new chart domains are chosen such that they are rotationally symmetric with respect to the arrangement of crease edges. Or expressed differently, the crease edges must partition the domain into equiangular sectors. The C 0 continuously connected polynomial pieces can subsequently be obtained by mapping a tensor-product basis, like the Lagrange or Bernstein basis, into each of the sectors of the chart domain. The new chart domains are parameterised with a quasi-conformal map so that each element on it can have a different shape depending on the arrangement of the crease edges. In contrast to the conformal map used in [24] the proposed quasi-conformal is not angle-preserving but still provides easily computable smooth transition functions. The proposed new construction leads to three different types of chart domains. One of them is specifically designed to deal with crease arrangements which lead to creased sectors with concave corners. We also introduce in this paper a new approach for assembling the blending functions from tensorproduct B-spline pieces. The new blending functions do not require normalisation so that they lead to non-rational manifold-based basis functions. The outline of this paper is as follows. In Section 2 the manifold-based basis functions introduced by Majeed et al. [24] are briefly reviewed. The treatment of sharp features is discussed in Section 3. Depending on the arrangement of crease edges we distinguish between rotationally symmetric and asymmetric chart domains. In addition, the case of crease arrangements leading to concave sectors on the chart domain is discussed. Each of the three cases requires a different quasi-conformal map for parameterisation. Section 4 introduces the finite element analysis of thin shells with normal control along boundaries and across crease edges. In Section 5 the new manifold-based basis functions are applied to several Bernoulli beams and linear and nonlinear Kirchhoff-Love shell problems. The numerical convergence of L 2 and energy norm errors with decreasing element sizes is demonstrated.
Review of manifold-based basis functions
In the following we briefly review the construction of univariate and bivariate manifold-based basis functions. The discussion is focused on their application in finite elements, so that the underlying manifold concepts from differential geometry and the partition of unity interpolation are only mentioned in passing. For a more comprehensive discussion on manifold-based basis functions and surface construction in geometry we refer to [18,20,24].
Univariate basis functions
It is instructive to first review the derivation of the univariate manifold-based basis functions. We consider the dash-dotted control polygon in Figure 2 representing a part of a finite element mesh consisting of vertices x I ∈ R 3 and elements between consecutive vertices. Our aim is to derive the basis functions for a representative element [x I , x I+1 ] ∈ R 3 , highlighted in Figure 2. As in conventional finite elements we define a reference element [0, 1] ∈ R that will serve as an integration domain for evaluating the finite element integrals. In the manifold-based approach the basis functions are obtained by smoothly blending local polynomials defined over sev-Figure 2: Univariate manifold construction over a reference element and the approximation of a given control mesh (dash-dotted). The reference element maps into the two elements (solid lines) in the two chart domainsΩ 1 andΩ 2 . The local polynomial basis onΩ 1 is a quadratic and onΩ 2 it is a piecewise linear Lagrange basis. The blending functions w 1 (ξ 1 ) = w 1 (Ψ 1 (η)) and w 2 (ξ 2 ) = w 2 (Ψ 2 (η)) sum up to one for η ∈ . The same vertex in different domains has the same colour. eral overlapping charts, or patches as they were called in [24] 1 . In the following two chart domainsΩ 1
[−1, 1] ∈ R andΩ 2 [−1, 1] ∈ R are introduced for the element [x I , x I+1 ]
.Ω 1 is associated with the vertex x I and its 1neighbourhood andΩ 2 is associated with the vertex x I+1 and its 1-neighbourhood. The 1-neighbourhood of a vertex is defined as the union of elements that contain the vertex. Hence, the basis functions in the element [x I , x I+1 ] will be obtained by blending local polynomials defined overΩ 1 andΩ 2 .
As shown in Figure 2, the reference element maps to elements in both chart domainsΩ 1 andΩ 2 . The respective maps are defined as
Ψ 1 : η ∈ → ξ 1 ∈ [0, 1] ∈Ω 1 , Ψ 2 : η ∈ → ξ 2 ∈ [−1, 0] ∈Ω 2 .(1)
In this specific univariate construction both maps are simple translations. The two maps imply the transition map between the two chart domains given by
Ψ 2 • Ψ −1 1 : ξ 1 ∈Ω 1 → ξ 2 ∈Ω 2 .(2)
Subsequently, we choose on each chart domain a polynomial approximant of the form
f j (ξ j ) = p T j (ξ j )α j with j ∈ {1, 2} ,(3)
where the vectors p j (ξ j ) contain a local polynomial basis, like the power, Lagrange or Bernstein basis, and the vectors α j contain their coefficients. In the present example a Lagrange basis is chosen, see Figure 2. The local basis on the chartΩ 1 is quadratic and on the chartΩ 2 it is piecewise linear. In addition to the polynomial approximant f j (ξ j ), on each chart a smooth blending, or weight function w j (ξ j ) is chosen. The two non-zero blending functions over the Figure 3: Construction of smooth blending functions w 1 (ξ 1 ) and w 2 (ξ 2 ) as the linear combination of cubic B-splines defined over a parameter space with knot-distance 1/4. The coefficients for w 1 (η) and w j (η) are indicated above the B-splines.
reference element must satisfy the partition of unity property, i.e.,
2 j=1 w j (ξ j ) = 1 ∀η ∈ with ξ 1 = Ψ 1 (η) , ξ 2 = Ψ 2 (η) .(4)
The blending functions can be conveniently combined from suitably defined B-spline basis functions. In Figure 3 the construction of C 2 -continuous blending functions from cubic B-splines is illustrated. The uniform knot interval length of the B-splines is 1/4. The blending functions are defined as the linear combinations
w 1 (η) = B 1 (η) + B 2 (η) + B 3 (η) + B 4 (η) 2 , w 2 (η) = B 4 (η) 2 + B 5 (η) + B 6 (η) + B 7 (η) ,(5)
and yield after mapping the blending functions
w j (ξ j ) = w j (Ψ −1 j (ξ j )) .(6)
Evidently, both functions satisfy the partition-of-unity property (4) given that a complete B-spline basis sums up to one. It is worth emphasising that the blending functions proposed here are polynomial in contrast to the rational ones introduced in [24], see also Appendix B. Moreover, the tensor products of the new univariate blending functions yield the corresponding multivariate blending functions. Finally, the approximant over the reference element is obtained by blending the approximants over the two charts
f (η) = 2 j=1 w j (ξ j ) p T j (ξ j )α j with ξ j = Ψ j (η) .(7)
Due to the choice of the Lagrange basis for p j (ξ j ) the coefficients α j can be interpreted as vertex coefficients. The coefficients α j of each chart are formally obtained with
α j = P j f ,(8)
where f is the array of coefficients of all control polygon vertices and P j is a gather matrix, filled with ones and zeros. This introduced in (7) yields the array of manifold-based basis functions where N(η) is the array of non-zero basis functions over the considered element. The number of non-zero basis functions in N(η) depends on the cardinality of the set of vertices in the two chart domainsΩ 1 andΩ 2 , and the specific local approximants on them.
f (η) = 2 j=1 w j (ξ j )p T j (ξ j )P j f = N T (η) f with ξ j = Ψ j (η) ,(9)
To investigate the smoothness of the basis functions N(η) it is necessary to consider their derivatives. According to definition (9), their smoothness depends on the blending functions w j (ξ j ), the local basis p j (ξ j ) and the maps Ψ j . To obtain C k -continuous basis functions each of these functions has to be k-times differentiable over each chart domain; and, in addition, the blending functions w j (ξ j ) and their up to k-th derivatives must vanish at the chart domain boundaries, i.e.,
d l w j (−1) dξ j = d l w j (1) dξ j = 0 ∀ l ≤ k .(10)
In addition, the differentiability of the mappings Ψ 1 (η) and Ψ 2 (η) requires that they satisfy at the centre of the chart domain
d l Ψ 1 (0) dη l = d l Ψ 2 (1) dη l ∀ l ≤ k .(11)
The smoothness of the basis functions N(η) can be pointwise reduced by selecting a suitable local polynomial basis p j (ξ j ), as in Figure 2 with piecewise linear basis on chartΩ 2 . The resulting basis functions N(η) and their first derivatives are plotted in Figure 4. The smoothness of the basis is reduced to C 0 at the vertex x I+1 . This has also an effect on the number of non-zero basis functions in an element. For instance, in the two elements [x I , x I+1 ] and [x I+1 , x I+2 ] with the linear piecewise C 0 continuous local basis p j (ξ j ) there are three non-zero basis functions. In elements with a quadratic local basis on each chart there are four non-zero basis functions. Finally, with the generated manifold-based basis functions points on the reference element η ∈ are mapped onto the manifold according to
x(η) = I N I (η)x I .(12)
Bivariate basis functions
The bivariate manifold-based basis functions provide smooth approximants even on unstructured surface meshes with extraordinary vertices. We consider the construction of the manifold-based basis functions for a representative element in the quadrilateral finite element mesh shown in Figure 5. The reference element is now defined as the unit square := [0, 1] × [0, 1] ∈ R 2 . Similar to the univariate case each of the four vertices of the considered element and their 1-neighbourhoods have an associated chart domainΩ j ∈ R 2 with j ∈ {1, 2, 3, 4}. The number of elements in a chartΩ j depends on the number of elements v j connected to the respective vertex, which is called the valence of the vertex. In quadrilateral meshes, the interior vertices with v j 4 are the extraordinary vertices. The basis functions will be obtained by smoothly blending local polynomials defined over each of the four overlapping chartsΩ j .
As shown in Figure 6, the reference element maps to elements in the four chart domainsΩ j according to Figure 6. The values j = 4, v j = 3, n j = 3 are substituted into (20) and (21) to obtain the expressions for Λ t r and Λ I qr .
Ψ I j : η = (η 1 , η 2 ) ∈ → ξ = (ξ 1 j , ξ 2 j ) ∈Ω j .(13)
The construction of these maps requires special care and will be detailed further below. The bivariate approximant over the reference element is obtained by blending the approximants over the four charts
f (η) = 4 j=1 w j (ξ j ) f j (ξ j ) = 4 j=1 w j (ξ j ) p T j (ξ j )α j with ξ j = Ψ I j (η) ,(14)
where w j (ξ j ) and p j (ξ j ) are the blending function and the vector of local basis functions on the chart domainΩ j , respectively. As mentioned in Section 2.1, the blending functions w j (ξ j ) are obtained by taking the tensor-product of the univariate blending functions (5). Next, the coefficients α j are to be expressed in dependence of the vertex coefficients collected in the array f . The number of unique vertices in the union of the four charts ∪ 4 j=1Ω j is not fixed and depends on the valences v j of the four vertices of the element. Hence, a least-squares projection is applied to express (14) in dependence of the vertex coefficients
f (η) = 4 j=1 w j (ξ j ) p T j (ξ j ) A j P j f ,(15)
where we used α j = A j P j f on each chartΩ j . Here, P j is the gather matrix and A j the least-squares projection matrix. It is evident that the number of polynomial coefficients in α j must not be greater than the number of vertices in the chart domainΩ j . It is worth mentioning that the projection matrix A j depends only on the valence v j of the vertex and the chosen local approximant p j so that it can be precomputed and tabulated. Finally, the array of basis functions are defined with
N T (η) = 4 j=1 w j (ξ j ) p T j (ξ j ) A j P j .(16)
As in the univariate case, the smoothness of the basis functions N(η) depends on the blending functions w j (ξ j ), the local basis p j (ξ j ) and the mappings Ψ I j (η). In the presented construction conformal maps are critical for the smooth parameterisation of the chart domains and the composition of the blending functions w j (ξ j ). Specifically, the map Ψ I j is composed of several conformal maps. To write conformal maps more succinctly, the coordinates η = (η 1 , η 2 ) of points in the reference element are expressed as a complex number
z = η 1 + iη 2 = |z|(cos φ + i sin φ) = |z|e iφ with |z| = (η 1 ) 2 + (η 2 ) 2 and φ = arctan(η 2 /η 1 ) ,(17)
where |z| is the radius and φ the angle in the complex plane. In the complex plane the coordinates of the four corners of the reference element are given by
z 1 z 2 z 3 z 4 T = 0 + 0i 1 + 0i 1 + 1i 0 + 1i T .(18)
As illustrated in Figure 7, the map Ψ I j : η ∈ → ξ j ∈Ω j is composed of two maps, i.e., The auxiliary linear map Λ t r is only responsible for translating and rotating the reference element and is given by
Ψ I j = Λ I qr • Λ t r .(19)Λ t r (z; j) = (z − z j )e −iπ( j−1)/2 ,(20)
where (z − z j ) is a translation and e −iπ( j−1)/2 a (rigid body) rotation. The quasi-conformal map Λ I qr maps the reference element into a wedge-shaped domain and is chosen as
Λ I qr (z; v j , n j ) = |z| β |z| 4/v j z 4/v j e i2π(n j −1)/v j = |z| β e iφ4/v j e i2π(n j −1)/v j ,(21)
where β is a free parameter, v j denotes the valence of the centre vertex and n j is the number of the sector onto the reference element is mapped. According to the first expression in (21), the map Λ I qr is composed of a standard conformal map z 4/v j , a scaling of the radius with |z| β−4/v j and a rotation e i2π(n j −1)/v j . When β = 4/v j the scaling term drops and the map Λ I qr is composed of a conformal map and a rotation as in [24]. The second expression in (21) shows that the radius |z| will remain constant when β = 1. Hence, in this paper we choose β = 1 to obtain a more uniform mapping Ψ I j with smaller entries in its Jacobian matrix. See Appendix A for a review on conformal maps and a discussion on the choice of β.
With the mapping Ψ I j at hand, we can now evaluate the manifold-based basis functions defined in (16). For instance, for a given integration point η in the reference element, first its image ξ j = Ψ I j (η) in each of the four overlapping charts is found and after that the blending functions w j (ξ j ) and local polynomials p j (ξ j ) are evaluated.
Creased bivariate manifold-based basis functions
We introduce domain boundaries and C 0 continuous creases by modifying the local polynomials p(ξ j ), and, as necessary, the chart domainsΩ j . Creases are introduced along the edges of the control mesh prescribed by the user. The local polynomials p(ξ j ) are chosen to be C 0 continuous across the creases. The new chart domains require new maps, i.e. Ψ II j : η ∈ → ξ j ∈Ω j , from the reference element to the chart domains. In turn, the blending functions w j (ξ j ) are combined from tensor-product B-splines using the new maps Ψ II j . The remaining steps in manifold-based basis construction are identical to the non-creased case.
Depending on the number and arrangement of creases meeting at a vertex three different crease types are possible. As introduced in the following each of them requires a different treatment. In Figure 8 the three different crease types are illustrated with the help of a sample geometry.
Crease Type 1: Rotationally symmetric chart domains
Type 1 creases have, as shown in Figure 9, a rotationally symmetric chart domain with respect to the arrangement of the creased edges. The centre vertex of a chartΩ j can have k j ≥ 2 attached edges tagged as crease 2 . The k j crease (a) v j = 6 and k j = 2 (b) v j = 6 and k j = 3 edges split the chart domain into k j equiangular sectors, and the number of elements in each of the sectors must be the same.
To introduce the basic idea in defining a suitable local approximant f j (ξ j ), at first the case with two creased edges, k j = 2 , as depicted in Figure 9a is considered. The crease splits the chart domain into the two sectors s j = 1 and s j = 2 along the ξ 1 j coordinate axis. The local approximant is chosen as the piecewise continuous function
f j (ξ j ) = f s j =1 j (ξ j ) if ξ 2 j ≥ 0 f s j =2 j (ξ j ) if ξ 2 j < 0 with f s j =1 j (ξ 1 j , ξ 2 j = 0) = f s j =2 j (ξ 1 j , ξ 2 j = 0) .(22)
As in the non-creased case both approximants are given by
f s j j (ξ j ) = p s j j (ξ j ) · α s j j .(23)
The coefficients α s j =1 j and α s j =2 j must be matched so that f j (ξ j ) is piecewise continuous. Especially for a tensorproduct Lagrange basis p s j j (ξ j ), it is straightforward to determine a new set of coefficients α j from the coefficients α s j =1 j and α s j =2 j which ensure piecewise continuity. The two Lagrange approximants must share the same coefficients along the common crease edge.
In chart domains with three or more creases, k j ≥ 3, the bases p s j j (ξ j ) are obtained by mapping a tensor-product Lagrange basis b(θ) to each sector. The respective local approximants f s j j (ξ j ) in two adjacent sectors meet C 0 continuously because they share the same coefficients along the common crease edge. The Lagrange basis b(θ) is mapped onto a sector s j using the quasi-conformal map (21) introduced earlier
p s j j (ξ j ) = b(θ) with θ = Λ I qr −1 (ξ j ; k j , s j ) .(24)
This mapping is best understood in conjunction with Figure 9b. Each of the three sectors in different colours represent the image of a square mapped with Λ I qr (θ; k j , s j ) : θ ∈ → ξ j ∈Ω j with s ∈ {1, 2, 3}. Obviously, the polynomial degree of b(θ) must be chosen such that the degrees of freedom is less than or equal to the number of vertices in each sector. More vertices have to be added, e.g. by refining the elements in each chart domain, if a higher degree basis is requested. Although we chose in our implementation Lagrange basis for b(θ) it is possible to use other boundary interpolating tensor-product basis, like the Bernstein basis.
Crease Type 2: Asymmetric chart domains
Type 2 creases have a rotationally asymmetric chart domain with respect to the arrangement of crease edges. Again, the centre vertex of a chartΩ j can have k j ≥ 2 attached edges tagged as crease. However, the k j crease edges split the chart domain into k j sectors, which are not equiangular and have different number of elements. This makes it difficult to establish C 0 continuous local approximants f j (ξ j ) following the approach introduced for Type 1 creases.
Therefore, Type 2 creases are first mapped onto a chart domain, which is equiangular with respect to the arrangement of crease edges, see Figures 10 and 11. This is possible in a manifold-based approach because the chart domainsΩ j can be freely chosen, subject to some smoothness and invertibility constraints. The respective map Ψ II j : η ∈ → ξ j ∈Ω j has the same structure like for non-creased charts (19), i.e.,
Ψ II j = Λ II qr • Λ t r ,(25)
where Λ t r consists as defined in (20) of a translation and rotation and Λ II qr is a quasi-conformal map with the arguments of Λ I qr in (21) replaced according to
Λ II qr (η; k j , s j , l s , m s ) = Λ I qr (η; k j l s , (s j − 1)l s + m s ) .(26)
Here, k j is the number of creases, s j ∈ {1, 2, . . . , k j } is the sector number, l s is the number of elements in the sector s j and m s ∈ {1, . . . , l s } is the local element number in the sector s j . The meaning of the introduced variables is further clarified in Figure 12. After the equiangular chart domain for Type 2 creases is established, as in Figures 10 and 11, the local approximants f s j j (ξ j ) in each sector are obtained following the approach for a Type 1 crease chart. That is, they are defined as (22) in case of two creases or otherwise mapped according to (24) from a tensor-product Lagrange basis b(θ). The local approximant f j (ξ j ), composed from the local approximants in each of the sectors f s j j (ξ j ), is C 0 continuous across the crease edges when the approximants share the same coefficients along the edges.
Crease Type 3: Chart domains with concave corners
The arrangement of creases in Type 2 chart domains can lead sometimes to non-convex sectors on the chart domain. There is, however, no regular mapping from a convex to a non-convex domain with a non-zero determinant of the Jacobian. Therefore, when the map Ψ II j defined in (25) is used in a non-convex sector it leads to an unwanted folding-over of the surface as visible in Figure 15. This behaviour is similar to the one observed with isoparametrically mapped non-convex Lagrange finite elements.
The folding-over can be avoided by composing the non-convex sector from several smoothly connected convex patches. To this end, the chart domain depicted in Figure 13 is introduced. The respective map Ψ III j : η ∈ → ξ j ∈Ω j has the same structure like the previously introduced maps, c.f. (19) and (25),
Ψ III j = Λ III qr • Λ t r(27)
with the quasi-conformal map Λ III qr obtained by replacing the arguments of Λ I qr in (21) in the following way
Λ III qr (η; k j , s j , l s , m s ) = Λ I qr (η; 4(k j − 1)l s , (s j − 1)l s + m s ) if s j < k j Λ I qr (η; 4l s /3, m s ) · e iπ/2 if s j = k j ,(28)
where k j is the number of creases, s j ∈ {1, 2, . . . , k j } is the sector number, l s is the number of elements in a sector and m s is the local element number in a sector. All the mentioned variables have the same meaning as in Section 3.2 and are clarified in Figure 14. As shown in Figure 13 with the introduced map Ψ III j the concave sector with s j = k j is mapped to a L-shaped domain and all other sectors are mapped to the first quadrant. For obtaining the local approximant f j (ξ j ) a similar approach as discussed for the Type 1 and 2 crease charts is followed with the exception of the non-convex sector. The local approximants f s j <k j j (ξ j ) on each convex sector are mapped according to (24) from a tensor-product Lagrange basis b(θ) using θ = (Λ I qr ) −1 (ξ j ; 4(k j − 1), s j ). The local approximant on the non-convex sector f s j =k j j (ξ j ) is composed out of three (cubic) Bézier patches that are smoothly connected within the sector while they are matched C 0 continuously with polynomial pieces in other sectors across the crease edges.
Examples of manifold surfaces with creases
We consider the construction of the manifold-based basis functions for one of the elements in the unstructured quadrilateral control mesh shown in Figure 16. The element is adjacent to a prescribed crease and belongs to four overlapping chart domains as illustrated in Figure 17. Two of the charts domains,Ω 1 andΩ 2 , contain each two crease (28), taking the example of the Type 3 crease chart shown in Figure 13. The number of creases k j = 3 takes the same value for every element. Figure 15: Behaviour of the manifold surface at concave corners. The close-up image in the blue box (middle) shows the behaviour of Type 2 mapping, which develops a fold at the concave corner. In contrast, the close-up image in the black box (right) has no fold and demonstrates that Type 3 mapping is required for concave corners. edges and the other two,Ω 3 andΩ 4 , contain no crease edges. In chart domainΩ 2 the arrangement of the crease edges is rotationally symmetric so that it is Type 1 and in chart domainΩ 1 it is asymmetric so that it is Type 2. For the chartΩ 1 the mapping (25) and for the chartΩ 2 the mapping (19) is to be used. Hence, in comparison to the smooth case illustrated in Figure 6 only the mapping for the chartΩ 1 is different while the other mappings are the same. The presence of creased edges only changes the chart parameterisation for the construction of local polynomials p j (ξ j ). The construction of blending functions w j (ξ j ) is the same for all charts regardless of the presence of crease edges. The tensor-product B-spline blending functions are defined in the reference element and mapped to each element in the chart domains, see (5).
In Figure 18 the application of the manifold-based basis functions in representing a surface with sharp features is demonstrated. The relatively coarse unstructured quadrilateral mesh has a number of tagged crease edges. The location and arrangement of creases has been chosen in order to obtain as many as possible distinct chart configurations. The eight charts with valence v = 6 include a smooth chart, four Type 1 crease charts and three Type 2 crease charts. The rendered manifold surface x(η) is obtained by first projecting the control vertices x I on a Catmull-Clark subdivision surface
x S I = I L I J x J ,(29)
where L I J is the limit matrix (or, mask) of the subdivision surface. The multiplication of the control vertices with the limit matrix involves only the one-neighbourhoods of the vertices x I and projects them onto a Catmull-Clark limit surface. The entries of the limit matrix depend on the valence of the vertex and can be found, for instance, in [15,27]. Moreover, note that Catmull-Clark subdivision surface reduces to cubic B-splines on structured meshes. With the projected control vertices the image of a reference element on the manifold surface is obtained as
x(η) = I N I (η)x S I ,(30)
where N I (η) are the introduced manifold-based basis functions. The manifold construction ensures that the images The reference element (centre) and its four overlapping chart domainΩ j with j ∈ {1, 2, 3, 4}. The reference element is mapped with Ψ j to the blue shaded elements in the four chartsΩ j . Notice that the parameter lines in the charts (with η 1 = const. and η 2 = const. ) are always orthogonal to the spoke edges which guarantees that the derivatives of the maps Ψ j are continuous over the entire chart.
Thin-shell formulation and discretisation
This section provides a brief review of the Kirchhoff-Love thin-shell equations and their discretisation with manifold-based basis functions. In addition, the boundary and crease constraints used in the presented numerical examples are introduced. Further details of our specific Kirchhoff-Love implementation may be found in [28][29][30].
We denote the reference and deformed mid-surfaces of the shell with Ω 0 and Ω, respectively, and the corresponding boundaries with Γ 0 = ∂Ω 0 and Γ = ∂Ω. Any configuration of the shell is assumed to be defined as
ϕ(η 1 , η 2 , η 3 ) = x(η 1 , η 2 ) + η 3 a 3 (η 1 , η 2 ) with η 3 ∈ − t 2 , t 2 ,(31)
where ϕ(η 1 , η 2 , η 3 ) is the position vector of a material point with the convective coordinates (η 1 , η 2 , η 3 ). Similarly, x(η 1 , η 2 ) is the position vector of a material point with the convective coordinates (η 1 , η 2 ) on the shell midsurface. Furthermore, a 3 (η 1 , η 2 ) is the unit normal to the mid-surface x(η 1 , η 2 ) and t is the shell thickness. The corresponding vectors in the reference configuration are denoted with uppercase letters Φ(η 1 , η 2 , η 3 ), X(η 1 , η 2 ) and A 3 (η 1 , η 2 ) so that the reference configuration reads
Φ(η 1 , η 2 , η 3 ) = X(η 1 , η 2 ) + η 3 A 3 (η 1 , η 2 ) .(32)
The covariant basis vectors of the tangent space of the mid-surface are defined as 3 A α = ∂X ∂η α and a α = ∂x ∂η α (33) with the corresponding normal vectors
A 3 = A 1 × A 2 | A 1 × A 2 | and a 3 = a 1 × a 2 |a 1 × a 2 | .(34)
The contravariant basis vectors A α and a α follow from the relations
A α · A β = δ β α and a α · a β = δ β α ,(35)
where δ β α is the Kronecker delta. The contravariant metric of the reference configuration needed in the following is defined as
A αβ = A α · A β .(36)
The deformation gradient can now be expressed as
F = ∂ϕ ∂Φ = ∂ϕ ∂η j ⊗ ∂η j ∂Φ = a j ⊗ A j ,(37)
and a straightforward calculation yields the Green-Lagrange strain tensor
E = 1 2 (F T F − I) = α + η 3 β + η 3 2 . . .(38)
with the membrane and bending strain tensors
α = 1 2 (a α · a β − A α · A β ) A α ⊗ A β (39a) β = 1 2 (a α · a 3,β + a β · a 3,α − A α · A 3,β − A β · A 3,α ) A α ⊗ A β . (39b)
As usual, the quadratic terms in the Green-Lagrange strain tensor E are neglected for thin shells. The potential energy of a hyper-elastic shell is given by
Π(x) = Ω 0 W(α, β) dΩ 0 + Π ext (x) = Ω 0 1 2 Et 1 − ν 2 H αβγδ α αβ α γδ + 1 2 Et 3 12(1 − ν 2 ) H αβγδ β αβ β γδ dΩ 0 + Π ext (x) ,(40)
where W(α, β) is the internal energy density and Π ext (x) is the potential of the external forces. The internal energy density depends, in addition to the two strain tensors α and β, on the Young's modulus E, the Poisson's ratio ν and a geometry related fourth-order tensor with the components
H αβγδ = νA αβ A γδ + 1 2 (1 − v)(A αγ A βδ + A αδ A βγ ) .
When rigid joints or clamped boundaries are present, it is necessary to constrain the surface normals. The introduced manifold-based basis functions are interpolating at the boundaries so that it is straightforward to impose displacement boundary conditions. Along the clamped boundaries Γ 0,c the change of the mid-surface normal (a 3 − A 3 ) is required to be zero. Along the joints Γ 0, j , the C 0 continuity of the basis functions ensures that the displacements to the left (l) and to the right (r) are the same. When the joints are structurally rigid the relative change of the mid-surface normals a (l) 3 · a (r) 3 − A (l) 3 · A (r) 3 is constrained to be zero. In our current implementation the rigid joint and clamped boundary constraints are enforced with the penalty method
Π C [x] = Π[x] + γ 1 2 Γ r (a l 3 · a r 3 − A l 3 · A r 3 ) 2 dΓ rigid joints + γ 2 2 Γ c (a 3 − A 3 ) · (a 3 − A 3 )dΓ clamped edges ,(41)
where γ 1 and γ 2 are penalty parameters. It is straightforward to consider alternative constrained enforcement techniques, like the Nitsche method [31,32] or Lagrange multipliers [29,33]. As usual, the discrete finite element equilibrium equations are derived by first writing the potential (41) as a sum over the set of reference elements { e } using the Jacobian |∂X/∂η|. Here, the index e denotes the number of an element. It is emphasised that each element corresponds to a quadrilateral in the control mesh, as in isogeometric analysis with B-splines. The charts considered in the manifold construction are only used for obtaining the basis functions. With the determined basis functions the reference and deformed mid-surface of a reference element e are approximated by
X h (η 1 , η 2 ) = I N I (η 1 , η 2 )X I and x h (η 1 , η 2 ) = I N I (η 1 , η 2 )x I ,(42)
where X I and x I are the coordinates of the control vertex coordinates. After introducing X h (η 1 , η 2 ) and x h (η 1 , η 2 ) into the element-wise expressed potential energy (41) and numerical integration the discrete equilibrium equations follow from the stationarity principle
∂Π(x h ) ∂x I = ∂Π int (x h ) ∂x I + ∂Π ext (x h ) ∂x I = 0 .(43)
We solve this set of nonlinear equations iteratively with the Newton-Raphson method. That is, at each iteration step (n) the linear equation
∂ 2 Π x (n−1) I ∂x I ∂x J x (n) J − x (n−1) J = − ∂Π x (n−1) I ∂x I(44)
is solved to determine x (n) J for a given x (n−1) J . The expression ∂ 2 Π/∂x I ∂x J on the left is the stiffness matrix whereas the expression ∂Π/∂x I on the right is the residual vector. The full expression for (44) can be found in [4,34].
Examples
We consider beam, plate and shell examples to demonstrate the utility and convergence of the proposed creased manifold-based basis functions in finite element analysis. In all the examples the Kirchhoff-Love model reviewed in Section 4 is discretised with manifold-based basis functions. The blending functions are chosen to be cubic B-splines and the local polynomial basis is a quadratic tensor-product polynomial. In convergence studies all the finite element integrals are integrated with 9 × 9 Gauss quadrature points. The used standard Gauss integration rules do not take into account that the introduced polynomial manifold-based basis functions consist in each element out of 4 × 4 piecewise polynomials. For the geometrically nonlinear simulation of a pinched tube we use 3 × 3 quadrature points to save computing time.
In constructing the basis functions all element edges on the domain boundaries are tagged as crease. This eliminates the need for ghost cells as originally used in [24]. As discussed, the number of basis functions in a chart domain has to be equal or less than the number of vertices. Hence, to be able to use a quadratic tensor-product basis in all charts, mid-face and mid-edge vertices are introduced. This increases the number of vertices in a chart domain from 2v j + 1 to 6v j + 1, where v j is the valence of the centre vertex. In all examples, the rigid joint and clamped boundary constraints are enforced with the penalty method. The penalty parameters are chosen to be γ = O(E), where E is the Young's modulus. Figure 19: Geometry and loading of the propped cantilever beam. The left end is clamped and the right end is simply supported. Different treatments are considered for the hinge at the midspan.
Propped cantilever beam
As an introductory example, we compute the deflection of a beam subjected to a uniformly distributed transversal loading, see Figure 19. To be able to use bivariate basis functions the beam is modelled as a plate with uniform width and is discretised with a structured quadrilateral mesh. The chosen boundary conditions, loading and the Poisson ratio of ν = 0 ensure that the structure deflects like a beam. Both the left and right ends of the beam are tagged as crease, eliminating the need for ghost cells. The left end is clamped so that the displacement and the change of the beam axis normal are required to be zero. The right end is simply supported so that only the displacement is required to be zero.
In the computations the beam midspan is modelled either as continuous, hinged or rigidly-jointed hinged. The considered meshes have always a vertex at the midspan. For the continuous beam (with no hinge) the chart domain corresponding to the midspan vertex is treated in the same way like the other chart domains. In the two cases with a hinge the midspan vertex is tagged as a crease. In geometric terms, across a hinge the displacement is continuous but its derivative may be discontinuous. And, in structural terms, across a hinge only shear forces may be transferred while the bending moment at the hinge is zero. In the case of the rigidly-jointed hinged beam, the two beam axis normals across the hinge are enforced to be weakly continuous with the penalty method so that it becomes possible to transfer moments.
As can be obtained by a straightforward integration the analytical deflection of the continuous propped cantilever is
u(x) = q 4Et 3 (2x 4 − 5Lx 3 + 3L 2 x 2 ) ,(45)
and the deflection of the propped of the cantilever with a hinge is
u(x) = q 2Et 3 (x 4 − 3Lx 3 + 3L 2 x 2 ) if x < L/2 q 2Et 3 (x 4 − 3Lx 3 + 3L 2 x 2 − 2L 3 x + L 4 ) if x ≥ L/2 .(46)
In Figure 20 the convergence of the relative L 2 norm and energy norm errors with decreasing mesh size is plotted. As can be seen, both L 2 and energy norm errors converge in all cases with the optimal rate. The errors for the beam with the hinge are slightly smaller. As to be expected, the continuous beam and rigidly-jointed hinged beam give very similar results. This confirms the soundness of the enforcement of rigid joint constraints with the penalty method.
Square plate
Next, we consider the deflection of a square plate with different boundary conditions subjected to uniformly distributed transversal loading, see Figure 21. Two different types of boundary conditions are considered. In the first case all boundary edges are simply supported and in the second case two opposite boundary edges are simply supported and the other two are clamped. At the simply supported boundary edges the displacements are constrained to be zero and at the clamped edges both the displacements and their derivatives are constrained to be zero. Similar to the propped cantilever example also a hinge line is introduced along the centre of the plate, which is considered to be present in only some of the computations. If a hinge line is considered to be present, it is always modelled as rigidly-jointed.
The structured and unstructured coarse meshes used in the computations are depicted in Figure 22. The unstructured mesh has eight extraordinary vertices, with four vertices of valence v j = 3 and the other four of valence v j = 5. In the convergence studies the refined meshes are obtained by subdividing the shown coarse meshes with the Catmull-Clark subdivision. The analytical deflections for both boundary conditions can be found in Timoshenko and Woinowsky-Krieger [35,Chapters 5 and 6]. Two representative finite element solutions for the considered two boundary conditions are shown in Figure 23. As to be expected the deflection of the plate with two clamped edges is much smaller than the deflection of the fully simply supported plate. In Figure 24 the convergence of the relative L 2 norm error with decreasing mesh size is plotted. In all cases the errors converge with the optimal rate of 2. Figure 24a depicts the convergence for the structured mesh and Figure 24b for the unstructured mesh. Both plots contain the convergence of the fully simply supported plate as well as the partially clamped plate. In addition, the convergence of the plates with rigidly-jointed hinges are included. The continuous and rigidly-jointed plates yield the same results providing an evidence for the soundness of the proposed weak enforcement of rigid joint constraints. Similar as in the propped cantilever example, the results for the less constrained fully simply supported plate are smaller than the errors for the partially clamped plate. It is noteworthy that the manifold-based basis functions can achieve the optimal convergence order even for an unstructured mesh. Moreover, the magnitude of the relative errors for the structured and unstructured meshes is almost the same.
Pinched square tube
This last example involves the geometrically nonlinear analysis of a pinched square tube subjected to two diametrically opposite concentrated forces, see Figure 25. The tube consists out of two horizontal and two vertical plates that are rigidly connected along their edges. It is discretised either with a uniform structured mesh with an element size h = L/16 or with an unstructured mesh as shown in Figure 26. To simulate the rigid joints between the plates the relevant edges in the mesh are tagged as a crease and the angle between two normals across a crease are constrained to remain constant during deformation. This example is an adaptation of the pinched cylinder benchmark example widely used for comparing the performance of shell elements. The square tube has also been previously considered in [29]. The deflected shapes at three different load values are shown in Figure 26. As can be inferred from the deflected shapes, the tube exhibits large membrane deformations and localised bending deformations around the rigid creases and two load application points. Furthermore, it is evident that the right angle between the plates is preserved during deformation. In the load-displacement curve depicted in Figure 27 the results for the structured and unstructured meshes are visually indistinguishable. Both are in close agreement with the results obtained with the commercial finite element software Abaqus.
Conclusions
We have presented, by extending Majeed et al. [24], new manifold-based basis functions for isogeometric design and analysis of surfaces with arbitrary smoothness, prescribed sharp features and boundaries. The surface is described with a quadrilateral control mesh and C 0 continuous creases are introduced along mesh edges tagged by the user. Manifold techniques are extremely versatile in the sense that the chart domains, their respective transition functions, local polynomial approximants, and blending functions can all be chosen to fit the specific needs of the application at hand. We introduce creases by choosing polynomial approximants that are piecewise C 0 continuous along the element edges and, in addition, modifying the geometry of the chart domains. Due to the similarities between the C 0 continuous creases and boundaries, the new basis functions also simplify the treatment of boundaries. That is, different from [24], the boundaries can be described without introducing additional layers of ghost elements outside the domain. The chart domains have the shape of polygonal disks with curved boundaries and are composed out of quasi-conformally mapped unit squares representing the reference finite elements. The respective transition functions are easily computed by expressing the parametric coordinates of a unit square as a complex number. Furthermore, a new type of blending function is proposed which is assembled from tensor-product cubic B-spline pieces. In contrast to the ones introduced in [24], the new blending functions do not require normalisation so that the partition of unity approximation leads to non-rational basis functions. Finally, in order to obtain a mesh-based approximation scheme, the coefficients of the local polynomials are expressed as vertex coefficients using a least-squares procedure. The degree of the local polynomials has to be chosen so that there are no more coefficients than the number of vertices in a chart domain. The number of vertices in a chart domain can be increased by introducing more vertices, i.e. refining the control mesh, while keeping the chart domain size constant. The obtained basis functions have a closed form analytic description, are locally supported and are polynomial in regular regions of the mesh. In closing, we note that the introduced manifold-based basis function construction may be interpreted as the extension of the partition of unity method to manifold surfaces. At the same time, with suitably chosen local polynomials, manifold-based basis functions can reproduce B-splines in mesh regions with a regular connectivity [36]. Hence, manifold-based constructions may serve as a bridge between isogeometric analysis using splines [37] and the many partition of unity method inspired discretisation techniques, such as the generalised finite element method [38], hp- clouds [39] or the extended finite element method [40]. Specifically, manifold-based constructions can facilitate the consideration of industrial CAD geometries, in the form of NURBS and other spline representations, in the partition of unity methods. In turn, the very many enrichment techniques developed for partition of unity methods can be applied to isogeometric analysis by utilising them as local polynomial approximants in the manifold construction. The exploration of these links suggests itself as a promising direction for future research.
Appendix A. Conformal maps
In this section we briefly motivate the quasi-conformal map used throughout this paper; see also [41] for a general visually orientated introduction to complex analysis. In contrast to the conformal map used in Majeed et al. [24] the quasi-conformal map is not angle-preserving, but it is infinitely smooth except at the extraordinary vertex. Or expressed differently, with a conformal map an infinitesimal circle is mapped to a circle whereas a quasi-conformal map maps it to an ellipse. The quasi-conformal map is essential for the introduced new chart domains, which are necessary for the construction of the creased basis functions. As discussed, three different quasi-conformal maps Ψ I j , Ψ II j and Ψ III j are needed to map the reference element onto elements in the chart domainsΩ j according to Ψ I j , Ψ II j , Ψ III j : η = (η 1 , η 2 ) ∈ → ξ = (ξ 1 j , ξ 2 j ) ∈Ω j .
In the complex plane the reference element coordinates η = (η 1 , η 2 ) can be, as be depicted in Figure A.28a, expressed either in Cartesian or polar form
z = η 1 + iη 2 = |z|(cos φ + i sin φ)
with the radius and the phase |z| = (η 1 ) 2 + (η 2 ) 2 , φ = arctan(η 2 /η 1 ) .
With the Euler's formula the polar form is equivalently expressed by
z = |z|e iφ .
Considering a complex number as a vector from the origin to the point with the coordinates z, a multiplication by a scalar represents a scaling and a multiplication by e iθ represents a rotation. For instance, Figure A.28b illustrates that ze iθ is obtained by rotating z anticlockwise by an angle θ and that 1/2z is obtained by scaling the radius of z by one half. In the quasi-conformal map introduced in (21), repeated here for convenience, Λ I qr (z; v j , n j ) = |z| β |z| 4/v j z 4/v j e i2π(n j −1)/v j = |z| β e iφ4/v j e i2π(n j −1)/v j , the parameter β controls how the radius of z is scaled. For β = 4/v j the mapping reduces to the conformal map used in Majeed et al. [24], i.e., Λ I qr (z; v j , n j ) = z 4/v j e i2π(n j −1)/v j = |z| 4/v j e i4φ/v j e i2π(n j −1)/v j , which scales the radius of z to |z| 4/v j . In the present paper we choose β = 1 so that the radius is not scaled. The choice of β on the conformal maps is illustrated in Figure A.29. As visible, for β = 1 the parameter lines are not orthogonal to each other within the elements and the map is not angle-preserving. Therefore, it is called a quasi-conformal map. For the introduced maps Ψ II j and Ψ III j , i.e. (26) and (28), it is important that the radius of z is not scaled. This is necessary because each creased sector may have different number of elements. Not scaling the radius ensures that the quasi-conformal map and its derivatives across crease edges are continuous c.f. Figures 10, 11 and 13.
Appendix B. Blending functions
The blending functions proposed in this paper are polynomial in contrast to the rational blending functions used in Majeed et al. [24]. The difference between the two constructions is best understood by comparing the two Figures 3 and B.30. In Figure B.30 the construction process of the rational blending functions is illustrated. Notice that the cubic B-spline basis used within the reference element is not complete so that it does not add up to one. Therefore, normalisation is necessary to satisfy the partition of unity property, resulting in rational blending functions. With the numbering introduced in Figure B.30 the rational blending functions are given by
w j (η) = B 2 j (η) 2 k=1 B 2k (η)
with w j (ξ j ) = w j (Ψ −1 j (ξ j )) and j ∈ {1, 2} .
It is straightforward to extend this construction to the bivariate case. As a final remark, the rational blending functions have one knot and the polynomial blending functions used in this paper have three knots within the reference element.
Figure 1 :
1Isogeometric analysis of a genus 2 surface with creased C 0 continuous edges and sharp corners. (a) Control mesh with all the edges to be creased marked in red. (b) The manifold surface with the faithfully reproduced creases. (c) The deformed surface obtained with (nonlinear) thin-shell finite element computation. In the computation the creases are modelled as rigid so that the angle across two adjacent surface pieces is maintained during the deformation.
Figure 4 :
4Univariate basis functions and their derivatives over four elements. Five basis functions (solid) are non-zero over the two elements of the centre chart.
Figure 5 :
5An unstructured mesh with extraordinary vertices of valence v ∈ {3, 5}. The shaded element is overlapped by four charts and the union of the four charts have 16 unique vertices in total. The reference element and the chart domains corresponding to the shaded element are shown inFigure 6.
Figure 6 :
6The reference element (centre) and its four overlapping chart domainsΩ j with j ∈ {1, 2, 3, 4}. The reference element is mapped with Ψ I j to the blue shaded faces in the four chartsΩ j . The variables n j denote the face number of the shaded face and v j the valence of the centre vertex. Notice that the parameter lines in the charts (with η 1 = const. or η 2 = const.) are always orthogonal to the spoke edges which guarantees that the derivatives of the maps Ψ I j are continuous over the entire chart domain.
Figure 7 :
7The map from the reference element to a chart domain corresponding to a vertex with valence v j = 3 as denoted Ψ I 4 in
Figure 8 :
8Genus 2 surface with creased edges and sharp corners. The creased edges are marked in red. Each subfigure indicates the type of crease treatment to be used around the marked vertices.
Figure 9 :
9Two examples of Type 1 creases. The k j creased edges tagged by the user (marked in red) divide the chart domain into k j equiangular sectors shaded in different colours. Notice the rotational symmetry of the crease sectors.
Figure 10 :
10Example of a Type 2 crease. The centre vertex has the valence v j = 5 and k j = 2 crease edges (marked in red). The number of elements in each of the two sectors is different, i.e. two and three respectively.
Figure 11 :
11Example of a Type 2 crease. The centre vertex has the valence v j = 5 and k j = 3 crease edges (marked in red). The number of elements in each of the three sectors is different, i.e. two, one and two respectively.
Figure 12 :
12Illustration of the numbering used to define the map Λ II qr in(26) with the help of the Type 2 crease chart shown inFigure 10. The number of creases k j = 2 takes the same value for every element.
Figure 13 :
13Example of Type 3 crease chart domain. The centre vertex has valence v j = 5 and k j = 3 creased edges (marked in red). One of the three sectors is non-convex (highlighted in red).
Figure 14 :
14Illustration of the numbering used to define the map Ψ III c in
Figure 16 :
16An unstructured mesh with extraordinary vertices of valence v ∈ {3, 5}. The shaded element is overlapped by four charts and the union of the four charts have 16 unique vertices in total. Edges tagged as crease are marked in red.
Figure 17 :
17Figure 17: The reference element (centre) and its four overlapping chart domainΩ j with j ∈ {1, 2, 3, 4}. The reference element is mapped with Ψ j to the blue shaded elements in the four chartsΩ j . Notice that the parameter lines in the charts (with η 1 = const. and η 2 = const. ) are always orthogonal to the spoke edges which guarantees that the derivatives of the maps Ψ j are continuous over the entire chart.
Figure 18 :
18Three different views of a manifold surface (top) approximating a given control mesh (bottom). In the control mesh, crease edges are marked as red. The mesh includes eight valence v = 6 vertices and the corresponding charts are categorised as a smooth chart (empty dot) and Type 1 (red dot) or Type 2 (blue dot) crease charts. of the reference elements on the surface are connected with the required smoothness. The projection of the control vertices onto a Catmull-Clark surface is only used to obtain a visually pleasing surface. It is inconsequential when manifold-based basis functions are used as finite element basis functions.
Figure 20 :
20Convergence of the relative L 2 and energy norm errors for the propped cantilever beam. The rigidly-jointed and hinged beams contain a hinge at the midspan while the continuous beam does not. For the rigidly-jointed beam the beam axis normals across the hinge are constrained to be weakly continuous.
Figure 21 :Figure 22 :Figure 23 :
212223Definition of the square plate problem. Two types of boundary conditions are considered, with either all edges simply supported (left), or two opposite edges are simply supported while the other two are clamped (centre). The dashed vertical line at the centre indicates a hinge, which is present in some of the computations. If a hinge is present, the mid-surface normals across the hinge line are constrained to be weakly continuous. Coarse control meshes used in square plate computations. Deflected shapes of the square plate with two different boundary conditions. The deflections are scaled with the same factor in both cases.
Figure 24 :
24Convergence of the relative L 2 norm error for the simply supported and the partially clamped square plates. The results for the continuous and rigidly-jointed plates are almost identical.
Figure 25 :
25Geometry and loading of the pinched square tube.
Figure 26 :
26Deflected shapes of the pinched tube at load values F = 10, F = 50 and F = 100 (from left to right).
Figure 27 :
27Load-displacement curve of the pinched square tube for the load attachment point.
and scaling in the complex plane Figure A.28: Illustration of a complex number in the complex plane and the operations of rotation and scaling.
Figure A. 29 :
29Comparison of the iso-parameter lines for the conformal and the quasi-conformal map depending on the value of the parameter β for valences v ∈ {3, 4, 6}.
Figure B. 30 :
30Construction of smooth rational blending functions from cubic B-splines defined over a parameter space with a knot-distance 1/2.
We refrain in this paper from using the term patches because in computer-aided design literature patches denote what are the elements in computational mechanics.
In this section, k is used for the number of creased spoke edges, which is different from its meaning in G k and C k continuity.
Throughout Section 4, the Greek indices take the values {1, 2}, the lowercase Latin indices take the values {1, 2, 3}, a comma denotes differentiation and the summation convention over repeated indices applies.
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| []
|
[
"Robust Pricing with Refunds *",
"Robust Pricing with Refunds *"
]
| [
"Toomas Hinnosaar ",
"Keiichi Kawai "
]
| []
| []
| We characterize a selling mechanism that is robust to the seller's uncertainty about the buyer's signal structure. We show that by offering a generous refund policy the seller can significantly reduce this type of uncertainty and regain market power. A simple mechanism that utilizes a generous refund policy and random discounts achieves the best guaranteed-profit among all possible mechanisms. JEL: D82, C79, D42 | 10.2139/ssrn.3165146 | [
"https://arxiv.org/pdf/1808.02233v1.pdf"
]
| 51,934,504 | 1808.02233 | 3d384b5eb48849bef086fd9749fef9aece781926 |
Robust Pricing with Refunds *
Toomas Hinnosaar
Keiichi Kawai
Robust Pricing with Refunds *
This version: August 6, 2018 First Draft: April 18, 2018optimal pricingrobustnessreturn policiesrefundsmonopolyinforma- tion designmechanism design
We characterize a selling mechanism that is robust to the seller's uncertainty about the buyer's signal structure. We show that by offering a generous refund policy the seller can significantly reduce this type of uncertainty and regain market power. A simple mechanism that utilizes a generous refund policy and random discounts achieves the best guaranteed-profit among all possible mechanisms. JEL: D82, C79, D42
Introduction
We analyze a seller's robust pricing problem in the face of uncertainty about the buyer's information and learning. Consumers learn about product characteristics from various sources, and firms are often not only uncertain about how much value buyers receive from consuming their products but also how well buyers know whether these products fit their needs. For example, an online shoe store does not know whether the shopper has been "showrooming" and already knows whether particular shoes fit well. Similarly, a booking agent does not know whether a consumer planning to book a flight has a specific travel itinerary in mind or not. Such uncertainty regarding the buyer's information and learning can severely limit the seller's ability to extract profits from trade.
What profit can the seller guarantee itself in the presence of such uncertainty? How should the seller set the price to achieve the best guaranteed-profit? We show that a simple mechanism that combines a generous refund policy with random non-refundable offers is the answer.
More specifically, we analyze a bilateral trade model, where the buyer's valuation of the product is either high or low, and the exact value is unknown to everyone including the buyer. The seller and the buyer share a common prior about the buyer's valuation. The buyer observes a private signal of her valuation but learns her valuation only after purchasing the product. The seller can allow the buyer to return the product but incurs a restocking cost for each returned product.
The key feature of the model is the seller's uncertainty about the buyer's information structure (i.e., the distribution of signals). The seller neither knows the buyer's signal distribution nor has a prior belief about possible signal distributions. He only knows that the signals satisfy Bayes' rule. 1 For example, the support of the signal distribution may only consist of moderate signals, i.e., the buyer's information structure may be quite uninformative. Then a sufficiently high price would result in no sales. Alternatively, the buyer's signal distribution may generate either a sufficiently favorable or unfavorable signal, i.e., the buyer's information structure may be quite informative. Then, setting a low price would leave a lot of money on the table.
Our goal is to identify the seller's selling mechanism that performs as well as possible regardless of the buyer's signal distribution. We show that a simple mechanism that randomizes between an offer with a generous refund and log-uniformly distributed nonrefundable discounted offers achieves the seller's best guaranteed-profit, i.e., the profit the seller can guarantee irrespective of the buyer's signal distribution.
To understand the intuition behind our result, we first note that by offering a generous refund the seller can reduce the importance of the buyer's signal in the purchasing decision. That is, for any given buyer's signal distribution, the buyer is more likely to buy the product at the same price with a refund than without. By making the buyer's demand less elastic this way, a generous refund enables the seller to charge a higher price without sacrificing the probability of trade. The seller, however, needs to be wary of costs associated with product return. For each returned product, the seller incurs the restocking cost. Thus, the seller must ensure that if an offer is refundable, it is only attractive to the buyer who is unlikely to return the product (i.e., the one with a sufficiently favorable signal). By doing so, the seller can ensure that the buyer who accepts the refundable offer always brings a positive profit in expectation. The seller achieves this goal by offering a generous but not full refund.
We also note that, loosely speaking, any pricing policy that brings a large profit when the buyer is relatively well-informed tends to perform poorly when the buyer is relatively uninformed, and vice versa. Therefore, to increase its guaranteed profit, the seller needs to hedge simultaneously against the buyer's signal distributions that are relatively informative and relatively uninformative.
With these observations at hand, we now explain why the seller can achieve the best guaranteed-profit through the mixture of a generous refund (that is only attractive to the buyer with a sufficiently favorable signal) and random non-refundable offers (that are attractive to the buyer only when her signal is moderate or favorable). The generous refund brings a large profit if and only if the buyer's signal distribution is likely to generate sufficiently favorable signals. A non-refundable offer brings a large profit if and only if the buyer's signal distribution is likely to generate a signal that is equal to or slightly above the price offered by the seller. Furthermore, for any buyer's signal distribution under which the refundable offer brings a small profit, the randomized non-refundable offer brings a sufficiently large profit and vice versa. 2 In this sense, the refundable offer hedges against the event that the signal distribution is informative, and the randomization over non-refundable offers hedges against the less informative distributions. This observation also explains why the seller who is interested in maximizing the guaranteed-profit would find it optimal to always use a generous refundable offer with a positive probability no matter how high the restocking cost to the seller is.
We then use our findings to derive the buyer-optimal information structure, as well as the sharp upper-bound of the buyer's payoff with respect to the possible information structure. More specifically, we analyze the buyer-optimal information design problem, where the buyer first chooses an information structure, and the seller chooses an optimal pricing and refund policy knowing the buyer's information structure. 3 We show that the buyer-optimal signal distribution is the worst-possible distribution to the seller (i.e., the distribution for which the seller's maximal profit is exactly the best guaranteed-profit). This finding implies that if we compare the highest payoffs the buyer can obtain (by choosing her information structure) when the seller can offer a refund and cannot offer a refund, then the former is strictly lower than the latter. In this sense, a refund policy increases the seller's market power against buyers who search information strategically.
Our findings are closely related to the results in Roesler and Szentes (2017) and Du (2018). Roesler and Szentes (2017) identify the information structure that maximizes the buyer's welfare when the seller best responds to the information structure via uniform pricing. Du (2018) shows that the information structure found in Roesler and Szentes (2017) indeed minimizes the profit the seller can obtain, and the seller can obtain the best guaranteed-profit by what he calls an exponential pricing regardless of the buyer's signal distribution. 4 Unlike the present paper, these papers do not consider the possibility of product returns. With a refund policy, the seller can potentially increase his profit through two channels. The seller can indirectly control the buyer's learning by incentivizing the buyer to learn through purchase. The seller also can sequentially screen the buyer, first by the signal distribution and the signal; and then by the realized valuation for the product. 5 Therefore, the selling mechanism that maximizes the seller's best guaranteed-profit may be complex. Nevertheless, we obtain results parallel to Roesler and Szentes (2017) and Du (2018): a generalization of the signal distribution identified in Roesler and Szentes (2017) is the worst possible buyer's signal distribution for the seller; and an extension of the exponential pricing found in Du (2018) achieves the seller's best guaranteed-profit against any possible buyer's signal distributions.
Our findings offer a novel rationale for generous return policies. The literature has identified various reasons that companies may use return policies: e.g., as costly signals for product quality and product fit for the consumer (Grossman (1981); Moorthy and Srinivasan (1995); Inderst and Ottaviani (2013)); as insurance for risk-averse consumers (Che (1996)); and as a tool for price discrimination (Zhang (2013); Escobari and Jindapon (2014); Inderst and Tirosh (2015)). 6 Among these, the closest to our paper is Inderst and Tirosh (2015). In an environment where the seller knows the buyer's signal distribution, Inderst and Tirosh (2015) show that return policies work as "metering devices", where refunds make different consumers more similar and thus allow the firm to capture more of the surplus by raising prices. 7 Consequently, the seller sets the refund amount above the restocking cost. In contrast, our results show that the seller offers a generous refund (i.e., "almost" full refund) when the seller is uncertain of the buyer's signal distribution.
Lastly, we note that the model analyzed in the present paper can be interpreted as a game between the seller, who aims to maximize his profit by choosing a price-refund pair, and the adversarial nature, who aims to minimize the seller's profit by choosing the buyer's signal distribution. Thus, the game is akin to Bayesian persuasion games with competing senders. We therefore fully utilize the concavification technique (Aumann et al. (1995); Kamenica and Gentzkow (2011)); and the properties of equilibrium payoff functions in the competitive Bayesian-persuasion settings (Boleslavsky and Cotton (2018); Au and Kawai (2017a,b)) to derive our results.
Model
There is a (male) seller who can produce a product at no cost, and a (female) buyer whose valuation for the product is ∈ {0, 1}. The buyer's valuation follows a commonly known distribution such that = Pr ( = 1). Neither the seller nor the buyer knows the realization of . However, the buyer receives a signal of her valuation prior to purchase. (Details will be explained below.)
We start our analysis with the case in which the seller's (pure) strategy is a contract ( , ) that specifies a price together with a refund . The seller may use a mixed strategy {( , )}. We call the seller's mixed strategy, i.e., a distribution over contracts, a policy; and the realized contract an offer. Based on the information she has, the buyer decides whether to buy after observing the contract ( , ). We analyze the more general setting in which the seller can offer any mechanism in Section 3.4.
If the buyer purchases the product, then she learns the realized value of . If the realized offer is non-refundable (i.e. = 0), then the game ends. If > 0, then the buyer decides whether or not to return the product. When the product is returned, the seller incurs a 6 Escobari and Jindapon (2014) also provide some empirical evidence on the use of refundable tickets by airlines. They show that a fully refundable ticket is typically about 50% more expensive than a non-refundable ticket, but the difference disappears in the last week before the departure. These facts fit well with our model predictions. 7 Similar ideas have been studied in other contexts, such as overbooking by airlines, e.g., Ely et al. (2017).
commonly known restocking cost > 0. We sometimes use ≡ 1+ ∈ (0, 1) to denote the normalized restocking cost. If the buyer purchases and keeps the product then her payoff is − , and the seller's profit is . If the realized offer is refundable (i.e., > 0) and she returns the product, then her payoff is − , and the seller's profit is − − = − − 1− . If the buyer does not buy the product, then her payoff and the seller's profit are both zero.
We can represent a buyer's signal as a posterior = Pr ( = 1) that is a random variable drawn from a distribution function ∈ F ≡ { ∶ [ ] = }. 8 For this reason, we use a distribution over posteriors to represent the buyer's information structure, and call it a signal distribution. Analogously, by a signal , we refer to the realization of the posterior from signal distribution .
Take an arbitrary buyer's signal distribution . We use (( , ) | ) to represent the seller's expected profit when the offer is ( , ), and {( , )} (( , ) | ) to represent the expected profit from policy {( , )}. We say policy {( , )} guarantees profit when
{( , )} (( , ) | ) ≥ for all ∈ F, i.e., min ∈F {( , )} (( , ) | ) ≥ . Our goal is to identify a strategy {( , )} that brings the best guaranteed-profit defined by * ≡ sup {( , )} min ∈F {( , )} (( , ) | ) .
The best guaranteed-profit is the supremum of the profits the seller can guarantee in the following game against a fictitious player called the (adversarial) nature whose objective is to minimize the seller's expected profit: 4. The buyer decides whether or not to buy. If she does not buy, the game ends.
5. The buyer learns the value if she buys. If = 0, the game ends.
6. If > 0, the buyer decides whether or not to return the product.
7. If the buyer returns the product, the seller refunds to the buyer, and incurs the restocking cost .
Best Guaranteed-Profit
We characterize the best guaranteed-profit * by identifying its lower and upper bounds. We first identify the seller's best guaranteed-profit from a deterministic policy (Lemma 1). That is, we identify * ≡ sup Next we derive a worst distribution * for the seller by analyzing an auxiliary game in which the seller chooses his pricing policy after nature chooses the buyer's signal distribution (Lemma 3). That is, we derive * ∈ arg min ∈F sup ( , ) (( , ) | ) and * ≡ sup
( , ) (( , ) | * ) .
While * ≤ * ≤ * by construction, there exists a threshold value such that if the (normalized) restocking cost ∈ (0, ] then * = * and therefore the best guaranteed-profit * = * = * can be achieved by a deterministic policy. In contrast, if the normalized restocking cost is higher, i.e. ∈ ( , 1), then * < * and therefore there is room for improvement from a randomized policy. In this case we show that a randomization over a refundable offer with a generous refund and log-uniformly distributed non-refundable offers achieves the upper bound * regardless of the signal structure (Theorem 1). Lastly, we show that there exists no mechanism that guarantees the seller a higher profit than * = * (Theorem 2).
Best Guaranteed-Profit with Deterministic Policies
We start our analysis by identifying the best guaranteed-profit by a deterministic policy. That is, we characterize * ≡ sup ( , ) min ∈F (( , )| ), which is a lower bound for * . Let ( | ( , )) denote the seller's expected profit when the offer is ( , ) and the buyer's signal is . Then, (( , ) | ) = [ ( | ( , ))]. Also for notational simplicity, we use ( , ) to denote its minimum over , i.e, ( , ) ≡ min ∈F (( , ) | ), so the best guaranteed-profit with deterministic offers is * = sup ( , ) ( , ).
Suppose that the seller makes a non-refundable offer ( , 0). Then, the buyer with signal buys if and only if ≥ , i.e.,
( | ( , 0)) = { 0 if ≤ , if > .
The seller's profit (( , ) | ) is minimized when the probability of the buyer's signal being larger than is minimized, i.e., when minimizes 1− ( ) subject to [ ] = . If > , then this occurs when the buyer's signal distribution induces two signals (which results in no trade) and 1 (which results in trade) with probabilities 1− 1− , and − 1− , respectively. In contrast, if ≤ , then this occurs when the buyer's signal does not disclose any additional information, i.e., induces signal (which results in no trade) with probability one.
More formally, we can derive ( , 0) by utilizing the concavification approach. 9 Let con [− (⋅| ( , 0))] ( ) be the value of concave closure of − (⋅| ( , 0)) at , which is represented by the red-dotted line in Figure 1. 10 Then ( , 0) = −con [− (⋅| ( , 0))] ( ). While a higher results in a higher profit margin should trade occur, it leads to a lower probability of trade. The seller balances this trade-off by offering = 1 − √ 1 − , so
( , 0) = −con [− (⋅| ( , 0))] ( ) = { 0 if ≥ × − 1− if < ≤ sup ( , 0) = sup ∈[0, ] − 1 − = (1 − √ 1 − ) 2 . (1) 0 0 1 1 ( , 0) ( |( , 0)) −con [− (⋅| ( , 0))] ( )(2) ( | ( , )) = { { { 0 if <̃( , ) , min {0,̃( ; , )} if =̃( , ) , ( ; , ) if >̃( , ) , wherẽ( ; , ) ≡ − (1 − ) ( + ) = − (1 − ) ( 1− + ).
Observe that if the marginal signal̃( , ) is low, i.e., if the refund is generous, then the buyer buys even when her signal is low. The buyer with a low signal is likely to return the product, and thereby is likely to bring the seller an (ex-post) negative profit. More specifically, as captured by the increasing blue-lines in Figures 2(a) and 2(b), the seller's profit from the buyer with signal conditional on sales̃( ; , ) is increasing in ; and the profit from the buyer with signal is negative, i.e.,̃( ; , ) < 0 if and only if̃( , ) < . Therefore, if̃( , ) < , then the seller's profit is minimized when the probability of the seller receiving a signal̃( , ) is maximized, as captured by the red dotted-line in Figure 2(a). The seller then can improve his guaranteed-profit by offering a less generous refund, and thereby increasing the marginal signal.
In contrast, if the marginal signal is sufficiently high, i.e.,̃( , ) > , then the buyer who buys the product always brings a positive (ex-post) profit to the seller. The seller's profit is thus minimized when the probability of the seller receiving a signal abovẽ( , ) is minimized, as captured by the red dotted line in Figure 2(b). The seller then can improve his guaranteed-profit by offering a more generous refund, and thereby by lowering the marginal signal.
Combining these observations, we conclude that if the seller were to offer a positive refund, then he would set the price as close to one as possible, and to − 1− so that the marginal signal is exactly at̃( , ) = . By doing so, the seller can guarantee himself − 1− . Therefore, when > 0,
( , ) = −con [− (⋅| ( , ))] ( ) ≤ sup , >0 ( , ) = max { − 1 − , 0} .
Recall that the seller's best guaranteed-profit without any refund is (1 − √ 1 − ) 2 , as derived in (1). Thus, the seller's best guaranteed-profit with a deterministic policy is
(3) * = sup , ( , ) = { { { − 1− if ≤̂( ) , (1 − √ 1 − ) 2 if >̂( ) , wherê( ) ≡ 2(1− √ 1− ) 2− √ 1− .
The resulting best guaranteed-profit * is depicted in Figure 3. Lemma 1. Suppose that the seller can only use a deterministic pricing policy ( , ). Then the seller's best guaranteed-profit is * defined in (3). For any > 0, either a generous refund ( , ) = (1 − , 1 − 1− ) (when ≤̂( )), or no refund ( , ) = (1 − √ 1 − , 0) (when ≥̂( )) guarantees profit of (1 − ) * .
Best Guaranteed-Profit with Policy-Independent Signals
We now consider an auxiliary game in which the seller chooses his pricing policy {( , )} after observing the nature's choice of the buyer's signal distribution . By analyzing this To facilitate the further discussion, for any small , we say ( , ) = (1 − , 1 − 1− ) is an offer with a generous refund. Notice that the marginal signal for an offer with a generous refund is , and hence the seller's profit from a generous refund is (1 − ) ∫ 1 − 1− ( ) ≥ 0. Also, when the buyer's signal distribution is , by the profit from a generous refund, which we denote by ( ), we refer to the supremum of the profits the seller can achieve by some offer with a generous refund ( ), That is,
(4) ( ) ≡ sup ((1 − , 1 − 1 − )| ) = ∫ 1 − 1 − ( ) .
We derive * in the following steps. We first characterize the seller's highest profit from a non-refundable offer for a given buyer's signal distribution , which we denote by ( ). We also identify a lower bound of the seller's profit from a generous refund when his profit from a non-refundable offer is , which we denote by ( ). Lastly, we show that a distribution that minimizes max { ( ( )) , ( )} is a worst distribution * for the seller, and * = ( ( * )). Take an arbitrary signal distribution . By applying an argument similar to Roesler and Szentes (2017), we find that the profit the seller can achieve by a non-refundable offer is
( ) ≡ inf { ∶ ( ) ≥ ( ) for all } , where (5) ( ) ≡ { { { { { 0 if ∈ [0, ) , 1 − if ∈ [ , 1) , 1 if = 1.
To see this, notice that if the buyer's distribution is , then for any non-refundable offer ( , 0) , ≥ , the seller's profit is because the buyer's demand is unit elastic. Therefore, if ( ) = ( ) for some and ≥ , then the seller's profit from a non-refundable offer is . 12 Next, take an arbitrary buyer's distribution such that ( ) = . We derive a lowerbound of the seller's profit from a generous refund, which we denote by ( ). We note that, by (4) Proof. In Appendix.
( ) ≡ { { { (ln −1)+ 1− if < , − 1− if ≥ .= { if < 1 − √ 1 − , if ≥ 1 − √ 1 − , and (̂) ≥̂. Now, define * by (6) * ( ) ≡ { { { { { { { { { 0 if ∈ [0, (̂)) , ( ) if ∈ [ (̂) , ) , 1 −̂if ∈ [max { (̂) , } , 1) , 1 if = 1.
Furthermore, for any distribution ≠ * , there exists a non-refundable offer or a generous refund that brings a profit higher than (̂). If ( ) < * ( ) for some ∈ [0, ), then the seller's profit from a non-refundable offer ( , 0) is higher than̂, i.e., ( ) >̂= (̂). Next, suppose that ( ) ≥ * ( ) for all ∈ [0, ) so that This proves that * = (̂), and 13
(7) * = { { { − 1− if < 1 − √ 1 − , − * 1− −ln * = − −1 (− −2 ) if ≥ 1 − √ 1 − .
Lemma 3. The distribution * , defined by (6), is a seller's worst distribution. If the buyer's signal distribution is * , then the highest profit the seller can achieve is * defined by (7), and it bounds the guaranteed-profit * from above.
An Optimal Stochastic Policy
We now identify the seller's best guaranteed-profit * using the lower bound * (the red dotted line in Figure 5) and the upper bound * (the blue solid line in Figure 5). For each > 0, we show that there exists a strategy {( , )} such that
(1 − ) * ≤ {( , )} (( , ) | ) for all ∈ F.
This establishes that the seller's best guaranteed-profit is * , i.e., * = * . To see this, we say a strategy is a mixture of log-uniform discounts and a generous refund if, for some > 0, the seller makes: 14 1. non-refundable discounted offers ( , 0), ∈ [ * , ) , with density
0 ( ) = { { { 0 if ≤ , 1 (1− −ln * ) if > , 2. a generous refund ( , ) = (1 − , 1 − 1− ) with probability = { { { 1 if ≤ 1 − ∫ * 0 ( ) = 1− 1− −ln * if > .
Below we show that the seller can guarantee himself the upper bound * identified in Section 3.2 by a mixture of log-uniform discounts and a generous refund. More specifically, the supremum of the guaranteed profit from all mixtures of log-uniform discounts and a generous refund is * . Loosely speaking, any pricing policy that brings a large profit when the buyer is relatively well-informed tends to perform poorly when the buyer is relatively uninformed, and vice versa. Therefore, to increase his guaranteed profit, the seller needs to hedge simultaneously against the buyer's signal distributions that are relatively informative and relatively uninformative. A mixture of log-uniform discounts and a generous refund achieves this goal: the randomization over non-refundable offers works as a hedge against the distributions that are not informative while the refundable offer works as a hedge against the distributions that are informative. As we formally show in the proof of Theorem 1, when he uses a mixture of log-uniform discounts and a generous refund and the buyer's signal distribution turns out to be , the seller's (expected) profit from some non-refundable offer̃( ) and from the generous refund̃( ) are, respectively,
( ) = − * − ∫ * ( ) 1 − − ln * and̃( ) = − + ∫ 0 ( ) 1 − − ln * + ,
The seller's profit from non-refundable offers,̃( ), is decreasing in ∫ * ( ) . Since
∫ 1 0 ( ) = 1 − and is increasing, if ∫ * ( ) is small, then ∫ 1 ( ) is large.
Loosely speaking, this occurs when ( ) is relatively large and flat on [ , 1], i.e., is relatively uninformative. In contrast, ∫ * ( ) is large, i.e.,̃( ) is small, when ( ) is relatively small and flat on [ , 1], i.e., is relatively informative. In this sense, the randomization over non-refundable offers (in the mixture of log-uniform discounts and a generous refund) brings a large profit when the buyer's signal distribution is uninformative, and a small profit when the buyer's signal distribution is informative. A similar argument show that in clear contrast tõ( ), the seller's profit from the generous refund,̃( ), brings a large profit when the buyer's signal distribution is informative, and a small profit when the buyer's signal distribution is uninformative.
Theorem 1.
For any , there exists a mixture of log-uniform discounts and a generous refund that guarantees profit of (1 − ) * . That is, the seller's best guaranteed-profit * is * .
Proof. In Appendix.
Observe that the seller's best guaranteed-profit * is strictly higher than the best profit the seller can guarantee without refunds sup min ∈F (( , 0) | ) (i.e., the brown dottedline in Figure 5). The reason is that without the opportunity of making refundable offers, which work as a hedge against informative signal distributions, the seller is forced to use a pricing policy that induces lower prices more frequently. Then, however, the resulting profit would be lower when the buyer's signal distribution turned out to be not so informative. 15 15 The result reported here is closely related to the finding in Du (2018) that analyzes the environment where the seller cannot make refundable offers (or equivalently the restocking cost is infinitely large). Du (2018) shows that the distribution * , where * ∈ (0, * ) that uniquely solves * (1 − ln * ) = , is the worst possible buyer's information structure for the seller; and that the log-uniformly randomized prices over [ * , 1] achieves the seller's best guaranteed-profit regardless of the buyer's signal distribution. We also note that Roesler and Szentes (2017) show that this distribution * maximizes the buyer's welfare when the seller best responds to this information structure via uniform pricing.
Best Guaranteed-Profit with General Mechanisms
We have identified the best guaranteed-profit * that the seller can achieve using a simple randomized pricing and refund policy. One may wonder if the seller can improve his best guaranteed-profit by using a more intricate mechanism that screens the buyer based on her signal distribution and her realized signal . Below we show that there exists no such mechanism. More specifically, for any (indirect or direct) mechanism, the seller's profit is bounded from above by * for some buyer's signal distribution.
To show this, we analyze an auxiliary game in which the nature chooses the buyer's signal distribution , and then the seller chooses a mechanism after observing the nature's choice. We show that if the buyer's signal distribution is * , then the seller's profit is bounded from above by * . Since the seller's best guaranteed-profit is bounded from below by * , the seller's equilibrium profit of this auxiliary game is * . Furthermore, the seller's equilibrium profit of this auxiliary game defines an upper bound of the seller's best guaranteed-profit.
We first note that, for any mechanism that the seller chooses after observing the nature's choice, there exists an outcome-equivalent simple static direct mechanism with refunds.
Definition 1.
We say a mechanism ≡ { ( ) , { 0 ( ) , ( )}} ∈[0,1] is a direct mechanism with refunds if, for each buyer's report ∈ [0, 1], the mechanism specifies (i) ( ): the transfer from the buyer to the seller; (ii) 0 ( ) ∈ [0, 1]: the probability that the buyer receives the product without an option to return; and (iii) ( ) ∈ [0, 1 − 0 ( )]: the probability that the buyer receives the product with an option to return with refund = 1. By adopting the standard argument, we can simplify the seller's problem to the one in which he chooses an increasing function 0 (⋅) (instead of , which is a triplet of functions, that satisfies IC and IR). More formally, the incentive compatibility condition (IC) is equivalent to 0 ( ) is increasing in and ( ; | ) = ∫ 0 0 (̃)̃. 1] is the seller's best-response to the nature's choice , then we can represent ( | * ), i.e., the seller's profit from buyer with signal , independent of * ( ) and * ( ).
Therefore, if * = { * ( ) , { * 0 ( ) , * ( )}} ∈[0,( | * 0 ) ≡ { * 0 ( ) − ∫ 0 * 0 (̃)̃if < , * 0 ( ) + − 1− (1 − * 0 ( )) − ∫ 0 * 0 (̃)̃if ≥ .(8)
Proof. In Appendix.
In the subgame where the nature has chosen * , the seller's payoff is * . Proof. In the appendix.
Therefore, the seller's equilibrium profit in this auxiliary game is * . Furthermore, the seller's equilibrium profit defines an upper bound of the seller's best-guaranteed profit, which is bounded from below by * . We thus have the required result.
Theorem 2. Suppose that the seller can guarantee himself using some mechanism. Then, for any > 0, there exists a mixture of log-uniform discounts and a generous refund that guarantees (1 − ) .
3. The buyer observes a signal generated by , and then decides whether to purchase or not.
The key assumption we impose is that the buyer commits to a signal distribution at the beginning of the game. In other words, the buyer commits not to learn any information that is not contained in the signal generated using distribution . 16 Note that if the buyer lacks such a commitment power, then she would choose to acquire a fully informative signal. Knowing this, the seller would always set the price to 1 and can capture the full surplus.
Given our focus on the buyer-optimal outcome, we assume that the seller makes an offer that induces a higher buyer's payoff in case he is indifferent between two or more offers. Therefore, the assumption that the seller only uses a deterministic policy is without loss. More formally, let (( , ) | ) be the buyer's payoff when her signal distribution ∈ F, and the offer is ( , ). Also, let (( , ) | ) be the corresponding seller's profit. Let ( * , * ) be an element in * ( ) ≡ arg max ( , )
(( , ) | ) such that (( * , * ) | ) = max ( * , * )∈ * ( ) (( * , * ) | ) .
Our goal is to characterize * ∈ arg max ∈F (( * , * ) | ) , and * ≡ (( * , * ) * | * ) .
Note that the seller's attempt to maximize his profit can cause inefficiency through two channels. First, the seller may make an offer that results in no sales even if the buyer's expected value is strictly positive. Second, even when the buyer accepts the seller's offer, the product may be returned. Nevertheless, there exists a buyer-optimal outcome that is efficient.
The total surplus from trade is bounded from above by . Furthermore, the seller can guarantee a profit of * irrespective of the choice the buyer makes (Theorem 1). Therefore, the buyer's payoff under the buyer-optimal information structure * is bounded from above − * . However, the preceding analysis implies that this bound is sharp, i.e., * = − * , and the worst distribution * for the seller identified in Section 3.2 is the buyer-optimal signal distribution. This is because when the buyer's signal distribution is * the seller's profit is maximized by offering ( , ) = ( * , 0). More precisely, sup ( , ) (( , ) | * ) = * = (( * , 0) | * ) .
Furthermore, when the seller offers ( * , 0), the trade occurs with probability one and there are no returns. Consequently, (( * , 0) | ) = − * .
Theorem 3. The seller's worst signal distribution * is a buyer-optimal signal distribution. Under the corresponding buyer-optimal outcome, the seller's profit and the buyer's payoff are * and − * , respectively.
As an immediate corollary, we can characterize pairs of buyer's payoff and seller's profit that can be supported by some signal structure. More precisely, we say (̂,) is a feasible outcome supported by a if (,) maximizes (( , ) | ) with respect to ( , );̂= ((,) | ); and̂= ((,) | ). We denote the set of all feasible outcomes by O. It follows from the previous discussion that O = {( , ) ∶ ∈ [ * , − ] , ∈ [0, − * ]}.
Discussion
We analyzed the seller's robust pricing problem with uncertainty about the buyer's information and learning, and showed that a simple mechanism that utilizes a generous refund achieves the best guaranteed-profit.
Our model hinges on the key, but restrictive, assumption of a binary buyer valuation, which significantly simplifies the analysis. This assumption is reasonable if buyers' primary concerns are of the exact product match: e.g., shoes and clothes either do or don't fit, a gadget is either compatible with the buyer's use or not, and a business traveler either needs to be in a particular location on a specific date or not. We note that the implication of this assumption is twofold. First, there exists a one-to-one mapping between signals and expected willingness to pays for the product (in the absence of a refund policy). Second, a buyer's return decision is independent of the amount of refund.
Those two properties fail to hold if more than two levels of product fit are present. Thus, the assumption of binary buyer valuation, whilst restrictive, is important for the tractability of the model. More precisely, in the absence of a refund policy, the buyer buys only if her signal , i.e., a posterior over possible levels of product fit , satisfies [ ] ≥ . That is, the seller's profit depends on the buyer's signal only through the value of expected willingness to pay it induces, i.e., [ ]. In this sense, if the seller cannot utilize a refund policy, then we still can represent the seller's uncertainty as the uncertainty over distributions over expected willingness to pays, which is a one-dimensional random variable. 17 In the presence of a refund policy ( , ), however, the buyer with signal buys only if the right-tail of is sufficiently fat, i.e., Pr ( > | ) [ | > ] + Pr ( ≤ | ) ≥ . Therefore, two signals with an identical willingness to pay can result in different outcomes for the seller. Consequently, we are no longer able to capture the seller's uncertainty in terms of the uncertainty over distributions on expected willingness to pays, rendering the generalization into this direction not straightforward.
Having said that, we conjecture that the relaxation of the binary buyer valuation assumption yields qualitatively similar results: for any given restocking cost, there exists a generous refund policy (or a randomization over refund policies) that guarantees the seller a non-negative ex-post (expected) profit; and hence the seller can strictly improve his profit against the worst signal distribution by utilizing such a refund policy.
Another aspect that we did not address is the seller's learning of demand through pricing. Our insight that a well-designed refund policy limits the significance of buyer learning on the seller's profit should apply even in a dynamic environment. However, in a dynamic environment, carefully designed dynamic pricing and resulting buyer's purchasing and re-turn decisions can also be used to learn about what/how buyers learn about the product fit, and hence to improve future pricing decisions. Investigating how the seller's learning motive would shape the intertemporal pricing with refunds would be an interesting venue for future research.
Another possible venue is the application to platform designs. A platform, such as eBay or Airbnb, could choose information that is revealed to the buyer. The seller chooses a pricing and return policy, and the buyer makes a purchasing and return decision. In the absence of competition among platforms, it is natural to expect that one of the Paretooptimal outcomes identified in the previous section would arise. Analyzing how competition affects the welfare and equilibrium information structure is an interesting question.
A Proofs for Section 3 (Best Guaranteed-Profit)
Proof of Lemma 2: First, suppose that < 1− √ 1 − . Then Recall that − * 1− −ln * = * by (7). We thus have .
Since ∫ * ( * 0 ( ) − ∫ * * 0 ()) * 2 = * ∫ * * 0 ( ) , we have * [ ( | * 0 )] = * .
1 .
1The seller chooses a policy {( , )}. 2. Nature chooses a signal distribution after observing the seller's choice of {( , )}. 3. The buyer observes a signal ∼ and an offer ( , ) ∼ {( , )}.
(
( , )| ) . 8 Notice that ∈ F if and only if ∫ 1 0 ( ) = 1 − .
Figure 1 :
1Profit of a non-refundable offer ( , 0)Next, consider a refundable offer, i.e., ( , ) such that > 0. Without loss of generality, we only consider the case where ≥ . 11 The payoff of the buyer with signal from buying is × 1 + (1 − ) × − . She thus buys only if her signal is above the marginal signal ( , ) ≡ − 1− ; and returns with probability 1 − if she buys. The seller's profit from the buyer with signal is thus
Figure 2 :
2Profit from a refundable offer ( , )
Figure 3 :
3Best guaranteed-profit * with a deterministic offer game, we identify a worst distribution for the seller * , i.e., * ∈ arg min∈F sup ( , ) (( , ) | ) . Then * ≡ sup ( , ) (( , ) | * ) defines an upper bound of * . As we discussed in Section 3.1, for a given offer ( , ), the marginal signal is̃( , ) = − 1− . Therefore, when the buyer's signal distribution is , the seller's profit from an offer ( , ) is (( , ) | ) = (1 − (̃( , ))) − 1 >0 (
, the seller's profit from a generous refund is equal to the case where the buyer's signal distribution is̃∈ F such that̃( ) = ( ) for all ∈ [0, ), and̃( ) =̃( ) for all ∈ [ , 1]. Therefore, if the buyer's signal distribution is , then the seller's profit from a generous refund is 1 The nature thus can lower the seller's profit from a generous refund (while keeping the highest profit from a non-refundable offer at ) by making the area under ( ) on [ , 1], i.e., ∫ 1 ( ) , as large as possible subject to the constraints for all . With this observation at hand, define
( )) = 1 − . Then, we can conclude that the seller's profit from a generous refund is at least ( ). Furthermore, the seller's profit from a generous refund is ( ) if ( ) = ( ) for all ∈ [0, ). Let̂be ∈ [0, ] that minimizes max { , ( )}. Observe that ( ) is strictly decreasing in on [0, ), and lim → ( ] such that (̃) =̃(when ≥ 1 − √ 1 − ). Therefore,
Figures 4 (
4a) and 4(b) illustrate * when < 1 − √ 1 − and ≥ 1 − √ 1 − , respectively. 12 More formally, the supremum of the seller's profit from a non-refundable offer, i.e., sup (( , 0) | ), is sup (1 − ( ) + ( )), where ( ) is the size of atom of at . Furthermore, since ( ) ≥ ( ) ( ) for all , we have (1 − ( ) + ( )) ≤ (1 − ( ) ( )) for all . Since (1 − ( ) ( )) = ( ) for all ∈ [ ( ) , 1], we have sup (( , 0) | ) = ( ).
Figure 4 :
4Worst-case distribution * Lemma 2. Suppose that the buyer's signal distribution is * . Then the seller's profit and profit from a generous refund are both (̂), i.e., sup ( , ) (( , ) | * ) = ( * ) = (̂).
, ) | * ) for all ∈ F.
Figure 5 :
5* * > * if > . Therefore, if ≤ , then by the analysis in Section 3.1, for any > 0, there exists a generous refund that achieves (1 − ) * . If ≥ , however, then for any deterministic pricing policy ( , ), there exists a signal distribution such that * > (( , ) | ). This implies that the seller cannot guarantee profits close to * by a deterministic policy. Nevertheless, for any > 0, there exists a pricing policy that guarantees (1 − ) * .
Lemma 4 .
4Take an arbitrary subgame following the nature's move. For any outcome the seller can induce by an indirect mechanism, there exists an outcome-equivalent direct mechanism with refunds that is individually rational and incentive compatible.Proof. In the appendix.Let be the nature's choice. Under a given direct mechanism with refunds = { ( ) , { 0 ( ) , ( )}} ∈[0,1] , if the buyer with signal reports ′ , then her payoff is 's profit is [ ( | )], where ( | ) is his profit from the buyer with signal , i.e., Let M be the set of all direct mechanisms with refunds that satisfy the following two conditions: ( ; | ) ≥ ( ′ ; | ) for all ′ and , (IC) ( ; | ) ≥ 0 for all . (IR)
Lemma 5 .
5Suppose that * = { * ( ) , { * 0 ( ) , * ( )}} ∈[0,1] is a seller's best-response to the nature's choice , then * 0 ( ) is increasing,
Lemma 6 .
6Suppose that * = { * ( ) , { * 0 ( ) , * ( )}} ∈[0,1] is a seller's best-response to the nature's choice * . Then * [ ( | * )] = * [ ( | * 0 )] = * .
Next, suppose that ≥ 1 − √ 1 − . Then ( * ) =̂= (̂). Consider a refundable offer ( , ). If̃( , ) > , then (( , ) | * ) ≤̂= (̂). If̃( , ) ≤ , then (( , ) | * ) < (̂) because the seller's profit from the buyer with signal ∈ [̃( , ) , ] is negative.Proof of Theorem 1: We limit our attention to the case where > . Fix arbitrary signal structure . The seller's payoff of using policy with is
.
Proof of Lemma 4:Any outcome the seller can induce by an indirect mechanism can be induced by an individually-rational and incentive-compatible direct mechanism = { , , {( 0 , 0 ) , ( 1 , 1 )}} that specifies, for each ∈ [0, 1], (i) : the probability that the buyer receives the product; (ii) : the transfer from the buyer to the seller; and (iii) { , } ∈{0,1} : the direct mechanism that specifies, for each buyer's reported realized valuation ∈ {0, 1}, (a) : the probability the buyer keeps the product; and (b) : the transfer from the seller to the buyer with the following properties: (IR) 1 + 1 ≥ 1 and 0 + 0 ≥ 0 and (IC) 1 + 1 ≥ 0 + 0 and 0 ≥ 1 . Notice that for any stochastic direct mechanisms (over { , , {( 0 , 0 ) , ( 1 , 1 )}}), there exists an outcome-equivalent deterministic direct mechanism. So without loss, we limit our attention to deterministic mechanisms.Under this direct mechanism = { , , {( 0 , 0 ) , ( 1 , 1 )}}, the payoff of the buyer with signal and the seller's profit from her are, denote 0 ( ) = (1 − )̃0, and ( ) = ( + (1 − ) (1 −̃0)) so that 0 ( ) + ( ) = , then we have the required result.Proof of Lemma 5: Since ( ; | )Observe that (0| ) ≤ 0. Therefore, if * is a solution and = 0, then the seller chooses * 0 ( ) = * ( ) = 0. Next, if ∈ (0, ), then since − 1− < 0, * ( ) = 0. If = , ( | * ) does not depend on * ( ). Therefore, Proof of Lemma 6: If < 1 − √ 1 − , then the support of * is { * , 1}. Thus, by (Since, * induces * and 1 with probability 1 − and , respectively, the seller's profit from using * is * [ , then the support of * is [ * , ] ∪ {1}. Therefore, ) on ( , 1). Furthermore, * has density * 2 over the interval [ * , ), and two mass points, (with probability * − * ) and 1 (with probability * ). Thus, the seller's profit from using * is * [
The environment where the seller chooses a price-refund pair without knowing the realized signal drawn from a commonly-known distribution is analyzed inInderst and Tirosh (2015) andKrahmer and Strausz (2015).
The Bayes-plausibility requires that the expected value of the posteriors induced by the buyer's signal distribution must be equal to the prior. Loosely speaking, this implies that when the buyer's signal distribution is likely to generate a sufficiently favorable signal, it also is likely to generate a sufficiently unfavorable signal.3 For example, the buyer may delegate the information gathering to a third party such as an algorithm or an employee to commit to a certain information structure.4 Libgober and Mu (2017) analyze a robust dynamic pricing problem where the product is durable and buyers learn about their value for the product over time.5 The literature on sequential screening and dynamic mechanism design has identified why and how advance sales to still-uninformed consumers can help the seller. See e.g.,Gale and Holmes (1992, 1993);Courty and Li (2000);Eső and Szentes (2007);Nocke et al. (2011);Gallego and Sahin (2010);Ely et al. (2017). The closest to our paper is von Wangenheim(2017), who studies a model where the buyer learns the value over time and the seller can offer flexible contracts, which can be interpreted as a (costless) refund policy. He finds that the seller obtains the static monopoly profit, which means capturing full surplus in our framework.
See Aumann et al. (1995) andKamenica and Gentzkow (2011).10 The concave closure of function is defined by con [ ] ( ) = sup{ |( , ) ∈ co( )}, where co( ) is the convex hull of the graph of .
Notice that if < , then the buyer buys irrespective of the value of the signal. Therefore, ( , ) < ( , − ) for a sufficiently small > 0.
The function −1 (⋅) ≤ −1 denotes the lower branch of the Lambert W function, i.e. function −1 ( ) = is defined as the smaller of the two real solutions to the equation = for < 0.
This is a variant of the exponential price auction found inDu (2018).
Buyer-Optimal OutcomesIn this section, we identify the buyer-optimal information structure and the outcomes that can be supported by some information structure. We analyze the following game:1. The buyer chooses a signal distribution ∈ F.2. The seller observes the signal structure and chooses a mechanism. By the arguments above, we can without loss in generality focus on (deterministic) policies ( , ).
An interpretation of our buyer-optimality result is as follows: the buyer ex-ante delegates the information gathering to a third-party, such as an agent or an algorithm, knowing that the seller best responds to the signal structure by choosing a pricing policy.
More precisely, the distributions (over signals) that minimize the seller's profit from a deterministic nonrefundable offer; and the seller's worst distribution over signals when the seller cannot utilize a refund policy can both be characterized in terms of distributions over expected willingness to pays.
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Revenue Maximizing with Return Policy When Buyers Have Uncertain Valuations. J Zhang, International Journal of Industrial Organization. 31≡ ( ( 1 + 1 ) + (1 − ) 0 ) −Zhang, J. (2013): "Revenue Maximizing with Return Policy When Buyers Have Uncertain Valuations, " International Journal of Industrial Organization, 31, 452-461. ( ) ≡ ( ( 1 + 1 ) + (1 − ) 0 ) − ,
̃0 − 1). Then, (̃) = ( (̃1 +̃1) + (1 − )̃0) −= ( + (1 − )) − = (. − ) (̃0 − 1). Then, (̃) = ( (̃1 +̃1) + (1 − )̃0) −= ( + (1 − )) − = ( )
| []
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"Jan Chorowski [email protected] \nUniversity of Wroclaw\nPoland\n\nNavAlgo\nFrance\n",
"Grzegorz Ciesielski \nUniversity of Wroclaw\nPoland\n",
"Jarosław Dzikowski \nUniversity of Wroclaw\nPoland\n",
"Adrian Łańcucki \nNVIDIA\n\n",
"Ricard Marxer \nUniversité de Toulon\nAix Marseille Univ\nCNRS\nLISFrance\n",
"Mateusz Opala \nUniversity of Wroclaw\nPoland\n",
"Piotr Pusz \nUniversity of Wroclaw\nPoland\n",
"Paweł Rychlikowski [email protected] \nUniversity of Wroclaw\nPoland\n",
"Michał Stypułkowski \nUniversity of Wroclaw\nPoland\n"
]
| [
"University of Wroclaw\nPoland",
"NavAlgo\nFrance",
"University of Wroclaw\nPoland",
"University of Wroclaw\nPoland",
"NVIDIA\n",
"Université de Toulon\nAix Marseille Univ\nCNRS\nLISFrance",
"University of Wroclaw\nPoland",
"University of Wroclaw\nPoland",
"University of Wroclaw\nPoland",
"University of Wroclaw\nPoland"
]
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| We present a number of low-resource approaches to the tasks of the Zero Resource Speech Challenge 2021. We build on the unsupervised representations of speech proposed by the organizers as a baseline, derived from CPC and clustered with the kmeans algorithm. We demonstrate that simple methods of refining those representations can narrow the gap, or even improve upon the solutions which use a high computational budget. The results lead to the conclusion that the CPC-derived representations are still too noisy for training language models, but stable enough for simpler forms of pattern matching and retrieval. | 10.21437/interspeech.2021-1465 | [
"https://arxiv.org/pdf/2106.11603v1.pdf"
]
| 235,593,288 | 2106.11603 | 77ba536360ba5e46d3072424264c02ce9d9223ab |
22 Jun 2021
Jan Chorowski [email protected]
University of Wroclaw
Poland
NavAlgo
France
Grzegorz Ciesielski
University of Wroclaw
Poland
Jarosław Dzikowski
University of Wroclaw
Poland
Adrian Łańcucki
NVIDIA
Ricard Marxer
Université de Toulon
Aix Marseille Univ
CNRS
LISFrance
Mateusz Opala
University of Wroclaw
Poland
Piotr Pusz
University of Wroclaw
Poland
Paweł Rychlikowski [email protected]
University of Wroclaw
Poland
Michał Stypułkowski
University of Wroclaw
Poland
22 Jun 2021Information Retrieval for ZeroSpeech 2021: The Submission by University of WroclawIndex Terms: ZeroSpeech Challengeunsupervised learninginformation retrievalspoken language modeling
We present a number of low-resource approaches to the tasks of the Zero Resource Speech Challenge 2021. We build on the unsupervised representations of speech proposed by the organizers as a baseline, derived from CPC and clustered with the kmeans algorithm. We demonstrate that simple methods of refining those representations can narrow the gap, or even improve upon the solutions which use a high computational budget. The results lead to the conclusion that the CPC-derived representations are still too noisy for training language models, but stable enough for simpler forms of pattern matching and retrieval.
Introduction
The Zero Resource Speech Challenge series (ZeroSpeech) [1,2] is an initiative with the ultimate goal of building from scratch a system that learns an end-to-end spoken dialogue system for an unknown language, using only sensory information mainly in the form of recordings, and does not use any linguistic resources or knowledge.
The high-level objective of the competition is to learn various qualities of a natural language at different levels of granularity directly from raw audio without any supervision. Ze-roSpeech 2021 evaluates speech understanding using the following tasks and datasets:
1. Phonetic ABX (Libri-Light dataset [3]), where the system has to judge whether two phonemes are identical 2. Lexical (sWUGGY dataset) -classifying whether a spoken utterance is a real or misspelled word 3. Semantic (sSIMI dataset) -assessing semantic similarity between two spoken words 4. Syntactic (sBLIMP dataset) -assigning a grammatical score to an utterance in such a way that erroneous sentences have lower scores than correct ones For all tasks it is assumed that spoken corpora (either Lib-riSpeech [4], or Libri-Light [3]) are the only sources of language knowledge.
The organizers have provided a baseline solution [2], which we adapt and modify in our submission. Raw audio is fed to a Contrastive Predicting Coding (CPC) model [5]. By trying to predict future representations, the CPC model learns useful intermediate representations of speech. These come in the form of an embedding vector emitted every 10 ms. Collected from a large training corpus, the embeddings are then clustered with the k-means algorithm into 50 pseudo-phones. With this pipeline, any unseen audio can be transformed into a stream of pseudo-phones, on which language models may be trained for downstream tasks.
In this paper we present our submission which tries to address all four tasks. We extend the baseline solution in several directions: we refine the intermediate representations, extracted with CPC, to directly improve the ABX scores. We show that such representations can be used to perform simple fuzzy lookups in a large dataset, and even extract some common patterns that serve as pseudo-words. Our approach to the semantic word similarity task is also based on pseudo-words. Instead of pooling the hidden states of the language model, we opt for a direct discovery of pseudo-words in the corpus. These can be embedded with a word2vec-like approach [6] to form a semantic vector for the entire word. Lastly, for the syntax modeling task we use a simple LSTM model similar to the baseline one.
We provide complete source code of our submission at https://github.com/chorowski-lab/zs2021.
Phonetic Task: Libri-Light ABX
In the ABX task two speech categories A and B are determined by two recordings (e.g., "bit" vs "bet"), and a third recording X has to be correctly assigned to either one of them.
The baseline representations for the ABX task are CPCderived embedding vectors. In order to improve upon those representations, we introduce two approaches described in the following section.
Improvements to CPC Representations
Factoring Out Speaker Identities The embeddings produced by CPC contain information about both the phonetic content and speaker identity. In case of ABX, which is a phoneme recognition metric, the latter is irrelevant. We therefore project the embeddings of the baseline model (CPC-big [2]) into the nullspace of a linear speaker classification model to render the embeddings less speaker-sensitive. We perform speaker classification on baseline CPC embeddings with a projection factorized into matrices A and B, where A ∈ R D inb ×D emb , B ∈ R D spk ×D inb , D emb is the dimensionality of embeddings and D inb is the linear bottleneck dimensionality. In order to compute ABX, we multiply the CPC-derived embeddings by A ′ ∈ R (D emb −D inb )×D emb , the nullspace matrix of A.
Averaging with Centroids
Higher-level tasks of the competition rely on pseudo-phones found by clustering these vectors with k-means. Doing so proves useful, so we incorporate some of the outcomes of the clustering back into the dense CPCderived vectors. Specifically, we take a weighted average of every dense CPC-derived embedding e in the embedding space with its cluster centroid ce:
ê = α ce + (1 − α) e.(1)
This averaging moves every dense embedding towards its assigned centroid proportionally to the distance from it. This aims to include information about the global characteristics of the embedding space coming from clustering without substantial loss of local information, and might be regarded as a simple form of denoising. It does not change the assignment to the closest centroid.
ABX Experiments
We evaluate both aforementioned improvements on the Ze-roSpeech ABX task. To begin with, we extract 512-dimensional embeddings from the second layer of the CPC's autoregressive module. We run each classification experiment for 10 epochs on the train-clean-100 subset of LibriSpeech. ZeroSpeech 2021 dev/test sets are subsets of LibriSpeech dev/test sets, respectively. However, for best results (and for replication of the baseline) we had to first compute the embeddings on the full LibriSpeech test set, to allow the model to keep latent state between consecutive utterances. After all embeddigs were computed, we have kept only the ones needed for ZeroSpeech ABX evaluations.
We investigate the variation in ABX scores with respect to the dimensionality of the resulting nullspace, by testing with different bottleneck sizes D inb of the speaker classifier. We achieve the best ABX result when the nullspace size is 448.
Next, we evaluate the influence of averaging with centroids on the ABX scores (Table 3). It also noticeably improves ABX results, and we achieve the highest error reduction when it is combined with a 448-dimensional nullspace projection. We have experimented with both Euclidean and cosine distance metrics when performing k-means and later choosing the closest centroid. Both yield similar results, and we select the centroid according to the cosine distance in subsequent experiments.
Lastly, we evaluate the influence of both methods on supervised speaker and phoneme classification accuracies. In the nullspace approach, both of them are low (Table 2), and increase with the size of the nullspace (D emb − D inb ). This indicates that after we attempt to make the representations speakerinsensitive, there is still some stray of speaker-related information present in the remaining dimensions. As seen in Table 4, averaging with centroids also reduces phoneme classification accuracy, proportionally to how much the embeddings are altered, both with and without the nullspace projection. Thus, both tested methods improve ABX, and degrade phoneme separation results. This can be because difficulties of those tasks and power of downstream models used differs -we discard less important parts of the information, which improves ABX results as the task is simple and there are no trained parameters, just representation distances (in which case removing less important parts of information helps) but degrades phoneme separation performance (as we still discard some information which classifier with trainable weights could use). In both cases the relative gain in ABX is bigger than the relative loss in phoneme separation performance.
Quantization for Higher-level Tasks
The baseline methods for the remaining tasks of higher linguistic levels require quantized inputs, that act as discrete input tokens for language models. This is achieved by clustering CPCderived vectors with k-means, and mapping every dense vector to its centroid. To achieve the best results on lexical and syntactic tasks (sWUGGY and sBLIMP datasets), we use the CPCnullspace embeddings instead of the raw CPC embeddings. In contrast to feature extraction for the ABX task, now the LSTM context in CPC is not kept between the files, as the datasets for specific tasks are not related to one another. Additionally, when computing the distances, we normalize L2-norm lengths of vectors and in effect switch from the Euclidean metric to the cosine metric for quantization. For the semantic task (sSIMI dataset), we use the baseline quantizations produced with the raw CPC embeddings, and kmeans clustering with the Euclidean metric.
Lexical Task: sWUGGY
The goal of the task is to distinguish a real word from a similar pseudo-word in a given pair. Pseudo-words were generated with Wuggy [7], and adhere to orthographic and phonological patterns of the English language. Such pairs make up the sWUGGY dataset, which has two parts: one with real words which appear in the LibriSpeech training set (base), and another one in which they do not (OOV).
The baseline solution takes pseudo-phones as input, and judges the likelihood of a word with a language model, following [8]. For the base pairs, our method performs a dictionary lookup of the quantized representations trying to spot the words in the entire LibriSpeech training corpus.
For the OOV pairs, we were trying to use a simple LSTM language model and to combine it with dictionary lookup. But since guessing whether a word is in vocabulary is somehow problematic, and LSTM yielded similar results to DTW, we decided to treat all words in the same way.
Dictionary Lookup
We build a corpus by pre-processing all LibriSpeech training set utterances to strings of pseudo-phones. For every query word, our goal is to find the closest matching subsequence in the corpus. The lookup is based on dynamic time warping (DTW) [9], and uses subsequence DTW which matches a given sequence to a contiguous subsequence of another, such that the matching cost is minimal across all subsequences. This can be done without any increase in complexity, and is easy to parallelize.
Query words and pseudo-phone representations of training utterances are strings of discrete centroid numbers. A simple similarity matrix between the elements of two sequences x, y would be a binary one. We take advantage of having Euclidean coordinates of the centroids, and compare two pseudo-phones by the similarity of their centroid vectors. Thus, every similarity matrix has real-valued entries, and we perform soft matching of sequences.
We estimate pseudo-log probability of a query word as the negative quotient of the minimum DTW cost to the mean DTW cost of this word. We normalize with the mean DTW cost because longer words tend to have higher costs, which would result in a bias towards shorter words.
Experiments for sWUGGY
We have tried different lookup methods, such as direct comparison of subsequences or measuring edit distances. Out of the tested methods, the best results were obtained with dynamic time warping. We have also tried to post-process the quantizations, but all attempts worsened the results. This is probably due to loss of useful information, so we run DTW on vanilla quantizations.
For the sWUGGY test set, it was not possible to differentiate between base and OOV, so we have used only DTW. For OOV words, the difference between the results obtained with DTW and LSTM was minor. A significant improvement to our DTW relies on a technical detail. If we match the word correctly, we expect the match cost to be spread evenly over the entire sequence. However, when we match the word incorrectly, we expect the cost to be high in some places and low in the others. Thus, we increase the cost of distant pseudo-phones, and decrease the cost of similar ones by raising the cost in distance matrix to some power, which sharpens the distances. In our case, 1.6 was the best for the cosine metric quantization and 2.0 for the baseline quantization.
Processing of the train set took 18 hours on conventional hardware for both baseline and nullspace quantizations. Results presented in Table 5 show that both normalization and modification of the distance matrix yielded a significant improvement in the score.
Semantic Task: sSIMI
The goal of the task is to judge the semantic similarity of pairs of words (see [10], [11]). That similarity is then compared with semantic judgments made by the human annotators, collected from a set of 13 existing semantic similarity and relatedness tests (including WordSimilarity-353 [12], and mturk-771 [13] which was used as a development set).
The submission format encouraged solutions which assign a vector at every temporal location, with a simple pooling method to aggregate them into a vector for the entire recording. The pooling methods included min, max, avg, last, etc. In our submission, we have computed a vector for an entire recording, and replicated it along the time axis, so that after pooling with aforementioned functions it would remain unchanged.
Preparation of the Corpus We rely on the baseline pseudophone units, extracted with CPC and quantized with k-means. Streams of recognized pseudo-phones contain symbol repetitions, and we treat such blocks as higher order units. We further simplify these sequences by heuristically collapsing subsequent occurrences of the same pseudo-phone, and unify blocks which occur in similar contexts. The treat the result of this procedure as an approximation of a phoneme-level transcription with no word boundaries.
Segmentation We apply segmentation into pseudo-words with SentencePiece [14], which maximizes the probability under a unigram language model [15]. Given a vocabulary V = w1, . . . , wn with associated occurrence probabilities (p1, . . . , pn) and an utterance x, the most probable segmenta-tion is determined with the Viterbi algorithm. The vocabulary V is refined iteratively by maximizing the probability of every utterance under a unigram language model: P (x) = i p(xi).
We apply SentencePiece with target vocabulary size 50k. By using ground-truth transcriptions, we found that the corpus has 18,705,420 words, which translates to 1.95 pseudo-word for every real word.
Embedding and Retrieval With the corpus segmented into pseudo-words, we train an ordinary word2vec model [6]. Every recording in the similarity task dataset comprises a single word, and we convert each of them to a sequence of pseudophones. Some of those sequences exist in our pseudo-word vocabulary, and already have a unique word embedding calculated with word2vec. Others need to be built from smaller pseudoword units. A simple way of doing so would be to split the sequence with SentencePiece into known pseudo-words. Knowing that the pseudo-representations tend to be noisy, we instead find the closest matches of each of them in the training corpus wrt. edit distance. Then, word2vec embeddings of these matched pseudo-words are averaged to a single embedding for every input recording.
Experiments for sSIMI
Since our approach differs from the ZeroSpeech baseline one, we decided to present other word-oriented toplines for sSIMI, that better suit our approach. We compare word vectors trained on a large corpus, LibriSpeech transcriptions, and on our tokenization of LibriSpeech transcriptions. In that last case, we delete spaces and tokenize into 50k units using SentencePiece. The results are presented in Table 6. The embeddings trained on word corpora outperform the RoBERTa topline, which suggests that the proposed approach might deserve further investigation. Table 7 presents our results in the ZeroSpeech contest together with other submissions. Our approach achieves the best score in the LibriSpeech subcategory, where the recordings are cut from LibriSpeech and not synthesized. This might indicate that our method is able to discover semantic shades of the known words from the corpus, but unable to generalize further.
Syntactic Task: sBLIMP
BLIMP [17] is a challenge dataset for evaluating to what extent language models can understand the English grammar. The dataset consists of pairs of similar sentences, and in every pair only one sentence is correct. In sBLIMP all sentences are synthesized. Since the aim of BLIMP was to evaluate how sentence likelihood is related to its grammatical correctness, there is a natural strategy of solving sBLIMP: use a language model to compute sentence probability and pick the most likely sentence from the pair as the correct one. To this end we use a LSTM language model trained on quantized nullspace features from LibriSpeech dev subset. In the competition, it had 53% accuracy both on dev and test sets, slightly outperforming the baseline, and being close to 54% of the best submission. However, this result is not impressive at all: LSTM with random weights has 52.9% accuracy. We hypothesize that this is caused by unbalanced utterance lengths in sBLIMP. We have discovered that incorrect sentences are typically longer than the correct ones. Nevertheless, it is worth saying that BLIMP even in text version is definitely a non-trivial task, as for instance a big LSTM trained on large text corpus achieves only 70% accuracy [18], and even large Transformer [19] model like GPT-2 don't exceed 82% [17,20].
Conclusions and Future Works
We have presented a low-resource, information-retrieval based approach to the tasks of the Zero Resource Speech Challenge 2021. We were able to outperform baselines on every task, and achieve best or close to the best results on all four tasks. Still, many issues deserve further investigation.
First, we can explore the relationship between the ability of neural networks to memorize words, and contrast that with fuzzy information retrieval system. Is it possible to discover the dictionary from recordings, using some combinations of these approaches?
Moreover, we believe that there is a potential synergy with computing semantic vector representation for pseudo-words using word2vec -mainly because it is inexpensive to compute, and moving to bigger datasets can lead to a substantial improvement of embedding quality.
Solving BLIMP in the Zero Resource regime is undoubtedly an ambitious task. We believe that it is worth to consider its simpler, artificially created variants. For instance, a variant in which the incorrect sentences were created by changing word order, or by replacing a randomly chosen word with another. Such simpler task can produce less noisy results.
Lastly, we can explore approach similar to averaging with centroids, but applied during training of CPC. For example by adding a loss based on the distance to simultaneously computed centroids, hoping that its denoising effect will improve the extracted representations.
Acknowledgments
Table 1 :
1ABX error rates (%, cosine distance) for multiple sizes of nullspaces of speaker classification models. The nullspace dimension complements the bottleneck dimension used to train the speaker recognizer.Nullspace dimensionality
Evaluation
None 464
448
416
320
256
dev clean within
3.38
3.28 3.25 3.29 3.26 3.31
clean across
4.17
3.98 3.94 3.92 3.98 3.99
other within
4.81
4.63 4.60 4.61 4.62 4.67
other across
7.53
7.34 7.24 7.24 7.21 7.26
Table 2 :
2Phoneme and speaker classification accuracies (%) of models after applying factorized projection heads (top) and nullspace matrices of the aforementioned speaker classification models (bottom) on the baseline embeddings (c.f. Table 1). . head bottleneck / Nullspace dimensionalityProjSetup
None / 512 48 / 464 64 / 448 96 / 416 192 / 320 256 / 256
Speakers (fact. proj. head)
84.85
91.17
91.32
91.59
92.02
92.02
Speakers + null space
84.85
29.42
27.91
24.95
19.50
15.56
Phonemes + null space
78.61
78.46
78.36
78.18
77.46
76.86
Table 3 :
3ABX error rates (%, cosine distance) for weighted averaging of CPC embeddings with centroids. The bottom half shows results combined with the best 448-dimensional nullspace setup. The nullspace dimension is equal to the difference of dimensions between the embeddings and the bottleneck used to train the speaker classifier. Phoneme classification results inTable 4Centroid weight
Table 4 :
4Phoneme classification accuracies (%) for averaging with centroids, both without the nullspace and after projection to the nullspace. ABX results inTable 3Centroid weight
Table 5 :
5Scores for DTW lookup on different quantizations, and with linear and optimal distance matrices. The results were computed for the base dev set of the lexical task (no OOV subset). We used the train-full-960 subset of the LibriSpeech as dictionary.Classification accuracy
Quantization Distance matrix no norm.
norm.
Baseline
none (constant)
68.47%
69.33%
Baseline
Euclidean
68.94%
70.98%
Baseline
Euclidean 2
71.00%
71.64%
Cosine
cosine
72.61%
73.36%
Cosine
cosine 1.6
73.12%
73.92%
Table 6 :
6Toplines for word-based sSIMI (dev set). First three methods used all word pairs from the dev set, the result for RoBERTa[16] are for the synthetized part of the data.Method
synth.
Google News word2vec
65.5
LibriSpeech 960 transcriptions (w2v)
36.3
Tokenized Librispeech
16.8
RoBERTa (ZeroSpeech2021 Topline)
32.28
Table 7 :
7Correlation between human judgments and system responses (× 100). For other contestants the best submission on the test part of the data is presented.synth.
libri.
Method
dev
test
dev
test
LSTM Baseline
4.42 7.35
7.07
2.38
BERT Baseline
6.25 5.17
4.35
2.48
Ours
5.90 2.42 10.20
9.02
van Niekerk et al.
4.29 9.23
7.69
-1.14
Liu et al.
3.16 7.30
1.79
-4.33
Maekaku et al.
-2.10 6.74
8.89
2.03
The authors thank Polish National Science Center for funding under the OPUS-18 2019/35/B/ST6/04379 grant and the PlGrid consortium for computational resources.
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]
| [
"Shuyuan Wu \nSchool of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina\n\nState Key Laboratory of Optoelectronic Materials and Technologies\nSun Yat-Sen University (Guangzhou Campus)\n510275GuangzhouChina\n",
"Xizhou Qin \nSchool of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina\n",
"Jun Xu \nSchool of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina\n\nCenter of Experimental Teaching for Common Basic Courses\nSouth China Agriculture University\n510642GuangzhouChina\n",
"Chaohong Lee \nSchool of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina\n\nState Key Laboratory of Optoelectronic Materials and Technologies\nSun Yat-Sen University (Guangzhou Campus)\n510275GuangzhouChina\n\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaChina\n"
]
| [
"School of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina",
"State Key Laboratory of Optoelectronic Materials and Technologies\nSun Yat-Sen University (Guangzhou Campus)\n510275GuangzhouChina",
"School of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina",
"School of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina",
"Center of Experimental Teaching for Common Basic Courses\nSouth China Agriculture University\n510642GuangzhouChina",
"School of Physics and Astronomy\nTianQin Research Center\nSun Yat-Sen University (Zhuhai Campus)\n519082ZhuhaiChina",
"State Key Laboratory of Optoelectronic Materials and Technologies\nSun Yat-Sen University (Guangzhou Campus)\n510275GuangzhouChina",
"Synergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaChina"
]
| []
| According to the famous Kibble-Zurek mechanism (KZM), the universality of spontaneous defect generation in continuous phase transitions (CPTs) can be understood by the critical slowing down. In most CPTs of atomic Bose-Einstein condensates (BECs), the universality of spontaneous defect generations has been explained by the divergent relaxation time associated with the nontrivial gapless Bogoliubov excitations. However, for atomic BECs in synthetic gauge fields, their spontaneous superfluidity breakdown is resulted from the divergent correlation length associated with the zero Landau critical velocity. Here, by considering an atomic BEC ladder subjected to a synthetic magnetic field, we reveal that the spontaneous superfluidity breakdown obeys the KZM. The Kibble-Zurek scalings are derived from the Landau critical velocity which determines the correlation length. In further, the critical exponents are numerically extracted from the critical spatial-temporal dynamics of the bifurcation delay and the spontaneous vortex generation. Our study provides a general way to explore and understand the spontaneous superfluidity breakdown in CPTs from a single-well dispersion to a double-well one, such as, BECs in synthetic gauge fields, spin-orbit coupled BECs, and BECs in shaken optical lattices.Introduction. Engineered synthetic gauge fields for neutral atoms [1-8] provide new opportunities to explore exotic collective quantum phenomena[9][10][11][12][13][14]. The dispersion relation plays an important role in the emergence of many collective quantum phenomena. Through controlling the applied external fields, the dispersion relation can be tuned from a single-well shape into a double-well one, such as, spin-orbit coupled quantum gases[15][16][17][18], ultracold atoms in shaken optical lattices[19][20][21]and Bose ladders within magnetic fields[22][23][24][25]. At the transition point, due to the interplay between the synthetic gauge fields and the atom-atom interactions, the Landau critical velocity vanishes[17,18,20]and thus the superfluid spontaneously breaks down. Such a spontaneous superfluidity breakdown is very different from the conventional Landau instability which requires the superfluid velocity exceeding a nonzero critical velocity[26][27][28]. Although the static phase transitions in synthetic gauge fields have been extensively studied, the underlying dynamics of phase transitions is still unclear. | 10.1103/physreva.94.043606 | [
"https://arxiv.org/pdf/1607.07524v2.pdf"
]
| 119,289,145 | 1607.07524 | 3ec2edaf4eb8f1934a0874bdcd863a70ff3a151d |
Universal spatiotemporal dynamics of spontaneous superfluidity breakdown in the presence of synthetic gauge fields
30 Sep 2016 (Dated: June 18, 2018)
Shuyuan Wu
School of Physics and Astronomy
TianQin Research Center
Sun Yat-Sen University (Zhuhai Campus)
519082ZhuhaiChina
State Key Laboratory of Optoelectronic Materials and Technologies
Sun Yat-Sen University (Guangzhou Campus)
510275GuangzhouChina
Xizhou Qin
School of Physics and Astronomy
TianQin Research Center
Sun Yat-Sen University (Zhuhai Campus)
519082ZhuhaiChina
Jun Xu
School of Physics and Astronomy
TianQin Research Center
Sun Yat-Sen University (Zhuhai Campus)
519082ZhuhaiChina
Center of Experimental Teaching for Common Basic Courses
South China Agriculture University
510642GuangzhouChina
Chaohong Lee
School of Physics and Astronomy
TianQin Research Center
Sun Yat-Sen University (Zhuhai Campus)
519082ZhuhaiChina
State Key Laboratory of Optoelectronic Materials and Technologies
Sun Yat-Sen University (Guangzhou Campus)
510275GuangzhouChina
Synergetic Innovation Center for Quantum Effects and Applications
Hunan Normal University
410081ChangshaChina
Universal spatiotemporal dynamics of spontaneous superfluidity breakdown in the presence of synthetic gauge fields
30 Sep 2016 (Dated: June 18, 2018)arXiv:1607.07524v2 [cond-mat.quant-gas]
According to the famous Kibble-Zurek mechanism (KZM), the universality of spontaneous defect generation in continuous phase transitions (CPTs) can be understood by the critical slowing down. In most CPTs of atomic Bose-Einstein condensates (BECs), the universality of spontaneous defect generations has been explained by the divergent relaxation time associated with the nontrivial gapless Bogoliubov excitations. However, for atomic BECs in synthetic gauge fields, their spontaneous superfluidity breakdown is resulted from the divergent correlation length associated with the zero Landau critical velocity. Here, by considering an atomic BEC ladder subjected to a synthetic magnetic field, we reveal that the spontaneous superfluidity breakdown obeys the KZM. The Kibble-Zurek scalings are derived from the Landau critical velocity which determines the correlation length. In further, the critical exponents are numerically extracted from the critical spatial-temporal dynamics of the bifurcation delay and the spontaneous vortex generation. Our study provides a general way to explore and understand the spontaneous superfluidity breakdown in CPTs from a single-well dispersion to a double-well one, such as, BECs in synthetic gauge fields, spin-orbit coupled BECs, and BECs in shaken optical lattices.Introduction. Engineered synthetic gauge fields for neutral atoms [1-8] provide new opportunities to explore exotic collective quantum phenomena[9][10][11][12][13][14]. The dispersion relation plays an important role in the emergence of many collective quantum phenomena. Through controlling the applied external fields, the dispersion relation can be tuned from a single-well shape into a double-well one, such as, spin-orbit coupled quantum gases[15][16][17][18], ultracold atoms in shaken optical lattices[19][20][21]and Bose ladders within magnetic fields[22][23][24][25]. At the transition point, due to the interplay between the synthetic gauge fields and the atom-atom interactions, the Landau critical velocity vanishes[17,18,20]and thus the superfluid spontaneously breaks down. Such a spontaneous superfluidity breakdown is very different from the conventional Landau instability which requires the superfluid velocity exceeding a nonzero critical velocity[26][27][28]. Although the static phase transitions in synthetic gauge fields have been extensively studied, the underlying dynamics of phase transitions is still unclear.
Introduction. Engineered synthetic gauge fields for neutral atoms [1][2][3][4][5][6][7][8] provide new opportunities to explore exotic collective quantum phenomena [9][10][11][12][13][14]. The dispersion relation plays an important role in the emergence of many collective quantum phenomena. Through controlling the applied external fields, the dispersion relation can be tuned from a single-well shape into a double-well one, such as, spin-orbit coupled quantum gases [15][16][17][18], ultracold atoms in shaken optical lattices [19][20][21] and Bose ladders within magnetic fields [22][23][24][25]. At the transition point, due to the interplay between the synthetic gauge fields and the atom-atom interactions, the Landau critical velocity vanishes [17,18,20] and thus the superfluid spontaneously breaks down. Such a spontaneous superfluidity breakdown is very different from the conventional Landau instability which requires the superfluid velocity exceeding a nonzero critical velocity [26][27][28]. Although the static phase transitions in synthetic gauge fields have been extensively studied, the underlying dynamics of phase transitions is still unclear.
The Kibble-Zurek mechanism (KZM) [29][30][31][32] describes the universality of real-time dynamics in continuous phase transitions. According to the KZM, universal scalings can be derived by comparing two characteristic time scales: the reaction time and the transition time. Not only the dynamics of thermodynamic phase transitions [33][34][35][36][37][38][39][40][41][42][43], but also the dynamics of quantum phase transitions [21,[44][45][46][47][48][49][50][51][52][53][54] obey the KZM. Due to their high controllability, atomic Bose-Einstein condensates (BECs) provide an excellent platform to examine KZM [21, 33-35, 37-42, 46-54]. Usually, the spontaneous defect generation are associated with the nontrivial gapless Bogoliubov excitations (such as Higgs modes) and thus the Kibble-Zurek scalings can be given by comparing the two characteristic time scales derived from the Bogoliubov excitation gap [46,49,50,54]. However, for atomic BECs in synthetic gauge fields [1-3, 5-8] whose dispersion relations are continuously tuned from a single-well shape to a double-well one, the spontaneous superfluidity breakdown is resulted from the spontaneous Landau instability. Two natural questions arise: Does the spontaneous superfluidity breakdown obey the KZM? If it obeys the KZM, can one explore the universal scalings via analyzing the Landau critical velocity?
In this Letter, we explore the universality of spontaneous superfluidity breakdown within synthetic gauge fields. We consider an atomic BEC ladder subjected to a synthetic magnetic field, which undergoes a continuous transition from a single-well dispersion to a doublewell one. By employing a variational ansatz, we analytically give the mean-field phase diagram. Due to the absence of nontrivial gapless Bogoliubov excitations, it is difficult to give the two characteristic time scales and the Kibble-Zurek scalings from the Bogoliubov excitation gap. Fortunately, we find that the universal scalings can be derived from the Landau critical velocity which determines the correlation length. In particular, the correlation length divergence and the spontaneous super-fluidity breakdown are resulted from the zero Landau critical velocity at the transition point. To extract the Kibble-Zurek scalings, we numerically simulate the realtime dynamics of the continuous transition from a singlewell dispersion to a double-well one, in which the ground state changes from the Meissner phase to the broken symmetry phase. Our numerical results show the bifurcation delay and the spontaneous vortex generation obeys the KZM. Our study opens a new avenue for exploring and understanding the universality of spontaneous superfluidity breakdown in various systems, such as, spin-orbit coupled BECs [15][16][17][18], BECs in shaken optical lattices [19][20][21], and BECs within magnetic fields [22][23][24][25].
Model and ground states. We consider an ensemble of Bose condensed atoms in a two-leg ladder subjected to a uniform synthetic magnetic field. The ladder potential can be created by a normal standing-wave along one direction and a bi-frequency standing-wave along the other direction [22], and the uniform synthetic magnetic field can be realized by laser-induced tunneling [1,3,5,6,8,9,11,12,22], see Fig. 1(a). The meanfield Hamiltonian reads as
H = − J l (ψ * l+1,L ψ l,L + ψ * l+1,R ψ l,R + h.c.) − K l (ψ * l,R ψ l,L e ilφ + h.c.) + g 2 l (|ψ l,L | 4 + |ψ l,R | 4 ).(1)
with ψ l,σ denoting the order parameters for the site (l, σ) and σ = {L, R} respectively labelling the {left, right} legs. Here, J is the intra-leg tunneling, Ke ilφ is the spatially dependent inter-leg tunneling and φ is the magnetic flux per plaquette. The on-site interaction g, which is proportional to the s-wave scattering length, can be tuned via Feshbach resonances [55]. The determine the ground states, we implement a variational procedure based on the ansatz
ψ l,L ψ l,R = C 1 cos θe i(k− φ 2 )l + C 2 sin θe −i(k+ φ 2 )l C 2 cos θe i(k+ φ 2 )l + C 1 sin θe −i(k− φ 2 )l ,(2)
with the total atomic number N , the ladder length L, the complex amplitudes (C 1 , C 2 ), the quasi-momentum k and the angle θ (0 ≤ θ ≤ π 2 ). By minimizing the Hamiltonian under the normalization condition |C 1 | 2 + |C 2 | 2 = n = N/L, the variational parameters (C 1 , C 2 , k, θ) can be determined. There are three different ground states:
(I) the vortex phase with {C 1 = C 2 , k = 0, θ = π/4} for 0 < K/J < R c 1 , (II) the biased ladder phase (BLP) with {C 1 = C 2 , k = 0, θ = 0, π/2} for R c 1 < K/J < R c 2 , and (III) the Meissner phase with {C 1 = C 2 , k = 0} for K/J > R c 2 .
The two critical points (R c 1 , R c 2 ) have to be determined numerically, however, the second critical point can be analytically given as R c 2 = √ 2 − gn/2 for
φ = π/2. Our numerical results show that the phase (II) disappears if there is no inter-particle interaction, that is, R c 1 = R c 2 = √ 2 for g = 0.
To clarify the Meissner-like effect, we calculate the chiral current j c = 1 2L l j || l,L − j || l,R , which is defined as the averaged difference between the local currents along the two legs j || l,σ = iJ ψ * l+1,σ ψ l,σ − ψ * l,σ ψ l+1,σ . The chiral current increases with K/J and then becomes saturated in the Meissner phase. This is analogy to the Meissner effect in the type-II superconductor [22]. The ground states and their chiral currents are consistent with the variational prediction [23], see Fig. 1
(b).
The phase transitions are characterized by the changes in the band structure (dispersion relation) and the chiral current. The transition between the vortex phase and the BLP is a first-order phase transition, which corresponds to a jump in the chiral current j c and the change from complete to single occupancy of the two band-minima at finite quasi-momentum ±k. The transition between the Meissner phase and the BLP is a continuous phase transition, in which the chiral current j c is continuous and the lowest band continuously changes from a singlewell shape to a double-well one. Below we concentrate on discussing the continuous phase transition.
Landau critical velocity and correlation length. In a continuous phase transition, the critical slowing down can be understood by the divergence of either the relaxation time τ r or the correlation length ξ. For our system, it is difficult to give the relaxation time τ r . Below, we show how to derive the universal scaling of ξ from the Landau critical velocity, v L = min q |ω/q| .
Here the Bogoliubov excitation gap ω is a function of the quasi-momentum q. According to the Landau criterion, if the superfluid velocity exceeds v L , elementary excitations appear due to the conservation of energy and momentum. This indicates that elementary excitation will take place spontaneously if v L = 0.
To give v L , we implement the Bogoliubov analysis to obtain the excitation gap ω. We express the perturbed ground-state as,
ψ l,L (t) ψ l,R (t) = (C 1 + δψ l,L (t)) e i(k− φ 2 )l (C 2 + δψ l,R (t)) e i(k+ φ 2 )l e −iµt ,(4)
with the perturbation terms
δψ l,σ (t) = q u q,σ e i(ql−ωt) + v * q,σ e −i(ql−ωt) .(5)
Here, the discrete quasi-momenta are given as q = 2mπ L with the integers m = {−L/2, −L/2 + 1, · · · , L/2 − 1}, and the perturbation amplitudes (u q,σ , v q,σ ) are complex numbers. Inserting the perturbed state into the timeevolution equation, i∂ψ l,σ /∂t = ∂H/∂ψ * l,σ , one can obtain the Bogoliubov-de Gennes (BdG) equation. Therefore, ω can be obtained by diagonalizing the BdG equation. At the critical point, K = K c = J( √ 2 − gn/2), the low-energy long-wavelength excitation behaves as lim q→0 ω(q) ≈ q 2 gn 2 √ 2 .
As ω(q) ∝ |q| z when q → 0 [56][57][58], we have the dynamical critical exponent z = 2. In Fig. 2 (a), we show the dependence of v L on K/K c . The Landau critical velocity v L gradually decreases to zero when K → K c . The notch of v L near K = K c originates from the softening phonon mode ω(q) ∝ q 2 , which implies |ω(q)/q| → 0 as q → 0 [17]. In particular, the vanishing critical velocity at the critical point will result the spontaneous superfluidity breakdown and the spontaneous elementary excitations.
Usually, the correlation length ξ is defined by the equality between the kinetic energy per particle 2 /(2mξ 2 ) and the interaction energy per particle gn. However, the Landau critical velocity v L provides another general definition according to ξ = /(mv L ), which is consistent with the usual definition [59]. This means that, near the critical point, the inverse critical velocity v −1 L scales as v −1 L ∼ ξ ∼ |ǫ| −ν .
In Fig. 2 with b = 0.4985 ± 0.0172 and 0.4983 ± 0.0160 for the BLP and the Meissner phase, respectively. This indicates that the static correlation-length critical exponent ν = 1/2.
Kibble-Zurek scalings. Now we discuss the universal scalings of the real-time dynamics across the critical point. To drive the system from the Meissner phase to the BLP, we quench the tunneling strength K according to
K(t) = K c (1 − t/τ Q )(8)
where τ Q is the quench time. In the vicinity of the critical point, both the relaxation time τ r and the correlation length ξ diverge as
τ r ∼ |ǫ| −zν , ξ ∼ |ǫ| −v ,(9)
where ǫ(t) = [K(t) − K c ] /K c is the dimensionless distance from the critical point and (z, ν) are the critical exponents. Due to the critical slowing down caused by the divergent relaxation time, a system driven across its critical point has no sufficient time to follow its instantaneous ground state no matter how slow it is driven. The Kibble-Zurek scalings can be derived by comparing the relaxation time τ r and the transition time τ t = |ǫ/ǫ| (whereǫ = dǫ/dt). The system evolves adiabatically if τ r < τ t , otherwise the adiabaticity breaks down. Defining the freezing timet with τ r (t) = τ t (t) = |t|, the two characteristic times change from τ r < τ t to τ r > τ t when the time t changes from |t| > |t| to |t| < |t|. Thus, at the freezing timet, the dimensionless distance ǫ(t) and the correlation length ξ(t) exhibit universal power laws,
ǫ = ǫ(t) ∼ τ −1/(1+zν) Q ,ξ = ξ(t) ∼ τ ν/(1+zν) Q ,(10)
with respect to τ Q . After the system is driven through the critical point, distant parts of the system choose to break the symmetry randomly and defects (discrete vortices) are spontaneously created. The total number of generated vortices scales as
N v ∼ξ −d ∼ τ −dν/(1+zν) Q ,(11)
with d the dimension of the system. Numerical scalings. Below we show how to numerically extract the Kibble-Zurek scalings from the realtime dynamics. According to the time-evolution equation i∂ψ l,σ /∂t = ∂H/∂ψ * l,σ , we simulate the quenching process [K(t) = K c (1 − t/τ Q )] for different τ Q . When the time increases from t < 0 to t > 0, the system goes from the Meissner phase into the BLP. In our simulation, the parameters are chosen as N = 5 × 10 4 , L = 200, J = 1, φ = π/2 and gn = 0.2. For each τ Q , we perform 150 runs of simulations under random initial fluctuations. In a single run, we calculate the bifurcation delay b d = |K(t) − K c | and the vortex number N v and then give their averaged values for each τ Q .
In Fig. 3, we show the universal scaling of the averaged bifurcation delay. As a signature of the impulse regime resulted from the critical slowing down, the dynamic chiral current j c (t) keep unchanged in the duration of −t < t <t. Unlike the static case, j c (t) does not decrease immediately after the critical point t = 0 but starts to decrease after the impulse-adiabatic transition at t =t, see the inset of Fig. 3. In a single run,t is numerically determined by δj c = (j c (t) − j max In Fig. 4, we show the universal scaling of the averaged vortex number. In a single run, we count the number of discrete vortices at a certain time t v after the impulseadiabatic transition at t =t. Ast is defined as the time where δj c = 0.005, we count the vortex numbers at different t v where 0.005 ≤ δj c ≤ 0.02 and find similar scalings of the averaged vortex numbers with respect to τ Q . The counting of vortex numbers is proceeded by analyzing the current patterns. To minimize numerical errors, we take the inter-leg current j ⊥ to be zero if |j ⊥ /j max ⊥ | is less than a threshold 0.01. We checked that the results are unaffected for other reasonable thresholds between 0.001 and 0.05. Our numerical results show that the averaged vortex numbers follows N v ∼ τ −0.2510±0.0387 Combining the scalings for the inverse critical velocity v −1 L , the averaged bifurcation delay b d and the averaged vortex number N v , one may determine the dimension d and the critical exponents (z, ν). From the numerical results shown in Figs. 2(b), 3 and 4, we give the dimension d = 1, the static correlation-length critical exponent ν = 1/2 and the dynamic critical exponent z = 2. Actually, z = 2 is a direct result of the unique quartic dispersion at the critical point. These critical exponents are in the same universal class for the continuous phase transitions, in which the dispersion continuously changes from a single-well shape to a double-well one [15][16][17][18][19][20][21][22][23][24][25].
Summary and discussion. To summarize, we reveal the universality of spontaneous superfluidity breakdown within synthetic gauge fields. We find that the spontaneous superfluidity breakdown obeys the KZM and extract the critical exponents from the Landau critical velocity and the critical spatial-temporal dynamics. The numerical scalings extracted from the critical spatialtemporal dynamics well agree with the analytical Kibble-Zurek scalings. Our study provide a general approach to explore and understand the dynamic universality of continuous phase transitions involving continuous variation from a single-well dispersion to a double-well one [15][16][17][18][19][20][21][22][23][24][25]. As an experimental evidence, the critical spatialtemporal dynamics of BECs in a shaken optical lattice [21] share the same critical exponents (z = 2, ν = 1/2) for ours.
At last, we briefly discuss the experimental feasibility. Based upon the recent experiment [22], our system can be realized and the considered phase transition can be observed if the interaction is sufficiently strong. The interaction strength can be enhanced via Feshbach resonance [55] or tuning the transverse confinement [60]. To drive the system across the considered phase transition, one may gradually decrease the inter-leg potential barriers via decreasing the laser intensity. The Kibble-Zurek scalings can be extracted by counting the vortex number and measuring the chiral current via the well-developed high-resolution imaging [61,62].
FIG. 1 :
1(a) Schematic diagram for a Bose ladder in a uniform magnetic field. Where, J and K are respectively the intra-and inter-leg tunneling strengthes, g is the interaction strength, and φ is the magnetic flux per plaquette. (b) The chiral current jc versus the ratio K/J. There are three typical phases: (I) the vortex phase, (II) the biased ladder phase (BLP), and (III) the Meissner phase. The thickness and length of the arrows denote the current strength, which is normalized to the maximum current for the chosen K/J. The two dash-dotted lines label the two critical points between different phases. The dispersion relations ε(k) for three typical phases are shown in insets. The parameters are chosen as N = 5 × 10 4 , L = 200, J = 1, gn/J = 0.2 and φ = π/2.
(b), we show the dependence of v −1 L on |ǫ| = |K(t) − K c | /K c . Our numerical results show v −1 L ∼ |ǫ| −b FIG. 2: (a) The Landau critical velocity vL versus K/Kc. Inset: the critical region of the Landau critical velocity. (b) The inverse critical velocity v −1 L versus |ǫ| = |K(t) − Kc| /Kc. The open triangles and circles correspond to the Meissner phase and the biased ladder phase, respectively. Here, to minimize finite-size effects, we choose L = 50000 and the other parameters as same as the ones for Fig. 1.
the maximum chiral current j max c , and the bifurcation delay is given as b d = K(t) − K c ∝ |ǫ|. Our numerical results show that the averaged bifurcation delay scales as b d = K(t) − K c ∼ τ −0.
FIG. 3 :
3The universal scaling of the averaged bifurcation delay b d . The error bars denote the standard deviation. Inset: the chiral current jc versus K(t)/Kc for different τQ. All parameters are chosen as same as the ones forFig. 1.
FIG. 4 :
4The universal scaling of the averaged vortex number Nv counted at the time where δjc = |(jc(t) − j max c )/j max c | = 0.005. The error bars denote the standard deviation. All parameters are chosen as same as the ones for Fig.1.
This work was supported by the National Basic Research Program of China (Grant No. 2012CB821305) and the National Natural Science Foundation of China (Grants No. 11374375 and No. 11574405).
. * Email, [email protected]; [email protected]* Email: [email protected]; [email protected]
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"Todas as afirmações verdadeiras são demonstráveis",
"Todas as afirmações verdadeiras são demonstráveis"
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"Jaime Gaspar "
]
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| 1 Motivação Todas as afirmações demonstráveis são verdadeiras. Mas teremos o recíproco: todas as afirmações verdadeiras são demonstráveis? Isto seria o sonho de uma matemática omnisciente capaz de demonstrar todas as afirmações verdadeiras. O teorema da completude dá vida a este sonho: todas as afirmações verdadeiras são demonstráveis. ( * ) Neste texto vamos enunciar e demonstrar o teorema da completude. Como sublinhado em ( * ), este teorema fala de afirmações, verdades e demonstrações pelo que, para o podermos enunciar e demonstrar, primeiro precisamos de definir as noções de afirmação, verdade e demonstração.2 Afirmação Começamos por definir a noção de afirmação. Informalmente, uma afirmação é uma expressão como ¬(L ∨ M) ∨ M ∨ L. Recordemos que algumas destas expressões são bem formadas como por exemplo ¬(L ∨ M) ∨ M ∨ L, enquanto outras são mal formadas como por exemplo L¬ ∨ M) ∨ ∨ ML. Definição. Fixemos uma lista de símbolos distintos L, M, N, . . . aos quais chamamos letras. Chamamos afirmação, e denotamos por A, B, C, . . ., a uma expressão bem formada construída a partir de letras por meio dos símbolos ¬, ∨ e dos parênteses. Exemplo. A expressão ¬(L ∨ M) ∨ M ∨ L é uma afirmação porque é bem formada e construída a partir de letras L e M por meio de ¬, ∨ e dos parênteses.O leitor talvez objete notando que alguns símbolos estão em falta, como por exemplo ⇒ e ∃. A omissão de ⇒ não é relevante: sempre que quisermos falar * INRIA Paris-Rocquencourt, πr 2 , Univ Paris Diderot, Sorbonne Paris Cité, F-78153 Le Chesnay, France. Apoiado financeiramente pela Fondation Sciences Mathématiques de Paris francesa. [email protected], www.jaimegaspar.com. Agradeço a Joana Cerveira, Fernando Ferreira, Gilda Ferreira e Rafael Pacheco; no entanto, este texto é da minha inteira responsabilidade. | null | [
"https://arxiv.org/pdf/1212.4665v1.pdf"
]
| 117,943,720 | 1212.4665 | 47b871c684d1d18ae404e7bd276c0b9fd4c4cb1b |
Todas as afirmações verdadeiras são demonstráveis
14 Dec 2012 14 de Dezembro de 2012
Jaime Gaspar
Todas as afirmações verdadeiras são demonstráveis
14 Dec 2012 14 de Dezembro de 2012arXiv:1212.4665v1 [math.HO]
1 Motivação Todas as afirmações demonstráveis são verdadeiras. Mas teremos o recíproco: todas as afirmações verdadeiras são demonstráveis? Isto seria o sonho de uma matemática omnisciente capaz de demonstrar todas as afirmações verdadeiras. O teorema da completude dá vida a este sonho: todas as afirmações verdadeiras são demonstráveis. ( * ) Neste texto vamos enunciar e demonstrar o teorema da completude. Como sublinhado em ( * ), este teorema fala de afirmações, verdades e demonstrações pelo que, para o podermos enunciar e demonstrar, primeiro precisamos de definir as noções de afirmação, verdade e demonstração.2 Afirmação Começamos por definir a noção de afirmação. Informalmente, uma afirmação é uma expressão como ¬(L ∨ M) ∨ M ∨ L. Recordemos que algumas destas expressões são bem formadas como por exemplo ¬(L ∨ M) ∨ M ∨ L, enquanto outras são mal formadas como por exemplo L¬ ∨ M) ∨ ∨ ML. Definição. Fixemos uma lista de símbolos distintos L, M, N, . . . aos quais chamamos letras. Chamamos afirmação, e denotamos por A, B, C, . . ., a uma expressão bem formada construída a partir de letras por meio dos símbolos ¬, ∨ e dos parênteses. Exemplo. A expressão ¬(L ∨ M) ∨ M ∨ L é uma afirmação porque é bem formada e construída a partir de letras L e M por meio de ¬, ∨ e dos parênteses.O leitor talvez objete notando que alguns símbolos estão em falta, como por exemplo ⇒ e ∃. A omissão de ⇒ não é relevante: sempre que quisermos falar * INRIA Paris-Rocquencourt, πr 2 , Univ Paris Diderot, Sorbonne Paris Cité, F-78153 Le Chesnay, France. Apoiado financeiramente pela Fondation Sciences Mathématiques de Paris francesa. [email protected], www.jaimegaspar.com. Agradeço a Joana Cerveira, Fernando Ferreira, Gilda Ferreira e Rafael Pacheco; no entanto, este texto é da minha inteira responsabilidade.
Motivação
Todas as afirmações demonstráveis são verdadeiras. Mas teremos o recíproco: todas as afirmações verdadeiras são demonstráveis? Isto seria o sonho de uma matemática omnisciente capaz de demonstrar todas as afirmações verdadeiras. O teorema da completude dá vida a este sonho: todas as afirmações verdadeiras são demonstráveis.
( * )
Neste texto vamos enunciar e demonstrar o teorema da completude. Como sublinhado em ( * ), este teorema fala de afirmações, verdades e demonstrações pelo que, para o podermos enunciar e demonstrar, primeiro precisamos de definir as noções de afirmação, verdade e demonstração.
Afirmação
Começamos por definir a noção de afirmação. Informalmente, uma afirmação é uma expressão como ¬(L ∨ M) ∨ M ∨ L. Recordemos que algumas destas expressões são bem formadas como por exemplo ¬(L ∨ M) ∨ M ∨ L, enquanto outras são mal formadas como por exemplo L¬ ∨ M) ∨ ∨ ML.
Definição. Fixemos uma lista de símbolos distintos L, M, N, . . . aos quais chamamos letras. Chamamos afirmação, e denotamos por A, B, C, . . ., a uma expressão bem formada construída a partir de letras por meio dos símbolos ¬, ∨ e dos parênteses.
L M ¬(L ∨ M) ∨ M ∨ L V V V V F V F V V F F V Definição.
Dizemos que uma afirmação A é verdadeira, e denotamos por A, se e só se todas as entradas da última coluna da tabela de verdade de A são V.
Demonstração
Finalmente, vamos definir a noção de demonstração. Informalmente, uma demonstração é um argumento em vários passos em que cada passo é um axioma ou resulta de passos anteriores por meio de uma regra. Recordemos que, informalmente: um axioma A é a asserção "A é verdadeiro"; uma regra A 1 ... An B é a asserção "se A 1 , . . . , A n são verdadeiros, então B é verdadeiro", onde os A i s chamam-se premissas e o B chama-se conclusão.
Notação. Associamos n k=1 A k = A 1 ∨ · · · ∨ A n , à direita; por exemplo, A 1 ∨ A 2 ∨ A 3 ∨A 4 = A 1 ∨(A 2 ∨(A 3 ∨A 4 )). Denotamos por A uma expressão da forma n k=1 A k (eventualmente n = 0, caso em que não há A i s, isto é, A é vazio). Denotamos por σ uma permutação de (isto é, uma bijeção de e para) {1, . . . , n}.
Definição. Consideremos os axiomas e as regras da forma
A ∨ ¬A ∨ B, (A) A 1 ∨ · · · ∨ A n A σ(1) ∨ · · · ∨ A σ(n) , (R 1 ) A ∨ (B ∨ C) (A ∨ B) ∨ C , (R 2 ) A ∨ B ¬¬A ∨ B , (R 3 ) ¬A ∨ C ¬B ∨ C ¬(A ∨ B) ∨ C . (R 4 )
Dizemos que uma afirmação A é demonstrável, e denotamos por ⊢ A, se e só se existe uma sequência de afirmações D 1 , . . . , D n , chamada demonstração de A, tal que A = D n , e cada D i é um dos axiomas, ou é a conclusão de uma das regras sendo a(s) premissa(s) da regra D j (s) com j < i.
Exemplo. Temos ⊢ ¬(L ∨ M) ∨ M ∨ L porque o seguinte é uma demonstração de ¬(L ∨ M) ∨ M ∨ L: D 1 (L ∨ M) ∨ ¬(L ∨ M) (A) com A = L ∨ M e B vazio , D 2 ¬(L ∨ M) ∨ L ∨ M conclusão de (R 1 ) com n = 2, A 1 = L ∨ M , A 2 = ¬(L ∨ M ), σ(1) = 2 e σ(2) = 1; a premissa é D 1 , D 3 ¬(L ∨ M) ∨ M ∨ L conclusão de (R 1 ) com n = 3, A 1 = ¬(L ∨ M ), A 2 = L, A 3 = M , σ(1) = 1, σ(2) = 3 e σ(3) = 2; a premissa é D 2 .
Teorema da completude
Depois de termos definido as noções de afirmação, verdade e demonstração, estamos finalmente em condições de enunciar e demonstrar o teorema da completude.
Teorema da completude. Todas as afirmações verdadeiras são demonstráveis (isto é, para toda a afirmação A tal que A temos ⊢ A).
Demonstração.
A cada afirmação A atribuímos uma pontuação A da seguinte forma: cada ∨ em A vale 1 ponto, e cada ¬ em A que não esteja imediatamente antes de uma letra vale 1 ponto. Demonstremos que (1) n k=1 A k implica (2) ⊢ n k=1 A k por indução completa em n k=1 A k ; o teorema é o caso n = 1.
Caso base n k=1 A k = 0. Temos A i = 0 para cada A i , logo cada A i é uma letra ou a negação de uma letra. Suponhamos (1). Então algum A i é uma letra L e um algum A j é ¬L, caso contrário dávamos valores de verdade às letras de modo que os A k s fossem falsos contradizendo (1). Seja A = n i,j =k=1 A k . Temos ⊢ L ∨ ¬L ∨ A por (A), isto é, ⊢ A i ∨ A j ∨ A, logo (2) por (R 1 ). (3) e (4) implica (2) por (R 1 ). (4).
Passo de indução n k=1 A k > 0. Temos A i > 0 para algum A i , logo A i é da forma B ∨ C ou ¬D, onde D não é uma letra logo é da forma ¬B ou B ∨ C. Seja A = n i =k=1 A k . Basta demonstrar que (3) A i ∨ A implica (4) ⊢ A i ∨ A porque (1) implicaCaso A i = B ∨ C. Se (3), isto é, (B ∨ C) ∨ A, então B ∨ C ∨ A, logo ⊢ B ∨ C ∨ A por hipótese de indução (que se aplica porque B + C + n i =k=1 A k < n k=1 A k ), portanto ⊢ (B ∨ C) ∨ A por (R 2 ), isto é, (4). Caso A i = ¬¬B. Se (3), isto é, ¬¬B ∨ A, então B ∨ A, logo ⊢ B ∨ A por hipótese de indução (que se aplica porque B + n i =k=1 A k < n k=1 A k ), portanto ⊢ ¬¬B ∨ A por (R 3 ), isto é,Caso A i = ¬(B ∨ C). Se (3), isto é, ¬(B∨C)∨ A, então ¬B∨ A e ¬C∨ A,
logo ⊢ ¬B ∨ A e ⊢ ¬C ∨ A por hipótese de indução (que se aplica porque ¬B + n i =k=1 A k , ¬C + n i =k=1 A k < n k=1 A k ), portanto ⊢ ¬(B ∨ C) ∨ A por (R 4 ), isto é, (4). [4] 6 Sugestões de leitura Se o leitor estiver interessado em saber mais, o passo seguinte é ler sobre:
Teorema da completude de Gödel. Estende o teorema aqui apresentado de modo a abranger afirmações com ∃. [1,2,4] Teoremas da incompletude de Gödel. Demonstram que existem afirmações verdadeiras (num sentido mais geral do que o usado neste texto) que são indemonstráveis. [2,3,5]
Exemplo. A expressão ¬(L ∨ M) ∨ M ∨ L é uma afirmação porque é bem formada e construída a partir de letras L e M por meio de ¬, ∨ e dos parênteses. O leitor talvez objete notando que alguns símbolos estão em falta, como por exemplo ⇒ e ∃. A omissão de ⇒ não é relevante: sempre que quisermos falar * INRIA Paris-Rocquencourt, πr 2 , Univ Paris Diderot, Sorbonne Paris Cité, F-78153 Le Chesnay, France. Apoiado financeiramente pela Fondation Sciences Mathématiques de Paris francesa. [email protected], www.jaimegaspar.com. Agradeço a Joana Cerveira, Fernando Ferreira, Gilda Ferreira e Rafael Pacheco; no entanto, este texto é da minha inteira responsabilidade. de A ⇒ B podemos em vez disso falar de uma afirmação equivalente que só use ¬ e ∨ tal como ¬A ∨ B; por exemplo, no exemplo anterior em vez de falarmos de L∨M ⇒ M ∨L, falamos de ¬(L∨M)∨M ∨L. Já a omissão de ∃ é relevante: ∃x A(x) não é equivalente a nenhuma afirmação só com ¬ e ∨. Assim, omitir ∃ empobrece de forma relevante as nossas afirmações, mas optámos por o fazer porque simplifica imenso o teorema da completude. 3 Verdade Vamos agora definir a noção de verdade. Denotemos por V (respetivamente, F) o valor de verdade verdadeiro (respetivamente, falso). Recordemos que podemos calcular o valor de verdade de uma afirmação por meio de uma tabela de verdade. Por exemplo, a tabela de verdade seguinte dá-nos o valor de verdade de ¬(L ∨ M) ∨ M ∨ L em função dos valores de verdade de L e M:
Exemplo. Temos ¬(L ∨ M) ∨ M ∨ L porque todas as entradas da última coluna da tabela de verdade de ¬(L ∨ M) ∨ M ∨ L acima são V.
Vilnis Detlovs e Karlis Podnieks. Introduction to mathematical logic. Vilnis Detlovs e Karlis Podnieks. Introduction to mathematical logic, 2000. http://www.ltn.lv/~podnieks/mlog/ml.htm.
. Juliette Kennedy Kurt Gödel, Edward N. Zalta, editorStanford Encyclopedia of PhilosophyJuliette Kennedy. Kurt Gödel. In Edward N. Zalta, edi- tor, Stanford Encyclopedia of Philosophy. Fevereiro 2007. http://plato.stanford.edu/entries/goedel/.
What is mathematics: Gödel's theorem and around. Karlis Podnieks, Karlis Podnieks. What is mathematics: Gödel's theorem and around, 1997. http://www.ltn.lv/~podnieks/gt.html.
The completeness theorem of Gödel. Resonance. S M Srivastava, 6Julho, AgostoS. M. Srivastava. The completeness theorem of Gödel. Resonance, 6(7, 8), Julho, Agosto 2001. http://www.ias.ac.in/resonance/.
Gödel's proof. S M Srivastava, Resonance. 122S. M. Srivastava. Gödel's proof. Resonance, 12(2, 3, 5), Fevereiro, Março, Maio 2007. http://www.ias.ac.in/resonance/.
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[
"Double Pomeron Opportunities at √ s = 1.8 TeV *",
"Double Pomeron Opportunities at √ s = 1.8 TeV *"
]
| [
"Jon Pumplin \nDepartment of Physics and Astronomy\nMichigan State University\n48824East Lansing, Bitnet: PUMPLIN@MSUPAMI\n"
]
| [
"Department of Physics and Astronomy\nMichigan State University\n48824East Lansing, Bitnet: PUMPLIN@MSUPAMI"
]
| []
| I describe possible ways to discover hard double pomeron exchange (HDPE) with the existing detectors at the Fermilab Tevatron, by using the small-angle "luminosity" counters as a veto. Estimates of the cross sections and backgrounds are made. In addition to the intrinsic importance of HDPE, its observation would be useful for calibrating the detectors, and for estimating the "survival probability" of rapidity gaps. | 10.1103/physrevd.47.r4820 | [
"https://export.arxiv.org/pdf/hep-ph/9301216v1.pdf"
]
| 18,235,408 | hep-ph/9301216 | be712d9d2185328f86e1acb74ddd660f58a99108 |
Double Pomeron Opportunities at √ s = 1.8 TeV *
7 Jan 1993 January 6, 1993
Jon Pumplin
Department of Physics and Astronomy
Michigan State University
48824East Lansing, Bitnet: PUMPLIN@MSUPAMI
Double Pomeron Opportunities at √ s = 1.8 TeV *
7 Jan 1993 January 6, 1993* Research supported in part by Texas National Research Laboratory Commission grant RGFY9240 to the CTEQ Collaboration.
I describe possible ways to discover hard double pomeron exchange (HDPE) with the existing detectors at the Fermilab Tevatron, by using the small-angle "luminosity" counters as a veto. Estimates of the cross sections and backgrounds are made. In addition to the intrinsic importance of HDPE, its observation would be useful for calibrating the detectors, and for estimating the "survival probability" of rapidity gaps.
Introduction
Typicalpp interactions at √ s = 1.8 TeV produce a large number of particles, which are distributed rather uniformly in pseudo-rapidity η = − ln tan θ 2 , with dN/dη ∼ 6 for |η| < 4. There can nevertheless exist final states with one or more "rapidity gaps", defined as intervals of length ∆η > 2 to 3 containing zero particles. Long rapidity gaps are by definition governed by the pomeron, which is believed related to the s-channel unitarity phenomenon of shadow scattering. A QCD-based understanding of the pomeron remains elusive, although a qualitative description as a two-gluon system with vacuum quantum numbers is promising [1,2,3,4].
The 'grand-daddy' of rapidity gap processes is elastic scattering, which makes up ∼ 20% of σ tot . Inelastic single diffraction, defined by a gap with a leading p orp at one end, is also responsible for a sizeable fraction of σ tot , and has been seen to exhibit hard-scattering effects [5]. Double pomeron exchange (DPE), defined by two rapidity gaps, has been observed using special detectors for small angle quasi-elastic protons at √ s = 0.063 TeV [6] and √ s = 0.63 TeV [7].
A complete study of rapidity gap physics demands detectors that cover a long range in η, to establish the absence of particles in one or more gap regions while detecting particles outside the gaps. Such detectors have been proposed for SSC ( √ s = 40 TeV) [8] and the FNAL Tevatron ( √ s = 1.8 TeV) [9]. The point of the present paper, however, is to consider some hard double-pomeron exchange (HDPE) processes that can be studied at the Tevatron in the working detectors CDF [10] and DØ [11], which cover roughly −4 < η < 4 .
The processes are defined by rapidity gaps of −4 < η < −2 and 2 < η < 4, with an experimentally clean high-Q 2 object in the central region. That object should be producible from two pomerons, which are assumed to have vacuum quantum numbers. Two jets separated by ∆η < 1. The odd-C bb states can be detected by Υ → e + e − or µ + µ − . The even-C states can be detected by χ → γΥ followed by Υ → e + e − or µ + µ − [12].
If any of these ΥΥ or χ b final states are observed, the DPE nature of their production can be demonstrated by showing that the cross section does not decrease drastically when the required rapidity gaps are extended into the central rapidity region. There is plenty of room for this in the case of the bb states. For the 2-jet final states, the requirement ∆η < 1 also leaves some room for extending the rapidity gaps into the central region to make this test. One can also check that single Υ states are strongly suppressed relative to χ b , since they cannot be made from two pomerons according to charge conjugation [13].
We will show that the above processes offer a reasonable opportunity to observe the hard scattering of two pomerons (HDPE). The Υ and χ b states are especially attractive because their bb wave functions are relatively well understood, which will facilitate attempts to compute the production. Because of their low multiplicity and accurately known masses, these states will also be valuable experimentally for calibrating the energy resolution and noise level of the detector.
Trigger and backgrounds
At current Tevatron luminosities, interactions occur at a rate of a few × 10 5 per second, while events can be recorded at a rate of a few per second. Background events must therefore be rejected by ∼ 10 −5 . The trigger decision is made in a series of stages, with the initial rejection of 10 −2 to 10 −3 based on rather incomplete information. I propose to cope with this trigger challenge as follows.
A minor part of the DØdetector consists of scintillation counters that cover approximately −4 < η < −2 and 2 < η < 4. Similar counters cover 3.24 < |η| < 5.90 in CDF.
Normal triggers require a coincidence between hits in these "luminosity"(DØ) or "beambeam"(CDF) counters to make a first estimate of the interaction vertex by timing, and to discriminate against beam+gas interactions. The crucial experimental proposal I make is to use the luminosity counters instead as a veto. A trigger defined by an absence of hits in these counters, in coincidence with some indication of hard scattering in the central region |η| < 2, will eliminate a large fraction of "ordinary" events to look for HDPE.
To study the trigger, I use the QCD Monte Carlo program HERWIG5.5 [14] to simulate events. Events that contain no charged particles in the veto regions −4 < η < −2 and A second opinion on the minimum bias physics can be obtained from the 2 → 2 QCD hard scattering mode of HERWIG, with the "background event" turned off and the minimum transverse momentum in hard scattering set to a small value (2.2 GeV) chosen to produce the entire inelastic cross section. This "mini-jet" model predicts a similar rate for the electromagnetic E T trigger, and a somewhat larger rate ∼ 2 µb for the hadronic trigger.
Leaving the "background event" on would give a lower rate.
It will be better to trigger on coincidences between two cells above an E T threshold in the central region. This will further reduce the trigger rate from minimum bias physics, and at the same time suppress non-physics backgrounds from detector noise, beam halo, and beam-gas collisions. In this way it will be possible to look for HDPE processes, which are of course not simulated by HERWIG, down to the smallest cross sections visible at the Tevatron luminosity.
Estimates of signals
I estimate diffractive ΥΥ production by a pole-dominance model, used long ago to calculate DPE π + π − production [15]. The amplitude from Fig. 1 is
M = T (p 1 q → p 3 p 4 ) (q 2 − M 2 Υ ) −1 [1 − A(q 2 − M 2 Υ )] −1 T (p 2 −q → p 6 p 5 )(1)
An exchanged graph is obtained by p 4 ↔ p 5 . The two-body Υp elastic amplitudes are
T (p 1 q → p 3 p 4 ) = i s 34 σ Υp e β t 13 /2(2)
where s 34 = (p 3 +p 4 ) 2 and t 13 = (p 1 −p 3 ) 2 . Reasonable guesses for the Υp total cross section and forward elastic slope are σ Υp = 2 mb and β = 6 GeV −2 . Eq.(1) includes an off-shell suppression factor controlled by the parameter A, which is hard to guess but expected to be O(1 GeV −2 ). For A = 1 GeV −2 , this model gives 4 nb for ΥΥ production. Allowing for the branching fractions Υ → e + e − or µ + µ − for both of two Υ(9460)s, and making the rapidity and E T cuts above reduces the estimate to 4 pb. At the anticipated Tevatron integrated luminosity of 50 pb −1 , this will lead to 200 events and be clearly visible.
The pole-dominance model predicts that DPE production of J/ψ J/ψ will also be observable. Assuming σ J/ψ p = 3 mb and again using A = 1 GeV −2 leads to 20 nb for J/ψ J/ψ production. When both J/ψ decay to e + e − or µ + µ − , at least one of the four leptons has E T > 3 GeV more than half of the time. This relatively large E T arises because the individual J/ψ transverse momenta are comparable to M J/ψ , even though their sum is small. The DPE J/ψJ/ψ leptonic final states will therefore also generally pass our proposed trigger.
We next attempt to estimate the diffractive production of single χ b states. First note that these states can be formed by gluon+gluon fusion with a coupling strength that is measured by their hadronic width:
σ gg→χ = (π 2 Γ χ→gg /16M χ ) δ(ŝ − M 2 χ ) ,(3)
which includes a factor 1/256 from spin and color averaging. The state χ b0 (2P ) has M χ = 10.23 GeV and Γ hadronic ≈ 400 KeV [12]. According to a simple parton-model calculation, it is formed by g + g fusion, with a production cross section integrated over |η| < 1.5 of ∼ 20 nb. Including the poorly known branching ratios to γΥ followed by Υ → e + e − or µ + µ − reduces the observable cross section for this non-pomeron process to ∼ 30 pb.
One can imagine a second gluon exchange that modifies the g + g fusion process and makes the overall exchange between beam and target a color singlet. This color singlet exchange does not necessarily produce a large average number of particles per unit of rapidity. It occasionally produce zero particles in the veto regions −4 < η < −2 and 2 < η < 4. If the price for the two gaps is less than a factor ∼ 1/300, the DPE production rate of the χ b state will be observable. For a discussion of this idea, see reference [16].
A similar estimate can be made for g + g → jet + jet. From HERWIG, the cross section for two jets with E T > 10 GeV, |η| < 1.3, and |η 1 − η 2 | < 1.0 is ∼ 3 × 10 7 pb . If a second gluon exchange can produce rapidity gaps as suggested above, the corresponding HDPE process will be observable even if the gap requirement suppresses this huge cross section by 10 −8 .
Conclusion
I have shown that a trigger based on two rapidity gaps can be used to look for hard double pomeron interactions in the existing detectors at the Tevatron. Trigger rates and non-diffractive backgrounds are small enough to look for these processes down to ∼ 0.1 pb.
Crude estimates suggest that several HDPE processes will be observable. I have emphasized processes with one or two bb states, because they have experimentally very clean e + e − and γe + e − decay modes with sufficient transverse energy to satisfy the trigger. The bb states are also attractive theoretically because one can use their known wave functions in attempting to calculate the production. Final states involving two J/ψ instead of two Υ, with nothing else visible in the detector, are also worth looking for.
Final states with two jets nearby in rapidity, with gaps on either side, will allow the most sensitive search for HDPE, since two-jet production must have a relatively large cross section compared to the other processes we consider. Requiring |η 1 − η 2 | < 1.0 for the jet axes, with jets defined by cones √ ∆η 2 + ∆φ 2 < 0.7, leaves an average of 2.8 units in η on either side of the jj system to define the rapidity gaps. Assuming one observes HDPE candidates, in which no particles (or no calorimeter cells above the noise level) appear in the gap regions, it will be important to study the distribution in the number of particles in the gap regions. One must see if there is a peak at 0 particles which signals HDPE; or if the 0-particle events appear to be simply fluctuations of normal hard scattering. This background due to fluctuation has a cross section ∼ 0.3 nb for jets defined by E T > 10 GeV, according to a HERWIG simulation.
Observing HDPE processes would also establish the survival of rapidity gaps, which offer important possibilities for Higgs and WW scattering physics at the SSC [17].
Promising candidates for the central object are (1) Pairs of bb bound states: Υ(1S)Υ(1S) or Υ(2S)Υ(2S); (2) Single bb bound states: χ b0 (1P ), χ b0 (2P ), χ b2 (1P ), or χ b2 (2P ); or (3)
2
< η < 4 are analyzed in a simple model of a calorimeter detector, consisting of cells 0.20 × 0.20 in η × azimuthal angle φ. Requiring a transverse energy E T > 2.5 GeV or an electromagnetic transverse energy E EM T > 1.5 GeV in at least one of the cells in the central region, and no electromagnetic or hadronic energy above 0.1 GeV in the veto regions, I find a cross section of ∼ 1 µb according to the "minimum bias" mode of HERWIG.
Acknowledgements I thank H. Weerts for discussions on DØand J. Huston for discussions on CDF.
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. J Pumplin, Phys. Rev. D. 282741J. Pumplin, Phys. Rev. D 28, 2741 (1983).
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FIGURE CAPTION. H Chehime, Phys. Lett. B. 286397H. Chehime et al., Phys. Lett. B 286, 397 (1992). FIGURE CAPTION
Pole model for DPE production of ΥΥ or J/ψJ/ψ. The blobs represent elastic scattering. which is dominated by pomeron exchangePole model for DPE production of ΥΥ or J/ψJ/ψ. The blobs represent elastic scat- tering, which is dominated by pomeron exchange.
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[
"Astronomical distances and velocities and special relativity",
"Astronomical distances and velocities and special relativity"
]
| [
"Germano D ' Abramo [email protected] \nMinistero dell'Istruzione\nUniverstà e della Ricerca\n00041Albano LazialeRMItaly\n"
]
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"Ministero dell'Istruzione\nUniverstà e della Ricerca\n00041Albano LazialeRMItaly"
]
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| We show that some primary special relativity effects, which are believed to be hardly detectable in everyday life, such as time dilation, relativistic Doppler effect, and length contraction, should tangibly and spectacularly show up here on the Earth. They should occur in ordinary observations of known astronomical phenomena, also when these phenomena involve astronomical systems that move with very low velocities relative to us but are very distant. We shall do that by providing a reanalysis of the so-called Andromeda paradox and by revisiting the standard explanation of the muon lifetime dilation given when this phenomenon is observed from muon's perspective. Ultimately, we shall show that if Lorentz transformations (and basically, special relativity) are meant to entail real physical consequences, then the observable Universe should appear very differently from what we see every clear night. | null | [
"https://arxiv.org/pdf/1711.03833v7.pdf"
]
| 118,957,855 | 1711.03833 | a1b4c91870a4abdeb147d40e5fbe266c9968983a |
Astronomical distances and velocities and special relativity
Germano D ' Abramo [email protected]
Ministero dell'Istruzione
Universtà e della Ricerca
00041Albano LazialeRMItaly
Astronomical distances and velocities and special relativity
Annales de la Fondation Louis de Broglie 45(1) (2020)special relativityLorentz transformationsAndromeda paradoxmuon decaylength contractionDoppler effect PACS: 0330+p
We show that some primary special relativity effects, which are believed to be hardly detectable in everyday life, such as time dilation, relativistic Doppler effect, and length contraction, should tangibly and spectacularly show up here on the Earth. They should occur in ordinary observations of known astronomical phenomena, also when these phenomena involve astronomical systems that move with very low velocities relative to us but are very distant. We shall do that by providing a reanalysis of the so-called Andromeda paradox and by revisiting the standard explanation of the muon lifetime dilation given when this phenomenon is observed from muon's perspective. Ultimately, we shall show that if Lorentz transformations (and basically, special relativity) are meant to entail real physical consequences, then the observable Universe should appear very differently from what we see every clear night.
Introduction
It is well known that most of the primary special relativity effects, such as time dilation, relativistic Doppler effect, and length contraction, become macroscopically observable only when the velocity v of the physical system, relative to the observer, approaches the speed of light c. There is one notable exception, though. According to Purcell's explanation of magnetic forces, the magnetic force acting upon a single charge moving parallel to a neutral current-carrying wire (Lorentz force) is, in fact, a macroscopic manifestation of the relativistic length contraction of the distances between the moving conduction electrons in the wire, even though the velocities involved are always v c. The contraction, which is observable only in the reference frame of the moving single charge, allegedly causes an unbalance in the charge density of the wire that results in the attraction (or repulsion) of the moving single charge. However, the present author has already shown [1,2] that this mechanical/dynamical approach to the explanation of magnetic forces is problematic and we do not deal with it here.
To the author's knowledge, it is less widely known that special relativity effects 1 should macroscopically show up also in other physical systems moving with very low relative velocities (v c), provided that they are placed at huge distances d from the observer (with d/c 2 1 s 2 /m). Astronomical objects, with their huge distances and fairly high velocities relative to the Earth, are thus good candidates to actually observe special relativity effects.
In the following two sections we describe two examples of relativistic effects which should allegedly show up in plain observations of astronomical objects (very distant and/or very fast) made here on the Earth: the first example is related to the so-called 'Andromeda paradox', while in the second one we compare the relativistic explanation of the muon retarded decay, given when the phenomenon is analyzed from the muon reference frame, to what we should see from the Earth when we observe relatively fast astronomical objects.
The Andromeda paradox
The Andromeda paradox, also known with the name of Rietdijk-Putnam-Penrose argument [3,4,5,6], gives a colorful demonstration that if special relativity is true, then observers moving at different relative velocities (any velocity, also non-relativistic) have different sets of events that are present for them. In particular, if two people walk past each other in the street and one of the people was walking towards the Andromeda galaxy, then the events in this galaxy that are simultaneous with the present time of this observer might be hours or even days advanced of the events on Andromeda simultaneous with the person walking in the other direction.
This argument has been introduced in the past to support the philosophical stance known as 'four-dimensionalism' (or 'block Universe' view), namely that an object's persistence through time is like its extension through space (for an entertaining and accessible presentation of the philosophical and physical theories of Time see, for instance, [7]).
Simple derivation of the paradox
The Andromeda paradox can be explained by recurring to the planes of simultaneity in the space-time diagram (Minkowski diagram). Here, instead, we make use of the plain Lorentz transformations
x = x−vt 1− v 2 c 2 , y = y, z = z, t = t− vx c 2 1− v 2 c 2 ,
where the non-primed coordinates (x, y, z, t) refer to the reference frame assumed to be at rest and v is the velocity of the primed frame relative to the non-primed one along the x-axis.
Consider an observer A here on the Earth who moves towards the Andromeda galaxy (relative distance d, the direction Earth-Andromeda being along the x-axis) at a relative velocity v with v c. For the sake of derivation, we shall equivalently consider the Andromeda galaxy as approaching the observer, and thus the velocity to be inserted in the Lorentz transformations is −v. Observer B is also on the Earth. He is initially close to the place where A starts walking, but he is at rest. It is further assumed that the relative velocity between the Earth and the Andromeda galaxy is negligible, and thus the relative velocity of observer B relative to Andromeda is taken as zero. According to the Lorentz transformations, if t A and t B are the present instants of time of observer A and observer B respectively (with t A t B , since v c and the observers' clocks can be considered as continuously synchronized), then the instant of time on Andromeda simultaneous with t A is
t A = t A + vd c 2 1 − v 2 c 2 ,(1)
while the instant of time on Andromeda simultaneous with t B ( t A ) can be taken as simply t B = t B .
Since the distance d between the Andromeda galaxy and the Earth is huge, we have that vd c 2 can be much greater than unity, even with v c, and then eq. (1) can be approximated to
t A t A + vd c 2 .(2)
This has the paradoxical consequence that although observer A and observer B always experience the same 'present instant' of time (t A t B ), the events on Andromeda simultaneous with observer A are events subsequent (instant of time t A t A + vd c 2 ) to the events on the same galaxy that are simultaneous with observer B (instant of time t B = t B t A ). For instance, it might well happen that, in the plane of simultaneity of observer A, a supernova has just exploded in some part of the Andromeda galaxy while, in the plane of simultaneity of observer B, the same event has not yet happened.
Going further
In the literature, the extent of the paradox's consequences has been partially downplayed by noticing that the observers cannot actually see what is happening in Andromeda since it is light-years away, and then the paradox is only that they have different ideas of what is happening "now" in Andromeda.
We believe that there is more to it. There is something that can be in principle physically measured. Suppose that, for the sake of argument, both observes can live for millions of years and both decide, starting at time t A t B , to wait an interval of time equal to d/c and see what happens. This interval is the time needed by a light signal emitted in the Andromeda galaxy to reach the Earth. Please note that observer A does not keep moving for the whole interval of time d/c: observer A is near observer B, moves a bit, and then comes back near to B almost immediately. Now, the problem is: what will observer A and observer B see after the interval of time d/c has passed? Will they see the same events or not?
What observer A sees after the interval d/c are the events that were simultaneous with the instant of time t A of observer A exactly d/c years ago and we have just seen that these events are surely different from the events that were simultaneous with the instant of time t B ( t A ) of observer B exactly d/c years ago. All this means that after the same interval of time d/c has passed, observer A and observer B, who are at rest and close to one another already for a time nearly equal to d/c, will actually see different events while observing the very same galaxy at the very same time here on the Earth (e.g. observer A detects the explosion of a supernova and observer B does not).
Let us linger over this with the following more direct representation. Observer Bob is at rest on the Earth, sitting on a bench and staring at the Andromeda galaxy (which is d away from the Earth). Observer Alice passes by with a velocity v c. After few meters traveled (or, equivalently, after few seconds), Bob shouts "Now!" at Alice. Both Alice and Bob start their stopwatches. Then, Alice suddenly stops walking away, makes a U-turn, and goes sitting close to Bob. They both shut their eyes and wait an interval of time equal to d/c before opening their eyes again. Since v c, their proper times are the same, their stopwatches are synchronized, their distance from Andromeda is the same (d) and the time they have to wait before opening their eyes is also the same (d/c). What will they see when they open their eyes? Bob will surely see events in Andromeda that were simultaneous with Bob's present when he yelled "Now!", and Alice will surely see events in Andromeda that were simultaneous with her present when Bob yelled "Now!". But, according to the Lorentz transformation of the time coordinate, the Andromeda events that were simultaneous with Alice's present when Bob yelled "Now" are vd/c 2 subsequent in time to those simultaneous with Bob's present when he yelled "Now!". This is a bizarre situation: people staying in the same place at the same time and staring at the same source in the sky see different events. But, this is a strict logical consequence of the accepted laws of physics 2 . Moreover, the very same logic could be applied to the past, namely to events that happened millions of years ago. Today, we should see a rainbow of different and simultaneous events while observing the Andromeda galaxy. Millions of years ago, we were not yet born and it would be difficult to define the velocity v of the observers then (and even just define the 'observers'), but we are sure that the reader has got the point.
By the way, eq. (2) should have an observable consequence today: it is possible to demonstrate that even the periodic movement of the Earth around the Sun should induce a sort of visible (wild and haphazard) 'Doppler oscillations' of the radiation coming from very distant astronomical sources. Note that the frequency shift we are referring to here is not the standard Doppler shift due to the (usually high) relative velocity between the source and the observer. It is an exclusively relativistic effect. For the sake of derivation, let us focus on the velocity variation of the Earth with respect to the Andromeda galaxy during the Earth revolution around the Sun. The galaxy is assumed to be at rest with respect to the Sun. Let us further consider only a small trait of the Earth orbit, where the change of velocity (acceleration a) can be taken as constant (and obviously a∆t c, for every ∆t considered). The acceleration is assumed to be directed along the line of sight. Suppose that at initial Earth time instant t E 1 the velocity of the Earth relative to Andromeda is equal to zero and then the simultaneous instant of time in Andromeda is t A 1 = t E 1 . At Earth time instant t E 2 , the relative velocity of the Earth has increased to a(t E 2 − t E 1 ) = a∆t E and thus, by making use of the differential form of eq. (2), we have the following result for the interval of time elapsed in Andromeda corresponding to the interval of time ∆t E elapsed on the Earth
dt A dt E + d c 2 dv → ∆t A ∆t E + d c 2 a∆t E = ∆t E 1 + ad c 2 ,(3)
where d is, as before, the distance between the Earth and the Andromeda galaxy. Now, if there is a star in Andromeda that emits radiation at a frequency ν 0 for an interval of time ∆t A , then it will emit a number of periods equal to ν 0 ∆t A . This number of periods will also be observed here on the Earth (after the traveling time d/c) but as emitted in the shorter time interval ∆t E and thus the frequency of the radiation seen here on the Earth will be higher, equal to
ν E = ν 0 1 + ad c 2 .(4)
Since the motion of the Earth is not uniform around the Sun and since there are other dynamical mechanisms that contribute to the relative motion of the Earth with respect to distant galaxies (e.g. motion of the Sun around the center of the Galaxy, relative motion of galaxies, not to mention the proper motion of the stars that emit radiation from inside the Andromeda galaxy), we should observe the light of distant galaxies weirdly and haphazardly Doppler shifted. The effects we would observe today would be due to radiation emitted a very long time ago (∼ d/c, where d is the astronomical distance of the source from the Earth), but this delay does not cancel out the phenomenon, we simply do not see it live.
Muon decay and length contraction
In the '40s, studies conducted on muons generated by cosmic rays in the upper atmosphere suggested that what was thought to be an anomalous absorption of these particles by the atmosphere itself was in fact due to their spontaneous decay and that the decay-rate depended upon muons' momentum [8,9]. The decay-rate dependence on momentum has been interpreted in the framework of special relativity as one of the neatest experimental verification of the time dilation of a 'moving clock'. Although muons mean lifetime τ 0 is of only ∼ 2.2 µs and thus not enough to guarantee their arrival at the lower atmosphere, their high abundance at this atmospheric depth is explained by the fact that their lifetime measured in the reference frame of the Earth is relativistically dilated to τ = τ 0 / 1 − v 2 /c 2 (due to their high relative velocity, v 0.99 c); this is just the amount needed to explain their anomalous lower atmosphere abundance.
But, how is the same phenomenon explained when it is seen in the reference frame of the traveling muon? In the muon's rest frame, the particle decays, on average, after a time τ 0 , and from its perspective the rate of clocks on the Earth is slowed down. Therefore, from its perspective, an observer on the Earth should measure a decrease and not an increase of its lifetime and thus a decrease and not an increase in the number of muons in the lower atmosphere. However, all relativists explain the phenomenon simply by invoking the length contraction of the atmosphere: for a muon, the atmosphere is thinner and the particle has the time to penetrate it deeper.
At first sight, this explanation appears quite neat and it is considered as a solid proof of the internal coherence and strength of special relativity. Under close inspection, however, it is a bit problematic and, apparently, it has never been recognized as such before.
For the sake of simplicity, consider the setup shown in Figure 1. With regard to the key aspects of the process, it is completely equivalent to the process observed in nature. The proper mean lifetime of a muon is τ 0 . This means that if we travel with the muon we will see it decay after an interval of time τ 0 . Observers on the Earth, instead, see the muon decay after a dilated interval of time τ = τ 0 / 1 − v 2 /c 2 , since v c. During this time, for the observers on the Earth, the muon travels a distance L = v · τ = v · τ 0 / 1 − v 2 /c 2 . For the sake of argument, the muon generator has been placed exactly at distance L from the surface of the Earth and thus muons can reach the surface just before decay.
In the reference frame of the muon, however, the particle sees the Earth approaching at speed v and thus, from its point of view, during that interval of time the distance covered by the Earth before muon decay is v · τ 0 < L. Namely, the muon disintegrates before touching the surface of the Earth. This result simply comes from elementary kinematics. Here, we only appeal to the principle of relativity by which the laws of physics (e.g. kinematics) are the same in every inertial frame.
The only possibility to reconcile these two different views is the widely known and accepted explanation (e.g. see [11]) that in the muon reference frame the distance that separates the muon (generator) from the surface of the Earth is Lorentz contracted,
L = L · 1 − v 2 /c 2 = v · τ 0 / 1 − v 2 /c 2 · 1 − v 2 /c 2 = v · τ 0 .(5)
This means that, from muon's perspective, the Earth's surface appears to be (and actually is) closer than L. This effect is also considered when an interstellar journey of a spaceship traveling at speeds close to that of light is analyzed from the perspective of the astronaut. From the perspective of the Earth, time on spaceship dilates and the astronaut can cover a huge distance in a relatively short period of his own time. From the perspective of the astronaut, though, his time rate does not change and the only possibility to match the observations made by the observers on the Earth is that the distance to travel actually shortens for the astronaut. Consider the situation depicted in Figure 2. A spaceship is located at a distance L from the Earth and heads towards our planet at constant velocity v c. The distance L is intended as measured from the Earth. Suppose that the time L/v needed by the spaceship to reach us is greater than 100 years. According to special relativity, if v is suitably high, the observers on the Earth will measure a time dilation within the spaceship that makes it possible for the astronaut to reach the Earth in a shorter period of his own time, say 8 years. Now, in the reference frame of the astronaut, the same outcome can only be explained with length contraction. In order for the astronaut to reach the Earth in 8 years of his own time, the distance L should be suitably shorter from his perspective. The other possibility, namely that the Earth appears faster to the astronaut, cannot be accepted owing to the principle of relativity. The very same principle of relativity, however, discloses a problem with the length contraction explanation. According to this principle, there is no reason to believe that the spaceship moves and the Earth is a rest. It may well be the other way around. In that case, it should be the distance seen by the Earth to be contracted. At any rate, the distance measured from the Earth should be equal to that measured by the astronaut in the reference frame of the spaceship because nobody can say who is moving and who is at rest. Now, in the previous paragraph change the words 'astronaut' or 'spaceship' with 'muon' and it should be evident why length contraction cannot be an acceptable explanation for the muon problem when it is analyzed in the muon rest frame.
To recapitulate, two observations follow in order. First, if the principle of relativity holds, one may equivalently assume that the Earth is actually moving towards the muon and thus the distance L that separates us from the place where the particle originated (generator) is already 'shrunk'. Or, better, for the principle of relativity if we measure a distance equal to L, then also the muon must see the Earth distant L from itself. Owing to the principle of relativity, the two distances (contracted or not) must be equal. Length contraction, like time dilation, is symmetrical 3 when the relative velocity is uniform, as is in this case. Thus, the standard explanation of the muon lifetime dilation from muon's perspective becomes inconsistent, to say the least.
Secondly, what about observations of astronomical objects (matter) moving towards our position at relativistic speeds? Consider, for instance, relativistic jets of particles moving towards our position from Active Galactic Nuclei (AGNs) 4 . According to the principle of relativity and the relativity of uniform motion, we may equivalently consider the Solar System (or our galaxy) as moving towards the jet particles at relativistic speeds and thus, according to special relativity, these jets should appear to us a lot closer than the AGNs that have generated them. If the length contraction explanation of the muon decay phenomenon has a real physical meaning (it is physically real), then we should observe a weird distribution of matter in deep space, due to the existence of objects with different (and relativistic) relative velocity with respect to our reference frame.
Conclusion
We have shown that some primary special relativity effects, which are believed to be hardly detectable in everyday life, like time dilation, relativistic Doppler effect, and length contraction, should tangibly and spectacularly show up here on the Earth: they should occur in ordinary observations of known astronomical phenomena, also when the observations involve astronomical systems that move with very low relative velocities (v c) but are placed at huge distances d from us (with d/c 2 1 s 2 /m). In that regard, we have offered two examples: the first involves the so-called Andromeda paradox, and the second, inter alia, calls into question the standard special relativity explanation of the muon lifetime dilation when the phenomenon is analyzed from muon's perspective. These two examples ultimately imply that if special relativity consequences (basically the consequences deriving from the application of the Lorentz transformations) are real physical consequences, then the observable Universe should appear very differently from what we actually see every clear night. Unfortunately, none of the effects described in this paper, and that necessarily and strictly follows from special relativity, seem to have been ever observed. Thus, there are concrete elements to believe that something is actually not as it should be in the physical interpretation of Lorentz transformations and the allegedly real physical consequences of special relativity. A discussion on this last aspect from different standpoints can be found in [1]. We want to end this paper with two quotes from the renowned physicist Mendel Sachs that appear to be particularly pertinent here: "I believe that Einstein's identification of the Lorentz transformation with a physical cause-effect relation, and the subsequent conclusion about asymmetric ageing, was a flaw, not in the theory of relativity itself, [...], but rather a flaw in the reasoning that Einstein used in this particular study-leading him to an inconsistency with the meaning of space and time, according to his own theory. [10]" [emphasis added] "The crux of my argument was that the essence of Einstein's theory implies that the space-time transformations between relatively moving frames of reference must be interpreted strictly kinematically, rather than dynamically. Thus, according to this theory, the transformations are not more than the necessary scale changes that must be applied to the measures of space and time, when comparing the expressions of the laws of nature in relatively moving frames of reference, so as to satisfy the principle of relativity-that is, to ensure that their expressions in the different reference frames are in one-to-one correspondence. [12]" [emphasis in the original text] between the inertial observer and the accelerating system is suitably large, neglecting again terms containing the 2nd power of v/c), from equation (8) we arrive at the following relation
∆t = ∆t 1 − ±ad c 2 ,(10)
which is the sought formula for the time dilation/contraction (generalization of equation (3) in the text).
In short, we have replicated Einstein's derivation of the time dilation for a clock arbitrarily moving with respect to a stationary clock [13]. Like Einstein, we started from the Lorentz transformation of the time coordinate. However, we have plugged in the equation an explicit and simpler type of motion for the moving clock: namely, the moving clock moves away from the stationary one on a straight line at constant velocity v 1 for a time t 1 , and then, for a time (t 2 − t 1 ), it accelerates with a low acceleration a. That is simpler than Einstein's motion in a polygonal or continuously curved line [13]. Therefore, if special relativity holds for non-uniform motion in a "continuously curved line", it does hold also for a body slightly accelerating in a straight line. By the way, what we have done so far is equivalent to mapping the considered set-up onto a continuous sequence of events which are analyzed with respect to instantaneous co-moving inertial frames. Now, suppose that during interval ∆t , the light source at rest in S emits a beam of light of frequency ν . That means that N wave crests are emitted with N = ν ∆t . The same number of crests must then be received by the observer in S exactly after the traveling time d/c, no matter how big d/c is. Moreover, the observer in S will receive the N wave crests within the shorter interval of time ∆t because, for S, the whole emission process in S has taken place within ∆t (the traveling time d/c cannot affect that duration since d/c is only a delay in receiving the wave train). That means that the observer in S receives a beam of light of frequency ν such that ν∆t = N = ν ∆t , and thus
ν = ν 1 − (±a)d c 2 ,(11)
which is the generalization of equation (4) in the text. Let us remind that the above formula gives a frequency shift that has nothing to do with the standard Doppler shift depending upon relative speed nor with the gravitational redshift depending upon gravity.
Figure 1 :
1Setup described in the text.
Figure 2 :
2Muons as interstellar travellers.
To make the picture clearer from here on out, with 'special relativity effects' we actually mean effects which are the mathematical consequence of the application of the Lorentz transformations.
Lorentz transformations intended as physical laws.
Unless we want to resort to Lorentz ether theory.4 Consider, for instance, the pulsar IGR J11014-6103: the estimated speed of its jet is 0.8c.
AcknowledgmentsThe author acknowledges the anonymous reviewer for valuable comments and suggestions.A Rigorous derivation of the equations(3)and(4)We take Einstein's derivation of the time dilation formula for a clock moving in arbitrary motion (clock moving in a polygonal or continuously curved line[13]) and apply it to the case of a system moving on a straight line but subject to a uniform acceleration a for a short period of time. Hereafter, without loss of generality, we assume that all the involved velocities are such that v c. We also adopt the same assumption made by Einstein in[14](and, implicitly, in[13]), namely that acceleration a has negligible physical effects on the rate of clocks in the accelerated frame. That is known as the 'clock hypothesis'[15].We shall see that when acceleration a goes to zero, one recovers the wellknown Einstein's time dilation formula. On the other hand, if the distance one obtains the sought formulas.Consider a moving reference frame S and an inertial (stationary) reference frame S. Primed quantities refer to the system S , while non-primed ones refer to S. Moreover, S moves in the positive x-direction of S, and all the three coordinate axes are parallel. Suppose that S initially moves with constant velocity v 1 , and at time t = t = 0, the origins of S and S overlap. Thus, the relation between the instants of time t 1 of S and t 1 of S is given by the Lorentz trasnformation of the time coordinate as followsAt instant t 1 , the system S starts to accelerate in the positive or negative x-direction with constant acceleration a, and at instant t 2 returns to uniform motion with the new constant velocity v 2 = v 1 ± a(t 2 − t 1 ).Thus, the relation between the instants of time t 2 of S and t 2 of S is now given bywhereNow, it is not difficult to see that if we set a = 0 in equation(8)and do not neglect terms containing the 2nd power of v/c, we recover Einstein's time dilation formulaOn the other hand, if we set v 1 t 1 = d, with d equal to an extremely large astronomical distance, and if we consequently adopt the natural approximations, v 1 (t 2 − t 1 ) ± 1 2 a(t 2 − t 1 ) 2 v 1 t 1 and [v 1 ±a(t 2 −t 1 )] 2 c 2 ≈ v 2 1 c 2 ≈ 0 (we are now
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Introducing Time: A Graphic Guide (Icon Books Ltd. C Callender, R Edeny, C. Callender and R. Edeny, Introducing Time: A Graphic Guide (Icon Books Ltd., 2012).
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| []
|
[
"Towards Interpretable and Robust Hand Detection via Pixel-wise Prediction",
"Towards Interpretable and Robust Hand Detection via Pixel-wise Prediction"
]
| [
"Dan Liu \nUniversity of Chinese Academy of Sciences\n100049China\n",
"Libo Zhang \nInstitute of Software\nChinese Academy of Sciences\n100190China\n",
"Tiejian Luo \nUniversity of Chinese Academy of Sciences\n100049China\n",
"Lili Tao \nUniversity of the West of England\nBS16 1QYBristolU.K\n",
"Yanjun Wu \nInstitute of Software\nChinese Academy of Sciences\n100190China\n"
]
| [
"University of Chinese Academy of Sciences\n100049China",
"Institute of Software\nChinese Academy of Sciences\n100190China",
"University of Chinese Academy of Sciences\n100049China",
"University of the West of England\nBS16 1QYBristolU.K",
"Institute of Software\nChinese Academy of Sciences\n100190China"
]
| []
| The lack of interpretability of existing CNN-based hand detection methods makes it difficult to understand the rationale behind their predictions. In this paper, we propose a novel neural network model, which introduces interpretability into hand detection for the first time. The main improvements include: (1) Detect hands at pixel level to explain what pixels are the basis for its decision and improve transparency of the model. (2) The explainable Highlight Feature Fusion block highlights distinctive features among multiple layers and learns discriminative ones to gain robust performance. (3) We introduce a transparent representation, the rotation map, to learn rotation features instead of complex and non-transparent rotation and derotation layers. (4) Auxiliary supervision accelerates the training process, which saves more than 10 hours in our experiments. Experimental results on the VIVA and Oxford hand detection and tracking datasets show competitive accuracy of our method compared with state-of-the-art methods with higher speed. (Libo Zhang)1 Dan Liu and Tiejian Luo were contributed equally and should be considered as co-first authors. Models and code are available at https://isrc.iscas.ac.cn/gitlab/research/pr2020phdn. | 10.1016/j.patcog.2020.107202 | [
"https://arxiv.org/pdf/2001.04163v1.pdf"
]
| 210,164,453 | 2001.04163 | cee7cf996f5ddf9df0bc263d204a5857824b1934 |
Towards Interpretable and Robust Hand Detection via Pixel-wise Prediction
13 Jan 2020
Dan Liu
University of Chinese Academy of Sciences
100049China
Libo Zhang
Institute of Software
Chinese Academy of Sciences
100190China
Tiejian Luo
University of Chinese Academy of Sciences
100049China
Lili Tao
University of the West of England
BS16 1QYBristolU.K
Yanjun Wu
Institute of Software
Chinese Academy of Sciences
100190China
Towards Interpretable and Robust Hand Detection via Pixel-wise Prediction
13 Jan 2020Preprint submitted to Pattern Recognition January 14, 2020Interpretabilityhand detectionpixel levelexplainable * Corresponding author
The lack of interpretability of existing CNN-based hand detection methods makes it difficult to understand the rationale behind their predictions. In this paper, we propose a novel neural network model, which introduces interpretability into hand detection for the first time. The main improvements include: (1) Detect hands at pixel level to explain what pixels are the basis for its decision and improve transparency of the model. (2) The explainable Highlight Feature Fusion block highlights distinctive features among multiple layers and learns discriminative ones to gain robust performance. (3) We introduce a transparent representation, the rotation map, to learn rotation features instead of complex and non-transparent rotation and derotation layers. (4) Auxiliary supervision accelerates the training process, which saves more than 10 hours in our experiments. Experimental results on the VIVA and Oxford hand detection and tracking datasets show competitive accuracy of our method compared with state-of-the-art methods with higher speed. (Libo Zhang)1 Dan Liu and Tiejian Luo were contributed equally and should be considered as co-first authors. Models and code are available at https://isrc.iscas.ac.cn/gitlab/research/pr2020phdn.
representation, rotation map
Introduction
Deep neural networks are widely adopted in many fields of study, e.g., computer vision and natural language processing, and achieve state-of-the-art results. However, as their inner workings are not transparent, the correctness and objectivity of the predicting results cannot be guaranteed and thus limit their development in industry. In recent years, some researchers have begun to explore interpretable deep leaning methods. [1] focuses on network interpretability in medical image diagnosis. [2] decomposes output into contributions of its input features to interpret the image classification network. There is also a clear need to develop an interpretable neural network in driving monitoring as the predicting results will directly affect the safety of drivers, passengers, and pedestrians.
In this paper, we present a highly interpretable neural network to detect hands in images, which is a basic task in driving monitoring.
Hand detection in natural scenes plays an important role in virtual reality, human-computer interaction, driving monitoring [3,4]. It is a critical and primary task for higher-level tasks such as hand tracking, gesture recognition, human activity understanding. Particularly, accurately detecting hand is a vital part in monitoring driving behavior [4,5]. Detecting hands in images is a challenging task. The illumination conditions, occlusion, and color/shape similarity will bring great difficulties to hand detection. Moreover, hands are highly deformable objects, which hard to detect due to their variability and flexibility.
Hands are not always shown in an upright position in images, so the rotation angle needs to be considered to locate the hand in images more accurately.
The problem of hand detection has been studied for years. Traditional methods extract features such as skin-related features [6], hand shape and background, Histograms of Oriented Gradients (HOG) [7] to build feature vector for each sample. Then these vectors are used to train classifiers such as SVM [8]. stand, they are too limited to meet the requirements for the accuracy of hand detection in the real world. With the increasing influence of Convolutional Neural Networks (CNNs) in the field of computer vision, many CNN-based object detection methods have emerged, Region-Based Convolutional Networks(R-CNNs) [9], Single Shot MultiBox Detector (SSD) [10], for example. Inspired by these advances, many CNN-based methods have been proposed to deal with hand detection. Features are extracted automatically by designed CNNs from the original images [11,12] or the region proposals [3] and then used to locate the hands in original images. In order to extract as many effective features as possible to detect hand more accurately, the network structure is always very complicated and therefore has a heavy computational burden. This limits its value in practical applications such as monitoring driving behavior and sign language recognition. The deep CNNs are used as black-boxes in the existing methods. Different from hand-crafted features, it is difficult to know the meaning of features extracted by CNNs. As a result, the stability and robustness of these methods cannot be guaranteed.
In view of the issues mentioned above, we propose an interpretable framework, Pixel-wise Hand Detection Network (PHDN), to detect hands more effi-ciently. The proposed method achieves better performance with faster computational speed. An explainable module named Highlight Feature Fusion (HFF) block is developed to get more discriminative features. With HFF block, PHDN performs effectively and stably in different image contexts. To the best of our knowledge, this is the first time to give reasonable explanations of learned features in the hand detection procedure. Popular deep convolutional neural networks VGG16 [13] or ResNet50 [14] is adopted as a backbone network in PHDN. Fig. 1(b)) to iteratively fuse multi-scale features, which greatly reduces computational overhead and saves time compared to the serial connection (see Fig. 1(a)). As PHDN makes hand region predictions with multiscale features, it is more robust to hands of different sizes. In other words, our model is scale-invariance.
As for the rotated hand detection, adding additional rotation and derotation layers [15] makes the network more complicated and thus increases the computational burden and time overhead. We propose the rotation map and the distance map to store the rotation angle and the geometry information of the hand region respectively, which handles the rotation hands without increasing complexity of the network and learns more interpretable representations of angles by recording angles of pixels directly.
In the training process, we add supervision to each HFF block. Deep supervision to the hidden layers makes the learned features more discriminative and robust, and thus the performance of the detector is better. The auxiliary losses accelerate the convergence of training in a simple and direct way compared with [16], which accelerates training by constraining the input weight of each neuron with zero mean and unit norm.
Existing detection methods make predictions for grid cells [17] or default boxes [10], which need to seek appropriate anchor scales. Alternatively, we predict hand regions at pixel resolution to avoid the adverse effects of improper • The rotation map is designed to predict hand rotation angles precisely. It learns and represents the angles in an interpretable way with less computational cost.
• Auxiliary losses are added to provide supervision to hidden layers of the network, leading to faster convergence of the training and higher precision. to store the rotation angle instead of adding rotation and derotation layers [15] to networks.
Related Work
Hand Detection
Current hand detection methods can be divided into two categories. One is based on the hand-crafted structured features, such as color, shape and so on.
The other is based on features extracted by CNNs. The methods based on handcrafted features have strong interpretability, but the detection performance is poor due to the limitations of features. On the contrary, CNNs-based methods tend to have good performance but poor interpretability.
Human-interpretable Features Based Methods
Hand detection methods that use human crafted features usually propose hand regions using features like skin color, hand shape, Histograms of Oriented Gradients (HOG) [24]. These features have specific meanings and are easy to understand. Then the features are used to train a classifier, such as Support
Vector Machine (SVM) [8], to generate the final detection results. [25] uses the skin and hand shape features to detect hands from images. Skin areas are extracted first using a skin detector and the hands are separated out using hand contour comparison. However, it may be confusing when distinguishing between face and fist since their contours are similar. [8] generates hand region proposals using a hand shape detector, a context-based detector and a skinbased detector. Then a SVM classifier, with the score vectors built by the three detectors as input, is trained to classify the hand and non-hand regions. To enhance the robustness of hand detection in cluttered background, [26] proposes three new features based on HOG, Local Binary Patterns (LBP) and Local
Trinary Patterns (LTP) descriptors to train classifiers, but it does not perform well if the image is low resolution and it cannot handle well with occlusion. [7] trains a SVM classifier with the HOG features, and extends it with a Dynamic
Bayesian Network for better performance. Due to the limitation of hand-crafted features, these methods are not robust to the change of illumination, background and hand shape. Moreover, the non-end-to-end optimization process is timeconsuming and the performance is often suboptimal.
Non-transparent CNNs Based Methods
Inspired by the progress of Convolutional Neural Networks (CNNs), many hand detection methods proposed recently are based on CNNs. [3] presents a lightweight hand proposal generation approach, of which a CNN-based method is used to disambiguate hands in complex egocentric interactions. Context information, such as hand shapes and locations, can be seen as prior knowledge, and they can be used to train a hand detector [27]. However, it is no doubt that additional context cues over-complicates the image preprocessing step. Inspired by these, [11] first generates hand region proposals with the Fully Convolutional Network (FCN) [28] and then fuses multi-scale features extracted from FCN into a large feature map to make final predictions, as a result of which the convolution operations are time-consuming in the later steps. Similarly, [12] concatenates the multi-scale feature maps from the last three pooling layers into a large feature map. Although different receptive fields are taken into consideration, simple concatenation of feature maps results in high computational cost.
In contrast to human-crafted features, the features extracted by CNNs are not interpretable and thus the rationality and validity of the model are difficult to verify. In order to provide interpretability to CNN-based hand detection models, we detect hands at pixel level. For any pixel in the image, we predict whether it belongs to a hand and the bounding box of the hand. In this way, we can know the basis for the model to make predictions. Under the fact that the high-level feature maps reflect the global features while the low-level feature maps contain more local information, the feature maps from different scales are weighted before merged so that the features from multiple scales can complement each other in the subsequent process. In view of the heavy computational burden caused by the fusion of multi-scale information, our model fuses multi-scale features iteratively rather than simultaneously.
Another issue of hand detection is to handle the rotation. Hands are rarely shown in upright positions in images. To accurately detect hands and estimate their poses, [15] designs a rotation network to predict the rotation angle of region proposals and a derotation layer to obtain axis-aligned rotating feature maps (see Fig. 2). However, the method is of great complexity as it includes two components for rotation, a shared network for learning features and a detection network for the classification task. It is also hard to find out what the rotation and derotation layers really learn. To handle rotated hand samples more effectively, we develop the rotation map to replace the complex rotation and derotation layers, as shown in Fig. 2. It is also more interpretable as each pixel value represents the rotation angle directly. The results on the Oxford hand detection dataset show that the rotation map brings a significant increase (about 0.30) in AP compared to using only the distance maps.
Multiple Hand Tracking in Vehicles
Tracking hands in the vehicle cabin is important for monitoring driving behavior and research in intelligent vehicles. Although hand tracking has been studied since the last century, there are few studies on tracking multiple hands simultaneously in naturalistic driving conditions. To the best of our knowledge, only [5] has given the research results on multiple hand tracking so far. [5] proposes a tracking-by-detection method, where each video frame is processed by the detector first and then integrates with a tracker to provide individual tracks online. The ACF detector [29] is used to generate hand detection results and the data association is performed using a bipartite matching algorithm. It reports the tracking results on the VIVA hand tracking dataset. To investigate the performance of our model in hand tracking, we apply PHDN to SORT tracker [20], deep SORT tracker [21], IOU tracker [22]. SORT
Interpretable Pixel-wise Hand Detection Network
The PHDN architecture is illustrated in Fig. 3 Input Image Fused feature maps, f s , s ∈ {0, 1, 2, 3};
Restore NMS f0 f1 f2 f3 M(f0, f1 ׳ ) f2 ׳ M(f1, f2 ׳ ) M(f 2 , f 3 ׳ ) 1×1, 1 1×1, 1 1×1, 4 f3 ׳ f1 ׳ f0 ׳ L3 L2 L1 L01: f 3 = f 3 ; 2: for s from 2 to 0 do 3: u s+1 = U psampling(f s+1 ); 4: masked = f s * (1 − Convolution(u s+1 , 1 × 1)); 5: Concate = Concatenate(masked, u s+1 ); 6: Conv1 = Convolution(Concate, 1 × 1, c s ); 7: Conv2 = Convolution(Concate, 3 × 3, c s ); 8: f s = Conv2 9: end for 10: return f s , s ∈ {0, 1, 2, 3};
Visually Interpretable and Robust Feature Fusion
The size of hands varies greatly in different images or even the same image.
The larger hand detection needs more global information. It is known that the higher the level of feature maps, the more global the information is presented.
Hence multi-scale feature maps should be merged to detect different sizes of hands. We propose to fuse the feature maps from multiple layers in an iterative way to reduce the computational cost, which can be achieved by cascaded feature fusion blocks as shown in Fig. 1(b) To reduce the interference of useless features and learn more discriminative features, we develop the Highlight Feature Fusion (HFF) block to fuse the features from different scales. Fig. 3 displays three cascaded HFF blocks, which are marked with red dotted rectangles.
The cascaded HFF blocks operate the fusion as Algorithm 1.
We generate a mask with the higher-level feature maps to filter the common features in the current level feature maps, which formulated as Line 4 above We visualize features extracted by HFF block and BFF block to interpret the robustness and effectiveness of HFF block in Section 4.5.1.
0 X Y X' Y' θ p p' dl -db y' x' x y p0 p1 p2 (p3')p3 0 x -x' y -y' p0' p1' p2' -(dt + db) (dl + dr) 0 X Y X' Y' ld td bd rd dl dt db dr
Pixel-wise hand detection
For each pixel in the image, we generate the confidence that it belongs to a hand region and the corresponding hand bounding box. In this way, the model can interpret what features the prediction is based on. The following paragraphs elaborate on this process.
After the last HFF block, the feature maps go through a 3 × 3 convolution and then be upsampled to the same size as the input image. Finally, 1 × 1, 1 × 1 and 3 × 3 convolutions are employed to generate the score map, rotation map and distance map respectively. The three kinds of map are the same size as the original images, and their pixels correspond one by one. Similar to the confi-dence map used in Fully Convolutional Networks (FCN) [28], each pixel value in the score map, a scalar between 0 and 1, represents the confidence that the corresponding pixel in the input image belongs to a hand region. The rotation map is developed for the rotated hand detection issue. It records the rotation angle of the hand bounding box and the range of the angle is (−π/2, π/2). Inspired by the work of [31], we use the distance map to store the geometry information of the hand box. The distance map has four channels, recording distances to the boundaries of the corresponding hand bounding box, denoted as d t , d r , d b , d l in Fig. 4.
Hand boxes are generated with the rotation map and distance map for pixels whose scores are higher than a given threshold in the score map. An example is given in Fig. 4 to illustrate the restoring process for pixel p. Based on the distance map we can obtain the distances d t , d r , d b , d l from p to the four boundaries (top, right, bottom, left) of the rectangle R p . In order to calculate the coordinates of p 0 , p 1 , p 2 , p 3 in image coordinate system (drawn in black in Fig. 4), an auxiliary coordinate system (drawn in red in Fig. 4) is introduced with p 3 as the origin. The directions of X-axis and Y-axis are the same as the image coordinate system. We rotate R p to the horizontal around p 3 . The corresponding position of p in the rotated rectangle R p is denoted as p . Let (x , y ), (x i , y i ), i ∈ {0, 1, 2} be the coordinates of p, p i , i ∈ {0, 1, 2} in the auxiliary coordinate system. For the clockwise rotation of rectangle R p , we
have M (θ) x y = d l −d b , M (θ) x 0 y 0 = 0 −(d t + d b ) , M (θ) x 1 y 1 = d l + d r −(d t + d b ) , M (θ) x 2 y 2 = d l + d r 0 ,(1)
where M (θ) is the rotation matrix in two-dimensional space, which can be formulated as
M (θ) = cos θ − sin θ sin θ cos θ .(2)
θ is the rotation angle with counter-clockwise as the positive direction, and it can be restored from the rotation map in our experiments.
Finally, the coordinates (x i , y i ), i ∈ {0, 1, 2, 3} of p i in the image coordinate system are calculated by
x 3 y 3 = x y − x y , x i y i = x i y i + x 3 y 3 , i ∈ {0, 1, 2}.(3)
Auxiliary Supervision
The detection loss function usually includes the confidence loss and the location loss. Specific to our method, the confidence loss is calculated with the score map, and the location loss consists the rotation loss and the geometry loss, related to the rotation map and distance map respectively. To learn a more discriminative mask in the HFF, deep supervision is added to the intermediate HFF blocks with auxiliary losses (L s , s = 1, 2, 3 in Fig. 3) besides the L 0 for the output. The overall objective loss function is formulated as
L = s∈S w s L s ,(4)
where S = {0, 1, 2, 3} represents the scale index of the HFF blocks as shown in Fig. 3 and the parameter w s adjusts the weight of the corresponding scale.
For scale s, the loss L s is a weighted sum of the losses for the score map L
L s = αL [s] sco + βL [s] rot + L [s] dis .(5)
The factors α and β control the weights of the three loss terms. We describe these three parts of the loss in detail below.
Loss Function of Score Map
Regarding the score map as a segmentation of the input image, we use the Dice Similarity Coefficient [32] (DSC) to construct the loss for score map. DSC measures the similarity between two contour regions. Let P, G be the point sets of two contour regions respectively, then the DSC is defined as
DSC(P, G) = 2|P G| |P | + |G| .(6)
|P | (. |G|) represents the number of elements in set P (G). As the ground truth of the score map is a binary mask, the dice coefficient can be written as
DSC(P, G) = 2 N i=1 p i g i N i=1 p 2 i + N i g 2 i ,(7)
where the sums run over all N pixels of the score map. p i is the the pixel in the score map P generated by the detection network, and g i is the pixel in the ground truth map G. Based on the dice similarity coefficient, the dice loss is proposed and proved to perform well in segmentation tasks [33,32,34].
Motivated by this strategy, the loss for the score map is formulated as
L sco = 1 − 2 N i=1 p i g i + ε 0 N i=1 p 2 i + N i=1 g 2 i + ε 0 ,(8)
where ε 0 is the smooth.
Loss Function of Rotation Map
The rotation map stores the predicted rotation angles for corresponding pixels in the input image. The cosine function is adopted to evaluate the distance between the predicted angleθ i and the ground truth θ i . Consequently, we can calculate the loss of rotation map by
L rot = 1 − 1 N N i=1 cos θ i − θ i .(9)
Loss Function of Distance Map
As for the regression of the object bounding box, the l 2 loss
L dis = − 1 N N i=1 ln I [i] + ε 1 U [i] + ε 1 ,I [i] = I [i] h * I [i] w , I [i] h = min(d t ,d t ) + min(d b ,d b )
,
I [i] w = min(d l ,d l ) + min(d r ,d r ), U [i] = X [i] +X [i] − I [i] , X [i] = (d t + d b ) * (d l + d r ), X [i] = (d t +d b ) * (d l +d r ),(10)
where N is the number of pixels in the distance map and ε 1 is the smooth term.
Experiments
We evaluate our detector on three benchmark datasets: the VIVA hand detection dataset [18], the Oxford hand detection dataset [8] and the VIVA hand tracking dataset [19].
Experimental Settings
All experiments are conducted on an Intel(R) Core(TM) i7-6700K @ 4.00GHz
CPU with a single GeForce GTX 1080 GPU. We try two backbone networks:
VGG16 [13] and ResNet50 [14] for feature extraction and use the pre-trained models on ImageNet [30]. We employ the network with the Base Feature Fusion In order to reduce the over-fitting risk and improve the generalization performance of the model, a variety of data enhancement strategies are employed. We randomly mirror and crop the images, as well as distort the hue, saturation and brightness for color jittering. Due to the limitation of the GPU capacity, the batch size is set as 12 and all the images are resized to 512 × 512 before fed into the network in training. When predicting on the test dataset, the original size of the input image is preserved as the network is a fully convolutional network that allows arbitrary sizes of input images. As presented in Table. 1, we compare our methods with MS-RFCN [11,37],
Evaluations on VIVA Hand Detection Dataset
Multi-scale fast RCNN [12], FRCNN [27], YOLO [17] and ACF Depth4 [18]. Apart from the accuracy, the detection speed is also an important metric.
As we can see in Table. 1, YOLO [17] performs hand detection in real-time, but its accuracy is unsatisfactory. On the contrary, MS-RFCN [11] performs against other detectors in accuracy but the detecting speed is very slow, i.e., 4.65 fps.
With our PHDN based on VGG16 and ResNet50, the detection speeds are up Losses) obtains competitive accuracy while a 4.23 times faster running speed compared to [11]. Therefore, it is of great significance that our model achieves a good trade-off between accuracy and speed.
Evaluations on Oxford Hand Detection Dataset
Oxford Hand Detection Dataset consists of three parts: the training set, the validation set and the testing set, with 1, 844, 406 and 436 images separately.
Unlike the VIVA dataset, the images in Oxford dataset are collected from various different scenes. Moreover, the ground truth is given by the four vertexes (x i , y i ), i ∈ {1, 2, 3, 4} of the box in the format of .mat and not necessarily to be axis-aligned but oriented with respect to the wrist. The rotation angle will be calculated furthermore in our experiments.
According to the official evaluation tool 3 on the Oxford dataset, we report the performance on all the "bigger" hand instances, those with more than 1, 500 pixels. As shown in To evaluate our detector, we employ the SORT tracker [20], deep SORT tracker [21] and IOU tracker [22] to associate our detection results to extend a trajectory on the VIVA hand tracking dataset. The results are reported in
Ablation Study
Ablation experiments are conducted to study the effect of different aspects of our model on the detection performance. We choose the ResNet50 as a default backbone network and Oxford hand detection dataset to do further analysis of our model.
Interpretable and Robust HFF Block
Some visual explanations for the effectiveness and robustness of HFF block are given in Fig. 8. The activation feature map is converted into a blue-yellowred color scale and then added to the original input image to see which pixels are activated in the detection procedure. We can see that the HFF block is good at locating discriminative pixels comparing with the BFF block. The HFF block keeps off confusing parts like faces and feet. It can also activate the hand pixels accurately even in clutter background as shown in the second example in Fig. 8
Influence of the Score Map and Rotation Map
We adjust the value of α in Eq. (4) to find appropriate weights of score map in training. The results are reported in Fig. 7(a). As α increases from 0.01 to 1, the AP increases first and then decreases. It reaches the maximum 0.7966 when α takes 0.10 in our experiments. As we can see, if weight the classification loss highly, the AP score will decline (0.7966 vs. 0.7738). In other words, over consideration of score map brings declines in AP score , which is consistent with the fact that the detection is not a simple classification task, but also involves bounding box regression.
The rotation map is designed to predict the rotation angle of the box and
Effectiveness of Auxiliary Supervision
In order to investigate the effectiveness of the auxiliary losses, we train models considering different numbers of scales. The variation of training time and AP score with the number of supervision scales is shown in Fig. 9 creases. The convergence of the network is accelerated significantly (more than 10 hours) by adding auxiliary losses into the total loss. At the same time, the AP score is stable regardless of the number of scales. It can be concluded that the auxiliary losses accelerate the training process without sacrificing the AP score. This is attributed to the multiple supervision to the intermediate layers
Visualization Results
We show several qualitative detection examples in Fig. 10. As these results show, our model can handle different scales of hands and shapes in various illumination conditions, even the blurred samples. Fig. 11 compares our detection results with Multi-scale fast RCNN and shows the tracking results and the corresponding ground truth on the VIVA hand tracking dataset. We can see that our model achieves fewer false positives and produces more accurate hand locations compared with the visualization results given in [12]. Besides, the model trained with rotated hand labels on the Oxford dataset is capable to predict hand rotation angle precisely. Further, applied into the hand tracking task, our model generates satisfactory trajectories as we can see in Fig tection, e.g., the hand obscured by the toy is not recognized in Fig. 12(b). Our model does not perform well in these situations possibly because the context information, such as surroundings and similar hand color or shape objects, is not thoroughly mined and integrated effectively. We will investigate the effect of context information in future work and try to address these issues.
Conclusion
Existing hand detection neural networks are "black box" models and people cannot understand how they make automated predictions. This hinders their application in areas such as driving monitoring. In this paper, we present the The rotation map is interpretable because it directly records the rotation angles of pixels as features. It makes the model more transparent. In addition, deep supervision is added with auxiliary losses to accelerate the training procedure.
Compared with the state-of-the-art methods, our algorithm shows competitive accuracy and runs a 4.23 times faster speed on the VIVA hand detection dataset and achieves an improvement of 5.5% in average precision at a speed of 62.5 fps on Oxford hand detection dataset. Our detector is practical, for which it can track hands better in naturalistic driving conditions compared with other methods on VIVA hand tracking dataset. For future work, we will enhance the transparency and robustness of our model and apply our detector to real-world scenarios such as driving monitoring and virtual reality.
Figure 1 :
1Although the hand-crafted features have clear meanings and are easy to under-Different connection modes of multi-scale features. (a) Serial mode. (b) Cascade mode.
Figure 2 :
2Novel and transparent representation of the rotation angle. We use the rotation map
Figure 4 :
4Restore hand bounding boxes from the rotation map and distance map. and * denotes element-wise multiplication. Masking f s with the complementary feature maps of u s+1 can highlight the fine-grained distinctive information contained in f s that u s+1 may not have. Conv1 is the result of conducting a 1 × 1 convolution on the concatenated feature maps. It is designed to reduce the output channels and thus lessen the computational burden. Then a 3 × 3 convolution is operated to further fuse the features of multiple scales. To investigate the effect of the mask, we remove the mask operation and concatenate f s and u s+1 directly as a Base Feature Fusion (BFF) block in our experiments.
(x, y) are the coordinates of p in the image coordinate system. According toEq. (1)∼(3), the hand bounding box R p = {(x i , y i )|i ∈ {0, 1, 2, 3}} correspond-ing pixel p can be restored with the rotation map and distance map.Many redundant detection bounding boxes are produced by the network. To generate pure detection results, we use the NMS to filter the boxes with low scores and high overlapping rates.
I
[i] and U [i] denote the intersection and union of the predicted box {d t ,d r ,d b ,d l } and the ground truth {d t , d r , d b , d l } respectively.
(
BFF) block as our base model and conduct ablation experiments to evaluate the performance of the Highlight Feature Fusion (HFF) block and the auxiliary losses.Training is implemented with a stochastic gradient algorithm using the ADAM scheme. We take the exponential decay learning rate, the initial value of which is 0.0001 and decays every 10, 000 iterations with rate 0.94. The weight parameters w s , s ∈ {1, 2, 3, 4} are all set to 1 for default. The hyper-parameters α, β are set to 0.01 and 20, respectively. Besides, the score map threshold is set to 0.8. In other words, all the pixels that obtain scores higher than 0.8 are considered in the bounding box restoration. Then the bounding boxes are filtered by the NMS with a threshold 0.2.
VIVA
Hand Detection Dataset is published by the Vision for Intelligent Vehicles and Applications Challenge [18] for hand detection subtask. The dataset includes 5, 500 training and 5, 500 testing images. The images are collected from 54 videos captured in naturalistic driving scenarios. There are 7 possible viewpoints in the videos. Annotations for the images are publicly accessible. The
format annotation file. x, y are the upper-left coordinates of the box and w, h are the width and height of the box, respectively. As the given annotations are axis-aligned, the rotation angles are set to 0 in training and the predictions are axis-aligned bounding boxes in our experiments on this dataset. We evaluate the algorithms on two levels according to the size of the hand instances using the evaluate kit provided by the Vision for Intelligent Vehicles and Applications Challenge. Level-1 focuses on the hand instances with a minimum height of 70 pixels, only over the shoulder (back) camera view, while Level-2 evaluates hand samples with a minimum height of 25 pixels in all camera views. Evaluation metrics include the Average Precision (AP) and Average Recall (AR). AP is the area under the Precision-Recall curve and AR is calculated over 9 evenly sampled points in log space between 10 −2 and 10 0 false positives per image. As performed in PASCAL VOC [38], the hit/miss threshold of the overlap between a pair of predicted and ground truth bounding boxes is set to 0.5.
Figure 5 :
5Precision-Recall curves and ROC curves (logarithmic scale for x-axis) on VIVA dataset. The Precision-Recall curves and ROC curves of these methods and our model (ResNet50+HFF+Auxiliary Losses) are shown in Fig. 5. Our model achieves 92.3%/89.1% (AP/AR) at Level-1 while 83.6%/68.8% (AP/AR) at Level-2 using VGG16 as the backbone network. The ResNet50 based PHDN network obtains more accurate performance, i.e., 94.8%/91.1% (AP/AR) at Level-1 and 86.3%/75.8% (AP/AR) at Level-2.
Figure 6 :
6Precision-Recall curve and ROC curve on oxford dataset. to 13.10 and 19.68 fps, respectively. The model (ResNet50+HFF+Auxiliary
•
MOTA (The Multiple Object Tracking Accuracy): A comprehensive metric combining the false negatives, false positives and mismatch rate. • MOTP (The Multiple Object Tracking Precision): Overlap between the estimated positions and the ground truth averaged by all the matches. • Recall: Ratio of correctly matched detections to ground truth detections. • Precision: Ratio of correctly matched detections to total result detections. • MT (Most Tracking): Percentage of ground truth trajectories which are covered by the tracker output for more than 80% of their length. • ML (Most Lost): Percentage of ground truth trajectories which are covered by the tracker output for less than 20% of their length. • IDS (ID Switches): Number of times that a tracked trajectory changes its matched ground truth identity. • FRAG (Fragments): Number of times that a ground truth trajectory is interrupted in the tracking result. For MOTA, MOTP, Recall, Precision and MT, greater values mean better performance, whereas the ML, IDS and FRAG are the smaller the better.
Figure 7 :
7The change of AP with α and β on the Oxford dataset.
a) PHDN with ResNet50 and Base Feature Fusion (BFF) block (b) PHDN with ResNet50 and Highlight Feature Fusion (HFF) block (a) PHDN with ResNet50 and Base Feature Fusion (BFF) block (b) PHDN with ResNet50 and Highlight Feature Fusion (HFF) block
Figure 8 :
8Visual explanations for predictions. The heatmap in the blue-yellow-red color scale is added to the original image to show the activated regions.
Figure 9 :
9Training time and AP score vs. different numbers of scales on the Oxford dataset. further locate the hand more accurately. To investigate the role it plays in the detection, we control the weights of rotation map in the training process by changing β in Eq. (4). We first set β to 0, i.e., ignore the rotation map in training, to obtain detection results. Then we try four different values (1, 5, 10 and 20) for β to train models and evaluate all the detection results on the Oxford test set. The AP score and corresponding β are plotted in Fig. 7(b) When considering the rotation angle in the optimization procedure, i.e., β > 0, the AP score is stable and larger than 0.78 for all the values of β tried in our experiments. Otherwise, there is a significant drop in the AP score (0.8061 vs. 0.4991) on Oxford dataset when β is set as 0. Therefore, the rotation map plays a very important role in optimizing the final model and can improve the locating accuracy greatly.
Figure 11 :
11Detection results comparisons. (a) and (b) compare the performance between our PHDN based on ResNet50 model (cyan bounding boxes) and Multi-scale fast RCNN [12] (red bounding boxes). (c) and (d) show the ground truth and our tracking results on the VIVA hand tracking dataset.
Figure 12 :
12Incorrectly detection examples using PHDN model with ResNet50 as backbone.
. 11 .
11Fig. 12 shows some false detected samples. The false detections can be divided into three types: (1) When the color or shape of the hand is very close to the background, it may mislead the model to make false predictions or result in missed detection. (2) The faces and feet with confusing colors and shapes are incorrectly detected as hand regions by the model. (3) Heavy occlusions cause missed de-
interpretable Pixel-wise Hand Detection Network (PHDN). To the best of our knowledge, this is the first study towards interpretable hand detection. The pixel-wise prediction shows the basis of detection and provides the model interpretability. Features from multiple layers are fused iteratively with cascaded Highlight Feature Fusion (HFF) blocks. This allows our model to learn better representations while reducing computation overhead. The proposed HFF block outperforms the Base Feature Fusion (BFF) block and improves the detection performance significantly. To gain insight into the reasonability of the HFF block, we visualize regions activated by the HFF block and BFF block respectively. The visualization results demonstrate that the HFF block highlights the distinctive features of different scales and learns more discriminative ones to achieve better performance. Complex and non-transparent rotation and derotation layers are replaced by the rotation map to handle the rotated hand samples.
The HFF block makes full use of multi-scale features by weighting the lowerlevel features with the higher-level features. In this way, the discriminative features, namely the effective ones for locating the hand, are highlighted in the detection procedure. Each HFF block fuses features from two layers. It first weights the lower-level features by the last higher-level feature maps and then fuses the features by convolution operations. Several HFF blocks are connected in cascade mode (see
anchor scales settings, for which we name our model as Pixel-wise Hand Detection Network. Detecting hands at pixel level also explains what pixels are the basis for its decision, which improves transparency of the model. The hand re-Recall (AR) on VIVA dataset with 4.23 times faster detecting speed, and obtains 5.5% AP improvement on Oxford dataset. Furthermore, we test the PHDN with the hand tracking task on VIVA hand tracking dataset[19], which is a higher application scenario of hand detection. We try three tracking-by-detection methods: SORT tracker[20], deep SORT tracker[21] and IOU tracker[22], where the PHDN acts as a detector. Experimental results show that using any of the aforementioned tracking algorithms based on our detector can achieve better results than existing methods. It indicates that PHDN is robust and practicable as the detector performance plays a crucial role in tracking-by-detection multiple object tracking methods.Part of the work has been introduced in[23]. The extensions made in this more detailed description of our model including related work in hand detection and multiple hand tracking in vehicles, network architecture, feature fusion processing, loss functions and the settings and results of conducted experiments.The main contributions of this paper are in four folds:gions predicted by PHDN are filtered by the Non-Maximum Suppression (NMS)
to yield the final detection results.
To evaluate our model, experiments are conducted on two authentic and pub-
licly accessible hand detection datasets, the VIVA hand detection dataset [18]
and the Oxford hand detection dataset [8]. Compared with the state-of-the-art
methods, our model achieves competitive Average Precision (AP) and Average
article compared to [23] are as follows: (1) We analyze the interpretability
of our model by visualizing the features extracted by HFF block to interpret
our model. It shows the mechanism of internal layers and demonstrates how
our method outperforms the others. (2) We integrate our detector with the
popular trackers to track hands in videos and achieve state-of-the-art results
on the authoritative VIVA hand tracking challenge dataset [19]. (3) We give a
• We give insight to the interpretability of the hand detection network for
the first time. Reasonable explanations for the feature activated in hand
detection procedure and the discriminative features learned by HFF block
are first given. The proposed Pixel-wise Hand Detection Network predicts
hand regions at pixel resolution rather than grid cells or default boxes. It
gets rid of the adverse effects of inappropriate anchor scales and can detect
different sizes of hands by fusing multi-scale features with the cascaded
HFF blocks.
in SORT tracker. The reported results show the deep SORT tracker has fewer identity switches than the SORT tracker. IOU tracker is an offline tracking method that can generate trajectories with all observations in the video. It associates the detection with the highest IOU to the last detection in previous frames to extend a trajectory. It can run at 100K fps as its complexity is very low. The tracking performance depends largely on the detector. Therefore, we conduct experiments on the VIVA hand tracking dataset with our detector and we use three trackers to evaluate our model in the practical tracking task.tracker and
deep SORT tracker are online tracking methods, where only the current and
previous frames are visible to the tracker. SORT tracker performs Kalman
filtering in image space and uses the Hungarian method to associate detections
across frames in a video sequence. Deep SORT tracker is developed for the many
identity switches in SORT tracker. It adopts a novel association metric with
more motion and appearance information compared to the IOU distance used
. To show our model more clearly, only the VGG16 backbone is presented in the figure for its simpler structure compared with ResNet50. The feature maps from four different scales extracted by the VGG16 extractor or ResNet extractor are fused iteratively in the cascaded HFF blocks. The final feature maps, containing multi-scale information, are upsampled and convoluted to get the score map, the rotation map and the distance map. With the three kinds of maps, we can restore the hand bounding boxes and filter them by the NMS to generate the final hand regions. In the following, we describe the pipeline in detail and construct the loss function for the training.512×512×3
Max Pooling, /2
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512×512×64
3×3, 64
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×2
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×3
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Max Pooling, /2
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Max Pooling, /2
Upsampling, ×2
Mask
Upsampling, ×2
Concatenate
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Upsampling, ×2
Concatenate, 512
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Mask
Concatenate, 256
1×1, 64
3×3, 64
3×3, 32
Upsampling, ×4
Score Map
Rotation Map
Distance Map
Figure 3: PHDN architecture with VGG16 as the backbone. The left is feature extracting stem, and the right is feature fusion branch and the output layers. Highlight Feature Fusion (HFF) block is marked with red dotted rectangle.3.1. Feature ExtractionWe try two popular deep convolutional networks, i.e., VGG16 and ResNet50, to extract features from the images. The pre-trained model on the ImageNet dataset[30] is used in our study. Feature maps from four layers are selected for the feature fusion module. For VGG16, we adopt the feature maps from pooling-2 to pooling-5. Similarly, the outputs of conv2 1, conv3 1, conv4 1 and conv5 1 are extracted in ResNet50. The feature maps extracted from VGG16 the size of input images, and represent information of different sizes of receptive fields.Algorithm 1 Feature Fusion Procedure Input: Feature maps extracted by VGG16 or Resnet50, f s , s ∈ {0, 1, 2, 3}; Channels of fused feature maps, c s , s ∈ {0, 1, 2, 3};or ResNet50 are ( 1
4 ) 2 , ( 1
8 ) 2 , ( 1
16 ) 2 , ( 1
32 ) 2 Output:
[35] performs the four distances d t , d r , d b , d l as independent variables, which may mislead the training when only one or two bounds of the predicted box are close to the ground truth. To avoid this,[36] proposes the IoU loss which treats the four distances as a whole. Besides, the IoU loss can handle bounding boxes with various scales as it uses the IoU to norm the four distances to [0, 1]. In other words, the IoU loss is scale-invariant, which is important to detect hands of different sizes. The IoU loss for the distance map is calculated as
Table 1 :
1Results on VIVA Hand Detection Dataset bounding boxes of hand regions in an image are given by (x, y, w, h) in the .txtMethods
Level-1
(AP/AR)/%
Level-2
(AP/AR)/%
Speed/fps Environment
MS-RFCN [11]
95.1/94.5
86.0/83.4
4.65
6 [email protected], 32GB RAM, Titan X GPU
MS-RFCN [37]
94.2/91.1
86.9/77.3
4.65
Multi-scale fast RCNN [12]
92.8/82.8
84.7/66.5
3.33
6 [email protected], 64GB RAM, Titan X GPU
FRCNN [27]
90.7/55.9
86.5/53.3
-
-
YOLO [17]
76.4/46.0
69.5/39.1
35.00
6 [email protected], 16GB RAM, Titan X GPU
ACF Depth4 [18]
70.1/53.8
60.1/40.4
-
-
Ours (VGG16+BFF)
88.9/82.8
72.6/56.7
13.88
4 [email protected], 32GB RAM, GeForce GTX 1080
Ours (VGG16+BFF+Auxiliary Losses)
92.9/88.3
80.9/62.7
13.16
Ours (VGG16+HFF+Auxiliary Losses)
92.3/89.1
83.6/68.8
13.10
Ours (ResNet50+BFF)
93.7/89.9
83.6/73.6
20.40
Ours (ResNet50+BFF+Auxiliary Losses) 94.0/90.1
85.7/74.0
20.00
Ours (ResNet50+HFF+Auxiliary Losses) 94.8/91.1
86.3/75.8
19.68
Table 2 :
2Results on Oxford Hand Detection DatasetMethods
AP/%
MS-RFCN [11]
75.1
Multiple proposals [8]
48.2
Multi-scale fast CNN [12]
58.4
Ours (VGG16+BFF)
68.7
Ours (VGG16+BFF+Auxiliary Losses)
77.8
Ours (VGG16+HFF+Auxiliary Losses)
78.0
Ours (ResNet50+BFF)
78.2
Ours (ResNet50+BFF+Auxiliary Losses) 78.6
Ours (ResNet50+HFF+Auxiliary Losses) 80.6
Table 3 :
3Results on VIVA Hand Tracking DatasetMethods
MOTA/% MOTP/% Recall/% Precision/% MT/% ML/% IDS FRAG
Online
TDC(CNN) [5]
25.1
64.6
-
-
39.1
18.8
34
415
TDC(HOG) [5]
24.6
64.5
-
-
35.9
17.2
39
426
Ours+SORT
83.4
78.4
90.4
92.8
87.5
3.13
2
88
Ours+Deep SORT 85.2
77.6
90.1
94.9
84.4
1.56
1
106
Offline
TBD [39]
6.75
65.96
-
-
50
12.5
29
320
Ours+IOU
83.6
77.1
90.0
93.3
84.4
3.13
5
159
Table . 2
., similar to the results on VIVA dataset, ResNet50performs better than VGG16 as a backbone network. Specifically, ResNet50
based PHDN achieves an improvement of 5.5% in AP score compared with the
state-of-the-art MS-RFCN [11]. VGG16 based PHDN still outperforms MS-
RFCN [11] by 2.9% in AP score. The Precision-Recall curve and ROC curve
are presented in Fig. 6. In addition, it is worth mentioning that the detecting
speed on the Oxford dataset is up to 62.5 fps using ResNet50 while 52.6 fps
using VGG16.
4.4. Evaluations on VIVA Hand Tracking Dataset
VIVA hand tracking dataset is built by the Vision for Intelligent Vehicles
and Applications Challenge for hand tracking sub contest. There are 27 training
and 29 test sequences captured under naturalistic driving conditions in this
dataset and 2D bounding box annotations of hands are provided with {frame,
id, bb left, bb top, bb width, bb height}. Evaluation metrics [5] follow standard
multiple object tracking and are listed as follows.
Table .
.3. The model (ResNet50+HFF+Auxiliary Losses) is used to generate
detection results. Note that, we present the Recall and Precision of our method
as they are metrics concerned with the detection performance in multiple object
tracking. Our model (ResNet50+HFF+Auxiliary Losses) performs much better
than the existing methods on this dataset. It indicates that our detector is
practicable and well-performed in hand tracking task.
(b). HFF block uses the mask to filter the redundant features of the corresponding layer while the BFF does not.FromTable.1 and 2, we can see that the HFF block outperforms the BFF block whether using the VGG16 or ResNet50 as the backbone. Specifically, with VGG16 as the backbone and evaluated at Level-2, HFF block achieves an improvement of 2.7% in AP and 6.1% in AR on VIVA hand detection dataset.With ResNet50, there are 0.6% in AP and 1.8% in AR respectively. The AR score is improved greatly, which indicates that the model with the HFF block produces less false negatives than the BFF block and makes better use of the distinctive features of different scales. The HFF block also show better performance on the Oxford dataset: It gains an improvement of 0.2% in AP score with VGG16 and 2.0% with ResNet50 comparing to the BFF block.
. The number of scales 1, 2, 3, 4 correspond to S = {0}, S = {0, 1}, S = {0, 1, 2}, S = {0, 1, 2, 3} in Eq. (4) respectively. From Fig. 9, we can see that the time it takes for the model to convergence decreases as the number of scales used in loss function in-Figure 10: Detection results visualization. Annotations of VIVA hand detection dataset and VIVA hand tracking dataset are horizontal bounding boxes. Images in Oxford hand detection dataset are labeled with wrist-oriented boxes.(a) Examples from VIVA
hand detection dataset
(a) Examples from VIVA
hand detection dataset
(b) Examples from Oxford
hand detection dataset
(c) Human annotations for VIVA
hand tracking dataset
(d) Tracking results by our
detector with SORT tracker on
VIVA hand tracking dataset
(b) Examples from Oxford
hand detection dataset
(c) Examples from VIVA
hand tracking dataset
(a)
(b)
(a) Examples from VIVA
hand detection dataset
(b) Examples from Oxford
hand detection dataset
(c) Manual annotations for
VIVA hand tracking dataset
(d) Tracking results by our
detector with SORT tracker on
VIVA hand tracking dataset
http://www.robots.ox.ac.uk/~vgg/data/hands/index.html
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| []
|
[
"On the Partition Dimension and the Twin Number of a Graph",
"On the Partition Dimension and the Twin Number of a Graph"
]
| [
"C Hernando ",
"M Mora ",
"I M Pelayo "
]
| []
| []
| A partition Π of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of Π. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u, v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G.In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n ≥ 9 having partition dimension n − 2. This set is formed by exactly 15 graphs, instead of 23, as was wrongly stated in the paper "Discrepancies between metric dimension and partition dimension of a connected graph" (Disc. Math. 308 (2008) 5026-5031). | null | [
"https://arxiv.org/pdf/1602.08907v3.pdf"
]
| 51,486,495 | 1602.08907 | bd104b4b293cef2f07f99b8b3ff5d830035ca497 |
On the Partition Dimension and the Twin Number of a Graph
January 1, 2018
C Hernando
M Mora
I M Pelayo
On the Partition Dimension and the Twin Number of a Graph
January 1, 2018locating setlocating partitionmetric dimensionpartition dimensiontwin number
A partition Π of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of Π. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u, v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G.In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n ≥ 9 having partition dimension n − 2. This set is formed by exactly 15 graphs, instead of 23, as was wrongly stated in the paper "Discrepancies between metric dimension and partition dimension of a connected graph" (Disc. Math. 308 (2008) 5026-5031).
Introduction
All the graphs considered are undirected, simple, finite and connected. The vertex set and edge set of a graph G are denoted by V (G) and E(G). Let v be a vertex of G. The open neighborhood of v is N G (v) = {w ∈ V : vw ∈ E}, and the closed neighborhood
of v is N G [v] = N (v) ∪ {v}. The degree of v is deg G (v) = |N G (v)|. If N G [v] = V (G)
Theorem 2 ( [17]). Let G = (V, E) be a graph of order n ≥ 9. Then β p (G) = n − 2 if and only if it belongs either to the family {H i } 15 i=1 , except H 7 , (see Figure 7) or to the family {F i } 8 i=1 (see Figure 1). K n−3 K n−3 K n−3 K n−3 K n−4 K n−4 K n−4 K n−4 K n−4 K n−4 Figure 1: The thick horizontal segment means the join operation ∨. For example: [17] G 5 K 2,n−2 − e K 1,n−1 + e G 11 K n − E(K 1,3 + e) G 3 G 7 G 12 Table 1: The second row contains the names used in [17] for the graphs shown in Figure 1.
F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8F 1 ∼ = K n−3 ∨ (K 2 + K 1 ), F 3 ∼ = K 1 ∨ (K n−3 + K 2 ) and F 5 ∼ = K n−4 ∨ (P 3 + K 1 ). Figure 1 F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 Paper
Thus, in particular, and according to [17], for every n ≥ 9, all the graphs G displayed in Figure 1 satisfy β p (G) = n−2. However, for all of them, it holds that β p (G) = n−3, as we will prove in this paper (see Corollaries 3,5 and 6). As a matter of example, we show next that β p (F 1 ) ≤ n − 3, for every n ≥ 7. First, notice that F 1 ∼ = K n−3 ∨ (K 2 + K 1 ). Next, if V (K n−3 ) = {v 1 , . . . , v n−3 }, V (K 2 ) = {v n−2 , v n−1 } and V (K 1 ) = {v n }, then consider the partition Π = {{v 1 , v n−2 }, {v 2 , v n−1 }, {v 3 , v n }, {v 4 }, . . . , {v n−3 }}. Finally, observe that Π is a locating partition of F 1 since 2 = d(v i , v 4 ) = d(v n+i−3 , v 4 ) = 1, for every i ∈ {1, 2, 3}.
The main contribution of this work is, after showing that the theorem of characterization presented in [17] is far for being true, finding the correct answer to this problem. Motivated by this objective, we introduce the so-called twin number τ (G) of a connected graph G, and present a list of basic properties, some of them directly related to the partition dimension β p (G).
The rest of the paper is organized as follows. Section 2 is devoted to introduce the notions of twin class and twin number, and to show some basic properties. In Section 3, subdivided in three subsections, a number of results involving both the twin number and the partition dimension of a graph are obtained. Finally, Section 4 includes a theorem of characterization presenting, for every n ≥ 9, which graphs G of order n satisfy β p (G) = n − 2.
Twin number
A pair of vertices u, v ∈ V of a graph G = (V, E) are called twins if they have exactly the same set of neighbors other than u and v. A twin set of G is any set of pairwise twin vertices of G. If uv ∈ E, then they are called true twins, and otherwise false twins. It is easy to verify that the so-called twin relation is an equivalence relation on V , and that every equivalence class is either a clique or a stable set. An equivalence class of the twin relation is referred to as a twin class. Definition 1. The twin number of a graph G, denoted by τ (G), is the maximum cardinality of a twin class of G. Every twin set of cardinality τ (G) will be referred to as a τ -set.
As a direct consequence of these definitions, the following list of properties hold. Proposition 1. Let G = (V, E) be a graph of order n. Let W be a twin set of G. Then (1) If w 1 , w 2 ∈ W , then d(w 1 , z) = d(w 2 , z), for every vertex z ∈ V \ {w 1 , w 2 }.
(2) No two vertices of W can belong to the same part of any locating partition.
(3) W induces either a complete graph or an empty graph.
(4) Every vertex not in W is either adjacent to all the vertices of W or non-adjacent to any vertex of W .
(5) W is a twin set of G.
(6) τ (G) = τ (G). (7) τ (G) ≤ β p (G). (8) τ (G) = β p (G) = n if and only if G is the complete graph K n . (9) τ (G) = n − 1 if and only if G is the star K 1,n−1 .
It is a routine exercise to check all the results showed in Table 2 (see also [5] and the references given in [2]).
We conclude this section by characterizing the set of graphs G such that τ (G) = n−2. Figure 2: Graphs of order n ≥ 4 such that τ (G) = n − 2.
G P n C n K 1,n−1 K k,k K k,n−k K n order n n ≥ 4 n ≥ 5 n ≥ 3 4 ≤ n = 2k 2 ≤ k < n − k n ≥ 2 β(G) 1 2 n − 2 n − 2 n − 2 n − 1 τ (G) 1 1 n − 1 k n − k n β p (G) 2 3 n − 1 k + 1 n − k nK n−2 K n−2 K n−2 K n−2 (a)K n−2 ∨ K 2 (b) K 1 ∨ (K 1 + K n−2 ) (c) K 2,n−2 (d) K n−2 ∨ K 2
Proposition 2. Let G = (V, E) be a graph of order n ≥ 4. Then, τ (G) = n − 2 if and only if G is one of the following graphs (see Figure 2):
(a) the complete split graph K n−2 ∨K 2 , obtained by removing an edge from the complete graph K n ;
(b) the graph K 1 ∨ (K 1 + K n−2 ), obtained by attaching a leaf to the complete graph K n−1 ;
(c) the complete bipartite graph K 2,n−2 ;
(d) the complete split graph K n−2 ∨ K 2 .
Proof. It is straightforward to check that the twin number of the four graphs displayed in Figure 2 is n − 2. Conversely, suppose that G is a graph such that
τ (G) = n − 2. Let x, y ∈ V such that W = V \ {x, y} is the τ -set of G.
Since G is connected, we may suppose without loss of generality that W ⊆ N (x). We distinguish two cases.
Case 1: G[W ] ∼ = K n−2 .
If xy / ∈ E, then N (y) = W , and thus G ∼ = K 2 ∨K n−2 . If xy ∈ E, then N (y) = {x}, as otherwise G ∼ = K n , a contradiction. Thus, G ∼ = K 1 ∨ (K n−2 + K 1 ).
Case 2: G[W ] ∼ = K n−2 .
If xy / ∈ E, then N (y) = W , and thus G ∼ = K 2,n−2 . If xy ∈ E, then N (y) = W , as otherwise G ∼ = K 1,n−1 , a contradiction. Hence, G ∼ = K n−2 ∨ K 2 .
Twin number versus partition dimension
This section, consisting of 3 subsections, is devoted to obtain relations between the partition dimension β p (G) and the twin number τ (G) of a graph G. In the first subsection, a realization theorem involving both parameters is presented, without any further restriction than the inequality τ (G) ≤ β p (G). The second subsection is devoted to study the parameter β p (G), when G is a graph of order n with "few" twin vertices, to be more precise, such that τ (G) ≤ n 2 . Finally, the last subsection examines β p (G), whenever G is a graph for which τ (G) > n 2 .
Realization Theorem for trees
A complete k-ary tree of height h is a rooted tree whose internal vertices have k children and whose leaves are at distance h from the root. Let T (k, 2) denote the complete kary tree of height 2. Suppose that x is the root, x 1 , . . . , x k are the children of x, and x i1 , . . . , x ik are the children of x i for any i ∈ {1, . . . , k} (see Figure 3(a)).
Proposition 3. For any integer k ≥ 2, τ (T (k, 2)) = k and β p (T (k, 2)) = k + 1.
Proof. Certainly, τ (T (k, 2)) = k, and thus β p (T (k, 2)) ≥ k. Suppose that β p (T (k, 2)) = k and Π = {S 1 , . . . , S k } is a locating partition of size k. In such a case, for every i ∈ {1, . . . , k} the vertices x i1 , . . . , x ik are twins, and thus each one belongs to a distinct part of Π. So, if x r , x s ∈ S i for some pair r, s ∈ {1, . . . , k}, with r = s, then r(x r |Π) = r(x s |Π) = (1, . . . , 1, 0 i) , 1 . . . , 1), which is a contradiction. Hence, the vertices x 1 , . . . , x k must belong to distinct parts of Π. We may assume that x i ∈ S i , for every i ∈ {1, . . . , k}. Thus, if x belongs to the part S i , then r(x|Π) = r(x i |Π) = (1, . . . , 1, 0 i) , 1 . . . , 1), which is a contradiction. Hence, β p (T (k, 2)) ≥ k + 1. Finally, consider the partition Π = {S 1 , . . . , S k , S k+1 } such that S k+1 = {x} and, for any i ∈ {1, . . . , k}, S i = {x i , x 1i , x 2i , . . . , x ki }. Then, for every u ∈ V (T (k, 2)) and for every i, j, h ∈ {1, . . . , k} such that j < i < h:
r(u|Π) = (2, . . . , 2, j) 1, 2, . . . , 2, i) 0, 2, . . . , 2, h) 2 , 2, . . . , 2, 2) if u = x ij) if u = x ii (2,) if u = x ih) if u = x i
Therefore, Π is a locating partition, implying that β p (T (k, 2)) = k + 1.
x (1,2) y (1,2) x (h−1,h−1)
y (h−1,h−1) z 1 z 2 z k x 12 x 1k x 22 x 2k x k2 x kk x x 1 x 2 x k x 11 x 21 x k1 z (a) (b) x
x (1,1) y (1,1) Figure 3: The trees (a) T (k, 2) and (b) T * (k, h).
For any k ≥ 1 and h ≥ 1, let T * (k, h) denote the tree of order k + 2 + 2(h − 1) 2 defined as follows (see Figure 3(b)):
V (T * (k, h)) = {x, z} ∪ {z 1 , . . . , z k } ∪ {x (i,j) : 1 ≤ i, j ≤ h − 1} ∪ {y (i,j) : 1 ≤ i, j ≤ h − 1}, E(T * (k, h)) = {xx (i,j) : 1 ≤ i, j ≤ h−1}∪{x (i,j) y (i,j) : 1 ≤ i, j ≤ h−1}∪{xz, zz 1 , . . . , zz k }. Proposition 4. Let k, h be integers such that k ≥ 1 and h ≥ k+2. Then, τ (T * (k, h)) = k and β p (T * (k, h)) = h. Proof. Certainly, τ (T * (k, h)) = k. Let β p (T * (k, h)) = t. Next, we show that t ≥ h. Let Π = {S 1 , . . . , S t } be a locating partition of T * (k, h).
If there exist two distinct pairs (i, j) and (i , j ) such that the vertices x (i,j) , x (i ,j ) are in the same part of Π and y (i,j) , y (i ,j ) are in the same part, then r(x (i,j) |Π) = r(x (i ,j ) |Π), which is a contradiction. Notice that this tree contains (h − 1) 2 pairs of vertices of the type (x (i,j) , y (i,j) ) and if t ≤ h − 2, we achieve at most (h − 2) 2 such pairs avoiding the preceding condition. Thus, t ≥ h − 1. Moreover, if t = h − 1, then for every pair (m, n) ∈ {1, . . . , h − 1} 2 , there exists a pair (i, j) ∈ {1, . . . , h − 1} 2 such that x (i,j) ∈ S m and y (i,j) ∈ S n . So, by symmetry, we may assume without loss of generality that
x ∈ S 1 . Consider the vertices x (i,j) , y (i,j) , x (i ,j ) , y (i ,j ) such that x (i,j) ∈ S 2 , y (i,j) ∈ S 1 and x (i ,j ) ∈ S 2 , y (i ,j ) ∈ S 2 . Then r(x (i,j) |Π) = r(x (i ,j ) |Π) = (1, 0, 2, . . . , 2), which is a contradiction. Hence, t ≥ h.
To prove the equality t = h, consider the partition Π = {S 1 , . . . , S h } such that:
S i = {x (i,m) : 1 ≤ m ≤ h − 1} ∪ {y (n,i) : 1 ≤ n ≤ h − 1} ∪ {z i }, if 1 ≤ i ≤ k S i = {x (i,m) : 1 ≤ m ≤ h − 1} ∪ {y (n,i) : 1 ≤ n ≤ h − 1}, if k < i ≤ h − 1 S h = {x, z}. Let i ∈ {1, . . . , h − 1}.
Then, for every m, n ∈ {1, . . . , h − 1}, m, n = i:
r(u|Π) = (2, . . . , 2, i) 0, 2, . . . , 2, m) 1 , 2, . . . , 2, 1) if u = x (i,m) (2, . . . , 2, 0, 2, . . . , 2, 2 , 2, . . . , 2, 1) if u = x (i,i) r(u|Π) = (3, . . . , 3, i) 0, 3, . . . , 3, n) 1 , 3, . . . , 3, 2) if u = y (n,i) (3, . . . , 3, 0, 3, . . . , 3, 3, 3, . . . , 3, 2) if u = y (i,i) Therefore, r(u, |Π) = r(v|Π) if u, v ∈ {x (i,m) : 1 ≤ m ≤ h − 1} ∪ {y (n,i) : 1 ≤ n ≤ h − 1} and u = v. Moreover, it is straightforward to check that, if i ∈ {1, . . . , k}, then for every u ∈ S i , u = z i , we have r(z i |Π) = (2, . . . , 2, i) 0, 2, . . . , k) 2 , 3, . . . , 3, 1) = r(u|Π) .
Finally, for x, z ∈ S h , we have
r(x|Π) = (1, . . . , k) 1 , 1, . . . , 1, 0) = (1, . . . , k) 1 , 2, . . . , 2, 0) = r(z|Π).
Therefore, Π is a locating partition, implying that β p (T * (k, h)) = h.
Theorem 3. Let a, b be integers such that 1 ≤ a ≤ b. Then, there exists a tree T such that τ (T ) = a and β p (T ) = b.
Proof. For a = b = 1, the trivial graph P 1 satisfies τ (P 1 ) = β p (P 1 ) = 1. For a = b ≥ 2, consider the star K 1,a . For a = 1 and b = 2, take the path P 4 . If 2 ≤ a and b = a + 1, consider the tree T (a, 2) studied in Proposition 3. Finally, if a ≥ 1 and b ≥ a + 2, take the tree T * (a, b) analyzed in Proposition 4.
Twin number at most half the order
In this subsection, we approach the case when G is a graph of order n such that τ (G) = τ ≤ n 2 . Concretely, we prove that, in such a case, β p (G) ≤ n − 3. Lemma 1. Let D be a subset of vertices of size k ≥ 3 of a graph G such that G[D] is neither complete nor empty. Then, there exist at least three different vertices u, v, w ∈ D such that uv ∈ E(G) and uw / ∈ E(G)
Proof. If G[D] is neither complete nor empty, then there is at least one vertex u such
that 1 ≤ deg G[D] (u) ≤ k − 2.
Let v (resp. w) be a a vertex adjacent (resp. non-adjacent) to u. Then, u, v, w satisfy the desired condition.
Lemma 2. If G is a nontrivial graph of order n with a vertex u of degree k, then β p (G) ≤ n − min{k, n − 1 − k}. Proof. Let N (v) = {x 1 , . . . , x k } and V (G)\N (v) = {y 1 , . . . , y n−1−k } and m = min{k, n− 1 − k}. Take the partition Π = {S 1 , . . . , S m } ∪ {{z} : z / ∈ S 1 ∪ . . . ∪ S m }}, where S i = {x i , y i } for i = 1, . . . , m. Observe that {v} resolves the vertices of S i = {x i , y i } for i = 1, . . . , m. Therefore, Π is a locating partition of G, impliying that β p (G) ≤ |Π| = n − m = n − min{k, n − 1 − k}. Corollary 1. If G is a graph of order n ≥ 7 with at least one vertex u satisfying 3 ≤ deg(u) ≤ n − 4, then β p (G) ≤ n − 3.
As a direct consequence of Theorem 1, we know that if G is a graph such that diam(G) ≥ 4, then β p (G) ≤ n − 3. Next, we study the cases diam(G) = 3 and diam(G) = 2.
Proposition 5. Let G be a graph of order n ≥ 9. If τ (G) ≤ n 2 and diam(G) = 3, then β p (G) ≤ n − 3.
Proof. By Corollary 1, and having also in mind that G has no universal vertex, since its diameter is 3, we may suppose that, for every vertex w, deg(w) ∈ {1, 2, n − 3, n − 2}. Let u be a vertex of eccentricity 3. Consider the nonempty subsets D i = {v : d(u, v) = i} for i = 1, 2, 3. If at most one of these three subsets has exactly one vertex, then there exist five distinct vertices x 1 , x 2 , x 3 , y 1 , y 2 such that, for i = 1, 2, 3, x i ∈ D i and vertices y 1 and y 2 do not belong to the same set D i . Consider the partition Π = {S 1 ,
S 2 }∪{{z} : z / ∈ S 1 ∪S 2 }}, where S 1 = {x 1 , x 2 , x 3 } and S 2 = {y 1 , y 2 }.
Then, {u} resolves every pair of vertices in S 1 and the vertices in S 2 . Therefore, Π is a locating partition, implying that β p (G) ≤ n − 3.
Next, suppose that |D i 0 | = n − 3 for exactly one value i 0 ∈ {1, 2, 3} and |D i | = 1 for i = i 0 . We distinguish two cases.
(1) G[D i 0 ] is neither complete nor empty. Then by Lemma 1, there exist vertices r, s, t ∈ D i 0 such that rs ∈ E(G) and rt / ∈ E(G). Consider the sets S 1 = {s, t} and S 2 = {x 1 , x 2 , x 3 }, where x i ∈ D i for i = 1, 2, 3, with the additional condition S 2 ∩{r, s, t} = ∅, which is possible since |D i 0 | ≥ 4. Take the partition Π = {S 1 , S 2 } ∪ {{z} : z / ∈ S 1 ∪ S 2 }}. Observe that {r} resolves the vertices in S 1 and {u} resolves every pair of vertices in S 2 . Therefore, Π is a locating partition, implying that β p (G) ≤ n − 3.
(2) G[D i 0 ] is either complete or empty. We distinguish three cases, depending on for
which i 0 ∈ {1, 2, 3}, |D i 0 | = n − 3.
(a) |D 3 | = n − 3. Then, D 3 is a twin set with n − 3 vertices, a contradiction as n ≥ 9.
(b) |D 1 | = n − 3. Let v be the (unique) vertex of D 2 . Then D 1 ∩ N (v) and D 1 ∩ N (v) are twin sets. If deg(v) = 2, then |D 1 ∩ N (v)| = n − 4, a contradiction. If n − 3 ≤ deg(v) ≤ n − 2, then |D 1 ∩ N (v)| ≥ n − 4, again a contradiction. (c) |D 2 | = n−3. Let v be the (unique) vertex of D 3 . Then, both N (v) and D 2 \N (v) are twin sets. Notice that deg(v) ∈ {1, 2, n − 3}. We distinguish cases. (c.i) If deg(v) = 1 (resp. deg(v) = n − 3) , then |D 2 \ N (v)| = n − 4 (resp. |N (v)| = n − 3), a contradiction. (c.ii) If deg(v) = 2, then |D 2 \ N (v)| = n − 5. Let N (v) = {a 1 , a 2 }, D 2 \ N (v) = {b 1 , . . . , b n−5 }, D 1 = {x}. Take the partition Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 }}, where S 1 = {a 1 , b 1 }, S 2 = {a 2 , b 2 } and S 3 = {x, b 3 }.
Observe that {v} resolves the vertices in S 1 and S 2 and {u} resolves the vertices in S 3 . Therefore, Π is a locating partition, implying that β p (G) ≤ n − 3.
Proposition 6. Let G be a graph of order n ≥ 9. If τ (G) ≤ n 2 and diam(G) = 2, then β p (G) ≤ n − 3.
Proof. By Corollary 1, we may suppose that, for every vertex w ∈ V (G), deg(w) ∈ {1, 2, n − 3, n − 2, n − 1}. We distinguish three cases. (2) G[D 2 ] is either complete or empty. Then the subsets of (ii) There exists at least one vertex u of degree 1 and there is no vertex of degree 2. In this case, the neighbor u of v is a universal vertex v. Let Ω be the set of vertices different from v that are not leaves. Notice that there are at most two vertices of degree 1 in G, as otherwise all vertices in Ω would have degree between 3 and n − 4, contradicting the assumption made at the beginning of the proof.
D 2 , A = N (x 1 ) ∩ N (x 2 ) ∩ D 2 , B = N (x 1 ) ∩ N (x 2 ) ∩ D 2 and C = N (x 1 ) ∩ N (x 2 ) ∩ D 2
If there are exactly two vertices of degree 1, then |Ω| = n − 3. In such a case, Ω induces a complete graph in G, as otherwise the non-universal vertices in G[Ω] would have degree at most n−4. So, Ω is a twin set, implying that τ (G) = n−3 > n 2 , a contradiction.
Suppose thus that u is the only vertex of degree 1, which means that Ω contains n − 2 vertices, all of them of degree n − 3 or n − 2. Consider the graph H = G[Ω].
Certainly, H has n − 2 vertices, all of them of degree 0 or 1. Let H i denote the set of vertices of degree i of H, for i = 0, 1. Observe that |H 0 | ≤ n 2 , since H 0 is a twin set in G. Hence, |H 1 | ≥ 4, as n ≥ 9 and the size of H 1 must be even. We distinguish two cases, depending on the size of H 1 .
(a) |H 1 | = 4. Notice that |H 0 | ≥ 3. Let {y 1 , y 2 , y 3 } ⊆ H 0 and
H 1 = {x 1 , x 2 , x 3 , x 4 } such that {x 1 x 2 , x 3 x 4 } ⊆ E(H 1 ). Consider the partition Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 }, where S 1 = {x 1 , y 1 }, S 2 = {x 2 , y 2 } and S 3 = {u, v}. Observe that d G (x 2 , x 1 ) = 2 = 1 = d G (x 2 , y 1 ), d G (x 4 , x 3 ) = 2 = 1 = d G (x 4 , y 3 ), d G (y 2 , u) = 2 = 1 = d G (y 2 , v).
Hence, {x 2 } resolves the vertices in S 1 , {x 4 } resolves the vertices in S 2 and {y 2 } resolves the vertices in S 3 . Therefore, Π is a locating partition of G, implying that β p (G) ≤ n − 3.
(b) |H 1 | ≥ 6. Let {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } ⊆ H 1 such that {x 1 x 2 , x 3 x 4 , x 5 x 6 } ⊆ E(H 1 ). Consider the partition Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 }, where S 1 = {u, v}, S 2 = {x 2 , x 4 } and S 3 = {x 3 , x 5 }. Observe that d G (u, x 1 ) = 2 = 1 = d G (v, x 1 ), d G (x 4 , x 1 ) = 2 = 1 = d G (x 2 , x 1 ), d G (x 3 , x 6 ) = 2 = 1 = d G (x 5 , x 6 )
. Hence, {x 1 } resolves the vertices in S 1 and in S 2 , and {x 6 } resolves the vertices in S 3 . Therefore, Π is a locating partition of G, implying that β p (G) ≤ n − 3.
(iii) There are no vertices of degree at most 2. In this case, all the vertices of G have degree n − 3, n − 2 or n − 1, that is to say, all the vertices of G have degree 0, 1 or 2. Since G has at most n 2 pairwise twin vertices, there are at most n 2 vertices of degree 0 in G. Let H i denote the set of vertices of degree i of G, for i = 0, 1, 2. Let Γ = H 1 ∪ H 2 and H = G[Γ]. We distinguish cases, depending on the size of Γ, showing in each of them a collection of three 2-subsets S 1 , S 2 , S 3 such that the corresponding partition Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 } is a locating partition for G, implying thus that β p (G) ≤ n − 3. (c.3) If H has 3 connected components, say C 1 , C 2 and C 3 , then we may assume that |V (C 1 )| ≥ 3 and |V (C 1 )| ≥ |V (C 2 )| ≥ |V (C 3 )| ≥ 2. Let x 1 x 2 x 3 , y 1 y 2 and z 1 z 2 be paths contained in C 1 , C 2 and C 3 respectively. Take S 1 = {x 1 , y 1 }, S 2 = {x 3 , y 2 } and S 3 = {z 2 , t}, where t is a vertex in C 1 ∪ C 2 different from x 1 , x 2 , x 3 , y 1 , y 2 that exists because |Γ| ≥ 8. Then, it is easy to check that {x 2 } resolves the vertices in S 1 and in S 2 , and {z 1 } resolves the vertices in S 3 . (c.4) If H has at least 4 connected components, say C 1 , C 2 , C 3 and C 4 , then we may assume that |V (
C 1 )| ≥ |V (C 2 )| ≥ |V (C 3 )| ≥ |V (C 4 )| ≥ 2. Let
x 1 x 2 , y 1 y 2 , z 1 z 2 and t 1 t 2 be edges of C 1 , C 2 , C 3 and C 4 respectively. Take S 1 = {x 1 , y 1 }, S 2 = {y 2 , z 2 } and S 3 = {t 1 , w}, where w is a vertex in C 1 ∪ (V \ Γ) different from x 1 , x 2 , that exists since G has order at least 9. Then, it is easy to check that {x 2 } resolves the vertices in S 1 , {z 1 } resolves the vertices in S 2 , and {t 2 } resolves the vertices in S 3 .
As a consequence of Theorem 1, Proposition 5 and Proposition 6, the following result is obtained.
Theorem 4. Let G be a graph of order n ≥ 9. If τ (G) ≤ n 2 , then β p (G) ≤ n − 3.
Twin number greater than half the order
In this subsection, we focus our attention on the case when G is a nontrivial graph of order n such that τ (G) = τ > n 2 . Notice that, in these graphs there is a unique τ -set W . Among others, we prove that, in such a case, τ (G) ≤ β p (G) ≤ n + τ (G) 2 .
Proposition 7. Let G be a graph of order n, other than K n . If W is a τ -set such that
G[W ] ∼ = K τ , then β p (G) ≥ τ (G) + 1.
Proof. Assume that Π = {S 1 , S 2 , . . . , S τ } is a locating partition of G. If W = {w 1 , w 2 , . . . , w τ }, then we can assume without loss of generality that, for every i ∈ {1, ..., τ },
w i ∈ S i . Let v be a vertex of N (W ) \ W . Take j ∈ {1, ..., τ } such that {w j , v} ⊆ S j . Certainly, r(v|Π) = (1, . . . 1, 0 j)
, 1, . . . , 1) = r(w j |Π), a contradiction.
Theorem 5. Let G = (V, E) be a graph of order n such that n 2 < τ (G) = τ = n − k and let W be its τ -set.
If G[W ] ∼ = K τ , then β p (G) ≤ n − k/2. Proof. Let W = {w 1 , . . . , w τ }, W 1 = N (W ) \ W = {x ∈ V (G) : d(x, W ) = 1} and W 2 = V \ N [W ] = {x ∈ V (G) : d(x, W ) ≥ 2}
, and denote r = |W 1 |, t = |W 2 |. Observe that {W, W 1 , W 2 } is a partition of V (G) and k = r + t. Since W is a set of twin vertices, we have that xy ∈ E(G) for all x ∈ W and y ∈ W 1 . Consider the subsets U 1 = {x ∈ W 1 : deg G[W 1 ] (x) = r − 1} and U 2 = W 1 \ U 1 of W 1 . If x ∈ U 1 , then there exists at least one vertex y ∈ W 2 such that xy ∈ E(G), otherwise x should be in W . Let us assign to each vertex x ∈ U 1 one vertex y(x) ∈ W 2 such that x y(x) ∈ E(G) and consider the set A 2 = {y(x) : x ∈ U 1 } ⊆ W 2 . Observe that for different vertices x, x ∈ U 1 , the vertices y(x), y(x ) are not necessarily different. By construction, |A 2 | ≤ |U 1 |.
W W 1 W 2 w 1 w h w h+1 w τ w j w j+1 v 1 v j v j+1 v h A 2 A 1 w l w l+1 v l v l+1 U 1 U 2
Next, consider the subgraph G[U 2 ] induced by the vertices of U 2 . If s = |U 2 |, then by definition, this subgraph has maximum degree at most s − 2, and hence, the complement G[U 2 ] has minimum degree at least 1. It is well known that every graph without isolated vertices contains a dominating set of cardinality at most half the order (see [13]). Let A 1 be a dominating set of G[U 2 ] with |A 1 | ≤ s/2.
If (W 1 ∪ W 2 ) \ (A 1 ∪ A 2 ) = {v 1 , . . . , v h },
we show that the partition Π defined as follows is a locating partition of G (see Figure 4):
Π = {{x} : x ∈ A 1 } ∪ {{y} : y ∈ A 2 } ∪ {{w i , v i } : 1 ≤ i ≤ h} ∪ {{w i } : h + 1 ≤ i ≤ τ }.
Observe that Π is well defined since h < k < n 2 ≤ τ . To prove this claim, it is sufficient to show that for every i ∈ {1, . . . , h} there exists a part of Π at different distance from w i and v i . We distinguish the following cases:
i) If v i ∈ U 1 , consider the vertex y(v i ) ∈ A 2 such that v i y(v i ) ∈ E(G). Then, d(w i , {y(v i )}) = 2 = 1 = d(v i , {y(v i )}). ii) If v i ∈ U 2 \ A 1 , consider a vertex x ∈ A 1 dominating v i in G[U 2 ], i.e., x v i / ∈ E(G). Then, d(w i , {x}) = 1 < d(v i , {x}). iii) If v i ∈ W 2 \ A 2 , then d(w i , {w τ }) = 1 < 2 ≤ d(v i , {w τ }).
Observe that |A 2 | ≤ |U 1 | = r − s and |A 2 | ≤ |W 2 | = t, so we can deduce that |A 2 | ≤ (r − s + t)/2. Therefore, the partition dimension of G satisfies: By Proposition 1(7), β p (G) ≥ τ . To prove the equality, consider the
β p (G) ≤ |Π| = n − |(W 1 ∪ W 2 ) \ (A 1 ∪ A 2 )| = n − [(r + t) − (|A 1 | + |A 2 |)] = n − k/2 Proposition 8. Let G = (V, E) be a graph of order n such that τ (G) = τ > n 2 . If its τ -set W satisfies G[W ] ∼ = K τ , then β p (G) = τ . Proof. Let W = {w 1 , . . . , w τ }, V \ W = {v 1 , . . . , v s } and N (W ) = {v 1 , . . . , v r }, where 1 ≤ r ≤ s < τ .partition Π = {S 1 , . . . , S τ }, where Si = {w i , v i } if 1 ≤ i ≤ s, and S i = {w i } if s < i ≤ τ .
Observe that for any i, j ∈ {1, . . . , τ } with i = j, h ∈ {1, . . . , r} and k ∈ {r + 1, . . . , s}, d(w i , w j ) = 2, d(v h , w j ) = 1 and d(v k , w j ) = 2. To prove that Π is a locating partition of G, we distinguish cases.
Case 1: 1 ≤ i ≤ r. Then, d(w i , S τ ) = d(w i , w τ ) = 2 = 1 = d(v i , w τ ) = d(v i , S τ ).
Case 2: r < i ≤ s. We distinguish two cases. Figure 5(a)). Figure 5(b)).
Case 2.1: For some k ∈ {1, . . . , r}, v i v k / ∈ E(G). Consider the part S k = {w k , v k }. On the one hand, d(w i , S k ) = 1 since d(w i , v k ) = 1. On the other hand, d(v i , S k ) ≥ 2 since d(v i , w k ) ≥ 2 and d(v i , v k ) ≥ 2. Therefore, d(w i , S k ) = d(v i , S k ) (seew 1 w r w i w τ v 1 v i W N (W ) w k v k v r (a) S k v r v 1 v i v k W w 1 w r w τ w i w k N (W ) (b) S k v s v r+1 v s v r+1 V \ N [W ] V \ N [W ]v i , S k ) = 1 since d(v i , v k ) = 1. Therefore, d(w i , S k ) = d(v i , S k ) (see
As a direct consequence of Proposition 7, Theorem 5 and Proposition 8, the following result is derived. Theorem 6. Let G be a graph of order n, other than K n , such that τ (G) = τ > n 2 . Then, τ ≤ β p (G) ≤ n+τ 2 . Moreover, if W is its τ -set, then
1. β p (G) = τ if and only if G[W ] ∼ = K τ . 2. τ < β p (G) ≤ n+τ 2 if and only if G[W ] ∼ = K τ .
Corollary 2. Let G be a nontrivial graph of order n, other than K n , such that β p (G) = n − h and τ (G) = τ > n 2 . Let W be its τ -set. Then,
n − 2h ≤ τ ≤ n − h − 1 if and only if G[W ] ∼ = K τ .
Corollary 3. For every n ≥ 7, the graphs F 1 , F 2 , F 3 and F 4 , displayed in Figure 1, satisfy β(F i ) = n − 3.
Partition dimension almost the order
Our aim in this section is to completely characterize the set of all graphs of order n ≥ 9 such that β p (G) = n − 2. This issue was already approached in [17], but, as remarked in our introductory section, the list of 23 graphs presented for every order n ≥ 9 turned out to be wrong.
As was shown in Proposition 1 (8), it is clear that the only graphs whose partition dimension equals its order, are the complete graphs. The next result, along with Proposition 1(9) and Proposition 2, allows us to characterize, in a pretty simple way, all connected graphs of order n with partition dimension n − 1, a result already proved in [5] for n ≥ 3.for the case β p (G) = n − 1, Proposition 9. Let G be a graph of order n ≥ 9 and twin number τ , and let W be a τ -set. Then, β p (G) = n − 1 if and only if G satisfies one of the following conditions:
(i) τ = n − 1. (ii) τ = n − 2 and G[W ] ∼ = K n−2 .
Proof. Suppose that β p (G) = n − 1. Then, by Theorem 4, τ > n 2 . Thus, by Theorem 6 and Corollary 2, n − 2 ≤ τ ≤ n − 1 and if τ = n − 2, then G[W ] ∼ = K n−2 .
If τ = n − 1, i.e., if G ∼ = K 1,n−1 , then β p (G) = n − 1. If τ = n − 2 and G[W ] ∼ = K n−2 then, according to Proposition 7, β p (G) ≥ τ +1 = n−1, which means that β p (G) = n−1, as G is not the complete graph. 2. the complete split graph K n−2 ∨K 2 obtained by removing an edge e from the complete graph K n (see Figure 2(a)).
3. the graph K 1 ∨(K 1 +K n−2 ) obtained by attaching a leaf to the complete graph K n−1 (see Figure 2(b)).
Next, we approach the case β p (G) = n − 2.
Definition 2. Let G = (V, E) a graph such that τ (G) = τ . Let W be a τ -set of G such that G[W ] ∼ = K τ . A vertex v ∈ V \ W is said to be a W -distinguishing vertex of G if and only if, for every vertex z ∈ N (W ) \ W , d(v, z) = d(v, W ).
Lemma 3. Let G = (V, E) be a nontrivial graph of order n such that τ (G) = τ > n 2 . Suppose that its τ -set W satisfies G[W ] ∼ = K τ . Then, the following statements hold:
(a) If G contains a W -distinguishing vertex, then β p (G) = τ + 1. then v would be a twin of any vertex in W , which is a contradiction.
As a straightforward consequence of item (b) of the previous lemma, the following holds.
Corollary 5. For every n ≥ 9, the graphs F 5 and F 6 , displayed in Figure 1, satisfy β(F i ) = n − 3. Case 3.2: |N (W ) \ W | = 3. According to Lemma 3, G[N (W ) \ W ] is a either C 3 or a P 3 . Suppose that G[N (W ) \ W ] is C 3 . Then, by Lemma 3(d), every vertex of N (W ) \ W is adjacent to the unique vertex z of V \ N (W ), a contradiction since in this case z would be a W -distinguishing vertex. Thus, G[N (W ) \ W ] is P 3 . According to Lemma 3(d), the central vertex w of P 3 is adjacent to the unique vertex z of V \ N (W ). Observe also that one of the remaining two vertices of this path may be adjacent to vertex z, but not both, since in this case z would be a W -distinguishing vertex. Hence, G is isomorphic to either H 14 or H 15 .
( i )
iThere exists a vertex u of degree 2. Consider the subsets D 1 = N (u) = {x 1 , x 2 } and D 2 = {v : d(u, v) = 2}. We distinguish two cases.(1) G[D 2 ] is neither complete nor empty. Then, Lemma 1, there exist three different vertices r, s, t ∈ D 2 such that rs ∈ E(G) and rt / ∈ E(G). Consider two different vertices y 1 , y 2 ∈ D 2 \ {r, s, t} and let S 1 = {x 1 , y 1 }, S 2 = {x 2 , y 2 }, S 3 = {s, t}. Then, {u} resolves the vertices in S 1 and in S 2 , and {r} resolves the vertices in S 3 . Hence, Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 } is a locating partition of G.
are twin sets. We distinguish cases.(a) If either deg(x 1 ) ≤ 2 or deg(x 2 ) ≤ 2, then either |C| ≥ n − 4 or |A| ≥ n − 4, in both cases a contradiction as n ≥ 9. (b) If both x 1 and x 2 have degree at least n − 3, then 0 ≤ |A| ≤ 2, 0 ≤ |C| ≤ 2 and n − 7 ≤ |B| ≤ n − 3. We distinguish cases, depending on the size of B.(b.1) |B| ≥ n − 4. Then, τ (G) ≥ n − 4, a contradiction as n ≥ 9. (b.2) |B| = n − 5 ≥ 4. If D 2 ∼ = K n−3 , then τ (G) = n − 4, as B ∪ {u} is a (maximum) twin set of G. Suppose that D 2 ∼ = K n−3 . Let A ∪ C = {y 1 , y 2 }, {b 1 , b 2 , b 3 , b 4 } ⊆ B. Consider the partition Π = {S 1 , S 2 , S 3 } ∪ {{z} : z / ∈ S 1 ∪ S 2 ∪ S 3 }}, where S 1 = {b 1 , y 1 }, S 2 = {b 2 , y 2 } and S 3 = {b 3 , u}.Observe that either {x 1 } or {x 2 } resolves the vertices of S 1 and S 2 . Notice also that {b 4 } resolves the vertices in S 3 . Hence, Π is a locating partition of G. (b.3) 2 ≤ n − 7 ≤ |B| ≤ n − 6. We may assume without loss of generality that |A| = 2 and 1 ≤ |C| ≤ 2.Let A = {a 1 , a 2 }, {b 1 , b 2 } ⊆ B, {c 1 } ⊆ C. Consider the partition Π = {S 1 , S 2 , S 3 }∪{{z} : z / ∈ S 1 ∪S 2 ∪S 3 }}, where S 1 = {a 1 , b 1 }, S 2 = {a 2 , b 2 } and S 3 = {x, c 1 }.Observe that {x 2 } resolves the vertices of S 1 and S 2 . Notice also that {u} resolves the vertices in S 3 . Hence, Π is a locating partition of G.
( a )
a|Γ| ∈ {5, 6}. Then, |H 0 | ≥ 3. Let {y 1 , y 2 , y 3 } ⊆ H 0 . It easy to check that in both cases H contains at least three edges either of the form (i) x 1 x 2 , x 3 x 4 , x 4 x 5 or of the form (ii) x 1 x 2 , x 3 x 4 , x 5 x 6 . Take S 1 = {x 1 , y 1 }, S 2 = {x 3 , y 2 }, S 3 = {x 5 , y 3 }. Notice that, in case (i), {x 2 } resolves the vertices in S 1 and {x 4 } resolves the vertices in S 2 and in S 3 , and in case (ii), {x 2 } resolves the vertices in S 1 , {x 4 } resolves the vertices in S 2 and {x 6 } resolves the vertices in S 3 .(b) |Γ| = 7. Then, |H 0 | ≥ 2. Let Γ = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 } such that {x 1 x 2 , x 2 x 3 , x 4 x 5 , x 6 x 7 } ⊆ E(G), x 4 x 6 / ∈ E(G) and {y 1 , y 2 } ⊆ H 0 . Take S 1 = {x 1 , y 1 }, S 2 = {x 3 , y 2 } and S 3 = {x 5 , x 6 }. Observe that {x 2 }resolves the vertices in S 1 and in S 2 , and {x 4 } resolves the vertices in S 3 . (c) |Γ| ≥ 8. Then, all the connected components of H are isomorphic either to a path or to a cycle. We distinguish cases, depending on the number of components of G[Γ]. (c.1) If H is connected, then H contains a path x 1 x 2 . . . x 8 of length 7. Take S 1 = {x 1 , x 2 }, S 2 = {x 4 , x 5 } and S 3 = {x 7 , x 8 }. Then, {x 3 } resolves the vertices in S 1 and in S 2 , and {x 6 } resolves the vertices in S 3 . (c.2) If H has 2 connected components, say C 1 and C 2 , let assume that |V (C 1 )| ≥ |V (C 2 )| ≥ 2. We distinguish two cases. (c.2.1) If one of the connected components has at most 3 vertices, then |V (C 1 )| ≥ 5 and 2 ≤ |V (C 2 )| ≤ 3. If x 1 x 2 x 3 x 4 x 5 and y 1 y 2 are paths contained in C 1 and C 2 , respectively, then consider S 1 = {x 1 , y 1 }, S 2 = {x 3 , y 2 } and S 3 = {x 5 , t}, where t is any vertex different from x 1 , x 2 , x 3 , x 4 , x 5 , y 1 , y 2 . Then, it is easy to check that {x 2 } resolves the vertices in S 1 and in S 2 , and {x 4 } resolves the vertices in S 3 . (c.2.2) If both connected components have at least 4 vertices, let x 1 x 2 x 3 x 4and y 1 y 2 y 3 y 4 be paths contained in C 1 and C 2 , respectively. Take S 1 = {x 1 , y 1 }, S 2 = {x 3 , y 2 } and S 3 = {x 4 , y 4 }. Then, it is easy to check that {x 2 } resolves the vertices in S 1 and in S 2 , and {y 3 } resolves the vertices in S 3 .
Figure 4 :
4In this figure, W 1 = N (W ) \ W and W 2 = V \ N [W ].
Figure 5 :
5In both cases, the part S k resolves the pair w i , v i . Solid lines hold for adjacent vertices and dashed lines, for non-adjacent vertices.Case 2.2:Vertex v i is adjacent to all vertices in {v 1 , . . . , v r }. As v i / ∈ W , v i v k ∈ E(G) for some k ∈ {r + 1, . . . , s}. Consider the part S k = {w k , v k }. On the one hand, d(w i , S k ) = 2 since d(w i , w k ) = 2 and d(w i , v k ) ≥ 2. On the other hand, d(
Corollary 4 .
4([5]) Let G be a graph of order n ≥ 9. Then, β p (G) = n − 1 if and only if G is one of the following graphs:1. the star K 1,n−1 .
G[N (W ) \ W ] contains an isolated vertex, then β p (G) = τ + 1.(c) If |N (W ) \ W | = 1, then β p (G) = τ + 1.(d) If G[N (W ) \ W ] contains a universal vertex v, then v is adjacent to at least one vertex of V \ N [W ]. Proof. (a) Let v be a W -distinguishing vertex. Set W = {w 1 , w 2 , . . . , w τ } and V \ W = {v, z 1 , . . . , z r }, where r = n − τ − 1 < τ . Take the partition Π = {S 1 , . . . , S τ +1 }, where: S 1 = {w 1 , z 1 }, . . . , S r = {w r , z r }, S r+1 = {w r+1 }, . . . , S τ = {w τ }, S τ +1 = {v}. Notice that if z i ∈ N (W ) \ W , then d(z i , S τ +1 ) = d(z i , v) = d(v, W ) = d(v, w i ) = d(w i , S τ +1 ),and if z i / ∈ N (W ), then for any j ∈ {1, . . . , τ } such that i = j we haved(z i , S j ) = d(z i , w j ) > 1 = d(w i , w j ) = d(w i , S j ).Thus, r(w i |Π) = r(z i |Π) for every i ∈ {1, . . . , r}, and consequently Π is a locating partition of G.(b) If v is an isolated vertex in G[N (W ) \ W ], then for every vertex z ∈ N (W ) \ W , d(v, z) = 2 = 1 = d(v, W ). Hence, v is a W -distinguishing vertex of G,and, according to item (a), β p (G) = τ + 1. (c) In this case, the only vertex in N (W )\W is isolated in G[N (W )\W ] and, according to item (b), β p (G) = τ + 1. (d) Notice that if v is universal in G[N (W ) \ W ] and has no neighbor in V \ N [W ],
Lemma 4 .
4Let G = (V, E) be a graph of order n ≥ 9 such that τ (G) = τ = n − 4 andits τ -set W satisfies G[W ] ∼ = K τ . If |N (W ) \ W | = 2, then β p (G) = n − 3. Proof. Let W = {w 1 , . . . , w n−4 }, V \ W = {z 1 , z 2 , z 3 , z 4 } and N (W ) \ W = {z 1 , z 2 }.If z 1 z 2 ∈ E, then both z 1 and z 2 are isolated vertices in G[N (W ) \ W ]. Hence, by Lemma 3(b), β p (G) = τ (G) + 1 = n − 3.
Table 2 :
2Metric dimension β, twin number τ and partition dimension β p of paths, cycles, stars, bicliques and cliques.
Case 3.1: |N (W ) \ W | = 4. According to Lemma 3, all vertices of G[N (W ) \ W ] have degree either 1 or 2. Thus, G[N (W ) \ W ] is a isomorphic to either C 4 or P 4 or 2K 2 . Hence, G is isomorphic to either H 11 or H 12 or H 13 .
AcknowledgementsResearch partially supported by grants MINECO MTM2015-63791-R, Gen. Cat. DGR 2014SGR46 and MTM2014-60127-P.Suppose that z 1 z 2 ∈ E. According to Lemma 3(b), both z 1 and z 2 are adjacent to atAssume thus that {z 1 z 3 , z 2 z 4 } ⊂ E and {z 1 z 4 , z 2 z 3 } ∩ E = ∅ (seeFigure 6(a)). Notice that none of the vertices of V \ W is W -distinguishing. Take the partition Π = {S 1 , . . . , S n−3 }, whereAs a straightforward consequence of this lemma, the following holds.Corollary 6. For every n ≥ 9, the graphs F 7 and F 8 , displayed inFigure 1, satisfy β(F i ) = n − 3.Theorem 7. Let G be a graph of order n ≥ 9. Then, β p (G) = n − 2 if and only if G belongs to the following family {H i } 15 i=1 (seeFigure 7):where e is an edge joining the vertex of K 1 with an endpoint of P 3 in (K n−4 + K 1 ) and e 1 is an edge joining the vertex of K 1 with a vertex of K 2 in H 6 ; e 2 is an edge joining the vertex of K 1 with a vertex of K 2 in H 8 ; e 3 is an edge joining the vertex of K 1 with an endpoint of P 3 in H 14 . Proof. (⇐=) First suppose that G is a graph belonging to the family {H i } 15 i=1 . We distinguish three cases.Case 1: G ∈ {H 1 , H 2 }. Hence, τ (G) = n − 2 and its τ -set W satisfies G[W ] ∼ = K n−2 . Thus, according to Proposition 8, β p (G) = n − 2.i=3 . Hence, τ (G) = n−3 and its τ -set W satisfies G[W ] ∼ = K n−3 . Thus, according to Proposition 7, β p (G) ≥ τ (G) + 1 = n − 2. Furthermore, from Proposition 9 we deduce that β p (G) = n − 2.Clearly, for all these graphs diam(G) = 2, τ (G) = n − 4 and its τ -set W satisfies G[W ] ∼ = K n−4 . According to Proposition 7 and Theorem 4, n − 3 ≤ β p (G) ≤ n − 2. Suppose that there exists a locating partition Π = {S 1 , . . . , S n−3 } of cardinality n − 3. If W = {w 1 , . . . , w n−4 }, assume that, for every i ∈ {1, . . . , n − 4}, w i ∈ S i . We distinguish two cases.
1, i) 0, 1, . . . , 1, h), with h ∈ {1, 2}. Hence, there are exactly three sets of Π of cardinality 2. We can suppose without loss of generality that S 1 = {w 1 , x}, S 2 = {w 2 , y}, S 3 = {w 3 , z} and S n−3 = {t}, where {x, y, z, t} = {a 1. 1, 0) for every z ∈ {a 1 , a 2 , b 1 , b 2 } ∩ S n−3 . Notice also that |S i | ≤ 2 for i ∈ {1, . . . , n − 4}, as for every x ∈ S i , we have r(x, Π) = (1. b 1 , b 2 }. Hence, d(t, x) = d(t, y) = d(t, z) = 2, a contradictionObserve that |S n−3 | = 1, since r(z, Π) = (1, . . . , 1, 0) for every z ∈ {a 1 , a 2 , b 1 , b 2 } ∩ S n−3 . Notice also that |S i | ≤ 2 for i ∈ {1, . . . , n − 4}, as for every x ∈ S i , we have r(x, Π) = (1, . . . , 1, i) 0, 1, . . . , 1, h), with h ∈ {1, 2}. Hence, there are exactly three sets of Π of cardinality 2. We can suppose without loss of generality that S 1 = {w 1 , x}, S 2 = {w 2 , y}, S 3 = {w 3 , z} and S n−3 = {t}, where {x, y, z, t} = {a 1 , a 2 , b 1 , b 2 }. Hence, d(t, x) = d(t, y) = d(t, z) = 2, a contradiction.
Note that |N (W ) \ W | = 3 and that there is a labelling V (G) \ W = {a, b, c, z} such that N (W ) \ W = {a, b, c}, d(a, b) = d(b, c) = d(b, z) = 1, d(c, a) = d(c. Case 3.2: G ∈ {H 14 , H 15 }2and d(a, z) ∈ {1, 2} (see Figure 6(c)Case 3.2: G ∈ {H 14 , H 15 }. Note that |N (W ) \ W | = 3 and that there is a labelling V (G) \ W = {a, b, c, z} such that N (W ) \ W = {a, b, c}, d(a, b) = d(b, c) = d(b, z) = 1, d(c, a) = d(c, z) = 2 and d(a, z) ∈ {1, 2} (see Figure 6(c)).
Notice that |S n−3 | ≤ 2, since for every x ∈ {a, b, c} ∩ S n−. 3r(x, Π) = (1, . . . , 1, 0)Notice that |S n−3 | ≤ 2, since for every x ∈ {a, b, c} ∩ S n−3 , r(x, Π) = (1, . . . , 1, 0).
. B Moreover, / ∈ S N−3, Moreover, b / ∈ S n−3 , otherwise a and c do not belong to S n−3 and we would have r(a, Π) =
So, we can assume without loss of generality that {w 1 , b} ⊆ S 1 . If {a, c} ∩ S n−3 = ∅, then r(w 1 , Π) = r(b, Π) = (1, . . . , 1, 1). Consequently, r(w i , Π) = r(c, Π) = (1, . . . , 1, 2) for every i ∈ {1, . . . , n − 4}, a contradiction. (=⇒) Now assume that G is a graph such that β p (G) = n − 2. By Theorem. c, Π) = (1, . . . , 1, 142and according to Corollary 2, we have n − 4 ≤ τ (G) ≤ n − 2. We distinguish three cases, depending on the cardinality of τ (G)r(c, Π) = (1, . . . , 1, 1). So, we can assume without loss of generality that {w 1 , b} ⊆ S 1 . If {a, c} ∩ S n−3 = ∅, then r(w 1 , Π) = r(b, Π) = (1, . . . , 1, 1). Consequently, r(w i , Π) = r(c, Π) = (1, . . . , 1, 2) for every i ∈ {1, . . . , n − 4}, a contradiction. (=⇒) Now assume that G is a graph such that β p (G) = n − 2. By Theorem 4, τ (G) > n 2 , and according to Corollary 2, we have n − 4 ≤ τ (G) ≤ n − 2. We distinguish three cases, depending on the cardinality of τ (G).
Case 1: τ (G) = n − 2. Thus, according to Proposition 2 and Theorem 4. G ∈ {H 1 , H 2 }Case 1: τ (G) = n − 2. Thus, according to Proposition 2 and Theorem 4, G ∈ {H 1 , H 2 }.
In thid case, from Proposition 8 we deduce that its τ -set W satisfies G[W ] ∼ = K n−3 . We distinguish three cases. 2depending on the cardinality of N (W ) \ WCase 2: τ (G) = n − 3. In thid case, from Proposition 8 we deduce that its τ -set W satisfies G[W ] ∼ = K n−3 . We distinguish three cases, depending on the cardinality of N (W ) \ W .
(W )\W ] is K 3 or P 3 , then τ (G) ≥ n−2, a contradiction. Case 2.1: |N (W ) \ W | = 3. In this case, G[N (W ) \ W ] ∈ {K 3 , P 3 , K 2 + K 1 , K 3 }. If G[NIf G[N (W )\W ] ∼ = K 2 +K 1 , then G ∼ = H 3 , and if G[N (W ) \ W ] ∼ = K 3 , then G ∼ = H 4 .Case 2.1: |N (W ) \ W | = 3. In this case, G[N (W ) \ W ] ∈ {K 3 , P 3 , K 2 + K 1 , K 3 }. If G[N (W )\W ] is K 3 or P 3 , then τ (G) ≥ n−2, a contradiction. If G[N (W )\W ] ∼ = K 2 +K 1 , then G ∼ = H 3 , and if G[N (W ) \ W ] ∼ = K 3 , then G ∼ = H 4 .
and deg(z) = 1, then τ (G) = n − 2, a contradiction. If G[N (W ) \ W ] ∼ = K 2 and deg(z) = 2, then G ∼ = H 5. V \ N (W ). If G[N (W ) \ W ] ∼ = K 2If G[N (W ) \ W ] ∼ = K 2 and deg(z) = 2, then G ∼ = H 6 . Finally, if G[N (W ) \ W ] ∼ = K 2 and deg(z) = 1, then G ∼ = H 7Let z be the vertex in V \ N (W ). If G[N (W ) \ W ] ∼ = K 2 and deg(z) = 1, then τ (G) = n − 2, a contradiction. If G[N (W ) \ W ] ∼ = K 2 and deg(z) = 2, then G ∼ = H 5 . If G[N (W ) \ W ] ∼ = K 2 and deg(z) = 2, then G ∼ = H 6 . Finally, if G[N (W ) \ W ] ∼ = K 2 and deg(z) = 1, then G ∼ = H 7 .
(W ) = {y, z}. If deg(y) = deg(z) = 2, then G ∼ = H 8 . If {deg(y), deg(z)} = {1, 2}, then G ∼ = H 9. Case 2.3: |N (W ) \ W | = 1. Let N (W ) \ W = {x} and V \ NIf deg(y) = deg(z) = 1, then G ∼ = H 10Case 2.3: |N (W ) \ W | = 1. Let N (W ) \ W = {x} and V \ N (W ) = {y, z}. If deg(y) = deg(z) = 2, then G ∼ = H 8 . If {deg(y), deg(z)} = {1, 2}, then G ∼ = H 9 . If deg(y) = deg(z) = 1, then G ∼ = H 10 .
Let W be its τ -set. In this case, from Proposition 8 we deduce that its τ -set W satisfies G[W ] ∼ = K n−4 . Moreover, from Lemmas 3 and 4, we deduce that G does not contain any W -distinguishing vertex and |N (W ) \ W | ≥ 3. Hence, 3 ≤ |N (W ) \ W | ≤ 4. Case. 34We distinguish two cases, depending on the cardinality of ReferencesCase 3: τ (G) = n − 4. Let W be its τ -set. In this case, from Proposition 8 we deduce that its τ -set W satisfies G[W ] ∼ = K n−4 . Moreover, from Lemmas 3 and 4, we deduce that G does not contain any W -distinguishing vertex and |N (W ) \ W | ≥ 3. Hence, 3 ≤ |N (W ) \ W | ≤ 4. We distinguish two cases, depending on the cardinality of References
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| []
|
[
"A three-level multi-continua upscaling method for flow problems in fractured porous media",
"A three-level multi-continua upscaling method for flow problems in fractured porous media"
]
| [
"Maria Vasilyeva ",
"Eric T Chung ",
"Yalchin Efendiev ",
"Aleksey Tyrylgin "
]
| []
| []
| Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method. | 10.4208/cicp.oa-2018-0219 | [
"https://arxiv.org/pdf/1810.01581v1.pdf"
]
| 54,514,280 | 1810.01581 | 6dd6ae81438182c2b15790990babc12441aa2392 |
A three-level multi-continua upscaling method for flow problems in fractured porous media
October 4, 2018
Maria Vasilyeva
Eric T Chung
Yalchin Efendiev
Aleksey Tyrylgin
A three-level multi-continua upscaling method for flow problems in fractured porous media
October 4, 2018
Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method.
upscaling techniques are necessary [19,15,32,26,20]. We will, in this paper, focus on a class of multiscale methods based on local multiscale basis functions. In typical two level methods, multiscale basis functions are constructed locally, namely, within a coarse block or a union of several coarse blocks of an underlying coarse mesh, which does not necessarily resolve any scale. Constructing multiscale basis functions involves solutions, using the fine grid, of some local problems, which can be expensive for the case when coarse grid size is much larger than the fine grid size [9]. Therefore, problems with very large disparate scales require some coarsening techniques or multilevel techniques [21]. The commonly used techniques for such problems are the re-iterated homogenization methods or multilevel multiscale methods [3,22,33,28,23,21,9]. In multilevel multiscale approaches, multiple levels of coarsening are constructed by a recursive application of the basic two level method with the aim of improving computational efficiency. The main advantage of multilevel methods is to avoid solving local problems of large dimensions.
In our previous works, we developed multiscale model reduction techniques based on the Generalized Multiscale Finite Element Method (GMsFEM) for flow in fractured porous media [2,7,16,1]. The general idea of GMsFEM is to design suitable spectral problems on some snapshot spaces to obtain dominant modes of the solutions. These dominant modes are used to construct the required multiscale basis functions [13,14,6,5]. The resulting multiscale space contains basis functions that take into account the microscale heterogeneities as well as high contrast and channelized effects, and the resulting multiscale scale solution provides an accurate and efficient approximation of the fine scale solution. We remark that the GMsFEM is related to the Proper Orthogonal Decomposition (POD) (c.f. [14]) in the way that the GMsFEM constructs multiscale basis functions that optimize an appropriate error within a finite dimensional space. The error of the GMsFEM has a spectral decay and is inversely proportional to the eigenvalues of the spectral problems used for constructing basis functions.
Recently, the authors in [8,10] proposed a new Constraint Energy Minimizing GMsFEM (CEM-GMsFEM) with the aim of finding a multiscale method with a coarse mesh dependent convergence. Constructing the multiscale space starts with an auxiliary space, which consists of eigenfunctions of a local spectral problem, and is defined for each coarse element. Using the auxiliary space, one can obtain the required multiscale basis functions by solving a constraint energy minimization problem. The resulting multiscale basis functions have an exponential decay away from the coarse element for which the basis functions are formulated. Therefore, the multiscale basis functions are only numerically computed in an oversampled region defined by enlarging the target coarse element by a few coarse layers. It has been shown that these basis functions are able to capture high contrast channel effects. Moreover, the convergence of this method depends only on the coarse grid size, and is independent of the scales and the heterogeneities of the coefficients of the PDE. We remark that the size of the oversampling domains depends on the coarse grid size and depends logarithmically on the contrast of the medium. Recently in [10], we introduced a non-local multi-continuum (NLMC) method for problems in heterogeneous fractured media. In the NLMC method, we construct multiscale basis functions based on the solution of some local constrained energy minimization problems as in the CEM-GMsFEM.
One key ingredient of the NLMC method is that we can specify the location of all continua within coarse elements, and we construct these multiscale basis functions so that they have mean value zero in all continua within all coarse elements, except one target continuum within a fixed coarse element. In this case, the degrees of freedoms of the resulting upscaled system have a physical meaning, namely, they are the mean value of the solution on each continuum within each coarse element. The NLMC has similar theoretical properties as that of the CEM-GMsFEM. As we mentioned above, two level multiscale methods can still suffer from large offline computational costs. In this work, we propose a new three level multiscale method based on both the GMsFEM and the NLMC with the aim of taking advantage of both methodologies. Overall speaking, the proposed technique is the three-level scheme (see Figure 1) described as follows:
• fine grid model for fractured porous media,
• intermediate grid model based on the NLMC method,
• coarse grid approximation using the GMsFEM.
Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct an NLMC method using an intermediate grid. As discussed before, the system resulting from the NLMC method gives solutions that have physical meaning, namely, mean values on local continua. We remark that by an intermediate grid, we mean that the grid size is between the fine and the coarse grids. In order to enhance locality, the grid size of the intermediate grid needs
to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of GMsFEM to the NLMC system to obtain a final reduced model. This paper contains several novel ideas. We present an extension of the GMsFEM for the NLMC models and show that the GMsFEM can work with any multicontinuum upscaled model. The NLMC method provides an accurate upscaled multicontinuum approximation that we use for intermediate grid approximation.
The second advantage of the proposed method is the acceleration of the GMsFEM model construction, when the solution of the local spectral problems are computationally expensive due to disparate scales and this requires coarsening [9,21]. Coarsening techniques should provide accurate and fast intermediate grid approximation. For this purpose, the NLMC method is applied for constructing the accurate upscaled intermediate grid model. The paper is organized as follows. In Section 2, we consider a fine grid model to approximate the flow problem in the fractures porous media. In Section 3, we discuss an intermediate grid upscaled model construction using the NLMC method. Next, we present a construction of the multiscale basis functions on an intermediate grid for the GMsFEM in Section 4 to obtain the final reduced model. Finally, we present numerical results and a conclusion in Section 5.
Fine grid model
First, we discuss the fine grid discretization of the flow system. We consider a mixed dimensional mathematical model for flow problem in fractured porous media. A common approach to model fracture media is to consider the fractures as lower-dimensional objects [27,12,17,11]. Let Ω ∈ R d (d = 2,3) be the computational domain for the porous medium and γ ∈ R d−1 be a reduced dimensional domain representing fracture networks. The flow model can be described as follows
a m ∂p m ∂t − ∇ · (b m ∇p m ) + η m σ(p m − p f ) = q m , x ∈ Ω, a f ∂p f ∂t − ∇ · (b f ∇p f ) − η f σ(p m − p f ) = q f , x ∈ γ,(1)a m = c m , a f = d c f , b m = k m /µ, b f = d k f /µ,
where µ is the fluid viscosity, c α , k α are the compressibility and permeability for porous matrix (α = m) and
fractured (α = f ), q α is the source term for α = f, m, d is the fracture thickness, p m is the pressure in the porous matrix denoted by Ω, p f is the pressure in the fractures γ. Coefficients η m and η f depend on mesh parameters and will be described later.
Let T F = ∪ i ς i be the fine grid with triangular or tetrahedral cells for the domain Ω. The fracture mesh, denoted by E γ = ∪ l ι l , is constructed on the fractures domain γ. The coupled system (1) is discretized using the embedded fracture model (EFM) [18,30,29]. For the approximation in space, we apply the cell-centered finite-volume method with two-point flux approximation [18,30,4,31,29]. Thus, we obtain the following discrete problem
a m p n+1 m,i − p n m,i τ |ς i | + j T ij (p n+1 m,i − p n+1 m,j ) + σ il (p n+1 m,i − p n+1 f,l ) = q m |ς i |, ∀i = 1, N m F , a f p n+1 f,l − p n f,l τ |ι l | + n W ln (p n+1 f,l − p n+1 f,n ) − σ il (p n+1 m,i − p n+1 f,l ) = q f |ι l |, ∀l = 1, N f F ,(2)
where T ij = b m |E ij |/∆ ij (|E ij | is the length of facet between cells ς i and ς j , ∆ ij is the distance between midpoint of cells ς i and ς j ), W ln = b f /∆ ln (∆ ln is the distance between points l and n), N m F is the number of cells in T F , N f F is the number of cells related to the fracture mesh E γ , σ il = σ if ι l ⊂ ς i and is zero otherwise.
Here, we choose η m = 1/|ς i |, η f = 1/|ι l | and use an implicit scheme for the time discretization, where n is the number of time steps and τ is the given time step size.
We can write the above scheme as the following system of equations for p n = (p n m , p n f ) T in matrix form
M p n − p n−1 τ + Ap n = F,(3)
where
M = M m 0 0 M f , A = A m + Q −Q −Q A f + Q , F = F m F f , and M m = {m m ij }, m m ij = a m |ς i | i = j, 0 i = j , M f = {m f ln }, m f ln = a f |ι l | l = n, 0 l = n , Q = {q il }, q il = σ i = l, 0 i = l , where A m = {T ij }, A f = {W ln }, F m = {f m i }, f m i = q m |ς i |, F f = {f f l }, f m i = q f |ι l |.
We note that the size of this fine-grid system is
N F = N m F + N f F .
The NLMC on intermediate grid
In this section, we will construct an upscaled system for the fine system (3) on an intermediate grid. In particular, we will construct an upscaled model using the nonlocal multicontinua (NLMC) upscaling approach [10]. In this method, the upscaled coefficients are based on the construction of multiscale basis functions.
To do so, we solve local problems in some oversample local regions subject to the constraints that the mean values of the local solution vanishes in all continua except the one for which it is formulated. It has been shown that these multiscale basis functions have a spatial decay property and separate background medium and fractures. For more details in the derivation, we refer the reader to [10]. Below, we will state a brief discussion of the derivation.
Let T I = ∪ i K i be a structured intermediate grid.
We consider a coarse cell K i and let K + i be its oversampling region obtained by enlarging K i with few coarse cell layers. For the fractures, we write
γ = ∪ L l=1 γ (l) ,
where γ (l) denotes the l-th fracture network and L is the total number of fracture networks. Let γ
(l) j = K j ∩ γ (l) be the fracture inside cell K j ∈ K +
i and L j be the number of fractures in K j . For each K j ⊂ K + i , we therefore need L j + 1 basis functions: one for K j and one for each γ (l) j . Following the framework of [10] and [8], we will construct the required multiscale basis functions by solving a local problem on K + i subject to some constraints to be specified in the following paragraph. We now define the constraints that will be used for multiscale basis construction. We use φ i,0 to denote the basis function corresponding to the porous matrix in the coarse element K i and use φ i,l to denote the basis function corresponding to the l-th continuum within the coarse element K i . We remark that these basis functions are supported in K + i and have zero trace on ∂K + i . The required constraints are defined as follows:
(1) porous matrix in K i , φ i,0 = (φ i,0 m , φ i,0 f ) : Kj φ i,0 m dx = δ i,j , γ (l) j φ i,0 f ds = 0, l = 1, L j . (2) l-th fracture network in K i , φ i,l = (φ i,l m , φ i,l f ): Kj φ i,l m dx = 0, γ (l) j φ i,l f ds = δ i,j δ m,l , l = 1, L j .
We remark that the constraints are defined for each K j ⊂ K + i . To construct the multiscale basis functions with the energy minimizing property, we solve the following local problems in K + i using a fine-grid approximation for flow in fractured porous media presented in Section 2. In particular, we solve the following coupled system in K + i :
A i,+ m + Q i,+ −Q i,+ B T m 0 −Q i,+ A i,+ f + Q i,+ 0 B T f B m 0 0 0 0 B f 0 0 φ m φ f µ m µ f = 0 0 G m G f (4)
with the zero Dirichlet boundary condition on ∂K + i for both φ m and φ f . Here A i,+ m , A i,+ f and Q i,+ denote the parts of the fine-scale matrices that are related to the local domain K + i . Note that we used Lagrange multipliers µ m and µ f to impose the constraints defined above.
For the construction of the multiscale basis function with respect to porous matrix φ i,0 = (φ i,0 m , φ i,0 f ), we set G m = δ i,j and G f = 0. For the multiscale basis function φ i,l = (φ i,l m , φ i,l f ) with respect to the l-th fracture network, we set G m = 0 and G f = δ i,j δ m,l . Combining these multiscale basis functions, we obtain the following multiscale space As an approximation, we use diagonal mass matrix directly calculated on the intermediate grid
V ms = span{(φ i,l m , φ i,l f ), i = 1, N c , l = 0, L i } and the projection matrix R = R mm R mf R f m R f f , where R T mm = φ 0,0 m , φ 1,0 m . . . φ Nc,0 m , R T f f = φ 0,1 f . . . φ 0,L0 f , φ 1,1 f . . . φ 1,L1 f , . . . , φ Nc,1 f . . . φ Nc,L Nc f , R T mf = φ 0,0 f , φ 1,0 f . . . φ Nc,0 f , R T f m = φ 0,1 m . . . φ 0,L0 m , φ 1,1 m . . . φ 1,L1 m , . . . , φ Nc,1 m . . . φ Nc,L Nc m ,
Finally, the resulting upscaled intermediate grid model reads
Mp n −p n−1 τ +Āp n =F ,(5)M = M m 0 0M f ,F = F m F f , whereM m = diag{a m |K i |},M f = diag{a f |γ i |},
The GMsFEM on coarse grid
In this section, we will present a model reduction technique based on the GMsFEM. We will form a reduced model on a coarse grid based on the NLMC system constructed in the previous section. Generally speaking, the GMsFEM is a systematic approach to identify multiscale basis functions via local spectral problems [14,13]. In the original GMsFEM, the method is constructed based on a fine grid discretization of the PDE.
In this paper, we will apply the GMsFEM idea to the system resulting from the NLMC method and this is a new idea. To obtain a reduced system using GMsFEM, we first identify the local matrices from the NLMC system corresponding to a set of overlapping coarse regions, typically called coarse neighborhoods [14]. Then for each coarse neighborhood, we solve a spectral problem using the local matrices, and select the dominant eigenfunctions corresponding to the small eigenvalues. The multiscale basis functions are then obtained by multiplying a suitable partition of unity function to the eigenfunctions. Finally, the GMsFEM system is obtained by forming a suitable projection matrix using the basis functions.
For completeness, we summarize below the main steps in GMsFEM:
Preprocessing (offline stage).
-The construction of the multiscale basis functions in local domains.
-The construction of the coarse grid system. Solver (online stage).
-Solution of the coarse grid system.
Postprocessing.
-Reconstruction of the fine grid solution.
In the following, we will describe the construction of the multiscale basis functions ψ ω k which is supported in a coarse neighborhood ω, where k represents the numbering of the basis functions.
Let T C = ∪ i Θ i be the structured coarse grid and assume that each coarse element is a connected union of fine grid and intermediate grid blocks. We use
{x i } N C vert i=1
to denote the vertices of the coarse mesh T C , where N C vert is the number of coarse nodes. We define the coarse neighborhood of the node x i by
ω i = ∪ j Θ j | x i ∈ Θ j .
We now consider a coarse neighborhood ω i . In order to construct the multiscale space V ωi ms with respect to ω i , we solve following local spectral problem in local domain ω i
AΨ i = λ i SΨ i ,(6)
where the matrix A is the restriction of the matrixĀ in the coarse neighborhood ω i and the matrixĀ is the matrix resulting from the NLMC method (5). Moreover, the matrix S is defined as follows:
S = S m 0 0S f ,S m = {s m ij }, s m ij = b m |K i | i = j, 0 i = j ,S f = {s f ln }, s f ln = b f |γ l | l = n, 0 l = n .
To define the required multiscale space, we choose eigenvectors Ψ i k (k = 1, ..., M i ) corresponding to the smallest M i eigenvalues and set
V C = span{ψ i k = χ i Ψ i k : 1 ≤ i ≤ N C vert and 1 ≤ k ≤ M i },(7)
where χ i are the standard linear partition of unity functions and M i denotes the number of eigenvectors that are chosen for each coarse node i. The construction in (7) Using a single index notation for the basis functions, we may write
V C = span{ψ 1 , ψ 2 , ..., ψ N C }, R T C = [ψ 1 , . . . , ψ N C ] ,
where R C is the projection matrix and N C = N C vert i=1 M i is the size of the coarse grid system. Finally, we can write the GMsFEM system as
M C p n C − p n−1 C τ + A C p n C = F C ,(8)
and p C ∈ V C and p C = i p C,i ψ i (x). In the above system, we have
M C = R CM R T C , A C = R CĀ R T C , F C = R CF , andp = R T C p C is
Numerical results
In this section, we present numerical results for our three level scheme. We consider the problem in domain Ω = [0, 1] × [0, 1]. As model problems, we consider two geometries with different fracture distribution:
• Geometry 1. Domain with 30 fracture lines.
• Geometry 2. Domain with 160 fracture lines.
In Figure 2, we show computational grids for Geometry 1 and Geometry 2. The implementation is based on the open-source simulation library FEniCS, where we use geometry objects and interface to the linear and spectral solvers [24,25]. We construct three grids for multiscale solver:
• Fine level with mesh 200 × 200.
• Intermediate level with mesh 40 × 40. Second row: downscaled fine grid solution.
• Coarse level with coarse grids 5 × 5 and 10 × 10.
For approximation on fine grid, we constrict finite volume approximation using embedded fracture model.
We note that, another approximation techniques can be used, for example, discrete fracture model with Intermediate grid approximation using NLMC method.
||p I −p|| 2 L 2 = K (p K I −p K ) 2 , p K I = 1 |K| K p dx.
In Figures 3 and 4, we present the pressure on mesh 40 × 40 with In Tables 3 and 4 Coarse grid approximation using GMsFEM.
Next, we consider the coarse grid approximation using GMsFEM using intermediate grid upscaled model.
We use an intermediate grid approximation projection matrix for reconstruction of the fine grid solution.
We calculate errors between reference fine grid and GMsFEM solutions on intermediate and fine grids
e IC I = ||p I − p C || L 2 ||p I || L 2 , e IC F = ||p − p F || L 2 ||p|| L 2 ,
where p C is the GMsFEM solution, p F = R T p C is the reconstructed fine grid GMsFEM solution, p is the reference fine grid solution, p I is the intermediate grid cell average for reference fine grid solution p.
We consider two coarse grids: 5×5 and 10×10. In Tables 2 and 3 coarse grid, respectively. We observe that for geometry with larger number of fractures, we should use more multiscale basis functions. For example, we obtain 3.9% of intermediate grid errors for Geometry 1, when we take 4 multiscale basis functions on 10 × 10 coarse grids. We obtain similar errors for Geometry 2, when we take 8 multiscale basis functions.
Finally, we discuss the computational advantages in terms of degrees of freedom. In GMsFEM method, we have offline and online stages. On online stage, we calculate multiscale basis functions and construct coarse grid matrices. On offline stage, we solve coarse grid system. We can consider proposed algorithm as an extension of the GMsFEM for the upscaled multicontinuum models. The advantage of the proposed method We proposed three-level technique for multiscale simulations for fractured porous media. On the fine grid we use embedded fracture model, but another methods can be used, for example, discrete fracture model. On intermediate grid, we use nonlocal multicontimuum method to construct an upscaled model. On coarse grid, we construct multiscale solver based on the Generalized Multiscale Finite Element Method. We perform numerical simulations for three-level method for model problems for two fractures geometries.
Figure 1 :
1Concept of three-level scheme.
whereĀ = RAR T ,p = (p m ,p f ) is the average cell solution on intermediate grid element for porous matrix (p m ) and for fractures (p f ). We can reconstruct the downscale solution by p = R Tp .
and for the right-hand side vectorF m = {q m |K i |},F f = {q f |γ i |}. We remark that the matrix A is non-local and provides a good approximation due to the coupling of various components in the basis construction. The resulting upscaled model has one degree of freedom (DOF) for each fracture network and the size of intermediate grid system is N I = N I cell + N I cell i=1 L i , where N I cell is the number of intermediate grid cells.
yields a counterpart of the continuous basis functions due to the multiplication of local domain eigenvectors with the continuous partition of unity functions.
the reconstructed intermediate grid solution and p = R Tp is the reconstructed fine grid solution.
Figure 2 :
2Computational grids (black color -coarse grid, red color -intermediate grid and blue color -fine grid). Fractures are depicted by white color. Left: Geometry 1 with 30 fracture lines. Right:
Figure 3 :
3Multiscale solutions on mesh 40×40 with K 4 using NLMC model for different time steps t 10 = 0.02, t 30 = 0.06 and t 50 = 0.1 (from top to bottom). Geometry 1. First row: upscaled intermediate grid solution.
unstructured grids. Fine grid for fractures domain for Geometry 2 contains 3216 cells. For Geometry 1, we use grid with 1042 cells for fractures. In Figure 2, the fine grid for Geometry 1 and Geometry 2 is depicted with blue color and contains 40000 cells. The intermediate grid is depicted by red color and contains 1600 cells. By black color, we depict the coarse grid that contains 36 and 121 vertices. Note that DOF C , DOF I and DOF F are the number of degrees of freedom for coarse, intermediate and fine grids approximations. We set following parameters for model problem: a m = 10 −5 , a f = 10 −6 , b m = 10 −6 , b f = 1.0 with σ = 10 −4 . We set p 0 = 0 as initial pressure and zero flux on boundary. We set a source term on the fractures inside cells K = [0.1, 0.15]×[0.05, 0.1] and K = [0.6, 0.65]×[0.9, 0.95] with q = 10 −3 . We simulate t max = 0.1 with 50 time steps.
Figure 4 :
4Multiscale solutions on mesh 40×40 with K 4 using NLMC model for different time steps t 10 = 0.02, t 30 = 0.06 and t 50 = 0.1 (from top to bottom). Geometry 2. First row: upscaled intermediate grid solution. Second row: downscaled fine grid solution. First, we consider relative errors for upscaled multicontinuum model using NLMC method on intermediate grid. To compare the results, we use the relative L 2 errors between fine grid in upscaled intermediate grid models e F I . We calculate errors on intermediate grid (e F I I ) and on fine grid (e F I F )e F I I = ||p I −p|| L 2 ||p I || L 2 , e F I F = ||p −p F || L 2 ||p|| L 2 ,wherep is the upscaled intermediate grid solution,p F = R Tp is the downscaled of fine grid intermediate grid solutionp, p is the reference fine grid solution, p I is the intermediate grid cell average for reference fine grid solution p and
K 4 41042 Figure 5 :
4410425using upscaled model for different time steps t 10 = 0.02, t 30 = 0.06 and t 50 = 0.1 Geometry 1 and Geometry 2, respectively. In the first row, we depict an upscaled medium grid solution. Using projection matrix, we can reconstruct fine grid solution from intermediate grid upscaled model (second row in figures). The fine-scale systems have DOF f = Relative errors vs time for upscaled intermediate grid solution with different number of oversampling layers K s , s = 2, 3, 4 and 6. Left: Geometry 1. Right: Geometry 2. for Geometry 1 and DOF f = 43216 for Geometry 2. Upscaled intermediate grid model has DOF c = 1965 for Geometry 1 and DOF c = 2428 for Geometry 2. NLMC method provides accurate meaningful intermediate grid solution with less then one percent errors on fine and intermediate grids.
10 Figure 6 :
106, we show relative errors on intermediate and fine grids for different number of multiscale basis functions, M . The construction of the multiscale basis functions performed on intermediate grid. For coarse grid approximation with 36 vertices for sufficient number of multiscale basis function, we obtain accurate solution with one percent of errors for Geometry 1 and Geometry 2. For finer coarse grid, we can use smaller number of miltiscale basis functions for accurate approximation. In Figures 6 and 7, we depict the relative errors vs time for GMsFEM with 5 × 5 and 10 × Relative errors vs time for GMsFEM with 5 × 5 coarse grid. First row: e IC I . Second row: e IC F . Left: Geometry 1. Right: Geometry 2.
Figure 7 :
7in the acceleration of the GMsFEM model construction by performing offline stage on the intermediate coarse grid for upscaled model, where nonlocal multicontinuum method used for construction an accurate model. Next, we consider the computational advantages of the offline computations. Let N C vert is the number local domains ω i , i = 1, ..., N C vert . We construct multiscale basis functions in each ω by solution of the Relative errors vs time for GMsFEM with 10 × 10 coarse grid. First row: e IC I . Second row: e IC F . Left: Geometry 1. Right: Geometry 2. local spectral problems. If we perform calculations of the fine grid, the number of degrees of freedom of local spectral problem is DOF ω = N ω F , where for finite volume approximation N ω F = N number of fine grid cells for porous matrix and for fractures grid, respectively. When we perform solution on the local spectral problem on intermediate grid using upscaled model, we have DOF ω = N I,ω cells + N I,ω cells j=1 L j , where N I,ω cells is the number of intermediate grid cells K j in ω and L ω j is the number of fractures in K j ∈ ω. If fine grid is 200×200 and intermediate grid is 40×40, then for local domain ω 26 and performing calculations on the fine grid, we have DOF ω = 6899 with N ωgrid 5 × 5. For same coarse grid and same local domain, for the case of intermediate grid based GMsFEM basis construction, we have DOF ω = 387 with N I,ω cells = 256. Furthermore, construction of the coarse grid system using intermediate upscaled model can also be done much faster. For online computation using GMsFEM on coarse grid 5 × 5, we have DOF C = 720 for 20 multiscale basis functions and for fine grid DOF F = 41042 for Geometry 1.
Table 1 :
1Relative errors for NLMC intermediate grid solution with different number of oversampling layers Geometry 1 with DOF I = 1965 and DOF F = 41042. Right: Geometry 2 with DOF I = 2428 and DOF F = 43216.K s , s = 1, 2, 3, 4 and 6. Left:
Table 2 :
2Relative errors for GMsFEM with 5 × 5 coarse grid solution with different number of multiscale
basis functions M . Left: Geometry 1. Right: Geometry 2.
M DOF C
e IC
I
e IC
F
1
121
48.473 48.616
2
242
15.437 16.173
4
484
3.949
4.531
8
968
1.177
1.446
12
1452
0.367
0.495
M DOF C
e IC
I
e IC
F
1
121
59.190 59.519
2
242
59.124 59.422
4
484
42.111 41.714
8
968
3.171
3.867
12
1452
1.336
1.772
Table 3 :
3Relative errors for GMsFEM with 10 × 10 coarse grid. solution with different number of multiscale basis functions M . Left: Geometry 1. Right: Geometry 2.
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| []
|
[
"Functional Integral in terms of the Field Strength: An Approach to Chiral Symmetry Breaking",
"Functional Integral in terms of the Field Strength: An Approach to Chiral Symmetry Breaking"
]
| [
"Naoki Tanimura [email protected] \nDepartment of Physics\nKyushu University\n812-8581FukuokaJAPAN\n"
]
| [
"Department of Physics\nKyushu University\n812-8581FukuokaJAPAN"
]
| []
| The chiral symmetry breaking in the 4-dimensional QED with the chirally invariant four-fermion interaction is discussed by using a novel path integral expression in terms of the field-strength tensor. In the local potential approximation, we find that the chiral symmetry is spontaneously broken for any nonzero gauge and four-fermion couplings on the tree level of an auxiliary field σ. The present approach allows us to easily include higher orders of the gauge coupling so that the effective potential up to the sixth order is obtained. | 10.1143/ptp.103.665 | [
"https://arxiv.org/pdf/hep-th/9908081v1.pdf"
]
| 16,762,364 | hep-th/9908081 | 7c49f0b9a5964c0e3714f59f6c01f0ae644410a6 |
Functional Integral in terms of the Field Strength: An Approach to Chiral Symmetry Breaking
Aug 1999 April 23, 2017
Naoki Tanimura [email protected]
Department of Physics
Kyushu University
812-8581FukuokaJAPAN
Functional Integral in terms of the Field Strength: An Approach to Chiral Symmetry Breaking
Aug 1999 April 23, 2017arXiv:hep-th/9908081v1 11
The chiral symmetry breaking in the 4-dimensional QED with the chirally invariant four-fermion interaction is discussed by using a novel path integral expression in terms of the field-strength tensor. In the local potential approximation, we find that the chiral symmetry is spontaneously broken for any nonzero gauge and four-fermion couplings on the tree level of an auxiliary field σ. The present approach allows us to easily include higher orders of the gauge coupling so that the effective potential up to the sixth order is obtained.
Introduction
The standard model of elementary particles has had good agreement with experiments, in which masses of quarks and leptons are generated by the Yukawa interactions with the Higgs bosons. There, however, have been no experimental observations of the Higgs boson, and moreover, there is a theoretical problem of the "naturalness." Therefore models without any elementary scalars have been considered [1,2] as effective field theories to understand the origin of fermion masses: Those are generated dynamically through the interaction
g 2 2 (ψψ) 2 + (ψiγ 5 ψ) 2 ,(1.1)
which is suppressed by inverse powers of a high energy scale Λ, at energies much below it, that is, g 2 Λ 2 ∼ O(1). It is called the Nambu-Jona-Lasinio (NJL) model [3] and has been generalized to the one with the gauge interaction [4]:
L = − 1 4 F µν F µν + ψ iγ µ (∂ µ − ieA µ )ψ + g 2 2 (ψψ) 2 + (ψiγ 5 ψ) 2 . (1.2)
Although, in this model, nonperturbative approaches such as the Schwinger-Dyson equation with the ladder approximation [5,6] or renormalization group [7] give a nontrivial phase structure for chiral symmetry in a plane of gauge and four-fermi couplings, they suffer from gauge dependence. In the perturbative framework the inversion method [8] does give a gauge-independent result [9], higher order calculations are tough tasks.
If we employ the perturbative effective action approach to the problem of chiral symmetry breaking(χSB), there is some room for improvement or simplification of the calculation from two viewpoints: (i) In view of the nonperturbative renormalization group approach [7], the critical behavior is successfully described in the local potential approximation: taking the lowest order of derivative expansion in the effective action. We, therefore, expect that it could also be good in the perturbative calculation. (ii) After integrating with respect to matter (fermionic) fields the result must be gauge invariant, that is, be written as a functional of the field-strength tensor F µν . Therefore it is preferable to rewrite the functional integral of the gauge potential A µ into that of F µν . This could be a great help to the gauge-invariant calculation.
Indeed in the lattice gauge theory, a change of variables from gauge potentials to field strengths has been given [10], whose recipe, however, is specialized to the lattice. In the continuum theory, the field-strength formulation has already been proposed [11]. It is, however, based on the special choice of gauge-the coordinate gauge x µ A µ = 0, which cannot fix the gauge at the origin and moreover yields complicated fermionic currents with line integrals in the lagrangian.
In this article we give a perturbative and gauge-independent method for calculating the dynamical fermion mass under the local potential approximation. Inclusion of higher orders of the gauge coupling constant is simpler in this method. We use the functional integration not as a method simply reproducing perturbative diagrams systematically, but as "integration" with nice features: Changing variables and introducing auxiliary variables can easily be made. We first construct a Euclidean path integral expression in terms of the gauge field strength for the gauge sector with a conserved source J µ : We start from the canonical formalism for quantization and find a suitable change of variables, which is the contents of Sec. 2. Second, in Sec. 3 the fermionic partition function, minimally coupled to the gauge potential, in the local potential approximation is obtained by introducing auxiliary fields and by utilizing the Fock-Schwinger proper-time method [12,13]. At last combining the above two partition functions and integrating with respect to the gauge field strength to obtain an effective potential of the dynamical fermion mass. The final Sec. 4 is devoted to a discussion.
Functional Integral in terms of the Field Strength
In this section we construct a Euclidean path integral expression in 4-dimensional abelian gauge theory coupled to a conserved current J µ , based on the canonical formalism [14],
L = − 1 4 F µν F µν + J µ A µ . (2.1)
In order to show the gauge independence manifestly we fix the gauge in terms of an arbitrary real function φ µ (x) satisfying [14,15]
∂ µ φ µ (x) = δ 4 (x); (2.2)
the gauge fixing condition is [14]
d 4 y φ µ (x − y)A µ (y) = 0, (2.3)
which is satisfied by
A µ (x; φ) = A µ (x) − ∂ µ x d 4 y φ ν (x − y)A ν (y),(2.4)
for an arbitrary A µ (x). Here we take a φ µ (x) whose support is 3-dimensional and spacelike:
φ µ (x) = (0, f i (x)δ(x 0 )), φ µ (p) = (0, f i (p)) (2.5)
with
∇ i f i (x) = δ 3 (x), p i f i (p) = −i. (2.6)
If it were not for this restriction, A µ (x; φ) in Eq. (2.4) is nonlocal in time, so that we cannot follow the canonical procedure. (Here and hereafter repeated indices i, j, k, etc., imply the summation over 1 to 3.) The relation (2.4) turns into
A 0 (x; φ) = A 0 (x) + f j (−i∇)Ȧ j (x), (2.7) A i (x; φ) = A i (x) − ∇ i f j (−i∇)A j (x), (2.8) where f j (−i∇)A j (x) ≡ d 3 p (2π) 3 e ip·x f j (p) A j (p) = d 3 y f j (x − y)A j (y) (2.9)
and the time argument has been omitted. Hereafter we use an abbreviation like Eq. (2.9) for simplicity; all the functions of −i∇ including 1/|∇| and 1/∇ 2 should be understood in the momentum space representation. The gauge-fixed variable A i (x; φ) must have two degrees of freedom out of three A i (x)'s, which can be singled out by considering the norm of the functional space
d 3 x δA i (x; φ)δA i (x; φ) = d 3 x δA j (x)M jk (−i∇)δA k (x), (2.10) where M jk (p) ≡ δ jk + i f j * (p)p k − ip j f k (p) + p 2 f j * (p) f k (p). (2.11)
This matrix can be diagonalized as
n i (α) (p)M ij (p)n j * (β) (p) = 1 p 2 | f (p)| 2 0 αβ ,(2.12)
where n k (α) 's are given by
n k (1) (p) ≡ ǫ klm n l * (2) (p) n m * (3) (p), n k (2) (p) ≡ ip k + p 2 f k (p) p 2 (p 2 | f (p)| 2 − 1), (2.13) n k (3) (p) ≡ ip k /|p|. and form an orthonormal base obeying 3 α=1 n j * (α) (p)n k (α) (p) = δ jk , n k * (α) (p) = n k (α) (−p).
(2.14)
In view of Eqs. (2.10) and (2.12), genuine physical variables are given as
A (α) (x) ≡ n k (α) (−i∇)A k (x) = n k (α) (−i∇)A k (x; φ), (α = 1, 2). (2.15)
With these variables the action is
S = d 4 x − 1 4 [∂ µ A ν (x; φ) − ∂ ν A µ (x; φ)] 2 + J µ (x)A µ (x; φ) = d 4 x 1 2 2 α=1 A (α) (x)∇ 2 A (α) (x) + 1 2 Ȧ (1) (x) 2 − 1 2Ȧ (2) (x)∇ 2 | f (−i∇)| 2Ȧ(2) (x) + 1 2 ∇A 0 (x; φ) 2 − A 0 (x; φ) ∇ 2 (∇ 2 | f (−i∇)| 2 + 1)Ȧ (2) (x) (2.16) + J 0 (x)A 0 (x; φ) − J (x)·[A(x; φ)] ,
where [A(x; φ)] in the last line is given by
A i (x; φ) = n i * (1) (−i∇)A (1) (x) (2.17) + n i * (2) (−i∇) + n i * (3) (−i∇) −∇ 2 |f (−i∇)| 2 − 1 A (2) (x).
The Hamiltonian is written as
H(t) = d 3 x 1 2 2 α=1 Π (α) (x) 2 + ∇A (α) (x) 2 + J 0 (x) ∇ 2 (∇ 2 | f (−i∇)| 2 + 1) ∇ 2 Π (2) (x) (2.18) + 1 2 J 0 (x)| f (−i∇)| 2 J 0 (x) + J (x)·[A(x; φ)]
in terms of four dynamical variables, two A (α) (x)'s and their canonical conjugate momenta
Π (1) (x) =Ȧ (1) (x), (2.19) Π (2) (x) =Ȧ (2) (x) − ∇ 2 (∇ 2 | f (−i∇)| 2 + 1) ∇ 2 J 0 (t, x), (2.20)
where A 0 (x; φ) has been eliminated by
∇ 2 (∇ 2 | f (−i∇)| 2 + 1) ∇ 2 | f (−i∇)| 2 Π (2) (x) + 1 | f (−i∇)| 2 A 0 (x; φ) + J 0 (t, x) = 0. (2.21)
The field strengths are given as
E i ≡ F i0 = −n i * (1) (−i∇)Π (1) − n i * (2) (−i∇)Π (2) + f i * (−i∇)J 0 , (2.22) B i ≡ − 1 2 ǫ ijk F jk = |∇|n i (1) (−i∇)A (2) − |∇|n i (2) (−i∇)A (1) . (2.23)
Following the standard canonical procedure we obtain the path integral representation of the partition function Z T [J] = Tr(e −T H ) [14],
Z T [J] = DΠ (α) DA (α) exp d 4 x E i 2 α=1 Π (α) (τ, x)Ȧ (α) (τ, x) − 1 2 2 α=1 Π (α) (τ, x) 2 + ∇A (α) (τ, x) 2 +iJ 4 (τ, x) ∇ 2 (∇ 2 | f (−i∇)| 2 + 1) ∇ 2 Π (2) (τ, x) + 1 2 J 4 (τ, x)| f (−i∇)| 2 J 4 (τ, x) − J (τ, x)·[A(τ, x; φ)] , (2.24) where A(τ, x; φ) is given by Eq. (2.17) with A(x; φ) → A(τ, x; φ), d 4 x E ≡ T 0 dτ d 3 x, J 4 ≡ iJ 0 , (2.25)
and the periodic boundary condition
A (α) (T, x; φ) = A (α) (0, x; φ) should be under- stood.
Our purpose is to rewrite the expression (2.24) of four variables Π α and A α into that of six variables E i and B i . In view of Eq. (2.14) quantities which are proportional
to n i * (3) (−i∇) or n i (3) (−i∇)
are missing in their expressions (2.22) and (2.23). To this end, we introduce ε and β with the aid of the delta functions in the functional measure,
E i = −n i * (1) (−i∇)Π (1) − n i * (2) (−i∇)Π (2) + n i * (3) (−i∇)ε − i f i * (−i∇)J 4 , (2.26) B i = |∇|n i (1) (−i∇)A (2) − |∇|n i (2) (−i∇)A (1) + n i (3) (−i∇)β, (2.27) or equivalently ε = n i (3) (−i∇)[E i + i f i * (−i∇)J 4 ] = 1 |∇| (∇ i E i − iJ 4 ), (2.28) β = n i * (3) (−i∇)B i = 1 |∇| ∇ i B i . (2.29)
The norm of the functional space is given by
d 3 x δE i δE i = d 3 x (δΠ (1) δΠ (1) + δΠ (2) δΠ (2) + δεδε), (2.30) d 3 x δB i δB i = d 3 x (−δA (1) ∇ 2 δA (1) − δA (2) ∇ 2 δA (2) + δβδβ), (2.31) so that DΠ (α) DA (α) DεDβ x δ(ε)δ(β) = DE i DB i x δ(∇ i E i − iJ 4 )δ(∇ i B i ). (2.32)
We then arrived at the desired result
Z T [J] = DE i DB i x δ(∇ i E i − iJ 4 )δ(∇ i B i ) × exp d 4 x E iE i (τ, x)ǫ ijk ∇ j ∇ 2Ḃ k (τ, x) − 1 2 E i (τ, x) 2 + B i (τ, x) 2 + J i (τ, x)ǫ ijk ∇ j ∇ 2 B k (τ, x) , (2.33)
which is apparently free from the choice of f i (x) as expected.
To carry out the integration in the next section, it is convenient to introduce E i as
E i ≡ E i − i ∇ i ∇ 2 J 4 , (2.34) so that ∇ i E i = ∇ i E i − iJ 4 , (2.35) giving Z T [J] = DE i DB i x δ(∇ i E i )δ(∇ i B i ) exp d 4 x E iE i (τ, x)ǫ ijk ∇ j ∇ 2Ḃ k (τ, x) − 1 2 E i (τ, x) 2 + B i (τ, x) 2 − 1 2 J 4 (τ, x) 1 ∇ 2 J 4 (τ, x) + J i (τ, x)ǫ ijk ∇ j ∇ 2 B k (τ, x) . (2.36)
Further introducing an auxiliary field ρ:
1 = Dρ exp − 1 2 d 4 x E ρ + 1 |∇| J 4 2 ,(2.E i = 3 α=1 n i * (α) (−i∇)ε α , B i = 3 α=1 n i (α) (−i∇)β α ,(2.38)
and integrating with respect to ε α 's and β 3 , we obtain
Z T [J] = Dβ α Dρ[Det|∇|] −2 exp d 4 x E − 1 2 2 α=1 1 |∇|β α (τ, x) 2 + β α (τ, x) 2 − 1 2 ρ(τ, x) 2 − J 4 (τ, x) 1 |∇| ρ(τ, x) −J i (τ, x) 1 |∇| n i * (2) (−i∇)β 1 (τ, x) − n i * (1) (−i∇)β 2 (τ, x) . (2.39)
This form with further changes of variables is used in the next section, where we regard the coefficients of the sources in Eq. (2.39) as gauge potentials:
A 4 ≡ 1 |∇| ρ, (2.40) A i ≡ 1 |∇| n i * (2) (−i∇)β 1 − n i * (1) (−i∇)β 2 . (2.41)
Therefore field strengths are given as
F i4 ≡ ∇ i A 4 −Ȧ i = n i * (1) (−i∇) 1 |∇|β 2 − n i * (2) (−i∇) 1 |∇|β 1 − n i * (3) (−i∇)ρ, (2.42) F ij ≡ ∇ i A j − ∇ j A i = ǫ ijk n k (1) (−i∇)β 1 + n k (2) (−i∇)β 2 ,(2.
χSB in QED 4 with the chirally invariant fourfermion interaction
In this section, we consider fermionic system coupled minimally to the "gauge potentials" (2.40) and (2.41) with the chirally invariant four-fermion interaction. The partition function is
Z[A] = DψDψ exp d 4 x E −ψγ µ (∂ µ − iA µ )ψ + g 2 2 (ψψ) 2 + (ψiγ 5 ψ) 2 , (3.1)
where the gauge coupling constant has been absorbed into A µ . Our scenario is as follows: introduce auxiliary fields to cancel the four-fermion interaction, and then integrate with respect to the fermionic fields and finally the gauge field strengths with the aid of the representation in the previous section. The result is a tree potential of the auxiliary field, with which we examine the dynamical mass generation of fermions. After introducing auxiliary fields σ and π, as usual, and integrating with respect to ψ and ψ, we have
Z[A] = DσDπ exp − d 4 x E σ 2 + π 2 2g 2 + ln Det γ µ (∂ µ − iA µ ) + σ + iγ 5 π . (3.2)
Shifting as σ → m + σ ′ and π → π ′ and ignoring σ ′ and π ′ , we obtain the tree level potential of σ(m):
Z[A] 0 = exp − d 4 x E m 2 2g 2 + ln Det γ µ (∂ µ − iA µ ) + m . (3.3)
The exponent of Eq. (3.3) must be a functional of the field strength F µν rather than A µ as far as a regularization preserves the gauge invariance. We here employ the local potential approximation: We adopt the lowest order of derivative expansion, that is, discard any terms with differentials like F µν E F µν , to obtain a polynomial of F µν . This approximation seems to be valid, since we are interested only in a low energy phenomena, the chiral symmetry breaking, where contributions from large p µ should be much less important.
The functional form of the effective action under this approximation can be obtained nonperturbatively by the Fock-Schwinger's proper time method [12,13]:
Z[A] 0 = exp − d D x E m 2 2g 2 + 1 2(2π) D 2 lim s→0 ∞ 0 dτ τ s− D 2 −1 e −τ m 2 G(τ F ) ,(3.4)
where
G(F ) = F + F − coth(F + ) coth(F − ), (3.5) F ± = 1 2 F µν F µν + F µν F µν 2 ± F µν F µν − F µν F µν 2 . (3.6)
To evaluate the τ -integration in Eq. (3.4), we expand G(F ) as
G(F ) = 1 + 1 3 (F 2 + + F 2 − ) − 1 45 (F 2 + + F 2 − ) 2 − 7F 2 + F 2 − + 1 945 2(F 2 + + F 2 − ) 3 − 13F 2 + F 2 − (F 2 + + F 2 − ) + O(F 8 ). (3.7)
We need some regularization: In order to reproduce the NJL result in the limit e → 0, we need an ultraviolet cutoff Λ which is introduced by a modification of the range of the τ -integration to [1/Λ 2 , ∞). However, this proper-time cutoff breaks the gauge invariance in the same way as the momentum-space cutoff, contrary to the dimensional regularization (D = 4−2ǫ). To overcome this difficulty, we employ both regularizations at the same time. We use the cutoff for the zeroth order in the gauge coupling, since it has nothing to do with gauge fields. While, for higher orders, the dimensional regularization is used 2 . The result is
Z[A] 0 = exp − d 4 x E m 2 2g 2 + 1 8π 2 Λ 4 2 1 − m 2 Λ 2 e − m 2 Λ 2 + m 4 2 E 1 (m 2 /Λ 2 ) + 1 3 1 ǫ + ln µ 2 m 2 F µν F µν 2 − 1 45m 4 F µν F µν 2 2 − 7 F µν F µν 4 2 + 2 315m 8 2 F µν F µν 2 3 − 13 F µν F µν 4 2 F µν F µν 2 + O(F 8 ) , (3.8) where E 1 (z) = ∞ 1 dt e −zt t , (3.9) 1 ǫ = 1 ǫ − γ + ln 2π,(3.10)
and µ is a renormalization scale. [Note that E 1 (z) > 0 for any real z(> 0).] Recall that the gauge action is written as (−1/4e 2 bare )F µν F µν to define a renormalized charge such that
1 e 2 R (µ) = Z 3 e 2 R (µ) + 1 12π 2 1 ǫ ,(3.
11)
2 If we adopt the dimensional regularization from the zeroth order we have a different critical coupling due to the lack of a quadratic term of the renormalization scale µ in the effective action. Compare two expressions of the zeroth order effective action, Eq. (3.13) by the proper-time cutoff after expanding in terms of m 2 /Λ 2 ,
V 0 (m) = m 2 2g 2 + m 4 16π 2 Λ 4 m 4 − 2 Λ 2 m 2 − γ + 3 2 + ln Λ 2 m 2 ,
and one by the dimensional regularization,
V 0 (m) = m 2 2g 2 + m 4 16π 2 1 ǫ + ln 2π − γ + 3 2 + ln µ 2 m 2 .
where the first term of the right-hand side is the bare part and Z 3 is the wave function renormalization constant. Now we turn our attention to the functional integration of the gauge field strength. The total partition function is given by combining Eq. (3.8) with the result in the preceding section:
Z[m] = exp − d 4 x E V 0 (m) Dβ α Dρ[Det|∇|] −2 (3.12) × exp − 1 8π 2 d 4 x E 4π 2 e 2 R (µ) + 1 3 ln µ 2 m 2 F µν F µν 2 + higher orders , where V 0 (m) = m 2 2g 2 + 1 8π 2 Λ 4 2 1 − m 2 Λ 2 e − m 2 Λ 2 + m 4 2 E 1 (m 2 /Λ 2 ) ,(3.F µν F µν = 2(S 2 + T 2 + U 2 ), (3.14) F µν F µν = 2(S 2 − T 2 ), (3.15) yielding Dβ α Dρ[Det|∇|] −2 = DS DT DU Det[− E ] −1 , (3.16)
from Eq. (2.44). As for F µν F µν , though it cannot be expressed by the total divergence any more, its expectation value in the space-time integral would vanish,
d 4 x E F µν F µν = d 4 x E 2(S 2 − T 2 ) = 0,(3.Z[x] = {site} 8 (32π) 3 2 [0,∞) 3 ds dt du × exp − v 0 (x) + 16π 2 e 2 R (µ) + 4 3 ln µ 2 xΛ 2 (s 2 + t 2 + u 2 ) − 1 45x 2 4(s 2 + t 2 + u 2 ) 2 − 7(s 2 − t 2 ) 2 + 2 315x 4 8(s 2 + t 2 + u 2 ) 3 − 13(s 2 + t 2 + u 2 )(s 2 − t 2 ) 2 , (3.18)
where the first coefficients are normalization factors to cancel the Gaussian integration; dimensionless parameters, x = m 2 /Λ 2 , s = S/Λ 2 , etc. have been introduced; and
v 0 (x) ≡ 32π 2 V 0 (m) Λ 4 = 16π 2 x g 2 Λ 2 + 2(1 − x)e −x + 2x 2 E 1 (x). (3.19)
Introducing a polar coordinate (r, θ, ϕ) with r 2 = s 2 +t 2 +u 2 and r 2 sin 2 θ cos 2ϕ = s 2 −t 2 and integrating with respect to θ and ϕ, we find
Z[x] = {site} 4π (32π) 3 2 ∞ 0 dr exp [− {v 0 (x) + v F (r; x)}] , (3.20) where v F (r; x) = A(x)r 2 − B(x)r 4 + C(x)r 6 − ln r 2 ,(3.
21)
A(x) = 16π 2 e 2 R (µ)
+ 4 3 ln µ 2 xΛ 2 = 16π 2 e 2 R (Λ) − 4 3 ln x, (3.22) B(x) = 32 675x 2 , (3.23) C(x) = 136 4725x 4 ,(3.24)
and O(r 8 ) terms in the exponent has been neglected. Let us first consider the lowest correction, up to the O(r 2 ) term. The r-integration is analytically performed to give
Z[x] = {site} 2 15/2 π 3 exp[−v(x)] (3.25) where v(x) = v 0 (x) + 3 2 ln A(x) = 4x G + 2(1 − x)e −x + 2x 2 E 1 (x) + 3 2 ln 4π α(Λ) − 4 3 ln x (3.26)
with G = g 2 Λ 2 /4π 2 and α(Λ) = e 2 R (Λ)/4π. The argument of the logarithm becomes negative at the Landau pole x = exp[3π/α(Λ)], but we do not care such a heavy fermion and thus assume that the argument is always positive. The stationary point x * defined by
1 4 ∂v(x) ∂x x=x * = 1 G − e −x * + x * E 1 (x * ) − 3α(Λ) 8(3π − α(Λ) ln x * )x * = 0 (3.27)
should satisfy the stability condition 1 4
∂ 2 v(x) ∂x 2 x=x * = E 1 (x * ) + 9πα(Λ) − 3α(Λ) 2 (ln x * + 1) 8(3π − α(Λ) ln x * ) 2 x * 2 > 0, (3.28)
for which x * < e −1 ≃ 0.368 is a sufficient condition. To solve Eq. (3.27) we set a condition:
1 ≫ −x ln x ≫ (1 − γ)x ≫ x 2 ,(3.29)
which is fulfilled if x < 1. × 10 −2 . (In the actual situation m ∼ 1 MeV and we should take Λ > 1 TeV, a lower bound of compositeness from experiments [16], to give x < 10 −12 .) Under this condition Eq. (3.27) becomes
1 G − 1 − x ln x − 3α(Λ) 8(3π − α(Λ) ln x)x = 0, (3.30)
where we have used the expansion of E 1 (x) for x ≪ 1,
E 1 (x) = −γ − ln x + O(x). (3.31)
For α(Λ) = 0 there exists a nonvanishing solution only if G ≥ 1; therefore the critical coupling G c is 1. The solution for G ≃ 1 is
x * ≃ − G − 1 ln[G − 1]
.
(3.32)
For α(Λ) > 0 the solution is obtained in two separate regions:
(i) x ≪ exp − 3π α(Λ) ; x * ≃ − 3G 8 ln( 3G 8 ) , (3.33) which is independent of α(Λ). (ii) exp − 3π α(Λ) ≪ x < α(Λ)/4; x * ≃ α(Λ)G 8π(1 − G)
.
(3.34)
Thus there always exists a nonvanishing solution for a given G, that is, G c = 0.
[Solutions at several α(Λ)'s are depicted in Fig. 1.] The actual situation, α(Λ) ≃ 1/137, x ≃ 10 −12 , and Λ ≃ 1 TeV, lies in the case (ii) (exp − 3π α(Λ) ≃ 1.7 × 10 −561 ) and from Eq. (3.34)
πG = g 2 Λ 2 4π ≃ 8π 2 x α(Λ) ≃ 1.1 × 10 −8 ; (3.35)
g 2 is highly suppressed even at this scale (Λ ≃ 1 TeV). Now we include the higher order terms. We evaluate the r-integration with the WKB approximation;
∞ 0 dr exp [−v F (r; x)] ≃ (2π) 1 2 exp − v F (r 0 (x); x) + 1 2 ln v ′′ F (r 0 (x); x) ,(3.[P (x) + Q(x)] 1 3 − [P (x) − Q(x)] 1 3 ,(3.Positivity of v ′′ F (r 0 (x); x) v ′′ F (r; x) = 2 15C(x)r 4 − 6B(x)r 2 + A(x) + 1 r 2 > 0 (3.42)Z[x] = {site} 2 10 π 3 exp[−v(x)], (3.43) with v(x) = v 0 (x) + v F (r 0 (x); x) + 1 2 ln v ′′ F (r 0 (x); x) = v 0 (x) + 1 3 + 2 3 A(x)r 0 (x) 2 − 1 3 B(x)r 0 (x) 4 (3.44) + 1 2 ln 12C(x){3 − 2A(x)r 0 (x) 2 + 2B(x)r 0 (x) 4 } 1 − A(x)r 0 (x) 2 + 2Br 0 (x) 4 ,
where Eq. (3.37) has been used. Differentiating this with respect to x gives us the gap equation, whose numerical solution is shown in Fig. 2. In comparison with Fig. 1, there is a large deviation in the region where both x = m 2 /Λ 2 and G = g 2 Λ 2 /4π 2 are small, and the mass is increased by higher order corrections.
polynomial of F 2 + + F 2 − [= F µν F µν /2] and F 2 + F 2 − [= (F µν F µν /4) 2 ]
and integrate with respect to the proper-time τ . The last task is to evaluate the triple integral of s, t, and u. In this article we employ the WKB approximation for it, which, however, cannot be performed in an elementary manner when higher orders are included further. Meanwhile the ordinary perturbative treatment, considering the Gaussian part the kernel and expanding the exponential of higher parts than the third order, can always be ensured.
As for the local potential approximation its efficacy is still open. In order to go beyond the local potential approximation, we must perform the functional integration of S, T , and U, instead of the ordinary integration like (3.18). For example, an O(α(Λ)) quantity after integrating with respect to the gauge fields would be
D p E (2π) D Π(p 2 )(D − 1) = d D p E (2π) D Π(p 2 ) F µν (p)F µν (−p) 2 . (4.3)
The local potential approximation is to expand Π(p 2 ) in terms of p 2 and keep the lowest. Since the change of variables, Eq. (3.14), gives F µν (p)F µν (−p) = S(p)S(−p) + T (p)T (−p) + U(p)U(−p), the leading correction to the local potential approximation is
d D x Π ′ (0)(−S E S − T E T − U E U). (4.4)
Therefore we must calculate the effective potential with the help of the Feynman graphs using the propagators of S, T , and U. Our result indicates that chiral symmetry is always broken for any α(Λ) and G as is seen from Figs. 1 and 2. This seems to be different from previous results [5,6,7,9] which claim that there is a nonzero G c if α(Λ) < π/3. It is, however, too early to conclude, since our present work is restricted only to the tree level of auxiliary fields and thus no higher orders of G are included. As is seen from the Figs. 1 and 2, the small negative correction, linear in G, to dynamical mass could easily swallow the broken region. From a recent study [17], the one-loop inclusion of the auxiliary fields is promising. Thus a further study must be necessary for a definite conclusion.
In QED the functional integral in terms of the field strength can be constructed, since the configuration space of the gauge potential as well as the field strength is trivial enough for the gauge to be completely fixed. It is challenging to generalize Eq. (2.33) to QCD where an obstacle for gauge fixing, the Gribov ambiguity [18], exists. In order to examine the dynamics of chiral symmetry breaking and color confinement in QCD, this must be done and the choice of convenient variables describing the low energy phenomena [19] must be necessary.
These directions of study are in progress.
17) due to a symmetry S ↔ T of the effective action(3.12); the dependence on S and T appears only in the forms S 2 + T 2 and (S 2 − T 2 ) 2 .Note that Z[m] (3.12) is a trivial product of integrals at each space-time point. We evaluate these integrals with the help of the WKB approximation. For a technical reason we discard O(F 8 ) terms in Eq.(3.8). (This enables us to obtain the stationary point analytically. See below.) Discretizing space-time with (lattice spacing) 4 = 32π 2 /Λ 4 and neglecting the irrelevant factor Det[− E ] −1 , we obtain
Figure 1 :
1Squared dynamical mass of fermion m 2 /Λ 2 shown as a function of the fourfermion coupling constant g 2 Λ 2 /4π 2 = G for several fixed gauge coupling constants α(Λ) = e 2 R (Λ)/4π, obtained from Eq. (3.27). where the superscript ′ denotes the r-differentiation and r 0 (x) is a solution of the stationary point equation v ′ F (r; x) = 2 r A(x)r 2 − 2B(x)r 4 + 3C(x)r 6 − 1 = 0, (3.37) which is the cubic equation of r 2 . There exists only one real-positive solution for r 2 ,
Figure 2 :
2Squared dynamical mass of fermion m 2 /Λ 2 shown as a function of the fourfermion coupling constant g 2 Λ 2 /4π 2 = G for several fixed gauge coupling constants α(Λ) = e 2 R (Λ)/4π, obtained from Eq. (3.44). is also guaranteed, since the sufficient condition 9B(x) 2 − 15A(x)C(x) < 0 is fulfilled. [This leads to a similar condition as Eq. (3.41).] The final form of the partition function is
37 )
37so as to cancel the (J 4 ) 2 term in Eq. (2.36), decomposing E i and B i as 1
are F µν F µν and F µν F µν , so we can change the integration variables from β α and ρ to S, T , and U:13)
"higher orders" are those of Eq. (3.8) with F µν → F µν , and F µν are given by Eqs. (2.42)
and (2.43). All variables in Eq. (3.12)
AcknowledgmentsThe author thanks T. Kashiwa and K. Harada for discussions and encouragements and also thanks D. McMullan for discussions.For sufficiently small x, xA(x) ≪ 1 or x ≪ α(Λ)/4π, Eq. (3.44) can be expanded:yielding the gap equationThe stability condition 1 4is fulfilled for x < 1 and α(Λ) < 2065500 2589043 π. Therefore an α-independent solution exists:for x obeying x ≪ α(Λ)/4π and Eq. (3.29). In this case g 2 is also highly suppressed:for x ≃ 10 −12 , that is, Λ ≃ 1 TeV.DiscussionWe give a gauge invariant recipe for calculating the effective action in QED with the four-fermion interaction. We use perturbation and the local potential approximation to study the dynamical fermion mass. In order to include the higher orders of the gauge coupling constant we just expand the completely known function (3.5) into a
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J See For Example, Schwinger, Particles, Sources and Fields. MassachusettsAddison-Wesley Publishing CompanyII123See for example, J. Schwinger, Particles, Sources and Fields Vol. II (Addison- Wesley Publishing Company, Massachusetts, 1973) p. 123.
C Itzykson, J B Zuber, Quantum Field Theory. New YorkMcGraw-Hill100C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980) p. 100.
. M Ishi-I, T Kashiwa, N Tanimura, Prog. Theor. Phys. 100353M. Ishi-i, T. Kashiwa, and N. Tanimura, Prog. Theor. Phys. 100 (1998), 353.
. T Kashiwa, N Tanimura, Phys. Rev. 562281T. Kashiwa and N. Tanimura, Phys. Rev. D56 (1997), 2281.
. E Emilio, M Mintchev, Fortschr. Phys. 32232Ann. Phys.E. d'Emilio and M. Mintchev, Fortschr. Phys. 32 (1984), 473; 503. O. Steinmann, Ann. Phys. 157 (1984), 232.
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. H D I Abarbanel, J N Bartels ; V, Gribov, Nucl. Phys. 1361Nucl. Phys.H. D. I. Abarbanel and J. Bartels, Nucl. Phys. 136 (1978), 237. V. N. Gribov, Nucl. Phys. 139 (1978), 1.
. L Faddeev, A J Niemi, Phys. Rev. Lett. 821624L. Faddeev and A. J. Niemi, Phys. Rev. Lett. 82 (1999), 1624.
| []
|
[
"ESTIMATION OF VEGETATION LOSS COEFFICIENTS AND CANOPY PENETRATION DEPTHS FROM SMAP RADIOMETER AND ICESAT LIDAR DATA",
"ESTIMATION OF VEGETATION LOSS COEFFICIENTS AND CANOPY PENETRATION DEPTHS FROM SMAP RADIOMETER AND ICESAT LIDAR DATA"
]
| [
"M Baur \nInstitute of Geoecology\nUniversity of Bayreuth\nUniversitätsstr. 3095447Bayreuth\n",
"T Jagdhuber \nGerman Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling\n",
"M Link \nGerman Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling\n\nDepartment of Geography\nLMU Munich\nLuisenstr. 3780333Munich\n",
"M Piles [email protected] \nInstitut de Ciencies del Mar CSIC\nPg. Marítim de la Barceloneta 37-4908003Barcelona\n",
"D Entekhabi \nDepartment of Civil and Environmental Engineering\nMIT\nVassar Street 1502139Cambridge\n",
"A Fink \nGerman Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling\n\nInstitute of Geography\nUniversity of Augsburg\nAlter Postweg 11886159Augsburg\n"
]
| [
"Institute of Geoecology\nUniversity of Bayreuth\nUniversitätsstr. 3095447Bayreuth",
"German Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling",
"German Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling",
"Department of Geography\nLMU Munich\nLuisenstr. 3780333Munich",
"Institut de Ciencies del Mar CSIC\nPg. Marítim de la Barceloneta 37-4908003Barcelona",
"Department of Civil and Environmental Engineering\nMIT\nVassar Street 1502139Cambridge",
"German Aerospace Center, Microwaves and Radar Institute\nP.O. BOX 111682234Wessling",
"Institute of Geography\nUniversity of Augsburg\nAlter Postweg 11886159Augsburg"
]
| []
| In this study the framework of the -model is used to derive vegetation loss coefficients and canopy penetration depths from SMAP multi-temporal retrievals of vegetation optical depth, single scattering albedo and ICESat lidar vegetation heights. The vegetation loss coefficients serve as a global indicator of how strong absorption and scattering processes attenuate L-band microwave radiation. By inverting the vegetation loss coefficients, penetration depths into the canopy can be obtained that is displayed for the global forest reservoirs. A simple penetration index is formed combining vegetation heights and penetration depth estimates. The distribution and level of this index reveal that for densely forested areas the soil signal is attenuated considerably, which can affect the accuracy of soil moisture retrievals. | 10.1109/igarss.2017.8127602 | [
"https://arxiv.org/pdf/2012.03318v1.pdf"
]
| 20,090,101 | 2012.03318 | 474744e33a7513483c79e6059d57d1b794919ec0 |
ESTIMATION OF VEGETATION LOSS COEFFICIENTS AND CANOPY PENETRATION DEPTHS FROM SMAP RADIOMETER AND ICESAT LIDAR DATA
M Baur
Institute of Geoecology
University of Bayreuth
Universitätsstr. 3095447Bayreuth
T Jagdhuber
German Aerospace Center, Microwaves and Radar Institute
P.O. BOX 111682234Wessling
M Link
German Aerospace Center, Microwaves and Radar Institute
P.O. BOX 111682234Wessling
Department of Geography
LMU Munich
Luisenstr. 3780333Munich
M Piles [email protected]
Institut de Ciencies del Mar CSIC
Pg. Marítim de la Barceloneta 37-4908003Barcelona
D Entekhabi
Department of Civil and Environmental Engineering
MIT
Vassar Street 1502139Cambridge
A Fink
German Aerospace Center, Microwaves and Radar Institute
P.O. BOX 111682234Wessling
Institute of Geography
University of Augsburg
Alter Postweg 11886159Augsburg
ESTIMATION OF VEGETATION LOSS COEFFICIENTS AND CANOPY PENETRATION DEPTHS FROM SMAP RADIOMETER AND ICESAT LIDAR DATA
Index Terms -Vegetation attenuationloss coefficientscanopy penetrationSMAPICESatmulti-sensor
In this study the framework of the -model is used to derive vegetation loss coefficients and canopy penetration depths from SMAP multi-temporal retrievals of vegetation optical depth, single scattering albedo and ICESat lidar vegetation heights. The vegetation loss coefficients serve as a global indicator of how strong absorption and scattering processes attenuate L-band microwave radiation. By inverting the vegetation loss coefficients, penetration depths into the canopy can be obtained that is displayed for the global forest reservoirs. A simple penetration index is formed combining vegetation heights and penetration depth estimates. The distribution and level of this index reveal that for densely forested areas the soil signal is attenuated considerably, which can affect the accuracy of soil moisture retrievals.
INTRODUCTION
As a very fundamental assumption all soil moisture retrieval algorithms are dependent on a sufficient soil signal after radiating trough the vegetation canopy [1]. Within the widely used -model [2], a zeroth order solution of the radiative transfer equation, the attenuation due to the vegetation canopy is expressed using the vegetation optical depth and the single scattering albedo . Besides the quantification of canopy attenuation both parameters can have biophysical importance and can be empirically linked to vegetation characteristics, like vegetation biomass [3] or vegetation water content [4]. In the following the -model will be used as a basis to derive vegetation loss coefficients and penetration depths into canopy. In a next step global L-band penetration will be calculated and analyzed.
DEFINITION OF VEGTATION LOSS COEFFICIENTS AND PENETRATION DEPTH
Generally microwave radiation within a vegetation medium is attenuated in proportion to the mediums extinction coefficient [ −1 ], which is composed out of scattering and absorption losses, represented by their loss coefficients [ −1 ] and [ −1 ]. Hence the extinction coefficient is defined as [5,6],
= + .(1)
and can be derived using the loss coefficients. The single scattering albedo [-] is derived from the scattering loss relative to the total extinction [7],
= = + .(2)
The vegetation optical depth [-] can be linked to in the following manner [8],
= • ℎ,(3)
with h [m] being the layer thickness, or vegetation height. This quantity describes total attenuation for a vegetation layer, dependent on and h. Following definitions in (1) to (3) all loss coefficients can be derived as:
= ℎ , (4) = ℎ • ,(5)= − = ℎ (1 − ).(6)
Inversion of the vegetation loss coefficients leads to vegetation/canopy penetration depth , a depth at which the intensity of the electromagnetic wave has fallen to 1/e of its initial intensity due to attenuation within the vegetation layer [9]. Canopy penetration depths
[m], [m],
[m] can be calculated for all three loss coefficients:
= 1 ,(7)= 1 ,(8)= 1 .(9)
Equation (9) and (10) indicate rather special cases of penetration depth compared to (8), as attenuation normally is caused by both, scattering and absorption, within the canopy and not only by one of the two. Dividing by h can give an estimate (index), if the microwave signal penetrates through the vegetation layer to the soil, or if the signal decays within the canopy due to attenuation.
Index of penetration
[-] → ℎ = 1(10)
Index of penetration values lower than one indicate that the signal is attenuated within the vegetation layer to an intensity below 1/e. Values higher than one indicate penetration through the vegetation layer. This index cannot give a definite answer, if a sufficient signal for the retrieval of soil parameters is available, but indicate that values might be spurious and should be quality flagged.
ESTIMATION OF LOSS COEFFICIENTS FROM SMAP AND ICESAT DATA
Both and can be estimated from satellite data using different retrieval approaches. In this case they were retrieved from SMAP radiometer data using a multi temporal dual channel algorithm (MT-DCA) [10]. Lidar vegetation heights were accessed from ICESat GLAS sensor [11]. They were processed for the SMAP active-passive period from 13.04. to 07.07.2015 and averaged over this time period. The input parameters for loss and penetration depth estimation are shown in Fig. 1. has highest values for tropical forest and shows accordance with vegetation height.
indicates high values in artic regions, what can be influenced by freeze-thaw dynamics, and in river deltas. Fig. 2 shows global plots of mean vegetation loss coefficients , and . All three parameters have different spatial patterns, where is mainly driven by , as it is one order of magnitude bigger than . The extinction coefficient generally shows high values for boreal forest areas. In addition to that there are several extinction hotspots e.g. the Gran Chaco, Southern Horn of Africa and Yucatán peninsula. All three parameters are dependent on seasonal dynamics and change of vegetation cover (especially crop cycle) and therefore only valid for the active-passive period of the SMAP mission. For further analysis of the presented methods, C-band data from AMSR-E could be used to compare vegetation loss/attenuation results from different frequencies. This can give further insights on the frequency dependence of the loss coefficients and eventually and .
PENETRATION DEPTHS AND A PENETRATION INDEX FOR GLOBAL FOREST AREAS
Vegetation loss coefficients were inverted to estimate the penetration depth into the vegetation canopy. Penetration depths have to be used with care, as they assume the target coefficients to be fully representative and constant for the entire attenuating vegetation layer. Hence penetration depths should have a higher representativeness for vegetated areas with sufficiently closed canopy. Therefore penetration depths in Fig. 3 were only calculated for the global forest areas indicated by the IGBP landcover classes 1-5 (evergreen needleleafevergreen broadleafdeciduous needleleafdeciduous broadleafmixed forest). Fig. 4 Distinct growth stages during the acquisition time could be a possible reason. Dividing by h can give an estimate (index) if the microwave signal penetrates through the vegetation layer to the soil, or if the signal is lost in the canopy due to attenuation. Values lower than one indicate that the signal is attenuated to an intensity below 1/e within the vegetation layer. Values higher than one indicate significant penetration through the vegetation canopy. Fig. 5 shows fairly well penetration globally, with exception of dense tropical forest. Furthermore the boxplots provided in Fig. 6 reveal concurrent results, but visualize a considerable spread for all classes with parts of the distributions having values below one (6.25% of all valid pixels show values below 1).
Fig. 3. Mean penetration depths
and for IGBP classes 1-5 (forested areas), inverted from the corresponding loss coefficients. The mean was calculated for SMAP activepassive period. Values lower than one indicate that the signal is attenuated to an intensity below 1/e within the vegetation layer. Values higher than one indicate significant penetration through the vegetation canopy. Fig. 5 shows fairly well penetration globally, with exception of dense tropical forest. Furthermore the boxplots provided in Fig. 6 reveal concurrent results, but visualize a considerable spread for all classes with parts of the distributions having values below one (6.25% of all valid pixels show values below 1). Nevertheless one may conclude, that at L-band even with dense vegetation cover a sufficient soil signal is obtained for most areas (93.75%)) and can be used for soil parameter retrieval purposes.
Fig. 1 .
1Global input data. and derived from SMAP time series using an MT-DCA algorithm and lidar vegetation heights h from ICESat lidar sensor.
Fig. 4 .
4Box-plots showing the median, quantiles, maximum and minimum values for penetration depths and . Within the whiskers 99.3% of the data is located, while outliers are not displayed.
Fig. 5 .
5Index areas). Values above one indicate a significant penetration through the vegetation into the soil, while values below one indicate an attenuation of the signal to a value below 1/e within the vegetation.
Fig. 6 .
6Boxplots showing the median, quantiles, maximum and minimum values for the penetration index. Within the Whiskers 99.3% of the data is located, while outliers are not displayed. The red line and areas above indicate that canopy penetration to the underlying soil occurs.
shows the distributions of the penetrations depths as boxplots. Considering the IGBP classes, all show fairly similar values, with deciduous broadleaf forests having the biggest spread Fig. 2. Mean vegetation loss coefficients , and calculated from SMAP vegetation optical depth ( ), single scattering albedo ( ) and lidar vegetation heights h. The mean was calculated for SMAP active-passive period.
Passive microwave remote sensing of soil moisture. E G Njoku, D Entekhabi, Journal of Hydrology. 184E.G. Njoku and D. Entekhabi, "Passive microwave remote sensing of soil moisture," Journal of Hydrology, 184, pp. 101-129, October 1996.
Passive microwave remote sensing of soil moisture under vegetation canopies. T J Jackson, T Schmugge, J R Wang, Water Resources Research. 18T.J. Jackson, T. Schmugge and J.R. Wang, "Passive microwave remote sensing of soil moisture under vegetation canopies," Water Resources Research, 18, pp. 1137-1142, August 1982.
Multifrequency microwave emission for estimating optical depth and vegetation biomass. S Paloscia, E Santi, P Pampaloni, S Pettinato, 2016 IEEE International Geoscience and Remote Sensing Symposium (IGARSS). BeijingS. Paloscia, E. Santi, P. Pampaloni and S. Pettinato, "Multifrequency microwave emission for estimating optical depth and vegetation biomass," 2016 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Beijing, pp. 5296-5299, July 2016.
Seasonal parameterizations of the tau-omega model using the ComRAD ground-based SMAP simulator. P O'neill, A Joseph, P Srivastava, M Cosh, R Lang, 2014 IEEE Geoscience and Remote Sensing Symposium. Quebec CityP. O'Neill, A. Joseph, P. Srivastava, M. Cosh and R. Lang, "Seasonal parameterizations of the tau-omega model using the ComRAD ground-based SMAP simulator," 2014 IEEE Geoscience and Remote Sensing Symposium, Quebec City, pp. 2423-2426, 2014.
Evaluating the First-Order Tau-Omega Model of Terrestrial Microwave Emission. B K Hornbuckle, T L Rowlandson, IGARSS 2008 -2008 IEEE International Geoscience and Remote Sensing Symposium. BostonB.K. Hornbuckle and T.L. Rowlandson, "Evaluating the First- Order Tau-Omega Model of Terrestrial Microwave Emission," IGARSS 2008 -2008 IEEE International Geoscience and Remote Sensing Symposium, Boston, pp. 193-196, 2008.
A composite discrete-continuous approach to model the microwave emission of vegetation. J P Wigneron, J C Calvet, A Chanzy, O Grosjean, L Laguerre, IEEE Transactions on Geoscience and Remote Sensing. 331J.P. Wigneron, J.C. Calvet, A. Chanzy, O. Grosjean and L. Laguerre, "A composite discrete-continuous approach to model the microwave emission of vegetation," IEEE Transactions on Geoscience and Remote Sensing, 33 (1), pp. 201-211, Jan 1995.
Simulating Lband emission of forests in view of future satellite applications. P Ferrazzoli, L Guerriero, J P Wigneron, IEEE Transactions on Geoscience and Remote Sensing. 4012P. Ferrazzoli, L. Guerriero and J. P. Wigneron, "Simulating L- band emission of forests in view of future satellite applications," IEEE Transactions on Geoscience and Remote Sensing, 40 (12), pp. 2700-2708, Dec 2002.
Multiple Scattering Effects With Cyclical Correction in Active Remote Sensing of Vegetated Surface Using Vector Radiative Transfer Theory. T H Liao, S B Kim, S Tan, L Tsang, C Su, T J Jackson, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing. 94T.H. Liao, S.B. Kim, S. Tan, L. Tsang, C. Su and T.J. Jackson, "Multiple Scattering Effects With Cyclical Correction in Active Remote Sensing of Vegetated Surface Using Vector Radiative Transfer Theory," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 9 (4), pp. 1414-1429, April 2016.
Penetration depth as a DInSAR Observable and Proxy for Soil Moisture. M Nolan, D R Fatland, IEEE Transactions on Geoscience and Remote Sensing. 413M. Nolan, D.R. Fatland, "Penetration depth as a DInSAR Observable and Proxy for Soil Moisture," IEEE Transactions on Geoscience and Remote Sensing, 41 (3), pp. 532-537, March 2003.
Vegetation optical depth and scattering albedo retrieval using time series and dual-polarized L-band radiometer observations. A G Konings, M Piles, K Rötzer, K A Mccoll, S K Chan, D Entekhabi, Remote Sensing of the Environment. 172A. G. Konings, M. Piles, K. Rötzer, K.A. McColl, S.K. Chan, D. Entekhabi, "Vegetation optical depth and scattering albedo retrieval using time series and dual-polarized L-band radiometer observations," Remote Sensing of the Environment, 172, pp. 178- 189, January 2016.
Mapping forest canopy height globally with spaceborne lidar. M Simard, N Pinto, J B Fisher, A Baccini, Journal of Geophysical Research. 116BiogeosciencesM. Simard, N. Pinto, J.B. Fisher, A. Baccini, "Mapping forest canopy height globally with spaceborne lidar," Journal of Geophysical Research: Biogeosciences, 116, pp. 2156-2202, November 2011.
| []
|
[
"Mass Transportation Proofs of Free Functional Inequalities, and Free Poincaré Inequalities",
"Mass Transportation Proofs of Free Functional Inequalities, and Free Poincaré Inequalities"
]
| [
"Michel Ledoux \nInstitut de Mathématiques de Toulouse\nUniversité de Toulouse\nF-31062ToulouseFrance\n\nIMAR 21\nIonel Popescu Georgia Institute of Technology\n686 Cherry Street, Calea Grivitei Street, Sector 130332, 010702Atlanta, BucharestGAUSA, Romania\n"
]
| [
"Institut de Mathématiques de Toulouse\nUniversité de Toulouse\nF-31062ToulouseFrance",
"IMAR 21\nIonel Popescu Georgia Institute of Technology\n686 Cherry Street, Calea Grivitei Street, Sector 130332, 010702Atlanta, BucharestGAUSA, Romania"
]
| []
| This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R + with the reference example of the Marcenko-Pastur distribution. arXiv:0903.3761v1 [math.FA] | 10.1016/j.jfa.2009.03.011 | [
"https://arxiv.org/pdf/0903.3761v1.pdf"
]
| 16,007,878 | 0903.3761 | e9a2afcc7e43b82be60ae1cdf427f97223d69f30 |
Mass Transportation Proofs of Free Functional Inequalities, and Free Poincaré Inequalities
Michel Ledoux
Institut de Mathématiques de Toulouse
Université de Toulouse
F-31062ToulouseFrance
IMAR 21
Ionel Popescu Georgia Institute of Technology
686 Cherry Street, Calea Grivitei Street, Sector 130332, 010702Atlanta, BucharestGAUSA, Romania
Mass Transportation Proofs of Free Functional Inequalities, and Free Poincaré Inequalities
This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R + with the reference example of the Marcenko-Pastur distribution. arXiv:0903.3761v1 [math.FA]
Introduction
A distinguished role in the world of functional inequalities is played by the logarithmic Sobolev (Log-Sobolev) inequality and the Talagrand or transportation cost inequality. There is an extensive literature dedicated to these inequalities in the classical setting of Euclidean and Riemannian spaces (cf. e.g. [2], [23], [29], [32]).
Given a probability measure ν on R d , the transportation cost inequality states that for some ρ > 0 and any other probability measure µ on R d , ρ W 2 2 (µ, ν) ≤ E(µ|ν).
(T (ρ))
Here W 2 (µ, ν) is the Wasserstein distance between µ and ν of finite second moment defined by W 2 (µ, ν) = inf π∈Π(µ,ν) |x − y| 2 π(dx, dy) 1/2 with Π(µ, ν) denoting the set of probability measures on R 2d with marginals µ and ν and E(µ|ν) = log dµ dν dµ is the relative entropy of µ with respect to ν if µ << ν and +∞ otherwise. The Log-Sobolev inequality is that for any µ E(µ|ν) ≤ 1 2ρ I(µ|ν) (LSI(ρ))
where
I(µ|ν) = ∇ log dµ dν 2 dµ
is the Fisher information of µ with respect to ν which is defined in the case µ << ν with dµ dν being differentiable. A more subtle inequality is the HWI inequality relating entropy ( notice that E(µ|ν) is H(µ|ν) in [25] which explains the H), Wasserstein distance W, and Fisher information I E(µ|ν) ≤ I(µ|ν) W 2 (µ, ν) − ρ 2 W 2 2 (µ, ν).
(HW I(ρ))
Poincaré's inequality in this classical context is that for any compactly supported and smooth function ψ on R d , ρ Var µ (ψ) ≤ |∇ψ| 2 µ(dx) (P (ρ)) where Var µ (ψ) = ψ 2 (x)µ(dx) − ψ(x)µ(dx)) 2 is the variance of ψ with respect to µ.
Starting with Gaussian measures ( [14], [28]), these inequalities were established for measures on R d with strictly convex potentials by the Bakry-Émery criterion ( [2], [23], [29], [32]). More precisely, if ν(dx) = e −V (x) dx, with V (x) − ρ|x| 2 convex on R d for some ρ > 0, both T (ρ) and LSI(ρ) hold true. Otto and Villani generated interest in this topic through their remarkable paper [25], in which they showed that the logarithmic Sobolev inequality implies the trasportation inequality, in a rather general setting. This connection was actually put further through the stronger HW I(ρ) inequality, which was shown in [25] to be valid in the case V (x) − ρ|x| 2 is convex for some ρ ∈ R, When ρ > 0, LSI(ρ) is a consequence of HW I(ρ). Subsequently the main result from [25] was simplified and extended, for example [5] and recently [13] to mention only two sources. Another interesting connection in these families of functional inequalities is that any of T (ρ), LSI(ρ) or HW I(ρ) imply the Poincaré inequality P (ρ).
The work [25] by Otto and Villani input in a powerful way the use of mass transportation ideas in the context of functional inequalities. Starting from this, Cordero-Erausquin used in [9] direct convexity arguments combined with mass transport methods to reprove the Log-Sobolev, transportation and HWI inequalities for measures with strictly convex potentials. The strategy is going back to the original approach of [28] to the transportation inequality (see also [4]).
In the world of free probability, as it was shown by Ben Arous and Guionnet in [1], one can realize the free entropy as the rate function of the large deviations for the distribution of eigenvalues of some n × n complex random matrix ensembles (see also [19]). To wit a little bit here, let V : R → R be a nice function with enough growth at infinity and define the probability distribution For a measure µ on R, its the logarithmic energy with external field V is defined by
E(µ) = V (x)µ(dx) −
log |x − y| µ(dx)µ(dy).
The minimizer of E(µ) over all probability measures on R is exactly the measure µ V . From [1] we learned that the distributions of {µ n } n≥1 under P n satisfy a large deviations principle with scaling n 2 and rate function given by
R(µ) = E(µ) − E(µ V )
The example of the quadratic potential V (x) = x 2 defining the paradigmatic Gaussian Unitary Ensemble in random matrix theory gives rise to the celebrated semicircular law as equilibrium measure. Within this random matrix framework, if V (x) − ρx 2 is smooth and convex for some ρ > 0, then the function φ(M ) = Tr n (V (M )) is strongly convex (Φ(M ) − nρ|M | 2 is convex) on R n 2 = H n . An application of the classical LSI(nρ) on H n for large n was used by Biane [3] to prove a Log-Sobolev inequality in the context of one-dimensional free probability which holds (cf. [18]) in the following form
E(µ) − E(µ V ) ≤ 1 4ρ I(µ) (1.1)
for any probability measure µ on R whose density with respect to the Lebegue measure is in L 3 (R), where
I(µ) = Hµ(x) − V (x) 2 µ(dx)
with Hµ = 2 1
x−y µ(dx) being the Hilbert transform of µ. More precisely, Biane and Voiculescu used the free Ornstein Uhlenbeck process and the complex Burger equation. Using the large random matrix strategy, Hiai Petz and Ueda [18] reproved and extended the result of Biane and Voiculescu in the following form. If V (x) − ρx 2 is convex for some ρ > 0, then for every probability measure µ on R, ρ W 2 2 (µ, µ V ) ≤ E(µ) − E(µ V ). (1.2) Later, the first author [24] gave a simpler proof of (1.1) and (1.2) based on a free version of the geometric Brunn-Minkowski inequality obtained as a random matrix limiting case of its classical counterpart. He also showed the free analog of the Otto-Villani theorem indicating that the free Log-Sobolev inequality implies the free transportation inequality (1.2).
The first scope of this paper is to provide direct proofs of the preceding functional inequalities in free probability without random matrix approximation. The second author of this paper in [26] gave a simple proof of the transportation inequality (1.2) on the same line of ideas as in [28] for the classical case where random matrix theory is entirely avoided.
In this paper, following the approach of Cordero-Erausquin [9] (see also [4]), we use a combination of mass transport and convex analysis which apply to strictly convex potentials. The methods allow us besides to enlarge the class of potentials under consideration, in particular in instances which lack a proper random matrix approximation. For example, we cover potentials V on the line such that V (x) − ρ|x| p is convex for some ρ > 0 and p > 1 as well as a class of bounded perturbations of convex potentials. Using this approach, we present here an HWI free inequality for various cases of potentials. For the case V (x) − ρx 2 convex for some ρ ∈ R, this is
E(µ) − E(µ V ) ≤ I(µ) W 2 (µ, µ V ) − ρ W 2 2 (µ, µ V ). (1.3)
Also a Brunn-Minkovski inequality receives a direct proof as well.
One interesting byproduct of our method is that some constants may be shown to be sharp. For the case of a quadratic V , equations (1.1), (1.2) and (1.3) are sharp.
Another topic discussed here in Section 3 is a free form of the transportation inequality which does not depend on the potential and that might be thought of as a version of the celebrated Pinsker inequality comparing total variation distance and entropy between probability measures. As opposed to the classical case, the free counterpart is more delicate.
The second part of this work is devoted to free one-dimensional Poincaré inequalities. Using random matrix approximations and the classical Poincaré inequality, we first give an ansatz to what could be a possible Poincaré inequality in the free probability world. In the case of V (x) − ρx 2 convex for some ρ > 0, such that the measure µ V has support [−1, 1], this states as,
φ (x) 2 µ V (dx) ≥ ρ 2π 2 1 −1 1 −1 φ(x) − φ(y) x − y 2 1 − xy √ 1 − x 2 1 − y 2 dxdy,(1.4)
for any smooth function φ on the interval [−1, 1]. There is also a second version of the Poincaré which is discussed in [3] for the case of the semicircular law. This inequality has a natural meaning in the context of free probability as the derivative ∇φ of a function from the classical P (ρ) is replaced by the noncommutative derivative φ(x)−φ(y)
x−y , and thus our second version takes the form
φ(x) − φ(y) x − y 2 µ(dx)µ(dy) ≥ C Var µ (φ) for every φ ∈ C 1 0 (R). (1.5)
As opposed to (1.4) which requires certain conditions on the measure µ V , it turns out that (1.5) is always satisfied for any compactly supported measure µ with some constant. As was shown in [3] for the semicircular law, one can completely characterize the distribution in terms of the constant C.
After the use of convexity, inequality (1.4) may actually be interpreted as a spectral gap as follows. On
L 2 1 [−2,2] (x)dx √ 4−x 2
take the Jacobi operator
Lf = −(1 − x 2 )f (x) + xf (x)
and the counting number operator defined by
N T n = nT n
where T n are the Chebyshev polynomials of the first kind, which are orthogonal in
L 2 1 [−2,2] (x)dx √ 4−x 2
. Then, (1.4) for V (x) = x 2 /2 is equivalent to L ≥ N.
Inequality (1.5) in the case of V (x) = x 2 /2 can also be seen as the spectral gap for the counting number operator on L 2 1 [−2,2] (x) √ 4 − x 2 dx with respect to the basis given by the Chebyshev polynomials of second kind. A more general situation is discussed in Section 9 which includes both versions of the Poincaré inequalities.
As we mentioned already, in the classical setting, the Log-Sobolev and the transportation inequality imply the Poincaré inequalities. We do not have a satisfactory picture of these implications in the free context, for any of the two versions of the Poincaré inequality discussed here.
In the final part, we investigate the preceding families of functional inequalities for probability measures supported on the positive real axis. The random matrix context is the one of Wishart ensembles with reference measure the Marcenko-Pastur distribution as opposed to the semicircular law, and the free functional inequalities correspond formally to the case of potentials V (x) = rx − s log(x) for r > 0, s ≥ 0 on R + . Using the mass transportation method, we prove transportation, Log-Sobolev and HWI inequalities which were not investigated previously. A version of the Poincaré inequality is also discussed.
The structure of the paper is as follows. Sections 2, 4, 5 and 6 deal with the mass transportation proofs of respectively the transportation, Log-Sobolev, HWI and Brunn-Minkowski inequalities. Section 3 studies transportation inequalities which involve some metric on the probabilities and which are independent of the potential V . Sections 7 and 8 are devoted to the two versions of the Poincaré inequality in the free context, related in Section 9 through Jacobi operators. Section 10 investigates the preceding inequalities with respect to the Marcenko-Pastur distribution and its convex extensions.
Transportation Inequality
Throughout this paper we consider lower semicontinuous potentials V :
R → R such that lim |x|→∞ V (x) − 2 log |x| = ∞. (2.1)
For a given Borel set Γ ⊂ R, denote by P(Γ) the set of probability measures supported on Γ. The logarithmic energy with external potential V is defined by
E V (µ) := V (x)µ(dx) − log |x − y| µ(dx)µ(dy).
whenever both integrals exist and have finite values. In particular for measures µ which have atoms, E V (µ) = +∞ because the second integral is +∞. It is known (see [27] or [11]) that under condition (2.1) there exists a unique minimizer of E V in the set P(R) and the solution µ V is compactly supported. The variational characterization of the minimizer µ V (cf. [27,Theorem 1.3]) is that for a constant C ∈ R,
V (x) ≥ 2 log |x − y| µ V (dy) + C for quasi-every x ∈ R V (x) = 2 log |x − y| µ V (dy) + C for quasi-every x ∈ supp(µ V ), (2.2)
where supp(µ V ) stands for the support of µ. If µ is such that E V (µ) < ∞, then Borel quasi-everywhere sets have µ measure 0 and thus the properties above hold almost surely with respect to µ. For simplicity of the notation, we will drop the subscript V from E V unless the dependence of the potential has to be highlighted. Now we summarize some known facts about the equilibrium measure and its support as one can easily deduce them from [27,Chapter IV] and [11,Chapter 6].
Theorem 1.
1. Let V be a potential satisfying (2.1) and
α = 0, β ∈ R. Set V α,β (x) = V (αx + β). Then, µ V α,β = ((id − β)/α) # µ V and E V (µ V ) = E V α,β (µ V α,β ) − log |α|. (2.3)
2. If V is convex satisfying (2.1), then the support of the equilibrium measure µ V consists of one interval [a, b] where a and b solve the system
1 2π b a V (x) x−a b−x dx = 1 1 2π b a V (x) b−x x−a dx = −1.
(2.4)
3. Let V be either a C 2 satisfying (2.1) whose equilibrium measure has support [a, b]. Then the equilibrium measure µ V has density g(x), given by
g(x) = 1 [a,b] (x) (x − a)(b − x) 2π 2 b a V (y) − V (x) (y − x) (y − a)(b − y) dy. (2.5) 4. If V is C 2 , then V (x) = p.v. 2 x − y µ V (dx) for µ V − a.s. all x ∈ supp(µ V ), (2.6)
where p.v. stands for the principal value integral. Notice that the principal value makes sense as µ V has a continuous density.
We mention as a basic example that if V (x) = ρx 2 is quadratic, then µ V is the semicircular law
µ V (dx) = 1 [− √ 2/ρ, √ 2/ρ] (x) 2ρ − ρ 2 x 2 dx 2π .
In this work, for p ≥ 1, we use W p (µ, ν) for the Wasserstein distance on the space of probability measures on R defined as
W p (µ, ν) = inf π∈Π(µ,ν) |x − y| p π(dx, dy) 1/p (2.7)
with Π(µ, ν) denoting the set of probability measures on R 2 with marginals µ and ν. Note here that if θ is the (non-decreasing) transport map such that θ # µ = ν, then
W p p (µ, ν) = θ(x) − x p ν(dx). (2.8)
For a detailed discussion on this topic we refer the reader to [29]. Our first result concerns the free version of the transportation cost inequality. As discussed in the introduction, the first assertion for strictly convex potentials was initially proved by large matrix approximation in [18]. The strategy of proof is inspired from [28], [4] and [9] (see [26]).
Theorem 2 (Transportation inequality).
1. If V is C 2 and V (x) − ρx 2 is convex for some ρ > 0, then for any probability measure µ on R,
ρ W 2 2 (µ, µ V ) ≤ E(µ) − E(µ V ). (2.9)
If V (x) = ρx 2 , then the equality in (2.9) is attained for measures µ = θ # µ V , with θ(x) = x + m, therefore the constant ρ in front of W 2 2 (µ, µ V ) is sharp. 2. Assume that V is C 2 , convex and V (x) ≥ ρ > 0 for all |x| ≥ r. Then, there is a constant C = C(r, ρ, µ V , V ) > 0, such that
C W 2 2 (µ, µ V ) ≤ E(µ) − E(µ V ). (2.10)
3. In the case V is C 2 and V (x) − ρ|x| p is convex for some real number p > 1, then, for any probability measure µ on R,
c p ρ W p p (µ, µ V ) ≤ E(µ) − E(µ V ) (2.11) where c p = inf x∈R |1 + x| p − |x| p − psign(x)|x| p−1 > 0.
Proof. 1. Since there is nothing to prove in the case E(µ) = ∞, we assume that E(µ) < ∞. In this case we also have that the measure µ and µ V both have second finite moments. Now we take the non-decreasing transportation map θ such that θ # µ V = µ which exists due to the lack of atoms of µ V . Using the transport map θ, we first write
E(µ) − E(µ V ) = V (θ(x)) − V (x) − V (x)(θ(x) − x) µ V (dx) (2.12) + θ(x) − θ(y) x − y − 1 − log θ(x) − θ(y) x − y µ V (dx)µ V (dy)
where in between we used the variational equation (2.6) to justify that
V (x) θ(x) − x µ V (dx) = 2 θ(x) − x x − y µ V (dy)µ V (dx) = (θ(x) − x) − (θ(y) − y) x − y µ V (dy)µ V (dx).
Since V (x) − ρx 2 is convex, for any x, y the following holds
V (y) − V (x) − V (x)(y − x) ≥ ρ y 2 − x 2 − 2x(y − x) = ρ(y − x) 2 .
On the other hand since a − 1 ≥ log(a) for any a ≥ 0, equations (2.12) and (2.8) yield (2.9). In the case V (x) = ρx 2 it is easy to see that for θ(x) = x + m, all inequalities involved become equalities, thus we attain equality in (2.9) for translations of µ V .
2. We start the proof with (2.12), whereas this time we need to exploit the logarithmic term to get our inequality. The idea is to use the strong convexity where ψ(x) := θ(x) − x takes large values and for small values of ψ(x) we try to compensate this with the second integral of (2.12). Notice in the first place that by Taylor's theorem we have that
V (y) − V (x) − V (x)(y − x) = (y − x) 2 1 0 V (1 − τ )x + τ y (1 − τ )dτ. (2.13)
Now, let us assume that the support of the equilibrium measure µ V is [a, b]. Next, V (x) ≥ 0 and V (x) ≥ ρ for |x| ≥ r, implies that for |y| ≥ 2r + 2 max{|a|, |b|}, we obtain that
V (y) − V (x) − V (x)(y − x) ≥ (y − x) 2 1 1/2 V (1 − τ )x + τ y (1 − τ )dτ ≥ ρ(y − x) 2 /8 for any x ∈ [a, b].
Now write θ(x) = x + ψ(x). Thus using (2.12), and denoting R = 2r + 2 max{|a|, |b|} we continue with
V (θ(x)) − V (x) − V (x)(θ(x) − x) µ V (dx) ≥ 1 2 ψ 2 (x) 1 0 V x + τ ψ(x) (1 − τ )dτ µ V (dx) ≥ ρ 16 |ψ|≥R ψ 2 (x)µ V (dx). (2.14)
This inequality provides a lower bound of the first term in (2.12). Further, it is not hard to check that
|ψ|≥R ψ 2 (x)µ V (dx) = 1 2 1 |ψ|≥R (x)ψ 2 (x)µ V (dx) + 1 2 1 |ψ|≥R (y)ψ 2 (y)µ V (dy) ≥ 1 8 1 |ψ(x)−ψ(y)|≥2R (x, y) ψ(x) − ψ(y) 2 µ V (dx)µ V (dy). (2.15)
Now we treat the second integral on the left hand side of (2.12). Use that t − log(1 + t) ≥ |t| − log(1 + |t|) for any t > −1 together with the fact that t − log(1 + t) is an increasing function for t ≥ 0 to argue that
ψ(x) − ψ(y) x − y − log 1 + ψ(x) − ψ(y) x − y µ V (dx)µ V (dy) ≥ |ψ(x) − ψ(y)| b − a − log 1 + |ψ(x) − ψ(y)| b − a µ V (dx)µ V (dy) (2.16)
Further, for s ≥ 0 and u, v > 0 we have
us 2 + s − log(1 + s) ≥ v−log(1+v) v 2 s 2 0 ≤ s ≤ v us 2 v ≤ s ≥ min u, v − log(1 + v) v 2 s 2 .
This inequality used for u = ρ(b−a) 2
128
and v = 2R b−a in combination with (2.15) and (2.16) yields for the choice of c = min{u,
(v − log(1 + v))/v 2 } that ρ 16 |ψ|≥R ψ 2 (x)µ V (dx) + ψ(x) − ψ(y) x − y − log 1 + ψ(x) − ψ(y) x − y µ V (dx)µ V (dy) ≥ c ψ(x) − ψ(y) 2 µ V (dx)µ V (dy) = c ψ 2 (x)µ V (dx) − ψ(x)µ V (dx) 2 . (2.17)
This shows that E(µ) − E(µ V ) is bounded below by a constant times the variance of ψ. Notice that W 2 2 (µ, µ V ) = ψ 2 (x)µ V (dx) and in order to complete the proof we have to replace the variance of ψ by the integral of ψ 2 with respect to µ V . This boils down to estimating the µ V integral of ψ in terms of the integral of ψ 2 . To this end, use Cauchy's inequality:
ψ(x)µ V (dx) 2 ≤ ψ 2 (x) 1 + 1 2c 1 0 V x + τ ψ(x) (1 − τ )dτ µ V (dx) × 1 1 + 1 2c 1 0 V (x + τ ψ(x))(1 − τ )dτ µ V (dx).
This inequality combined with equations (2.12), (2.14) and (2.17), results with
E(µ) − E(µ V ) ≥ ψ 2 (x) c + 1 2 1 0 V x + τ ψ(x) (1 − τ )dτ µ V (dx) × 1 0 V (x + τ ψ(x))(1 − τ )dτ 2c + 1 0 V (x + τ ψ(x))(1 − τ )dτ µ V (dx) ≥ c 1 0 V (x + τ ψ(x))(1 − τ )dτ 2c + 1 0 V (x + τ ψ(x))(1 − τ )dτ µ V (dx)W 2 2 (µ, µ V ),
where here we used the convexity encoded into V ≥ 0 and the fact that W 2 2 (µ, µ V ) = ψ 2 (x)µ V (dx) to get the lower bound of the first integral. From the previous inequality, it becomes clear that we are done as soon as we prove that the quantity in front of W 2 2 (µ, µ V ) is bounded from below by a positive constant uniformly in ψ. To carry this out, notice that V can not be identically zero on [a, b]. Indeed, if V were identically zero on [a, b], then we would have that V (x) = K for all x ∈ [a, b], and this plugged into equation (2.4), yields that K(b − a) = 2 and
K(b − a) = −2, a system without a solution. Therefore V is not identically 0 on [a, b]. If |ψ(x)| > R, then V (x + τ ψ(x)) ≥ ρ for 1/2 ≤ τ < 1, which implies 1 0 V (x + τ ψ(x))(1 − τ )dτ ≥ ρ/8. On the other hand, if |ψ(x)| ≤ R, then 1 0 V x + τ ψ(x) (1 − τ )dτ ≥ δ 0 V x + τ ψ(x) (1 − τ )dτ ≥ δ 2 inf |y−x|≤δR V (y)
for all 0 ≤ δ ≤ 1. Define
w(x) = sup δ∈[0,1] min ρ 8 , δ 2 inf |y−x|≤δR V (y) .
Since V is not identically 0 on [a, b], it follows that w is not identically zero on [a, b]. With this we obtain that
1 0 V x + τ ψ(x) (1 − τ )dτ ≥ w(x) ≥ 0,
and then that
c 1 0 V (x + τ ψ(x))(1 − τ )dτ 2c + 1 0 V (x + τ ψ(x))(1 − τ )dτ µ V (dx) ≥ C = cw(x) 2c + w(x) µ V (dx) > 0
which finishes the proof of (2.10) with this choice of C. 3. For the inequality (2.11), we follow the same route as in the proof of (2.9), the only change this time being that V (x) − ρ|x| p is convex, and thus we obtain
V (y) − V (x) − V (x)(y − x) ≥ ρ |y| p − |x| p − psign(x)|x| p−1 (y − x) . (2.18)
Writing θ(x) = x + ψ(x), and using (2.12) together with a − 1 ≥ log(a) for a ≥ 0, one arrives at
E(µ) − E(µ V ) ≥ ρ |x + ψ(x)| p − |x| p − psign(x)|x| p−1 ψ(x) µ V (dx).
Now we use the fact that for all a, b ∈ R,
|a + b| p − |b| p − p sign(b)|b| p−1 a ≥ c p |a| p ,(2.19)
which applied to the above inequality in conjunction to (2.8), yields inequality (2.11).
Remark 1.
1. The C 2 regularity of V for (2.9) can be dropped (see [26]) but to simplify the presentation here we decided to consider only this case.
2. If V (x) − ρ|x| p is convex, then using inequalities (2.11), (2.10) and Young's inequality we obtain that for
any 2 ≤ k ≤ p, there exists a constant c = c(k, p, ρ, µ V , V ) such that c W k k (µ, µ V ) ≤ E(µ) − E(µ V ).
3. We want to point out that the inequalities (2.11) and (2.10) are somehow complementary to each other. For example, if we take V (x) = ρ|x| p with p > 1 and the measure µ = θ # µ V for θ(x) = x + m, then equation (2.11) takes the form
c p m p ≤ |x + m| p − |x| p µ V (dx) (2.20)
while equation (2.10) becomes
Cm 2 ≤ |x + m| p − |x| p µ V (dx),
which, because it is easy to check that µ V is symmetric, is the same as
Cm 2 ≤ |x + m| p − |x| p − p sign(x)|x| p−1 m µ V (dx). (2.21)
Notice here that (2.20) is in the right scale for large m as (2.21) is in the right scale for m close to 0, because in this case the integrand is of the size m 2 . It seems that Talagrand's transportation inequality in this context has two aspects, one is the large W p (µ, µ V ) which is dictated by the potential V for large values and results with equation (2.11) and the small W 2 (µ, µ V ) regime which is dictated by the repulsion effect of the logarithm and results with equation (2.10).
4. It is not clear whether inequality (2.10) still holds for the case of a potential V which is not convex. Of interest would be the particular case V (x) = ax 4 + bx 2 for some a > 0 and b < 0. This example actually raises the question of the stability of transportation inequality under bounded perturbations.
5. Very likely the constant c p in (2.11) is not sharp.
Potential Independent Transportation Inequalities
In this section, we investigate some potential independent transportation inequalities. A transportation inequality in the form of (2.10) can not possibly hold without a quadratic growth at infinity. Also, the proof of (2.10) might lead to the conclusion that the logarithmic term plays a more important role. Therefore the natural question one may ask is whether there is a manifestation of this fact in some sort of transportation type inequality which is independent of the potential involved. The main question reduces to hint some appropriate distance one needs to use to replace the Wasserstein distance in Theorem 2. We investigate in this section several possibilities, starting with the free version of the classical Pinsker's inequality. The Pinsker's inequality classically states that (cf. [10] and [21])
2 µ − ν 2 v ≤ E(µ|ν) for any µ, ν probability measures on R,
where µ − ν v is the total variation distance between µ and ν and E(µ|ν) is the relative entropy between µ and ν. This in particular shows that if µ n convergence to µ in entropy, then µ n converges to µ is a very strong sense. The same natural question can be posed in the logarithmic entropy context. For a given potential V , is there an inequality of the form
C µ − µ V 2 v ≤ E(µ) − E(µ V )
for a given constant C > 0 and any probability distribution µ on R?
It turns out that these inequalities do not hold for the logarithmic energy. In fact, we will show that even a weaker inequality of the form
C |F µ − F µ V | 2 u ≤ E(µ) − E(µ V ) (3.1)
does not hold, where F µ denotes the cumulative function of a probability measure µ on the line. Even though the uniform distance does not have the same widespread use in probability it appears for example in the Berry-Esseen type estimates for the convergence in the central limit theorem. This is the reason why we consider this distance as the first next best candidate wherever the total variation fails. Clearly this metric gives a stronger topology as the topology of weak convergence. Will construct a counterexample to (3.1) in the case of V (x) = 2x 2 , for which the equilibrium measure is
µ V (dx) = 1 [−1,1] (x) 2 √ 1 − x 2 π dx,
the semicircular law on [−1, 1]. Consider now the sequence
µ n (dx) = 1 [−1,1] (x) 2 √ 1 − x 2 π dx + 2n−1 k=2 (−1) k T 2k+1 (x) 4(n 2 − 1)π √ 1 − x 2 dx
where T k is the k th Chebyshev polynomial of the first kind. With these choices we have that
E(µ n ) − E(µ V ) ≤ π 2 log(n/3) |F µn − F µ V | 2 u for all n ≥ 4. (3.2)
Let us point out that µ n is indeed a probability measure. This requires a little proof but it's entirely elementary and is left to the reader.
To prove (3.1), notice that since the support of µ n is the same as the support of µ V , we have from (2.2) that
E(µ n ) − E(µ V ) = − log |x − y|(µ n − µ V )(dx)(µ n − µ V )(dy). (3.3) Next remark that µ n = cos # (f n λ) and µ V = cos # (gλ), where λ is the Lebesgue measure on [0, π] and f n (t) = 1 − cos(2t) π + 1 4π(n 2 − 1) 2n−1 k=2 (−1) k cos((2k + 1)t), g(t) = 1 − cos(2t) π . and further − log |x − y|(µ n − µ V )(dx)(µ n − µ V )(dy) = − π 0 π 0 log | cos t − cos s|h n (t)h n (s)dtds where h n = f n − g.
Now we provide a formula for the logarithmic energy we learnt from [15] and have not seen it elsewhere.
Here is a quick description. Write first cos t = (e it + e −it )/2 and cos s = (e is + e −is )/2 so | cos t − cos s| = |(e it + e −it )/2 − (e is + e −is )|/2 = |1 − e i(t+s) ||1 − e i(t−s) |/2 and so, for t = s, and t or s not equal to π,
log | cos t − cos s| = − log 2 + Re log(1 − e i(t+s) ) + log(1 − e i(t−s) ) = − log 2 − ∞ =1 Re e i (t+s) / + e i (t−s) / = − log 2 − ∞ =1 2 cos( t) cos( s).
From this, one gets to
− π 0 π 0 log | cos t − cos s|h n (t)h n (s)dtds = ∞ =1 2 π 0 cos( t)h n (t)dt 2 . (3.4) But now, π 0 cos( t)h n (t)dt = 1 4π(n 2 − 1) 2n−1 k=2 (−1) k π 0 cos( t) cos((2k + 1)t)dt = (−1) ( −1)/2 8(n 2 −1)
4 ≤ ≤ 4n and odd 0 otherwise and thus
− π 0 π 0 log | cos t − cos s|h n (t)h n (s)dtds = ∞ =1 2 π 0 cos( t)h n (t)dt 2 = 1 32(n 2 − 1) 2 2n−1 =2 1 2 + 1 . (3.5) On the other hand |F µn − F µ V | u = |F fnλ − F gλ | u = sup x∈[0,π]
x 0 h n (t)dt and
x 0 h n (t)dt = 1 4π(n 2 − 1) 2n−1 =2 (−1) sin((2 + 1)x) 2 + 1 ,
from which for x = π/4, we obtain
|F µn − F µ V | u = sup x∈[0,π] x 0 h n (t)dt ≥ 1 4π(n 2 − 1) 2n−1 =2 1 2 + 1 . (3.6)
Combining (3.5) and (3.6) we get
π 2 2 2n−1 =2 1 2 +1 |F µn − F µ V | 2 u ≥ − log |x − y|(µ n − µ V )(dx)(µ n − µ V )(dy) (3.7)
which together with the fact that
2n−1 =2 1 2 +1 ≥ 1 2 log(n/3) for n ≥ 4 and (3.
3), we finally arrive at (3.2). The example shown above has the property that E(µ n ) − E(µ V ) converges to 0 when n goes to infinity, and also that |F µn − F µ V | u converges to zero. Despite the fact that (3.1) does not hold, we will see below in
Corollary ?? that if E(µ n ) − E(µ V ) converges to 0, then |F µn − F µ V | u always converges to 0.
We consider now a weak form of (3.1). To do this we define the distance
d(µ, ν) = sup a,b∈R e −|ax+b| µ(dx) − e −|ax+b| ν(dx) . (3.8)
With this definition we have the following result.
Theorem 3. For any potential V satisfying (2.1), we have that for any compactly supported measure µ,
4π 3 d 2 (µ, µ V ) ≤ E(µ) − E(µ V ). (3.9)
Proof. Using equations (2.1) and (2.2), we get for any compactly supported measure µ with E(µ) finite,
E(µ) − E(µ V ) ≥ − log |x − y|(µ − µ V )(dx)(µ − µ V )(dy).
We will prove that for any measures µ and ν with compact support such that − log |x − y|µ(dx)µ(dy) < ∞ and − log |x − y|ν(dx)ν(dy) < ∞, we have that
4π 3 d 2 (µ, ν) ≤ − log |x − y|(µ − ν)(dx)(µ − ν)(dy),(3.10)
which shows that (3.10) implies (3.9). Now we use [11, equation 6.45] to write
− log |x − y|(µ − µ V )(dx)(µ − µ V )(dy) = ∞ 0 |μ(t) −μ V (t)| 2 t dt (3.11)
where the hat stands for the Fourier transform, and continue with
∞ 0 |μ(t) −ν(t)| 2 t dt = 1 2 ∞ −∞ |μ(t) −ν(t)| 2 |t| dt ≥ |a| ∞ −∞ |μ(t) −ν(t)| 2 a 2 + t 2 dt ≥ a 2 π ∞ −∞ (μ(t) −ν(t)) e −ict a 2 + t 2 dt 2
for any a, c ∈ R with a = 0. Further, using the inversion formula for the Fourier transform, one has
∞ −∞ (μ(t) −ν(t)) e −ict a 2 + t 2 dt = 2π φ (x)(µ − ν)(dx) = 2π 2 |a| e −|a(x+c)| (µ − ν)(dx) (3.12) because for φ(t) = e ict a 2 + t 2 ,φ (x) = e i(x+c)t t 2 + a 2 dt = πe −|a(x+c)| |a| .
From here, (3.10) follows immediately.
Remark 2. From equation (3.11) it seems that the distance one should consider should be the Sobolev norm with exponent −1/2. This is another possible candidate to the role of d played here, however not always finite. We chose the metric d as it's definition is somehow close to uniform norm of the difference of the Laplace transforms of the measures. It is also always defined and bounded by 1, thus resembling the total variation distance.
The next result is collecting facts about how strong the topology induced by d is.
Proposition 1. 1. d is a distance on P(R) and if d(µ n , µ) − −−− → n→∞ 0, then µ n − −−− → n→∞
µ in the weak topology. In addition d(δ a , δ b ) = 1 for a = b, thus the topology induced by d is strictly stronger than the weak convergence topology.
2. For any two probability measures µ and ν,
d(µ, ν) ≤ 2 |F µ − F ν | u . (3.13) 3. If V satisfies condition (2.1), then E V (µ n ) − −−− → n→∞ E V (µ V ) implies |F µn − F µ V | u − −−− → n→∞ 0.
Proof.
1. To prove that d is a distance the only non trivial fact is that for two probability measures µ and ν, d(µ, ν) = 0 implies µ = ν. Thus from equation (3.12), we obtain for a = 1 that for all c ∈ R,
∞ −∞ (μ(t) −ν(t)) e −ict 1 + t 2 dt = 0.
Since this holds true for any c ∈ R, it implies that the Fourier transform of the function t →μ (t)−ν(t) 1+t 2 is 0, which means that the function in discussion must be 0. This means thatμ =ν, or equivalently that µ = ν. Let L(µ, ν) stand for the Levy distance which induces the weak topology on P(R).
Let d(µ n , µ) − −−− → n→∞ 0.
Assume now that there exists > 0 and a subsequence such that L(µ n k , µ) ≥ . Otherwise said, the sequence µ n has a subsequence which is not convergent to µ. Since, we are dealing with probability measures, there is a subsequence µ n k l which is vaguely convergent to a measure ν with total mass less than 1. This means that for any continuous function φ which is vanishing at infinity, we have that
φdµ n k l − −− → l→∞ φdν.
We can apply this for functions φ(x) = e −|ax+b| where a = 0 and infer that
e −|ax+b| µ n k l (dx) − −− → l→∞ e −|ax+b| ν(dx) for all a = 0, b ∈ R.
On the other hand, because d(µ n k l , µ) − −− → l→∞ 0, these considerations result with
e −|ax+b| µ(dx) = e −|ax+b| ν(dx) for all a = 0, b ∈ R.
Further, using the dominated convergence for b = 0 and a → 0, we obtain that ν is a probability measure. From the discussion at the beginning of this proof, it also follows that ν = µ and this in turn results with µ n k l being weakly convergent to µ, a contradiction. This proves that the convergence in the metric d implies weak convergence. It is obvious that d(µ, ν) ≤ 1 for any measures µ and ν. For the case of discrete measures, we also have
that 1 ≥ d(δ a , δ b ) ≥ e −α|x−a| δ a (dx) − e −α|x−a| δ b (dx) for any α > 0, which yields that 1 ≥ d(δ a , δ b ) ≥ 1 − e −α|b−a| for all α > 0. Letting α → ∞, we get that d(δ a , δ b ) = 1 for a = b
which shows that convergence in d is strictly stronger than convergence in the weak topology.
2. From the fact that for any finite positive measure µ,
(0,∞) (1 − e −αy )µ(dx) = (0,∞) αe −αy µ((y, ∞))dy, we deduce that e −α|x−a| (µ − ν)(dx) = (0,∞) αe −αy F µ (a − y) − F µ (a + y) − F ν (a − y) + F ν (a + y) dy
which easily yields (3.13).
3. We actually show that if µ n and µ are compactly supported probability measures such that (3.10) and the first part, we obtain that µ n converges weakly to µ. In addition, none of the measures µ n or µ have atoms. Thus F µn and F µ are continuous functions which combined with the weak convergence implies that F µn converges pointwise to F µ . Since the functions F µn and F µ are distributions of probability measures, it is an easy matter to check that the convergence is actually uniform.
− log |x − y| µ(dx)µ(dy) < ∞, − log |x − y| µ n (dx)µ n (dy) < ∞ and lim n→∞ log |x − y|(µ n − µ)(dx)(µ n − µ)(dy) = 0, then |F µn − F µ | u − −−− → n→∞ 0. From
Remark 3. We do not know if the topology of convergence in d is the same as the one defined by the metric
|F µ − F ν | u .
This result might leave one wondering if a stronger convergence takes place. In other words, is it true that
E V (µ n ) − −−− → n→∞ E V (µ V ) implies µ n − µ V v − −−− →µ V (dx) = 1 [−1,1] (x) dx π √ 1 − x 2 , µ n (dx) = 1 [−1,1] (x) (1 − T n (x))dx π √ 1 − x 2 ,
then, using the same argument which led us to (3.4), with h n there replaced by h n (x) = cos(nx) here, one arrives at E(µ n ) − E(µ V ) = 1 n while the total variation distance is µ n − µ V v ≥ 1/4.
Log-Sobolev Inequality
In this section, we develop similarly the mass transportation method to prove the Log-Sobolev inequality in the free context. Note again that, as discussed in the introduction, the first assertion for strictly convex potentials was initially proved by large matrix approximation in [3]. Before we state the main result, we define inspired by Voiculescu [31], the relative free Fisher information as
I(µ) = Hµ(x) − V (x) 2 µ(dx) with Hµ(x) = p.v. 2 x − y µ(dy). (4.1)
for measures µ on R which have density p = dµ/dx in L 3 (R). In this case the principal value integral is a function in L 3 . Otherwise we let I(µ) be equal to +∞.
Theorem 4 (Log-Sobolev).
1. If V is C 2 and V (x) − ρx 2 is convex for some ρ > 0, then for any probability measure µ on R,
E(µ) − E(µ V ) ≤ 1 4ρ I(µ). (4.2)
Equality is attained for the case V (x) = ρx 2 and µ = θ # µ V , where θ(x) = x + m. Thus the inequality (4.2) is sharp for translations of µ V .
2. If V is C 2 and V (x) − ρ|x| p is convex for some ρ > 0 and p > 1, then for any probability measure µ on R,
E(µ) − E(µ V ) ≤ k p ρ q/p I q (µ) where I q (µ) = Hµ(x) − V (x) q µ(dx) (4.3)
where here q is the conjugate of p i.e. 1/q + 1/p = 1 and the constant k p = (pc p ) q/p /q, with c p from (2.11).
Proof.
1. We will assume that the measure µ has a smooth compactly supported density as the general case follows via approximation arguments discussed in details in [18]. Take the (increasing) transport map θ from µ V into µ. We write the inequality (4.2) in the following equivalent way 1 4ρ
Hµ
(θ(x)) − V (θ(x)) 2 µ V (dx) + V (x) − V (θ(x)) − V (θ(x)) x − θ(x) µ V (dx) − Hµ(θ(x)) − V (θ(x)) (x − θ(x))µ V (dx) + Hµ(θ(x)) x − θ(x) µ V (dx) − log x − y θ(x) − θ(y) µ V (dx)µ V (dy) ≥ 0. (4.4)
Notice now that from the convexity of V (x) − ρx 2 , one obtains that
V (x) − V (θ(x)) − V (θ(x)) x − θ(x) ≥ ρ x 2 − θ(x) 2 − 2θ(x)(x − θ(x)) = ρ x − θ(x) 2 . (4.5)
Now,
Hµ(θ(x)) x−θ(x) µ V (dx) = x−θ(x) 2 θ(x) − θ(y) µ V (dy)µ V (dx) = x − y θ(x) − θ(y) − 1 µ V (dx)µ V (dy)
(4.6) where one has to interpret the second integral here in the principal value sense, however since θ is increasing, the last integral is actually taken in the Lebesgue sense. Using these, equation (4.4) may be rewritten as
1 4ρ (Hµ(θ(x)) − V (θ(x)) − 2ρ(x − θ(x)) 2 µ V (dx) + x − y θ(x) − θ(y) − 1 − log x − y θ(x) − θ(y) µ V (dx)µ V (dy) ≥ 0 which is seen to hold since u − 1 − log(u) ≥ 0 for u ≥ 0.
Equality is attained for the case V (x) = ρx 2 and θ(x) = x + c, which corresponds to the translations of the measure µ V .
2. With the same arguments used in the above proof and the proof of Theorem 2, we use equations (2.18) and (2.19) to argue that
k p ρ q/p Hµ(x) − V (x) q µ(dx) − E(µ) + E(µ V ) ≥ k p ρ q/p Hµ(θ(x)) − V (θ(x) q + V (θ(x)) − Hµ(θ(x)) x − θ(x) + c p ρ|x − θ(x)| p µ V (dx) + x − y θ(x) − θ(y) − 1 − log x − y θ(x) − θ(y) µ V (dx)µ V (dy) ≥ 0
where we used Young's inequality a q /q + b p /p ≥ ab for a, b ≥ 0 and the constant k p = (pc p ) q/p /q.
Remark 4.
It was proved in [24] that a Log-Sobolev inequality always implies a transportation inequality.
HWI Inequality
This section is devoted to the free analog of the HWI inequality of Otto and Villani [25] in the classical context, connecting thus the (free) entropy, Wasserstein distance and Fisher information. As we will see, the HWI implies the Log-Sobolev inequality for strictly convex potentials. This free HWI inequality was not considered before, and in particular it is not clear whether there is a random matrix proof, delicate points involving the Wasserstein distance entering into the proof.
Theorem 5 (HWI inequality).
1. Assume that V is C 2 such that for some ρ ∈ R, V (x) − ρx 2 is convex. Then, for any measure µ ∈ P(R),
E(µ) − E(µ V ) ≤ I(µ) W 2 (µ, µ V ) − ρ W 2 2 (µ, µ V ). (5.1)
In the case V (x) = ρx 2 , the inequality is sharp.
2. If V is C 2 and V (x) − ρ|x| p is convex for some ρ ≥ 0 and p > 1, then for the same constant c p appearing in Theorem 2, we have that
E(µ) − E(µ V ) ≤ I 1/q q (µ) W p (µ, µ V ) − ρc p W p p (µ, µ V ), (5.2)
where 1/p + 1/q = 1.
Proof.
1. We employ here the notations used in Theorem 4 and we will give a proof of the inequality for the case of a measure µ with smooth and compactly supported density, the general case follows through careful approximations pointed in [18]. The inequality to be proved can be restated as (5.3) + (5.4) + (5.5) ≥ 0, where
(5.3) = Hµ(θ(x)) − V (θ(x)) 2 µ V (dx) θ(x) − x 2 µ V (dx) 1/2 − Hµ(θ(x)) − V (θ(x)) (x − θ(x))µ V (dx) (5.3) (5.4) = V (x) − V (θ(x)) − V (θ(x)) x − θ(x) − ρ θ(x) − x 2 µ V (dx) (5.4) (5.5) = Hµ(θ(x)) x − θ(x) µ V (dx) − log x − y θ(x) − θ(y) µ V (dx)µ V (dy). (5.5)
A simple application of Cauchy's inequality shows that (5.3) ≥ 0. Using convexity of V (x) − ρx 2 we have from equation (4.5), that (5.4) ≥ 0. Finally, using (4.6), we have that
(5.5) = x − y θ(x) − θ(y) − 1 − log x − y θ(x) − θ(y) µ V (dx)µ V (dy) ≥ 0,
which finishes the proof of (5.1). For the case V (x) = ρx 2 , we have equality if θ(x) = x + m.
2. The inequality we want to prove is equivalent to the statement that (5.6) + (5.7) + (5.8) ≥ 0, where
(5.6) = Hµ(θ(x)) − V (θ(x)) q µ V (dx) 1/q (θ(x) − x) p µ V (dx) 1/p − Hµ(θ(x)) − V (θ(x)) x − θ(x) µ V (dx) (5.6) (5.7) = V (x) − V (θ(x)) − V (θ(x)) x − θ(x) − ρc p θ(x) − x p µ V (dx) (5.7) (5.8) = Hµ(θ(x)) x − θ(x) µ V (dx) − log x − y θ(x) − θ(y) µ V (dx) µ V (dy). (5.8)
Now, (5.6) is non-negative thanks to Hölder's inequality, equation (5.7), follows from the convexity of V (x) − ρ|x| p and the combination of (2.18) and (2.19), while equation (5.8) is the same as (5.5).
As pointed out in [25], HWI inequalities for ρ > 0 always implies Log-Sobolev. We give here the following formal corollary of HWI inequality. Corollary 1.
1. If ρ > 0, then inequality (5.1) implies (4.2) and (5.2) implies (4.1).
2. If V (x) − ρx 2 is a convex for some ρ ∈ R, then Talagrand's free transportation inequality with constant C > max{0, −ρ} implies free Log-Sobolev inequality with constant K = max{ρ, (C+ρ) 2 32C }. More precisely,
∀µ ∈ P(R), C W 2 2 (µ, µ V ) ≤ E(µ) − E(µ V ) =⇒ ∀µ ∈ P(R), E(µ) − E(µ V ) ≤ 1 4K I(µ).
3. In particular, if V is convex and C 2 such that V (x) ≥ ρ > 0 for |x| ≥ r, then free Log-Sobolev inequality holds with the constant C > 0 from (2.10).
Proof.
1. It follows as an application of Young's inequality a p /p + b q /q ≥ ab for a, b ≥ 0.
2. For ρ > 0, everything is clear. In the case ρ ≤ 0, then, from (5.1) and Talagrand's transportation inequality, one has for δ > 0, that
E(µ) − E(µ V ) ≤ I(µ) W 2 (µ, µ V ) − ρ W 2 2 (µ, µ V ) ≤ 4δI(µ) + 1 Cδ − ρ C E(µ) − E(µ V )
which yields for any δ > 1
C+ρ E(µ) − E(µ V ) ≤ 4Cδ 2 (C + ρ)δ − 1 I(µ).
Taking minimum over δ > 1 C+ρ gives the conclusion. 3. In the case V is convex, C 2 and strongly convex for large values, part 2 of Theorem 2 does the rest.
Brunn-Minkowski Inequality
The (one-dimensional) free Brunn-Minkowski inequality was put forward in [24] again through random matrix approximation. We provide here a direct mass transportation proof similar to the one of its classical (onedimensional) counterpart (see e.g. [12]). As discussed in [24], this inequality may be used to deduce in an easy way both the Log-Sobolev and transportation inequalities.
The main result of this section is the following theorem.
Theorem 6. Assume that V 1 , V 2 , V 3 are some potentials satisfying (2.1) such that for some a ∈ (0, 1),
aV 1 (x) + (1 − a)V 2 (y) ≥ V 3 (ax + (1 − a)y) for all x, y ∈ R. (6.1) Then aE V1 (µ V1 ) + (1 − a)E V2 (µ V2 ) ≥ E V3 (µ V3 ). (6.2)
Proof. Take the (increasing) transportation map θ from µ V1 into µ V2 . This certainly exists as the measure µ V1 has no atoms.
Noticing that for any measure with finite logarithmic energy, we have the obvious equality log |x − y|µ(dx)µ(dy) = 2 x>y log(x − y)µ(dx)µ(dy).
Using this we argue that
aV 1 (x) + (1 − a)V 2 (θ(x))µ V1 (dx) − 2 x>y a log(x − y) + (1 − a) log(θ(x) − θ(y)) µ V1 (dx)µ V1 (dy) ≥ V 3 ax + (1 − a)θ(x) µ V1 (dx) − 2 x>y log (ax + (1 − a)θ(x)) − (ay + (1 − a)θ(y) µ V1 (dx)µ V1 (dy) = E V3 (ν) ≥ E V3 (µ V3 )
where ν = (aid + (1 − a)θ) # µ V1 and we used (6.1) and the concavity of the logarithm on (0, ∞). The proof is complete.
Random Matrices and a First Version of Poincaré Inequality
In the next three sections, we investigate Poincaré type inequalities in the free (one-dimensional) context. We discuss two versions of it. The first one is suggested by large matrix approximations and the classical Poincaré inequality for strictly convex potentials, but will be proved directly. Recall first the classical Poincaré inequality (cf. e.g. [2], [23], [29], [32]...).
Theorem 7. Let µ(dx) = e −W (x) dx be a probability measure on R d such that W (x) − r|x| 2 is convex. Then for any compactly supported and smooth function φ : R d → R, we have that
|∇φ| 2 dµ ≥ r Var µ (φ). (7.1)
Assume now that V is a potential on R with enough growth at infinity. Consider the matrix models on H n , the space of Hermitian n × n matrices with the inner product A, B = T r(AB * ) and the probability measure given by
P n (dM ) = 1 Z n (V ) e −nTr(V (M )) dM
where here dM is the standard Lebesgue measure on H n . We have that for any bounded continuous function [19], it is known that this is universal in the sense that the limit in distribution of the fluctuations is Gaussian and, at least in the case of polynomial V (for which V (x) − ρx 2 fulfills the conditions in there), the variance of the Gaussian limit depends only on the endpoints of the support of µ V . Moreover, in the particular case of V (x) = 2x 2 , the variance of the distribution was computed for example in [22] and [19]
F : R → R, 1 n Tr F (M ) P n (dM ) − −−− → n→∞ F (x)µ V (dx).as 1 2π 2 1 −1 1 −1 φ(t) − φ(s) t − s 2 1 − ts √ 1 − t 2 √ 1 − s 2 dtds. (7.4)
This variance is interpreted in [8] in terms of the number operator of the arcsine law. We will come back to this aspect in Section 9.
Dividing the inequality in equation (7.3) by n and taking the limit when n → ∞, these heuristics (after a simple rescaling) suggest the following result.
Theorem 8. Assume that V (x) − ρx 2 is convex for some ρ > 0. Then for any smooth function φ, one has that
φ (x) 2 µ V (dx) ≥ ρ 2π 2 b a b a φ(x) − φ(y) x − y 2 −2ab + (a + b)(x + y) − 2xy 2 (x − a)(b − x) (y − a)(b − y) dxdy. (7.5)
where supp(µ V ) = [a, b]. Equality is attained for V (x) = ρ(x − α) 2 + β and φ(x) = c 1 + c 2 x for some constants c 1 , c 2 .
The reader may wonder if the numerator in the second fraction of (7.5) is nonnegative. This is so because
−2ab + (a + b)(x + y) − 2xy = 2 b − a 2 2 − x − a + b 2 y − a + b 2 ≥ 0 for any x, y ∈ [a, b].
Proof. Using a simple rescaling we may assume without loss of generality that a = −1 and b = 1 and the inequality we have to show reduces to
φ (x) 2 µ V (dx) ≥ ρ 2π 2 1 −1 1 −1 φ(x) − φ(y) x − y 2 1 − xy √ 1 − x 2 1 − y 2 dxdy. (7.6)
Then, based on equation (2.5), we have that
g(x) = √ 1 − x 2 2π 2 1 −1 V (y) − V (x) 1 − y 2 (y − x) dy.
From the convexity of V (x) − ρx 2 , we learn that V (y)−V (x)
y−x ≥ 2ρ and thus that
g(x) ≥ ρ π 1 − x 2 ,(7.7)
which implies
φ (x) 2 µ V (dx) ≥ ρ π 1 −1 φ (x) 2 1 − x 2 dx.
Therefore it is enough to check that
1 −1 φ (x) 2 1 − x 2 dx ≥ 1 2π 1 −1 1 −1 φ(x) − φ(y) x − y 2 1 − xy √ 1 − x 2 1 − y 2 dxdy (7.8)
for any smooth φ. Now, we make the change of variables x = cos t to justify
1 −1 φ (x) 2 1 − x 2 dx = π 0 φ (cos t) 2 sin 2 (t)dt = π 0 ψ (t) 2 dt where ψ(t) = φ(cos t).
On the other hand, using the change of variable x = cos t, y = cos s on the right hand side, inequality (7.8) becomes
π 0 ψ (t) 2 dt ≥ 1 2π π 0 π 0 ψ(t) − ψ(s) cos t − cos s 2 (1 − cos t cos s)dtds. (7.9)
To show this, we write ψ(t) = ∞ k=0 a k cos kt and then, because ψ is a smooth function, we can differentiate term by term to get ψ (t) = − ∞ k=1 ka k sin kt, therefore To compute the integrals on the right hand side of the above equation, we take the generating function of these numbers and with a little algebra one can show that
= π 0 π 0 (u − u 3 )(v − v 3 )(1 − cos t cos s) (1 + u 2 − 2u cos t)(1 + u 2 − 2u cos s)(1 + v 2 − 2v cos t)(1 + v 2 − 2v cos s) dtds = π 2 uv (1 − uv) 2 = π 2 ∞ k=1 ku k v k (7.10)
for all u, v ∈ (−1, 1). The last integral can be computed as follows. First use partial fractions to justify
π 0 (A + B cos t)dt (1 + u 2 − 2u cos t)(1 + v 2 − 2v cos t) = π 0 Cdt 1 + u 2 − 2u cos t + π 0 Ddt 1 + v 2 − 2v cos t = C/2 1 − u 2 + D/2 1 − v 2
where the constants C, D are linear combinations of A and B. Further, taking A = 1 and B = − cos s and repeating once more the partial fractions argument, one can cary out the proof of (7.10). The main consequence of the above calculation is that Therefore inequality (7.9) becomes equivalent to
π 2 ∞ k=1 k 2 a 2 k ≥ π 2 ∞ k=1 ka 2 k
which is obviously true. Notice that equality in this inequality is attained for the case a k = 0 for all k ≥ 2 and arbitrary a 1 . This corresponds to the case ψ(t) = c 1 + c 2 cos t or φ(x) = c 2 x + c 1 for some c 1 , c 2 . Finally we point out that equality in (7.6) is attained if the equality is attained in (7.7) and (7.9). From there one can easily see from rescaling that equality in (7.5) is attained for V (x) = ρ(x − α) 2 + β and φ(x) = c 1 + c 2 x. The proof of Theorem 8 is complete.
In the above proof we showed a direct calculation for equation (7.11) which is natural in the course of the above proof. However, there is another way of looking at it which will appear below in Section 9 as the kernel of the number operator.
A Second Version of Poincaré Inequality
The second version of the Poincaré inequality is motivated by the free calculus and the noncommutative derivative. It was already investigated by Biane [3] for the case of the semicircular law.
Definition 1. For a given probability measure µ on R, we say that it satisfies a Poincaré inequality if there is a constant C > 0 such that
φ(x) − φ(y) x − y 2 µ(dx)µ(dy) ≥ C Var µ (φ) for every φ ∈ C 1 0 (R). (8.1)
By the best constant we mean the largest C > 0 for which the above inequality is satisfied and we denote it by Poin(µ) or λ 1 (µ) or SG(µ).
In the noncommutative setting for a given function φ, we can think of Dφ(x, y) = φ(x)−φ(y)
x−y as the noncommutative derivative of φ. As pointed out by Voiculescu in [30], this is the unique map D :
C x → C x ⊗ C x such that 1. D1 = 0 2. D(f g) = D(f )g + f D(g) for any f, g ∈ C x .
First we collect a couple of obvious properties of the Poincaré constant.
Proposition 2.
1. For any a = 0, Poin (ax + b) # µ = 1 a 2 Poin(µ) where here and elsewhere, for a given function f : R → R, f # µ is the push forward measure given by (f # µ)(A) = µ(f −1 (A)).
2. If f : R → R is a differential map such that |f (x)| ≥ c > 0 for all x ∈ R, then
Poin(µ) ≥ c 2 Poin(f # µ).
3. If {µ n } n≥1 is a sequence of probability measures which converges weakly to µ, then
Poin(µ) ≥ lim sup n→∞ Poin(µ n ).
Next we describe some bounds for the Poincaré constant.
Theorem 9. Assume that the measure µ has compact support and is not concentrated at one point. Then µ satisfies a Poincaré inequality with 2
d 2 (µ) ≤ Poin(µ) ≤ 1 Var(µ) (8.2)
where d(µ) = diam(supp(µ)) is the diameter of the support of µ and Var(µ) = x 2 µ(dx)− xµ(dx) 2 . Equality on the left in (8.2) is attained only for the case
µ = αδ a + (1 − α)δ b , a < b, 0 < α < 1.
Equality on the right of (8.2) is attained only for the case of a semicircular law (a ∈ R, r > 0)
µ(dx) = 1 2πr 2 1 [a−2r,a+2r] (x) 4r 2 − (x − a) 2 dx.
In addition, assume that V is a C 2 potential on R such that for some integer p and real ρ > 0, V (x) − ρx 2p , is convex and µ is the minimizer of
V (x)µ(dx) − log |x − y| µ(dx)µ(dy)
over all probability measures of R. Then pρ 2p
p 1 p 8 ≤ Poin(µ). (8.3)
In particular if p = 1, we get that ρ 4 ≤ Poin(µ).
Proof. For a given function φ ∈ C 1 0 (R), the left hand side of (8.2) follows from
Var µ (φ) = 1 2 (φ(x) − φ(y)) 2 µ(dx)µ(dy) = 1 2 (x − y) 2 φ(x) − φ(y) x − y 2 µ(dx)µ(dy) ≤ d 2 (µ) 2 φ(x) − φ(y) x − y 2 µ(dx)µ(dy). (8.4)
The right hand side of (8.2) follows from (8.1) for a φ ∈ C 1 0 (R) such that φ(x) = x on the support of µ.
For measures µ = αδ a + (1 − α)δ b , condition (8.1) is equivalent to Cα(1 − α) φ(b) − φ(a) 2 ≤ α 2 (φ (a)) 2 + (1 − α) 2 (φ (b)) 2 + 2α(1 − α) φ(b) − φ(a) b − a 2
for any φ ∈ C 1 0 (R).
Since for any function φ ∈ C ∞ 0 (R) we can find another function ψ ∈ C 1 0 (R) so that φ(a) = ψ(a) and φ(b) = ψ(b) and ψ(a) = 0, ψ(b) = 0, this is also equivalent to
Cα(1 − α) ψ(b) − ψ(a) 2 ≤ 2α(1 − α) ψ(b) − ψ(a) b − a 2
for any ψ ∈ C 1 0 (R).
This amounts to C ≤ 2/(b − a) 2 and therefore, in this case, Poin(µ) = 2 d 2 (µ) . Conversely, if µ is a measure so that Poin(µ) = 2 d 2 (µ) , then, for 1 > > 0, there is a function φ ∈ C 1 0 (R) such that 2 d 2 (µ)
+ 2 Var µ (φ ) > φ (x) − φ (y) x − y 2 µ(dx)µ(dy).
Without loss of generality we can assume that 0 = inf supp(µ), 1 = sup supp(µ) and φ dµ = 0, φ 2 dµ = 1 where we recall that supp(µ) stands for the support of µ. In this case, the above inequality implies
2 + 2 ≥ |x−y|≥1− φ (x) − φ (y) x − y 2 µ(dx)µ(dy) + |x−y|<1− φ (x) − φ (y) x − y 2 µ(dx)µ(dy) ≥ |x−y|≥1− φ (x) − φ (y) 2 µ(dx)µ(dy) + 1 (1 − ) 2 |x−y|<1− φ (x) − φ (y) 2 µ(dx)µ(dy) = − (2 − ) (1 − ) 2 |x−y|≥1− φ (x) − φ (y) 2 µ(dx)µ(dy) + 2 (1 − ) 2 ,
which results with
|x−y|≥1− φ (x) − φ (y) 2 µ(dx)µ(dy) ≥ 2 − (1 − ) 2 2 − . (8.5) Now, |x−y|≥1− φ (x) − φ (y) 2 µ(dx)µ(dy) ≤ |x−1/2|≥1/2− |y−1/2|≥1/2− φ (x) − φ (y) 2 µ(dx)µ(dy) ≤ 2µ |x − 1/2| ≥ 1/2 − . (8.6)
Thus (8.5) and (8.6) give
µ |x − 1/2| ≥ 1/2 − ≥ 1 − (1 − ) 2 4 − 2 for any 1 > > 0.
This shows that µ((0, 1)) = 0 and therefore µ = αδ 0 + (1 − α)δ 1 . The other extreme case of inequality (8.2) is contained in Biane's paper [3] in the more general context of several noncommutative variables. For completeness we will provide here a selfcontained proof. In the first place, using Proposition 8.1, we may assume that
µ(dx) = 1 2π 1 [−2,2] (x) 4 − x 2 dx
is the semicircular law on [−2, 2]. Take U n to be the Chebyshev polynomials of second kind defined by U n (cos(θ)) = sin(n+1)θ sin θ
. With this choice, we have that U n ( x 2 ) are the orthogonal polynomials with respect to µ. The generating function of U n is given by
∞ n=0 r n U n (x) = 1 1 − 2rx + r 2 for |x|, |r| < 1 from which one gets ∞ n=0 r n U n (x) − U n (y) x − y = 2r (1 − 2rx + r 2 )(1 − 2ry + r 2 ) = 2 ∞ n=0 r n n−1 k=0 U k (x)U n−1−k (y),
and then U n (x) − U n (y)
x − y = 2 n−1 k=0 U k (x)U n−1−k (y). (8.7)
Now, for a given φ ∈ C 1 0 (R), we can write in L 2 (µ) sense,
φ(x) = ∞ n=0 α n U n x 2 ,
yielding from orthogonality and (8.7) that
Var µ (φ) = φ 2 dµ − φdµ 2 = ∞ n=1 α 2 n and φ(x) − φ(y) x − y 2 µ(dx)µ(dy) = ∞ n=1 nα 2 n .
It follows that in this case Poin(µ) = 1 = 1/Var(µ) and equality is attained only for φ(x) = c 1 +c 2 U 1 (x) = c 1 +c 2 x for some constants c 1 , c 2 .
To prove the converse, take a compactly supported measure µ and assume that xµ(dx) = 0 and x 2 µ(dx) = 1. In order to show that µ is the semicircular distribution, it suffices to show that U n x 2 µ(dx) = 0 for all n ≥ 1. We use induction to this task. Assuming true for U 1 , U 2 , . . . , U n , and using U n+1 (x) = 2xU n (x)−U n−1 (x), we need to show that xU n x 2 integrates to 0 against µ. Applying Poincaré's inequality to U n x 2 + rU 1
x 2 together with the induction hypothesis and equation (8.7), we get that for any r ∈ R,
U 2 n x 2 µ(dx) + r xU n x 2 µ(dx) ≤ U n x 2 − U n y 2 x − y 2 µ(dx)µ(dy),
which implies that xU n x 2 µ(dx) = 0.
In the case of the equilibrium measure of a convex potential V , we have the support of the measure consists of one interval [a, b] and a, b solve the system (cf. equation (2.4
)) 1 2π b a V (x) x − a b − x dx = 1 and 1 2π b a V (x) b − x x − a dx = −1.
If we denote c = (b − a)/2 and β = (a + b)/2, the system above can be rewritten in terms of β and c as c 2π
1 −1 V (β + ct) 1 + t √ 1 − t 2 dt = 1 and c 2π 1 −1 V (β + ct) 1 − t √ 1 − t 2 dt = −1 which is equivalent to c 2π 1 −1 V (β + ct) t √ 1 − t 2 dt = 1 and 1 −1 V (β + ct) 1 √ 1 − t 2 dt = 0.
Since V is C 2 the first equation can be integrated by parts to get that
c 2 2π 1 −1 V (β + ct) 1 − t 2 dt = 1.
On the other hand we know that
V (x) ≥ 2p(2p − 1)ρx 2p−2 , hence 1 ≥ 2p(2p − 1)ρc 2 2π 1 −1 (ct + β) 2p−2 1 − t 2 dt ≥ 2p(2p − 1)ρc 2p 2π 1 −1 t 2p−2 1 − t 2 dt = p(2p − 1)ρc 2p 2p p 4 p (2p − 1) = pρ 2p p c 2p 4 p .
This yields
c ≤ 2 mρ 2p p − 1 2p .
Finally, because d(µ) = b − a = 2c, we arrive at (8.3).
To conclude this section, we present an inequality which relates the equilibrium measure of a strong convex potential and the arcsine law.
Theorem 10. Assume that V (x) − ρx 2 is a convex for some ρ > 0 and the equilibrium measure µ V has support
[a, b]. Let arcsine a,b = 1 [a,b] (x) 1 π √ (b−x)(x−a)
dx be the arcsine law with support [a, b]. Then for any smooth
function supported on [a, b], φ (x) 2 µ V (dx) ≥ ρ Var arcsine a,b (φ), (8.8)
where the variance is considered with respect to the arcsine a,b law.
Proof. It suffices to deal with the case a = −1, b = 1, the rest following by simple rescaling. Recall that in the proof of Theorem 8, we use convexity to get that the density g(x) of µ V satisfies g(x) ≥ ρ π √ 1 − x 2 . Thus the proof reduces to 1 π
1 −1 φ (x) 2 1 − x 2 (dx) ≥ Var arcsine (φ). (8.9)
For this, write φ = ∞ n=0 α n T n (x) the expansion of φ in terms of Chebyshev polynomials of the first kind. Now, T n = nU n−1 and thus the above inequality reduces to the obvious inequality ∞ n=1 n 2 α 2 n ≥ ∞ n=1 α 2 n .
We will actually see below that inequality (8.9) is simply the spectral gap for the Jacobi operator associated to the arcsine law.
Poincaré Inequalities and Jacobi Operators
In this section we show how the two versions of the Poincaré inequalities can be viewed as spectral gaps for some Jacobi operators. This discussion is mainly driven from the work [8] by Cabanal-Duvillard and his interpretation of the variance in (7.4) in terms of the number operator of the Jacobi operator associated to the arcsine law. This viewpoint allows for an unified perspective of the Poincaré inequalities presented in the preceding sections.
For our purpose we consider here the Jacobi operators given, for smooth functions on (−1, 1), by
L λ f (x) = −(1 − x 2 )f (x) + (2λ + 1)xf (x) (9.1)
for λ ≥ 0. We consider the Gegenbauer polynomials C λ n , λ > 0, defined by the generating function
∞ n=0 r n C λ n (x) = 1 (1 − rx + r 2 ) λ .
For λ = 0 we set C λ n (x) = T n (x)/n, n ≥ 1, where T n are the Chebyshev polynomials of the first kind. It is known that C λ n are eigenfunctions of L λ , with eigenvalue n(n + 2λ), i.e.
L λ C λ n = n(n + 2λ)C λ n On the other hand the Gegenbauer polynomials are orthogonal with respect to the probability measure
ν λ = 2 2λ Γ 2 (λ + 1) πΓ(2λ + 1) 1 [−1,1] (x)(1 − x 2 ) λ−1/2 .
Notice that in the case of λ = 0, this becomes the arcsine law and for λ = 1, this is the semicircular law, while for λ = 1/2, this becomes the uniform measure on [−1, 1]. Take now the normalized Gegenbauer polynomials φ λ n = G λ n / c λ n , where c λ n = G λ n (x) 2 ν λ (dx). Then φ λ n form an orthonormal basis of L 2 (ν λ ) and thus the operator L λ is diagonalized in this basis. Consider N λ to be the counting number operator with respect to the basis φ λ n , i.e.
N λ φ λ n = nφ λ n . (9.2)
This implies that L λ = N 2 λ + 2λN λ . Therefore we have the following two inequalities
L λ ≥ (2λ + 1)N λ and N λ ≥ 1 − P λ (9.3)
where P λ here stands for the projection on constant functions in L 2 (ν λ ). In other words, P λ φ = φν λ . Notice that equation (9.3) include two statements. The first one is the comparison of L and N , with the spectral gap 2λ + 1 while the second one is the spectral gap of the counting number operator with the spectral gap 1. In the sequel we want to translate these spectral gaps in terms of Poincaré type inequality. For this matter we need to find the kernel of the operator N .
Then we have for any function in the domain of definition of L λ , that φ = ∞ n=0 α n φ λ n , and then
Lφ, φ L 2 (ν λ ) = ∞ n=0 n(n + 2λ)α 2 n .
On the other hand, using integration by parts, we can justify that
Lφ, φ L 2 (ν λ ) = φL λ φdν λ = φ (x) 2 (1 − x 2 )ν λ (dx).
For the number operator, we have that
φN λ φdν λ = ∞ n=0 nα 2 n = lim r↑1 ∞ n=0 nr n−1 α 2 n . Now, for −1 < r < 1, ∞ n=0 nr n−1 α 2 n = φ(x)φ(y) ∞ n=0
nr n−1 φ λ n (x)φ λ n (y)ν λ (dx)ν λ (dy).
Furthermore, since φ λ n dν λ = 0 for n ≥ 1, we also obtain that φ 2 (x)φ λ n (y)ν λ (dx)ν λ (dy) = 0 for n ≥ 0 and thus, denoting K λ (r, x, y) = − ∞ n=0 nr n−1 φ λ n (x)φ λ n (y),
φ(x)φ(y) ∞ n=0 nr n−1 φ λ n (x)φ λ n (y)ν λ (dx)ν λ (dy) = 1 2 (φ(x) − φ(y)) 2 K λ (r, x, y)ν λ (dx)ν λ (dy).
The following formula is essentially due to Watson [33] and valid for λ > 0,
∞ n=0 r n φ λ n (x)φ λ n (y) = (1 − r 2 )Γ(2λ) 2 2λ−1 Γ 2 (λ) 1 −1 (1 − z 2 ) λ−1 (1 − 2r(xy + z (1 − x 2 )(1 − y 2 )) + r 2 ) 1+λ dz.
For λ = 0, we have to deal with the Chebyshev polynomials of the first kind which was more or less what appeared in the proof of Theorem 8. For this case, we have that (denoting x = cos t and y = cos s),
∞ n=0 r n c n T n (x)T n (y) = 1 − r cos(t + s) 1 − 2r cos(t + s) + r 2 + 1 − r cos(t − s) 1 − 2r cos(t − s) + r 2
where c n = T 2 n dν 0 = 1 for n = 0 and 1/2 otherwise. Thus, we obtain, after differentiation with respect to r and then limit over r ↑ 1, that
K λ (x, y) = lim r↑1 K λ (r, x, y) = Γ(2λ) 2 3λ−1 Γ 2 (λ) 1 −1 (1 − z 2 ) λ−1 1 − xy − z (1 − x 2 )(1 − y 2 ) 1+λ dz, λ > 0 1 − xy (x − y) 2 , λ = 0 1 2(x − y) 2 , λ = 1. (9.4)
The integrand is not a rational function. In some cases, it is algebraic since λ ≥ 0 need not be an integer.
To reveal the singularity of this kernel, we make the change of variable
1 − xy − z (1 − x 2 )(1 − y 2 ) = t 1 − xy − (1 − x 2 )(1 − y 2 ) .
Then, after simple algebraic manipulations, setting f λ : (0, 1) → R,
f λ (u) = 1/u 1 [(t − 1)(1 − ut)] λ−1 t λ+1 dt, and H λ (x, y) = Γ(2λ) 1 − xy + (1 − x 2 )(1 − y 2 ) λ 2 3λ−1 Γ 2 (λ) ((1 − x 2 )(1 − y 2 )) λ−1/2 f λ (x − y) 2 1 − xy + (1 − x 2 )(1 − y 2 ) 2 , λ > 0 1 − xy, λ = 0, 1 2 , λ = 1,(9.5)
we can rewrite equation (9.4) for |x|, |y| < 1 as
K λ (x, y) = H λ (x, y) (x − y) 2 (9.6)
where H λ (x, y) is a continuous function of x, y ∈ [−1, 1]. Now, from (9.3), we obtain the following result.
Theorem 11. For any λ ≥ 0, one has for all λ ≥ 0 and any φ ∈ C 1 ([−1, 1]), that
φ (x) 2 (1 − x 2 )ν λ (dx) ≥ 2λ + 1 2 φ(x) − φ(y) x − y 2 H λ (x, y)ν λ (dx)ν λ (dy). (9.7)
and φ(x) − φ(y) x − y Remark 5.
1. Equation (9.7) for λ = 0 is the statement of Theorem 8 for the case V (x) = 2x 2 and for λ = 1 (more precisely, equation (7.8)) while equation (9.8) is the statement of the second Poincaré inequality contained in Theorem 9 for the semicircular law. The combination of these two inequalities is equation (8.9). In other words, for measures ν λ , the first Poincaré type inequality is driven by the comparison of the Jacobi and counting number operators defined in (9.1) and (9.2), as the second Poincaré type is the spectral gap of the counting number operator.
2. Combining equations (9.7) and (9.8), we also get a Brascamp-Lieb type inequality:
φ (x) 2 (1 − x 2 )ν λ (dx) ≥ (2λ + 1) Var ν λ (φ). (9.9) For λ ≥ 1/2, the measure ν λ is of the form e −V (x) dx, where V (x) = −c λ − (λ − 1/2) log(1 − x 2 )
, a strictly convex function on (−1, 1) and according to the classical Brascamp-Lieb inequality [6],
φ (x) 2 (1 − x 2 ) 2 (1 + x 2 ) ν λ (dx) ≥ (2λ − 1) Var ν λ (φ). (9.10)
Notice here that neither (9.9) not (9.10) implies the other which means that they complement each other in some sense. For example if φ has support in [− 1 2λ , 1 2λ ], (9.9) implies (9.10), while if φ is supported on
[−1, 1]\[− 1
2λ , 1 2λ ], (9.10) implies (9.9).
Wishart Ensembles and Marcenko-Pastur Distributions
In this section, we address the preceding functional inequalities for probability measures on the real positive axis in the context of the Wishart Ensembles from random matrix theory and their associated Marcenko-Pastur distributions.
We start with the random matrix heuristics although, as far as we know, it has not been used towards functional inequalities as before. The problems of large deviations principle for the distribution of the eigenvalues of Wishart ensembles is discussed in [16]. The model is as follows. Take T (n) a n × p(n) random matrix with all the entries being iid N (0, 1) random variables. Then T (n)T (n) t for n < p(n) is known as the nonsingular Wishart random ensemble. According to [17, page 129], the distribution of the Wishart ensembles is given by
C np e − p(n) 2 TrM (det M ) (p−n−1)/2 dM.
where the measure dM = i≤j dM ij the restriction of the Lebegue measure on the set of n × n non-negative matrices. It is also known (for example [17, page 129]) that the joint distribution of eigenvalues (λ 1 , λ 2 , . . . , λ n ) of 1 p(n) T (n)T (n) t is given by
1 Z n e − p(n) 2 P n i=1 ti n i=1 λ (p(n)−n−1)/2 i 1≤i<j≤n |λ i − λ j |.
Our interest is in the limit distribution of µ n = 1 n n i=1 δ λi . The classical result states that if n/p(n) − −−− → n→∞ α ∈ (0, 1], then the limit distribution of µ n is the so called Marcenko-Pastur distribution given by
1 [(1− √ α) 2 ,(1+ √ α) 2 ] (x) 4α − (x − 1 − α) 2 2παx dx.
This is a particular model for the standard Wishart ensembles. However one can consider a more general example with potentials for which the distribution of the matrix is driven by a potential Q : [0, ∞) → R,
C n e −p(n)TrQ(M ) (det M ) γ(n) dM
where dM stands for the Lebesgue measure on n × n positive definite matrices. The distribution of eigenvalues of M is given by 1 Z n e −p(n)
P n i=1 Q(ti) n i=1 t γ(n) i 1≤i<j≤n |t i − t j |.
The main result of [16] is that the distribution of the random measures µ n = 1 p(n) p(n) i=1 δ λi under the conditions n/p(n) − −−− → n→∞ α ∈ (0, 1], γ(n)/n − −−− → n→∞ γ > 0, ν n satisfy a large deviation principle with scale n −2 and the rate function given by
R(µ) =Ẽ Q (µ) − inf µ∈P([0,∞))Ẽ Q (µ), whereẼ Q (µ) = α Q(x) − γ log(x) µ(dx) − α 2 2 log |x − y|µ(dx)µ(dy).
This gives the following motivation. Assume that V : [0, ∞) → R∪{+∞} is a lower semi-continuous potential such that lim |x|→∞ (V (x) − 2 log |x|) = ∞. Then, according to the results in [27], we know that there is a unique minimizer of inf
µ∈P([0,∞)) E V (µ).
In addition the equilibrium measure µ V has compact support. A particular case of interest is V (x) = rx − s log(x) with r > 0, s ≥ 0 for which we know [27, page 207] that the equilibrium measure is given by In addition, the minimizer of EṼ is µṼ =μ V Further, for the non-decreasing transportation map θ of µ V into µ, defineθ (x) = sign(x) θ(x 2 ), (10.4) which transportsμṼ intoμ. In addition, as it was pointed out in [18], the relative free Fisher information I V (µ) is defined for measures µ on [0, ∞) with density p = dµ/dx in L 3 ([0, ∞), xdx) as
µ V (dx) = 1 [a,b] (x) r (x − a)(b − x) 2πx dx where a = s + 2 − 2 √ s + 1 r , b = s + 2 + 2 √ s + 1 r .(10.I V (µ) = ∞ 0 x Hµ(x) − V (x) 2 µ(dx) with Hµ(x) = p.v. 2 x − y µ(dy). (10.5)
Otherwise we take I V (µ) = +∞. The main reason for defining this in this way is because, cf. [18,Lemma 6.3] and the discussion following, one has I V (µ) = 2IṼ (μ), (10.6) where IṼ is defined by (4.1).
To state the transportation cost result, we define the appropriate distance. For any µ, ν ∈ P([0, ∞)), set the distance as
W (µ, ν) = inf π∈Π(µ,ν) √ x − √ y 2 π(dx, dy) 1/2 (10.7)
where Π(µ, ν) is the set of probability measures on R 2 with marginals µ and ν.
In this context we have the following transportation cost inequality.
Theorem 12. Assume that V : (0, ∞) → R is C 2 ((0, ∞)) such that V (x 2 ) − ρx 2 is convex on (0, ∞) for some ρ > 0 and let µ V be the equilibrium measure of V on [0, ∞). Then, for any probability measure µ on [0, ∞), we have that
ρ W 2 (µ, µ V ) ≤ E V (µ) − E V (µ V ),(10.
8)
In the case of V (x) = rx − s log(x) with r > 0 and s ≥ 0, this inequality with ρ = r is sharp.
Proof. As announced, the idea is to interpret this inequality as an inequality for potentials on the whole real line instead of [0, ∞). Using the measuresμ andμ V from equation (10.2) together with (10.3), we have that
E V (µ) − E V (µ V ) = 2 EṼ (μ) − EṼ (μ V ) .
On the other hand, if θ is the (increasing) transportation map of µ V into µ, then it is not hard to check that
W 2 (µ, ν) = √ x − θ(x) 2 µ V (dx) = x −θ(x) 2μ V (dx).
In this framework the inequality (10.8) translates as
ρ 2 W 2 2 (μ,μ V ) ≤ EṼ (μ) − EṼ (μ V ). (10.9)
From here we will use the same argument as in the proof of Theorem 2. Start with
EṼ (μ) − EṼ (μ V ) = Ṽ (θ(x)) −Ṽ (x) −Ṽ (x)(θ(x) − x) μ V (dx) + θ (x) −θ(y) x − y − 1 − logθ (x) −θ(y) x − y μ V (dx)μ V (dy).
and notice that the second line of this is non-negative. For the first line we point out that becauseṼ (x) − ρ 2 x 2 is convex and x andθ(x) have the same sign, for any x,
V (θ(x)) −Ṽ (x) −Ṽ (x) θ (x) − x ≥ ρ 2 (θ(x) − x) 2 ,
which implies (10.8).
In the case V (x) = rx − s log(x), take θ(x) = ( √ x + m) 2 for large m and notice thatθ(x) = x + msign(x). Therefore inequality (10.9) becomes
rm 2 ≤ rm 2 +2rm |x|μ(dx)−2s log |x + msign(x)| |x| μ(dx)− log 1 + m sign(x) − sign(y) x − y μ(dx)μ(dy)
which is sharp for large m.
The next result is the Log-Sobolev type inequality, which was conjectured by Cabanal-Duvillard in [7, page 140] for the case of Marcenko-Pastur distribution.
Theorem 13. Let V be as in the previous theorem. Then, with the definition from (10.5) and for any measure µ ∈ P([0, ∞)),
E V (µ) − E V (µ V ) ≤ 1 2ρ I V (µ). (10.10)
In the case V (x) = rx − s log(x), r > 0 and s ≥ 0 inequality (10.10) with ρ = r is sharp.
Proof. We will discuss here the proof only in the case when µ has a smooth compactly supported density, careful approximations being described in [18]. From (10.6), we have I V (µ) = 2IṼ (μ), where IṼ (μ) = (Hμ(x) −Ṽ (x)) 2μ (dx). Rewriting everything in terms ofμ and the associated quantities, the inequality to be proven can be written in the same way as we did in the proof of Theorem 4, 1 2ρ
Hμ
(θ(x) −Ṽ (θ(x))) 2μṼ (dx) + Ṽ (x) −Ṽ (θ(x)) −Ṽ (θ(x))(x −θ(x)) μṼ (dx) − Hμ(θ(x)) −Ṽ (θ(x)) x −θ(x) μṼ (dx) + Hμ(θ(x)) x −θ(x) μṼ (dx) − log x − ỹ θ(x) −θ(y)μṼ (dx)μṼ (dy) ≥ 0. (10.11)
Notice thatṼ (x) − ρ 2 x 2 is not convex on the whole real line but it is convex on the intervals (0, ∞) and (−∞, 0). The key to everything here is thatθ(x) has the same sign as x and this allows us to apply convexity ofṼ (x)− ρ 2 x 2 on each of the intervals (−∞, 0) and (0, ∞) to conclude that
V (x) −Ṽ θ (x) −Ṽ (θ(x))(x −θ(x)) ≥ ρ 2 x 2 −θ(x) 2 − 2θ(x)(x −θ(x)) = ρ 2 x −θ(x) 2 . (10.12)
From here we can follow word by word the proof of Theorem 4. For the case V (x) = rx, we have equality in (10.10) ifθ(x) = x + msign(x) and thus this means
θ(x) = ( √ x + m) 2 .
In the case V (x) = rx − s log(x), we look atθ(x) = x + m for large m. In this caseṼ (x) = rx 2 /2 − s log |x| and then a simple calculation shows that (10.10) is equivalent to
rm 2 +2mr |x|μ V (dx) − 2s log |x + msign(x)| |x| μ V (dx) − 2 log 1 + m sign(x) − sign(y) x − y μ(dx)μ(dy) ≤ m 2 ρ r − s x(x + msign(x)) 2μ V (dx).
Dividing both sides by m 2 and taking the limit of m to infinity implies that ρ ≤ r. On the other hand ρ = r validates (10.10), hence ρ = r is the best constant.
Next in line is the HWI inequality which is the content of the following statement.
Theorem 14. Assume V is as in Theorem 12 and the distance W given by (10.7). Then for any measure µ ∈ P([0, ∞)),
E V (µ) − E V (µ V ) ≤ 2I V (µ)W (µ, µ V ) − ρ W 2 (µ, µ V ). (10.13)
For the case of V (x) = rx − s log(x), r > 0, s ≥ 0, this inequality for ρ = r is sharp.
Proof. As it was made clear in the previous two theorems, we translate this inequality in terms of the associated symmetric measures on R. Following upon the proofs of above theorems, we can rewrite (10.13) in the following form:
(Hμ(θ(x)) −Ṽ (θ(x))) 2μ
V (dx) (θ(x) − x) 2μ V (dx) 1/2 − Hμ(θ(x)) −Ṽ (θ(x)) (x −θ(x))μ V (dx) + Ṽ (x) −Ṽ (θ(x)) −Ṽ (θ(x)) x −θ(x) − ρ(θ(x) − x) 2 μ V (dx) + Hμ(θ(x)) x −θ(x) μ V (dx) − log x − ỹ θ(x) −θ(y)μ V (dx)μ V (dy) ≥ 0.
Using the fact thatṼ (x) − ρ 2 x 2 is convex on each interval (−∞, 0) and (0, ∞) combined with the fact that x and θ(x) have the same sign, the rest of the proof is the same as the one of Theorem 5.
For the case V (x) = rx − s log(x), using θ(x) = ( √ x + m) 2 , one can show that ρ = r is sharp.
At last, we would like to discuss a Poincaré type inequality in this context. As in Section 7, for the heuristics, we consider the general model of random matrices with distribution On the other hand, from [20] or [8] the variance of Φ(M ) converges to 1 4 Var arcsine [a,b] (φ), where we recall that arcsine [a,b]
= dx π √ (x−a)(b−x)
is the arcsine law on the support [a, b] of µ V . Next, 1 n Tr((∇ 2 Ψ(M )) −1 φ (M ) 2 ) = 1 sn Tr((φ (M )M ) 2 ), whose integral against P n converges to the integral of 1 s x 2 φ (x) 2 against the equilibrium measure µ V from equation (10.1). These considerations suggest that
x 2 φ (x) 2 µ V (dx) ≥ s 4
Var arcsine [a,b] (φ). (10.15) Notice here that one can actually make this heuristic into an actual proof of this inequality. Motivated by these heuristics and also inspired by Theorem 8, we have the following stronger result.
Theorem 15. Assume that Q : [0, ∞) → R is a convex potential and let V (x) = Q(x) − s log(x) for s > 0 satisfy lim x→∞ (V (x) − 2 log(x)) = ∞. Assume that the support of µ V is [a, b]. Then for any smooth function φ on [a, b], the following holds,
x 2 φ (x) 2 µ V (dx) ≥ s 4π 2 b a b a φ(x) − φ(y) x − y 2 −2ab + (a + b)(x + y) − 2xy 2 (x − a)(b − x) (y − a)(b − y) dxdy. (10.16)
If Q(x) = rx + t, equality is attained for φ(x) = c 1 + c2 x , therefore (10.16) is sharp. In particular, combining (10.16) with (9.8) for λ = 0, we get an improvement of (10.15) as
x 2 φ (x) 2 µ V (dx) ≥ s 2 Var arcsine [a,b] (φ).
Equality though is attained only for φ identically 0.
In the case V (x) = rx, r > 0, on [0, ∞), there is no constant C > 0 such that inequality (10.16) holds with C instead of s/4π 2 . Nevertheless, for every smooth φ on [a, b], the following holds,
xφ (x) 2 µ V (dx) ≥ r 4π 2 b a b a φ(x) − φ(y) x − y 2 −2ab + (a + b)(x + y) − 2xy 2 (x − a)(b − x) (y − a)(b − y) dxdy,(10.17)
with equality for φ(x) = c 1 + c 2 x.
As remarked after the statement of Theorem 8, the numerator in (10.17) is nonnegative.
Proof. The same argument as in the proof of Theorem 8, shows that the density g(x) of µ V satisfies
g(x) ≥ s (x − a)(b − x) 2πx √ ab ,
therefore it suffices to show that
1 π √ ab b a xφ (x) 2 (x − a)(b − x) dx ≥ 1 2π 2 b a b a φ(x) − φ(y) x − y 2 −2ab + (a + b)(x + y) − 2xy 2 (x − a)(b − x) (y − a)(b − y) dxdy.
Next, making the change of variable x = (a + b)/2 + u(b − a)/2 and denoting ζ(u) = φ((a + b)/2 + u(b − a)/2), we reduce the problem to showing that for any smooth function φ on [−1, 1], we have
1 π √ ab 1 −1 a + b 2 + b − a 2 u ζ (u) 2 1 − u 2 du ≥ 1 2π 2 1 −1 1 −1 ζ(u) − ζ(v) u − v 2 1 − uv √ 1 − u 2 √ 1 − v 2 dudv.
Denoting β = b−a b+a , we have that a+b 2 √ ab = 1 √ 1−β 2 , and the preceding inequality reformulates as
(1 + βu)ζ (u) 2 1 − u 2 du ≥ 1 − β 2 2π 1 −1 1 −1 ζ(u) − ζ(v) u − v 2 1 − uv √ 1 − u 2 √ 1 − v 2 dudv.(10.18)
To show this, take ψ(t) = ζ(cos(t)) and then after the change of variable u = cos(t) we need to check π 0 (1 + β cos(t))ψ (t) 2 dt ≥ 1 − β 2 2π π 0 π 0 ψ(t) − ψ(s) cos(t) − cos(s) Writing ψ(t) = ∞ n=0 a n cos(nt) and using that ψ (t) = − ∞ n=1 na n sin(nt), together with the fact that π 0 cos(t) sin(nt) sin(mt)dt = π 4 for |m − n| = 1 0 otherwise, and equation (7.11), the inequality becomes n≥1 (n 2 a 2 n + βn(n + 1)a n a n+1 ) ≥ 1 − β 2 n≥1 na 2 n . (10.19)
Let δ = 1− √ 1−β 2 β
be the solution 0 < δ < 1 of βδ 2 − 2δ + β = 0. Notice that for any n ≥ 1, we have a n a n+1 ≥ − δ 2 a 2 n − 1 2δ a 2 n+1 which implies that n≥1 n 2 a 2 n +βn(n+1)a n a n+1 ≥ n≥1 n 2 a 2 n − βn(n + 1) 2
δa 2 n + 1 δ a 2 n+1 = n≥1 nβ(1 − δ 2 ) 2δ a 2 n = 1 − β 2 n≥1 na 2 n ,
what we had to prove. Notice here that equality is attained in this inequality if and only if a n+1 = −δa n for all n ≥ 1, which means that a n = (−1) n−1 δ n−1 a 1 . This corresponds to the function ψ(t) = a 1 δ+cos t 1+δ 2 +2δ cos t , or ζ(u) = a 1 δ+u 1+δ 2 +2δu which means that φ(x) = a 1 (r − s/x). Therefore equality holds also for φ(x) = c 1 + c 2 /x. For the second part, in the case V (x) = rx with r > 0, notice that if there is a C > 0 so that (10.16) holds with C instead of s/4π 2 , then, following the same argument as above, we would have the equivalent of (10.19) as n≥1 n 2 a 2 n + n(n + 1)a n a n+1 ≥ C n≥1 na 2 n .
Taking in this a n = (−γ) n n for 0 < γ < 1, we have that γ 2 /(γ + 1) ≥ −C log(1 − γ 2 ), and this is certainly false for γ close to 1.
For equation (10.17), notice that in this case the equilibrium measure is µ V (dx) = r √ b−x 2π √
x and then after a simple rescaling this follows from equation (7.8). This complete the proof of the theorem.
It is interesting to look at this inequality as a spectral gap result as in Section 9. For example in the case of the Marcenko-Pastur measure (Q(x) = rx), the inequality (10.16) is actually equivalent to inequality (10.18). Using the interpretation from Section 9, we can rephrase this as, for a given β ∈ (0, 1),
(1 + βx)(1 − x 2 )φ (x) 2 ν 0 (dx) ≥ 1 − β 2 N φ, φ ν0
where ν 0 is the arcsine law on [−1, 1] and N is the number operator. Now we can define the operator
L β φ(x) = −(1 + βx)(1 − x 2 )φ (x) − (β − x − 2βx 2 )φ (x).
With this definition,
L β φ, φ ν0 = 1 π (1 + βx)φ (x) 2 1 − x 2 dx
and then inequality (10.18) becomes
L β φ, φ ν0 ≥ 1 − β 2 N φ, φ ν0
for any smooth function φ on [−1, 1]. In particular this means that L β ≥ 1 − β 2 N . On the other hand it is clear that the operator L β can not be diagonalized by the Chebyshev polynomials of the first kind, therefore the orthogonal polynomial approach given in Section 9 does not work the same way here.
Remark 6. We want to point out that for the case V (x) = rx − s log(x) for r > 0 and s ≥ 0, the parameter r appears in the transportation, Log-Sobolev and HWI, while the parameter s plays the dominant role in the Poincaré inequality.
e −nTrn(V (M )) dM on the set H n of complex Hermitian n × n matrices where dM is the Lebesgue measure on H n . For a matrix M , let µ n (M ) = 1 n n k=1 δ λ k (M ) be the distribution of eigenvalues of M . These are random variables with values in P(R), the set of probability measures on R which converge almost surely to a non-random measure µ V on R.
see[27, page 46]) that µ V is the arcsine law of [−1, 1]. Thus if we consider
addition that V (x) − ρx 2 is a convex function on R. Then, consider Φ(M ) = Trφ(M ), where φ : R → R is a compactly supported and smooth function. Notice that ∇Φ(M ) = φ (M ) and thus |∇Φ(M )| 2 = |φ (M )| 2 = Tr(φ (M ) 2 ). Since nTr(V (M )) − nρ|M | 2 is convex, we can apply Poincaré's inequality on H n to obtain that Tr φ (M ) 2 P n (dM ) ≥ nρ Var Pn Tr(φ(M )) . (7.3) The first term in this inequality (cf. equation (7.2)) converges to φ (x) 2 µ V (dx). To understand the second term in the above equation, notice that Var(Tr(φ(M ))) = E (Tr(φ(M )) − E[Tr(φ(M ))]) 2 . The study of the asymptotic of the linear statistics, Tr(φ(M )) − E[Tr(φ(M ))] in the literature of random matrix is known as "fluctuations". From Johansson's paper
kt − cos ks)(cos lt − cos ls)(1 − cos t cos s) (cos t − cos s)2 dtds.
kt − cos ks)(cos lt − cos ls)(1 − cos t cos s) (cos t − cos s)2 dtds
kt − cos ks)(cos lt − cos ls)(1 − cos t cos s) (cos t − cos s) 2 dtds = π 2 kδ kl and that π 0 π 0 ψ(t) − ψ(s) cos t − cos s 2 (1 − cos t cos s)dtds = π 2 ∞ k=1 ka 2 k . (7.11)
the Marcenko-Pastur distribution for V (x) = rx − s log(x), r > 0, s ≥ 0, with r = 1/α and s = (1 − α)/α. The natural way to deal with functional inequalities in the context of measures on the positive axis [0, ∞) is to transfer measures from [0, ∞) into measures on the whole R. For a measure µ on [0, ∞), consider thus the associated symmetric measureμ on R defined as µ(F ) =μ {x : x 2 ∈ F } (10.2) for any measurable set F of [0, ∞). DefiningṼ (x) = V (x 2 )/2, it is then an easy exercise to check that E V (µ) = 2EṼ (μ). (10.3)
P n (dM ) = C n e −nrTrM (det M ) sn dM = C n e −nTr rM −s log(M ) dM = C n e −nTr(V (M )) dM (10.14) where dM stands for the Lebesgue measure on n × n positive definite matrices and s ≥ 0. For a given smooth compactly supported function φ : [0, ∞) → R, we want to apply the Brascamp-Lieb inequality [6] to the function Φ(M ) = Trφ(M ) on the space of positive definite matrices. Now, ∇Φ(M ) = φ (M ). The Hessian of Ψ(M ) := Tr(V (M )) can be interpreted as a linear map from H n (n×n Hermitian matrices) into itself which is given by ∇ 2 Ψ(M )X = sM −1 XM −1 . Hence the inverse of the Hessian is then (∇ 2 Ψ(M )) −1 X = 1 s M XM . Thus we obtain from Brascamp-Lieb that 1 n Tr (∇ 2 Ψ(M )) −1 φ (M ) 2 P n (dM ) ≥ Var Pn Φ(M ) .
H λ (x, y)ν λ (dx)ν λ (dy) ≥ 2 Var ν λ (φ).(9.8)
(1 − cos(t) cos(s))dtds.
Acknowledgements. We would like to thank D. Cabanal-Duvillard for pointing to us the formula of the fluctuation for Wishart ensembles and for informing us about his Log-Sobolev conjecture in[7]. Many thanks to the anonymous referee for the pertinent and scholarly comments which pointed several shortcomings of the submitted version and led to an overall improvement of this paper.
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| []
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[
"Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture",
"Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture"
]
| [
"Neda Bagherifard ",
"Eaman Eftekhary "
]
| []
| []
| The first author introduced a notion of equivalence on a family of 3-manifolds with boundary, called (simple) balanced 3-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence. Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced 3-manifolds which are not in the trivial equivalence class (i.e. the equivalence class of S 2 × [−1, 1]). | null | [
"https://arxiv.org/pdf/2205.14706v2.pdf"
]
| 249,191,665 | 2205.14706 | 09c18af0a8e6f648302d72dfc52e4ce5d05e03de |
Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture
Neda Bagherifard
Eaman Eftekhary
Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture
The first author introduced a notion of equivalence on a family of 3-manifolds with boundary, called (simple) balanced 3-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence. Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced 3-manifolds which are not in the trivial equivalence class (i.e. the equivalence class of S 2 × [−1, 1]).
Introduction
Suppose that R = {b 1 , . . . , b m } is a finite subset of the free group F (X ) generated by the finite set X = {a 1 , . . . , a n }. We denote by (X |R) the quotient G of F (X ) by the normal subgroup generated by R. The pair (X , R) is then called a presentation of G with generators X and relators R, which is balanced if |X | = |R|. An extended Andrews-Curtis transformation (EACtransformation for short) on (X , R) is defined as one of the following transformations, or its inverse, which of course results in another presentation of G [Wri75] (see also [HAM93]):
1. Composition: Replace b ∈ R with bb for some b = b in R; 2. Inversion: Replace b ∈ R with b −1 ; 3. Cancellation: Replace b = b aa −1 b ∈ R with b b
, where a ∈ X or a −1 ∈ X ; 4. Stabilization: Add a new element a to both X and R; 5. Replacement: Replace a a or a a −1 for a in all the relators for some a = a in X .
Stable Andrews-Curtis transformations (or SAC transformations) consist of the first 4 transformations and their inverses. The presentations P = (X , R ) and P = (X , R) are called EAC equivalent (respectively, SAC equivalent) if P is obtained from P by a finite sequence of EAC transformations (respectively, SAC transformations). For the trivial group, the SAC equivalence class of a presentation is the same as its EAC equivalence class ( [Wri75]). The stable Andrews-Curtis conjecture (or SAC conjecture) states that every balanced presentation of the trivial group is SAC equivalent to the trivial presentation, i.e. (a, a) (c.f. [AC65]). Most experts expect that the SAC conjecture is not true and there are potential counterexamples ( [Bro84,BM93,MS99,MMS02]). One of the simplest potential counterexamples for the SAC conjecture is given by P 0 = (X 0 , R 0 ), where X 0 = x, y and R 0 = r = x −1 y 2 x y −3 , s = y −1 x 2 y x −3 ,
(see [MMS02]). The group presentation P 0 is considered in this paper in correspondence with a notion of equivalence for balanced 3-manifolds, as explained below.
A compact oriented 3-manifold N with boundary is called balanced if each component of N has two boundary components of the same genus. Let ∂ ± N denote boundary components of N where the orientation of ∂ + N (resp. ∂ − N ) matches with (resp. is the opposite of) the orientation inherited as the boundary of N . Let ι ± : ∂ ± N → N denote the inclusion maps and H ± denotes the normalizer of ι ± * (π 1 (∂ ± N )) in π 1 (N ). A balanced 3-manifold is called simple if for each connected component N of it as above, both quotient groups π 1 (N )/H ± are trivial.
Associated with each Heegaard diagram H = (Σ, α, β) of N , there are two balanced presentations P α (H) and P β (H) for the latter quotient groups where for P α (H) (resp. P β (H)) the generators are in correspondence with the α (resp. β) and the relators are in correspondence with the β (resp. α) (see [Bag]). Let p α (N ) and p β (N ) denote the EAC equivalence classes of the presentations P α (H) and P β (H), respectively. Note that these EAC equivalence classes are independent of the choice of the Heegaard diagram H for N . Similarly, we may define p α (N ) and p β (N ) for a balanced 3-manifold N which is not connected. If N is a simple balanced 3-manifold, p α (N ) and p β (N ) are both EAC equivalence classes of presentations for the trivial group.
A notion of equivalence in the family of balanced 3-manifolds was introduced in [Bag]. We say that a balanced 3-manifold N simplifies to another balanced 3-manifold N if there is an embedded cylinder C ∼ S 1 × [−1, 1] in N , with ∂ ± C ∼ S 1 × {±1} ⊂ ∂ ± N , such that N is obtained by cutting N along C and gluing two copies of D 2 ×[−1, 1] to the resulting boundary cylinders in N \ C . We then write N C − → N . We say that a balanced 3-manifold N admits a simplifier if there is a sequence of simplifications
N = N n C n − − → N n−1 C n−1 − −− → · · · C 2 −→ N 1 C 1 −→ N 0 ,
such that N 0 is a disjoint union of copies of S 2 × [−1, 1]. The inverse of a simplification is called an anti-simplification. Two balanced 3-manifolds are called equivalent if they may be changed to one-another by a finite sequence of simplifications, anti-simplifications and homeomorphisms. The equivalence of the balanced 3-manifolds N and N implies that p α (N ) = p α (N ) and p β (N ) = p β (N ). Therefore, a pair of well-defined EAC equivalence classes (of group presentations) are assigned to each equivalence class of balanced 3-manifolds and in this sense, the equivalence notion between balanced 3-manifolds is weaker than the EAC equivalence for group presentations. In the family of simple balanced 3-manifolds, both EAC equivalence classes are presentations of the trivial group. Motivated by the SAC conjecture, it is thus natural to ask if there is a simple balanced 3-manifold N which is not equivalent to the trivial simple balanced 3-manifold S 2 × [−1, 1]? In this paper, we combine the main result of [Bag] with tools from Heegaard Floer theory (see [OS04b]) to prove the following theorem.
Theorem 1.1. There is a simple balanced 3-manifold N with
p α (N ) = p β (N ) = P 0 = [(X 0 , R 0 )],
where P 0 is given in (1), which is not equivalent to S 2 × I .
As mentioned above, besides Heegaard Floer theory, the main tool used in proving Theorem 1.1 is a fundamental result about the equivalence class of the simple balanced 3-manifold S 2 × [−1, 1], which is proved in [Bag] and may be stated as follows.
Theorem 1.2. [Bag, Theorem 1.6] Every balanced 3-manifold N which is equivalent to S 2 × I admits a simplifier. N admits a simplifier. We have ∂N = ∂ + N −∂ − N where ∂ ± N are surfaces of genus 1. If N admits a simplifier, there is a nontrivial cylinder C in N such that ∂ ± C in ∂ ± N are essential curves. Let f : ∂ + N → ∂ − N be the homeomorphism from ∂ + N to ∂ − N which makes the following diagram commutative:
H 1 (∂ + N , Z) f * > H 1 (∂ − N , Z) H 1 (N , Z) ι − * < ι + * >
This criteria determines f upto isotopy. Since ∂ + C is homologous to ∂ − C , we may further assume that f maps ∂ + C to ∂ − C . Let N f denote the closed 3-manifold obtained from N by identifying ∂ + N with ∂ − N using f . LetC denote the torus in M which is obtained from C by identifying ∂ + C with ∂ − C . ThusC and ∂ + N ∼ f ∂ − N represent linearly independent homology classes in H 2 (N f , Z) = Z ⊕ Z ⊕ Z with zero Thurston semi-norm. Recall that the Thurston semi-norm of a closed 3-manifold M is defined on H 2 (M , Z) by
Θ : H 2 (M , Z) → Z ≥0 , Θ(ξ) := min χ + (Σ) Σ → M and [Σ] = ξ ,
where the minimum is taken over all compact, oriented surfaces Σ = i Σ i embedded in M and representing the homology class ξ, while χ + (Σ) is defined by g (Σ i )>0 (2g (Σ i ) − 2) (see [Thu86]). Heegaard Floer homology groups with twisted coefficients detect the Thurston semi-norm. More precisely, for a closed 3-manifold M , let H F (M ) denote the Heegaard Floer homology group of M with twisted coefficients, which is a Z/2Z-graded Z 2 [H 1 (M , Z)]module defined in [OS04b]. There is a decomposition of this group by Spin c structures, A remark about the above argument may be appropriate here. Let us assume that N 1 and N 2 are balanced 3-manifolds and that N 1 simplifies to N 2 , i.e. N 1 C − → N 2 . For k = 1, 2, let M k denote the closed 3-manifold obtained by taking two copies N 1 k and N 2 k of N k , iden-
H F (M ) =tifying ∂ + N 1 k with ∂ + N 2 k and identifying ∂ − N 1 k with ∂ − N 2 k .
Then the cylinder C gives the torus T ⊂ M 1 , while M 2 is obtained by cutting M 1 along T and gluing two solid tori to the resulting boundary components. Theorem 1.3 is then helpful in detecting T . Nevertheless, the equivalence of N 1 and N 2 is yet not well-translated to Heegaard Floer theory, e.g. to a practical correspondence between H F (M 1 ) and H F (M 2 ). If T is 2-sided, the problem is studied in [Eft15] and [Eft18], and a relatively powerful machinery is developed in [HRW16]. For non-separating T , it is interesting to develop such a correspondence.
About the proof. In Section 2, we construct a Heegaard diagram H for the closed manifold M from H, following the approach of [Lek13]. The number of generators for the Heegaard diagram H is 7936, and it is thus not feasible to find the Spin c structures s 1 and s 2 and compute the groups H F (M , s i ) without computational assistance (from computers). We prove a simple lemma from linear algebra in Section 3, in the sprit of the general discussion in [Eft15, Section 2]. The lemma is used, in combination with a computer program, to obtain a short-list of potential Spin c structures s with H F (M , s) = 0 (although obtaining the short-list is not an official part of our argument). Among the potential candidates, two specific Spin c structures s 1 and s 2 are considered in Section 5 and Section 6. The chain complexes associated with these Spin c structures are 8-dimensional and 72-dimensional, respectively. The homology groups of the chain complexes C F (M , s i ) (for i = 1, 2) are studied using the lemma proved in Section 3, a series of computer assisted computations and explicit computations of the contribution of moduli spaces associated with certain classes of Whitney disks. Since the Heegaard diagram is not nice (in the sense of [SW10]), such explicit computations are necessary and appear in Section 4.
A Heegaard diagram for the mapping torus
In this section, we obtain a Heegaard diagram for M = N f , using the construction of [Lek13].
Let us assume that the diagram H is obtained from a Morse function h : N → [−1, 1]. Then h gives a circle-valued Morse functionh : M → S 1 with two critical points x 1 and x 2 of index 1 and two critical points y 1 and y 2 of index 2, such that
N u x 1 (h) ∩ Σ = α 1 , N u x 2 (h) ∩ Σ = α 2 , N s y 1 (h) ∩ Σ = β 1 , N s y 2 (h) ∩ Σ = β 2 , h −1 (1) = ∂ + N ∼ f ∂ − N = Σ mi n andh −1 (−1) =Σ = Σ max .
Here N s x and N u x denote the stable and unstable manifold of x ∈ N with respect to the flow of a gradient-like vector field for h. Following [Lek13], let p 1 and p 2 be disjoint points in Σ \ α ∪ β and γ 1 and γ 2 denote two gradient flow lines disjoint from N u x i and N s y i such that
γ i ∩Σ = {p i } and γ i ∩ ∂ + N = {p i }, for i = 1, 2.
Furthermore, γ 1 (resp. γ 2 ) is mapped onto the northern (resp. southern) semi-circle of S 1 . Let N (γ i ), i = 1, 2, denote the normal neighborhood of γ i that intersectsΣ and ∂ + N in the small disks D p i and Dp i , respectively. By removing D • p i and D •p i and gluing ∂D p i to ∂Dp i along ∂N (γ i ) we obtain the Heegaard surface Σ. Let α 5 = ∂Dp 1 and β 5 = ∂Dp 2 . Let α 3 and α 4 (resp. β 3 and β 4 ), be disjoint arcs in ∂ + N such that ∂α 3 and ∂α 4 are disjoint points on β 5 and ∂β 3 and ∂β 4 are disjoint points on α 5 , while |α 3 ∩ β 4 | = |α 4 ∩ β 3 | = 1 and |α 3 ∩ β 3 | = |α 4 ∩ β 4 | = 0. Flowing the arcs β 3 and β 4 through the gradient flow ofh above the northern semi-circle, we obtain disjoint arcs β 3 and β 4 in Σ\∂ + N which are disjoint from β 1 and β 2 . Similarly, flowing the arcs α 3 and α 4 , we obtain α 3 and α 4 which are disjoint from α 1 and α 2 . This determines the sets of α and β curves:
α = {α 1 , α 2 , α 3 = α 3 ∪ α 3 , α 4 = α 4 ∪ α 4 , α 5 } and β = {β 1 , β 2 , β 3 = β 3 ∪ β 3 , β 4 = β 4 ∪ β 4 , β 5 }.
Having fixed a marked point z, finger-move isotopies may be used to make (Σ, α, β, z) weakly admissible. If we apply the procedure to the Heegaard diagram of Figure 1, we arrive at the admissible Heegaard diagram illustrated in Figure 2 with 21392 generators. Handleslides of α 4 over α 3 (10 times) and isotopies on α 3 give an alternative (more suitable) weakly admissible Heegaard diagram H with 7936 generators, as illustrated in Figure 3. We use x i , j ,k to label the intersection point of α i and β j which is labeled k in the diagram of Figure 3.
The set of periodic domains for H is generated by three domains P 1 , P 2 and P 3 . The first two generators are shown in Figure 3. The periodic domains P 1 and P 2 are of the form
P 1 = −D 52 + 57 i =53 D i + i ∈I 1 D i and P 2 = −D 49 + i ∈I 2 D i + i ∈I 1 D i ,
where the domains D i with i in I 1 and I 2 are colored gray and green in Figure 3, respectively. If ∂ b P denote the β-boundary of a periodic domain P , we then have ∂ b (P 1 ) = −β 5 and
β 1 β 2 β 3 α 4 α 1 α 2 α 5 β 5 β 4 α 3 Figure 2. A weakly admissible Heegaard diagram of N f with 21392 generators ∂ b (P 2 ) = β 3 .
We may choose the third generator P 3 of the space of periodic domains so that
∂ b (P 3 ) = β 4 + 2β 3 − 3β 1 + 2β 2 . Let H (P i ) ∈ H 2 (M , Z)
denote the homology classes associated with the periodic domains P i for i = 1, 2, 3, which form a basis for H 2 (M , Z) (see [OS04b], Proposition 2.15). Correspondingly, we obtain a bijection
c : Spin c (M ) → Z ⊕ Z ⊕ Z, c(s) := 1 2 〈c 1 (s), H (P 1 )〉, 〈c 1 (s), H (P 2 )〉, 〈c 1 (s), H (P 3 )〉 ,
which gives an identification of Spin c (M ) with Z 3 . To compute s z :
T α ∩ T β → Spin c (M ) = Z 3 under this identification, define s r z : T α ∩ T β → Z 3 by setting s r z (x 0 ) = (0, 0, 0) for x 0 = (x 0 i ) 5 i =1 = (x 1,1,1 , x 2,2,1 , x 3,4,1 , x 4,5,1 , x 5,3,2 ). Let (y 0 i ) 5 i =1 denote a permutation of (x 0 i ) 5 i =1 so that y 0 i ∈ β i .
Fix a connected path γ 0 on α ∪ β in the diagram such that for each α ∈ α and β ∈ β, γ 0 ∩α and γ 0 ∩β are connected and x 0 i ∈ γ 0 , 1 ≤ i ≤ 5, (the yellow path in Figure 3 satisfies these properties). Fix (a, b, c). In the definition of s r z , note that the intersection numbers take place over the Heegaard surface. The map s r z is used instead of s z for the purposes of this paper.
x = (x i ) 5 i =1 = (y i ) 5 i =1 ∈ T α ∩ T β with x i ∈ α i and y i ∈ β i , i.e. (y i ) i is just a permutation of (x i ) i . Let (x 0 , x) denote the closed 1-cycle in Σ obtained by connecting y i to y 0 i through β i , connecting y 0 i to x 0 i through γ 0 , and connecting x 0 i to x i through α i for i = 1, . . . , 5. Note that for j = 1, 2, 3, the evaluation 〈P D[ (x 0 , x)], H (P j )〉 is the algebraic intersection number of (x 0 , x) with ∂ b P j . Therefore, if we set s r z (x) := − 〈 (x 0 , x), β 5 〉, 〈 (x 0 , x), β 3 〉, 〈 (x 0 , x), β 4 + 2β 3 − 3β 1 + 2β 2 〉 , there is a fixed triple (a, b, c) = (0, −1, −4) ∈ Z 3 such that s z = s r z (x) +z D 34 D 15 D 30 D 12 D 2 D 1 D 28 D 20 D 29 D 19 D 24 D 31 D 13 D 33 D 25 D 23 D 0 D 48 D 21 D 22 D 17 D 35 D 3 D 26 D 4 D 37 D 36 D 38 D 51 D 50 D 49 D 18 D 14 β 2 D 54 D 55 D 53 D 52 D 56 D 58 D 59 D 60 D 61 D 62 D 63 D 64 D 65 D 66 D 67P 1 = −D 52 + 57 i =53 D i + i ∈I 1 D i , P 2 = −D 49 + i ∈I 2 D i + i ∈I 1 D i
and a third periodic domain P 3 , where D i is colored gray for i ∈ I 1 and green for i ∈ I 2 . We have ∂ b (P 1 ) = −β 5 and ∂ b (P 2 ) = β 3 . The periodic domain P 3 may be chosen so that ∂ b (P 3 ) = β 4 + 2β 3 − 3β 1 + 2β 2 .
Simplifying computations using algorithmic calculations
All our computations are performed with coefficients in Z 2 [H 1 (M , Z)]. In the discussions of this section, we have the diagram H = (Σ, α, β, z) from Figure 3 in mind. Nevertheless, the strategy works for many of the chain complexes associated with sutured manifold diagrams in the sense of [Juh06], or even [AE15]. Since there is a large number of generators associated with H, we break the computation of H F (M ) into a computer-assisted part and a human part using the following observation.
Let z 2 ⊂ z 1 denote two sets of marked points containing z. Most of the time, we take z 2 = {z}. If z 1 is sufficiently large so that it contains a marked point in each one of the periodic domains, we may choose a decomposition C F (Σ, α, β, z 1 ) = A⊕B ⊕H , so that the differentials d z 1 and d z 2 = d z 1 + d are determined by the matrices
d z 1 = 0 0 0 I 0 0 0 0 0 and d = f h m k g n p q l .
(2)
Lemma 3.1. Suppose that with the above notation in place, I + k is invertible. Then
H * ( C F (Σ, α, β, z 2 )) = H * (H , l + p(I + k) −1 n).
Proof. The proof follows from two base changes. The first base change is given by
I + k (I + k) f (I + k) −1 0 0 I 0 0 0 I f h m I + k g n p q l (I + k) −1 f (I + k) −1 0 0 I 0 0 0 I = 0 * m I * n p(I + k) −1 * l = 0 m A m I n A n A l A l ,
where A = p(I + k) −1 and the last equality follows from d 2 z 2 = 0. The second base change is
I 0 0 0 I 0 0 A I 0 m A m I n A n A l A l I 0 0 0 I 0 0 A I = 0 0 m I 0 n 0 0 l + An .
In applications of Lemma 3.1, we choose an area assignment A for the regions in Σ\α∪β such that A(P i ) = 0 for i = 1, 2, 3. Moreover, z 1 , z 2 and A are chosen so that the regions not touched by z 1 have very small areas and the regions containing marked points from z 1 \ z 2 have very large areas. Under these assumptions, in each Spin c class s, A descends to an energy filtration on C F (Σ, α, β, z 2 , s) (see [OS04b]). We may further assume that A = 〈a 1 , . . . , a r 〉, B = 〈b 1 , . . . , b r 〉, with A(a 1 ) < A(a 2 ) < · · · < A(a r ), while the differential d z 1 is given by sending a i to b i . With respect to the energy filtration,
A(a i )− A(b i )
is then a small positive number and k is a lower triangular matrix with zeros on the diagonal. Therefore, I +k is an invertible matrix with (I +k) −1 = ∞ i =0 k i . This allows us use Lemma 3.1. Of course, the use of Lemma 3.1 is not restricted to the aforementioned situation.
In our search for the Spin c classes s with the property that H F (M , s) = 0, we may first restrict our attention to the Spin c classes which satisfy 〈c 1 (s), H (P 1 )〉 = 0, since P 1 corresponds to ∂ + N and is represented by an embedded surface of genus 1. We may then enlarge the set z 2 = {z} of punctures in the Heegaard diagram to a bigger set z 1 , so that (Σ, α, β, z 1 ) is nice, while the criteria discussed in the previous two paragraphs is satisfied. If the group H * ( H F (Σ, α, β, z respectively. As we will see in Section 5 and Section 6, there are 8 generators x 1 and 72 generators x 2 of the above type.
Non-polygonal disks with holomorphic representatives
In this section, we study the moduli spaces associated with three classes of Whitney disks with non-polygonal domains, which will be encountered in Section 6. First, let D k,l ,n = D(φ k,l ,n ) denote the genus zero domain of a Whitney disk φ k,l ,n , with two boundary components having 2k-edges and 2l -edges, respectively. The edges on each boundary component consist of alternating arcs from distinct α and β curves. For such a disk to have Maslov index 1, it is necessary that all the 2(k + l ) angles on the boundary are acute angles, except for precisely one of them. We further assume that the obtuse angle is on the boundary component with 2l edges, where α 1 and β 1 meet at x n and enter the interior of D k,l ,n , and intersect each other at x n−1 , . . . , x 1 in D • k,l ,n . There is some extra freedom in choosing the domain D k,l ,n (up to isotopy of the curves) which corresponds to the edges where α 1 and β 1 exit D k,l ,n and is dropped from the notation (see Figure 4 (left)).
Lemma 4.1. Let φ k,l ,n be a disk with a domain as described above. Then # M (φ k,l ,n ) = 1.
Proof. First, consider the case k = l = 1. Consider the triply punctured Heegaard diagram Figure 4 (right). Here Σ 1 is a surface of genus three and the sutured manifold determined by H 1 is the same as the sutured manifold determined by a Heegaard diagram (T = S 1 × S 1 , α, β = {b} × S 1 , z 1 , z 2 ), where α is homotopically trivial and cuts β twice, one of the punctures is located in one of the two bigons in T − α − β, and two of the punctures are located in the cylindrical component of T − α − β. Therefore, the Heegaard Floer group associated with H 1 is trivial. With the notation of Figure 4, the generators in T α ∩ T β are
H 1 = H n 1 = (Σ 1 , α 1 = {α 1 , α 2 , α 3 }, β 1 = {β 1 , β 2 , β 3 }, z 1 , z 2 , z 3 ) illustrated inR i = (x i , r, r ), S i = (x i , r, s ), T i = (x i , s, r ), U i = (x i , s, s ), i = 1, . . . , n + 1 V = (t , t , r ), W = (t , t , s ), X = (u, r, u ), Y = (u, s, u ).
Most Whitney disks with positive domain and index 1 which contribute to the differential are of the form φ 1,1,k for some k = 1, . . . , n. In fact, there are Whitney disks ψ 1 k ∈ π 2 (U k , S k+1 ), ψ 2 k ∈ π 2 (T k , R k+1 ), ψ 3 n+1−k ∈ π 2 (R k+1 , S k ) and ψ 4 n+1−k ∈ π 2 (T k+1 ,U k ) for k = 1, . . . , n, where each ψ i k is of type φ 1,1,k . Other than these classes, there are also disks ψ 1 0 ∈ π 2 (V, R 1 ), ψ 2 0 ∈ π 2 (W, S 1 ), ψ 3 0 ∈ π 2 (X , S n+1 ) and ψ 4 0 ∈ π 2 (Y ,U n+1 ),
with Maslov index one, and the domain of every one of them is a rectangle. Therefore, # M (ψ i 0 ) = 1 for i = 1, . . . , 4. Moreover, there are disks φ ∈ π 2 (T 1 ,W ) and φ ∈ π 2 (T n+1 , X ) with domains of type φ 1,2,n . If we set it follows that m i 0 = m i 1 = 1 and that the differential of the chain complex is given by
T k m 2 k > R k+1 T 1 1 > R 2 T n+1 m > X Y V U k−1 m 4 n+2−k ∨ m 1 k−1 > S k m 3 n+1−k ∨ W m ∨ 1 > S 1 m 3 n ∨ U n 1 ∨ m 1 n > S n+1 1 ∨ U n+1 1 ∨ R 1 1 ∨
for k = 2, . . . , n. Therefore, we conclude that m = m 3 n , m = m 1 n and
m 2 k · m 3 n+1−k = m 4 n+2−k · m 1 k−1 for k = 2, . . . , n.(3)
For n = 2, (3) implies m 2 2 = m 4 2 . Moreover, since the homology is trivial, m 2 2 = m 4 2 = 1. This proves the claim for φ 1,1,2 . Having established the proof for φ 1,1, j with j = 1, . . . , n − 1 (where n > 2), Equation (3) for k = 2 implies that m 4 n = 1, proving the claim for φ 1,1,n .
Next, we consider the case l = 1 while k is arbitrary. Let
H 2 = (Σ 2 , α 2 = {α 0 , α 1 , . . . , α k+1 }, β 2 = {β 0 , β 1 , . . . , β k+1 }, z)
be the Heegaard diagram shown in Figure 5. Here Σ 2 is a surface of genus k + 3. With the notation of Figure 5 in place and refreshing the notation set for the case k = l = 1, the generators in T α ∩ T β are where i belongs to {1, . . . , n + 1}. Consider the Whitney disks φ 1,1,n ∈ π 2 (R n , S n+1 ), φ ∈ π 2 (S n+1 , T n+1 ), φ ∈ π 2 (R n ,U n ), φ k,1,n ∈ π 2 (U n , T n+1 ).
W = (v,
such that D(φ ) is the green domain, D(φ 1,1,n ) is the union of yellow domains, D(φ) is the union of grey and green domains and D(φ k,1,n ) is the union of grey and yellow domains. Then ψ = φ 1,1,n * φ = φ * φ k,1,n has index 2, while these are the only degenerations of ψ as a juxtaposition of two positive Whitney disks of Maslov index 1. This implies
# M (φ k,1,n ) = # M (φ k,1,n ) · # M (φ ) = # M (φ 1,1,n ) · # M (φ) = # M (φ 1,1,n ) = 1,
completing the proof for the case where l = 1, while k and n are arbitrary. Similarly, the argument above may be used to conclude # M (φ k,l ,n ) = 1 for arbitrary values of k, l and n.
Let D(φ n,m ) denote the genus zero domain of a Whitney disk φ n,m which has three boundary components, each consisting of 2 edges on α i and β i for i = 1, 2, 3. Let α 3 have n intersection points {x 1 , . . . , x n } with β 3 and α 2 have m intersection points {y 1 , . . . , y m } with β 2 in D(φ n,m ). The union of the yellow regions and the grey regions in Figure 6 illustrates the domain of such a disk. We assume that all the corners of the boundary edges in D(φ n,m ) are acute except for two, where α 2 intersects β 2 in an obtuse angle in y m−1 and α 3 intersects β 3 in an obtuse angle in x n−1 (see Figure 6).
Proof. Consider the Heegaard diagram
H 3 = (Σ, α = {α 1 , α 2 , α 3 , α 4 }, β = {β 1 , β 2 , β 3 , β 4 }, z)
which is illustrated in Figure 6. Here Σ is a surface of genus six which is obtained by attaching six one-handles that each one connects the boundary circles of disks with the same color. There are 4nm + 3m + 2n + 2 intersection points in T α ∩ T β . With the notation of Figure 6 in place, these intersection points are P = (t 4 , u 1 , r 3 , s), Q = (t 4 , u 3 , r 1 , s)
R i = (t 1 , u 3 , x i , s), S i = (t 3 , u 1 , x i , s), T i , j = (t 2 , u 1 , x i , y j ), U i , j = (t 5 , u 1 , x i , y j ), V i , j = (t 1 , u 2 , x i , y j ), W i , j = (t 1 , u 4 , x i , y j )
X j = (t 4 , u 2 , r 1 , y j ), Y j = (t 4 , u 4 , r 1 , y j ) and Z j = (t 4 , u 1 , r 2 , y j ), for i = 1, . . . , n and j = 1, . . . , m. Consider the Whitney disks of index 1 φ 1,1,m−1 ∈ π 2 (V n−1,m−1 ,W n−1,m ), φ 2,1,n−1 ∈ π 2 (W n−1,m ,U n,m ), φ ∈ π 2 (V n−1,m−1 , T n−1,m−1 ) and φ n,m ∈ π 2 (T n−1,m−1 ,U n,m )
Here, the domains D(φ 1,1,m−1 ) is the union of the regions colored yellow, D(φ) is the union of the regions colored green,D(φ 2,1,n−1 ) is the union of the regions colored grey and green, and D(φ n,m ) is the union of the regions colored yellow and grey in Figure 6. Then ψ = φ 1,1,m−1 * φ 2,1,n−1 = φ * φ n,m , determined by D(ψ) which is the union of all colored regions, is a Whitney disk of Maslov index 2 in π 2 (V n−1,m−1 ,U n,m ). The disk ψ degenerates as the juxtaposition of two disks of Maslov index 1 only in the above two ways. Therefore, we conclude that
# M (φ n,m ) = # M (φ n,m ) · # M (φ) = # M (φ 1,1,m−1 ) · # M (φ 2,1,n−1 ) = 1.
The last equality, which follows from Lemma 4.1, completes the proof of the lemma.
For φ ∈ π 2 (x, y), let D(φ) be a surface of genus one with one boundary component consisting of 2 edges that contains a unique intersection point u in the interior which belongs to both x and y (see Figure 7). The grey domains on the left illustrate D(φ). Lemma 4.3. Let φ be a disk with a domain as described above. Then # M (φ) = 1.
Proof. Consider the triply punctured Heegaard diagram
H 4 = (Σ, α = {α 1 , α 2 }, β = {β 1 , β 2 }, z = {z 1 , z 2 , z 3 })
of genus 1 which is illustrated in Figure 7 (left). With the notation of Figure 7 in place, there are 6 intersection points in T α ∩ T β , which may be listed as
R 1 = (x, u), R 2 = (y, u), S 1 = (t , v), S 2 = (t , w), T 1 = (s, v) and T 2 = (s, w).
The differential is shown in Figure 7, on the right. A black arrow which connects a generator X to a generator Y denotes that there is a disk from X to Y with a unique holomorphic representative. The arrow in purple denotes the disk with the domain D(φ). By doing an isotopy which removes the two intersection points of β 2 with α 1 and then doing a destabilization which removes α 2 and β 2 , we obtain the standard genus zero Heegaard diagram for the closed three manifold S 1 ×S 2 . Therefore the Heegaard Floer homology group associated with H is Z 2 . This proves that # M (φ) = 1. Figure 7. Left: A Heegaard diagram with three marked points and a Heegaard surface of genus one. Right: the differential associated with this diagram.
α 2 α 1 β 1 β 2 x z 1 z 2 z 3 y t s u v w R 2 R 1 T 2 S 2 T 1 S 1
The first non-trivial Heegaard-Floer group
Let us assume that s 1 corresponds to the triple (0, 1, 7). The chain complex C F (M , s 1 ) is then generated by the following 8 generators:
1 = x 1,1,2 , x 2,2,5 , x 3,3,1 , x 4,5,2 , x 5,4,2 2 = x 1,1,2 , x 2,2,5 , x 3,3,2 , x 4,5,2 , x 5,4,2 3 = x 1,1,2 , x 2,4,2 , x 3,2,1 , x 4,5,2 , x 5,3,2 4 = x 1,1,3 , x 2,4,2 , x 3,2,2 , x 4,5,2 , x 5,3,2 5 = x 1,2,1 , x 2,1,2 , x 3,5,1 , x 4,3,1 , x 5,4,2 6 = x 1,2,1 , x 2,4,2 , x 3,5,1 , x 4,1,2 , x 5,3,2 7 = x 1,3,1 , x 2,1,2 , x 3,2,1 , x 4,5,2 , x 5,4,2 8 = x 1,4,2 , x 2,1,2 , x 3,2,2 , x 4,5,2 , x 5,3,2
Let z 1 consist of marked points in all domains except for D 12 , D 13 , D 30 , D 49 . The differentials for the Heegaard diagram (Σ, α, β, z 1 ) along with the domains of the connecting disks are shown in Figure 8 on the left. In this figure, a black arrow from a generator x to a generator y denotes that there is a disk from x to y with a unique holomorphic representative. In fact, the domains associated with all the disks are polygons. The group H * ( C F (Σ, α, β, z 1 ), s 1 ) is thus isomorphic to Z 2 2 and is generated by C = { 5 , 7 }. To compute H F (M , s 1 ), we need to determine the matrices l , n, p, and k in Lemma 3.1. Define the disks φ ∈ π 2 3 , 7 and φ 1 ∈ π 2 5 , 7 , φ 2 ∈ π 2 5 , 6 , φ 3 ∈ π 2 2 , 7 , φ 4 ∈ π 2 2 , 6 , φ 5 ∈ π 2 5 , 1 , Then all the disks of index 1 and positive domain between the generators of this complex are the disks ψ i for i = 1, 2, 3, the disk φ, the disks φ i for i = 1, . . . , 5, the disks φ i − P 1 for i = 1, . . . , 4 and the disks φ i + P 1 for i = 1, 2, 5, see Figure 8 Σ, α, β, z). The contributions from the disks φ i for i = 1, . . . , 5, the disks φ i − P 1 for i = 1, . . . , 4 and the disks φ i + P 1 for i = 1, 2, 5 are denoted with green, red and blue arrows, respectively.
Setting K = b 4 + c 4 e −P 1 , N 1 = b 5 + c 5 e P 1 , N 2 = b 2 + c 2 e −P 1 + d 2 e P 1 , P = b 3 + c 3 e −P 1 and L = b 1 + c 1 e −P 1 + d 1 e P 1 , it then follows that
k = 0 0 0 K 0 0 0 0 0 , n = N 1 0 N 2 0 0 0 , p = 0 0 0 P 1 0 , l = 0 0 L 0 ⇒ l + p(I + k) −1 n = 0 0 L + N 1 (P + K ) + N 2 0 .
Note that in this matrix we have
= L + N 1 (P + K ) + N 2 =(b 1 + b 2 + b 5 b 3 + b 5 b 4 + c 5 c 3 + c 5 c 4 ) + (c 1 + c 2 + b 5 c 3 + b 5 c 4 )e −P 1 + (d 1 + d 2 + c 5 b 3 + c 5 b 4 )e P 1 .(4)
The computation of H F (M , s 1 ) is thus reduced to a computation of . Consider the disks λ 1 ∈ π 2 7 , 5 , λ 2 ∈ π 2 6 , 5 , λ 3 ∈ π 2 7 , 2 , λ 4 ∈ π 2 6 , 2 , λ 5 ∈ π 2 1 , 5 , and λ 6 ∈ π 2 1 , 2 .
which correspond to the domains Consider the Whitney disk classes of index 2 η 1 , η 1 , η 1 ∈ π 2 ( 5 , 5 ), η 2 , η 2 , η 2 ∈ π 2 ( 1 , 1 ) andη 3 , η 3 , η 3 ∈ π 2 ( 2 , 2 ) which correspond to the periodic domains D(η i ) = Σ, D(η i ) = Σ − P 1 and D(η i ) = Σ + P 1 , for i = 1, 2, 3.
D(λ i ) = Σ − D(φ i ), for i = 1,
The possible degenerations of η 1 , η 1 and η 1 to positive disks of Maslov index 1 are:
η 1 = φ j * λ j = (φ 5 + P 1 ) * (λ 5 − P 1 ), for j = 1, 2, 5, η 1 = (φ i − P 1 ) * λ i = φ 5 * (λ 5 − P 1 ), for i = 1, 2 and η 1 = (φ i + P 1 ) * λ i = (φ 5 + P 1 ) * λ 5 , for i = 1, 2 Therefore, the following three equations follow:
b 1 + b 2 + b 5 b 5 + c 5 c 5 = 0, c 1 + c 2 + c 5 b 5 = 0 and d 1 + d 2 + b 5 c 5 = 0.(5)
The possible degenerations of η 2 , η 2 and η 2 into positive disks of Maslov index 1 are η 2 = λ 6 * ψ 1 = λ 5 * φ 5 = (λ 5 − P 1 ) * (φ 5 + P 1 ) η 2 = (λ 6 − P 1 ) * ψ 1 = (λ 5 − P 1 ) * φ 5 and η 2 = (λ 6 + P 1 ) * ψ 1 = λ 5 * (φ 5 + P 1 ).
Therefore, we obtain the following 3 equations
b 6 + b 5 b 5 + c 5 c 5 = 0, c 6 + c 5 b 5 = 0 and d 6 + b 5 c 5 = 0.(6)
Similarly, the possible degeneration of η 3 , η 3 and η 3 into positive disks of Maslov index 1 are
η 3 = φ i * λ i = (φ i − P 1 ) * (λ i + P 1 ) = ψ 1 * λ 6 η 3 = (φ i − P 1 ) * λ i = ψ 1 * (λ 6 − P 1 ) and η 3 = φ i * (λ i + P 1 ) = ψ 1 * (λ 6 + P 1 ) for i = 3, 4.
Therefore, we obtain the following 3 equations as well
b 3 b 3 + b 4 b 4 + c 3 c 3 + c 4 c 4 + b 6 = 0, b 3 c 3 + b 4 c 4 + c 6 = 0 and c 3 b 3 + c 4 b 4 + d 6 = 0. (7)
Let z 1 contain a marked point in all the regions of Σ − α − β except for those appearing in D(λ 3 ), D(λ 4 ), D(φ 5 ), and D 13 and ∂ 1 denote the corresponding differential. Note that P 1 , P 2 and P 3 − Σ may still be considered as a basis for the space of periodic domains. Therefore, the diagram remains admissible for this choice of marked points. Then
∂ 2 1 3 = (b 3 + b 4 ) 2 and ∂ 2 1 7 = (b 5 + b 3 ) 1 ⇒ b 3 = b 4 = b 5 .(8)
Similarly, let z 2 contain a marked point in all the regions of Σ − α − β except for those appearing in D(λ 3 + P 1 ), D(λ 4 + P 1 ), D(φ 5 + P 1 ), and D 13 and ∂ 2 denote the corresponding differential. Then
∂ 2 2 3 = (c 3 + c 4 ) 2 and ∂ 2 2 7 = (c 5 + c 3 ) 1 ⇒ c 3 = c 4 = c 5 .(9)
If follows from Equations 4-9 that the matrix l + p(I + k) −1 n = 0. Thus H F (M , s 1 ) = 0. latter domains is illustrated in Figure 9, where the aforementioned domains are colored green.
The differential corresponding to the Heegaard diagram (Σ, α, β, z 1 ) is illustrated in Figure 10. In fact, most of the positive Whitney disks of Maslov index 1 for (Σ, α, β, z 1 ), which connect two of the aforementioned 72 generators have polygonal domains, and their contribution to the differential is thus equal to 1. There are precisely 12 disks φ i for i = 1, . . . , 7, and φ j for j = 1, 2, 5, 6, 7, with non-polygonal domains, where we have φ 1 ∈ π 2 21 , 45 , φ 1 ∈ π 2 22 , 46 , φ 2 ∈ π 2 13 , 37 , φ 2 ∈ π 2 14 , 38 , φ 3 ∈ π 2 3 , 1 , φ 4 ∈ π 2 71 , 64 , φ 5 ∈ π 2 47 , 43 , φ 5 ∈ π 2 48 , 44 , φ 6 ∈ π 2 39 , 33 , φ 6 ∈ π 2 40 , 34 , φ 7 ∈ π 2 35 , 31 , φ 7 ∈ π 2 36 , 32 . Thus, the differential is as illustrated in Figure 10 and H * ( C F (Σ, α, β, z 1 )) is generated by C = C 1 = 49 ,C 2 = 50 ,C 3 = 53 ,C 4 = 69 .
To compute H F (M , s 2 ), we need to determine the matrices l , n, p, and k in the Lemma 3.1. All possible positive disks with Maslov index 1 between the generators in C are ψ 1 , ψ 1 ∈ π 2 ( 49 , 50 ), ψ 2 , ψ 2 ∈ π 2 ( 53 , 69 ), ψ 3 ∈ π 2 ( 49 , 53 ), ψ 4 ∈ π 2 ( 50 , 69 ).
For i = 1, 2, the domain associated with ψ i is a polygon. The domain associated with ψ 1 is shown in Figure 11 as the union of yellow, blue and pink regions. By Lemma 4.3, # M (ψ 1 ) = 1. The domain associated with ψ 3 is shown in Figure 11 as the union of blue, brown and pink Let A and B denote the chain complexes generated by all the generators colored in light green and yellow in Figure 10, respectively. Denote the generators of A and B by A i and B i , respectively. We may choose the labeling of the aforementioned generators of A and B such that k is a lower triangular matrix. Therefore, p(I + k) −1 n = pn + pkn + pk 2 n + . . . .
For j ≥ 0, since the coefficients are in Z 2 , each non-zero entry in pk j n is of the form (pk j n) w v = r 1 ,...,r j +1 p wr 1 k r 1 r 2 . . . k r j r j +1 n r j +1 v , and implies the existence of positive disks λ t of Maslov index 1 for t = 1, . . . , j + 1, where λ 1 ∈ π 2 (C v , B r j +1 ), λ j +2 ∈ π 2 (A r 1 ,C w ) and λ t ∈ π 2 (A r t , B r t −1 ), t = 2, . . . , j + 1. and # M (λ t ) = 1. In particular, D(λ t ) > 0 for all t and
D(λ t ) ⊂ j +2 t =1 D(λ t ) = j +1 t =1
D(λ t ) + D(λ) ± D(P 1 ), µ(λ t ) = 1, D(λ t ) > 0,
for some positive Whitney disks λ t ∈ π 2 (A r t , B r t ) and λ ∈ π 2 (C v ,C w ) of Maslov index 1. Potentially, there are only two such sequences satisfy (10), which are shown in Figure 12. Here ψ 5 ∈ π 2 ( 49 , 51 ) is a disk with a polygonal domain. The domain associated with the disk ψ 6 ∈ π 2 ( 23 , 53 ) is shown in Figure 11 as the union of grey, brown and blue regions.
Lemma 6.1. With the above notation in place, we have # M (ψ 3 ) = # M (ψ 6 ).
Proof. The domains associated with ψ 3 and ψ 6 are extended in two ways in Figure 13
η • = φ • 1 * σ • 1 = σ • 2 * φ • 2 = σ • 3 * φ • 3 • = a, c, η • = φ • 1 * σ • 1 = σ • 2 * φ • 2 = φ • 3 * σ • 3 • = b, d ,
where the corresponding domains are given by Potential sequences of disks corresponding to non-zero summands p wr 1 k r 1 r 2 . . . k r j r j +1 n r j +1 v in (pk j n) w v .
D(φ • 1 ) = 5 j =1 D • j , D(φ • 2 ) = D • 1 + 8 j =3 D • j , D(φ • 3 ) = D • 1 + D • 2 + D • 3 + D •
Figure 1 .
1The group presentation P 0 of Equation 1 is realized by the Heegaard diagram H = (Σ,ᾱ = {α 1 , α 2 },β = {β 1 , β 2 }), illustrated inFigure 1. In fact, the Heegaard diagram H determines a simple balanced 3manifold N with p α (N ) = p β (N ) = [P 0 ]. If N is equivalent to S 2 × I , Theorem 1.2 implies that The Heegaard surface is a surface of genus three which is obtained by identifying the boundaries of disks with the same color. The curves are oriented in a way that the balanced presentation associated with this Heegaard diagram is P 0 .
s∈Spin c (M ) H F (M , s). Let us consider the case where M = N f is given as above. Extend [∂ + N ] to a basis for H 2 (N f , Z) ∼ = Z 3 and consider a corresponding identification of Spin c (M ) with Z 3 (by evaluation of the first Chern class of the Spin c structures over the generators of the homology group H 2 (M , Z) ∼ = Z 3 ). In order to prove Theorem 1.1, we show that there are two linearly independent Spin c structures s 1 and s 2 , with the property that 〈c 1 (s i ), [∂ + N ]〉 = 0 and H F (M , s i ) = 0 for i = 1, 2. Since Θ([C ]) = 0, we thus have [C ] = λ[∂ + N ], for some integer λ, which contradicts our assumption. This shows that N does not have a simplifier and is thus not equivalent to S 2 × I .
Figure 3 .
3A weakly admissible Heegaard diagram for M with 7936 generators. The connected components of Σ\α ∪ β are labeled D i , for i = 0, . . . , 67. The periodic domains are generated by
1 , s) is trivial, it follows that H F (M , s) is also trivial. Trying different sets z 1 of marked points allows us exclude many of the Spin c classes s from the the Thurston polytope, consisting of Spin c structures t with H F (M , t) = 0. Among the remaining Spin c classes, we combine Lemma 3.1, computer assisted computations and the study of certain classes of Whitney disks (from next section) to show that H F (M , s i ) = 0 for i = 1, 2, where s 1 and s 2 are the classes of generators x 1 and x 2 of C F (H) with s r z (x 1 ) = (0, 1, 7) and s r z (x 2 ) = (0, −1, −8),
Figure 4 .
4Part of a Heegaard diagram which illustrates the domain associated with the disk φ k,1,n is illustrated (left). The red curves are the α curves and the blue curves are the β curves. A Heegaard diagram of genus 3 containing the domain D 1,1,n is illustrated on the right. The domain of the Whitney disk ψ ∈ π 2 (R n ,U n ) is shaded.
m
= # M (φ), m = # M (φ ) and m i k = # M (ψ i k ) for i = 1, . . . , 4 and k = 0, . . . , n,
Figure 5 .
5r, u 2 , . . . , u k , r ), V = (s, t k , u 2 , . . . , u t −1 , s , t t , . . . , t k−1 , r ), R i = (u 1 , r, u 2 , . . . , u k , x i ),S i = (t 1 , r, u 2 , . . . , u k , x i ), T i = (s, t k , t 1 , . . . , t k−1 , x i )and U i = (s, u 1 , u 2 , . . . , u k , x i ), A Heegaard surface of genus k +3.The colored region denotes the domain associated with ψ, which degenerates in two ways as φ 1,1,n * φ and φ * φ k,1,n , where D(φ ) is the region colored green and D(φ 1,1,n ) is the union of regions colored yellow.
Lemma 4. 2 .
2If the domain of φ n,m is as described above, we have # M (φ n,m ) = 1.
Figure 6 .
6A Heegaard surface of genus 6. The colored regions denote the domain associated with a Whitney disk ψ of index 2 which degenerates in two ways.
of Maslov index 1 by specifying their domains. If I = {4, 16, 18, 26, 34, 36, 38, 51, 52} and D is the formal sum i ∈I D i , we have D(φ) = D 0 , D(φ 1 ) = Σ − {D 12 + D 14 + D 30 }, D(φ 2 ) = Σ − {D 0 + D 14 }, D(φ 3 ) = Σ − (D + D 12 + D 14 + D 30 ), D(φ 4 ) = Σ − (D + D 0 + D 14 ), D(φ 5 ) = D + D 49 .
D(ψ 1 )
1(right). Let b i = # M (φ i ), for i = 1, . . . , 5, c i = # M (φ i − P 1 ), for i = 1, . . . , 4, d i = # M (φ i + P 1 ), for i = 1, 2, and c 5 = # M (φ 5 + P = D 49 , D(ψ 2 ) = D 12 + D 30 , D(ψ 3 ) = D 13 .
Figure 8 .
8Left: The differential for (Σ, α, β, z 1 ), Right: The differential for (
. . . , 5 and D(λ 6 ) = Σ − D(ψ 1 ). The domains of λ 1 and λ 2 are polygons. Then all the positive disks of index 1 in π 2 ( 1 , x), π 2 (x, 5 ), and π 2 (x, 2 ), with x a generator of C F (M , s 1 ), which the region containing the marked point z are λ i for i = 1, . . . , 6, λ i + P 1 for i = 3, 4, 6, and λ i − P 1 for i = 5, 6. Let b i = # M (λ i ), for i = 3, . . . , 6, c i = # M (λ i + P 1 ), for i = 3, 4, c i = # M (λ i − P 1 ), for i = 5, 6, and d 6 = # M (λ 6 + P 1 ).
Figure 9 .
9Part of the Heegaard diagram, where the marked points z 1 are in the domains other than those in green.
Figure 10 .
10The differential corresponding to the diagram (Σ, α, β, z 1 ).The domains associated with these disks areD(φ 1 ) = D(φ 1 ) = D 4 + D 16 + D 58 + D 59 + D 60 + D 61 , D(φ 2 ) = D(φ 2 ) = D(φ 1 ) + D 62 + D 63 + D 64 , D(φ 3 ) = D 4 + D 17 + D 21 + D 22 + D 23 + D 26 + D 36 , D(φ 4 ) = D 4 + D17 , D(φ 5 ) = D(φ 5 ) = D 4 + D 16 + D 17 + D 27 + D 58 + D 59 , D(φ 6 ) = D(φ 6 ) = D(φ 5 ) + D 60 + D 61 + D 62 , D(φ 7 ) = D(φ 7 ) = D(φ 5 ) + D 60 . By Lemma 4.1, # M (φ i ) = # M (φ j ) = 1, for i = 1, . . . , 4 and j = 1, 2. Moreover, by Lemma 4.2, # M (φ i ) = # M (φ i ) = 1 for i = 5, 6, 7.
Figure 11 .
11If I 1 , I 2 , I 3 , I 4 and I 5 denote the set of indices of domains colored yellow, blue, pink, grey, and brown respectively, the domains associated with ψ 1 , ψ 3 and ψ 6 are given byD(ψ 1 ) = i ∈I 1 ∪I 2 ∪I 3 D i , D(ψ 3 ) = i ∈I 2 ∪I 3 ∪I 5 D i and D(ψ 6 ) = i ∈I 2 ∪I 4 ∪I 5 D i .regions. Setting V = # M (
. The domains D • i for • = a, b, c and d denote the components of the regions colored pink, grey, yellow and green, respectively. They determine the domains of Whitney disks η a , η b , η c and η d , respectively, with domains D(η • ) = i D • i . Note that for • = a, c the values of i are in {1, . . . , 8}, while for • = b, d the values of i are in {1, . . . , 4}. The possible degenerations for the disks η a , η b , η c and η d are given by
6 D(σ • 3 )Figure 12 .
6312= D • 4 + D • 5 + D • 7 + D • 8 , D(φ 1 ) = D 1 + D 2 + D 3 , D(φ 2 ) = D 1 + D 3 + D 4 , and D(φ 3 ) = D 1 + D 2 + D
for • = a, c and = b, d , while the domains associated with σ a j , σ c j , σ b i and σ d i are polygons for j = 1, 2 and i = 1, 2, 3. By Lemma 4.1, we also have # M (σ • 3 ) = 1, • = a, c. ThereforeOn the other hand, we haveThus by 11, we have V = # M (ψ 3 ) = # M (ψ 6 ).Having established Lemma 6.1, we conclude thatThis means that 49 survives in H F (M , s 2 ), and the latter group is thus non-trivial.
The chain complex C F (M , s 2 ) is bigger, in comparison with C F (M , s 1 ), and is generated by the following 72 generators: 1 = {x 1. 1,2 , x 2,2,4 , x 3,4,1 , x 4,5,1 , x 5,3,2 } 2Let s 2 be the Spin c class which corresponds to (0, −1. = {x 1,1,3 , x 2,2,5 , x 3,4,1 , x 4,5,1 , x 5,3,2 } 3 = {x 1,1,2 , x 2,2,5 , x 3,5,1 , x 4,3,1 , x 5,4,1 }Let s 2 be the Spin c class which corresponds to (0, −1, −8). The chain complex C F (M , s 2 ) is bigger, in comparison with C F (M , s 1 ), and is generated by the following 72 generators: 1 = {x 1,1,2 , x 2,2,4 , x 3,4,1 , x 4,5,1 , x 5,3,2 } 2 = {x 1,1,3 , x 2,2,5 , x 3,4,1 , x 4,5,1 , x 5,3,2 } 3 = {x 1,1,2 , x 2,2,5 , x 3,5,1 , x 4,3,1 , x 5,4,1 }
. = {x, x 5,5,1 } 20 = {x 1,3,2 , x 2,1,1 , x 3,2,1 , x 4,4,10 , x 5,5,2 } 21 = {x 1,3,2 , x 2,1,1 , x 3,2,2 , x 4,4,8 , x 5,5,1 }= {x 1,3,2 , x 2,1,1 , x 3,2,1 , x 4,4,10 , x 5,5,1 } 20 = {x 1,3,2 , x 2,1,1 , x 3,2,1 , x 4,4,10 , x 5,5,2 } 21 = {x 1,3,2 , x 2,1,1 , x 3,2,2 , x 4,4,8 , x 5,5,1 }
. = {x, x 5,5,2 } 29 = {x 1,3,1 , x 2,2,2 , x 3,1,1 , x 4,4,10 , x 5,5,1 } 30 = {x 1,3,1 , x 2,2,2 , x 3,1,1 , x 4,4,10 , x 5,5,2 }= {x 1,3,1 , x 2,2,1 , x 3,1,2 , x 4,4,10 , x 5,5,2 } 29 = {x 1,3,1 , x 2,2,2 , x 3,1,1 , x 4,4,10 , x 5,5,1 } 30 = {x 1,3,1 , x 2,2,2 , x 3,1,1 , x 4,4,10 , x 5,5,2 }
. = {x, x 5,5,1 } 44 = {x 1,3,2 , x 2,2,4 , x 3,1,1 , x 4,4,9 , x 5,5,2 } 45 = {x 1,3,2 , x 2,2,5 , x 3,1,1 , x 4,4,7 , x 5,5,1 }= {x 1,3,2 , x 2,2,4 , x 3,1,1 , x 4,4,9 , x 5,5,1 } 44 = {x 1,3,2 , x 2,2,4 , x 3,1,1 , x 4,4,9 , x 5,5,2 } 45 = {x 1,3,2 , x 2,2,5 , x 3,1,1 , x 4,4,7 , x 5,5,1 }
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Neda Bagherifard email: neda.bagherifard@gmail. Neda Bagherifard email: [email protected]
email: [email protected] for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746. Tehran, IranEaman Eftekhary School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran email: [email protected]
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"Search for solar axions in XMASS, a large liquid-xenon detector",
"Search for solar axions in XMASS, a large liquid-xenon detector"
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"K Masuda \nSolar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan\n",
"Y Nishitani \nSolar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan\n",
"H Takiya \nSolar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan\n",
"H Uchida \nSolar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan\n",
"N Y Kim \nDepartment of Physics\nSejong University\n143-747SeoulSouth Korea\n",
"Y D Kim ",
"F Kusaba \nDepartment of Physics\nSejong University\n143-747SeoulSouth Korea\n\nDepartment of Physics\nTokai University\n259-1292HiratsukaKanagawaJapan\n",
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"Now at Research Center for Neutrino Science\nTohoku University\n980-8578SendaiJapan",
"Kamioka Observatory\nInstitute for Cosmic Ray Research\nthe University of Tokyo\nHigashi-Mozumi\n506-1205Kamioka, Hida, GifuJapan",
"Kamioka Observatory\nInstitute for Cosmic Ray Research\nthe University of Tokyo\nHigashi-Mozumi\n506-1205Kamioka, Hida, GifuJapan",
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"Department of Physics\nKobe University\n657-8501KobeHyogoJapan",
"Department of Physics\nKobe University\n657-8501KobeHyogoJapan",
"Department of Physics\nKobe University\n657-8501KobeHyogoJapan",
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"Korea Research Institute of Standards and Science\n305-340DaejeonSouth Korea",
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"Solar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan",
"Solar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan",
"Solar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan",
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"Solar Terrestrial Environment Laboratory\nNagoya University\n464-8602NagoyaAichiJapan",
"Department of Physics\nSejong University\n143-747SeoulSouth Korea",
"Department of Physics\nSejong University\n143-747SeoulSouth Korea",
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"School of Science and Technology\nTokai University\n259-1292HiratsukaKanagawaJapan",
"Now at Research Center for Neutrino Science\nTohoku University\n980-8578SendaiJapan",
"Department of Physics\nTokai University\n259-1292HiratsukaKanagawaJapan",
"Department of Physics\nFaculty of Engineering\nYokohama National University\n240-8501YokohamaKanagawaJapan",
"Department of Physics\nFaculty of Engineering\nYokohama National University\n240-8501YokohamaKanagawaJapan",
"Department of Physics\nFaculty of Engineering\nYokohama National University\n240-8501YokohamaKanagawaJapan"
]
| []
| XMASS, a low-background, large liquid-xenon detector, was used to search for solar axions that would be produced by bremsstrahlung and Compton effects in the Sun. With an exposure of 5.6 ton days of liquid xenon, the modelindependent limit on the coupling for mass ≪ 1 keV is |g aee | < 5.4 × 10 −11 (90% C.L.), which is a factor of two stronger than the existing experimental limit. The bounds on the axion masses for the DFSZ and KSVZ axion models are 1.9 and 250 eV, respectively. In the mass range of 10-40 keV, this study produced the most stringent limit, which is better than that previously derived from astrophysical arguments regarding the Sun to date. | 10.1016/j.physletb.2013.05.060 | [
"https://arxiv.org/pdf/1212.6153v3.pdf"
]
| 54,678,714 | 1212.6153 | 81303996e8c4338ed66b0a13d49e64d0ab13e363 |
Search for solar axions in XMASS, a large liquid-xenon detector
29 May 2013
K Abe
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
K Hieda
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
K Hiraide
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
S Hirano
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Y Kishimoto
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
K Kobayashi
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
S Moriyama
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
K Nakagawa
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
M Nakahata
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
H Ogawa
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
N Oka
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
H Sekiya
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
A Shinozaki
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Y Suzuki
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
A Takeda
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
O Takachio
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
K Ueshima
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Now at Research Center for Neutrino Science
Tohoku University
980-8578SendaiJapan
D Umemoto
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
M Yamashita
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
B S Yang
Kamioka Observatory
Institute for Cosmic Ray Research
the University of Tokyo
Higashi-Mozumi
506-1205Kamioka, Hida, GifuJapan
S Tasaka
Information and multimedia center
Gifu University
501-1193GifuJapan
J Liu
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
K Martens
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
K Hosokawa
Department of Physics
Kobe University
657-8501KobeHyogoJapan
K Miuchi
Department of Physics
Kobe University
657-8501KobeHyogoJapan
A Murata
Department of Physics
Kobe University
657-8501KobeHyogoJapan
Y Onishi
Department of Physics
Kobe University
657-8501KobeHyogoJapan
Y Otsuka
Department of Physics
Kobe University
657-8501KobeHyogoJapan
Y Takeuchi
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
the University of Tokyo
277-8582KashiwaChibaJapan
Department of Physics
Kobe University
657-8501KobeHyogoJapan
Y H Kim
Korea Research Institute of Standards and Science
305-340DaejeonSouth Korea
K B Lee
Korea Research Institute of Standards and Science
305-340DaejeonSouth Korea
M K Lee
J S Lee
Korea Research Institute of Standards and Science
305-340DaejeonSouth Korea
Y Fukuda
Korea Research Institute of Standards and Science
305-340DaejeonSouth Korea
Department of Physics
Miyagi University of Education
980-0845SendaiMiyagiJapan
Y Itow
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe
Nagoya University
Furo-cho, Chikusa-ku464-8602NagoyaAichiJapan
Solar Terrestrial Environment Laboratory
Nagoya University
464-8602NagoyaAichiJapan
K Masuda
Solar Terrestrial Environment Laboratory
Nagoya University
464-8602NagoyaAichiJapan
Y Nishitani
Solar Terrestrial Environment Laboratory
Nagoya University
464-8602NagoyaAichiJapan
H Takiya
Solar Terrestrial Environment Laboratory
Nagoya University
464-8602NagoyaAichiJapan
H Uchida
Solar Terrestrial Environment Laboratory
Nagoya University
464-8602NagoyaAichiJapan
N Y Kim
Department of Physics
Sejong University
143-747SeoulSouth Korea
Y D Kim
F Kusaba
Department of Physics
Sejong University
143-747SeoulSouth Korea
Department of Physics
Tokai University
259-1292HiratsukaKanagawaJapan
D Motoki
School of Science and Technology
Tokai University
259-1292HiratsukaKanagawaJapan
Now at Research Center for Neutrino Science
Tohoku University
980-8578SendaiJapan
K Nishijima
Department of Physics
Tokai University
259-1292HiratsukaKanagawaJapan
K Fujii
Department of Physics
Faculty of Engineering
Yokohama National University
240-8501YokohamaKanagawaJapan
I Murayama
Department of Physics
Faculty of Engineering
Yokohama National University
240-8501YokohamaKanagawaJapan
S Nakamura
Department of Physics
Faculty of Engineering
Yokohama National University
240-8501YokohamaKanagawaJapan
Search for solar axions in XMASS, a large liquid-xenon detector
29 May 2013Preprint submitted to Physics Letters B May 6, 2014arXiv:1212.6153v3 [astro-ph.CO]AxionSunxenon
XMASS, a low-background, large liquid-xenon detector, was used to search for solar axions that would be produced by bremsstrahlung and Compton effects in the Sun. With an exposure of 5.6 ton days of liquid xenon, the modelindependent limit on the coupling for mass ≪ 1 keV is |g aee | < 5.4 × 10 −11 (90% C.L.), which is a factor of two stronger than the existing experimental limit. The bounds on the axion masses for the DFSZ and KSVZ axion models are 1.9 and 250 eV, respectively. In the mass range of 10-40 keV, this study produced the most stringent limit, which is better than that previously derived from astrophysical arguments regarding the Sun to date.
Introduction
The axion is a hypothetical particle invented for solving the CP problem in strong interactions [1]. As the initial Peccei-Quinn-Weinberg-Wilczek model of axions is directly tied to the electroweak symmetry-breaking scale, an experimental search was relatively easy and the model was ruled out early. However, invisible axion models such as DFSZ [2] and KSVZ [3], whose symmetry-breaking scale is separated from the electroweak scale, are still viable. The DFSZ axions have direct couplings to leptons whereas the KSVZ axions (hadronic axions) do not have tree-level couplings to leptons. In these models, the mass of axions is m a = √ z
1 + z f π m π f a = 6.0 eV f a /10 6 GeV ,
where f a , f π , and m π are the axion decay constant [4], the pion decay constant, and pion mass, respectively, and z = m d /m u ∼ 0.56 is the quark mass ratio. At present, the search for axions as well as axion-like particles (ALPs) focuses on couplings to photons (g aγγ ), nucleons (g aN N ) and electrons (g aee ).
There are three types of searches: (1) laboratory-based experiments in which sources and detectors are prepared, (2) astrophysical investigations that examine any significant deviations in the properties of stars from theoretical predictions due to extra emission of energy, and (3) using laboratory detectors to look for axion signals from the Sun or cosmological relics. Experiments searching for axions have so far produced null results, but sensitivities continue to improve.
In experimental searches that utilize g aγγ , a series of experiments using strong magnets [5,6,7] successfully improved sensitivities by increasing the magnetic field strength and the conversion length. The suggestion [8] to use Bragg scattering to improve sensitivity for solar axions in crystalline detectors was used in [9,10,11,12]. Another way to enhance sensitivity is to exploit resonant absorption on nuclei [13]. To date, several experimental results are obtained in this scheme [14,15,16,17,18,19,20,21,22,24,23]. Significant improvement can be achieved if the signals can be read out efficiently. On the other hand, an efficient experimental search with g aee has not been performed. A pioneering experiment used a Ge detector (710 g) [25] and a recent search used a Si(Li) detector (1.3 g) to search for signals from axions generated by the bremsstrahlung and Compton effect via the axioelectric effect [26].
The choice of target material strongly affects the reach of a solar axion experiment using axion coupling to electrons. Liquid Xe is both dense and has a high atomic numbers [27]. The XMASS detector, which uses 835 kg of liquid xenon in its sensitive volume, is suitable for this purpose. Its low energy threshold (0.3 keV) is also useful as the predicted energy spectrum is very soft and has a peak at less than 1 keV for light axions. Its low background (a few keV −1 kg −1 day −1 ) makes it particularly useful when searching for solar axions.
Expected Signal
The signals we searched for are produced by the Compton scattering of photons on electrons e + γ → e + a and the bremsstrahlung of axions from electrons e + Z → e + a + Z in the Sun. The expected fluxes and spectra are derived as follows.
The solar axion flux produced by Compton scattering was calculated in [28,29]. The axion differential flux is expressed as
dΦ c a dE a = 1 A 2 R ⊙ 0 ∞ Ea dN γ dE γ dσ c dE a dE γ N e (r)r 2 dr,(1)
where E a is the total energy of the axions, A is the average distance between the Sun and the Earth, R ⊙ is the radius of the Sun, dN γ /dE γ is the blackbody spectrum of photons, dσ c /dE a is the cross section for the Compton effect, and N e (r) is the electron density at the radius r. Since m a and E γ is assumed to be much smaller than m e , the differential cross section is approximately a product of δ(E a − E γ ), and the total cross section [29] is expressed as
σ c = α g 2 aee E 2 γ v a 4m 4 e 1 + v 2 a 3 1 + m 2 a 2E 2 γ − m 2 a E 2 γ 1 − m 2 a 2E 2 γ ,(2)
where α is the fine structure constant, m e is the electron mass, g aee is the axion's coupling to electrons [4] which is (1/3)(cos 2 β)m e /f a in the DFSZ axion model [26], and v a = (1 − m 2 a /E 2 γ ) 1/2 is the velocity of the outgoing massive axion. cot β is the ratio of the two Higgs vacuum expectation values of the model [4].
The energy spectrum of solar axions produced by the bremsstrahlung effect was calculated in [28,30]. The differential energy spectrum is
dΦ b a dE a = 1 A 2 R ⊙ 0 ∞ Ea dN e dE e v e dσ b dE a dE e Z,A Z 2 N(r)r 2 dr,(3)
where v e is the velocity of the electrons, dN e /dE e is the energy spectrum of the electrons, dσ b /dE A is the cross section for the bremsstrahlung effect, and N Z,A (r) is the atom density at radius r. The cross section dσ b /dE a is calculated by considering the energy conservation of the electron and axion system [30]. The temperature, electron density, and atomic density are given by the standard solar model BP05(OP) [31]. Figure 1 in Ref. [26] shows the energy spectra for various masses of axions. The bremsstrahlung component dominates below 10 keV, whereas the Compton contribution dominates at higher energy.
The expected energy spectrum to be observed with a detector is
dN obs dE = σ ae (E a ) dΦ c a dE a + dΦ b a dE a Ea=E ,(4)
where σ ae (E a ) is the cross section for the axioelectric effect [32]. For the cross section, the expression of Eq.
(3) in Ref. [26] is used for v a σ ae (E a ) = σ pe (E a ) g 2 aee v a 3E 2 a 16παm 2 e 1 − v a 3 ,(5)
where σ pe (E a ) is the photoelectric cross section of the detector medium for gamma rays with energy E a . The photoelectric cross section is available in Ref. [33,34]. The predicted energy spectra for a xenon target for various axion masses are shown in Fig. 1.
The predicted energy spectra calculated above are used to generate Monte Carlo simulation samples. Axion signal samples can be simulated by injecting gamma rays whose energy is the same as the total energy of the incoming axions. This is because (1) there is a relationship between the cross section of the axioelectric effect and the photoelectric effect as in Eq. (5), (2) the photoelectric effect is dominant in this energy range (<100 keV), and (3) the process after the axioelectric effect is exactly the same as that for the photoelectric effect. In the simulation, we considered the nonlinearity of the scintillation yield for gamma rays, the optical processes of the scintillation photons in the detector, the photoelectron distributions and discrimination threshold of photomultipliers, and the trigger conditions of the data acquisition system. The detailed description of the simulation and efficiencies were previously reported [36,35]. After taking into account the reduction efficiency described in the next section, the expected energy spectra for various masses of axions are obtained.
The Data
The XMASS detector is a large liquid-xenon detector located underground (3000 m water equivalent) at the Kamioka Observatory, Japan. It contains an 835-kg liquid-xenon target with a surface of a pentakis-dodecahedron that is tiled with inward looking photomultiplier tubes (PMT), 630 of which have hexagonal and 12 have round photocathodes. The PMTs (R-10789, Hamamatsu) are specially developed for this low-background detector. The photoelectron yield at the center of the detector is evaluated at 14.7 photoelectrons (p.e.)/keV using an internal 57 Co source. The positional dependence (maximum 15%) of the photoelectron yield caused by the angular acceptance of PMTs and absorption of scintillation light are taken into account in the Monte Carlo simulations. Data acquisition is triggered if four or more PMTs have more than 0.2 p.e. within 200 ns. The trigger efficiency around the trigger threshold was examined by LEDs placed at the detector wall. The observed behavior was well reproduced by the Monte Carlo simulations. Signals from each PMT are fed into charge ADCs and TDCs whose resolution is around 0.05 p.e. and 0.4 ns, respectively. The liquid-xenon detector is surrounded by a water Cherenkov veto counter, which is 10.5 m in height and 10 m in diameter. It is equipped with 72 20-inch PMTs whose signals are fed into the ADCs and TDCs. Data acquisition is triggered if eight or more 20-inch PMTs have hits. The detector is described in detail in Ref. [36].
The data set used in the solar axion search experiments covers February 21-27, 2012. A sequence of standard data reduction is applied to remove events caused by afterpulses and electronic ringing. The standard reduction consists of a series of cuts: (1) the event is triggered only by the liquid-xenon detector; (2) the time difference to the previous event is more than 10 ms;
(3) the root mean square of the hit timing is less than 100 ns and is used to reject events caused by afterpulses of PMTs due to bright events; and (4) the number of PMT hits in the first 20 ns divided by the total number of hits is less than 0.6 for events in which the number of photoelectrons is less than 200. The fourth cut was applied to remove Cherenkov events originated from 40 K in photocathodes (Cherenkov cut). The energy threshold of this analysis is low (0.3 keV) because of our exceptional photoelectron yield, which is the largest among current low-background detectors. A more detailed description of the reduction can be found in Ref. [35]. Figure 2 shows the observed energy spectra. The total livetime is 6.7 days after considering the dead time caused by the cut (2). The effect of trigger cut (1) is visible below 0.4 keV as shown in Fig. 3 in Ref. [35] and is considered in our Monte Carlo simulations. The same samples show that the cut (3) has negligible effect on the signals. The signal efficiency due to the Cherenkov cut, which is drawn in the same figure, was conservatively evaluated using low-energy gamma-ray sources such as 55 Fe and 241 Am sources at various positions. Because the efficiency weakly depends on the radial position of the events and gradually decreases outward, the efficiency adopted in the analysis was mostly evaluated at a radius of 40 cm where 93% of the mass was contained inside. The Monte Carlo samples were compared with the observed energy spectra after weighting this efficiency.
Limit on g aee
The observed spectra do not have any prominent features to identify axion signals with respect to the background. Instead, strong constraints on g aee can be obtained from the observed event rate in the relevant energy range. In order to set a conservative upper limit on the axion-electron coupling constant g aee , the coupling is adjusted until the expected event rate in XMASS does not exceed the one observed in any energy bin above 0.3 keV. Figure 3 shows the expected energy spectra with the coupling constants ob- [26] and references therein.) The dashdotted lines show astrophysical limits from red giant stars [4] and the solar neutrino flux [37]. The dashed lines are theoretical predictions for the DFSZ (cos 2 β = 1) and KSVZ (E/N = 8/3) models. This study gives a stronger constraint by a factor of two over previous direct experimental limits for axion mass ≪1 keV, and the best constraint absolute between 10 and 40 keV.
tained by the procedure above. Figure 4 shows the summary of the bounds of g aee . For small axion masses, a g aee value of 5.4 × 10 −11 is obtained. This is the best direct experimental limit to date and is close to that derived from astrophysical considerations based on measured solar neutrino fluxes: g aee = 2.8 × 10 −11 [37]. For axion masses > 10 keV the energetics in the Sun are no longer sufficient to effectively produce such axions. A systematic uncertainty inherent to our method of comparing bin contents arises from the specific choice of binning. This and systematic uncertainties for energy scale including energy threshold, Cherenkov cut efficiency, and energy resolution are evaluated to be 2%, 1%, 2%, and 1%, respectively. The total systematic error, 3%, is obtained by summing these contributions in quadrature, and the limit in Fig. 4 (90% C.L.) takes this error into account.
The calculated limit depends on the interaction processes considered in our detector as well as the processes considered for solar axion production in the Sun. Processes such as the inverse Primakoff effect and nuclear absorption on the detection side, and the Primakoff effect and nuclear deexcitation on the production side can be neglected because the constraints on g gγγ and g aN N are tight. A possible additional contribution caused by g aee on the detection side is the inverse Compton effect. This can be neglected because of its small cross section [38]. On the production side, there are other known contributions such as electron-electron bremsstrahlung [39] and the axiorecombination effect [40]. However, the expected fluxes for these processes are only known in the limit of massless axions. For this reason and in order to directly compare our results with the most relevant previously published ones we restrict the production processes we consider to the electron-nuclei bremsstrahlung and the Compton effect. As omitting production mechanisms lowers the flux estimate, all the limits thus derived will have to be considered conservative.
The nature of the events surviving the analysis cuts is also of interest. According to our study on these events, most of them originate on the inner surface of the detector [41]. These events are attributed to radioactive contamination in the aluminum seal of the PMT entrance windows, 14 C decays in the GORE-TEX R sheets between the PMTs and the copper support structure, and light leaking from gaps in between the triangular elements of this support structure.
Conclusion
In summary, solar axions produced through axion-electron coupling were searched for in XMASS, a large liquid-xenon detector. The energy threshold is low (0.3 keV) because of our exceptional photoelectron yield, which is the largest among current low-background detectors. As our observed spectrum does not show any indications of axion signals, we derive constraints on the g aee coupling. Our limit on g aee for axions with mass much smaller than 1 keV is 5.4 × 10 −11 . The bounds on the axion masses for the DFSZ and KSVZ axion models are 1.9 and 250 eV, respectively. For axion masses between 10 and 40 keV, our new limits are the most stringent that are currently available.
Figure 1 :
1Expected energy spectra of events observed using the liquid-xenon detector. No resolution effects are included. Different curves are for axion masses with 0, 1, 2, 4, 8, and 16 keV. The inset shows spectra of axion masses with 32 and 64 keV. Due to a cross section enhancement for nonrelativistic axions, an increase at E ∼ m a can be seen. The step around 5 keV corresponds to the L-shell absorption edge of the axioelectric effect.
Figure 2 :Figure 3 :
23Observed energy spectra. The horizontal axis shows the "scaled energy" calculated by dividing the number of photoelectrons by the photoelectron yield at the center of the detector, 14.7 p.e./keV. Error bars are statistical only. In this figure we also show the efficiencies for the Cherenkov cut (closed circles with horizontal bars for the applicable range; 1 for 100%) and for the combination of all our cuts (open circles). Only at the trigger threshold is the overall efficiency not dominated by the Cherenkov cut efficiency. The inset shows the same quantities for energies extending up to 100 keV. Comparison between the observed data (points with error bars) and expected spectrum (solid histogram) for axion masses of 0, 5, 10, and 50 keV. The solid histograms are scaled to the maximum coupling allowed at 90% C.L.
Figure 4 :
4Limits on g aee . The thick solid line shows the limit obtained in this study. The other solid lines are limits obtained by laboratory experiments: Ge[25], Si(Li), 169 Tm, reactors, o-Ps, and beam-dump experiments (see
AcknowledgementsWe gratefully acknowledge the cooperation of Kamioka Mining and Smelting Company. This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research, and partially by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-220-C00006). We thank J. Redondo for useful discussion.
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| []
|
[
"Where Infinite Spin Particles are Localizable",
"Where Infinite Spin Particles are Localizable"
]
| [
"Roberto Longo [email protected] ",
"Vincenzo Morinelli [email protected] ",
"Karl-Henning Rehren [email protected] ",
"\nDipartimento di Matematica\nDipartimento di Matematica\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica, 1I-00133RomaItaly\n",
"\nInstitut für Theoretische Physik\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica, 1, I00133RomaItaly\n",
"\nUniversität Göttingen\n37077GöttingenGermany\n"
]
| [
"Dipartimento di Matematica\nDipartimento di Matematica\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica, 1I-00133RomaItaly",
"Institut für Theoretische Physik\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica, 1, I00133RomaItaly",
"Universität Göttingen\n37077GöttingenGermany"
]
| []
| Particles states transforming in one of the infinite spin representations of the Poincaré group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher-Haag-Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincaré group containing infinite spin, then it has infinite statistics.These results hold under the natural assumption of the Bisognano-Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s + 1 where infinite spin representations exist, namely s ≥ 2. | 10.1007/s00220-015-2475-9 | [
"https://arxiv.org/pdf/1505.01759v3.pdf"
]
| 119,569,712 | 1505.01759 | 2173762658be405d7f2a1a5dd6408c55d3ce34f6 |
Where Infinite Spin Particles are Localizable
27 Nov 2015
Roberto Longo [email protected]
Vincenzo Morinelli [email protected]
Karl-Henning Rehren [email protected]
Dipartimento di Matematica
Dipartimento di Matematica
Università di Roma Tor Vergata
Via della Ricerca Scientifica, 1I-00133RomaItaly
Institut für Theoretische Physik
Università di Roma Tor Vergata
Via della Ricerca Scientifica, 1, I00133RomaItaly
Universität Göttingen
37077GöttingenGermany
Where Infinite Spin Particles are Localizable
27 Nov 2015* Supported in part by the ERC Advanced Grant 669240 QUEST "Quantum Algebraic Structures and Models", PRIN-MIUR, GNAMPA-INdAM, and Alexander von Humboldt foundation. † Supported in part by PRIN-MIUR and GNAMPA-INdAM. 1
Particles states transforming in one of the infinite spin representations of the Poincaré group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher-Haag-Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincaré group containing infinite spin, then it has infinite statistics.These results hold under the natural assumption of the Bisognano-Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s + 1 where infinite spin representations exist, namely s ≥ 2.
Introduction
The classical notion of particles as pointlike objects is meaningless in Quantum Mechanics. Here the wave function satisfies the Schrödinger equation and the Heisenberg uncertainty relation prevents a sharp localization; increasing energy is needed for better localization. We are going to discuss the intrinsic particle localization properties, and show why infinite spin particles exhibit an essential difference from finite spin particles in this respect.
Wigner particles and classification of Poincaré group representations. In Relativistic Quantum Mechanics, one better defines a particle through its symmetry, rather than localization, property. The Schrödinger equation is replaced by the Lorentz invariant Klein-Gordon equation and this point of view led to define a particle as an irreducible, positive energy, projective unitary representation of the Poincaré group P ↑ + , hence to an irreducible, positive energy, unitary representation of the double (universal) cover of P ↑ + . These are the "minimal" Poincaré covariant objects, the building blocks of any more complete theory.
Within this point of view, E. Wigner [34] obtained his famous classification of the irreducible, positive energy, unitary representations of the double cover of P ↑ + , which is isomorphic to R 4 ⋊ SL(2, C).
We briefly recall that a unitary, positive energy representation U is classified, up to unitary equivalence, by two parameters m and s. The mass m takes values in [0, ∞) (the lower point in the energy spectrum). If m > 0, then the values of the spin s are 0, 1 2 , 1, 3 2 , 2, . . . (the unitary representations of the cover of the rotation subgroup).
In the mass zero case, the representations fall in two distinct classes according to the representations of the little group, which is the double cover of E(2), the Euclidean group of the plane. The representation with trivial E(2)-translations are representations of the (double) torus, labelled by the helicity, a parameter s that takes the place of the spin, s = 0, 1 2 , 1, 3 2 , 2, . . . . The remaining massless representations correspond to infinite-dimensional, irreducible representations of the double cover of E (2) and are labelled by a parameter κ > 0 (the radius of the circle that is the joint spectrum of the E(2)-translations) and a ± sign (Bose/Fermi alternative). They are called infinite spin (or continuous spin) representations.
Infinite spin particles have so far not been observed in nature, although they are compatible with all physical first principles, and are usually disregarded without further explanation. A result by Yngvason [36] shows that they cannot appear in a Wightman theory [36] since no Wightman fields (which have pointlike localization) transforming under an infinite spin representation can exist. One of the main aims of this paper is to study the peculiar localization property of these particles so as to explain why they are not observable in finite space and time. They are however localizable in certain unbounded spacetime regions [4] (cf. also [17]). Indeed, the authors of [27] have constructed such fields Φ(x, e) which are localized along rays x + R + · e where e is a spacelike direction.
We mention at this point that more general notions of particle are necessary to describe situations where, for example, infrared clouds are present, cf. [6]; these will not be considered in the present paper.
The main body of this article deals with the issue of localization of (one-particle) states. In Sect. 9, we present some consequences for the localization of algebras of observables (in the sense of spacelike commutation relations). The one-particle results directly pass to free fields by second quantization, and we shall discuss general results in the interacting case.
Localized particle states. Given a particle, namely an irreducible, positive energy representation U of P ↑ + , what are the localized states of of U ? If we restrict our attention to finite spin particles, the answer is well known in the Quantum Field Theory context, where one assumes the existence of a local free field transforming in a given representation. In the scalar case, for simplicity, the one-particle Hilbert space H can be obtained by equipping the Schwartz function space S(R s+1 ) with a scalar product, given by the two-point function (f, g) = (Φ(f )Ω, Φ(g)Ω) of the field. Its Hilbert space closure H can be viewed as the space of positive energy solutions to the Klein-Gordon wave equation, and carries an irreducible representation U of P ↑ + with zero spin/helicity. The localization of one-particle states is given by the support of the Schwartz functions: by assigning to an open region X the closed real linear subspace H Φ (X) ⊂ H, the closure of the space of real smooth functions with support in X, one obtains a local U -covariant net of standard subspaces of H (see below). The locality of the field, together with the identity iℑ(f, g) = (Ω, [Φ(f ) * , Φ(g)]Ω), imply that two subspaces H Φ (X) and H Φ (Y ) are symplectically orthogonal whenever X and Y are spacelike separated.
In the sequel, we describe the procedure of modular localization, which intrinsically associates with a given representation the states localized in a region X, without referring to a local field. Terminology. A wedge region W is a Poincaré transform of the standard wedge W 0 = {x ∈ R 4 : x 3 > |x 0 |}, and W is the set of all wedge regions. The standard one-parameter family of boosts preserving W 0 is called Λ W 0 (t), and we put Λ
W (t) := gΛ W 0 (t)g −1 if W = g(W 0 ). A double cone O
is the open intersection of a future and a backward light cone, and O is the set of all double cones. A spacelike cone is a region of the form C = x + t>0 t · O where x ∈ R 4 and O ∈ O is a double cone spacelike to the point 0, and C is the set of all spacelike cones. Two regions X, Y are spacelike separated if every pair of points (x, y) ∈ X × Y is spacelike separated. The spacelike complement of a region X is denoted by X ′ . Standard subspaces. Let H be a Hilbert space. A standard subspace H of H is a closed, real linear subspace which is cyclic (H + iH is dense) and separating (H ∩ iH = {0}). Standard subspaces of the one-particle space naturally appear in the above free field construction: the standardness of H Φ (O) is equivalent to the Reeh-Schlieder property that the vacuum vector is cyclic and separating for the corresponding local von Neumann algebras A(O) [1].
If H is a standard subspace, the Tomita operator S : ξ +iη → ξ −iη, ξ, η ∈ H, remembers H as H = Ker(S − 1), and its polar decomposition S = J∆ 1/2 gives the modular operator ∆ and the modular conjugation J that satisfy the one-particle version of the Tomita-Takesaki theorem:
∆ it H = H ∀ t ∈ R , JH = H ′ ,(1)
where H ′ is the symplectic complement of H. The Bisognano-Wichmann property. Now, let U be a positive energy representation of P ↑ + on H, and W ∋ W −→ H(W ) a net of standard subspaces on the wedge regions of the Minkowski spacetime R 4 , which is U -covariant:
U (g)H(W ) = H(gW ) .
The Bisognano-Wichmann property [2] asserts that the modular group of H(W ) is related to the boost transformations Λ W preserving W :
∆ it W = U Λ W (−2πt) .(2)
If H(C) is cyclic for all cones C, then J W acts geometrically as a reflection around the edge of the wedge, so there exists an anti-unitary PCT operator
Θ ≡ U (R W )J W ,(3)
where R W is the spatial π-rotation mapping W onto W . In Quantum Field Theory, the Bisognano-Wichmann property pertains to the standard subspaces A(W ) s.a. Ω where A(W ) s.a. is the selfadjoint part of the von Neumann algebra of local observables in a wedge. It was established model-independently for large classes of quantum field theories, cf. Sect. 10.3. Because the modular group is characterized by the KMS property, its physical meaning is that the vacuum state is a KMS state for the boost subgroup, when restricted to the algebra of a Rindler wedge; in other words, the restriction of the vacuum state is a thermal state for the geodesic observer on the Rindler spacetime. By this feature it is closely related to the Hawking-Unruh effect. We therefore believe the Bisognano-Wichmann property to be of a most fundamental character, and refer to the final comment 10.3 and to [14] for a discussion of this important point. Modular localization. The paper [4] provided a canonical construction of a local net H U of standard subspaces on the wedge regions of the Minkowski spacetime R 4 associated with any unitary, positive energy, representation U of the Poincaré group (with anti-unitary PCT operator Θ). One defines ∆ W and J W by the equations (2)
, (3), then sets S W ≡ J W ∆ 1/2 W and H U (W ) ≡ ξ ∈ H : S W ξ = ξ .(4)Isotony of the assignment W −→ H U (W ) (i.e., H U (W 1 ) ⊂ H U (W 2 ) whenever W 1 ⊂ W 2 )
follows from positivity of the energy. Moreover H U is local (or twisted-local if we consider representations of the cover of the Poincaré group), indeed H U is wedge dual:
H U (W ′ ) = H U (W ) ′ .
This construction is intrinsic, depending only on the representation U without reference to a quantum field. By construction, H U satisfies the Bisognano-Wichmann property.
Notice that any net W −→ H(W ) on wedges defines closed, real linear subspaces associated with any region X that is contained in some wedge:
H(X) ≡ W∋W ⊃X H(W ) .(5)
Obviously, these definitions respect isotony (H( A general result [4] shows furthermore that H U (C) defined as in (5) from the canonical net (4) is standard for spacelike cones C ∈ C, for every representation U .
X 1 ) ⊂ H(X 2 ) whenever X 1 ⊂ X 2 ),
If U is a representation with finite spin/helicity, then the modular localization subspace H U (X) as in (5) agrees with the standard subspace H Φ (O) defined by the free field oneparticle construction recalled above, therefore H U (O) is standard for any double cone O, and in this case we explicitly see how the space H(X) of particle states localized in a bounded region X is cyclic.
We should also comment that the paper [4] deals with the bosonic case (true representation of P ↑ + ), however the fermionic case can be treated analogously with usual modifications (and quantization on the anti-symmetric Fock space).
Infinite spin particles cannot be localized in bounded regions. As recalled, in Wigner's classification of unitary, positive energy, irreducible representations of the Poincaré group [34], massless representations fit in two classes, the ones with finite spin (helicity) and the ones with infinite spin, according to the representations of the "little group", the double cover E(2) of the Euclidean group of the plane E(2).
Let U be a massless representation with infinite spin; the space H U (C) was shown to be standard (cyclic) for spacelike cones but it remained open whether there are non-zero vectors localized in bounded regions [12]. Generalized (stringlike) Wightman fields associated with U were later constructed [27], but the above localization problem remained unsettled.
We shall show here that
H U (X) is trivial if X is bounded, say X = O a double cone, namely H U (O) ≡ W∋W ⊃O H U (W ) = {0} .
Quantum Field Theory, I. An immediate consequence is that the free field net A of local von Neumann algebras associated with a representation U of P ↑ + with infinite spin is well defined, the vacuum vector is cyclic for A(C) if C is a spacelike cone, but A(O) = C · 1 if O is a double cone: there is no non-trivial observable localized in a bounded spacetime region.
It also follows that there are no compactly localized observables on the same Hilbert space that are relatively local w.r.t. the infinite spin free field net. The absence of such observables was recently also demonstrated within an explicit field theoretic ansatz [18].
An important more general corollary is that, if B is any (Fermi-)local net of von Neumann algebras on a Hilbert space, covariant under a unitary positive energy representation U of the Poincaré group, with the vacuum vector being cyclic (Reeh-Schlieder property) for double cone algebras, then no infinite spin representation can appear in the irreducible direct integral decomposition of U (up to measure zero), provided that B satisfies the fundamental Bisognano-Wichmann property [2].
This shows why infinite spin particles do not appear in a theory of local observables. They are however compatible with stringlike localization. At this point it is worth mentioning that localization in spacelike cones is natural in Quantum Field Theory, indeed massive charges may always be localized in spacelike cones [5]. Low-dimensional non-trivial models with trivial local algebras are exhibited in [20].
Strategy of proof. Let U be a unitary, massless irreducible representation of P ↑ + . The starting point is the observation that U is dilation covariant if and only if it has finite spin. Assuming H U (O) to be standard for double cones O, we infer by the Huygens principle that H U (V + ) is standard, where V + is the forward light cone. By standard subspace analysis, in particular by using an analogue of Borchers' theorem [3,24], ∆ H U (V + ) has dilation commutation relations with U . So U must have finite spin.
Extensions of results. Our results hold in any space dimension s ≥ 2. As is known, if s is even the Huygens principle doesn't hold and we need to work with a corresponding property of the wave equation that we haven't found in the literature. The case s = 2 is peculiar as the infinite spin representations are not "infinite", namely they are associated with onedimensional representations of the little group. The Fermi case, namely representations of a cover of P ↑ + , is also studied. We treat the case s = 3 (the physical Minkowski spacetime) in detail and add a further section with the necessary analysis in different spacetime dimensions.
Quantum Field Theory, II. For interacting theories satisfying the Bisognano-Wichmann property, we show that the subspace B(O)Ω (independent of the double cone O) cannot carry an infinite spin representation. Thus, if the theory possesses infinite spin particles, then the vacuum vector cannot be cyclic for B(O), where B is the local net of von Neumann algebras describing our theory, i.e., the infinite spin particles states cannot be generated from the vacuum by operations in bounded spacetime regions.
Indeed, no infinite spin particle state can be obtained by adding to B a finite charge localized in a bounded spacetime region. In other words, no infinite spin representation can appear in the irreducible disintegration of the covariance unitary representation of DHR sectors of B (with finite statistics).
We emphasize that the Bisognano-Wichmann property is essential in the argument, by providing a counter-example without this property, in which free infinite spin particles exist with cyclic double cone algebras.
Thus, at least one (artificial) way to accomodate New Physics involving observable infinite spin particles would consist in relaxing the Bisognano-Wichmann property -in spite of its very fundamental nature. More interesting is the picture (Sect. 10.1) that we obtain when we start with a (compactly) local observable net; we have a field algebra net that generates a non-trivial but non-cyclic subspace, an interacting theory with infinite spin particles; this structure exactly complies with the picture envisaged in [30].
Standard subspaces
We begin by recalling some definitions and results on standard subspaces and their modular structures. Further details can be found in [24,25]. If H is a closed, real linear subspace of H, the symplectic complement of H is defined by
H ′ ≡ {ξ ∈ H ; ℑ(ξ, η) = 0 ∀η ∈ H} = (iH) ⊥ R ,
where ⊥ R denotes the orthogonal in H viewed as a real Hilbert space with respect to the real part of the scalar product. H ′ is a closed, real linear subspace of H and
H = H ′′ . H is cyclic (separating) iff H ′ is separating (cyclic), thus H is standard iff H ′ is standard and we have S H ′ = S * H .
The fundamental properties of the modular operator and conjugation are
∆ it H H = H, J H H = H ′ , t ∈ R ,
and t → ∆ it H is called the one-parameter unitary modular group of H (cf. [33,29]).
Let H be a real linear subspace of H and V a one-parameter group of unitaries on H such that V (t)H = H, t ∈ R. V satisfies the KMS condition with inverse temperature β > 0 on H if, for every given ξ, η ∈ H, there exists a function F , analytic on the strip {z ∈ C : 0 < ℑz < 1}, bounded and continuous on its closure, such that:
F (t) = η, V (t)ξ , t ∈ R , F (t + iβ) = V (t)ξ, η , t ∈ R .
Since the uniform limit of holomorphic functions is holomorphic, it follows that if the KMS condition holds on H, then it holds on the closure H of H.
Lemma 2.1. [24,25]. If H ⊂ H is a standard subspace, then t → ∆ −it H satisfies the KMS condition at inverse temperature 1.
Conversely, if H is a closed, real linear, cyclic subspace of H and V a one-parameter unitary group on H with V (t)H = H, t ∈ R, satisfying the KMS condition on H at inverse temperature 1, then H is standard and V (t) = ∆ −it H . The following lemma is a consequence of the KMS condition for the modular group.
Lemma 2.2. [24,25]. Let H ⊂ H be a standard subspace, and K ⊂ H a closed, real linear subspace of K.
If ∆ it H K = K, ∀t ∈ R, then K is a standard subspace of K ≡ K + iK and ∆ H | K is the modular operator of K on K. If moreover K is a cyclic subspace of H, then H = K.
We shall also need the following basic lemma. The following is the one-particle analogue of Borchers' theorem [3].
Theorem 2.4. [24,25]. Let H ⊂ H be a standard subspace, and U a one-parameter unitary group on H with positive generator, such that
U (t)H ⊂ H, t ≥ 0. Then ∆ is H U (t)∆ −is H = U (e −2πs t).
We now want to study the tensor product of standard subspaces. Let H and K be standard subspaces of the Hilbert spaces H and K respectively, and S H , S K the associated Tomita operators. Then S ≡ S H ⊗ S K is a closed, densely defined anti-linear involution. Therefore
S = S M where M ≡ ξ ∈ Dom(S) : Sξ = ξ is a standard subspace of H ⊗ K.
We define the tensor product of H and K by
H ⊗ K ≡ M ; in other words H ⊗ K is defined through the formula S H⊗K ≡ S H ⊗ S K .
Proposition 2.5. If H and K are standard subspaces of H and K respectively, we have
(H ⊗ K) ′ = H ′ ⊗ K ′ .
Proof. Immediate from the equality
S (H⊗K) ′ = S * H⊗K = S H ⊗ S K * = S * H ⊗ S * K = S H ′ ⊗K ′ .
With H, K real linear subspaces of H and K respectively we denote by H ⊙ K the real linear span of {ξ ⊗ η : ξ ∈ H, η ∈ K}.
Proposition 2.6. Let H and K be standard subspaces of H and K. We have:
H ⊗ K = H ⊙ K . Proof. H ⊙ K is cyclic since H ⊙ K + iH ⊙ K = (H + iH) ⊙ (K + iK), which is dense in H ⊗ K. Clearly H ⊙ K is in the domain of S H ⊗ S K = S H⊗K , thus H ⊙ K ⊂ H ⊗ K. Now ∆ it H⊗K = ∆ it H ⊗ ∆ it K leaves globally invariant H ⊙ K, hence H ⊙ K. By Lemma 2.2 we conclude that H ⊙ K is equal to H ⊗ K.
By Prop. 2.6, we may equivalently define the tensor product of the closed, real linear subspaces H and K of H and K by
H ⊗ K ≡ H ⊙ K.
Given a family of real linear subspaces H a of H, we shall denote by a H a the real linear span of the H a 's.
Lemma 2.7. Let {H a } be a family of closed, real linear subspaces of H. Then a H a ′ = a H ′ a . Proof. We have a H a ′ = i a H a ⊥ R = a iH a ⊥ R = a (iH a ) ⊥ R = a H ′ a .
Lemma 2.8. Let {H a } and {K b } be families of standard subspaces of H and K respectively, and suppose both the intersections H ≡ a H a and K ≡ b K b to be cyclic. We have:
H ⊗ K = a,b H a ⊗ K b .
Proof. By Lemma 2.7 we have to show that
(H ⊗ K) ′ = a,b H a ⊗ K b ′ .
By Prop. 2.5, we have indeed:
H ⊗ K ′ = H ′ ⊗ K ′ = a H ′ a ⊗ b K ′ b = a,b H ′ a ⊗ K ′ b = a,b H a ⊗ K b ′ .
Massless representations of the Poincaré group
For the benefit of the reader, we first deal within the case of the four-dimensional spacetime, later extending our results to different dimensions. If G is a locally compact group, H ⊂ G a closed subgroup, and V a unitary representation of H, we denote by Ind H↑G V the unitary representation of G induced by V .
The Poincaré group P ↑ + is the semi-direct product R 4 ⋊ L ↑ + of the proper orthochronous Lorentz group L ↑ + and the translation group R 4 , where L ↑ + acts naturally on R 4 . The universal cover L ↑ + of L ↑ + is a double cover, isomorphic to SL(2, C). Accordingly, the universal cover P ↑ + of P ↑ + is isomorphic to R 4 ⋊ SL(2, C). One can choose the covering map σ : SL(2, C) → L ↑ + , so that σ maps the one-parameter subgroup α
α(t) = e t/2 0 0 e −t/2 , t ∈ R ,(6)
to the one-parameter group of boosts in the x 3 -direction, and σ restricts to the usual covering [32].
map SU(2) → SO(3). Explicitly, one identifies a vector x = (x 0 , x 1 , x 2 , x 3 ) ∈ R 4 with the matrix X x = x 0 +x 3 x 1 −ix 2 x 1 +ix 2 x 0 −x 3 and defines the Lorentz transformation σ(A) ∈ L ↑ + acting on x through X σ(A)x = AX x A * , A ∈ SL(2, C), see
The translation group R 4 is thus also a normal subgroup of P ↑ + . According to the Mackey machine (see [37]), if U is an irreducible unitary representation of P ↑ + , then U is induced by an irreducible unitary representation U 0 of Stab p :
U = Ind Stab p ↑ P ↑ + U 0 ;(7)
here the momentum p ∈ R 4 is a point in the dual group of the translations (i.e., a character), Stab p is the stabilizer of p for the action of P ↑ + on the characters given by the adjoint action on their arguments, and U 0 | R 4 is the one-dimensional representation p.
Notice that L ↑ + acts naturally on R 4 and R 4 acts trivially on itself, so one has
Stab p = R 4 ⋊ Stab p ,
where Stab p ⊂ L ↑ + is the stabiliser of p in L ↑ + acting naturally on R 4 (the little group). Points p in the same L ↑ + -orbit give rise to equivalent representations. We are interested in a positive energy, massless representation U , thus p ∈ ∂V + the boundary of the forward light cone. We assume U is not the identity, thus p = 0 and we shall choose and fix p = q with
q ≡ (1, 0, 0, 1) ∈ ∂V + (∂V + {0} is a L ↑ + -orbit)
. Then Stab q , the little group of (1, 0, 0, 1), is isomorphic to E(2), the double cover of the Euclidean group of the plane E(2):
Stab q = u z 0ū : u, z ∈ C, |u| = 1 .(8)
The irreducible representation U 0 of Stab p in (7) has the form
U 0 (g, x) = V (g)q(x) , g ∈ Stab q , x ∈ R 4 ,(9)
where V is an irreducible representation of Stab q = E(2) and q is the character of R 4 . Now E(2) is the semi-direct product R 2 ⋊ T and an irreducible representation V of E(2) fits in one of the following two classes:
(a) The restriction of V to R 2 is trivial; (b) The restriction of V to R 2 is non-trivial.
Irreducible representations of E(2) in class (a) are thus labelled by the integers, the dual of T, while irreducible representations in class (b) are labelled by κ > 0, the radius of a circle in R 2 , the joint spectrum of the E(2)-translations.
We say in case (a) that U has finite spin (or finite helicity); in case (b) that U has infinite spin. Therefore an irreducible, infinite spin representation U of P ↑ + has the form
U κ,ε = Ind Stabq↑ P ↑ +V κ,ε(10)
whereV κ,ε is given by (9):
V κ,ε (g, x) = V κ,ε (g)q(x) , g ∈ E(2) , x ∈ R 4 , with V = V κ,ε is the representation of E(2)
in which the spectrum of the translations is the circle of radius κ > 0, and the rotation by 2π is represented by +1 (bosonic case, ε = 0) resp. by −1 (fermionic case, ε = 1 2 ); so infinite spin representations are labelled by κ > 0 and ε = 0, 1 2 . We shall denote by τ (z), z ∈ C, the element of E(2) ⊂ SL(2, C) given by
τ (z) = 1 z 0 1 ;
the two translation one-parameter subgroups of E(2) are R ∋ x → τ (x), and R ∋ y → τ (iy) and we have the commutation relations
α(t)τ (z)α(t) −1 = τ (e t z) .(11)
4 Infinite spin representations are not dilation covariant
As is known, an irreducible, massless finite helicity unitary representation extends, on the same Hilbert space, to a representation of the group of transformations of the Minkowski spacetime generated by P ↑ + and dilations (indeed to a unitary representation of the conformal group). We show here that irreducible infinite spin representations are not dilation covariant in this sense. We suppress the Bose/Fermi label ε which is irrelevant for the issue at hand.
U · β = Ind H↑G V · β 0 where β 0 ≡ β| H .
Proof. The lemma follows by the unicity of the induced representation, a consequence of the unicity of the measure class of a quasi-invariant Borel measure on H\G.
Corollary 4.2. Let U κ = Ind R 4 ⋊ E(2)↑ P ↑ +V
κ be an infinite spin, irreducible unitary representation of P ↑ + , and β an automorphism of P ↑ + preserving the element q of (the dual of ) the translation subgroup. Then β(Stab q ) = Stab q and
U κ · β = U κ β ≡ Ind R 4 ⋊ E(2)↑ P ↑ +V κ β , where κ β is given by V κ β = V κ · β 0 with β 0 the automorphism of E(2) given by β 0 = β| Stabq
Proof. This follows from Lemma 4.1.
We shall say that a unitary representation U of P ↑ + on the Hilbert space H is dilation covariant if U extends to a unitary representation on H of the group generated by P ↑ + and dilations. Namely there exists a one-parameter unitary group D(t) on H such that D commutes with U | L + and
D(t)U (x)D(−t) = U (e t x)
,
for x in the translation group R 4 . Proposition 4.3. Let U be an irreducible, positive energy, unitary representation of P ↑ + . Then U is dilation covariant iff U is massless with finite spin. Proof. Let δ t the the automorphism of P ↑ + given by δ t (g) = g if g ∈ L + and δ t (p) = e t p if p ∈ R 4 . We want to show that U is inequivalent to U · δ t , t = 0, if U is irreducible with infinite spin.
Let then U = U κ be given by (7),
namely U κ = Ind R 4 ⋊ E(2)↑ P ↑ + V κ . We shall show that U κ · δ t = U e −t κ .
This will prove the Proposition because U κ and U κ ′ are inequivalent if κ = κ ′ . Now let α t be the lift to P ↑ + of the inner one-parameter automorphism group of P ↑ + implemented by the boost in 3-direction, namely α is given by eq. (6). Then
α t (q) = δ t (q) = (e t , 0, 0, e t ) ,
where q = (1, 0, 0, 1) as above. Thus the automorphisms
β t ≡ α −t · δ t (12) fix q. Since α −t is inner, we have U κ · α −t = U κ , thus U κ · δ t = U κ · α −t · δ t = U κ · β t .
We now apply Corollary 4.2 and see that
U κ · δ t = U κ ′
where κ ′ is given by 4. Bisognano-Wichmann property:
V κ ′ = V κ · β t | E(2) = V κ · α −t | E(∆ it H(W ) = U Λ W (−2πt) , ∀ W ∈ W ;
5. Twisted locality: For every wedge W ∈ W we have
ZH(W ′ ) ⊂ H(W ) ′ with Z unitary, Z = 1 + iΓ 1 + i .
Due to twisted locality, each H(W ) is indeed a standard subspace, so the modular operators in Property 4 are defined.
Here Γ ≡ U (2π), the unitary corresponding to a 2π spatial rotation in the representation U , namely Γ is the image under U of the non-trivial element in the centre of L ↑ + . Clearly Γ, hence Z, commutes with U .
Notice that if U is bosonic (Γ = 1), then Z = 1, and twisted locality is locality. If U is fermionic (Γ = −1), then Z = −i and H(W ′ ) ⊂ iH(W ). Lemma 2.2 then implies twisted duality for wedges:
H(W ′ ) = ZH(W ) ′ .
Starting with a U -covariant net H on W as above, one gets a net of closed, real linear subspaces on double cones O defined by The following proposition is proved in [4], (ii) ⇒ (i), and in [12], (i) ⇒ (ii), for nets of von Neumann algebras; yet the same argument gives a proof in the standard subspace setting.
H(O) ≡ W∋W ⊃O H(W ) .(13)
Proposition 5.1. [4,12]. Let H be a U -covariant net of standard subspaces of H as above (properties 1-5). The following are equivalent:
(i) H(C) ≡ W∋W ⊃C H(W )
is cyclic for all spacelike cones C;
(ii) U extends to an (anti-)unitary representationÛ of P + on H and H is the canonical net HÛ associated withÛ (eq. (4)).
Thus (in even spacetime dimension), with the above cone cyclicity assumption, there is an anti-unitary PCT operator.
The
D(2πt) = ∆ −it H(V + ) , t ∈ R . Then, by Lemma 2.3, D(t) commutes with U (g) if g is in the Lorentz group, because gV + = V + , so U (g)H(V + ) = H(V + ).
Thanks to positivity of the energy, the one-particle version of Borchers' theorem (Thm. The statement for U irreducible then follows immediately by Prop. 5.2.
Infinite spin states are not localized in bounded regions
We give here our main result. The consequences of this theorem in Quantum Field Theory will be discussed in Section 9.
A counter-example
In this section, we are going to see how dilation covariance and the double cone Reeh-Schlieder property for infinite spin (reducible) representations may both hold if the Bisognano-Wichmann property fails. We shall indeed show that a multiple of the direct integral ⊕ R + U κ dκ over all irreducible representations U κ of P ↑ + of infinite spin κ is dilation covariant and admits a local covariant net of standard subspaces, cyclic on double cones. Similar examples were put forward in [28,35].
For the sake of the example, it is sufficient to consider representations V of SL(2, C) that factor through L ↑ + , i.e., V (1) = V (−1). Namely, V is a true representation of L ↑ + . Since the choice of the pre-image of the covering map σ does not matter in true representations, we shall identify A ∈ SL(2, C) with σ(A) ∈ L ↑ + in this section, and again suppress the corresponding label ε = 0.
The subgroup E(2) ⊂ SL(2, C), the pre-image of E(2) through σ, is given by (8).
Let U 0 be the unitary, massless, zero helicity, representation of the Poincaré group and V a real unitary representation L ↑ + on the Hilbert spaces H and K respectively. With J an anti-unitary involution on K commuting with V, the vectors fixed by J form a standard subspace K of K and V (L ↑ + )K = K, J K = J, ∆ K = 1. In particular the constant net of standard subspaces K(W ) ≡ K is V -covariant.
We consider V as a representations of P ↑ + where the translation group acting identically. Consider the following net of standard subspaces of K ⊗ H
H I : W ∋ W −→ H I (W ) ≡ K ⊗ H(W ) ⊂ K ⊗ H
where H ≡ H U 0 is the canonical net associated with U 0 . There are two unitary representations of the P ↑ + on K ⊗ H:
U V ≡ V ⊗ U 0 and U I ≡ I ⊗ U 0 ,
where I is the identity representation of P ↑ + on K. Clearly U V and U I are massless representations, as the energy-momentum spectrum is that of U 0 .
H I is the canonical net associated with U I . The net H I is both U V -covariant and U I -covariant. Only U I satisfies the Bisognano-Wichmann property as, by Lemma 2.6, the modular operator of K ⊗ H(W ) is 1 ⊗ ∆ H(W ) . Then by Lemma 2.8
H I (O) = W ⊃O H I (W ) = K ⊗ W ⊃O H(W ) is cyclic, since W ⊃O H(W ) is cyclic in H.
So we have shown the following. We notice that the canonical net H V associated with U V is not covariant under the representation U I . We will now show that U V decomposes in a direct integral of infinite spin representations if V does not contain the trivial representation.
Let V + \{0} ∋ p → B p ∈ L ↑ + be a continuous map, with B p a Lorentz transformation mapping q = (1, 0, 0, 1) to p.
We can identify as usual the elements of H with L 2 -functions on ∂V + {0} w.r.t. the Lorentz invariant measure, thus elements of K ⊗ H with K 0 -valued L 2 -functions.
The following unitary operator
K ⊗ H ∋ p → φ(p) −→ p → V (B −1 p )φ(p) ∈ K ⊗ H intertwines U V with the representation U ′ V given by U ′ V (a, A)φ (p) = e ia·p V (B −1 p AB A −1 p )φ(A −1 p), φ ∈ H.(15)
Since B −1 p AB A −1 p ∈ Stab q = E(2) we may consider the irreducible disintegration of V | E(2) , then U ′ V , thus U V , will accordingly disintegrate. Since SL(2, C) is a simple, connected, non-compact Lie group with finite centre, the vanishing of the matrix coefficients theorem by Howe-Moore [37] ensures that lim g→∞ ξ, V (g)η = 0, for all ξ, η ∈ K, if V does not contain the identity representation. Proof. By the vanishing of the matrix coefficients theorem, there is no non-zero vector fixed by V · τ , thus no radius zero representation appears in the irreducible direct integral decomposition of V | E(2) , namely V | E(2) = R m(κ)V κ dµ(κ), where m(κ) is the multiplicity function and µ is a Borel measure on R + .
The one-parameter subgroup α of SL(2, C) given in (8) acts as dilation on the translations τ , eq. (11), thus
V | E(2) = ⊕ R m(κ)V κ dµ(κ) = ⊕ R m(κ)V e t κ dµ(κ) = ⊕ R m(e −t κ)V κ dµ t (κ)(16)
where µ t (κ) ≡ µ(e −t κ), and this implies that µ t is equivalent to µ (thus µ is equivalent to the Lebesgue measure) and m constant µ-almost everywhere.
The following Proposition is a consequence of the above Lemma.
Proposition 7.3. U V is a multiple of ⊕ R + U κ dκ,
where U κ is the infinite spin, radius κ representation of P ↑ + .
Proof. One considers the disintegration of V | E(2) obtained in Lemma 7.2 and concludes the thesis by formula (15).
Extensions to spacetime dimension s ≥ 2
In this section we are going to extend Propositions 4.3 and 5.4, and hence also Theorem 6.1, in any spacetime dimensions s ≥ 2.
Dilation covariance
We begin by discussing the dilation covariance property. The proper Lorentz group is L + ≡ L + (s) = SO(1, s), i.e., the group of d×d real matrices A preserving the Minkowski metric 1, −1, . . . , −1 . L + has two connected components and we denote by L ↑ + the connected component of the identity. L ↑ + is not simply connected when s > 1. Any element in L ↑ + is the product of a rotation and a boost, so L ↑ + is homotopy equivalent to SO(s), whose first homotopy group is Z 2 if s > 2 and Z if s = 2 (see [19]). Therefore the universal covering L ↑ + of L ↑ + is a double covering for s > 2, whereas it is an infinite sheet covering if s = 2. We shall thus treat the case s = 2 separately.
The proper orthochronous Poincaré group P ↑ + ≡ P ↑ + (s) is the semi-direct product of P ↑ + ≡ R s+1 ⋊ L ↑ + , with the natural action of L ↑ + on R s+1 . We shall consider unitary representations of the universal covering group P ↑ + = R s+1 ⋊ L ↑ + , as they correspond to the projective unitary, positive energy representations of P ↑ + . We are interested here in an irreducible, positive energy, massless representation U of P ↑ + . We choose and fix the point q ≡ q s = (1, 0, . . . , 0, 1) in the Lorentz orbit ∂V + \{0}. If U is non-trivial, then U is associated with a unitary, irreducible representation of the little group of q, by inducing representations as in Sect. 3.
The little group of q, namely the stabiliser subgroup of L ↑ + for the action of L ↑ + on R s+1 , is isomorphic to E(s − 1), the double cover of the Euclidean group E(s − 1) on R s−1 , s > 2, i.e., E(s − 1) is the semi-direct product R s−1 ⋊ SO(s − 1). If s = 2, the little group is the abelian group R. We now assume s > 2, afterwords we shall indicate the modifications in the s = 2 case.
Every unitary representation V of E(s−1) = R s−1 ⋊ SO(s−1) is now induced by a unitary representation of the stabiliser of a point in R s−1 (for the adjoint action of E(s − 1)). Points in the same orbit give equivalent representations. The orbits in R s−1 under the natural SO(s − 1) action are spheres of radius κ ≥ 0. Such radii define inequivalent classes of unitary representations. As in the 3 + 1-dimensional case, there are two cases:
• the restriction V | R s−1 is trivial (κ = 0); • the restriction V | R s−1 is non-trivial (κ > 0).
If U is associated, by induction, with a representation V of the little group E(s − 1) with κ = 0 we say that U has finite helicity, in the case κ > 0 we say that U has infinite spin.
With V an irreducible representation of E(s − 1) of radius κ > 0 as above, s > 2, the joint spectrum of the E(s − 1)-translation generators is the sphere in R s−1 of radius κ. Therefore spec (iV (X)) = [−κ, κ]
where X is any generator of the E(s − 1)-translations and V (X) the corresponding translation generator in the representation V .
We show now that infinite spin representations are not dilation covariant:
β t (X) = e −t X(18)
where β t , the automorphisms of P ↑ + defined in (12), here acting on the Lie algebra of P ↑ + , and X is a translation generator on the Lie algebra lie (E(2)). Now assume s ≥ 3. The inclusion P ↑ + (3) ⊂ P ↑ + (s) restricts to an inclusion E(2) ⊂ E(s−1) hence we have inclusions of Lie algebras lie(E(2)) ⊂ lie(E(s − 1)) ⊂ lie(P ↑ + ). We consider the automorphisms β t of P ↑ + (s) analogously defined w.r.t. the zero and s coordinates (the natural extension of β t from P ↑ + (3) to P ↑ + (s), we keep the same notation). Let now U be an irreducible, positive energy, massless, unitary representation U of P ↑ + (s) with infinite spin κ > 0. Then U is associated as above by induction with an irreducible representation V of the little group E(s − 1) of radius κ. As in Proposition 4.3 we have to show that V · β t | E(s−1) is a representation of radius e −t κ.
Indeed, due to the relation (18), with X ∈ lie(E(2)) ⊂ lie(E(s − 1)) we have
spec (iV (X)) = [−e −t κ, e −t κ](19)
so U is not dilation covariant by the above comment. An analogous discussion shows that finite helicity representations are dilation covariant. The case s = 2 is discussed here below.
Case s = 2. In 2 + 1 spacetime dimensions, the Lorentz group L + (2) is isomorphic to SL(2, R)/{1, −1}. The little group of the point q = (1, 0, 1) is R, which is simply connected, and lifts uniquely to a one-parameter subgroup of the universal (infinite sheet) cover L + (2). The pre-image of the little group in L + (2) is thus isomorphic to R × Z, with Z the centre of L + (2). The irreducible representations of the little group R × Z are thus one-dimensional, given by a pair (κ, z) where κ belongs to R (the dual of R) and z ∈ T (the dual of Z).
Denote by U κ,z the representation of P ↑ + (2) associated with the representation (κ, z) of the little group. In analogy with the higher-dimensional case, we say that a unitary representation U κ,z of P ↑ + (2) has "infinite spin" if κ = 0. Yet, in this case, the name "infinite spin" does not refer to any infinite-dimensional representation.
Again, equation (18) holds, thus the representation (κ, z) composed with the restriction of β t to the little group is equal to (e −t κ, z). It follows that U κ,z is dilation covariant iff κ = 0.
We also notice that the conjugate representation of (κ, z) is (−κ,z), thus U κ,z extends to a (anti-)unitary representation of P + (2), iff κ = 0 and z = ±1. The other irreducible massless representations of P + (2) are given by U κ,z ⊕ U −κ,z , with κ = 0 or z = ±1.
Twisted timelike locality
The second step consists of showing an analogous of Proposition 5.4 in any spacetime dimension s ≥ 2.
We start with a unitary massless representation U acting covariantly on a net on wedges W ∋ W −→ H(W ) ⊂ H s.t. assumptions 1-5 hold. Furthermore, suppose that for some double cone, the subspace H(O), defined as in (13), is not trivial. In this setting the proof of Proposition 5.2 straightforwardly extends to every spacetime dimension.
Case s odd
When the space dimension s is odd, the Huygens principle holds and the proof of Proposition 5.4 easily extends in this case.
Case s even, s ≥ 2
In this case, timelike commutativity does not hold. Our results hold true, but Lemma 5.3, necessary to show that H(V + ) is separating, needs a variation.
As is well known, the Huygens principle is not satisfied in odd space dimensions, due to reverberations, yet we show here a version of this principle that holds if s is even.
Let f be a tempered distribution on R s+1 ; we define h(f ) by its Fourier transform
h(f )(p) = −i sign(p 0 )f (p),
provided this expression is well defined. h is the Hilbert transform with respect to the time variable, thus
h(f )(x) = 1 π ∞ −∞ f (t, x 1 , . . . x s ) x 0 − t dt
(integral in the principal value sense for a continuous function). Clearly, if f 1 ∈ S(R s+1 ), f 2 ∈ S ′ (R s+1 ), the convolution product satisfies
h(f 1 * f 2 ) = h(f 1 ) * f 2 = f 1 * h(f 2 )
If f is a function which is the boundary value of an analytic function on the tube R s+1 −iV + , f = ℜf + iℑf , then h(ℜf ) = ℑf ; we assume here thatf (0) is defined and equal to zero (to rule out the non-zero constants), namely f has zero mean. We are interested in the case f is a solution of the wave equation f = 0, then also h(f ) = 0. Let ∆ + be the massless, scalar two-point function, namely the Fourier antitransform of the Lorentz invariant measure on ∂V + \{0}. We have (up to a real proportionality constant), see e.g. [15],
∆ + (x) = 1/|x| s−1 if x 2 ≡ x 2 0 − x 2 1 − · · · x 2 s = 0 ,
where |x| = √ −x 2 (with opposite square root determination in V ± ) and
∆ + (x) real, ∆ + (x) = ∆ + (−x) , x spacelike (x 2 < 0) ∆ + (x) imaginary, ∆ + (x) = −∆ + (−x) , x timelike (x 2 > 0).
The commutator function
∆ 0 (x) = ∆ + (x) − ∆ + (−x)
vanishes for x spacelike, while the function
∆ ′ 0 (x) = −i ∆ + (x) + ∆ + (−x)
vanishes for x timelike. Notice that we have Moreover
∆ ′ 0 = h(∆ 0 ) .h(f ) = h(h * ∆ 0 ) = h * h(∆ 0 ) = h * ∆ ′(f * j ε ) = h(f ) * j ε .
We are now ready to prove the version of Lemma 5.3 in odd spacetime dimensions. Our choice of D is canonical as it is given by modular unitaries. To show that D(t) ∈ U ( P ↑ + ) ′′ , notice that this trivially holds if U is irreducible. Recall now that finite helicity representations are dilation covariant. Assume first thatÛ is an irreducible, finite non-zero helicity h representation of P + , thenÛ restricts to
U = U h ⊕ U −h on P ↑ + , where U h is the helicity h irreducible representation of P ↑ + .
There is a unitary implementation of dilations T (s) which decomposes according to U . As T (s)D(−s) ∈ U ( P ↑ + ) ′ and U h and U −h are disjoint, also D(s) decomposes according to U , and D(t) ∈ U ( P ↑ + ) ′′ holds. In the general case with U reducible, U extends as above to a representationÛ of P + , and so the net HÛ disintegrates according toÛ . In particular, H(V + ) and its modular unitaries disintegrate according toÛ and we have D(t) ∈ U ( P ↑ + ) ′′ as stated.
We note that, by Lemma 2.2, if s is even we have
H(V + ) = iZH(V − ) ′ .
General result
We indicate in this section the modifications that are necessary to extend our results in any spacetime dimension s + 1 ≥ 3. Let U be a unitary, positive energy representation of P ↑ + (s) on a Hilbert space H. We assume here that a 2π-rotation in space gives a selfadjoint operator Γ ≡ U (2π), i.e., the eigenvalues of Γ are ±1. In other words U is a representation of the double cover of P ↑ + (s) which coincides with the universal cover P ↑ + (s) if s > 2; and Γ is the image under U of the non-trivial element in the centre of the double cover.
A U -covariant (twisted-local) net of standard subspaces H is defined as a map
W ∋ W −→ H(W ) ⊂ H
as in Section 5. Note that the proof Proposition 5.2 does not use the twisted locality property, and is valid also here. We have: (c): Either U is irreducible, and we apply (b), or U is the direct sum of two irreducible, inequivalent representations of P ↑
+ , U = U 1 ⊕ U 2 on H 1 ⊕ H 2 .
In this case, let K ⊂ H be the complex Hilbert subspaces generated by H. Then K is U -invariant. If K = H we apply (a). Otherwise U | K = U 1 (or U | K = U 2 ). Then H(O) is cyclic on K for some double cone as in Prop. 5.2, so U 1 extends to an (anti-)unitary representation of P + on H 1 [11]; thuŝ U is easily seen to be reducible, contrary to our assumption. Therefore K = H, and the conclusion follows from (a).
Quantum Field Theory: Nets of von Neumann algebras
In this section the Minkowski spacetime dimension is s ≥ 2.
Given a positive energy (anti-)unitary representation of the proper Poincaré group P + on a Hilbert space H, the paper [4] provides a canonical construction of a U -covariant local net of standard subspaces of H on wedges with the properties 1-5. Similarly, this construction gives a twisted-local canonical U covariant net if one considers a representation U of the the universal cover P + . The above Theorems 6.1, 8.5 apply to this net, hence to the net of von Neumann algebras obtained via second quantisation on the Bose/Fermi Fock space, depending on U (2π) = ±1.
A twisted-local, U -covariant net of von Neumann algebras on wedges F is an isotonous map
W −→ F(W )
that associates a von Neumann algebra F(W ) on a fixed Hilbert space H with every W ∈ W, with the following properties:
• Poincaré covariance: U (g)F(W )U (g) * = F(gW ), g ∈ P ↑ + ; • Vacuum with Reeh-Schlieder property: there exists a unique (up to a phase) U -invariant
vector Ω ∈ H and F(W ) is cyclic on Ω for all W ∈ W;
• Bisognano-Wichmann property:
∆ it W = U Λ W (−2πt) , W ∈ W ,
where ∆ W is the modular operator of (F(W ), Ω);
• Twisted locality: For every wedge W ∈ W we have
ZF(W ′ )Z * ⊂ F(W ) ′
where Z is unitary and Z = 1 + iΓ 1 + i , Γ = U (2π) as above.
Due to twisted locality, Ω is indeed also separating for each F(W ), so the modular operators ∆ W are defined.
Given F as above, we define the von Neumann algebra associated with the region O as We now start with a local net A of von Neumann algebras on double cones, with the double cone Reeh-Schlieder property and the Bisognano-Wichmann property. Let (norm closure) be the quasi-observable C * -algebra. We shall say that a representation π of A is cone localizable if, for every spacelike cone C, π| A(C ′ ) is unitarily equivalent to id| A(C ′ ) , where A(C) is the C * -algebra generated by A(O) as O runs in the double cones contained in C. Similarly π is double cone localizable if π| A(O ′ ) ≃ id| A(O ′ ) , for all double cones O.
F(O) ≡ W∋W ⊃O F(W ) .(21)
A Doplicher-Haag-Roberts (DHR) (resp. a Buchholz-Fredenhagen) representation [7,5] is a Poincaré covariant representation with positive energy, which is double cone (resp. cone) localizable. (Poincaré covariance with positive energy follows by general assumptions [11]). Theorem 9.4. Let π be a DHR representation of A with finite statistics [7]. Then the unitary representation U π of P ↑ + in the representation π does not contain infinite spin subrepresentations.
Proof. By considering the dual net, we can assume Haag duality for double cones. We consider the Doplicher-Roberts twisted-local field net F. We have A(O) ⊂ F(O) and the restriction of the vacuum representation of F to A is the direct sum (with multiplicity) of all DHR representations of A with finite statistics.
The representation U F of P ↑ + restricts accordingly to the representations of A. Thus we have to show that U F does not contain an infinite spin sub-representation. This will follow from Theorem 9.3 once we show the Bisognano-Wichmann property. Now the Bisognano-Wichmann property for F is a consequence of the Bisognano-Wichmann property for A as one can identify the Connes-Radon-Nikodym cocycles, see [23,16].
As a consequence, let π be a Poincaré covariant representation of A. If π contains infinite spin particles (i.e., U π contains an infinite spin sub-representation) then:
π is localizable in a double cone =⇒ π has infinite statistics.
This indicates an intimate relation among infinite spin, infinite statistics and localization in infinitely extended regions.
10 Final comments 10.1 Field algebra structure
We now describe the field algebra structure that we obtain starting from the observable algebra and adding all charges with finite statistics, including the ones with infinite spin (space dimension s > 2).
Let A be as a above a local net with the double cone Reeh-Schlieder property and the Bisognano-Wichmann property. Let T be the family of all irreducible representations, up to unitary equivalence (sectors), of A of Buchholz-Fredenhagen type with finite statistics.
The Doplicher-Roberts construction [8] yields a field net F of von Neumann algebras on a larger Hilbert space with F(C) ⊃ A(C) for every cone C, and the identity representation of A on the Hilbert space of F decomposes into the direct sum of elements of T , with multiplicity.
By the spin-statistics theorem [12], F is a twisted-local net. If infinite spin sectors exist, then by Theorem 9.3 F(O) cannot be cyclic on the vacuum vector if O is a bounded region. If one restricts F to the cyclic Hilbert space generated by F(O), one gets the field algebra associated with DHR charges. We discuss a physical interpretation of this structure in the outlook.
We mention also that, in two space dimensions, cone localizable representations may have braid group statistics. If we consider only those ones with Bose or Fermi statistics, then the above field algebra description still holds (the spin-statistics theorem in 2 + 1 dimensions is treated in [22]). However, with general statistics, no field algebra exists that describes an analogue of the above picture.
de Sitter spacetime
If A is a local net on spacelike cones of the Minkowski spacetime R s+1 , one can associate a local net B on double cones of the s-dimensional de Sitter spacetime dS s (and similarly in the twisted-local case). As usual, one views dS s as an hyperboloid of R s+1 , which is the manifold of spacelike directions of Minkowski spacetime. With E any region of dS s , one sets B(E) ≡ A(C E ), where C E ⊂ R s+1 is the spacelike cone with apex in the origin spanned by E.
This construction has been made in [4]. In particular, in the free field case (finite or infinite spin), one gets the canonical modular construction on dS s associated with the restriction of Poincaré unitary representation to the Lorentz subgroup.
We emphasize here that the de Sitter picture is natural in the presence of infinite spin particles. These particles have no bounded spacetime localization on the Minkowski spacetime, yet they are localized in bounded spacetime regions of the de Sitter spacetime.
The role of the Bisognano-Wichmann property
In this paper, we rely on the Bisognano-Wichmann property as a first principle (cf. [11,12]), so we briefly comment here on its roots.
The Bisognano-Wichmann property implies the positivity of the energy (see [4]), and is slightly stronger than that; it reflects the stability of the vacuum state. It always holds in a Wightman theory [2], including string localized fields [27]. In the local algebra framework one can find a counter-example (see Sect. 7), that has however a pathological nature (with continuous degeneracy) and is built on the non-uniqueness of the covariance unitary representation of the Poincaré group: if one chooses the wrong (non-canonical) representation, one obviously violates the Bisognano-Wichmann property. So we may expect the latter to always hold when the Poincaré representation is unique, say by assuming the split property.
In a massive theory, the Bisognano-Wichmann property can be derived by asymptotic completeness [26]. It always holds in the conformal case. It is equivalent to a sub-exponential growth estimate on the energy density levels of localized states for the Rindler Hamiltonian, namely (ξ, e −2πK ξ) < ∞ for all vector states ξ localized in a given cone C contained in a wedge W , with K the generator of the unitary one-parameter group of boosts associated with W [13].
A further argument for the Bisognano-Wichmann property is its mentioned equivalence with the Hawking-Unruh effect for Rindler black holes [31] (the Hawking temperature is the KMS temperature). An illustration of this fact goes beyond the purpose of this paper and we refer to the book [14] for more insight on this point and related aspects.
Outlook
Infinite spin particle states cannot be localized in a bounded spacetime region. This corresponds to the fact that no local observables exist that can generate these states from the vacuum. These results, obtained in the present paper in the Operator Algebraic intrinsic setup [4], extend the no-go theorem on local fields with infinite spin obtained previously in the Wightman setting [36]. The string-localized free fields constructed in [27], that correspond to and generate the von Neumann algebras in [4], cannot thus be compactly localized.
As described in Sect. 10.1 our results provide the following picture in a theory of local observables.
A quantum field theory on a Hilbert space including infinite spin states is described by a net W −→ F(W ) of von Neumann algebras for wedge regions, and the vacuum vector is cyclic for the von Neumann algebras for spacelike cones (defined by intersections of wedge algebras), and has the Bisognano-Wichmann property. The algebras for double cone regions are non-trivial, forming a covariant subnet O −→ A(O), but the vacuum is not cyclic for A(O). The full Hilbert space therefore splits into representations of A, with the infinite spin states absent from the vacuum representation. The representations containing the infinite spin states are massless sectors of the Buchholz-Fredenhagen type, i.e., localized in spacelike cones, and the net F serves as a field algebra for these sectors. This picture complies with the scenario proposed by Schroer [30] with a hindsight on "dark matter".
One may reasonably expect that A contains local generators of Poincaré transformations, i.e., a stress-energy tensor subnet (which could couple to gravity). As infinite spin states are localized in spacelike cones, their Lorentz transforms will be localized in different cones. Thus, the obstruction against infinite spin states to be present in the vacuum repre-sentation is necessary because they cannot be Lorentz transformed by local generators. But the representatives of local generators in a cone-localized representation may well Lorentz transform infinite spin states present in these sectors.
By our result (Cor. 9.2), if F is the free field net associated with an infinite spin representation, then the subnet A would be trivial. Hence, the above scenario necessarily requires a self-interaction of some unknown sort. It is an exciting challenge to describe such an interaction, and the possible interaction of infinite spin fields with "ordinary matter" fields.
A linear, real, closed subspace H of a complex Hilbert space H is called cyclic if H + iH is dense in H, separating if H ∩ iH = {0} and standard if it is cyclic and separating. Given a standard subspace H one defines the Tomita operator S H , the closed, antilinear involution with domain H + iH, given by S H : ξ + iη → ξ + iη, ξ, η ∈ H. The polar decomposition S H = J H ∆ 1/2 H defines the positive selfadjoint modular operator ∆ H and the anti-unitary modular conjugation J H . ∆ H is invertible and J H ∆ H J H = ∆ −1 H . Pairs (J, ∆), where J is an anti-unitary involution and ∆ a selfadjoint positive invertible operator s.t. J∆J = ∆ −1 are in 1-1 correspondence with closed, anti-linear, densely defined involutions S = J∆ 1/2 and in 1-1 correspondence with standard subspaces H = Ker(S − 1).
Lemma 2.3.[24,25]. Let H ⊂ H be a standard subspace, and U a unitary on H such that U H = H. Then U commutes with ∆ H and J H .
Lemma 4 . 1 .
41Let G be a locally compact group, H ⊂ G a closed subgroup and β an automorphism of G such that β(H) = H. If V is a unitary representation of H and U ≡ Ind H↑G V , then
2) , thus κ ′ = e −t κ by the commutation relation (11) [36, Lemma 4].5 Double cone localization implies dilation covariance Let U be a unitary, positive energy representation of the cover P ↑ + of the Poincaré group on a Hilbert space H. A U -covariant net of standard subspaces H on the set W of wedge regions of the Minkowski spacetime is a map H : W ∋ W −→ H(W ) ⊂ H that associates a closed real linear subspace H(W ) with each W ∈ W, satisfying: 1. Isotony: if W 1 ⊂ W 2 then H(W 1 ) ⊂ H(W 2 ); 2. Poincaré covariance: U (g)H(W ) = H(gW ), g ∈ P ↑ + ; 3. Reeh-Schlieder property: H(W ) is cyclic ∀ W ∈ W;
following proposition ensures a variant of the Reeh-Schlieder property. If O, O are double cones, we write O ⋐ O if the closure of O is contained in the interior of O. Proposition 5.2. Let H(O) be defined as above in (13), with U irreducible. If H(O) = {0} for some double cone O, then H( O) is cyclic for every double cone O ⋐ O. Proof. Let O ⋐ O be double cones with H(O) = {0} and ξ a vector orthogonal to H( O). We can find a δ > 0 s.t. x + O ⊂ O, so f (x) ≡ ξ, U (x)Zη = 0, for |x| < δ and η ∈ H(O), where U (x) is the unitary translation by x. By positivity of the energy, f has an analytic continuation on the tube R 4 − iV + . Since f (x) = 0 on an open subset of the boundary, by the Edge of the Wedge theorem f is identically zero. Thus ξ is orthogonal to all translates H(O + x).We consider now a wedge W ⊃ O and the corresponding boost one-parameter group Λ W . By the KMS property entailed by the Bisognano-Wichmann property, there exists an analytic extension of the function h:h(s) ≡ ξ, U Λ W (2πs) Zη , on the strip {z ∈ C : 0 < ℑ z < 1}. Because O ⋐ O, h(s)is zero for small real values of s. Thus the whole extension of h has to be zero. It follows that ξ ⊥ H(gO) , ∀g ∈ P ↑ + .Now the closed, complex linear span generated by H(gO) : g ∈ P ↑ + is a U -invariant, non-zero, closed linear subspace of H, that must be equal to H since U is irreducible. Thus ξ = 0 and H( O) is cyclic.Lemma 5.3. Assume that U is a massless, unitary representation of P ↑ + acting covariantly on a twisted-local net of closed, real linear subspaces on double cones. Let O 1 , O 2 be double cones with O 2 in the timelike complement of O 1 , then H(O 2 ) ⊂ ZH(O 1 ) ′ . Proof. Let O r be the double cone of radius r > 0 centred at the origin, namely O r is the causal envelope of the ball of radius r centred at the origin in the time zero hyperplane. Consider the two point function f (x) = ξ, U (x)Zη , ξ, η ∈ H(O r ) . Then f = 0, namely f is a solution of the wave equation, since the Fourier transform of f (w.r.t. the Minkowski metric) is a measure with support in ∂V + . In particular ℑf = 0. Now ℑf (x) = 0 if x ∈ O ′ 2r , because O r + x ⊂ O ′ r . Thus, by the Huygens principle for solutions of the wave equations, also ℑf (x) = 0 if x belongs to the timelike complement of O 2r . Thus H(O r + x) ⊂ ZH(O r ) ′ for such x, namely for x such that O r + x is contained in the timelike complement of O r . This entails the thesis as r > 0 is arbitrary. Proposition 5.4. Let U be a massless representation of P ↑ + , acting covariantly on a net H of standard subspaces on wedges satisfying properties 1-5. If H(O) is cyclic for some double cone O, then U is dilation covariant. If U is irreducible, the same conclusion holds by assuming that H(O) = {0} for some double cone O. Proof. Let H(V + ) be the closed, real linear subspace generated by H(O) as O runs in the double cones contained in V + , and similarly for H(V − ). H(V + ) (and H(V − )) is cyclic as it contains a cyclic real linear subspace H(O) by assumptions (if H(O) is cyclic, all its translated are cyclic). Since H(V + ) ⊂ ZH(V − ) ′ by Proposition 5.4, H(V + ) and H(V − ) are also separating, hence standard subspaces. Set
2.4) applies to all one-parameter groups of timelike translations. Since the latter generate all translations, we conclude that D(s) scales the translations: D(s)U (x)D(−s) = U (e s x) , s ∈ R , if x is in the translation group. Thus U is dilation covariant, with dilation unitaries D(t).
Theorem 6. 1 .
1Let U be an irreducible unitary, positive energy, massless, infinite spin representation of P ↑ + on a Hilbert space H, and H : W ∋ W −→ H(W ) ⊂ H a U -covariant net of standard subspaces satisfying properties 1double cone O ∈ O. Proof. If H(O) = {0} for some double cone O, then by Proposition 5.4 U must be dilation covariant, which is not possible by Proposition 4.3.
Proposition 7. 1 .
1The net H I of standard subspaces is local, U V -covariant, and cyclic on double cones. U V decomposes into a direct integral of infinite spin representation. U V does not satisfy the Bisognano-Wichmann property.
Lemma 7 . 2 .
72Let V be a unitary representation of L ↑ + not containing the identity representation. Then V | E(2) is a multiple of ⊕ R + V κ dκ,where V κ is the unitary irreducible representation of E(2) with radius κ.
Proposition 8. 2 .
2Let f be a bounded continuous function on R s+1 with f = 0, and O a double cone. If f (x) = 0 for x in the spacelike complement of O, then h(f )(x) = 0 for x in the timelike complement of O. Proof. Let h be a smooth function with supp(h) ⊂ O. Then f ≡ h * ∆ 0 satisfies f = 0 and f (x) = 0 if x ∈ O ′ .
0 vanishes on the timelike complement of O. Now any smooth function f with f = 0 and supp(f ) ⊂ O can be written f = h * ∆ 0 as above, hence the proposition holds true for every smooth solution of the wave equation f . For a general continuous f , one can approximate as usual f by f * j ε by a smooth approximate identity j ε , and get the thesis because h
Lemma 8 . 3 .
83Let U be a massless, unitary representation of the double cover of the Poincaré group P ↑ + (s), s even, s ≥ 2, on a Hilbert space H. Assume that H is a twisted-local, Ucovariant net of standard subspaces of H on wedges.Let O 1 , O 2 ∈ O with O 2 in the timelike complement of O 1 , then H(O 2 ) ⊂ iZH(O 1 ) ′ .(20)Proof.With O r and f (x) = ξ, U (x)Zη , ξ, η ∈ H(O r ) as in the proof of Lemma 5.3, we have ℑf = 0 andℑf (x) = 0 if x ∈ O ′ 2r . Thus, by Proposition 8.2, h(ℑ(f )) = −ℜ(f ) vanishes in the timelike complement of O 2r ; but ℜ(f )(x) = ℑ(if )(x) = ℑi ξ, ZU (x)η = ℑ ξ, ZU (x)iηand we get the thesis.We may now extend Proposition 5.4 in any spacetime dimension. Note that, in the following Prop. 8.4, the cyclicity assumption for H(O) follows from H(O) = {0} by Prop. 5.2. Proposition 8.4. Let U be a massless representation of P ↑ + , acting covariantly on a net H of standard subspaces of H, satisfying 1-5, on wedges on the s + 1-dimensional Minkowski spacetime, with s ≥ 2. If H(O) is cyclic for some double cone O, then U is dilation covariant. Moreover the dilation one-parameter unitary group D can be chosen canonically, and D(t) ∈ U ( P ↑ + ) ′′ , t ∈ R. Proof. H(V + ), the closed linear span of all spaces H(O) with O ⊂ V + , is a standard subspace of H by Lemma 8.3, so, by positivity of the energy and Theorem 2.+ ) implements dilations on U -translations, and commutes with the Lorentz unitaries by Lemma 2.3. Namely D implements the dilations on U . Now, by Proposition 5.1, U extends to an (anti)-unitary representationÛ of P + on H, U maps the reflection around the edge of W to J H(W ) , and H(W ) = HÛ (W ), the standard subspace associated byÛ with W .
In particular, if H is local, we have twisted timelike duality H(V + ) = iH(V − ) ′ , and if H is purely Femi-local (Z = −i) we have timelike duality H(V + ) = H(V − ) ′ .
Theorem 8 . 5 .
85Let U be a unitary, positive energy representation of the cover of the Poincaré group P ↑ + (s), acting covariantly on a net H of standard subspaces of H on wedges satisfying 1-5 as above, s ≥ 2.(a) If H(O) is cyclic for some double cone O, then U does not contain an infinite spin subrepresentation (namely there is no infinite spin fibre in the irreducible direct integral decomposition). (b) If U is irreducible and H(O) = {0} for some double cone O, then U is not massless with infinite spin. (c) If U extends to an (anti-)unitary, irreducible representationÛ of P + on H and H(O) = {0} for some double cone O, then U does not contain an infinite spin subrepresentation. Proof. (b) follows from (a) by Prop. 5.2, so we prove the statement (a). By restricting to the massless component, we may assume that U is massless. By Proposition 8.4, U is dilation covariant; U = ⊕ X U λ dµ(λ) is the irreducible direct integral decomposition of U , by Prop. 8.4 the dilation unitary group D decomposes accordingly, D = ⊕ X D λ dµ(λ), so U λ is dilation covariant for µ-almost all λ. Thus U λ has not infinite spin by Proposition 8.1.
is cyclic for double cones O, then H(W ) defined by additivity from the double cones coincides with the original H(W ) (assuming the Bisognano-Wichmann property).and
locality.
If H(O)
Proposition 8.1. Let U be an irreducible, positive energy, unitary representation of P ↑ + (s), s ≥ 2. Then U is dilation covariant iff U is massless with finite spin.Proof. We have seen in Proposition 4.3 in the case s = 3 that
A twisted-local, U covariant net O −→ F(O) on double cones is analogously defined, by requiring the U -covariance and the cyclicity of the algebras F(O). Then F(W ) is defined by additivity and W −→ F(W ) is a twisted-local, U -covariant net on wedges. The von Neumann algebras F(O) defined by(21) are, in general, larger than the original F(O) (they define the dual net).The free Bose (resp. Fermi) field net F ± is defined by second quantization on the symmetric/anti-symmetric Fock space F ± (H) as (a + ) S + Theorem 9.3. Let F be a twisted-local net of von Neumann algebras F(O) on double cones on a Hilbert space H, covariant w.r.t. a unitary representation U of P ↑ + with vacuum vector Ω ∈ H. As above, we assume the double cone Reeh-Schlieder property and the Bisognano-Wichmann property.Then U does not contain an infinite spin sub-representation.Proof. For every wedge W ∈ W we setH(W ) = F(W ) s.a. Ω ,where F(W ) s.a. is the selfadjoint part of F(W). By assumptions, H : W −→ H(W ) is then a twisted-local, U -covariant net of standard subspaces of H satisfying Properties 1-5. With O a double cone, we have that H(O) ≡ W∋W ⊃O H(W ) ⊃ F(O) s.a. Ω is cyclic. We thus infer from Theorem 8.5 (a) that U does not contain an infinite spin subrepresentation.
Acknowledgements. We thank S. Carpi, D. Guido, G. Morsella, and J. Yngvason for useful discussions.where H = H U is the canonical net of standard subspaces of the one-particle Hilbert space H associated with the unitary representation U of the cover of Poincaré group with U (2π) = ±1, and R ± (H(W )) are defined as follows.With H a Hilbert space and H ⊂ H a real linear subspace, R ± (H) is the von Neumann algebra on F ± (H) generated by the CCR/CAR operators:with w(ξ) the Weyl unitaries on F + (H) and Ψ(ξ) the Fermi field operators on F − (H). Note that, by continuity,Moreover the vacuum vector Ω is cyclic (resp. separating) for R ± (H) iffH is cyclic (resp. separating).If H is standard, we denote by S ± H , J ± H , ∆ ± H the Tomita operators associated with (R ± (H), Ω), and by Γ ± (T ) the Bose/Fermi second quantization of a one-particle operator T on H, defined by tensor products on F ± (H).This assignment(22)respects the lattice structure, as originally proven in[1](Bose case) and[10](Fermi case). The modular operators were computed in[9,21,10]. For convenience, we state these properties in the following proposition with a sketch of proof.where denotes the von Neumann algebra generated, Z = 1 (resp. Z = −i) on the nparticle subspace, n even (resp. odd), and H is standard in (a ± ).Proof. (a ± ) S + H = Γ + (S H ) due to the relation S + H w(ξ)Ω = w(−ξ)Ω (see[21]), whilewith Ω the Fock vacuum vector (see[10]). By the uniqueness of the polar decomposition, we then have We state now the following consequence of Theorems 6.1, 8.5 for free fields. Proof. Immediate by Theorem 6.1, the results in[4], and the fact (Theorem 9.1) that the intersection of closed real linear spaces of the one-particle Hilbert space corresponds to the intersection of the associated von Neumann algebras on the Fock space:for X = C a cone, resp. X = O a double cone.The following theorem shows why infinite spin representations do not occur in a theory of local observables.We shall say that a unitary representation U of P ↑ + has infinite spin if U is a direct integral of irreducible, infinite spin representations. Thus U does not not contain an infinite spin sub-representation iff no infinite spin representation appears in the irreducible direct integral decomposition of U (up to a measure zero set).
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[
"How to produce quantum entanglement for ascertaining incompatible properties in double-slit experiments",
"How to produce quantum entanglement for ascertaining incompatible properties in double-slit experiments"
]
| [
"G Nisticò [email protected] ",
"A Sestito [email protected] ",
"\nDipartimento di Matematica\nIstituto Nazionale Fisica Nucleare\nItaly\n",
"\nUniversità della Calabria\nvia P. Bucci 30b87036RendeItaly\n"
]
| [
"Dipartimento di Matematica\nIstituto Nazionale Fisica Nucleare\nItaly",
"Università della Calabria\nvia P. Bucci 30b87036RendeItaly"
]
| []
| Double-slit experiment very well lends itself in describing the problem of measuring simultaneously incompatible properties. In such a context, we theoretically design an ideal experiment for spin-7/2 particles, able to produce the entanglement which makes possible the detection. | null | [
"https://arxiv.org/pdf/1001.1927v1.pdf"
]
| 115,458,788 | 1001.1927 | 00a9597b9038a6e8f2c2f0708b0ff9e110965edd |
How to produce quantum entanglement for ascertaining incompatible properties in double-slit experiments
12 Jan 2010
G Nisticò [email protected]
A Sestito [email protected]
Dipartimento di Matematica
Istituto Nazionale Fisica Nucleare
Italy
Università della Calabria
via P. Bucci 30b87036RendeItaly
How to produce quantum entanglement for ascertaining incompatible properties in double-slit experiments
12 Jan 2010numbers: 0365Ca0365Db0365Ta
Double-slit experiment very well lends itself in describing the problem of measuring simultaneously incompatible properties. In such a context, we theoretically design an ideal experiment for spin-7/2 particles, able to produce the entanglement which makes possible the detection.
I. INTRODUCTION
In Quantum Mechanics the algebraic structure of the set of observables is mirrored in the algebraic structure of the self-adjoint operators, which is not commutative; then, whereas in Classical Mechanics all the observables can be measured together on a physical system, this is not the case in Quantum Mechanics [1]. Here such a possibility is restricted to those observables corresponding to commutative self-adjoint operators.
The double-slit experiment is a very effective example in describing such a phenomenon: the incompatibility between the position observables at different times makes impossible to measure simultaneously which slit each particle passes through (WS property) and the localization on the final screen (F (∆)). This notwithstanding, over the years a lot of devices have been conceived in order to reach "indirect" knowledge about WS property by measuring a different property, T , correlated with it [2,3]. More recently, experiments have been proposed which can give indirect knowledge about either WS property or an incompatible one, according to their set-up [4,5]. The theoretical investigation performed in [6] in the context of double-slit experiment, establishes that, under suitable hypothesis and exploiting entanglement, the question whether experimental situations may be conceived, which make possible to obtain "indirect" knowledge about two incompatible properties (WS and an incompatible one, G), is positively answered, in spite of the predicted impossibility of measuring them simultaneously. Following such a result, in [7] an ideal double-slit experiment for atoms is designed such that the values of two non-commuting quantum properties are inferred simultaneously by revealing photons emitted by single atoms within micro-maser cavities.
Other approaches in literature face the problem of ascertaining simultaneously incompatible properties. The claimed impossibility [8] of producing inferences for more than three observables urges to consider the situation of detecting two incompatible properties, besides WS, together with the measurement of the final impact point. The theoretical investigation in [9] shows that also this question has affirmative answer within this approach; an ideal experiment realizing such a detection is presented in [7].
In the present work, after introducing the problem the simultaneous detection of three incompatible properties, WS, G and L, together with the final impact point (section 2), an ideal double slit experiment is designed which realizes such a detection (section 3). We notice that this ideal experiment correspond to a particular solution of the theoretical investigation in [9] and with the assumptions therein, properties L and G turn out always correlated. Moreover, it must be stressed that such a detection requires an entanglement between the particle and the detector. Section 3 is devoted to show how such an entanglement can be ideally realized.
II. DETECTING INCOMPATIBLE PROPERTIES SIMULTANEOUSLY
We consider a system (e.g. the nucleus of an atom) which can travel towards the two slits, but not elsewhere, whose position is represented, in Heisenberg picture, by an operator in the Hilbert space H I . It possesses further degrees of freedom (e.g. spin) described in the Hilbert-space H II ; hence, H = H I ⊗ H II is the Hilbert space describing the entire system. In general, we denote by A I (A II ) an operator of H I (H II ). The projection operator representing WS property "the particle passes through slit 1 (2)" has the form E = E I ⊗ 1 II (E ′ = (1 I − E I ) ⊗ 1 II ). Given any interval ∆ on the final screen, we denote the projection operator representing the property "the particle hits the final screen in a point within ∆" by F (∆). Though E and F (∆) are both localization projections, they refers to different times -the time t 1 of the slits' crossing for E and the time t 2 > t 1 of the final impact for F (∆). We suppose that the Hamiltonian operator in independent of the degrees of freedom described by H II , so that it has the form H = H I ⊗ 1 II ; hence, it can be shown that F (∆) = F I (∆) ⊗ 1 II follows, but [E, F (∆)] = 0 must hold too [9]. Thus, it is not generally possible to ascertain WS property and the final impact point, by direct localization measurements. Rather than measuring E, a property T = 1 I ⊗ T II acting on H II can be measured together with F (∆), whose outcomes are correlated with the outcome of E, as expressed by the following general definition of detector [10]:
Definition 1. A projection operator S of H is called a detector of a property R with respect to a state ψ if (i) [S, F (∆)] = 0, (ii) [S, R] = 0 and Sψ = Rψ .
Condition (i) ensures that S can be measured together with F (∆); condition (ii) allows us to infer the outcome of R (albeit not measured) from the outcome of S [11]. According to Def. 1, if for a given state Ψ, a projection operator T = 1 I ⊗ T II exists such that T Ψ = EΨ, then it is possible to detect which slit each particle passes through by means of a measurement of T ; indeed, since if for a given state Ψ, two commuting projection operators Y = 1 I ⊗ Y II and W = 1 I ⊗ W II exist such that Y is a detector for G and W is a detector for L, then the outcomes of Y and W reveal the occurrence of the properties G and L respectively; therefore, the problem is:
E = E I ⊗ 1 II and F (∆) = F I ⊗ 1 II ,Problem (P ′ ). Given WS property E = E I ⊗ 1 II we have to find -two projection operators G I and L I of H I , -three projection operators T II , Y II and W II of H II , -a state vector Ψ ∈ H I ⊗ H II ,
such that the following conditions are satisfied:
(C.1) [E, G] = 0 i.e [E I , G I ] = 0 (C.2) [E, L] = 0 i.e [E I , L I ] = 0, (C.3) [L, G] = 0 i.e [L I , G I ] = 0, (C.4) [T, E] = 0 and T Ψ = EΨ (C.5) [Y, G] = 0 and Y Ψ = GΨ (C.6) [W, L] = 0 and W Ψ = LΨ (C.7) [T, Y ] = 0 (C.8) [T, W ] = 0 (C.9) [Y, W ] = 0 (C.10) Ψ = EΨ = 0, Ψ = GΨ = 0 and Ψ = LΨ = 0.
The ideal experiment described in the next section represents a solution of problem (P ′ ).
III. AN IDEAL EXPERIMENT FOR SIMULTANEOUS DETECTION ON INCOMPATIBLE PROPERTIES
The system of our ideal experiment is a spin-7/2 particle whose position observable is described in a Hilbert space H I , while the spin observables are described in H II ≡ C 8 . Let ψ i , (resp., ψ i+5 ), i = 1, ..., 5 be 5 mutually orthonormal vectors of H I localized in slit 1 (resp., slit 2) when the particle crosses the slits' support, i.e. such that E I ψ i = ψ i (resp., E I ψ i+5 = 0). No further condition is required to these vectors. These ten vectors form an orthonormal set. Then we take the Hilbert space H I as the space generated by them. This implies that
E I ϕ = ψ 1 | ϕ I ψ 1 + ψ 2 | ϕ I ψ 2 + ψ 3 | ϕ I ψ 3 + ψ 4 | ϕ I ψ 4 + ψ 5 | ϕ I ψ 5 for every ϕ ∈ H I .(1)
There are 8 eigenvectors α 1 = |7/2 , α 2 = |5/2 , α 3 = |3/2 , α 4 = |1/2 , α 5 = | − 1/2 , α 6 = | − 3/2 , α 7 = | − 5/2 , α 8 = | − 7/2 ∈ H II corresponding to the 8 possible values (in units) of the spin along direction z, represented by the hermitian operator S z of C 8 .
Let the particle be prepared in the entangled state represented by Ψ = 1 32 (−ψ 1 − 2ψ 2 + ψ 3 + ψ 4 + ψ 5 ) |7/2 + + 1 8 7 10 (−ψ 1 + ψ 2 − 2ψ 3 + 3ψ 5 ) |3/2 + √ 35 32 (−ψ 6 − 2ψ 7 + ψ 8 + ψ 9 + ψ 10 ) |1/2 + + 1 16
35 11 (ψ 1 + ψ 2 + 3ψ 4 ) + (4ψ 1 + ψ 2 + 3ψ 3 + 3ψ 5 ) | − 1/2 + + 1 8 7 30 (−ψ 6 + ψ 7 − 2ψ 8 + 3ψ 9 ) | − 5/2 + + 1 16 1 √ 11 (ψ 6 + ψ 7 + 3ψ 9 ) + 1 √ 35 (4ψ 6 + ψ 7 + 3ψ 8 + 3ψ 10 ) | − 7/2 .(2)
The 8 projection operators A i II = |j i j i |, i = 1, ..., 8 represent spin observables pertaining H II (if A = 1 I ⊗ A i II has outcome 1, then the particle has spin j i = 7 2 −(i−1)" along z and so on). They trivially commute with both F (∆) and E. Then also the projection operator T = A 1 +A 2 +A 3 +A 5 = 1 I ⊗(|7/2 7/2|+|5/2 5/2|+|3/2 /2|+|−1/2 −1/2|) commute with F (∆) and E. Now, straightforward calculations show that EΨ = T Ψ. Therefore, T turns out to be a WS detector. Now we introduce a property G = G I ⊗ 1 II incompatible with E, which can be detected by means of a suitable detector Y without renouncing to the WS knowledge provided by T . Given any ϕ ∈ H I , we define
G I ϕ = ψ (1) | ϕ I ψ (1) + ψ (2) | ϕ I ψ (2) + ψ (3) | ϕ I ψ (3)(3)
where
ψ (1) = 1 6 ψ 1 − 1 6 ψ 2 − 1 6 ψ 1 − √ 3 2 ψ 7 + 1 2 √ 3 ψ 9 + 1 2 √ 3 ψ 10 ψ (2) = − √ 3 2 ψ 1 + 1 2 √ 3 ψ 2 + 1 2 √ 3 ψ 3 − √ 6 4 ψ 6 + √ 6 4 ψ 8 + 1 2 √ 6 ψ 9 + 1 2 √ 6 ψ 10 ψ (3) = − √ 2 4 ψ 1 − √ 2 2 ψ 2 + √ 2 4 ψ 3 + √ 2 4 ψ 4 + √ 2 4 ψ 5 .
A straightforward calculation based on (1) and (3) shows that [G I , E I ]ϕ = 0 so that G and E are incompatible with each other. However, the projection operator Y = A 1 + A 2 + A 4 + A 6 = 1 I ⊗ (|7/2 7/2| + |5/2 5/2| + |1/2 1/2| + | − 3/2 −3/2|) satisfies condition Y Ψ = GΨ and it trivially commutes with F (∆) = F I (∆) ⊗ 1 II ; therefore Y is a detector of G. Now we introduce a further property L = L I ⊗ 1 II incompatible with E and G, which can be detected by means of a suitable detector W without renouncing to the WS knowledge provided by T and to the knowledge of G provided by L. Given any ϕ ∈ H I , we define [3] | ϕ I ψ [3] + ψ [4] | ϕ I ψ [4] + ψ [5] | ϕ I ψ [5] (4)
L I ϕ = ψ [1] | ϕ I ψ [1] + ψ [2] | ϕ I ψ [2] + ψ
where
ψ [1] = 1 5 11 15 −2ψ 1 + 2ψ 2 + 2ψ 3 − 9 √ 11 ψ 6 + 9 √ 11 ψ 7 + 9 √ 11 ψ 10 ψ [2] = 1 10 11 65 3ψ 1 − 3ψ 2 − 3ψ 3 − 24 √ 11 ψ 6 − 51 √ 11 ψ 7 + 25 √ 11 ψ 9 + 49 √ 11 ψ 10 ψ [3] = 1 5 33 26 −ψ 1 + ψ 2 + ψ 3 − 17 6 √ 11 ψ 6 − 14 3 √ 11 ψ 7 + 65 6 √ 11 ψ 8 + 5 2 √ 11 ψ 9 − 11 2 √ 11 ψ 10 ψ [4] = 1 √ 15 (−ψ 1 + ψ 2 − 2ψ 3 + 3ψ 5 ) ψ [5] = 1 2 √ 2 (−ψ 1 − 2ψ 2 + ψ 3 + ψ 4 + ψ 5 ) .
A straightforward calculation shows that [G I , L I ]ϕ = 0 so that G and L are incompatible with each other; furthermore, [E I , L I ]ϕ = 0 so that E and L are incompatible with each other, too. However, the projection operator W = A 1 + A 3 + A 4 + A 7 = 1 I ⊗ (|7/2 7/2| + |3/2 3/2| + |1/2 1/2| + | − 5/2 −5/2|) satisfies condition W Ψ = LΨ and it trivially commutes with F (∆) = F I (∆) ⊗ 1 II ; therefore W is a detector of L. Nevertheless, we have that W , Y and T pairwise commute with each other. Then all, W , Y and T can be simultaneously measured together with the position of the final impact; in other words, properties L, G and E, incompatible with each other, can be detected together on each particle localized on the final screen. In the present work, we are not concerned with the question of the physical meaning of G and L, which evidently depends upon the choice of vectors ψ i , ψ i+5 , i = 1, ..., 5. Thus, we have a solution of problem (P ′ ).
IV. EXPERIMENTAL ISSUES FOR ENTANGLEMENT
In the perspective of a realization of the detection of E, G and L, a crucial experimental task is to create the entanglement, encoded in the state vector Ψ in (2), between the particle and the detectors, before the time t 1 when the particle reaches the screen supporting the slits. This can be (ideally) realized in three steps. In the first step only particles with the x component of the spin equal to 7/2 are selected, for instance by means of a suitable Stern-Gerlach apparatus. Hence, in Schroedinger picture, at this stage -time t 0 < t 1 -the state vector is of the kind ψ|s , with ψ ∈ H I , ψ = 1, and S x |s = 7/2|s , so that we can take |s = 1
8 √ 2 {|7/2 + √ 7|5/2 + √ 21|3/2 + √ 35|1/2 + √ 35| − 1/2 + √ 21| − 3/2 + √ 7| − 5/2 + | − 7/2 }.
In the second step, during their flight between times t 0 and t 1 , the particles undergo the action of another Stern-Gerlach magnet, able to spatially separate the particles with respect to S z : the particles with S z = 7/2, 5/2, 3/2 or −1/2 (resp. −7/2, −5/2, −3/2 or 1/2) are forced to travel towards slit 1 (resp. 2), but through two alternative spatial channels according to the value of S z , so that the dynamical evolution between times t 0 and a time t 1/2 < t 1 is represented by a unitary operator U such that
U (ψ|7/2 ) = ψ [ 7 2 ] 1 |7/2 , U (ψ| − 7/2 ) = ψ [− 7 2 ] 2 | − 7/2 , U (ψ|5/2 ) = ψ [ 5 2 ] 1 |5/2 , U (ψ| − 5/2 ) = ψ [− 5 2 ] 2 | − 5/2 , U (ψ|3/2 ) = ψ [ 3 2 ] 1 |3/2 , U (ψ| − 3/2 ) = ψ [− 3 2 ] 2 | − 3/2 , U (ψ| − 1/2 ) = ψ [− 1 2 ] 1 | − 1/2 , U (ψ|1/2 ) = ψ [ 1 2 ] 2 |1/2 . where ψ [J]
k are vectors of H I representing the alternative spatial channels taken by the particles to reach slit k; hence ψ k |j i . Before reaching the slits, between the times t 1/2 , t 1 , the particles undergo the action of a filter blocking the beams of particles corresponding to S z = 5/2 and −3/2: the state Ψ is of the kind Ψ 0 + Ψ 1 , with Ψ 0 = ψ
[ 5 2 ] 1 |5/2 + ψ [− 3 2 ] 2 | − 3/2 and Ψ 1 = ji =5/2,−3/2,k ψ [ji]
k |j i ; the effect of the filter is represented by the projection operator P = |Ψ 1 Ψ 1 | so that the final state vector outcoming the whole preparing procedure is (2) which carries the right entanglement allowing for a simultaneous detection of the mutually incompatible properties L, G, E, together with the measurement of the final impact point.
Ψ = P Ψ = 1 8 √ 2 {ψ [ 7 2 ] 1 |7/2 + √ 21ψ [ 3 2 ] 1 |3/2 + √ 35ψ [ 1 2 ] 1 |1/2 + √ 35ψ [− 1 2 ] 1 | − 1/2 + √ 7ψ [− 5 2 ] 1 | − 5/2 + ψ [− 7 2 ] 1 | − 7/2 }.
conditions [T, F (∆)] = 0 and [T, E] = 0 are automatically satisfied. Hence, in other words, outcome 1 (0) for T reveals the passage of the particle through slit 1 (2). We seek for the possibility of detecting two further properties, G and L: let G = G I ⊗ 1 II and L = L I ⊗ 1 II be other properties, incompatible with each other and with WS property E, i.e [L, G] = 0, [L, E] = 0 and [E, G] = 0;
= 0 .
0k1,k2 · δ J1,J2 and E I ψ J 1 = ψ Then the outcoming state must be Ψ = U (ψ|s ) = ji,k ψ[ji]
Now, if dim(E I H I ), dim((1 I −E I )H I ) ≥ 5, then five mutually orthonormal vectors ψ i ∈ E I H I , ψ i+5 ∈ (1 I −E I )H I , with i = 1, . . . , 5, exist such that
− 2ψ 7 + ψ 8 + ψ 9 + ψ 10 ) , + ψ 7 + 3ψ 8 + 3ψ 10 ) , then the state Ψ outcoming from this dynamical preparing process must be + ψ 7 + 3ψ 8 + 3ψ 10 ) | − 7/2 , which is just the state vector[ 7
2 ]
1
=
1
2
√
2
(−ψ 1 − 2ψ 2 + ψ 3 + ψ 4 + ψ 5 ) ,
ψ
[ 3
2 ]
1
=
1
√
15
(−ψ 1 + ψ 2 − 2ψ 3 + 3ψ 5 ) ,
ψ
[ 1
2 ]
2
=
1
2
√
2
(−ψ 6 ψ
[− 1
2 ]
1
=
1
√
22
(ψ 1 + ψ 2 + 3ψ 4 ) +
1
√
70
(4ψ 1 + ψ 2 + 3ψ 3 + 3ψ 5 ) ,
ψ
[− 5
2 ]
2
=
1
√
15
(−ψ 6 + ψ 7 − 2ψ 8 + 3ψ 9 ) ,
ψ
[− 7
2 ]
2
=
1
√
22
(ψ 6 + ψ 7 + 3ψ 9 ) +
1
√
70
(4ψ 6 Ψ = P (U (ψ|s =
1
32
(−ψ 1 − 2ψ 2 + ψ 3 + ψ 4 + ψ 5 ) |7/2 +
+
1
8
7
10
(−ψ 1 + ψ 2 − 2ψ 3 + 3ψ 5 ) |3/2 +
√
35
32
(−ψ 6 − 2ψ 7 + ψ 8 + ψ 9 + ψ 10 ) |1/2 +
+
1
16
35
11
(ψ 1 + ψ 2 + 3ψ 4 ) + (4ψ 1 + ψ 2 + 3ψ 3 + 3ψ 5 ) | − 1/2 +
+
1
8
7
30
(−ψ 6 + ψ 7 − 2ψ 8 + 3ψ 9 ) | − 5/2 +
+
1
16
1
√
11
(ψ 6 + ψ 7 + 3ψ 9 ) +
1
√
35
(4ψ 6
J Neumann, Mathematical Foundation of Quantum Mechanics. Princeton University PressJ. von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton University Press, 1955.
N Bohr, Albert Einstein: Philosopher-Scientist, P.A. Schilpp. Evanston200Library of Living PhilosophersN. Bohr, in Albert Einstein: Philosopher-Scientist, P.A. Schilpp, ed., p. 200, Library of Living Philosophers, Evanston 1949.
R P Feynman, A R Hibbs, Quantum mechanics and path integrals. New YorkMc Graw-Hill incR.P. Feynman, A.R. Hibbs, Quantum mechanics and path integrals, Mc Graw-Hill inc., New York 1965.
. M O Scully, B.-G Englert, H Walther, Nature. 351111M.O. Scully, B.-G. Englert, H. Walther, Nature 351(1991) 111.
. P D D Schwindt, P G Kwiat, B.-G Englert, Phys. Rev. A. 604285P.D.D. Schwindt, P.G. Kwiat, B.-G. Englert, Phys. Rev. A, 60 (1999) 4285.
. G Nisticò, J. Phys. A: Math. Theor. 41125302G. Nisticò, J. Phys. A: Math. Theor. 41 (2008) 125302.
. G Nisticò, A Sestito, Journal of Modern Optics. 562-3374G.Nisticò, A. Sestito, Journal of Modern Optics, 56, n.2-3 (2009) 374.
. L Vaidman, Y Aharonov, D Z Albert, Phys.Rev.Lett. 581385L. Vaidman, Y. Aharonov and D.Z. Albert, Phys.Rev.Lett., 58 (1987) 1385.
. A Sestito, Found. Phys. 38935A. Sestito, Found. Phys. 38 (2008) 935.
G Nisticò, Foundations of Probability and Physics. 4, G. Adenier, C. Fuchs, A.Y. KhrennikovMelville889374AIP Conferences ProceedingsG. Nisticò, in Foundations of Probability and Physics-4, G. Adenier, C. Fuchs, A.Y. Khrennikov, eds, AIP Conferences Proceedings, 889, p. 374 Melville 2007.
. G Nisticò, M C Romania, J. Math. Phys. 3594534G. Nisticò, M.C. Romania, J. Math. Phys., 35 (9) (1994) 4534.
| []
|
[
"Double-ringed debris discs could be the work of eccentric planets: explaining the strange morphology of HD 107146",
"Double-ringed debris discs could be the work of eccentric planets: explaining the strange morphology of HD 107146"
]
| [
"Tim D Pearce \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n",
"Mark C Wyatt \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n"
]
| [
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK"
]
| [
"Mon. Not. R. Astron. Soc"
]
| We investigate the general interaction between an eccentric planet and a coplanar debris disc of the same mass, using analytical theory and n-body simulations. Such an interaction could result from a planet-planet scattering or merging event. We show that when the planet mass is comparable to that of the disc, the former is often circularised with little change to its semimajor axis. The secular effect of such a planet can cause debris to apsidally anti-align with the planet's orbit (the opposite of what may be naïvely expected), leading to the counter-intuitive result that a low-mass planet may clear a larger region of debris than a highermass body would. The interaction generally results in a double-ringed debris disc, which is comparable to those observed in HD 107146 and HD 92945. As an example we apply our results to HD 107146, and show that the disc's morphology and surface brightness profile can be well-reproduced if the disc is interacting with an eccentric planet of comparable mass (∼ 10 − 100 Earth masses). This hypothetical planet had a pre-interaction semimajor axis of 30 or 40 au (similar to its present-day value) and an eccentricity of 0.4 or 0.5 (which would since have reduced to ∼ 0.1). Thus the planet (if it exists) presently resides near the inner edge of the disc, rather than between the two debris peaks as may otherwise be expected. Finally we show that disc self-gravity can be important in this mass regime and, whilst it would not affect these results significantly, it should be considered when probing the interaction between a debris disc and a planet. | 10.1093/mnras/stv1847 | [
"https://arxiv.org/pdf/1507.04367v1.pdf"
]
| 119,169,161 | 1507.04367 | 348c42813f3b0606be2659c17217a7b1d97637d8 |
Double-ringed debris discs could be the work of eccentric planets: explaining the strange morphology of HD 107146
2002
Tim D Pearce
Institute of Astronomy
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
Mark C Wyatt
Institute of Astronomy
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
Double-ringed debris discs could be the work of eccentric planets: explaining the strange morphology of HD 107146
Mon. Not. R. Astron. Soc
0002002Printed 17 July 2015 Released 2002 Xxxxx XX(MN L a T E X style file v2.2)planets and satellites: dynamical evolution and stability -planet- disc interactions -circumstellar matter -stars: individual: HD 107146
We investigate the general interaction between an eccentric planet and a coplanar debris disc of the same mass, using analytical theory and n-body simulations. Such an interaction could result from a planet-planet scattering or merging event. We show that when the planet mass is comparable to that of the disc, the former is often circularised with little change to its semimajor axis. The secular effect of such a planet can cause debris to apsidally anti-align with the planet's orbit (the opposite of what may be naïvely expected), leading to the counter-intuitive result that a low-mass planet may clear a larger region of debris than a highermass body would. The interaction generally results in a double-ringed debris disc, which is comparable to those observed in HD 107146 and HD 92945. As an example we apply our results to HD 107146, and show that the disc's morphology and surface brightness profile can be well-reproduced if the disc is interacting with an eccentric planet of comparable mass (∼ 10 − 100 Earth masses). This hypothetical planet had a pre-interaction semimajor axis of 30 or 40 au (similar to its present-day value) and an eccentricity of 0.4 or 0.5 (which would since have reduced to ∼ 0.1). Thus the planet (if it exists) presently resides near the inner edge of the disc, rather than between the two debris peaks as may otherwise be expected. Finally we show that disc self-gravity can be important in this mass regime and, whilst it would not affect these results significantly, it should be considered when probing the interaction between a debris disc and a planet.
INTRODUCTION
Given the calm order of the Solar System today, where most planets and minor bodies occupy near circular and coplanar orbits, one could be forgiven for forgetting that planetary systems can be violent places. Indeed, our own system probably had a tumultuous youth; planets may have scattered off each other and collided (Hartmann & Davis 1975), switched places with one-another (Tsiganis et al. 2005) and ploughed into regions of debris (Walsh et al. 2011). This turbulent picture is also inferred from extrasolar planets; many of these objects are eccentric ), a possible hallmark of previous scattering events (Jurić & Tremaine 2008). Some, such as Fomalhaut b, may also pass through regions of debris (Kalas et al. 2013;Pearce, Wyatt & Kennedy 2015).
[email protected] Furthermore, major orbital evolution is required to explain some classes of extrasolar planets, such as the Hot Jupiters (Rasio & Ford 1996;Wu & Lithwick 2011). So if planet-planet scattering, mergers and dynamical instabilities could be the norm then it is pertinent to ask how planets affected by these processes interact with other bodies in the system, and whether we can use this information to probe their past or ongoing dynamical evolution.
In this paper we examine the general evolution of a system hosting a debris disc interacting with an equalmass, coplanar, eccentric planet, assuming the planet's eccentricity was rapidly driven up by one of the above processes. Our Solar System hosts two debris discs, the Asteroid and Kuiper Belts, and many extrasolar discs have been detected as infrared excesses in the spectra of stars (e.g. Rhee et al. 2007;Eiroa et al. 2013). So given that debris discs are reasonably common, it is likely that dynamically evolving planets interact with these structures in at least some systems. Instruments such as Spitzer, the HST and ALMA have resolved some extrasolar debris discs (e.g. Backman et al. 2009;Schneider et al. 2009;Dent et al. 2014), and the eccentricity or clumpiness of these discs can be used to infer the presence of planets which would be otherwise undetectable. Hence one aim of this work is to characterise this interaction in general, to ascertain the signatures an eccentric planet leaves on a disc which can then be compared to observations. This part of the paper is similar in its goals and methodology to our previous investigation (Pearce & Wyatt 2014), in which we considered only planets much more massive than the disc; the main difference now that the planet and disc are of equal mass is the disc's ability to significantly alter the planet's orbit, which can have major effects on the system evolution.
Our second aim is to apply the general results to the debris disc of HD 107146, which has been resolved in both infrared emission and scattered light (Ardila et al. 2004;Williams et al. 2004;Carpenter et al. 2005;Corder et al. 2009;Hughes et al. 2011;Ricci et al. 2015). This disc is seen nearly face on, and is broad (∼ 100 au) and axisymmetric with a ∼ 30 au inner hole. The curiosity of this system is that the 1.25mm debris surface density profile appears to either first decrease and then increase with radius, or generally increase but with a gap at around 80 au (Ricci et al. 2015). This differs from the surface density profiles of protoplanetary discs, which decrease with radius (e.g Andrews & Williams 2007); it has been suggested that the unusual profile could be the result of planetary perturbations. Given the large disc mass (possibly ∼ 100M⊕ in total; Ricci et al. 2015), this system is a prime application of our results. We will show that the strange disc morphology can be explained as the aftermath of the interaction between an eccentric planet and a coplanar debris disc of the same mass.
The layout of this paper is as follows. We examine the general outcomes of this interaction in Section 2, using a combination of theory and n-body simulations. We then apply these results to HD 107146 in Section 3, where we run simulations over a broad region of parameter space to replicate the disc structure and surface brightness profile observed with ALMA. We discuss the implications of the results in Section 4, and conclude in Section 5.
GENERAL INTERACTION OUTCOMES
In this section we describe the general outcomes of the interaction between an eccentric planet and a comparablemass, coplanar debris disc. We first summarise the dynamical processes involved in the interaction in sections 2.1 to 2.3, and describe their effects on the orbits of both the debris particles and the planet. Wherever possible we give quantitative predictions from dynamical theory. In Section 2.4 we present an n-body simulation as an example of the general outcomes of this interaction, and explain these in the context of the theoretical results.
Secular effects on debris
Secular interactions are long-term angular momentum exchanges between bodies, which can cause a particle's eccentricity and orbital plane (but not semimajor axis) to evolve. As secular timescales are much longer than orbital timescales then, providing the objects are not in a mean-motion resonance, each interacting body may be thought of as an extended "wire" of material in the shape of the object's orbit, with a density at each point inversely proportional to the body's velocity there (this approach to secular perturbations is known as Gauss averaging). The influence of other masses causes this wire to change shape and orientation. There exists no general analytic evaluation of this interaction (although semi-analytic solutions can be constructed in some cases; see Beust et al. 2014), so a common approach is to derive an analytical form complete up to second order in eccentricities and inclinations, and disregard all higher terms. This is valid for small eccentricities and inclinations, but introduces errors if larger values are considered; however, we showed in Pearce & Wyatt (2014) that this still produces qualitatively correct results for very large eccentricities so long as the inclinations are small. We now summarise the behaviour of a particle undergoing a secular interaction with a planet according to second-order secular theory, and apply the results to the interaction between an eccentric planet and a debris disc.
Second-order secular theory predicts that a body's eccentricity e and longitude of pericentre are coupled (for details see Murray & Dermott 1999). Specifically, for a test particle undergoing secular perturbations from one or more massive bodies, these quantities satisfy the equations e cos = ep cos p + e f cos f ,
(1) e sin = ep sin p + e f sin f .
(
Here ep and p denote the "free" parameters of the particle; ep is constant, and p increases linearly with time t such that
p = At + β,(3)
where A and β are constants. The quantities e f and f are the "forced" values, which depend on the parameters of the massive bodies in the system and may evolve in time. Hence the test particle moves around a circle on the e cos , e sin plane, of radius ep and at a rate A. Meanwhile the centre of this circle also moves as the perturbing bodies evolve over time.
We now consider a system comprised of a star of mass M * , a test particle at semimajor axis a and a planet of mass M plt at semimajor axis a plt (where a > a plt ). We also assume that some mechanism causes the planet's pericentre to precess (˙ plt = 0). We set both the test particle's eccentricity and the planet's longitude of pericentre to zero at time zero for simplicity, i.e. β = π, f = plt , ep = |e f0 | (where e f0 is e f evaluated at t = 0). The forcing eccentricity is now
A < ϖ plt A > ϖ plt e sin(ϖ -ϖ plt ) -0.5 0.0 0.5 e cos(ϖ -ϖ plt ) -0.5 0.0 0.5 A > ϖ plt x / aplt −2 0 2 A < ϖ plt y / aplt −2 0 2 x / aplt −2 0 2 Figure 1
. Evolution of a test particle's orbit under the influence of a single, precessing planet, according to second-order secular theory. Shown are the cases where its orbit precesses more quickly than that of the planet (A >˙ plt ) and more slowly (A <˙ plt ), for a planet of a given eccentricity. Left plot: the coupled evolution of eccentricity e and longitude of pericentre , where the large black circle denotes the planet. A particle of a given semimajor axis will move around one of the two paths in the direction indicated, depending on the sign of A −˙ plt . Right plots: the corresponding physical areas swept out by the particles, in a frame instantaneously aligned with the planet's orbit. The asterisk and thick red line denote the star and the planet's orbit respectively, and the dashed lines show the extreme orbits of the particle. They grey region is the area swept out by the particle, resulting from the superposition of all orbits between the two extremes. Both plots show systems with identical parameters, but with differing signs of A −˙ plt . Hence a particle which would never cross the planet's orbit if A >˙ plt may do so if A <˙ plt . Also note that the planet is precessing, so the structures on the right hand plots will rotate whilst remaining aligned with the planet's orbit.
e f = |A plt | A −˙ plt e plt ,(4)
where
A plt = − 1 4 GM * a 3 M plt M * a plt a b(2)
3/2 (a plt /a),
A = 1 4 GM * a 3 M plt M * a plt a b (1) 3/2 (a plt /a),(5)
G is the gravitational constant and b
(j) s (α) is a Laplace coefficient, such that b (j) s (α) ≡ 1 π 2π 0 cos(jψ)dψ (1 − 2α cos ψ + α 2 ) s .(7)
Equation 4 shows that the orbit of the test particle will evolve differently in the regimes A >˙ plt and A <˙ plt . Firstly, if A >˙ plt then the precession rate of the particle is faster than that of the planet, and e f > 0. The particle will therefore move anticlockwise about a circle on the e cos( − plt ), e sin( − plt ) plane, which crosses the origin and is offset from the origin in the direction of the planet. This is shown by the solid line on the left hand plot of Figure 1. Hence the particle's eccentricity will be maximised when its orbit is aligned with that of the planet, and small when antialigned. Superimposing all intermediate orbits shows that the particle will sweep out a broad, eccentric disc aligned with the planet's orbit (central plot of Figure 1). This allows the particle to attain high eccentricities and yet be shielded from scattering by the planet. For a complete summary of particle evolution in this regime, see Pearce & Wyatt (2014) and Faramaz et al. (2014).
If A <˙ plt , the planet precesses more quickly than the particle and the forcing eccentricity becomes negative. The particle will now move clockwise about a circle on the e cos( − plt ), e sin( − plt ) plane, again crossing the origin but offset in the direction opposing the planet. This is the dashed line on the left hand plot of Figure 1. The particle's eccentricity will now be maximised when its orbit is antialigned with that of the planet, and superimposing all intermediate orbits results in a broad, eccentric disc antialigned with the planet's orbit (right hand plot of Figure 1). Hence a particle which would never cross the planet's orbit if A >˙ plt may now do so, and could therefore be ejected by the latter. Equation 6 shows that the particle's precession rate A is set by the perturber's mass and the semimajor axes of the particle and perturber, all of which are constants in the secular problem. On Figure 2 we plot A as a function of test particle semimajor axis for an example system, which contains a precessing planet. Generally, A → 0 if the test particle's semimajor axis is very small or very large, and A → ∞ if the semimajor axes of the particle and planet are similar. Hence if˙ plt = 0 then there will always be a region in particle semimajor axis space where A >˙ plt , and two regions where A <˙ plt . In this case a sufficiently broad disc of test particles will cover both the A >˙ plt and A <˙ plt regimes. Figure 3 shows a schematic of the resulting debris structure, for particles external to the planet. The result is a superposition of the debris structures on Figure 1, with the innermost particles in the A >˙ plt regime and the outermost in the A <˙ plt regime. There exists a crescent-shaped region devoid of debris, and also a location where A >˙ plt and A <˙ plt particles overlap. However this overlap does not necessarily correspond to debris overdensity, and it is unlikely to be a site of increased dust production; the . Precession rate of a test particle as a function of its semimajor axis, derived using second-order secular theory.
Here the planet (black circle) has a semimajor axis of 40 au and precesses at a rate˙ plt = 4 × 10 −5 • yr −1 . Particles precessing more rapidly than the planet form an eccentric disc apsidally aligned with the planet's orbit (central plot on Figure 1), whilst those with slower precession rates will antialign with the planet's orbit (right hand plot on Figure 1). Note that this plot does not take account of mean motion resonances, the effect of which could dominate over secular behaviour. . General shape of a debris disc undergoing a secular interaction with a interior, precessing planet. The grey regions are disc material, the red ellipse denotes the planet's orbit and the asterisk shows the star. There are two distinct debris populations; an inner population apsidally aligned with the planet, and an outer one which is antialigned. The inner population may or may not be present depending on the parameters of the system, and likewise for the outer population.
collision velocities between particles are actually greater within the inner (aligned) disc than those in this innerouter disc overlap region. One mechanism which may cause planet precession is the secular effect of the debris. If the planet is much more massive than the total disc mass, the precession rate of the former will be slower than that of the debris, unless the disc extends very close to or far from the star. Hence for all but the broadest discs, if the planet is much more massive than the disc then the result will be an eccentric debris structure apsidally aligned with the planet's orbit. This was the case investigated in Pearce & Wyatt (2014). Alternatively if the planet mass is comparable to that of the disc (ignoring planet evolution for now), the planet will likely precess more rapidly than the outermost debris. Thus the farthest debris will assume an eccentric structure antialigned with the planet's pericentre. Whether the innermost debris also forms this structure, or forms a structure apsidally aligned with the planet (as on Figure 3), depends on the parameters of the system; a sufficiently eccentric planet will eject all particles with similar semimajor axes, leaving only distant debris which may be slowly precessing. In this case only the antialigned debris structure would be formed. This also leads to a counter-intuitive result; a planet of comparable mass to the disc will clear a larger region of debris than a much more massive planet. This is because particle orbits antialign with that of a low mass (i.e. rapidly precessing) planet, and are therefore more likely to be ejected than if under the influence of a more massive planet (which their orbits would align with).
Thus far we have only considered the secular evolution of debris which does not intersect the planet's orbit. Particles with orbits crossing that of the planet will eventually be scattered unless in a mean-motion resonance, however secular evolution may still occur before then. Beust et al. (2014) simulated the evolution of debris under the influence of an eccentric planet, when the planet's orbit crosses that of the debris. They showed that the secular interaction still drives up particle eccentricities as before. However their orbits do not preferentially align or antialign with that of the planet, but rather initially orientate themselves such that for much of the time their pericentres are misaligned with the planet's by ∼ 70 • . We observed similar evolution of particles on planet-crossing orbits in our simulations using high mass planets (Pearce & Wyatt 2014), suggesting that this behaviour arises because the orbits intersect. That we are now concerned with comparable planet and disc masses does not make a difference; the particles affected by this mechanism have semimajor axes similar to the planet's, and therefore still precess faster than the latter even if the planet is rapidly precessing. Thus in addition to the long-term secular structures described above, particles crossing the planet's orbit will have their eccentricities driven up and their orbits misaligned with the planet's by ∼ 70 • , before eventually being scattered.
The effect of scattering on debris
Material which regularly crosses the planet's orbit (again, if not in a mean-motion resonance) may be scattered by the planet. A particle's post-scattering orbit may differ significantly from its pre-scattering orbit, but both must pass through the scattering point. Hence a planet which scatters material at all points around its orbit will form an overdensity of debris tracing its orbit, caused by the overlapping orbits of scattered particles. This overdensity is often strongest at planet apocentre, where scattering is most efficient; this is because the planet spends more time around apocentre than at other points in its orbit, and the relative velocity between the planet and (nonscattered) disc particles is smallest here.
Objects repeatedly scattered by the planet will eventually leave the system or collide with other bodies. In the meantime, scattered objects may attain very large semimajor axes and eccentricities, and hence their orbits could extend far beyond the initial outer edge of the disc. Inclinations are typically excited less than eccentricities in repeated scattering encounters (Ida & Makino 1992), so scattered material will form a broad disc superimposed on the secular debris structure discussed in Section 2.1.
The surface density Σ of such a scattered disc will follow an r −3.5 profile. This is an empirical result which is observed in all of our simulations regardless of parameters (as well as those of Duncan, Quinn & Tremaine 1987), but it can also be obtained using the following semi-analytic method. According to the model of Yabushita (1980), particles repeatedly scattered by a planet will diffuse in semimajor axis space, such that the number of particles n with x in the range x → x + dx (where x ≡ 1/a) at time t is given by
n(x, τ ) = 4 xτ exp − 8 τ 1 + x/x0 I2 16 τ 4 x/x0 . (8) Here x0 is the initial value of x, I2 is the modified Bessel function, and τ ≡ t/tD(x0) where tD(x) is the diffusion timescale: tD(x) ≡ 0.01T plt √ a plt x M plt M * −2 ,(9)
where T plt is the orbital period of the perturbing planet. A reasonable approximation is that scattered particles diffuse in x whilst their pericentre distance and inclination remain constant (Duncan, Quinn & Tremaine 1987). Hence we may build a simple model of the system, whereby debris particles initially on circular orbits with semimajor axes equal to a plt diffuse in x, whilst their pericentres remain at a plt . We calculate the distribution of x values at time t, where t tD(x0), using Equation 8. We then create a virtual scattered disc consisting of a large number of particles, with semimajor axes drawn from the above distribution and pericentres equal to a plt . For each orbit we calculate the instantaneous radial distance r of the particle at a randomised mean anomaly, and calculate the surface density profile resulting from the summation of these r values for all particles. Regardless of the parameters used (planet mass, semimajor axis and stellar mass) this profile always goes as r −3.5 . Hence an r −3.5 profile appears to be a natural consequence of scattering, and a population of scattered material could potentially be identified from such a slope.
Planet evolution
Unlike when the planet is much more massive than the disc, if the two are of comparable mass then the planet's orbit may undergo significant evolution. It was noted in Section 2.1 that secular perturbations from the disc cause the planet's orbit to precess, and its orbital plane will also evolve if the planet and disc planes are initially misaligned (although no significant plane evolution will occur if the two are roughly coplanar at the start of the interaction). However the planet's eccentricity may evolve significantly, through planet-particle scattering and secular interactions with the disc. These mechanisms individually affect eccentricity in different ways, so overall the planet's eccentricity behaviour combines two effects. Secular perturbations from the disc will cause the planet's eccentricity to increase and decrease periodically, whilst scattering damps the eccentricity and will circularise the orbit (given enough scattering events). Hence the planet's eccentricity will undergo a long-term decline, with additional oscillatory behaviour in the meantime. If the planet scatters sufficient material before circularisation then there may be too little debris remaining to continue the damping process; in this case, the planet's eccentricity may not tend to zero but to some higher value.
Scattering will also change the planet's semimajor axis. For a single planet scattering debris, the lack of interior planets to remove material scattered inwards means than particles may only leave the system through collisions or ejection. The former mechanism will be rare as the star and planet pose small targets, hence the eventual location of scattered material is likely exterior to its initial orbit. The planet will lose energy to counter this increase in particle energy, hence its semimajor axis will tend to decrease. In section 5.3 of Pearce & Wyatt (2014) we derived a theoretical upper limit on this semimajor axis change, and showed that a planet cannot undergo significant migration if much more massive than the disc except for a contrived set of circumstances. The same arguments still apply even when the planet and disc are of comparable mass; in the context of the parameters in equations 14-16 of Pearce & Wyatt (2014), we require Γ ∼ 1 for significant migration, which is unlikely for broad discs. Hence scattering is unlikely to cause any significant change in planet semimajor axis. Recalling that secular interactions also have no effect on this quantity, we conclude that the semimajor axis of an eccentric planet interacting with a comparable-mass debris disc is unlikely to evolve significantly.
General numerical simulations
We now present n-body simulations of an eccentric planet interacting with a comparable-mass coplanar debris disc, to demonstrate the physical effects described in Sections 2.1 to 2.3. We ran almost 100 n-body simulations using the Mercury 6.2 integrator (Chambers 1999), covering a broad region of parameter space. The general simulation setup is as follows. A planet of mass M plt orbits a star of mass M * , with an initial semimajor axis a plt and eccentricity e plt . The planet's pericentre is typically of order 1-10 au; this is roughly the location of the water snow line for solar-type stars, and hence the region where giant planets may be expected to form. The planets have initial eccentricities ranging from 0.1 to 0.9. We fix M * = 1M ; changing this parameter affects the timescales in the interaction, but not the nature of the evolution.
The star also hosts a debris disc exterior to the planet's pericentre, of mass M disc (where M plt = M disc ), composed of N equal-mass particles. The disc midplane lies in the planet's orbital plane. We consider discs with initial inner and outer radii (r1 and r2 respectively) of the order of 10 − 100 au. The semimajor axes a of disc particles have initial values between r1 and r2, and are distributed such that
n(a) ∝ a 1−γ ,(10)
where γ is the surface density index. These particles are initially on circular orbits, which are randomised in longitude of ascending node and have inclinations up to an opening angle I with respect to the disc midplane. We use a pre-interaction opening angle of I = 5 • , that of the classical Kuiper Belt (Bernstein et al. 2004), and γ = 1.5, that of the Minimum Mass Solar Nebula (Hayashi 1981), in our simulations. We probe disc and planet masses from 0.1 Earth masses (0.1 M⊕) to 3 Jupiter masses. The discs contain N = 10 3 − 10 4 equal-mass debris particles, representing the more massive bodies (i.e. those unaffected by radiation pressure and PR-drag); hence only gravitational forces are included. Each particle exerts a force on the planet and vice-versa, but does not perturb other debris. Thus we ignore the self-gravity of the disc; this is discussed in Section 4.1. Each simulation lasts of the order of 10 − 100 Myr.
Despite the broad range of parameters tested, the interaction always produced the same qualitative results. We found four distinct evolutionary stages occurring on logarithmic timescales, which we describe below. We also present an example simulation; we show the disc surface density at each evolutionary stage on Figure 4, and the planet's eccentricity evolution on Figure 5. The example shows a 10 M⊕ planet interacting with a comparablemass disc, with a plt = 40 au, e plt = 0.6, r1 = 50 au, r2 = 150 au and N = 10 3 .
Stage 1: The planet begins to scatter material from the inner regions of the disc, depleting the debris surface density inwards of planet apocentre. Non-scattered particles with orbits crossing that of the planet have their eccentricity increased via the planet's secular influence; these orbits are preferentially misaligned by ± ∼ 70 • to the planet's orbit, due to the effect described in Beust et al. (2014). Particles beyond the planet's apocentre still have roughly circular orbits, and the similar secular phases of neighbouring particles cause the formation of a spiral-shaped overdensity beyond the planet's orbit (Wyatt 2005). The planet's eccentricity undergoes its most rapid decline due to debris scattering, and secular effects may also cause this eccentricity to oscillate.
Stage 2: All debris initially crossing the planet's orbit has been scattered at least once; an overdensity of scattered material forms along the planet's orbit, and this overdensity is strongest around planet apocentre. A population of scattered material with surface density going as ∼ r −3.5 begins to form beyond the planet's orbit, extending beyond the initial outer edge of the disc. This population is initially small, and hence is not visible on Figure 4 until the final panel. However a logarithmic surface density plot demonstrates that a ∼ r −3.5 population has begun forming by the second stage. Material originating just exterior to the planet's orbit may form a coherently eccentric disc apsidally aligned with the planet, depending on the planet's precession rate and eccentricity (see Section 2.1). The spiral-shaped overdensity continues to develop beyond the planet's orbit. The debris surface density profile hence has two peaks: a broad peak of scattered material stretching between the planet's initial pericentre and apocentre distances, and sharp peak farther out corresponding to the spiral overdensity.
Stage 3: At least one secular period has elapsed for particles initially orbiting just exterior to the planet's apocentre; some initially stable material has been driven onto eccentric orbits apsidally antialigned with the planet, crossed its orbit and been scattered. Hence the surface density of scattered material tracing the planet's orbit is increased, and a large crescent shaped gap forms in the disc in the direction of planet pericentre. The spiral overdensity exterior to the planet continues to move outwards, as more distant particles are still in secular phase with their neighbours. The planet's eccentricity may by now have reduced significantly. Note that the planet's location corresponds to a region of overdensity in the disc, rather than the region of underdensity (as might naïvely be expected).
Stage 4: The planet has scattered all material crossing its orbit, and its eccentricity evolution essentially ceases.
Hence the innermost peak of the surface density profile has been reduced or even removed. An overdensity of scattered material may still exist just exterior to the planet's orbit; this material no longer comes close to the planet since the latter's eccentricity decreased, so this debris is now stable. Particles driven to high eccentricities by secular effects early on may now have their eccentricities frozen, as the forcing eccentricity becomes small owing to the decrease in planet eccentricity. If the planet circularisation timescale is much longer than the secular timescale of the outermost particles then surviving non-scattered debris forms a smooth disc apsidally antialigned with the planet; otherwise, the spiral overdensity may still be present in the outer debris and remain there indefinitely.
Generally, whilst the planet's eccentricity and longitude of pericentre evolve significantly throughout the interaction, its other orbital elements remain roughly constant. In the example simulation a plt changes by less than 5 per cent, and i plt never exceeds 0.6 • (from an initial value of 0 • ).
The qualitative results presented in this section are general. The quantitative results will differ for specific systems, but rough scaling rules can be applied. For example, increasing the mass of the planet and disc simultaneously will decrease the interaction timescales, whilst increasing the planet semi-major axis and disc radii will increase timescales. Increasing the planet eccentricity will decrease the circularisation timescale, and moving the disc mass inwards (either through reducing r1 or increasing γ) makes the planet circularise faster and to a greater degree.
Whilst this paper primarily considers the case where M plt = M disc , the results are applicable to the M plt > M disc regime too. Even if the planet were orders of magnitude more massive than the disc, the general secular behaviour is the same as in the equal mass case. A difference between the two mass regimes is that the transition between aligned and antialigned particles occurs farther from the star if the planet is more massive than the disc; this is because the planet would precess more slowly relative to debris than the equal mass case, so the location where particles precess more slowly than the planet is farther from the star (see Figure 2). This is why we did not observe this secular behaviour in Pearce & Wyatt (2014); our discs simply did not extend far enough outwards to probe this regime. The main qualitative evolutionary difference between the equal mass case and that when the planet is much more massive is that the planet will not undergo the same degree of orbital evolution in the latter regime.
An interesting result may occur if 1 M plt /M disc 10, whereby particles can change between the aligned and antialigned secular regimes. This occurs because the planet initially precesses rapidly (leading to antialignment of some particle orbits), yet the planet is massive enough to eject a significant fraction of the disc particles. The declining disc mass causes the planet precession to slow, meaning that the precession rate of some particles can "overtake" that of the planet. The final result of such an interaction is the formation of a coherently eccentric disc (as in Pearce & Wyatt 2014), but with a more messy structure due to additional particles which have changed their secular behaviour. We do not wish to comment on the case where M plt < M disc , as disc self gravity would be very important in this regime and thus the results of this paper probably do not apply there (see Section 4.1).
Our results may be used to predict the outcome of an eccentric planet interacting with a coplanar debris disc of the same or greater mass. They may also be used to infer the presence of an unseen perturber from the structure of an imaged debris disc, as we will now demonstrate for HD 107146.
APPLICATION TO HD 107146
HD 107146 is a 80-200 Myr old G2V star, located 27.5 pc from the Sun (van Leeuwen 2007; Williams et al. 2004). In 2000, IRAS imaging revealed excess infrared emission in the stellar spectrum, indicative of a debris disc (Silverstone 2000). As noted in Section 1, further observations resolved the disc in both infrared emission and scattered light (Ardila et al. 2004;Williams et al. 2004;Carpenter et al. 2005;Corder et al. 2009;Hughes et al. 2011). For an excellent summary of work on HD 107146 up until 2011, see Ertel et al. (2011).
The recent 1.25mm ALMA image reveals the disc at millimetre wavelengths in unprecedented detail (Ricci et al. 2015). These data show that the disc spans 30−150 au from the star and, assuming it is circular, inclined by 21 • to the sky plane at a position angle (E of N) of 140 • . These observations detected 0.2M⊕ of dust at 1.25mm, and by extrapolating this up to bodies of diameter D = 1000 km (with the number of bodies of a given diameter n(D) ∝ D −3.6 ) the authors inferred a total disc mass of 100M⊕. The ALMA image and corresponding surface brightness profile are shown on the top two plots of Figure 6.
It is clear from these plots that the disc has an unusual morphology. The outer regions are brighter than the inner, and the brightness profile decreases with radius before increasing again farther out. Lower-resolution 880µm SMA observations also show that the surface brightness does not decrease with radius as expected (Hughes et al. 2011). These data have been interpreted as the disc's surface density profile either being doublepeaked or increasing with radius with a gap at 80 au (Ricci et al. 2015), and these models are indistinguishable at the observation resolution. Possible causes of these profile include embedded Pluto-sized objects inducing collisions between large debris bodies, or perturbations from an unseen planetary companion. The ALMA observations failed to detect any CO gas, suggesting that a dust-gas interaction is not responsible for the disc morphology.
We wish to ascertain whether a past (or ongoing) interaction between the disc and a hypothetical eccentric planet can explain the disc features, and if so, estimate the pre-interaction orbit of the planet as well as its present day location. We assume the planet originated interior to the disc, where some event placed it onto an eccentric orbit; this could have been a planet-planet scattering or merger event, for example (Lin & Ida 1997;Ford & Rasio 2008). We aim to reproduce the 1.25mm ALMA observations with an n-body simulation of such an interaction. Millimetre grains are unaffected by radiation pressure and PR-drag, so should act as tracers of the parent debris bodies (those most important for the dynamics of the system). Hence the ALMA data is well-suited to modelling with purely gravitational n-body simulations.
Simulation setup
In addition to the simulations described in Section 2.4, we ran a further ∼ 150 simulations specifically aimed at reproducing the HD 107146 debris disc. We describe their setup now. HD 107146 is a G2V star, so we fix its mass at 1M . We also fix the disc mass to the 100M⊕ value of Ricci et al. (2015). However this still leaves nine physical variables: the planet's mass, its initial semimajor axis, eccentricity, inclination and argument of pericentre, and the disc's initial inner and outer radii, opening angle and surface density profile. Computational limitations prevent us from exploring this whole parameter space, so we make several assumptions about the pre-interaction system to reduce the number of variables.
We again fix the pre-interaction disc opening angle at I = 5 • and γ = 1.5, and again assume the planet initially orbits in the disc midplane. These assumptions leave five physical parameters: M plt , a plt , e plt , r1 and r2. However we can use physical reasoning to fix a further two of these. Firstly, the disc of HD 107146 appears to be roughly axisymmetric. In Pearce & Wyatt (2014) we showed that if M plt M disc , the planet's eccentricity will not be significantly damped by the disc, and external debris will form a coherently eccentric disc aligned with the planet's orbit. Conversely, in Sections 2.3 and 2.4 we showed that if M plt ∼ M disc then the planet's eccentricity will be significantly damped, and hence the outer edge of the disc will remain roughly circular. Thus if HD 107146's disc is interacting with a reasonably eccentric planet (or did so in the past) then M plt ∼ M disc , so we fix the planet mass to be 100M⊕ in our simulations. This means that the outer edge of the disc will be largely unchanged by the interaction, so we fix r2 = 151 au (which best fits the data in the outermost regions). Again, we use N = 10 3 − 10 4 equal-mass debris particles to simulate the disc, and omit disc self-gravity (see Section 4.1).
We are thus left with three free parameters: the initial values of a plt and e plt , and the initial inner disc radius r1. These three are somewhat degenerate, so we cannot fix any at a single value. Instead, we run simulations with a plt = 20, 30, 40 and 50 au (noting that the inner peak of the observed surface density profile is at 50 au, and that the planet's initial semimajor axis is typically interior to this peak in our general simulations), and for each a plt we run simulations with various values of e plt and r1. We disfavour simulations where the planet's initial pericentre is within 3 Hill radii of the disc inner edge, or external to this location; in these cases the disc would be unstable before the interaction started. Once our simulations are complete we compare them to the observations, using the method described below. (Ricci et al. 2015). Note that we use a different colour scale to that in the aforementioned paper. The white ellipse represents the beam size and orientation. Top right: points show the normalised, radially-averaged surface brightness profile of the disc as observed by ALMA, measured using elliptical apertures. The solid line is the profile from our best-fitting n-body simulation, at the time (19 Myr after the start of the interaction) of the best fit; the two agree with a reduced χ 2 value of 0.4. The simulation parameters and the method used to compare the data and simulation are described in Section 3. Bottom left: positions of debris particles in the best-fitting n-body simulation at 19 Myr. The x − y plane is the initial disc midplane, with planet pericentre initially pointing along the x axis. The orbit of each particle has been populated with 100 points with randomised mean anomalies, to increase the effective number of particles plotted. The white point is the star, and the white ellipse the planet's orbit. Bottom right: simulated ALMA image of the n-body disc. The particles have been scaled for emission, the image rotated, and smoothed with a 2D Gaussian representing the ALMA beam (white oval). Compare this to the ALMA observation in the top left, noting that we have not added noise and hence our image is smoother.
Constructing simulated observations
Throughout each simulation we compare the instantaneous distribution of debris to that observed by Ricci et al. (2015). This requires the simulated debris to be converted into an image and surface brightness profile as would be observed by ALMA, for which we use the following method. Firstly, we populate each particle's orbit with 100 points at randomised mean anomalies, to increase the effective number of particles simulated. We then scale for emission by weighting each point by a black body; a point at radial distance r from the star is weighted to have a luminosity L, where
L(r) ∝ Bν (λ, T ).(11)
Here Bν (λ, T ) is the spectral radiance of a body of temperature T at a wavelength λ, given by Planck's law:
Bν (λ, T ) ∝ exp hc λkBT − 1 −1 ,(12)
where h is the Planck constant, c is the speed of light and kB is the Boltzmann constant. The temperature of the body is determined by the flux it receives from the star (again assuming black body behaviour), hence
T = L * 4πσ 1/4 r −1/2(13)
where L * is the star's luminosity and σ is the Stefan-Boltzmann constant. For HD 107146, we use L * = L and λ = 1.25mm, that of the ALMA observations. For these parameters, Bν (λ, T ) roughly scales as r −1/2 . For this analysis we have assumed the disc is optically thin; this is valid here since, whilst the disc is massive, it covers a broad region. Whilst the optical depth could be high if the disc were extremely thin, even a moderate 5 • opening angle would be large enough that we do not have to consider flux attenuation here. To produce images for comparison with the ALMA observations, we rotate the simulated (emission scaled) disc to an inclination of 21 • and a position angle of 143 • . We then convolve our image with a two-dimensional Gaussian to simulate the ALMA point spread function (PSF); this Gaussian has a standard deviation along its major axis of 13.4 au, along its minor axis of 9.8 au, and its major axis has a position angle of 19.8 • . We also calculate the radially averaged surface brightness profile of the simulated disc, using elliptical apertures on the simulated image as in Ricci et al. (2015). We may then compare our simulations to the ALMA observations of HD 107146.
Fitting the HD 107146 disc
We identified the simulations which best replicate the HD 107146 disc using a χ 2 analysis. At 100 time intervals throughout each simulation we calculated the χ 2 value comparing the observed radial surface brightness profile with the simulated profile at this time (found using the method in Section 3.2). On Figure 7 we plot the a plt , e plt , r1 parameter space tested in our simulations, and colour each point by min(χ 2 red ) (the minimum value of reduced χ 2 , that is the minimum value of χ 2 attained during that simulation, divided by the number of degrees of freedom). If the data are independent, a reduced χ 2 of order 1 means that the obs1ervations are consistent with the model, and the smaller the value, the better the fit (although values much smaller than 1 imply the data is overfitted). Here the observed surface brightness profile points are correlated with each other, so little should be inferred from the exact value of min(χ 2 red ) itself; however this value does allow a comparison between simulations, to identify that which best reproduces the HD 107146 disc. Figure 7 shows that there are several regions of tested parameter space which produce discs consistent with ALMA observations. The best fit is attained using a planet with initial semimajor axis a plt = 40 au and eccentricity e plt = 0.4, interacting with a disc with initial inner radius r1 = 50 au. For this case the simulated surface brightness profile is most similar to the ALMA observations 19 Myr after the start of the interaction, and we plot the simulated ALMA image and surface brightness profile at this time on the lower two plots of Figure 6. These well reproduce the observations; the surface brightness profile yields a min(χ 2 red ) value of 0.4, and the simulated image resembles the ALMA observation by eye. Note that globular structures in the observed image are probably noise, which has not been accounted for in the simulated image and hence the latter appears smoother than the observation. This best fit occurs when the simulated system is at stage 3 or 4 in its evolution (as described in Section 2.4); the planet has removed most of the material crossing its orbit, and its orbital evolution has essentially stalled. By this point the planet's eccentricity has decreased from 0.4 to 0.05, whilst its semimajor axis (initially 40 au) has only reduced by 4 au. The simulation first reaches χ 2 red ∼ 1 at 10 Myr, and this parameter remains less than 1 until the end of the simulation (at 30 Myr); hence the simulation also provides a good fit to the observations over a long time interval.
However the best-fitting simulation is not unique in reproducing the observations. A well-defined χ 2 minimum also exists for a planet semimajor axis of 30 au, centred on e plt = 0.55 and r1 = 50 au, and this is almost as good as the best-fitting 40 au solution (min(χ 2 red ) = 0.5). Again, the planet in the a plt = 30 au simulation undergoes minimal semimajor axis evolution whilst its eccentricity is significantly reduced, and the simulation fits best once it has evolved to stage 3 or 4 and resembles the observations for a long time. Conversely, we find that planets with initial semimajor axes of 20 and 50 au do not reproduce the observed disc well. Also note that our well-fitting simulations have the disc's initial inner edge exterior to its present day value, and material has since been scattered inwards by the planet. In conclusion, a planet with an initial semimajor axis of 30 or 40 au and an eccentricity of 0.4-0.5, interacting with a comparablemass debris disc with initial inner edge at 50 au, can well reproduce the disc of HD 107146. At present, the planet is most likely on a roughly circular orbit at 30-40 au.
DISCUSSION
We have examined the general interaction between an eccentric planet and a coplanar, comparable-mass debris disc, and applied our results to HD 107146 in an attempt to explain its unusual disc. In this section we discuss a possible limitation of our work: the omission of disc selfgravity. We also discuss the timescale of the HD 107146 interaction. Finally, we examine the implications of this paper for planet searches, both for general systems with debris discs and also for HD 107146.
Disc self-gravity
Our simulated debris particles exert a force on the planet (and vice-versa), but do not interact with each other; hence we do not include disc self-gravity in our simulations. This omission dramatically increases computational efficiency, allowing us to run several hundred simulations for this paper. However whilst self-gravity does not affect the interaction outcome if the planet is much more massive than the disc (as in Pearce & Wyatt 2014), if the two are of comparable mass then this effect could become important.
Debris in a self-gravitating disc would undergo additional secular and scattering evolution from the influence of other disc particles. Secular interactions work over large distances on timescales scaling inversely with the object masses, whilst scattering works over short distances on timescales going as the inverse-square of the masses (see . Reduced χ 2 of our simulated radially-averaged surface density profiles compared to that of HD 107146, at the time in each simulation when this parameter is minimised. We varied the initial inner disc radius r 1 and the initial planet eccentricity e plt for four initial planet semimajor axes a plt , fixing all other parameters as described in the text. The points show our simulations, with the colourmap and contours interpolated between them. Contours show log 10 [min(χ 2 red )] = 0, 0.5 and 1 respectively. The hatched regions show an unphysical area of parameter space, where the planet's initial pericentre is closer than three Hill radii to the disc inner edge (see Section 3.1).
Therefore given the small debris particle masses, the major effect of self-gravity is likely to be on the secular evolution of the disc.
We investigate the possible secular effect of selfgravity by analytically calculating the precession rate of a test particle embedded in a disc. This gives us a feel for how the initial disc in our best-fitting HD 107146 simulation would evolve due to self-gravity alone (in the absence of any planetary perturbations), and allows us to compare the magnitude of the self-gravity effect to that of the planet.
To calculate the precession rate, we consider a test particle at position (R, φ, z) in cylindrical coordinates, which experiences a force from a 2 dimensional, axisymmetric disc in the z = 0 plane. Equation 2-146 in Binney & Tremaine (1987) gives the radial acceleration of the particle due to the disc as
Fr(R) = − G R 3/2 r 2 r 1 K(k) − 1 4 k 2 1 − k 2 × R R − R R + z 2 R R E(k) kΣ(R ) √ R dR ,(14)
where R is the radial location of a point in the disc,
k 2 ≡ 4RR (R + R ) 2 + z 2 ,(15)
and K(k) and E(k) are the complete elliptical integrals of the first and second kind respectively. We wish to consider a particle in the disc midplane, i.e. z = 0. However Binney & Tremaine (1987) note that Equation 14 has an unphysical singularity at R = R if z = 0, because k = 1 here and so the K(k) and (1 − k 2 ) −1 terms become undefined. This issue can be resolved by setting 0 < z R, so we use z = 10 −4 au in our evaluation. A particle in the midplane experiences no vertical acceleration, and its tangential acceleration is also zero because the disc is axisymmetric. Hence the self-gravity of an axisymmet-ric disc exerts only a radial force on a disc particle. The precession rate of a particle at true anomaly f moving in a Keplerian potential and perturbed by an additional radial force Fr iṡ
ω = − 1 e a(1 − e 2 ) µ Fr cos f(16)
(section 2.9 of Murray & Dermott 1999), and we average this over the orbital period T :
ω ≡ 1 T T 0ω dt ≈ 1 2πa 2 √ 1 − e 2 2π 0ω r 2 df,(17)
where we have assumed that a and e are constant over one orbital period. Finally, if e 1 then Fr will not vary significantly over the particle's orbit. In this case ω ≈ a µ Fr(a),
and similar analyses for semimajor axis and eccentricity yieldȧ ≈ė ≈ 0. We evaluated Equation 14 numerically for a disc with the initial parameters of that in our bestfitting HD 107146 simulation. The force, and the resulting precession rate, are shown as functions of radius by the black lines on Figure 8. The plot shows that the precession rate of particles inwards of 75 au is still dominated by the planet, even when disc self-gravity is considered. In the n-body simulation (without self-gravity), the planet drives up the eccentricities of these particles and scatters the majority of them, with the remainder forming an eccentric disc aligned with the planet's orbit. Hence this would still occur with the inclusion of self-gravity. However the disc's gravity may initially dominate beyond this region; in the n-body simulation, particles beyond 80 au preferentially antialign with the planet's orbit, so debris out to 100 au is removed. Self-gravity would effectively cause these Figure 8. The evolution of a test particle under the influence of a massive axisymmetric disc. Top plot: the radial force imparted by the disc on a coplanar particle at radius R, from Equation 14. The black line shows a 100 M ⊕ disc with an r −1.5 surface density profile, with inner and outer radii of 50 and 150 au respectively. These are the initial disc parameters in the best-fitting HD 107146 simulation. The red line shows the same disc but with all mass inwards of 75 au removed, representing the truncation of the disc by the planet. Bottom plot: the magnitudes of the resulting particle precession rates. The green dotted line shows the precession rate due to the secular influence of a 100 M ⊕ planet with a plt = 40au (Equation 6). The plot shows that the planet dominates particle evolution in the inner regions of the disc, whilst disc self-gravity may be more important in the outer regions.
particles' precession rates to be uncorrelated with the planet's evolution, preventing both preferential antialignment and also significant eccentricity excitation. So were disc self-gravity included, the depletion of the disc in the best-fitting HD 107146 simulation would initially extend out to 75 au rather than 100 au. This would not fit the observational data. However we have not yet considered the evolution of the disc self-gravity. Particles inwards of 75 au would be depleted even with self-gravity, and this would change the disc potential. The red lines on Figure 8 show a disc with the same parameters as discussed above, but with all mass inwards of 75 au removed. Now the planet is still influential out to about 95 au, so much of this debris may still eventually undergo scattering. Beyond this region the disc self-gravity will always dominate, although in the simulation these particles were not significantly perturbed by the planet anyway. Hence the inclusion of self-gravity will not affect the overall simulation results in the outermost regions.
Our analysis suggests that, for our best-fitting HD 107146 simulation, the inclusion of disc self-gravity would not qualitatively affect the resultant disc structure interior to 75 au and exterior to 95 au. In the region between these radii, the potential effect of self-gravity is unclear. This region might not undergo the same level of depletion as in the simulations, and the spiral structure visible in Figure 6 might not be present. Hence our observational fit might not be as good as that on Figure 6. More generally, depending on the simulation parameters, self-gravity may affect our predicted outcomes for the interaction investigated in this paper. The main effect of self-gravity would probably be the reduction of debris depletion in the region immediately interior to the outermost peak of the disc. However we stress that a more sophisticated selfgravity analysis is required to fully explore its potential effect, which is beyond the scope of this paper.
HD 107146 interaction timescale
Our best-fitting HD 107146 simulations all reproduce the observed disc ∼ 10 Myr after the start of the interaction, compared to the 100 Myr age of the star. These two timescales are compatible, but scenarios in which they are comparable would be preferable. The interaction timescales are set by the disc (and hence planet) mass and, since the disc mass derived from observations is uncertain (Ricci et al. 2015), there is scope to change this in our simulations. The secular interactions between the planet and disc are the dominant effects in the simulations, and Equations 4 -7 show the secular precession rate to scale linearly with mass whilst the forcing eccentricity is independent of mass. Equation 14 shows that the effect of disc self-gravity also scales linearly with disc mass, so scaling both M plt and M disc simultaneously will not change the importance of self-gravity relative to planetary perturbations. Hence changing the disc and planet masses should affect the secular interaction timescales, but not the nature of this interaction.
Reducing the masses will make the planet less efficient at ejecting debris, but seeing as the main effects of the interaction are secular in nature, this should not affect the outcome too much. Hence if we reduce the disc and planet masses in our simulations by an order of magnitude (so M disc ∼ 10M⊕), then roughly the same interaction will occur over a timescale comparable with the stellar lifetime. Hence whilst our interaction timescales are by no means incompatible with the system age, if we assume that this interaction is responsible for the observed disc structure then our results might suggest the disc mass is closer to 10M⊕ than 100M⊕. Alternatively the planet may have only recently been placed on an eccentric orbit, and we happen to have observed the system at this stage in its evolution.
Implications for planet searches
Our findings have interesting implications for the inference of unseen planets from debris disc features, both generally and for HD 107146. An important result is that the planets in this interaction generally circularise with little change in semimajor axis, having ejected much of the debris interior to their final orbital distance. Hence the eventual location of the planet is typically near the inner edge of the disc, on an orbit traced by a debris overdensity, beyond which lies a gap followed by another overdense ring. This configuration would not otherwise be expected; having observed a double-peaked debris disc, the naïve assumption would be that any perturbing planet lies in the underdense region between the two peaks.
Hence future planet finding missions should not be discouraged if no planets are found in a debris disc gap; indeed, the absence of planets in this region may hint at a violent dynamical history, and could motivate the search for planets near the inner edge of the disc instead. If our hypothesis on the evolution of the HD 107146 system is correct, then a ∼ 100M⊕ planet currently orbits near the inner edge of the disc, with a semimajor axis of 30 − 40 au and an eccentricity of ∼ 0.1. This planet originated interior to the disc; assuming it was scattered out of its original location by another body then, based on its initial pericentre, a second companion with mass at least equal to that of the scattered planet exists at 10−25 au from the star. Companions of less than 10 Jupiter masses (3000M⊕) have not been ruled out anywhere in the system by imaging (Apai et al. 2008), so this scenario is possible and could be tested with deeper planet searches. Furthermore, our simulations suggest that the outermost debris peak actually forms a thin spiral, rather than a continuous ring. This structure would be detectable in observations with ∼ 3 times the resolution of the ALMA image, and such a detection would favour our hypothesis on the history of the system (although with the caveat that the disc self-gravity would also have an effect, and may partially or completely wash-out this spiral). Such a resolution may well be possible with current instrumentation.
Another potential application of this work is to HD 92945; this system may also harbour a double-peaked debris disc (Golimowski et al. 2011), so our results could be used to invoke a perturbing planet in that system too. However the HD 92945 disc was imaged in scattered light, so the emitting dust would be affected by radiation forces. Hence it is unclear without more detailed analysis whether the more massive debris also follows this doublepeaked profile (as in HD 107146), or whether the observed morphology is a consequence of non-gravitational forces on small dust.
CONCLUSIONS
Broad, double-ringed debris discs could potentially have evolved to their present state under the influence of an eccentric, comparable-mass planet. We investigate this interaction in general, and show that it follows four distinct stages on logarithmic timescales. A key result is that planet precession may cause distant debris orbits to anti-align with that of the planet, whilst the innermost debris orbits align with the planet's. This results in distinct inner and outer debris regions with a gap or depletion between them, akin to the double-peaked debris structures potentially observed in HD 107146 and HD 92945. It also produces the counter-intuitive result that a low-mass planet may clear a larger region of debris than a higher-mass body. In general the planet undergoes a rapid eccentricity decrease whilst its semimajor axis remains constant; thus if the planet initially scattered off another body then the two would quickly decouple, so our results still hold in the presence of additional massive planets (providing the eccentricity damping is fast enough).
We then modelled the HD 107146 system in detail, confirming that the debris disc's unusual morphology can be well explained by this interaction. If an unseen eccentric planet did sculpt debris into the structure seen today, then this hypothetical planet initially had pericentre in the inner regions of the system and apocentre within the disc itself; based on our best-fitting model, the planet is currently on a low-eccentricity orbit 30-40 au from the star. This is below the companion detection thresholds of current observations of the system, but could potentially be found by future imaging projects.
Figure 2
2Figure 2. Precession rate of a test particle as a function of its semimajor axis, derived using second-order secular theory. Here the planet (black circle) has a semimajor axis of 40 au and precesses at a rate˙ plt = 4 × 10 −5 • yr −1 . Particles precessing more rapidly than the planet form an eccentric disc apsidally aligned with the planet's orbit (central plot on Figure 1), whilst those with slower precession rates will antialign with the planet's orbit (right hand plot on Figure 1). Note that this plot does not take account of mean motion resonances, the effect of which could dominate over secular behaviour.
Figure 3
3Figure 3. General shape of a debris disc undergoing a secular interaction with a interior, precessing planet. The grey regions are disc material, the red ellipse denotes the planet's orbit and the asterisk shows the star. There are two distinct debris populations; an inner population apsidally aligned with the planet, and an outer one which is antialigned. The inner population may or may not be present depending on the parameters of the system, and likewise for the outer population.
Figure 4 .Figure 5 .
45Example n-body simulation of an interaction between an eccentric planet and an equal mass, coplanar debris disc, with the simulation parameters described in the text. The left panels show the debris surface density and the planet's orbit (white ellipse), at time zero and then at subsequent evolutionary stages. The right panels show the radially averaged surface density at these times; the thick black lines are the surface density profiles, the dashed lines are the analytic surface density at t = 0, and the points show the planet's semimajor axis and pericentre/apocentre distances. The red line on the Stage 3 and 4 surface density plots shows an r −3.5 profile, typical of scattered debris, and material beyond 150 au follows this profile. The evolutionary stages are common to all our simulations, and are described in Section 2.4. Eccentricity evolution of the planet in the simulation shown onFigure 4, as an example of the general behaviour observed in all our simulations. At early times (up to ∼ 10 Myr) secular eccentricity oscillations are noticeable, on top of the long-term decline from debris scattering. The dotted lines and numbers in boxes refer to the stages of system evolution, for comparison withFigure 4.
Figure 6 .
6ALMA observations of HD 107146, along with our best-fitting simulation. Top left: ALMA 1.25mm continuum image
Figure 7
7Figure 7. Reduced χ 2 of our simulated radially-averaged surface density profiles compared to that of HD 107146, at the time in each simulation when this parameter is minimised. We varied the initial inner disc radius r 1 and the initial planet eccentricity e plt for four initial planet semimajor axes a plt , fixing all other parameters as described in the text. The points show our simulations, with the colourmap and contours interpolated between them. Contours show log 10 [min(χ 2 red )] = 0, 0.5 and 1 respectively. The hatched regions show an unphysical area of parameter space, where the planet's initial pericentre is closer than three Hill radii to the disc inner edge (see Section 3.1).
c 2002 RAS, MNRAS 000, 1-14
ACKNOWLEDGEMENTSWe thank Luca Ricci for allowing us the use of his ALMA image, and Mher Kazandjian for discussions concerning the modelling of disc self-gravity. We also thank Hervé Beust for his very constructive and helpful review. TDP acknowledges the support of an STFC studentship, and MCW is grateful for support from the European Union through ERC grant number 279973.
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| []
|
[
"Block Neural Network Avoids Catastrophic Forgetting When Learning Multiple Task",
"Block Neural Network Avoids Catastrophic Forgetting When Learning Multiple Task"
]
| [
"Guglielmo Montone [email protected] \nLaboratoire Psychologie de la Perception\nUniversité Paris Descartes\n75006ParisFrance\n",
"J Kevin O'regan [email protected] \nLaboratoire Psychologie de la Perception Université Paris Descartes\n75006ParisFrance\n",
"Alexander V Terekhov [email protected] \nLaboratoire Psychologie de la Perception Université Paris Descartes\n75006ParisFrance\n"
]
| [
"Laboratoire Psychologie de la Perception\nUniversité Paris Descartes\n75006ParisFrance",
"Laboratoire Psychologie de la Perception Université Paris Descartes\n75006ParisFrance",
"Laboratoire Psychologie de la Perception Université Paris Descartes\n75006ParisFrance"
]
| []
| In the present work we propose a Deep Feed Forward network architecture which can be trained according to a sequential learning paradigm, where tasks of increasing difficulty are learned sequentially, yet avoiding catastrophic forgetting. The proposed architecture can re-use the features learned on previous tasks in a new task when the old tasks and the new one are related. The architecture needs fewer computational resources (neurons and connections) and less data for learning the new task than a network trained from scratch | null | [
"https://arxiv.org/pdf/1711.10204v1.pdf"
]
| 28,653,334 | 1711.10204 | 6a467722a3a0336f284549d14c181bd272c21a30 |
Block Neural Network Avoids Catastrophic Forgetting When Learning Multiple Task
Guglielmo Montone [email protected]
Laboratoire Psychologie de la Perception
Université Paris Descartes
75006ParisFrance
J Kevin O'regan [email protected]
Laboratoire Psychologie de la Perception Université Paris Descartes
75006ParisFrance
Alexander V Terekhov [email protected]
Laboratoire Psychologie de la Perception Université Paris Descartes
75006ParisFrance
Block Neural Network Avoids Catastrophic Forgetting When Learning Multiple Task
In the present work we propose a Deep Feed Forward network architecture which can be trained according to a sequential learning paradigm, where tasks of increasing difficulty are learned sequentially, yet avoiding catastrophic forgetting. The proposed architecture can re-use the features learned on previous tasks in a new task when the old tasks and the new one are related. The architecture needs fewer computational resources (neurons and connections) and less data for learning the new task than a network trained from scratch
Introduction
Two recently suggested architectures, the block neural network [4,6] and the progressive neural network [5], tested respectively in a supervised learning paradigm and a reinforcement learning paradigm have shown impressive results in multi-task learning. The block neural network is created by training several Deep Feed Forward networks (DNNs) on different tasks. The networks are then connected using new neurons and connections, forming a bigger network that is trained on a new task by allowing just the new added connections to be updated. Block neural networks and progressive neural networks have both been shown to benefit from the advantages of transfer learning. Whereas in the past different forms of pre-training [2,3] and multi-task learning [1] have also achieved this, block neural networks and progressive networks do so without suffering from the disadvantage of catastrophic forgetting of old tasks in the case of pre-training and the necessity of a persistent reservoir of data for the multi-task learning. In this paper, after quickly revisiting the block network architecture, we propose a set of binary classification tasks and show that the block architecture learns more simply (the network needs less computational resources: neurons and connections) and more quickly (the train set can be much smaller) than a network trained from scratch.
Merging DNNs
We defined a set of tasks T 1 , . . . , T M and trained a DNN N 1 , . . . , N M (base models) on each task. After the first training phase, we used some of the trained networks, say N 1 , . . . , N m , to build a block architecture that was then trained on one of the remaining tasks, say T m+ . The block architecture was formed by adding a set of new neurons (block neurons) to the previously trained networks N 1 , . . . , N m . The block neurons were connected to the base models as follows: the first hidden layer of the block neurons received the input for the task T m+1 . The same input was sent to all networks N 1 , . . . , N m . The second hidden layer was fully connected to both the first hidden layer of the block neurons and the first hidden layer of each network N 1 , . . . , N m . This pattern was repeated for all the layers. This architecture was tested with two variations. In the two variations respectively the first and the second layer of the block neurons were removed. When training on the task T m+1 none of the parameters in the base model networks was allowed to change. Figure 1 provides a representation of the block neural network.
The tasks
We used six binary classification tasks, which the networks were trained on. The tasks all involved the concepts of line and angle. We wished to show that the networks N 1 , . . . , N m , when trained on such tasks, would develop features that could be reused by the block architecture to solve another task involving the same concepts. In each task the stimuli were gray scale images, 32 × 32 pixels ang_crs: requires classifying the images into those containing an angle (between 20 • and 160 • ) and a pair of crossing line segments (the crossing point must lie between 20% and 80% along each segment's length).
ang_crs_ln: the same as ang_crs, but has an additional line segment crossing neither of the other line segments.
ang_tri_ln: distinguishes between images containing an angle (between 20 • and 160 • ) and a triangle (with each angle between 20 • and 160 • ); each image also contains a line segment crossing neither angle nor triangle.
blt_srp: requires classifying the images into those having blunt (between 100 • and 160 • ) and those having sharp (between 20 • and 80 • ) angles in them.
blt_srp_ln: the same as blt_srp, but has an additional line segment, crossing neither of the line segments forming the angle. crs_ncrs: distinguishes between a pair of crossing and a pair of non-crossing lines (the crossing point must lay between 20% and 80% of each segment length).
Results
In this section, we first report the results obtained by training a DNN on each of the previously described tasks. Then we report the results of training different block neural networks on the same tasks. The number of possible architectures that can be built by changing the base models, the number of block neurons and the task on which the block network is trained, is very large, and exploring all possibilities was not feasible. A more detailed analysis of the configurations tried can be found in our previous studies [4,6]. Here we summarize the results obtained with two kinds of block network architectures that are particularly interesting because they are obtained by adding a very small number of block neurons. Moreover in this paper we focus on the ability of such architectures to learn using a much smaller dataset. We will in fact present the performance obtained by several block architectures when such architectures are trained on a dataset of almost half the size of the dataset used for training a network from scratch. The performance of the networks was evaluated by computing the percentage of misclassified samples on the test dataset. Each architecture was trained five times, randomly initializing its weights. The mean performance over the five repetitions and the best and worst performance are reported in the tables.
Block Architecture
In figure 3 we present the percentage of block networks outperforming a network trained from scratch as a function of the number of base models present in the block network. Here we focus on two kinds figure 3 were obtained as follow. We built several instantiations of the two kinds of block architectures by using randomly selected base models. Each architecture was then trained on each of the tasks if the task was not used to train any of its base models. For example a block network built using the base model trained on blt_srp and ang_crs was trained on all the other tasks excepts those two. The performance obtained by each block network was compared with the performance obtained with a network (NN-200-100-50) trained from scratch on the same task. The percentage of times the block network obtained a better score was evaluated. The plot in the figure clearly shows an increase in the performance of the block network as the number of base models grows. On the one hand this result was expected simply because the bigger the number of base models, the more parameters are trained; on the other hand it is important to stress that the number of parameters trained on the block network is in any case much smaller than that of a network trained from scratch. The architecture BA-0-50-50 with five base models, for example, has about 60K parameters compared to the 300K of the network 200-100-50. In table 2 and table 3 we show the performance of the block network when the network
Conclusions
The block architecture proves to be a very effective solution for approaching the problem of multi-task learning in DNN. The architecture can be a first step toward the construction of DNN architectures which, in an unsupervised fashion, are able to profit from training on prior tasks when learning a new task.
Figure 1 :
1(a) The architecture is built by adding a block of neurons with three hidden layers to one base model. (b) Adding a block of neurons with two hidden layers to one base model. (c) Adding a block of neurons with one hidden layer to one base model. (d) Adding a block of neurons to two base models. The dashed boxes indicate the layers of the two base models and the block of neurons added. An arrow connecting two boxes indicates that all the neurons in the first box are connected to all the neurons in the second box.
Figure 2 :
2Examples of stimuli: (a) ang_crs; (b) ang_crs_ln; (c) ang_ tri ln; (d) blt_srp; (e) blt_srp_ln; (f) crs ncrs in size. Each image contained two to four line segments, each at least 13 pixels long (30% of the image diagonal). The segments were white on a dark random background or black on a light random background. The 6 tasks were (see examples in figure 2):
Figure 3 :
3Percentage of block architectures outperforming a network trained from scratch as a function of the number of base models present in the block architecture Original Network Prior to building block architectures, we trained a DNN on each task. The networks used were of type NN-200-100-50, with 200, 100, and 50 nodes in the first, second, and third layers, respectively. Networks of this type were used as base models for all of the block networks. The percentages of misclassified test examples for these networks are shown in table 1 together with the results for another architecture, namely NN-60-40-30. Such networks had approximately the same number of parameters (weight of the networks) as some of the block networks, making interesting performance comparisons possible. The networks were trained on datasets with 350, 000 examples.
architecture, namely BA-0-50-50 and BA-0-0-50. The architecture BA-0-50-50 (BA-0-0-50) is obtained by connecting the base models to a DNN with 0(0) units in the first hidden layer, 50(0) units in the second hidden layer, and 50 units in the third hidden layer. The plots in
3 :
3Block network with five base models. Dataset of 200.000 examples Condition BA-0-50-50 (75K params) BA-0-0-50 (25K params) ang_crs (all model used except ang_crs) on a dataset of 200, 000 examples, almost half of the size of the dataset used to train the network NN-200-100-50. The percentages of misclassified test examples for block architectures with four and five base models are presented in the tables. The architectures that performed better than (or equal to) the network NN-200-100-50, which was trained from scratch, are shown in bold. In these tables, the tasks on which the block architectures were trained are listed together with the tasks on which the base models were trained (in parentheses).
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. arXiv:1711.10204v1 [cs.NE] 28 Nov 2017original
original
block
original
block
original #1
original #2
block
a
b
c
d
T 1
T 2
T 2
T 3
Table 1 :
1Original network results
Condition
200-100-50 (300K params) 60-40-20 (65K params)
ang_crs
5.5(5.4-5.9)
9.4(8.9-9.8)
ang_crs_ln
13.6(12.5-15.2)
18.3(16.7-18.8)
ang_tri_ln
6.1(5.5-6.8)
11.4(10.6-14.0)
blt_srp
2.0(1.8-2.3)
3.7(3.4-4.2)
blt_srp_ln
6.5(6.4-6.9)
12.5(11.6-14.1)
crs_ncrs
2.8(2.3-2.9)
4.5(4.1-5.2)
Table
Table
AcknowledgmentsThis work was funded by the ERC proof of concept grant number 692765 "FeelSpeech"
Multitask learning. Rich Caruana, Learning to learn. SpringerRich Caruana. Multitask learning. In Learning to learn, pages 95-133. Springer, 1998.
Why does unsupervised pre-training help deep learning. Dumitru Erhan, Yoshua Bengio, Aaron Courville, Pierre-Antoine Manzagol, Pascal Vincent, Samy Bengio, Journal of Machine Learning Research. 11Dumitru Erhan, Yoshua Bengio, Aaron Courville, Pierre-Antoine Manzagol, Pascal Vincent, and Samy Bengio. Why does unsupervised pre-training help deep learning? Journal of Machine Learning Research, 11(Feb):625-660, 2010.
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The usefulness of past knowledge when learning a new task in deep neural networks. Guglielmo Montone, Alexander V Kevin O'regan, Terekhov, Cognitive Computation Workshop. Guglielmo Montone, J Kevin O'Regan, and Alexander V Terekhov. The usefulness of past knowledge when learning a new task in deep neural networks. Cognitive Computation Workshop, NIPS, 2015.
A Andrei, Rusu, C Neil, Guillaume Rabinowitz, Hubert Desjardins, James Soyer, Koray Kirkpatrick, Kavukcuoglu, arXiv:1606.04671Razvan Pascanu, and Raia Hadsell. Progressive neural networks. arXiv preprintAndrei A Rusu, Neil C Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive neural networks. arXiv preprint arXiv:1606.04671, 2016.
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| []
|
[
"Graphical models in Macaulay2",
"Graphical models in Macaulay2"
]
| [
"Luis David García-Puente \nDepartment of Mathematics and Statistics\nSam Houston State University\n77341HuntsvilleTX\n",
"Sonja Petrović \nDepartment of Statistics\nPennsylvania State University\n16802University ParkPA\n",
"Seth Sullivant \nDepartment of Mathematics\nNorth Carolina State University\n27695RaleighNC\n"
]
| [
"Department of Mathematics and Statistics\nSam Houston State University\n77341HuntsvilleTX",
"Department of Statistics\nPennsylvania State University\n16802University ParkPA",
"Department of Mathematics\nNorth Carolina State University\n27695RaleighNC"
]
| []
| The Macaulay2 package GraphicalModels contains algorithms for the algebraic study of graphical models associated to undirected, directed and mixed graphs, and associated collections of conditional independence statements. Among the algorithms implemented are procedures for computing the vanishing ideal of graphical models, for generating conditional independence ideals of families of independence statements associated to graphs, and for checking for identifiable parameters in Gaussian mixed graph models. These procedures can be used to study fundamental problems about graphical models. | null | [
"https://arxiv.org/pdf/1208.6550v3.pdf"
]
| 88,512,750 | 1208.6550 | 59b22fabf0ee847699278c6e62d223a563e5db70 |
Graphical models in Macaulay2
Aug 2012
Luis David García-Puente
Department of Mathematics and Statistics
Sam Houston State University
77341HuntsvilleTX
Sonja Petrović
Department of Statistics
Pennsylvania State University
16802University ParkPA
Seth Sullivant
Department of Mathematics
North Carolina State University
27695RaleighNC
Graphical models in Macaulay2
Aug 2012
The Macaulay2 package GraphicalModels contains algorithms for the algebraic study of graphical models associated to undirected, directed and mixed graphs, and associated collections of conditional independence statements. Among the algorithms implemented are procedures for computing the vanishing ideal of graphical models, for generating conditional independence ideals of families of independence statements associated to graphs, and for checking for identifiable parameters in Gaussian mixed graph models. These procedures can be used to study fundamental problems about graphical models.
Graphical models
A graphical model is a statistical model associated to a graph, where the nodes of the graph represent random variables and the edges of the graph encode relationships between the random variables. Graphical models are an important class of statistical models used in many applications (see standard textbooks [5,8]) because of their ability to model complex interactions between many random variables, by specifying interactions using only local information about connectivity between the vertices in a graph.
There are two natural ways to specify a graphical model, through either conditional independence statements specified by the graph or via a parametric representation (often called a "factorization"). Every distribution which factors according to the graph satisfies the conditional independence statements implied by the graph. This leads to the question: Which distributions satisfy the conditional independence statements implied by the graph, but do not factor?
Once we specify the types of random variables under consideration (e.g. discrete random variables or Gaussian random variables) it is possible to address the questions in the preceding paragraph using (computational) algebraic geometry. Indeed, in these cases, the set of all probability distributions satisfying a family of conditional independence constraints is a semialgebraic set. For discrete random variables, that semialgebraic set is a subset of the probability simplex, and can be represented by a certain homogeneous ideal generated by quadrics. For Gaussian random variables, this set of distributions corresponds to a semialgebraic subset of the cone of positive definite matrices. Similarly, the parametrized family of probability distributions also is a semialgebraic set (of the probability simplex for discrete random variables, and of the cone of positive definite matrices for Gaussian random variables). This algebraic perspective has been studied by many authors [3,4,6], and the book [1] provides many details.
The Macaulay2 package GraphicalModels allows the user to compute the ideals of conditional independence statements for any collection of statements for discrete or Gaussian random variables. It can also compute the vanishing ideal of a graphical model in these cases. A number of auxiliary functions are useful for doing further analyses of graphical models.
To be explicit, consider the directed acyclic graph G with five vertices {a, b, c, d, e} and edge set {a → d, b → d, c → d, c → e, d → e}. The following commands compute the associated conditional independence ideal for the set of global Markov statements, CI global(G) , and the vanishing ideal I G of the Gaussian graphical model on G. GraphicalModels uses the package Graphs and a number of fundamental constructs and relationships associated with graphs. First we create a polynomial ring for carrying out the computations, which contains the entries of the covariance matrix Σ of a jointly normal random vector as its indeterminates. Information about the underlying graph is stored with the polynomial ring. Hence many functions in this package take only ring as input, but require that it be created with gaussianRing, or markovRing in the discrete case.
For directed acyclic graphs it is known that V (CI global(G) ) ∩ P D m = V (I G ) ∩ P D m , in particular, the set of positive definite matrices satisfying the conditional independence constraints equals the set of covariance matrices in the image of the parametrization. Unfortunately, this does not imply that CI global(G) = I G . In the case of Gaussian random variables, a larger ideal, the trek ideal T G , generated by all subdeterminants of the covariance matrix that vanish on the model, and satisfying CI global(G) ⊆ T G ⊆ I G is sometimes equal to I G (see [7]), which happens for our graph of interest G in this case.
i7 : isSubset(I,J) o7 = true i8 : I == J o8 = false i9 : J == trekIdeal(R,G) o9 = true Similar computations can also be performed for graphical models with discrete random variables, and with other graph families. The mathematical explanation of these graphical models and their associated ideals appear in the remaining sections.
Computing conditional independence ideals
Conditional independence constraints on discrete or Gaussian random variables translate to rank conditions on certain matrices associated to the probability densities. We briefly explain these constructions here and how to generate these constraints in Macaulay2 using GraphicalModels. See [1, Ch. 3] for more detail.
Let X = (X 1 , . . . , X n ) be a discrete random vector where each random variable X i has state space
[d i ] = {1, 2, . . . , d i }. Let d = (d 1 , . . . , d n ). A probability distribution for X is a tensor in R d1 ⊗ · · · ⊗ R dn ,
all of whose coordinates are nonnegative and sum to one. The set of all such distributions is the probability simplex ∆ d . Let p i1···in = P(X 1 = i 1 , . . . , X n = i n ) denote the probability of a primitive event. The polynomial ring in these quantities is created using the command markovRing.
P(X A = i A , X B = i B |X C = i C ) = P(X A = i A |X C = i C ) · P(X B = i B |X C = i C )
for all i A , i B , i C . This translates into vanishing 2×2 minors of certain matrices in the probabilities p i1···in . Those matrices are computed with the function markovMatrices, and the ideal generated by the 2 × 2 minors is computed with conditionalIndependenceIdeal. In the following example, the two conditional independence statements are X 1 ⊥ ⊥ X 2 |X 3 and X 1 ⊥ ⊥ X 3 (:= X 1 ⊥ ⊥ X 3 |X ∅ ). The ideal of vanishing minors has 7 quadratic generators. | p_(2,1,1) p_(2,2,1) p_(2,3,1) | | p_(2,1,2) p_(2,2,2) p_(2,3,2) | | p_(1,1,1)+p_(1,2,1)+p_(1,3,1) p_(1,1,2)+p_(1,2,2)+p_(1,3,2) |} | p_(2,1,1)+p_(2,2,1)+p_(2,3,1) p_(2,1,2)+p_(2,2,2)+p_(2,3,2) | i14 : I = conditionalIndependenceIdeal(R,s); flatten degrees I o14 : Ideal of R o15 = {2, 2, 2, 2, 2, 2, 2}
There are many different sets of conditional independence statements associated to a graph G, whose nodes correspond to random variables. For example, the local Markov statements of an undirected graph G is the set of conditional independence statements of the form X i ⊥ ⊥ X V \{i∪N (i)} |X N (i) , where N (i) is the set of neighbors of i in the graph G. The functions pairMarkov, localMarkov, and globalMarkov compute the pairwise, local, and global Markov statements, respectively, for both directed and undirected graphs. For example, the first conditional independence statement produced is X 1 ⊥ ⊥ (X 3 , X 4 )|(X 2 , X 5 ). In the context of conditional independence, the graphical model consists of all distributions satisfying one of these collections of independence statements associated to the graph G.
A Gaussian random vector, X = (X 1 , . . . , X n ) ∼ N (µ, Σ), is an n-dimensional random vector with state space R n and density function
f (x) = 1 (2π) n/2 (det Σ) 1/2 exp − 1 2 (x − µ) T Σ −1 (x − µ) ,
where µ ∈ R n and Σ = (σ s,t ) ∈ P D n , the cone of n×n symmetric positive definite matrices. The Gaussian random vector X satisfies the conditional independence statement X A ⊥ ⊥ X B |X C if and only if the submatrix Σ A∪C,B∪C := (σ s,t ) s∈A∪C,t∈B∪C has rank ≤ #C. Hence the set of all Gaussian random vectors satisfying a given collection of conditional independence statements naturally corresponds to a subset of P D n , and are studied algebraically by investigating the corresponding determinantal ideals in the polynomial ring in the σ s,t indeterminates. The corresponding ring is generated using the command gaussianRing. Computations involving conditional independence ideals with Gaussian random variables were exemplified in Section 1.
Computing the vanishing ideal of a model
The fact that graphical models can be described in two possible ways (either by a recursive factorization of probability distributions or by conditional independence statements) corresponds to the algebraic principle that varieties can be presented either parametrically or implicitly. The vanishing ideal of a model is the set of homogeneous polynomial relations in the joint probability distributions (for discrete random variables) or in the variance-covariance parameters (for Gaussian random variables). GraphicalModels has the capability of computing the vanishing ideals of graphical models on directed graphs (for discrete random variables) and also of graphical models on directed, undirected, or mixed graphs (for Gaussian random variables). The vanishing ideal of an undirected graphical model for discrete random variables is a toric ideal and should be computed using the Macaulay2 package FourTiTwo.
The method discreteVanishingIdeal implements this capability for graphical models on discrete random variables. For a directed acyclic graph G on discrete random variables, the graphical model consists of all distributions satisfying the recursive factorization property
p(X = i) = v P(X v = i v |X pa(v) = i pa(v) ),
where the product runs over all vertices v of G and pa(v) is the set of parents of v. Our implementation of this method does not compute the kernel of the corresponding ring map. Instead, the vanishing ideal is computed recursively using the factorization P(X = i) = P(X 1 = i 1 , . . . , X n−1 = i n−1 ) · P(X n = i n |X pa(n) = i pa(n) ), where 1, . . . , n is a topological ordering of the vertices of the directed acyclic graph G.
The following example computes the vanishing ideal of the graphical model 1 → 2 → 3 → 4 on four binary random variables. The vanishing ideal is minimally generated by 20 quadratic binomials. According to [3], the vanishing ideal of a graphical model on discrete random variables is the distinguished component of the conditional independence ideal described by the Markov statements of the model. For the directed path in our previous example, the conditional independence ideal of the local Markov statements is a radical ideal with 3 associated primes. However, since G is a directed tree, the conditional independence ideal of the global Markov statements is a prime ideal and it equals the vanishing ideal of G.
i22 : J = conditionalIndependenceIdeal (R, localMarkov G); o22 : Ideal of R i23 : I == J o23 = false i24 : K = conditionalIndependenceIdeal (R, globalMarkov G); o24 : Ideal of R i25 : I == K o25 = true
The method gaussianVanishingIdeal computes the vanishing ideal of a Gaussian graphical model on a graph, digraph, or mixed graph. If G is a mixed graph, the vanishing ideal I G corresponds to kernel of the polynomial parametrization of the variance-covariance parameters σ ij in terms of the direct causal effect parameters λ ij associated to the directed edges of G, and the noise parameters ψ ij associated to the bidirected edges of G. This parametrization is given by
Σ = (I − Λ) −T Ψ(I − Λ) −1 ,
where Σ is the variance-covariance matrix, Λ is the strictly upper triangular matrix with Λ ij = λ ij if i → j is a directed edge in G and 0 otherwise, and Ψ is a positive definite matrix of parameters ψ ij with zeros in each entry Ψ ij if there is no bidirected edge in G between i and j, and i = j.
The following example computes the vanishing ideal of the Gaussian graphical model on the mixed graph with directed edges {1 → 2, 1 → 3, 2 → 3, 3 → 4} and bidirected edges {1 ↔ 2, 2 ↔ 4}. This ideal is a principal ideal generated by one quartic polynomial with 8 terms. This ideal is not determinantal, i.e., it is not generated by the determinantal equations defining the trek ideal, which in this case is the zero ideal.
i26 : G = mixedGraph (digraph {{1,{2,3}},{2,{3}},{3,{4}}},bigraph {{1,2},{2,4}}); i27 : R = gaussianRing G; i28 : I = gaussianVanishingIdeal R; o28 : Ideal of R i29 : J = trekIdeal (R,G) o29 = 0
An important problem in these models consists in finding which parameters are identifiable or generically identifiable, see [2]. The method identifyParameters can be used to solve the identifiability problem for Gaussian graphical models on mixed graphs (also known as structural equation models). The following example shows that the parameter ψ 24 is generically identifiable by the formula ψ 24 = (σ 13 σ 24 − σ 14 σ 23 )/σ 13 .
i30 : H = identifyParameters R; i31 : H#(p_(2,4))_0 o31 = p s + s s -s s 2,4 1, 3 1,4 2,3 1,3 2,4
In this model there are three non-generically identified parameters. identifyParameters produces a hash table whose entries are indexed by the parameters and contain ideals that can be used to find explicit rational functions for every parameter that is generically identifiable.
s_(a,a) s_(a,b) s_(a,c) s_(a,d) s_(a,e) s_(b,b) s_(b,c) s_(b,d) s_(b,e) s_(c,c) s_(c,d) s_(c,e) s_(d,d) s_(d,e) s_(e,
i10
i16 : G = graph{{1,2},{2,3},{3,4},{4,5},{1,5}}; i17 : localMarkov G o17 = {{{1}, {3, 4}, {5, 2}}, {{1, 2}, {4}, {5, 3}}, {{1, 5}, {3}, {4, 2}}, {{2, 3}, {5}, {4, 1}}, {{2}, {4, 5}, {1, 3}}}
For A ⊆ [n], let X A = (x a ) A be the subvector indexed by A. Let A, B, C be disjoint subsets of [n]. The conditional independence statement X A ⊥ ⊥ X B |X C holds if and only if the conditional distribution satisfies: d = (2,3,2); R = markovRing d
o10 : PolynomialRing
i11 : gens R
o11 = {p
, p
, p
, p
, p
, p
, p
, p
, p
,
1,1,1
1,1,2
1,2,1
1,2,2
1,3,1
1,3,2
2,1,1
2,1,2
2,2,1
p
, p
, p
}
2,2,2
2,3,1
2,3,2
AcknowledgmentsThe following people generously contributed their time to the development of the package: Alexander Diaz, Shaowei Lin, David Murrugarra, and Mike Stillman. Work on the package was carried out during the 2010 and 2011 Macaulay2 workshops, which were partially supported by the US National Science Foundation and the Institute for Mathematics and Its Applications.LGP was partially supported by a 2012 SHSU Faculty Research Grant (29001). SP was partially supported by Grant #FA9550-12-1-0392 from the U.S. Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA). SS was partially supported by the US National Science Foundation (DMS 0954865) and the David and Lucille Packard Foundation.
. M Drton, B Sturmfels, S Sullivant, Lectures on Algebraic Statistics. 39Birkhäuser VerlagM. Drton, B. Sturmfels and S. Sullivant, Lectures on Algebraic Statistics, Oberwolfach Seminars, 39. Birkhäuser Verlag, Basel, 2009.
Identifying causal effects with computer algebra. L D Garcia-Puente, S Spielvogel, S Sullivant, Proceedings of the 26th Conference of Uncertainty in Artificial Intelligence. the 26th Conference of Uncertainty in Artificial IntelligenceL. D. Garcia-Puente, S. Spielvogel and S. Sullivant: Identifying causal effects with computer algebra, Proceedings of the 26th Conference of Uncertainty in Artificial Intelligence.
Algebraic geometry of Bayesian networks. L D Garcia, M Stillman, B Sturmfels, J. Symbolic Comput. 39L. D. Garcia, M. Stillman and B. Sturmfels: Algebraic geometry of Bayesian networks, J. Symbolic Comput. 39 (2005) 331-355.
On the toric algebra of graphical models. D Geiger, C Meek, B Sturmfels, Ann. Statist. 343D. Geiger, C. Meek and B. Sturmfels: On the toric algebra of graphical models, Ann. Statist. 34 (2006), no. 3, 1463-1492.
Graphical models. S Lauritzen, Oxford Statistical Science Series. 17Oxford University PressS. Lauritzen, Graphical models, Oxford Statistical Science Series, 17. Oxford University Press, New York, 1996.
Algebraic geometry of Gaussian Bayesian networks. S Sullivant, Adv. in Appl. Math. 40S. Sullivant: Algebraic geometry of Gaussian Bayesian networks, Adv. in Appl. Math. 40 (2008) 482-513.
Trek separation for Gaussian graphical models. S Sullivant, K Talaska, J Draisma, Ann. Statist. 38S. Sullivant, K. Talaska, and J. Draisma: Trek separation for Gaussian graphical models, Ann. Statist. 38 (2010) 1665-1685.
. Whitaker, Graphical Models in Applied Multivariate Statistics. Wiley Series in Probability and Mathematical Statistics. John Wiley & SonsWhitaker. Graphical Models in Applied Multivariate Statistics. Wiley Series in Probability and Math- ematical Statistics. John Wiley & Sons, Ltd., Chichester, 1990.
| []
|
[
"Ambipolar High Mobility Hexagonal Transistors on Hydrogen-Terminated Silicon (111) Surfaces",
"Ambipolar High Mobility Hexagonal Transistors on Hydrogen-Terminated Silicon (111) Surfaces"
]
| [
"Binhui Hu ",
"Mohamad M Yazdanpanah ",
"Joyce E Coppock ",
"B E Kane ",
"\nJoint Quantum Institute\nLaboratory for Physical Sciences\nUniversity of Maryland\n20740College Park, Maryland\n",
"\nUniversity of Maryland\n20742College Park, Maryland\n"
]
| [
"Joint Quantum Institute\nLaboratory for Physical Sciences\nUniversity of Maryland\n20740College Park, Maryland",
"University of Maryland\n20742College Park, Maryland"
]
| []
| We have fabricated ambipolar transistors on chemically prepared hydrogen-terminated Si(111) surfaces, in which a two-dimensional electron system (2DES) or a two-dimensional hole system (2DHS) can be populated in the same conduction channel by changing the gate voltage of a global gate applied through a vacuum gap. Depending on the gate bias, ion implanted n + and p + regions function either as Ohmic contacts or as in-plane gates, which laterally confine the carriers induced by the global gate. On one device, electron and hole densities of up to 7.8×10 11 cm −2 and 7.6×10 11 cm −2 respectively are obtained. The peak electron mobility is 1.76 × 10 5 cm 2 /Vs, and the peak hole mobility is 9.1 × 10 3 cm 2 /Vs at 300 mK; the ratio of about 20 is mainly due to the very different valley degeneracies (6:1) of electrons and holes on the Si(111) surface. On another device, the peak electron mobility of 2.2 × 10 5 cm 2 /Vs is reached at 300 mK. These devices are hexagonal in order to investigate the underlying symmetry of the 2DESs, which have a sixfold valley degeneracy at zero magnetic field. Three magnetoresistance measurements with threefold rotational symmetry are used to determine the symmetry of the 2DESs at different magnetic field. At filling factor 1 < ν < 2, the observed anisotropy can be explained by a single valley pair occupancy of composite fermions (CFs). Qualitatively the CFs preserve the valley anisotropy, in addition to the twofold valley degeneracy. At magnetic field up to 35 T, the 2/3 fractional quantum Hall state is observed with a well developed hall plateau; at ν < 2/3, the three magnetoresistances show a large anisotropy (50:1). We also show that device degradation is not a serious issue for our measurements, if the device is kept in vacuum or a nitrogen gas environment and its time in air is minimized. | null | [
"https://arxiv.org/pdf/1509.03849v1.pdf"
]
| 119,304,093 | 1509.03849 | 4ee889fd96f2d2d9840a877ad96f19eb7d8957b5 |
Ambipolar High Mobility Hexagonal Transistors on Hydrogen-Terminated Silicon (111) Surfaces
13 Sep 2015
Binhui Hu
Mohamad M Yazdanpanah
Joyce E Coppock
B E Kane
Joint Quantum Institute
Laboratory for Physical Sciences
University of Maryland
20740College Park, Maryland
University of Maryland
20742College Park, Maryland
Ambipolar High Mobility Hexagonal Transistors on Hydrogen-Terminated Silicon (111) Surfaces
13 Sep 2015(Dated: September 15, 2015)
We have fabricated ambipolar transistors on chemically prepared hydrogen-terminated Si(111) surfaces, in which a two-dimensional electron system (2DES) or a two-dimensional hole system (2DHS) can be populated in the same conduction channel by changing the gate voltage of a global gate applied through a vacuum gap. Depending on the gate bias, ion implanted n + and p + regions function either as Ohmic contacts or as in-plane gates, which laterally confine the carriers induced by the global gate. On one device, electron and hole densities of up to 7.8×10 11 cm −2 and 7.6×10 11 cm −2 respectively are obtained. The peak electron mobility is 1.76 × 10 5 cm 2 /Vs, and the peak hole mobility is 9.1 × 10 3 cm 2 /Vs at 300 mK; the ratio of about 20 is mainly due to the very different valley degeneracies (6:1) of electrons and holes on the Si(111) surface. On another device, the peak electron mobility of 2.2 × 10 5 cm 2 /Vs is reached at 300 mK. These devices are hexagonal in order to investigate the underlying symmetry of the 2DESs, which have a sixfold valley degeneracy at zero magnetic field. Three magnetoresistance measurements with threefold rotational symmetry are used to determine the symmetry of the 2DESs at different magnetic field. At filling factor 1 < ν < 2, the observed anisotropy can be explained by a single valley pair occupancy of composite fermions (CFs). Qualitatively the CFs preserve the valley anisotropy, in addition to the twofold valley degeneracy. At magnetic field up to 35 T, the 2/3 fractional quantum Hall state is observed with a well developed hall plateau; at ν < 2/3, the three magnetoresistances show a large anisotropy (50:1). We also show that device degradation is not a serious issue for our measurements, if the device is kept in vacuum or a nitrogen gas environment and its time in air is minimized.
I. INTRODUCTION
Investigations of two-dimensional systems (2DSs) on Si surfaces began with metal-oxide-semiconductor fieldeffect transistors (MOSFETs) [1][2][3][4][5]. However, the amorphous SiO 2 /Si interface severely limits the quality of the 2DSs. The development [6][7][8][9][10] of hydrogen-terminated silicon (111) [H-Si(111)] vacuum field effect transistors (FETs) is based on a simple but extraordinary fact: a wet chemical treatment of a Si(111) surface with an ammonium fluoride (NH 4 F) solution can produce an ideal H-passivated Si surface, which is both atomically flat and possesses a very low surface state density (≤ 10 10 cm −2 ) [11][12][13][14], leading to much higher carrier mobilities. In recent years, both a high quality two-dimensional electron system (2DES) with electron mobility of 325,000 cm 2 /Vs [10] and a high quality two-dimensional hole system (2DHS) with hole mobility of 10,000 cm 2 /Vs [9] have been realized in this system. They are comparable to the best Si/SiGe heterostructures [15][16][17][18][19], which are limited to the (100) oriented surfaces because of much higher threading dislocation densities on the other surface orientations [20,21].
In this work we have fabricated ambipolar hexagonal devices, which can switch between a 2DES and a 2DHS in the same device by changing a gate voltage. Compared to other ambipolar devices [22][23][24][25][26][27][28], the 2DES on the Si(111) surface has a sixfold valley degeneracy, while the valley degeneracy of holes is one. The ambipolar devices provide a direct comparison between electrons and holes with very different valley degeneracies in the same conduction channel, which allows a new perspective to the fundamental research, such as the properties of 2D metals [29]. In addition, the hexagonal devices enable the exploration of the sixfold valley-degenerate electron system.
In the H-Si(111) vacuum FET device, a global Si/SiO 2 gate piece is used to induce a 2DS, which is further confined into a well-defined 2D region by PN junction isolation and trench isolation. Similarly, future nanoelectronic devices, like quantum point contacts (QPCs) and quantum dots (QDs), may be realized on the Si(111) surfaces using the in-plane depleting gates based on PN junctions. Moreover, because of the sixfold valley degeneracy of the 2DES, the valley degree of freedom can provide extra resources for practical applications in the same way as spintronic devices utilize the spin degree of freedom. This will open up new opportunities for another class of devices -valleytronic devices [30][31][32].
The remainder of this paper is organized as follows. In Sec. II, we describe the device structure and operation principles, followed by detailed fabrication processes. In Sec. III, we characterize the devices to make sure that they work as intended, including surface topography measurement, a gate leakage check, PN junction isolation and leakage tests, and contact resistance measurements. In Sec. IV, we discuss the transport measurement results on these devices. We determined the 2D carrier mobilities in the experimentally accessible range of densities, which show that electron and hole mobilities are very different. Magnetotransport measurements show rich phenomena, including Shubnikov-de Haas (SdH) oscillations, integer quantum Hall effect (IQHE), fractional quantum Hall effects (FQHE) and large anisotropy in three magnetoresistance measurements at ν < 2/3. In Sec. V, we discuss the device degradation with time and with exposure to ambient air, including the mobility deterioration and the increase of contact resistance. We conclude in Sec. VI with a discussion of further improvements and future directions.
II. DEVICE STRUCTURE AND FABRICATION
The vacuum FET device is fabricated from two pieces. One is a SiO 2 /Si(100) remote gate piece, and the other is a H-terminated high purity Si(111) piece, as shown in Fig. 1(a). Just like a MOSFET, when a positive (negative) gate voltage is applied between the remote gate and a n + (p + ) source Ohmic contact through an encapsulated vacuum cavity, a 2DES (2DHS) will be induced on the H-Si(111) surface. Since the 2DES has a sixfold valley degeneracy [ Fig. 1(b)], the hexagonal device is designed to investigate its underlying symmetry, with six n + Ohmic contacts placed along the major axes of the constant energy ellipses on the Si(111) surface, labeled as 2,4,. . . , 12. It also has six p + Ohmic contacts placed between the n + contacts, labeled as 1,3,. . .,11. The remote gate piece with a hexagonal cavity [ Fig. 2 (b)] is contact bonded on the H-Si(111) piece in a vacuum chamber. When a positive (negative) gate voltage is applied, the n + (p + ) Ohmic contacts are used to access the 2DES (2DHS) at the center, and the p + (n + ) contacts act as lateral confinement. Figure 1 (b)-(d) show the arrangement of the n + and p + Ohmic contacts relative to crystallographic directions.
Two different types of devices are investigated. One has shallow trench isolation (STI) to confine the center 2DS [the center blue hexagon in Fig. 1(c)]; the other doesn't have the STI, and the center 2DS is defined by the hexagonal cavity [ Fig. 1(d)]. There is typically a ∼ 100 µm misalignment between the remote gate piece and the H-Si(111) piece when they are contact bonded. Because the distance between the trenches and the edges of the hexagonal cavity is more than 200 µm, and the trenches are etched on the Si(111) piece, the misalignment doesn't change the center 2DS in the device with the STI; the misalignment can affect the center 2DS in the device without the STI when the n + and p + contacts are not concentric and aligned to the hexagonal cavity. However, the measurement results show that there is no significant difference between these two types of devices, because the relative placement of the contacts is fixed, and the bonding alignment is not critical in defining the device with the edges of the cavity far outside the contact perimeters. We have fabricated six devices (194)(195)(196)(197)(198)(199) with peak electron mobilities of 2.2 × 10 5 cm 2 /Vs, 5.6 × 10 4 cm 2 /Vs, 2.0 × 10 5 cm 2 /Vs, 5.2 × 10 4 cm 2 /Vs, 1.76 × 10 5 cm 2 /Vs and 5.8 × 10 4 cm 2 /Vs at 300 mK respectively, in which four (194)(195)(196)(197) of them have the STI, and the other two (198,199) do not have it. Most data presented here are from two devices, sample 194 (with the STI) and sample 198 (without the STI).
A. Gate piece
The remote gate piece is fabricated from a Si(100) wafer (float zone, Si:B, resistivity > 10, 000Ω·cm). First, a p + gate conducting layer is formed by ion implantation with 2.4×10 15 cm −2 , 15 keV boron ions, as shown in Fig. 1(a). Then a 2 µm deep Si mesa is dry etched around the edges of each die using reactive ion etching (RIE). Next, a ∼ 350 nm thick SiO 2 layer is thermally grown on the top: dry oxidation at 1050 • C for 15 min, followed by wet oxidation at 1050 • C for 27 min and another dry oxidation at 1050 • C for 10 min. The dry-wet-dry cycle is used to improve the gate oxide quality, and also activates the implanted dopants. After that, a cavity with a ∼ 30 nm thick oxide layer left behind is formed by dry etching and wet etching. In order to make sure the oxide thickness is reduced to ∼ 30 nm in the cavity, a test wafer and the device wafer are loaded into a RIE machine and dry etched at the same time. Both have the same ∼ 350 nm thick oxide layer, and the oxide layer is dry etched by ∼ 270 nm. Then the depth of the cavity in the test wafer is measured using a profilometer after removing photoresist. The time of the wet etching to leave the ∼ 30 nm thick oxide layer is calculated based on the oxide thickness, the depth of the cavity and the wet etching rate. The fully SiO 2 encapsulated gate piece with the oxide reduced cavity drastically reduces the likelihood of gate leakage, compared to the gate piece where the doped regions were exposed. We have tested remote gate pieces with either ∼ 350 nm or ∼ 200 nm thick thermal oxide, and both of them work well. Finally, the wafer is diced into 5.6 × 10 mm 2 remote gate pieces, as shown in Fig. 2 (b).
B. Si(111) piece
The H-Si(111) piece is also 5.6 × 10 mm 2 made from a p − high purity Si(111) wafer (float zone, resistivity > 10, 000Ω·cm ). First, a 30 nm thick sacrificial SiO 2 layer is thermally grown on the top of the Si(111) substrate by dry oxidation at 950 • C for 30 min in a CMOS grade furnace to reduce the channeling effect during ion implantation and protect the clean Si(111) surface. Next, the six n + contact regions are patterned by photolithography with the primary flat of the Si(111) wafer (oriented to [112] [33]) aligned (< 5 • ) along the long side of the dies. Then the wafer is ion implanted with 4.5 × 10 14 cm −2 phosphorus ions at 50 keV. Alignment marks are formed at the same time. Similarly, the six p + contact regions are defined by photolithography and ion implantation with 9 × 10 14 cm −2 boron ions at an energy of 15 keV. The ion implantation parameters are based on previous works [6,9] and simulation results from SUPREM-IV process simulator [34]. The parameters are selected so that after thermal annealing, the peak doping concentration is located near the Si(111) surface, and about one order of magnitude higher than that of the three-dimensional (3D) metal-insulate transition (MIT), as shown in Fig. 9. (The 3D MIT occurs at doping concentration of 3.74 × 10 18 cm −3 for phosphorus [35] and 3.95×10 18 cm −3 for boron [36].) After each ion implantation, the photoresist is removed by acetone cleaning and piranha cleaning. First the wafer is immersed in boiling acetone for 30 min to remove most of the photoresist. Then the wafer is rinsed with isopropyl alcohol (IPA) and de-ionized (DI) water. Next, the wafer is immersed in a piranha solution (4 H 2 SO 4 (98%) : 1 H 2 0 2 (30%)) at 100 • C for 30 min to remove the residual photoresist. Right before rapid thermal annealing (RTA), the wafer is RCA-1 cleaned for 5 min with a H 2 O 2 /NH 4 OH/H 2 O solution, then thoroughly rinsed in DI water, and spindried. The wafer is annealed in a CMOS grade rapid thermal annealer at 950 • C for 60 sec. Alignment marks from ion implantations are visible before the RTA process, but invisible after it. RCA-1 clean etches the ion implanted SiO 2 regions before the RTA process. They serve as alignment marks to do mesa etching, but the contrast is very low. It is helpful to define global alignment marks before the RTA process using CMOS compatible processes. Next, if STI is desired, the 200 nm deep and 10 µm wide shallow trenches are formed by dry etching. After that, a 2 µm deep Si mesa is dry etched around the edges of each die. This mesa prevents particles from accumulating on the surface while handling the substrate and ensures a clean edge for bonding [6]. It also defines each die for dicing. Finally, the wafer is diced into individual Si(111) pieces, as shown in Fig. 2(a).
C. Bonding and wiring
After standard cleaning (acetone cleaning, IPA rinse, DI water rinse, piranha cleaning, and DI water rinse) in clean room, one Si(111) piece and one Si/SiO 2 remote gate piece are transferred into an oxygen-free glove box with a typical O 2 concentration of 2 ppm. The Si(111) piece is put in a diluted solution of HF/H 2 O (1:20) for 2 min to remove the sacrificial SiO 2 layer, and then immersed in a 40% NH 4 F solution for 10 min to create an atomically flat H-passivated Si(111) surface, with DI water rinse after each step. It is crucial that each of these solutions is oxygen-free to ensure the high quality of the final device [37]. Usually they are deoxygenated with a constant agitation from a magnetic stir rod for 72 hours in the glove box. Finally, these two pieces are transferred into a vacuum chamber, and pushed against a sapphire boss to apply a pressure between them, sufficient to initiate bonding through van der Waals forces .
The bonded device is loosely placed in a customized holder machined from polyimide material, which is designed to facilitate handling of the device without applying stress. It is wired in a nitrogen glove box by hand soldering using indium at 320 • C and installed on a standard 8-pin dual in-line package (DIP) header, which is compatible with sample holders at the National High Magnetic Field Laboratory (NHMFL). If only the 2DES is to be investigated, eight contacts are sufficient, including two gate contacts and six Ohmic contacts to the 2DES. There are 12 wires in our cryostat. In order to measure both the 2DES and the 2DHS, the 8 pin DIP header is inserted into another 20 pin DIP header, and four Ohmic contacts to the 2DHS are directly wired to the 20 pin DIP header. Two p + Ohmic contacts 3, 9 are not wired. The wired sample is transferred from the nitrogen glove box to the cryostat, and the cryostat is pumped to vacuum (p < 10 −3 mbar) in less than one hour. Figure 2(d) shows one of the final devices without wiring. Similar fabrication processes have been discussed elsewhere in detail [38,39].
III. DEVICE CHARACTERIZATION
A. Topography
Because the Si(111) piece and the Si/SiO 2 remote gate piece are contact bonded through van der Waals forces, successful bonding requires that the surfaces are clean, flat and smooth. Atomic force microscopy (AFM) is used to investigate the topography of the Si(111) surface, particularly the differences in height and flatness of doped and undoped areas. Figure 3 shows that the Si(111) surface is clean with atomic steps. Typically the doped contacts on the Si(111) surface are at different ele- when Vg is swept between -50 V (p 2d = 7.6 × 10 11 cm −2 ) and 50 V (n 2d = 7.8 × 10 11 cm −2 ) at a sweep rate of 0.2 V/s with all wired n + , p + contacts grounded. Arrows indicate the voltage sweep direction. Current peaks at Vg ∼ -12 V and -3 V are due to the formation or depletion of 2DSs. Ig ∼ 0.5 nA is from the gate capacitance.
vations from the undoped Si(111) surface after wet chemical cleaning and passivation. Specifically, n + Ohmic contacts (phosphorus doped) are ∼ 1 nm below the undoped region, as shown in Fig. 3(a); p + Ohmic contacts (boron doped) are ∼ 1.5 nm above the undoped region, as shown in Fig. 3(b). We have found that the RCA-1 clean with a H 2 O 2 /NH 4 OH/H 2 O solution etches the SiO 2 region exposed to ion implantation much faster than the unimplanted oxide region. Once it etches through the sacrificial oxide layer completely, it etches the underlying doped silicon. This results in larger height difference, which may impair the Ohmic contact to the 2DS. Therefore, the time in the RCA-1 clean should be minimized, and under 20 minutes [39]. Piranha clean with a H 2 SO 4 /H 2 O 2 solution does not have this issue and thus is preferred when cleaning Si(111) pieces.
B. Gate leakage
Samples are intended for measurements at low temperature; consequently even sub-nanoampere gate leakage can potentially affect the data. All electrical measurements made in this section (Sec. III) were performed at T = 300 mK in a helium 3 refrigerator (Oxford Instruments, HelioxVL). Gate leakage was checked when all Ohmic contacts were grounded and the gate voltage V g was swept between -50 V (p 2d = 7.6 × 10 11 cm −2 ) and 50V (n 2d = 7.8×10 11 cm −2 ) at a sweep rate of 0.2 V/sec. The measured gate current was less than 0.5 nA at -50 V < V g < -20 V and 0 V < V g < 50 V, as shown in Fig. 4. (The capacitance between the gate and the ground was measured to be about 3 nF, so the gate current of 0.6 nA was expected.) The static leakage current was less than 50 pA at V g = ±50 V. There is no measurable leakage current for these devices. In Fig. 4, the peak current at V g ∼ -12 V and -3 V is due to the formation or depletion (b) Leakage current I ds vs. V ds at n 2d = 6 × 10 11 cm −2 with either p + isolation lines (S1, S2) grounded or floated. The leakage due to forward biasing happens at V ds < −0.4 V when S1, S2 are grounded. Arrows show the voltage sweep direction.
of the 2DES or 2DHS.
C. PN junction isolation and leakage
In earlier generation devices, a silicon-on-insulator (SOI) remote gate piece was used to define the 2DS on the H-Si(111) surface with a shield layer to restrict the electric field of the gate [6]. With the introduction of the Si/SiO 2 global gate, the electric field is all over the H-Si(111) surface, as shown in Fig.1 (c) (d). PN junction isolation and trench isolation are used to confine the 2DS. Using the SUPREM-IV process simulator, the depth of the p + and n + Ohmic contacts, defined as the region where doping concentration is higher than that of the 3D MIT (∼ 4 × 10 18 cm −3 [35,36]), is determined to be about 100 nm, as shown in Fig. 9(a)(b). For the trench isolation, 200 nm deep and 10 µm wide trenches are used. Since the depth of the 2DS is less than 10 nm for 2D carrier density higher than 10 11 cm −2 in the H-Si(111) vacuum FET [40,41], both PN junction isolation and trench isolation should work well.
A test device [ Fig. 5(a)] was fabricated to verify the effectiveness of the PN junction isolation. A 2DES [shaded area in Fig. 5(a)] is separated by a 10 µm wide boron ionimplanted line (the vertical line). At an electron density of 6 × 10 11 cm −2 , the resistance between n + contacts C1 and C3 were measured, while p + contacts S1 and S2 were either grounded or floated. In both cases, Fig. 5(b) shows that the resistance between C1 and C3 was larger than 25 FIG. 6. PN junction leakage current on sample 198. (a) P + contact leakage current Ic at Vg = 50 V (n 2d = 7.8 × 10 11 cm −2 ) when a voltage Vc is applied at each wired p + contact, while all n + contacts are grounded. (b) N + contact leakage current Ic at Vg = −50 V (p 2d = 7.6 × 10 11 cm −2 ) when Vc is applied at each n + contact, while all wired p + contacts are grounded. In both cases, the resistance is larger than 100 GΩ at −0.5 < Vc < 0.4 V.
GΩ at −0.2 < V ds < 1 V, likely dominated by inter-wire leakage, so the PN junction isolation is effective. When S1 and S2 were grounded, the PN junction began to leak at V ds = −0.4 V because of forward bias.
For the device under investigation (sample 198), we also measured the PN junction leakage current. At an electron density of 7.8 × 10 11 cm −2 , the leakage current from p + contacts 1, 5, 7, 11 to the 2DES was measured by applying a voltage V c at each contact, while all n + contacts were grounded. (Here p + contacts 3, 9 were not wired, and thus floated.) The resistance was larger than 100 GΩ at −0.5 < V c < 0.4 V, as shown in Fig. 6(a). Similarly, the leakage current from n + contacts 2, 4, 6, 8, 10, 12 to the 2DHS was measured by applying a voltage V c at each n + contact, while all wired p + contacts were grounded. Figure 6(b) shows that the resistance was also larger than 100 GΩ at −0.5 < V c < 1 V. In our transport measurements, lateral confining gates were floated with respect to the 2DS, and the source-drain voltage is less than ±0.1 V with the source-drain current equal or larger than 10 nA, so the leakage current from the PN junction isolation has only a negligible effect on the measurements.
We also measured the conductance between n + contacts C1 and C2 while negatively biasing p + contacts S1 and S2. The results show that the conductance decreases with increasing negative bias voltage, which suggests that depletion is occurring and reverse biasing lateral gates to confine carriers should be possible.
D. Ohmic contacts
Contacts to the 2DSs on H-Si(111) surfaces have been problematic especially at low temperature (< 1 K) and low carrier density, which limits the lowest accessible 2D carrier density in experiments. The main issues are large contact resistances (on the order of 1 MΩ) and nonlinearity in current-voltage (IV) curves [38].
For the device without STI, the contact problem can be mitigated by the presence of SiO 2 over the contact regions. At the center 2DS region, the dielectric is vacuum with a dielectric constant of 1; at the contact regions, SiO 2 is the dielectric with a dielectric constant of 3.9. So at the same gate voltage, the contact regions have about four times higher carrier density than the center 2DS region when the threshold voltages are negligible, and the contact resistance is reduced.
The contact resistance was measured at T = 300 mK, using a homemade voltage amplifier with input bias current ∼ 1 pA and input impedance > 200 GΩ. Figure 7(a) shows the circuit diagram measuring contact 6, where current I ds is injected from contact 12 to contact 6, and voltage V d is measured between contacts 2 and 6. The IV curves were nonlinear at V g = 5 V (n 2d = 0.87 × 10 11 cm −2 ) and became linear at V g = 6 V (n 2d = 1.02 × 10 11 cm −2 ), as shown in Fig. 7(b). So the lowest accessible 2D electron density was about 0.9×10 11 cm −2 at 300 mK for this device. Figure 7(c) shows contact resistance R c as a function of gate voltage V g . The sheet resistance of the n + regions was measured to be 129 Ω/ at 4.2 K, and this 3D metallic resistance should not change too much at 300 mK because of the lack of phonon scattering with the doping concentration well above the 3D MIT density. The n + contact resistance is calculated to be about 2.5 kΩ if only considering this sheet resistance, which is much smaller than the measured contact resistance at V g < 10 V. The commercial software package COMSOL Multiphysics [42] is used to calculate the contact resistance including both the n + contact regions and the 2DS region. For example, at V g = 10 V, the sheet resistances of the n + contact regions and the 2DES region are 129 Ω/ and 807 Ω/ respectively, and the calculated contact resistance is about 2.9 kΩ, which is significantly less than measured contact resistance of 4.9 kΩ. The difference is likely related to the highly resistive transition region between the metallic n + contact and the 2DES, where the doping concentration is less than the 3D MIT density (∼ 3.74 × 10 18 cm −3 [35]), as shown in Fig. 9(a). The 2DES in the transition region can undergo a 2D MIT and become an insulator at low carrier density and low temperature, resulting in high contact resistance [43].
The p + contact resistance was also measured. Figure FIG. 7. N + contacts on sample 198 measured at 300 mK.
(a) Circuit diagram measuring contact 6, with current I ds injected from contact 12 to contact 6 and voltage V d measured between contacts 2 and 6. (b) Current-voltage curves of n + contacts at Vg = 5 V (n 2d = 0.87 × 10 11 cm −2 )) and 6V (n 2d = 1.02 × 10 11 cm −2 ). (c) Contact resistance Rc as a function of gate voltage Vg. At Vg < 6 V, Rc is calculated when |I ds | 4 nA; otherwise, Rc is determined at |I ds | 100 nA.
8(a) shows that the IV curves were nonlinear at V g = −20 V (p 2d = 2.94 × 10 11 cm −2 ) and became linear at V g = −25 V (p 2d = 3.72 × 10 11 cm −2 ), which is much higher than the corresponding gate voltage for the n + contacts. The lowest accessible 2D hole density was about 3 × 10 11 cm −2 at 300 mK. Figure 8(b) shows contact resistance R c as a function of gate voltage V g . We measured the sheet resistance of the p + regions to be 103 Ω/ at 4.2 K, which is comparable to the n + regions. If we compare the doping profile of the phosphorus doped region and the boron doped region (Fig. 9), especially the top ∼ 10 nm area where 2DSs are populated, there is no significant difference. It is quite puzzling why the p + contacts are much worse than the n + contacts. There are probably two reasons. One is that electrons have much stronger Coulomb disorder screening strength than holes due to the very different valley degeneracy (6:1) [29]; the other is that there could be very different lateral defect profiles, such as end-of-range (EOR) dislocation loops [44]. The very different defect profiles can be caused by two reasons: first, phosphorus and boron ions have very different critical doses for amorphization of silicon [45]. At room temperature, the critical dose of phosphorus ions is ∼ 10 15 cm −2 , about twice as high as the dose used here; while it is ∼ 10 17 cm −2 for boron ions, which is about 100 times higher than the current dose. It is often easier to regrow the crystal from an amorphous layer via solid state epitaxy (activation energy ∼ 2.3 eV in Silicon) than it is to anneal out defects (activation energy ∼ 5 eV). Thus, two schemes for ion implantation are usually used: either perform ion implantation above the critical dose and use low temperature annealing to regrow material or perform ion implantation below the critical dose and use high temperature annealing to get rid of defects [44]. Second, phosphorus and boron ions have different mask edge effects, such as lateral straggle and shadowing effect, resulting in different lateral doping profile [46,47].
It is probably beneficial to do the ion implantation at higher dose and cryogenic temperature. We observed improvement of the p + contacts when the dose of the boron ion implantation was increased from 6 × 10 14 cm −2 to 2.4 × 10 15 cm −2 , then the contacts were annealed at 1000 • C for 10 min [9]. But there is a possibility that the contact bonding could be difficult, because of the larger height difference between the p + contact region and the undoped region after wet chemical etching than that shown in Fig. 3 (b). In order to access lower 2D carrier densities, a structure like a bi-layer gate structure using SOI, as we will discuss in Sec. V, can be used to solve the contact problem.
IV. TRANSPORT MEASUREMENTS
A. 2D carrier densities vs. gate voltage Longitudinal (R xx ) and Hall (R xy ) resistances were measured at 300 mK using standard low-frequency AC lock-in techniques with an excitation current of 100 nA at 7 Hz. Sheet resistance was determined by standard Van der Pauw method. Magnetotransport measurements were performed at different gate voltages in a perpendicular magnetic field (B ⊥ ) up to 12 T. From the SdH oscillations, the 2D hole densities are determined at each gate voltage. The 2D electron densities n 2d are calculated from the magnetic field B of R xx minima at the filling factor ν = 6: n 2d = νeB/h, where e is the electron charge, and h is Planck's constant. The 2D densities and their linear fits for sample 198 are shown in Fig. 10(a). For electrons, the slope is 1.563 ± 0.004 × 10 10 cm −2 /V, and the threshold voltage is V e th = 0.4 ± 0.1 V. For holes, the slope is 1.566 ± 0.001 × 10 10 cm −2 /V, and the threshold voltage is V h th = −1.21 ± 0.03 V. If a parallel plate capacitor model is used, the equivalent depth of the vacuum cavity is 355 ± 1 nm, which is consistent with the measurement result using a profilometer. The interface trap density in the band gap is determined to be: D it ≈ (V e th − V h th − E gap ) × 1.56 × 10 10 ≈ 0.8 × 10 10 (cm −2 ), where the band gap E gap is 1.17 V for silicon at 0 K [48].
B. 2D carrier mobilities 2D carrier mobilities are calculated from the sheet resistance, and plotted in Fig. 10(b). For sample 194, the peak electron mobility was 2.2 × 10 5 cm 2 /Vs at 300 mK. For sample 198, the peak electron mobility and the peak hole mobility were very different, 1.76 × 10 5 cm 2 /Vs and 9.1 × 10 3 cm 2 /Vs respectively at 300 mK, although the electrons and holes were populated in the same conduction channel. The ratio between the peak electron and hole mobilities is about 20, which cannot be explained by the different effective mass of electrons and holes. In fact, the effective mass is comparable (∼ 0.38m e for electrons [3,49,50] and ∼ 0.34m e for holes [51], where m e is the electron mass) at the investigated densities. This is quite different from the results of GaAs ambipolar devices, in which the ratio of the peak electron mobility to the peak hole mobility is about 5, consistent with the different effective mass of electrons and holes [22]. The large difference in the electron and hole mobilities is a direct consequence of the very different Coulomb disorder screening strength due to the different valley degeneracies of electrons (g v = 6) and holes (g v = 1) at the Si(111) surface [29].
In addition, the electron mobility increases monotonically with the electron density, and the mobility saturation is not observed up to the highest electron density measured (n = 7.8 × 10 11 cm −2 ), which is limited by the gate breakdown voltage. From the electron conductivity σ vs. electron density n data, the fitting exponent α in the relation of σ ∼ n α is about 1.3 at n = 7 × 10 11 cm −2 in these devices [29], which is consistent with the case (α ≈ 1.3) where the electron mobility is mainly limited by 3D bulk impurities, while deviating from the case (α ≈ 1.05) where 2D interface impurities limit the electron mobility [52]. We have developed a rebonding technique, which can revive a sample by taking it apart, cleaning the two pieces and H-passivating the Si(111) piece, then rebonding them together. The piranha cleaning and the diluted HF etching can remove the top few nm of silicon from the Si(111) piece. We tested it on sample 195 with an initial electron mobility of 5.6 × 10 4 cm 2 /Vs at n = 6 × 10 11 cm −2 and T = 300 mK. After repeated piranha cleaning and diluted HF etching three times, the electron mobility increased to 7.2×10 4 cm 2 /Vs (n = 6 × 10 11 cm −2 , T = 300 mK), which is likely due to the removal of the surface layers with higher 3D impurity density from earlier processes. All these suggest that the electron mobility is still mainly limited by 3D bulk impurities for current generation devices.
C. Magnetotransport measurements
All magnetotransport measurements were carried out in B ⊥ in this paper. Figure 11 shows the magnetoresistance for similar electron and hole densities in the same device (sample 198) at T = 300 mK. The frequency of the SdH oscillations was much higher for holes than electrons at low field, because electrons and holes have different areas of the Fermi surface resulting from the different valley degeneracies. For the 2DHS, the characteristic beating node was observed at B = 1.25 T for p 2d = 6.8 × 10 11 cm −2 [9], and the IQHE was also observed at high field (B > 5 T). For the 2DES, the SdH oscillations began at B ≈ 0.25 T, and R xx minima showed clearly the 12fold periodicity [the insert of Fig. 11(a)] from the sixfold valley degeneracy and the twofold spin degeneracy. In contrast to previous investigations of Si(111) transport on MOSFETs with peak electron mobility µ 2, 500 cm 2 /Vs which have shown conflicting valley degeneracies of two [50] and six [53], all six current generation H-Si(111) devices with electron mobility of more than 50,000 cm 2 /Vs show exclusively the sixfold valley degeneracy at low magnetic field.
D. Three magnetoresistance measurements with threefold rotational symmetry
Because electrons have six equivalent valleys on the Si(111) surface, the six n + Ohmic contacts are placed along the major axes of the constant energy ellipses, labeled as 2,4,. . . , 12, as shown in Fig. 12 (a)(b). The six equivalent valleys form three pairs with opposite momentum, labeled as A, B and C. The degeneracy of valley pairs with ± k symmetry on the Si(111) surface cannot be broken within the effective mass approximation or by a confinement potential [54,55]. Similarly, longitudinal magnetoresistance R xx , measured with current contacts on opposite sides of the hexagonal devices, also has three different directions defined by the direction of current flow. Here four-terminal resistance is defined as R i−j,l−m = V l−m /I i−j (i, j, l, m = 2,4,. . . , 12), where current I i−j is injected from contact i to contact j, and voltage V l−m is measured between contacts l and m. If the current flows from 2 to 8, it is called trace A (R 2−8,12−10 ). If the current flows from 10 to 4, it is called trace B (R 10−4,8−6 ), and if the current flows from 12 to 6, it is called trace C (R 12−6,10−8 ). These three magnetoresistances R xx with threefold rotational symmetry can be used to determine the underlying symmetry of the 2DES, and help identify the valley occupancies of the 2DES. For example, if electrons only occupy valley pair A, trace A will show higher resistance, while trace B and C will show similar lower resistance, due to the different effective mass. Figure 12(c)-(e) show the three magnetoresistance measurements on sample 194 and 198. Data were taken at 300 mK and a density of 3.6 × 10 11 cm −2 or 7.8 × In a Drude model with noninteracting electrons, assuming an isotropic scattering time, the resistance ratio is equal to the mass ratio m l /m t , where m l = 0.67m e and m t = 0.19m e [54] are the longitudinal effective mass and the transverse effective mass respectively. For CFs, all physical quantities are determined by the Coulomb interaction [56]. In an isotropic 2DS, the Coulomb interaction is ∝ 1/ x 2 + y 2 , where x and y are components of the distance between two electrons. According to Ref. 57, in a system with an anisotropic Fermi surface, it can be mapped to a system with an isotropic Fermi surface with an anisotropic Coulomb interaction ∝ 1/ x 2 γ 2 + y 2 /γ 2 , where γ = (m l /m t ) 1/4 , x and y are along the major and minor axes of the constant energy ellipse. If the transport mass of CFs along some direction is determined by the strength of the Coulomb interaction along this direction, the mass-anisotropy ratio of CFs is given by γ 2 = m l /m t . We have calculated two cases using COMSOL assuming the isotropic scattering time: in the first case, the mass-anisotropy ratio of CFs is assumed to be m l /m t ; in the second case, it is assumed to be m l /m t . Using a hexagonal geometry similar to Fig. 1(c) and assuming the anisotropic resistance solely from the anisotropic effective mass, the calculated ratios between the higher and lower resistances are 3.58 and 2.07 respectively for these two cases. The measured ratios 2.43 and 2.12 fall between these two cases and are closer to the second case. Our results are similar to Ref. 57, but deviate from a theoretical study which predicts almost isotropic CF effective mass [58]. Because the scattering time anisotropy is not well known, the above discussion becomes more complicated. Nevertheless, in a simplest picture, electrons occupy the lowest valley pair at filling The three magnetoresistances are highly anisotropic at ν < 2/3, with two similar traces higher than the other one. This may indicate the change of the underlying symmetry. factor 1 < ν < 2. Figure 12(c) and (e) show that the corresponding CFs occupy valley pair A, and qualitatively preserve the valley anisotropy, in addition to the twofold valley degeneracy manifested by observed exclusive even numerator fractional quantum Hall (FQH) states (8/5 and 4/3) [10].
Moreover, at filling factor ν between 4 and 6, the devices also show anisotropic magnetoresistances. For sample 194, trace C is highest, while trace A is highest for sample 198. Both cases can be qualitatively explained when the Fermi level is located at valley pair C or valley pair A respectively. For filling factor ν between 3 and 4, the three magnetoresistances are similar for both devices, indicating an isotropic distribution of electrons in the three valley pairs. This cannot be explained by a valley occupancy of noninteracting electrons, and may related to the quantum Hall nematic phase [59,60]. When filling factor ν is between 2 and 3, the three magnetoresistances are quite different for sample 194, but similar for sample 198. At lower B (ν > 6), the three magnetoresistances are similar especially for sample 198, consistent with the approximation that six valleys are equally occupied.
The three magnetoresistance measurements indeed provide a powerful tool to investigate the underlying symmetry of the 2DES on the Si(111) surface. In addition, a tilted magnetic field can re-distribute electrons between the three pairs of valleys through the application of an in-plane magnetic field [54]. If these two techniques are combined, it will further our understanding of this 2DES and the nature of the observed anisotropy. measured at T = 25 mK and density of 3.6 × 10 11 cm −2 . As shown in Fig. 13(a), the 2/3 FQH state is clearly observed with a well developed hall plateau. Moreover, Fig. 13(b) shows that the three magnetoresistances are highly anisotropic at ν < 2/3. It is less than 1 kΩ along trace B, and about 50 kΩ along traces A and C. This 50:1 anisotropy cannot arise solely from the different effective mass. The three magnetoresistances also change from one higher resistance trace with two similar lower resistance traces at ν > 2/3, to two similar higher resistance traces with one lower resistance trace at ν < 2/3. This may indicate the change of the underlying symmetry of the 2DES.
At filling factor ν between 1 and 2, only even numerator FQH states (8/5 and 4/3) are observed with no 5/3 FQH state, consistent with twofold valley-degenerate CFs [10]. It is quite interesting to find out the underling symmetry of the 2/3 FQH state, i.e. whether it also has the twofold valley degeneracy. Because valley degenerate 2/3 FQH state may be related to non-Abelian states which are being intensively studied for possible applications in intrinsically fault tolerant quantum computation [61][62][63][64], the question becomes more important. We have determined the activation energies at ν = 2/3 and 4/3 from the temperature dependence of magnetoresistance R xx (Fig. 14). The activation energies are comparable, 1.64 K for ν = 2/3 and 1.34 K for ν = 4/3. In a twofold valley-degenerate system, 4/3 (= 2 − 2/3) state can be related to 2/3 state by using particle-hole symmetry [10,56,65,66]; their activation energies are proportional to √ B, and E 2/3 /E 4/3 = 22.94/11.15 = 1.4. The measured ratio is about 1.2, which may imply that the 2/3 FQH state also has the twofold valley degeneracy.
V. DEVICE DEGRADATION
The H-Si(111) surface is encapsulated in a vacuum cavity in the H-Si(111) vacuum FET [ Fig. 1(a)], but the seal is not perfectly hermetic. Consequently H-Si(111) vacuum FETs still suffer from device degradation in air owing to degradation of H-terminated surfaces [67][68][69], which may limit their applications. However, if care is taken to limit the time of the device spent in air, while keeping it mostly in vacuum or a nitrogen gas environment, the device degradation is manageable.
The degradation of the two devices has been recorded. Sample 194 was cooled down five times in a period of 246 days, as shown in Fig. 15. The mobility was mostly measured at 300 mK, except the fourth cooldown (at 25 mK). As described in Sec. II, after the device was bonded and wired in the nitrogen glove box, it was rapidly loaded into the cryostat at the first cooldown. There was about one hour exposure in air before pumping down the cryostat at each cooldown. After the device was measured for about two weeks, it was warmed up, put into a nitrogen gas filled barrier foil ziplock storage bag with oxygen absorbers, and then the bag was heat sealed. After one week, it was cooled down and measured a second time at day 26. Figure 15 shows that the change of the peak electron mobility was about 2%. The device was then warmed up and stored in a nitrogen glove box (O 2 < 10 ppm) for about 6 months. Right before transferring the device to the NHMFL, the device was measured a third time. The peak electron mobility increased by about 3%, which may be due to the different temperatures or the different contact resistances. So within 211 days, the device mobility did not change much when keeping it mostly in a nitrogen gas environment. However, the mobility fluctuations were pronounced at lower electron densities at the third cooldown, which was caused by the degradation of the contacts. For transport to NHFML, the device was again placed in a nitrogen gas filled storage bag, as described above. After the NHMFL experiment, the device was exposed in air for about one week, and the peak electron mobility decreased by 35% at the fifth cooldown. Similarly, sample 198 was cooled down four times in 193 days, as shown in Fig. 16. After the first measurement, the device was stored in the nitrogen glove box for about 3 months. The peak electron mobility decreased by about 17% at the second cooldown, which degraded faster than that of sample 194. After the same NHMFL experiment, the device was in air for about one week, and the peak electron mobility decreased by another 18% at the fourth cooldown. The hole mobility was decreased by about 26% between the first and fourth cooldown, measured at 4.2 K [ Fig. 16(b)].
After ∼ 200 days, the peak electron mobility decreased by about 35% for both devices. After one week exposure in air, the peak electron mobility decreased by about 35% and 18% for sample 194 and 198 respectively. Roughly it can be estimated that the peak electron mobility decreased by ∼ 0.2%/day in vacuum or a nitrogen gas environment, and by ∼ 5%/day in air.
In addition to the mobility degradation, the contacts also degrade with time. The contact degradation makes accurate resistance measurements difficult, and shows up FIG. 17. N + contact resistance Rc as a function of gate voltage Vg on sample 198 at 300 mK. For 4th cooldown, at Vg < 15 V, Rc is calculated when |I ds | 4 nA; otherwise, Rc is determined at |I ds | 100 nA. Compared to 1st cooldown, the contact resistance increased dramatically at low electron density (5 < Vg < 25 V).
as the conductance (mobility) fluctuations both for electrons and holes at lower densities [9,38]. In sample 198 (without STI), the contact problem had been mitigated, but it resurfaced at the 4th cooldown. The contacts were linear only at above V g = 16 V (n 2d = 2.5 × 10 11 cm −2 ) rather than V g = 6 V (n 2d = 1.02 × 10 11 cm −2 ) at the first cooldown, and the contact resistances were also much worse than those at the 1st cooldown (Fig. 17). Correspondingly, the mobility fluctuated at n 2d < 2.5 × 10 11 cm −2 for the 4th cooldown in Fig. 16(a).
Although resistance measurements are possible with pure large contact resistances (∼ 10 MΩ) using 4terminal DC method, there probably exist charge trapping and emission processes in these devices like single electron transistors [38]. The 4-terminal DC method is not effective in this case. In order to solve the contact problem, the bi-layer gate structure using SOI, as shown in Fig. 18 can be adopted. Similar to a split-gate geometry, the bi-layer gate structure which permits high densities around the contacts while allowing independent control of the density of the 2DSs under investigation [70], is quite important going forward.
FIG. 18. A vacuum FET with a bi-layer gate structure using SOI. The 2D carrier densities around the contacts and at the investigated area can be independently controlled. The gate is also fully covered by a SiO2 layer.
VI. PROSPECTS AND CONCLUSIONS
We have investigated a process to fabricate high mobility ambipolar H-Si(111) vacuum FETs with electron mobility of ∼ 200, 000 cm 2 /Vs and hole mobility of ∼ 10, 000 cm 2 /Vs. From the electron conductivity vs. electron density analysis, the electron mobility is still mainly limited by 3D bulk impurities, which implies that the mobility can be further improved with attention to high temperature steps. At a minimum, electron mobility of ∼ 300, 000 cm 2 /Vs should be able to realize in the near future, since we have already fabricated a device with this mobility using previous sample fabrication techniques [10]. Combined with a bi-layer gate structure to solve the contact issue, the device quality should be comparable to the best SiGe quantum well devices. In addition, the 2DES has a sixfold valley degeneracy, which will open up many opportunities for fundamental research and practical applications.
Three magnetoresistance measurements are a very useful tool to determine the underlying symmetry of the 2DES. They can be used to detect possible phase transition when the associate symmetry changes [59,60]. Further experiments with a tilted magnetic field to re-distribute electrons between valleys can dynamically change the underlying symmetry. It will surely improve our understanding of this 2DES and the nature of the observed anisotropy.
The devices were also investigated at filling factor ν < 1 in a magnetic field up to 35 T. We clearly observed the 2/3 FQH state with a well developed hall plateau, and a high magnetoresistance anisotropy at ν < 2/3 on both sample 194 and 198. On sample 194, there were some hints of possible 3/5, 4/7, 5/9 and 6/11 FQH states (data not shown). With better devices, we should be able to verify these additional states. It is quite possible to observe new phases in this unique sixfold valley-degenerate system.
Because of the distinct device structure, the 2D systems are resident at the surface in the vacuum cavity. Atoms, molecules, superconductors and other systems can easily couple to the 2DSs and form hybrid systems [6], and novel functional devices can be implemented in the future.
FIG. 1 .
1(a) Schematic cross section of an ambipolar vacuum FET (not to scale), with a SiO2/Si(100) remote gate piece and a H-Si(111) piece contact bonded in a vacuum. The remote gate piece is used to induce 2D carriers on the H-Si(111) surface by applying a gate voltage on a p + gate layer through an encapsulated vacuum cavity. It also has a fully covered SiO2 layer with a thin layer of SiO2 (∼ 30 nm) left in the cavity. (b) Six degenerated valleys of the 2DES with crystallographic orientations. (c) (d) In the H-Si(111) piece, n + (p + ) Ohmic contacts are used to access a 2DES (2DHS) at the center, with p + (n + ) regions acting as lateral confinement to restrict current flow between contacts to the hexagonal center 2DS. Two different configurations are investigated. (c) One has trench isolation, and (d) the other doesn't have trench isolation. The dash lines in (c) show the cross section depicted in (a).
FIG. 2 .
2(a) A H-Si(111) piece with six n + contacts (yellow) and six p + contacts (purple). (b) A Si/SiO2 remote gate piece which is fully covered by SiO2, except the two grey crosses where the gate contacts are made. There is a ∼ 30 nm oxide at the bottom of the hexagonal cavity. (c) 3D view of a bonded device. (d) A bonded device placed in a customized holder and installed on an 8-pin dual in-line package (DIP) header without wiring.
FIG. 3 .
3AFM images of a H-Si(111) surface with (a) a phosphorus-doped contact edge and (b) a boron-doped contact edge. They also show atomic steps in undoped areas and the crystallographic directions. Bottom graphs show height profiles along white dash lines in the top graphs.
FIG. 4 .
4Gate current Ig vs. gate voltage Vg on sample 198
FIG. 5 .
5(a) A test device with four n + Ohmic contacts (yellow) and p + isolation lines (purple). A center vertical p + isolation line separates the 2DES (shaded area) into two halves.
FIG. 8 .
8P + contacts on sample 198 measured at 300 mK. (a) Current-voltage curves of p + contacts at Vg = −20 V (p 2d = 2.94 × 10 11 cm −2 ) and -25 V (p 2d = 3.72 × 10 11 cm −2 ). (b) Contact resistance Rc as a function of gate voltage Vg. At −Vg < 25 V, Rc is calculated when |I ds | 4 nA; otherwise, Rc is determined at |I ds | 100 nA.
FIG. 9 .
92D doping profile of (a) phosphorus-doped region and (b) boron-doped region in Si using SUPREM-IV simulator. In (a), 4.5 × 10 14 cm −2 , 50 keV phosphorus ions are implanted through a 30 nm oxide; in (b), 9 × 10 14 cm −2 , 15 keV boron ions are implanted through a 30 nm oxide. Both are annealed at 950 • C for 1 min. Ions are implanted at 0 • with vertical photoresist edge at Length=0. (c) Doping concentrations of phosphorus and boron along the Si surface at (a) and (b).
FIG. 10 .
10(a) Electron and hole densities as a function of gate voltage for sample 198. The electron density is determined from the magnetic field of Rxx minima at the filling factor ν = 6, and the hole density is calculated from the Shubnikovde Haas oscillations. (b) 2D carrier mobility of sample 194 and 198 vs. carrier density at 300 mK. For the ambipolar device 198, the ratio between the peak electron and hole mobilities is about 20.
FIG. 11 .
11Magnetoresistance and Hall resistance measured at 300 mK for similar (a) electron and (b) hole densities on sample 198. The frequency of the SdH oscillations is much higher for holes than electrons owing to different valley degeneracy (1:6). The insert in (a) clearly shows the 12-fold periodicity from the sixfold valley degeneracy of the 2DES and the twofold spin degeneracy. The insert in (b) shows a beating pattern with a beating node at the arrow.
FIG. 12 .
12(a) Top view of a device with six electron contacts and six hole contacts in relation to (b) the 3 pairs of the valleys, labeled as A, B and C. (c) Three magnetoresistance measurements on sample 194 at n 2d = 3.6 × 10 11 cm −2 with the insert showing the low field behavior. (d) (e) Three magnetoresistance measurements on sample 198 at n 2d = 7.8 × 10 11 cm −2 . For both samples 194 and 198, one trace is higher than the other two similar traces at 1 < ν < 2.
10 11 cm −2 for samples 194 and 198 respectively. The two simplest cases are B = 0 and B at ν = 3/2 where composite fermions (CFs) experience a zero effective magnetic field. At B = 0, both devices show more or less isotropic resistances. For sample 194, the three resistances are 37.2 Ω, 30.4 Ω, and 30.3 Ω; for sample 198, they are 16.0 Ω, 17.9 Ω, and 17.9 Ω, as shown in Fig. 12(c)(d). At B 3/2 , the devices are anisotropic. For sample 194, the three magnetoresistances are 558.1 Ω, 226.3 Ω, and 232.8 Ω. For sample 198, they are 1356.9 Ω, 615.4 Ω, and 664.7 Ω. The observed anisotropy can be explained by CFs' occupying valley pair A. Because of the effective mass anisotropy, trace A shows higher resistance, and traces B and C show similar lower resistance. The ratios between the higher and lower resistances are 2.43 and 2.12 for sample 194 and 198 respectively.
FIG. 13 .
13(a) Three magnetoresistances and Hall resistance measured on sample 194 at 25 mK with the clearly observed 2/3 fractional quantum Hall state. (b)
E. Magnetotransport measurements up to 35 T Sample 194 was further investigated at filling factor ν < 1 in the portable dilution refrigerator (PDF) at NHMFL with a magnetic field up to 35 T. The device wasFIG. 14. Arrhenius plots for ν = 2/3, 4/3 showing activated behavior on sample 194.
FIG. 15 .
15Electron mobility of sample 194 vs. electron density during five cooldowns, measured at 300 mK except the 4th cooldown (at 25 mK). There was one week of air exposure between the 4th and 5th cooldowns.
FIG. 16 .
16(a) Electron mobility of sample 198 as a function of electron density during four cooldowns, measured at 300 mK except the 3rd cooldown (at 25 mK). One week air exposure happened between the 3rd and 4th cooldowns. (b) Hole mobility of sample 198 vs. hole density at 1st and 4th cooldowns, measured at 4.2 K.
ACKNOWLEDGMENTSThis work was funded by the Laboratory for Physical Sciences. Research was performed in part at the NIST Center for Nanoscale Science and Technology, and the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida. We also acknowledge the support of the Maryland NanoCenter and its FabLab. The authors are grateful to Dr. Neil Zimmerman and Dr. Michael Stewart for their help on the RTA process.
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[
"Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions",
"Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions"
]
| [
"Helena Jasiulewicz ",
"Wojciech Kordecki "
]
| []
| []
| In this paper a quantitative analysis of the ruin probability in finite time of a discrete risk process with proportional reinsurance and investment of financial surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -exponential distribution and a heavytailed distribution -Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover, for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs. | 10.5277/ord150302 | [
"https://arxiv.org/pdf/1306.3479v2.pdf"
]
| 55,638,131 | 1306.3479 | 95957045c49b334f7f95e3f49b77891263cad2b3 |
Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions
15 Mar 2015
Helena Jasiulewicz
Wojciech Kordecki
Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions
15 Mar 2015arXiv:1306.3479v2 [q-fin.RM]discrete time risk processruin probabilityproportional reinsuranceLundberg's inequalityregularly varying tail
In this paper a quantitative analysis of the ruin probability in finite time of a discrete risk process with proportional reinsurance and investment of financial surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -exponential distribution and a heavytailed distribution -Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover, for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.
Introduction
In the risk theory, works concerning the financial surplus of insurance companies in a continuous time have been proceeding for nearly a century. Very advanced models of the classical continuous risk process were established. Although such a model is more natural in the description of reality, the research on the discrete process of financial surplus is considerably more modest. The review of the results concerning the discrete process of financial surplus one can find in the paper [6]. This paper is one of the series of papers which try to bring closer of the classical discrete process of financial surplus to the reality of insurance companies. Namely, the analysis of the investment of financial surplus enhances the security of an insurance company. These problems are considered in the papers [1,2,3,9,10]. Reinsurance has a considerable influence on increasing the security of an insurance company. The results concerning a discrete risk process with investment and reinsurance can be found in [4,7].
In this paper we consider the ruin probability in finite time of a discrete risk process with proportional reinsurance and investment of financial surplus. Moreover, we obtain numerical results for particular cases: exponential and Pareto distribution of a total loss and some asymptotic results.
In the paper by Cai and Dickson [3] the ruin probability in a discrete time risk process with a Markov interest model is studied. Recursive equations for the ruin probabilities, generalised Lundberg inequalities and an approximating approach to the recursive equations are given in that paper. Diaspara and Romera [4] introduced a proportional reinsurance in the discrete risk process with an investment.
For any reinsurance, not only proportional, Jasiulewicz [7] obtained recursive equations and Lundberg inequality for the ruin probability in the discrete-time risk process with Markovian chain interest rate model. Moreover, for the proportional reinsurance and the reinsurance of stop-loss an optimal level of retention was considered, assuming the maximisation of Lundberg adjustment coefficient as an optimising criterion. This paper is a continuation of the research initiated by Jasiulewicz [7]. For the given theoretical results we conduct a detailed quantitative analysis for particular distributions of the total loss in a unit period and proportional reinsurance. We consider the ruin probability for a light-tailed distribution (exponential pdf) and a heavy-tailed distribution (Pareto pdf) taking into account an investment of finance surplus according to a random interest rate. Based on these considerations we give practical conclusions concerning connections between the initial capital level and the reinsurance level. We pointed out the level of reinsurance of a loss in order to set a ruin probability at the level low enough to be accepted by an insurer and vice versa i.e. how high his own capital should be.
The quality of the upper bound of ruin probability in finite time with the use of Lundberg coefficient was illustrated by the example of exponential distribution.
We observe that if an insurer and a reinsurer use the same security loading then the adjustment coefficient as a function of the reinsurance level is convex, which considerably improves an upper estimation of the ruin probability. However, if loading of a reinsurer is greater than loading of insurer, the adjustment coefficient is not a convex function, which lowers the quality of an upper estimation. This observation was not taken into account in the numerical examples in Diaspara and Romera [4].
It is known that for heavy-tailed distributions Lundberg adjustment coefficient does not exist. For distributions of that type we give the theorem about the approximation of the ruin probability if the initial capital is sufficiently large. The example of Pareto distribution shows that such an approximation is appropriate and quickly tends to the limit value.
In the paper we assume the expectation of a loss in a unit period as a monetary unit. For that reason we assume that the expected values in both considered distributions are equal to 1. For the assumed values of parameters in Pareto distribution a variance does not exist. To compare numerical results for both distributions we also take such parameters in order to obtain the same geometric means as well as geometric variances.
Concluding, below we list the new elements, ideas and results which are introduced in this article:
1. In the continuous risk process the level of retention is optimal if it minimises the ruin probability which can be determined by maximising an adjustment coefficient relative to the level of retention (see Dickson and Waters [5]). Then we can pose the following natural question: does the discrete risk process hold the same?
2. The upper bound of the ruin probability obtained by Lundberg coefficient in the case of proportional reinsurance is given by Diaspara and Romera [4]. The numerical example for ξ = θ shows that this estimation is reasonable. Is that estimation also good for the more natural case ξ > θ ?
3. In the case of heavy tailed claims we give the approximation of the ruin probability. The question is: is the sequence of approximations is fast convergent for sufficiently large initial capital?
Notations and theorems
Further notations, assumptions and theorems 1 and 2 given below come from the paper by Jasiulewicz [7]. In that paper the following notations and assumptions were taken.
1. Let Z n denote the total loss in unit period (n − 1, n]. The loss is calculated at the end of each period. Let us assume that {Z n , n = 1, 2, . . .} is a sequence of independent and identically distributed random variables with a common distribution function W (z).
2. The premium is calculated by the expected value principle with the loading factor θ > 0. Constant premium c = (1 + θ ) E Z n is paid at the end of every unit period (n − 1, n].
3. The insurer's surplus at the moment n is denoted by U n and is calculated after the payoff. The surplus U n is invested at the beginning of the period (n, n + 1] at a random rate I n .
4. Let us assume that the interest rates {I n , n = 0, 1, . . .} follow a time-homogeneous Markov chain. We further assume that for all n = 0, 1, . . ., the rate I n takes possible values i 1 , i 2 , . . ., i l . For all n and all states, the transition probability is denoted by Pr (I n+1 = i t |I n = i s ) = p st ≥ 0 and the initial distribution is denoted by
Pr (I 0 = i s ) = π s .
5. Suppose that the insurer effects reinsurance and that the amount paid by the insurer when the loss Z n occurs is h (Z n , b) where a parameter b > 0 denotes a retention level. The meaning of the parameter b will be explained in two examples of the most frequent reinsurancies applied in the insurance practice.
(a) Proportional reinsurance, if a function h (x, b) has the form
h (x, b) = bx, where b ∈ (0, 1]. (b) Stop loss reinsurance, if a function h (x, b) has the form h (x, b) = x, x ≤ b, b, x > b, where b > 0.
The
following assumption 0 ≤ h (x, b) ≤ x about h is obvious.
A part of the loss Z n retained by the insurer is denoted by Z ce n = h (Z n , b) and its distribution function by V (z). Therefore Z re n = Z n − Z ce n is a reinsured part of the loss Z n .
6. Le us assume that a reinsurer calculates a premium rate c re according to the expected value rule with a loading factor η, i.e.
c re = (1 + η) E (Z n − h (Z n , b)) .
We assume that η ≥ θ > 0, so an insurer does not earn without risk if he retains only zero value of claims.
7. The premium rate retained by an insurer in a unit period is denoted by c (b) and is given by
c (b) = c − c re = (1 + η) E h (Z n , b) − (η − θ ) µ.
8. Let U b n denote a financial surplus of an insurer at the end of the unit period (n − 1, n] after the payment of premium and after the payoff. The process U b n considered in the paper is given by
U b n = U b n−1 (1 + I n ) + c (b) − h (Z n , b) .
9. The ultimate ruin probability for this risk process in the finite time is denoted by Ψ b n (u, i s ) and is defined by
Ψ b n (u, i s ) = Pr n i=1 U b i < 0 |U b 0 = u, I 0 = i s = Pr U b i < 0 for some i ≤ n|U b 0 = u, I 0 = i s .
The ultimate ruin probability in the infinite time is given by
Ψ b (u, i s ) = Pr ∞ i=1 U b i < 0 |U b 0 = u, I 0 = i s = Pr U b i < 0 for some i ≥ 1|U b 0 = u, I 0 = i s . Obviously Ψ b (u, i s ) = lim n→∞ Ψ b n (u, i s ) .
The further research is conducted for a proportional reinsurance. The premium rate retained by an insurer is
c (b) = ((1 + η) b − (η − θ )) µ. (2.1)
To avoid such an event that the ruin could occur with probability 1 it is assumed that
E h (Z 1 , b) < c (b) . (2.2)
To write the self-contained paper, we give theorems from Jasiulewicz [7] (Theorems 1 and 2), which will be used in the analysis of the ruin probability. In the special case of reinsurance, namely proportional reinsurance, the theorems analogous to Theorems 1 and 2 were given in the paper by Diaspara and Romera [4].
Theorem 1. Ruin probability of an insurer in finite time is given recursively in the following way:
Ψ b 1 (u, i s ) = l ∑ j=1 p s j V u 1 + i j + c (b) , (2.3) Ψ b n+1 (u, i s ) = l ∑ j=1 p s j V u 1 + i j + c (b) + u(1+i j) +c(b) 0 Ψ b n u 1 + i j + c (b) − z, i j dV (z) .
(2.4)
Ruin probability in infinite time:
Ψ b (u, i s ) = l ∑ j=1 p s j V u 1 + i j + c (b) + u(1+i j )+c(b) 0 Ψ b u 1 + i j + c (b) − z, i j dV (x) , where c (b) = (1 + η) E h (Z n , b) − (η − θ ) µ. (2.5) Proof. Let Z ce 1 = z, I 1 = i j . If z > u 1 + i j + c (b)
, then a ruin will occur in the first period (0, 1]. Therefore
Ψ b 1 (u, i s ) = l ∑ j=1 p s j Pr Z b 1 > u 1 + i j + c (b) |I 1 = i 1 , I 0 = i s = l ∑ j=1 p s j V u 1 + i j + c (b) .
The ruin in first n + 1 periods can occur in two excluding ways:
• the ruin will occur in the first period or
• the ruin will not occur in the first period but it will occur in next periods.
Since the process U b n is stationary with independent increments then
Ψ b n+1 (u, i s ) = l ∑ j=1 p s j ∞ 0 Pr n+1 k=1 u b k < 0|Z b 1 = z, I 1 = i s dV (z) = l ∑ j=1 p s j V u 1 + i j + c (b) + u(1+i j) +c(b) 0 Ψ b n u 1 + i j + c (b) − z, i j dV (z) .
The probability of the ruin in infinite time is obtained by taking a two-sided limit in the above formula for n → ∞.
Recurrence formulas for the ruin probability can be presented in a matrix form, which simplifies calculations using several computer programs 1 .
Let
Ψ Ψ Ψ b n (u) = Ψ b n (u, i 1 ) , Ψ b n (u, i 2 ) , . . ., Ψ b n (u, i l ) and V n = v (n) 1 , v (n) 2 , . . . , v (n) l , where v (1) j = V u 1 + i j + c (b) and for n ≥ 2 v (n+1) j = v (1) j + u(1+i j )+c(b) 0 Ψ b n u 1 + i j + c (b) − z, i j , dV (z) .
Then we can write equations (2.3) and (2.4) in a matrix form
Ψ Ψ Ψ b n (u) = V n P T . Theorem 2. If E h (Z 1 , b) < c (b) and there exists a positive constant R (b) fulfill- ing the equation E e R(b)h(Z 1 ,b) = e R(b)c(b) , (2.6)
the upper estimation of the ruin probability in finite and infinite time is in the form
Ψ b n (u, i s ) ≤ Ψ b (u, i s ) ≤ ξ (b) E e −R(b)u(1+I 1 ) |I 0 = i s , (2.7) where ξ (b) = sup x≥c(b) e R(b)x V (x) ∞ x e R(b)z dV (z) , 0 < ξ (b) ≤ 1. (2.8) Proof. For every x ≥ 0 we have V (x + c (b)) = e R(b)x V (x + c (b)) ∞ x e R(b)z dV (z + c (b)) e −R(b)x ∞ x e R(b)z dV (z + c (b)) = e R(b)(x+c(b)) V (x + c (b)) ∞ x+c(b) e R(b)y dV (y) e −R(b)x ∞ x+c(b) e R(b)(y−c(b)) dV (y) . (2.9) Let g (t) = e R(b)(t) V (t) ∞ t e R(b)y dV (y)
.
Then
V (x + c (b)) ≤ sup x≥0 {g (x + c (b))} e −R(b)x ∞ x+c(b) e R(b)(y−c(b)) dV (y) = β e −R(b)x ∞ x+c(b) e R(b)(y−c(b)) dV (y) , (2.10) where β = sup y≥c(b) g (y)
.
From Equation (2.6) we obtain
V (x + c (b)) ≤ β e −R(b)x ∞ −∞ e R(b)(y−c(b)) dV (y) = β e −R(b)x . (2.11)
Whereas the inequality (2.8) follows from the fact that for z ≥ t an inequality
exp (R (b) z) ≥ exp (R (b)t) occurs. Therefore ∞ t e R(b)z dV (z) e R(b)t V (t) ≥ e R(b)t ∞ t dV (z) e R(b)t V (t) = 1.
From the conversion of this inequality the inequality (2.8) is obtained.
In the next step we prove (2.7) inductively. From Theorem 1 and inequality (2.11) we have
Ψ b 1 (u, i s ) ≤ l ∑ j=1 p s j β e −R(b)u(1+i j ) = β E e −R(b)u(1+I 1 ) |I 0 = i s .
From a inductive assumption
Ψ b n (u, i s ) ≤ β E e −R(b)u(1+I 1 ) |I 0 = i s and Theorem 1 we have Ψ b n+1 (u, i s ) ≤ l ∑ i= j p s j β e −R(b)u(1+i j ) ∞ u(1+i j )+c(b) e R(b)(y−c(b)) dV (y) + u(1+i j) +c(b) 0 β E e −R(b)(u(1+i j )−z+c(b))(1+I1) |I 0 = i s . Since E e −R(b)(u(1+i j )−z+c(b))(1+I1) |I 0 = i s ≤ e −R(b)(u(1+i j )−z+c(b)) , (2.12) then Ψ b n+1 (u, i s ) ≤ l ∑ i= j p s j β e −R(b)u(1+i j ) ∞ −∞ e R(b)(y−c(b)) dV (y) = β E e −R(b)u(1+I 1 ) |I 0 = i s .
Taking limits for n → ∞ we obtain the inequality (2.7).
Theorem 1 gives recurrence formulae for the ruin probability and Theorem 2 gives an upper estimation of the ruin probability using Lundberg adjustment coefficient, which exists only for a light-tailed distribution. Therefore one cannot use Theorem 2 to estimate the ruin probability for heavy-tailed distributions. In that case we will use an asymptotic ruin probability in the respect of an initial capital tending to infinity, whereas the total loss has the distribution with a regularly varying tail.
Definition 1.
A distribution F on (−∞, ∞) has a regularly varying tail if there exists some constant α ≥ 0 such that for every y > 0 is
lim x→∞ F (xy) F (x) = y −α .
The class of such distributions is denoted by R −α . with an initial condition c 0 (i s ) = 0 for n = 1, 2, . . .
Proof.
In the paper by Cai and Dickson [3] the above theorem was proved in the case where an insurer does not apply reinsurance but invests the financial surplus. It is sufficient to remark that with proportional reinsurance Z ce n = bZ n , if Z n has a distribution with a regularly changing tail with an index α, then Z ce n has also the distribution with a regularly varying tail with an index α. This follows from
lim x→∞ V (xy) V (x) = lim x→∞ W (yx/b) W (x/b) = lim z→∞ W (yz) W (z) = y −α , where z = x/b → ∞, if x → ∞, because b > 0.
Therefore our Theorem 3 is fulfilled for Z ce by Theorem 5.1 from the paper Cai and Dickson [3]. Our proof repeats the arguments given in Theorem 5.1 from that paper if we substitute V with G.
In the next sections we will consider particular cases if the total loss in the unit period has an exponential distribution with mean 1, i.e. W (x) = 1 − e −x and has Pareto distribution with the same mean:
W (x) = 1 − (β /x) α , x > β , α > 1, β = (α − 1) /α.
In Section 3 we give analytical formulae only for the cases l = 1,
i 1 = 0 (i.e.
financial surplus is not invested) and small values of the parameter n.
To determine these formulae we use the program Maxima assigned to symbolic calculations.
Numerical results will be presented for the case l = 2 and for selected values of the parameters α, β , η, θ and b.
Ruin probability
Calculations of values of function Ψ b (u, i s ) given by Theorem 1 were conducted for b = 0.2, 0.3, . . ., 1.0, u = 0, 1, 2, 3 , 4, 5 and n = 1, 2, . . . , 10. We considered the cases
c (b) = (1 + η) b − (η − θ ) = 1.25b − 0.05. The condition (2.2) is fulfilled for b > 1 − θ /η = 0.2.
Exponential distribution
Let us assume that Z n has the exponential distribution with mean 1. Hence Z ce n = bZ n has the distribution function
V (x) = 1 − e −x/b (3.1)
for x ≥ 0 and E Z ce n = b, Var Z ce n = b 2 . The explicit formulae for function Ψ b 1n (u, i s ) for n ≥ 2 are too complicated to present. We take l = 1 and i 1 = 0.
Ψ b 1 (u) =e −u−θ +(−b)(η+1)+η b (3.2) Ψ b 2 (u) = e 2η/b u + e 2η/b θ + ((b − 1) η + b) e 2η/b e −u/b−2θ /b−2η−2 b +e (−u−θ −b(η+1)+η)/b (3.3)
Formulae for Ψ b n (u) for n ≤ 5 obtained from Maxima were used to verify the correctness of numerical algorithms which are used for greater n and l.
From Table 1 we obtain the following conclusions. • If the initial capital grows, the part of the insurer's retained loss also grows with the constant level of risk of the company bankruptcy for any time horizon n.
• If the initial invention rate grows then the level of retention b also grows with the constant ruin probability for any time horizon n.
• If time horizon n grows, then the ruin probability grows for every fixed u ≥ 0.2 and interest rate I 0 = i s . The greater u, the smaller ruin probability. Table 2 implies that with initial capital u ≥ 4 and interest rate I 0 = i s = 0.03 for every b the ruin probability does not exceed 0.05 for time horizon n = 5 and n = 10. This means that without using an insurance the insurer is exposed to bankruptcy with a small probability not exceeding 5%.
In Table 2 the number 1 means that without reinsurance an insurer will have the level of bankruptcy below 5%.
ξ (b) = sup x≥c(b) e R(b)x V (x) ∞ x e R(b)z dV (z) = sup x≥c(b) e R(b)x e −x/b ∞ x e R(b)z 1 b e −z/b dz . (3.4)
We calculate the integral under assumption that bR (b) < 1:
∞ x e R(b)z 1 b e −z/b dz = 1 R (b) − 1/b e (R(b)−1/b)z z=∞ z=x = 1 1 − bR (b) e (R(b)−1/b)x .
After substitution to (3.4) we have
ξ (b) = sup x≥c(b) e (R(b)−1/b)x 1 1−bR(b) e (R(b)−1/b)x . Hence ξ (b) = 1 − bR (b) .1 1 − bR (b) = e R(b)c(b)
from which we determine R (b).
Based on Theorem 2, the upper estimation of the ruin probability has the form
Ψ b n (u, i s ) ≤ (1 − bR (b)) l ∑ t=1 p st e −R(b)u(1+i t ) , n = 1, 2, . . . (3.6)
Let us denote the right-hand-side of the inequality (3.6) by g b (u, i s ). Figure 1 depictes graphs of Ψ b n (u, i s ) for an exponential distribution for n = 5 and n = 10, for each one for b = 0.2, 0.4, . . ., 1.0 and for i 2 = 0.05. In Figure 2 graphs of Ψ b n (u, i s ) for n = 5 and n = 10 were depicted, for u = 1, 2, 3, 4, 5 and for i 2 = 0.05. Graphs for i 1 = 0.03 are almost the same so we omit them. The differences are easy to observe in Table 1.
Pareto distribution
We assume that the total loss Z n has Pareto distribution with the distribution function
W (x) = 1 − β x α (3.7)
for x ≥ β > 0. The random variable Z n has the expectation E X = αβ α − 1 for α > 1 and a variance
Var X = αβ 2 (α − 1) 2 (α − 2) for α > 2.
We assume that E Z n = 1. Hence the parameter β must be in the form
β = α − 1 α .
The loss Z ce n = bZ n retained by insurer has cdf In the numerical calculations we assume α = 1.25 similarly to the paper by Palmowski [8]. In this paper it was show that the greatest losses which came out at the end of eighties and nineties of XX century have Pareto distribution with the parameter approximately equal to 1.24138. With such a value of α the variance is infinite.
V (x) = 1 − bβ x α (3.8) for x ≥ bβ . ✻ ✲ u Ψ b n (u,From (2.5) we have c (b) = (1 + η) b − (η − θ ) .
The function Ψ b 1 (u, i s ) can be set by (2.3) in explicit form only for n = 1, l = 1 and
i 1 = 0. Ψ b 1 (u) = bβ u + θ + b (η + 1) − η α (3.9)
The cases n > 1 need numerical integrations. Let us consider the case n = 2. In this case it is necessary to calculate the integral
α(bβ ) α x+c(b) b β bβ u + θ + b (η + 1) − η − z α z −(α+1) dz.
Substituting A = u + θ + b (η + 1) − η we come to the problem of the calculation of the integral
1 (A − z) α z α+1 dz = − (1 − z/A) 2 F 1 (−α, α; 1; 1 − α, x/A) α (A − z) α z α ,
where 2 F 1 (a, b; c; z) is the hypergeometric function. Table 3 gives the same conclusion as for exponential distribution. Word "lack" in Table 4 means that for any level of retention b ∈ (0.2, 1] with initial capital u = 1, the ruin probability exceeds 0.05 both for a five-years-time horizon and for a ten-year-time horizon. In Figure 3 graphs of Ψ b n (u, i s ) for n = 5 and n = 10 for Pareto distribution were depicted for i 2 = 0.05. In Figure 4 graphs of Ψ b n (u, i s ) for n = 5 and n = 10, for u = 1, 2, 3, 4, 5 and i 2 = 0.05. Graphs for i 1 = 0.03 are almost the same so we omit them. The differences are easy to observe in Table 3.
Taking an advantage from Theorem 3 we will present the results concerning an approximation of ruin probability for Pareto distribution. In Figure 5 the ratio
Ψ b n (u, i s ) c n (i s )V (u)
Conclusions
In the continuous risk process the optimal level of retention can be determined by maximising of an adjustment coefficient relative to the level of retention. In the discrete risk process the above statement is not true. For the fixed initial capital u ≥ 1 the probability of ruin is an increasing function of the retention level b. Therefore the probability of the ruin is minimal if the retention level is minimal. It means that an insurer retains only very low losses which causes very low income and is very unfavourable for him. It seems that the right approach relies on fixing an acceptable level of the ruin probability, and appropriately to this probability, determining the retention level. bound is very imprecise, and basically it is worthless. For the heavy tailed claims we give the theorem about the approximation of the ruin probability if the initial capital is sufficiently large. The example of Pareto distribution shows that such an approximation is appropriate and quickly tends to the limit value.
Theorem 3 .
3Let total loss Z n have cdf W ∈ R −α for some α > 0. If 1 + I n > 0 for any fixed I 0 = i s there exists a finite positive moment of rank α of discounting factor (1 + I 1 ) −1 , then for a proportional reinsurance for every I 0 = i s and every n we haveΨ b n (u, i s ) ∼ c n (i s )V (u) ,(2.13) if u → ∞, where c n (i s ) are given recursively c n (i s ) = E (1 + c n−1 (I 1
The values η = 0.25 and θ = 0.2 were taken. For E h (Z n , b) = b from (2.5) we obtain the formula
2 :
2Maximal level of retention b, for which the ruin probability does not exceed 0.05 for exponential distribution. the parameter ξ (b) from Equation (2.8) for V (x) defined by (3.1):
parameter b > 1 − θ /η the adjustment coefficient R (b) is the positive solution of Equation (2.6). Since the moment generating function V (x) has the form M (z) = 1 1 − bz ,where z < 1/b, then Equation (2.6) has the form
Fig. 1 :
1Ruin probability for exponential distribution as a function of u. Ψ b 5 (u, 0.05) -thin line, Ψ b 10 (u, 0.05) -thick line, from the lowest to the highest for b = 0.2, 0.4 , 0.6 , 0.8, 1.0 respectively.
Fig. 2 :
2Ruin probability for exponential distribution as a function of b. Ψ b 5 (u, 0.05) -thin line, Ψ b 10 (u, 0.05) -thick line, from the highest do the lowest for u = 1, 2, 3, 4, 5 respectively.
for n = 3, b = 0.2, 0.4, . . ., 1.0 and 0 ≤ u ≤ 20 was depicted.
Fig. 3 :
3Ruin probability for Pareto distribution as a function of u. Ψ b 5 (u, 0.05) -thin line, Ψ b 10 (u, 0.05) -thick line, from the lowest do the highest for b = 0.2, 0.4 , 0.6 , 0.8, 1.0 respectively.
Fig. 4 :Fig. 5 :
45Ruin probability for Pareto distribution as a function of b. Ψ b 5 (u, 0.05)thin line, Ψ b 10 (u, 0.05) -thick line, from the highest do the lowest for 1, 2, 3, 4, 5 respectively.If loading of a reinsurer is greater than loading of an insurer (ξ > θ ), the adjustment coefficient is not a convex function, which lowers the quality of upper estimation. Basing on our numerical examples we conclude that such an upper Asymptotic approximation of the ruin probability for Pareto distribution -graphs Ψ b n (u, i s )/c n (i s )V (u). From the highest do the lowest for b =
Table 1 :
1Values of ruin probabilities for exponential distribution
Table
Table 3 :
3Values of ruin probabilities for Pareto distribution
Table 4 :
4Maximal level of retention b, for which the ruin probability does not exceed 0.05 for Pareto distribution. = 5 i s = 3% 0.2190 0.4052 0.5907 0.7696 0.9621 i s = 5% 0.2468 0.4379 0.6209 0.8133 0.9996 n = 10 i s = 3%Initial capital u
1
2
3
4
5
n lack
0.2567 0.3582 0.4588 0.5588
i s = 5%
lack
0.2884 0.3933 0.4958 0.5974
In this paper the calculations were made by program Maxima: http://maxima.sourceforge.net/ .
AcknowledgementsThe research by Helena Jasiulewicz was supported by a grant from the National Science Centre, Poland.
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"New characterizations for the variation of the spectrum of an arbitrary matrix",
"New characterizations for the variation of the spectrum of an arbitrary matrix"
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"Xuefeng Xu "
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| The celebrated Hoffman-Wielandt theorem reveals the strong stability of the spectrum of a normal matrix under perturbations. Over the past decades, some analogs of the Hoffman-Wielandt theorem have been developed to characterize the stability of the spectrum of an arbitrary matrix. In this paper, we establish new perturbation bounds to characterize the variation of the spectrum of an arbitrary matrix. The counterparts of the existing results are also given, which are sharper than the existing ones. Moreover, our results have generalized some perturbation bounds for the spectrum of a normal matrix. | null | [
"https://arxiv.org/pdf/1703.02422v2.pdf"
]
| 119,141,512 | 1703.02422 | 4ed2888541f9b6b71a86e6cfd4f6fdf63174f88d |
New characterizations for the variation of the spectrum of an arbitrary matrix
October 25, 2018
Xuefeng Xu
New characterizations for the variation of the spectrum of an arbitrary matrix
October 25, 2018Hoffman-Wielandt theoremspectrumperturbation AMS subject classifications: 15A1847A5565F15
The celebrated Hoffman-Wielandt theorem reveals the strong stability of the spectrum of a normal matrix under perturbations. Over the past decades, some analogs of the Hoffman-Wielandt theorem have been developed to characterize the stability of the spectrum of an arbitrary matrix. In this paper, we establish new perturbation bounds to characterize the variation of the spectrum of an arbitrary matrix. The counterparts of the existing results are also given, which are sharper than the existing ones. Moreover, our results have generalized some perturbation bounds for the spectrum of a normal matrix.
Introduction
Let C m×n and U n be the set of all m × n complex matrices and the set of all unitary matrices of order n, respectively. The identity matrix of order n is denoted by I n . For any X ∈ C m×n , let X * , X 2 , and X F denote the conjugate transpose, the spectral norm, and the Frobenius norm of X, respectively. For any M ∈ C n×n , its diagonal part, strictly lower triangular part, and strictly upper triangular part are denoted by D(M ), L(M ), and U(M ), respectively.
For any M ∈ C n×n , we define Let A ∈ C n×n and A = A + E ∈ C n×n (E ∈ C n×n is a perturbation) have the spectra {λ i } n i=1 and { λ i } n i=1 , respectively. For any permutation π of {1, . . . , n}, we define
D 2 := n i=1 λ π(i) − λ i 2 1 2 . (1.3)
If both A ∈ C n×n and A = A + E ∈ C n×n are normal, Hoffman and Wielandt [6] proved that there exists a permutation π of {1, . . . , n} such that
D 2 ≤ E F ,(1.4)
which is the well known Hoffman-Wielandt theorem. Over the past decades, various analogs of the Hoffman-Wielandt theorem have been established to characterize the variation of the spectrum of a matrix (see, e.g., [9,17,2,4,7,10,16,8,12,11,13,14,18,3]
D 2 ≤ √ n E F ,(1.5)
provided that A ∈ C n×n is normal and A = A+E ∈ C n×n is non-normal. In view of the quantity s(·) defined in (1.2), Li and Sun [12, Theorem 2.3] refined the estimate (1.5) and derived that
D 2 ≤ n − s( A) + 1 E F . (1.6)
Recently, Xu and Zhang [18, Theorems 3.6 and 3.10] established that
D 2 ≤ E 2 F + (n − 1)δ(E) 2 , (1.7) D 2 ≤ E 2 F + n − s( A) δ(E) 2 . (1.8)
It is easy to see that the estimates (1.7) and (1.8) are sharper than (1.5) and (1.6), respectively. For more theories on the variation of the spectrum of a normal matrix, we refer to the recent paper [18]. As is well known, for any A ∈ C n×n , there is a nonsingular matrix Q ∈ C n×n such that
Q −1 AQ = diag J 1 , . . . , J p , where each J i ∈ C m i ×m i ( p i=1 m i = n)
is a Jordan block. For any A ∈ C n×n and A = A + E ∈ C n×n , using (1.5), Song [16, Theorem 2.1] derived that
D 2 ≤ √ n √ n − p + 1 E Q 1 m F , if E Q F < 1, √ n √ n − p + 1 E Q F , if E Q F ≥ 1,(1.9)
where m := max 1≤i≤p m i and E Q := Q −1 EQ. Some applications of the estimate (1.9) can be found, e.g., in [5,1,15]. On the basis of (1.6), Li and Chen [11, Theorem 2.1] restudied the variation of the spectrum of an arbitrary matrix and proved that
D 2 ≤ s 1 n − p + 1 + 2 √ n − p E Q F E Q 1 m F , if E Q F < 1, s 2 n − p + 2 √ n − p + E Q F E Q 1 2 F , if E Q F ≥ 1,(1.10)
where
s 1 = n + 1 − s(T −1 Q −1 AQT ), s 2 = n + 1 − s(Q −1 AQ), T = diag T 1 , . . . , T p , T i = diag 1, ε, . . . , ε m i −1 with ε = E Q 1 m
F . In this paper, we establish some new perturbation bounds for the spectrum of an arbitrary matrix based on the estimates (1.7) and (1.8). For comparison, we here exhibit the counterparts of (1.9) and (1.10) (see Theorems 3.1 and 3.4), i.e.,
D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 , if E Q F < 1, n √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if E Q F ≥ 1, (1.11) D 2 ≤ s 1 n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 , if E Q F < 1, s 2 √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if E Q F ≥ 1.
(1.12)
Theoretical analysis suggests that our results (1.11) and (1.12) are sharper than (1.9) and (1.10), respectively; see Remarks 3.1 and 3.4 for details. Furthermore, if the original matrix A is normal, the estimates (1.11) and (1.12) will reduce to (1.7) and (1.8), respectively. The rest of this paper is organized as follows. In Section 2, we introduce some auxiliary results, which play an important role in our analysis. In Section 3, we develop some new upper bounds to characterize the variation of the spectrum of an arbitrary matrix. Finally, some conclusions are given in Section 4.
Preliminaries
In this section, we introduce some useful lemmas, which play a fundamental role in our analysis.
The first lemma gives an upper bound for L(M ) 2
F + U(M ) 2 F (see [18, Lemma 3.1])
, which is invariant under a unitary similarity transformation.
Lemma 2.1. Let M ∈ C n×n . Then L(M ) 2 F + U(M ) 2 F ≤ δ(M ) 2 , (2.1) where δ(M ) is defined by (1.1).
From the inequality (2.1), we can readily observe that M is a diagonal matrix if δ(M ) = 0. Indeed, M must be a scalar matrix (i.e., M = µI n for some µ ∈ C), which can be seen from the following lemma.
Conversely, if δ(M ) = 0, then M 2 F = 1 n | tr(M )| 2 , that is, i =j |m ij | 2 + n i=1 |m ii | 2 = 1 n n i=1 m ii 2 . Due to n i=1 |m ii | 2 ≥ 1 n n i=1 |m ii | 2 ≥ 1 n n i=1 m ii 2 ,
it follows that m ij = 0 (∀i = j) and m ii = const (∀i = 1, . . . , n), i.e., M is a scalar matrix.
In order to analyze the variation of the spectrum of an arbitrary matrix, we need the following perturbation bounds for the spectrum of a normal matrix (see [18, Theorems 3.6, 3.10, and 4.2]).
Lemma 2.3. Let A ∈ C n×n be a normal matrix with spectrum {λ i } n i=1 , and let A = A+E ∈ C n×n with spectrum { λ i } n i=1 ,
where E ∈ C n×n is a perturbation. Then there exists a permutation π of {1, . . . , n} such that
D 2 ≤ E 2 F + (n − 1)δ(E) 2 , (2.2) D 2 ≤ E 2 F + n − s( A) δ(E) 2 . (2.3)
In particular, if A is Hermitian, then there exists a permutation π of {1, . . . , n} such that
D 2 ≤ E 2 F + δ(E) 2 . (2.4)
For any A ∈ C n×n , there exists a nonsingular matrix Q ∈ C n×n such that
Q −1 AQ = diag J 1 , . . . , J p , (2.5) where each J i ∈ C m i ×m i ( p i=1 m i = n) is a Jordan block with the form J i = λ i 1 0 · · · 0 0 λ i 1 · · · 0 . . . . . . . . . . . . . . . 0 0 · · · λ i 1 0 0 · · · 0 λ i . Let 0 < ε ≤ 1, and let T = diag T 1 , . . . , T p ,
where T i = diag 1, ε, . . . , ε m i −1 for all i = 1, . . . , p. Then we have
T −1 (Q −1 AQ)T = diag T −1 1 J 1 T 1 , . . . , T −1 p J p T p = Λ + Ω,(2.6)
where Λ = diag λ 1 I m 1 , . . . , λ p I mp and Ω = diag Ω 1 , . . . , Ω p with
Ω i = 0 ε 0 · · · 0 0 0 ε · · · 0 . . . . . . . . . . . . . . . 0 0 · · · 0 ε 0 0 · · · 0 0 ∈ C m i ×m i ∀ i = 1, . . . , p.
Under the above settings, we can show that the following lemma holds, which is the foundation of our analysis.
Lemma 2.4. Let A ∈ C n×n have the decomposition (2.5), and let A = A + E, where E ∈ C n×n is a perturbation. Let Λ = diag λ 1 I m 1 , . . . , λ p I mp and T = diag T 1 , . . . , T p , where T i = diag 1, ε, . . . , ε m i −1 for all i = 1, . . . , p. Then, for any 0 < ε ≤ 1, it holds that
T −1 Q −1 AQT − Λ 2 F ≤ Φ(ε), (2.7) where Φ(ε) := ε 2(1−m) δ(E Q ) 2 + 2ε 2 √ n − p δ(E Q ) + (n − p)ε 2 + 1 n | tr(E)| 2 with m = max 1≤i≤p m i .
Proof. From (2.6), we have that
T −1 Q −1 AQT − Λ = T −1 E Q T + Ω,
which yields
T −1 Q −1 AQT − Λ 2 F = T −1 E Q T 2 F + 2 Re tr(Ω * T −1 E Q T ) + Ω 2 F . (2.8) (i) Partitioning E Q as the block form E Q = ( E ij ) p×p with E ij ∈ C m i ×m j , we have T −1 E Q T 2 F = p i=1 p j=1 T −1 i E ij T j 2 F . Then T −1 E Q T 2 F = p i=1 p j=1 m i k=1 m j =1 ε 2( −k) |( E ij ) k, | 2 ≤ ε 2(1−m) i =j E ij 2 F + p i=1 D( E ii ) 2 F + ε 2 U( E ii ) 2 F + ε 2(1−m i ) L( E ii ) 2 F ≤ ε 2(1−m) i =j E ij 2 F + p i=1 U( E ii ) 2 F + p i=1 L( E ii ) 2 F + D(E Q ) 2 F = ε 2(1−m) E Q 2 F − D(E Q ) 2 F + D(E Q ) 2 F = ε 2(1−m) E Q 2 F − ε 2(1−m) − 1 D(E Q ) 2 F . Since D(E Q ) 2 F ≥ 1 n | tr(E)| 2 ,
we get
T −1 E Q T 2 F ≤ ε 2(1−m) δ(E Q ) 2 + 1 n | tr(E)| 2 . (2.9) (
ii) It is easy to see that
Re tr(Ω * T −1 E Q T ) = Re p i=1 tr(Ω * i T −1 i E ii T i ) = Re p i=1 m i j=2 ε(T −1 i E ii T i ) j−1,j . Because (T −1 i E ii T i ) j−1,j = ε( E ii ) j−1,j for all i = 1, .
. . , p and j = 2, . . . , m i , we obtain
Re tr(Ω * T −1 E Q T ) = Re p i=1 m i j=2 ε 2 ( E ii ) j−1,j ≤ ε 2 p i=1 m i j=2 |( E ii ) j−1,j | ≤ ε 2 √ n − p p i=1 m i j=2 |( E ii ) j−1,j | 2 1 2 ≤ ε 2 √ n − p p i=1 U( E ii ) 2 F 1 2 ≤ ε 2 √ n − p U(E Q ) F . Due to U(E Q ) F ≤ δ(E Q ) (see (2.1)), it follows that Re tr(Ω * T −1 E Q T ) ≤ ε 2 √ n − p δ(E Q ).
(2.10)
(iii) Furthermore, we can easily see that
Ω 2 F = (n − p)ε 2 . (2.11)
Combining (2.8), (2.9), (2.10), and (2.11), we can derive the inequality (2.7) immediately.
Main results
In this section, we develop some new upper bounds to characterize the variation of the spectrum of an arbitrary matrix based on the estimate (2.7).
Complex eigenvalues case
In this subsection, we consider the (general) case that the eigenvalues of A ∈ C n×n are complex. Using (2.2) and (2.7), we can obtain the following estimate for D 2 , which is sharper than (1.9).
Theorem 3.1. Let A ∈ C n×n have the decomposition (2.5), and let A = A + E, where E ∈ C n×n is a perturbation. Assume that the spectra of A and A are {λ i } n i=1 and { λ i } n i=1 , respectively. Then there exists a permutation π of {1, . . . , n} such that Proof. Observe that Λ = diag(λ 1 , . . . , λ n ) is normal and the spectrum of
D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 , if E Q F < 1, n √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if E Q F ≥ 1.T −1 Q −1 AQT is { λ i } n i=1
. In view of (2.2), we obtain
D 2 ≤ n T −1 Q −1 AQT − Λ 2 F − n − 1 n | tr(E)| 2 .
By (2.7), we have
D 2 ≤ nΦ(ε) − n − 1 n | tr(E)| 2 . Set ε = E Q 1 m F , if E Q F < 1, 1, if E Q F ≥ 1.
The estimate (3.1) then follows immediately by using the following results:
Φ E Q 1 m F = n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 , Φ(1) = √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 .
This completes the proof.
Remark 3.1. If E Q F < 1, then (3.1) reads D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 .
Due to
n δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 = n E Q 2 F − | tr(E)| 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 ≤ n E Q 2 m F ,
it follows that
D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + 1 E Q 2 m F ≤ √ n √ n − p + 1 E Q 1 m F ,
which is the first estimate in (1.9). On the other hand, if E Q F ≥ 1, then (3.1) reads
D 2 ≤ n √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 .
Then
D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + E Q 2 F ≤ √ n √ n − p + 1 E Q F ,
which is the second estimate in (1.9). Therefore, the estimate (3.1) is sharper than (1.9).
The following two estimates for D 2 are based on the different constraints for E Q .
D 2 ≤ n n − p + 2 √ n − p δ(E Q ) + 1 δ(E Q ) 2 m + 1 n | tr(E)| 2 , if δ(E Q ) < 1, n √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if δ(E Q ) ≥ 1. (3.2) Proof. Set ε = δ(E Q ) 1 m , if δ(E Q ) < 1, 1, if δ(E Q ) ≥ 1.
Direct computation yields
Φ δ(E Q ) 1 m = n − p + 2 √ n − p δ(E Q ) + 1 δ(E Q ) 2 m + 1 n | tr(E)| 2 .
Using the similar argument as in Theorem 3.1, we can derive the estimate (3.2).
D 2 ≤ mn n−p+2 √ n−p δ(E Q ) m−1 1− 1 m δ(E Q ) 2 m + 1 n | tr(E)| 2 , if C 1 holds, n √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if C 2 holds, (3.3)
where the conditions C 1 and C 2 are given by
C 1 : n − p + 2 √ n − p δ(E Q ) > (m − 1)δ(E Q ) 2 , C 2 : n − p + 2 √ n − p δ(E Q ) ≤ (m − 1)δ(E Q ) 2 .
Proof. We first note that A is diagonalizable if and only if n = p, or, equivalently, m = 1.
(i) If A is diagonalizable, then T = I n , n = p, and m = 1. In this case, (2.7) reduces to
Q −1 AQ − Λ 2 F ≤ E Q 2 F .
Using (2.2), we obtain
D 2 ≤ n Q −1 AQ − Λ 2 F − n − 1 n | tr(E)| 2 ≤ n E Q 2 F − n − 1 n | tr(E)| 2 . (3.4)
(ii) If A is not diagonalizable, then n > p and m > 1. Straightforward calculation yields
Φ (ε) = 2ε n − p + 2 √ n − p δ(E Q ) − (m − 1)δ(E Q ) 2 ε 2m ,
where Φ (ε) denotes the derivative of Φ(ε) with respect to ε. Evidently,
Φ (ε) > 0, if ε > (m−1)δ(E Q ) 2 n−p+2 √ n−p δ(E Q ) 1 2m , Φ (ε) < 0, if 0 < ε < (m−1)δ(E Q ) 2 n−p+2 √ n−p δ(E Q ) 1 2m .
Then we set
ε = (m−1)δ(E Q ) 2 n−p+2 √ n−p δ(E Q ) 1 2m , if n − p + 2 √ n − p δ(E Q ) > (m − 1)δ(E Q ) 2 , 1, if n − p + 2 √ n − p δ(E Q ) ≤ (m − 1)δ(E Q ) 2 .
Direct computation yields
Φ (m − 1)δ(E Q ) 2 n − p + 2 √ n − p δ(E Q ) 1 2m = m n − p + 2 √ n − p δ(E Q ) m − 1 1− 1 m δ(E Q ) 2 m + 1 n | tr(E)| 2 .
The rest of the proof is similar to Theorem 3.1.
Remark 3.2. We remark that (3.3) has contained the diagonalizable case (i). More specifically, if A is diagonalizable, then the condition C 2 is satisfied. From (3.3), we have that
D 2 ≤ n δ(E Q ) 2 + 1 n | tr(E)| 2 ,
which is consistent with (3.4).
Remark 3.3. If A ∈ CD 2 ≤ s 1 n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 , if E Q F < 1, s 2 √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if E Q F ≥ 1,(3.
5)
where
s 1 = n + 1 − s(T −1 Q −1 AQT ) with ε = E Q 1 m
F and s 2 = n + 1 − s(Q −1 AQ).
Remark 3.4. If E Q F < 1, then (3.5) reads
D 2 ≤ s 1 n − p + 2 √ n − p δ(E Q ) + δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 .
Because
s 1 δ(E Q ) 2 E Q 2 F E Q 2 m F + 1 n | tr(E)| 2 ≤ s 1 E Q 2 m F − s 1 − 1 n | tr(E)| 2 ≤ s 1 E Q 2 m F , it follows that D 2 ≤ s 1 n − p + 2 √ n − p E Q F + 1 E Q 1 m
F , which is the first estimate in (1.10). On the other hand, if E Q F ≥ 1, then (3.5) reads
D 2 ≤ s 2 √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 . Since n − p + 2 √ n − p δ(E Q ) ≤ n − p + 2 √ n − p E Q F and s 2 δ(E Q ) 2 + 1 n | tr(E)| 2 ≤ s 2 E Q 2 F , we obtain D 2 ≤ s 2 n − p + 2 √ n − p + E Q F E Q 1 2
F . which is the second estimate in (1.10). In conclusion, the estimate (3.5) is sharper than (1.10).
Theorem 3.5. Under the assumptions of Theorem 3.1, there exists a permutation π of {1, . . . , n} such that
D 2 ≤ s 3 n − p + 2 √ n − p δ(E Q ) + 1 δ(E Q ) 2 m + 1 n | tr(E)| 2 , if δ(E Q ) < 1, s 2 √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if δ(E Q ) ≥ 1, (3.6) where s 3 = n + 1 − s(T −1 Q −1 AQT ) with ε = δ(E Q ) 1 m and s 2 = n + 1 − s(Q −1 AQ).
Theorem 3.6. Under the assumptions of Theorem 3.1, there exists a permutation π of {1, . . . , n} such that
D 2 ≤ ms 4 n−p+2 √ n−p δ(E Q ) m−1 1− 1 m δ(E Q ) 2 m + 1 n | tr(E)| 2 , if C 1 holds, s 2 √ n − p + δ(E Q ) 2 + 1 n | tr(E)| 2 , if C 2 holds,
Real eigenvalues case
If the eigenvalues of A ∈ C n×n are all real, then we can get the following more accurate estimates by applying (2.4).
Theorem 3.7. Let A ∈ C n×n have the decomposition (2.5), and let A = A + E, where E ∈ C n×n is a perturbation. Let {λ i } n i=1 and { λ i } n i=1 be the spectra of A and A, respectively. If λ i are all real, then there exists a permutation π of {1, . . . , n} such that
s
: U * M U = diag(M 1 , . . . , M s ) and each M i is square , (1.2) where tr(M ) denotes the trace of M . Obviously, δ(M ) ≤ M F and 1 ≤ s(M ) ≤ n. Moreover, δ(M ) = M F if and only if tr(M ) = 0, and s(M ) = n if and only if M is normal, namely, M M * = M * M .
Lemma 2 . 2 .
22Let M = (m ij ) ∈ C n×n . Then δ(M ) = 0 if and only if M is a scalar matrix. Proof. If M = µI n , then M 2 F = n|µ| 2 and tr(M ) = nµ. By definition (1.1), we have δ(M ) = 0.
Theorem 3. 2 .
2Under the assumptions of Theorem 3.1, there exists a permutation π of {1, . . . , n} such that
Theorem 3. 3 .
3Under the assumptions of Theorem 3.1, there exists a permutation π of {1, . . . , n} such that
4 = n + 1 − s(T −1 Q −1 AQT ) with ε = s 2 = n + 1 − s(Q −1 AQ),and the conditions C 1 and C 2 are the same as in Theorem 3.3.Remark 3.5. If A ∈ C n×n is normal, then (3.5), (3.6), and (3.7) will reduce toD 2 ≤ n + 1 − s( A) δ(E) 2 + 1 n | tr(E)| 2 ,which is exactly the estimate (2.3).
Example 3. 1 .
1For any A ∈ C n×n , taking E = tI n with 0 < |t| < 1 √ n , we have that D 2 ≡ √ n|t| for any permutation π of {1, . . . , n}. In this case, the aforementioned upper bounds for D 2 are listed as follows:
we can derive some deductive estimates for D 2 . In addition, using D ∞ ≤ D 2 , we can obtain the corresponding estimates for D ∞ .4 ConclusionsIn this paper, we have developed some novel perturbation bounds for the spectrum of an arbitrary matrix, which include the counterparts of the existing ones. Theoretical analysis shows that these counterparts are sharper than the existing estimates. Furthermore, our results have generalized some perturbation bounds for the spectrum of a normal matrix.
AcknowledgmentsThe author is grateful to Professor Chen-Song Zhang for his helpful suggestions.
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A note on unifying absolute and relative perturbation bounds. I C F Ipsen, Linear Algebra Appl. 358I. C. F. Ipsen. A note on unifying absolute and relative perturbation bounds. Linear Algebra Appl., 358:239-253, 2003.
Spectra of nearly Hermitian matrices. W Kahan, Proc. Amer. Math. Soc. 48W. Kahan. Spectra of nearly Hermitian matrices. Proc. Amer. Math. Soc., 48:11-17, 1975.
Relative perturbation theory: I. eigenvalue and singular value variations. R.-C Li, SIAM J. Matrix Anal. Appl. 19R.-C. Li. Relative perturbation theory: I. eigenvalue and singular value variations. SIAM J. Matrix Anal. Appl., 19:956-982, 1998.
The eigenvalue perturbation bound for arbitrary matrices. W Li, J.-X Chen, J. Comput. Math. 24W. Li and J.-X. Chen. The eigenvalue perturbation bound for arbitrary matrices. J. Comput. Math., 24:141-148, 2006.
The perturbation bounds for eigenvalues of normal matrices. W Li, W Sun, Numer. Linear Algebra Appl. 12W. Li and W. Sun. The perturbation bounds for eigenvalues of normal matrices. Numer. Linear Algebra Appl., 12:89-94, 2005.
Combined perturbation bounds: I. eigensystems and singular value decompositions. W Li, W Sun, SIAM J. Matrix Anal. Appl. 29W. Li and W. Sun. Combined perturbation bounds: I. eigensystems and singular value decompositions. SIAM J. Matrix Anal. Appl., 29:643-655, 2007.
On the variation of the spectrum of a Hermitian matrix. W Li, S.-W Vong, Appl. Math. Lett. 65W. Li and S.-W. Vong. On the variation of the spectrum of a Hermitian matrix. Appl. Math. Lett., 65:70-76, 2017.
Computing characteristic polynomials from eigenvalues. R Rehman, I C F Ipsen, SIAM J. Matrix Anal. Appl. 32R. Rehman and I. C. F. Ipsen. Computing characteristic polynomials from eigenvalues. SIAM J. Matrix Anal. Appl., 32:90-114, 2011.
A note on the variation of the spectrum of an arbitrary matrix. Y Song, Linear Algebra Appl. 342Y. Song. A note on the variation of the spectrum of an arbitrary matrix. Linear Algebra Appl., 342:41-46, 2002.
On the variation of the spectrum of a normal matrix. J.-G Sun, Linear Algebra Appl. 246J.-G. Sun. On the variation of the spectrum of a normal matrix. Linear Algebra Appl., 246:215-223, 1996.
New perturbation bounds for the spectrum of a normal matrix. X Xu, C.-S Zhang, J. Math. Anal. Appl. 455X. Xu and C.-S. Zhang. New perturbation bounds for the spectrum of a normal matrix. J. Math. Anal. Appl., 455:1937-1955, 2017.
| []
|
[
"Fluctuations of the largest fragment size in percolation and multifragmentation",
"Fluctuations of the largest fragment size in percolation and multifragmentation"
]
| [
"J Brzychczyk \nInstitute of Physics\nJagiellonian University\n30-059KrakówPoland\n",
"T Pietrzak \nInstitute of Physics\nJagiellonian University\n30-059KrakówPoland\n",
"A Wieloch \nInstitute of Physics\nJagiellonian University\n30-059KrakówPoland\n",
"W Trautmann \nD-64291Darmstadt, DarmstadtGermany\n"
]
| [
"Institute of Physics\nJagiellonian University\n30-059KrakówPoland",
"Institute of Physics\nJagiellonian University\n30-059KrakówPoland",
"Institute of Physics\nJagiellonian University\n30-059KrakówPoland",
"D-64291Darmstadt, DarmstadtGermany"
]
| []
| Aladin data on fragmentation of 197 Au projectiles are remarkably well reproduced by a bond percolation model. A critical behavior is identified on the basis of fluctuations of the largest fragment size. | null | [
"https://arxiv.org/pdf/1003.2327v1.pdf"
]
| 118,918,385 | 1003.2327 | 774e9f60fe04052b872e24aec95f60de5600c3ee |
Fluctuations of the largest fragment size in percolation and multifragmentation
11 Mar 2010
J Brzychczyk
Institute of Physics
Jagiellonian University
30-059KrakówPoland
T Pietrzak
Institute of Physics
Jagiellonian University
30-059KrakówPoland
A Wieloch
Institute of Physics
Jagiellonian University
30-059KrakówPoland
W Trautmann
D-64291Darmstadt, DarmstadtGermany
Fluctuations of the largest fragment size in percolation and multifragmentation
11 Mar 2010
Aladin data on fragmentation of 197 Au projectiles are remarkably well reproduced by a bond percolation model. A critical behavior is identified on the basis of fluctuations of the largest fragment size.
The present work is motivated by percolation studies [1] which demonstrated that a cumulant analysis of the largest fragment size distributions is a valuable tool in searching for a phase transition (critical behavior) in fragmenting systems. This method is applicable even to very small systems and therefore is well suited for applications to nuclear multifragmentation.
The simulations were performed with a three-dimensional bond percolation model on simple cubic lattices with free boundary conditions to account for the surface presence. Given a bond probability value p (control parameter) and the total number of sites Z 0 (system size), the probability distribution P (Z max ) of the largest cluster size Z max is determined from a large sample of events. The statistical measures as the mean, variance, skewness and kurtosis contain the most significant information about the distribution. Of particular interest are the following cumulant ratios
K 2 ≡ µ 2 / Z max 2 = κ 2 /κ 2 1 K 3 ≡ µ 3 /µ 3/2 2 = κ 3 /κ 3/2 2 K 4 ≡ µ 4 /µ 2 2 − 3 = κ 4 /κ 2 2 ,(1)
where Z max denotes the mean value, µ i = (Z max − Z max ) i is the ith central moment, and κ i is the ith cumulant of P (Z max ). K 2 is the variance normalized to the squared mean, K 3 is the skewness which indicates the distribution asymmetry, and K 4 is the kurtosis excess measuring the degree of peakedness with respect to the normal distribution. In the transition region, these quantities obey with good accuracy finite-size scaling relations even for very small systems with open boundaries. This allows to identify universal (independent of the system size) features of K i at the percolation transition. The transition in finite systems (the pseudocritical point) is associated with the broadest and most symmetric P (Z max ) distribution, which is indicated by K 3 = 0 and the minimum value of K 4 of about −1. This criticality signal is approximately preserved when events are sorted by measurable variables correlated with the control parameter (e.g. Z bound ) [1]. The present work compares percolation predictions with the Aladin S114 data on fragmentation of projectile spectators in 197 Au + Au, In, Cu collisions at the incident energies of 600-1000 AMeV. Details of the experiment and general characteristics of the data were presented in [2]. Figure 1 examines the cumulants ratios K i of the largest fragment size distribution P (Z max ). The percolation results are plotted in the left diagrams as a function of Z bound normalized to the system size Z 0 for three different system sizes that span over a range expected in the transition region. Percolation events were generated for the bond probabilities uniformly distributed in the interval [0, 1], and then sorted according to Z bound . As one can see, the pseudocritical point is located at Z bound /Z 0 = 0.84 independently of the system size. The experimental results are shown in the right diagrams for the Au + Au systems at 600 and 1000 AMeV and for the summed data sets (all targets and energies). Here, K i are plotted as a function of Z bound . The percolation and experimental patterns of the cumulants are very similar. In particular, the percolation pseudocritical point is well reflected in the data at Z bound = 54. Based on this comparison, the (mean) system size at Z bound = 54 can be estimated as Z 0 = Z bound /0.84 ≃ 64. Once the system size is established, we can examine other observables related to fragment charge partitions. In Fig. 2 we compare percolation results with the data at the pseudocritical point Z bound = 54. Panel (a) shows the fragment size distribution. The model well describes the data over four orders of magnitude. As expected for the percolation pseudocritical point, the distribution follows in some range the asymptotic power-law dependence shown by the dotted line [1]. Panel (b) shows the Zipf-type plot, i.e. the mean size of the largest, second largest,... r-largest fragments plotted against their rank r. The percolation model very well reproduces not only mean values but also event-to-event fluctuations. For example, the next panels show the multiplicity distribution of fragments with Z > 2 and the distribution of the second largest fragment charge.
Similar comparisons performed at other Z bound values in the range from 36 to 66 have also shown an almost perfect resemblance between percolation predictions and the data. The system sizes determined in the analysis are in a good agreement with the experimental estimates [3].
In conclusions, fluctuations of the largest fragment charge observed in the 197 Au spectator fragmentations show the percolation pattern. In analogy to percolation, the pseudocritical point is identified at Z bound = 54 which corresponds to the He-Li isotope temperature corrected for secondary decays of 5.2 ± 0.4 MeV [4]. Detailed comparisons have demonstrated that the experimental fragment charge partitions are remarkably well reproduced by the bond percolation model with no free parameters and no corrections for secondary decays in a wide range of Z bound . In the context of the lattice gas model which is equivalent to bond percolation at the normal density, the success of percolation suggests that clusters are formed at the dense medium and isolated fragments are cold, in line with the "Little big bang" scenario of multifragmentation [5].
This work has been supported by the Polish Ministry of Science and Higher Education grant N202 160 32/4308 (2007-2009).
Figure 1 :
1The cumulant ratios K i of the P (Z max ) distribution as a function of Z bound . Bond percolation calculations versus the experimental data.
Figure 2 :
2Percolation predictions versus the experimental data at Z bound = 54: (a) the mean fragment multiplicity as a function of the fragment size (the largest fragment excluded), (b) the mean fragment size as a function of the fragment rank, (c) the multiplicity distribution of fragments with Z > 2, (d) the distribution of the second largest fragment charge.
. J Brzychczyk, Phys. Rev. C. 7324601BRZYCHCZYK J., Phys. Rev. C 73 (2006) 024601.
. Et Schüttauf A, Al, Nucl. Phys. 607457SCHÜTTAUF A. ET AL., Nucl. Phys., A607 (1996) 457.
. Pochodzalla J. Et Al, Phys. Rev. Lett. 751040POCHODZALLA J. ET AL., Phys. Rev. Lett. 75 (1995) 1040.
. Et Trautmann W, Al, Phys. Rev. C. 7664606TRAUTMANN W. ET AL., Phys. Rev. C 76 (2007) 064606.
. Campi X, H Krivine, E Plagnol, Sator N, Phys. Rev. C. 6744610CAMPI X., KRIVINE H., PLAGNOL E., SATOR N., Phys. Rev. C 67 (2003) 044610.
| []
|
[
"HOMOGENIZATION FOR A MULTI-SCALE MODEL OF MAGNETORHEOLOGICAL SUSPENSION",
"HOMOGENIZATION FOR A MULTI-SCALE MODEL OF MAGNETORHEOLOGICAL SUSPENSION"
]
| [
"Grigor Nika ",
"Bogdan Vernescu "
]
| []
| []
| Using the homogenization method we obtain a model describing the behavior of the suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. We use the quasi-static Maxwell equations coupled with the Stokes equations to capture the magnetorheological (MR) effect. The model generalizes the one introduced by Neuringer and Rosensweig[14], for quasistatic phenomena. The macroscopic constitutive properties are given explicitly in terms of the solutions of the local problems. We determine the homogenized constitutive parameters for an aqueous MR fluid with magnetite particles using the finite element method. The Poiseuille flow, for the solution of our homogenized coupled system, approaches the Bingham flow profile for large values of the magnetic field. The stress-strain curves obtained for the Couette flow exhibit a yield stress close to the one determined experimentally.Date: 2018/04/09. | null | [
"https://arxiv.org/pdf/1804.02066v1.pdf"
]
| 55,980,852 | 1804.02066 | db07c4bfc5a263ad102e9d71d86323907592ed48 |
HOMOGENIZATION FOR A MULTI-SCALE MODEL OF MAGNETORHEOLOGICAL SUSPENSION
5 Apr 2018
Grigor Nika
Bogdan Vernescu
HOMOGENIZATION FOR A MULTI-SCALE MODEL OF MAGNETORHEOLOGICAL SUSPENSION
5 Apr 20181and phrases SuspensionsStokes flowMR fluids
Using the homogenization method we obtain a model describing the behavior of the suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. We use the quasi-static Maxwell equations coupled with the Stokes equations to capture the magnetorheological (MR) effect. The model generalizes the one introduced by Neuringer and Rosensweig[14], for quasistatic phenomena. The macroscopic constitutive properties are given explicitly in terms of the solutions of the local problems. We determine the homogenized constitutive parameters for an aqueous MR fluid with magnetite particles using the finite element method. The Poiseuille flow, for the solution of our homogenized coupled system, approaches the Bingham flow profile for large values of the magnetic field. The stress-strain curves obtained for the Couette flow exhibit a yield stress close to the one determined experimentally.Date: 2018/04/09.
Introduction
Magneto-rhelogical (MR) fluids are a suspension of non-colloidal, ferromagnetic particles in a non-magnetizable carrier fluid. The particles are often of micron size ranging anywhere from 0.05 − 10 µm with particle volume fraction from 10 − 40 %. They were discovered by J. Rabinow in 1948 [16]. Around the same time W. Winslow discovered electrorheological (ER) fluids, a closely related counterpart.
MR fluids respond to an external magnetic field by a rapid, reversible change in their properties. They can transform from a liquid to a semi solid state in a matter of milliseconds. Upon the application of a magnetic field, the dipole interaction of adjacent particles aligns the particles in the direction of the magnetic field lines. Namely particles attract one another along the magnetic field lines and repel one another in the direction perpendicular to them. This leads to the formation of aggregate structures. Once these aggregate structures are formed, the MR fluid exhibits a yield stress that is dependent and controlled by the applied external magnetic field [11], [5].
The formation of these aggregates means that the behavior of the fluid is non-Newtonian. In many works, the Bingham constitutive law is used an an approximation to model the response of the MR and ER fluids, particularly in shear experiments [15], [4], [6]. Although the Bingham model has proven itself useful in characterizing the behavior of MR fluids, it is not always sufficient. Recent experimental data show that true MR fluids exhibit departures from the Bingham model [22], [6].
Another member of the magnetic suspensions family are ferrofluids. Ferrofluids are stable colloidal suspensions of nanoparticles in a non-magnetizable carrier fluid. The initiation into the hydrodynamics of ferrofluids began with Neuringer and Rosensweig in 1964 [14] and by a series of works by Rosensweig and co-workers summarized in [17]. The model introduced in [14] assumes that the magnetization is collinear with the magnetic field and has been very useful in describing quasi-stationary phenomena. This work was extended by Shliomis [20] by avoiding the collinearity assumption of the magnetization and the magnetic field and by considering the rotation of the nanoparticles with respect to the fluid they are suspended in.
The models mentioned above have all been derived phenomenologically. The first attempt to use homogenization mechanics to describe the behavior of MR \ ER fluids was carried out in [8], [9] and [15]. In the works [8], [9] the influence of the external magnetic field is introduced as a volumic density force acting on each particle and as a surface density force acting on the boundary of each particle. The authors in [15] extend the work in [9], for ER fluids, by presenting a more complete model that couples the conservation of mass and momentum equations with Maxwell's equations through the Maxwell stress tensor. As an application they consider a uniform shearing of the ER fluid submitted to a uniform electric field boundary conditions in a two dimensional slab and they recover that the stress tensor at the macroscopic scale has exactly the form of the Bingham constitutive equation.
The authors in [15], [18], [17] use models that decouple the conservation of mass and momentum equations from the Maxwell equations. Thus in principle one can solve the Maxwell equations and use the resulting magnetic or electric field as a force in the conservation of mass and momentum equations.
The present work focuses on a suspension of rigid magnetizable particles in a Newtonian viscous fluid with an applied external magnetic field. We assume the fluid to be electrically non-conducting. Thus, we use the quasi-static Maxwell equations coupled with the Stokes equations through Ohm's law to capture the magnetorheological effect. In doing so we extend the model of [15]. Thus the Maxwell and the balance of mass and momentum equations must be simultaneously solved.
In Section 1. we introduce the problem in the periodic homogenization framework. The particles are periodically distributed and the size of the period is of the same order as the characteristic length of the particles. We assume the fluid velocity is continuous across the particle interface and that the particles are in equilibrium in the presence of the magnetic field.
The two scale expansion is carried out in Section 2. where we obtain a decoupled set of problems at order O( −1 ).
In Section 3. and in Section 4. we study the local problems that arise from the contribution of the bulk magnetic field as well as the bulk velocity and provide new constitutive laws for Maxwell's equations.
In Section 5. we provide the governing effective equations of the MR fluid which include, in addition to the viscous stresses, a "Maxwell type" stress. Furthermore, we provide formulas for the effective viscosity and effective magnetic permeabilities for the Maxwell type stress that generalize those in [9]. Section 6. is devoted to comparing the results of the proposed model against experimental data. We compute the constitutive coefficients for an aqueous MR fluid with magnetite particles using the finite element method, we obtain the velocity profiles of both Poiseuille and Couette flows for this MR fluid and plot the stress vs shear rate curve for different values of the applied magnetic field, that exhibit a yield stress comparable to the one obtained in experiments (e.g. [22]) Notation. Throughout the paper we are going to be using the following notation: I indicates the n × n identity matrix, e(u u u) will indicate the strain rate tensor defined by, e(u u u) = 1 2 ∇u u u + ∇u u u , where often times we will use subscript to indicate the variable of differentiation. The inner product between matrices is denoted by A:B = tr(A B) = ij a ij b ji and throughout the paper we employ the Einstein summation notation for repeated indices.
Problem statement
For the homogenization setting of the suspension problem we define Ω ⊂ R n , n = 2, 3, to . We now define the following subsets of Ω:
Ω 1 = ∈N T , Ω 2 = Ω\Ω 1 .
In what follows T will represent the magnetizable rigid particles, Ω 1 is the domain occupied by the rigid particles and Ω 2 the domain occupied by the surrounding fluid of viscosity ν. By n n n we indicate the unit normal on the particle surface pointing outwards and by · we indicate the jump discontinuity between the fluid and the rigid part.
Ω Ω 1 Figure 1. Schematic of the periodic suspension of rigid magnetizable particles in non-magnetizable fluid
Ω 2 x c T Y
The description of the problem is, where ρ is the density of the fluid, v v v represents the velocity field, p the pressure, e(v v v ) the strain rate, f f f the body forces, n n n the exterior normal to the particles, H H H the magnetic field, µ is the magnetic permeability of the material, µ (x x x) = µ 1 if x x x ∈ Ω 1 and µ (x x x) = µ 2 if x x x ∈ Ω 2 , η the electric conductivity of the rigid particles, and b b b is an applied constant magnetic field on the exterior boundary of the domain Ω. When the MR fluid is submitted to a magnetic field, the rigid particles are subjected to a force that makes them behave like a dipole aligned in the direction of the magnetic field. This force can be written in the form,
ρ ∂ v v v ∂ t + ρ (v v v · ∇)v v v − div σ = ρ f f f , where σ = 2 ν e(v v v ) − p I in Ω 2 , (1.1a) div v v v = 0, div B B B = 0, curl H H H = 0 0 0 in Ω 2 , (1.1b) e(v v v ) = 0, div B B B = 0, curl H H H = η v v v × B B B in Ω 1 ,(1.F F F = − 1 2 |H H H | 2 ∇µ ,
where | · | represents the standard Euclidean norm. The force can be written in terms of the Maxwell stress
τ ij = µ H i H j − 1 2 µ H k H k δ ij as F F F = div τ + B B B × curl H H H .
Since the magnetic permeability is considered constant in each phase, it follows that the force is zero in each phase. Therefore, we deduce that
div τ = 0 if x x x ∈ Ω 2 −B B B × curl H H H if x x x ∈ Ω 1 . (1.4)
Lastly, we remark that unlike the viscous stress σ , the Maxwell stress is present in the entire domain Ω. Hence, we can write the balance of forces and torques in each particle as,
T ρ du u u dt dx x x = S (σ n n n + τ n n n ) ds + T B B B × curl H H H dx x x + T ρ f f f dx x x, T ρ(x x x − x x x c ) × du u u dt dx x x = S (σ n n n + τ n n n ) × (x x x − x x x c ) ds + T (B B B × curl H H H ) × (x x x − x x x c ) dx x x + T ρ f f f × (x x x − x x x c ) dx x x, (1.5)
where x x x c is the center of mass of the rigid particle T .
Dimensional Analysis.
Before we proceed further we non-dimensionalize the problem. Denote by t
* = t/ L V , x * = x/L, v v v * = v v v/V , p * = p/ν V L , H H H * = H H H/H, f f f * = f f f / V 2 L , and µ * = µ /µ 2 .
Here L is a characteristic length, V is a characteristic velocity, p is a characteristic pressure, f f f is a characteristic force and H is a characteristic unit of the magnetic field. Substituting the above expressions into (1.1) as well as in the balance of forces and torques, and using the fact that the flow is assumed to be at low Reynolds numbers, we obtain
Re ∂ v v v * ∂ t + (v v v * · ∇)v v v * − div * σ * = Re f f f * , where σ * = 2 e(v v v * ) − p * I in Ω 2 , div * v v v * = 0, div * B B B * = 0, curl * H H H * = 0 0 0 in Ω 2 , e * (v v v * ) = 0, div * B B B * = 0, curl H H H * = R m v v v * × B B B * in Ω 1 ,
where B B B * = µ * H H H * and with boundary conditions on the surface of each particle T ,
v v v * = 0 0 0, B B B * · n n n = 0, n n n × H H H * = 0 0 0 on S , v v v * = 0 0 0, H H H * = b b b * on ∂Ω.
together with the balance of forces and torques,
Re T du u u * dt * dx x x * = S σ * n n n ds * + α S τ * n n n ds * + α T B B B * × curl H H H * dx x x * + Re T f f f * dx x x * , Re T (x x x * − x x x * c ) × du u u * dt * dx x x * = S σ * n n n × (x x x * − x x x c * ) ds * + α S τ * n n n × (x x x * − x x x c * ) ds * + α T (B B B * × curl H H H * ) × (x x x * − x x x c * ) dx x x * + Re T f f f * × (x x x * − x x x c * ) dx x x * , where Re = ρ V L ν is the Reynolds number, α = µ 2 H 2 L ν V is the Alfven number, and R m = η µ 1 L V is the magnetic Reynolds number.
In what follows we drop the star for simplicity. Moreover, for low Reynolds numbers the preceding equations become,
−div σ = 0 0 0, where σ = 2 e(v v v ) − p I in Ω 2 , (1.6a) div v v v = 0, div H H H = 0, curl H H H = 0 0 0 in Ω 2 , (1.6b) e(v v v ) = 0, div H H H = 0, curl H H H = R m v v v × B B B in Ω 1 , (1.6c) with boundary conditions v v v = 0 0 0, B B B · n n n = 0, n n n × H H H = 0 0 0 on S , v v v = 0 0 0, H H H = b b b on ∂Ω, (1.7)
together with the balance of forces and torques, 0 = S σ n n n ds + α S τ n n n ds + α
T B B B × curl H H H dx x x, 0 = S σ n n n × (x x x − x x x c ) ds + α S τ n n n × (x x x − x x x c ) ds + α T (B B B × curl H H H ) × (x x x − x x x c ) dx x x.
(1.8)
In the next section we will use a two scale expansion on the velocity, pressure and the magnetic field.
Two scale expansions
We assume the particles are periodically distributed in Ω and thus consider the two scale
expansion on v v v , H H H and p , v v v (x x x) = +∞ i=0 i v v v i (x x x, y y y), H H H (x x x) = +∞ i=0 i H H H i (x x x, y y y), p (x x x) = +∞ i=0 i p i (x x x, y y y) with y y y = x x x .
where x x x ∈ Ω and y y y ∈ R n . One can show that v v v 0 is independent of y y y and can thus obtain the following problem at order −1 ,
− ∂σ 0 ij ∂y j = 0 in Y f , (2.1a) σ 0 ij = −p 0 δ ij + 2 ν (e ijx (v v v 0 ) + e ijy (v v v 1 )) (2.1b) ∂v 0 j ∂x j + ∂v 1 j ∂y j = 0 in Y f , (2.1c) e ijx (v v v 0 ) + e ijy (v v v 1 ) = 0 in T, (2.1d) ∂B 0 j ∂y j = 0, ijk ∂H 0 k ∂y j = 0 where B 0 i = µH 0 i in Y, (2.1e) with boundary conditions v v v 1 = 0 0 0, B B B 0 · n n n = 0, n n n × H H H 0 = 0 0 0 on S , v v v 1 , H H H 0 are Y − periodic. (2.2)
Here Y f and T denote the fluid, respectively the particle part of Y ; and S denotes the surface of T . At order of 2 and 3 we obtain from (1.8) the balance of forces and torques for the particle T respectively,
0 = S σ 0 n n n ds + α S τ 0 n n n ) ds − α T B B B 0 × curl y H H H 0 dy y y, 0 = S y y y × σ 0 n n n ds + α S y y y × τ 0 n n n ds − α T y y y × B B B 0 × curl y (H H H 0 ) dy y y,(2.3)
where by τ 0 ij :
τ 0 ij = µ H 0 i H 0 j − 1 2 µ H 0 k H 0 k δ ij ,(2.4)
we denote the Maxwell stress. We remark that since from (2.1e) curl y (H H H 0 ) = 0 0 0 in Y , the balance of forces and torques simplify to the following, 0 = S σ 0 n n n + α S τ 0 n n n ds and 0 = S y y y × σ 0 n n n ds + α S y y y × τ 0 n n n ds. where · = 1 |Y | Y · dy y y. Using the fact div y B B B 0 = 0 in Y , B 0 i = µ H 0 i and the boundary conditions (1.7) we have,
H 0 i = − ∂ψ(x x x, y y y) ∂y i + H 0 i (x x x),(3.− ∂ ∂y i µ − ∂ψ ∂y i + H 0 i = 0 in Y , µ − ∂ψ ∂y i + H 0 i n i = 0 on S , ψ is Y − periodic, ψ = 0. (3.2)
Introducing the space of periodic functions, with zero average
W per (Y ) = w ∈ H 1 per (Y ) | w = 0 , then the variational formulation of (3.2) is Find ψ ∈ W per (Y ) such that Y µ ∂ψ ∂y i ∂v ∂y i dy y y = H 0 i Y µ ∂v ∂y i dy y y for any v ∈ W per (Y ). (3.3)
Since we have imposed that ψ has zero average over the unit cell Y , the solution to In principle, once H 0 k is known, we can determine ψ up to an additive function of x x x. Hence, combining (3.1) and the above relationship between ψ and φ k we obtain the following constitutive law between the magnetic induction and the magnetic field,
B 0 i = µ ik H 0 k , where µ ik = Y µ − ∂φ k ∂y i + δ ik dy y y. (3.5)
One can show (see [19]) that the homogenized magnetic permeability tensor is symmetric, µ ik = µ ki . Moreover, if we denote by A i (y y y) = − ∂φ (y y y) ∂y i + δ i one can see from (3.1) that H 0 i = A i H 0 and thus the Maxwell stress (2.4) takes the following form,
τ 0 ij = µ A i A jm H 0 H 0 m − 1 2 µ A mk A k δ ij H 0 m H 0 = µ A m ij H 0 m H 0 .
Here Here, χ χ χ ml satisfies where C ijm = 1 2 (δ im δ j + δ i δ jm ) − 1 n δ ij δ m . 8 The variational formulation problem of (4
A m ij = 1 2 (A i A jm + A j A im − A mk A k δ ij )− ∂ ∂y j ε m ij = 0 in Y f , ε m ij = −p m δ ij + 2 (C ijm + e ijy (χ χ χ m )) − ∂χ m i ∂y i = 0 in Y f , χ χ χ m = 0 on S , C ijm + e ijy (χ χ χ m ) = 0 in T , χ χ χ m is Y − periodic, χ χ χ m = 0 0 0 in Y,
.3)-(4.4) is
Find χ χ χ m ∈ U such that Y f 2 e ijy (χ χ χ m ) e ijy (φ φ φ − χ χ χ m ) dy y y = 0, for all φ φ φ ∈ U ad , (4.5)
where U is the closed, convex, non-empty subset of H 1 per (Y ) n defined by
U = u u u ∈ H 1 per (Y ) n | div u u u = 0 in Y f , e ijy (u u u) = −C ijm in T, u u u = 0 0 0 on S, u u u = 0 0 0 in Y
We remark that if we define B ij k = 1 2 (y i δ jk +y j δ ik )− 1 n y k δ ij , then e ij (B B B m ) = C ijm . Existence and uniqueness of a solution follows from classical theory of variational inequalities [7].
In similar fashion we can derive the local problem for ξ ξ ξ ml , We can formulate (4.6)-(4.8) variationally as
− ∂ ∂y j Σ m ij = 0 in Y f , Σ m ij = −π m δ ij + 2 e ijy (ξ ξ ξ m ) − ∂ξ m i ∂y i = 0 in Y f , ξ ξ ξ m = 0 on S , e ijy (ξ ξ ξ m ) = 0 in T , ξ ξ ξ m is Y −periodic, ξ ξ ξ m = 0.Find ξ ξ ξ m ∈ V per (Y ) such that Y f 2 e ijy (ξ ξ ξ m ) e ijy (φ φ φ) dy y y + Y A m ij e ijy (φ φ φ) dy y y = 0, for all φ φ φ ∈ V per (Y ), (4.9) where V per (Y ) = v v v ∈ H 1 per (Y ) n | div v v v = 0 in Y f , e ijy (u u u) = 0 in T, v v v = 0 0 0 on S, v v v = 0 0 0 in Y ,
is a closed subspace of H 1 per (Y ) n . Existence and uniqueness follows from an application of the Lax-Milgram lemma. Below we plot the streamlines of the solutions χ χ χ m of (4.5) and ξ ξ ξ m of (4.9). 9 Streamline for χ χ χ 11
Streamline for χ χ χ 12 Streamline for χ χ χ 22
Streamline of ξ ξ ξ 11 Streamline for ξ ξ ξ 12 Streamline of ξ ξ ξ 22 Remark 2. From the second line of the plots above, we can observe that the only driving force that makes the solution ξ ξ ξ m non zero in (4.9) are the rotations induced by the magnetic field through the fourth order tensor A m ij .
Homogenized equations of the magneto-rheological fluid
At the 0 order we obtain the following problems, In each period, we consider a Taylor expansion, around the center of mass of the rigid particle, both of the viscous stress and the Maxwell stress of the form (see [10]),
−div x σ 0 − div y σ 1 = 0 0 0 in Y f , (5.1a) div x v v v 1 + div y v v v 2 = 0 in Y f , (5.1b) div x B B B 0 + div y B B B 1 = 0 in Y , (5.1c) curl x H H H 0 + curl y H H H 1 = 0 in Y f , (5.1d) curl x H H H 0 + curl y H H H 1 = R m v v v 0 × B B B 0 in T ,σ (x x x) = σ 0 (x x x c , y y y) + ∂σ 0 (x x x c , y y y) ∂x α (x α − x c,α ) + σ 1 (x x x c , y y y) + ∂σ 1 (x x x c , y y y) ∂x α (x α − x c,α ) + · · · τ (x x x) = τ 0 (x x x c , y y y) + ∂τ 0 (x x x c , y y y) ∂x α (x α − x c,α ) + τ 1 (x x x c , y y y) + ∂τ 1 (x x x c , y y y) ∂x α (x α − x c,α ) + · · ·
where the expansion of the Maxwell stress occurs both inside the rigid particle and the fluid. Using this method we can expand the balance of forces, (1.8), and obtain at order 3 ,
0 = S ∂σ 0 ij ∂x k y k + σ 1 ij n j ds + α S ∂τ 0 ij ∂x k y k + τ 1 ij n j ds − α T (B B B 0 × (curl x H H H 0 + curl y H H H 1 )) i dy y y. (5.3)
Integrate (5.1a) over Y f and add to (5.3) obtain the following,
0 = Y f ∂σ 0 ij ∂x j dy y y + S ∂σ 0 ij ∂x k y k n j ds + α S ( ∂τ 0 ij ∂x k y k + τ 1 ij ) n j ds − α T (B B B 0 × (curl x H H H 0 + curl y H H H 1 )) i dy y y.σ 0 ij = −p 0 δ ij + ε m ij e mlx (v v v 0 ) + Σ m ij H 0 m H 0 , τ 0 ij = µ A m ij H 0 m H 0 .
Moreover, equations (2.1b), (2.4), (4.3) and (4.6) allow us to retain the only symmetric part of (5.5).
Hence the homogenized fluid equations (5.5) become,
0 = ∂ ∂x j −p 0 δ ij + Y f 2 e ijy (B B B m + χ χ χ m ) dy y y + S ε m pk B ij p n k ds e m x (v v v 0 ) (5.6) + Y f 2 e ijy (ξ ξ ξ m ) dy y y + S Σ m pk B ij p n k ds + α Y µ A m ij dy y y + α S µA m pk B ij p n k H 0 m H 0 .
Furthermore, using (2.1c)-(2.1d) and the divergence theorem we can obtain the incompressibility condition, div x v v v 0 = 0.
Denote by
ν ijm = Y f
2 e ijy (B B B m + χ χ χ m ) dy y y + S ε m pk B ij p n k ds ,
and β ijm = Y f 2 e ijy (ξ ξ ξ m ) dy y y + S Σ m pk B ij p n k ds + α Y µ A m ij dy y y + α S µA m pk B ij p n k .
then the homogenized equation (5.6) becomes
0 = ∂ ∂x j −p 0 δ ij + ν ijm e m x (v v v 0 ) + β ijm H 0 m H 0 .
Using local problem (4.3) we can re-write the ν ijm the following way,
ν ijm = Y f 2 e pq (B B B ml + χ χ χ ml ) e pq (B B B ij + χ χ χ ij ) dy y y. (5.7)
In a similar fashion, using local problem (4.6) and the kinematic condition in (4.3) we can re-write β ijm as follows
β ijm = Y f 2 e pq (ξ ξ ξ ml )e pq (B B B ij + χ χ χ ij ) dy y y + α Y f µ A m pq e pq (B B B ij + χ χ χ ij ) dy y y + α Y µ A m ij dy y y. (5.8)
It is now clear that ν ijm possesses the following symmetry, ν ijm = ν jim = ν m ij . While for β ijm , we have β ijm = β jim = β ij m .
To obtain the homogenized Maxwell equations, average (5.1c), (5.1d), and (5.1e) over Y , Y f , and T respectively and use equation (3.5) to obtain,
∂ (µ ik H 0 k ) ∂x j = 0, ijk ∂ H 0 k ∂x j = R m ijk v 0 j µ S kp H 0 p in Ω, where µ S ik = T µ − ∂φ k ∂y i + δ ik dy y y (5.9)
with boundary conditions,
H 0 i = b i , v 0 i = 0 on ∂Ω.
The effective coefficients are computed as the angular averaging of the tensors ν ijm and β ijm . This is done by introducing the projection on hydrostatic fields, P b , and the projection on shear fields P s (see [12]). The components of the projections in three dimensional space are given by:
(P b ) ijk = 1 n δ ij δ k , (P s ) ijk = 1 2 (δ ik δ j + δ i δ jk ) − 1 n δ ij δ k
Let us make the following notations:
ν b = tr(P b ν) = 1 n ν ppqq , ν s = tr(P s ν) = ν pqpq − 1 n ν ppqq , β b = tr(P b β) = 1 n β ppqq , β s = tr(P s β) = β pqpq − 1 n β ppqq .
Then we can re-write the homogenized coefficients ν ijm and β ijm as follows:
ν ijm = 1 n (ν b − ν s )δ ij δ m + 1 2 ν s (δ ik δ j + δ i δ jk ), β ijm = 1 n (β b − ν s )δ ij δ m + 1 2 β s (δ ik δ j + δ i δ jk ).
Gathering all the equations we have that the homogenized equations governing the MR fluid form the following coupled system between the Stokes equations and the quasistatic Maxwell equations,
∂ ∂x j σ H ij + τ H ij = 0, ∂v 0 i ∂x i = 0 in Ω, σ H ij + τ H ij = −p 0 δ ij +ν s e ij (v v v 0 ) + 1 n (β b − β s ) δ ij H H H 0 2 + β s H 0 i H 0 j ∂(µ jk H 0 k ) ∂x j = 0, ijk ∂ H 0 k ∂x j = R m ijk v 0 j µ S kp H 0 p in Ω, v 0 i = 0, H 0 i = b i in Ω.
(5.10)
Remark 3. We should remark here that the effective constitutive properties consist of the homogenized viscosity, ν ijm , and three homogenized magnetic permeabilities, µ ij , µ s ij , and β ijm , which all depend on the geometry of the suspension, the volume fraction, and the magnetic permeability µ. In addition the new coefficient β ijm depends also on the Alfven number α.
Velocity profile of the magneto-rheological fluid
In this section we compute the cross sectional velocity profiles of Poiseouille and Couette flow for spherical suspensions of rigid particles. We denote by v v v = (v 1 , v 2 ) the two dimensional velocity and by H H H = (H 1 , H 2 ) the two dimensional magnetic field. We remark that in two dimensions the tensors C ijmm = 0 and B B B mm = 0 0 0. Then due to the linearity of local problem (4.3) we have χ χ χ mm = 0 0 0. Thus, ν mmii = 0 which implies that ν b = 0. Using a similar argument, we further note that β mmii = 0 which implies that β b = 0. Hence, the two dimensional stresses of (5.10) reduce to
σ H ij + τ H ij = −p 0 δ ij + ν s e ij (v v v 0 ) − 1 2 β s δ ij H H H 0 2 + β s H 0 i H 0 j
Thus, the two dimensional MR equations in (5.10) reduce to the following:
ν s 2 ∂ 2 v 1 ∂x 2 1 + ∂ 2 v 1 ∂x 2 2 − ∂ π 0 ∂x 1 + ∂ ∂x 1 1 2 β s (H 2 1 − H 2 2 ) + ∂ ∂x 2 (β s H 1 H 2 ) = 0, (6.1a) ν s 2 ∂ 2 v 2 ∂x 2 1 + ∂ 2 v 2 ∂x 2 2 − ∂ π 0 ∂x 2 + ∂ ∂x 1 (β s H 1 H 2 ) + ∂ ∂x 2 1 2 β s (H 2 2 − H 2 1 ) = 0, (6.1b) ∂ ∂x 1 (µ H 1 ) + ∂ ∂x 2 (µ H 2 ) = 0, (6.1c) ∂H 2 ∂x 1 − ∂H 1 ∂x 2 = η µ S (v 1 H 2 − v 2 H 1 ), (6.1d) ∂v 1 ∂x 1 + ∂v 2 ∂x 2 = 0. (6.1e)
6.1. Poiseuille flow. We consider the problem of a steady flow due to a pressure gradient between two infinite, parallel, stationary plates that are non-conducting and nonmagnetizable with one plate aligned along the x 1 -axis while the second plate is of distance one unit apart. We apply a stationary magnetic field H H H on the bottom plate. Since we are dealing with infinite plates, the velocity v v v depends only on x 2 . Using (6.1e) we immediately obtain that v 2 is constant and since the plates are stationary v 2 = 0. Since the flow is unidirectional, we expect that the the magnetic field will depend only on the height x 2 . Hence, using (6.1c) we obtain H 2 (x 2 ) = K, while the component parallel to the flow depends on the fluid velocity. Therefore the equations in (6.1) reduce to the following, ν s 2
∂ 2 v 1 ∂x 2 1 + β s K ∂H 1 ∂x 2 = ∂ π 0 ∂x 1 , (6.2a) − ∂ π 0 ∂x 2 − 1 2 β s ∂H 2 1 ∂x 2 = 0, (6.2b) − ∂H 1 ∂x 2 = η µ S K v 1 . (6.2c)
Making use of (6.2b) we obtain that π 0 (x 1 , x 2 ) + 1 2 β s H 1 (x 2 ) 2 is a function of only x 1 and therefore by differentiating the expression with respect to x 1 we get that ∂ π 0 ∂x 1 is a function only x 1 . Therefore, on (6.2a) the left hand side is a function of x 2 and the right hand side is a function of x 1 . Thus they have to be constant. Substituting (6.2c) in (6.2a) we obtain the following differential equations,
d 2 v 1 d x 2 2 − λ 2 v 1 = C p , (6.3a) ∂ π 0 ∂x 1 = C p , (6.3b) with λ = 2 η µ s β s ν s K.
The general solution of (6.3a) is
v 1 (x 2 ) = c 1 e λ x 2 + c 2 e −λ x 2 + C p ν λ 2 . Given that v 1 (0) = v 1 (1) = 0 we have, v 1 (x 2 ) = C p ν λ 2 sinh(λ x 2 ) − sinh(λ (x 2 − 1)) sinh(λ) − 1 . (6.4)
Once the velocity v 1 (x 2 ) is known, we can use (6.2c) to compute H 1 (x 2 ) with boundary condition H 1 (0) = K 1 and obtain,
H 1 (x 2 ) = η µ s K C p ν λ 3 sinh(λ) (− cosh(λ x 2 ) + cosh(λ (x 2 − 1)) − cosh(λ) + 1) + K 1 .
Remark 4.
As K tends to zero, λ also tends to zero and we have
lim K→0 v 1 (x 2 ) = C p 2 ν x 2 (x 2 − 1),
which is precisely the profile of Poiseuille flow with stationary plates at x 2 = 0 and x 2 = 1.
v 1 (x 2 ) =γ ν λ sinh(λx 2 ) + C p cosh(λ (x 2 − 1)) νλ 2 cosh(λ) − C p ν λ 2 (6.5)
Remark 5. Again, as before, we note that as K approaches zero, λ also approaches zero and
lim λ→0 C p x 2 2 + 2γ x 2 ν s − 2 C p x 2 2 ν s
To compute H 1 we use (6.2c) to obtain 4). We can see that the damping force increases with B 2 ; the profile is close to flat in the middle region for high B 2 , but is not parabolic close to the walls as in the case of Bingham flows. Likewise, in Couette flow regime we can plot and compare the velocity profile (6.5) of MR Couette flow against the Couette flow in the absence of a magnetic field, for zero pressure gradient. The plots in FIG. 5 show that as b 2 increases the flow region is smaller close to the upper plate. Thus an "apparent" yield stress is present. However, the velocity profile is not linear like in the case of Bingham fluids.
H 1 (x 2 ) =γ λ 2 cosh(λ) (cosh(λ x 2 )−1)+ C p ν λ 3 cosh(λ) (sinh(λ (x 2 −1))−sinh(λ))− C p x 2 ν λ 2 +K 1 .
Velocity profile for MR Couette flow Velocity profile for plane Couette flow Remark 6. For shear experiments, the response of magneto-rheological fluids is often modeled using a Bingham constitutive law [4], [5], [15]. Although the Bingham constitutive law measures the response of the magneto-rheological fluid quite reasonably, actual magnetorheological fluid behavior exhibits departures from the Bingham model [6], [22]. In Fig. 4 and FIG 5 we see that for low values of the magnetic field, the Bingham constitutive law is not adequate, however, it appears that for higher values of the magnetic field the flow gets closer to resembling a Bingham fluid.
The plot in FIG. 6 depicts the stress vs shear rate curve relationship measured at x 2 = 1. When K 1 = 0 there is no yield stress present. However, for very small non-zero values of K 1 we obtain the results of [22] for the linear portion of the stress vs shear rate curve at high shear rates. Additionally, we are able to match their extrapolated Bingham yield stress values.
Conclusions
We consider a suspension of rigid magnetizable particles in an non-magnetizable, nonconducting aqueous viscous fluid. In (3.4), (4.5), (4.9) we derive the local problems that arise from the Maxwell equations, the bulk velocity and the bulk magnetic field and obtain new constitutive laws. The effective equations governing the behavior of the MR fluid are presented in (5.10). The proposed model generalizes the one in [14] by coupling the velocity field with the magnetic field intensity. Moreover, we obtain formulae for the effective coefficients that can be numerically computed and identify three different magnetic permeabilities governing the effective behavior. Unidirectional velocity profiles of Poiseuille and Couette flows are computed for magnetite nanoparticles of volume fraction φ = 0.14 to validate against experimental data for the stress-strain relationship of MR flows.
R n , and Z n is the set of all n-dimensional vectors with integer components. For every positive , let N be the set of all points ∈ Z n such that ( + Y ) is strictly included in Ω and denote by |N | their total number. Let T be the closure of an open connected set with sufficiently smooth boundary, compactly included in Y . For every > 0 and ∈ N we consider the set T ⊂⊂ ( + Y ), where T = ( + T ). The set T represents one of the rigid particles suspended in the fluid, and S = ∂T denotes its surface (seeFIG. 1)
1c) 3 where
3B B B = µ H H H with boundary conditions on the surface of each particle T ,
.
At first order, in the problem (2.1)-(2.5) the Stokes and Maxwell equations are decoupled. Hence, in principle one could solve the Maxwell equations (2.1e) and once a solution is obtained then solve the Stokes problem (2.1a)-(2.1c), albeit with an extra known force added to the balance of forces and torques (2.5). 3. Constitutive relations for Maxwell's equations 3.1. Study of the local problem. Using the results from the two scale expansions, (2.1e), we can see that curl y (H H H 0 ) = 0 0 0 in Y and thus there exists a function ψ = ψ(x x x, y y y) with average ψ = 0 such that
Plot of φ 1 Plot of φ 2 Figure 2 .
122(3.3) can be determined uniquely by a simple application of the Lax-Milgram lemma.Let φ k be the unique solution of Find φ k ∈ W per (Y ) such that dy y y for any v ∈ W per (Y ). Plot of the solution φ k in (3.4) for magnetite nanoparticles of volume fraction φ = 0.14 with magnetic permeability µ = 8.41946×10 −6 N/A 2 using FreeFem++.By virtue of linearity of (3.3) we can write ψ(x x x, y y y) = φ k (y y y) H 0 k (x x x) + C(x x x).
.
and has the following symmetry, A m ij = A m ji = A m ij . Recall that the div τ = 0 in Ω 2 and div τ = −B B B × curl H H H in Ω 1 . From the two scale expansion, at order −1 from equation (1.4) we obtain, div y τ 0 = 0 in Y. Study of the local problems. Problem (2.1)-(2.2), (2.5) is an elliptic problem in the variable y y y ∈ Y with forcing terms v v v 0 (x x x) and H H H 0(x x x) at the macroscale. We can decouple the contributions of v v v 0 (x x x) and H 0 (x x x) and split v v v 1 and p 0 in two parts: a part that is driven by the bulk velocity, and a part that comes from the bulk magnetic field.
v 1 k
1(x x x, y y y) = χ m k (y y y) e m (v v v 0 ) + ξ m k (y y y) H 0 m H 0 + A k (x x x),(4.1) p 0 (x x x, y y y) = p m (y y y) e m (v v v 0 ) + π m (y y y) H 0 m H 0 +p 0 (x x x),
the balance of forces and torques, S ε m ij n j ds = 0 and S ijk y j ε m kp n p ds = 0, (4.4)
Figure 3 .
3On the top are the streamlines of the solution χ χ χ m in (4.5) and on the bottom are the corresponding streamlines of the solution ξ ξ ξ m in (4.9) for spherical magnetite nanoparticles of volume fraction φ = 0.14 generated using FreeFem++.
v 2 = 0 0 0, B B B 1 · n n n = 0 n n n × H H H 1 = 0 0 0 on S , v v v 2 , H H H 1 are Y − periodic. (5.2)
0 we obtain, div x τ 0 +div y τ 1 = 0 in Y f and div x τ 0 +div y τ 1 = −B B B 0 ×(curl x H H H 0 + curl y H H H 1 ) in T .Combining the aforementioned results and the divergence theorem we can rewrite (5.4) the following waydecomposition of v v v 1 and p 0 in (4.1) and (4.2) we can re-write σ 0 ij and τ 0 ij ,
6. 3 .
3Magnetite nanoparticles. In this section we consider a suspension of spherical magnetite nanoparticles in de-ionized water of viscosity 0.001 Pa with volume fraction φ = 0.07. The electrical conductivity of the nanoparticles is assumed to be 20, 000 S/m, while the magnetic permeability is 8.41946 × 10 −6 N/A 2 for the nanoparticles and 1.25662 × 10 −6 N/A 2 for the water. Carrying out explicit computations of the effective coefficients in (5.7), (5.8) and (5.9) we obtain ν s = 0.006 P a, β s = 2.59 × 10 −6 N/A 2 , µ s = 3.28 × 10 −7 N/A 2 . In the case of Poiseuille flow we can plot the profile (6.4) of the MR flow, for a constant pressure gradient and different values of the magnetic field and compare them with the Poiseuille flow profile in the absence of a magnetic field (FIG.
profile for MR Poisseuille flow Velocity profile for regular Poisseuille flow
Figure 4 .
4The plots on the left represent the velocity profile for B 2 =0.05, 0.02, 0.01, 0.0075, 0.005 T (left to right). The plot on the right is the velocity profile for B 2 =0 T.
Figure 5 .
5The plots on the left represent the velocity profile for B 2 =0.05, 0.02, 0.01, 0.0075, 0.005 T (left to right). The plot on the right is the velocity profile for B 2 =0 T.
Figure 6 .
6The stress versus the shear rate curve for four different magnetic fields, B 2 = 0.288, 0.230, 0.173, 0.058.
1) 6
6.2. Couette flow. The setting and calculations for the unidirectional Couette flow are the same as Poiseuille flow. In a similar way, we can carry out computations for the plane Couette flow. For simplicity we assumed the bottom plate is the x 1 axis and the top plate is at x 2 = 1 and the pressure gradient is zero. A shear stressγ is applied to the top plate while the bottom plate remains fixed. Thus, we solve (6.3a) with initial conditions v 1 (0) = 0 and v 1 (1) =γ and obtain
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[
"Two-particle non-local Aharonov-Bohm effect from two single-particle emitters",
"Two-particle non-local Aharonov-Bohm effect from two single-particle emitters"
]
| [
"Janine Splettstoesser \nDépartement de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland\n",
"Michael Moskalets \nDépartement de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland\n\nDepartment of Metal and Semic. Physics\nNTU \"Kharkiv Polytechnic Institute\"\n61002KharkivUkraine\n",
"Markus Büttiker \nDépartement de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland\n"
]
| [
"Département de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland",
"Département de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland",
"Department of Metal and Semic. Physics\nNTU \"Kharkiv Polytechnic Institute\"\n61002KharkivUkraine",
"Département de Physique Théorique\nUniversité de Genève\nCH-1211Genève 4Switzerland"
]
| []
| We propose a mesoscopic circuit in the quantum Hall effect regime comprising two uncorrelated single-particle sources and two distant Mach-Zehnder interferometers with magnetic fluxes, which allows in a controllable way to produce orbitally entangled electrons. Two-particle correlations appear as a consequence of erasing of which path information due to collisions taking place at distant interferometers and in general at different times. The two-particle correlations manifest themselves as an Aharonov-Bohm effect in noise while the current is insensitive to magnetic fluxes. In an appropriate time interval the concurrence reaches a maximum and a Bell inequality is violated. PACS numbers: 73.23.-b, 72.10.-d, 73.50.TdIntroduction.-Interference phenomena are the most prominent feature of quantum mechanics. Of particular interest are interference effects in multi-particle states. For example in optics, the Hanbury Brown-Twiss effect [1] and the Hong-Ou-Mandel (HOM) effect [2] both result from two-particle interference of photons emitted by two independent sources. In mesoscopics, electrons can play a role similar to photons in optics. In an electrical circuit with currents incoming from different (uncorrelated) equilibrium contacts the noise can show interference even if the currents exhibit no interference contribution[3].Recently a single-particle emitter [4] was experimentally realized on the basis of a quantum capacitor in a two-dimensional electron gas in the integer quantum Hall effect regime. Subject to an appropriate large amplitude potential the capacitor emits a single electron during the first half-cycle and a single hole during the second halfcycle. With such an emitter it is possible to inject single electrons and holes in a non-equilibrium state into an electrical conductor. Injected particles can be guided by edge states and encounter splitters realized by quantum point contacts (QPC). These states can be considered as an analogue of short photon pulses produced by a laser. By using two such sources and tuning the times when they emit particles one can force emitted particles to collide at some QPC. Tuning can be achieved by varying the phase difference between the two potentials acting on the capacitors. Such a collision erases which-path information for particles leaving the QPC and it promotes the appearance of two-particle correlation effects. Based on this simple idea an electronic analogue of an optical HOM interferometer was suggested[5]. This interferometer shows a noise suppression due to two-particle correlations arising locally when particles collide at a QPC.In this Letter we propose a set-up, where two-particle correlations arise non-locally in time and in space. This is particularly intriguing, when the currents are magneticflux independent, namely when the width of the electron pulses is small with respect to the arm length differences of the Mach-Zehnder interferometers (MZIs), seeFig. 1. | 10.1103/physrevlett.103.076804 | [
"https://arxiv.org/pdf/0905.0787v2.pdf"
]
| 41,419,270 | 0905.0787 | a0cfbedd41593653973f69b77248752e9214da95 |
Two-particle non-local Aharonov-Bohm effect from two single-particle emitters
24 Aug 2009
Janine Splettstoesser
Département de Physique Théorique
Université de Genève
CH-1211Genève 4Switzerland
Michael Moskalets
Département de Physique Théorique
Université de Genève
CH-1211Genève 4Switzerland
Department of Metal and Semic. Physics
NTU "Kharkiv Polytechnic Institute"
61002KharkivUkraine
Markus Büttiker
Département de Physique Théorique
Université de Genève
CH-1211Genève 4Switzerland
Two-particle non-local Aharonov-Bohm effect from two single-particle emitters
24 Aug 2009(Dated: August 24, 2009)
We propose a mesoscopic circuit in the quantum Hall effect regime comprising two uncorrelated single-particle sources and two distant Mach-Zehnder interferometers with magnetic fluxes, which allows in a controllable way to produce orbitally entangled electrons. Two-particle correlations appear as a consequence of erasing of which path information due to collisions taking place at distant interferometers and in general at different times. The two-particle correlations manifest themselves as an Aharonov-Bohm effect in noise while the current is insensitive to magnetic fluxes. In an appropriate time interval the concurrence reaches a maximum and a Bell inequality is violated. PACS numbers: 73.23.-b, 72.10.-d, 73.50.TdIntroduction.-Interference phenomena are the most prominent feature of quantum mechanics. Of particular interest are interference effects in multi-particle states. For example in optics, the Hanbury Brown-Twiss effect [1] and the Hong-Ou-Mandel (HOM) effect [2] both result from two-particle interference of photons emitted by two independent sources. In mesoscopics, electrons can play a role similar to photons in optics. In an electrical circuit with currents incoming from different (uncorrelated) equilibrium contacts the noise can show interference even if the currents exhibit no interference contribution[3].Recently a single-particle emitter [4] was experimentally realized on the basis of a quantum capacitor in a two-dimensional electron gas in the integer quantum Hall effect regime. Subject to an appropriate large amplitude potential the capacitor emits a single electron during the first half-cycle and a single hole during the second halfcycle. With such an emitter it is possible to inject single electrons and holes in a non-equilibrium state into an electrical conductor. Injected particles can be guided by edge states and encounter splitters realized by quantum point contacts (QPC). These states can be considered as an analogue of short photon pulses produced by a laser. By using two such sources and tuning the times when they emit particles one can force emitted particles to collide at some QPC. Tuning can be achieved by varying the phase difference between the two potentials acting on the capacitors. Such a collision erases which-path information for particles leaving the QPC and it promotes the appearance of two-particle correlation effects. Based on this simple idea an electronic analogue of an optical HOM interferometer was suggested[5]. This interferometer shows a noise suppression due to two-particle correlations arising locally when particles collide at a QPC.In this Letter we propose a set-up, where two-particle correlations arise non-locally in time and in space. This is particularly intriguing, when the currents are magneticflux independent, namely when the width of the electron pulses is small with respect to the arm length differences of the Mach-Zehnder interferometers (MZIs), seeFig. 1.
We propose a mesoscopic circuit in the quantum Hall effect regime comprising two uncorrelated single-particle sources and two distant Mach-Zehnder interferometers with magnetic fluxes, which allows in a controllable way to produce orbitally entangled electrons. Two-particle correlations appear as a consequence of erasing of which path information due to collisions taking place at distant interferometers and in general at different times. The two-particle correlations manifest themselves as an Aharonov-Bohm effect in noise while the current is insensitive to magnetic fluxes. In an appropriate time interval the concurrence reaches a maximum and a Bell inequality is violated. Introduction.-Interference phenomena are the most prominent feature of quantum mechanics. Of particular interest are interference effects in multi-particle states. For example in optics, the Hanbury Brown-Twiss effect [1] and the Hong-Ou-Mandel (HOM) effect [2] both result from two-particle interference of photons emitted by two independent sources. In mesoscopics, electrons can play a role similar to photons in optics. In an electrical circuit with currents incoming from different (uncorrelated) equilibrium contacts the noise can show interference even if the currents exhibit no interference contribution [3].
Recently a single-particle emitter [4] was experimentally realized on the basis of a quantum capacitor in a two-dimensional electron gas in the integer quantum Hall effect regime. Subject to an appropriate large amplitude potential the capacitor emits a single electron during the first half-cycle and a single hole during the second halfcycle. With such an emitter it is possible to inject single electrons and holes in a non-equilibrium state into an electrical conductor. Injected particles can be guided by edge states and encounter splitters realized by quantum point contacts (QPC). These states can be considered as an analogue of short photon pulses produced by a laser. By using two such sources and tuning the times when they emit particles one can force emitted particles to collide at some QPC. Tuning can be achieved by varying the phase difference between the two potentials acting on the capacitors. Such a collision erases which-path information for particles leaving the QPC and it promotes the appearance of two-particle correlation effects. Based on this simple idea an electronic analogue of an optical HOM interferometer was suggested [5]. This interferometer shows a noise suppression due to two-particle correlations arising locally when particles collide at a QPC.
In this Letter we propose a set-up, where two-particle correlations arise non-locally in time and in space. This is particularly intriguing, when the currents are magneticflux independent, namely when the width of the electron pulses is small with respect to the arm length differences of the Mach-Zehnder interferometers (MZIs), see Fig. 1.
Our geometry resembles the optical Franson interferometer [6] but the underlying physics is different. The uncorrelated electrons (holes) emitted by the sources A and B and propagating along different arms of the same interferometer can collide at the interferometer exit if the times of emission, defined by the potentials U A (t) and U B (t), are properly chosen. If the difference of the arm lengths for both interferometers is the same then the electron collisions take place at both interferometers (possibly at different times). This makes the two two-particle amplitudes (with one electron going through the left and another going through the right interferometers) indistinguishable hence interfering. Corresponding amplitudes are shown in Fig. 1 in dotted and dashed lines. Such an interference results in two-particle correlations arising non-locally in space and time. As a direct manifestation of this interference, the noise shows an Aharonov-Bohm (AB) effect [7] with the total flux determined by both the magnetic fluxes Φ L and Φ R threading the distant interferometers. We analyze the two-electron state emitted by our set-up and find it to be fully entangled with collisions in place and completely disentangled in the case that no collisions are present. We arrive at the same conclusion by analyzing a Bell inequality [8], which can be violated even at partial overlap of electron wave-packets. The two-particle AB effect in electrical conductors was discussed theoretically [9] and investigated experimentally [10] in a Hanbury Brown-Twiss interferometer geometry [11]. The novelty of our proposal is the quantized injection of electrons and the possibility of controlling the appearance of two-particle correlations. In comparison to proposals discussed in the literature for a dynamical generation of entanglement [12], the present scheme deals with electron-electron (and hole-hole) rather then electron-hole entanglement.
|1 R |2 L |2 R |1 L 2 Φ R 1 U A (t) A B 4 3 Φ L U B (t) C
Model and formalism.-The system is schematically shown in Fig. 1. Two mesoscopic capacitors, A and B, are contacted by QPCs, with reflection (transmission) coefficients r A , (t A ) and r B (t B ), to chiral edge states. The mesoscopic capacitors are driven by time-dependent, homogeneously applied potentials U A (t) and U B (t), with equal frequency Ω and a large amplitude such that the capacitors serve as sources of single electrons and holes. The particles emitted are transmitted or reflected at the center QPC (C), with reflection (transmission) coefficients r C , (t C ). Before reaching the contacts 1 to 4 the signals traverse the lower (d) or upper (u) arms of two MZIs [13,14], L and R, pierced by magnetic fluxes, Φ L and Φ R . The beam splitters of the MZI have the reflection (transmission) coefficients r l β (t l β ) and r r β (t r β ). We use a scattering matrix approach [15,16] and describe the mesoscopic capacitor by a Fabry-Perot like amplitude [15], S α (t, E) = r α + t 2 α ∞ q=1 r α q−1 e iqEτα−iΦ q α (t) , α = A, B, depending on the energy of an incoming particle and the time it exits. Here τ α = h/∆ α and ∆ α is the capacitor's level spacing; Φ q
α (t) = ē h t t−qτα dt ′ U α (t ′ )
. The scattering matrix of the full system also depends on the central QPC and the MZIs. A phase Φ u(d)
βα (E) = Eτ u(d) βα /h +Φ u(d) β
is accumulated when a particle coming from source α traverses the upper (lower) arm of the interferometer β = L, R. The time for this traversal is τ u βα (τ d βα ) and the phaseΦ u βα (Φ d βα ) depends on the magnetic flux. We are interested in the case of slow driving, meaning that the frequency Ω is much smaller than the inverse of the lifetime of particles in the cavity, without requesting restrictions on Ω with respect to the time scales related to the entire system. In order to calculate the currents into contact 1 and 2, we need to know the scattering matrix elements for scattering from contacts 1 and 2 into contacts 1 and 2. We find
S 11 (t, E) = r C t l L t r L S A (t − τ u LA , E)e iΦ u LA (E) +r l L r r L S A (t − τ d LA , E)e iΦ d LA (E) . (1)
The elements S 12 (t, E), S 21 (t, E) and S 22 (t, E) are found analogously, while the elements which describe the scattering from contact 3 or 4 are time-independent. The current.-A periodic potential with period T = 2π/Ω, results in a current at contact j, [17]
I j (t) = e h dE ∞ n=−∞ 2 l=1 [f (E) − f (E +hnΩ)] (2) × T 0 dt ′ T e inΩ(t−t ′ ) S jl (t, E)S * jl (t ′ , E) .
This equation is valid for finite frequency and arbitrary amplitude driving. The current at contact, say, 1,
I 1 (t) = R C I 1A (t)+ T C I 1B (t) (here T C = 1 − R C = |t C | 2 ),
consists of a current coming from source A (B) reflected (transmitted) at the central QPC and passed through the interferometer L. At zero temperature, the partial current I 1α (t), α = A, B, comprises a classical part due to the current generated by the capacitor,
I α (t) = e 2 2πi ∂Uα ∂t S (0) * α (t) ∂ ∂E S (0)
α (t), and an interference part,
I 1α (t) = R l L R r L I α (t − τ d Lα ) + T l L T r L I α (t − τ u Lα )(3)+ eγ L /π τ u Lα − τ d Lα Im S (0) * α (t − τ u Lα )S (0) α (t − τ d Lα )e −iΦL .
The current in contact 2 is found analogously. Here we introduced the instantaneous scattering matrix of the cavity α at the Fermi energy µ, S
α (t) = S α (µ − eU α (t)(0)
). The transmission probability at each MZI beam splitter is T j β = 1 − R j β = |t j β | 2 and γ β = (R l β R r β T l β T r β ) 1/2 is a product of the reflection and transmission coefficients. Furthermore the flux enclosed by the interferometer β is given by Φ β =Φ u β −Φ d β , in units of Φ 0 /(2π), where Φ 0 = h/e is the magnetic flux quantum. As the coherent emission of quantized charge attracts our special attention, we now treat the case of small transmission of the cavities' QPCs, where the current emitted by a cavity is a series of well separated pulses of opposite sign for the emission of electrons and holes. We resort to a description in the Breit-Wigner regime, and assume that one electron and one hole are emitted per period from cavity α at times t e α and t h α , described by current pulses, having a Lorentzian shape with half width Γ α [17].
The last term in Eq. (3) is due to single-particle interference in one of the MZIs. Single-particle interference appears, when the wavefunctions traveling through the upper and the lower arm of the interferometer have an overlap at the interferometer exit, which is only possible if the path difference of the interferometer arms is at most of the order of the usual first-order coherence length. Thus this interference term is suppressed as soon as the difference in the traversal times of the respective interferometer β = L, R, given by ∆τ β = τ u βα −τ d βα is large compared to the half width Γ α of a current pulse emitted by the source α = A, B. The suppression of the singleparticle interference can therefore be used as a measure for the spreading of the wavepacket emitted by a driven capacitor. The path difference is tunable in experiment by side gates applied to the edge states [13]. We are now interested in the situation where the flux-dependence of the currents vanishes, and interference effects can be attributed to two-particle correlations.
The noise.-Two-particle correlations can be observed in the noise properties. We calculate the symmetrized zero-frequency noise power (shot noise) [18] for currents flowing into contacts 1 and 2. The equation for noise complementary to Eq. (2) reads,
P 12 = e 2 h ∞ n=−∞ sign(n) µ µ−nhΩ dE T 0 dt T T 0 dt ′ T e inΩ(t−t ′ ) × 4 l,j=1 S 1l (t, E)S * 1j (t, E n )S * 2l (t ′ , E)S 2j (t ′ , E n ) ,(4)
where E n = E + nhΩ. If the arm length differences of the MZIs are commensurate, ∆τ := ∆τ L = ∓∆τ R , we find in the zeroth order in ΩΓ α ,
P 12 = −P 0 s=e,h T L T R [1 − L(∆t s )] (5) −γ L γ R cos (Φ L ± Φ R ) [L(∆t s − ∆τ ) + L(∆t s + ∆τ )] ,
with the classical MZI transmission probability
T β = T l β T r β + R l β R r β .
Here ∆t s = t s A − t s B + ∆τ AB depends on the difference of emission times t s α and the time delay ∆τ AB = τ u βA − τ u βB due to the asymmetry of the set-up; the Lorentzians are defined by L(X) = 4Γ A Γ B /[X 2 + (Γ A + Γ B ) 2 ] and P 0 = (2e 2 /π)T C R C Ω is (minus) the shot noise produced by the central QPC alone [5].
The second line in Eq. (5) is a magnetic flux-dependent contribution, appearing under two conditions. First, the interferometers have to be commensurate, ∆τ L = ∓∆τ R . Second, the emission times of the cavities are such that a collision of two electrons (and/or two holes [19]) can take place at the interferometer outputs, |∆t s ± ∆τ | ≤ Γ α . Both conditions together imply that the collisions take place at both interferometers. That results in an appearance of non-local two-particle correlations irrelevant for the current but with a pronounced effect in the noise. In Fig. 2 we show the shot noise for ∆τ L = ∆τ R , as a function of the magnetic flux difference and the phase shift ϕ between potentials U A (t) = U A cos(Ωt) and U B (t) = U B cos(Ωt + ϕ) acting onto the capacitors A and B. We choose the difference between electron and hole emission times from the two sources to be equal. Each source emits one electron and one hole during the period T . Varying the phase ϕ we change the time t e B when the capacitor B emits an electron. At ϕ = ϕ 0 the condition ∆t e − ∆τ = 0 is satisfied and an electron emitted by the capacitor A and moving along the lower arm of an interferometer can collide (overlap) with an electron emitted by the capacitor B and moving along the upper arm of the same interferometer (vice versa for ϕ = −ϕ 0 ). Therefore, a mere variation of the phase difference between the driving potentials can switch on or switch off the two-particle AB-effect. Note the dip at ϕ = 0 is the fermionic HOM effect [5].
Entanglement.-Now we show that these correlations are quantum, i.e., the emitted electrons are orbitally entangled in pairs. The same, of course, is valid with respect to holes. The degree of entanglement of the two-particle state, expected whenever the shot-noise of the system becomes flux dependent, can be measured in terms of the concurrence [20]. A wavepacket created at the source A by the operatorŝ A † = dk w(k)e ikvF (tA−t)â † (k) acting on the Fermi sea |0 is scattered intô
A † out (t) = dk w(k) t C t l R F u Rĝ † R2 (k) (6) +t C r l R F d Rĝ † R1 (k) + r C t l L F u Lĝ † L2 (k) + r C r l L F d Lĝ † L1 (k) , where F u/d β = e iΦ u/d Aβ e ikvF(tA+τ u/d Aβ −t)
, (analogously for B † andB † out (t); for simplicity, we suppose that the wavepackets have the same shape, w(k)). The creation operatorsĝ(k) are related to the wavepacket detected at the four interferometer outcomes, representing the states |1 L , |2 L , |1 R , |2 R (see Fig. 1). We consider a part of the outgoing state corresponding to events with one particle to the left and one particle to the right. Decomposing this outgoing two-particle state, |Ψ out = A † out (t)B † out (t)|0 = 2 n,m=1 χ nm |n L |m R , we choose the two observation times, t L and t R such that each basis state consists of a product of single particle states detected at the left and at the right side of the system. The concurrence is calculated from C = 2 det(χχ † ) [21]. Whenever the collision conditions leading to a flux de-pendence of the noise are not fulfilled at least three elements of χ vanish and the concurrence is therefore zero, C = 0. We consider now the time differences of the interferometer paths to be equal for the two interferometers and choose the times fulfilling the collision conditions as measuring times,
t L = t B + τ d BL = t A + τ u AL and t R = t A + τ u AR = t B + τ d BR .
The concurrence is then
C = 2T C R C T l R R l L T l L R l R /(T 2 C T l R R l L + R 2 C T l L R l R ). (7)
Full entanglement, C = 1, is obtained by tuning the capacitors and choosing the appropriate measuring times, when the transmission and reflection probabilities are all equal.
To witness the entanglement found above, which is produced in definite bins of time, we use the violation of a Bell inequality (BI). Following Glauber [22] we introduce the joint probability N 12 to detect one electron (a wave packet) at the contact 1 at the time of collision t L and the other electron at the contact 2 at the time of collision t R . For simplicity we make no distinction between the time when a particle passed the interferometer output and the time of detection. This quantity can be calculated as follows, N 12 = δN 12 + N 1 N 2 , where N j is the mean number of electrons detected at the contact j = 1, 2 at the corresponding time, and δN 12 is a correlator. To calculate, e.g., eN 1 we integrate the current I 1 (t), Eq. (2), near t L over a time interval τ m longer than the width of a current pulse Γ α but shorter than the distance between different pulses. We find,
N 1 = R C T l L T r L + T C R l L R r L and N 2 = T C T l R T r R + R C R l R R r R .
To calculate δN 12 , which is proportional to the shot noise, we modify Eq. (4) in the following way. The total shot noise P 12 comprises contributions from particles arriving at contacts at any time during the driving period T . Since we are interested in the contribution of electrons, we restrict the integral over t (t ′ ) to the interval τ m around t L (t R ). The time-resolved shot noise P 12 (t L , t R ) obtained thus defines δN 12 = (π/Ω)P 12 (t L , t R )/e 2 . Calculations yield, δN 12
= δN A 12 + δN B 12 + δN AB 12 , where δN A 12 = −R C T C T l L T r L T l R T r R and δN B 12 = −R C T C R l L R r L R l R R r R
are due to the capacitors A and B alone, and the correlation contribution,
δN AB 12 = 2R C T C γ L γ R L(∆t e − ∆τ ) cos (Φ L − Φ R ) ,(8)
depends on the wave-packet overlap L(∆t e − ∆τ ).
To test a BI [8,23] we use the four joint probabilities to detect one electron at the contacts 1 (or 3) at the time t L and another electron at the contacts 2 (or 4) at time t R . For the normalized correlation function, E = (N 12 + N 34 − N 14 − N 32 ) / (N 12 + N 34 + N 14 + N 32 ), we find, E = L(∆t e −∆τ ) cos (Φ L − Φ R ), if the transmissions at all the QPCs are 1/2. The magnetic fluxes Φ L(R) have no effect on the collision times t L(R) . Therefore, we can choose four sets of Φ L − Φ R , obtaining four different values of E, see, e.g., Ref. [23], to maximally violate a BI. This inequality holds for L(∆t e − ∆τ ) > 1/ √ 2, i.e. even at partial overlap of wave-packets: for Γ := Γ A = Γ B , we find the condition |∆t e − ∆τ | < ∼ 1.2Γ. Conclusion.-Two uncorrelated but synchronized single-particle emitters can produce entangled pairs of electrons (holes). The orbital entanglement exists in bins of time with a well defined position within the driving period. Its key signature is a non-local Aharonov-Bohm effect in distant loops. Simply changing the phase difference between the potentials driving the capacitors switches the entanglement on or off in a controlled manner. A successful realization of our proposal would open up new perspectives for a coherent quantum electronics.
We thank P. Samuelsson for useful discussion. We acknowledge the support of the Swiss NSF, the program for MANEP, and the EU project SUBTLE.
PACS numbers: 73.23.-b, 72.10.-d, 73.50.Td
FIG. 1 :
1(color online). Emitters A and B driven by potentials UA(t) and UB(t) inject single particles into edge states (solid lines). After scattering at a center QPC (C) particles reach Mach-Zehnder interferometers with Aharonov-Bohm fluxes ΦL and ΦR. The colored (dotted and dashed) lines show possible two-particle amplitudes which lead to particle collisions erasing which path information.
FIG. 2 :
2Two-particle Aharonov-Bohm oscillations in the shot noise correlation P12 as a function of the difference in magnetic fluxes ΦA−ΦB and difference of phase ϕ of the potentials UA(t) and UB(t).
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[
"DECISION PROBLEMS FOR LINEAR RECURRENCES INVOLVING ARBITRARY REAL NUMBERS",
"DECISION PROBLEMS FOR LINEAR RECURRENCES INVOLVING ARBITRARY REAL NUMBERS"
]
| [
"Eike Neumann [email protected] \nMax Planck Institute for Software Systems\nSaarland Informatics CampusSaarbrückenGermany\n"
]
| [
"Max Planck Institute for Software Systems\nSaarland Informatics CampusSaarbrückenGermany"
]
| [
"Logical Methods in Computer Science"
]
| We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.CCCreativeCommons 16:2 E. Neumann Vol. 17:3 either Positivity or Ultimate Positivity at order six would entail major breakthroughs in the field of Diophantine approximation, making it highly unlikely for existing mathematical methods to allow for further progress to be made on these problems. For linear recurrences with simple characteristic roots the Positivity Problem is known to be decidable up to order nine [OW14a] and the Ultimate Positivity Problem is known to be decidable for all orders [OW14c]. These and related decision problems have numerous applications in theoretical computer science and beyond, see [OW14b] and references therein. See the survey [OW15]for a more detailed historical overview and further related results.In this paper we study the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences whose coefficients and initial values are arbitrary real numbers which are given as fast converging Cauchy sequences of rational numbers. For topological reasons no non-trivial problem is decidable in this setting. Instead one should ask if there exists a maximal partial algorithm for deciding a given problem. When such an algorithm is given a problem instance as input it either diverges or halts and outputs the correct answer for the decision problem. It is required to halt on every problem instance for which the answer for the decision problem is locally constant. We will call such problem instances robust instances. This ensures that its halting set contains the halting set of any correct partial algorithm for deciding the problem.Besides being mathematically interesting in their own right, the real number versions of the Skolem Problem and its variants can be motivated by practical applications. In most applications to engineering and the natural sciences the assumption that the input be given as an exact integer, rational number, or real algebraic number is quite unrealistic. There it is usually more appropriate to assume that the inputs be known only approximately to finite accuracy, but with a known error bound. This can be modelled by assuming that one is given a rational box which contains the problem instance of interest. A maximal partial algorithm for deciding the real number version of a decision problem can be automatically translated to an algorithm which takes as input such a box, halts and outputs 1 if the box is contained in the set of robust "yes"-instances, halts and outputs 0 if the box is contained in the set of robust "no"-instances, halts and outputs −1 if the box contains both robust "no"and robust "yes"-instances, and diverges in all other cases. Such an algorithm need not exist when the problem is decidable on rational inputs: There exist sets A ⊆ R such that A ∩ Q is decidable but A is not maximally partially decidable. One can for instance take the union of any decidable subset of Q with a singleton {x} that is not co-c.e. closed. It is of course conversely true that there exist sets A ⊆ R that are maximally partially decidable such that A ∩ Q is not decidable. One can take for instance any undecidable co-c.e. set of integers.We will show that the real number versions of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem are maximally partially decidable for linear recurrences of any order. Thus, the real number versions of all three problems are, in a sense, as close to decidable as one can expect them to be. That this is achievable for real linear recurrences despite the hardness results for rational ones is perhaps not too surprising. The existing decidability proofs for low orders only fail to generalise to higher orders due to the presence of "critical" problem instances where the dominant part of the exponential polynomial solution does not admit an exponential lower bound. Since this situation is unstable under small perturbations one will not be obliged to decide the problems in these critical instances for real number inputs, thus avoiding the aforementioned hardness results.Indeed, our proof consists almost entirely of translating the known decision methods for low-orders to the real number setting and noting that these already suffice to establish Vol. 17:3 DECISION PROBLEMS FOR LINEAR RECURRENCES 16:3 16:6 E. Neumann Vol. 17:3 | 10.46298/lmcs-17(3:16)2021 | [
"https://arxiv.org/pdf/2008.00583v4.pdf"
]
| 220,936,483 | 2008.00583 | 283f6f1f69a9709a347590e98a4e9557f45453ca |
DECISION PROBLEMS FOR LINEAR RECURRENCES INVOLVING ARBITRARY REAL NUMBERS
2021
Eike Neumann [email protected]
Max Planck Institute for Software Systems
Saarland Informatics CampusSaarbrückenGermany
DECISION PROBLEMS FOR LINEAR RECURRENCES INVOLVING ARBITRARY REAL NUMBERS
Logical Methods in Computer Science
17326202110.46298/LMCS-17(3:16)2021Submitted Nov. 04, 2020
We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.CCCreativeCommons 16:2 E. Neumann Vol. 17:3 either Positivity or Ultimate Positivity at order six would entail major breakthroughs in the field of Diophantine approximation, making it highly unlikely for existing mathematical methods to allow for further progress to be made on these problems. For linear recurrences with simple characteristic roots the Positivity Problem is known to be decidable up to order nine [OW14a] and the Ultimate Positivity Problem is known to be decidable for all orders [OW14c]. These and related decision problems have numerous applications in theoretical computer science and beyond, see [OW14b] and references therein. See the survey [OW15]for a more detailed historical overview and further related results.In this paper we study the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences whose coefficients and initial values are arbitrary real numbers which are given as fast converging Cauchy sequences of rational numbers. For topological reasons no non-trivial problem is decidable in this setting. Instead one should ask if there exists a maximal partial algorithm for deciding a given problem. When such an algorithm is given a problem instance as input it either diverges or halts and outputs the correct answer for the decision problem. It is required to halt on every problem instance for which the answer for the decision problem is locally constant. We will call such problem instances robust instances. This ensures that its halting set contains the halting set of any correct partial algorithm for deciding the problem.Besides being mathematically interesting in their own right, the real number versions of the Skolem Problem and its variants can be motivated by practical applications. In most applications to engineering and the natural sciences the assumption that the input be given as an exact integer, rational number, or real algebraic number is quite unrealistic. There it is usually more appropriate to assume that the inputs be known only approximately to finite accuracy, but with a known error bound. This can be modelled by assuming that one is given a rational box which contains the problem instance of interest. A maximal partial algorithm for deciding the real number version of a decision problem can be automatically translated to an algorithm which takes as input such a box, halts and outputs 1 if the box is contained in the set of robust "yes"-instances, halts and outputs 0 if the box is contained in the set of robust "no"-instances, halts and outputs −1 if the box contains both robust "no"and robust "yes"-instances, and diverges in all other cases. Such an algorithm need not exist when the problem is decidable on rational inputs: There exist sets A ⊆ R such that A ∩ Q is decidable but A is not maximally partially decidable. One can for instance take the union of any decidable subset of Q with a singleton {x} that is not co-c.e. closed. It is of course conversely true that there exist sets A ⊆ R that are maximally partially decidable such that A ∩ Q is not decidable. One can take for instance any undecidable co-c.e. set of integers.We will show that the real number versions of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem are maximally partially decidable for linear recurrences of any order. Thus, the real number versions of all three problems are, in a sense, as close to decidable as one can expect them to be. That this is achievable for real linear recurrences despite the hardness results for rational ones is perhaps not too surprising. The existing decidability proofs for low orders only fail to generalise to higher orders due to the presence of "critical" problem instances where the dominant part of the exponential polynomial solution does not admit an exponential lower bound. Since this situation is unstable under small perturbations one will not be obliged to decide the problems in these critical instances for real number inputs, thus avoiding the aforementioned hardness results.Indeed, our proof consists almost entirely of translating the known decision methods for low-orders to the real number setting and noting that these already suffice to establish Vol. 17:3 DECISION PROBLEMS FOR LINEAR RECURRENCES 16:3 16:6 E. Neumann Vol. 17:3
Introduction
A real linear recurrence sequence is a sequence (u k ) k of real numbers satisfying a linear recurrence relation of the form u k+n = c 1 u k+n−1 + · · · + c n u k . The Skolem-Mahler-Lech theorem asserts that the zero set {k ∈ N | u k = 0} of such a sequence is of a particularly simple form: it is the union of a finite set F and an empty or infinite set I which is the union of a finite number of arithmetic progressions. For given rational or integer coefficients c 1 , . . . , c n and initial values u 1 , . . . , u n the set I can be effectively computed [BM76]. The same is not known of the set F . It is well known [EvdPSW03] that the problem of computing the set F is equivalent to the problem of deciding whether a given linear recurrence has a zero. The latter problem, often referred to as the Skolem Problem in the literature, has proven to be infamously difficult. It is generally believed to be decidable for linear recurrences of any order, but known to be decidable only up to order four [TMS84,Ver85]. Two closely related decision problems of note are the Positivity Problem, which asks for given a linear recurrence to decide whether all of its terms are non-negative, and the Ultimate Positivity Problem, which asks for given a linear recurrence to decide whether all but finitely many of its terms are non-negative. In [OW14b] it was shown that both Positivity and Ultimate Positivity are decidable up to order five. At the same time it was shown that a feasible algorithm for solving maximal partial decidability for all orders. The proof is considerably more elementary than its counterparts for integer coefficients. The use of Baker's theorem and similar deep results from analytic number theory can be entirely avoided. On the other hand new problems appear in the real number setting that are absent from the discrete setting. The study of the asymptotic behaviour of a linear recurrence relies heavily on the study of its exponential polynomial solution. It is easy to see that in the real number setting the exponential polynomial solution is not in general computable from the linear recurrence. This is where new ideas are required. It is relatively easy to see that one can still compute those coefficients of the exponential polynomial solution which belong to simple characteristic roots. This will suffice to computably recognise all robust instances of the Skolem Problem and the Positivity Problem, and all robust "yes"-instances of the Ultimate Positivity Problem. For the Ultimate Positivity Problem there exist robust "no"-instances whose dominant characteristic roots are not simple, which implies that the corresponding coefficients in the exponential polynomial solution do not depend continuously on the input. These instances are by far the most difficult ones to handle. They will be treated by reduction to the first-order theory of the reals. Recall that the first-order theory of the reals is the first-order theory of the structure R, 0, 1, +, ×, −, >, = . By the Tarski-Seidenberg theorem [BPR06,Theorem 2.77] this theory is decidable. The main ideas in this case are best illustrated with the help of a simple example:
Example 1.1. Consider the linear recurrence u 1 = π, u 2 = 2π, u 3 = π, u k+1 = 3u k − 3u k−1 + u k−2 . Its characteristic polynomial is P (x) = x 3 − 3x 2 + 3x − 1 = (x − 1) 3 and its exponential polynomial solution is u k = π + kπ − k(k − 1)π. It is hence a "no"-instance of the Ultimate Positivity Problem. Let us show how we can verify this computationally when the coefficients and initial values are given as sequences of approximations. This is not completely straightforward since, as mentioned earlier, the exponential polynomial solution does not depend continuously on the input.
Choose a small rational number ε > 0. We can compute the roots of the characteristic polynomial to error ε to verify that all complex roots are contained in the open disk B(1, ε) of radius ε > 0 centred at 1. We can numerically count the roots in this disk with multiplicity, to find that there are three roots counted with multiplicity. Therefore there are only finitely many possibilities for the configuration of these roots:
(1) R 1 : There is one real root ρ with multiplicity 3.
(2) R 2 : There is a real root ρ 0 with multiplicity 2 and a real root ρ 1 with multiplicity 1 and ρ 0 > ρ 1 . (3) R 3 : There is a real root ρ 0 with multiplicity 1 and a real root ρ 1 with multiplicity 2 and ρ 0 > ρ 1 . (4) R 4 : There are three simple real roots ρ 0 , ρ 1 , ρ 2 with ρ 0 > ρ 1 > ρ 2 . (5) R 5 : There is one simple real root ρ and two complex conjugate roots λ,λ.
Call R 1 , . . . , R 5 the possible root configurations. To each root configuration we can assign a characteristic polynomial, for instance P 1 (x) = (x − ρ) 3 and P 5 (x) = (x − ρ)(x − λ)(x −λ). This gives rise to an associated linear recurrence. For instance, the linear recurrence associated with R 1 is u k+1 = 3ρu k − 3ρ 2 u k−1 + ρ 3 u k−2 and the linear recurrence associated with R 5 is u k+1 = (ρ + λ +λ)u k − (λρ +λρ + λλ)u k−1 + ρλλ We can symbolically compute the exponential polynomial solutions of each of these linear recurrences.
Each possible root configuration R j can be assigned a domain D R j which is the set of all "valid" assignments to the variables that occur in the root configuration. For instance, D R 2 = (ρ 0 , ρ 1 ) ∈ (1 − ε, 1 + ε) 2 | ρ 0 > ρ 1 and D R 5 = (1 − ε, 1 + ε) × (B(1, ε) ∩ H) where H = {z ∈ C | Im z > 0}. Note that up to identifying C with R 2 each domain is a definable set in the first-order theory of the reals. We can substitute a point in the domain of a possible root configuration R for the variables of R to obtain a linear recurrence, which we call the associated linear recurrence of that point. Any such linear recurrence is a "small perturbation" of our original linear recurrence, enriched with additional information about the algebraic multiplicities of its characteristic roots. In particular our original linear recurrence can be recovered as the associated linear recurrence of some point in the domain of some possible root configuration -in this concrete case it is the linear recurrence associated with 1 ∈ D R 1 .
To establish that our instance is not ultimately positive it would hence suffice to show that for all points in the domain of all possible root configurations the coefficient of the dominant real term in the exponential polynomial solution of the associated linear recurrence is negative. This is a well-known result about linear recurrences. See Lemma 3.2 below for a formal statement.
Let us carry this out for the root configuration R 2 . The dominant term in the exponential polynomial solution of the associated linear recurrence is kρ k−1 0 . An explicit symbolic calculation shows that its coefficient is equal to
u 3 − (ρ 0 + ρ 1 )u 2 + ρ 0 ρ 1 u 1 ρ 0 − ρ 1 .
We can compute rational approximations v 1 , v 2 , v 3 of the initial values to error ε. It then suffices to show that the following sentence holds true:
∀ρ 0 .∀ρ 1 .∀u 1 .∀u 2 .∀u 3 . |ρ 0 − 1| < ε ∧ |ρ 1 − 1| < ε ∧ ρ 0 > ρ 1 ∧ |u 1 − v 1 | < ε ∧ |u 2 − v 2 | < ε ∧ |u 3 − v 3 | < ε → u 3 − (ρ 0 + ρ 1 )u 2 + ρ 0 ρ 1 u 1 ρ 0 − ρ 1 < 0
This sentence can be formulated in the first-order theory of the reals. Its truth is hence decidable by the Tarski-Seidenberg theorem. It is clear that the sentence is indeed true for sufficiently small ε > 0. This yields a semi-decision procedure for showing that for all points in D R 2 the coefficient of the dominant real term in the exponential polynomial solution of the associated linear recurrence is negative. A similar reduction to the first-order theory of the reals can be carried out for the remaining root configurations R 1 , R 3 , . . . , R 5 to show that the given linear recurrence is a robust "no"-instance of Ultimate Positivity.
Of course, we have merely verified a particular sufficient condition which happened to hold true for this specific instance. The full algorithm will be slightly more involved, but it will follow the same ideas.
Decision Problems for continuous data
We work within the framework of represented spaces as introduced by Kreitz and Weihrauch [KW85] for countably based spaces and extended by Schröder [Sch02a,Sch02b] to quotients of countably based spaces. See [Wei00,BP03,Pau16] for introductions to this approach to computable analysis at varying levels of abstraction. It will suffice for our purpose to work with admissibly represented countably based T 0 spaces. We will mainly use the notation from [Pau16]. In particular for a represented space X we denote by K(X) the represented space of compact sets with the sequentialisation of the upper Vietoris topology, by (X) the space of open sets with the Scott topology, and by A(x) the space of closed sets where a closed set A is identified with its complement A C ∈ (X). For the benefit of readers unfamiliar with computable analysis we will briefly recall the very basic ideas for computing on the space R n .
A rational interval is a closed interval with rational endpoints. A rational box is a finite product of rational intervals. A point x ∈ R n can be represented by a sequence (B j ) j of rational boxes such that each box contains x and j∈N B j = {x}. Any such sequence is called a name for x. The point x is called computable if it has a computable name, i.e., there exists an algorithm which on input j ∈ N outputs a box B j such that the resulting infinite sequence (B j ) j is a name for x. A function f : R n → R m is computable if there exists an algorithm which takes as input 1 a name (B j ) j of a point x and a natural number k ∈ N and outputs a rational box C k , such that for every fixed name (B j ) j the resulting infinite sequence (C k ) k is a name for f (x). It is easy to see that any computable function is necessarily continuous with respect to the usual topology on R n .
One may object that this notion of computability models an unrealistic situation in which one has arbitrarily good approximations to a real vector available. However, it is easy to prove that f : R n → R m is computable if and only if there exists an algorithm which takes as input a rational box B and a positive rational number ε > 0, and returns as output a finite list of boxes C 1 , . . . , C s such that each C j has width at most ε, each C j intersects the range f (B), and C 1 ∪ · · · ∪ C s ⊇ f (B). Moreover, such an algorithm can be effectively computed from an algorithm which computes f in the above sense and vice versa. Thus, this notion of computability captures precisely the idea that one can compute arbitrarily good information on f (x) when x is given to finite accuracy.
Let us now discuss decision problems in this context. With any subset A ⊆ R n one can associate a decision problem: given x ∈ R n as input halt in finite time and output 1 if and only if x ∈ A or output 0 if x / ∈ A. It is easy to see that for continuity reasons the only subsets of R n that are decidable in this sense are the empty set and R n itself.
The next best thing one can hope for is to find an algorithm which decides the problem in as many points as possible. This is somewhat of a folklore idea in computable analysis, but there does not appear to be an established standard terminology in the literature. Let X be a represented space. A partial 2 algorithm for deciding a set A ⊆ X is an algorithm which takes as input a name of a point x ∈ X and either diverges or halts in finite time and 1 An infinite input sequence can for instance be implemented as an infinite stream that is written on a special input tape or as an oracle.
2 The term "partial algorithm" is used here in the traditional sense of theoretical computer science. In breach with the usual convention in computable analysis the algorithm's behaviour is constrained on the entire space. A computable analyst may hence prefer to view such an algorithm as a total algorithm with values in Kleene space K = {0, 1, ⊥}.
outputs 0 or 1. We require that such an algorithm be extensional, i.e., that its termination and output on termination depend only on the point x but not on the choice of name 3 . We further require that it be correct, i.e., that it halt and return 1 only when x ∈ A and that it halt and return 0 only when x / ∈ A. Its halting set is the set of points for which it halts. This set is well-defined by the extensionality assumption. A partial algorithm for deciding A is called maximal if its halting set contains the halting set of all other partial algorithms for deciding A. We call A maximally partially decidable if there exists a maximal partial algorithm for deciding A.
Proposition 2.1. Let X be an admissibly represented countably based T 0 space. A partial algorithm for deciding A is maximal if and only if its halting set is equal to the set of points of continuity of the characteristic function χ A : X → {0, 1}, where {0, 1} carries the discrete topology.
Proof. It is easy to see that the halting set of an algorithm for deciding A must be contained in the set of points of continuity of the characteristic function. Conversely, if x is a point of continuity of χ A then there exists a basic open set B which contains x such that χ A is constant on B. There exists an algorithm which halts on B and outputs the constant value of χ A . It follows that any maximal partial algorithm for deciding A must contain x in its halting set.
In other words, a maximal partial algorithm for deciding A is an algorithm that takes as input x ∈ X, halts and outputs 1 if x is contained in the interior of A, halts and outputs 0 if x is contained in the interior of the complement of A, and diverges if x is contained in the boundary of A. In particular, every set is maximally partially decidable relative to some oracle, and if X is a discrete space then A ⊆ X is maximally partially decidable if and only if it is decidable. Therefore maximal partial decidability seems to be, in some sense, a more appropriate generalisation of decidability over N than "naive" decidability. One should however bear in mind that this is a relative notion: The set of rational numbers Q, say, is a maximally partially decidable subset of R. A maximal partial correct algorithm is given by the algorithm that never halts. Once maximal partial decidability of a problem is established one is hence naturally lead to the study of the "absolute size" of the halting set.
It will be convenient to introduce the following terminology for decision problems: Let A ⊆ X be a set. Call any point x ∈ X an instance of (the decision problem associated with) A. If x ∈ A then x is called a "yes"-instance. If x / ∈ A then x is called a "no"-instance. If x is a point of continuity of the characteristic function χ A , i.e., if x is not contained in the boundary of A, then x is called a robust instance of A.
Finally, the following convention will be very useful: We say that a property P holds true for a point p ∈ X up to an arbitrarily small perturbation if for every open set U ∈ (X) which contains x there exists y ∈ U such that P holds true for y.
As mentioned in the introduction, maximal partial decidability is equivalent to "almost deciding" a trichotomy when the input is known only to finite accuracy. We formulate this only for the case of R n but it generalises easily to all locally compact spaces.
Proposition 2.2. Let A ⊆ R n be a set. Then A is maximally partially decidable if and only if there exists an algorithm which takes as input a rational box B, halts and outputs 1 if B is contained in the set of robust "yes"-instances of A, halts and outputs 0 if B is contained in the set of robust "no"-instances of A, halts and outputs −1 if B contains robust "yes"-instances as well as robust "no"-instances, and diverges in all other cases.
Moreover, such an algorithm can be effectively computed from a maximal partial algorithm for deciding A and vice versa.
Proof. It is obvious that the existence of such an algorithm implies maximal partial decidability.
Assume that A is maximally partially decidable. Then the set of robust "yes"-instances is computable as an element of the represented space (R n ) of open sets with the Scott topology. The same holds true for the set of "no"-instances.
An encoding of a rational box B can be effectively translated to a name of the same box as an element of the space K(R n ) of compact subsets with the sequentialisation of the upper Vietoris topology. One can therefore semi-decidable if all instances contained in B are robust "yes"-instances or if all instances contained in B are robust "no"-instances.
We can also effectively compute a name of the interior of B as an element of the space (R n ) of open sets with the Scott topology. Now, if B contains a robust "yes"-instance then the interior B • of B contains a robust "yes"-instance. We can thus semi-decide if B contains a robust "yes"-instance by computing the intersection of B • with the set of robust "yes"-instances as an element of (R n ) and semi-deciding if the result is non-empty. By symmetry the same is true for robust "no"-instances. The claim follows.
Throughout this section we have focussed our attention on countably based spaces. This will be sufficient for the purpose of this paper. It should be pointed out however, that Proposition 2.1 may fail for non-countably-based spaces. To see this, let
2 = (x n ) n ∈ R N | n∈N x 2 n < +∞
be the computable metric space of square-summable real sequences with the metric induced by the usual inner product ·, · . Let 2 ⊆ R 2 denote the admissibly represented space of linear functionals on 2 , with the representation inherited from the exponential R 2 in the category of represented spaces. As usual, the space 2 can be identified with 2 (with a weaker topology) by virtue of the self-duality of Hilbert spaces. In particular, the inner product ·, · can be defined on 2 . It is shown in [BS05] that the space 2 is computably
isomorphic to the space (x n ) n ∈ R N | x 0 = n≥1 x 2 n .
It is further shown that the space 2 is computably isomorphic to the quotient space
(x n ) n ∈ R N | x 0 ≥ n≥1 x 2 n / ∼
where (x n ) n ∼ (y n ) n if and only if (x n ) n≥1 = (y n ) n≥1 . In other words, a point x ∈ 2 can be represented by the sequence (x n ) n and its 2 -norm, while a point x ∈ 2 can be represented by the sequence (x n ) n and some upper bound on its 2 -norm. Now, let p ∈ 2 be a computable point with uncomputable 2 -norm (see e.g. [Neu15, Theorem 5.9] for a construction of a such a point). Consider the hyperplane H = x ∈ 2 | p, x = 0 . Then the complement of H is open and dense but contains no non-empty semi-decidable subset. To see the last claim, assume that there exists an algorithm whose halting set is a non-empty subset U of the complement of H.
Identify 2 with (x n ) n ∈ R N | x 0 ≥ n≥1 x 2 n / ∼ as above. Let x = (x 0 , x 1 , . . . ) be a point in U .
Then there exists an integer m such that the algorithm halts on all points of the form (N,
x 1 , . . . , x m , 0, 0, . . . ) with N ≥ x 2 1 + · · · + x 2 m .
From this it follows that for every N > x 2 1 + · · · + x 2 m there exists a positive integer ν(N ) such that the algorithm halts on all sequences in 2 that start with N, x 1 , . . . , x m , followed by ν(N ) zeroes. In particular the algorithm halts on all sequences
s(α) = N, x 1 , . . . , x m , 0, . . . , 0 ν(N ) times , α · p m+ν(N )+1 , α · p m+ν(N )+2 , . . . with x 2 1 + · · · + x 2 m + α 2 n>m+ν(N ) p 2 n ≤ N .
By solving the equation p, s(α) = 0 for α and using that the halting set is assumed not to intersect H, we find that
n>m+ν(N ) p 2 n < (p 1 x 1 + · · · + p m x m ) 2 N − x 2 1 + · · · + x 2 m .
Since N may be chosen arbitrarily large, this yields an algorithm for computing the norm of p, contradicting our initial assumption. It follows that H is maximally partially decided by the algorithm that never halts. The halting set of this algorithm is clearly much smaller than the points of continuity of the characteristic function of H.
This suggests that a reasonable alternative approach to the study of the "partial decidability" of subsets of admissibly represented spaces may be obtained by taking the characterisation given in Proposition 2.1 as the definition directly. Letting K = {0, 1, ⊥} be endowed with the topology generated by the sets {0} and {1}, one can study the computability of the continuous function χ A : X → K which sends the interior of A to 1, the complement of the closure of A to 0, and the boundary of A to ⊥. For countably based X one recovers the definition of maximal partial decidability via Proposition 2.1. Since the hyperplane H above is maximally partially decided by the algorithm that never halts, the function χ H is "nowhere computable" in the sense that its restriction to every open subset of 2 is uncomputable. The function χ A arises quite naturally as the continuous coreflection of the characteristic function χ A : X → K in the sense of Escardó [Esc98, Proposition 2.6.1] -an idea that already goes back to Scott [Sco72]. It is also the best continuous approximation to χ A in the very strong sense of [Neu19].
Linear Recurrences
A real linear recurrence sequence is a sequence (u k ) k such that there exists a positive integer n ∈ N and real numbers c 1 , . . . , c n ∈ R such that for all k > n. This sequence can hence be encoded by the vector
u k = c 1 u k−1 + c 2 u k−2 + · · · + c n u k−n (3.1) Vol(c, u) = (c 1 , . . . , c n , u 1 , . . . , u n ) ∈ R 2n .
Note that this encoding is not unique. The order of a linear recurrence sequence is the smallest possible n such that (u k ) k satisfies a relation of the above form. From a computational point of view it is important to treat different encodings of the same sequence as different problem instances. This will be illustrated in Example 3.1 below. We define a linear recurrence to be a vector (c, u) ∈ R 2n for some n ∈ N. The number n is called the order of (c, u). Note that the sequence (u k ) k generated by the linear recurrence (c, u) could satisfy a linear recurrence relation of strictly lower order. In other words, the order of the linear recurrence is in general not the same as the order of the linear recurrence sequence (u k ) k it generates. The space of all linear recurrences is identified with the represented space n∈N R 2n .
The companion matrix of a linear recurrence (c, u) is the matrix
c 1 c 2 . . . c n−1 c n 1 0 . . . 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 . . . 1 0
The characteristic polynomial of (c, u) is the polynomial
P (x) = x n − c 1 x n−1 − · · · − c n ,
i.e., up to sign, the characteristic polynomial of its companion matrix. The complex roots of the characteristic polynomial are called the characteristic roots of (c, u).
The exponential polynomial solution plays a crucial role in the study of the asymptotic behaviour of linear recurrences. Its definition is a bit subtle in our context. A formal complex polynomial is a vector Q = (a 0 , . . . , a m ) ∈ C m , which we also write as Q(x) = a 0 +· · ·+a m x m . The formal degree of Q is the number m. We do not assume here that a m = 0, hence the name "formal degree".
A formal exponential polynomial is a function f : N → C of the form
f (k) = P 1 (k)λ k 1 + · · · + P s (k)λ k s
where the P j 's are formal polynomials and the λ j 's are distinct complex numbers. A formal exponential polynomial is assumed to be encoded as a vector in C d 1 × · · · × C ds × C s , where d 1 , . . . , d s are the formal degrees of P 1 , . . . , P s . The space of formal exponential polynomials is the co-product over all spaces of this form. A formal exponential polynomial is formally real-valued if for every λ j ∈ C \ R there exists an index l with λ l =λ j and P l =P j . Note that this is stronger than to require for the function f (k) to be real-valued. The formal exponential polynomial 1 k + 0 · i k is real-valued but not formally real-valued.
The exponential polynomial solution of a linear recurrence (c, u) is the unique formal exponential polynomial f (k) = P 1 (k)λ k 1 + · · · + P s (k)λ k s satisfying u k = f (k) for all k, where λ 1 , . . . , λ s are the distinct characteristic roots of the characteristic polynomial P of (c, u) and the formal degree of the formal polynomial P j is the multiplicity of λ j as a root of P . Its existence and uniqueness follow from existence and uniqueness of the Jordan normal form of the companion matrix of (c, u). Note that the encoding of f as a vector in C d 1 ×· · ·×C ds ×C s is unique only up to permutation. The exponential polynomial solution is clearly formally real valued. Conversely, every formally real-valued exponential polynomial in C d 1 × · · · × C ds × C s is the exponential polynomial solution of a unique linear recurrence (c, u) ∈ R 2(d 1 +···+ds) .
The dominant characteristic roots are the roots of the characteristic polynomial of maximal modulus. Let d be the greatest multiplicity among dominant characteristic roots. The formal dominant part of the exponential polynomial solution is the formal exponential polynomial
a j 1 k d λ k j 1 + · · · + a jt k d λ k jt
where λ j 1 , . . . , λ jt are the dominant characteristic roots of multiplicity d and a j 1 , . . . , a jt are the coefficients of x d in the formal polynomials P j 1 , . . . , P jt . Note that it is possible for any of the a j l 's to vanish, so that the formal dominant part could vanish everywhere as a function.
We can now introduce the problems of interest more formally. The Positivity Problem is the decision problem for the set
(c, u) ∈ n∈N R 2n | u k ≥ 0 for all k ∈ N .
The Ultimate Positivity Problem is the decision problem for the set
(c, u) ∈ n∈N R 2n | there exists K ∈ N such that u k ≥ 0 for all k ≥ K .
The Skolem Problem is the decision problem for the set
(c, u) ∈ n∈N R 2n | u k = 0 for some k ∈ N .
Here we have chosen to treat different encodings of the same linear recurrence sequence as different problem instances. Since the above decision problems pertain to extensional properties of the sequences themselves, it may seem more natural to consider them on the quotient of the space n∈N R 2n under the identification of linear recurrences that encode the same sequence. Such identifications are commonly performed in the computable analysis literature in analogous situations, such as in the definition of the space of polynomials [CH20,Hoy20,dBPS20] or the space of analytic functions [PS18].
Example 3.1. The following example should illustrate why one should discuss decidability on the level of encodings rather than on the level of sequences, why one should treat the exponential polynomial solution as a formal exponential polynomial, and why the asymptotic behaviour of the formal dominant part is more relevant than the asymptotic behaviour of the exponential polynomial as a function.
Consider the sequence (u k ) k with u k = 1 for all k. This sequence is strictly positive and therefore a-fortiori ultimately positive and without zeroes.
The sequence can be viewed as a first-order linear recurrence satisfying the linear recurrence relation u k = u k−1 . It can therefore be encoded as the vector (1, 1) ∈ R 2 . Its characteristic polynomial is P (x) = x − 1, its only characteristic root is 1 with multiplicity 1, and its exponential polynomial solution is f (k) = 1 · 1 k . The formal exponential polynomial f is equal to its formal dominant part. It is easy to see that this linear recurrence remains strictly positive under small perturbations of the coefficients and initial values. It is therefore a robust "yes"-instance of Positivity and Ultimate Positivity and a robust "no"-instance of the Skolem Problem. The same sequence can also be viewed as a second-order linear recurrence satisfying the linear recurrence relation u k = 2u k−1 − u k−2 . This yields a different encoding as the vector (2, −1, 1, 1) ∈ R 4 . The characteristic polynomial of this linear recurrence is Q(x) = x 2 − 2x + 1 = (x − 1) 2 , its only characteristic root is 1 with multiplicity 2, and its exponential polynomial solution is g(k) = (0 · k + 1) · 1 k . The formal dominant part of g is 0 · k · 1 k . Note that the formal exponential polynomial g is equal as a function to the formal exponential polynomial f above. For N ∈ N, consider the linear recurrence encoded by
(2, −1, 1 − 1 N , 1 − 2 N ).
The exponential polynomial solution becomes (− 1 N k + 1) · 1 k . For k = N this new linear recurrence has a zero and for k > N it becomes strictly negative. Since N can be chosen arbitrarily large, we obtain arbitrarily small perturbations of the initial values such that the resulting linear recurrence is a "no"-instance of Positivity and Ultimate Positivity and a "yes"-instance of the Skolem Problem. Therefore, this linear recurrence is not a robust instance of any of these problems.
It is easy to extend the idea of Example 3.1 to show that the problems of interest become trivial on the represented space n∈N R 2n / ∼, where ∼ is the relation that identifies linear recurrences which encode the same sequence, in the sense that all three problems are maximally partially decided by the algorithm that never halts. The same holds true if one is given a linear recurrence sequence (u k ) k as an element of the represented space R N together with a bound n on its order. Another sensible way of encoding a linear recurrence sequence (u k ) k is to provide a matrix A and two vectors v and w such that u k = vA k w. But this is easily seen to be equivalent to the encoding that we have chosen. Finally, it also makes sense to encode a linear recurrence directly as a formal exponential polynomial. This is clearly a strictly stronger representation than the one we have chosen. It is relatively straightforward to show based on the proofs given in this paper that the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for exponential polynomials are maximally partially decidable. This is in fact much easier than the analogous problem for linear recurrences given by a vector of coefficients and a vector of initial values.
We will make frequent use of the following well-known result on linear recurrences. For a proof see e.g. [BG07, Theorem 2].
Lemma 3.2. Let (c, u) be a linear recurrence which is not identically zero. Assume that (c, u) has a dominant characteristic root which is not a positive real number. Let λ 1 , . . . , λ n ∈ C \ [0, +∞) be the non-positive dominant characteristic roots of (c, u). Let P 1 , . . . , P n be their respective coefficients in the exponential polynomial solution. Then the exponential polynomial P 1 (k)λ k 1 + · · · + P n (k)λ k n is either identically zero or admits infinitely many positive and infinitely many negative values.
The next result says that a linear recurrence depends continuously on its exponential polynomial solution. In other words, small perturbations of the exponential polynomial solution induce small perturbations of the linear recurrence. This fact will be extremely useful.
Proposition 3.3. Let n ∈ N. Let d 1 , . . . , d s ∈ N with d 1 + · · · + d s = n. There exists a surjective computable map
ψ : ⊆ C d 1 × · · · × C ds × C s → R 2n
which sends a vector that encodes a formally real-valued formal exponential polynomial f to the unique linear recurrence (c, u) with f (k) = u k for all k ∈ N. Proof. Assume we are given as input a vector (P 1 , . . . , P s , (λ 1 , . . . , λ s )) ∈ C d 1 × · · · × C ds × C s , where the P j 's are interpreted as formal polynomials, such that the exponential polynomial f (k) = P 1 (k)λ k 1 + · · · + P s (k)λ k s is formally real-valued for all k. To obtain the vector u we can simply evaluate f for k = 1, . . . , n. A priori this yields u 1 , . . . , u n as complex numbers, but since these numbers are guaranteed to be real we can compute their real part to obtain the same numbers as real numbers.
To obtain the coefficients c of the linear recurrence, we can effectively compute the vector of coefficients of the polynomial (x − λ 1 ) d 1 · · · · · (x − λ s ) ds as a vector of complex numbers. Since all coefficients are guaranteed to be real we can compute the real part of each entry to obtain c as a real vector.
Similarly, the coefficients of a linear recurrence depend continuously on its characteristic roots in the following sense:
Proposition 3.4. Let n ∈ N. Let D ⊆ C n be the set of all complex vectors whose entries constitute the roots of some monic polynomial with real coefficients, counted with multiplicity. Let h : D → R n be the function that sends a complex vector (λ 1 , . . . , λ n ) to the unique vector (c 1 , . . . , c n ) ∈ R n such that (x − λ 1 ) · · · · · (x − λ n ) = x n − c 1 x n−1 − · · · − c n . Then the map h is computable.
On the computability of the exponential polynomial solution
The aim of this section is to establish that the coefficients in the exponential polynomial solution of any simple dominant characteristic root are computable. First we recall that the characteristic roots are computable:
Theorem 4.1 [Spe69]. There exists an algorithm which takes as input a complex vector (a 0 , . . . , a d ) ∈ d≥1 C d+1 with a d = 0 and outputs a vector (λ 1 , . . . , λ d ) ∈ C d such that (λ 1 , . . . , λ d ) contains the roots of the polynomial a 0 +a 1 z +· · ·+a d z d , counted with multiplicity.
It is worth mentioning that the algorithm in Theorem 4.1 is non-extensional: the output vector depends on the name of the input vector and not just on the vector itself. This is not surprising, since it is well known that there is no continuous single-valued function which assigns to the coefficient vector of a complex polynomial its vector of complex roots.
Lemma 4.2. Let (c, u) be a linear recurrence. Let λ be a characteristic root of (c, u). Then the geometric multiplicity of λ in the companion matrix of (c, u) is equal to 1. Let (x 1 , . . . , x n ) be an eigenvector of A. Then for all j = 2, . . . , n we have the equation x j−1 = λx j . Since any eigenvector is by definition non-zero it follows that x n = 0. It then follows that (λ n−1 , . . . , λ, 1) is an eigenvector and that every eigenvector is a multiple of this one. Hence the geometric multiplicity of λ is equal to 1. Proof. The union of all generalised eigenspaces for any eigenvalue λ of A is uniformly computable from λ as an element of A(C n ). It follows that the union of all generalised eigenspaces for all λ = µ can be computed as an element of A(C n ). By intersecting with the unit sphere we can compute the set of all normalised generalised eigenvectors for all eigenvalues λ = µ as an element of K(C n ). Since linear independence is semi-decidable it follows that the linear span of the generalised eigenvectors which belong to eigenvalues other than µ is computable as an element of A(C n ). Since this is a linear space whose dimension is known to be n where I is the m × m identity matrix and M is some invertible (n − m) × (n − m)-matrix.
Proof. The companion matrix is
A = c 1 c 2 . . . c n 1 0 . . . 0 . . . . . . . . . 1 0 Hence A − λI = c 1 − λ c 2 . . . c n 1 −λ 0 . . . . . . 1 −λ
The exponential polynomial solution of the linear recurrence (c, u) is thus given by
v T T −1 J k 0 0 C k T w,
where C is an (n − m) × (n − m)-matrix in Jordan normal form whose eigenvalues are different from µ. It follows that the term belonging to µ k is indeed (v 1 , . . . , v m ) T J k (w 1 , . . . , w m ). It is clear that this can be computed as a complex vector. If µ is guaranteed to be real we can compute the real part of each entry of the vector to obtain the same vector as a real vector.
Lemma 4.5. There exists an algorithm which takes as input a vector (a 0 , . . . , a n ) ∈ R n+1 with a n = 0 and a rational number ε > 0 and halts if and only if the polynomial P (z) = a 0 + · · · + a n z n has a real root and the largest real root of P is simple, and on halting returns a rational approximation of the largest real root of P to accuracy ε.
Proof Sketch. Compute a list of intervals I 1 , . . . , I s of width at most ε such that every real root of P is contained in some interval I j . Let I k be such that the left endpoint of I k is larger than the right endpoint of every interval I j with j = k. Test to accuracy ε if P changes its sign on the endpoints of I k . If this is the case evaluate P on I using interval arithmetic with precision ε. If the resulting interval does not contain zero halt and output the centre of I k .
If any of the above tests fail, rerun the algorithm with ε/2 instead of ε.
Lemma 4.6. There exists an algorithm which takes as input a linear recurrence (c, u) and a rational number ε > 0, halts if and only if the linear recurrence has a unique and simple dominant characteristic root ρ which in addition is a positive real number, and on halting outputs ρ together with a rational approximation to error ε of its coefficient a ∈ R in the exponential polynomial solution of (c, u).
Proof. Given any linear recurrence (c, u) we can compute a vector in C n containing all the complex roots of its characteristic polynomial. We can then semi-decide if the vector contains a unique element µ of maximal modulus. Moreover, we can semi-decide if there exists a rational box B which is symmetric about the real axis and contains µ and no other characteristic roots. This then establishes that µ is a simple root of the characteristic polynomial and a real number. By computing Re µ we obtain a name of µ as a real number. We can then semi-decide if µ > 0. By Corollary 4.4 we can compute the coefficient a of µ in the exponential polynomial solution.
"Yes"-instances of Ultimate Positivity
Proposition 5.1. A linear recurrence (c, u) ∈ R 2n is a robust "yes"-instance of the Ultimate Positivity Problem if and only if it has a unique and simple dominant characteristic root which in addition is a positive real number whose coefficient in the exponential polynomial solution is a positive real number. Moreover, there exists an algorithm which takes as input a real linear recurrence and halts if and only if the linear recurrence is a robust "yes"-instance of the Ultimate Positivity Problem.
Proof. Clearly any instance of the described form is a "yes"-instance. We can apply the algorithm from Lemma 4.6 to check whether there exists a unique and simple dominant characteristic root which in addition is a positive real number, and if so compute its coefficient in the exponential polynomial solution. We can then semi-decide if this coefficient is positive. This shows that the set of instances of the described form is semi-decidable, which implies that any such instance is robust. It now remains to show that there are no further robust "yes"-instances. Let (c, u) be a "yes"-instance of order n. Assume that (c, u) is not of the described form. Then either (c, u) does not have a unique and simple dominant characteristic root ρ which in addition is a positive real number, or it does and the coefficient of ρ k in the exponential polynomial solution is non-positive.
Assume first that (c, u) has a unique and simple dominant characteristic root ρ which in addition is a positive real number, but its coefficient c in the exponential polynomial solution is non-positive. By Proposition 3.3 it suffices to show that there exist arbitrarily small perturbations of the exponential polynomial solution which fail to be ultimately positive. By an arbitrarily small perturbation of the formal exponential polynomial we can ensure that ρ is the only dominant characteristic root and that its coefficient is strictly negative. This slightly perturbed sequence will be negative for all large values of k. It follows that (c, u) is not robust.
It remains to consider the case where (c, u) does not have a unique and simple dominant characteristic root ρ which is a positive real number. We can further assume that the dominant characteristic roots of (c, u) have strictly positive modulus. Otherwise we have c = 0, which immediately implies that (c, u) is not robust. Since we assume that (c, u) is a "yes"-instance it follows from Lemma 3.2 that (c, u) has to have a dominant characteristic root ρ which is a positive real number. Hence, either ρ is not simple or there exists another dominant characteristic root λ ∈ C \ [0, +∞).
If there exists a dominant characteristic root λ ∈ C \ [0, +∞) then under arbitrarily small perturbations of the exponential polynomial solution the root λ becomes strictly larger in modulus than ρ and its coefficient can be ensured to be non-zero. By Proposition 3.3 this induces an arbitrarily small perturbation of the instance (c, u). It follows from Lemma 3.2 that the perturbed instance is a "no"-instance of the Positivity Problem. Hence (c, u) is not robust.
Finally assume that ρ is not simple. Let D ⊆ C n and h : D → R n be as in Proposition 3.4. Choose p ∈ D with h(p) = c. By continuity of h a small perturbation of p within D induces a small perturbation of the coefficients of c and therefore a small perturbation of the input (c, u). Since ρ is not simple the vector p contains at least two entries equal to ρ. By an arbitrarily small perturbation of p we can make these two entries into two complex conjugate numbers whose modulus is strictly larger than ρ. By an arbitrarily small perturbation of the exponential polynomial solution of the resulting perturbed instance we can further ensure that the coefficients of these complex conjugate characteristic roots in the exponential polynomial solution are non-zero. It follows from Proposition 3.3 that this induces an arbitrarily small perturbation of the original instance. By Lemma 3.2 the resulting perturbed instance is not ultimately positive.
Positivity
Lemma 6.1. Let (c, u) be a linear recurrence. Assume that (c, u) has a unique and simple dominant characteristic root ρ, which in addition is a positive real number. Let a ∈ R be 16:16
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Proof. By Lemma 4.6 we can compute ρ and its coefficient a in the exponential polynomial solution uniformly in (c, u) subject to the promise that (c, u) is of the required form.
We can compute a real number M such that |λ| < M < ρ for all characteristic roots λ of (c, u) with λ = ρ. We can then compute an integer p such that
(n − 1) M ρ p 1 + M ρ p n−2 < 1 2 . (6.1)
For all q = 0, . . . , p − 1 we can thus compute a linear recurrence of order n − 1 which generates the sequence v (q)
k = u pk+q ρ pk+q − a.
(6.2)
Its characteristic roots are of the form (µ/ρ) p , where µ is a characteristic root of (c, u) which is distinct from ρ. Letting µ 1 , . . . , µ n−1 denote the characteristic roots, counted with multiplicity, the coefficients of the linear recurrence are given by e 1 (µ 1 , . . . , µ n−1 ), . . . , e n−1 (µ 1 , . . . , µ n−1 ), where e j (x 1 , . . . , n − 1) = 1≤k 1 ≤···≤k j ≤n
x k 1 · · · · · x k j is the j th elementary symmetric polynomial in n−1 variables. Using the estimate |µ j | < M ρ p we obtain:
|e j (µ 1 , . . . , µ n−1 )| ≤ n − 1 j M ρ pj .
We can thus estimate v (q) k+1 ≤ (|e 1 (µ 1 , . . . , µ n−1 )| + · · · + |e n−1 (µ 1 , . . . , µ n−1 )|) max v
(q) k , . . . , v (q) k−n+1 ≤ n−1 j=1 n − 1 j M ρ pj max v (q) k , . . . , v (q) k−n+1 ≤ (n − 1) M ρ p n−2 j=0 n − 2 j M ρ pj max v (q) k , . . . , v (q) k−n+1 ≤ (n − 1) M ρ p 1 + M ρ p n−2 max v (q) k , . . . , v (q) k−n+1 < 1 2 max v (q) k , . . . , v (q) k−n+1 .
The last inequality follows from (6.1). By induction it follows for all k > 1 that v (q) We can hence compute an index N such that for all q = 0, . . . , p − 1 and all j ≥ N p − 1 we have v (q) j < |a|. Using (6.2) we obtain u k ρ k − a < |a| for all k ≥ N . Multiplication of this inequality with ρ k yields the result.
n−1+k < 1 2 k max v(
Proposition 6.2. A "yes"-instance (c, u) of the Positivity Problem is robust if and only if it is a robust "yes"-instance of the Ultimate Positivity Problem and satisfies u k > 0 for all k. Moreover, there exists an algorithm which takes as input a linear recurrence and halts if and only if it is a robust "yes"-instance of the Positivity Problem.
Proof. Since Positivity implies Ultimate Positivity it is clear that any robust "yes"-instance of the Positivity Problem must be a robust "yes"-instance of the Ultimate Positivity Problem. We claim that if (c, u) is a "yes"-instance of the Positivity Problem with u k = 0 for some k, then there exists an arbitrarily small perturbation of the instance with u k < 0. Indeed, we can choose k to be minimal with this property. If u k is an initial value then we can slightly perturb u k to make it negative. Otherwise we have u k = c 1 u k−1 + · · · + c n u k−n with u k−j > 0 for j = 1, . . . , n by minimality of k. Then an arbitrarily small perturbation of c 1 , say, will make u k negative. This proves the claim. Now consider the following algorithm: given an instance (c, u) of the Positivity Problem, semi-decide if it has a positive real dominant characteristic root ρ which is simple and strictly larger than the modulus of any other characteristic root. If this semi-decision procedure halts, compute the coefficient a of ρ k in the exponential polynomial solution. This is possible thanks to Lemma 4.6 Test if a > 0. If this test halts then compute an index N such that aρ k > |u k − aρ k | for all k ≥ N . This is possible thanks to Lemma 6.1. Then test for all k < N if u k > 0.
Together with the characterisation of robust "yes"-instances of the Ultimate Positivity Problem given in Proposition 6.2 it is now clear that this algorithm halts if and only if (c, u) is a robust "yes"-instance of the Positivity Problem.
Theorem 6.3. The Positivity Problem is maximally partially decidable.
Proof. It is obvious that all "no"-instances of Positivity are robust and algorithmically recognisable. Proposition 6.2 establishes that all robust "yes"-instances are algorithmically recognisable.
The Skolem Problem
Proposition 7.1. No "yes"-instance of the Skolem Problem is robust.
Proof. Let (c, u) be a "yes"-instance of the Skolem Problem. Let us assume that (c, u) has a dominant characteristic root λ with non-zero imaginary part. An analogous but simpler proof establishes the claim in case that there exists a real dominant characteristic root.
Since the vector of coefficients c depends continuously on the vector of characteristic roots we can assume, up to slightly perturbing (c, u), that λ is simple, that λ andλ are the only dominant characteristic roots, and that λ/|λ| is a root of unity. The exponential polynomial solution of (c, u) is then of the form cλ k +cλ k + r(k), with |r(k)| < ab k for some a ≥ 0 and b < M . Writing λ = M e iϕ and c = α + iβ we have cλ k +cλ k = 2M (α cos(kϕ) − β sin(kϕ)) .
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Thus, cλ k +cλ k does not vanish so long as the point (α, β) lies outside the set S = (x, y) ∈ R 2 | ∃k.x cos(kϕ) − y sin(kϕ) = 0 . Since e iϕ is a root of unity the expressions cos(kϕ) and sin(kϕ) admit only finitely many different values for k ∈ N. Additionally, the expressions cos(kϕ) and sin(kϕ) never vanish simultaneously. Therefore the set S is a finite union of straight lines in R 2 . It follows that there exist arbitrarily small perturbations of α and β such that α cos(kϕ) − β sin(kϕ) never vanishes. Since this expression only admits finitely many values it follows that |u k | > εM k − ab k with ε > 0 and b < M .
In particular up to arbitrarily small perturbation (c, u) has only finitely many zeroes k 1 < k 2 < · · · < k m . As above we can find arbitrarily small γ and δ such that (γ+iδ)λ k j +(γ−iδ)λ k j is non-zero for j = 1, . . . , m. Replace the coefficient c in the exponential polynomial solution of (c, u) by c + (γ + iδ) andc withc + (γ − iδ). Then the zeroes k 1 , . . . , k m are removed by construction. By choosing γ and δ sufficiently small we can ensure that no new zeroes are added. By Proposition 3.3 this induces an arbitrarily small perturbation of the instance (c, u).
Proposition 7.2. A "no"-instance of the Skolem Problem is robust if and only if one of the two following conditions is met:
(1) It has a simple real dominant characteristic root ρ with |ρ| > |µ| for all characteristic roots µ = ρ, and the coefficient of ρ in the exponential polynomial solution is non-zero. Proof. Assume that a "no"-instance (c, u) of the Skolem Problem has a complex dominant characteristic root λ ∈ C \ R. By Proposition 3.4 a small perturbation of the characteristic roots induces a small perturbation of the instance (c, u), so that we can assume up to an arbitrarily small perturbation of (c, u) that λ is simple, not a root of unity, and that that λ and λ are the only dominant characteristic roots of (c, u). Let M = |λ|. It follows from Dirichlet's approximation theorem that the sequence (λ/M ) k is dense in the unit circle S 1 ⊆ C. In particular, for every ε > 0 there exist infinitely many k ∈ N such that |(λ/M ) k | < ε. If follows that there exists an index K such that |u K /M K | < ε. Now, the linear recurrence sequence
w k = u k − u K 2λ K λ k − u K 2λ Kλ k satisfies w K = 0.
It is represented by a formal exponential polynomial which is ε-close to the exponential polynomial solution of (c, u). Since ε can be chosen to be arbitrarily small, this yields an arbitrarily small perturbation of (c, u) which is a "yes"-instance of the Skolem Problem. It follows that (c, u) is not robust. If (c, u) has a real dominant characteristic root which is not simple then a small perturbation of (c, u) has a complex dominant characteristic root. It equally follows that (c, u) is not robust.
Thus, if (c, u) is a robust "no"-instance of the Skolem Problem then its dominant characteristic roots are all simple and real. Assume that (c, u) has only one dominant characteristic root ρ. Let M be its absolute value. Note that M > 0 since we assume that the instance is a "no"-instance. If the coefficient of ρ k in the exponential polynomial solution is zero, then for all ε > 0 there exists an index K such that |u K /M K | < ε. Then the linear recurrence sequence w k = u k − u K ρ K ρ k has a zero and is represented by a formal exponential polynomial which is ε-close to the exponential polynomial solution of (c, u). It follows that (c, u) is not robust. On the other hand, if the coefficient of ρ k is non-zero then it is easy to see that the instance is a robust "no"-instance. Now assume that (c, u) has exactly two dominant characteristic roots ρ > 0 and −ρ < 0. If the coefficient in the exponential polynomial solution of ρ k , say, is zero then by an arbitrarily small perturbation we can ensure that ρ is the only dominant characteristic root of (c, u) and it follows that (c, u) is not robust. Hence the coefficients of ρ k and of (−ρ) k must be non-zero. If they have the same absolute value then the formal dominant part of the exponential polynomial solution vanishes either for all odd indexes or for all even indexes. Again it follows that for all ε > 0 there exists K such that |u K /ρ K | < ε and by the same argument as before it follows that (c, u) is not robust. On the other hand, if the coefficients of ρ k and (−ρ) k are non-zero and have different absolute values then it is easy to see that the resulting instance is a robust "no"-instance of the Skolem Problem. This concludes the characterisation of the robust "no"-instances.
To semi-decide if a given instance (c, u) is a "no"-instance, run the following two tests in parallel:
(1) (c, u) has a simple real dominant characteristic root with |ρ| > |µ| for all characteristic roots µ = ρ, and the coefficient of ρ in the exponential polynomial solution is non-zero.
(2) (c, u) has a simple positive real characteristic root ρ + and a simple negative real characteristic root ρ − such that |ρ + | > |µ| and |ρ − | > |µ| for all characteristic roots µ / ∈ {ρ + , ρ − }. The coefficients of ρ + and ρ − in the exponential polynomial solution are distinct and both non-zero.
That the first test is effective follows essentially from Lemma 4.6. The second test can be carried out effectively by similar ideas: Choose a rational number ε > 0. Compute a list of rational boxes B 1 , . . . , B s ⊆ C of width ε such that each box is guaranteed to contain a complex root of the characteristic polynomial and each root of the characteristic polynomial is contained in a box. Test if there exist boxes B + and B − which intersect the real axis and contain a unique root, such that all real numbers contained in B + are positive, all real numbers contained in B − are negative, the modulus of all numbers contained in B + is strictly larger than the modulus of all numbers contained in boxes other than B + and B − , and the same is true for the modulus of all numbers contained in B − . Clearly this can be tested effectively in finite time. If this test does not succeed then retry with ε/2 replacing ε. If the test succeeds then there is a unique characteristic root ρ + ∈ B + and a unique characteristic root ρ − ∈ B − , both of which are simple and real. We can effectively compute the respective coefficients of ρ + and ρ − in the exponential polynomial solution of (c, u) thanks to Corollary 4.4 and test if they are distinct and non-zero.
Whenever one of the tests terminates we can effectively compute positive real numbers δ and r such that the absolute value of the formal dominant part of the exponential polynomial solution is bounded from below by δr k . It then follows as in the proof of Lemma 6.1 that we can compute an index K ∈ N such that |u k | > 0 for all k ≥ K. To verify that the given instance is a "no"-instance it hence suffices to verify that u k > 0 for all k < K.
It is clear that if this algorithm halts then the given instance is a "no"-instance. Conversely, if the instance is a robust "no"-instance then it meets one of the two criteria above. If it meets the fist criterion then the first test will terminate. If it meets the second criterion then the second test will terminate.
Approximate root clusterings and possible root configurations
Finally we turn to the problem of computably recognising robust "no"-instances of Ultimate Positivity. This will be considerably more involved than the previous results, and further preparatory work is required. The ideas we introduce here have been motivated in Example 1.1 in the introduction. Let P ∈ R[x] be a non-constant univariate real polynomial. An approximate root clustering for p is a finite list
(B 1 , N 1 ), . . . , (B s , N s )
where each B j ⊆ C is a rational box, i.e., a product of intervals with rational endpoints, and each N j is a positive integer, such that (1) Every complex root of P is contained in one of the boxes B j .
(2) For all j = 1, . . . , s the number N j is the number of roots of p in B j counted with multiplicity. where ρ 1 , . . . , ρ d and λ 1 , . . . , λ e are distinct variables and r 1 , . . . , r d , m 1 , . . . , m e are positive integers with r 1 + · · · + r d + 2m 1 + · · · + 2m e = N j . For every j = 1, . . . , d the number r j is called the multiplicity of the variable ρ j and for j = 1, . . . , e the number m j is called the multiplicity of the variable λ j . The intention is that the variables ρ 1 , . . . , ρ d represent real roots with multiplicities r 1 , . . . , r d and λ 1 , . . . , λ e represent complex roots with positive imaginary part and multiplicities m 1 , . . . , m e . Let (B j , N j ) be a complex cluster. A possible root configuration for (B j , N j ) is a list We call the variables of the form ρ j the real variables and the variables of the form λ j the complex variables of the possible root configuration. Formally, a real variable is a variable that occurs in the first list associated with some R j and any other variable is a complex variable.
Proposition 8.1. Given a non-constant real polynomial P ∈ R[x], encoded as a vector (a n , . . . , a 0 ) ∈ R n+1 where p(x) = a n x n + · · · + a 0 and a n = 0, and a natural number p ∈ N we can compute an approximate root clustering
(B 1 , N 1 ), . . . , (B s , N s )
for P such that each B j has width at most 2 −p .
Proof Sketch. By Theorem 4.1 we can compute a complex vector (µ 1 , . . . , µ n ) ∈ C n which contains all roots of P such that a root of multiplicity m occurs precisely m times in this vector. Choose a small rational number δ > 0. Approximate each µ j to accuracy δ/2 and put a rational box C j of width δ around this rational approximation. Arrange these boxes into clusters of the form C j 1 , . . . , C j k such that the unions C j 1 ∪ · · · ∪ C j k are connected and each C j i is disjoint from all boxes C k with k / ∈ {j 1 , . . . , j k }. Initialise an empty list L = . For each such cluster compute a rational box B that contains C j 1 ∪ · · · ∪ C j k and add the element (B, k) to the list L. The boxes B in the list L may have width larger than 2 −p and the resulting list L may not be an approximate root clustering since the reflection of a real cluster about the real line may intersect another cluster, or the reflection of a complex cluster about the real line may intersect more than one other cluster. If either of these cases occurs repeat the algorithm with δ/2 replacing δ. It is easy to see that for sufficiently small δ this algorithm will produce an approximate root clustering as desired.
Proposition 8.2. Given an approximate root clustering (B 1 , N 1 ), . . . , (B s , N s ) we can compute a list containing all possible root configurations for this clustering.
Let P ∈ R[x] be a non-constant real polynomial. Let C = (B 1 , N 1 ), . . . , (B s , N s ) be an approximate root clustering. Let R be a possible root configuration for C. Let ρ 1 , . . . , ρ d be its real variables and let λ 1 , . . . , λ e be its complex variables. Let r 1 , . . . , r d and m 1 , . . . , m e be their respective multiplicities. Introduce new variablesλ 1 , . . . ,λ e . The characteristic polynomial of R is the polynomial
(x − ρ 1 ) r 1 · · · · · (x − ρ d ) r d · (x − λ 1 ) m 1 · (x −λ 1 ) m 1 · · · · · (x − λ e ) me · (x −λ e ) me
with coefficients in Z[x, ρ 1 , . . . , ρ d , λ 1 ,λ 1 , . . . , λ e ,λ e ]. Writing this polynomial in the form x n − c 1 x n−1 − · · · − c n−1 x − c n we obtain a linear recurrence relation u k+1 = c 1 u k + · · · + c n u k+1−n with coefficients c j ∈ Z[ρ 1 , . . . , ρ d , λ 1 ,λ 1 , . . . , λ e ,λ e ]. This yields a "formal" linear recurrence (c, u), where u 1 , . . . , u n are fresh variables. Call this the linear recurrence associated with R. Call its companion matrix the companion matrix of R and call its exponential polynomial where the φ j 's and ψ j 's are rational functions with rational number coefficients. It is clearly computable from R. Let R be a possible root configuration for an approximate root clustering C. Let (B a+1 , N a+1 ), . . . , (B b , N b ) be the list of complex clusters in C in the upper half plane. Let (B 1 , N 1 ), . . . , (B a , N a ) be the list of real clusters in C. Then R consists of possible root configurations R 1 , . . . , R a , C 1 , . . . , C b−a for the individual clusters. Let ρ 1 , . . . , ρ d be the real variables which occur in R and let λ 1 , . . . , λ e be the complex variables. Assume without loss of generality that there exists an integer w such that λ 1 , . . . , λ w belong to root configurations for real clusters and that λ w+1 , . . . , λ e belong to root configurations for complex clusters. Let β : {1, . . . , d} → {1, . . . , a} be the function that assigns to an integer j the root configuration R β(j) in which the variable ρ j occurs. Let γ : {1, . . . , w} → {1, . . . , a} be the function that assigns to an integer j the root configuration R γ(j) in which the variable λ j occurs. Let δ : {w + 1, . . . , e} → {1, . . . , b − a} be the function that assigns to an integer j the root configuration C δ(j) in which the variable λ j occurs. The domain D R ⊆ R d × C e of R is the set of all vectors (x 1 , . . . , x d , z 1 , . . . , z e ) ∈ R d × C e such that x 1 > · · · > x d , z j = z k for j = k, Im(z j ) > 0 for all j = 1, . . . , e, x j ∈ B β(j) for all j = 1, . . . , d, z j ∈ B γ(j) for all j = 1, . . . , w, and z j ∈ B a+δ(j) for all j = w + 1, . . . , e.
Thus, the domain of R is the set of all "valid assignments" to the variables ρ 1 , . . . , ρ d , and λ 1 , . . . , λ e . To any point (α, u) ∈ D R × R n corresponds a real linear recurrence which is obtained by substituting α for ρ 1 , . . . , ρ d , λ 1 , . . . , λ e and u for the variables representing initial values in the linear recurrence associated with R. We call this the linear recurrence associated with α with initial values u.
We conclude with two obvious observations:
Lemma 8.3.
There exists a computable function ϕ : n∈N R 2n × (0, 1) → (0, +∞) with the following property: Let (c, u) be a linear recurrence with characteristic polynomial P . Let C be an approximate root clustering for P to accuracy ε > 0. Let R be a possible root configuration for C. Let α ∈ D R and let v ∈ R n with |v − u| < ε. Then the linear recurrence associated with α with initial values v is ϕ(c, u, ε)-close to (c, u). Moreover, for all (c, u) ∈ n∈N R 2n we have ϕ(c, u, ε) → 0 as ε → 0.
Lemma 8.4. Let C be an approximate root clustering for a polynomial P . Let R be a possible root configuration for C. Then the domain D R is definable in the first-order theory of the reals as a subset of R e × R 2d . Proof. Consider the following algorithm. For all p ∈ N do the following:
(1) Compute an approximate root clustering C of the characteristic polynomial of (c, u) to accuracy 2 −p .
(2) Compute a rational box B of width 2 −p that contains the vector u of initial values.
(3) For all possible root configurations R of C do the following:
(a) Symbolically compute the exponential polynomial solution of the linear recurrence associated with R. (b) Use the symbolic exponential polynomial solution to construct a sentence in the first-order theory of the reals which expresses that for all points α ∈ D R and all v ∈ B the formal leading coefficient of the largest real characteristic root in the exponential polynomial solution of the linear recurrence associated with α with initial values v is negative or there exists a characteristic root λ ∈ C \ [0, +∞) whose modulus is larger than any positive real characteristic root such that the coefficient of λ is non-zero. If there exist no real roots then only include the second condition. (c) Employ the Tarski-Seidenberg theorem to decide whether the above sentence is true.
If it is true continue with the next root configuration, or leave the loop if there are no root configurations left. If the sentence is false break out of the loop and continue with the next p. (4) If all sentences constructed in the above loop are true, then halt.
We claim that this algorithm halts on a given instance (c, u) if and only if the instance is a robust "no"-instance. By construction if the algorithm halts then the instance (c, u) either has no positive real characteristic roots or the formal leading coefficient of the largest positive real root ρ in the exponential polynomial solution of (c, u) is negative, or there exists a characteristic root λ ∈ C \ [0, +∞) with |λ| > ρ such that the coefficient of λ is non-zero. It follows from Lemma 3.2 that (c, u) is a "no"-instance of Ultimate Positivity.
Conversely, assume that (c, u) is a robust "no"-instance. Suppose for the sake of contradiction that the algorithm does not halt. Then by Lemma 8.3 there exists an arbitrarily small perturbation (d, v) of (c, u) such that either (d, v) has no positive real characteristic roots and the coefficients of all characteristic roots are zero or the formal leading coefficient of the largest positive real characteristic root ρ in the exponential polynomial solution of (d, v) is non-negative and the coefficient of every complex or negative characteristic root with strictly larger modulus is zero. If the former is the case then the sequence generated by (d, v) is identically equal to zero and hence a "yes"-instance of Ultimate Positivity. Let us now assume that (d, v) has a positive real characteristic root. By Proposition 3.3 a small perturbation of the exponential polynomial solution of (d, v) induces a small perturbation of the linear recurrence (d, v). By an arbitrarily small perturbation of the exponential polynomial we can ensure that the largest positive characteristic root ρ has a positive leading coefficient and that every complex or negative characteristic root with a non-zero coefficient has strictly smaller modulus than ρ. The resulting instance is then clearly a "yes"-instance of the Ultimate Positivity Problem. It follows that the original instance is not robust. Unlike the previous results Theorem 9.1 falls short of providing a satisfactory characterisation of the robust "no"-instances of Ultimate Positivity. It is therefore worth pointing out that at least the robust simple instances of the Ultimate Positivity Problem admit a nice characterisation. A simple instance is one whose characteristic roots are all simple. With the help of Corollary 4.4 these instances are much easier to computably recognise than the general ones.
Proposition 9.2. A simple instance of the Ultimate Positivity Problem is robust if and only if it satisfies one of the two following conditions:
(1) It has a positive real dominant characteristic root ρ with |ρ| > |λ| for all other characteristic roots, and the coefficient of ρ in the exponential polynomial solution is non-zero. If the coefficient is negative then it is a robust "no"-instance. Otherwise it is a robust "yes"-instance.
(2) It has a characteristic root λ ∈ C \ [0, ∞) with |λ| > ρ for all positive characteristic roots ρ whose coefficient in the exponential polynomial polynomial is non-zero. In this case it is a robust "no"-instance.
On the measure of the robust instances
We observe that the simple "yes"-instances of the Positivity Problem and the Ultimate Positivity Problem have full measure. This is certainly false for the Skolem Problem, since the robust instances of the Skolem Problem are not even dense in R 2n . Proof. Consider all spaces R s,t = R s × C t with s + 2t = n. These can all be identified with the space R n . On each of these spaces we have a map h s,t : R s,t → R n which sends the vector (ρ 1 , . . . , ρ s , λ 1 , . . . , λ t ) to the n lowest coefficients of the monic polynomial (x − ρ 1 ) · · · · · (x − ρ s )(x − λ 1 )(x − λ 1 ) · · · · · (x − λ t )(x − λ t ).
This map is clearly differentiable as a map R n → R n . It hence maps null sets to null sets. Now, the sets A s,t = (ρ 1 , . . . , ρ s , λ 1 , . . . , λ t ) ∈ R s × C t | |ρ i | = |ρ j | for some i = j , B s,t = (ρ 1 , . . . , ρ s , λ 1 , . . . , λ t ) ∈ R s × C t | |ρ i | = |λ j | for some i = j , and C s,t = (ρ 1 , . . . , ρ s , λ 1 , . . . , λ t ) ∈ R s × C t | |λ i | = |λ j | for some i = j all have measure zero. As there are only finitely many spaces R s,t , we may remove the images h s,t (A s,t ∪ B s,t ∪ C s,t ) from R n and are still left with a set of full measure. This implies that the set of instances (c, u) ∈ R 2n whose characteristic roots are simple and distinct in modulus have full measure. Now consider the spaces E s,t ⊆ R 2s × C 2t with s + 2t = n, where a point (ρ 1 , . . . , ρ s , α 1 , . . . , α s , λ 1 , . . . , λ t , β 1 , . . . , β t )
is an element of E s,t if and only if (ρ 1 , . . . , ρ s , λ 1 , . . . , λ t ) ∈ R s,t \ (A s,t ∪ B s,t ∪ C s,t ). On each of these spaces we have a map m s,t : E s,t → R 2n which sends the data (ρ 1 , . . . , ρ s , α 1 , . . . , α s , λ 1 , . . . , λ t , β 1 , . . . , β t ) to a code (c 1 , . . . , c n , u 1 , . . . , u n ) for the linear recurrence which defines the exponential polynomial α 1 ρ k 1 + · · · + α s ρ k s + β 1 λ k 1 + β 1 λ 1 k + · · · + β t λ k t + β t λ t k .
As this map is again differentiable, it sends null sets to null sets. The set D s,t = (ρ 1 , . . . , ρ s , α 1 , . . . , α s ,λ 1 , . . . , λ t , β 1 , . . . , β t ) ∈ E s,t | α i = 0 for some i or β i = 0 for some i is clearly a null set, so that its image in R 2n under m s,t is again a null set. It follows that we may remove all sets of the form m s,t (D s,t ) from R 2n to be left with a set of full measure. In summary, the set of linear recurrences (c, u) whose characteristic roots are all distinct and have non-zero coefficients in the exponential polynomial solution has full measure. All of these linear recurrences are robust instances of the Ultimate Positivity Problem. The remaining non-robust instances of the Positivity Problem are "yes"-instances which have a zero. Within the space E s,t the linear recurrences which have a zero can be identified with the union of the sets Z k = (ρ 1 , . . . , ρ s , α 1 , . . . , α s , λ 1 , . . . , λ t , β 1 , . . . , β t ) ∈ E s,t | α 1 ρ k 1 + · · · + α s ρ k s + β 1 λ k 1 + β 1 λ 1 k + · · · + β t λ k t + β t λ t k = 0 .
Under the identification of E s,t with an open subspace of R 2n , each Z k is the zero set of a differentiable real-valued function f . Its gradient does not vanish in E s,t \ D s,t , so that Z k ∩ (E s,t ∩ D s,t ) has measure zero. It follows that the remaining non-robust instances also have measure zero. Thus everything is shown.
Lemma 4. 3 .
3Let A ∈ R n×n . Let µ be an eigenvalue of A with geometric multiplicity 1 and algebraic multiplicity m. Let v 1 , v 2 , . . . , v m be a Jordan chain of length m for A, i.e., v 1 is an eigenvector with eigenvalue µ and v i−1 = Av i − µv i for i > 1. Then we can uniformly compute in v 1 , . . . , v m and A an invertible matrix S ∈ C n×n such thatSAS −1 = J 0 0 Bwhere J is a Jordan block for µ of size m and all generalised eigenvectors of SAS −1 for eigenvalues λ of A with λ = µ lie in the span of the standard unit vectors e m+1 , . . . , e n .
− m it follows from [ZB04, Theorem 11] that we can compute a basis b 1 , . . . , b n−m of this space 4 . We can thus compute the matrix S which sends v 1 , . . . , v m to e 1 , . . . , e m and b 1 , . . . , b n−m to e m+1 , . . . , e n . The result follows immediately.
Corollary 4. 4 .
4Let (c, u) be a linear recurrence. Let µ be an eigenvalue of its companion matrix A with algebraic multiplicity m. Then coefficients of the formal polynomial coefficient of µ k in the exponential polynomial solution of (c, u) are uniformly computable in u, c, and m as a vector in C m . Moreover, if µ is real, then the coefficients are computable as a vector in R m .Proof. By Proposition 4.2 the geometric multiplicity of µ is equal to 1. By Lemma 4.3 we can thus compute a matrix S such thatSAS −1 = J 0 0 Bwith J being a Jordan block for µ of size m and all further generalised eigenspaces lying in the span of the standard unit vectors e m+1 , . . . , e n .Let v = (0, . . . , 0, 1)S −1 . Let w = Su. We claim that the term belonging to µ k in the exponential polynomial solution of the linear recurrence is equal to (v 1 , . . . , v m ) T J k (w 1 , . . . , w m ).Since e 1 , . . . , e m form a Jordan chain for the eigenvalue µ of SAS −1 and all remaining Jordan chains are by assumption contained in the span of e m+1 , . . . , e n the matrix SAS −1 can be put into Jordan normal form by conjugation with a matrix T = I 0 0 M 4 Theorem 11 in[ZB04] is formulated over R n but uses only Hilbert space properties of the reals and therefore carries over to C n without modification (cf. the remark in the second paragraph of page 191 in[ZB04]).
( 2 )
2It has exactly two dominant characteristic roots, both of which are simple and real, and their respective coefficients in the exponential polynomial solution are non-zero and have different absolute values. Moreover, there exists an algorithm which takes as input a linear recurrence (c, u) and halts if and only if (c, u) is a robust "no"-instance of the Skolem Problem.
( 3 )
3If a B j intersects the real line then its reflection about the real line is disjoint from all boxes B k with k = j. (4) If a B j does not intersect the real line then there exists a unique B k with k = j and B j ∩ B k = ∅, whereB j denotes the reflection of B j about the real axis. (5) There exists an index 0 ≤ a ≤ s such that the boxes B 1 , . . . , B a intersect the real line and no box B j with j > a intersects the real line. (6) For all 1 ≤ j < a with a as in the previous item, we have min (B j ∩ R) > max (B j+1 ∩ R). (7) If a = s then there exists and index a < b ≤ s such that the boxes B a+1 , . . . , B b are contained in the upper half-plane and the boxes B j with j > b are contained in the lower half-plane.The elements (B j , N j ) of an approximate root clustering are called clusters. A (B j , N j ) where B j intersects the real line is called a real cluster. Otherwise it is called a complex cluster. If (B j , N j ) is a complex cluster and (B k , N k ) is the unique cluster with k = j andB j ∩ B k = ∅ then we call (B j , N j ) and (B k , N k ) complex conjugate clusters and write (B j , N j ) = (B k , N k ). Note that it follows from the definition that for complex conjugate clusters (B j , N j ) and (B k , N k ) we have N j = N k .Let (B j , N j ) be a real cluster. A possible root configuration for (B j , N j ) is a pair of lists (ρ 1 , r 1 ), . . . , (ρ d , r d ) , (λ 1 , m 1 ), . . . , (λ e , m e )
(λ 1
1, m 1 ), . . . , (λ e , m e )where λ 1 , . . . , λ e are distinct variables and m 1 , . . . , m e are positive integers with m 1 + · · · + m e = N j . Again, for j = 1, . . . , e the number m j is called the multiplicity of the variable λ j . Now let (B 1 , N 1 ), . . . , (B s , N s ) be an approximate root clustering. Let a be the last index of a real cluster and b be the last index of a cluster in the upper half-plane. A possible root configuration for this root clustering is a list R 1 , . . . , R a , C 1 , . . . , C b−a where R 1 , . . . , R a root configurations for the real clusters(B 1 , N 1 ), . . . , (B a , N a )and C 1 , . . . , C b−a are possible root configurations for the complex clusters (B a+1 , N a+1 ), . .. , (B b , N b ), such that all variables occurring in the possible root configurations for different clusters are distinct.
j
exponential polynomial solution of R. The exponential polynomial solution of R is a term of the formd j=1 φ j (k, u, ρ 1 , . . . , ρ d , λ 1 ,λ 1 , . . . , λ e ,λ e j (k, u, ρ 1 , . . . , ρ d , λ 1 ,λ 1 , .. . , λ e ,λ e (k, u, ρ 1 , . . . , ρ d , λ 1 ,λ 1 , . . . , λ e ,λ e )λ k j ,
. 1 .
1There exists an algorithm which takes as input a linear recurrence (c, u) and halts if and only if (c, u) is a robust "no"-instance of the Ultimate Positivity Problem.
Proposition 10. 1 .
1For all n ∈ N the sets of simple robust instances of the Positivity Problem and the Ultimate Positivity Problem each have full measure.
In the case of countably based spaces this is an inessential restriction as these admit open representations.
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This work is licensed under the Creative Commons Attribution License. Martin Ziegler, Vasco Brattka, Theoretical Computer Science. 3261-3Suite. or Eisenacher Strasse 2, 10777Martin Ziegler and Vasco Brattka. Computability in linear algebra. Theoretical Computer Science, 326(1-3):187-211, 2004. This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany
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"Towards Network Traffic Monitoring Using Deep Transfer Learning",
"Towards Network Traffic Monitoring Using Deep Transfer Learning"
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"Harsh Dhillon [email protected] \nDepartment of Computer Science\nWestern University\nLondonOntarioCanada\n",
"Anwar Haque [email protected] \nDepartment of Computer Science\nWestern University\nLondonOntarioCanada\n"
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"Department of Computer Science\nWestern University\nLondonOntarioCanada",
"Department of Computer Science\nWestern University\nLondonOntarioCanada"
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| Network traffic is growing at an outpaced speed globally. The modern network infrastructure makes classic network intrusion detection methods inefficient to classify an inflow of vast network traffic. This paper aims to present a modern approach towards building a network intrusion detection system (NIDS) by using various deep learning methods. To further improve our proposed scheme and make it effective in real-world settings, we use deep transfer learning techniques where we transfer the knowledge learned by our model in a source domain with plentiful computational and data resources to a target domain with sparse availability of both the resources. Our proposed method achieved 98.30% classification accuracy score in the source domain and an improved 98.43% classification accuracy score in the target domain with a boost in the classification speed using UNSW-15 dataset. This study demonstrates that deep transfer learning techniques make it possible to construct large deep learning models to perform network classification, which can be deployed in the real world target domains where they can maintain their classification performance and improve their classification speed despite the limited accessibility of resources. | 10.1109/trustcom50675.2020.00144 | [
"https://arxiv.org/pdf/2101.00731v1.pdf"
]
| 230,437,924 | 2101.00731 | c3f6c198da1ffe345c00538aa55a0ec422d56cc4 |
Towards Network Traffic Monitoring Using Deep Transfer Learning
Harsh Dhillon [email protected]
Department of Computer Science
Western University
LondonOntarioCanada
Anwar Haque [email protected]
Department of Computer Science
Western University
LondonOntarioCanada
Towards Network Traffic Monitoring Using Deep Transfer Learning
Keywords-Network Intrusion Detection System, Deep Neural Network, Convolutional Neural Network, Long Short-Term Memory, Deep Transfer Learning.
Network traffic is growing at an outpaced speed globally. The modern network infrastructure makes classic network intrusion detection methods inefficient to classify an inflow of vast network traffic. This paper aims to present a modern approach towards building a network intrusion detection system (NIDS) by using various deep learning methods. To further improve our proposed scheme and make it effective in real-world settings, we use deep transfer learning techniques where we transfer the knowledge learned by our model in a source domain with plentiful computational and data resources to a target domain with sparse availability of both the resources. Our proposed method achieved 98.30% classification accuracy score in the source domain and an improved 98.43% classification accuracy score in the target domain with a boost in the classification speed using UNSW-15 dataset. This study demonstrates that deep transfer learning techniques make it possible to construct large deep learning models to perform network classification, which can be deployed in the real world target domains where they can maintain their classification performance and improve their classification speed despite the limited accessibility of resources.
I. INTRODUCTION
Internet Service Providers (ISP) globally have witnessed a fast expansion in their network traffic over the last few decades. Our societies are now closely intertwined with various networking services to perform its many day-to-day functions. Promising technologies such as 5G are currently making headway to increase the speed of connections between our machines swiftly. This trend will continue to evolve our network communication systems to become much rapid with each passing year. With so much reliance on communication networks, the infrastructure becomes a key target for cybercrimes, which can now have a more significant impact. Network security systems also need rapid improvements as we cannot rely on classic intrusion detection and scanning techniques, which have become quite obsolete to give accurate classifications in a dynamic and massive volume network traffic scenario.
Deep learning models excel at learning from a large dataset of labeled examples. Training a sizeable deep learning model also requires significant computational resources. Over the last decade, a lot of research and effort has been given to the space of supervised learning. In the real-world domain, supervised models usually generalize well if the environment is similar to the one where the end-to-end model was trained in as it expects similar data and computational resources to maintain its performance. The real world is non-deterministic and can present an infinite number of possibilities and patterns unseen by the deep learning algorithm. To navigate through such scenarios and perform with ideal accuracy and speed, learning models will need a prior understanding of the tasks and related domain at hand.
In this paper, we propose and demonstrate a novel deep transfer learning-based Intrusion Detection System architecture, which uses the prior knowledge of the models trained on larger datasets to perform at high speed while maintaining an optimal accuracy, despite the low availability of both data and computational resources. We will use the current USNW-15 dataset to demonstrate the proposed architecture. This research experimentally showcases that integrating transfer learning techniques in the core design of Intrusion Detection Systems can improve their overall efficacy in realworld settings. The novel methods outlined in this paper enable the IDS to utilize large and powerful deep learning models without the need for high computing power and extensive data. Using the defined methods, we can also effectively boost the classification speed of an IDS model, which enables it to maintain its accuracy while performing at real-time processing speeds.
This paper is organized as follows. Section II describes the historical background of modern intrusion detection systems and related work in the field of machine learning and deep learning. Section III describes the foundational concepts of transfer based learning. Section IV presents the proposed methodology to architect the deep transfer learning-based IDS. Section V presents the benchmark performance results of the applied deep learning algorithms in the architecture. Section VI concludes our paper and outlines the ideas for future research undertakings.
II. RELATED WORK
The earliest sketch of a real-time intrusion detection system was proposed by Dorothy E. Denning in 1986 [1]. Her work was inspired by the prior study of Jim Anderson in 1980, which formulated a way to audit a computer's data to identify abnormal usage patterns at the end of each day using a statistical analysis approach [2]. This research further augmented into IDES, abbreviated for Intrusion Detection Expert System developed by Teresa F. Lunt at SRI International in 1988 [3]. IDES had two main components. The first component adaptively learns the user's normal behavior pattern and detects patterns that deviate from them. The second component uses a rule-based approach to encode the encountered system vulnerabilities and store them in a knowledge base. Lunt proposed integrating an artificial neural network in the expert system as a third component, which was not fully implemented in the follow-up derivations of IDES. By the 1990s, intrusion detection systems were started to get implemented by various research labs and business computing firms, including AT&T Bell Labs, who built their own versions of detection systems, using IDES as a base on various other hardware and different programming languages [4]. The introduction of well-labeled KDD-99 intrusion detection dataset enabled researchers to work in the field of computer security to apply data mining and machine learning algorithms to build much efficient and generalized IDS [5]. In 2001, Tamas Abraham used data mining techniques to formulate the IDDM, abbreviated for Intrusion Detection Using Data Mining architecture [6]. Traditionally, data mining systems operated on large off-line data sets. IDDM architecture was designed to use data mining in real-time environments to identify anomalies and misuse. Li Jun et al. proposed a hierarchical network intrusion detection system, which used a Perceptron-Backpropagation hybrid model to classify anomalous and normal network traffic to recognize UDP flood attacks [7]. In 2002, Eskin et al. proposed an unsupervised intrusion detection framework using SVM, K-Nearest Neighbor, and clustering algorithms [8]. Weiming Hu et al. used an Adaboost-based algorithm with an adaptive weight strategy to build a detection model reporting low computational complexity and error rates [9]. J. Zhang et al. used random forest algorithm-based data mining techniques to build a hybrid IDS, which is capable of functioning as both a misuse and anomaly detection system [10]. Chandrasekhar et al. applied fuzzy neural networks to build their variation of IDS [11], which claimed better experimental results than the Backpropagation Neural Networks and other well-known machine learning methods.
The 2012 ImageNet victory led by Hinton et al. demonstrated that deep neural networks were able to outperform complex machine learning models in image recognition tasks [12] by beating the state-of-the-art algorithms by a whopping 10.8 percentage point margin rate and creating a renewed interest in the field of deep learning. In the proceeding years, academics working in computer security also started integrating deep neural networks in their research. In 2014, Wang et al. applied deep belief networks, a class of DNN, which reported the lowest published false-positive results with the KDD-99 dataset [13]. N. Moustafa et al. [14] reinvigorated the field with their UNSW-NB15 network data set, which is much superior for evaluating NIDS performance, as it reflects the modern traffic scenarios more fittingly than decade-old intrusion datasets such as KDD-99 and NSLKDD. In 2018, N. Moustafa et al. used the UNSW-NB15 dataset to create NIDS for IoT traffic data for classifying normal and suspicious instances by applying AdaBoost ensemble techniques [15]. A. Ahim et al. [16] combined different classifier approaches based on decision trees and various rules-based concepts to build a novel IDS using the CICIDS2017 dataset. In 2019, Y. Xiao et al. [17] implemented a CNN based IDS using Batch Normalization with KDD99 Dataset, the demonstrated CNN-IDS displayed fast classification times than non-CNN algorithms, making a deep neural network approach ideal to construct a real-time IDS. Vinayakumar et al. [18] created a hybrid IDS to monitor network and host level activities. Upon conducting an exhaustive comparative study with various machine learning and deep learning classifiers, DNN was demonstrated to outperform other applied classifiers. B. Riyaz et al. [19] designed an IDS using deep learning approach in wireless networks using the combination of novel coefficientbased feature selection algorithm (CRF-LCFS) and a CNN using KDD-99 dataset. Their proposed method demonstrated a 98.9% detection accuracy. In our research, we will be using a hybrid CNN-LSTM model to train our IDS to filter network traffic in a source domain at a high classification accuracy. Using transfer learning techniques, we will transfer the domain knowledge learned by our model in a simulated real-world environment. Our experiments demonstrate that using the novel method outlined in this paper, the applied unified model improves its classification accuracy in the real world target domain as well as efficiently increases its performance speed.
III. TRANSFER LEARNING
Transfer learning is a concept where a learning algorithm reuses the knowledge from the past related tasks to ease the process of learning to perform a new task [20]. The ability to transfer the knowledge gained from previous tasks has a wide range of real-world applications, including building real-time intrusion detection systems that can perform optimally even with scarcity of data and computing resources. Using deep transfer learning alleviates the massive data dependency of deep learning algorithms, which they require to learn the underlying patterns in the data. In general terms, using transfer learning, we aim to transfer the knowledge from a source domain to a target domain by relaxing the assumption that the training data and the test data must be independent and identically distributed, which is rare for the real-world data. Fig. 1 shows the process of transferring a model's network architecture and learned weights from a source domain with large dataset and higher computational resources to a target domain with a smaller dataset and limited computational resources.
A domain can be represented as, = { , ( )}, which consists of two parts: the feature space and a margin distribution P(X), Where X = { , , . . . , }, ∈ . Whereas A task can be represented as,
= { , ( | )} = { , }, Y = { , , . . . , }, ∈ , where
is a label space, and represents the predictive function which can be learned from the training data including pairs { , }, where ∈ , ∈ ; for each feature vector in the domain, predicts its corresponding label as ( ) = [21].
In this paper, we consider our source domain as , and target domain as . The source domain data is denoted as = {( , ), . . . , ( , )}, where ∈ is the data instance and ∈ is the corresponding class label. In our IDS, is the set of term vectors together with their associated attack and malicious labels. Similarly, we denote the target domain data as = {( , ), . . . , ( , )}, where the input is in and ∈ is the corresponding output [21]. We can now give the transfer learning definitions as follows, Given a source domain , learning task , a target domain and learning task , transfer learning aims to help improve the learning of the target predictive function by using the knowledge in source domain and learning task , where ≠ , or ≠ . The size of is much bigger than in various applied situations.
Additionally, when there exists some relationship, explicit or implicit, between the feature spaces of the two domains, we say that the source and target domains are related. In this paper, the two domains are related as they share a similar feature space from intrusion datasets. A transfer learning task defined by ( , , , , ) becomes a deep transfer learning task if is a non-linear function represented by a deep neural network.
Chuanqi Tan et al. [21] classified deep transfer learning approach into four main categories, namely instance-based, mapping-based, network-based, and adversarial-based transfer learning. In this paper, we utilize the network-based transfer learning approach. Network transfer learning refers to the transfer of partial network trained in the source domain, which includes its network structure and learned weights to the target domain, where it becomes the part of its existing architecture. The network-based transfer learning architecture works with the notion that the neural networks should become as iterative as human brains. Human brains use prior knowledge even when they are performing new tasks and often perform well with the new tasks by using the previously learned concepts.
IV. PROPOSED METHODOLOGY
A. Database
For architecting our transfer learning-based IDS model, we will use the USNW-15 dataset. This dataset is relatively modern when compared to other widely used datasets like KDD99 and NSL-KDD in network security research. USNW-15 was created at Cyber Range Lab of the Australian Centre for Cyber Security (ACCS). The dataset contains nine types of malicious attacks, namely Analysis, Backdoors, DoS, Exploits, Fuzzers, Generic, Reconnaissance, Shellcode, and Worms. In total, the UNSW-15 dataset contains 100 GB worth of raw network packet observations. We will use a partition set of this data which includes 257,673 records and will further divide the selected partition into a training set with 154,603 records. We will also use a validation set and a testing set, both with 51,535 records, respectively, to aptly evaluate the performance of the applied deep learning models in the separate domains. The raw packet data in the UNSW-15 dataset was recorded over the modern network infrastructure, which helps in building an IDS that can generalize well in the real-world environment as well. We will only use the training set and validation set in our source domain for training and validating our learning models and to benchmark their performance. In our target domain, we will use the testing set, which is the unseen partition of data, which simulates the real-world unobserved data for our models to test their performance.
B. Feature Engineering
The features we select to a model the IDS architecture form the core of the classifier, aiming to infer and differentiate between normal and malicious packets. USNW-15 dataset has in total of 49 features. To optimize our system further, we made feature selection to choose features that are substantial to the classification task. randomized decision trees to identify and select the best features in the dataset. Table I describes the most significant features selected during the feature selection step. Based on the results from the tree-based feature importance technique, we dropped 15 features from our dataset, which had the least scoring performance. We observed that filtering the dataset using such methodology improved the overall accuracy and speed performance of the applied deep learning techniques. Fig. 2 shows the top ten identified features in the dataset.
Feature selection enables the model to allocate its computational resources appropriately, which also leads to an increase in the training time because we are reducing the data to process and construct the model. The presence of irrelevant and redundant data makes the ultimate goal of knowledge discovery much harder as well.
C. Data Normalization
Data normalization or feature scaling is a data preprocessing technique where we convert all input values used in the learning model to a common scale. Without scaling the data present in the dataset, the features with large value will have a greater impact on the output of the learning model. Such scale difference leads to important features with smaller range become less effective to the overall inferences drawn by the classifier. To make all the features equal, it is important to normalize or scale our data, which also helps the algorithm reach convergence faster. Fig. 3 visualizes how normalization changed the natural range of raw feature named dbytes to a standard range of [0,1] as an example.
Normalization independently rescales the data feature-wise from its natural range into a standard range where for every feature the minimum value gets transformed into the value of zero, and the maximum value gets transformed into the value of one, hence giving all the features in data an equal footing for drawing the statistical inference. We used min-max normalization whose formula can be expressed as,
́= ( − ) ( − )
where represents the scaling data point, is the minimum and is the maximum absolute value of . The min-max normalization retains the shape of the feature intact during scaling which helps the model avoid overfitting as compared to other normalization techniques we tested during our experiments. This particular data pre-processing step is vital as various algorithms such as logistic regression and neural networks etc. assume that the input data for drawing the inference from will be duly scaled and normalized.
D. Learning Algorithms
Deep learning algorithms are capable of achieving higher accuracy in terms of classification when compared to other shallow networks and machine learning models. After training the chosen deep learning algorithm in the source domain, we use transfer learning methodology to transfer the model's architecture and learned weights to a target domain and test the performance of the model in the new domain with the unseen dataset partition. To choose the deep learning algorithm for our source domain architecture, we did a benchmark study on three deep learning algorithms, namely Deep Neural Network (DNN), Convolutional Neural Network (CNN), and Long Short-Term Memory architecture fitted with Convolutional Neural Network in its hidden layers (CNN-LSTM). The three types of learning algorithms are briefly summarized as follows.
A Deep Neural Network consists of multiple fullyconnected layers, which passes information from numerous layers in a feed-forward manner such as the input layer, several hidden layers, and an output layer to learn its weights iteratively using backpropagation algorithm, which computes the partial derivatives of the cost function with respect to each neuron unit with respect to its weights and biases to reach the local minima of the function. LeCun et al. [22] demonstrated one of the earliest practical implementations of backpropagation algorithm to build a handwriting recognition OCR.
Convolutional Neural Network is a class of deep neural networks which are applied to image recognition tasks as they can learn highly representative and hierarchical features from their input. Each layer inside a CNN is composed of several neuron units. The neurons are organized in a style that the output of neurons at layer l becomes the input of neurons at layer l + 1, such that where ( ) is the weight matrix of layer l, ( ) is the bias term, and represents the activation function. The activation for layer l is denoted by ( ) . A CNN consists of a convolutional layer which extracts the features from an input vector using filters, a ReLU unit for introducing non-linearity in the network, a pooling layer which reduces the dimensionality of the data while keeping the useful information and finally a fully connected layer which computes the potential output using a softmax function.
Long Short-Term Memory networks are a variant of recurrent neural networks, which in addition to feedforward connections, also have looping feedback connections that allow the model to store persistent information over a period of time. LSTM first proposed by S. Hochreiter et al. [23], are capable of learning long-term and short-term dependencies without losing or over accumulating information. LSTM is capable of adding or removing information in their cell states by using regulation structures called gates, which control the flow of data for each cell unit present in its architecture. Such design gives LSTM an advantage over conventional feed-forward neural networks because of their ability to selectively retain or drop information.
E. Proposed IDS Model
In this paper, our chosen deep learning architecture for the IDS consists of a CNN with LSTM present in its hidden layers and fully connected layer units to predict the classification labels. As shown in Fig. 3, the proposed unified IDS model can use the advantages of the three distinguished deep learning models and combines their latent feature extraction, memory retention, and classification abilities to give a higher accuracy score as compared to the models applied separately.
A CNN is capable of learning and recognizing patterns over an input space, whereas LSTM units can learn and recognize patterns across time. A DNN or a fully connected layer, on the other hand, is capable of learning mappings from an input vector to give precise class wise outputs. Both CNN and DNN belong to the class of feedforward networks where data can only flow in the forward direction. CNN can use a 2D input and transform it into internal vector representations to further extract its features. In contrast, when we apply LSTM with CNN, LSTM provides the capability of using the feature vector output of the CNN and further build internal states whose weights can repeatedly be updated because data in LSTM flows in a recurrent manner. During this entire process, the CNN extracts the inherent features from the input. In contrast, LSTM interprets those features across various time steps, which makes the architecture more efficient to learn deeper representations and relationships in the data, in contrast with any network architecture applied separately.
Combining DNN, CNN and LSTM have been explored in the past in [24], where the models are being trained separately, and then their outputs are later combined. In our approach, we are training the unified model jointly with each model providing their processed feature outputs as an input to the subsequent models in the scheme. Table Ⅱ shows the summary of our candidate CNN-LSTM model, where we are first using CNN layers to extract the contextual features in the training set. The utility of CNN's to downsample the input while conserving the important features during the extraction process reduces the overall dimension of the feature parameters. The output of CNN is then fed into the LSTM layers to model the signal in time and train the weights using backpropagation in time (BPTT) algorithm. Finally, after the signal is modeled in the LSTM layers, the output is passed into fully connected layers, which are used to learn higher-order feature representations that are suitable for separating the output into different class labels.
F. Experiments Setup
For our experimentation, we provisioned a VM cluster in the Google Cloud Platform (GCP). We used the VM cluster instance type n1-standard-16, which was configured with 16 vCPUs and a 30GB RAM allocation. For our deep learning libraries, we used Keras Framework with TensorFlow1.15 in the back-end. For data-preprocessing and manipulation, we used Pandas and Scikit-learn libraries. The experiments were ran using Jupyter notebook IDE and Python 3.7. To further simulate a resource sparse target domain, we used another provisioned VM in GCP. We used the VM cluster instance type n1-standard-1, which is configured with one vCPU and 3GB RAM allocation.
G. Performance Evaluation
To evaluate the performance of our applied model, we will use measures namely Classification Accuracy, Confusion Matrix and ROC curve briefly described as follows, a. Classification Accuracy is a metric used for classification models, where we compare the number of correct predictions drawn by the model with the total number of predictions made by the model. The classification accuracy can be expressed as,
V. RESULTS
A. Source Domain Architecture
For selecting the candidate source domain IDS model, we performed a benchmark study on three deep learning models, namely DNN, CNN, and CNN-LSTM. Our results show that CNN architecture demonstrated a 92.16% accuracy score on the source domain validation dataset, whereas DNN architecture demonstrated an accuracy score of 87.66%. The CNN architecture with LSTM layers present in its hidden layers demonstrated a 98.30% accuracy score outperforming other applied models. Fig. 4 shows the bar plot of model accuracies. Based on these results, we chose CNN-LSTM model as our candidate model for IDS architecture.
The results show that CNN-LSTM was able to learn more representational features in the training data and was able to generalize well to the validation dataset. As discussed previously, combining the three model architectures enhanced the overall classification accuracy. DNN is suitable for the task for generating higher-order feature representations, which can be separated into distinctive classes, but they don't enforce any structure or local information in the data. CNN, on the other hand, is suitable for extracting important features by condensing the input data to find the inherent structures and representations in the data, which gives a performance edge over DNN for the packet classification task. Adding LSTM units in hidden layers of CNN further improves the architecture. With the utility of LSTM, the output being parsed from CNN can further be modeled temporally using recurrence before feeding the results into DNN, which is vital for classifying the features into their appropriate class labels.
Accuracy by itself may not be the best evaluation metric for performance. Hence, we used confusion matrix for CNN-LSTM architecture to study the classification results in-depth, as shown in Fig. 5.
As per the confusion matrix, the CNN-LSTM model demonstrates a 1.03% false-positive rate and an 0.67% falsenegative rate. False classifications have been a major area of concern for IDS, as systems can incorrectly classify malicious packets as normal ones leading to successful intrusion scenarios that can debilitate the infrastructure. The normal packets can also be classified as malicious packets, which leads to false alarms and the dropping of good packets, which might be useful, reducing the overall quality of service of the network.
The applied models were also evaluated using the ROC metrics, as shown in Fig. 6.
B. Target Domain Architecture
To apply the learned knowledge in the target domain, we will use the source domain's CNN-LSTM model architecture as well as its learned weights and transfer them to the model in our target domain. We will use the unseen testing data-set in this domain to simulate the IDS model being in a real environment where it encounters new data. This helps in evaluating how the model will essentially react when it is deployed in a real-world network infrastructure.
Our experiments show that using the learned weights from the source domain improved the overall performance and speed of models in the target domain. The DNN architecture demonstrated an improved accuracy score of 88%, whereas CNN architecture reported a 91.88% accuracy score. Our candidate model CNN-LSTM architecture demonstrated an accuracy of 98.43%, which is higher than other applied models in this domain. The Fig. 7 shows the bar plot of accuracy score in the target domain.
The CNN-LSTM model also demonstrated better binary classification results with a general reduction in the false positive and false negative rates. Overall, the model has a 0.95% false-positive rate and a 0.62% false-negative rate, improving the packet classifications, as shown in Fig. 8.
To further study our results in the target domain, we used ROC curve metrics plotted in Fig. 9, which is noted to be comparable to the source domain ROC. It reflects the maintenance of the classification ability by the applied model with the unseen testing dataset in the target domain.
In essence, because we are using the transfer learning methodologies, the model is not being trained from scratch in the target domain. We are applying the knowledge learned prior to the source domain to perform the task of packet classification again in the target domain. Since the model has learned this task beforehand, it becomes easier for it to perform the same task again in the target domain. Various weights parameters between numerous units present in the models are not being modified; hence we don't need a large training set to retrain the model again. The low computational resources in the target domain are enough for the model to work with reasonable efficiency on the unseen data. We also observe that the knowledge transferred from the source domain improved the classification ability of IDS in the target domain. Overall, the testing speed of the applied deep learning models improved as well by a large margin in the target domain. DNN had the fastest testing speed in both domains but gave a low accuracy score because it is a simpler feedforward model with limited hierarchical feature extraction ability. CNN-LSTM had the slowest testing speed due to the inherent complexity of its architecture as well as the presence of more units and parameters as compared to both DNN and CNN models, but the model had a better accuracy score and classification results. Table Ⅲ shows the summarized results of our benchmark study with three learning algorithms, namely DNN, CNN, and CNN-LSTM. From our results, we observed that CNN-LSTM outperformed other applied models by giving more accurate classification results with a 98.30% accuracy score in the source domain and a 98.43% accuracy score in the target domain. The maintenance of accuracy reflects the model's ability to utilize the weights learned from the source domain and apply them in the target domain to generalize well on an unseen testing dataset. The testing speed also improved despite the target domain's simulated data and computational resource scarcity.
VI. CONCLUSION AND FUTURE WORK
Our study demonstrates that deep transfer learning approach can be highly effective in developing an efficient, unified network intrusion detection system that maintains and improves its classification accuracy and speed in a simulated real-world setting via knowledge transfer. Using the proposed method, we can train a large and powerful deep learning IDS model in a source domain with a high allocation of data and computational resources. After validating our model's performance, we can then transfer its architecture and learned weights in a target domain with reduced computational resources, where we observe that the model maintains its efficiency as well as improves its testing speed. The target domain is aimed at simulating the real-world environment where we are using a partition of the dataset, which is entirely unseen by our models during their training and development.
This research showcases that high powered deep learning based IDS architectures can be deployed on real-world devices with lesser resources, which can maintain their efficiency and improve their speed using the transfer learning approach. Applying transfer learning in the overall design of an IDS not only enhances its performance in a real-world setting but also essentially increases its speed of classification, which is a tremendously required feature demanded by an IDS to protect and secure modern network infrastructures. Our research is one of the earliest practical implementations of integrating transfer learning techniques in the core architecture of an IDS.
As future work, we would like to integrate dimensionality reduction techniques in the existing IDS architecture and use different modern network intrusion datasets to further validate the proposed model and techniques.
Fig. 1 .
1Transfer Learning Concept
Fig. 2 .
2Feature Importance Bar Chart Fig. 3. Data Normalization Visualization
Matrix is a visual representation for the performance of a classification model where the outcome of the model is expressed using four key categories. True Positive refers to the values which were predicted to be positive, and they are indeed positive and hence true. False-positive refers to the values which were predicted to be positive but are negative and thus false. True Negative refers to the values which were predicted to be negative and indeed are negative and hence true. False Negative refers to the values which were predicted to be negative but are, in fact, positive and thus false.c. Receiver Operating Characteristic (ROC) curve, is a curve plot where we compare two parameters, True Positive Rate and False Positive Rate. True positive rate, also known as sensitivity of the model, determines the proportion of values that are positive and were correctly identified as positive by the model. TPR can be expressed as, = + The false-positive rate, also known as specificity of the model, determines the proportion of the values which are negative and were also identified as negative by the model. FPR can be expressed as, = + ROC curve plots the True positive rate of a model with respect to its False positive rate at various thresholds.
Fig. 4 .
4Accuracy Score -Source Domain Fig. 5. Confusion Matrix -Source Domain
Fig. 7 .Fig. 8 .
78Accuracy Score -Target Domain Fig. 6. ROC Curve -Source Domain Confusion Matrix -Target Domain Fig. 9. ROC curve -Target Domain
TABLE I DATASET
IKEY FEATURE DESCRIPTIONWe use feature importance to judge each feature based on a scoring value, which represents how important and relevant the feature is to the output variable and vice versa. We used a treebased extra-trees classifier, which is an ensemble model ofFeature Description
sload
Source bits per second.
dload
Destination bits per second.
stcpb
Source TCP base sequence number.
dtcpb
Destination TCP base sequence number.
sbytes
Source to destination transaction bytes.
dbytes
Destination to source transaction bytes.
sttl
Source to destination time to live value.
dttl
Destination to source time to live value.
swin
Source TCP window advertisement value.
dwin
Destination TCP window advertisement value.
sjit
Source jitter (millisecond).
djit
Destination jitter (millisecond).
spkts
Source to destination packet count.
TABLE Ⅱ
ⅡCNN-LSTM IDS MODEL ARCHITECTURE
TABLE Ⅲ MODEL
ⅢACCURACY AND TEST SPEEDSource Domain
Target Domain
DL Model
Accuracy
Test Speed
Accuracy
Test Speed
DNN
87.66%
32.8s
88.00%
1.67s
CNN
92.16%
134.2s
91.88%
18.4s
CNN-LSTM
98.30%
189.5s
98.43%
22.3s
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| []
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[
"Oracle Importance Sampling for Stochastic Simulation Models",
"Oracle Importance Sampling for Stochastic Simulation Models"
]
| [
"Yen-Chi Chen \nDepartment of Statistics\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\nUniversity of Washington\n\n",
"Youngjun Choe \nDepartment of Statistics\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\nUniversity of Washington\n\n"
]
| [
"Department of Statistics\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\nUniversity of Washington\n",
"Department of Statistics\nDepartment of Industrial and Systems Engineering\nUniversity of Washington\nUniversity of Washington\n"
]
| []
| We consider the problem of estimating an expected outcome from a stochastic simulation model using importance sampling. We propose a two-stage procedure that involves a regression stage and a sampling stage to construct our estimator. We introduce a parametric and a nonparametric regression estimator in the first stage and study how the allocation between the two stages affects the performance of final estimator. We derive the oracle property for both approaches. We analyze the empirical performances of our approaches using two simulated data and a case study on wind turbine reliability evaluation. | 10.1214/19-ejs1604 | [
"https://arxiv.org/pdf/1710.00473v1.pdf"
]
| 88,521,966 | 1710.00473 | 48d917a13ecb235b701ab38aff5257f2000d0596 |
Oracle Importance Sampling for Stochastic Simulation Models
October 3, 2017
Yen-Chi Chen
Department of Statistics
Department of Industrial and Systems Engineering
University of Washington
University of Washington
Youngjun Choe
Department of Statistics
Department of Industrial and Systems Engineering
University of Washington
University of Washington
Oracle Importance Sampling for Stochastic Simulation Models
October 3, 2017Monte Carlononparametric estimationoracle propertyvariance reduc- tion * Emails: yenchic@uweduychoe@uwedu; both authors equally contributed to this work
We consider the problem of estimating an expected outcome from a stochastic simulation model using importance sampling. We propose a two-stage procedure that involves a regression stage and a sampling stage to construct our estimator. We introduce a parametric and a nonparametric regression estimator in the first stage and study how the allocation between the two stages affects the performance of final estimator. We derive the oracle property for both approaches. We analyze the empirical performances of our approaches using two simulated data and a case study on wind turbine reliability evaluation.
Introduction
The 2011 Fisher lecture (Wu, 2015) features the landscape change in engineering, where computer simulation experiments are replacing physical experiments thanks to the advance of modeling and computing technologies. An insight from the lecture highlights that traditional principles for physical experiments do not necessarily apply to virtual experiments on a computer. The virtual environment calls for new modeling and analysis frameworks distinguished from those developed under the constraint of physical environment. In this context, this study considers a new problem that emerged along with the recent development of stochastic simulation-based engineering. Choe et al. (2015) reports the problem of estimating a system failure probability based on stochastic simulations. A system configuration, X, is randomly sampled from a known density p and passed on to a stochastic simulation model. The simulation model, regarded as a stochastic black-box, produces V that follows an unknown distribution depending on X. When V is greater than a threshold, say ξ, the system fails. Thus, the goal is to estimate the probability P (V ≥ ξ) when X is from the density p. Because the simulation model is designed to mimic the real system accurately, it takes roughly 1-min wall-clock time to simulate 10-min real operation of the system on a computer commonly available nowadays. The engineering goal is estimating the probability of system failure during 50-year operation, which is computationally challenging even with U.S. national labs' supercomputers (Manuel et al., 2013;Graf et al., 2016).
Such computational challenges are commonly observed in engineering simulations.
Finite element simulations, which are used widely in various engineering applications, can take hours of computing time to obtain a single data point (e.g., Qian et al., 2006). Despite the computational expense, highly accurate simulations are costeffective alternatives to physical experiments and used widely in industry (e.g., Ford
Motor Company's crash simulation (Wang and Shan, 2007)) and in government (e.g., NASA's rocket booster simulation (Gramacy and Lee, 2012)).
The overarching goal of this study is to minimize the necessary computational burden while maintaining the same level of estimation accuracy. A flip-side of the same problem is minimizing the estimation variance given fixed computational resource.
Variance reduction techniques (VRTs) in the simulation literature aim to reduce the variance of estimator in simulation experiments. Traditional VRTs are well studied for the simulation model that outputs V given X in a deterministic fashion, also known as the deterministic simulation model (see Chapter 9 of Kroese et al. (2011) for survey of such VRTs), when the input X is sampled from a known distribution.
For stochastic simulation models, if their underlying processes have known properties (e.g., Markovian), Glynn and Iglehart (1989) and Heidelberger (1995) provide VRTs.
For black-box stochastic simulation models, few studies (e.g., Choe et al., 2015;Graf et al., 2017) consider VRTs. The research on VRTs for block-box stochastic simulations is still underdeveloped despite the rapid growth of such simulations being used in real systems, for example, chemical systems (Gillespie, 2001), biological systems (Henderson et al., 2012), and engineering systems (Ankenman et al., 2010;Picheny et al., 2013;Plumlee and Tuo, 2014).
Among VRTs, importance sampling (Kahn and Marshall, 1953) is known to be one of the most effective methods and has been used widely in various applications such as communication systems (Heidelberger, 1995;Bucklew, 2004), finance (Owen and Zhou, 2000;Glasserman and Li, 2005), insurance (Blanchet and Lam, 2011), and reliability engineering (Au and Beck, 1999;Lawrence et al., 2013;Choe et al., 2016) to name a few.
In the vast majority of literature, importance sampling takes a parametric form tailored to a problem at hand for both deterministic simulation model (e.g. Lawrence et al., 2013) and stochastic counterpart (Choe et al., 2015). Nonparametric approaches are also proposed for deterministic simulation models (Zhang, 1996;Neddermeyer, 2009). To the best of our knowledge, no nonparametric approach is developed for stochastic simulation models. This study particularly considers the black-box stochastic simulation model whose output takes an unknown stochastic relationship with the input.
The main contributions of this paper to the existing body of literature are as follows:
• We introduce an importance sampling approach to estimate the expectation of black-box stochastic simulation output and study the optimal importance sampler (Section 2).
• We design a two-stage procedure that uses a parametric or a nonparametric regression estimator to approximate the optimal importance sampler ( Figure 1 and 2).
• We analyze the allocation of the resources in both stages (Theorem 3 and 7) and study the convergence of the two-stage procedure toward the oracle importance sampler (Corollary 4 and 8).
• We conduct extensive simulation study to investigate empirical performances of the two-stage importance samplers (Section 4.2.1 and 4.2.2).
• We apply our methods to a case study on wind turbine reliability evaluation (Section 4.3) to validate our results. This paper is organized as follows. In Section 2 we formulate the stochastic simulation-based estimation problem and introduce a two-stage procedure to estimate the expected simluation output. In Section 3 we study the theoretical performance of the proposed procedure and derive the corresponding oracle properties. In Section 4 we study the empirical performance of our approach using two toy examples and a wind turbine simulator. We discuss our result in Section 5.
2 Importance Sampling for the Stochastic Simulation Model A stochastic simulation model takes an input configuration value and then returns a random number representing the outcome of this simulation result. The input configuration determines the distribution of the (random) outcome. Thus, the outcome of a stochastic simulation model can be represented by a random variable V conditioned on the input configuration x and the CDF of V is
F V |config=x (v|x) = P (V ≤ v|config = x),
where config = x denotes choosing the configuration to be x. For simplicity, we denotes the random variable V conditioned on config = x as V (x).
In many scientific or engineering problems (e.g., Heidelberger, 1995;Au and Beck, 2003;Bucklew, 2004;Graf et al., 2017), we assume the nature generates the configuration from a known density p and we are interested in evaluating the quantity
E = E(g(V (X))) = E(g(V )|config = x)p(x)dx,(1)
where g is a known function.
Example 1. A common example for equation (1) is the case when g(v) = v, often considered in the literature on two-level nested simulation (Sun et al., 2011), where the outer level generates a scenario (or configuration) according to a known density p and conditioning on the scenario, the inner level simulates a random outcome whose mean is of interest. Applications include decision theory (Brennan et al., 2007), financial engineering (Staum, 2009), and queuing system (Sun et al., 2011).
Example 2. Another example takes g(v) = 1(v ∈ S ξ ) for some set S ξ parametrized by ξ. Specifically, Choe et al. (2015) considers a reliability evaluation problem where V stands for an instability measurement of a system that fails when V falls into
S ξ = [ξ, ∞).
The goal is to estimate the failure probability when the system is exposed to the nature. In the natural environment, the configuration behaves like a random variable from a density p.
To estimate E, we can choose several configurations x 1 , · · · , x n and then run the simulation to obtain realizations v 1 = V (x 1 ), · · · , v n = V (x n ). However, generating V from a given configuration x is often computationally expansive for stochastic simulation models. Therefore, we would like to run the simulation as few as possible.
To put this constraint into consideration, we assume that we run the simulation only n times but we are able to choose the configuration for each simulation. We choose n configurations and evaluate the corresponding value of V . Namely, we only have pairs (x 1 , V 1 ), · · · , (x n , V n ), where each V i is a realization of the random variable V (x i ). Such a constraint on the number of simulations, n, is sometimes called a computational budget.
Under such situation, a natural question is: How do we choose the configurations x 1 , · · · , x n ? Here we use the idea from importance sampling -we choose x 1 , · · · , x n from a density function q. In other words, we first sample X 1 , · · · , X n from q and then use x i = X i as the configuration to run the i-th stochastic simulation. The density q is called sampling density. Note that each configuration does not necessarily have to be from the same density function.
When we generate X 1 , · · · , X n from q and then obtain V 1 , · · · , V n accordingly, a simple estimator of E is
E q = 1 n n i=1 g(V i ) p(X i ) q(X i ) .(2)
It is easy to see that E q is an unbiased estimator under the assumption that q(x) = 0
implies g(V (x))p(x) = 0 for all x, i.e., E E q = E
when the support of q covers the support of g(V (x))p(x). We call this type of estimator an importance sampling estimator. Throughout this paper, we will focus on importance sampling estimators.
Using an importance sampling estimator (2), a key problem we want to address is: what will be the optimal sampling density q * that minimizes the estimation error?
Because the estimator (2) is unbiased, we only need to find the minimal variance estimator. Thus, the above question is equivalent to: what will be the optimal sampling density q * that minimizes the variance of E q ? The following lemma provides an answer to the above questions:
Lemma 1. Assume X 1 , · · · , X n are IID from a density function q. Let r † (x) = E(g(V (X))|X = x) and r(x) = E(g 2 (V (X))|X = x). Then variance of E q equals to
Var E q = 1 n E X i ∼q r(X i ) p 2 (X i ) q 2 (X i ) − E 2 (r † (X * )) ≥ 1 n E 2 r(X * ) − E 2 (r † (X * )) ≡ 1 n V min(3)
where X * is a random variable from the density p. The equality holds when we choose q to be q(x) = q * (x) ∝ E(g 2 (V (X))|X = x) · p(x). Namely, the optimal sampling density is q * (x).
A special case of Lemma 1 with g(v) = 1(v ≥ ξ) appears in Choe et al. (2015).
We call the quantity V min the oracle variance of the importance sampling. It is the minimal variance that an importance sampler can achieve. A widely studied special case in engineering is the deterministic simulation model where Var(V |X = x) = 0
for all x, which implies V min = 0 for any nonnegative function g(v) (e.g., Kahn and Marshall, 1953;Au and Beck, 1999;Kroese et al., 2011). The density that leads to the oracle variance, q * , is called the optimal sampling density. This density is a modification from the natural configuration density p; q * puts more weight on the regions with higher r(x) (e.g., higher probability of system failure).
However, we cannot directly generate configurations from q * because it involves the unknown quantity r(x) = E(g 2 (V (X))|X = x). A remedy to this problem is to apply a two-stage sampling. In the first stage, we generate part of configurations and evaluate the corresponding values of V (x). Using the sample in the first stage, we obtain a pilot estimator r of r. In the second stage, we generate configurations based on an estimate of q * using the pilot estimator r and use the remaining computational budget to evaluate values of V (x). Finally, we use samples from both stages to form the final estimator of E.
Here is a useful insight in estimating r(x). Let Y be a random variable such that Y = g 2 (V (X)). Then r(x) = E(g 2 (V (X))|X = x) = E(Y |X = x). Thus, estimating r(x) is equivalent to estimating the regression function with the observations (X 1 , Y 1 = g 2 (V 1 )), · · · , (X n , Y n = g 2 (V n )) assuming we have n observations.
Thus the two-stage procedure can be summarized as follows. We first generate a size m sample
(X 1 , V 1 ), · · · , (X m , V m )
where X i , i = 1, · · · , m, are from an initial sampling density q 0 . Then we transform
V i into Y i = g 2 (V i )
, which leads to a sample (X 1 , Y 1 ), · · · , (X m , Y m ). Now we estimate the regression function r(x) by a regression estimator r(x) and
compute the corresponding estimator q * of the oracle sampling density q * . Finally, we generate the remaining data points (X m+1 , V m+1 ), · · · , (X n , V n ) from q * and pool both samples together to form the final estimator of the quantity E.
Because q * will tend to be closer to q * compared to the initial sampling density q 0 , estimating E using the sample in the second stage is more efficient (lower variance).
The sample size m in the first stage is a crucial quantity in our analysis. The quantity m is called the allocation. When m is too small, the estimator of q * is inaccurate, so that the overall estimation efficiency is suboptimal. When m is too large, we only have a small budget for the second stage so that the overall estimation efficiency is low as well. As a result, to balance the estimation accuracy of q * and the size of efficient sample in the second stage, there will be an optimal value of m depending on the total sample size n. In what follows we propose two different models to estimate r and q * and analyze the optimal value of the allocation m.
Parametric Importance Sampling
As in the regression analysis, a straightforward approach of estimating the regression function is to assume a parametric model and estimate the corresponding parameters.
Namely, we assume r(x) = r θ (x) for some θ ∈ Θ and use the first part of the sample to estimate θ.
To estimate r θ (x), we use a classical approach-the least square method :
θ m = argmin θ∈Θ m i=1 Y i − r θ (X i ) 2 .(4)
Then the estimator r(x) = r θm (x). Note that one can also assume a parametric form for the distribution of Y i |X i and then use a maximum likelihood estimator. Using the estimator θ m , we can then estimate the regression function r θm and construct the estimated optimal sampling density
q * θm (x) ∝ r θm (x) · p(x).
For the remaining (n − m) data points, we generate the configurations from q * θm , run the simulation, and estimate E accordingly.
Parametric Importance Sampling (S1) We choose an initial sampling density q 0 and generate the first part of the sample
(X 1 , V 1 ), · · · , (X m , V m ). (S2) Transform (X 1 , V 1 ), · · · , (X m , V m ) into (X 1 , Y 1 ), · · · , (X m , Y m ) using Y i = g 2 (V i ). (S3)
Use the least square method (4) to obtain θ m and the estimator r θm (x). (S4) We then change the sampling density to q * θm and generate the remaining sample
(X m+1 , V m+1 ), · · · , (X n , V n ), where q * θm (x) ∝ r θm (x) · p(x). (S5) The final estimator is E θm = 1 n m i=1 g(V i ) p(X i ) q 0 (X i ) + n i=m+1 g(V i ) p(X i ) q * θm (X i )
.
(5) Figure 1: Parametric importance sampling for the stochastic simulation model.
We summarize our Parametric Importance Sampling method in Figure 1. Later in Section 3.1 we will derive the variance of this approach and show that the optimal allocation is to choose m = O(n 2 3 ).
Nonparametric Importance Sampling
Now we consider estimating r(x) nonparametrically. For simplicity, we use the kernel regression (Nadaraya, 1964;Watson, 1964). Note that other nonparametric regression approach, such as the local polynomial regression (Wasserman, 2006), also works. The kernel regression uses the estimator
r h (x) = m i=1 Y i K x−X i h m i=1 K x−X i h ,(6)
where K is a smooth function (known as the kernel function) such as a Gaussian, and h > 0 is the smoothing bandwidth. Similar to the parametric approach, we then use this estimator to construct an estimated optimal sampling density
q * h (x) ∝ r h (x) · p(x),
generate the remaining data points from it, and construct the final estimator using the procedure described previously.
Nonparametric Importance Sampling (S1) We choose an initial sampling density q 0 and generate the first part of the sample
(X 1 , V 1 ), · · · , (X m , V m ). (S2) Transform (X 1 , V 1 ), · · · , (X m , V m ) into (X 1 , Y 1 ), · · · , (X m , Y m ) using Y i = g 2 (V i ). (S3) Based on (X 1 , V 1 ), · · · , (X m , V m )
, use the nonparametric regression to obtain the estimator r h and q * h . (S4) We then change the sampling density to q * h to generate the remaining sample
(X m+1 , V m+1 ), · · · , (X n , V n ), where q * h (x) ∝ r h (x) · p(x). (S5) The final estimator is E h = 1 n m i=1 g(V i ) p(X i ) q 0 (X i ) + n i=m+1 g(V i ) p(X i ) q * h (X i ) .(7)
Figure 2: Nonparametric importance sampling for the stochastic simulation model.
Theoretical Analysis
Throughout our analysis, we assume that the natural configuration density p has a compact support K ⊂ R d and the support of the initial sampling density q 0 contains K.
Parametric Importance Sampling
Assumptions.
(P1) There exists an unique θ 0 ∈ Θ such that r(x) = r θ 0 (x) and sup x∈K Var(Y 1 −
r θ 0 (X 1 )|X 1 = x) ≤ σ 2 max < ∞. The support of r θ (x) contains the support of p(x) for every θ ∈ Θ and r(x) > 0 for all x ∈ K. (P2) Let (θ) = E Y 1 − r θ (X 1 ) 2 . The Hessian matrix H(θ) = ∇ θ ∇ θ (θ) is positive definite at θ ∈ B(θ 0 , R 0 ) for some R 0 < ∞ and θ 0 is the one in (P1). Note that B(x, r)
is a ball centered at x with radius r.
(P3) There exists a positive L 0 < ∞ such that for any θ 1 , θ 2 ∈ B(θ 0 , R 0 ),
sup x∈K |r θ 1 (x) − r θ 2 (x)| ≤ L 0 · θ 1 − θ 2 ,
where θ 0 , R 0 are defined in (P2).
(P1) means that the model is correctly specified-the regression function can be parametrized in the parametric model we consider. (P2) is a common assumption in the M-estimation theory (van der Vaart and Wellner, 1996;van der Vaart, 2000) to derive the convergence rate. The extra assumption (P3) is a mild assumption that converts the convergence rate of parameter estimation to the convergence rate of function estimation. As long as r θ (x) is smooth within an open set around θ 0 , (P3) holds.
The following theorem describes the estimation error when the parametric family contains the true regression function.
Theorem 2. Assume (P1-3). The error rate for the estimator r θm (x) is
sup x∈K r θm (x) − r(x) = O P 1 m .
Theorem 2 presents the error rate for estimating r(x) when the model is correct. Based on this error rate, we can further derive the variance of the parametric importance sampler in Figure 1.
Theorem 3. Assume (P1-3). Let V q 0 = E X i ∼q 0 r(X i ) p 2 (X i ) q 2 0 (X i ) − E 2
r(X * ) be the excess variance from using q 0 compared to q * . The variance of the estimator E θm is
Var E θm = 1 n V min + 1 n 2 m · V q 0 + (n − m) · O 1 m .
Theorem 3 has three components. The first component V min is the oracle variance we have mentioned previously. It is the minimal variance that can be achieved by an importance sampling estimator. The second component 1 n 2 · m · V q 0 is the excess variance due to the initial sampling density. The third component 1
n 2 · (n − m) · O 1 m
is the excess variance due to the error of the estimator r θm (x).
By optimizing m with respect to the second and third components, we obtain the optimal rate of m as a function of sample size n:
m · V q 0 = (n − m) · O 1 m =⇒ m 3 2 n =⇒ m n 2 3 ,
where the notation means that the two quantities will be of the same order, i.e., a n b n ⇔ lim n→∞ an bn ∈ (0, ∞). Thus, the optimal allocation is to choose m n 2 3 , which leads to the following:
Corollary 4. Assume (P1-3). When m n 2 3 , the variance of the estimator E θm is
Var E θm = 1 n V min 1 + O n − 1 3 .
That is, if the model is correctly specified, the excess variance shrinks at rate
O n − 1 3
under the optimal allocation.
The key assumption of the parametric method is (P1): the actual r(x) belongs to the parametric family. However, if this assumption is violated, then the excess variance in the parametric method will never shrink to 0.
Theorem 5. Assume (P2-3). If r(x) = r θ (x) for all θ ∈ Θ, the variance of the parametric estimator Var E θm ≥ 1 n V min + 1 n V θ * where V θ * = inf θ∈Θ E r(X θ ) p 2 (X θ ) q 2 θ (X θ ) − E 2 r(X * ) > 0, X θ ∼ q θ (x) ∝ r θ (x) · p(x).
The proof of this theorem is trivial, so we omit it. Theorem 5 proves that when the model is incorrectly specified, there is an additional variance V θ * that never disappears. Thus, the variance of the parametric importance sampler will not converge to the optimal variance. Later we will see that this implies that the parametric importance sampler does not have the oracle inequalities when the model is incorrectly specified.
Nonparametric Importance Sampling
In this section, we study the properties of the nonparametric importance sampler in Figure 2. Similarly as the parametric importance sampler, we first derive the convergence rate of estimating r(x), then derive a variance decomposition for Var E h , and finally study the optimal allocation.
Assumptions.
(N1) sup x∈K Var(Y 1 − r(X 1 )|X 1 = x) ≤ σ 2 max < ∞ and r(x) > 0 for all x ∈ K.
(N2) For all x, the function r(x) has bounded second derivative and q 0 (x) has bounded first derivative and sup x∈K q 0 (x) ≥ q min > 0.
(K1) The kernel function K(x) is symmetric and
K(x)dx = 1, x 2 K(x)dx < ∞, K 2 (x)dx < ∞. (K2) The collection of functions K = y → K x − y h : x ∈ K, h > 0 , is a VC-type class. i.e. there exists constants A, v and a constant envelope b 0 such that sup Q N (K, L 2 (Q), b 0 ) ≤ A v ,(8)
where N (T, d T , ) is the -covering number for a semi-metric set T with metric d T and L 2 (Q) is the L 2 norm with respect to the probability measure Q.
(N1) and (N2) are common assumptions for nonparametric regression; see, e.g., Wasserman (2006) and Györfi et al. (2006). (K1) is a standard condition on kernel function (Wasserman, 2006;Scott, 2015). (K2) regularizes the complexity of kernel functions so that we have a uniform bound on the stochastic variation. This assumption was first proposed in Giné and Guillou (2002) and Einmahl and Mason (2005) and later was used in various studies such as Genovese et al. (2014); Chen et al. (2015bChen et al. ( , 2017.
Based on the above assumptions, the uniform convergence rate of the kernel estimator r h (x) is given by the following.
Theorem 6. Assume (N1-2), (K1-2). The error rate of the kernel estimator r h (x)
is
sup x∈K r h (x) − r(x) = O(h 2 ) + O P log m mh d .r h * (x) − r(x) = O P log m m 2 d+4 .(9)
Under such an optimal error rate, we again obtain the variance decomposition for the nonparametric importance sampler.
Theorem 7. Assume (N1-2), (K1-2). Let V q 0 = E X i ∼q 0 r(X i ) p 2 (X i ) q 2 0 (X i ) −E 2 r(X * )
be the excess variance from using q 0 compared to q * . The variance of the estimator E h * under the optimal smoothing bandwidth is
Var E h * = 1 n V min + 1 n 2 m · V q 0 + (n − m) · O log m m 2 d+4
. Similar to Theorem 3, the variance in Theorem 7 has three components: the oracle variance V min , the excess variance due to the initial sampling density 1 n 2 · m · V q 0 , and the excess variance from the estimator r h * (x).
To obtain the rate of the optimal allocation, we equate the two excess variances:
m · V q 0 = (n − m) · O log m m 2 d+4 =⇒ m · m log m 2 d+4 n =⇒ m n log n d+4 d+6
(ignoring the log log n and multi-logarithm terms).
This choice of m yields the following variance reduction rate.
Var E h * = 1 n V min + O 1 n 2 · n · log 2 d+4 n · n log n d+4 d+6 × −2 d+4 = 1 n V min 1 + O log (4d+20)/(d+4) n n 2 d+6 = 1 n V min 1 + O log 5 n n 2 d+6
Note that in the last equality in Corollary 8, we use the fact that a n = O(log (4d+20)/(d+4) n) implies a n = O(log 5 n) to simplify the expression. Corollary 8 shows that under the optimal allocation, the excess variance in the nonparametric importance sampler shrinks at rate O log 5 n n 2 d+6
. When the dimension is small, say d = 1 or d = 2, the nonparametric method has an excess variance at rate O Although the parametric method enjoys a fast convergence rate, it depends on a very restrictive assumption: the model has to be correctly specified. This assumption is generally not true in most applications. Thus, even if the nonparametric importance sampler has a slower variance reduction rate, the nonparametric approach still has its own merit in applicability.
Remark 1. Note that the nonparametric rate can be improved if the regression function is very smooth and we use a higher order kernel (Wasserman, 2006 and under the optimal allocation, the variance of nonparametric importance sampler will be
1 n V min 1 + O log d+2.5β n n β d+3β .
When β → ∞, the variance of nonparametric importance sampler becomes 1 n V min 1 + O n −1/3 (ignoring the log n term), which recovers the parametric rate in Corollary 4.
Oracle Properties
In Lemma 1, we see that the oracle sampler E q * has the minimal variance. Let
G = f : f (x)dx = 1, f (x) = 0 ⇒ g(V (x))p(x) = 0 ∀x
be the collection of density functions that leads to an unbiased importance sampler and let Ξ = E q : q ∈ G be the collection of all possible unbiased importance samplers. Because all importance samplers from G is unbiased, any estimator E ∈ Ξ satisfies
E E − E 2 = Var E(10)
so the oracle sampler E q * satisfies
E E q * − E 2 = Var E q * = inf E∈Ξ Var E = inf E∈Ξ E E − E 2 .(11)
Because of equation (11), E q * is called the oracle for Ξ with respect to the mean square error in nonparametric theory; see, e.g., page 60-61 in Tsybakov (2009).
We say E † ∈ Ξ satisfies the oracle inequalities if
E E † − E 2 inf E∈Ξ E E − E 2 = E E † − E 2 E E q * − E 2 = 1 + o(1).(12)
Note that the estimators in the above expressions are all based on a size n sample and we do not include the subscript n to abbreviation. In nonparametric theory, an estimator with the oracle inequalities implies that the estimator is asymptotically as good as the optimal estimator.
The crude Monte Carlo sampler E p (which samples only from the natural configu-ration density p (Kroese et al., 2011)) obviously does not satisfy the oracle inequalities because
E E p − E 2 inf E∈Ξ E E − E 2 = Var E p Var E q * = V min + V p V min = 1 + V p V min > 1, where V p = E(r(X * )) − E 2 r(X * ) > 0.
The parametric importance sampler E θm satisfies the oracle inequalities when the model is correctly specified (i.e. r(x) = r θ (x) for some θ ∈ Θ). To see this, recall
Corollary 4 and equation (10):
E E θm − E 2 inf E∈Ξ E E − E 2 = Var E θm Var E q * = 1 + O n − 1 3 = 1 + o(1).
However, when the model is incorrect, Theorem 5 proves that E θm does not have the oracle inequalities:
E E θm − E 2 inf E∈Ξ E E − E 2 = Var E θm Var E q * = 1 + V θ * V min > 1.
The nonparametric importance sampler has a good advantage that it satisfies the oracle inequalities in most cases. By Corollary 8 and equation (10),
E E h * − E 2 inf E∈Ξ E E − E 2 = Var E h * Var E q * = 1 + O log n n 2 d+6 = 1 + o(1).
Thus, without any further information about the structure of r(x), we recommend to use the nonparametric importance sampler since it behaves asymptotically as good as the oracle (optimal) importance sampler.
Remark 2. How we obtain the oracle property is very different from the classical approach. Many estimators with oracle properties are constructed by minimizing an estimated risk (Tsybakov, 2009). That is, for a collection of estimators, the risk of each of them is estimated and the one that minimizes the (estimated) risk is chosen. When the risk is consistently estimated uniformly for all estimators, this procedure leads to an estimator with the oracle property. However, in our case, we do not consider any risk estimator nor do we choose an estimator from many possible candidates, but we still obtain the oracle property.
Empirical Analysis
To analyze the empirical performances of the importance samplers, this section presents an implementation guideline, a numerical study, and a case study.
Implementation Guideline
To implement parametric or nonparametric importance sampling, we can follow the procedure in Figure 1 or Figure 2, respectively. In practice, n is typically determined based on the available computational budget. We can choose m according to the optimal allocation rate, m n Once we build a regression model r(x) for the unknown conditional expectation r(x) = E(g 2 (V (X))|X = x), we can exactly sample from q(x) ∝ r(x) · p(x) using the acceptance-rejection method (Kroese et al., 2011, p.59). Also, for an importance sampling estimator (e.g., in (5) or (7)), the normalization constant of q(x) can be calculated to a desired level of accuracy by using a numerical integration such as quadrature for low-dimensional x and Monte Carlo integration for high-dimensional
x.
Numerical Study
Our numerical study considers two examples, one with normal distributions for X and V |X and the other with exponential distributions for X and V |X. Motivated by our case study, we estimate the probability E = P (V > ξ) = E(g(V (X))) for g(V ) = 1(V > ξ) and a pre-specified ξ > 0. We set ξ such that P (V > ξ) is equal to 0.5 regardless of the input configuration dimension d because it is known that the performance of importance sampler often depends on the probability being estimated (Heidelberger, 1995;Kroese et al., 2011;Choe et al., 2015).
We vary the total sample size n = 1000, 2000, 4000, 8000 and the input configuration dimension d = 1, 2, 4 to see their impacts on the mean squared error (MSE)
MSE = 1 n M C n M C i=1 E i − E 2 ,
where E i is an estimate of the ith replication and the total number of Monte Carlo replications, n M C , is set as 10,000. We use high-performance computing (Lenovo NextScale E5-2680 v4, total 112 cores with 1TB RAM) for our simulation experiments, and they take several weeks in total. The R codes are available as a supplementary material.
We consider two parametric importance samplers, one with a correct model of
r(x) = E(g 2 (V (X))|X = x) = E(1(V (X) > ξ)|X = x) = P (V > ξ | X = x)
and the other with an incorrect model, and a nonparametric importance sampler. To build the parametric models of r(x), we use the sample of size m = 2n To sample from the importance sampling density q(x) ∝ r(x) · p(x) using the acceptance-rejection method, we use p(x) as the envelope density because p(x) ≥ r(x)p(x) in the examples: We sample x from p(x) and accept x with the probability r(x). To compute the normalizing constant of q(x), we use Monte Carlo integration.
Since we know the true r(x), which is unknown in practice, in the examples, we calculate the true E = P (V > ξ) and V min using Monte Carlo integration and use them to calculate MSEs and demonstrate how empirical results conform to the theoretical predictions made in Section 3.
Example 1: normal-normal data generating model
As a modification of an example in Ackley (1987), we use the data generating model where the d-dimensional input vector X = (X (1) , . . . , X (d) ) follows a multivariate normal distribution with zero mean and identity covariance matrix, and the output V at X follows N (µ(X), 1) with
µ(X) = 20 1 − exp −0.2 1 d X 2 + exp (1) − exp 1 d d i=1
cos(2πX (i) ) .
Thus, we have
r(x) = P (V > ξ | X = x) = 1 − Φ(ξ − µ(x)),
where Φ(·) is the CDF of a standard normal distribution. As parametric models of r(x), we consider two models:
(i) Correct model: We use r θ (x) = 1 − Φ(ξ − µ(x)), where µ(x) = 20 θ 0 − exp −0.2 1 d d i=1 θ 2 i (x (i) ) 2 + θ 0 exp (1) − exp 1 d d i=1 θ i cos(2πx (i) ) ,
such that r θ (x) = r(x) for some θ = (θ 0 , . . . , θ d ) ∈ Θ. For fitting with a least square method, the initial parameters are set at the correct values, i.e., by Corollary 4. Recalling that MSE is equal to the variance of an importance sampler, because of its unbiasedness, we see that nMSE approaches V min as n increases, with roughly the same rate regardless of d. For the incorrect parametric model, as foreseen by Theorem 5, nMSE fails to approach V min as n increases, because nVar E θm ≥ V min (1 + O(1)). As anticipated by Corollary 8, nMSE for the nonparametric model approaches V min as n increases, with an apparently slower rate for If X was sampled only from the natural configuration density p instead of an importance sampling density q in the estimator in (2), then this simple baseline approach, commonly called crude Monte Carlo (CMC) (Kroese et al., 2011), results in the estimator having the theoretical nMSE of 0.25 (= P (V > ξ)(1 − P (V > ξ)).
larger d, because nVar E h * = V min 1 + O log n n 2 d+6 . q q q q V min V min V min V min V min V min V min V min V min V min V min VV min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V min V
In this example, the incorrect parametric importance sampler essentially does not improve over the baseline.
As d increases, V min approaches the baseline nMSE of 0.25 in Figure 3(d). This increasing inefficiency of optimal importance sampling with respect to d is regarded as peculiar to this example, because V min in (3) depends on d only through g(V (X)).
Thus, this observation should not be interpreted as a manifestation of curse of dimensionality known in the importance sampling literature (e.g., Au and Beck, 2003), which may occur when the approximation of optimal density q * becomes harder as d increases. In contrast, it is known that the optimal importance sampler theoretically attains V min of zero regardless of d for deterministic simulation models with any nonnegative function g(v) (Kahn and Marshall, 1953).
Example 2: exponential-exponential data generating model
Here, we consider a data generating model where both X and V |X follow exponential distributions that have heavier tails than normal distributions and allow analytical calculations of key objects of interest such as the estimand E, the conditional expectation r(x) = E(g 2 (V (X))|X = x), the optimal sampling density q * (x), and the oracle variance V min .
Let X = (X (1) , . . . , X (d) ) be a vector of d independent exponential random vari-ables with the identical mean 1/λ > 0 so that the natural configuration density
p(x) = λ d e −λ(x (1) +...+x (d) ) .
Given a configuration X, let V follow an exponential distribution with a mean 1/ X (1) + . . . + X (d) . In our simulation experiment, we fix λ = 1.
With the given data generating model, we can analytically calculate
E = P (V > ξ) = λ ξ + λ d , r(x) = E(g 2 (V (X))|X = x) = P (V > ξ | X = x) = e −ξ(x (1) +...+x (d) ) , and q * (x) ∝ r(x) · p(x) ∝ e −( ξ 2 +λ)(x (1) +...+x (d) ) ,
which implies that q * is the joint density of d independent exponential random variables with the identical mean 1/ (ξ/2 + λ). We determine
ξ = λ [P (V > ξ)] (1/d) − λ
by plugging P (V > ξ) = 0.5. We also know
r † (x) = E(g(V (X))|X = x) = P (V > ξ | X = x)
and calculate
V min = E 2 r(X) − E 2 (r † (X)) = λ ξ/2 + λ 2d − λ ξ + λ 2d .
Similar to the normal-normal example, we consider two parametric models of r(x):
(i) Correct model: We use r θ (x) = e θ 0 +θ 1 x (1) +...+θ d x (d) . For least square fitting, the initial parameters are set at θ 0 = . . . = θ d = 0. We set r θ (x) = 1 if r θ (x) > 1.
(ii) Incorrect model: We use the logistic regression model, r θ (x) = 1 + e θ 0 +θ 1 x (1) +...+θ d x (d) −1 , with the initial parameters θ 0 = . . . = θ d = 0 for least square fitting.
As a nonparametric model, we use the kernel regression model r h (x) as in the normalnormal example. Figure 4(a) plots nMSE versus n for d = 1 with respect to the three importance samplers. The behaviors of correct and incorrect parametric importance samplers echo what we see in the normal-normal example. In contrast, the nonparametric importance sampler behaves irregularly (note that nMSE in Figure 4(a) is calculated after discarding the nonparametric estimates exceeding one). Figure 4(b) shows the Tukey box plots of the nonparametric estimates over different n. The interquartile range (i.e., box height) decreases as n increases (note that the interquartile range of estimates may be comparable to the square root of MSE, but not directly to nMSE in Figure 4(a)). The magnitudes of outliers (e.g., estimates greater than one, presented as numbers at the top of Figure 4(b) for each n) suggest that the sampling distribution of the nonparametric estimator might be heavy-tailed for this example.
We attribute the erratic nMSE of nonparametric importance sampler in Figure 4(a) to the severe violation of the assumption, made as a basis of our theoretical analysis in Section 3, that the natural configuration density p has a compact support K ⊂ R d . In this example, the tail of p(x) ∼ exp(−x) decays even more slowly than the tail of p(x) ∼ exp(−x 2 ) in the normal-normal example. The simulation experiment results for d = 2 and 4 are not presented here, as they repeat the same pattern as for d = 1. We note that the natural configuration density p in the case study in Section 4.3 has the compact support, indicating that the assumption is still practical.
Case Study
Wind energy is one of the fastest growing renewable energy sources (You et al., 2017). Yet, harvesting wind energy remains expensive, compared with fossil energy sources such as oil, coal, and natural gas, due to the high capital cost in installing wind turbines. A utility-scale wind turbine whose blade diameter is commonly greater than 100 ft typically costs more than one million U.S. dollars. Therefore, wind energy industry pays the utmost attention on ensuring the structural reliability of the wind turbine to prevent its failure (e.g., Moriarty, 2008;Graf et al., 2017). Figure 4: For the exponential-exponential data generating model, we compare the three importance samplers in terms of nMSE against n for d = 1 in (a), where an error bar represents the 95% confidence interval based on the Monte Carlo error with 10,000 replications. In (b), the Tukey box plots are drawn based on the 10,000 nonparametric estimates for each n. The ends of the whiskers represent the most extreme data points which are not exceeding 1.5 times the interquartile range from the box.
q q q q V min V min V min V min V min V min V min V min V min V min V min V
At the design stage of wind turbine, evaluating its reliability based on physical experiments is very limited due to the associated costs. Alternatively, the international standard, IEC 61400-1 (International Electrotechnical Commission, 2005), requires wind turbine manufacturers to use stochastic simulation models. For this purpose, the most widely used simulation models in the U.S. include TurbSim (Jonkman, 2009) and FAST (Jonkman and Buhl Jr., 2005), which are developed and maintained by the National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy. TurbSim simulates a 3-dimensional time marching wind profile, which becomes an input to FAST that, in turn, simulates a wind turbine's structural response to the wind. This case study focuses on two types of bending moments at the root of a turbine blade, namely, edgewise and flapwise bending moments, which represent two perpendicular structural responses of the blade root due to an external force or moment causing it to bend. We use the same benchmark turbine model ) and setup as Moriarty (2008) and Choe et al. (2015).
In this case study, the input configuration X is a 10-min average wind speed (unit:
meter per second, m/s), which is fed into TurbSim. X is sampled from the truncated Rayleigh distribution with the support of [3,25] and the scale parameter 10 2/π.
The simulation output of interest, V , is the 10-min maximum bending moment (unit: killonewton meter, kNm) at the blade root, which is produced by FAST based on the simulated wind from TurbSim. Because V is random even for a fixed X due to the randomness of wind profile generated in TurbSim, we regard TurbSim and FAST together as a black-box stochastic simulation model.
To compare the nonparametric importance sampler proposed in this paper with a parametric importance sampler, we take a parametric method in Choe et al. (2015) as a benchmark, which also approximates the optimal sampling density q * (x) ∝ r(x) · p(x) in Lemma 1, and use the same simulation experiment setup therein: For the edgewise bending moment, we use n = 3600, m = 600, and ξ = 9300 kNm; for the flapwise bending moment, we use n = 9600, m = 600, and ξ = 14300 kNm.
The parametric model is built as follows: As a pilot sample, X is sampled m = 600 times from a uniform distribution with the support of [3,25], and the corresponding V 's are generated from the NREL simulators. To build a model of r(x), the generalized additive model for location, scale and shape (GAMLSS) (Rigby and Stasinopoulos, 2005) is fitted to the pilot sample. Specifically, the GAMLSS model assumes that the conditional distribution of V given X is a generalized extreme value distribution whose location and scale parameters are cubic spline functions of X while the shape parameter is constant over X. The model parameters are estimated using the backfitting algorithm (Rigby and Stasinopoulos, 2005).
We implement the nonparametric importance sampler as in the numerical study in Section 4.2: To model r(x) based on the pilot sample, we fit the kernel regression model with the Gaussian kernel and choose the smoothing bandwidth by crossvalidation.
For both parametric and nonparametric importance samplers, we repeat estimating the failure probability E = P (V > ξ) 50 times (in contrast to 10,000 times in the numerical study in Section 4.2). Recall that running the NREL simulators once takes about 1-min wall-clock time, implying that obtaining the pilot sample of size m = 600 takes roughly 10 hours. We use the same pilot sample in all 50 replications, as in Choe et al. (2015), because repeating 50 times of the simulation experiment with n − m = 3000 or 9000 itself requires several days of computation even if we use high-performance computing described in Section 4.2.
The parametric importance sampler in Choe et al. (2015) uses the failure probability estimatorẼ
θm = 1 n − m n i=m+1 g(V i ) p(X i ) q * θm (X i ) ,(13)
where the pilot sample is not used, compared with the estimator in (5). In their procedure, the pilot sample is only used to build the model of r(x). For fair comparison, we report both estimation results with and without using the pilot sample in the estimator. The computational (comp.) saving is (n (CM C) − n)/n (CM C) , where n (CM C) = P (1 − P )/(Standard error) 2 is the theoretical sample size required for CMC to attain the same standard error when the true failure probability is equal to P , which is the sample mean of the parametric importance samplers using the pilot sample. The 95% bootstrap confidence interval (CI) is constructed based on 100,000 bootstrap replicates. Table 2 shows the estimation results for flapwise bending moments, which convey the similar message with the results in Table 1. Note that ξ is set to roughly yield the similar failure probability E = P (V > ξ) of 0.01 for both structural load types, because the magnitude of E tends to impact the computational saving (Choe et al., 2015). Yet, we see that the computational saving of importance sampling over CMC for flapwise bending moments is, albeit substantial, not as large as that for edgewise bending moment, as shown and explained in Choe et al. (2015), namely due to the fact that the natural configuration density p is not very different from the optimal sampling density q * for flapwise bending moments so that the benefit of changing the sampling density is not enormous.
Discussion
We consider the problem of estimating the average output from a stochastic simulation model and propose two-stage estimators using either a parametric approach or a nonparametric approach. Theoretically, both estimators satisfy the oracle inequalities but they achieve the oracle variance asymptotically under different rates and assumptions. As expected, the parametric approach needs a strong assumption but its variance converges to the oracle variance faster than the nonparametric approach.
The nonparametric approach, however, requires weak assumptions but the variance reduction rate is not as fast as the parametric approach. Empirically, our numerical study confirmed the theoretical results and our case study indicated that the proposed importance samplers perform well in practice, saving 50%-95% computational resources over a standard Monte Carlo estimator.
In what follows we discuss possible future research directions.
• Manifold support case. In reality, the dimension of the configuration d can be large but the support of p may be concentrated around a low dimensional manifold. In this case, the nonparametric importance sampling in Section 2.2 may not work well because d is large. However, if the dimension of the manifold is low, fast convergence rate of a nonparametric estimator is possible (Balakrishnan et al., 2013;Chen, 2016) so we may be able to design a modified nonparametric importance sampling procedure that achieves oracle variance much faster than the rate in Corollary 8. The construction of such a procedure is left as a future work.
• Multi-stage sampling. In this paper we only consider splitting the computational budget into two stages. We can generalize this idea into a k-stage sampling procedure, where at each stage, we use samples in all previous stages to design our estimator and sampler for the current stage (e.g., Choe, 2017). In this case, the allocation problem becomes more complicated since we may assign different sample sizes to different stages. Also, the number of stage k will be another quantity that we want to optimize. Because the two-stage approach is a special case of a multi-stage sampling procedure, the latter will have a higher variance reduction rate than the proposed methods in this paper.
• Confidence interval. In the current paper, we focus on the construction of an estimator of E. In practice, we often report not only a point estimator but also a confidence interval attached to it. Here we briefly describe two potential methods of constructing the confidence interval. The first method is to derive asymptotic normality of E and then find a consistent variance estimator. Note that this is a non-trivial task because when we use a two-stage approach, the observations are no longer IID. Moreover, estimating the variance could be another challenging task. The other approach is to use the bootstrap (Efron, 1982(Efron, , 1992 to obtain a confidence interval. If we choose to use the bootstrap, we need to prove the validity of such a bootstrap procedure.
A Proofs
Proof of Lemma 1. Now by the following variance formula:
Var(Y ) = E(Var(Y |X)) + Var(E(Y |X)) and choose Y = g(V i ) p(X i ) q(X i ) and X = X i , we have Var g(V i ) p(X i ) q(X i ) = E Var (g(V i )|X i ) p 2 (X i ) q 2 (X i ) + Var r † (X i ) p(X i ) q(X i ) = E r(X i ) − r †2 (X i ) p 2 (X i ) q 2 (X i ) + E r †2 (X i ) p 2 (X i ) q 2 (X i ) − E 2 r † (X i ) p(X i ) q(X i ) = E r(X i ) p 2 (X i ) q 2 (X i ) − E 2 (r † (X * )),(14)
when q(x) = 0 implies r † (x)p(x) = 0. Note that X * is from density p. Thus, the sampling density q affects the variance only via the quantity E r(X i ) p 2 (X i ) q 2 (X i ) . The quantity E r(X i ) p 2 (X i ) q 2 (X i ) has a lower bound from the Cauchy-Schwarz inequality:
E r(X i ) p 2 (X i ) q 2 (X i ) = r(x) p 2 (x) q(x) dx = r(x) p(x) q(x) 2 dx · ( q(x)) 2 dx =1 ≥ r(x)p(x)dx 2 = E 2 r(X * ) .(15)
And the equality holds when r(x) p(x) √ q(x) ∝ q(x), which implies the optimal sampling density is q * (x) ∝ r(x) · p(x).
Thus, when we choose the sampling density to be q * , by equation (14) and (15), the variance
Var E q * = 1 n E r(X i ) p 2 (X i ) q 2 (X i ) − E 2 (r † (X * )) = 1 n E 2 r(X * ) − E 2 r † (X * ) .
Proof of Theorem 2. By assumptions (P1-2) and the M-estimation theory (van der Vaart and Wellner, 1996;van der Vaart, 2000),
θ m − θ 0 = O P 1 m ,
where θ 0 is the parameter such that r(x) = r θ 0 (x). For our estimator E θm , we decompose it into two parts
E θm = A m + B n , where A m = 1 n m i=1 g(V i ) p(X i ) q 0 (X i ) , B n = 1 n n i=m+1 g(V i ) p(X i ) q * θm (X i ) ,
Thus,
Var E θm = Var(A m + B n ) = Var(A m ) + Var(B n ) + Cov(A m , B n ).
Note that
E(A m ) = m n E X i ∼q 0 g(V i ) p(X i ) q 0 (X i ) = m n E E(B n ) = n − m n · E X i ∼ q * θm g(V i ) p(X i ) q * θm (X i ) = n − m n · E X i ∼ q * θm E g(V i ) p(X i ) q * θm (X i ) | q * θm = n − m n · E.(17)
We first bound the covariance. Let D m = {(X 1 , V 1 ), · · · , (X m , V m )} be the collection of the first part of the data. Then
Cov(A m , B n ) = E(A m B n ) − E(A m )E(B n ) = E(A m E(B n |D m )) − (n − m) · m n 2 · E 2 = E(A m E(B n | q * θm )) − (n − m) · m n 2 · E 2 = E A m · n − m n · E − (n − m) · m n 2 · E 2 = (n − m) · m n 2 · E 2 − (n − m) · m n 2 · E 2 = 0.(18)
Therefore, we only need to focus on the variance of each part.
Let V min = E 2 r(X * ) − E 2 (r † (X * )) be the minimal variance under the optimal sampling density. By Lemma 1,
Var(A m ) = m n 2 E r(X i ) p 2 (X i ) q 2 0 (X i ) − E 2 (r † (X * )) = m n 2 V min + m n 2 E r(X i ) p 2 (X i ) q 2 0 (X i ) − E 2 r(X * ) = m n 2 V min + m n 2 V q 0 .(19)
And the variance of the second part is
Var(B n ) = E (Var(B n |D m )) + Var (E(B n |D m )) = E (Var(B n |D m )) + Var (E)
=0 = n − m n 2 · E E r(X i ) p 2 (X i ) q 2 θm (X i ) | q θm − E 2 (r † (X * )) = n − m n 2 · E E r(X * ) p(X * ) q θm (X * ) | q θm − n − m n 2 · E 2 (r † (X * )).(20)
So the key part is in the quantity E E r(X * ) p(X * ) q θm (X * ) | q θm . By Theorem 2 we have
E r(X * ) p(X * ) q θm (X * ) | q θm = r(x) p 2 (x) q θm (x) dx = r(x) p 2 (x) q * (x) + ∆ m · p(x) dx = r(x) p 2 (x) q * (x) dx + O (∆ m ) = E 2 r(X * ) + O P 1 m .(21)1 mh d m i=1 K x−X i h = q m (x)
, which is the difference between the kernel density estimator (KDE) q m (x) and q 0 (x). By assumption (N2)
For the second term, it equals to
sup x∈K r h (x) − r h (x) = sup x∈K 1 mh d m i=1 Y i K x−X i h q 0 (x) − E 1 h d Y i K x−X i h q 0 (x) = sup x∈K 1 q 0 (x) 1 mh d m i=1 Y i K x − X i h − E 1 h d Y i K x − X i h .
Now using Theorem 2.3 in Giné and Guillou (2002) and assumption (N1) and (K1-2), we can bound
sup x∈K 1 mh d m i=1 Y i K x − X i h − E 1 h d Y i K x − X i h = O P log m mh d .
Assumption (N2) implies that the density q 0 (x) is lower bounded by q min . Thus, we obtain the bound
sup x∈K r h (x) − r h (x) ≤ 1 q min sup x∈K 1 mh d m i=1 Y i K x − X i h − E 1 h d Y i K x − X i h = O P log m mh d .(26)
The third term sup x∈K r h (x)−r(x) involves the bias in nonparametric regression which is known to be at rate O(h 2 ) under assumption (N2). Based on this rate and equations (25) and (26), by equation (24) we obtain
sup x∈K r h (x) − r(x) = O(h 2 ) + O P log m mh d ,
which is the desired result.
Proof of Theorem 7. This proof follows the same way as the proof of Theorem 3: we decompose E h * = A m + B n and control the variance of A m and B n and
show that the covariance is 0.
The only difference is in the variance of B n . Because the estimation error now becomes (see equation (9) , which proves the desired result.
Figure 2
2summarizes the procedure of nonparametric importance sampling. There are two tuning parameters we need to select: the smoothing bandwidth h and the allocation size m. The smoothing bandwidth can be chosen by either cross-validation or a reference rule. In Section 3.3, we will derive the optimal rate for the smoothing bandwidth h = O
Corollary 8 .
8Assume (N1-2), (K1-2). When m n log n d+4 d+6 , and h * log m m 1 d+4 , the variance of the estimator E h * is
are just slightly slower than the rate of the parametric importance sampler under correct model (the rate is O n
Corollary 8 for nonparametric importance sampler. In the range of experiments we present below, any choice of multiplicative constant between two and six results in similar empirical performances of the importance samplers.
θ 0 =
0. . . = θ d = 1, in the implementation.(ii) Incorrect model: We use the logistic regression model r θ (x) = 1 + e θ 0 +θ 1x (1) +...+θ d x (d) −1 such that r θ (x) = r(x) for all θ ∈ Θ.For least square fitting, the initial parameters are set at θ 0 = . . . = θ d = 0. As a nonparametric model of r(x), we use the kernel regression model r h (x) with the Gaussian kernel and the smoothing bandwidth h chosen by cross-validation. Figures 3(a)-3(c) show how nMSE varies as n increases for d = 1, 2, 4. For the correct parametric model, nVar E θm is predicted to be V min 1 + O n − 1 3
Figure 3 :
3For the normal-normal data generating model, we compare the three importance samplers in terms of their scaled estimation error, nMSE, against the total sample size n for the input dimension d = 1 in (a), d = 2 in (b), and d = 4 in (c). While fixing n = 8000, d is varied in (d). An error bar represents the 95% confidence interval based on the Monte Carlo error with 10,000 replications.
Figure 3 (
3d) shows nMSE against d for fixed n = 8000. Regardless of d, nMSE of the correct parametric importance sampler stays close to V min . In contrast, nMSE for the incorrect parametric importance sampler essentially remains the same as d varies in this example, although this observation cannot be taken as a general pattern because the input configuration dimension d impacts how incorrect the model is. While the nonparametric importance sampler performs almost as well as the correct parametric importance sampler when d = 1, the performance gap widens as d increases since n is fixed.
nMSE (calculated after removing nonparametric estimates greater than one) vs.
θm (x) − r(x) = sup x∈K r θm (x) − r θ 0 (x) ≤ sup x∈K ( θ m − θ 0 )
θm (x) − q * (x) = ∆ m · p(x), where ∆ m = O P 1 m . Thus,
(x) − q 0 (x) = O(h 2 ) + O P log m mh d ;see, e.g., Lemma 5 inChen et al. (2015a) and Lemma 10 inChen et al. (x) −r h (x) = O(h 2 ) + O P log m mh d .
.
X * ) − E 2 (r † (X * )Thus, the total variance isVar E h * = Var(A m ) + Var(B n ) + Cov(A m , B n )
The error in Theorem 6 can be decomposed into two parts: the bias part O(h 2 ) and the stochastic variation O P (which is related to the variance). In many nonparametric studies, similar bounds appear for density estimation; see, e.g.,Giné and Guillou (2002);Einmahl and Mason (2005);Genovese et al. (2014); Chen et al.log m
mh d
(2015b, 2016).
By Theorem 6, the optimal bandwidth h *
log m
m
1
d+4 leads to the optimal error
rate
Table 1
1summarizes the simulation experiment results for edgewise bending moments. We see that the standard errors of parametric importance samplers and those of nonparametric importance samplers are not significantly different. Computational savings of both methods against CMC are remarkable and, at the same time, comparable with each other.
Table 1 :
1Estimation of the failure probability E = P (V > ξ) for the edgewise bending
moment V and the threshold ξ = 9300 kNm
Without the pilot sample
With the pilot sample
Sample
Standard error
Comp.
Sample
Standard error
Comp.
Method
mean (95% bootstrap CI) saving
mean (95% bootstrap CI) saving
Parametric
0.01005
0.00044
93%
0.01016
0.00036
95%
(0.00036, 0.00051)
(0.00030, 0.00042)
Nonparametric 0.00998
0.00046
92%
0.01010
0.00038
95%
(0.00034, 0.00056)
(0.00029, 0.00047)
Note:
Table 2 :
2Estimation of the failure probability E = P (V > ξ) for the flapwise bending moment V and the threshold ξ = 14300 kNm Note: Refer to the note ofTable 1.Without the pilot sample
With the pilot sample
Sample
Standard error
Comp.
Sample
Standard error
Comp.
Method
mean (95% bootstrap CI) saving
mean (95% bootstrap CI) saving
Parametric
0.01037
0.00063
64%
0.01079
0.00059
69%
(0.00046, 0.00078)
(0.00043, 0.00073)
Nonparametric 0.01061
0.00075
49%
0.01101
0.00070
56%
(0.00057, 0.00090)
(0.00053, 0.00084)
P. Zhang. Nonparametric importance sampling. Journal of the American StatisticalAssociation, 91(435):1245Association, 91(435): -1253Association, 91(435): , 1996
Putting this back to equation(20), we obtainm because the random quantity in the O P term is from ∆ m and is from the difference | θ m − θ 0 |, which is the absolute value of an asymptotic normal distribution so the expectation of the O P leads to the same convergence rate.Now putting altogether, we obtainProof of Theorem 6. Recall that r h (. Define the following two quantitiesWe can bound the difference sup x∈K r h (x) − r(x) byNow we separately bound each term.The first term sup x∈K r h (x) −r h (x) involves the difference between q 0 (x) and
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| []
|
[
"A CHARACTERIZATION OF SPARSE NONSTATIONARY GABOR EXPANSIONS",
"A CHARACTERIZATION OF SPARSE NONSTATIONARY GABOR EXPANSIONS"
]
| [
"Emil Solsbaek ",
"Morten Nielsen "
]
| []
| []
| We investigate the problem of constructing sparse time-frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames in the framework of decomposition spaces. Given a painless nonstationary Gabor frame, we construct a compatible decomposition space and prove that the nonstationary Gabor frame forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space.2010 Mathematics Subject Classification. 42B35, 42C15, 41A17. | 10.1007/s00041-017-9546-6 | [
"https://arxiv.org/pdf/1606.08647v2.pdf"
]
| 119,339,844 | 1606.08647 | bbf86f4f7c18a8548e8578e26b66e5362ab3b21e |
A CHARACTERIZATION OF SPARSE NONSTATIONARY GABOR EXPANSIONS
19 Apr 2017
Emil Solsbaek
Morten Nielsen
A CHARACTERIZATION OF SPARSE NONSTATIONARY GABOR EXPANSIONS
19 Apr 2017arXiv:1606.08647v2 [math.FA]
We investigate the problem of constructing sparse time-frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames in the framework of decomposition spaces. Given a painless nonstationary Gabor frame, we construct a compatible decomposition space and prove that the nonstationary Gabor frame forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space.2010 Mathematics Subject Classification. 42B35, 42C15, 41A17.
Introduction
Redundant Gabor frames play an essential role in time-frequency analysis as these frames provide expansions with good time-frequency resolution [6,28]. Gabor frames are based on translation and modulation of a single window function according to lattice parameters which largely determine the redundancy of the frame. By varying the support of the window function one can change the overall resolution of the frame, but it is in general not possible to change the resolution in specific regions of the time-frequency plane. For signals with varying time-frequency characteristics, a fixed resolution is often undesirable. To overcome this problem, the usage of multi-window Gabor frames has been proposed [10,11,35,43]. As opposed to standard Gabor frames, multi-window Gabor frames use a whole catalogue of window functions of different shapes and sizes to create adaptive representations. A recent example is the nonstationary Gabor frames (NSGF's) which have shown great potential in capturing the essential time-frequency information of music signals [1,12,33,34]. These frames use different window functions along either the timeor the frequency axes and guarantee perfect reconstruction and an FFT-based implementation in the painless case. Originally, NSGF's were studied by Hernández, Labate & Weiss [31] and later by Ron & Shen [41] who named them generalized shift-invariant systems. We choose to work with the terminology introduced in [1] as we will only consider frames in the painless case for which several practical implementations have been constructed under the name of NSGF's [1,12,33]. We consider painless NSGF's with flexible frequency resolution, corresponding to a sampling grid in the time-frequency plane which is irregular over frequency but regular over time at each fixed frequency position. This construction is particularly useful in connection with music signals since the NSGF can be set to coincide with the semitones used in Western music. Based on the nature of musical tones [9,39], we expect music signals to permit sparse expansions relative to the redundant NSGF dictionaries.
The main contribution of this paper is a theoretical characterization of the signals with sparse expansions relative to the NSGF dictionaries. By a sparse expansion we mean an expansion for which the original signal can be approximated at a certain rate by thresholding the expansion coefficients. To prove such a characterization, we follow the approach in [23,24,29] and search for a smoothness space compatible with the structure of the frame. Classical smoothness spaces such as modulation spaces [17] or Besov spaces [42] cannot be expected to be linked with sparse expansions relative to the NSGF dictionaries since these smoothness spaces are not compatible with the flexible frequency resolution of the NSGF's. Modulation spaces correspond to a uniform partition of the frequency domain while Besov spaces correspond to a dyadic partition. Therefore, we study NSGF's in the framework of decomposition spaces. Decomposition spaces were introduced by Feichtinger & Gröbner in [18], and further studied by Feichtinger in [16], and form a large class of function spaces on R d including smoothness spaces such as modulation spaces, Besov spaces, and the intermediate α−modulation spaces as special cases [2,3,26]. We construct the decomposition spaces using structured coverings, as introduced by Borup & Nielsen in [3], which leads to a partition of the frequency domain obtained by applying invertible affine transformations {A k (·) + c k } k∈N on a fixed set Q ⊂ R d .
Given a painless NSGF, we provide a method for constructing a compatible structured covering and the associated decomposition space. We then show that the NSGF forms a Banach frame for the decomposition space and prove that signals belong to the decomposition space if and only if they permit sparse frame expansions. Based on the sparse expansions, we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space. All these results are based on the characterization given in Theorem 5.1 which is the main contribution of this article. This theorem yields the existence of constants 0 < C 1 , C 2 < ∞ such that all signals f , belonging to the decomposition space D(Q, L p , ℓ q ω s ), satisfy
C 1 f D(Q,L p ,ℓ q ω s ) ≤ f, h p T,n T,n d(Q,ℓ p ,ℓ q ω s ) ≤ C 2 f D(Q,L p ,ℓ q ω s ) ,
with {h p T,n } T,n denoting L p −normalized elements from the NSGF and d(Q, ℓ p , ℓ q ω s ) an associated sequence space. In this way we completely characterize the decomposition space using the frame coefficients from the NSGF.
The outline of the article is as follows. In Section 2 we define decomposition spaces based on structured coverings and in Section 3 we define NSGF's in the notation of [1]. We construct the compatible decomposition space in Section 4 and in Section 5 we prove Theorem 5.1. In Section 6 we show that the NSGF forms a Banach frame for the compatible decomposition space and in Section 7 we provide the link to nonlinear approximation theory.
Let us now introduce some of the notation used throughout this article. We let f (ξ) := R d f (x)e −2πix·ξ dx denote the Fourier transform with the usual extension to L 2 (R d ). By F ≍ G we mean that there exist two constants 0 < C 1 , C 2 < ∞ such that C 1 F ≤ G ≤ C 2 F . For two (quasi-)normed vector spaces X and Y , X ֒→ Y means that X ⊂ Y and f Y ≤ C f X for some constant C and all f ∈ X. We say that a non-empty open set Ω ′ ⊂ R d is compactly contained in an open set Ω ⊂ R d if Ω ′ ⊂ Ω and Ω ′ is compact. We denote the matrix norm max{|a ij |} by A ℓ ∞ (R d×d ) and we call {ξ i } i∈I ⊂ R d a δ−separated set if inf j,k∈I,j =k ξ j − ξ k 2 = δ > 0. Finally, by I d we denote the identity operator on R d and by χ Q we denote the indicator function for a set Q ⊂ R d .
Decomposition Spaces
In order to construct decomposition spaces, we first need the notion of a structured covering with an associated bounded admissible partitions of unity (BAPU) as defined in Section 2.1. A BAPU defines a (flexible) partition of the frequency domain corresponding to the structured covering. We use the notation of [3] but with slightly modified definitions for both the structured coverings and the BAPU's.
Structured covering and BAPU.
For an invertible matrix A ∈ GL(R d ), and a constant c ∈ R d , we define the affine transformation
T ξ := Aξ + c, ξ ∈ R d .
For a subset Q ⊂ R d we let Q T := T (Q), and for notational convenience we define |T | := | det(A)|. Given a family T = {A k (·) + c k } k∈N of invertible affine transformations on R d , and a subset Q ⊂ R d , we set Q := {Q T } T ∈T and
(2.1) T := T ′ ∈ T Q T ′ ∩ Q T = ∅ , T ∈ T .
We say that Q is an admissible covering of R d if T ∈T Q T = R d and there exists n 0 ∈ N such that | T | ≤ n 0 for all T ∈ T . We note that the (minimal) number n 0 is the degree of overlap between the sets constituting the covering.
Definition 2.1 (Q−moderate weight). Let Q := {Q T } T ∈T be an admissible cov- ering. A function u : R d → (0, ∞) is called Q−moderate if there exists C > 0 such that u(x) ≤ Cu(y) for all x, y ∈ Q T and all T ∈ T . A Q−moderate weight (derived from u) is a sequence {ω T } T ∈T := {u(ξ T )} T ∈T with ξ T ∈ Q T for all T ∈ T .
For the rest of this article we shall use the explicit choice u(ξ) := 1 + ξ 2 for the function u in Definition 2.1. We now define the concept of a structured covering, first considered in [3]. To ensure that the resulting decomposition spaces are complete, we consider an extended version of the definition given in [3]. (2) There exists K > 0, such that
A −1 k ′ A k ℓ ∞ (R d×d ) ≤ K holds whenever (A k ′ Q + c k ′ ) ∩ (A k Q + c k ) = ∅. (3) There exists K * > 0, such that A −1 k ℓ ∞ (R d×d ) ≤ K * holds for all k ∈ N. (4) There exists a δ−separated set {ξ T } T ∈T ⊂ R d , with ξ T ∈ Q T for all T ∈ T ,
such that {ω T } T ∈T := {u(ξ T )} T ∈T is a Q−moderate weight. (5) There exists γ > 0, such that |Q T | ≤ ω γ T for all T ∈ T . Then we call Q = {Q T } T ∈T a structured covering. Remark 2.1. Definition 2.2(3)-(5) are new additions compared to the definition given in [3] and are necessary for proving Theorem 2.1 page 6. We note that Definition 2.2(2) implies |Q T ′ | ≍ |Q T | uniformly for all T ∈ T and all T ′ ∈ T , and Definition 2.2(3) implies a uniform lower bound on |Q T |.
For a structured covering we have the associated concept of a BAPU, first considered in [3,18]. With a small modification of the proof of [3, Proposition 1] we have the following result.
} T ∈T ⊂ C ∞ c (R d ) satisfying (1) supp(ψ T ) ⊂ Q T for all T ∈ T . (2) T ∈T ψ T (ξ) = 1 for all ξ ∈ R d . (3) sup T ∈T |Q T | 1/p−1 F −1 ψ T L p < ∞ for all 0 < p ≤ 1. (4) For all α ∈ N d 0 , there exists C α > 0 such that |∂ α ψ T (ξ)| ≤ C α χ QT (ξ), for all ξ ∈ R d and all T ∈ T .
We say that {ψ T } T ∈T is a BAPU subordinate to Q.
Remark 2.2. Proposition 2.1(3) is necessary to ensure that the decomposition spaces under consideration will be well-defined for 0 < p < 1. This case is of specific interest since it plays an essential role in connection with nonlinear approximation theory (cf. Section 7). The proof of [3, Proposition 1] is constructive and provides a method for constructing the associated BAPU. Given a structured covering {Q T } T ∈T (with P being compactly contained in Q), the method goes as follows:
(1) Pick a non-negative function Φ ∈ C ∞ c (R d ) with Φ(ξ) = 1 for all ξ ∈ P and supp(Φ) ⊂ Q.
(2) For all T ∈ T , define
ψ T (ξ) = Φ(T −1 ξ) T ′ ∈T Φ(T ′−1 ξ) . (3) Then {ψ T } T ∈T is a BAPU subordinate to Q = {Q T } T ∈T .
In the next section we define decomposition spaces based on structured coverings. For s ∈ R and 0 < q ≤ ∞, we define the associated weighted sequence space ℓ q ω s (T ) as the sequences of complex numbers {a T } T ∈T satisfying
{a T } T ∈T ℓ q ω s := {ω s T a T } T ∈T ℓ q < ∞. Given {a T } T ∈T ∈ ℓ q ω s (T ), we define {a + T } T ∈T by a + T := T ′ ∈ T a T ′ . Since {ω T } T ∈T is Q−moderate, {a T } T ∈T → {a + T } T(2.2) a + T T ∈T ℓ q ω s ≤ C + {a T } T ∈T ℓ q ω s , ∀ {a T } T ∈T ∈ ℓ q ω s (T ).
We will use (2.2) several times throughout this article. Using the notation of [3] we define the Fourier multiplier ψ T (D) by
ψ T (D)f := F −1 (ψ T F f ), f ∈ L 2 (R d ).
Combining
f ∈ L p (R d ) satisfy ψ T (D)f L p ≤ C f L p ,
for all T ∈ T and all 0 < p ≤ ∞. That is, ψ T (D) extends to a bounded operator on the band-limited functions in L p (R d ), uniformly in T ∈ T . Let us now give the definition of decomposition spaces on the Fourier side.
Definition 2.3 (Decomposition space). Let Q = {Q T } T ∈T be a structured cov- ering of R d with corresponding Q−moderate weight {ω T } T ∈T and subordinate BAPU {ψ T } T ∈T . For s ∈ R and 0 < p, q < ∞, we define the decomposition space D(Q, L p , ℓ q ω s ) as the set of distributions f ∈ S ′ (R d ) satisfying f D(Q,L p ,ℓ q ω s ) := { ψ T (D)f L p } T ∈T ℓ q ω s < ∞.
Remark 2.4. According to [18,Theorem 3.7], two different BAPU's yield the same decomposition space with equivalent norms so D(Q, L p , ℓ q ω s ) is in fact well defined and independent of the BAPU. Actually, the results in [18] show that decomposition spaces are invariant under certain geometric modifications of the covering Q, but we will not go into detail here. [15,30,40]. However, Wiener amalgam spaces are based on bounded uniform partitions of unity, which corresponds to a uniform upper bound on the size of the members of the covering. We do not find such an assumption natural in relation to NSGF's (cf. Section 3) and have therefore chosen the more general framework of decomposition spaces.
In Theorem 2.1 below we prove that D(Q, L p , ℓ q ω s ) is in fact a (quasi-)Banach space. Before presenting this result, let us first consider some examples of familiar decomposition spaces. By standard arguments, one can easily show that D(Q, L 2 , ℓ 2 ) = L 2 (R d ) with equivalent norms for any structured covering Q. The next two examples are not as straightforward and demand some structure on the covering. Recall that {ξ T } T ∈T denotes the δ−separated set from Definition 2.2(4).
= E d 2 \ E d 1 . For j ∈ N and k ∈ E define c j,k := 2 j (v(k 1 ), . . . , v(k d )), where v(x) := sgn(x) · 1/2 for x = ±1 3/2 for x = ±2
Let Q ⊂ R d be an open cube with center 0 and side length r > 2. Define T := {I, T j,k } j∈N,k∈E , with T j,k ξ := 2 j ξ + c j,k , and set ξ T j,k := c j,k for all j ∈ N and
k ∈ E. With Q := {Q T } T ∈T then D(Q, L p , ℓ q ω s ) = B s p,q (R d ) for s ∈ R and 0 < p, q < ∞, see [42, Section 2.5.4] for further details. △
Let us now study some important properties of decomposition spaces, in particular completeness.
Theorem 2.1. Given a structured covering Q = {Q T } T ∈T with Q−moderate weight {ω T } T ∈T and subordinate BAPU {ψ T } T ∈T . For s ∈ R and 0 < p, q < ∞, (1) S(R d ) ֒→ D(Q, L p , ℓ q ω s ) ֒→ S ′ (R d ). (2) D(Q, L p , ℓ q ω s ) is a quasi-Banach space (Banach space if 1 ≤ p, q < ∞). (3) S(R d ) is dense in D(Q, L p , ℓ q ω s )
. Remark 2.6. As was pointed out in [22], the definition of decomposition spaces given in [3] cannot guarantee completeness in the general case. However in [4], this problem was fixed by imposing certain weight conditions on the structured covering. Our proof of Theorem 2.1 is based on the approach taken in [4].
In Appendix A we have provided a sketch of the proof for Theorem 2.1. The underlying ideas for the proof are similar to those of [4, Proposition 5.2] and several references are made to results in [4]. However, in [4] the authors considered only coverings made up from open balls and not all arguments carry over to the general case of an arbitrary structured covering.
Nonstationary Gabor Frames
In this section we define nonstationary Gabor frames with flexible frequency resolution using the notation of [1]. Given a set of window functions {h m } m∈Z d in L 2 (R d ), with corresponding time sampling steps a m > 0, for m, n ∈ Z d we define atoms of the form
h m,n (x) := h m (x − na m ), x ∈ R d .
The choice of Z d as index set for m is only a matter of notational convenience; any countable index set would do.
If m,n | f, h m,n | 2 ≍ f 2 2 for all f ∈ L 2 (R d )
, we refer to {h m,n } m,n as a nonstationary Gabor frame (NSGF). For an NSGF {h m,n } m,n , the frame operator
Sf = m,n∈Z d f, h m,n h m,n , f ∈ L 2 (R d ),
is invertible and we have the expansions
f = m,n∈Z d f, h m,n h m,n , f ∈ L 2 (R d ),
with {h m,n } m,n := {S −1 h m,n } m,n being the canonical dual frame of {h m,n } m,n . An NSGF with flexible frequency resolution corresponds to a grid in the time-frequency plane which is irregular over frequency but regular over time at each frequency position. This property allows for adaptive time-frequency representations as opposed to standard Gabor frames. According to [1, Corollary 2], we have the following important result for NSGF's with band-limited window functions.
Theorem 3.1. Let {h m } m∈Z d ⊂ L 2 (R d ) with time sampling steps {a m } m∈Z d , a m > 0 for all m ∈ Z d . Assuming supp(ĥ m ) ⊆ [0, 1 am ] d +b m , with b m ∈ R d for all m ∈ Z d , the frame operator for the system h m,n (x) = h m (x − na m ), ∀m, n ∈ Z d , x ∈ R d ,
is given by
Sf (x) = F −1 m∈Z d 1 a d m ĥ m 2 * f (x), f ∈ L 2 (R d ). The system {h m,n } m,n∈Z d constitutes a frame for L 2 (R d ), with frame-bounds 0 < A ≤ B < ∞, if and only if (3.1) A ≤ m∈Z d 1 a d m ĥ m (ξ) 2 ≤ B, for a.e. ξ ∈ R d ,
and the canonical dual frame is then given by
(3.2)h m,n (x) = F −1 ĥ m l∈Z d 1 a d l ĥ l 2 (x − na m ), x ∈ R d .
Remark 3.1. We note that the canonical dual frame in (3.2) posses the same structure as the original frame, which is a property not shared by general NSGF's. We also note that the canonical tight frame can be obtained by taking the square root of the denominator in (3.2).
Traditionally, an NSGF satisfying the assumptions of Theorem 3.1 is called a painless NSGF, referring to the fact that the frame operator is simply a multiplication operator (in the frequency domain) and therefore easily invertible. This terminology is adopted from the classical painless nonorthogonal expansions [7], which corresponds to the painless case for standard Gabor frames. By slight abuse of notation we use the term "painless" to denote the NSGF's satisfying Definition 3.1 below. In order to properly formulate this definition, we first need some preliminary notation. Let {h m } m∈Z d ⊂ L 2 (R d ) satisfy the assumptions in Theorem 3.1. Given C * > 0 we denote by {I m } m∈Z d the open cubes
(3.3) I m := −ε m , 1 a m + ε m d + b m , m ∈ Z d ,
with ε m := C * /a m for all m ∈ Z d . We note that supp(ĥ m,n ) ⊂ I m for all m, n ∈ Z d . For m ∈ Z d we define
m := m ′ ∈ Z d I m ′ ∩ I m = ∅ ,(1) {ĥ m } m∈Z d ⊂ C ∞ c (R d ) and for β ∈ N d 0 there exists C β > 0, such that sup ξ∈R d ∂ β ξĥ m (ξ) ≤ C β a d/2+|β| m , for all m ∈ Z d .
(2) sup m∈Z d a m := a < ∞. In fact, compact support of the window functions is not a necessary assumption for characterizing modulation spaces [28], Besov spaces [21], or even general decomposition spaces [38]. However, a certain structure of the dual frame is needed and general NSGF's does not provide such structure. We choose to work with the painless case and base our argument on the fact that the dual frame posses the same structure as the original frame. We expect that it is possible to extend the theory developed in this paper to a more general setting by applying existence results for general NSGF's [13,14,32] or generalized shift invariant systems [31,36,37,41]. In particular, the paper [32] by Holighaus seems to provide interesting results in this regard. In this paper, it is shown that for compactly supported window functions, the sampling density in Theorem 3.1 can (under mild assumptions) be relaxed such that the dual frame posses a structure similar to that of the original frame. However, it is outside the scope of this paper to include such results and we will not go into further details.
We now provide a simple example of a set of window functions satisfying Definition 3.1(1).
Example 3.1. Let ϕ ∈ C ∞ c (R d ) \ {0} with supp(ϕ) ⊆ [0, 1] d and for m ∈ Z d definê h m (ξ) := a d/2 m ϕ (a m (ξ − b m )) , ∀ξ ∈ R d , with b m ∈ R d and a m > 0. Then supp(ĥ m ) ⊆ [0, 1 am ] d + b m . Furthermore, with w := a m (ξ − b m ) the chain rule yields ∂ β ξĥ m (ξ) = ∂ β ξ ϕ (w) a d/2+|β| m ≤ C β a d/2+|β| m χ [0, 1 am ] d +bm (ξ), ∀ξ ∈ R d .
This shows Definition 3.1(1). △
In the next section we consider painless NSGF's in the framework of decomposition spaces in order to characterize signals with sparse expansions relative to the NSGF dictionaries.
Decomposition Spaces Based on Nonstationary Gabor Frames
We first provide a method for constructing a structured covering which is compatibly with a given painless NSGF Proof. Define the set
P := C * 2C * + 1 , C * + 1 2C * + 1 d .
By straightforward calculations, it is easy to show that P is compactly contained in Q and P :
= {P T } T ∈T = {(0, 1 am ) d + b m } m∈Z d .
Let us now show that P and Q satisfy the five conditions of Definition 2.2 page 3.
(1) First we show that P covers R d . We note that this immediately implies that Q also covers R d . Assume P does not cover R d , i.e. that there exists some
ξ ′ ∈ R d such that ξ ′ / ∈ (0, 1 am ) d + b m for all m ∈ Z d . Since supp(ĥ m ) ⊆ [0, 1 am ] d + b m ,
andĥ m is continuous, we getĥ m (ξ ′ ) = 0 for all m ∈ Z d . This contradicts the inequality in (3.1) concerning the lower frame bound and thus shows that P covers R d . Now, Definition 3.1(3) is precisely the admissibility condition for Q and thus guarantees that both P and Q are admissible coverings. This shows Definition 2.2(1).
(2) If (A m ′ Q + c m ′ ) ∩ (A m Q + c m ) = ∅, then a m ′ ≍ a m according to Definition 3.1(3). Furthermore, since A −1 m ′ A m is a diagonal matrix and ε m = C * /a m ,
A −1 m ′ A m ℓ ∞ (R d×d ) = a m ′ a m ≤ Ka m a m = K,
for some K > 0, so Definition 2. Since Q is a structured covering, Proposition 2.1 applies and we obtain a BAPU {ψ T } T ∈T subordinate to Q. Given parameters s ∈ R and 0 < p, q < ∞ we may, therefore, construct the associated decomposition space D(Q, L p , ℓ q ω s ). For notational convenience, we change notation and write {h T,n } T ∈T ,n∈Z d , such that supp(ĥ T,n ) ⊂ Q T for all T ∈ T and all n ∈ Z d . Since A T = (2ε T + a −1 T ) · I d , the chain rule and Definition 3.1(1) yield
2(2) is satisfied. (3) To show Definition 2.2(3) we note that
A −1 m ℓ ∞ (R d ×R d ) = a m 2C * + 1 ≤ a 2C * + 1 , ∀m ∈ Z d ,∂ β ξ ĥ T (T ξ) = ∂ β ξĥ T (T ξ) · 2ε T + 1 a T |β| ≤ C β a d/2 T · (2C * + 1) |β| χ QT (T ξ) = C ′ β a d/2 T χ Q (ξ), ∀ξ ∈ R d . (4.1)
Using (4.1) we can prove the following decay property of {h T,n } T,n . Proposition 4.1. For every N ∈ N there exists a constant C N > 0 such that for T = A T (·) + c T ∈ T and n ∈ Z d ,
|h T,n (x)| ≤ C N |T | 1/2 (1 + A T (x − na T ) 2 ) −N , ∀x ∈ R d .
Proof. We will use the fact that
(4.2) u(ξ) N = (1 + ξ 2 ) N ≍ |β|≤N ξ β , ξ ∈ R d , for any N ∈ N with β ∈ N d 0 . Letĝ T (ξ) :=ĥ T (T ξ) such that supp(ĝ T ) ⊂ Q for all T ∈ T . Using (4.2) we get |g T (x)| ≤ C 1 (1 + x 2 ) −N |β|≤N x β g T (x) = C 1 (1 + x 2 ) −N |β|≤N F −1 ∂ β ξĝ T (x) ≤ C 1 (1 + x 2 ) −N |β|≤N R d ∂ β ξĝ T (ξ) dξ, x ∈ R d .
Applying (4.1) we may continue and write
(4.3) |g T (x)| ≤ C 2 a d/2 T (1 + x 2 ) −N |β|≤N R d χ Q (ξ)dξ = C 3 a d/2 T (1 + x 2 ) −N . Now, since ε T = C * /a T , (4.4) |T | |Q| = |Q T | = 2ε T + 1 a T d = (2C * + 1) d (a T ) −d .
Hence, a
d/2 T = C|T | −1/2 so (4.3) yields (4.5) |g T (x)| ≤ C 4 |T | −1/2 (1 + x 2 ) −N , x ∈ R d .
Using the fact that A T is a diagonal matrix, we obtain the relationship
h T (x) = R dĥ T (ξ)e 2πiξ·x dξ = |T | R dĝ T (u)e 2πi(AT u+c)·x du = e 2πic·x |T | R dĝ T (u)e 2πiu·AT x du = e 2πic·x |T | g T (A T x), x ∈ R d . (4.6)
Combining (4.6) and (4.5) we arrive at
|h T,n (x)| = |h T (x − na T )| = |T | |g T (A T (x − na T ))| ≤ C 4 |T | 1/2 (1 + A T (x − na T ) 2 ) −N , x ∈ R d .
This proves the proposition.
As a direct consequence of Proposition 4.1 we can prove the following lemma.
Lemma 4.2. For 0 < p < ∞, we have sup x∈R d {h T,n (x)} n∈Z d ℓ p ≤ C |T | 1/2 , and (4.7) sup n∈Z d h T,n L p ≤ C ′ |T | 1/2−1/p , (4.8) with constants C, C ′ > 0 independent of T ∈ T .
Proof. We will use the fact that
(4.9) R d u(ξ) −m dξ = R d (1 + ξ 2 ) −m dξ < ∞,
for any m > d. Choosing N > d/p in Proposition 4.1, then (4.9) yields
{h T,n (x)} n∈Z d ℓ p ≤ C 1 |T | 1/2 n∈Z d (1 + A T (x − na T ) 2 ) −N p 1/p ≤ C 2 |T | 1/2 (a d T |T |) −1/p .h T,n L p ≤ C 1 |T | 1/2 R d (1 + A T (x − na T ) 2 ) −N p dx 1/p ≤ C 2 |T | 1/2−1/p .
This proves (4.8).
In the next section we use the painless NSGF {h T,n } T,n to prove a complete characterization of the corresponding decomposition space D(Q, L p , ℓ q ω s ).
Characterization of Decomposition Spaces
The main result of this section is the characterization given in Theorem 5.1. To prove this result, we follow the approach taken in [3] where the authors proved a similar result for a certain type of tight frames for R d (see [3,Proposition 3]). Since the frames we consider are not assumed to be tight we need to modify the arguments given in [3]. We start with the following observations. Lemma 5.1. For 0 < p < ∞, the Fourier multiplier
(5.1) ψ h T (D)f := F −1 ψ h T F f := F −1 ψ T l∈ T 1 a d l ĥ l 2 F f
is bounded on the band-limited functions in L p (R d ) uniformly in T ∈ T . Further,
(5.2) sup x∈R d ψ h T (D)h T ′ ,n n∈Z d ℓ p ≤ C |T | 1/2 , T ∈ T , T ′ ∈ T ,
with a constant C > 0 independent of T ∈ T .
Proof. Let ψ h ′ T (ξ) := ψ h T (T (ξ)). For N > d/p, (4.9) and (4.2) imply
F −1 ψ h ′ T L p ≤ C 1 u(·) N F −1 ψ h ′ T L ∞ ≤ C 2 |β|≤N (·) β F −1 ψ h ′ T L ∞ = C 2 |β|≤N F −1 ∂ β ψ h ′ T L ∞ ≤ C 2 |β|≤N ∂ β ψ h ′ T L 1 . (5.3)
Since ε T = C * /a T , the chain rule yields
(5.4) ∂ β ψ h ′ T (ξ) = ∂ β ψ h T (T ξ) 2ε T + 1 a T |β| = Ca −|β| T ∂ β ψ h T (T ξ).
For estimating ∂ β ψ h T we use the quotient rule. Because all derivatives of ψ T are bounded according to Proposition 2.1(4), we need only to consider the derivatives of the denominator of ψ h T . The sum in the denominator consists of at most n 0 terms and for each term in the sum, the chain rule and Definition 3.1(1)- (2) imply an upper bound of Ca |β| T . Therefore, (5.4)
yields |∂ β ψ h ′ T (ξ)| ≤ C ′ χ Q (ξ), since supp(ψ h ′ T ) ⊂ Q for all T ∈ T . Combing this with (5.3) we get F −1 ψ h ′ T L p ≤ C 3 . It now follows from Lemma A.2 page 19 that f → F −1 (ψ h ′ T F f )
defines a bounded operator on the band-limited functions in L p (R d ) uniformly in T ∈ T . Finally, applying [3, Lemma 1] we obtain the same statement for ψ h T (D). We now prove (5.2). Repeating the arguments from the proof of Proposition 4.1 (using (3.1) and Definition 3.1(1)) we can prove the same decay property for ψ h T (D)h T ′ ,n . The result therefore follows from the arguments in the proof of (4.7).
The statement in Theorem 5.1 follows directly once we have proven the following technical lemma. We use the notation ψ T := T ′ ∈ T ψ T ′ .
Lemma 5.2. Given f ∈ S(R d ) and 0 < p < ∞. For all T ∈ T ,
{ f, h T,n } n∈Z d ℓ p ≤ C |T | 1/p−1/2 ψ T (D)f L p , and (5.5) ψ T (D)f L p ≤ C ′ |T | 1/2−1/p T ′ ∈ T { f, h T ′ ,n } n∈Z d ℓ p , (5.6) with constants C 1 , C 2 > 0 independent of T ∈ T .
Proof. The proof of (5.5) follows directly from (4.7) and the arguments for the first part of the proof for [3, Lemma 2]. To prove (5.6) we first assume p ≤ 1 and note
ψ T (D)f L p ≤ C 1 T ′ ∈ T n∈Z d | f, h T ′ ,n | ψ T (D)h T ′ ,n L p (5.7) ≤ C 2 T ′ ∈ T n∈Z d | f, h T ′ ,n | p ψ T (D)h T ′ ,n p L p 1/p ,
with {h T,n } T,n being the dual frame given in (3.2) page 7. Applying (5.1) and (4.8) this proves (5.6) for the case p ≤ 1. For p > 1, we note that Hölder's inequality (with p ′ being the conjugate index of p) yields
n∈Z d f, h T ′ ,n ψ T (D)h T ′ ,n p L p ≤ R d n∈Z d | f, h T ′ ,n | p ψ T (D)h T ′ ,n (x) n ′ ∈Z d ψ T (D)h T ′ ,n ′ (x) p/p ′ dx ≤ C 1 |T | p/2p ′ −1/2 n∈Z d | f, h T ′ ,n | p ,
according to Lemma 5.1 and (4.8). Taking the p'th root on both sides and applying (5.7) finishes the proof of (5.6) for p > 1.
Using the notation of [3] we define L p −normalized atoms h p T,n := |T | 1/2−1/p h T,n , for all T ∈ T , n ∈ Z d and 0 < p < ∞. We also define the coefficient space d(Q, ℓ p , ℓ q ω s ) as the set of coefficients {c T,n } T ∈T ,n∈Z d ⊂ C satisfying
{c T,n } T ∈T ,n∈Z d d(Q,ℓ p ,ℓ q ω s ) := {c T,n } n∈Z d ℓ p T ∈T ℓ q ω s < ∞.
Combining Lemma 5.2 with the fact that S(R d ) is dense in D(Q, L p , ℓ q ω s ) we obtain a characterization similar to that of [3, Proposition 3]. , for all f ∈ D(Q, L p , ℓ q ω s ). Remark 5.1. The characterization in Theorem 5.1 differs from the one given in [3,Proposition 3] in two ways. In [3] the frame elements are obtained directly from the structured covering such that the resulting system forms a tight frame. In our framework we take the "reverse" approach and explicitly state sufficient conditions which guarantee the existence of a compatible decomposition space for a given NSGF (cf. Definition 3.1). More importantly, we show that the assumption on tightness of the frame can be replaced with the structured expression for the dual frame given in (3.2) page 7.
In the next section we use the characterization given in Theorem 5.1 to prove that {h p T,n } T,n forms a Banach frame for D(Q, L p , ℓ q ω s ) with respect to d(Q, ℓ p , ℓ q ω s ) for s ∈ R and 0 < p, q < ∞.
Banach Frames for Decomposition Spaces
Let us start by giving the general definition of a Banach frame [27,28]. Traditionally, Banach frames are only defined for Banach spaces but we will also use the concept for quasi-Banach spaces.
Definition 6.1 (Banach Frame). Let X be a (quasi-)Banach space and let X d be an associated (quasi-)Banach sequence space on N. A Banach frame for X, with respect to X d , is a sequence {y n } n∈N in the dual space X ′ , such that
(1) The coefficient operator C X : f → { f, y n } n∈N is bounded from X into X d . (2) Norm equivalence: f X ≍ { f, y n } n∈N X d , ∀f ∈ X.
(3) There exists a bounded operator R X d from X d onto X, called a reconstruction operator, such that
R X d C X f = R X d ({ f, y n } n∈N ) = f, ∀f ∈ X.
Remark 6.1. We will actually prove that {h p T,n } T,n forms an atomic decomposition [5,19,20] for D(Q, L p , ℓ q ω s ) as the reconstruction operator takes the form f = T,n f, h p T,n x T,n with {x T,n } ⊂ D(Q, L p , ℓ q ω s ) (see Theorem 6.1 below). In order to show that that {h p T,n } T,n forms a Banach frame for D(Q, L p , ℓ q ω s ), we first note that
{h p T,n } T ∈T ,n∈Z d ⊂ S(R d ) ⊂ D ′ (Q, L p ,
ℓ q ω s ) as required by Definition 6.1. Furthermore, the equivalence in Theorem 5.1 implies that Definition 6.1(2) is satisfied and the corresponding proof reveals that Definition 6.1(1) is satisfied. What remains to be shown is the existence of a bounded reconstruction operator such that Definition 6.1(3) holds. For {c T,n } T,n ∈ d(Q, ℓ p , ℓ q ω s ), we define the reconstruction operator as
(6.1) R d(Q,ℓ p ,ℓ q ω s ) {c T,n } T,n = T ∈T ,n∈Z d c T,n |T | 1/p−1/2h T,n ,
with {h T,n } T ∈T ,n∈Z d being the dual frame given in (3.2) page 7. We now provide the main result of this section.
Theorem 6.1. Given s ∈ R and 0 < p, q < ∞, {h p T,n } T ∈T ,n∈Z d forms a Banach frame for D(Q, L p , ℓ q ω s ). Furthermore, we have the expansions
(6.2) f = T ∈T ,n∈Z d f, h T,n h T,n , ∀f ∈ D(Q, L p , ℓ q ω s ),
with unconditional convergence.
Proof. Let R and C denote the reconstruction-and coefficient operator, respectively. We first prove that R is bounded from d(Q, ℓ p , ℓ q ω s ) onto D(Q, L p , ℓ q ω s ). For {c T,n } T,n ∈ d(Q, ℓ p , ℓ q ω s ) we let g := R({c T,n } T,n ). For T ∈ T , Lemma 5.1 implies Applying (2.2) page 5 to (6.3) and then using (6.4) we get
ψ T (D)g L p = F −1 ψ T l∈ T 1 a d l ĥ l 2 · ψ T · T ′ ∈T ,n∈Z d c T ′ ,n |T ′ | 1/p−1/2ĥ T ′ ,n L p ≤ C 1 ψ T (D) T ′ ∈T ,n∈Z d c T ′ ,n |T ′ | 1/p−1/2 h T ′ ,n L p .g D(Q,L p ,ℓ q ω s ) ≤ C 2 T ∈T ,n∈Z d c T,n |T | 1/p−1/2 h T,n D(Q,L p ,ℓ q ω s ) ≤ C 3 {c T,n } T ∈T ,n∈Z d d(Q,ℓ p ,ℓ q ω s ) . (6.5)
This proves that R is bounded from d(Q, ℓ p , ℓ q ω s ) onto D(Q, L p , ℓ q ω s ). Let us now show the unconditional convergence of (6.2). Given f ∈ D(Q, L p , ℓ q ω s ), we can find a sequence {f k } k≥1 , with f k ∈ S(R d ) for all k ≥ 1, such that f k → f in D(Q, L p , ℓ q ω s ) as k → ∞. Furthermore, since {h T,n } T,n forms a frame for L 2 (R d ), for each k ≥ 1 we have the expansion
f k = T ∈T ,n∈Z d f k , h T,n h T,n = RC(f k ), with unconditional convergence. Since RC : D(Q, L p , ℓ q ω s ) → D(Q, L p , ℓ q ω s ) is continuous, letting k → ∞ yields (6.6) f = RC(f ) = T ∈T ,n∈Z d f, h T,n h T,n .
Given ε > 0, (6.5) implies that we can find a finite subset
F 0 ⊂ T × Z d , such that f − (T,n)∈F f, h T,n h T,n D(Q,L p ,ℓ q ω s ) ≤ C { f, h T,n } (T,n) / ∈F d(Q,L p ,ℓ q ω s ) < ε,
for all finite sets F ⊇ F 0 . According to [28,Proposition 5.3.1], this property is equivalent to unconditional convergence.
We close this section by discussing the implications of the achieved results. According to Theorem 5.1 and Theorem 6.1, every f ∈ D(Q, L p , ℓ q ω s ) has an expansion of the form
f = T ∈T ,n∈Z d f, h p T,n |T | 1/p−1/2h T,n , with f D(Q,L p ,ℓ q ω s ) ≍ f, h p T,n T,n d(Q,ℓ p ,ℓ q ω s )
. Now, assume there exists another set of reconstruction coefficients {c T,n } T,n ∈ d(Q, ℓ p , ℓ q ω s ) which is sparser than { f, h p T,n } T,n when sparseness is measured by the d(Q, ℓ p , ℓ q ω s )-norm. Since the reconstruction operator R is bounded we get when sparseness of the coefficients is measured by the d(Q, ℓ p , ℓ q ω s )-norm. Furthermore, f ∈ D(Q, L p , ℓ q ω s ) if and only if f permits a sparse expansion relative to the dictionary {|T | 1/p−1/2 h T,n } T,n .
Application to Nonlinear Approximation Theory
In this section we provide the link to nonlinear approximation theory. An important property of the sparse expansions obtained in Theorem 6.1 is that we can obtain a good compression by simply thresholding the coefficients from the expansion. As mentioned in the introduction, NSGF's can create adaptive time-frequency representations as opposed to standard Gabor frames. Such adaptive representations can be constructed to fit the particular nature of a given signal, thereby producing a more precise (and hopefully sparser) time-frequency representation. In particular, NSGF's have proven to be useful in connection with music signals. For instance, in [1,12] the authors use NSGF's to construct an invertible constant-Q transform with good frequency resolution at the lower frequencies and good time resolution at the higher frequencies. Such a time-frequency resolution is often more natural for music signals than the uniform resolution provided by Gabor frames.
The main result of this section is given in (7.2) below. The corresponding proof follows directly from the results obtained in Sections 5 and 6 together with standard arguments from nonlinear approximation theory [24]. Let f ∈ D(Q, L τ , ℓ τ ω s ), with 0 < τ < ∞, and let 0 < p < ∞ satisfy α := 1/τ − 1/p > 0. Write the frame expansion of f with respect to the L p −normalized coefficients
(7.1) f = T ∈T ,n∈Z d f, h p T,n |T | 1/p−1/2h T,n .
Let {θ m } m∈N be a decreasing rearrangement of the frame coefficients and let f N be the N -term approximation to f obtained by extracting the coefficients in (7.1) corresponding to the N largest coefficients {θ m } N m=1 . Then, we can prove the existence of C > 0 such that for f ∈ D(Q, L τ , ℓ τ ω s ) and N ∈ N,
(7.2) f − f N D(Q,L p ,ℓ p ω s ) ≤ CN −α f D(Q,L τ ,ℓ τ ω s )
. In other words, for f ∈ D(Q, L τ , ℓ τ ω s ) we can obtain good approximations in D(Q, L p , ℓ p ω s ) by thresholding the L p −normalized frame coefficients. We note that for 0 < τ < 2 we obtain good approximations in L 2 (R d ) with respect to the original coefficients { f, h T,n } T,n .
We now explain the obtained results in the general framework of Jackson-and Bernstein inequalities [8]. Let D denote the dictionary {|T | 1/p−1/2 h T,n } T,n and define the nonlinear set of all linear combinations of at most N elements from D as
Σ N (D) := T,n∈∆ c T,n |T | 1/p−1/2h T,n #∆ ≤ N
.
For any f ∈ D(Q, L p , ℓ p ω s ), the error of best N -term approximation to f is
σ N (f, D) := inf h∈ΣN (D) f − h D(Q,L p ,ℓ p ω s ) . Since f N ∈ Σ N (D), (7.2) yields σ N (f, D) ≤ CN −α f D(Q,L τ ,ℓ τ ω s ) .
This is a so-called Jackson inequality for nonlinear N -term approximation with D. It provides us with an upper bound for the error obtained by approximating f with the best possible choice of linear combinations of at most N elements from the dictionary. The converse inequality is called a Bernstein inequality and is in general much more difficult to obtain for redundant systems [25]. The existence of a Bernstein inequality would provide us with a lower bound and hence a full characterization of the error of best N -term approximation to f with respect to the dictionary D. However, for this particular system (and for many other redundant systems), the existence of a Bernstein inequality is still an open question.
f D s p,∞ ≤ Cp N (f ) and f D s 1,1 ≤ C ′ p N ′ (f ),
for sufficiently large N and N ′ . This proves that S(R d ) ֒→ D s p,∞ and S(R d ) ֒→ D s 1,1 . To show that D s p,∞ ֒→ S ′ (R d ) we need to take a different approach than in [4]. Setting ψ T := T ′ ∈ T ψ T ′ , we first note that for f ∈ D s p,∞ and ϕ ∈ S(R d ),
| f, ϕ | ≤ T ∈T ψ T (D)f ψ T (D)ϕ L 1 ≤ T ∈T ψ T (D)f L ∞ ψ T (D)ϕ L 1 .
Using Lemma A.1 below (with g = F −1 {ψ Tf (T ξ)}) we thus get ≤ C 4 f D s p,∞ p N (ϕ), for sufficiently large N since S(R d ) ֒→ D s 1,1 . We conclude that D s p,∞ ֒→ S ′ (R d ) which proves Theorem 2.1 (1).
The proof of Theorem 2.1(2) follows directly from Theorem 2.1(1) and the arguments in [4,Page 150].
To prove Theorem 2.1(3) we let f ∈ D s p,q and choose I ∈ C ∞ c (R d ) with 0 ≤ I(ξ) ≤ 1 and I(ξ) ≡ 1 in a neighbourhood of ξ = 0. Also, we define ( f ) := If and
f ε := F −1 ϕ ε * f ∈ S(R d ),
with ϕ ε (ξ) := ε −d ϕ(ξ/ε) and ϕ being a compactly supported mollifier. Since supp(I) is compact, we may choose a finite subset T * ⊂ T , such that supp(I) ⊂ ∪ T ∈T * Q T and T ∈T * ψ T (ξ) ≡ 1 on supp(I). Using Lemma A.2 below we obtain
f L p = F −1 IF F −1 T ∈T * ψ T ·f L p ≤ C T ∈T * F −1 I Lp ψ T (D)f L p < ∞,= T ∈T c • ω sq T F −1 ψ T f − If q L p ≤ C 1 T ∈T c • ω sq T ψ T (D)f L p + F −1 IF (ψ T (D)f ) L p q ≤ C 2 T ∈T c • ω sq T ψ T (D)f q L p .
Finally, since f ∈ D s p,q we can choose supp(I) large enough, such that f − f D s p,q < ε, for any given ε > 0. This proves Theorem 2.1(3).
In the proof of Theorem 2.1 we used the following two lemmas. A proof of Lemma A.1 can be found in [3, Lemma 3] and a proof of Lemma A.2 can be found in [42, Proposition 1.5.1].
Lemma A.1. Let g ∈ L p (R d ) and supp(ĝ) ⊂ Γ, with Γ ⊂ R d compact. Given an invertible affine transformation T , letĝ T (ξ) :=ĝ(T −1 ξ). Then for 0 < p ≤ q ≤ ∞, g T Lq ≤ C |T | 1/p−1/q g T Lp ,
for a constant C independent of T .
Lemma A.2.
Let Ω and Γ be compact subsets of R d . Let 0 < p ≤ ∞ andp = min{1, p}. Then there exists a constant C such that
F −1 M F f L p ≤ C F −1 M Lp f L p for all f ∈ L p (R d )
with supp(f ) ⊂ Ω and all F −1 M ∈ Lp(R d ) with supp(M ) ⊂ Γ.
Definition 2. 2 (
2Structured covering). Given a family T = {A k (·) + c k } k∈N of invertible affine transformations on R d , suppose there exist two bounded open sets P ⊂ Q ⊂ R d , with P compactly contained in Q, such that (1) {P T } T ∈T and {Q T } T ∈T are admissible coverings.
Proposition 2 . 1 .
21Given a structured covering Q = {Q T } T ∈T , there exists a family of non-negative functions {ψ T
Remark 2 . 3 .
23Proposition 2.1(4) is a new addition compared to [3, Proposition 1] and is necessary for proving Theorem 2.1 page 6. The proof of Proposition 2.1(4) follows easily from the arguments in the proof of [3, Proposition 1] and Definition 2.2(3). Finally, it should be noted that the assumptions in Definition 2.2(4)-(5) are not necessary for proving Proposition 2.1, however, these assumptions are needed for the proof of Theorem 2.1.
2. 2 .
2Definition of decomposition spaces. Given a structured covering Q = {Q T } T ∈T with corresponding Q−moderate weight {ω T } T ∈T and BAPU {ψ T } T ∈T .
Remark 2 . 5 .
25In their most general form, decomposition spaces D(Q, B, Y ) are constructed using a local component B and a global component Y [18]. This construction is similar to the construction of Wiener amalgam spaces W (B, C) with local component B and global component C
Example 2 . 1 (
21Modulation spaces). Let Q ⊂ R d be an open cube with center 0 and side length r > 1. Define T := {T k } k∈Z d , with T k ξ := ξ − k, and set ξ T k := k for all k ∈ Z d . With Q := {Q T } T ∈T then D(Q, L p , ℓ q ω s ) = M s p,q (R d ) for s ∈ R and 0 < p, q < ∞, see [17, Section 4] for further details. △ Example 2.2 (Besov spaces). Let E 2 := {±1, ±2}, E 1 := {±1} and E :
using the notation of (2.1). With this definition, | m| denotes the number of cubes overlapping with I m . Finally, we recall the choice u(ξ) := 1 + ξ 2 for the function u in Definition 2.1.
Definition 3. 1 (
1Painless NSGF). Let {h m } m∈Z d ⊂ S(R d ) satisfy the assumptions in Theorem 3.1 and assume that
( 3 )
3There exists C * > 0 and n 0 ∈ N, such that the open cubes {I m } m∈Z d satisfy | m| ≤ n 0 and a m ′ ≍ a m uniformly for all m ∈ Z d and all m ′ ∈ m. (4) The centerpoints {b m } m∈Z d forms a δ−separated set and the sequence {ω m } m∈Z d := {u(b m )} m∈Z d constitutes a {I m } m∈Z d −moderate weight. (5) There exists γ > 0 such that |I m | ≤ ω γ m for all m ∈ Z d . Then we refer to {h m,n } m,n∈Z d as a painless NSGF.
Remark 3. 2 .
2Definition 3.1(2) implies a uniform lower bound on |I m | and Definition 3.1(4) guarantees a minimum distance between the center of the cubes. Furthermore, Definition 3.1(3) implies that each cube I m has at most n 0 overlap with other cubes and that the side-length of I m is equivalent to the side-length of any overlapping cube.
Remark 3. 3 .
3The assumptions in Definition 3.1 are natural in relation to decomposition spaces and are easily satisfied. However, the support conditions for {ĥ m } m∈Z d given in Theorem 3.1 are rather restrictive and deserves a discussion.
{h m,n } m,n∈Z d ⊂ S(R d ). We recall the definition of ε m = C * /a m used in the construction of {I m } m∈Z d in (3.3). Define Q := (0, 1) d together with the set of affine transformations T := {A m (·) + c m } m∈Z d with A m := 2ε m + 1 a m · I d and (c m ) j := −ε m + (b m ) j , 1 ≤ j ≤ d. Then Q := {Q T } T ∈T = {I m } m∈Z d and, furthermore, we have the following result.
Lemma 4. 1 .
1Q is a structured covering of R d .
(4.4), a d T = C|T | −1 , which inserted into (4.10) yields (4.7). To show (4.8), we again let N > d/p in Proposition 4.1, so (4.9) yields
Theorem 5 . 1 .
51For s ∈ R and 0 < p, q < ∞ we have the equivalencef D(Q,L p ,ℓ q ω s ) ≍ f, h p T,n T ∈T ,n∈Z d d(Q,ℓ p ,ℓ q ω s )
T
∈T ,n∈Z d c T,n |T | 1/p−1/2 h T,n D(Q,L p ,ℓ q ω s ) ≤ C {c T,n } T,n d(Q,ℓ p ,ℓ q ω s ) .
=
C 1 R({c T,n } T,n ) D(Q,L p ,ℓ q ω s ) ≤ C 2 {c T,n } T,n d(Q,ℓ p ,ℓ q ω s ) .We conclude that the canonical coefficients { f, h p T,n } T,n are (up to a constant) the sparsest possible choice for expanding f as f = T ∈T ,n∈Z d c T,n |T | 1/p−1/2h T,n ,
proof is done if we can show that f − f D s p,q can be made arbitrary small by choosing f appropriately. To show this, we define the set T • := {T ∈ T | I(ξ) ≡ 1 on supp(ψ T )}. Denoting the complement T c • , Lemma A.
∈T defines a bounded operator on ℓ q ω s (T ) [18, Remark 2.13 and Lemma 3.2]. Denoting its operator norm by C + , we have
AcknowledgementsWe thank the two anonymous reviewers for their constructive comments on the original manuscript. Their valuable suggestions have helped improve the manuscript considerably.Appendix A. Proof of Theorem 2.1 Proof. To simplify notation we let D s p,q := D(Q, L p , ℓ q ω s ). Let us first prove Theorem 2.1(1). Allowing the extension q = ∞, and repeating the arguments from the proof of [4, Proposition 5.7], we can show that D s+ε p,∞ ֒→ D s p,q ֒→ D s p,∞ , ε > d/q, for any s ∈ R and 0 < p < ∞ using Definition 2.2(4). It therefore suffice to show that S(R d ) ֒→ D s p,∞ ֒→ S ′ (R d ) for any s ∈ R and 0 < p < ∞. For N ∈ N, we define semi-norms on S(R d ) bywith u(ξ) = 1+ ξ 2 as usual. Following the approach in [4, Page 149], and applying Proposition 2.1(4), we get
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| []
|
[
"-a High Amplitude δ Scuti star with peculiar pulsational properties",
"-a High Amplitude δ Scuti star with peculiar pulsational properties"
]
| [
"Ernst Paunzen \nDepartment of Theoretical Physics and Astrophysics\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic\n",
"Klaus Bernhard \nBundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)\nBerlin, CambridgeGermany, USA\n",
"Moriz Frauenberger \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Santiago Helbig \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Andreas Herdin \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Stefan Hümmerich \nBundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)\nBerlin, CambridgeGermany, USA\n",
"Jan Janík \nDepartment of Theoretical Physics and Astrophysics\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic\n",
"Andreas Karnthaler \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Richard Komžík \nAstronomical Institute\nSlovak Academy of Sciences\n059 60Tatranská LomnicaSlovakia\n",
"Beatrice Kulterer \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Hans-Michael Maitzen \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Stefan Meingast \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Sebastian Miksch \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Theodor Pribulla \nAstronomical Institute\nSlovak Academy of Sciences\n059 60Tatranská LomnicaSlovakia\n",
"Monika Rode-Paunzen \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Wolfgang Sakuler \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Carla Schoder \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n",
"Eugene Semenko \nSpecial Astrophysical Observatory of the Russian Academy of Sciences\n369167Nizhnii ArkhyzRussia\n",
"Nikolaus Sulzenauer \nInstitute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria\n"
]
| [
"Department of Theoretical Physics and Astrophysics\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic",
"Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)\nBerlin, CambridgeGermany, USA",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)\nBerlin, CambridgeGermany, USA",
"Department of Theoretical Physics and Astrophysics\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Astronomical Institute\nSlovak Academy of Sciences\n059 60Tatranská LomnicaSlovakia",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Astronomical Institute\nSlovak Academy of Sciences\n059 60Tatranská LomnicaSlovakia",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria",
"Special Astrophysical Observatory of the Russian Academy of Sciences\n369167Nizhnii ArkhyzRussia",
"Institute for Astrophysics\nUniversity of Vienna\nTürkenschanzstrasse 171180ViennaAustria"
]
| []
| In some δ Scuti stars, only one or two radial modes are excited (usually the fundamental mode and/or first overtone mode) and the observed peak-to-peak amplitudes exceed 0.3 mag (V ). These stars are known as High Amplitude Delta Scuti (HADS) variables.We here present a detailed photometric and spectroscopic analysis of the HADS star TYC 3637-1152-1. We have derived a metallicity close to solar, a spectral type of F4 V and an age of log t = 9.1. Employing archival time series data from different sources, two frequencies f 0 = 10.034 c/d and f 1 = 12.681 c/d and their harmonics and linear combinations were identified. The period ratio of f 0 /f 1 = 0.791 puts this star into a peculiar position in the Petersen diagram, from which we conclude that TYC 3637-1152-1 is a unique object with peculiar pulsational properties that indicate a transitional state between HADS stars pulsating in the fundamental and first overtone modes and stars pulsating in higher overtones. | 10.1016/j.newast.2018.10.001 | [
"https://arxiv.org/pdf/1811.05219v1.pdf"
]
| 119,516,617 | 1811.05219 | b7d925781a5b660abf738bf15ec34fa8fd094422 |
-a High Amplitude δ Scuti star with peculiar pulsational properties
13 Nov 2018
Ernst Paunzen
Department of Theoretical Physics and Astrophysics
Masaryk University
Kotlářská 2611 37BrnoCzech Republic
Klaus Bernhard
Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)
Berlin, CambridgeGermany, USA
Moriz Frauenberger
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Santiago Helbig
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Andreas Herdin
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Stefan Hümmerich
Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV)
Berlin, CambridgeGermany, USA
Jan Janík
Department of Theoretical Physics and Astrophysics
Masaryk University
Kotlářská 2611 37BrnoCzech Republic
Andreas Karnthaler
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Richard Komžík
Astronomical Institute
Slovak Academy of Sciences
059 60Tatranská LomnicaSlovakia
Beatrice Kulterer
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Hans-Michael Maitzen
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Stefan Meingast
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Sebastian Miksch
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Theodor Pribulla
Astronomical Institute
Slovak Academy of Sciences
059 60Tatranská LomnicaSlovakia
Monika Rode-Paunzen
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Wolfgang Sakuler
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Carla Schoder
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
Eugene Semenko
Special Astrophysical Observatory of the Russian Academy of Sciences
369167Nizhnii ArkhyzRussia
Nikolaus Sulzenauer
Institute for Astrophysics
University of Vienna
Türkenschanzstrasse 171180ViennaAustria
-a High Amplitude δ Scuti star with peculiar pulsational properties
13 Nov 2018stars: variables: delta Scuti; stars: individual: TYC 3637-1152-1
In some δ Scuti stars, only one or two radial modes are excited (usually the fundamental mode and/or first overtone mode) and the observed peak-to-peak amplitudes exceed 0.3 mag (V ). These stars are known as High Amplitude Delta Scuti (HADS) variables.We here present a detailed photometric and spectroscopic analysis of the HADS star TYC 3637-1152-1. We have derived a metallicity close to solar, a spectral type of F4 V and an age of log t = 9.1. Employing archival time series data from different sources, two frequencies f 0 = 10.034 c/d and f 1 = 12.681 c/d and their harmonics and linear combinations were identified. The period ratio of f 0 /f 1 = 0.791 puts this star into a peculiar position in the Petersen diagram, from which we conclude that TYC 3637-1152-1 is a unique object with peculiar pulsational properties that indicate a transitional state between HADS stars pulsating in the fundamental and first overtone modes and stars pulsating in higher overtones.
Introduction
The multiperiodic δ Scuti (DSCT) stars have long been established as a class of variable stars (Fath 1935). They usually belong to luminosity classes V to III and are located inside the classical instability strip between spectral types A0 to F5 (Breger & Pamyatnykh 1998). δ Scuti stars show variability on timescales between about 15 minutes and 5 hours (Holdsworth et al. 2014), which are caused by multiple radial and non-radial low-order pressure (p) modes that are excited by the κ mechanism (e.g. Breger 2000).
Several open questions about the nature of δ Scuti stars remain, e.g. their relationship to the class of γ Doradus variables. The theoretical instability regions of both classes on the Hertzsprung-Russell diagram (HRD) overlap (Dupret et al. 2004), and hybrid pulsators exhibiting both p and gravity (g) modes are encountered (e.g. Uytterhoeven et al. 2011). On the basis of space-based data, Bowman et al. (2016) discussed the diversity in the pulsational behaviour of δ Scuti stars, describing several subgroups with 1) constant amplitudes and phases; 2) amplitude modulation caused by the beating of close-frequency pulsation modes; 3) pure amplitude modulation (with no associated phase variation); 4) phase modulation caused by binarity; and 5) amplitude modulation caused by nonlinearity.
As their name implies, the High-amplitude δ Scuti (HADS) stars are set apart by the large amplitudes of their light variations (A V > 0.3 mag) (McNamara 2000). They usually exhibit one or two stable frequencies (Rodriguez 2004) associated with radial pulsation in fundamental or loworder overtone modes (Petersen & Christensen-Dalsgaard 1999). Furthermore, they are usually slow rotators, which seems to be a prerequisite for the observed high-amplitude pulsation (Breger 2000). Evidence of period changes in a significant fraction of HADS stars were presented by Breger & Pamyatnykh (1998), who found continuous period decreases and increases, period jumps and even cyclic period variations due to a possible binary light-time effect. However, none of these effects could be correlated with the evolutionary status (i.e. age, life time on the mainsequence, and mass) of the stars (Breger & Pamyatnykh 1998).
SX Phoenicis (SX Phe) stars are the Population II counterparts of the δ Scuti variables (Eggen & Iben 1989) and present phenomenologically similar light curves. Field Table 1: Observed quantities and astrophysical parameters of TYC 3637-1152-1. Kinematic data and radial velocity information were taken from Gaia DR2.
Parameter
Value RA (J2000) 23h 30m 37s.26
Dec (J2000) +46 • 24' 04".3 l (deg) 108.7086 b (deg) −14.2266 D (pc) 525(15) Z (pc) −129(15) V (mag) 10.398(47) E(B − V ) (mag) 0.10(2) T eff (K) 6 800(250) M bol (mag) 1.46(8) π (mas) 1.905(54) µ α cos δ (mas yr −1 ) −6.998(74) µ δ (mas yr −1 ) +15.697(68) υ r (km s −1 ) −21.73(4.76) U (km s −1 ) −16 V (km s −1 ) +5 W (km s −1 ) +53 T (km s −1 ) 56
SX Phe stars generally exhibit the kinematic properties of halo (or thick disk) stars, asymmetric and large-amplitude light curves, and low metallicities (McNamara 1995). From studies of globular clusters, it was found that SX Phe stars have pulsation amplitudes A V > 0.1 mag, and simple light curves consistent with radial pulsation and one or two dominant frequencies. However, as more accurate photometric data became available, SX Phe variables with much lower amplitudes have also been discovered (Cohen & Sarajedini 2012). Current pulsation models imply that the classical κ mechanism in the He II partial ionization zone is the corresponding driving mechanism in these stars (Fiorentino et al. 2014). For both HADS and SX Phe stars, period-luminosity (PL) relations have been established (Petersen & Christensen-Dalsgaard 1999;Cohen & Sarajedini 2012) which makes them interesting targets for distance estimations. Here, we present a detailed photometric and spectroscopic study of TYC 3637-1152-1, a neglected HADS variable, investigating its location in the HRD and its pulsation characteristics.
Target star and its astrophysical parameters
The variability of TYC 3637-1152-1 (NSVS 3564994, V0670 And) was first reported by Dimitrov & Popov (2007), who used Northern Sky Variability Survey (NSVS) data to search for variable stars in the Andromeda constellation. The star was classified as a δ Scuti variable with a period of 0.09966 d and an amplitude of 0.29 mag. 1 Since then, no further studies of TYC 3637-1152-1 have been published.
As an initial step, we determined our target star's kinematic properties and location in the HRD to decide whether it is a Population I (HADS) or II (SX Phe) star. The observed quantities and the final results are listed in Table 1.
Data Release 2 (DR2) of the Gaia satellite mission 2 lists a parallax of π = 1.905(54) mas for our target star. This converts to a distance between 510 and 540 pc, respectively. Employing the 3D dust maps of the Pan-STARRS1 project (Green et al. 2018), we find that TYC 3637-1152-1 is located in a region only marginally affected by interstellar absorption and deduce a reddening E(B − V ) between 0.09 and 0.11 mag for the above listed distance range. In the following, we have therefore adopted a value of 0.31 mag for the total absorption. Because neither observations in Johnson U nor in Strömgren u (Paunzen 2015) are available, we have not been able to apply any additional photometric dereddening procedure.
Using parallax, total absorption and bolometric correction from Flower (1996) and an apparent magnitude of V = 10.398(47) mag (Henden et al. 2018), we have derived a bolometric magnitude of M bol = 1.46(8) mag for TYC 3637-1152-1. For the estimation of T eff , we employed photometry from APASS DR9 (Henden et al. 2018) and the 2MASS catalogue (Skrutskie et al. 2006) and scaled the total absorption to the corresponding filter bands. As 1 NSVS data consist of unfiltered CCD observations that have been placed onto a V -equivalent scale; the resulting passband is approximately 4 000-9 000Å (Akerlof et al. 2000) (cf. Section 4).
2 http://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=I/345/gaia2 next step, independent photometric calibrations for different colours (Pinsonneault et al. 2012;Pecaut & Mamajek 2013;Paunzen et al. 2017) were employed. All these calibrations yield a consistent value of 6 800(250) K. To test this T eff , the VOSA (VO Sed Analyzer) tool v5.1 (Bayo et al. 2008) was applied to fit the Spectral Energy Distribution (SED) to the available photometry. The best fit was achieved for a main-sequence 6 750 K SED of solar metallicity, supporting our results. Figure 1 illustrates the location of TYC 3637-1152-1 and other HADS variables (McNamara 2000) in the M Bol versus log T eff diagram. Also indicated are isochrones for different ages and solar metallicity ([Z] = 0.019) taken from the PARSEC database (Bressan et al. 2012). With an age of log t = 9.1, our target lies well within the borders of the δ Scuti instability strip (Breger & Pamyatnykh 1998), although it is significantly cooler than the other HADS variables -a fact which might also have a bearing on its pulsational behaviour (cf. Sect. 5).
Galactic U V W space velocities and the total space velocity T (often referred as "Toomre" energy) are important and useful for discriminating halo, thick disk and thin disk stars (Sandage & Fouts 1987). According to Fuhrmann (2004), thin disk stars have T values up to 50 km s −1 , whereas a mixture between thin and thick disk stars is observed between 50 and 75 km s −1 . A value of 180 km s −1 separates thick disk from halo stars.
The list of SX Phe variables published by Nemec et al. (2017), for example, includes one star with T = 144 km s −1 , while all other stars have much higher values up to 1 274 km s −1 . We here employ a left-handed Galactic system, i.e. U positive in the anti-centre direction, V positive in the direction of Galactic rotation, and W positive in the direction of the North Galactic Pole. Calculations are done according to the formulae presented by Johnson & Soderblom (1987) with respect to the Local Standard of Rest (Coskunoglu 2011).
The space velocity values for TYC 3637-1152-1 are listed in Table 1. Radial velocity information and proper motions were taken from Gaia DR2. The derived T value of 56 km s −1 is not in agreement with a Population II object and strongly supports that our target star belongs to Population I and hence is an HADS star. Proper motion and distance Z to the Galactic plane (Table 1) are compatible with the kinematics of the thin disk within the solar neighbourhood (Pasetto et al. 2012). The W component (+53 km s −1 ) is rather large compared to other stars in the solar vicinity (Holmberg et al. 2009) but still within the 5σ range of the velocity distribution for the thin disk (Anguiano et al. 2017).
To sum up, from its kinematic properties, we conclude that TYC 3637-1152-1 belongs to Population I and is an HADS variable.
Spectroscopic observations
To shed more light on the nature of TYC 3637-1152-1, spectroscopic observations have been secured. Both high and classification resolution spectroscopy guarantee the best possible analysis. Details on the spectroscopic observations and employed instrumentations are provided below. The reduction of the raw frames and extraction of the 1D spectra using the IRAF package tasks, Linux shell scripts, and FORTRAN programs has been described by Pribulla et al. (2015).
The classification resolution spectrum of TYC 3637-1152-1 was classified using MKK standards as proposed by Gray & Garrison (1989). The S/N ratio of the spectrum is sufficient to classify the spectral type (temperature) and the overall appearance of the metallic line spectrum, i.e. a Population II object can be identified with certainty. We classify the spectrum as F4 V (Fig. 2) which is in line with the star's location in the HRD (Fig. 1). The observed metallic line spectrum is as strong as in the standard stars.
To investigate the metallicity of TYC 3637-1152-1 in more detail, we computed a synthetic spectrum using the program SPECTRUM 3 (Gray & Corbally 1994) and modified versions of the ATLAS9 code taken from the Vienna New Model Grid of Stellar Atmospheres, NEMO 4 (Heiter et al. 2002). We used a stellar atmosphere with the following parameters: T eff = 6800 K, log g = 3.8, and υ mic = 2 km s −1 . The synthetic spectrum was first folded with the instrumental profile and then with different rotational profiles, which yielded a best fit for 50 km s −1 , with an uncertainty of about 5 km s −1 . To test these parameters, a grid of atmospheres with effective temperatures and surface gravities around the input values were applied. The Hα and Hβ lines are best fitted with the original values (T eff , log g, and υ mic as listed above) under the constraint that they are not sensitive to log g. To estimate the [M/H] value, we used different models from +0 to −2 dex, as compared to solar values. We have investigated the region between 5 000 and 5 600Å because it boasts the highest S/N ratio and the most prominent metallic lines. Within the noise level, the solar abundance synthetic spectrum reproduces the observed spectrum very well. From a comparison of synthetic spectra with −0.2 and −0.5 dex, we estimate that the overall elemental abundance of TYC 3637-1152-1 is not lower than −0.2 dex.
In summary, the star's spectroscopic characteristics corroborate that it belongs to Population I. (Table 2).
Photometric data sources and time series analysis
In order to investigate the light variability of our target star, photometric observations from the All-Sky Automated Survey for Supernovae (ASAS-SN), Northern Sky Variability Survey (NSVS), and Wide Angle Search for Planets (SuperWASP) archives were procured.
The ASAS-SN survey is imaging the entire visible sky every night to a depth of V < 17 mag (Kochanek et al. 2017). The available data span up to five years of observations. As of end-2017, ASAS-SN consists of five stations equipped with four 14 cm aperture Nikon telephoto lenses. Observations are made using V (two stations) or g (three stations) band filters and three dithered 90 s exposures. ASAS-SN saturates at 10 to 11 mag; the exact limit depends on the camera and the image position. However, a procedure inherited from the ASAS survey is applied which corrects for saturation but introduces additional noise in the corresponding data sets (Pojmanski 2002).
NSVS data were collected with the ROTSE-I instruments (Woźniak et al. 2004) and comprise unfiltered CCD observations that have been placed onto a V -equivalent scale (Akerlof et al. 2000); the resulting passband is approximately 4 000 − 9 000Å. The time base is about one year and 100 to 500 measurements are typically available per object.
The SuperWASP survey started in 2004 and covers both hemispheres. It provides long-term photometric time series in a broadband filter (4 000 -7 000Å) with an accuracy better than 1% for objects in the magnitude range 8 < V < 11.5 mag (Pollacco et al. 2006). Observations consist in general of two consecutive 30 s integrations followed by a 10-minute gap. Here, we have used data from the first and only WASP public data release (Butters et al. 2010) 5 . Measurements with an error larger than 0.05 mag have been excluded, and each camera has been treated separately. All data sets with more than 100 points were cleaned of obvious outliers by visual inspection and searched for periodic signals in the frequency range of 5 < f (c/d) < 50 using Peranso (Paunzen & Vanmunster 2016) and Pe-riod04 (Lenz & Breger 2005). Consecutive prewhitening with the most significant frequency was applied until no significant frequencies remained. Neither time binning nor smoothing was applied to the data sets. Table 2 lists the results of the time series analysis of the employed data sets. The detection of the frequencies f 0 (10.034 c/d) and f 1 (12.681 c/d) is completely without doubt. Furthermore, harmonics and linear combinations of these frequencies are also clearly present. Interestingly, no harmonics of f 1 in any of the data sets was detected. Fig. 3 exemplarily illustrates the Fourier spectrum of the SuperWASP data set for CCD camera #144 in the investigated frequency range of 5 < f (c/d) < 50. In Fig. 4, the corresponding phased light curves are shown.
Analysis and Conclusions
For the two main frequencies, we calculate a ratio of f 0 /f 1 = 0.791. This is well above the theoretical limit 5 https://wasp.cerit-sc.cz/ (0.775) for stars pulsating in the fundamental and first overtone modes but also below the value (0.780) for the first and second overtone (Petersen & Christensen-Dalsgaard 1996). To further investigate this matter, we have employed the models by Stellingwerf (1979) and calculated the mean density (log ρ/ρ ⊙ = −1.2) of TYC 3637-1152-1 using the stellar parameters listed in Table 1. For the given mean density, the observed frequencies almost exactly match the first and second overtones from the models.
We also have to emphasize that non-radial modes were found in HADS and SX Phe stars (Zhou et al. 1999;Poretti et al. 2011). Normally, they have smaller amplitudes than radial modes and have been only observed together with radial modes. We cannot rule out that f 1 is a non-radial mode, but it is highly unlikely. For a definite identification, multicolour photometry or time resolved high resolution spectroscopy is needed.
Seven HADS stars are known that do not pulsate in the fundamental and first overtone modes. The observed period ratios for these objects range from 0.797 to 0.834. Only three of these stars have been studied in more detail, viz. V798 Cyg (Musazzi et al. 1998), V1719 Cyg (Poretti & Antonello 1988), and VZ Cnc (Fu & Jiang 1999). Only basic data like periods and amplitudes are available for the other four stars, viz. BD+08 4583 (Khruslov 2011), HD 77906 (Khruslov 2011), HD 146953 (Khruslov 2009), and TYC 2706-1244-1 (Khruslov 2014.
As an aside, we would like to comment on the possible misclassification of some stars currently included in the group of HADS variables. The following five stars show light variations with amplitudes less than 0.3 mag in V : BD+08 4583 (semi-amplitude of the main period, 0.071 mag), HD 77906 (0.128 mag), HD 146953 (0.072 mag), TYC 2706-1244-1 (0.093 mag), and V798 Cyg (0.091 mag). Therefore, according to the most commonly used definition (cf. Sect. 1), these stars do not qualify as HADS variables. In this respect, it is interesting to point out that Musazzi et al. (1998) write in their introduction that HADS stars are defined by amplitudes larger than ∼0.2 mag (V ). As reference, they cite Petersen & Christensen-Dalsgaard (1999), who, however, do not comment on a lower amplitude limit but state that HADS stars exhibit amplitudes of 0.4 mag (V ). Even if we adopt a limit of ∼0.2 mag, out of the above-listed objects, only HD 77906 would qualify as an HADS star. This demonstrates the need for a critical assessment of all published HADS variables, which might eventually lead to a refinement of the definition of HADS and SX Phe stars. This, however, is out of the scope of the present study. Figure 5 shows the Petersen diagram for double-mode HADS stars taken from Furgoni (2016), who compiled a catalogue of stars classified as such in the AAVSO International Variable Star Index (VSX) (Watson 2006) by the end of December 2015. We did not check the amplitudes of the stars from this catalogue and whether they qualify as HADS variables by the standard criteria. The Furgoni (2016) sample was expanded by the stars apparently pulsating in higher overtones discussed above. The only exception is TYC 2706-1244-1, which has been excluded for the sake of clarity. With a period ratio of P 1/P 0 = 0.834, this star might be pulsating in the second and third overtones, as indeed suggested by Khruslov (2014). The general trend of the pulsational behaviour as described in Poretti et al. (2005) -an almost linear behaviour of P 1/P 0 versus log P 0 -is clearly visible. Interestingly, our target star lies in the transition region between objects pulsating in the fundamental and first overtone modes and objects pulsating in higher overtones, which makes it quite unusual.
The models predict an upper limit for P 1/P 0 of about 0.775 for log P 0 < −0.85 and solar metallicity. Stars of lower metallicity, e.g. Population II objects, tend to have larger P 1/P 0 values up to 0.780. Six objects boast P 1/P 0 values larger than 0.782 and are therefore promising SX Phe star candidates: GSC 00010-00276, GSC 07460-01520, LINEAR 2653935, LINEAR 9328902, LINEAR 16586778, and NSV 7805. For LINEAR 9328902 and LINEAR 2653935, light curves are available in the VSX which clearly demonstrate an HADS star-like variability.
To investigate this matter, we have procured parallax information and proper motions from Gaia DR2. The as-trometric and kinematic results are summarized in Table 3 and indicate that all these stars belong to Population II and are consequently SX Phe variables. The observed high P 1/P 0 values are therefore due to the lower metallicity of these objects and are not caused by high overtone mode pulsation.
This scenario, however, is not suited to explain the peculiar position in the Petersen diagram of the HADS star TYC 3637-1152-1, which is of near solar abundance (cf. Sect. 3). Employing the available PL relations, we have calculated absolute or bolometric magnitudes. We note that the bolometric correction from Flower (1996) accounts for only ∼0.02 mag, and thus does not have a big impact on the derived values. First of all, we used the relations for SX Phe stars derived by Santolamazza et al. (2001). Assuming that f 0 is the fundamental mode and using the observed effective temperature, we derive M Bol = 1.6 mag, which is in the range of the here derived value of 1.46(8) mag (cf. Table 1).
Assuming that f 0 is the first overtone mode, we are faced with the problem that, to our knowledge, no PL relation for the first overtone periods of δ Scuti or HADS stars exists. We have therefore employed the PL relation of McNamara (2000) for the fundamental pulsation mode and scaled f 0 with 0.78. From this, we get M V = Table 3: Astrometric and kinematic data, and G magnitudes from Gaia DR2 for HADS stars with P 1/P 0 ratios larger than 0.782. The ranges for the distance from the Sun (D) and the Galactic plane (Z) were calculated taking into account the error of the parallax (π). 1.4 mag, which agrees well with the here derived value of 1.46(8) mag. If we assume that f 0 is the fundamental mode, we derive M V = 1.8 mag is not in line with the observation. This is another strong argument that f 0 is indeed the first overtone mode. We thus conclude that TYC 3637-1152-1 follows the classical PL relation and it is not outstanding in that respect.
In summary, we conclude that with a metallicity close to solar, a spectral type of F4 V and an age of log t = 9.1, TYC 3637-1152-1 is an unique object with peculiar pulsational properties that indicate a transitional state between HADS stars pulsating in the fundamental and first overtone modes and stars pulsating in higher overtones. This result is supported by the star's position in the HRD and Petersen diagram.
From an assessment of the literature, it is also evident that the definition of the HADS and SX Phe variables has to be verified on the basis of newly available photometric, astrometric and kinematic data, and there is the need for a critical assessment of all published HADS variables.
Figure 1 :
1The location of TYC 3637-1152-1 in the M Bol versus log T eff diagram. The asterisks denote the positions of HADS variables fromMcNamara (1997). The dashed lines indicate the borders of the δ Scuti instability strip according toBreger & Pamyatnykh (1998). Also shown are isochrones for solar metallicity ([Z] = 0.019) and different ages, which have been taken from the PARSEC database(Bressan et al. 2012).
Figure 2 :
2The SAO classification resolution spectrum of TYC 3637-1152-1 in comparison to two MKK standard stars.
telescope at Vienna University Observatory (Austria), LISA spectrograph (Shelyak), covering 3 900 to 8 000Å with a resolving power of about 1 500, S/N of about 20, spectrum taken during night of 19./20.01.2018 • 1.0 m telescope of the Special Astrophysical Observatory (SAO, Russia), UAGS spectrograph, covering 3 900 to 5 300Å with a resolving power of about 5 000, S/N of about 100, spectrum taken during night of 05./06.06.2018 • 1.3 m telescope at Tatranská Lomnica (Slovakia), MU-SICOS spectrograph, covering 4 200 to 7 300Å with a resolving power of about 35 000, S/N of about 25, spectrum taken during night of 19./20.02.2018
Figure 3 :
3Frequency analysis of one SuperWASP data set (CCD camera #144) for TYC 3637-1152-1 in the investigated range of 5 < f (c/d) < 50. The upper and lower panels illustrate the Fourier spectra for unwhitened data and data that has been prewhitened with f 0 , respectively.
3Figure 4 :
4http://www.appstate.edu/∼grayro/spectrum/spectrum.html 4 http://www.univie.ac.at/nemo The phased light curves of TYC 3637-1152-1, based on the SuperWASP data set for CCD camera #143
Figure 5 :
5Petersen diagram for double-mode HADS stars afterFurgoni (2016). Open circles denote stars pulsating in the fundamental and first overtone modes. Filled asterisks refer to stars pulsating in higher overtone modes. These are BD+08 4583(Khruslov 2011), HD 77906(Khruslov 2011), HD 146953(Khruslov 2009), V798 Cyg(Musazzi et al. 1998), V1719 Cyg(Poretti & Antonello 1988), and VZ Cnc(Fu & Jiang 1999). Open asterisks are possible SX Phe star candidates (Section 5). The position of TYC 3637-1152-1 is indicated by the filled circle. For the sake of clarity, the star TYC 2706-1244-1 (Khruslov 2014) with P 1/P 0 = 0.834 (probably pulsating in the second and third overtones) has not been included.
Table 2 :
2Characteristics of the employed time series (time baseline and number of observations) and the derived frequencies and amplitudes along with their error estimates.Source
Frequency
Semi-Amplitude Time-base
N
Designation
(c/d)
(mag)
(d)
ASAS-SN
10.034019(1)
0.1543(5)
1653.27057 671
f 0
20.068035(4)
0.0391(5)
2f 0
12.681113(9)
0.0182(5)
f 1
30.10203(1)
0.0118(5)
3f 0
22.71513(1)
0.0113(5)
f 0 +f 1
32.74916(2)
0.0071(5)
2f 0 +f 1
39.13593(3)
0.0045(5)
4f 0 -1
NSVS (set1)
10.03378(9)
0.099(4)
260.73834
286
f 0
20.0676(2)
0.038(4)
2f 0
NSVS (set2)
10.0354(3)
0.108(12)
226.76294
228
f 0
SWASP (143) 10.03403(1)
0.1325(3)
139.90112 5424
f 0
20.06804(3)
0.0369(3)
2f 0
12.68123(7)
0.0173(3)
f 1
30.1019(1)
0.0111(3)
3f 0
22.7152(1)
0.0102(3)
f 0 +f 1
32.7498(3)
0.0045(3)
2f 0 +f 1
40.1356(3)
0.0042(3)
4f 0
SWASP (144) 10.03404(1)
0.1310(4)
133.90130 5064
f 0
20.06786(4)
0.0364(4)
2f 0
12.68105(9)
0.0177(4)
f 1
30.1020(1)
0.0116(4)
3f 0
22.7155(1)
0.0108(4)
f 0 +f 1
32.7492(3)
0.0051(4)
2f 0 +f 1
40.1354(4)
0.0040(4)
4f 0
AcknowledgementsThis project was supported by the grants 7AMB17AT030 (MŠMT). This paper makes use of data from the DR1 of the WASP data(Butters et al. 2010) as provided by the WASP consortium, and the computing and storage facilities at the CERIT Scientific Cloud, reg. no. CZ.1.05/3.2.00/08.0144 which is operated by Masaryk University, Czech Republic. This article was created by the realisation of the project ITMS No. 26220120029, based on the supporting operational Research and development program financed from the European Regional Development Fund.
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| []
|
[
"Electromagnetic wave scattering by Schwarzschild black holes",
"Electromagnetic wave scattering by Schwarzschild black holes"
]
| [
"Luís C B Crispino ",
"Sam R Dolan ",
"Ednilton S Oliveira ",
"\nFaculdade de Física\nSchool of Mathematical Sciences\nUniversidade Federal do Pará\n66075-110BelémPABrazil\n",
"\nInstituto de Física\nUniversity College Dublin\nBelfield, Dublin 4Ireland\n",
"\nUniversidade de São Paulo\n05315-970, São Paulo66318CP, SPBrazil\n"
]
| [
"Faculdade de Física\nSchool of Mathematical Sciences\nUniversidade Federal do Pará\n66075-110BelémPABrazil",
"Instituto de Física\nUniversity College Dublin\nBelfield, Dublin 4Ireland",
"Universidade de São Paulo\n05315-970, São Paulo66318CP, SPBrazil"
]
| []
| We analyze the scattering of a planar monochromatic electromagnetic wave incident upon a Schwarzschild black hole. We obtain accurate numerical results from the partial wave method for the electromagnetic scattering cross section, and show that they are in excellent agreement with analytical approximations. The scattering of electromagnetic waves is compared with the scattering of scalar, spinor and gravitational waves. We present a unified picture of the scattering of all massless fields for the first time. | 10.1103/physrevlett.102.231103 | [
"https://arxiv.org/pdf/0905.3339v1.pdf"
]
| 32,897,419 | 0905.3339 | 567ca40d5737a9f5c11d9e832e3b7d32fad456f3 |
Electromagnetic wave scattering by Schwarzschild black holes
(Dated: May 20, 2009) 20 May 2009
Luís C B Crispino
Sam R Dolan
Ednilton S Oliveira
Faculdade de Física
School of Mathematical Sciences
Universidade Federal do Pará
66075-110BelémPABrazil
Instituto de Física
University College Dublin
Belfield, Dublin 4Ireland
Universidade de São Paulo
05315-970, São Paulo66318CP, SPBrazil
Electromagnetic wave scattering by Schwarzschild black holes
(Dated: May 20, 2009) 20 May 2009
We analyze the scattering of a planar monochromatic electromagnetic wave incident upon a Schwarzschild black hole. We obtain accurate numerical results from the partial wave method for the electromagnetic scattering cross section, and show that they are in excellent agreement with analytical approximations. The scattering of electromagnetic waves is compared with the scattering of scalar, spinor and gravitational waves. We present a unified picture of the scattering of all massless fields for the first time.
Black holes are thought to be efficient catalysts for the liberation of rest-mass energy. As such, black holes are implicated in the most energetic phenomena in the known universe (e.g. gamma ray bursts). On the other hand, after a turbulent youth, many black holes settle into a quiescent old age. Some estimates suggest there may be up to a billion quiescent stellar-mass black holes within our galaxy [1]. Their existence may be inferred from, for example, the transient lensing of background sources; a handful of events have so far been observed [2]. A possibility for future consideration is that quiescent black holes may be indirectly identified from the 'fingerprint' they leave on radiation that impinges upon them.
Over the last four decades, some clues about the properties of any such 'fingerprint' have been uncovered. For example, a time-dependent perturbation incident upon a black hole will excite characteristic damped ringing in response. The frequencies and decay rates of the ringing are linked to the well-studied quasinormal mode spectrum [3]. Black holes illuminated by long-lasting planar radiation will create interference patterns, and rotating black holes will create distinctive polarization patterns [4]. Both effects depend strongly on the ratio of horizon size to wavelength. Hence, it is conceivable that future gravitational-wave detectors may be able to identify the fingerprint from rapid and distinctive variations across a narrow frequency band. Nevertheless, inferring the presence of quiescent black holes from such clues must remain a challenge for future decades.
Scattering by black holes is of foundational interest in both black hole physics [5] and scattering theory [6]. Many authors have studied the simplest timeindependent scenario, in which a black hole is subject to a long-lasting, monochromatic beam of radiation. Here, the key dimensionless quantity is the ratio r h /λ where r h is the horizon size of the black hole, and λ is the wave-length of the incident wave. The interference pattern depends also on the spin s of the perturbing field, with s = 0, 1/2, 1 and 2 corresponding to scalar, neutrino, electromagnetic and gravitational fields, respectively.
To the best of our knowledge, the first paper outlining a calculation of wave scattering cross section in the spacetime of a black hole was published by Matzner [7] in the late sixties. Since then, planar wave scattering from black holes has received much attention, especially in Schwarzschild and Kerr spacetimes (see Refs. [5,8,9] for comprehensive accounts on the subject). Let us briefly review a sample of the literature for the simplest case, the Schwarzschild black hole, for which the scattering of monochromatic fields of all spins (s = 0, 1/2, 1 and 2) has been studied through the years. The case of scalar waves (s = 0) was extensively studied by Sanchez [10,11], both analytically and numerically, and an accurate numerical study was later performed by Andersson [12]. Fermion (s = 1/2) scattering by a Schwarzschild black hole was the subject of a recent study [4], in which the authors also elucidated the effect of non-zero field mass. The case of electromagnetic waves (s = 1) was studied analytically by Mashoon [13] and Fabbri [14], and some results were obtained in the low-and high-frequency limits. Gravitational waves (s = 2) were the first to be studied in black hole scattering [15], and are the subject of old [5,16] and new [17,18] works.
In this letter we present the first detailed numerical investigation of the scattering of an electromagnetic plane wave by a Schwarzschild black hole. This work fills a gap in the literature, and complements recent numerical studies of the scalar [19], fermionic [4], and gravitational [18] cases. We take this opportunity to present a unified picture of all four fields, for the first time.
We use natural units with c = G = 1 and the metric signature (+ − −−).
arXiv:0905.3339v1 [gr-qc] 20 May 2009
The line element of Schwarzschild spacetime can be written as (1) where f (r) = 1 − 2M/r, with M being the black hole mass. The Schwarzschild solution describes static and chargeless black holes, with event horizon at r h = 2M .
ds 2 = f (r)dt 2 − [f (r)] −1 dr 2 − r 2 (dθ 2 + sin 2 θdφ 2 ),
The Lagrangian density of the electromagnetic field in the modified Feynman gauge is [20]
L = − √ −g 1 4 F µν F µν + 1 2 G 2 with g = det (g µν ), G ≡ ∇ µ A µ + K µ A µ and K µ = (0, df /dr, 0, 0)
. The equations of motion are found to be
∇ ν F µν + ∇ µ G − K µ G = 0.(2)
The two physical polarizations in Schwarzschild spacetime can be written as
A (Iωlm) µ = (0 , ϕ I ωl (r) r 2 Y lm , f l (l + 1) d dr ϕ I ωl (r) ×∂ θ Y lm , f l (l + 1) d dr ϕ I ωl (r) ∂ φ Y lm )e −iωt ,(3)A (IIωlm) µ = 0, 0, ϕ II ωl (r) Y lm θ , ϕ II ωl (r) Y lm φ e −iωt ,(4)
with ω > 0, and l 1. (See, e. g., Ref. [20] for a discussion on the possible solutions of Eq. (2).) In Eqs. (3) and (4), Y lm and Y lm a (a = θ, φ) are the scalar and vector spherical harmonics [21], respectively, and ϕ λ ωl (r) satisfy the following equation
ω 2 − V (r) ϕ λ ωl (r) + f d dr f d dr ϕ λ ωl (r) = 0, (5) with λ = I, II, where the effective potential is V (r) = f [l(l + 1)/r 2 ].
To evaluate the solutions of the Eq. (5) in the asymptotical limits we use the Wheeler coordinate, defined as x = r + r s ln (r/r s − 1), and rewrite Eq. (5) as
ω 2 − V ϕ λ ωl (x) + d 2 dx 2 ϕ λ ωl (x) = 0.(6)
For the computation of the scattering cross section we need only to consider modes incoming from the past null infinity J − . For these modes, the asymptotic solutions of Eq. (6) are [22]
ϕ λ ωl (x) ≈ A λ ωl T λ ωl e −iωx ,(7)
for x → −∞ (r → r s ) and
ϕ λ ωl (x) ωx ≈ A λ ωl (−i) l+1 h (1) * l (ωx) + R λ ωl i l+1 h (1) l (ωx) ,(8)
for x r s (r r s ). Here h (1) l (x) denote the spherical Bessel functions of the third kind [23], and |R λ ωl | 2 and |T λ ωl | 2 are the reflexion and transmission coefficients, respectively, which satisfy |R λ ωl | 2 + |T λ ωl | 2 = 1. A λ ωl is a normalization constant which is not important for the scattering properties.
The phase shifts are related to the reflexion coefficient by
e 2iδ λ l (ω) = (−1) l+1 R λ ωl .(9)
For Schwarzschild black holes, the phase shifts of the two different physical polarizations are the same, i. e. δ I l (ω) = δ II l (ω) = δ l (ω) [14]. The differential electromagnetic scattering cross section is [24]
dσ dΩ = 1 4ω 2 ∞ l=1 2l + 1 l(l + 1) e +2iδ l (ω) P 1 l (cos θ) sin θ + d dθ P 1 l (cos θ) 2 ,(10)
where P m l (cos θ) are the associated Legendre functions. Note that Eq. (10) takes into account the contributions from the two physical polarizations, and it is valid for both linearly and circularly polarized waves. The polarization properties of the initial wave remain unchanged in the scattering by non-rotating black holes [13].
For small angles, this scattering cross section is the same for the massless scalar and electromagnetic fields, and it is given by [13,14]
dσ dΩ ≈ 16M 2 θ 4 .(11)
In fact, the same behavior for small angles is obtained for massless fermionic and gravitational fields scattered in Schwarzschild spacetime [4,5,17]. The glory approximation for scattering of electromagnetic waves by a Schwarzschild black hole can be determined using the strong field approximation for the deflection angle (which was first obtained by Darwin [25]) together with the general glory formula [26], namely
dσ dΩ θ≈π ≈ 2πωb 2 g db dθ θ=π [J 2s (ωb g sin θ)] 2 ,(12)
where b is the impact parameter of the incident particle, J l (x) are the Bessel functions of first kind, b g is the impact parameter for which the scattering angle is π, and s is the particle spin. For the electromagnetic field (s = 1), the glory scattering cross section is given by
1 M 2 dσ dΩ θ≈π ≈ 30.75M ω[J 2 (5.36M ω sin θ)] 2 ,(13)
where the coefficients (30.75 and 5.36) were obtained by solving the geodesic equation numerically [27]. In order to evaluate the electromagnetic scattering cross section numerically, we first solve Eq. (5) and match the solution with Eqs. (8) and (9). The scattering cross section for arbitrary frequencies and angles is obtained through Eq. (10). The numerical method used here is analogous to the one described in Ref. [28]. We have employed an iterative method similar to that used in Refs. [18,29] to improve the numerical convergence of the partial wave series.
In Fig. 1 we show the differential electromagnetic scattering cross section of Schwarzschild black holes computed numerically for different values of the incident wave frequency (M ω = 1, 2, 3, 4). We also show the results for the glory scattering [given by Eq. (13)] in each case. Our numerical results are in excellent agreement with the glory approximation for θ ≈ π.
The zero in the backward direction (Fig. 1) is a consequence of the parallel-transport of the polarization vector along a geodesic. Consider an incoming geodesic ray in the z-direction which orbits the hole once, to return in the opposite direction (θ = π). Assume, without loss of generality, it has an electric-field vector in the x-direction. If the ray orbits in the x-z plane then the vector will be reversed, whereas if the ray orbits in the y-z plane the vector remains unchanged. Hence, by integrating over the circular degeneracy (all orbital planes), there is perfect cancellation. Similar arguments hold for other spins [5,30].
In Fig. 2 we plot the differential scattering cross section of Schwarzschild black holes for massless scalar (s = 0), massless spinor (s = 1/2), electromagnetic (s = 1) and gravitational (s = 2) waves. As expected, in the backward direction all non-zero spin massless fields have vanishing cross section, whereas the zero-spin (scalar) massless field has a glory maximum at θ = π. We see that scalar (s = 0) and electromagnetic (s = 1) scattering cross sections are very similar in the angular region 45 • < θ < 160 • . All integer spin fields (s = 0, 1, 2) behave similarly for 45 • < θ < 120 • . Bosonic (s = 0, 1, 2) and fermionic (s = 1/2) scattering cross sections oscillate in antiphase throughout almost all the angular range of Fig. 2 (except near θ ∼ π).
The regular oscillations in the cross sections of Fig. 2 can be understood semi-classically. They arise from the interference of rays passing in opposite senses around the hole. There is a 'path difference' between the rays passing through angles θ and 2π −θ. A maximum (minimum) occurs when the path difference is an integer (half-integer) multiple of the wavelength λ. Hence the angular width of the oscillations is inversely proportional to M ω.
In summary, we have studied the scattering of a monochromatic planar electromagnetic wave by a Schwarzschild black hole. We have applied the partial wave method to obtain the differential scattering cross section numerically, for different values of the frequency of the incident plane wave and for different values of the scattering angle. We have presented graphs with accurate numerical results for massless fields of all spins, that is, scalar (s = 0), fermionic (s = 1/2), electromagnetic (s = 1) and gravitational (s = 2) fields. All non-zero spin massless fields have a vanishing scattering cross section in the backward direction (θ = π), whereas the scattering cross section of the massless scalar field has a local maximum.
= 4 FIG. 1 :
41cross section for the electromagnetic field at M$ # (deg) Schwarzschild scattering cross section for the electromagnetic field at M$ Electromagnetic scattering cross section of Schwarzschild black holes for different choices of M ω. We compare our numerical results (solid lines) with the glory approximation (dashed lines) given by Eq. (13), obtaining excellent agreement for θ ≈ π.
FIG. 2 :
2Scattering cross section of Schwarzschild black holes for massless scalar (s = 0), electromagnetic (s = 1), gravitational (s = 2) and massless fermionic fields (s = 1/2) at M ω = 4.0. Note the log scale on the vertical axis of the lower plot. We see that, as all other non-zero spin fields, the electromagnetic wave has a vanishing scattering cross section in the backward direction.
The authors would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for partial financial support, and Roberto Fabbri for email correspondence. S. D. acknowledges financial support from the Irish Research Council for Science, Engineering and Technology (IRCSET). S. D. and E. O. thank the Universidade Federal do Pará (UFPA) in Belém for kind hospitality. L. C. and E. O. acknowledge partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
* Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
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| []
|
[
"Velocities of an Erupting Filament",
"Velocities of an Erupting Filament"
]
| [
"Shuo Wang \nDepartment of Astronomy\nNew Mexico State University\nMSC 4500P.O. Box 3000188003Las CrucesNMUSA\n",
"Jack M Jenkins \nDepartment of Mathematics\nCentre for mathematical Plasma Astrophysics\nKU Leuven\nCelestijnenlaan 200BB-3001LeuvenBelgium\n",
"Karin Muglach \nCatholic University of America\n20064WashingtonDCUSA\n\nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA\n",
"PilletValentin Martinez \nNational Solar Observatory\n3665 Discovery Drive80303BoulderCOUSA\n",
"Christian Beck \nNational Solar Observatory\n3665 Discovery Drive80303BoulderCOUSA\n",
"David M Long \nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK\n",
"Debi Prasad Choudhary \nDepartment of Physics and Astronomy\nCalifornia State University Northridge\n91330-8268NorthridgeCAUSA\n",
"James Mcateer \nDepartment of Astronomy\nNew Mexico State University\nMSC 4500P.O. Box 3000188003Las CrucesNMUSA\n\nSunspot Solar Observatory\n88349SunspotNMUSA\n"
]
| [
"Department of Astronomy\nNew Mexico State University\nMSC 4500P.O. Box 3000188003Las CrucesNMUSA",
"Department of Mathematics\nCentre for mathematical Plasma Astrophysics\nKU Leuven\nCelestijnenlaan 200BB-3001LeuvenBelgium",
"Catholic University of America\n20064WashingtonDCUSA",
"NASA Goddard Space Flight Center\n20771GreenbeltMDUSA",
"National Solar Observatory\n3665 Discovery Drive80303BoulderCOUSA",
"National Solar Observatory\n3665 Discovery Drive80303BoulderCOUSA",
"Mullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK",
"Department of Physics and Astronomy\nCalifornia State University Northridge\n91330-8268NorthridgeCAUSA",
"Department of Astronomy\nNew Mexico State University\nMSC 4500P.O. Box 3000188003Las CrucesNMUSA",
"Sunspot Solar Observatory\n88349SunspotNMUSA"
]
| []
| Solar filaments exist as stable structures for extended periods of time before many of them form the core of a coronal mass ejection (CME). We examine the properties of an erupting filament on 2017 May 29-30 with high-resolution He i 10830Å and Hα spectra from the Dunn Solar Telescope, full-disk Dopplergrams of He i 10830Å from the Chromospheric Telescope, and EUV and coronograph data from SDO and STEREO. Pre-eruption line-of-sight velocities from an inversion of He i with the HAZEL code exhibit coherent patches of 5 Mm extent that indicate counter-streaming and/or buoyant behavior. During the eruption, individual, aligned threads appear in the He i velocity maps. The distribution of velocities evolves from Gaussian to strongly asymmetric. The maximal optical depth of He i 10830Å decreased from τ = 1.75 to 0.25, the temperature increased by 13 kK, and the average speed and width of the filament increased from 0 to 25 km s −1 and 10 to 20 Mm, respectively. All data sources agree that the filament rose with an exponential acceleration reaching 7.4 m s −2 that increased to a final velocity of 430 km s −1 at 22:24 UT; a CME was associated with this filament eruption. The properties during the eruption favor a kink/torus instability, which requires the existence of a flux rope. We conclude that full-disk chromospheric Dopplergrams can be used to trace the initial phase of on-disk filament eruptions in real-time, which might potentially be useful for modelling the source of any subsequent CMEs. | 10.3847/1538-4357/ac3a04 | [
"https://arxiv.org/pdf/2111.07830v1.pdf"
]
| 244,117,672 | 2111.07830 | f0b36a45287a6438ac8c0146c8af31eeee01f1df |
Velocities of an Erupting Filament
15 Nov 2021 November 16, 2021
Shuo Wang
Department of Astronomy
New Mexico State University
MSC 4500P.O. Box 3000188003Las CrucesNMUSA
Jack M Jenkins
Department of Mathematics
Centre for mathematical Plasma Astrophysics
KU Leuven
Celestijnenlaan 200BB-3001LeuvenBelgium
Karin Muglach
Catholic University of America
20064WashingtonDCUSA
NASA Goddard Space Flight Center
20771GreenbeltMDUSA
PilletValentin Martinez
National Solar Observatory
3665 Discovery Drive80303BoulderCOUSA
Christian Beck
National Solar Observatory
3665 Discovery Drive80303BoulderCOUSA
David M Long
Mullard Space Science Laboratory
University College London
Holmbury St. Mary
RH5 6NTDorkingSurreyUK
Debi Prasad Choudhary
Department of Physics and Astronomy
California State University Northridge
91330-8268NorthridgeCAUSA
James Mcateer
Department of Astronomy
New Mexico State University
MSC 4500P.O. Box 3000188003Las CrucesNMUSA
Sunspot Solar Observatory
88349SunspotNMUSA
Velocities of an Erupting Filament
15 Nov 2021 November 16, 20211 Draft version Typeset using L A T E X default style in AASTeX63Sun: filaments, prominences -Sun: infrared -Sun: activity -Sun: coronal mass ejections (CMEs)
Solar filaments exist as stable structures for extended periods of time before many of them form the core of a coronal mass ejection (CME). We examine the properties of an erupting filament on 2017 May 29-30 with high-resolution He i 10830Å and Hα spectra from the Dunn Solar Telescope, full-disk Dopplergrams of He i 10830Å from the Chromospheric Telescope, and EUV and coronograph data from SDO and STEREO. Pre-eruption line-of-sight velocities from an inversion of He i with the HAZEL code exhibit coherent patches of 5 Mm extent that indicate counter-streaming and/or buoyant behavior. During the eruption, individual, aligned threads appear in the He i velocity maps. The distribution of velocities evolves from Gaussian to strongly asymmetric. The maximal optical depth of He i 10830Å decreased from τ = 1.75 to 0.25, the temperature increased by 13 kK, and the average speed and width of the filament increased from 0 to 25 km s −1 and 10 to 20 Mm, respectively. All data sources agree that the filament rose with an exponential acceleration reaching 7.4 m s −2 that increased to a final velocity of 430 km s −1 at 22:24 UT; a CME was associated with this filament eruption. The properties during the eruption favor a kink/torus instability, which requires the existence of a flux rope. We conclude that full-disk chromospheric Dopplergrams can be used to trace the initial phase of on-disk filament eruptions in real-time, which might potentially be useful for modelling the source of any subsequent CMEs.
INTRODUCTION
Solar filaments are condensations of relatively cool plasma suspended at coronal heights within the solar atmosphere. When projected against the solar disk, their cool properties render them in absorption whereas their off-limb counterpart, prominences, appear bright against the dark background of space (Labrosse et al. 2010;Mackay et al. 2010;Vial & Engvold 2015;Gibson 2018). With lengths of several tens to hundreds of Mm, and heights and widths of only a few to several ten Mms they are amongst the longest structures in the solar atmosphere, often appearing as elongated channels of chromospheric plasma that snake across the solar disk.
Filaments and prominences exist within the solar atmosphere for periods ranging from a few hours to a few months. The shortest-lived samples tend to be ejected from the solar atmosphere during eruptions, many of which are cotemporal with flares and CMEs (Green et al. 2018). The longer-lived samples are relatively slow to evolve and their end-of-life 1 DKIST Ambassador Corresponding author: Shuo Wang [email protected] dynamics can vary from weak, partial eruptions (e.g., Choudhary & Moore 2003) to large interplanetary CMEs (e.g., Wood et al. 2016), thermal disparitions brusques (Sakai & Nishikawa 1983), or a complete decay of the structure as the topology of the host magnetic field evolves in such a way as to no longer provide support for the filament material against gravity (e.g., Jing et al. 2003).
Despite the slow global evolution of the latter cases, they often exhibit a highly dynamic structure at smaller scales. The wide range of small Mm-scale, presumably thermodynamically-driven plasma evolutions within stable filaments and prominences have been studied for many decades (Leroy & Priest 1989;Engvold et al. 1990). Zirker et al. (1998) reported on counter-streaming i.e., oppositely-oriented flows within filaments with velocities as high as 20 km s −1 (see also Litvinenko & Martin 1999;Wang 1999;Alexander et al. 2013;Ahn et al. 2010;Diercke et al. 2018). Similar observational signatures have also been interpreted as buoyant or gravitational flows with velocities of the order of 10 km s −1 (e.g., Berger et al. 2011;Hillier 2018), or small-scale oscillations in the host magnetic field itself (e.g., Lin et al. 2007). On intermediate ≈ 10 Mm scales, the motions of plasma within filaments and prominences have historically been related to the evolution of the host magnetic field rather than a thermodynamic driver (see the reviews by Tripathi et al. 2009;Arregui et al. 2018). For example, the particularly large amplitude oscillations are reserved for those filaments in the vicinity of a flare, wherein the filaments are subjected to the expanding magnetic pressure bubble of the nearby eruption. The amplitudes of such dynamics are also significantly larger than those at the smaller scales, with velocities and displacements in the region of 30 -100 km s −1 and 110 Mm, respectively (e.g., Luna & Karpen 2012;Luna et al. 2014;Liakh et al. 2020). More recently, similarly large-scale and correlated mass motions occurring in the lead-up to a filament eruption have been added to the conditions for global flux rope stability (e.g., Bi et al. 2014;Reva et al. 2017;Jenkins et al. 2018Jenkins et al. , 2019Fan 2020), alongside the more commonly-considered stability conditions (e.g., torus/kink instability, breakout reconnection, tether cutting, etc.; Antiochos et al. 1999;Moore et al. 2001;Török & Kliem 2005;Kliem & Török 2006).
Unlike the motions of plasma within stable on-disk filaments, the study of the behaviour of filament plasma within the early stages of an on-disk eruption is far less common due to the scarcity of spectral observations for such events, although some examples do exist (e.g., Muglach et al. 1997;Penn 2000;Sasso et al. 2011Sasso et al. , 2014Doyle et al. 2019). Once the eruptive filaments and prominences have propagated further out into the upper corona, the motion of the associated plasma is routinely tracked using automated algorithms (e.g., Byrne 2015), although many of these methods focus more on the white-light CME component than the embedded filament. Where possible, some authors have previously aimed to obtain a more-complete picture (e.g., eruption trigger mechanism) by also including a more detailed analysis of the evolution of the associated filament (e.g., mass evolution or the relationship between the 3D global shape and the inferred background decay index; Seaton et al. 2011;Rees-Crockford et al. 2020).
Authors have previously obtained observations of eruptive filaments using the optically thin He i 10830Å or He D 3 lines (López Ariste 2015), which enabled them to adopt the assumption of a Gaussian absorption profile in their inversion methods. In each case, the authors concluded that the measured spectral profile for He i yielded a satisfactory fit only if multiple summed Gaussians were employed, which indicates multiple velocity components within the line of sight of the erupting structure (see specifically Sasso et al. 2014). Velocities extracted from these fits range between 60 -300 km s −1 , comparable to the velocities of prominences in the plane-of-sky depending on the eruption stage (e.g., McCauley et al. 2015). Doyle et al. (2019) recently used similar assumptions to characterise the evolution of an erupting filament recorded in the more-readily observed optically thick Hα line, also measuring speeds of ≈ 60 km s −1 . However, the study of Chae et al. (2006) suggests such approximations applied to the analysis of optically-thick spectral lines are only first-order accurate.
Erupting filaments that lead to CMEs are one of the main drivers of space weather near Earth. A major goal of the solar community is to establish a network of ground-based facilities that enables routine observations of the Sun at wavelengths that permit the extraction of crucial parameters for space weather modelling tools (Martinez Pillet et al. 2019). The two main parameters are the velocity to infer the travel time of a given eruption from the Sun to the Earth and the magnetic field orientation to ascertain whether an interaction between the associated CME and the Earth's magnetosphere will be geo-effective (Singh et al. 2018;Owens et al. 2020). The aforementioned automated methods for extracting the velocities of CMEs are well-suited for those events where the CME propagates close to the plane-of-sky and therefore the observed projected 2D CME speed is close to the actual 3D speed of the CME (e.g., Byrne 2015). On the other hand, these methods typically fail for those eruptions which have a significant component along the line of sight (LOS), i.e., towards the Earth. This problem can of course be mitigated with suitable observations from an angle away from the Sun-Earth line, e.g., using instruments on board the Solar Terrestrial Relations -700" -600" -500" -400" -300" Observatory (STEREO) Kaiser et al. (2008) spacecraft Barnard et al. 2020). Furthermore, we are yet to routinely measure the magnetic field of the corona, including CMEs, although some preliminary efforts have been made, e.g., Bak-Stȩślicka et al. (2013). Fortunately, and as already indicated, eruptive filaments embedded within these CMEs may prove to be ideal candidates for providing the initial velocity and magnetic field properties of their host eruptive structures (e.g., Kuckein et al. 2020;Hanaoka et al. 2020).
In Wang et al. (2020, hereafter Paper I) we have already demonstrated that the magnetic field may be routinely extracted from erupting filaments observed in He I 10830Å. Paper I focused primarily on the derivation of the magnetic field structure of the erupting filament that we will further study here, and found it to be consistent with a flux rope. The magnetic maps exhibited a large variation of field strengths, peaking above the 90th percentile value of 435 G, with average values of 24, 70, and 45 G during the eruption. At the same times, the magnetic field azimuth and inclination (to the vertical) were found to gradually increase from 48 to 54 degrees and decrease from 80 to 63 degrees, respectively. Schwartz et al. (2019) presented a non-local thermodynamic equilibrium (NLTE) inversion study of Hα plasma parameters for the pre-eruptive phase of the same filament on 2017 May 29, where they found a temperature range from 6-14 kK and non-thermal velocities from 4-9 km s −1 over six different locations inside the filament. This paper is the continuation of Paper I with the objective to derive He I 10830Å plasma diagnostics from the further application of the Hanle and Zeeman Light (HAZEL) Asensio Ramos et al. (2008) code to measure e.g., the velocities within the erupting filament, which are complemented by a variety of measurements from other ground-based and space-based sources. In Section 2 we briefly describe our data sets. The methods for analysing spectral and image data are provided in Section 3. We present the results of the application of these methods to the spectral and imaging observations in Section 4. Section 5 and 6 give the discussion and conclusions, respectively.
OBSERVATIONS
From 23-29 May 2017, multiple Earth-positioned observatories recorded a long (≈ 660 ′′ ) and stable quiescent filament stretching across the south-eastern quadrant of the solar surface. At approximately 12:00 UT on 30 May 2017, the filament erupted, for instance seen in Atmospheric Imaging Assembly (AIA) Lemen et al. (2012) 304Å images, propagating to the south east (as projected on the solar disk from Earth view). An animation of the 304Å observations is available in the online material. It shows the filament eruption on 2017 May 30 from 00:06 UT to 18:56 UT in AIA and Extreme Ultraviolet Imager (EUVI) Wuelser et al. (2004) 304Å images in top and side view. Solid white curves in the animation indicate the location of the solar limb as seen by STEREO-A.
The Facility Infrared Spectropolarimeter (FIRS) Jaeggli (2011) and Interferometric Bidimensional Spectropolarimeter (IBIS) Cavallini (2006); Reardon & Cavallini (2008) instruments installed at the Dunn Solar Telescope (DST) recorded this filament before and during the eruption on 2017 May 29 and 30, respectively. Over the two observing days, the FIRS instrument observed the He i 10830Å spectra and completed four full rasters across the width of the filament Figure 2 for the associated contour definitions and explicit timestamps). The telescope pointing at the DST covered three different parts of the filament, one part towards the northern end of the filament twice on May 29 and two different sections along the filament body on May 30. The field of view (FOV) for the IBIS instrument was centred on the same location as the FIRS FOV. The IBIS instrument observed during the same four time windows as FIRS, continuously recording both Hα 6562.8Å and Ca ii 8542Å intensity spectra at a cadence of 12 s. The Ca ii 8542Å spectra did not show a clear signature of this quiescent, high (>> 10 Mm; Paper I) filament, especially during the eruption, as for instance also seen in Beck et al. (2018, their Figure 6) for another quiescent filament . The off-center location of the filament also led to an inclined LOS, which should reduce the opacity in the presumably more vertical structures closer to the photosphere that would be seen in Ca ii. Since the Ca ii spectra do not capture the filament body and only show traces of the filament foot points on May 29 one day prior to the eruption, we thus discarded them in the current investigation as they do not provide additional information on the conditions within the filament body above and beyond that provided by He i 10830Å. The orientation of the rotating coudé table was adjusted such that the slit of the FIRS instrument was roughly aligned with the main axis of the filament, as can be seen in Figure 2. Descriptions of the full setup for both the FIRS and IBIS instruments, including their data reduction, may be found in Paper I and Schwartz et al. (2019), respectively. The Chromospheric Telescope (ChroTel) Kentischer et al. (2008) was also observing during this period from 07:15 to 17:09 UT on 30 May 2017. The ChroTel instrument observes the full-disk of the Sun with Lyot filters centred on Ca ii K 3933, Hα 6562.8, and He i 10830Å. The Lyot filter at 10830Å can be tuned to obtain Dopplergrams of the He i line (Bethge et al. 2011). Finally, at 12:00 UT on 30 May 2017, the STEREO-A spacecraft was positioned approximately 136 • behind the Earth in its orbit and viewed the eruption of the filament from the side with its EUVI and Coronagraph 2 (COR2) Howard et al. (2008) instruments.
DATA ANALYSIS
He i 10830Å Inversion with Hazel
The telluric H 2 O line at 10832.108Å was first used to determine an accurate rest wavelength for the solar spectral lines. The HAZEL code was then used to invert the He i 10830Å intensity spectra from FIRS. The HAZEL model assumes a slab with constant physical parameters at a fixed altitude above the solar surface. The height of this slab was set to 15 Mm on May 29, and to 33 and 79 Mm on May 30 during the eruption as in Paper I. The region-of-interest (ROI) for the inversion is indicated by the green contour in Figure 2. The full-resolution Stokes I data were used as the input and the input magnetic field was set to zero. Figure 3 shows that this one-component inversion can sufficiently reproduce the observations. For a portion of the filament there are multiple spectral components observed in the scan taken at 14:29 UT on May 30. The less dominant components have usually a shallower line depth and large Doppler shifts. We will, however, not discuss these features in more detail in the current study that focuses specifically on the properties of the erupting front of the filament.
Fit of Beckers' Cloud Model (BCM) to Hα 6562.8Å spectra
We employed a cloud model following Beckers (1964) to fit the IBIS Hα spectra. The simplifying approximations adopted by Beckers (1964) reduce the number of dependent variables of the radiative transfer equation (RTE) to four;
(1) constant background intensity I 0 -The assumption that the background light incident across the studied pixel is constant, an assumption that may be less well-satisfied in more dynamic environments e.g., active regions, (2) constant source function S -The assumption that the source function does not vary along the LOS, (3) Gaussian-like optical thickness in wavelength -the assumption that the studied cloud is isotropic along the LOS, and (4) a constant LOS velocity (Maltby 1976;Raadu et al. 1987;Kuckein et al. 2016). Each of these parameters may, of course, vary across the FOV. As such, the RTE reduces to the form,
I(λ) = I 0 (λ)e −τ (λ) + S 1 − e −τ (λ) ,(1)τ (λ) = τ 0 e − λ 0 −λ λ D 2 ,(2)
where τ 0 is the optical thickness of the line centre (assumed constant), λ and λ 0 are the measured and rest wavelength (calibrated as in Schwartz et al. 2019, i.e., quiet-Sun spectral averages), respectively, and λ D is the total (thermal + non-thermal) Doppler width,
λ D = λ 0 c 2k B T m + ξ 2 ,(3)
with k B Boltzmann's constant, T and m the temperature and mass of the cloud, respectively, and ξ the non-thermal velocity (NTV). In practice, and owed to the limited constraints on the components when using a single spectral line, we only solve for the total value of the Doppler width and hence Equation (3) does not explicitly feature within the Beckers' cloud model (BCM) Beckers (1964) method. Beckers (1964) introduced a further, seemingly arbitrary, simplification of Equation (1) to,
C(λ) ≡ I(λ) − I 0 (λ) I 0 (λ) = S I 0 (λ) − 1 [1 − exp(−τ (λ))],(4)
referred to as the so-called contrast profile. We used a constant source function and Doppler velocity, while the profile I 0 (green spectra in the lower right panels of Figure 4) was derived from a quiet-Sun region within the IBIS FOV but away from the filament (black rectangle in the upper left panel of Figure 4). The source function, optical thickness, velocity, and line width model parameters were permitted to vary across the FOV within the bounds [0.01,0.4] W m −2 sr −1 Hz −1 , [0,3], [-45,45] km s −1 , and [0.09,0.71]Å, respectively (see e.g., Alissandrakis et al. 1990;Chae et al. 2006;Kuckein et al. 2016). The inversion procedure initially centers on the deepest portion of the profile to the blue side of the rest wavelength before solving the contrast equation (4) using the common iterative Levenberg-Marquardt least-squares fitting algorithm (Levenberg 1944;Marquardt 1963) implemented in the Interactive Data Language. Examples of the fitting results are shown in the bottom-right of Figure 4.
We restricted the cloud model fit to only the filament area. We defined a mask based on the average residual between normalised intensity and average profile I(λ) − I 0 (λ) at 6561.8, 6561.95, and 6562.1Å with a 3-2-1 weighting. For each snapshot in time between 13:47 -15:00 UT the intensity value for the contour varied in line with atmospheric seeing between -0.008 and -0.014. The resulting mask contour was then visually inspected to ensure that it did not include regions clearly not associated with the filament spine. We then extracted the average of the velocities within the mask that were both negative and had a corresponding optical thickness of less than 0.5 as a measure of the filament speed according to the Hα observations. Regions within the mask that contained optical thicknesses > 0.5 corresponded to much smaller, or even zero velocity, and were often located at the outer boundary of the filament (upper right panel of Figure 4). This is suggestive of either a complicated internal structure/evolution within the erupting filament or that these regions, although isolated within the mask, may be more related to the properties of the background, low-altitude structures rather than those of the filament body.
Derivation of Temperature Estimates from He i 10830Å
From Equation (3) one can derive estimates of the temperature T and the NTV ξ when simultaneous observations of two spectral lines from chemical elements with a significantly different molecular weight m i are available (e.g., Bendlin et al. 1988;Beck et al. 2016). The approach is valid for spectral lines that form in an optically thin medium. The roughly Gaussian shape of the He i 10830Å spectra ( Figure 3) and the optical depth in the HAZEL inversion results (see Section 4.2 below) support this for the He line. As we cannot reliably confirm it for the Hα spectra in an on-disk filament observation and because the molecular weights of helium and hydrogen are rather close, we only used the full-width at half maximum (FWHM) of the He I 10830Å spectra to estimate the temperature in the filament material at a given NTV.
To get estimates for the magnitude of the NTV in a reasonable range, we evaluated Equation (3) with the average FWHM of the He i line on May 29 prior to eruption of 0.75Å by assuming average temperatures in the filament of T = 6, 10, and 20 kK and solving for the corresponding values of the NTV. This yielded three possible values ξ i = 11.3, 10.5 and 8.3 km s −1 , respectively. With those NTVs, we then converted the FWHM of the individual He i spectra on each pixel to three temperature estimates T i (x, y) and calculated average, minimal and maximal temperatures within the filament for the three different NTVs in each of the four FIRS maps.
ChroTel Dopplergrams
The ChroTel He I 10830Å observations covered the filament eruption from 07:15 UT to 17:09 UT on May 30, 2017. However, after 16:00 UT, the filament line depth decayed rapidly until the absorption signature was no longer present at around 16:30 UT. The Dopplergrams of ChroTel observations on May 30, 2017 were derived using the center-of-mass method described in Bethge et al. (2011, BE11) with some modifications. The median filtergram intensities I i for the filters i = 1...6 were normalized to the median intensity in the seventh filtergram centered at 10833.15Å that is least affected by solar spectral lines (see Figure 4 of BE11). The median value in a square covering about 6 % of the solar disk around disk center was calculated in each filtergram. Each filtergram was then multiplied by the ratio of the median intensity of the seventh to the actual filtergram:
I i = I 7 I i I i for i = 1...6.(5)
-1000" -800" -600" -400" -200" The line-shift maps were then derived according to
∆λ = i (I 7 −Ĩ i )λ i i (I 7 −Ĩ i ) − λ 0 ,(6)
where λ 0 is the rest wavelength of the He i line at 10830Å, which corresponds to an equal weight for each filter position (α j ≡ 1 in Equation 3 of BE11). The Doppler velocities are then derived as
v = (∆λ/λ 0 ) · c.
The filament appeared in five Lyot filtergrams with the range of center wavelength [-2.7Å, +0.7Å] (i = 1...5) during the eruption. For each observation, the filtergrams used to reconstruct the line shift value were selected dynamically based on their signal strength, i.e. filtergrams without a recognizable filament shape were rejected and not included in the calculation of Equation (6). Blue shifts of up to −50 km s −1 (filter i = 2) are deemed reliable, but not the values beyond that result from filter position 1 (≡ −80 km s −1 ) that is strongly affected by the presence of the photospheric Si i line at 10827Å (e.g., BE11, Kuckein et al. 2020).
Velocity Derivation from Imaging Data
Example AIA 304Å, EUVI-A 304Å, and COR2 white light imaging data that captured the eruption of the studied filament are shown in Figure 5. For each instrument, one slice was selected in the direction from the disk center to the filament front to construct the time slices shown later in Figure 11. For the AIA observations, the position where the filament first appears above the limb was used to set the slice direction. For both EUVI and COR2 observations, the feature point that is farthest away from the disc center was used to set the slice direction. The filament front positions were determined by a point-and-click method in the time slice of AIA 304Å, and were automatically selected based on the gradient along the slice with manual correction for points before 12:00 UT in the time slice of EUVI-A 304Å. The front positions of the CME in the STEREO-A COR2 white-light data were automatically selected based on the gradient along the slice. A Savitzky-Golay filter was applied to all three observations to smooth the position results (Byrne et al. 2013). The uncertainties are estimated to be two pixels for both the AIA and the EUVI data. The velocities in the plane of sky were then obtained from the spatial derivative of the smoothed distance with time. Figure 6 shows maps of the FWHM for the He I 10830Å spectra and the corresponding histograms. On May 29, the average line width was 0.75Å with a range from 0.55 to 1.07Å (Table 1). On May 30, the mean value increased by about 15 % to 0.86Å, while the maximum value increased by up to 50 % to 1.53Å. Using the three values of ξ i = 11.3, 10.5 and 8.3 km s −1 , the maximal derived temperatures on May 29 were 46-60 kK and increased to 125-139 kK on May 30. The average temperatures found on May 30 are 13 kK higher than on the previous day independent of which value of ξ is used. For ξ > 8.3 km s −1 , the minimum FWHM of 0.56Å on both May 29 and 30 would have to correspond to negative temperatures. From this simple estimate of temperature, an average temperature of about 20 kK on May 29 and a mean increase by 13 kK on May 30 with a non-thermal velocity below 8.3 km s −1 are the most likely results, while several small-scale areas forming elongated separate threads show significantly higher temperatures on May 30 (rightmost top panel of Figure 6). Spectra with a large line width often only have a small line depth and show asymmetric line profiles with extended red wings. The inclusion of the extended red wing led to spurious large FWHM values for some of these spectra. The maximal derived temperatures are thus less reliable than the average values. observations on this day have a time difference of 26 min at the same location. Despite being described as globally stable, the LOS velocity values may of course vary slightly or even reverse sign at any given local position. The general patterns remain, nonetheless, similar and may be explained as counter streaming flows along magnetic field lines with changing speed, oscillations perpendicular to magnetic field lines, or perhaps even signatures of individual magneto-thermal convection events. The fact that there are always white regions (v ≈ 0) between red and blue patches indicates velocity changes at the border of the patches with a smooth continuous transition; there is no imposition of lateral-atmosphere, pixel-to-pixel, coherency within the HAZEL inversion tool. Along the direction of the magnetic fields there appears to be no change of sign in the LOS velocities apart from a few assumed threads where the velocity changes sign. This may indicate a slight curvature of the magnetic field line relative to the LOS direction, as would be expected for the concave-up topology present within a magnetically-dipped portion of a flux rope.
Line Width and Temperature
In the first observation at 13:46 UT on May 30, the filament was exhibiting blue shifts of about -11.0 km s −1 across its entire area. Most convincingly, the relationship between the velocity structure and the thin, elongated individual threads is now as apparent and clearly visible as in the corresponding line-core intensity map of Figure 2. The filament width at this time was 20 arcsec, twice the width as on May 29. Maps of the LOS velocity magnitude relative to the mean value of the filament within the FOV (middle bottom panel in Figure 7) show that the rising speed varies along the filament axis, with the south-east end rising faster than the other end at 13:46 UT on May 30.
For the second observation at 14:29 UT, the mean LOS velocity of the filament has increased to -22.9 km s −1 with yet further distinctive, elongated threads than earlier. The region east of the filament spine marked with the red rectangle in the middle-rightmost panel of Figure 7 shows nearly zero velocities. The two regions marked with black rectangles in the same panel show a small line depth and a LOS velocity of about -23 km s −1 . The threads with the highest LOS velocities of -36.2 km s −1 are found around the region marked with an orange rectangle on the south-west side of the filament axis. The LOS velocities relative to the mean value of the filament within the FOV show that the rising speed is different perpendicular to the filament axis, with the north-west edge rising fastest at 14:29 UT on May 30.
In Figure 8 the distribution and evolution of velocities at different distances from the filament axis on May 30 is further highlighted. The axis was determined by connecting two points which are centroids of the ends of the filament within the FOV. These two points are shown in green at the top and bottom of the LOS velocity panels of Figure 7. The velocity at 13:46 UT appears to have been symmetric to the filament axis, whereas the velocity at 14:29 UT was asymmetric with increasing values along the positive direction of distance from the filament axis. For the observation at 13:46 UT, the velocities were in the range of -5 to -18 km s −1 . For the observation at 14:29 UT, the mean velocity on the left (right) of the axis was -20.4 (-26.2) km s −1 . As such, there is a clear increase in the average velocity with time of 5.8 km s −1 . The two outer edges of the filament are about 20 arcseconds apart and were observed with a time difference of four minutes because of the sequential spatial scanning. Assuming a constant acceleration of 3.6 m s −2 during the observation (the acceleration is obtained from the second derivative of the fitting line in Figure 12), a velocity difference of 0.8 km s −1 would be explained. The remaining difference of 5.0 km s −1 between the right and left half of the filament indicates that the velocity distribution in the direction perpendicular to filament axis is skewed during the observation. This is explored in more detail in the next section. At 14:29 UT, the velocity distribution at both edges of the filament, where the coloured boxes of Figure 7 were previously located, is broader than the central part.
Histograms of LOS velocities for the three scans are shown in the upper panel of Figure 9. The width of the distributions increases significantly during the eruption in comparison with the pre-eruptive state. The mean value of LOS velocities is close to 0 on May 29, while on May 30, the values are -11.0 and -22.9 km s −1 , respectively. To describe the range of physical parameters in each observation, the range is defined as the difference between the 95th percentile and 5th percentile. The ranges of the LOS velocities for the three observations are −6.5, −17.9, and −14.5 km s −1 , respectively.
The maps of the optical depth are shown in the top row of Figure 7, while the distributions of optical depth are presented in the right panel of Figure 9. On May 29, the spine of the filament shows a continuous enhanced, relative to the background, optical depth along its full length with only a few short threads to the east at about the middle of the FOV. The LOS velocity pattern has no discernible correlation with the optical depth. During the eruption on May 30, the optical depth maps show individual elongated threads that partially align with corresponding structures in the velocity maps. The optical depth values monotonically decreased during the eruption. The mean values of optical Figure 9) shows that the data points of the three observations are separated from each other with small overlap. The filament was stable at 14:41 UT on May 29, with large widespread optical depths τ from about 0.1 up to 1.6 and LOS velocities around zero. During the eruption on May 30, the optical depth decreases to a maximum of τ = 1.2 at 13:48 UT and τ = 0.3 at 14:29 UT while the average velocities reach -11.0 and -22.9 km s −1 , respectively.
Hα LOS Velocities
In Figure 4 we present the BCM inversion results for the Hα spectra observed by the IBIS instrument at the DST. Velocities derived within the mask of the filament were primarily negative, i.e., towards the observer. Inspection of the fitting examples presented in the bottom-right of the Figure demonstrates that this is not imposed by the initial fitting procedure outlined in Section 3.2; the deepest portions of the observed profiles lie far into the blue wing of the Hα profile. These plots also demonstrate that those profiles inverted within the absorption mask were generally shallower than the assumed-average profile within the FOV, i.e., consistent with the weak absorption signature presented in the top-left intensity image of the same figure and Figure 9. Finally, the BCM approach yields that the filament velocity increased from ≈ -10 to -22 km s −1 between 13:47 UT and 15:00 UT on May 30 for material with an optical thickness less than 0.5.
ChroTel He i 10830Å Velocities
Finally, Figure 10 After that the visibility of the filament gradually decreased. The ChroTel observations thus provided a continuous, uninterrupted measure of the filament's speed at a 3 minute cadence from 7:15 UT until 16:00 UT that could be used to derive its acceleration.
Evolution of the Filament Speed in the Plane of Sky
In addition to the FIRS and IBIS instruments the filament eruption was also observed by several other instruments which can be used to derive velocities; SDO/AIA and STEREO-A/EUVI at 304Å and STEREO-A COR2 in whitelight (see Figure 5 and its associated online animation) . The results for the tracking of the leading edge of the filament as observed by the three instruments are summarised in the three panels of Figure 11.
The filament started moving at 10:30 UT (corresponding to t = 2.5 hrs in the left two panels of Figure 11) according to the 304Å observations taken by the AIA. By 16:10 UT on May 30, the projected filament front reached the solar limb, and the portions of the filament that project against the background of space are no longer visible (see the animation in the online material). In the EUVI 304Å channel the filament front reached the edge of the field of view at 16:10 UT (corresponding to t = 8 hrs in the left two panels of Figure 11) with a velocity of 60.1 km s −1 .
The STEREO-A COR2 observations show that the filament eruption was associated with a CME. The CME observed in STEREO-A COR2 white-light can be seen in the right panel of Figure 5. The direction of the slice in the COR2 white-light observation is the same as the direction of the slice observed from the STEREO-A/EUVI 304Å shown in the middle panel of Figure 5. The resulting time-slice image through the center of the CME is shown in the right panel of Figure 11. As can be seen in Figure 11, both the EUVI and the COR2 white-light coronagraph observed a propagating intensity decrease which implies a density depletion of the associated CME as it expands outward. Starting at 18:24 UT the CME had a velocity of 145.6 km s −1 that increased to a final velocity of 430 km s −1 at 22:24 UT after which it became too faint in the COR2 coronagraph images (details of the CME velocity can be found in Figure 12).
Velocity Evolution of the Erupting Filament
With the assumption that the direction of the filament eruption was radial, we derived the deprojected height of the filament/CME front (top panel of Figure 12). The heights derived from EUVI 304Å and AIA 304Å are consistent, and the COR2 instrument tracks a much later stage in the CME evolution. For the velocity diagnostics, all LOS data were converted to a rising speed also in the radial direction. The velocity of the erupting filament according to EUVI and COR2 were derived from its height (see bottom panel of Figure 12) where the results are consistent across both instruments. Then, the continuous observations of ChroTel He i 10830Å are over an extended period of time that subsequently enabled us to fit both the early and late eruption phases.
The mean velocity derived from the ChroTel He i 10830Å data, and both an exponential and a linear fit are shown in the lower panel of Figure 12, with a reduced χ 2 of 0.19 and 1.81, respectively. Hence, the velocity curve during the eruption appears most-consistent with an exponential growth. The exponential fitting gives a value of 6.3 km s −1 for its horizontal asymptote. The uncertainty of the velocity of the ChroTel He i 10830Å data was estimated based on the difference between the observed and fitted values to be about 1.8 km s −1 . The mean filament velocity reached 46.6 km s −1 on May 30 at 16:00 UT. The CME velocities reached 350 km s −1 on May 30 at 22:20 UT. The acceleration value was 1.7 (2.8) m s −2 at 14:05 UT (14:45 UT) when the first (last) FIRS observation on May 30 was halfway through, and subsequently increased to 7.4 m s −2 at 16:00 UT, when the filament was about to disappear in the 10830Å observation due to decreased line depth. The acceleration in the CME phase was derived from the COR2 heights to have been 12 m s −2 on May 30 at 21:20 UT. Kuckein et al. (2020) analyzed an eruption of part of a quiescent filament with blue-shifted line profiles exhibiting different shapes. They advocated convincingly for the use of k-means clustering to avoid inverting physically different spectra with a single model. However, we do not find any regions that show line profiles containing significant asymmetric line wings in this event. Of course, we already selected a subset of available profiles within the observations with the use of an inversion mask that isolated the deepest profiles believed to be related exclusively to the erupting filament. All line profiles of Stokes I for this filament observed by FIRS show one dominant component with symmetric line wings during the eruption. Many previous reports of events with LOS velocities > 20 km s −1 observed in He i 10830Å spectra are also accompanied by a distinct component at rest (< 8 km s −1 ) (e.g. Muglach & Sütterlin 1998;Schmidt et al. 2000;Sasso et al. 2011Sasso et al. , 2014Schad et al. 2016). However, this erupting quiescent filament did not exhibit a component at rest. Crucially, the aforementioned papers studied targets predominantly within active regions and performed inversions across their entire FOV, while the event presented here occurred within the quiet Sun and only specific regions of the FOV were analysed. We therefore concur with a possible conclusion suggested before that the component at rest observed by these previous authors is likely associated with stronger photospheric magnetic field beneath the filament that is absent for this event (cf. Díaz Baso et al. 2016Baso et al. , 2019c. Another explanation could be that because the position of the filament is far from disk center, the inclined LOS does not scan the lower part of the filament but a quiet region far from the position of the eruption source.
DISCUSSION
Derivation and Comparison of Filament Velocities During Eruption
It is worth noting that some threads in one region showed much lower rising speeds of 2-6 km s −1 . Assuming that the plasma has some average velocity during the eruption, and neglecting the possibility that this signature is sourced below the erupting filament, plasma at a LOS velocity of about zero must correspond to downward flows along threads relative to the rising body of the flux rope. The location of these threads hints at a potential relation to a barb that previously connected the filament to the photosphere, although this is purely a spatial correlation (Jenkins 2020).
The fit for deriving the velocity profile in Figure 12 was applied to the ChroTel He i 10830Å full-disk chromospheric Dopplergrams. Only synoptic full-disk instruments such as ChroTel or the Solar Flare Telescope (Hanaoka et al. 2020) can currently provide the data needed for measuring LOS velocities of on-disk eruptions that might be used to refine an estimated time of arrival for space weather prediction purposes in near real-time. Nevertheless, the rising speed of the erupting filament was derived from different data sources with consistent results. Numerical simulations show that an eruption driven by breakout reconnection exhibits a height profile best fit with a quadratic function (e.g., Lynch et al. 2004), while a kink/torus instability requires an exponential function (cf. O'Kane et al. 2019). A quadratic function of the height profile is often seen in prominence eruptions (e.g., Gopalswamy et al. 2003;Gopalswamy 2015;Cheng et al. 2020), which would correspond to a linear function fitted to the velocity profile in the lower panel of Figure 12. In this observation, the velocity profile derived using the He i 10830Å Dopplergrams is fitted well with an exponential function, thus the observed eruption is consistent with a kink/torus instability as the driving mechanism.
The ChroTel data at 07:15 UT on May 30 indicate that the filament had been perturbed prior and had already gained a mean upward velocity of 6.3 km s −1 by that time. Unfortunately, the ChroTel data do not extend further back in time than 07:15 UT, and the high-resolution observations of either FIRS or IBIS on May 29 preceded the initiation, as indicated by their observations of a stable filament, by some 10 hours or so. As such, we are unable to suggest which of the many possible trigger mechanisms was responsible for the slow evolution preceding the eruption. There are some filament eruption events associated with CMEs and ICMEs reported with upward velocities observed hours before the eruptions, similar to the event that we have presented here (Hanaoka et al. 2020). Telescopes with off-band Hα 6562.8Å or He i 10830Å capability are able to detect this kind of filament eruption through Dopplergrams about half a day before its motion shows up in chromospheric line-center observations. It is therefore unfortunate that in most cases the instrumentation at telescopes that have a synoptic program currently lack the ability to perform such observations.
The analysis of the high resolution Hα observations of IBIS yields velocities that are consistent with those obtained from both FIRS and ChroTel. Quantitatively, the velocity in Hα is observed to have increased from rest to -10 --22 km s −1 during the period of observation. Furthermore, and most crucially, the results of all of the spectroscopicallyderived velocities are in agreement, to at least the same order of magnitude or better, with those velocities derived using the imaging instruments of SDO/AIA and Solar Terrestrial Relations Observatory (Ahead) (STEREO) Kaiser et al. (2008)/EUVI.
The methods employed to extract the velocities from both the He i and Hα observations may be considered simplistic in their handling of the radiative transfer theory. Nevertheless, authors such as Mein et al. (1996) have shown that, for Hα, the discrepancies between the results of the BCM and a fully-NLTE model may be of the order of only a few tens of % and only critical for those filaments with an optical thickness much larger than one, i.e., larger than measured for the filament studied here. However, although a valid conclusion for comparatively stable filaments, such a relationship may become of second-order importance when considering eruptive geometries; the assumption of a 1D, plane-parallel atmosphere with zero lateral photon-loss will undoubtedly become increasingly invalid with increasing altitude and internal structural complexity. It is imperative to understand the finer details of conditions present within the filament plasma in general, however, it appears from Figure 12 that the addition of such considerations (e.g., Heinzel et al. 1999;Tziotziou 2007;Schwartz et al. 2019) to the simple models used in this study are not necessary to extract complimentary information (agreement with other models to within a few km s −1 ) so as to consistently characterise the early velocity evolution within an erupting filament. Naturally, this does not exclude the consideration that each of the spectral inversion methods may be similarly incomplete.
Additional Points of Interest
Our study here focuses primarily on the evolution of plasma velocity within an erupting quiescent filament, measured using a combination of spectroscopic and monochromatic observations and their associated analysis tools. Nevertheless, these tools also provide additional parameters, and the observations contain additional features, that we consider to be of interest to the wider community.
To begin, the properties of the plasma within the pre-eruptive filament have previously been studied in detail by Schwartz et al. (2019), where the authors performed a careful analysis of the Hα absorption at six positions along the filament spine. Although a less focused approach, the more-general HAZEL tool has enabled us here to invert the entire FOV and as such we have access to the spatial variation of the radial velocity on a scale of about ≈ 100 -200 km. A general one-to-one, pixel-to-pixel comparison of these maps to the parameters inverted by Schwartz et al. (2019) would require a separate, dedicated study. Nevertheless, the global, striped pattern in the radial velocity is intriguing for a number of reasons. Similar observations have previously been interpreted as signatures of counter-streaming material along the host magnetic field (e.g., Zirker et al. 1998, and many subsequent citations). In this case, we find coherent, ≈ 5 Mm width plasma motions aligned with the azimuth field vector as deduced in paper I. The occasional reversal in sign of the motions may thus represent the projected velocity of material flowing coherently away from the observer on one side of the filament and towards on the other. The consideration of a flux rope topology, as deduced in paper I, then points to the hypothesis that filament material was flowing in different directions (counterstreaming) around the inside of a flux rope. The occasional reversal in sign along a given flux tube (cf. Paper I) thus illuminates the concave-up shape inherent to the magnetic configuration (cf. simulations of Jenkins & Keppens 2021). Alternatively, assuming the motions of the plasma were oriented parallel to the LOS they thus describe material flowing towards and away from lower heights. If so, such undular velocity patterns may be the filament counterpart of the magneto-thermal convection frequently recorded within quiescent prominences above the limb i.e., radial striations induced by the Rayleigh-Taylor instability ( see Hillier 2018, and references therein). The closely-arranged red-and blue-shifted regions would thus correspond to the 'falling fingers' and 'rising plumes', respectively. However, the ability to confidently distinguish either behaviour from general small-scale oscillations would require a more detailed study that lies outside of the scope of the current work.
Returning to the eruptive phase, the FIRS inversion results of the LOS velocities on May 30 indicate that the filament motion may be decomposed into three categories: the erupting translational motion in the radial direction which has the largest magnitude, the flow motion along magnetic field lines that highlights the thread structure, but also a possible third, rotational motion about the main axial field. For material flowing around a cylinder, one would expect to observe a velocity gradient across the centre of the cylinder associated with a smooth variation in the alignment between the LOS and the cylinder edge. Presuming that we may consider the magnetic 'cage' in which the filament material is evolving to be both symmetric and translationally invariant along its axis, the cross-axis gradient signature is indeed suggested in the bottom-right panel of Figure 7. However, the relative velocity gradient (green dotted lines in the bottom row middle and right panels of Figure 7) is distinctly different between 13:46 and 14:29 UT on May 30, with the gradient clearly being along the axis, rather than across it, for the earlier scan. At 13:46 UT and for the position of the FOV, the filament and its bounding magnetic field will have been both closer to the surface and had more curvature to its axial field. The gradient along the axis may be explained by this assumed curvature in the same direction (cf. Titov & Démoulin 1999;Xia & Keppens 2016;Kaneko & Yokoyama 2018).
The position of the FIRS FOV changed between the two scans in an attempt to follow the erupting structure and as such there is no guarantee that the two scans observed the same portion of the erupting structure. Consider, first, the possibility that the tracking was successful and the regions observed by the two snapshots are related. The observed expansion would presumably involve a straightening of the legs of the erupting structure, an evolution in the gradient of the velocity along the same portion of the filament axis would then be expected. This would not, however, necessarily explain the shift in the gradient direction from along to across the axis. Such a shift would require either a sudden and significant flow along the assumed-helical magnetic field, or a rotation of the magnetic field around the axis itself. In the absence of a reasonable hypothesis for such a sudden and bulk flow of plasma, we speculate that it is instead more likely that this change in gradient orientation is a consequence of an unravelling motion associated with the expanding magnetic field. The untwisting of filaments and prominences during eruption has previously been reported by, e.g., Koleva et al. (2012), Xue et al. (2016) and Kuckein et al. (2020). Figure 13 presents a time slice of the AIA 304Å observations showing that the absorption signature of the filament started to expand around 15:00 UT on May 30 i.e., at the end of the last FIRS scan. After 15:30 UT, there are many dark stripes that are parallel and appear to show the motion of the filament threads (cf. Figure 1 of Xue et al. 2016). The gradient of the blue dashed line overlaid on this Figure equates to ≈ 17 km s −1 , slightly larger than the magnitude of the relative velocity on either side of the assumed-axis shown in Figure 7. Then, assuming the material captured in 304Å absorption is located in the underside of the flux rope, a common assumption for a stable filament, the extension of these 'threads' towards the bottom of Figure 13 is also consistent with the position of the (relative) red-shifted portion of the filament in the bottom-left of the bottom-right panel of Figure 7. A second possibility would be that on May 30 the region captured in the second FIRS snapshot at 14:29 UT was closer to the top of the erupting filament than the first observation at 13:46 UT. The apparent velocity gradient across the axis recorded in Figure 7 may then instead indicate the flow of material associated in some way with the expanding bow of the eruptive structure. Nevertheless, the independent observation of these moving 'threads' in Figure 13 remains, wherein the associated cut (white line) in Figure 5 is positioned across the western leg of the erupting structure. Unfortunately, we are unable to confidently distinguish between these two possibilities without additional information. Figure 9 shows how the optical depth decreased during the eruption, while both the velocity ( Figure 12) and the lateral width (Figure 2) of the filament increased instead. In addition, we estimated a consistent temperature increase of 13 kK between the quiescent and rising phase in Table 1 for non-thermal velocities between 8 and 11 km s −1 with a most likely average pre-eruptive temperature of 20 kK. In some regions of their erupting prominence, Zhang et al. (2019) found NTVs below 9 km s −1 along with a smaller temperature increase of a few hundred K during the activation phase. Observational determinations of temperature and NTV in limb spicules range from 6-20 kK and 5-24 km s −1 (Bendlin et al. 1988;Socas-Navarro & Elmore 2005;Beck et al. 2016;Alissandrakis et al. 2018), where Beck et al. (2016) found up to 50 kK in a macrospicule. For filaments, a temperature range of 10 kK for the core and up to 200 kK for the prominence-corona-transition-region layer, i.e., the outer boundary of a filament thread, has previously been reported (Labrosse et al. 2010;Parenti 2014;Vial & Engvold 2015). The microturbulent velocity within prominences has commonly been assumed to be approximately 5 km s −1 with only a limited number of corroborating observational studies (Gouttebroze et al. 1993;Tziotziou et al. 2001;Schwartz et al. 2019), while Rezaei & Beck (2015) found values > 15 km s −1 in an Ellerman bomb. The corresponding values in the current study thus align with previous findings. The increase in the average temperature suggests that an increased degree of ionization of Helium may be at least partly responsible for the reduction of the opacity, while the aforementioned expansion (lateral and symmetric or involving an untwisting) would contribute to the same effect by spreading the mass contained in the filament over a larger volume.
From the perspective of AIA, the filament reached the solar limb around 16:00 UT on May 30, 2017. The AIA 304Å observations show that the filament disappeared once projected above the limb (see the animation). This is different from stable filaments which often appear clearly as prominences when rotated above the limb. The absorption signature of this eruptive filament had a mean intensity of 1.3 DN as it approached the limb. According to the upper panel of Figure 12, the filament was at a height of about 500 Mm at this time. Assuming the absorbed light was subsequently re-emitted isotropically, the dilution factor takes a value of 0.094 (see 5.4.2.2 of Heinzel 2015). As such, the mean intensity of the filament once it rises above the limb and transitions to a prominence is expected to be a maximum of 0.13 DN. This expected value is an order of magnitude lower than the AIA 304Å read noise (see Table 6 of Boerner et al. 2012), and so it is not surprising that the prominence signal is not detected in the AIA 304Å data.
Finally, it is of interest that there was a small coronal hole (CH) close to the disk center visible in the AIA 193Å data on May 30. CHs are the source of high-speed streams (HSS) in the solar wind. The solar wind speed observed by the Advanced Composition Explorer showed an increase of wind speed on June 3, going up to around 500 km s −1 . This would correspond to a transit time of around 3.5 days, compatible with the CH close to disk center on May 30. Most of the in-situ solar wind characteristics of this event between June 3 to June 5 is that of a HSS. Nevertheless, the magnetic field data from 11:00 UT to 18:00 UT on June 3 indicate its components are smooth and switch sign, which is not usually the case in a HSS but typical for a magnetic flux rope. A possible explanation is that the western flank of the CME got embedded in the HSS originating from the small disk center CH and both the HSS and the CME flank arrived together on June 3, travelling closer to a speed of 500 km s −1 .
We have derived the propagation velocity, in addition to a variety of additional parameters, for an erupting largescale filament from a series of multi-instrument imaging and spectroscopic data. Importantly, we have successfully demonstrated consistency between the ejection velocity measured spectroscopically and the speed inferred using the propagation of filament material from monochromatic images. The velocity profile during the eruption is better reproduced by an exponential than a linear function. This behavior is in favour of a kink/torus instability, which requires a flux rope. The existence of a flux rope is consistent with the corresponding results concerning the magnetic topology found in Paper I (Wang et al. 2020). We conclude that synoptic full-disk chromospheric Doppler measurements can provide a near real-time determination of the rise speed of on-disk erupting filaments which might be used in future data-driven CME propagation models.
Figure 1 .
1SDO/AIA 304Å maps on 2017 May 29 and 30. Contours mark the parts of filament observed by FIRS at 10830Å and IBIS at Hα 6563Å. The image intensity is in units of digital numbers (DN) and scaled logarithmically. These images show a subfield of the FOV of the animation of the prominence eruption that is available in the online material. The animation covers the evolution at this part of the Sun between 00:06 UT and 18:56 UT on 2017 May 30 with a changing cadence of 2 -10 minutes. The duration of the animation is seven seconds.
Figure 2 .
2Line-core intensity map of He I 10830Å as observed by FIRS on 2017 May 29 and 30. Green contours show the borders of the regions of interest (ROIs) used for the inversion in Figure 7. Red points mark the positions of the pixels in the spine shown in Figure 3. The absorption by the He I line decreases during the eruption leading to rising line-core intensities. The two vertical bright stripes at x ∼ 15 ′′ in the rightmost panel with the data on May 30 at 14:29 UT were caused by a temporarily loss of the lock point of the adaptive optics system. at 14:41 & 15:07 UT on 29 May, and 13:46 & 14:29 UT on 30 May. The position of these rasters relative to the entire filament is shown in Figure 1 as the green contour overlaid on full-disk He ii 304Å observations provided by the AIA instrument on board the Solar Dynamics Observatory (SDO) Pesnell et al. (2012) (see also
Figure 3 .
3Profiles of pixels in the filament spine at different times. The positions of the three pixels are marked with red dots in Figure 2. The blue dashed vertical lines mark the positions of the wavelengths for the He i 10830Å triplet at rest.
Figure 3
3shows examples of the fitting wherein positions with a large line depth have been selected to show clear line profiles with small noise. Each of the three line profiles exhibits a single dominant component with symmetric line wings. The line profiles were observed at three different times and at three different locations in the filament as a consequence of the change in the position of the FOV relative to the filament (Figure 1). As time progresses, the LOS velocity is observed to have increased, showing a stronger blueshift and a decrease in the line depth.
Figure 4 .
4Method and results of the application of the cloud model to the IBIS Hα observations. Top-left: normalised intensity at 6561.95Å wherein the absorption signature of the erupting filament is bordered by the red contour. Top-right & bottom-left: optical thickness and LOS velocity as derived from the BCM. Sample profiles are presented in the bottom-right panel. The green line indicates the average profile I0 of Hα derived as an average of all quiet-Sun profiles within the black box of the top-left panel. The orange line is the Hα profile measured at a randomly chosen position within the FOV. Red dots are the values of the contrast profile at the narrowband filter positions, and the blue line is the result of the BCM fitting to these points. The horizontal, dashed-black line highlights the level of zero normalised intensity.
Figure 5 .
5Left panel: map of SDO/AIA 304Å. Middle panel: STEREO-A/EUVI 304Å. Blue curves show the solar limb observed from STEREO-A. Right panel: STEREO-A COR2 white-light image. Green and white lines represent the position of slices. An animation of the left and middle panels of this figure is available in the online material. It shows maps of SDO/AIA 304Å and STEREO-A/EUVI 304Å during the filament eruption from 00:06 to 18:56 UT on 2017 May 30. In both panels of the animation, white solid curves show the solar limb observed from STEREO-A.
Figure 6 .
6FWHM maps (top row) and histograms (bottom row) of the FIRS observations at He I 10830Å. The top left panel shows the same portion of the filament observed twice on May 29 with a starting time of 14:41 UT at x = 0, and 15:07 UT at x = 33 arcsec, while the right two panels show the maps on May 30 at 13:46 UT and 14:29 UT.
4. 2 .Figure 7 .
27LOS Velocities and Optical Depths from He i and Hα Spectra 4.2.1. He i 10830Å LOS Velocities In Figure 7 we present the LOS velocity maps from the HAZEL inversion. On May 29, the filament was stable as summarised in Paper I. The pattern in the LOS velocity map consists of elongated patches with widths of about five Mm that have their long axis parallel to the magnetic field lines whose directions have been provided already in Paper I. The average magnetic field direction forms an acute angle to the filament axis and is indicated by a green dashed line in the left middle panel of Figure 7. Adjacent patches tend to have oppositely directed velocities. Optical depth and line-of-sight velocity maps of the filament observed by FIRS at 10830Å. Left to right: on May 29, May 30 at 13:46 UT and 14:29 UT. Top row: optical depth. Only the map at 14:41 UT is shown for May 29. Both observations on May 30 share the color bar at the right. First column, bottom two panels: LOS velocities for the two maps on 29 May 2017 at 14:41 UT and 15:07 UT. Middle row, right two panels: LOS velocities on May 30 at 13:46 UT and 14:29 UT. Bottom row, right two panels: LOS velocity on May 30 at 13:46 UT and 14:29 UT relative to the mean value of the whole filament within the FOV in each map. The green dashed line in the middle left panel shows the average magnetic field azimuth from Paper I. In the middle rightmost panel, two protrusions are marked with black rectangles, while red/orange rectangles indicate threads with small/large velocities. Dashed-black lines with green dots mark the positions of the filament axis used inFigure 8. In the middle and right panels of the bottom row, the centroids of blue/red shifted regions are marked as crosses in reversed color. They are connected with green dotted lines to show the direction of the gradient. Uniform gray pixels were not inverted.
Figure 8 .
8Scatter plots of velocity and distance from the filament axis marked in Figure 7 on May 30 at 13:46 UT (left panel) and 14:29 UT (right panel). The zero positions along the axis are marked with green dots at the bottom of the axis line in each FOV in Figure 7. depth are 0.51, 0.34, 0.13 on May 29, at 14:41 UT, May 30 at 13:46 UT and 14:29 UT, respectively, with the ranges of optical depth also measured to have decreased, with values of 0.80, 0.46, 0.20. A scatter plot of optical depth and LOS velocities (lower left panel in
Figure 9 .
9shows three panels of the ChroTel Dopplergrams. The left panel is at the beginning of the ChroTel observations on May 30 at 07:15 UT with a mean LOS velocity of the filament of -5.3 km s −1 . The two other panels were obtained during the two FIRS scans on May 30 at 13:57 UT and 14:45 UT. The mean value of the LOS velocities Scatter plot of optical depth and LOS velocities with the corresponding histograms. Red dots and histograms: on May 29 at 14:41 UT. Green (blue) dots and histograms: on May 30 at 13:46 UT (14:29 UT). The ROI used is marked in Figure 2.are -14.7 and -19.5 km s −1 , respectively. At 16:00 UT, the mean value of the LOS velocities reached -50.3 km s −1 .
Figure 10 .Figure 11 .
1011ChroTel He I 10830Å Dopplergrams on May 30, 2017. Quiet sun regions are noisy due to the shallow line depth. Left panel: the filament had LOS velocities close to zero at 07:24 UT. The filament is still discernible due to its better signalto-noise ratio than its surroundings. Middle panel: the whole filament was rising during the first FIRS observation on May 30 at 13:57 UT. Right panel: the whole filament was rising faster during the second FIRS observation on May 30 at 14:45 UT. Left panel: Time slices of the SDO/AIA 304Å observations on May 30. Middle panel: time slices of the STEREO-A/EUVI 304Å observations. Both 304Å observations are shown on an inverted colorscale. The blue vertical lines indicate the time range of the DST observations. Right panel: time slices of the STEREO-A/COR2 white-light observations. Blue dots show the position of the filament/CME front.
Figure 12 .
12Upper panel: evolution of filament height. Error bars represent 3σ. The orange rectangle at the top shows the time range of the animation that overlaps with the panel, while blue and green rectangles indicate the time range of the DST observations. Lower panel: evolution of filament velocity. The black vertical bars representing FIRS velocity show 10th/90th percentile at lower/upper ends. The horizontal bars mark the observing time with median value of velocities. Error bars for EUVI-A, COR2-A, and ChroTel represent 1σ.
Figure 13 .
13Time slices of the SDO/AIA 304Å observations. The slice position is shown as white line and a white dot marking the starting point in the left panel of Figure 5. The green solid line marks the time of the end of the last FIRS observation at 15:02 UT. After 15:30 UT, there are a number of dark stripes that are parallel to the blue dashed line that may indicate the untwisting motion of the filament. The gradient of the blue dashed line is ≈ 17 km s −1 .
Table 1 .
1FWHM and Temperature ValuesT [kK] / ξ [km/s]
6 / 11.3
10 / 10.5
20 / 8.3
Day
FWHM [Å]
T [kK]
T [kK]
T [kK]
min ave max min ave max min ave max min ave max
29
0.55 0.75 1.07
-
6
46
-
10
50
4
20
60
30
0.56 0.86 1.53
-
19
125
-
23
129
4
33
139
. CONCLUSIONS
Acknowledgements. We wish to thank the anonymous referee for their constructive comments that helped with the clarity of the arguments presented, and generally improved the quality of the manuscript. This work was funded by NSF grant 1839306. Sunspot Solar Observatory is a multi institution consortium that is funded by multiple entities including NSF (1649052, 1945705) Pre-EST (no. 739500) as well as support from the Leverhulme Trust for an Early-Career Fellowship (ECF-2014-792) and is grateful to the Science Technology and Facilities Council for the award of an Ernest Rutherford Fellowship (ST/R003246/1). D.P.C. was partially supported through NSF grant AGS-1413686.
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| []
|
[
"Towards fully-fledged quantum and classical communication over deployed fiber with up-conversion module",
"Towards fully-fledged quantum and classical communication over deployed fiber with up-conversion module"
]
| [
"Davide Bacco \nDepartment of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark\n",
"Ilaria Vagniluca \nDepartment of Physics \"Ettore Pancini\"\nUniversity of Naples \"Federico II\"\n80126NaplesIT\n\nCNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly\n",
"Daniele Cozzolino \nDepartment of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark\n",
"Søren M M Friis \nNLIR ApS\nHirsemarken 1 1st floor3520FarumDenmark\n",
"Lasse Høgstedt \nNLIR ApS\nHirsemarken 1 1st floor3520FarumDenmark\n",
"Andrea Giudice \nMicro Photon Devices S.r.l\nvia Antonio Stradivari 439100BolzanoItaly\n",
"Davide Calonico \nI.N.Ri.M\nIstituto Nazionale di Ricerca Metrologica\nTorinoItaly\n",
"Francesco Saverio Cataliotti \nCNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly\n\nLENS and Dipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone1 -50019Sesto FiorentinoItaly\n\nQTI SRL\nLargo Enrico Fermi6 -50125FirenzeItaly\n",
"Karsten Rottwitt \nDepartment of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark\n",
"Alessandro Zavatta \nCNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly\n\nLENS and Dipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone1 -50019Sesto FiorentinoItaly\n\nQTI SRL\nLargo Enrico Fermi6 -50125FirenzeItaly\n"
]
| [
"Department of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark",
"Department of Physics \"Ettore Pancini\"\nUniversity of Naples \"Federico II\"\n80126NaplesIT",
"CNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly",
"Department of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark",
"NLIR ApS\nHirsemarken 1 1st floor3520FarumDenmark",
"NLIR ApS\nHirsemarken 1 1st floor3520FarumDenmark",
"Micro Photon Devices S.r.l\nvia Antonio Stradivari 439100BolzanoItaly",
"I.N.Ri.M\nIstituto Nazionale di Ricerca Metrologica\nTorinoItaly",
"CNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly",
"LENS and Dipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone1 -50019Sesto FiorentinoItaly",
"QTI SRL\nLargo Enrico Fermi6 -50125FirenzeItaly",
"Department of Photonics Engineering\nCenter for Silicon Photonics for Optical Communication (SPOC)\nTechnical University of Denmark\n2800Kgs. LyngbyDenmark",
"CNR -Istituto Nazionale di Ottica (CNR-INO)\nLargo E. Fermi6 -50125FirenzeItaly",
"LENS and Dipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone1 -50019Sesto FiorentinoItaly",
"QTI SRL\nLargo Enrico Fermi6 -50125FirenzeItaly"
]
| []
| Quantum key distribution (QKD), the distribution of quantum secured keys useful for data encryption, is expected to have a crucial impact in the next decades. However, although the notable achievements accomplished in the last twenty years, many practical and serious challenges are limiting the full deployment of this novel quantum technology in the current telecommunication infrastructures. In particular, the copropagation of quantum signals and high-speed data traffic within the same optical fiber, is not completely resolved, due to the intrinsic noise caused by the highintensity of the classical signals. As a consequence, current co-propagation schemes limit the amount of classical optical power in order to reduce the overall link noise. However, this ad-hoc solution restrains the overall range of possibilities for a large scale QKD deployment. Here, we propose and demonstrate a new method, based on up-conversion assisted receiver, for co-propagating classical light and QKD signals. In addition, we compare the performances of this scheme with an off-the-shelf quantum receiver, equipped with a standard InGaAs detector, over different lengths of an installed fiber link. Our proposal exhibits higher tolerance for noise in comparison to the standard receiver, thus enabling the distribution of secret keys in the condition of 4 dB-higher classical power. arXiv:2106.05073v1 [quant-ph] 9 Jun 2021 | 10.1002/qute.202000156 | [
"https://arxiv.org/pdf/2106.05073v1.pdf"
]
| 235,376,815 | 2106.05073 | 207131cf1a21876c8132a28cc535d488871f3fd5 |
Towards fully-fledged quantum and classical communication over deployed fiber with up-conversion module
Davide Bacco
Department of Photonics Engineering
Center for Silicon Photonics for Optical Communication (SPOC)
Technical University of Denmark
2800Kgs. LyngbyDenmark
Ilaria Vagniluca
Department of Physics "Ettore Pancini"
University of Naples "Federico II"
80126NaplesIT
CNR -Istituto Nazionale di Ottica (CNR-INO)
Largo E. Fermi6 -50125FirenzeItaly
Daniele Cozzolino
Department of Photonics Engineering
Center for Silicon Photonics for Optical Communication (SPOC)
Technical University of Denmark
2800Kgs. LyngbyDenmark
Søren M M Friis
NLIR ApS
Hirsemarken 1 1st floor3520FarumDenmark
Lasse Høgstedt
NLIR ApS
Hirsemarken 1 1st floor3520FarumDenmark
Andrea Giudice
Micro Photon Devices S.r.l
via Antonio Stradivari 439100BolzanoItaly
Davide Calonico
I.N.Ri.M
Istituto Nazionale di Ricerca Metrologica
TorinoItaly
Francesco Saverio Cataliotti
CNR -Istituto Nazionale di Ottica (CNR-INO)
Largo E. Fermi6 -50125FirenzeItaly
LENS and Dipartimento di Fisica e Astronomia
Università di Firenze
Via G. Sansone1 -50019Sesto FiorentinoItaly
QTI SRL
Largo Enrico Fermi6 -50125FirenzeItaly
Karsten Rottwitt
Department of Photonics Engineering
Center for Silicon Photonics for Optical Communication (SPOC)
Technical University of Denmark
2800Kgs. LyngbyDenmark
Alessandro Zavatta
CNR -Istituto Nazionale di Ottica (CNR-INO)
Largo E. Fermi6 -50125FirenzeItaly
LENS and Dipartimento di Fisica e Astronomia
Università di Firenze
Via G. Sansone1 -50019Sesto FiorentinoItaly
QTI SRL
Largo Enrico Fermi6 -50125FirenzeItaly
Towards fully-fledged quantum and classical communication over deployed fiber with up-conversion module
† These authors contributed equally to this work * [email protected]
Quantum key distribution (QKD), the distribution of quantum secured keys useful for data encryption, is expected to have a crucial impact in the next decades. However, although the notable achievements accomplished in the last twenty years, many practical and serious challenges are limiting the full deployment of this novel quantum technology in the current telecommunication infrastructures. In particular, the copropagation of quantum signals and high-speed data traffic within the same optical fiber, is not completely resolved, due to the intrinsic noise caused by the highintensity of the classical signals. As a consequence, current co-propagation schemes limit the amount of classical optical power in order to reduce the overall link noise. However, this ad-hoc solution restrains the overall range of possibilities for a large scale QKD deployment. Here, we propose and demonstrate a new method, based on up-conversion assisted receiver, for co-propagating classical light and QKD signals. In addition, we compare the performances of this scheme with an off-the-shelf quantum receiver, equipped with a standard InGaAs detector, over different lengths of an installed fiber link. Our proposal exhibits higher tolerance for noise in comparison to the standard receiver, thus enabling the distribution of secret keys in the condition of 4 dB-higher classical power. arXiv:2106.05073v1 [quant-ph] 9 Jun 2021
Introduction
Quantum key distribution (QKD) aims at distributing unconditional secure keys, useful for protecting our data communications [1,2]. By exploiting the properties of quantum mechanics, it is possible to deliver information theoretic secure keys which can be used for the encryption and decryption of data and messages. A conspicuous number of implementations (e.g., discrete variable, continuous variable, differential phaseshift modulation) and in-field experiments have been carried out in the last decades, demonstrating the feasibility of such technology [3][4][5][6][7][8][9]. However, multiples open problems are still limiting the full deployment of QKD in real-world applications. For example, the low key generation rate and the co-existence with the already-existing infrastructure for optical communication, are the central points for a successful implementation of this technology on a large-scale. The first limitation can be surmounted either by improving the state-of-art devices (i.e., by employing higher repetition rate transmitters or better performing singlephoton detectors), or by adopting novel quantum communication schemes with multidimensional modulation [10]. In particular, multiple degrees of freedom of light can be employed simultaneously to enlarge the Hilbert space dimensions, thus increasing the information capacity of single photons and enhancing the secret key rate [11][12][13][14][15][16]. Among the various degrees of freedom to be adopted for quantum communication, time-bin encoding is the most suitable for propagation in single-mode fiber links [17,18]. The second open problem, the compatibility with existing telecommunication infrastructures, is still very challenging although many solutions have been tested. For example, in order to co-propagate classical and quantum signals within the same fiber, various approaches can be adopted: time-division multiplexing, space-multiplexing, polarization multiplexing and wavelength division multiplexing (WDM) [19][20][21][22]. Both time and space approaches are interesting and promising for future development, but the most common and versatile in the current telecommunication networks is the frequency multiplexing for serving multiple users [23]. In the WDM technique, different wavelengths in the C-band are used for sending different data or to distribute communication between different users. The general idea is depicted in Figure 1, in which multiple transmitters (Classical TX λ 1,...,N ) are combined in a dense wavelength division multiplexer filter and sent over an optical fiber link. Within the N transmitters, λ 3 is used for quantum communication and in particular for QKD. At the other end of the fiber link, a similar filter is used to separate all the different wavelengths. Although this configuration is very convenient from a practical point of view, the quantum signal seriously suffers from the proximity of the high-intensity classical light, which generates a large amount of extra noise and thus it limits the overall performance of the quantum communication.
To be more precise, the interaction of high-intensity laser with the optical devices and fiber link generates photons at different wavelengths (scattering Raman, Brillouin and Rayleigh) which can survive to the DWDM filter, thus resulting in a source of noise for the fragile quantum signal [24]. In addition, the standard components for optical communication (i.e., optical filters, attenuators, isolators, etc..) are designed and tested for classical intense signals, which have less requirements in terms of loss and extinction ratio. We believe that a proper design of new components (quantum custom components) would help in the development of large-scale quantum networks. As a matter of that, the co-propagation of classical and quantum light is very challenging and the most common and practical solution is lowering the amount of classical power which is injected in the fiber, in order to reduce the scattering effects. However, this solu-tion cannot always be adapted since in many classical networks the required input power is 0 dBm, which is the standard value for classical optical communication. An alternative solution is represented by the wide-range wavelength multiplexing, in which is possible to combine the already existing data traffic in the C-band with a quantum transmitter in the O-band, as demonstrated in previous works [21]. Another effective approach is to employ continuous-variable (CV) QKD protocols, where homodyne and heterodyne detection schemes are exploited instead of single-photon detectors, thus enabling a powerful and intrinsic filter of the optical mode related to quantum signals [25][26][27]. However, CV-QKD is still limited in terms of secret key rate achievable and transmission distances, as well as of practical security proofs, with respect to the more advanced and long-studied discrete-variable (DV) QKD. Therefore, the challenge of exploiting the existing infrastructure for dense wavelength division multiplexing of quantum and classical signals in the C-band, for DV-QKD protocols, is still unsolved. In this work, we propose and demonstrate the possibility of co-propagating classical and quantum light through adjacent channels in a dense-wavelength multiplexing approach, by exploiting the intrinsic filter of a frequency up-conversion receiver at the single-photon level, combined with the timing performances of silicon single-photon avalanche detector. In addition, we compared our new scheme with an off-the-shelf quantum receiver, equipped with a standard indium gallium arsenide (InGaAs) detector, over different channel lengths of a metropolitan deployed fiber .
Protocol
The QKD protocol used in this experiment is the threestate BB84 protocol with time-bin encoding [9,28]. The quantum states belonging to Z basis are used for the key generation process, and the X basis is used for the security check. More specifically, in the Z basis, one of the two time bins (early and late) is occupied by a weak coherent state, while the third state in the X basis is the equal superposition of the Z basis states with zero relative phase, as reported in the top-left corner of Figure 2. The security of the three-state protocol using finite-key analysis against general attacks has been demonstrated and combined with a very efficient onedecoy state scheme, in order to detect photon number splitting attacks [29,30]. The secret key rate (SKR) length ( ) per privacy amplification block is given by:
≤ D Z 0 + D Z 1 1 − h φ Z 1 − λ EC − 6 log 2 (19/ sec ) − log 2 (2/ corr )(1)
where D Z 0 and D Z 1 are the lower bounds of vacuum events and single-photon events in the Z basis, h(·) is the binary entropy function, φ Z 1 is the upper bound on the phase error rate and λ EC is the number of bits that are publicly announced during error correction [29]. Finally, sec and corr are the secrecy and correctness parameters. In our computations we used a block size of 10 7 bits and sec = corr = 10 −9 .
Experimental Setup
In order to validate our hypothesis (i.e. up-conversion detection is more robust to spurious effects), we have tested the three-state time-bin protocol (as described above), exploiting weak coherent pulses and one-decoy method, combined with a classical laser co-propagating in the same fiber link. As illustrated in Figure 2, the experimental setup consists of two optical transmitters (classical CC and quantum-Alice) and three receivers (one classical, and two quantum: Bob 1 and Bob 2) connected by a metropolitan dark-fiber link in a loop-back configuration. A fiber mirror is used to reflect the light back to the European Laboratory for Non-linear Spectroscopy (LENS), where the transmitters and receivers are located [9]. The installed fiber link is part of a fiber backbone provided by the Italian National Institute of Metrological Research (INRIM). In particular, we have used a QKD transmitter composed of a telecom laser at 1555.70 nm (channel 27 of the ITU-T 200 GHz grid) followed by two intensity modulators and a phase modulator [31]. The two intensity modulators (IMs) are used for carving out the different time-bins and to implement the decoy state method. The phase modulator (PM) is used to set a random phase between different time-slots in order to assure the security against coherent attacks. The electrical outputs used to drive the IMs are provided by a field programmable gate array (FPGA), giving a state preparation rate of 595 MHz. Electrical pulse width is approximately 80 ps, whereas the obtained optical pulse width is around 100 ps. The PM is driven by a digital-to-analog converter which uses 8 bit to obtain 2 8 − 1 different phase values. Furthermore, a pseudo random binary sequence of 2 12 − 1 bits is used as a key generator, although a quantum random number generator should be adopted in a real implementation [32,33].
Subsequently to the IMs and PM, a variable optical attenuator (ATT) is employed to decrease the mean photon number per pulse to the quantum regime. The second transmitter (classical, CC in Figure 2) consists of a continuous wave (CW) laser at 1557.36 nm (channel 25 of the ITU-T 200 GHz grid) to emulate a classical communication link. The CC transmitter is then composed by an optical isolator, a variable optical attenuator and a beam-splitter (BS), which allows to monitor in real-time the optical power co-propagated in the fiber link. To be noted that although we did not encode any data transmission on the classical channel, a CW laser is perfectly able to emulate such a system for our proof-of-concept experiment. The classical and the quantum light are then combined by means of a DWDM filter, whose output is connected to the deployed fiber. After being reflected back by the fiber mirror, the light passes again through the same DWDM device, which separates the two different wavelengths (quantum and classical). The classical light is then measured at the output of the BS using a photodiode. As illustrated in Figure 2, the experimental setup consists of two optical transmitters (classical CC and quantum-Alice) and three receivers (one classical, and two quantum: Bob 1 and Bob 2). On the contrary, the quantum light passes through the optical circulator (port 2) and is propagated through the exit (port 3). Here, we performed projective measurements in the Z and X bases. In particular, in the Z basis, the fiber is connected directly to one of the two detection schemes (Bob 1 or Bob 2), while and in the X basis the quantum states are sent to a freespace delay-line interferometer (with 4 dB loss) before reaching the detector, in order to monitor the relative phase within the two time-bins. The two single-photon detection schemes are, respectively, an up-conversion module [1] (from 1555.70 nm to 631.90 nm), followed by a silicon-based photon counter from Micro Photon Devices [2] and a free-running fiber-based InGaAs single-photon detector from ID Quantique (ID221). The detailed scheme of the up-conversion setup is reported in the Supplementary Material. Finally, a time-tagging unit, which is synchronized with the FPGA through an electrical clock signal, collects the measurement outputs from the detectors. Although the total length of a round-trip in the fiber is about 40 km, with a overall loss of 21 dB, we have decided to use only a portion of this fiber as a quantum channel, in order to emulate different link configurations (i.e., 3, 5 and 8 dB of channel loss). In addition, the loss of the two detection systems, combined with the loss of the interferometer, has limited our ability to distribute the quantum signals over the entire link. It is important to notice that these restrictions are not limiting the overall idea, but the same principle can be tested in longer fiber links by accurately designing the QKD setup. Another important point to be mentioned is that although the actual quantum channel is shorter than the overall length of the fiber, the noise introduced by the the classical light is generated over the entire fiber link. In this condition, the noise level is overestimated with respect to the typical case of application. The up-conversion module is built with a high-finesse laser cavity (confined by mirrors DM1-DM3) in which a Nd:YVO4 crystal emitting at 1064 nm is pumped by an external laser diode at 808 nm [36]. Inside the cavity, a 40 mm nonlinear crystal is located in such a way that the intra-cavity field propagates in the direction of the poling. The quantum light at 1555.70 nm (corresponding to channel 27 of the ITU-T grid) is focused into the nonlinear crystal, where it is up-converted into 631.90 nm, which exits the cavity through DM2. More details on the up-conversion setup are reported in the Supplementary Material. In Figure 3 we report the phase-matching profile of the up-conversion process, which acts as an intrinsic wavelength filter with a 3 dB bandwidth of 0.8 nm. In order to filter out the noisy nonlinear emission generated by the pump laser, four off-the-shelf optical filters (short-pass 650 nm, long-pass 600 nm, and two band-pass with 10 nm and 5 nm of bandwidth) are inserted before the free-space silicon-based single-photon counter (Micro Photon Devices, with a quantum efficiency of 40% around 632 nm [37]). In our experiment, the overall efficiency of the up-conversion detector (including the conversion efficiency, filtering, coupling and silicon detector) is approximately 2%, with an overall dark count rate of 11 kHz. With respect to our working point, the pump power at 1064 nm could be further increased to enhance the conversion efficiency (although, in this way, the dark count noise would be raised as well). On the contrary, the commercial InGaAs detector ex-hibits 20% efficiency and 700 Hz of intrinsic dark count rate. However, even though the conversion process adversely affects the signal-to-noise ratio of the silicon detector, the up-conversion receiver still outperforms the InGaAs detector in terms of timing performances, thanks to the higher count rate (77 ns dead time) and ultra-low timing jitter (34 ps) of the Micro Photon Device module [37]. By contrast, the InGaAs detector requires a longer dead time (20 µs) in order to avoid the high after-pulsing noise, which is further enhanced in the working condition of 20% detection efficiency (condition which, on the other hand, is necessary to optimize the timing jitter to ∼ 200 ps).
Up-conversion assisted detector
Noise evaluation and filtering
In order to set the wavelength of the classical laser, we have decided to characterize the noise generated in our experimental setup, including the metropolitan fiber link and the DWDM device. Based on the assumption that the quantum laser was fixed at channel 27 due to the up-conversion module (see Figure 3), we have decided to evaluate the amount of spurious light scattered within the bandwidth of channel 27, as a function of the wavelength of the classical laser in input. The experimental setup for this characterization is reported in Figure 4a). By using the 200 GHz DWDM filter and a tunable laser source, we have tested one-by-one all the different ITU-T channels from 21 to 51 (at 0 dBm of input power) and we have measured the count rate at the output of channel 27 with an InGaAs single-photon detector. The normalized noise counts (after removing the averaged dark counts of the detector) are reported in Figure 4b). It is clear that channel 25 was found to introduce the highest noise counts in the quantum channel. The reason of this could be the specific configuration of our experimental setup, including device imperfections of the DWDM filter. Anyway, the noise counts were found to be independent from polarization. Based on this result, channel 25 was selected as the wavelength of the classical laser in our experiment, in order to test the QKD protocol under the worst condition of noise. Furthermore, since the up-conversion unit is both polarization dependent (due to the nonlinear process) and wavelength dependent (see Figure 3), we have decided to include analogous advantages also in the InGaAs receiver, as depicted in Bob 1 setup, by adding off-the-shelf devices in front of the InGaAs single-photon detector.
Specifically, we employed a polarizing beam splitter (PBS) and a 100 GHz band-pass filter of channel 27, exhibiting 0.64 nm of 3 dB bandwidth and 80 dB of extinction ratio between channels 25 and 27. In addition, we decided to test both receivers in the condition of orthogonal polarization directions of quantum signals and residual classical noise incoming at the QKD detector. To do so, we have included a polarization controller (PC) in the CC transmitter, and we have tuned it in order to minimize the amount of noise counts into both detection systems. In Bob 1 setup, used to filter out the noise from classical light before the InGaAs detector, with an overall insertion loss of 6 dB. In Bob 2 setup, polarization and wavelength filtering are provided inherently by the up-conversion process. Another PC is put in front of both receivers, in order to align the polarization of quantum light with the filtered direction. To be noted that the polarization drift in a deployed fiber is slow with respect to the typical QKD acquisition time, as demonstrated by the long-term acquisitions reported in previous works [9,38]. Finally, temporal filtering of the time-bin windows is used to post-select the acquired clicks from both detectors.
Results
In order to test the QKD protocol, the experimental parameters (such as the mean photon number of signal and decoy states, and their probability of preparation) were mathematically optimized for the two types of receiver, in order to maximize the secret key rate achievable at the different channel lengths. The detailed values that we set are reported in the Supplementary Material. We experimentally measured the quantum bit error rate (QBER) exhibited by the two receivers in the two mutually unbiased bases, for different classical power levels at the DWDM input, ranging from -20 dBm to -8 dBm. The results are reported in Figure 5a), 5b), 5d), and 5e). After the collection of the data, by using Eq. (1) we estimated the secret key rate achievable with the two receivers, that is reported in Figure 5c) and 5f). The black line represents the numerical simulation of the QKD performance in the back-to-back configuration, i.e., without using the metropolitan fiber as transmission channel and without co-propagating the classical laser. The experimental data acquired in this configuration are given by the filled circles in Figure 5.
Discussion
In this proof-of-concept experiment, we tested the ability of the up-conversion-assisted QKD to tolerate more noise in the quantum channel, as compared to a standard detector. The final figure of merit is represented by the secret key rate, which is a fundamental parameter in the telecommunication system (the higher the key generation rate, the faster is the key refresh rate for data encryption). More specifically, we reported in Figure 5c) and 5d) the amount of secret key (bit/s) as a function of the channel losses for different classical power levels. It is quite clear that, although the upconversion scheme is inherently affected by high dark count rate (about 11 kHz) and low overall detection efficiency (around 2%), which also limit the transmission distance of QKD, the amount of tolerable classical power is 4 dB larger for this receiver, both at 3 and 5 dB channel loss. As an example, by considering 5 dB channel loss, Bob 1 can tolerate up to -12 dBm of input power, as shown in Figure 6b). Since the 5-dB quantum channel is only a portion of the overall 21-dB attenuation in the loop-back fiber link, the actual launch power of classical light in the quantum channel is -28 dBm, corresponding to -12 dBm of power at the DWDM input. On the other hand, at the same channel loss of 5 dB, the up-conversion assisted Bob 2 can tolerate up to -8 dBm of input power (corresponding to -24 dBm of launch power in the quantum channel). An overall gain of 4 dB in telecommunication systems is a big step since it makes possible at least to double the amount of data transmitted into the single mode fiber, in order to enhance the signal-to-noise ratio, thus decreasing the amount of errors. The same behaviour is reported in the case of 3 dB quantum channel, as reported in Figure 6a). This figure gives a clear indication that the up-conversion receiver for QKD can tolerate more optical power in the quantum channel, at least for short-link configurations. The advantage of the up-conversion unit, as compared to the standard QKD receiver, is the intrinsic filter in polarization and wavelength provided by the nonlinear crystal, that is combined also with the siliconbased single-photon detector, exhibiting far better timing performances than the InGaAs detector. In particular, the ultra-low timing jitter (35 ps) provided by the silicon detector, allows for a more efficient postselection filter of the time-bin windows, thus reducing the impact of dark and noise counts. In addition, the shorter dead time of silicon detector (77 ns) enables a higher click rate than the InGaAs detector, which is typically limited by saturation effect. Conversely, the up-conversion detector exhibits very high intrinsic noise, due to the multiple nonlinear effects and scattering generated by the intense laser pump at 1064 nm. With this setup, our up-conversion receiver can work only for short-distance QKD, below 6 dB channel loss, as shown in Figure 5f). Nonetheless, state-of-the art systems for up-conversion detectors of single-photon signals in the C band, have demonstrated a very high overall efficiency above 30%, with a dark count rate as low as 100 Hz, thus enabling quantum communication up to 45 dB of channel loss [39][40][41]. Furthermore, it is important to notice that although we have used a bulky and custom home-made system (which requires the pre-alignment of the free-space silicon detector), the nonlinear crystal could be integrated in photonic platforms for a more efficient and stable solution [42]. Regarding the InGaAs detector, we would like to stress once again that we have used a commercial device, equipped with an off-the-shelf band-pass filter of 100 GHz bandwidth, combined with a polarization beam splitter for polarization filtering. Ad-hoc components could be used and designed for improving the signalto-noise ratio of the InGaAs detector, but our idea was to compare the up-conversion unit, which intrinsically offers a filter action, with a standard commercial quantum receiver. Finally, another important point to be considered in the key generation process, is the amount of time required for the key establishment. In particular, the acquisition time necessary to collect a block size of raw key bits depends on the click rate of the detector, thus the acquisition is expected to be faster with the upconversion receiver. In our experiment, to collect a block size of 10 7 bits at 5 dB channel loss, the acquisition takes about 3 minutes with the up-conversion unit, and 10 minutes with the standard receiver.
In conclusion, we have demonstrated that by exploiting an up-conversion unit, in a quantum key distribution scheme, is possible to tolerate higher classical power in the optical channel compared to a standard InGaAs detector equipped with off-the-shelf filtering devices. This proof-of-concept experiment represents a pivotal step towards the full integration of quantum and classical light within the same infrastructure. Furthermore, we believe it can pave the way to wider quantum applications in the deployed infrastructure. Figure 5c) and 5f), as a function of the classical launch power that is actually injected into the quantum channel, a) at 3 dB channel loss and b) at 5 dB loss.
Supplementary Information
Towards fully-fledged quantum and classical communication over deployed fiber with up-conversion module Supplementary Here are reported the values that we set at the transmitter for each fiber channel, such as the mean photon number for signal and decoy states (µ1, µ2) and their relative probability (pµ 1 ). The state preparation rate is 595 MHz, while the probability to prepare and measure the Z basis at the transmitter and the receiver are fixed to 90% and 50%, respectively, for both detection systems.
Pulse width
One of the key point of the single photon detector is the timing jitter. In the case of time-bin encoding, detection jitter is important to determine the maximum repetition rate of the source and to decrease the amount of timing errors at the receiver side. In this specific configuration, the properties of the up-conversion module combined with the free-space single photon detector allows for a very sharp pulse-shape, as reported in Figure 2. In Figure 2 a) we reported the normalized count of our weak coherent state acquired by the InGaAs detector in non saturation regime and in the back-to-back configuration. The full-width-half maximum value is around 250 ps. On the contrary, the pulse acquired with the up-conversion unit combined with the silicon detector presents a FWHM value of about 130 ps, as reported in Figure 2 b). Thus, the silicon single photon detector allows an ultra precise time filtering technique, which limits the amount of spurious clicks in the time-bin window both from the dark counts event but also from the classical signals.
Supplementary Figure 2: Pulse width acquired with two different single-photon detector.
Figure 1 :
1Schematic of classical and quantum communication channels in a dense wavelength multiplexing scheme. Multiple classical transmitters are combined with a quantum one in a dense wavelength division multiplexing scheme. After the propagation through the communication channel, the different wavelengths are separated and measured.
Figure 2 :
2Setup of the experiment. In the top left corner we report the encoding scheme adopted in the experiment. Legend: Laser C: classical laser, ISO: isolator, ATT: attenuator, PC: polarization controller, BS: beam splitter, PD: photodiode, DWDM: dense wavelength division multiplexing filter, M: fiber mirror, Laser Q: quantum laser, IM: intensity modulator, PM: phase-modulator, CIRC: circolator, PBS: polarization beam splitter, PC: polarization controller, F: filter, UC: up-conversion scheme (full setup in the Supplementary information), InGaAs: single-photon detector based on indium gallium arsenide, Si: silicon based single-photon detector.
Figure 3 :
3Phase matching profile. Up-converted power (normalized) as a function of the input wavelength in the nonlinear crystal. The phase matching condition of the up-conversion process is optimized by 1555.70 nm of wavelength (channel 27 of the ITU-T grid), with a 3 dB bandwidth of 0.8 nm.
Figure 4 :
4Noise characterization of the experimental setup. Noise counts within the bandwidth of channel 27 are reported as a function of the different wavelength of a tunable laser entering the DWDM device.
Figure 5 :
5Experimental results. Each point represents the averaged quantum bit error rate and secret key rate, as a function of channel loss, experimentally acquired in the back-to-back configuration (B2B) and for the different power levels of the classical laser entering in the DWDM device, ranging from -20 dBm to -8 dBm. a), b), e), f ): quantum bit error rate in the Z and X basis, measured with the InGaAs detector and the up-conversion unit, respectively. c), d): secret key rate achievable by the two different QKD receivers.
Figure 6 :
6Secret key rate as a function of the classical launch power. Here are reported the experimental data from
Table 1 :
1Experimental parameters of the QKD test.
AcknowledgementsThe authors would like to thank D. RuscaCompeting financial interestsThe authors declare that there are no competing interests.Data availabilityThe data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.
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* † , Ilaria Vagniluca, 3 † , Daniele Cozzolino 1 , Søren M. M. Friis. 7,8 , Karsten Rottwitt 1 , Alessandro Zavatta 3,7,82Davide CalonicoDavide Bacco 1 * † , Ilaria Vagniluca 2,3 † , Daniele Cozzolino 1 , Søren M. M. Friis 2 ,Lasse Høgstedt 2 ,Andrea Giudice 3 ,Davide Calonico 6 , Francesco Saverio Cataliotti 3,7,8 , Karsten Rottwitt 1 , Alessandro Zavatta 3,7,8
Center for Silicon Photonics for Optical Communication (SPOC). Kgs. Lyngby, DenmarkDepartment of Photonics Engineering, Technical University of Denmark1 Center for Silicon Photonics for Optical Communication (SPOC), Department of Photonics Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark.
80126 Naples, IT 3 CNR -Istituto Nazionale di Ottica (CNR-INO). Largo E. Fermi, 6 -50125 Firenze, Italy; Farum, Denmark3520Department of Physics "Ettore Pancini", University of Naples "Federico IIHirsemarken 1 1st floorDepartment of Physics "Ettore Pancini", University of Naples "Federico II", 80126 Naples, IT 3 CNR -Istituto Nazionale di Ottica (CNR-INO), Largo E. Fermi, 6 -50125 Firenze, Italy. 4 NLIR ApS, Hirsemarken 1 1st floor, 3520 Farum, Denmark.
. Qti Srl, Largo Enrico Fermi, Firenze, Italy† These authors contributed equally to this work * [email protected] SRL, Largo Enrico Fermi, 6 -50125 Firenze, Italy. † These authors contributed equally to this work * [email protected]
Up-conversion scheme. Up-conversion scheme
In addition, the DM2 is AR coated for near-visible light up to 1000 nm, and DM3 is AR coated for 808 nm. Inside the cavity there is a 40 mm long nonlinear crystal (PPLN) which is located such that the intra-cavity field propagates in the direction of the poling. The nonlinear crystal is AR coated for 1064 nm on both end facets and AR coated for MIR wavelengths (left side on Figure1) and AR coated for the up-converted wavelengths (the right side Figure1). The quantum light at 1555.70 nm (channel 27 of the ITU-T grid) is focused into the non-linear crystal where it is up-converted to the visible regime, and it exits the cavity through DM2 at 631.90 nm. Furthermore, in order to increase the signal-to-noise ratio and to limit the different noise generated in the up-conversion unit, we have used four off-the-shelf optical filters (low-pass, high-pass filter, band-pass 5 nm and band-pass 10 nm). After the filters the light is collected by a free-space silicon-based photon counter from Micro Photon Devices, with a peak efficiency of 40% around 632 nm [2]. The overall efficiency of the up-conversion unit, including the filter stage and the single-photon detector efficiency and coupling. One of the two receivers presented in the main text, is realized with a frequency up-conversion unit which is depicted in Figure 1. The up-conversion module is built with a high-finesse laser cavity confined by mirrors DM1-DM3 in which the gain medium is a Nd:YVO4 crystal. The filter (F) inside the cavity consists of four band-stop mirrors centered at 1064 nm that remove at eight orders of magnitude of residual laser diode light and fluorescence generated from the crystal. More specifically, one mirror is concave with a focal length of 75 mm in order to make the entire cavity stable and to reduce the spot size of the incoming beam size of 180 µm inside the nonlinear crystal. The three mirrors DM1-DM3 are all high-reflection (HR) coated for 1064 nm and additionally the first mirror (DM1) is anti-reflection (AR) coated for mid-infrared wavelengths (MIR). is approximately 5% at the maximum pump power. In our experiment, the working point of the up-conversion detector was set to 2% efficiency, in order to decrease the dark count rate generated by the intense pump laserOne of the two receivers presented in the main text, is realized with a frequency up-conversion unit which is depicted in Figure 1. The up-conversion module is built with a high-finesse laser cavity confined by mirrors DM1-DM3 in which the gain medium is a Nd:YVO4 crystal, emitting at 1064 nm and pumped by an external laser diode at 808 nm [1]. The filter (F) inside the cavity consists of four band-stop mirrors centered at 1064 nm that remove at eight orders of magnitude of residual laser diode light and fluorescence generated from the crystal. More specifically, one mirror is concave with a focal length of 75 mm in order to make the entire cavity stable and to reduce the spot size of the incoming beam size of 180 µm inside the nonlinear crystal. The three mirrors DM1-DM3 are all high-reflection (HR) coated for 1064 nm and additionally the first mirror (DM1) is anti-reflection (AR) coated for mid-infrared wavelengths (MIR). In addition, the DM2 is AR coated for near-visible light up to 1000 nm, and DM3 is AR coated for 808 nm. Inside the cavity there is a 40 mm long nonlinear crystal (PPLN) which is located such that the intra-cavity field propagates in the direction of the poling. The nonlinear crystal is AR coated for 1064 nm on both end facets and AR coated for MIR wavelengths (left side on Figure1) and AR coated for the up-converted wavelengths (the right side Figure1). The quantum light at 1555.70 nm (channel 27 of the ITU-T grid) is focused into the non-linear crystal where it is up-converted to the visible regime, and it exits the cavity through DM2 at 631.90 nm. Furthermore, in order to increase the signal-to-noise ratio and to limit the different noise generated in the up-conversion unit, we have used four off-the-shelf optical filters (low-pass, high-pass filter, band-pass 5 nm and band-pass 10 nm). After the filters the light is collected by a free-space silicon-based photon counter from Micro Photon Devices, with a peak efficiency of 40% around 632 nm [2]. The overall efficiency of the up-conversion unit, including the filter stage and the single-photon detector efficiency and coupling, is approximately 5% at the maximum pump power. In our experiment, the working point of the up-conversion detector was set to 2% efficiency, in order to decrease the dark count rate generated by the intense pump laser.
Supplementary Figure 1: Detailed setup of the up-conversion scheme. Laser Q: quantum laser, DM1,2,3: dichroic mirrors, M: mirror, LPF: low-pass filter, HPF: high-pass filter, PH: pinhole, NF: notch filters. Supplementary ReferencesSupplementary Figure 1: Detailed setup of the up-conversion scheme. Laser Q: quantum laser, DM1,2,3: dichroic mirrors, M: mirror, LPF: low-pass filter, HPF: high-pass filter, PH: pinhole, NF: notch filters. Supplementary References
Upconversion-based mid-infrared spectrometer using intra-cavity LiNbO 3 crystals with chirped poling structure. M M Søren, Lasse Friis, Høgstedt, Optics letters. 44Søren MM Friis, Lasse Høgstedt, "Upconversion-based mid-infrared spectrometer using intra-cavity LiNbO 3 crystals with chirped poling structure" Optics letters, vol. 44.17, p. 4231-4234, 2019.
High-rate photon counting and picosecond timing with silicon-SPAD based compact detector modules. A Giudice, M Ghioni, R Biasi, F Zappa, S D Cova, P Maccagnani, A Gulinatti, Journal of Modern Optics. 542A. Giudice, M. Ghioni, R. Biasi, F. Zappa, S. D. Cova, P. Maccagnani, and A. Gulinatti, "High-rate photon counting and picosecond timing with silicon-SPAD based compact detector modules" Journal of Modern Optics, vol. 54, no. 2, pp. 225-237, 2007
| []
|
[
"Dimensional hierarchy of fermionic interacting topological phases",
"Dimensional hierarchy of fermionic interacting topological phases"
]
| [
"Raquel Queiroz \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n",
"Eslam Khalaf \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n",
"Ady Stern \nDepartment of Condensed Matter Physics\nWeizmann Institute of Science\n76100RehovotIsrael\n"
]
| [
"Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany",
"Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany",
"Department of Condensed Matter Physics\nWeizmann Institute of Science\n76100RehovotIsrael"
]
| []
| We present a dimensional reduction argument to derive the classification reduction of fermionic symmetry protected topological phases in the presence of interactions. The dimensional reduction proceeds by relating the topological character of a d-dimensional system to the number of zero-energy bound states localized at zero-dimensional topological defects present at its surface. This correspondence leads to a general condition for symmetry preserving interactions that render the system topologically trivial, and allows us to explicitly write a quartic interaction to this end. Our reduction shows that all phases with topological invariant smaller than n are topologically distinct, thereby reducing the non-interacting Z classification to Zn. | 10.1103/physrevlett.117.206405 | [
"https://arxiv.org/pdf/1601.01596v2.pdf"
]
| 20,523,849 | 1601.01596 | e684871f7b69cb21b2aef3282ce26985928d228e |
Dimensional hierarchy of fermionic interacting topological phases
14 Nov 2016
Raquel Queiroz
Max-Planck-Institut für Festkörperforschung
Heisenbergstrasse 1D-70569StuttgartGermany
Eslam Khalaf
Max-Planck-Institut für Festkörperforschung
Heisenbergstrasse 1D-70569StuttgartGermany
Ady Stern
Department of Condensed Matter Physics
Weizmann Institute of Science
76100RehovotIsrael
Dimensional hierarchy of fermionic interacting topological phases
14 Nov 2016(Dated: November 15, 2016)
We present a dimensional reduction argument to derive the classification reduction of fermionic symmetry protected topological phases in the presence of interactions. The dimensional reduction proceeds by relating the topological character of a d-dimensional system to the number of zero-energy bound states localized at zero-dimensional topological defects present at its surface. This correspondence leads to a general condition for symmetry preserving interactions that render the system topologically trivial, and allows us to explicitly write a quartic interaction to this end. Our reduction shows that all phases with topological invariant smaller than n are topologically distinct, thereby reducing the non-interacting Z classification to Zn.
Topological phases of matter are presently one of the main research topics in condensed matter physics. The discovery of time-reversal invariant topological insulators (TIs) [1,2] and superconductors (TSCs) has led, among other results, to a systematic classification of topological phases of non-interacting fermions in a general spatial dimension and symmetry class [3][4][5]. Similar to the quantum Hall states, TIs and TSCs are gapped systems hosting gapless modes on their surfaces, insensitive to small perturbations [6,7]. However, the robustness of these surface states relies on the existence of discrete antiunitary symmetries, either time-reversal (TRS), T , and (or) particle-hole symmetry, C, being denoted symmetry protected topological (SPT) phases. A natural question is whether SPT phases are robust in the presence of interactions that do not break explicitly or spontaneously their protecting symmetries. Fidkowski and Kitaev [8] provided the first example in which an interaction could adiabatically connect two states that are topologically distinct in its absence. This example involved one dimensional (1D) spinless p-wave superconductor with TRS, where it was shown that in the presence of an interaction involving eight Majorana operators the noninteracting topological classification is reduced from Z to Z 8 . Two gapped quadratic Hamiltonians whose topological invariants differ by eight can be connected by a gap-preserving trajectory that involves the interaction. Consequently, eight Majorana zero modes localized at the interface between these two phases will be gapped by the interaction. Subsequently, the full topological classification of interacting 1D SPT phases was obtained [9][10][11], as well as some examples in two [12][13][14] and three dimensions [15][16][17]. An exhaustive classification of SPT phases still remains subject of intense research [18][19][20][21][22][23][24], where methods such as cobordism, group (super)cohomology and non-linear sigma model have been recently used.
In this work, we study the reduction of the topological classification of interacting fermionic phases in a given dimension and symmetry class, when this classification in the absence of interactions is Z. That is, we classify interacting fermionic phases which are adiabatically connected to noninteracting ones. We derive and employ a correspondence be-tween the topological invariant ν of the d-dimensional bulk and the number n 0D of zero-energy bound states localized at zero-dimensional (0D) topological defects on its surface. We argue that when the zero modes, localized in any 0D topological defect, are gapped by an interaction, the state becomes topologically trivial under the same interaction. This constitutes a general criterion on any interaction that allows for a change in the topological sector. The argument is made explicit by piercing the surface with a lattice of defects and presenting a concrete example of a quartic interaction that gaps a general surface. Our analysis reproduces the classification obtained in [23], without making an assumption on the form of the interaction. Thus, we conclude that lifting the restriction of quartic interactions does not alter the classification.
There is an important distinction between classes with Z topology in even and odd dimensions. The former host chiral modes on the boundaries whose gapping is forbidden by the conservation of energy; while the latter host nonchiral (helical) boundary modes whose protection depends on the presence of chiral symmetry. Following Refs. [12,13], we construct nonchiral SPT phases in even dimensions by combining two systems with opposite chirality and Chern invariant, adding a Z 2 unitary symmetry R preventing the coupling of modes with opposite chirality. The resulting classes, which we refer to as prime ( ) classes, naturally generalize SPT phases to even dimensions, making our analysis equally applicable to all dimensions.
We pursue an analogous path to Ref. [25], in its description of the ten-fold classification of non-interacting topological states using a eight-or two-hour "Bott clock". We relate the topological character of a SPT class with symmetry s in dimension d to one with s + 1 and d + 1 (Fig. 1), and find that in the presence of interactions, Z is reduced to Z n , with n violating the clock periodicity. This is represented as a spiral clock in Figs
where n 0 = {4, 2, 4} for (a) to (c), respectively, and µ = 2 arXiv:1601.01596v2 [cond-mat.str-el] 14 Nov 2016 for classes BDI, D and DIII, when they are realized by triplet superconductors. For all other cases µ = 1.
We start by reviewing how the 0D zero modes at the ends of 1D systems reflect their topological invariants, and how they may be gapped by interactions [10,11,16]. We then introduce the 0D surface defects in two and three dimensions, and their reflection of the bulk topological invariant. Following that, we discuss the possible interaction-induced gapping of zero modes within such a defect, the gapping of a lattice of such defects and the derivation of the parameter n. Finally, we generalize to any dimension.
One dimension -We focus on Dirac Hamiltonians (DH) as representatives of the different topological sectors [26]. The minimal DH in class AIII reads
H AIII = dx c † (iσ z ∂ x + m(x)σ y )c, c = (c L , c R ) T , (2)
for c L,R complex fermion operators. Eq. (2) is invariant under the chiral symmetry defined by S −1 c L,R S = c † R,L and S −1 iS = −i. A 0D edge can be implemented by forcing the mass m(x) to change sign at x = 0. Choosing m(x) = tanh x without loss of generality, we obtain the zero energy localized operator ψ = dx(c L + c R )sech x, obeying S −1 ψ S = ψ † . For a set of AIII chains there is one zero mode at the end of each chain, so that n 0D = ν, i.e., the number of zero energy end modes equals the topological invariant. Inter-chain coupling is restricted by the symmetry. Mass terms M ij ψ † i ψ j are forbidden since hermiticity requires M ij = M * ji , while chiral symmetry requires M ij = −M * ji . Quartic terms, on the other hand, allow for a fully symmetric interaction between the chains
H int = V ψ † 1 ψ 2 ψ † 3 ψ 4 + h.c..(3)
Here the subscript enumerates the chains. The interaction (3) has a unique S-symmetric ground state separated by a gap V from the remaining states given by |0101 − |1010 , with 0, 1 the eigenvalues of ψ † i ψ i . Due to the absence of edge modes, we conclude that a 1D AIII system with ν = 4 becomes topologically trivial once the interaction (3) is included. Thus, n = 4 for d = 1 and class AIII.
Analogously, we can describe a chain in class BDI by Eq. (2), where the fermionic operators c are replaced by Majorana operators η. In this basis, charge conjugation C is the identity, and T = S. The edge Majorana bound states are formed by γ = dx(η L +η R )sech x, localized at x = 0. Pairing the operators γ i into complex fermions ψ i = γ 2i−1 + iγ 2i , we see that, Eq. (3) satisfies the symmetries of class BDI and can gap out groups of 8 Majoranas, reducing Z → Z 8 (n = 8). Similarly, we find n = 2 in class CII, where the Hamiltonian acts on spinful fermions with an intrinsic double degeneracy.
From one to two dimensions -In 2D we focus on class D . The analysis for A and C proceeds similarly. In all cases we find n 0D = ν, such that the step from d = 1 to FIG. 1. Classification of interacting fermionic SPT phases with Zn topology, with n given by Eq. (1) (blue for µ = 2). The symmetry classes are arranged to form a eight-or two-hour "Bott clock", starting at the 1D symmetry classes, (a) BDI, (b) CII and (c) AIII, tracing each series clockwise for increasing dimensions. At each even to odd dimension step, n is increased by a factor of 2. The 8(2)-hour periodicity of the real(complex) classes is not satisfied, leading to an infinite spiral. d = 2 does not increase n. Class D can be constructed by adding two copies of 2D p-wave superconductors with broken TRS, bearing electrons of opposite spin direction and gapless edge modes of opposite chirality. The resulting system has TRS with T 2 = −1, but an additional Z 2 unitary symmetry, R, distinguishes it from the class DIII, in which an even number of pairs of counter-propagating edge modes may be gapped without violating TRS [27]. The symmetry R can be physically implemented either as a conservation of S z modulo 2 (conservation of the parity (−1) N ↑ ) satisfying R 2 = 1 and {T , R} = 0 [12,13] or as a mirror symmetry satisfying R 2 = −1 and [T , R] = 0 [12] in which case D corresponds to the crystalline phase DIII+R. In both options, the combined operatorT = T R is antiunitary and satisfiesT 2 = 1. Here, we choose the first R realization. The D system has zero Chern number N ↑ + N ↓ = 0 but a non-zero spin Chern number 1 2 (N ↑ − N ↓ ), resulting in a Z topological classification at the non-interacting level.
d = 2 … AIII 4 4 8 8 A 0 d = 1 d = 2 d = 3 (a) (b) (c) 4 (8)(8)
The Hamiltonian for ν pairs of Majorana 1D edge modes is
H D = i dx ν a=1 η † a σ z ∂ x η a , η = (η ↑ , η ↓ ) T ,(4)
with T and R defined by
T −1 (η ↑ , η ↓ )T = (η ↓ , −η ↑ ) and R −1 (η ↑ , η ↓ )R = (η ↑ , −η ↓ ).
We relate it to the BDI Majorana chain studied above by adding the mass term m(x)η † σ y η at the edge [13]. Locally, the mass term breaks both T and R, but preserves the combinationT = T R. Consequently, the resulting edge Hamiltonian has the antiunitary symmetryT squaring to +1, representing spinless fermions (class BDI). As in the previous section, we choose m(x) to change sign at x = 0 to form a 0D mass defect and find ν Majorana bound states at x = 0. Thus, in the noninteracting limit, the 1D BDI system constructed at the boundary inherits the topology of the bulk 2D system of class D . If the mass oscillates in space m(x) = m 0 cos(qx), a lattice of 0D defects is formed with a separation of 2π/q. When each defect contains eight zero modes, the interaction (3) between the zero modes may be used to induce the topological transition from m 0 > 0 to m 0 < 0 without closing the energy gap [13] leading to n = 8. From two to three dimensions -We now follow the same construction to relate class DIII in 3D to its descendant 2D system class D and to its 0D surface defects. In this step we find a doubling of the ratio ν/n 0D , giving rise to a doubling of n. Class DIII represents superconductors with T 2 = −1. At the free fermion level, DIII has a Z classification where a bulk topological invariant ν is associated with ν helical gapless surface modes. The surface Hamiltonian reads
H DIII = i d 2 r ν a=1 η † a [σ z ∂ x + σ x ∂ y ]η a ,(5)
with ν taken to be even in the discussion below. A naive introduction of a surface mass term of the form m(r)η † σ y η results in a gapped surface of class D, with ν chiral edge modes along a 1D defect (a line along which m = 0), which cannot be gapped. Gapping requires two time-reversed copies of class D at the surface, obtained by grouping the ν surface modes into pairs defining an isospin doublet χ a = (η T 2a−1 , η T 2a ), and adding the mass term mχ † σ y ⊗ τ z χ, with τ i the Pauli matrices in isospin space. Taking the mass to have an opposite sign for the two isospin directions leads to a class D surface with counter-propagating chiral modes. Combining T with an isospin flip results in a modified TRS (Fig. 2(a)). The 2D surface Hamiltonian reads,
H = d 2 r ν/2 a=1 χ † a [i(σ z ∂ x + σ x ∂ y ) ⊗ τ 0 + m(r)σ y ⊗ τ z ]χ a .
(6) H is invariant under the modified TRS T = iσ y τ x K and under R = τ z satisfying T 2 = −1, R 2 = 1 and {T , R} = 0, which implements the D system described above. Choosing m(r) to change sign along the y-direction, m(y) = tanh y, we find the 1D gapless modes η ± = dy sech y v ± χ with v + = (1, 0, 0, 0) and v − = (0, 0, 0, 1). The Hamiltonian acting on η ± is given exactly by Eq.(4) with ± replacing the spin index. We can then conclude that on the non-interacting level, the 2D boundary D system inherits the topology of the 3D DIII system with an invariant ν/2 rather than ν. Adding an xdependent mass m(x) = tanh x, the 1D edge mode becomes gapped and a 0D defect is introduced with ν/2 zero modes.
The surface Hamiltonian (5) may be equivalently written in terms of Dirac matrices, to be conveniently generalized to higher dimensions. We construct the DH using Γ 1,...,5 matrices satisfying the usual Clifford algebra {Γ i , Γ j } = 2δ ij . The matrices are chosen to be symmetric and antisymmetric for odd and even i, respectively. The kinetic part of Eq. (6) becomes Γ 1 ∂ x + Γ 3 ∂ y , and TRS acts as T = Γ 2 K with T 2 = −1. By adding the mass term m(x)Γ 2 + m(y)Γ 4 , we obtain a 0D defect localized at points where both m(x) and m(y) change sign. The mass term breaks T of the original Hamiltonian but leads to an emergent antiunitary symmetry given byT = Γ 5 K withT 2 = 1. That is, the new Hamiltonian is in class BDI, hosting Majorana bound states at the 0D topological defect [25]. For m(x) = (−1) sx tanh x and m(y) = (−1) sy tanh y, the zero modes are given explicitly by γ = d 2 r sech x sech y vχ, with v the non-zero eigenvector of the projection operator P = 1
4 (1 + i(−1) sx Γ 1 Γ 2 )(1 + i(−1) sy Γ 3 Γ 4 ).
Here s x , s y are integers. The sign changes they introduce emerge naturally when the mass oscillates in space to introduce a lattice of defects, for example using the mass functions m(x) = cos qx and m(y) = cos qy.
A local interaction that renders the system topologically trivial must guarantee that the 0D defect zero modes are gapped by acting in the projected space of zero modes. This is required since the interaction matrix elements which couple the zero and high energy modes can be made arbitrarily small by an appropriate choice of m, and the elements coupling zero modes and bulk states vanish by locality. Hence, we conclude that when the defect contains less than 8 zero modes the topological sector is stable to any interaction. In the case where a 0D defect cannot be constructed, for example for odd values of ν, it is always possible to reduce the 2D gapless modes to a single 1D gapless mode coupled to a number of 0D zero modes. The 1D mode can never be gapped out by interactions, guaranteeing the stability of the topological sector. Put together, these considerations imply the stability of the 2D DIII surface with ν < 16 to any interaction.
For ν = 16, 0D defects containing 8 zero modes can be constructed and gapped by interactions. We can explicitly find one interaction that gaps the full surface by combining pairs of Majorana surface states into four complex fermions ψ i = χ 2i−1 + iχ 2i in analogy to the 0D case,
H int = d 2 r V (ψ † 1 Γ 5 ψ 2 )(ψ † 3 Γ 5 ψ 4 ),(7)
with Γ 5 = Γ 1 Γ 2 Γ 3 Γ 4 . This interaction reduces to the interaction (3) upon projecting to the space of zero modes in the defect. It respects both T andT symmetries.
The gapping procedure of the 2D surface is similar to the one employed to gap the 1D surface of the class D system at d = 2 [13]. In the absence of interaction within each 0D defect, tunnel coupling between zero modes in neighboring defects creates a spectrum that is gapless at zero energy. This spectrum is not identical to the one obtained in the absence of the mass terms, but its low energy characteristics are identical. The zero modes within each defect all share the same values of s x , s y and thus the interaction (7) respects the symmetry. When the interaction is stronger than the hopping terms between defects, the surface is gapped. This procedure bears similarity to the proliferation of monopoles used in Refs. [24,28].
Higher dimensions -We now generalize beyond d = 1, 2, 3 to obtain the complete classification of fermionic SPTs , as summarized in Fig. 1 and Eq. (1). We start with the complex series (Fig. 1(c)), and comment on the extension to real classes. We first identify the correspondence between ν and n 0D , and then introduce an interaction that gaps the surface once it is pierced with a lattice of 0D defects. The (d − 1)dimensional surface is described by
H = d d−1 r χ † (iα · ∇ + β · M) χ.(8)
Here, α = (α 1 , . . . , α d−1 ), β = (β 1 , . . . , β d−1 ) are Dirac matrices satisfying the Clifford algebra. In d spatial dimensions, the minimal massless DH with chiral symmetry has dimension 2 d 2 . This can be understood from the doubling of the bulk DH dimension from odd to even d, due to the intrinsic doubling of class A , and the fact that the surface and bulk DHs differ in dimension by a factor of two.
A gapless edge Hamiltonian of a system with a topological invariant ν may be constructed by ν copies of the edge Hamiltonian (8) with β = 0, enlarging the size of the matrix by a factor ν, giving the combined dimension ν2 d 2 . We seek the value ν for which a mass term that allows the formation of a 0D defect may be introduced.
A zero mode is an operator that commutes with (8). For a single 0D defect, the i-th component of the mass vector is chosen to satisfy M i = (−1) si tanh r i . The zero mode is then
γ = d d−1 r d−1 i=1 sech r i vχ,
where v corresponds to the non-zero eigenvector of the d − 1 commuting projection oper-
ators P si i = 1 2 (1 + i(−1) si α i β i ).
To obtain a single non-zero eigenvalue to this set of operators, we need an Hamiltonian of dimension 2 d−1 . The ratio of this dimension to the dimension ν2 for odd d.
As before, a group of four complex fermions in a single defect may be gapped by interactions. The surface may be gapped by piercing it by a lattice of 0D defects, each of which containing four zero modes, hence reducing the Z classification to Z n , with n = 2 for odd d.
When there are four zero modes in a single 0D defect, one interaction that reduces to Eq. (3) for any choice of s i 's (and therefore for any 0D defect in the lattice) reads
H int = d d−1 r V (ψ † 1 Γ 2d−1 ψ 2 )(ψ † 3 Γ 2d−1 ψ 4 ),(9)
where
Γ 2d−1 = d−1 i=1 α i β i .
For values of ν < n, a topologically protected 0D defect cannot be formed, or will host less than four zero modes that cannot be gapped by any interaction. Thus, any interaction involving less that n fermions will be either projected to one that cannot locally gap the surface, or one that is not relevant in the low energy subspace.
The analogous derivation of n for the real classes finds the same doubling of n 0D whenever d is increased by two, as described here for the complex classes. As explained below Eq. (1), however, the three Z-class series differ in n 0D in the d = 1 case, and this difference carries over to all larger dimensions. Furthermore, the value of n depends also on the nature of the zero modes being complex or Majorana fermions.
Conclusion -We showed that under interactions the classification of topological phases with chiral symmetry is reduced from Z to Z n , and reproduced the value of n derived in [23], summarized in Eq. (1) and Fig. 1. In our approach, the gapless surface separating topologically distinct phases is replaced by a gapless lattice of coupled 0D defects. These defects enclose zero modes, whose number n 0D is determined by dimension, bulk symmetries and bulk topological invariant ν. We identify the relation between ν and n 0D and find it to double with an increase of the dimension by two. The number of zero modes for d = 1 depends on the symmetry of the problem, leading to a difference between the the series (a)−(c) (Fig. 1). Our construction establishes the stability of topological phases with ν < n for any symmetry-preserving interaction, and provides a necessary and sufficient condition for ones that gap the nboundary modes. It turns out it is always possible to find a quartic interaction satisfying this condition for an arbitrary dimension and symmetry class, Eq. (9).
We acknowledge stimulating discussions with A.P. Schnyder, J.S. Hofmann and B.A. Bernevig. We acknowledge financial support by the European
.1(a)-(c), distinguished by their d = 1 symmetry classes: (a) BDI (b) CII and (c) AIII. Generally,
FIG. 2 .
2Illustration of the reduction scheme from a 3D DIII system to a 1D BDI system, hosting zero energy modes γ: (a) Iso-spin multiplet χa, Eq. (6), of gapped surface modes (light grey) with protected 1D counter-propagating chiral modes η± (blue and red lines), localized at y = 0. (b) Effective D system with helical mode η, Eq. (4). (c) Gapped 1D BDI system (darker gray) with zero mode (blue and red dots) at x = y = 0.
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This excludes the Sz conserving spin quantum Hall effect, for which the edge states cannot be gapped by an interaction. 29In the generalization to class A , we recover chiral symmetry by imposing an additional Z2 symmetry (e.g. Sz mod 2)In the generalization to class A , we recover chiral symmetry by imposing an additional Z2 symmetry (e.g. Sz mod 2). This excludes the Sz conserving spin quantum Hall effect, for which the edge states cannot be gapped by an interaction [29].
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[
"The consensus problem for opinion dynamics with local average random interactions",
"The consensus problem for opinion dynamics with local average random interactions"
]
| [
"M Gianfelice [email protected] \nDipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende\n",
"G Scola [email protected] \nDipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende\n",
"Roma Tre \nDipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende\n",
"Largo S Leonardo Murialdo \nDipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende\n",
"Roma \nDipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende\n"
]
| [
"Dipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende",
"Dipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende",
"Dipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende",
"Dipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende",
"Dipartimento di Matematica e Informatica Università della Calabria\nDipartimento di Matematica e Fisica Università degli Studi\nCampus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende"
]
| []
| We study the consensus formation for an agents based model, generalizing that originally proposed by Krause [Kr], by allowing the communication channels between any couple of agents to be switched on or off randomly, at each time step, with a probability law depending on the proximity of the agents' opinions. Namely, we consider a system of agents sharing their opinions according to the following updating protocol. At time t+1 the opinion X i (t + 1) ∈ [0, 1] of agent i is updated at the weighted average of the opinions of the agents communicating with it at time t. The weights model the confidence level an agent assign to the opinions of the other agents and are kept fixed by the system dynamics, but the set of agents communicating with any agent i at time t + 1 is randomly updated in such a way that the agent j can be chosen to belong to this set independently of the other agents with a probability that is a non increasing function of |X i (t) − X j (t)| . This condition models the fact that a communication among the agents is more likely to happen if their opinions are close. We prove that the system reaches consensus, i.e. as the time tends to infinity the agents' opinions will reach the same value exponentially fast. | 10.48550/arxiv.2204.05689 | [
"https://arxiv.org/pdf/2204.05689v1.pdf"
]
| 248,118,686 | 2204.05689 | 5cdc5374f7866716e6ffa040e558b5164634350b |
The consensus problem for opinion dynamics with local average random interactions
12 Apr 2022 April 13, 2022
M Gianfelice [email protected]
Dipartimento di Matematica e Informatica Università della Calabria
Dipartimento di Matematica e Fisica Università degli Studi
Campus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende
G Scola [email protected]
Dipartimento di Matematica e Informatica Università della Calabria
Dipartimento di Matematica e Fisica Università degli Studi
Campus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende
Roma Tre
Dipartimento di Matematica e Informatica Università della Calabria
Dipartimento di Matematica e Fisica Università degli Studi
Campus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende
Largo S Leonardo Murialdo
Dipartimento di Matematica e Informatica Università della Calabria
Dipartimento di Matematica e Fisica Università degli Studi
Campus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende
Roma
Dipartimento di Matematica e Informatica Università della Calabria
Dipartimento di Matematica e Fisica Università degli Studi
Campus di Arcavacata Ponte P. Bucci -cubo30B I-87036Arcavacata di Rende
The consensus problem for opinion dynamics with local average random interactions
12 Apr 2022 April 13, 2022
We study the consensus formation for an agents based model, generalizing that originally proposed by Krause [Kr], by allowing the communication channels between any couple of agents to be switched on or off randomly, at each time step, with a probability law depending on the proximity of the agents' opinions. Namely, we consider a system of agents sharing their opinions according to the following updating protocol. At time t+1 the opinion X i (t + 1) ∈ [0, 1] of agent i is updated at the weighted average of the opinions of the agents communicating with it at time t. The weights model the confidence level an agent assign to the opinions of the other agents and are kept fixed by the system dynamics, but the set of agents communicating with any agent i at time t + 1 is randomly updated in such a way that the agent j can be chosen to belong to this set independently of the other agents with a probability that is a non increasing function of |X i (t) − X j (t)| . This condition models the fact that a communication among the agents is more likely to happen if their opinions are close. We prove that the system reaches consensus, i.e. as the time tends to infinity the agents' opinions will reach the same value exponentially fast.
Introduction, notations and results
Opinion dynamics is a topic in applied mathematics which has witnessed a growing interest in the last decades. This is due to the possibility to describe the emergence of collective phenomena such as the reaching of consensus in a community of peers by means of simple models of interacting agents [CFL]. The literature on the subject concentrates mainly on two families of models: those cast in the framework of interacting particle systems (IPS) e.g. the voter model and the majority-vote process [Li], the Axelrod model and its generalizations [La], the Deffuant-Weisbuch model [DNAW], [Ha], [HH], and those belonging to the family of coupled dynamical systems (CDS) e.g. [Kr] (see [FF] for linear models).
The common feature of these models is that the interactions among the agents are designed in such a way that an agent adjust its opinion to that of its neighbours and that agents are more likely to interact with those sharing similar opinions. On the other hand, the main difference between IPS type models and CDS type models, aside form the fact that IPS type models typically evolve in continuous time while CDS type models evolve in discrete time, is that in CDS type models all the agents can change the value of their opinion at each time step while, in IPS type models, at a given time (usually at the tick of a Poisson clock), only the elements of a randomly chosen subset of agents are allowed to change the value of their opinion, and the rule under which the opinion of the selected agents are updated, being stochastic in general (see [CF] for a stochastic version of the Deffuant model), can be deterministic too [DNAW], [HH].
In this paper we present a model of consensus formation which, although it represents a modification of that originally proposed by Krause, it is defined by an updating rule of the agents opinions that recalls those characterizing IPS type models. In particular, at each time step, firstly the set of neighboring peers of any agent i is selected at random in such a way that the events that any two distinct agents belong to the neighbourhood of i are independent. Then, each agent update the value of its opinion to the average value of the opinions of its neighbours.
More precisely, we consider a collection of agents which form the set of vertices of a directed graph G := (V, E) , E ⊆ V × V. We assume that agent u communicate with agent v if the directed edge (u, v) is in E. Each agent hold an opinion (belief ) represented by a variable taking values in [0, 1] . Agent's beliefs evolve in time in such a way that the opinion X u (t + 1) of the agent labelled by the graph vertex u at time t + 1 is updated at the weighted average of the beliefs X v (t) of the agents communicating with u at time t. The weights appearing in the just mentioned average represent the quality of the information exchange among the agents and do not change in time. On the other hand, the set of the agents communicating with agent u at time t is randomly chosen according to a probability distribution which gives more chance to a communication exchange between agent u and agent v to happen if the value of their opinion are close. It is also natural to assume that the information exchange an agent has with itself is always maximal.
In the rest of the section we introduce the notation used throughout the paper, formally display the definition of the model and present the results obtained in the following sections about the emergence of consensus for the finite size system as well as the extension of these results when the size of the system is very large.
We stress that, as in [Kr], in this work, the dynamics of the state of the edges of G, representing the communication channels amid the agents, is synchronous, i.e. they are all updated at each time step. On the other hand, one may wish to modify the system's dynamics by letting the state of the edges of G to evolve under an asynchronous dynamics. In the case of a finite size system, this can be realized, for example, by updating at each time step the state of just one edge sampled uniformly at random among the elements of E and leaving unchanged the states of the other edges. In fact, it seems that the techniques used to analyse the emergence of consensus in the synchronous case do not apply to this particular case. Therefore, the discussion about the possibility to reach consensus for the opinion exchange model characterized by the same updating rule for the values of the agents' opinions presented in this paper, but subject to the asynchronous evolution of the communication exchange among the agents just described, are deferred to a forthcoming paper.
We also remark that the differences between the way, synchronous or asynchronous, one chooses to update the state of the communication links shared by the agents reflects in the large system limit analysis. As a matter of fact, for the model where the state of just one communication channel is updated at a time, as the one just described, with a suitable modification of the interaction among the agents, the evolution of the system, in the large system limit, can be analysed looking at the evolution of the empirical distribution of the agents' opinions, in the spirit of the kinetic limit for models of flocking (see e.g. [GO], [CCH]), as it has already been discussed for the stochastic version of Deffuant model in [CF] and more recently for other IPS models [AM]. On the contrary, this is not possible in models where the state of all the communication channels are update at once as for the one studied in this paper. However, if the size of the neighborhood of the agents is finite it is possible to define directly the evolution of the system on very large, possibly infinite, graphs as in [La] and [HH].
Notations
If A is a set and B ⊆ A, 1 B denotes the indicator function of A and B c := A\B. Let P (A) the set of the subsets of A. For any k ≥ 1, we set P k (A) := {B ∈ P (A) : |B| = k} and denote by P 0 (A) := k≥1 P k (A) the set of finite subsets of A.
If A is a metric space, B (A) denotes its Borel σalgebra. We also denote by BM (A) the Banach space of bounded measurable functions on A, by C (A) the Banach space of real-valued continuous functions on A and by · the sup-norm. Moreover, if Lip (A) is the Banach space of real-valued bounded Lipschitz functions on A, for any ϕ ∈ Lip (A) , we denote its norm by ϕ L .
For A, A ′ metric spaces, BL (A, A ′ ) denotes the space of bounded linear operators on A with values in A ′ . If A ′ = A we set BL (A, A) := BL (A) . In particular, if V is a finite set, we denote by St (V) the convex subset of BL R V of stochastic matrices.
We denote by E the expected value of a random element when there is no need to specify the probability space on which it is defined and consequently write P {B} for the expected value of the indicator function 1 B of an event B ⊆ A. The same notation will be also kept when considering conditional expectations and conditional probabilities. Besides, given a σalgebra A of subsets of A, we denote by P (A, A) the set of probability measures on (A, A) . If µ ∈ P (A, A) , sptµ denotes the support of µ, and, for µ, ν ∈ P (A, A) , µ − ν denotes the total variation distance between the two measures.
Let A := A V and denote by a :
= {a v } v∈V . If A is a poset w.r.t. the partial order: a ≤ a ′ if a v ≤ a ′ v , for any v ∈ V, we say that a real-valued function ϕ on A is non-decreasing if ϕ (a) ≤ ϕ (a ′ ) whenever a ≤ a ′ .
Given two probability measures P, P ′ on A, A ⊗V we say that P is stochastically dominated by P ′ , and denote this property by P st ≤ P ′ , if for any bounded non-decreasing function ϕ,
E [ϕ] = dP (a) ϕ (a) ≤ dP ′ (a) ϕ (a) = E ′ [ϕ] .
Moreover, if A is finite, a probability measure P on (A, P (A)) is called irreducible if starting from any element of A with positive P-probability one can reach any other element with positive P-probability via successive coordinate changes without passing through elements with zero P-probability [GHM], [Gr].
If A := A N , for any N ∈ N, we set a N := (a i , .., a N ) and denote by C (A) the cylinder σalgebra that is the σalgebra generated by the cylinder subsets
C N (B) := {a ∈ A : a N ∈ B} ,(1)with B ⊆ A N if A is a discrete set, while B ∈ B A N = B (A) ⊗N if A is a metric space.
If L (A) denotes the algebra of real-valued bounded local (cylinder) functions on A we denote byL (A) the space of real-valued bounded quasilocal functions on A that is is the closure in topology of uniform convergence of the algebra of cylinder functions [Ge].
If V is denumerable we denote by E A the product σalgebra A ⊗V on A := A V .
Graphs
We recall some basic definition of graph theory useful to give a mathematical definition of consensus for the system. The connection of graph theory with Markov chains will be exploited in the next section. We refer the reader to basic textbooks such as [Bo] and [St] for an account on this subject. A directed graph G is a ordered pair of sets (V, E) where V is a finite set called set of vertices and E ⊆ V × V is called set of edges or bonds.
G ′ = (V ′ , E ′ ) such that V ′ ⊆ V and E ′ ⊆ (V ′ × V ′ ) ∩ E
is said to be a subgraph of G and this property is denoted by G ′ ⊆ G. If G ′ ⊆ G, we denote by V (G ′ ) and E (G ′ ) respectively the set of vertices and the collection of the edges of
G ′ . |V (G ′ )| is called the order of G ′ while |E (G ′ )| is called its size. Given G 1 , G 2 ⊆ G, we denote by G 1 ∪ G 2 := (V (G 1 ) ∪ V (G 2 ) , E (G 1 ) ∪ E (G 2 )) ⊂ G the graph union of G 1 and G 2 . Moreover, we say that G 1 , G 2 ⊆ G are disjoint if V (G 1 ) ∩ V (G 2 ) = ∅. For any E ′ ⊆ E, we denote by G (E ′ ) := (V, E ′ ) the spanning graph of E ′ . We also define V (E ′ ) := e∈E ′ e ⊂ V .
(2)
Given V ′ ⊆ V, we set E (V ′ ) := {e ∈ E : e ⊂ V ′ }(3)
and denote by G [V ′ ] := (V ′ , E (V ′ )) that is called the subgraph of G induced or spanned by V ′ . Two vertices u, v are said to be adjacents if belong to the same bond i.e. if V (e) = {u, v} . If e = (u, v) , e is said to be outgoing from u and ingoing in v. Let
E − v := {e ∈ E : e = (u, v) , u ∈ V } , E + v := {e ∈ E : e = (v, u) , u ∈ V }(4)
be the set of edges respectively ingoing, outgoing from v. We denote by
N − (v) := ∪ e∈E − v V (e) ⊆ V the closed ingoing neighborhood of v and by N + (v) := ∪ e∈E + v V (e) ⊆ V the closed outgoing neighborhood of v. Moreover, for any W ⊂ V, we set N + (W ) := ∪ v∈W N + (v) to be the closed outgoing neighborhood of W. Given v ∈ V, we set N + 1 (v) := N + (v) and, for k ≥ 2, N + k (v) := N + N + k−1 (v)
to be the outgoing k-neighborhood of v. Given two vertices u and v, v is said to communicate with u if there exists k ≥ 1 such that u ∈ N + k (v) . Therefore, u, v ∈ V are said to be connected if one communicates with the other. Indeed, since if u ∈ N + k (v) for some k ≥ 1, then u ∈ N + l (v) , ∀l > k, for u and v to be connected there must be k 1 , k 2 ≥ 1 such that u ∈ N + k 1 (v) and v ∈ N + k 2 (u) , that is u ∈ N + k 1 ∨k 2 (v) , v ∈ N + k 1 ∨k 2 (u) . G is then said to be strongly connected if any two distinct vertices are connected. The maximal connected subgraphs of G are called components of G and to denote that G ′ ⊂ G is a component of G we write G ′ ⊏ G.
An example of directed graph is the one which can be associated to a Markov chain. In this case, V coincides with the set of states of the chain and, denoting by P the transition matrix associated to the chain,
E = E (P ) := {(u, v) ∈ V × V : P u,v > 0} .
Then, the directed graph associated to the Markov chain with transition matrix P is denoted by G (P ) . Hence, the Markov chain and therefore P are said to be irreducible if and only if G (P ) is strongly connected.
In the following, if e = (u, v) is an edge of a directed graph (V, E) , we will occasionally noteē for the edge (v, u) ∈ E.
Description of the model and results
In the following, unless differently specified, we will be concerned only with graphs G being subgraphs of the complete directed graph G = (V, E) of finite order where (u,v) and defining
E := (V × V) . A bond (or edge) configuration is a map E ∋ e −→ ω e ∈ {0, 1} so that a bond e is said to be open if ω e = 1. Setting ∀u, v ∈ V, ω u,v := ω e δ e,Ω := ω ∈ {0, 1} E : ∀u ∈ V , ω u,u = 1 ,(5)
we define
Ω ∋ ω −→ E (ω) := {e ∈ E : ω e = 1} ∈ P (E)(6)
and consequently
G (ω) := G (E (ω)) ⊆ G .(7)
We also set, ∀v ∈ V,
E − v (ω) := {e ∈ E (ω) : e = (u, v) , u ∈ V} , (8) E + v (ω) := {e ∈ E (t) : e = (v, u) , u ∈ V} ,(9)N ± (v, ω) := ∪ e∈E ± v (ω) V (e) ,(10)N + 1 (v, ω) := N + (v, ω) ; N + k (v, ω) := N + N + k−1 (v, ω) , k ≥ 2 .(11)
Moreover, given a Ω-valued random sequence
{ω (t)} t≥0 we set E (t) := E (ω (t)) , G (t) := G (ω (t)) as well as, ∀v ∈ V, E ± v (t) := E ± v (ω (t)) and ∀k ≥ 1, N ± k (v, t) := N ± k (v, ω (t)) . A belief configuration is a map V ∋ v −→ X v ∈ [0, 1] .
We set Ξ := [0, 1] V and consider the sequence {X (t)} t≥0 representing the beliefs evolution in time.
Beliefs dynamics
The beliefs evolution is given by the system of equations
X v (t + 1) := u∈N − (v,t) ru,vXu(t) u∈N − (v,t) ru,v = u∈V ru,vωu,v(t)Xu(t) u∈V ru,vωu,v(t) = X v (t) + u∈V ru,vωu,v(t)[Xu(t)−Xv(t)] u∈V ru,vωu,v(t) X v (0) = X 0 v , v ∈ V , t ≥ 0 ,(12)
which, by (8), can be rewritten as
X v (t + 1) = e∈E − v (t) re u∈V δ e,(u,v) Xu(t) e∈E − v (t) re = X v (t) + e∈E − v (t) re∆eX(t) e∈E − v (t) re = X v (t) + e∈E − v reωe(t)∆eX(t) e∈E − v reωe(t) = e∈E − v reωe(t) u∈V δ e,(u,v) Xu(t) e∈E − v reωe(t) X v (0) = X 0 v , v ∈ V , t ≥ 0 ,(13)where, ∀e ∈ E, ∆ e X (t) := (X u (t) − X v (t)) 1 (v,u) (e)(14)
and r e ∈ [0, 1] is the communication rate between the agents labelled by the the vertices incident in e, namely r u,v := r e δ e, (u,v) represents the confidence level assigned by the agent v to the belief of the agent u.
Communication channels dynamics
For any v ∈ V, the P (E)-valued sequence {E − v (t)} t≥0
, as well as the G-valued sequence {G (t)} t≥0 , are constructed by {ω (t)} t≥0 through the random evolution described by the collection of regular conditional probabilities
P {ω e (t + 1) = ω ′ e |X (t)} = δ ω ′ e ,1 p (|∆ e X (t)|) + δ ω ′ e ,0 (1 − p (|∆ e X (t)|)) (15) = ω ′ e p (|∆ e X (t)|) + (1 − ω ′ e ) (1 − p (|∆ e X (t)|)) , e ∈ E ,
where X (t) ∈ Ξ is the belief configuration at time t ≥ 0 and p : [0, 1] is a nonincreasing function such that p (0) = 1.
Notice that, for any t ≥ 0, given e, f ∈ E such that ∆ e X (t) = ∆ f X (t) , the r.v.'s ω e (t + 1) and ωē (t + 1) have the same conditional probabilities w.r.t. X (t) . In particular this holds for e = (u, v) and f =ē = (v, u) , although the edge configurations ω e and ωē are different in general.
In the following we will consider ∀e ∈ E, r e > 0. As a matter of fact, since the r e 's are fixed, we can restrict ourselves to consider instead of G each component of its spanning
subgraph G r := G (E r ) , where E r := {e ∈ E : r e > 0} ,(16)
because, by (12), if G 1 , G 2 ⊏ G r the evolution of the beliefs labeled by the vertices of G 1 is never affected by those labeled by the vertices of G 2 . Moreover, since it is reasonable to assume that the agents put maximal confidence on their own beliefs, we can set r (u,u) = 1, for any u ∈ V.
Results for the finite system
Let V be of finite order. If X 0 ∈ Ξ is such that ∀v ∈ V, X 0 v = x ∈ [0, 1] , then by (12) X (t) = X 0 , ∀t ≥ 0. Hence these configuration, called consensus configurations, are stationary for the system evolution.
In the next section we will prove the following result.
Theorem 1 The agents system reaches consensus for any realization of the initial value of the noise ω 0 ∈ Ω and any initial configuration X 0 ∈ {X ∈ Ξ : Γ (X) > 0} .
Where Γ : Ξ −→ [0, 1) is defined in (41).
Moreover, we will also prove that, the random sequence {(X (t) , ω (t))} t≥0 started at (X 0 , ω 0 ) ∈ {X ∈ Ξ : Γ (X) > 0}×Ω in the limit as t tends to infinity weakly converges at geometric rate to (X ∞ ,1) , where X ∞ ∈ Ξ is such that, for any v ∈ V, X ∞ v = x, for some x ∈ [0, 1] , and1 is the element of Ω such that all its entries are equal to 1. As a byproduct of this result we will obtain that the random sequence {(X (2t) , X (2t + 1))} t≥0 started at (X 0 , X 1 ) ∈ {X ∈ Ξ : Γ (X) > 0} × Ξ will also weakly converge, in the limit as t tends to infinity, to (X ∞ , X ∞ ) at geometric rate.
Results for the large system
Let V := N, E := {(u, v) ∈ N × N} and set Ξ := [0, 1] N and Ω as in 5. Given N ∈ N, let V N := {1, .., N} ⊂ N and E N := {(u, v) ∈ V N × V N } we denote by X N := (X 1 , .
., X N ) the element of Ξ N := [0, 1] N representing the restriction of the beliefs configuration X ∈ Ξ to V N and by ω N the restriction of the configuration ω ∈ Ω to Ω N := {0, 1} E N and, by (6)
, if E := E (ω) , we set E N := E ∩ E N . Assuming that R := {(u, v) ∈ V × V : r u,v > 0} is finite, in the last section we prove that the random sequence {(X (2t) , X (2t + 1))} t≥0 started at (X 0 , X 1 ) ∈ {X ′ ∈ Ξ : inf N ∈N Γ (X ′ N ) > 0}×Ξ
, in the limit as t tends to infinity, weakly converges at geometric rate to (X ∞ , X ∞ ) .
Finite system evolution
Therefore, the evolution of the system is given by the following algorithm:
Algorithm 2
1. Label the elements of V from 1 to N in such a way that V := {1, .., N} and consequently label (i, j) the elements of E := V × V, then go to the next step.
Set
t := 0, X (0) = (X 1 (0) , .., X N (0)) := (X 0 1 , .., X 0 N ) ∈ [0, 1] N , ω (0) := ω 0 i,j (i,j)∈E ∈ {0
, 1} E such that ∀i = 1, .., N, ω 0 i,i = 1, and go to the next step. 3. Set i := 1 and go to the next step.
(a) Set j := 1 and go to the next step.
(b) Compute p i,j (t) := p (|X i (t) − X j (t)|) and form the vector p (t) := (p 1,1 (t) , .., p 1,N (t) , p 2,1 (t) , .., p 2,N (t) .., p i,1 (t) , .., p i,j (t))(17)
and go to the step.
(c) Set j := j + 1. If j + 1 ≤ N go back to step 3.b, otherwise go to the next step.
(d) Set i := i + 1. If i + 1 ≤ N go back to step 3.a, otherwise go to the next step.
4. Set i := 1 and go to the next step.
(a) Compute X i (t + 1) according to (12) and form the vector X (t + 1) := (X 1 (t + 1) , .., X i (t + 1)) , then go to the next step.
(b) Set i := i + 1. If i + 1 ≤ N go back to step 4.a, otherwise go to the next step.
5. Read X (t) = (X 1 (t) , .., X N (t)) . If X (t + 1) = X (t) stop, otherwise go to the next step.
6. Set i := 1 and go to the next step.
(a) Set j = 1 and go to the next step.
(b) Read the p i,j (t) entry of the vector p (t) . Sample a random variable U uni- formly distributed on [0, 1] . If U ≤ p i,j (t) then set ω i,j (t + 1) := ω i,j (t) if ω i,j (t) = 1, otherwise set ω i,j (t + 1) := 1 − ω i,j (t) . If U > p i (t) then set ω i,j (t + 1) := 1 − ω i,j (t) if ω i,j (t) = 1, otherwise set ω i,j (t + 1) := ω i,j (t) .
Form the vector ω (t + 1) := (ω 1,1 (t + 1) , .., ω 1,N (t + 1) , .., ω i,1 (t + 1) , .., ω i,j (t + 1)) (18) and go to the next step.
(c) Set j := j + 1. If j + 1 ≤ N go back to step 6.b, otherwise go to the next step.
(d) Set i := i + 1. If i + 1 ≤ N go back to step 6.a, otherwise go to the next step.
7. set t := t + 1, X (0) := X (t + 1) , ω (0) := ω (t + 1) and go to step 3.
In terms of stochastic process the system evolution can be described as follows.
Let Ω ∋ ω −→ P (ω) ∈ St (V) the stochastic matrix-valued function on R V such that, for any ω ∈ Ω,
P v,u (ω) := e∈E − v (ω) δ e,(u,v) r e e∈E − v (ω) r e = r u,v 1 N − (v,ω) (u) u∈N − (v,ω) r u,v = r u,v ω u,v u∈V r u,v ω u,v , u, v ∈ V . (19)
Remark 3 We remark that, given ω ∈ Ω, v ∈ V, by (19) u ∈ N − (v, ω) iff P v,u (ω) > 0. Therefore, denoting by G (ω) := G (P (ω)) the graph associated to P (ω) , this is the spanning graph of E (ω) :
= {e ∈ E : e = (u, v) if (v, u) ∈ E (ω)} .
Considering Ξ := [0, 1] V ⊂ R V endowed with the norm X := X ∞ = sup v∈V |X v | , let (X, F ) be the measurable space such that X := Ξ × Ω and, if V is a finite set,
F := B (Ξ) ⊗ P (Ω) .
For any ω ∈ Ω, P (ω) ∈ BL (Ξ) , therefore we set
X ∋ (X, ω) −→ T v (X, ω) := u∈V P v,u (ω) X u ∈ [0, 1] ,(20)
and consider the measurable map
X ∋ (X, ω) −→ T (X, ω) := {T v (X, ω)} v∈V ∈ Ξ .(21)
Defining, by (15), the probability kernel from (Ξ, B (Ξ)) to (Ω, P (Ω))
X ∋ (X, ω) −→ Π (ω|X) := e∈E [δ ωe,1 p (|∆ e X|) + δ ωe,0 (1 − p (|∆ e X|))](22)= e∈E [ω e p (|∆ e X|) + (1 − ω e ) (1 − p (|∆ e X|))] ∈ [0, 1] ,
we introduce the positive linear operator on BM (X) such that
BM (X) ∋ ϕ −→ Tϕ (X, ω) := ω ′ ∈Ω ϕ (T (X, ω) , ω ′ ) Π (ω ′ |X) ∈ BM (X) .(23)
Let P 0 be the probability distribution on X Z + , C , where C := C (Ξ) ⊗ C (Ω) , describing the homogeneous discrete time Markov process started at (X 0 , ω 0 ) defined by the one-step transition probability kernel associated to T. We denote by {χ t } t≥0 the random process on X Z + , C, P 0 such that, ∀t ≥ 0,
X Z + ∋ x −→ χ t (x) = (X (t) , ω (t)) ∈ X(24)
and by {F t } t≥0 , with F t := t s=0 χ −1 s (B (Ξ) ⊗ P (Ω)) , the associated natural filtration. Therefore, denoting by E 0 the expectation value w.r.t. P 0 , for any bounded measurable function ϕ on X,
E 0 [ϕ • χ t+1 |F t ] = E 0 [ϕ • χ t+1 |χ t ] = (Tϕ) (χ t ) P 0 − a.s. .(25)
Notice that, by (20), T : C (X, R) , that is {χ t } t≥0 is a Feller process. Setting π ω : X −→ Ω, π X : X −→ Ξ, we denote by {w t } t≥0 , {x t } t≥0 the random processes on X Z + , C, P 0 such that, ∀t ≥ 0,
X Z + ∋ x −→ w t (x) := π ω • χ t (x) = ω (t) ∈ Ω(26)
and
X Z + ∋ x −→ x t (x) := π X • χ t (x) = X (t) ∈ Ξ .(27)Hence, {χ t } t≥0 can be represented as {(x t , w t )} t≥0 . We also set {F ω t } t≥0 , with F ω t := t s=0 w −1 s (P (Ω)) , and F X t t≥0 , with F X t := t s=0 x −1 s (B (Ξ)) .
Remark 4 Notice that neither {w t } t≥0 nor {x t } t≥0 are Markov processes. Indeed, by (15), for any t ≥ 0, w t+1 is independent of w t . Moreover, since ∀t ≥ 0, F X t and F ω t are subσalgebras of F t , for any B ∈ P (Ω) ,
P 0 ({w t+1 ∈ B} |F ω t ) = E 0 [E 0 [1 B • π ω • χ t+1 |F t ] |F ω t ](28)= E 0 [E 0 [1 B • π ω • χ t+1 |χ t ] F ω t ] = E 0 [T (1 B • π ω ) (χ t ) |F ω t ] = E 0 ω ′ ∈Ω 1 B (ω ′ ) Π (ω ′ |x t ) F ω t = E 0 ω ′ ∈B Π (ω ′ |x t ) F ω t = P 0 ({w t+1 ∈ B} |w t ) = P 0 {w t+1 ∈ B} = T t+1 1 B • π ω X 0 , ω 0
while, by (15), (23) and (25), ∀ϕ ∈ BM (Ξ, R) , since for any t ≥ 0, F X t is a subσalgebra of F t ,
E 0 ϕ • x t+1 |F X t = E 0 ϕ • π X • χ t+1 |F X t = E 0 E 0 [ϕ • π X • χ t+1 |F t ] |F X t (29) = E 0 E 0 [ϕ • π X • χ t+1 |χ t ] |F X t = E 0 (T (ϕ • π X )) (χ t ) |F X t = ω ′ ∈Ω wt∈Ω ϕ (T (x t , w t )) Π (ω ′ |x t ) Π (w t |x t−1 ) = ω∈Ω ϕ (T (x t , ω)) Π (ω|x t−1 ) = E 0 [ϕ • x t+1 |x t , x t−1 ] P 0 − a.s. .
In particular, by (29), we get that {y t } t≥0 such that ∀t ≥ 0, y t := (x 2t , x 2t+1 ) is a homogeneous Markov process on X Z + , C, P 0 . Indeed, denoting by {F y t } t≥0 the filtration generated by {y t } t≥0 , since ∀t ≥ 0, F y t = F X 2t+1 , for any bounded measurable function ϕ on Ξ 2 ,
E [ϕ • y t+1 |F y t ] = E ϕ (x 2t+2 , x 2t+3 ) |F X 2t+1 = E [ϕ (x 2t+2 , x 2t+3 ) |x 2t+1 , x 2t ] (30) = E [ϕ • y t+1 |y t ] .
Therefore, the transition operator associated to {y t } t≥0 is
(Tϕ) (X 1 , X 2 ) := ω,ω ′ ∈Ω ϕ (T (X 2 , ω) , T (T (X 2 , ω) , ω ′ )) Π (ω|X 1 ) Π (ω ′ |X 2 )(31)
and, setting
Ξ 2 ∋ (X 1 , X 2 ) −→ π i (X 1 , X 2 ) := X 1 δ i,1 + X 2 δ i,2 ∈ Ξ , i = 1, 2 ,(32)
for any bounded measurable function ϕ on Ξ, we have
E 0 [ϕ • x t+1 |x t , x t−1 ] = E 0 [(ϕ • π 1 ) (y t+1 ) |y t ](33)
= E 0 [T (ϕ • π 1 ) (y t )] P 0 − a.s. .
Consensus
If X 0 ∈ Ξ is such that ∀v ∈ V, X 0 v = x ∈ [0, 1]
, then by (12) X (t) = X 0 , ∀t ≥ 0. Hence these configuration, called consensus configurations, are stationary for the system evolution.
We denote by
I := x∈[0,1] I x(34)
where
I x := {X ∈ Ξ : X v = x , ∀v ∈ V}(35)
and by M : Ξ −→ Ξ the consensus projection map, that is the map associating to each belief configuration X the consensus configuration MX such that ∀v ∈ V, (MX) v := 1 |V| u∈V X u . It is easy to see that M is a projection operator on I, moreover an orthogonal projection if Ξ is endowed with the Euclidean structure ·, · of R V . Indeed, ∀X ∈ Ξ, M 2 X = MX. Therefore, ∀X ∈ Ξ, we set
dist (I, X) := inf Y ∈I X − Y ≤ [I − M] X ,(36)
where we denote by I the identity operator on R V . Consequently, since if X = MX, ∀ (u, v) ∈ E, X u − X v = 0, we can modify the algorithm 2 erasing the line 5 and adding the line
3.e If N i=1 N j=1
(1 − δ i,j ) p i,j (t) = N (N − 1) stop, otherwise proceed to the next step.
Invariant measures for T and T
Setting X := (I − M) Ξ, we can represent Ξ as I ⊕ X . Moreover, for any ω ∈ Ω, I is invariant under T (·, ω) , since X ∈ Ξ, by (21) we get T (MX, ω) = MX. Therefore
T (X, ω) = T (MX + (I − M) X, ω) = MX + T ((I − M) X, ω) .(37)
Moreover, by (22), for any ω ∈ Ω, Π (ω|X) = Π (ω| (I − M) X) . Hence, denoting by δ ω 1 the Dirac measure at1, by the definition of p and by (15), given X ∈ I, ∀ω ∈ Ω, Π (ω|X) = e∈E δ ωe,1 = δ ω 1 . Denoting by δ X the probability measure on (Ξ, B (Ξ)) concentrated on the beliefs configuration X ∈ Ξ, let δ X I be the probability measure on (Ξ, B (Ξ)) putting mass 1 on the configuration X ∈ I. It is easy to see that the probability measure δ X I ⊗δ ω 1 on (X, F ) is invariant for the evolution given by T. Indeed, if X ∈ I, by (34) and (35), there exists x ∈ [0, 1] such that X ∈ I x . Hence, by (20), for any ω ∈ Ω, T (X, ω) = X. Therefore, given any bounded measurable function ϕ on X, by (21)
, ∀ω, ω ′ ∈ Ω, δ X I [ϕ (T (·, ω) , ω ′ )] = δ X I [ϕ (·, ω ′ )] .
Thus, by (23),
δ X I ⊗ δ ω 1 [Tϕ] = δ ω 1 δ X I ω ′ ∈Ω ϕ (T (·, ω) , ω ′ ) Π (ω ′ |·) (38) = δ ω 1 ω ′ ∈Ω δ X I [ϕ (T (·, ω) , ω ′ ) Π (ω ′ |·)] = δ ω 1 ω ′ ∈Ω δ X I ϕ (·, ω ′ ) e∈E δ ω ′ e ,1 = δ X I ⊗ δ ω 1 ω ′ ∈Ω ϕ (·, ω ′ ) e∈E δ ω ′ e ,1 = δ X I ⊗ δ ω 1 [ϕ (·,1)] = δ X I ⊗ δ ω 1 [ϕ] .
Thus the set I T of invariant probability measures under T is the weak limit of convex combinations of elements of the set δ ω 1 ⊗ δ X I X∈I ⊂ P (X, F ) . Since for any X 2 ∈ I and any µ ∈ P (Ξ, B (Ξ)) , from (31) it follows that
µ ⊗ δ X 2 I [Tϕ] = µ (dX 1 ) δ X 2 I ω,ω ′ ∈Ω ϕ (T (·, ω) , T (T (·, ω) , ω ′ )) Π (ω|X 1 ) Π (ω ′ |·)(39)= µ (dX 1 ) ω,ω ′ ∈Ω ϕ (X 2 , , T (X 2 , ω ′ )) Π (ω|X 1 ) e∈E δ ω ′ e ,1 = µ (dX 1 ) ω∈Ω ϕ (X 2 , , T (X 2 ,1)) Π (ω|X 1 ) = µ (dX 1 ) ω∈Ω ϕ (X 2 , X 2 ) Π (ω|X 1 ) = ϕ (X 2 , X 2 ) ,
we have that the set I T of invariant probability measures under T is the weak limit of convex combinations of elements of the set δ
(X,X) I X∈I ⊂ P (Ξ 2 , B (Ξ 2 )) , where δ (X,X) I := δ X I ⊗ δ X I .
Emergence of consensus
Definition 5 Given E ⊆ E, consider the spanning graph G (E) = (V, E) . We call pivots the elements w of V such that N + (w) = V and denote their collection by P (E) . Moreover, for any ω ∈ Ω, we set P (ω) := P (E (ω)) and define Ω P := {ω ∈ Ω : P (ω) = ∅} .
Let us denote by γ the r.v. 1
Ω ∋ ω −→ γ (ω) := min u,v∈V : u =v w,z∈V P u,w (ω) P v,z (ω) ∧ P u,z (ω) P v,w (ω) ∈ [0, 1) (40)
and by Γ the r.v.
Ξ ∋ X −→ Γ (X) := E [γ|X] = ω∈Ω Π (ω|X) γ (ω) ∈ [0, 1) .(41)
Lemma 6 Given ω ∈ Ω, γ (ω) > 0 if and only if P (ω) is not empty.
Proof. Let ω ∈ Ω be such that P (ω) = ∅. Denoting by u = u (ω) , v = v (ω) the elements of V such that w,z∈V P u,w (ω) P v,z (ω) ∧ P u,z (ω) P v,w (ω) = (42) min u ′ ,u ′′ ∈V w,z∈V P u ′ ,w (ω) P u ′′ ,z (ω) ∧ P u ′ ,z (ω) P u ′′ ,w (ω) ,
for anyw ∈ P (ω) , we have
γ (ω) = w,z∈V P u,w (ω) P v,z (ω) ∧ P u,z (ω) P v,w (ω) = w∈V P u,w (ω) P v,w (ω) + (43) w,z∈V : w =z P u,w (ω) P v,z (ω) ∧ P u,z (ω) P v,w (ω) ≥ (P u,w (ω) ∧ P v,w (ω)) 2 > 0 .
Conversely by (19), γ (ω) > 0 iff, for any u, v ∈ V such that u = v, N − (u, ω) ∩ N − (v, ω) = ∅, which is equivalent to say that γ (ω) > 0 implies that there exists at least onew =w (ω) in V such that, by (50), N + (w, ω) = V, or, in other words, by Definition 5, that P (ω) is not empty.
In the next section we will prove the following result.
Theorem 7 The agents system reaches consensus for any realization of the initial value of the noise ω 0 ∈ Ω and any initial configuration X 0 ∈ {X ∈ Ξ : Γ (X) > 0} . More precisely, for any X 0 ∈ {X ∈ Ξ : Γ (X) > 0} and any ω 0 ∈ Ω, the sequence of probability measures {µ t 0 } t≥0 on (Ξ, B (Ξ)) such that
B (Ξ) ∋ A −→ µ t 0 (A) := P 0 x ∈ X Z + : π X • χ t (x) ∈ A ∈ [0, 1] ,(44)
converges to a probability measure µ ∞ 0 supported on I.
Remark 8 We stress that this result give no information on the common value of the beliefs when consensus is reached.
Given X ∈ Ξ, let W (X) := max u,v∈V |X u − X v | .(45)
Since
W ([I − M] X) = W (X) ,(46)
W is a seminorm on R V and therefore induces a norm on W := R V /RanM. Hence, because MI = I, for any Y ∈ I, we have
X − Y = W (X − Y ) = W ([I − M] (X − Y )) = W (X) ,(47)
which implies dist (I, X) = W (X) .
For any t ≥ 0, let W (t) := W (X (t)) . In the following we will prove that the random sequence {W (t)} t≥0 converges to zero w.p.1 w.r.t. the noise, hence proving Theorem 7.
Proposition 9 The sequence {W (t)} t≥0 is non-increasing hence bounded. Moreover, {W (t)} t≥0 is a non-negative L 1 -supermartingale w.r.t. F X t t≥0 , therefore P 0 -a.s. convergent to a L 1 (X, F , P 0 ) r.v. which we denote by W.
Proof. By (12), given u, v ∈ V such that u = v, for t ≥ 0,
X u (t + 1) − X v (t + 1) = (X u (t + 1) − X u (t)) − (X v (t + 1) − X v (t)) + X u (t) − X v (t) (49) = X u (t) − X v (t) + e∈E − u (t) r e 1 (u,w) (e) [X w (t) − X u (t)] e∈E − u (t) r e − e ′ ∈E − v (t) r e ′ 1 (v,z) (e ′ ) [X z (t) − X v (t)] e ′ ∈E − v (t) r e ′ = e∈E − u (t) r e 1 (u,w) (e) e∈E − u (t) r e X w (t) − e ′ ∈E − v (t) r e ′ 1 (v,z) (e ′ ) e ′ ∈E − v (t) r e ′ X z (t) .
By (19), setting
P u,v (t) := P u,v (ω (t)) = e∈E − v (t) δ e,(v,u) r e e∈E − u (t) r e = r v,u 1 N − (u,t) (v) v∈N − (u,t) r v,u(50)
we can rewrite the previous expression as
X u (t + 1) − X v (t + 1) = w∈V P u,w (t) X w (t) − z∈V P v,z (t) X z (t) .(51)
Since, ∀t ≥ 0, v∈V P u,v (t) = 1, we have
X u (t + 1) − X v (t + 1) = w,z∈V P u,w (t) P v,z (t) [X w (t) − X z (t)](52)
and, since [
X w (t) − X z (t)] = − [X z (t) − X w (t)] ,we obtain X u (t + 1) − X v (t + 1) = 1 2 w,z∈V {P u,w (t) P v,z (t) − P u,z (t) P v,w (t)} [X w (t) − X z (t)] .
(53) Hence
|X u (t + 1) − X v (t + 1)| ≤ 1 2 w,z∈V |P u,w (t) P v,z (t) − P u,z (t) P v,w (t)| |X w (t) − X z (t)| . (54) Since ∀a, b ∈ R, a ∧ b = a+b−|a−b| 2 , |X u (t + 1) − X v (t + 1)| ≤ w,z∈V P u,w (t) P v,z (t) + P u,z (t) P v,w (t) 2 (55) −P u,w (t) P v,z (t) ∧ P u,z (t) P v,w (t)} |X w (t) − X z (t)| ≤ w,z∈V P u,w (t) P v,z (t) + P u,z (t) P v,w (t) 2 −P u,w (t) P v,z (t) ∧ P u,z (t) P v,w (t)} max w,z∈V |X w (t) − X z (t)| ≤ 1 − w,z∈V P u,w (t) P v,z (t) ∧ P u,z (t) P v,w (t) max w,z∈V |X w (t) − X z (t)| . Therefore, choosing u, v ∈ V such that |X u (t + 1) − X v (t + 1)| = max w,z∈V |X w (t + 1) − X z (t + 1)| ,(56)
by (40) we get
W (t + 1) ≤ 1 − min u,v∈V : u =v w,z∈V P u,w (t) P v,z (t) ∧ P u,z (t) P v,w (t) W (t)(57)
= (1 − γ (ω (t))) W (t) ; (57) we get
hence, ∀t ≥ 0, W (t) ≤ W (0) ≤ 1. Thus, representing the random sequence {W (t)} t≥0 as {W • x t } t≥0 , fromE 0 W (t + 1) |F X t = E 0 W • x t+1 |F X t = E 0 E 0 [W • π X • χ t+1 |F t ] |F X t (58) ≤ E 0 [E 0 [[{1 − γ • π ω } W • π X | χ t ]| F X t = E 0 [E 0 {1 − γ (π ω • χ t )} W • x t | F X t = 1 − E 0 [γ (π ω • χ t )| F X t W (t) ≤ W (t) , that is {W (t)} t≥0 is a L 1 -supermartingale w.r.t. F X t t≥0 .
Asymptotic estimate of E 0 [W (t)]
Lemma 10 The sequence E 0 γ|F X t t≥0 is predictable w.r.t. the filtration F X t t≥0 .
Proof. For any t ≥ 1, by (19),
E 0 γ|F X t = E 0 [γ (π ω • χ t )| F X t = E 0 [[γ (w t )| x t−1 ] = (59) ω∈Ω Π (ω|x t−1 ) min u,v∈V : u =v w,z∈V P u,w (ω) P v,z (ω) ∧ P u,z (ω) P v,w (ω) = Γ (x t−1 ) .
Let us set
X ∋ (X, ω) −→ Π (ω|W (X)) := e∈E [δ ωe,1 p (W (X)) + δ ωe,0 (1 − p (W (X)))]
= e∈E(60)
[ω e p (W (X)) + (1 − ω e ) (1 − p (W (X)))] ∈ [0, 1] .
Since Ω is a poset w.r.t. the partial order relation: ω ≤ ω ′ if, ∀e ∈ E, ω e ≤ ω ′ e , we have Lemma 11 For any X ∈ Ξ, Π (·|X) st ≥ Π (·|W (X)) . Moreover, for any t ≥ 0, Π (·|W (t + 1)) st ≥ Π (·|W (t)) .
Proof. Let us consider first the statement Π (·|X) st ≥ Π (·|W (X)) . For X ∈ I, by (22) and (60) Π (·|X) and Π (·|W (X)) coincide. Let now X ∈ X . By (22) and (60) Π (·|·) is irreducible, then to prove Π (·|X) st ≥ Π (·|W (X)) is enough to prove that the Holley inequality is satisfied, namely
Π (ω ∨ ω ′ |X) Π (ω ∧ ω ′ |W (X)) ≥ Π (ω|X) Π (ω ′ |W (X)) , ω, ω ′ ∈ Ω ,(61)
where ω ∨ ω ′ ∈ Ω is such that ∀e ∈ E, (ω ∨ ω ′ ) e = ω e ∨ ω ′ e and ω ∧ ω ′ ∈ Ω is such that ∀e ∈ E, (ω ∧ ω ′ ) e = ω e ∧ ω ′ e . This is equivalent to prove that, for any e, f ∈ E,
Π ω {e} |X Π ω {e} |W (X) ≥ Π ω {e} |W (X) Π ω {e} |X(62)
and
Π ω {ef } |X Π ω {ef } |W (X) ≥ Π ω {e} {f } |W (X) Π ω {f } {e} |X ,(63)
where, for any E ⊂ E, ω E ∈ Ω is such that ∀e ∈ E, ω E e := ω e 1 E c (e) + 1 E (e) and ω E ∈ Ω is such that ∀e ∈ E, (ω E ) e := ω e 1 E c (e) (see e.g. [Gr] Theorem 2.3). But, by (22) and (60), Π (·|X) and Π (·|W (X)) are product measures, then (62) becomes Π (ω e = 1|X) Π (ω e = 0|W (X)) ≥ Π (ω e = 1|W (X)) Π (ω e = 0|X) , which can be rewritten as
p (|∆ e X|) (1 − p (W (X))) ≥ p (W (X)) (1 − p (|∆ e X|))(64)
and (63) becomes
Π (ω e = 1|X) Π (ω f = 1|X) Π (ω e = 0|W (X)) Π (ω f = 0|W (X)) ≥ (65) Π (ω e = 1|W (X)) Π (ω f = 0|W (X)) Π (ω e = 0|X) Π (ω f = 1|X)
which is again (64). Since by (45), for any e ∈ E,W (X) ≥ ∆ e X and since p : [0, 1] is non increasing, we have, for any e ∈ E, p (|∆ e X|) ≥ p (W (X)) and consequently (1 − p (W (X))) ≥ (1 − p (|∆ e X|)) which proves (64).
The proof of the statement Π (·|W (t + 1)) st ≥ Π (·|W (t)) , t ≥ 0, follow the same lines of the proof of Π (·|X) st ≥ Π (·|W (X)) since, by (57), W (t + 1) ≤ W (t) , which implies p (W (t + 1)) ≥ p (W (t)) .
Proof of Theorem 7
Since γ is an non-decreasing function, by (41) and by the previous lemma we have that
Γ (X) ≥ ω∈Ω P (ω|W (X)) γ (ω) = Γ (W (X))(66)
and, for any t ≥ 0, Γ (W (t + 1)) ≥ Γ (W (t)) . Then, by (58) and Lemma 10 we get
E 0 [W (t)] = E 0 E 0 W • π X • χ t |F X t−1 ≤ E 0 1 − E 0 γ|F X t−1 W • π X • χ t−1 (67) = E 0 [(1 − Γ (X (t − 2))) W (t − 1)] ≤ E 0 [(1 − Γ (W (t − 2))) W (t − 1)] = E 0 (1 − Γ (W • π X • χ t−2 )) E 0 W • π X • χ t−1 |F X t−2 ≤ E 0 (1 − Γ (W • π X • χ t−2 )) 1 − E 0 γ|F X t−2 W • π X • χ t−2 = E 0 [(1 − Γ (W (t − 2))) (1 − Γ (X (t − 2))) W (t − 2)] ≤ E 0 (1 − Γ (W (t − 2))) 2 W (t − 2) ≤ E 0 (1 − Γ (W (t − 3))) 2 W (t − 2) .
Iterating this inequality, after k steps, with k ≤ t, we obtain
E 0 [W (t)] ≤ E 0 (1 − Γ (W (t − k))) k−1 W (t − k + 1)(68)
which, by (57) implies
E 0 [W (t)] ≤ E 0 [W (1)] 1 − Γ W X 0 t−1 ≤ W X 0 1 − Γ W X 0 t .(69)
Therefore, for any X 0 ∈ {X ∈ Ξ : Γ (X) > 0} , since W (X 0 ) and for any ε > 0 the Markov inequality implies
P 0 {W (t) > ε} ≤ E 0 [W (t)] ε ≤ ε −1 1 − Γ W X 0 t ,(70)
by the Borel-Cantelli Lemma {W (t)} t≥0 converges to zero P 0 -a.s., that is µ ∞ 0 := lim t→∞ P 0 {X (t) ∈ ·} is supported on I.
Convergence to the stationary measure of {χ t } t∈Z + and {y t } t∈Z +
We can rephrase (69) and therefore the content of Theorem 7 in terms of exponential (more precisely geometric since t ∈ Z + ) convergence to an element of the set of the invariant measures of the Markov chains defined by the transition operators T and T. More precisely, for any ε > 0 and t > 0, given χ 0 = (X 0 , ω 0 ) ∈ {X ∈ Ξ : Γ (X) > 0} × Ω, by (36) { (I − M) x t > ε} ⊆ {W (t) > ε} . Hence, by Theorem 7, {v t } t∈Z + converges to zero P 0 − a.s. and, by (71), {w t } t∈Z + converges to1P 0 − a.s.. But, since, by (23), for any Y ∈ I, ω ∈ Ω, Tϕ (Y, ω) = ϕ (Y,1) , {u t } t∈Z + converges P 0 − a.s. to an element of I which we denote by X ∞ . Given X ∈ Ξ, let us set X = (U, V ) such that U := MU, V := (I − M) X and consider the random processes {u t } t∈Z + and {u t } t∈Z + such that ∀t ≥ 0, u t := Mx t and v t := (I − M) x t . From (22), (23) and (37), for any bounded measurable function ϕ on Ξ 2 × Ω,
Tϕ (U, V, ω) = ω ′ ∈Ω ϕ (U + MT (V, ω) , (I − M) T (V, ω) , ω ′ ) Π (ω ′ |V ) .(71)
Hence, {z t } t∈Z + such that, ∀t ≥ 0, z t := (v t , w t ) , is an homogeneous Markov process. Let us introduce on Ω the metric
Ω × Ω ∋ (ω, ω ′ ) −→ d (ω, ω ′ ) := 1 |E| e∈E 1 − δ ωe,ω ′ e ∈ [0, 1] .(72)
Lemma 12 From (45), for any (X 0 , ω 0 ) ∈ X and t ≥ 1, we have
E d (w t ,1) | X 0 , ω 0 ≤ E (1 − p (W • x t−1 )) | X 0 , ω 0 .(73)
Proof. For any ω ∈ Ω, we get
d (ω,1) = 1 |E| e∈E (1 − δ ωe,1 ) = 1 |E| e∈E (1 − ω e ) .(74)
Hence, by the Markov property, from (23) and (22) we have
E 0 [d (w t ,1)] = E 0 [E [d (w t ,1) | (x t−1 , w t−1 )]](75)= E 0 ω ′ ∈Ω 1 |E| e∈E (1 − ω ′ e ) Π (ω ′ |x t−1 ) = 1 |E| e∈E E 0 ω ′ ∈Ω (1 − ω ′ e ) (ω ′ e p (|∆ e x t−1 |) × × (1 − ω ′ e ) (1 − p (|∆ e x t−1 |)))] = 1 |E| e∈E E 0 [(1 − p (|∆ e v t−1 |))] ≤ 1 |E| e∈E E 0 [(1 − p (W (t − 1)))] ≤ E 0 [(1 − p (W (t − 1)))] .
Let Ξ I := Ξ\I, Given a bounded measurable function ϕ on Ξ I × Ω ⊂ X, let
∇ V ϕ 1 := sup (V ′ ,ω)∈Ξ I ×Ω v∈V ∂ ∂V v ϕ (V ′ , ω) ,(76)
ϕ Ω := sup
V ∈Ξ I sup ω,ω ′ ∈Ω : ω =ω ′ |ϕ (V, ω) − ϕ (V, ω ′ )| d (ω, ω ′ )(77)
and consider the Banach space L of measurable functions ϕ on Ξ I × Ω, with norm
ϕ L := sup (V,ω)∈Ξ I ×Ω |ϕ (V, ω)| + ∇ V ϕ 1 + ϕ Ω .(78)
Since for any X ∈ Ξ,
(I − M) X = sup v∈V |X v − (MX) v | = sup v∈V X v − 1 |V| u∈V X u (79) = sup v∈V 1 − 1 |V| X v − 1 |V| u∈V : u =v X u = sup v∈V |V| − 1 |V| X v − 1 |V| u∈V : u =v X u = sup v∈V 1 |V| u∈V : u =v (X v − X u ) ≤ W (X) ,
then, for ϕ ∈ L, by (79), we have
|ϕ ((I − M) X, ω) − ϕ (0, ω)| = 1 0 ds v∈V ∂ ∂V v ϕ (s (I − M) X, ω) ((I − M) X) v(80)= 1 0 ds v∈V ∂ ∂V v ϕ (s (X − MX) , ω) 1 |V| u∈V : u =v (X v − X u ) ≤ ∇ V ϕ 1 W (X) .
Proposition 13 Starting from an initial state χ 0 = (X 0 , ω 0 ) ∈ {X ∈ Ξ : Γ (X) > 0} × Ω ⊂ X, the Markov chain {χ t } t≥0 weakly converges to the degenerate random vector (X ∞ ,1) ∈ I × Ω, where X ∞ is the P 0 − a.s. limit of the random process {u t } t∈Z + .
Moreover, if p is concave function and lim x↓0
1−p(x) x
exists, the rate of convergence is exponential.
Proof. Given ϕ ∈ L, by (80) and (75) we have
|E 0 [ϕ (v t , w t )] − ϕ (0,1)| ≤ E 0 [|ϕ (v t , w t ) − ϕ (0,1)|] (81) ≤ E 0 [|ϕ (v t , w t ) − ϕ (0, w t )|] + E 0 [|ϕ (0, w t ) − ϕ (0,1)|] ≤ ∇ X ϕ 1 E 0 [W (t)] + ϕ Ω E 0 [(1 − p (W (t − 1)))] ≤ ϕ L (E 0 [W (t)] ∨ E 0 [(1 − p (W (t − 1)))])
which tends to zero in the limit t → ∞ by Theorem 7. Clearly, if p is concave, 1 − p is convex, hence, by (69),
E 0 [(1 − p (W (t − 1)))] ≤ (1 − p (E 0 [W (t − 1)]))(82)≤ 1 − p W X 0 1 − Γ X 0 t−1 , therefore, if lim x↓0 1−p(x) x
is finite the rate of convergence is exponential. Similar conclusions hold for the Markov chain {y t } t≥0 . Indeed, by (31) and (37), setting X 1 = (U 1 , V 1 ) , X 2 = (U 2 , V 2 ) , for any bounded measurable ϕ : Ξ 4 × Ω → R,
(Tϕ) (U 1 , V 1 , U 2 , V 2 ) := ω,ω ′ ∈Ω ϕ (U 2 + MT (V 2 , ω) , (I − M) T (V 2 , ω) ,(83)U 2 + MT (T (V 2 , ω) , ω ′ ) , (I − M) T (T (V 2 , ω) , ω ′ )) × × Π (ω|V 1 ) Π (ω ′ |V 2 ) .
Hence, {Z t } t≥0 such that ∀t ≥ 0, Z t := (v 2t , v 2t+1 ) is an homogeneous Markov process. Moreover, from (57), for any t ≥ 0 we get
E 0 W • x t+1 |F X t−1 = E 0 E 0 W • x t+1 |F X t F X t−1(84)= E 0 (1 − Γ • x t−1 ) W • x t |F X t−1 = E 0 W • x t |F X t−1 (1 − Γ • x t−1 ) ≤ (1 − Γ • x t−1 ) (1 − Γ • x t−2 ) W • x t−1 ≤ (1 − Γ • W • x t−1 ) (1 − Γ • W • x t−2 ) W • x t−1 ≤ (1 − Γ • W • x t−2 ) 2 W • x t−1 .
Hence, by (32), defining Ξ 2 ∋ (X 1 , X 2 ) −→Γ (X 1 , X 2 ) := (Γ • π 1 ) (X 1 , X 2 ) ∈ (0, 1) ,
Ξ 2 ∋ (X 1 , X 2 ) −→W (X 1 , X 2 ) := (W • π 1 ) (X 1 , X 2 ) ∈ [0, 1] ,(85)
from (84) we have
E 0 W • y t+1 |F y t ≤ 1 −Γ • y t 2W • y t , t ≥ 0 .(87)
Therefore, proceeding as in (67), by (87) we get
E 0 W • y t ≤ E 0 1 −Γ • y t−1 2W • y t−1(88)≤ E 0 1 −Γ •W • y t−1 2W • y t−1 ≤ E 0 1 −Γ •W • y t−2 2W • y t−1 .
Thus, iterating,
E 0 T tW ≤ 1 −Γ • y 0 2tW • y 0 .(89)
Let us denote by L be the Banach space of bounded measurable functions ϕ on Ξ 2 I with norm ϕ L := sup
(V 1 ,V 2 )∈Ξ 2 I |ϕ (V 1 , V 2 )| + ∇ϕ 1 ,(90)
where
∇ϕ 1 := sup (V ′ ,V ′′ )∈Ξ 2 I v∈V ∂ ∂ (V 1 ) v ϕ (V ′ , V ′′ ) + ∂ ∂ (V 2 ) v ϕ (V ′ , V ′′ ) .(91)
Corollary 14 The Markov chain {y t } t≥0 started at (X 0 , X 1 ) ∈ {X ∈ Ξ : Γ (X) > 0}×Ξ converges weakly to the degenerate random vector (X ∞ , X ∞ ) ∈ Ξ 2 with exponential rate.
Proof. Given ϕ ∈ L, proceeding as in (80), by (58) and (89), we have
|E 0 [ϕ (Z t )] − ϕ (0, 0)| ≤ E 0 [|ϕ (Z t ) − ϕ (0, 0)|](92)
≤ ϕ L E 0 W • y t .
Large system evolution
Given N ∈ N, let V N := {1, .., N} ⊂ N and denote by E N the subset of E := {(u, v) ∈ N × N} such that E N := {(u, v) ∈ V N × V N } .
In this section we set Ξ := [0, 1] N and, denoting by X N := (X 1 , .., X N ) the element of Ξ N := [0, 1] N representing the restriction of the beliefs configuration X ∈ Ξ to V N , we endow Ξ with the metric induced by the norm X := N ∈N 2 −N X N ∞ .
Setting Ω := {0, 1} E , we denote by ω N the restriction of the configuration ω ∈ Ω to Ω N := {0, 1} E N and, by (6), if E := E (ω) , we set E N := E ∩ E N .
Then, for any X ∈ Ξ, Π (·|X) denotes the random field on (Ω, C (Ω)) such that, for any cylinder event C N (ω ′ ) = {ω ∈ Ω : ω N = ω ′ } , N ≥ 1, ω ′ ∈ Ω N , ω∈Ω Π (ω|X) 1 C N (ω ′ ) (ω) = Π (ω ′ |X N )
where Π (ω ′ |X N ) is given by (22). Moreover, for any M ≥ N,
ω∈Ω M Π (ω|X M ) 1 {ω∈Ω M : ω N =ω ′ } (ω) = ω∈Ω Π (ω|X) 1 C N (ω ′ ) (ω) = Π (ω ′ |X N )(94)
Setting V := N for notational convenience, let K (V) be the set of the transition probability kernels on (V, P (V)) . From (19), given the C (Ω)-measurable function Ω ∋ ω −→ P (ω) ∈ K (V) such that ∀ω ∈ Ω,
we denote as in (21) and (20) X ∋ (X, ω) −→ T · (X, ω) ∈ Ξ such that, for any v ∈ V and any (X, ω) ∈ X, T v (X, ω) := u∈V p v,u (E (ω)) X u . Then, the operator BM (X) ∋ ϕ −→ Tϕ (X, ω) := ω ′ ∈Ω ϕ T (X, ω) , ω ′ Π (ω ′ |X) ∈ BM (X)
represents the transition probability kernel of the homogeneous Markov chain {χ t } t≥0 on X Z + , C, P 0 with initial condition (X 0 , ω 0 ) ∈ X such that, by (23), for any ω ′ ∈ Ω N , B ∈ B [0, 1] N ,
P 0 ({χ t+1 ∈ C N (B) × C N (ω ′ )} |χ t ) = T1 C N (B)×C N (ω ′ ) (χ t )(97)
= Π (ω ′ |X N (t)) 1 B T N (X (t) , ω (t)) .
Consequently, the operator
L Ξ 2 ∋ ϕ −→ Tϕ (X, Y ) := ω,ω ′ ∈Ω ϕ T (Y, ω) , T T (Y, ω) , ω ′ Π (ω|X) Π (ω ′ |Y ) ∈L Ξ 2 ,(98)
defined as in (31), represents the transition probability kernel of the homogeneous Markov chain {y t } t≥0 on X Z + , C, P 0 such that, for any B ∈ B [0, 1] 2N , P 0 ({y t+1 ∈ C N (B)} |y t ) = T1 C N (B) (y t ) = (99) ω,ω ′ ∈Ω 1 B T N (X (2t + 1) , ω) , T N T (X (2t + 1) , ω) , ω ′ Π (ω|X (2t)) Π (ω ′ |X (2t + 1)) = ω,ω ′ ∈Ω 1 B T N (X (2t + 1) , ω) , T N T (X (2t + 1) , ω) , ω ′ Π (ω|X (2t)) Π (ω ′ |X (2t + 1)) = ω,ω ′ ∈Ω 1 B T (X N (2t + 1) , ω) , T T N (X (2t + 1) , ω) , ω ′ Π (ω|X (2t)) Π (ω ′ |X (2t + 1)) = ω,ω ′ ∈Ω 1 B (T (X N (2t + 1) , ω) , T (T (X N (2t + 1) , ω) , ω ′ )) Π (ω|X (2t)) Π (ω ′ |X (2t + 1)) .
Remark 15 Notice that if the cardinality of the set R := {(u, v) ∈ V × V : r u,v > 0} is finite, there exists M > N such that P 0 ({χ t+1 ∈ C N (B) × C N (ω ′ )} |χ t ) = Π (ω ′ |X N (t)) 1 B (T (X N (t) , ω M (t)))
and P 0 ({y t+1 ∈ C N (B)} |y t ) = (101) ω M ,ω ′ M ∈Ω M 1 B (T (X N (2t + 1) , ω) , T (T (X N (2t + 1) , ω) , ω ′ )) × ×Π (ω M |X M (2t)) Π (ω ′ M |X M (2t + 1)) = T1 (X (1) ,X (2) )∈Ξ 2 M : X
(1) N ,X
(2) N ∈B (y t ) .
In the following we make this assumption.
Proposition 16 Let the initial datum X 0 ∈ Ξ be such that α := inf N ∈N Γ (X 0 N ) > 0. Then, for any ϕ ∈ C (Ξ) and any X 1 ∈ Ξ,
lim t→∞ E ϕ • Z t |Z 0 = X 0 , X 1 − ϕ (0, 0) = 0 .(102)
Proof. For any ϕ ∈ C (Ξ) there exists a sequence {ϕ N } N ∈N such that, ∀N ≥ 1, ϕ N is a continuous B (Ξ N )-measurable function uniformly convergent to ϕ [Ge]. Hence, given ε > 0, there exists N ε ≥ 1 such that for any N > N ε , ϕ − ϕ N ∞ < ε. Moreover, denoting by I N := x∈ [0,1] {X N ∈ Ξ N : (X N ) v = x , ∀v ∈ V N } and Ξ N I := Ξ N \I N , by the Stone-Weierstass theorem there exists φ ∈ L N (with L N defined as L in the previous section) such that φ N − ϕ N ∞ < ε. Then, the thesis follows from Corollary 14. Indeed, E ϕ • y t |y 0 = X 0 , X 1 − ϕ (0, 0) ≤ 2ε + E ϕ N • y t |y 0 = X 0 , X 1 − ϕ N (0, 0)
≤ 4ε + E φ N • y t |y 0 = X 0 , X 1 − φ N (0, 0) = 4ε + δ (X 0
M ,X 1 M ) T t (φ N − φ N (0, 0)) , where M ≥ N. But, δ ( X 0 M ,X 1 M ) T t |φ N − φ N (0, 0)| ≤ φ N L N (1 − α) 2t .(104)
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| A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the q-oscillators (q-Weyl-Heisenberg algebra) and for the su q (2) and su q (1, 1) algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the q → 1 limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras. | 10.1063/1.530591 | [
"https://arxiv.org/pdf/hep-th/9307083v1.pdf"
]
| 14,793,197 | hep-th/9307083 | d4e82a7b8b81a9aee71b8c296cac1dc2c63f2d05 |
COMPLEX ANALYTIC REALIZATIONS FOR QUANTUM ALGEBRAS 1
Jul 1993
J A De Azcárraga
Departamento de Física Teórica and IFIC
Centro Mixto
Universidad de Valencia -CSIC
E-46100Burjasot, ValenciaSpain
D Ellinas
Departamento de Física Teórica and IFIC
Centro Mixto
Universidad de Valencia -CSIC
E-46100Burjasot, ValenciaSpain
COMPLEX ANALYTIC REALIZATIONS FOR QUANTUM ALGEBRAS 1
Jul 1993arXiv:hep-th/9307083v1 12 FTUV/93-21 IFIC/93-09
A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the q-oscillators (q-Weyl-Heisenberg algebra) and for the su q (2) and su q (1, 1) algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the q → 1 limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras.
I.-Introduction
The representation theory of quantum algebras and groups [1,2,3,4], constitutes an open field of research. Some of the features specific to quantum algebras, for example those appearing when the deformation parameter q becomes a root of unity, have been studied already [5,6]. It will be shown here for some of the simplest types of deformed algebras that also for q real there are some interesting realizations which can elucidate the relation between deformation and non-linearity.
We will confine our scope in this paper to three of the simplest quantum algebras i.e., the so called q-oscillator and the su q (2) and su q (1, 1) algebras [7,8,9,10,11]. We will seek complex analytic realizations of the above algebras which, unlike the ones existing in the literature which are based on the so called q-coherent states [12,13,14] and involve q-deformed (Jackson) derivatives [13,15], will be given instead in terms of a series of higher powers of ordinary derivatives. The representation spaces of these deformed realizations will be the ordinary Hilbert spaces of square-integrable analytic functions L 2 ( G H , dµ(ζ)), built on the corresponding cosets of the non-deformed Lie groups, i.e G/H = W eyl−Heisenberg U (1) , SU (2) U (1) and SU (1,1) U (1) . The invariant measure of integration dµ(ζ), the so-called Bargmann measure, will be explicitly given below for each case. One feature of the obtained realizations of the quantum algebra generators is that they constitute a deformation of the ordinary Lie algebra generators in the sense that they involve a series in powers of ordinary derivatives (the coefficients of which depend on the deformation parameter q) that reproduces in the 'classical' q → 1 limit the Lie algebra vector field generators.
The outline the paper is as follows: in Section II, the required coherent states (CS) formulae for each of the non-deformed Weyl-Heisenberg (wu), su(2) and su(1, 1) algebra are given. Also the deformation mappings relating the quantum versions of the above algebras to the generators of the respec-tiveΥ non-deformed ones will be provided, as they will be important in the next Section for the analytic realizations. Section IV will extend the method of obtaining the generator realizations, described in Section III, to the coproducts realized on complex functions depending on two arguments. Finally some conclusions are given in Section V.
II.-Coherent states and deforming mappings
Let G + , G − and G 0 be the generic expressions for the generators of the three-dimensional algebras G = wh, su(2) and su(1, 1). The (unnormalized, see below) coherent states (see [16,17,18,19,20] to which we refer for details on the general group definition of coherent states) can be defined generically as:
|ζ) = eζ G + |φ > , (ζ| =< φ|e ζG − ,(1)
where |φ > is the corresponding lowest weight state of the different algebras,
i.e. |φ >≡ |n = 0 > is the Fock vacuum state for the oscillator; |φ > is given by |j, m = −j > for su (2) with j = 1/2, 1, 3/2, ... and |φ >≡ |k, l = 0 > for su(1, 1) with k = 1, 3 2 , 2, 5 2 , .... The round ket indicates that the CS are unnormalized; the normalized ones are given by |ζ >= 1 √ (ζ|ζ) |ζ). One of the generic relations that will be extensively used below reads
[G ± , f (G 0 )] = (f (G 0 ∓ 1) − f (G 0 ))G ±(2)
for any analytic function f of G 0 ; this relation is common to the three algebras considered and is a consequence of the first commutator in eqs. (4),(5) and (6) below. It also follows from the definition (1) that G + |ζ) = ∂ζ |ζ), (ζ|G − = ∂ ζ (ζ|.
Let us now turn to the deforming mapping by which the generators of the quantum q-oscillator, su q (2) and su q (1, 1) are written uniquely in terms of their non-deformed counterparts. Generically (see eg. [21,22,23,24,11]),
G q ± = G ± F ± (G 0 ) (3a) G q 0 = G 0 (3b)
where G q 0 ≡ N, J 3 , K 3 and G q ± ≡ a ± q , J q ± , K q ± and F ± (G 0 ) is given for each algebra by wh q su q (2) su q (1, 1) Table 1 The deformed generators have the respective commutation relations:
F + (G 0 ) [N +1] N +1 [J 3 +j+1][J 3 −j] (J 3 +j+1)(J 3 −j) [K 3 −k+1][K 3 +k] (K 3 −k+1)(k 3 +k) F − (G 0 ) [N ] N [J 3 −j−1][J 3 +j] (J 3 −j−1)(J 3 +j) [K 3 −k][K 3 +k−1] (K 3 −k)(K 3 +k−1)[N, a ± q ] = ±a ± q , [a − q , a + q ] = [N + 1] − [N] (q − oscillator); (4) [J q 3 , J q ± ] = ±J q ± , [J q + , J q − ] = [2J q 3 ]) (su q (2)) ;(5)
and
[K q 3 , K q ± ] = ±K q ± , [K q + , K q − ] = −[2K q 3 ] (su q (1, 1)) ,(6)
where the square bracket is defined by
[x] ≡ q x −q −x q−q −1 .
III.-Realizations of the q-algebra generators
We shall now develop a method which will enable us to specify complex analytic realizations of the generators of the above quantum algebras. This method will utilize the factorization provided by the deforming mappings (3) by which a deformed generator is given by a non-deformed one times a deformation operator factor F (Table 1).
Let us consider the action of G q i on a generic state |Ψ > leading to another state |Φ i >,
G q i |Ψ >= |Φ i > , (i = +, −, 0) .(7)
Then,
(ζ|G q i |Ψ >= (ζ|Φ i >(8)
may be used to define the representative π ξ (G q i ) of the generators G q i acting on functions Ψ(ζ) defined on the general ζ-Bargmann space,
π ζ (G q i )Ψ(ζ) ≡ Φ i (ζ)(9)
with Ψ(ζ) = (ζ|Ψ > and Φ i (ζ) = (ζ|Φ i > .
Since G q 0 = G 0 (eq. (3b)), in what follows we shall concentrate on G q ± only and give the result for π ζ (G q 0 ) at the end. Using eq.(3) the l.h.s. of eq.(8) is written
(ζ|G q ± |Ψ >= (ζ|G ± F ± (G 0 )|Ψ >= τ ± (ζ|f ± (G 0 )|Ψ > ,(10)
where the actions (ζ|G ± have been evaluated using the following formulae for the wh, su(2) and su(1, 1) coherent states:
(α|a + = α(α| and (α|a = ∂ α (α| (ordinary oscillator) ;
(z|J + = (z|(j − J 3 )z and (z|J − = (j + J 3 )z −1 (su(2)) ;(11)
and
(ξ|K + = (ξ|(K 3 + k)ξ and (ξ|K − = (ξ|(K 3 − k)ξ −1 (su(1, 1)) ,(13)
and τ + (τ − ) are given by α, z, ξ (∂ α , z −1 , ξ −1 ). Then, by virtue of the above relations, the factorizations (3a) and Table 1, we obtain for the f ± (G 0 ) defined by (10) the following values: Table 2 where we have also introduced the definitions of τ ± .
wh q su q (2) su q (1, 1) τ + α z ξ τ − ∂ α z −1 ξ −1 f + (G 0 ) [N +1] N +1 (j − J 3 ) [J 3 +j+1][J 3 −j] (J 3 +j+1)(J 3 −j) (K 3 + k) [K 3 −k+1][K 3 +k] (K 3 −k+1)(K 3 +k) f − (G 0 ) [N ] N (j + J 3 ) [J 3 +j][J 3 −j−1] (J 3 +j)(J 3 −j−1) (K 3 − k) [K 3 −k][K 3 +k−1] (K 3 −k)(K 3 +k−1)
Using the CS defined by (1), the r.h.s of eq.(10) can be cast in the following form:
τ ± (ζ|f ± (G 0 )|Ψ >= τ ± < φ|e ζG − f ± (G 0 )e −ζG − .e ζG − |Ψ > = τ ± < φ|{f ± (G 0 ) + ζ[G − , f ± (G 0 )] + ζ 2 2! [G − , [G − , f ± (G 0 )]] + ...}e ζG − |Ψ > .(14)
Using now relation (2)
τ ± (ζ|f ± (G 0 )|Ψ >= τ ± < φ|{B ± 0 + ζB ± 1 G − + ζ 2 2! B ± 2 G 2 − + ...}e ζG − |Ψ > = τ ± < φ|{ m ζ m m! B ± m G m − }e ζG − |Ψ > ,(15)
where
B ± m ≡ ( m m )f ± (G 0 + m) − ( m m−1 )f ± (G 0 + m − 1) + ...(−1) m ( m 0 )f ± (G 0 ) = m p=0 (−1) m−p ( m p )f ± (G 0 + p)(16)
and m = 0, 1, 2, ...
As the G 0 generators are all diagonal in the basis of the states constructed from their lowest weight |φ >, we will have generically that
< φ|B ± m =< φ|b ± m ,(17)
where the numerical eigenvalues b ± m will be evaluated explicitly below for each of the three algebras considered. Using (17), (1) and that (ζ|G − = ∂ ζ (ζ|, eq.
(15) now leads to
τ ± (ζ|f ± (G 0 )|Ψ >= τ ± {b ± 0 + b ± 1 ζ∂ ζ + b ± 2 2! ζ 2 ∂ 2 ζ + ...}(ζ|Ψ >≡ π ζ (G q ± )Ψ(ζ),(18)
where π ξ (G q ± ) defines the realization of G q ± on the functions Ψ(ζ). Let us notice that in the su q (2) case, the expansions (15) and (18) Collecting now all the above results and replacing τ ± by their values ( Table 2) the following realizations for the deformed generators are obtained:
a) Quantum deformed oscillator π α (a + q ) = ∂ α ∞ m=0 b + m m! α m ∂ m α (19a) π α (a − q ) = α ∞ m=0 b − m m! α m ∂ m α (19b) π α (N) = α∂ α ,(19c)where b ± m = ( m m )f ± m − ( m m−1 )f ± m−1 + ...(−1) m ( m 0 )f ± 0 = m p=0 (−1) m−p ( m p )f ± p (20a) with f + p = [p + 1] p + 1 and f − p = [p] p (20b) since for instance < 0|f + (N + p) =< 0| [p+1] p+1 ≡< 0|f + p .
The carrier space for this realization L 2 (C, 1 π d 2 α (α|α) ) possesses an orthonormal basis formed by the monomials {γ n (α) ≡ (α|n >= α n √ n! ; n = 0, 1, 2, ...}, where
< γ n (α), γ n ′ (α) >= δ nn ′ = 1 π d 2 αe −αᾱγ n (α)γ n ′ (α) (21) since 1 = 1 π d 2 αe αᾱ |α)(α| .(22)
The action π α of the generators a ± q , N on the basis vectors is given by the usual expressions
π α (a + q ) α n √ n! = [n + 1] α n+1 (n + 1)! (23a) π α (a − q ) α n √ n! = [n] α n−1 (n − 1)! (23b) π α (N) α n √ n! = n α n √ n! .(23c)
In the q → 1 limit eq. (20b) shows that f ± p → 1 and it is simple to check in eq. (20a) that b ± 0 → 1 and b ± i =0 → 0. As a result, the previous expressions yield the expected 'classical' limit relations π α (a + q ) → α, π α (a − q ) → ∂ α (and of course π α (N) = α∂ α ) of the oscillator Bargmann algebra.
b) su q (2) algebra
Proceeding similarly, we find
π z (J q + ) = z 2j m=0 b + m m! z m ∂ m z (24a) π z (J q − ) = z −1 2j m=0 b − m m! z m ∂ m z (24b) π z (J q 3 ) = −j + z∂ z ,(24c)
where now
b ± m = m p=0 (−1) m−p ( m p )f ± p (25a) and f + p = (2j − p) [2j − p][p + 1] (2j − p)(p + 1) , f − p = p [p][p − 2j − 1] p(p − 2j − 1) (25b)
which follow from computing < j, −j|f ± (J 3 + p) using Table 2.
The Hilbert representation space L 2 (C − {∞}, (2j+1) π d 2 z (1+|z| 2 ) 2j+2 ) admits the following basis {γ n (z) ≡ (z|n >= ( 2j j+n ) 1/2 z j+n , n = −j, ..., j} with orthonormality conditions,
< γ n (z), γ n ′ (z) >= δ nn ′ = (2j + 1) π d 2 z (1 + |z| 2 ) 2j+2γ n (z)γ n ′ (z) ,(26)
where the new measure is designed so that the γ n (z) states are orthonormal since 1 = (2j + 1) π
d 2 z (1 + |2| 2 ) 2j+2 |z)(z| .(27)
The action of the deformed generator on the unit vector of the representation space reproduces the familiar q-Fock expressions
π z (J q + ) γ m (z) = [j − m][j + m + 1] γ m+1 (z) (28a) π z (J q − ) γ m (z) = [j + m][j − m + 1] γ m−1 (z)(28b)
and
π z (J q 3 ) γ m (z) = mγ m (z) .(28c)
In the zero deformation limit, q → 1, f + p → (2j − p) and f − p → p (eq. (25b)). Then, b + 0 → 2j, b + 1 → −1 and all the other b + ′ s become zero. Thus, π z (J q + ) → 2jz − z 2 ∂ z . Similarly b − 1 → 1 and the other b − 's are zero in the same limit and we obtain π z (J q − ) → ∂ z , the realization π z (J 3 ) = −j + z∂ z remaining unaffected. Thus, the reduction to the ordinary coset representation (see e.g.
[17, 18,19]) of the su(2) algebra is provided by the classical q → 1 limit.
c) su q (1, 1) algebra
Finally, for the su q (1, 1) generators we find the realization
π ξ (K + q ) = ξ ∞ m=0 b + m m! ξ m ∂ m ξ (29a) π ξ (K − q ) = ξ −1 ∞ m=0 b − m m! ξ m ∂ m ξ (29b) π ξ (K 3 q ) = k + ξ∂ ξ , (29c) with b ± m = m p=0 (−1) m−p ( m p )f ± p , (30a)
where now Table 2 gives for < k, 0|f ± p (K 3 + p) the expressions
f + p = (2k + p) [p + 1][2k + p] (p + 1)(2k + p) and f − p = p [p][2k + p − 1] p(2k + p − 1) . (30b)
The representation space is now the Hilbert space of square integrable functions with support on the open unit disk on the complex plane, D = {ξ ∈ C :
|ξ| 2 < 1}, denoted by L 2 (D, (2k−1) π d 2 ξ
(1−|ξ| 2 ) −(2k−2) ) which has the basis of vectors {γ n (ξ) ≡ (ξ|n >= Γ(2k+n) n!Γ(2k) 1 2 ξ n , n = 0, 1, 2, ...}, satisfying the orthonormality
condition < γ n (ξ), γ n ′ (ξ) >= δ nn ′ = (2k − 1) π |ξ|<1 d 2 ξ (1 − |ξ| 2 ) −(2k−2)γ n (ξ)γ n ′ (ξ) (31a)
corresponding to the completeness relation
1 = (2k − 1) π |ξ|<1 d 2 ξ (1 − |ξ| 2 ) −(2k−2) |ξ)(ξ| . (31b)
The action on the above basis monomials is given by
π ξ (K q + ) γ n (ξ) = [2k + n][n + 1] γ n+1 (ξ) ,(32a)π ξ (K q − ) γ n (ξ) = [2k + n + 1][n] γ n−1 (ξ) ,(32b)
and
π ξ (K q 3 )γ n (ξ) = (k + n)γ n (ξ) .(32c)
If there is no deformation f + p → 2k + p and f − p → p (eq. (30b)). Then, eq. (30a) gives b − n=0 = b − n≥2 = 0 and b − 1 = 1 so π ξ (K q − ) → π ξ (K − ) = ∂ ξ , and π ξ (K q + ) → π ξ (K + ) = 2kξ + ξ 2 ∂ ξ ; π ξ (K 3 ) is again given by (32c). Having derived a complex analytic realization of the three algebra generators we shall give in the next Section the realizations of the corresponding co-products.
IV.-Analytic realizations for the co-products
In order to construct a realization of the co-products we shall write them generically in the form
∆G q ± = G q ± ⊗ g(G 0 ) + g ′ (G 0 ) ⊗ G q ± (33a) ∆G q 0 = G 0 ⊗ 1 + 1 ⊗ G 0 ,(33b)
where g(G 0 ) and g ′ (G 0 ) will be explicitly found for each algebra separately.
As the co-product of each algebra generator acts in the tensor product of the representation space of the algebra, we shall look for a analytic functional realization carried by functions of two variables. To this aim consider
(ζ 1 | ⊗ (ζ 2 |∆G q ± |Ψ >= (ζ 1 | ⊗ (ζ 2 |G q ± ⊗ g(G q 0 )|Ψ > + +(ζ 1 | ⊗ (ζ 2 |g ′ (G q 0 ) ⊗ G q ± |Ψ > .(34)
Using the deformation mapping (3) and eq. (10) this equation gives
(ζ 1 | ⊗ (ζ 2 |∆G q ± |Ψ >= τ 1 ± (ζ 1 | ⊗ (ζ 2 |f ± (G 0 ) ⊗ g(G 0 )|Ψ > + +τ 2 ± (ζ 1 | ⊗ (ζ 2 |g ′ (G 0 ) ⊗ f ± (G 0 )|Ψ >≡ I 1 + I 2 ,(35)
where τ 1 ± and τ 2 ± , where the indices refer to the first or second term in the tensor product, are the same as in Table 2 for the different algebras. We shall now compute I 1 and limit ourselves to give the formulae for I 2 , since they are derived in exactly the same fashion. Using again definition (1) we find
I 1 = τ 1 ± < φ|⊗ < φ|(e ζ 1 G − ⊗ e ζ 2 G − )(f ± (G 0 ) ⊗ g(G 0 ))(e −ζ 1 G − ⊗ e −ζ 2 G − ) (e ζ 1 G − ⊗ e ζ 2 G − )|Ψ >= = τ 1 ± < φ|⊗ < φ|(e ζ 1 G − f ± (G 0 )e −ζ 1 G − ⊗ e ζ 2 G − g(G 0 )e −ζ 2 G − ) (e ζ 1 G − ⊗ e ζ 2 G − )|Ψ > .(36)
Expanding the exponentials in the above formula and rearranging the ensuing nested commutators for f ± (G 0 ) and g(G 0 ) as we did for eqs. (14,15) we obtain
I 1 = τ ± 1 < φ|⊗ < φ| l n=0 ζ n 1 n! B ± n G n − ⊗ l m=0 ζ m 2 m! C m G m − e ζ 1 G − ⊗ e ζ 2 G − |Ψ > ,(37)
where l is the appropriate limit for each algebra, B ± n are the functions of G 0 introduced in (15), (16) and (17), and the operators C m are defined (cf. (16)) by the expansion
C m = ( m m )g(G 0 + m) − ( m m−1 )g(G 0 + m − 1) + ... + (−1) m ( m 0 )g(G 0 ) = m p=0 (−1) m−p ( m p )g(G 0 + p) .(38)
On the vacuum, < φ|C m =< φ|c m (the numerical eigenvalues will be specified later for each algebra separately). Then, using that (ζ|G − = ∂ ζ (ζ|, eq.(37) may be written as
I 1 = τ ± 1 (ζ 1 | ⊗ (ζ 2 | l n,m=0 b ± n n! c m m! ζ n 1 ζ m 2 G n − ⊗ G m − |Ψ > = τ ± 1 l n,m=0 b ± n n! c m m! ζ n 1 ζ m 2 ∂ n ζ 1 ∂ m ζ 2 (ζ 1 | ⊗ (ζ 2 |)|Ψ > = π ζ 1 (G q ± )π ζ 2 (g(G 0 ))Ψ(ζ 1 , ζ 2 ) ,(39)
where we have used (18) for the identification
π ζ 1 (G q ± ) = τ ± 1 l n=0 b ± n n! ζ n 1 ∂ n ζ1 (40a)
and written
π ζ 1 (g(G 0 )) = l m=0 c m m! ζ m 2 ∂ m ζ 2 .(40b)
Collecting all results for eq.(33) the realization of the co-product is obtained,
π ζ 1 ζ 2 (∆G q ± )Ψ(ζ 1 , ζ 2 ) ≡ (ζ 1 | ⊗ (ζ 2 |∆G q ± |Ψ >= = [π ζ 1 (G q ± )π ζ 2 (g(G 0 )) + π ζ 1 (g ′ (G 0 ))π ζ 2 (G q ± )] Ψ(ζ 1 , ζ 2 ) ,(41)
and, similarly,
π ζ 1 ζ 2 (∆G q 0 ) = [π ζ 1 (G 0 ) + π ζ 2 (G 0 )] Ψ(ζ 1 , ζ 2 ) .(42)
As there is no satisfactory co-product and bialgebra structure for the qoscillator, we shall now specialize to the su q (2) and su q (1, 1) cases for which we have, respectively,
∆J q ± = J q ± ⊗ q J q 3 + q −J q 3 ⊗ J q ± (43a) ∆J q 3 = J q 3 ⊗ 1 ⊗ 1 ⊗ J q 3 ,(43b)
and
∆K q ± = K q ± ⊗ q K q 3 + q −K q 3 ⊗ K q ± ,(44a)∆K q 3 = K q 3 ⊗ 1 + 1 ⊗ K q 3 . (44b) π z 1 z 2 (∆J q 3 )γ n,m (z 1 , z 2 ) = (n + m)γ n,m (z 1 , z 2 ) . (49b) b) su q (1, 1)
Finally, for the su q (1, 1) case, eq. (41), we find
π ξ 1 ξ 2 (∆K q ± ) = π ξ 1 (K q ± )π ξ 2 (q K q 3 ) + π ξ 1 (q −K q 3 )π ξ 2 (K q ± ) (50a) π ξ 1 ξ 2 (∆K q 3 ) = π ξ 1 (K q 3 ) + π ξ 2 (K q 3 ) ,(50b)
where
π ξ 1,2 (K q ± ) = ξ ±1 1,2 ∞ m=0 b ± m m! ξ m 1,2 ∂ m ξ 1,2 (51a) π ξ 1,2 (K q ± ) = k + ξ 1,2 ∂ ξ 1,2 (51b) π ξ 1,2 (q ±K q 3 ) = ∞ m=0 c ± m ξ m 1,2 ∂ m ξ 1,2 ,(51c)
where the coefficients b ± m are given in eq.(28) and the c ± m in this case read (cf. eq. (47))
c ± m = ( m m )q ±(k+m) − ( m m−1 )q ±(k+m−1) + ... + (−1) m q ±k = m p=0 (−1) m−p q ±(k+p) .(52)
As for su q (2), the orthonormal basis which spans the representation space of the above co-product realization inherits its orthonormality properties from eq.(31) and reads γ nm (ξ 1 , ξ 2 ) ≡ (ξ 1 | ⊗ (ξ 2 |.|n > ⊗|m >= = Γ(2k + n) n!Γ(2k) 1 2 Γ(2k + m) m!Γ(2k)
Extending the action expressed by eqs. (32) to the co-product realization of the su q (1, 1) generators we obtain π ξ 1 ξ 2 (∆K q + )γ n,n (ξ 1 , ξ 2 ) = = [n + 1][n + 2k]q k+n γ n+1,m (ξ 1 , ξ 2 ) + q −(k+m) [m + 1][m + 2k]γ n,m+1 (ξ 1 , ξ 2 ) (51a) π ξ 1 ξ 2 (∆K q − )γ n,m (ξ 1 , ξ 2 ) = [n][n − 1 + 2k]q k+n γ n−1,m (ξ 1 , ξ 2 )+ q −(k+m) [m][m − 1 + 2k]γ n,m−1 (ξ 1 , ξ 2 ) (54b) and π ξ 1 ξ 2 (∆K q 3 )γ n,m (ξ 1 , ξ 2 ) = (2k + n + m)γ n,m (ξ 1 , ξ 2 ) .
(54c)
V.-Conclusions
In this paper we have looked for realizations of quantum algebras in terms of ordinary differential operators. We have found their expression for certain q-algebras for which there exist functional deforming mappings relating the deformed and non-deformed generators which allow us to write explicitly the deformed generators as elements of the original enveloping algebra. The qoscillator, su q (2) and su q (1, 1) quantum algebras are particular cases of this class, and the method of constructing realizations relies on the ordinary coherent states for the undeformed Lie algebras.
Finally, we recall that the realization for the deformed algebra generators obtained here is given in terms of a series of powers of derivatives with qdependent coefficients which in the classical q → 1 limit reproduces the vector field generators of the Lie algebra. Thus, the appearance of ordinary but higher order derivatives provides an alternative way of describing the deformation process.
The complex variables ζ (ζ = α, z, ξ) label the CS; ζ andζ are the projective coordinates of the respective coset spaces G/H where H is the isotropy group of the vacuum state |φ >, namely G/H = W H/U(1) ≈ R 2 , SU(2)/U(1) ≈ S 2 ; SU(1, 1)/U(1) ≈ S 1,1 , i.e. the two-dimensional plane, sphere and hyperboloid respectively. Moreover G + stands for the generic creation operator which together with the two other generators G − , G 0 , close into the respective Lie algebras G.
we may arrange the nested commutators in the last equation in such a way that the functions of the generators G 0 always appear at the left of the monomials of G − . This yields
on any vector of the representation space.
Supported by DGICYT, Spain. *azcarrag @ evalvx.ific.uv.es. **ellinas @ evalvx.ific.uv.es.
so that g(G 0 ) = q Go and g ′ (G 0 ) = q −G 0 with G 0 = J 3 , K 3 respectively. a) su q(2)For the su q (2) case eq. (41) and(42)givewhere the b's are given by eq.(25) and (cf. eq. (38))The realization of the co-product acts on the tensor product of two representation spaces which is spanned by the basisand which satisfies an orthonormality relation induced by that in each of the factors (eq. (26)). On the basis elements γ nm (z 1 , z 2 ) the action of π z 1 z 2 is given by (cf. eq.(28)), π z 1 z 2 (∆J q ± )γ n,m (z 1 , z 2 ) = = [j ∓ n][j ± n + 1]q m γ n±1,m + q −n [j ∓ m][j ± m + 1]γ n,m±1 ,
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| []
|
[
"Harnessing the power of Social Bookmarking for improving tag-based Recommendations",
"Harnessing the power of Social Bookmarking for improving tag-based Recommendations"
]
| [
"Georgios Pitsilis [email protected] \nComputer Science Research\nAthensGreece\n",
"Wei Wang [email protected] \nSchool of Computer and Information Technology\nBeijing Jiaotong University\nBeijingChina\n"
]
| [
"Computer Science Research\nAthensGreece",
"School of Computer and Information Technology\nBeijing Jiaotong University\nBeijingChina"
]
| []
| Social bookmarking and tagging has emerged a new era in user collaboration. Collaborative Tagging allows users to annotate content of their liking, which via the appropriate algorithms can render useful for the provision of product recommendations. It is the case today for tag-based algorithms to work complementary to rating-based recommendation mechanisms to predict the user liking to various products. In this paper we propose an alternative algorithm for computing personalized recommendations of products, that uses exclusively the tags provided by the users. Our approach is based on the idea of using the semantic similarity of the user-provided tags for clustering them into groups of similar meaning. Afterwards, some measurable characteristics of users' Annotation Competency are combined with other metrics, such as user similarity, for computing predictions. The evaluation on data used from a real-world collaborative tagging system, citeUlike, confirmed that our approach outperforms the baseline Vector Space model, as well as other state of the art algorithms[19][18], predicting the user liking more accurately. | 10.1016/j.chb.2015.03.045 | [
"https://arxiv.org/pdf/1410.5072v1.pdf"
]
| 13,405,297 | 1410.5072 | 305e3f4a71d79b65fb3a9aad1757f89c51b7b119 |
Harnessing the power of Social Bookmarking for improving tag-based Recommendations
19 Oct 2014
Georgios Pitsilis [email protected]
Computer Science Research
AthensGreece
Wei Wang [email protected]
School of Computer and Information Technology
Beijing Jiaotong University
BeijingChina
Harnessing the power of Social Bookmarking for improving tag-based Recommendations
19 Oct 2014Recommender SystemsCollaborative TaggingAffinity Propaga- tionciteUlikeTaxonomy
Social bookmarking and tagging has emerged a new era in user collaboration. Collaborative Tagging allows users to annotate content of their liking, which via the appropriate algorithms can render useful for the provision of product recommendations. It is the case today for tag-based algorithms to work complementary to rating-based recommendation mechanisms to predict the user liking to various products. In this paper we propose an alternative algorithm for computing personalized recommendations of products, that uses exclusively the tags provided by the users. Our approach is based on the idea of using the semantic similarity of the user-provided tags for clustering them into groups of similar meaning. Afterwards, some measurable characteristics of users' Annotation Competency are combined with other metrics, such as user similarity, for computing predictions. The evaluation on data used from a real-world collaborative tagging system, citeUlike, confirmed that our approach outperforms the baseline Vector Space model, as well as other state of the art algorithms[19][18], predicting the user liking more accurately.
Introduction
Collaborative tagging, a web-based service that is representive of the new Web 2.0 technology, allows users to store and share various kinds of web resources, such as news, blogs, and photos into social data repositories. Resources are stored into self-emerging structures called folksonomies, in the form of a post that combines a) an identifier of the resource, b) the user who posted it and c) a set of tags. Many web-based resource sharing and publishing services, like youtube 3 , flickr 4 , and Amazon 5 have already adopted such model, allowing user-generated tags to facilitate user information search. The concept of using tags for on-line annotation of objects, also known as Social Bookmarking or Collaborative Tagging, constitutes tags as a novel source of information. Although the use of tags has been found very convenient for managing and organizing people's digital material, from the research perspective it seems to have attracted much interest in Recommender Systems (RS) in the recent years, with literature rapidly expanding.
Despite Collaborative Filtering (CF) algorithms being the most adopted techniques for Recommender Systems, the increasing popularity of collaborative tagging systems pushed towards to tags being integrated into the process of recommendation production. Mechanisms which employ the tags alone for computing item recomendations are less common [8], while wherever numeric ratings are additionally provided, they are used complementary to tags for computing item recommendations [31]. Relying exclusively on the user-provided tags for computing recommendations, it requires that such information is exploited in the best way for achieving satisfactory quality of predictions. This is the case for digital publication services, like flickr, and in general for social networking services, since they provide no-mechanism for numeric ratings-based evaluation of the published content by the users.
Different from numeric ratings, tags also carry sementic information that can be further exploitable. In addition, tag-words can be classified into hierrarchical ordered systems, called taxonomies, structured upon the natual relationships between their elements. Measurements like, Semantic Distance and Relatedness between tags are computable using the taxonomies. Knowing such distance can prove very important when needing to group similar tags together. In some cases grouping can help to overcome issues like polysemy of tags or the use of synonyms by users for annotating the same item. Such issues exist because users behave differently as far as the way they do annotations, expressing their own style on this task, that differs from one user to another.
While exploring the information hidden on tags for improving recommendations has already been a topic for investigation by the research community in the past, the Annotation Competency of users has not been taken into account yet. In this paper we attempt to utilize the power of taxonomies through tag clustering, along with giving a useful insight into the Annotation Competency of the users.
The rest of the paper is organized as follows: In Section 2 we explain our motivation and related work in the field. In Section 3 we reason about the idea of tag clustering in more detail and describe our algorithm. Section 4 is referred to the evaluation tests we performed and the results received, and finally in section 5 we present our conclusions.
Related Work and Motivation
Based on the existing literature, a simple taxonomy of the tag recommender systems could be as depicted in Fig.1 and it is explained as follows: We can distinguish two major types of algorithms, a) tag recommendation algorithms and b) tag-oriented resource recommenders. The first category comprizes solutions aiming to ease the process of annotation by providing personalized recommendations of tags to users about specific items [11][20] [27]. Mechanisms that belong to the first category can either exist as part of a larger concept for resource recommendion, or they can stand as independent services, enabling social network applications to providing automated annotation of various kinds [9] [15]. The tag-oriented category regard prediction models exclusively for resource recommendations, which can be further divided into two sub-categories. For better reference we will call these categories: Tag-assisted CF and Tag-based CF. In tag-assisted CF belong those models which require both item rating values and tags to be available for working out predictions. On the other hand, tag-based CF comprises those models in which the computation of predictions can be performed using the tags alone. The first category has been more explored than the second one, hence the more literature available. It is interesting to note that almost all proposed models that belong to this category perceive the task of recommendation production as a two step process. First, computing the nessesary similarity correlations, and then performing the item predictions using the pre-computing similarities from the first step.
Limitations of the Classical Approaches
There has been a growing number of research efforts that could be classified as tag-assisted CF model. For instance, in [10] [33] [21] there has been an effort to user and tag similarities to be combined together into a single expression of similarity, while the work by Parra et.al [16], employs a type of tag-based similarity. Finally, Tso-shutter et.al in [28] introduce a method of fusing userbased with item-based CF, treating the user tags as additional data.
Nevertheless, we focus our interest on tag-based CF model mainly for two reasons. First, because it is very common for tags to exist as the only available source of information that users provide in a RS, and second, the less constraints imposed by this model extends its suitability to a wider range of applications.
For instance, we will refer to one key model from the literature for tag-based CF. In the work by Peng et al. in [19], each tag is viewed as a distinct topic, while the liking of a user to an item is seen as the probability of this user to experience that item. The value of this probability is computed by summing the transition probability through all tags used for tagging this item. The formula they introduced for computing the probability p(i|u) that a user u would like item i is given as follows:
p(i|u) = t∈T p(t|u) · p(i|t)(1)
where: T is the set of tags used by user u, and p(t|u) is the probability that user u uses tag t for item annotation, and p(i|t) is the conditional probability of experiencing item i when tag t is given.
The intuition behind their formula can be phrased as follows: The liking of a user u to item i is highly related to the probability that a particular tag is used by that user, as well as the popularity of this tag when used for annotating item i. That probability p(i|u) is more or less characteristic to the way that a user makes his own selection of tags for annotating objects. To capture this for a user we introduce the notion of Annotation Competency. We hypothesize that the quality of predictions received by a user would certaintly be affected to some degree by his/her such Competency.
We should also point out that the use of Eq.1 becomes inefficient for the reason that users maintain some own collection of words they use for annotating. For that reason it becomes less likely for the probability p(i|u) to be computable, thus strongly affecting the avaibalility of predictions. Such formula imposes a serious limitation for the model to work as it requires a significant overlap to exist between the words that various users have used. For the reason that it is very much the case for users' Annotation Competency to be as such, we argue that the above requirement expressed in Eq.1 would result to the algorithm performing poorly.
In our opinion, a good model should consider as well any differences that might exist in the Annotation Competency from one user to another.
Another evidence that supports our argument, is the fact that most systems which belong to the above two Tag-oriented categories consider the relationship between the available sources of data as a tripartite structure of items-tags-users. Employing such structure has the weakness of being able to capturing only the two out of the three binany associations at a time, among the tags, entities and users, yet something easy to observe and well mentioned by other researchers in the field [21]. As such, that would result to sparse user-tag and item-tag matrices, which in addition to the above mentioned requirment we set regarding the Annotation Competency would further degrade the recommendation quality.
Solutions for Overcoming the Limitations
The above remarks suggest the need for taking into account as well the Annotation Competency in the recommendation process. We distinguish the following two central attributes to describe the Annotation Competency of a user.
-The Diversity of Concepts used by a user throughout his/her tagging excercise -The Annotation Contribution on items.
The attribute of Diversity of Concepts accounts the variety of topics which characterize the profile of a user's interests.
The intuition behind this attribute is driven from the fact that not all users have the same level of experience nor they show the same willingness in their annotation exercise. We consider this attribute to be highly important for the quality of recommendations, mainly because the more experienced or eager users, compared to other users, are meant to be more influencing on the computed predictions. That is because a user who selects tags out from a corpus that includes words from many disciplines, accually provides more data for the system training, and hence he should be regarded to be more contributing than another user whose tags regard only a small area of topics.
As far as the second attribute, Annotation Contribution, while it captures the same requirement of Annotation Competency, as the first one, it though works on item scale. Concerning Annotation Contribution, we perceive that attribute as being expressed by quantitative criteria and it varies from one user to another for a particular item. The quantitative aspect in our case interprets as: The more tags provided for annotating an object the better it is. That is nessesary for distinguishing the contribution of a user who has used few, but identical tags, from another user who has used more descriptive tags for annnotating the same item. For example the tags 'cat' and 'hungry' add more information to the context of the annotated subject than if tags 'cat' and 'hungry' were used. While the first attribute regards the whole reprtoire of tags used by some user in overall, the second one is refered to only some particular annotation experience of that user.
Next we present in more detail our design considerations concerning the two attributes we introduced.
Design Considerations concerning the Diversity of Concepts
To implement the first attibute which expresses the Variety in the Concepts used in a tagging excersize, it requires indentifying all subjects incorprorated into the Annotation Competency of a user.
A straightforard approach to identify the concepts is to partition the tags into distinct Subjects. There are models found in the literature which incorprorate such idea of distincted Subjects. For instance, the work by Peng et al. in [18], follows the concept of organizing the tags into groups. In their work is attempted a refinement of the technique proposed by the same authors and is summed in eqn.1. The refined method employs a type of soft clustering called Consistent Nonegative Matrix Factrorization (CNMF), that is a method of applying multivariate analysis onto the tags, for categorizing them into Subjects. Their refined technique is described in the following formula:
p(i|u) = s∈S p(s|u) · p(i|s)(2)
where p(i|u) is the predicted liking of user u for item i, S is the set of all identified subjects which tags belong to. p(s|u) is the probability of user's u interest in subject s and p(i|s) is the probability of experiencing item i when users are interested in subject s. Similarly to their model described in eqn.1, the refined one follows the central intuition of social annotation that is: If user u has used tag t (or a tag of a subject s) for many times, and tag t (or the subject s) has been annotated on the article i for many times then it is very likely that user u has strong interest on the article i, which should finally be recommended to him/her. From now on, when referring to this formula we will be using the term Topic-Based Variation, for short, while the term Tag-Based Variation will be used when refering to their original method in eq.1
A quite similar approach that works on the same idea of distinuishing topics of interests from the tags, has also been proposed by Shepitsen et al. [25]. Their idea works exactly on the same principle expressed in eqn.2, but it employs hierarchical clustering onto the set of tags for extracting topics. While we consider the solution of partitioning as a very useful one, nevertheless it does not suffciently capture the users' Annotation Competency in the way we expressed it in the two central attributes, for making recommendations. For example, to be in line with the first attribute we set for capturing the diversion in the areas of interest for a user, if applying partitioning alone would not be enough for achieving this.
Furthermore, for the reason that it is quite common for two different people to may have chosen different tags for annotating the same item, it is reasonable to assume that an algorithm that would work on exact matches on the set of tags used, would not be that efficient for identifying similar users. Therefore, the benefit aquired from Collaborative Tagging would not be exploited if using such algorithm.
The logical path to follow would be to group together those tags which are closer to each other, as far as the contextual meaning they carry. From then on in the recommendation proccess, tags would be identified by the cluster ID which they belong to. In this way, it would suffice to simply knowing the user-tocluster associations rather the user-to-item associations, as the existing models require. That would make easier to spot hidden similarities, and any likely common interests between users, which otherwise, due to the sparce user-tag-item relationships, such similarities would not be easily distinguisable.
Design Considerations concerning the Annotation Contribution of a user
As we mentioned earlier, the second attribute we indruduced requires that Annotation Contribution has also been described using quantitative criteria. Next we will refer to the Vector Space model [24], which is itself a basic tag-based recommendation algorithm, that takes as input the frequencies of tags for computing the distance or similarity between users and items.
We mention Vector Space model here as a good starting point of our consideration as it employs qualitative criteria onto the collection of words used by users for finding potentially interesting items for them.
Roughly, the vector space model works as follows: Each user is represented by a tag vector u = [w t1 , w t2 , ..., w tn ] with w t denoting the weight of the patricular tag t on that user. Vector weights may be expressed through many ways, with the frequency of tags to be the most common. Likewise, each resource can be modeled as a vector r = [u t1 , u t2 , ..., u tn ] over the same set of tags. Next, the user's profiles and the resources can be matched over those tag expressions by computing the similarity value between them. Cosine Similarity can be used to obtain these similarity scores between user profiles and rated resources. Then, by sorting the similairites in decending order we eventually get the T op N list of personalized recommendations of resources for a specific user. The cosine of the two vectors in eqn.3 is derived from the Eucledian dot product formula, with ||t n || denote as the length of the vector t n , and t 1 · t 2 is the inner product of the two vectors.
cos(t 1 , t 2 ) = t 1 · t 2 ||t 1 || ||t 2 ||(3)
Since frequency values cannot be negative, cosine similarity will range from 0 to 1, with 1 denoting a perfect match. Adapted versions of the vector space model to work with folksonomies has made this algorihm dominant in the Top-N tag based recomendations models. While Vector Space model takes into account the individual annotation scores of users on items for computing personalized recommendations, in our opinion it only exploits a single dimension of these data.
Another interesting work which also employs cosine similarity is that of Xu et al. [32]. They proposed a Tag-based CF system, which approaches the concept of tag-clustering, in which the cosine similarity of the frequency of each tag over the set of users is used to express a form of distance between the tags. In this way, a resource-tag matrix is composed out of the accumulated occurience rate of each tag, and is then used as input to a clustering algorithm. While the motivation of their approach is rational, finally allocating the similar tags into the same partition, however it does not take into account the semantic information carried by each tag. We mention this approach in our survey mainly because it follows a similar principle for similarity with that of the standard retrieval model for social tagging systems. Different from our concept, the work by Gemmell et.al [5], while it incoprorates the concept of clustering, it finally follows the well established principle used in the Vector Space model. In that one, item suggestions are made upon the computed relevance between users and items. Similarly to the work by Xu et al. [32] they perform clustering on the tags using as input their frequencies of usage in annotation excersizes.
In our opinion, it would not suffice if using alone the frequencies of tags in a metric of distance for tag clustering, because, frequencies can prove not enough to eliminate the implications of Synonimy and Polysemy. Polisemy exists because a tag might have multiple related meanings, and Synonimy exists when different tags sharing the same or similar meaning.
We should also mention the existence of works which explored the idea of studying the relationships between the tags in tag clouds. In the approach by Venetis et.al in [29] new metrics were introduced that capture those relationships. Nevertheless, in these works the concept of Annotation Competency is not approached towards a recommendation model.
Semantic Similarity for Clustering
The above points highlight the need to take into account the Semantic Similarity of tags in our model we introduce for personalized recommendations.
In the way it works, taxonomic similarity for tags works on static knowledge, which tags are as such by their nature. On the contrary, the computation of Cosine Similarity on the Vector Space model, being based on dynamic data, such as the item-tag data structure, requires recomputation upon the arrival of new data. The fact of taxonomic similarity of tags not being dependent on dynamic data, offers the practical advantage of such metric to work efficiently with complex algorithms, like Clustering. That means Clustering should need to run only once, since the distance between the tags does not change. Instead, in the Vector Space model, the frequency of each tag does change, as the RS system develops, requiring frequent recomputation of its value. In a real system, it would suffice to run Clustering just once, during the system initialization, no matter how much data have been provided by the users for annotation. Furthermore, cold start issues would be avoided for the reason that the complete semantic network is established early on, during system initialization.
We believe that the concept of applying Clustering on the tag corpus, using the Semantic similarity as a metric for measuring the distance between the words used as tags, has truly strong potential. To the best of our knowledge, applying the above concept of Clustering for improving the performance of tag-oriented RS, while taking into account the users Annotation Competency, has not been investigated before. As far as the concept of Clustering is concerned, it has itself been the subject of research in RS for either improving the performance of predictions [2] [22] or for securing RS against threats, such as profile injection attacks [14][26] [34].
In our opinion, to achieve a substantial benefit in terms of accuracy from the appplication of Semantic Clustering, it requires a model that would capture the characteristics of social tagging systems more sufficiently and that would incorporate the concept of Annotation Competency. As opposed to existing models, with our prediction model we attempt to utilize the potential of Collaborative Tagging by fusing our introduced properties of Diversity of Concepts and Annotation Contribution into the concept of Annotation Competency.
We sum up the main novelties introduced in our work to the following:
-A semantic-similarity based concept for partitioning the user tags.
-A model for computing personalized item recommendations that uses as input the Annotation Competency of users.
Considering the above requirements, in the next section we propose a model suitable for the tag-based CF type.
Proposed Model
In this section, before we elaborate the details of our concept we will refer to useful knowledge about the components incorporated in our design. Moreover, we present the design considerations of our approach derived from the attributes and their requirements we set in the previous section.
Concept Similarity
In this section we describe in more detail the concept of Semantic Distance we will incorporate in our Similarity model. For computing the distance between any pair of tags we follow the intuitive idea of using the semantic similarity in a taxonomy. That is, the shorter the distance from one tag to another, the more similar the tags are. In our case, tags are regarded as nodes in the taxonomy tree. For computing the distance we used a metric introduced by Resnick [23], which is based on the notion of Information Content. According to this theory, the higher in the hierrarchy a concept is, the more abstract it is, and hence the less information it contains. Resnick's metric assumes the association of probabilities with concepts in a taxonomy and there is also an IS-A relatioship between them in the hierarchy. In that metric the similarity between two concepts c 1 and c 2 is given by:
sim(c 1 , c 2 ) = max c∈U(c1,c2) [− log p(c)],(4)
with function p : C → [0, 1] such that for any concept c ∈ C the value p(c) to be the likelihood of encountering an instance of c. p is monotonic such that, if c 1 IS-A c 2 then p(c 1 ) ≤ p(c 2 ). U (c 1 , c 2 ) denote as the set of concepts that subsume both c 1 and c 2 in the hierrarchy. The actual meaning of that equation is that the more infromation two concepts share in common, the more similar they are.
The information shared by two concepts is indicated by the information content of the concepts that subsume them in the taxonomy.
To enchance clarity and provide a better understanding on how such model could adapt to the issue we come to address, we give an example. In the taxonomy of Fig.2, the similarity between felines and reptiles equals to similarities between tigers and snakes, as well as between tigers and reptiles (sim=5. 19), for the reason that the set of concepts that subsumes both of them is {animals}, and it is common for them. On the contrary, tigers are found to be more similar with cats (sim=10.15), than with bovines (sim=5.61). Tigers and cats are subsumed by felines, that is a concept of higher information content than mammals which subsumes tigers and bovines. Therefore, the similarity of the first pair has higher value than that of the second pair.
The Clustering Algorithm
In order to partition the set of tags into clusters first and formost we needed a metric to express the distance between a pair of tags. We considered as distance the similariy values derived from the application of the Resnick similarity computed onto every pair of tags from a lexical database. WorldNet [3] is a large lexical database of the English language we chose for our experimentation. In WordNet the various parts of speech are grouped together into sets of cognitive synonyms, making up a network of meaningfully related words and concepts.
As far as the clustering algorithm to be used, we chose Affinity Propagation (AP), a newly developed clustering algorithm proposed by Frey et al. [4]. AP was chosen as it is more efficient than other conventional approaches, such as k-means [12], and it has shown to achieve remarkably better clustering quality in various applications. For instance, when AP is applied onto a model for clustering the users of a collaborative filtering system, it helped to improve the prediction quality [22]. AP is also known to achieve better performance than if using Kmeans for Abstracting data in anomaly Intrusion Detection Systems (IDS) [30]. Contrary to k -means algorithm, in which the number of clusters is predefined, in AP the quantity of clusters is dependent on the input data to be clustered, and it can also be affected by a value called Preference. That is a global value applied to each point expressing its suitability to serve as an exemplar. A big Preference value would cause AP to find many exemplars, while a small value would lead to a small number of clusters. Hence, the exact number of clusters emmerges deterministically after a few iterations of the algorithm. In our particular case, AP takes as input the similarities of the pairs of tags in the form of (tag1, tag2, similarity) which we considered as the data points to be clustered. For initial Preference value we chose the minimum similarity value, as that is the one suggested by the authors of the AP algorithm. For our dataset of 3162 tags, after the application of AP, 239 clusters were finally emmerged, 112 of which had been allocated one element only. We provide a graph of the distribution of the sizes of clusters in Fig.3. Since the internals of tag clustering is out of the scope of our paper we will not describe the AP algorithm in more detail. The details of the AP can be referred to [4].
System Model
Assuming that all tags used by a user would portray his/her personal taste in annotation, it means that a proper analysis on the tags used by each user would reveal any hidden similarities that might exist between a user with others. Different from the existing approaches mentioned in the previous section, which mainly exploit the static knowledge provided by the users, in our proposed model instead there is an attempt to capture the characteristics of social tagging via the concept of Annotation Competency we set in section 2.2.
Examining the problem from a data perspective, the operations on those data can be divided into those applied onto the static ones, like the corpus of tags, and those applied onto the dynamic ones, which are related to the users' excercise on tagging.
In the high level view of our proposed idea that is shown pictorially in fig.4, can be distinguished, the available tags being clustered into separate groups, along with the distances between the users, denoted as similarity value S(u1, u2). In the same figure, the relatedness of a user u with some cluster c is denoted as G(u, c). Considering the above perception on data, Clusters represent the static, while the Sim and G refer to the dynamic part, which is derived from the user-item-tag association with the static data.
Next we define new metrics we use for capturing the central attributes of our concept of Annotation Competency. The metric of User Similarity shown in Def.1 captures the Diversity of Concepts expressed by the differences in the tastes of people and in their interests. The Diversity Value shown in Def.2 is another metric that captures the Diversity of Concepts on the user level. The metric of Tagging Effort in Def.3 captures the Annotation Contribution of users on particular items. With these 3 metrics combined together can be determined the suitable items to be recommended to a user. Let's call T the set of all tags used by all users. Tags are partitioned into N clusters c 1 , c 2 , ..., c N , with C the set of all clusters C = {c 1 , c 2 , ..., c N }.
We call E(u i , t j , p k ) = {0, 1} a function that specifies whether a user u i has tagged item p k using tag t j .
We call F a function that specifies whether a tag t j belongs to a cluster c k as:
F (t j , c k ) = 1 t j ∈ c k 0 otherwise
We introduce a function G, we call relatedness and it specifies whether a user u i is a member of cluster c k as:
G(u i , c k ) = 1 ∃t j , p k : F (t j , c k ) = 1, E(u i , t j , p k ) = 1 0 otherwise
The value of 1 is received only when the user in question has used at least one tag that belongs to that particular cluster in his annotation excersize.
Let the set of clusters by user u i be:
C ui = {c k |c k ∈ C : G(u i , c k ) = 1}, C ui ⊆ C
To express the similarity between two users u i and u j we adapted a function proposed by Jaccard [7] to the needs of our concept. The formula of user similarity is given in Eqn.5. Jaccard metric is typically used in the field of data mining to measure the diversity or similarity in sample sets.
S(u i , u j ) = |C ui ∩ C uj | |C ui ∪ C uj |(5)
The || indicates cardinality, (i.e., the number of clusters in the set). S(u i , u j ) has a range value in the interval [0, 1] and maximizes when the two sets C ui and C uj match. The intuition behind this formula is that the larger the number of common clusters the tags of the two users belong to, the more similar the users are, with regard to their taste in annotation and interests. For example, two users who are both interested in cars and machinery are expected to have used tags which belong to clusters most relevant to cars and machinery. Instead, if the first user has used tags that belong to a cluster that is more relevant to housing, it would be expected to have very low similarity with another user who has used tags which belong to both clusters of sports and leasure.
The central attribute of Diversity of Concepts within the tagging excersize of a single user is captured with the Diversity function we introduce next. The intuition behind this function is that, users whose interests comprise many subjects are meant to provide more valuable contribution, in comparison with other users, as far as the tagging excersize is concerned. Being classified as the most important ones, the opinions of those users will be taken into account for rec-ommending articles to others. Definition 2. We propose the following function for Diversity Value w(u i ), which returns a binary quantity, by which we classify whether a user's tagging contribution is valuable or not.
We define U h ⊂ U , U h = {u 1 , u 2 , ..., u k } : k ∈ [1, .., |U |], an ordered subset of all users set U , such that for any two users u f ,u g ∈ U h and ∀f, g ∈ [1, ..., k], with k = |U |, for which |C u f | < |C ug | ⇒ f < g. We call U h1 an ordered subset of U h , so that U h1 = {u 1 , ..., u h } with h = |U| 2 . The Diversity function returns the value:
w(u i ) = 1, u i ∈ U h1 0 otherwise
The binary value received from Diversity function w(u i ) is finally used for filtering out the poorly experienced users, judged on objective criteria. More particularly, a user that belongs to the top 50% of the most experienced ones, in terms of diversity in the subjects of interest, would be considered as a highly contributing user. In our concept, every distinct area of interest is assumed to belonging to a different cluster.
As we mentioned, the above two metrics of Diversity in Def.2 and Similarity in Def.1, are computed upon both the static and dynamic portion of data, and therefore their values require recomputation as the user experiences grow.
In order to be in line with the second property of our design, which we called Annotation Contribution, we adopt the following intuition: We consider those users who have put more effort in annotating some particular items, as being the strongest candidates for recommeding these items to other users. To be consistent with our desing principals, the prediction mechanism should be more sensitive to the quantity and the diversity of tags used by some user for annotating a particular item. Therefore, we find nessasary to introduce the notion of Tagging Effort, that we express here in the form of a metric and we use it as a complementary criterion for filtering out the items of lower interest from being recommended to users. Definition 3. We define the metric of Tagging Effort, f on some item p k as follows. We call T the set of all tags used by all users and T ui ∈ T the set of tags used by a particular user in his annotation excersize. We call T (ui,p k ) ∈ T ui the subset of tags used by u i for tagging item p k . Tagging Effort f (u i , p k ) =
|T (u i ,p k ) | |Tu i |
is defined as the fraction of tags used by user u i for tagging the candidate item p k over all tags used by that user.
To investigate the level of contribution for each of the two criteria of Diversity of Concepts and Annotation Contribution into the quality of predictions, we introduce the contribution factor d. We use the following formula to combine together the above two criteria expressed as per metrics of User Similarity and Tagging Effort. The probability of an item p k to be recommended to user u i by another user u j is computed as:
p(u i , u j , p x ) = d · S(u i , u j ) + (1 − d) · f (u j , p k )(6)
Finally, the probability of an item p k to be liked by user u i is given in equation 7, and it represents the normalized liking of the particular item over all m users who have also experienced the item p k .
p(u i , p x ) = m j=1 [p(u i , u j , p x ) · w(u j )] m(7)
Evaluation
For computing the similarity between the tags of users we used Wordnet::Similarity, a freely available software package by Pedersen et al. [17], that is written in perl. This package provides various measures of relatedness including Resnick's metric which we finally chose to use in our experiment. WordNet::Similarity implements the similarity proposal for IS-A relationships in [23].
We chose the CiteUlike dataset as the most appropriate set for our evaluation. CiteUlike is a public social bookmarking site aiming to promote and develop the sharing of scientific references amongst researchers. One can add a scientific references and then add tags of his choice, allowing to other users to search for references by keywords. The data we used was taken from an available snapshot retrieved in 2009 from the CiteUlike website, and that is provided for research purposes [1]. This dataset is available in the form of a single file, every line of which is consisted of four elements: a) the id of an article annotated, b) the ID of the user who annotated the article and c) the tag word used for annotation, and d) the time of the annotation. From the above fields we can easily build the associations between users, tags and articles. For the needs of our experiment, and due to the fact that the original dataset was very large and sparse, we finally chose a subset of 1000 users, randomly selected out of the 46444 users contained in the original set.
For computational efficiency and recommendation quality we applied filtering onto the selected articles so that only those which have been annotated by at least 15 users were finaly considered in the evaluation. In addition, articles that had been annotated for more than 75 times were excluded. For the same reason we applied filtering on the tags too, considering only those that have been used for at least 10 times in the training set. Finally, we also applied filtering on the users set. Thus, the final 1000 user dataset we used, contained only users who had annotated at least 20 articles. We chose these filtering values in order to minimize the impact of the use of reduced dataset on the tested algorithms.
We performed 5-fold cross validation over the 1000 user data set to test all algorithms. That is, we randomly divided the user dataset into 5 subsets of 200 users each, where in each fold we kept the annotations of one subset of users hidden and tried to predict the liking of those users, using the remaining 4 folds. The former and the latter subsets are known as test set and training set respectivelly. In the prediction phase we used our algorithm to recommend the top 20 articles for each user, and compare them with the actual articles found in the test set for the same user.
To distinguish the probable articles we set a Threshold Probability value of zero as the probability of an article must not be equal to, in order to be counted as a probable article overall. Then, for every user we compiled a top 20 list which includes those articles whose predicted probabilities to be liked by that user would have exceeded the Threshold Probability value. We call a hit an article which has been selected in the top 20 list of a user and for which it trully happens to be one of the items which the user has annotated. We assume that users only annotate items which they trully like. We measure the number of correct recommendations, or hits with the symbol N hit . N rec is the total number of recommendations and it counts those cases in which the computed probability p(u i , p x ) in eq. 7 has received a positive value. N test is the number of articles in the test set. After applying the filtering there were in total 845 articles found to meet the criteria, which composed our test set (N test = 845). More details about the data used can be found in table 1. Moreover, we performed further analysis onto the results for the proposed algorithm, measuring the performance for the highly active and the least active users in separate. For classifying the highly active users we used the Median of the number of clusters |C ui | of each individual user as a threshold value. As defined earlier in Def.1, C ui is the the number of clusters which the tags of user u i are spanning to, meaning that a user with |C ui | value larger than the chosen threshold would be considered as a highly contributing user and hence as a highly active one.
For measuring the ability of our proposed algorithm to provide a list of recommendations of articles that users actually like, we used the evaluation method called Precision and Recall. This method measures this ability in terms of Classification Accuracy and it is widely used in Information Retrieval [6] [13]. The metrics used in Classification accuracy are Precision (P), Recall (R) and F Score (F). For the case of systems that generate Top N recommedations, like ours, the definitions of Precision and Recall are slightly adjusted from the standard way used in Information Retrieval.
Precision indicates the success of the algorithm regarding whether some recommendation provided by the algorithm for some particular user matches a real liking of that user. Precision is defined as the ratio of size of hit set size of top N set . The relative success in retrieving all items liked by individual users is expressed with Recall. Finally, the trade-off between P and R is measured with the F score, which is the harmonic mean of the two values. The metrics used are shown in table 2. For reference we also tested the classical Vector Space method onto the same data, which employs no clustering. To perform this we computed the cosine similarity as in eqn.3 between every pair of users and items and we finally selected a list of Top 20 most similar items to recommend for each user. We also compared against the while simple, but powerful alternative Tag-Based recommendation method by Peng et al., expressed in eqn. 1. For comparison and for showing whether the use of the transition probability over all subjects might work better with clustering, we also evaluated the Topic-Based variation by Peng et al., expressed in eqn.2. Finally, in our evaluation we used our clustering approach for deriving the various subject categories.
To reason whether prediction schemes do actually worth over doing selections without definite aim, we also tested our method against a random selection scheme. The main idea of random selection scheme is to build up the lists of recommended items for each user, in which the top items (20 in our experiment) will be randomly selected out of the N test items of the whole set. Next, the number of correct recommendations N hit is counted for each user as normal.
Results -Discussion
We report the most interesting results of our experimentation. The data presented in table 3 shows the average values of 10 measurements. The largest values of P,R and F score are highlighted in bold. We tested our scheme for various values of the d factor, ranging from 0.0 to 1.0. This was mainly done to study whether a mixture of the two criteria has any effect on the prediction quality. We also include in our report the performance figures of Tag-based Recommendation method by Peng et.al [19], and its variation we called Topic-based Recommendation [18], that we compare ours against to. All comparative results are shown pictorially in fig.5,6 and 7. We make the following observations on the results. As can be seen in figures 5,6 and 7, our method outperforms all the alternative algorithms we compared against it. More specifically, our method performs best when the d factor receives extreme values (b → 0 or b → 1). Moreover, the proposed method outperforms all the other alternatives in terms of F Score, when d > 0.7, with performance reaching its peek for d = 0.9. According to our results, the Vector Space model is the second best performing alternative, with the third best to be the Topicbased recommendation method by Peng et.al., which our method outperforms for almost all values of d, (except for d = 0.3 and d = 0.4). The observed peak value for F score at d = 0.9 can be interpreted as saying: in our proposed method the Tagging Effort criterion on particular items is less significant than that of Profile Similarity.
More precisely, at that peak value of F Score, our method (d = 0.9) appears to be 17.17% better than the second best (Vector Space model) achieving F score=0.00432 vs 0.00368.
From the diagrams of P and R in fig.5 and 6, we observe that the good performance of our method is terms of F, in relation to the other methods we compared it with, is due to the high R values achieved. If considered the Precision values alone, the Vector Space model would have been the best performed. On the contrary, in terms of Precision, the value of d does not appear to have the same strong impact on the performance as it does for Recall.
Moreover, looking at the diagrams of Precision and Recall more carefully, we can observe a significant drop in the Precision values for a long range of To investigate the distribution of the F Score in the number of users, we also demonstrate the Cumulative Distribution Function (CDF) of F Score in figure 8. CDF describes the probability that F Score receives a value less or equal to x ( P r(F score ≤ x) ). In a good model, F score would receive as large as possible values, meaning that the CDF curve should go up as less quickly as possible for a good model, meaning that a curve close to the right-bottom corner of the diagram indicates a good model. In general, a CDF curve that is away from the left-top corner of the diagram indicates a good model. In figure 8 we present the CDF of the F Score of our proposed method, as well as the two variations of the Peng et al. technique, and the Vector Space model. We also include the performnce of the Random selection scheme for recommending items.
In total we test our proposed technique for 3 different values of the d factor, 0.0, 0.9 and 1.0. The choice of values for the d factor was done using the following reasoning: 1.0 and 0.0 were chosen as the extreme values which indicate the sole application of either of the two criteria of Tagging Effort or Profile Similarity
The value 0.9 was chosen as an intermediate case in which the method behaves best in terms of Classification Accuracy. We observe that our proposed method performed best in terms of CDF, only for the case that a mixture of criteria was applied. More particularly, the best results achieved for d = 0.9.
Compared to the alternative methods we included in our evaluation, we observe that our proposed technique has shown the best behaviour, with Vector Space and Topic-Based model by Peng et. al to have achieved poor results. Given a scenario that F Score would not fall below than 0.0075, in our proposed technique only the 99.27% (for d = 0.9) of the users would behave as such. That is to say: the F Score has 99.27% chance to not exceed the value of 0.0075, while for the vector space model, the chance of exceeding the same value is even larger (99.55%). In the method by Peng et al, as well as in their Topic-Based variation that makes use of subjects, the chance is also higher, (99.9% for both). For the random policy model, that chance reaches the 100.0%. Compared to each other in therms of CDF, our technique in overall (Considering all cases where F score< x) produces better results, by just 0.126% than the vector space model, 0.412% than the peng et al. method, and 0.389% than the subjects version of the algorithm by the same authors.
Despite the marginal superiority of our method, we can intrepret the promishing results of our approach as saying: In our method it is more likely for the F Score to receive higher value than in any other model used in our experiment. We also present separate performance figures in table 3 and fig. 9 for specific classes of users, like the Most active and the Least active ones.
For the class of Most Active users our method does significalty better than for the mixed population and it outperforms all the approaches it was tested against in terms of F Score in the whole range of d values. As can be seen the best performance is achieved when a mixture of criteria is applied (d = 0.9), achieving 19.50% better accuracy than the second best approach, (Vector Space model). It is interesting to note that this value is the highest ever counted for all categories of users we tested (Most active, Least active and the mixed group of users). On the contrary, for the Least actrive users, the performance of our method produced a lower figure, but it still outperformed the Vector Space for d > 0.8. More specifically F score ranged from performance levels as low as that of the random choice (d =[0.3,...,0.5], F=0.003084), but finally achieving the best performance (F=0.004070) for that category of users for d = 1.0. Beside Classification Accuracy, it is equally important to know the success of the method as far as the population of users that can actually receive the recommendation service. With Covered Population we refer to the users who were able to receive at least one recommendation for their articles included in the test set. Data sparsity is the reason that not all 1000 users from the sample could finally receive recommendations. We present the results of Covered Population for all algorithms we compared against in table 4 and in fig.10. As can be seen the Covered Population is indeed affected by the use of clusters. On the contrary, the Topic-based variation of the algorithm by Peng et al. is affected the most, allowing only to the 5.6% of the total population (56 over 1000 users) to receive recommendations. Instead the Vector Space method is the least affected in terms of coverage. As we can observe, our method is becoming more sensitive with the increase of d, achieving Coverage Population that ranges from 103.9 (d = 0) to 56.6 (d = 1). At the peak performance in terms of F Score (d = 0.9), the Coverage of our method shows a significant drop, as opposed to the Vector Space model, with the number of users who can accually receive recommendations to have been reduced in half (62.3 vs 120.7). In addition, the figures show that the clustering algorithm we chose (Affinity Propagation) applied partitioning onto the tags in a way that resulted to receiving a large number of small clusters, making the computation of Similarity betweed users less probable. That posed a serious implication on the number of the computed recommendations. As such, the chance for the topics of users' interests to overlap is reduced. For that reason we conclude that clustering with Affinity Propagation is not suitable for the Topic-based variation method by The computational cost of our approach to generate recommendations for a user comes from the participation of the following 3 factors:
-User Similarity computation -(as per eqn.5)
-Tagging Effort of a user to particular item -given by f (u i , p k ) (as per Def.3)
-Probability for an item to be liked by a user -(as per eqn.7). We used the following notation, with m denoting the users, n the items, l the tags and c the clusters. As such, the time complexity of the similarity computation is O(c(m+m)). Similarly, the time complexity of the Tagging effort is O(l + nl) and the complexity of computing the probability of liking for an item is O(m). In overall, the complexity of our model is O(m[2c(m + m) + l + nl]) = O(m(cm+nl)) = O(m 2 c+mnl). Respectivelly, for the conventional method (Tagbased ) by Peng et al. expressed in eqn.1, the complexity is O(l(n + nl + m + ml)) = O(l 2 n+l 2 m+ln+lm). As can be seen, the second parts in the two expressions (ours and Tag-Based Pend et al.) denote complexity of equal degree (3rd). The complexity of Topic-based variation of Peng et al. method is O(c(n+nc+m+mc)) = O(c n + c 2 m + cn + cm), and it is nearly equal to that of the first variation, if not including the cost of clustering.
Likewise, the complexity of the vector space method is computed as follows: The time compexity of the task of computing the tag frequency tables over all users and all articles are: O(lm) and O(ln) respectivelly. The cosine similarity computation itself adds another O(3lmn) time complexity to the method, when applied onto all pairs of users and articles, while adding up another O(nm) for the construction of the top lists. In overall for the Vector Space model the complexity is O(lm + ln + 3lnm + nm) = O(lnm). As can be seen, our method again does not exceed the complexity levels of the classical vector space method.
In our opinion, the large overhead generated by the tag clustering process is not that serious for causing any applicability issues in our method. Such overhead is mainly caused by the fact that the input data used for expessing the distances between the tags are not derived directly from the user's tagging experience, as it is the case for other traditional methods, such as Vector Space, or the method by Peng et al. In our method instead, clustering is computed upon the semantic similarity of tags, and for that reason clustering data remains constant thereafter. Therefore, such cost does not contribute to the computational complexity of the recommendation process. For that reason it sufficies if applying pre-clustering once, upon system initialization, and then using the clustering data to any predictions computed thereafter. On the contrary, any approaches based on Vector Space model would require re-computation of clusters on a regular basis, as the user data change, resulting to significant overhead in the system.
Conclusions and Future work
Annotation Competency of users has very little been explored in Recommender Systems. In this paper we attempted to explore the potential of using the information derived from the Annotation Competency of users for improving the prediction accuracy of a Tag-based Recommender system. Such type of systems use alone the tags provided by users for computing personalized item recommendations. Prior works on tag-based recommendations have indicated that there was still space for improvement. Our work is motivated by the need to better understand how users' annotation works and it provides a new insight on how such knowledge could be incorprorated into the mecanism of producing personalized recommendations. We introduced a new approach which applies clustering onto the set of tags that works in succession with our proposed formula for predicting recommendations. Our formula takes into account the properties of Diversity of Concepts and Annotation Contribution we introduced for describing the notion of Annotation Competency. We attempted evaluation on our proposed model using data from citeUlike, a public annotation system for scientific documents. Our experimentation showed that, if the above two properties are put together, it can help substantially to increase the benefit expressed in terms of recommenations quality for users. At the same time, the proposed method was found to be equally computational efficient with other baseline approaches.
We believe that our work will make significant impact on on-line Searching and Recommendation services as its simplicity and its low overhead makes it suitable for such services. We note the importance of getting a better undestanding of the mechanism of the users' annotation excersize. A wider comparison against more Tag-based recommendation algorithms is left as future work. Another important future work is to confirm our conclusions on more annotation datasets. Investigating our method from the security point of view is also an interesting research direction.
Fig. 1 .
1Taxonomy of algorithms
Fig. 2 .
2Example taxonomy
Fig. 3 .
3Distribution of cluster sizes
Fig. 4 .
4Pictorial representation of our concept showing the entities involved.
Definition 1 .
1To capture the central attribute of Diversity of Concepts in the tagging excercises on a pair of users u i and u j we introduce the metric of Similarity.
Peng et al. (topic-based)
Fig. 5 .
5Comparison of the proposed method against other algorithms in terms of Pre
Fig. 6 .Fig. 7 .
67Comparison of the proposed method against other algorithms in terms of Recall Comparison of the proposed method against other algorithms in terms of F score the d factor. Nevertheless, this drop seems to be not enough to eradicate the advantage of Recall values in our method, which achieved for large values of d. This observation can be interpreded as saying that the Profile Similarity criteria is more important for achieving good predictions. On the contrary, in terms of Precision, the value of d does not have a strong impact on the performance.Comparing against the method by Peng et.al alone, we conlcude that,in overall, their both variations produced significantly lower figures of performance than ours, in all aspects.
Fig. 8 .
8CDF of F.Score in the proposed method against other algorithms
F
-Score (Proposed method vs other schemes) for various classes of users Proposed method most active Proposed method least active Peng et al. (Tag-based) Peng et al. (Topic-based) Vector space Random Choice
Fig. 9 .
9Comparison of the proposed method against other algorithms for various classes of users
Peng et al. (see eqn.2), while for our method it worked beneficially for a whole range of values of d. For example, for d ≥ 0.5 our method does better both in terms of Coverage and F Score, against the Topic-based by Peng et. al. The higher performance achieved by our technique, which also employs partitioning for distinguishing the subjects used, justifies the importance of the notions of Diversity of Concepts and Annotation Contribution we introduced.
Fig. 10 .
10Comparison of the proposed method against other algorithms in terms of Coverage
Table 1 .
1Data description table. *A transaction indicates an instance of a single tag out of all tags used by some user for annotating an article.Metric
Value
Number of users after applying filtering
518
Number of articles after applying filtering 845
Number of total tags used by filtered users 3162
Number of transactions *
4931
Avg. Number of articles per user
1.631
Avgerage frequency of selected tags
3.062
Number of clusters of Tags
239
Table 2 .
2Evaluation MetricsMetric
Formula used
Precision size of hit set
size of top N set = N hit
Nrec
Recall
size of hit set
size of test set = N hit
Ntest
F.score
2·P recision·Recall
P recision+Recall
Table 4 .
4Coverage over 1000 users populationMethod
Covered
Population
Random Choice
47.0
Peng el al.
106.9
Peng et al. (Topic-based)
56.0
Vector Space
120.7
proposed, d = 0.0
103.9
proposed, d = 0.1
103.7
proposed, d = 0.2
93.0
proposed, d = 0.3
91.8
proposed, d = 0.4
92.0
proposed, d = 0.5
85.2
proposed, d = 0.6
75.9
proposed, d = 0.7
74.0
proposed, d = 0.8
69.4
proposed, d = 0.9
62.3
proposed, d = 1.0
56.6
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Toast: a topic-oriented tagbased recommender system. G Xu, Y Gu, Y Zhang, Z Yang, M Kitsuregawa, Proceedings of the 12th international conference on Web information system engineering, WISE'11. the 12th international conference on Web information system engineering, WISE'11Berlin, HeidelbergSpringer-VerlagG. Xu, Y. Gu, Y. Zhang, Z. Yang, and M. Kitsuregawa. Toast: a topic-oriented tag- based recommender system. In Proceedings of the 12th international conference on Web information system engineering, WISE'11, pages 158-171, Berlin, Heidelberg, 2011. Springer-Verlag.
How useful are tags? -an empirical analysis of collaborative tagging for web page recommendation. D Zeng, H Li ; H. Chen, M Chau, K Chang, S.-D Lang, P S Chen, R Hsieh, D Zeng, F.-Y Wang, K M Carley, W Mao, J Zhan, ; Paisi, Paccf Soco, Intelligence and Security Informatics. C. C. Yang,Taipei, TaiwanSpringer5075IEEE ISID. Zeng and H. Li. How useful are tags? -an empirical analysis of collabora- tive tagging for web page recommendation. In C. C. Yang, H. Chen, M. Chau, K. Chang, S.-D. Lang, P. S. Chen, R. Hsieh, D. Zeng, F.-Y. Wang, K. M. Carley, W. Mao, and J. Zhan, editors, Intelligence and Security Informatics, IEEE ISI 2008 International Workshops: PAISI, PACCF, and SOCO 2008, Taipei, Taiwan, June 17, 2008. Proceedings, volume 5075 of Lecture Notes in Computer Science, pages 320-330. Springer, 2008.
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| []
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[
"THE SIMPLIFIED WEIGHTED SUM FUNCTION AND ITS AVERAGE SENSITIVITY",
"THE SIMPLIFIED WEIGHTED SUM FUNCTION AND ITS AVERAGE SENSITIVITY"
]
| [
"Jiyou Li ",
"Chu Luo "
]
| []
| []
| In this paper we simplify the definition of the weighted sum Boolean function which used to be inconvenient to compute and use. We show that the new function has essentially the same properties as the previous one. In particular, the bound on the average sensitivity of the weighted sum Boolean function remains unchanged after the simplification. | 10.1016/j.ipl.2016.01.002 | [
"https://arxiv.org/pdf/1412.6268v1.pdf"
]
| 14,541,172 | 1412.6268 | aa57d6b1702bfd45eb5a6e2cb87c361ccfc30eea |
THE SIMPLIFIED WEIGHTED SUM FUNCTION AND ITS AVERAGE SENSITIVITY
19 Dec 2014
Jiyou Li
Chu Luo
THE SIMPLIFIED WEIGHTED SUM FUNCTION AND ITS AVERAGE SENSITIVITY
19 Dec 2014
In this paper we simplify the definition of the weighted sum Boolean function which used to be inconvenient to compute and use. We show that the new function has essentially the same properties as the previous one. In particular, the bound on the average sensitivity of the weighted sum Boolean function remains unchanged after the simplification.
Introduction
In previous study, the weighted sum function has a complicated structure. With a residue ring modulo a prime, the explicit definition of this function can be given using the weighted sum as follows [16]. Let m ∈ Z + = {1, 2, 3, . . . } and prime number p ≥ m where no other prime numbers are between p and m. For vector X = (x 1 , x 2 , . . . , x m ) ∈ Z m 2 , where Z 2 = {0, 1}, let u(X) be the least positive integer which satisfies u(X) = m k=1 kx k (mod p), 1 ≤ u(X) ≤ p.
Then the weighted sum function g(X) is defined as g(X) =
x u(X) , 1 ≤ u(X) ≤ m; x 1 , otherwise.
This function was used to study read-once branching programs by P. Savický and S.Žák [16]. It was also used to demonstrate the exponential improvement from conventional read-once branching programs to quantum ones by M. Sauerhoff in [14], see also [15].
To simplify the definition of the previous weighted sum function, we define a new function f (X) as follows. For It is worth noting that this new function f (X) is more convenient to compute and use than g(X). One particular reason for the prime modulus in the previous function g(X) is that there are nice results and structures in prime fields. In this This work is supported by the National Science Foundation of China (11001170) and the National Science Foundation of Shanghai Municipal (13ZR1422500).
paper we call such f (X) the simplified weighted sum function. Note that when m is prime then the two definitions are the same.
In this paper it is shown that the simplified function f (X) has many similar properties as the previous one g(X). For instance, in [16] the authors used g(X) to establish the lower bound of read-once branching programs. One of the key ingredients in their proof is that Theorem 1.1 (Dias da Silva and Hamidoune, [7]). Let ǫ > 0 be fixed. Then, for every large enough p and A ⊆ Z p with |A| > (2 + ǫ) √ p, and for every b ∈ Z p , there is a subset B ⊆ A such that the sum of the elements of B is equal to b.
We note that the work of Freeze, Gao and Geroldinger [8] implies the similar result in Z m . Theorem 1.2 (Freeze, Gao and Geroldinger, [8]). Let d be the smallest prime divisor of m. Then, for every A ⊆ Z m with |A| > m d + d − 2, and for every b ∈ Z m , there is a subset B ⊆ A such that the sum of the elements of B is equal to b.
We then determine the average sensitivity of this newly defined function f (X) and show that it also satisfies the Shparlinski's conjecture [19] which says that the average sensitivity of f (X) is asymptotically m/2. We introduce the main concepts of this conjecture in the following.
For an input X = (x 0 , x 1 , . . . , x m−1 ), the sensitivity σ s,X (f ) on X denotes the number of variables such that flipping one of these variables will shift the value of f . Explicitly,
σ s,X (f ) = m−1 i=0 f (X) − f (X (i) ) ,
where X (i) = (x 0 , . . . , x i−1 , 1 − x i , x i+1 . . . , x m−1 ) is the vector assignment after flipping the i-th coordinate in X. The sensitivity σ s (f ) of f (X) denotes the maximum of σ s,X (f ) on vector X in Z m 2 and the average sensitivity σ av (f ) is the mean value of sensitivity on every possible input, i.e.,
σ av (f ) = 2 −m X∈Z m 2 m−1 i=0 f (X) − f (X (i) ) .
Sensitivity, together with a more general concept called block sensitivity, is a useful measure to predict the complexity of Boolean functions. It has recently drawn extensive attention, for instance [1,2,3,4,5,6,10,13,17,18,19]. For a good survey on the main unsolved problems on sensitivity, please refer to [9].
In [19] Shparlinski addressed the average sensitivity problem of the previous weighted sum function g(X) and obtained a lower bound from a nontrivial bound on its Fourier coefficients using exponential sums methods. He also developed several conjectures on the average sensitivity of the weighted sum function and the bounds of the Fourier coefficients. Explicitly, one conjecture was that the average sensitivity of g(X) on m variables is not less than ( 1 2 + o(1))m. In the same paper he gave a proof that the average sensitivity is greater than γm, where constant γ satisfies γ ≈ 0.0575.
By applying a new sieving technique, in [10] the first author gave an asymptotic counting formulas of the subset sums over prime fields and thus confirmed the Shparlinski's conjecture on the average sensitivity of the weighted sum function.
In this paper we extend this result for the simplified weighted sum function f (X). That is, for f (X) with m variables, the average sensitivity of f (X) is exactly
(1/2 + o(1))m.
In addition, we also compute the weight of f (X). We prove that the weight of f (X) on m variables is exactly 2 m−1 (1 + o (1)). Thus, f (X) is an asymptotically balanced function.
This paper is organized as follows. In Section 2 we present a sieve formula. By applying this formula, we give a series of formulas for counting subsets sums over cyclic groups in Section 3. The proof of the main results is given in Section 4. We also list several further questions in Section 5.
Notation. For x ∈ R, let (x) 0 = 1 and (x) k = x(x − 1) · · · (x − k + 1) for k ∈ Z + = {1, 2, 3, . . .}. For k ∈ N = {0, 1, 2, . . .} define the binomial coefficient x k = (x) k k! .
A distinct coordinate sieving formula
For the purpose of our proof, we briefly introduce a sieving formula discovered by Li-Wan [11], which significantly improves the classical inclusion-exclusion sieving. We cite it here without any proof. For details and related applications, we refer to [11,12].
Let S k be the symmetric group on k elements. It is well known that every permutation τ ∈ S k factorizes uniquely as a product of disjoint cycles and each fixed point is viewed as a trivial cycle of length 1. For τ ∈ S k , define sign(τ ) = (−1) k−l(τ ) , where l(τ ) is the number of cycles of τ including the trivial cycles.
Theorem 2.1. Suppose X is a finite set of vectors of length k over an alphabet set
D. Define X = {(x 1 , x 2 , · · · , x k ) ∈ X | x i = x j , ∀i = j}. Let f (x 1 , x 2 , . . . , x k ) be a complex valued function defined over X and F = x∈X f (x 1 , x 2 , . . . , x k ).. Then F = τ ∈S k sign(τ )F τ , (2.1) where X τ = (x 1 , . . . , x k ) ∈ X, x i1 = · · · = x ia 1 , · · · , x l1 = · · · = x la s ,(2.
2)
for a permutation τ = (i 1 i 2 · · · i a1 )(j 1 j 2 · · · j a2 ) · · · (l 1 l 2 · · · l as ) with 1 ≤ a i , 1 ≤ i ≤ s and F τ = x∈Xτ f (x 1 , x 2 , . . . , x k ).
Note that the symmetric group S k acts on D k naturally by permuting coordinates. That is, for τ ∈ S k and x = (
x 1 , x 2 , . . . , x k ) ∈ D k , τ •x = (x τ (1) , x τ (2) , . . . , x τ (k) ).
A subset X in D k is said to be symmetric if for any x ∈ X and any τ ∈ S k , τ •x ∈ X. In particular, if X is symmetric and f is a symmetric function under the action of S k , we then have the following formula which is simpler than (2.1).
Corollary 2.2. Let C k be the set of conjugacy classes of S k . If X is symmetric and f is symmetric, then
F = τ ∈C k sign(τ )C(τ )F τ ,(2.
3)
where C(τ ) is the number of permutations conjugate to τ .
For the purpose of evaluating the above summation, we need several combinatorial formulas. Recall that a permutation τ ∈ S k is said to be of type (c 1 , c 2 , · · · , c k ) if τ has exactly c i cycles of length i and that k i=1 ic i = k. Let N (c 1 , c 2 , . . . , c k ) be the number of permutations in S k of type (c 1 , c 2 , . . . , c k ) and it is well-known that
N (c 1 , c 2 , . . . , c k ) = k! 1 c1 c 1 !2 c2 c 2 ! · · · k c k c k ! .
Lemma 2.3. Define the generating function
C k (t 1 , t 2 , . . . , t k ) = ici=k N (c 1 , c 2 , . . . , c k )t c1 1 t c2 2 · · · t c k k .
If t 1 = t 2 = · · · = t k = q, then we have
C k (q, q, . . . , q) = ici=k N (c 1 , c 2 , . . . , c k )q c1 q c2 · · · q c k = (q + k − 1) k .
In another case, if t i = q for d | i and t i = s for d ∤ i, then we have
C k ( d−1 s, · · · , s, q, d−1 s, · · · , s, q, · · · ) = ici=k N (c 1 , c 2 , · · · , c k )q c1 q c2 · · · s c d q c d+1 · · · = k! ⌊k/d⌋ i=0 q−s d + i − 1 q−s d − 1 s + k − di − 1 s − 1 ≤ k! s + k + (q − s)/d − 1 k = (s + k + (q − s)/d − 1) k .
Subset Sum Problem in a Subset of the Cyclic Groups
Let Z m be the cyclic group of m elements. Let D ⊆ Z m be a nonempty subset of cardinality n. Let Z m be the group of additive characters of Z m , i.e, all the homomorphisms from Z m to the nonzero complex numbers C * . Note that Z m is isomorphic to Z m . Define s χ (D) = a∈D χ(a) and Φ(D) = max χ∈ Zm,χ =χ0 |s χ (D)|. Let N (k, b, D) be the number of k-subsets T ⊆ D such that x∈S x = b. In the following theorem we will give an asymptotic bound for N (k, b, D) which ensures N (k, b, D) > 0 when Z m − D is not too large compared with Z m . Theorem 3.1. Let N (k, b, D) be defined as above.
N (k, b, D) − m −1 n k ≤ 1 m 1<r≤m r|m φ(r) n+Φ(D) r + k − 1 k , (3.1)
where d is the smallest prime divisor of m.
Proof. Let X = D × D × · · · × D be the Cartesian product of k copies of D. Let X = (x 1 , x 2 , . . . , x k ) ∈ D k | x i = x j , ∀i = j} . It is clear that |X| = n k and |X| = (n) k . Applying the orthogonal relation ψ∈ Zm ψ(a) = 0 for a ≡ 0(mod m) and ψ∈ Zm ψ(a) = m for a ≡ 0(mod m), we have
k!N (k, b, D) = m −1 (x1,x2,...x k )∈X χ∈ Zm χ(x 1 + x 2 + · · · + x k − b) = m −1 (n) k + m −1 χ =χ0 (x1,x2,···x k )∈X χ(x 1 )χ(x 2 ) · · · χ(x k )χ −1 (b) = m −1 (n) k + m −1 χ =χ0 χ −1 (b) (x1,x2,...x k )∈X k i=1 χ(x i ). Denote f χ (x) = f χ (x 1 , x 2 , . . . , x k ) = k i=1 χ(x i ). For τ ∈ S k , let F τ (χ) = x∈Xτ f χ (x) = x∈Xτ k i=1 χ(x i ),
where X τ is defined as in (2.2). Obviously X is symmetric and f χ (
x 1 , x 2 , . . . , x k ) is normal on X. Applying (2.3) in Corollary 2.2, we get k!N (k, b, D) = m −1 (n) k + m −1 χ =χ0 χ −1 (b) τ ∈C k sign(τ )C(τ )F τ (χ),
where C k is the set of conjugacy classes of S k , C(τ ) is the number of permutations conjugate to τ . If τ is of type (c 1 , c 2 , . . . , c k ), then
F τ (χ) = x∈Xτ k i=1 χ(x i ) = x∈Xτ c1 i=1 χ(x i ) c2 i=1 χ 2 (x c1+2i ) · · · c k i=1 χ k (x c1+c2+···+ki ) = k i=1 ( a∈D χ i (a)) ci = n cimi(χ) s χ (D) ci(1−mi(χ)) ,
where m i (χ) = 1 if χ i = 1 and otherwise m i (χ) = 0. Now suppose order(χ) = r with d ≤ r | m. Note that C(τ ) = N (c 1 , c 2 , . . . , c k ) and by Lemma 2.3 we have
τ ∈C k sign(τ )C(τ )F τ (χ) ≤ τ ∈C k C(τ )n cimi(χ) Φ(D) ci(1−mi(χ)) ≤ k! n+Φ(D) r + k − 1 k .
Similarly, if order(χ) is greater than k, then
τ ∈C k sign(τ )C(τ )F τ (χ) ≤ k! Φ(D) + k − 1 k .
Note that there are φ(r) characters of order r. Summing over all nontrivial characters, we obtain
N (k, b, D) − m −1 n k ≤ 1 m 1<r≤m r|m φ(r) n+Φ(D) r + k − 1 k ,
where φ(r) is the Euler function. This completes the proof.
Corollary 3.2. We have
N (k, b, D) − m −1 n k ≤ n+Φ(D) d + k − 1 k ,
where d is the minimum prime divisor of m.
N (k, b, D) − m −1 m − c k ≤ m d + k − 1 k ,
where d is the minimum prime divisor of m.
A simple combinatorial arguments on sums of binomial coefficients gives (1)).
Average sensitivity
In [10], the weight and the average sensitivity of g(X) are computed. We now generalize these results to the simplified function f (X). We first compute the weight of f (X). In other words, f (X) is an asymptotically balanced function.
Proof. By applying Corollary 3.4 we have
wt(f (X)) = X∈Z m 2 f (X) = m−1 s=0 X∈Z m 2 ,s(X)=s,xs=1 1 = m−1 s=0 N (0, Z m \{s}) = m−1 s=0 1 m 2 m−1 (1 + o(1)) = 2 m−1 (1 + o(1))
In [19] Shparlinski studied σ av (g(X)) and raised the following conjecture:
Conjecture 4.2.
Is it true that for the function given by (1) we have σ av (g(X)) ≥ 1 2 + o(1) m?
In the same paper Shparlinski gave a lower bound by obtaining a nontrivial bound on the Fourier coefficients of g(X) via analytical methods. He proved in the same paper that this value is greater than γm, where γ ≈ 0.0575 is a constant. Li [10] solved this conjecture. Here we prove that this conjecture still holds for f (X): Proof. Since we have the symmetry between the bits 1 and 0, for simplicity we just need to consider the number of bit changes from 0 to 1. Thus by Corollary 3.4, (1)).
2 m−1 σ av (f (X)) = X∈Z m 2 m−1 i=0 f (X) − f (X (i) ) = m−1 s=0 X∈Z m 2 ,s(X)=s,xs=1 m−1 i=0 1 − f (X (i) ) + s∈D X∈Z
Finally we have
σ av (f (X)) = 1 2 + o(1) m.
Further questions
In [19] Shparlinski studied the Fourier coefficients of the weighted sum function. Shparlinski raised the following conjecture which is stronger than his conjecture on the average sensitivity.
Conjecture 5.2 (Shparlinski, [19]). For the previous weighted sum function g(X), we have max | g(a)| = 2 (− 1 2 +o(1))m .
Shparlinski proved in [19] Note that ρ ≈ 0.1587. Currently this is still the best result. We compute the value of max | f (a)| over 0 < m < 22 using a computer program. The results are shown in Table 1. Note again that when m is prime then f (X) = g(X). The experimental results indicate that Shparlinski's conjecture may not be true. It will be amazing to obtain the true bounds on both max | g(a)| and max | f (a)|.
Instead of the Shparlinski's conjecture, we propose a new conjecture:
X = (x 0 , x 1 , . . . , x m−1 ) ∈ Z m k (mod m),and define the new weighted sum function f (X) = x s(X) .
Corollary 3. 3 .
3If |D| = m − c and c is a positive constant, noting that Φ(D) ≤ c we have
Corollary 3. 4 .
4Let n = m − o(m). Let N (b, D) = n k=0 N (k, b, D) be the number of subsets in D which sums to b. Then N (b, D) = 2 n m (1 + o
Theorem 4 . 1 .
41Let f (X) be defined as above. Then we have wt(f ) = 2 m−1 (1 + o(1)).
Theorem 4. 3 .
3Let σ av (g) be the average sensitivity of the previous weighted sum function g(X). Then σ av (g(
Theorem 4. 4 .
4Let σ av (f ) be the average sensitivity of the simplified weighted sum function f (X). Then σ av (f (
N ( 0 ,N ( 0 ,
00Z m \{i, s + i, s}) + Z m \{i, s − i, s})
Conjecture 5. 3 .
3For the newly defined weighted sum function f (X), we have max | f (a)| = 2 (−ρ+o(1))m .
Table 1 .
1The relationship between the variable number m and the maximal Fourier coefficients of f (X) over 0 < m < 22. Decimals are rounded to three decimal places.m max | f (a)| m −1 log 2 max | f (a)| Definition 5.1. Let h(X) be a Boolean function from {0, 1} m to {0, 1}. The Fourier coefficient of h(X) at a is defined by1
1
0
2
0.5
-0.5
3
0.5
-0.333
4
0.75
-0.104
5
0.75
-0.083
6
0.438
-0.199
7
0.469
-0.156
8
0.281
-0.229
9
0.305
-0.191
10
0.227
-0.214
11
0.146
-0.252
12
0.209
-0.188
13
0.093
-0.264
14
0.086
-0.253
15
0.159
-0.177
16
0.067
-0.244
17
0.059
-0.240
18
0.119
-0.171
19
0.053
-0.224
20
0.050
-0.216
21
0.089
-0.166
h(a) =
1
2 m
X∈{0,1} m
(−1) h(X)+a·X .
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| []
|
[
"A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge",
"A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge"
]
| [
"Y Meurice \nDepartment of Physics and Astronomy\nThe University of Iowa\n52242Iowa CityIowaUSA\n"
]
| [
"Department of Physics and Astronomy\nThe University of Iowa\n52242Iowa CityIowaUSA"
]
| []
| We show that for two non-trivial λφ 4 problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For λ larger than some critical value, the method outperforms Padé's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff. | 10.1103/physrevlett.88.141601 | [
"https://arxiv.org/pdf/hep-th/0103134v3.pdf"
]
| 19,069,087 | hep-th/0103134 | 65be4e3d7d0bf4c9137b21624b6a40b904a1ea77 |
A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge
arXiv:hep-th/0103134v3 8 Feb 2002
Y Meurice
Department of Physics and Astronomy
The University of Iowa
52242Iowa CityIowaUSA
A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge
arXiv:hep-th/0103134v3 8 Feb 20021110-z1115Bt1238Cy3115Md
We show that for two non-trivial λφ 4 problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For λ larger than some critical value, the method outperforms Padé's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff.
Perturbative series associated with Feynman diagrams are commonly used in particle physics, solid state physics, optics and chemistry [1]. One remarkable success of this method is the prediction of the values of the anomalous magnetic moments of the electron and the muon with an incredible accuracy. Perturbative methods are also used to perform precision tests of the standard model of electro-weak and strong interactions [2]. Despite these successes, it has been known for a long time [3,4] that series calculated from Feynman diagrams are not convergent but asymptotic. In other words, the range of validity shrinks with the order. One can improve this situation by using Padé approximants [5], either on the original series or a Borel sum [6] of the series, if meaningful. However, even in the cases where the convergence of these alternate procedures can be proven, the convergence is very slow when the coupling is too large. In addition, for short series, it is difficult to estimate the error and to choose the best approximants.
In this Letter, we construct improved perturbative series which converge to values which are exponentially close to the exact ones. The error can be estimated analytically. The method can be applied on the lattice and in the continuum and works well when the methods mentioned above are inefficient or not applicable. The method is a perturbative version of recent numerical calculations performed for various λφ 4 models, namely the anharmonic oscillator [7] and the Landau-Ginzburg model in the hierarchical approximation [8]. In these calculations, we were led to introduce large φ cutoffs and realized that as λ increases, the field cutoff can be decreased without affecting the accuracy of the result. The calculations presented here are perturbative series calculated with a large field cutoff. We only consider λφ 4 problems, however the procedure should extend to any kind of model where large field configurations are suppressed at positive coupling. This is in agreement with the general argument [9] linking the large field configurations to the impossibility of applying Lebesgue dominated convergence to the path integral expression of the perturbative series.
In order to give an idea about the efficiency of existing methods, we consider the well-known example of the ground state energy of the anharmonic oscillator [10]. The solid lines of Fig. 1 represent the number of significant digits obtained with perturbation theory for various λ. As the order increases, the approximate lines rotate clockwise while moving left, forming an approximate envelope. For a fixed coupling, there is an order of perturbation for which the error is minimized. On the other hand, the number of digits obtained with Padé approximants increases with the order. This is a consequence of Carleman's theorem which can be used to show [11] that diagonal sequences of Padé approximants converge to the ground state energy in an appropriately restricted domain of the complex plane. However, the convergence rate becomes slower as the coupling increases. This can be explained [11] from the fact that, when the coupling increases, a [L/L] approximant tends to a constant while the energy increases like λ 1/3 . If Padé approximants are used for the Borel transform instead of the series, one obtains results qualitatively similar which are discussed later. The method that we propose provides a systematic and controllable improvement to regular perturbation and a better convergence than the Padé based methods in the right hand part of Fig. 1. It can also be used in cases where the Borel sum has singularities on the positive real axis. In particle physics, the experimental error bars of precision measurements are shrinking and higher order terms of perturbative series are being calculated. Firmly established discrepancy or agreement between theory and experiment provides valuable information regarding the laws of nature at shorter distances. A common practice [12] to estimate at which order, for a given coupling, we reach the envelope illustrated in Fig. 1, is to determine when the ratio of successive contributions reaches one. This method works quite accurately for the three examples considered below. It seems reasonable to interpret ratios of successive contributions close to one as a signal that one needs to go beyond regular perturbative theory. We are getting close to this situation. For instance, the electro-weak corrections to g µ − 2, give a contribution [13] of 151(4) × 10 −11 and in this calculation, the twoloop effects reduce the one-loop prediction by 35 percent. The total electroweak contribution is about one third of the discrepancy of 43(16) × 10 −10 found by the recent Brookhaven experiment [14]. The problem is more serious in the case of QCD corrections. For instance, in the calculation [15] of the hadronic width of the Z 0 , the term of order α 3 s is more than 60 percent of the term of order α 2 s and contributes to one part in 1,000 to the total width.
We claim that introducing large field cutoffs leads to significantly improved perturbative series. An important reference to understand the general mechanism and to interpret the results presented below is the well studied [4,9] integral
Z(λ) = +∞ −∞ dφe −(1/2)φ 2 −λφ 4 .(1)
If we expand e −λφ 4 , the integrand for the order p contribution is e −(1/2)φ 2 φ 4p /p! and has its maximum when φ 2 = 4p. On the other hand, the truncation of e −λφ 4 at order p is accurate provided that λφ 4 << p. Requiring that the peak of the integrand for the p-th order term is within the range of values of φ for which the pth order truncation provides an accurate approximation, yields the condition λ << (16p) −1 . One sees that the range of validity for λ shrinks as one increases the order. We can avoid this problem by restricting the range of integration in Eq. (1) to |φ| < φ max . We call the truncated integral Z(λ, φ max ). As the order increases, the peak of the integrand moves across φ max and the contribution is suppressed. It is easy to show that the coefficients of the modified series satisfy the bound |a p | < √ 2πφ 4p max /p! and the modified series defines an entire function. However, we are now constructing a perturbative series for a problem which is slightly different than the original one. This procedure is justified from the fact that the error is controlled by the inequality
|Z(λ) − Z(λ, φ max )| < 2e −λφ 4 max ∞ φmax dφe −(1/2)φ 2 . (2)
We have applied large field cutoffs to the anharmonic oscillator with an Hamiltonian H = p 2 /2 + φ 2 /2 + λφ 4 . We use units such that the mass, the frequency andh are unity. The method of Ref. [7] was used to obtain a solution of the time-independent Schroedinger equation for an arbitrary value of the energy E. The eigenvalues are determined by using Sturm-Liouville theorem to monitor the "entrance" of the zeroes of the wave function in a region 0 ≤ φ ≤ φ max as E increases. If φ max is large enough, one obtains excellent numerical values of E n for n not too large by requiring that the n + 1-th zero occurs exactly at φ max . This numerical procedure can be converted into a perturbative expansion order by order in λ. By taking φ max large enough, namely 8, we were able to reproduce accurately the first twenty terms of the series for the ground state calculated by Bender and Wu [10] without using diagrammatic techniques. We have also applied large field cutoffs to Dyson's hierarchical model, an approximation of lattice scalar field theory where the renormalization group transformation can be calculated numerically with great accuracy [8]. We have used a local Lagrangian density of the Landau-Ginzburg form −Aφ 2 − λφ 4 such that when λ = 0, the mass is unity.
The field cutoff appears in the calculation of the Fourier transform of the local measure necessary for the numerical procedure. The free parameter (called c/4 in Ref. [8]) appearing in the kinetic term was chosen in a such a way that a free massless field scales as it would for a nearest neighbor model in 3 dimensions.
We now present numerical results concerning the perturbative series for Z(λ, φ max ), the ground state of the anharmonic oscillator and the zero-momentum two point function of the Landau-Ginzburg model. In Fig. 2, we compare the accuracy of regular perturbation theory with what is obtained with various field cutoffs. The curves for the modified series reach an asymptotic value on the left and drop on the right with the same slope as regular perturbation theory but with an intercept on the coupling axis shifted to the right. In between the two regimes, the curve reaches a maximum. As the order increases, the maximum moves up and right making the convergence apparent in this region. In all cases, the modified models provide an accuracy that goes far beyond the envelope of regular perturbation theory. We have then compared our results with those obtained with Padé approximants. For the clarity of the figure, we have limited ourselves to order 3 and 4 but similar situations are observed for higher orders. At each order, we have selected the best approximant for the series or its Borel sum. These sums are obtained by dividing the l-th coefficient by Γ[l+1+b]. We have followed the procedure of Ref. [6], except that at the end the inverse integral transform was performed numerically at fixed value of λ. We call this procedure the Padé-Borel method. HIER. MODEL FIG. 3. Number of significant digits obtained with field cutoffs given in the text, at order 3 (blue line) and 4 (red line) compared to the best approximants for the regular series at order 3 (blue dots) and 4 (red dots) or the best results obtained with Padé-Borel method at order 3 (purple) and 4 (green).
The values of b were adjusted to get the best possible result. At order 4, the best approximant is [2/2] in the 6 cases considered, but at order 3, the situation is more complicated. In summary, we have used our knowledge of the exact result to get the best possible result for the methods to which we compare our results. Random choices of approximants or of the value of b lead to significantly worse results. The results are shown in Fig. 3. We have used field cutoffs of 1.5 and 1 for the integral and 2 and 1.5 for the two other models. As the field cutoff decreases, the curve moves right as in Fig. 2. By comparing the three methods at the same order, we see that beyond a certain value of λ, the method used here outperforms the two methods based on Padé approximants within a certain range (which broadens when the coupling increases). Number of significant digits for the double-well at order 3 to 6 for regular perturbation (black) compared to series obtained with ymin = −3 and ymax = 3 (blue) or ymax = 2.5 (green). As the order increases, the black curves reach the one-instanton contribution (red) over wider regions to the left while the two other sets reach the accuracy level obtained numerically for ymax = 3 (purple) or ymax = 2.5 (brown).
In all the examples considered above, the Borel sum has no singularity on the positive real axis. One can introduce such singularities by adding a cubic interaction with an appropriate coupling. However, for all examples worked out, this modification can be handled properly with the proposed method. This is due to the fact that as in the previous examples, the exponentials converge uniformly in a compact neighborhood of the origin, and it is legitimate to interchange the sum and the integral. We report here the case of the double-well potential in quantum mechanics as discussed in Ref. [16]. In shifted coordinates, the potential reads (1/2)y 2 − gy 3 + (g 2 /2)y 4 . By imposing the vanishing of the wave function at y = ±10, we were able to reproduce all the significant digits of the first 10 coefficients for the ground state given in Table I of Ref. [16]. This series is not Borel-summable. We have then constructed a modified series by imposing the wavefunction to vanish at y min < 0 and its derivative to vanish at y max > 0. If this prescription is used for numerical purposes, one obtains arbitrarily accurate results when g = (1/2y max ) and y min negative enough. The numerical results are shown in Fig. 4 for values of g where the one-instanton contribution accounts for most of the discrepancy obtained with the regular series. The modified series converge rapidly to the numerical value obtained with the corresponding y min and y max . It takes into account instanton effects and significantly improves the regular perturbative answer. Significant improvements have also been obtained for larger g by decreasing y max .
The striking resemblance among the three models appearing in Figs. 2 and 3 suggests that, in general, the corrections due to the field cutoffs can be expressed as simple one-dimensional integrals. The perturbative expansion of the partition function of an arbitrary lattice scalar field theory with a large field cutoff can be obtained by writing the truncated integral at each site as the integral over the whole real axis minus the integral over |φ| > φ max . Regrouping the contributions with 0, 1, . . . large field contributions, we obtain the partition function
Z[J] = Ce −λ x ( ∂ ∂J(x) ) 4 e 1 2 y,z J(y)G(y,z)J(z) × (1 − A 1 y |φy|>φmax e −A2(φy− z G(y,z)Jz) 2 + . . .),(3)
with A 1 = (2πG(0, 0)) −1/2 , A 2 = (2G(0, 0)) −1 , G(x, y) being the two-point function at λ = 0 (with no field cutoff) and all quantities being written in lattice spacing units. The dots in Eq. (3) are calculable and exponentiate in the limit of a coarse lattice where the correlations among the sites are small. In general, Eq. (3) can be interpreted in terms of Feynman diagrams. A continuum version of this expression can be obtained by using a dilute-gas approximation for configurations with one "lump" of large values. We carried the detail of this calculation in the case of the anharmonic oscillator using the classical configuration φ max e −|τ −τ0| and adapting the arguments of Ref. [17]. The result for the zero-th order correction to the ground state reads:
δE (0) 0 ≃ 4π −1/2 φ 2 max +∞ φmax dφe −φ 2 .(4)
This prediction fits the numerical data over a wide range of φ max . We can estimate the optimal value of φ max without knowing the numerical answer. The left part of the curves shown in Fig. 2 can be estimated by semi classical methods while the right part is given by the next order contribution. In the case of the anharmonic oscillator, using the classical configuration mentioned above, we obtain the error at order n :
|δE o (λ)| ≃ δE (o) o e −(1/2)λφ 4 max + |a n+1 |λ n+1(5)
This approximate formula fits the data very well if φ max is not too small and allows a good estimate of the value of the coupling where the accuracy peaks. The semi-classical calculation performed for the anharmonic oscillator can be extended to other scalar theories with exponentially decaying two-point functions and we expect an exponential control of the error for these models. In the case of lattice gauge theory, the integration over the fields is already reduced to a compact space. In the literature on lattice perturbation theory (with the exception of van Baal [18]), one usually replaces dg by +∞ −∞ dA i µ since in the continuum limit the range becomes infinite. We claim that this lattice artifact can be used to obtain a smooth truncation of the perturbative series as in the scalar case. Different approximations need to be developed to solve the massless quadratic theory with a field cutoff. We expect that, as in the case of the double-well, this approach will lead to a quantitative understanding of the instanton effect.
This research was supported in part by the Department of Energy under Contract No. FG02-91ER40664.
FIG. 1 .
1Number of correct significant digits obtained with regular perturbation theory (solid lines) at order 1, 2, 3, ..., 15 and with Padé approximants [2/2], [3/3], .... [7/7] (dots) for the anharmonic oscillator, vs. log10λ. The various orders can be identified from the explanations in the text. In all the graphs, the logarithms are in base 10.
FIG. 2 .
2Number of significant digits obtained with regular perturbation theory at order 1, 3, 5, ...., 15 (black) and with φmax = 3 (green), 2.5 (blue) and 2 (red), at order 1, 3, ..., 11 , as a function of λ, for the three quantities described in the text. Even orders have high cusps and are not displayed.-0.75 -0.5 -
FIG. 4. Number of significant digits for the double-well at order 3 to 6 for regular perturbation (black) compared to series obtained with ymin = −3 and ymax = 3 (blue) or ymax = 2.5 (green). As the order increases, the black curves reach the one-instanton contribution (red) over wider regions to the left while the two other sets reach the accuracy level obtained numerically for ymax = 3 (purple) or ymax = 2.5 (brown).
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| []
|
[
"Ground state heavy tetraquark production in heavy quark fragmentation",
"Ground state heavy tetraquark production in heavy quark fragmentation"
]
| [
"S Mohammad ",
"Moosavi Nejad \nSchool of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nP.O.Box19395-5531TehranIran\n",
"Nahid Amiri ",
"\nFaculty of Physics\nYazd University\nP.O. Box 89195-741YazdIran\n"
]
| [
"School of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nP.O.Box19395-5531TehranIran",
"Faculty of Physics\nYazd University\nP.O. Box 89195-741YazdIran"
]
| []
| During recent years, the study of exotic hadrons including tetraquarks and pentaquarks has attracted a lot of interests and more studies are in progress experimentally and theoretically. It is well-known that at sufficiently large transverse momentum the dominant production mechanism for standard heavy hadrons (mesons/baryons) is actually the fragmentation so that the same mechanism is also proposed for the production of heavy exotic hadrons. This work is the first attempt to study the direct fragmentation of a heavy quark into ground state heavy tetraquarks in leading order of perturbative QCD. In this regard, we will present an analytical expression for the fragmentation production of neutral hidden flavor tetraquarks (QqQq) which includes most of the kinematical and dynamical properties of the process.I. INTRODUCTIONSince the early days of the quark model it has been assumed the possibility of the existence of exotic multiquark hadrons with the content of the valence quarks/antiquarks different from those in standard mesons and baryons. In this context, the exotic hadrons such as tetraquarks (composed of two quarks and two antiquarks) and pentaquarks (including four quarks and one antiquark) had been anticipated by Murray Gell-Mann and George Zweig in their 1964 papers [1]. The only necessary condition for these new multiquark states is to be color singlets. However, the absence of convincing experimental evidences for such exotic structures made their investigation of marginal interest for several decades. Nevertheless, the situation dramatically changed in the last two decades when the first explicit experimental evidence of the existence of these exotic hadrons became available, for recent reviews see Refs.[2][3][4]and references therein. Within the past decade several exotic hadrons, the majority pertaining to the class of heavy tetraquarks with at least two heavy quarks, were clearly confirmed by the BELLE, BESIII and LHCb experimental collaborations. The first experimental evidence for a heavy tetraquark was the X(3872) state [5] observed in 2003 by the Belle collaboration in exclusive B ± → K ± π + π − J/ψ decays. The decay of this state as well as the state Z c (3900) ± into the J/ψ meson (observed in BESIII) confirmed that these exotic particles are formed of two quarks and two antiquarks. Very recently, a narrow structure around 6.9 GeV, named as X(6900), is observed by the CERN LHCb in the process pp → J/ψJ/ψX [6]. This is the first candidate of fully-heavy tetraquarks observed in experiment which is proposed to be either the first radial (2S) excitation or the second orbital (1D) excitation of the hidden-charm tetraquark cccc. Nowadays, convincing candidates for both the exotic tetraquark (qqqq) and pentaquark (qqqqq) states are found. In the literature, there is no definite consensus about the composition of these states and different interpretations for the tetraquark candidates are proposed, for example: molecules composed from two mesons loosely bound by the meson exchange, compact tetraquarks composed from a diquark and antidiquark bound by strong forces, hadroquarkonia composed of a heavy quarkonium embedded in a light meson, kinematic cusps, etc. The distinction between various models is a very complicated experimental task. Among exotic hadrons (tetraquarks, pentaquarks, hexaquarks, etc), heavy tetraquarks are of particular interest since the presence of a heavy quark increases the binding energy of the bound system and, in conclusion, the possibility that such tetraquarks will have masses below the thresholds for decays to mesons with open heavy flavor. In this case the strong decays, which proceed through the quark and antiquark rearrangements, are kinematically forbidden and the corresponding tetraquarks can decay only weakly or electromagnetically and thus they should have a tiny decay width.When the strong interactions are concerned, the investigation of production mechanism of heavy standard/exotic hadrons is a powerful tool to understand the dynamics of strong interactions. Basically, two different mechanisms are assumed for the production of heavy hadrons: recombination and fragmentation[7]. In the second mechanism, the fragmentation refers to the process of a parton which carries large transverse momentum and subsequently forms a jet containing the expected hadron[8]. At sufficiently large transverse momentum of heavy hadron production, the * Electronic address: [email protected] | 10.1103/physrevd.105.034001 | [
"https://arxiv.org/pdf/2110.15251v2.pdf"
]
| 240,070,960 | 2110.15251 | a8633b01465cb5b265e9f3082c3c7d215cec6a14 |
Ground state heavy tetraquark production in heavy quark fragmentation
16 Jan 2022
S Mohammad
Moosavi Nejad
School of Particles and Accelerators
Institute for Research in Fundamental Sciences (IPM)
P.O.Box19395-5531TehranIran
Nahid Amiri
Faculty of Physics
Yazd University
P.O. Box 89195-741YazdIran
Ground state heavy tetraquark production in heavy quark fragmentation
16 Jan 2022(Dated: January 19, 2022)
During recent years, the study of exotic hadrons including tetraquarks and pentaquarks has attracted a lot of interests and more studies are in progress experimentally and theoretically. It is well-known that at sufficiently large transverse momentum the dominant production mechanism for standard heavy hadrons (mesons/baryons) is actually the fragmentation so that the same mechanism is also proposed for the production of heavy exotic hadrons. This work is the first attempt to study the direct fragmentation of a heavy quark into ground state heavy tetraquarks in leading order of perturbative QCD. In this regard, we will present an analytical expression for the fragmentation production of neutral hidden flavor tetraquarks (QqQq) which includes most of the kinematical and dynamical properties of the process.I. INTRODUCTIONSince the early days of the quark model it has been assumed the possibility of the existence of exotic multiquark hadrons with the content of the valence quarks/antiquarks different from those in standard mesons and baryons. In this context, the exotic hadrons such as tetraquarks (composed of two quarks and two antiquarks) and pentaquarks (including four quarks and one antiquark) had been anticipated by Murray Gell-Mann and George Zweig in their 1964 papers [1]. The only necessary condition for these new multiquark states is to be color singlets. However, the absence of convincing experimental evidences for such exotic structures made their investigation of marginal interest for several decades. Nevertheless, the situation dramatically changed in the last two decades when the first explicit experimental evidence of the existence of these exotic hadrons became available, for recent reviews see Refs.[2][3][4]and references therein. Within the past decade several exotic hadrons, the majority pertaining to the class of heavy tetraquarks with at least two heavy quarks, were clearly confirmed by the BELLE, BESIII and LHCb experimental collaborations. The first experimental evidence for a heavy tetraquark was the X(3872) state [5] observed in 2003 by the Belle collaboration in exclusive B ± → K ± π + π − J/ψ decays. The decay of this state as well as the state Z c (3900) ± into the J/ψ meson (observed in BESIII) confirmed that these exotic particles are formed of two quarks and two antiquarks. Very recently, a narrow structure around 6.9 GeV, named as X(6900), is observed by the CERN LHCb in the process pp → J/ψJ/ψX [6]. This is the first candidate of fully-heavy tetraquarks observed in experiment which is proposed to be either the first radial (2S) excitation or the second orbital (1D) excitation of the hidden-charm tetraquark cccc. Nowadays, convincing candidates for both the exotic tetraquark (qqqq) and pentaquark (qqqqq) states are found. In the literature, there is no definite consensus about the composition of these states and different interpretations for the tetraquark candidates are proposed, for example: molecules composed from two mesons loosely bound by the meson exchange, compact tetraquarks composed from a diquark and antidiquark bound by strong forces, hadroquarkonia composed of a heavy quarkonium embedded in a light meson, kinematic cusps, etc. The distinction between various models is a very complicated experimental task. Among exotic hadrons (tetraquarks, pentaquarks, hexaquarks, etc), heavy tetraquarks are of particular interest since the presence of a heavy quark increases the binding energy of the bound system and, in conclusion, the possibility that such tetraquarks will have masses below the thresholds for decays to mesons with open heavy flavor. In this case the strong decays, which proceed through the quark and antiquark rearrangements, are kinematically forbidden and the corresponding tetraquarks can decay only weakly or electromagnetically and thus they should have a tiny decay width.When the strong interactions are concerned, the investigation of production mechanism of heavy standard/exotic hadrons is a powerful tool to understand the dynamics of strong interactions. Basically, two different mechanisms are assumed for the production of heavy hadrons: recombination and fragmentation[7]. In the second mechanism, the fragmentation refers to the process of a parton which carries large transverse momentum and subsequently forms a jet containing the expected hadron[8]. At sufficiently large transverse momentum of heavy hadron production, the * Electronic address: [email protected]
direct leading order production scheme (recombination mechanism) is normally suppressed while the fragmentation mechanism becomes dominant, though it is formally of higher order in the strong coupling constant α s [8,9]. In Refs. [10][11][12][13][14][15][16][17], we calculated the perturbative QCD fragmentation function (FF) for a heavy quark to fragment into the S-wave heavy mesons and baryons at leading-order (LO) and next-to-leading order (NLO). In Ref. [18], it has been pointed out that the dominant production mechanism for pentaquarks consisting of a heavy quark is heavy quark fragmentation, similar to heavy-light mesons and baryons consisting of a heavy quark. So, this mechanism is also adopted in our work for the production of heavy tetraquark states and we, for the first time, study the production mechanism of heavy tetraquarks in the fragmentation of a heavy quark at lowest order of perturbative QCD. In our calculation, tetraquarks are assumed as four-quark QqQq states which are tightly bound by the color forces. Our analytical results are presented for the fragmentation function of heavy quark Q into neutral hidden flavor tetraquarks QqQq with Q = c, b and q = u, d, s. Having these FFs it would be possible to evaluate the production rate of these heavy states in e + e − annihilation and hadron colliders. In fact, the specific importance of FFs is for their model independent predictions of the cross sections at colliders. However, due to a large number of decay modes the results from e + e − annihilation seem to be small but sizable rates are expected in energetic hadron colliders where a large number of them is produced. To assess the prospects for the experimental study of heavy tetraquarks in present and future colliders, it is important to know the accurate predictions for their production rates and, in conclusion, their FFs.
This paper is organized as follows. In Sec. II, we present the theoretical formalism for the heavy tetraquark FF in the Suzuki model and present our numerical results, and our discussions and conclusions are given in the last section.
II. HEAVY TETRAQUARK FRAGMENTATION
A motivation for studying the production of multiquark structures is for better understanding of the dynamics of strong interactions and the confinement mechanism. It is well known that the dominant mechanism to produce standard heavy hadrons at sufficiently large transverse momentum is fragmentation so this mechanism is valid for the production of heavy tetraquarks too. Fragmentation refers to the production of a parton with a large transverse momentum which subsequently decays to form a jet containing an expected hadron. It is hence important to obtain the corresponding fragmentation function in order to properly estimate the production rate of a specific hadronic state. According to the QCD factorization theorem, the cross section for the production of hadron H in the typical scattering process A + B → H + X, can be expressed as
dσ = a,b,cˆ1 0 dx aˆ1 0 dx bˆ1 0 dzf a/A (x a , Q)f b/B (x b , Q)dσ(a + b → c + X)D H c (z, Q),(1)
where a and b are incident partons in the colliding initial hadrons A and B, respectively, f a/A and f b/B are the nonperturbative parton distribution function (PDFs) at the scale Q 2 of the partonic subprocess a + b → c + X, c is the fragmenting parton and X stands for any unobserved jet. Here, D H c (z, Q) is the FF at the scale Q which can be obtained by evolving from the initial FF D H c (z, µ 0 ) using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) renormalization group equations [19]. In electron-positron annihilation, one does not need to deal with the nonperturbative PDFs as in hadron collisions. This process has in general less contributions by background processes compared to hadron collisions and the uncertainties introduced by parton density functions are not appeared. The process of inclusive heavy hadron production in electron-positron annihilation, i.e., e + e − → (γ, Z) → H + X, has the cross section as [20][21][22]
dσ dx (x, s) = aˆ1 x dy y dσ dy (y, µ R , µ F )D H a ( x y , µ F ),(2)
where, a stands for the partons a = g, u,ū, · · · , b,b and µ F and µ R are the factorization and renormalization scales, respectively. Generally, one can choose two different values for these scales, but a common choice consists of setting
µ R = µ F = √ s.
Here, x = 2E H / √ s measures the energy of hadron in units of the beam energy in the center-of-mass frame. At lowest-order of perturbative QCD, the cross section of relevant partonic subprocesses is given by [23] dσ dy
(e + e − → qq) = N c σ 0 (V 2 qi + A 2 qi )δ(1 − y).(3)
Here, N c = 3 is the number of quark colors, V qi and A qi are the effective vector and axial-vector couplings of quark q i to the photon and Z boson. For small energies ( √ s ≪ M Z ), for the summation of squared effective electroweak Q(p) charges we have V 2 qi + A 2 qi = e 2 qi where e qi is the electric charge of the quark q i [24]. In Eq. (3), σ 0 = 4πα 2 /(3s) is the leading order total cross section of e + e − → µ + µ − for the massless muons, in which α is the Sommerfeld fine-structure constant. Therefore, for the cross section of e + e − → H + X at LO, on has
Q(t) Q(s ) Q(s) q(r) q(t ) g(q ) q(q ) g(q )dσ LO dx (x, s) = N c σ 0 q e 2 q [D H q (x, µ) + D H q (x, µ)] = 8πα 2 s q e 2 q D H q (x, µ),(4)
where we assumed D H q = D H q for q = u, d, s, c, b. Conventionally, the fragmentation mechanism is described by the function D H i (z, µ) which refers to the probability for a parton i at the fragmentation scale µ to fragment into a hadron H carrying away a fraction z of its momentum. The FFs are related to the low-energy part of the hadroproduction processes and forms the nonperturbative aspect of QCD but, fortunately, it was found that these functions for heavy hadron productions are analytically calculable by virtue of perturbative QCD (pQCD) with limited phenomenological parameters. Historically, the first theoretical effort to describe the procedure of heavy hadron production was made by Bjorken [25] by using a naive quark-parton model and in the following the pQCD scheme was applied by Suzuki [26], Amiri and Ji [27]. Among them, an elaborate model of fragmentation which does contain spin information has been proposed by Suzuki [28]. This model includes most of the kinematical and dynamical properties of the splitting process and gives a detailed insight about the fragmentation process. In this approach, Suzuki calculates the fragmentation function of a heavy quark using the convenient Feynman diagrams for the parton level of the process and also by considering the wave function of heavy bound state which contains the nonperturbative dynamic of hadroproduction process. In fact, the Suzuki model mixes a perturbative picture with nonperturbative dynamics of fragmentation and not only predicts the z-dependence of the FFs, but also their dependence on the transverse momentum of the hadron relative to the jet.
Here, using the Suzuki model we focus on the heavy tetraquark FF for which the only Feynman diagram at leading order in α s is shown in Fig. 1. Any other possible Feynman diagram is related to higher orders in α s . Considering this diagram we first set the relevant four-momenta as
p µ = [p 0 , p T , p L ] , s µ = [s 0 , 0, s L ] s ′ µ = [s ′ 0 , p T , s ′ L ] , t µ = [t 0 , 0, t L ] (5) t ′ µ = [t ′ 0 , 0, t ′ L ] , r µ = [r 0 , 0, r L ]
where we assumed that the produced tetraquark X moves along theẑ-axes (fragmentation axes). The four-momentum of tetraquark is also written asP µ = [P 0 , 0,P L ] so thatP L = s L + r L + t L + t ′ L . In the Suzuki model the FF for the production of unpolarized ground state tetraquark is defined as [28] D X Q (z, µ 0 ) =
1 1 + 2s Q spin colorˆ| T X | 2 δ 3 (P + s ′ − p)d 3P d 3 s ′ ,(6)
where µ 0 is the fragmentation scale, s Q refers to the spin of the fragmenting quark and the summation is going over the spins and colors. Four momenta are as labeled in Fig. 1. In computing Eq. (6), following Ref. [28], we adopt the infinite momentum frame where the fragmentation variable in the usual light-cone form, i.e. z = (P 0 +P L )/(p 0 + p L ), is reduced to a more popular one as z =P 0 /p 0 = E X /E Q . From experimental point of view, this definition is more suitable since refers to the energy fraction of the fragmenting heavy quark which is taken away by the produced tetraquark. This scaling variable ranges as 0 ≤ z ≤ 1.
In Eq. (6), T X is the probability amplitude of the tetraquark production which, at the large momentum transfer, is expressed in terms of the hard scattering amplitude T H and the process-independent distribution amplitude Φ B as [29,30]
T X (P , s ′ ) =ˆ[dx]T H (P , s ′ , x i )Φ B (x i , Q 2 ),(7)
where
[dx] = Π 4 i=1 dx i δ(1 − 4 i=1 x i ).
Here, x i 's stand for the tetraquark energy fractions carried by its constituent quarks which are defined as
x 1 = s 0 P 0 , x 2 = r 0 P 0 , x 3 = t ′ 0 P 0 , x 4 = t 0 P 0 · (8)
The energy conservation law requires that 4 i=1 x i = 1. The scheme applied to describe the probability amplitude T X as the convolution in Eq. (7) is convenient to absorb the soft behavior of the bound state (tetraquark in our example) into the hard scattering amplitude T H [29]. The short-distance part of the amplitude, i.e. T H , can be computed perturbatively from quark-gluon subprocesses at leading-order or higher-order QCD approximations. The long-distance distribution amplitude Φ B which contains the bound state nonperturbative dynamic of the outgoing tetraquark, is the probability amplitude for a QqQq-set to evolve into a particular bound state. The underlying link between hadronic phenomena in QCD at long-and short-distances is the hadronic wave function. In fact, the nonperturbative aspect of the hadroproduction processes is contained in the bound state of the hadron which is described by the wave function Φ B . Following Ref. [28] and according to the Lepage-Brodsky's approach [31] we neglect the relative motion of the constituent quarks inside the tetraquark therefore we assume, for simplicity, that the constituent quarks are emitted collinearly with each other and move along theẑ-axes. In fact, in the Suzuki model a multiquark state is replaced by collinear constituents with neglecting their Fermi motion and the nonperturbative aspect of the hadroproduction is included in the wave function of the heavy tetraquark state. By this simplicity, the relative motion of the constituent quarks is effectively nonrelativistic and this allows one to estimate the nonrelativistic wave function of heavy tetraquark as a delta function form, see also [13]. Therefore, the distribution amplitude for a S-wave heavy tetraquark with neglecting the Fermi motion reads [32] Φ
B ≈ f B δ(x i − m i m X ).(9)
where m X is the tetraquark mass and f B refers to the decay constant of hadron. In Ref. [33], we studied the effect of meson wave function on its FF by considering a typical mesonic wave function which is different of the delta function and is the nonrelativistic limit of the solution of Bethe-Salpeter equation with the QCD kernel [29]. There, we showed that the Fermi motion effect can be neglected with a reasonable approximation. With the delta function approximation for the tetraquark wave function (9), we are assuming that the contribution of each constituent from the tetraquark energy is proportional to its mass, i.e.
x i = m i /m X where m X = 2(m q + m Q ).
Considering the definitions of fragmentation variable (z =P 0 /p 0 ) and the fractions x i 's presented in Eq. (8) one also may write the parton energies in terms of the initial heavy quark energy (p 0 ) as: Eq. (7), the amplitude T H is, in essence, the partonic cross section to produce a set of heavy quarks with certain quantum numbers that in the QCD perturbation theory considering the Feynman rules is written as
s 0 = x 1 zp 0 , r 0 = x 2 zp 0 , t ′ 0 = x 3 zp 0 , t 0 = x 4 zp 0 and s ′ 0 = (1 − z)p 0 where x 1 = x 4 = m Q /m X and x 2 = x 3 = m q /m X . InT H = 16π 2 α s (2m Q )α s (2m q )m X m 2 Q C F 2 2P 0 s ′ 0 p 0 Γ D 0 ,(10)
where, D 0 =P 0 + s ′ 0 − p 0 is the energy denominator, α s (µ) is the strong coupling constant at the renormalization scale µ and C F is the color factor which is calculated using color line counting rule.
In Eq. (10), Γ represents an appropriate combination of the propagators and the spinorial parts of the amplitude. Considering Fig. 1 for the transition Q → X QqQq the transition amplitude at leading order is written as
Γ = Gū(t)γ µ u(p) ū(t ′ )γ µ ( q 2 + m q )γ ν v(r) v(s)γ νū (s ′ )(11)
where G is related to the denominator of propagators as
G = 1 q 2 1 (q 2 2 − m 2 q )q 2 3 = 1 8(m 2 Q − p · t) 2 (m 2 Q + s · s ′ ) 2 (m 2 Q + r · s + r · s ′ + s · s ′ ) 2 .(12)
In squaring the amplitude Γ, one needs the following dot products of the relevant four-momenta:
r · t ′ = m 2 q , s · t = m 2 Q r · s = t · t ′ = s · t ′ = m q m Q , p · s = p · t = m X m Q 2z + zm Q 2m X (m 2 Q + p 2 T ), p · s ′ = 1 2 (m 2 Q + p 2 T )(1 − z + 1 1 − z ) − p 2 T ,(13)r · s ′ = s ′ · t ′ = zm q 2m X (1 − z) (m 2 Q + p 2 T ) + m X m q (1 − z) 2z , s · s ′ = t · s ′ = m Q m X (1 − z) 2z + zm Q 2m X (1 − z) (m 2 Q + p 2 T ), p · t ′ = p · r = m q m X 2z + zm q 2m X (m 2 Q + p 2 T ).
To obtain the FF for an unpolarized S-wave heavy tetraquark X QqQq , considering Eqs. (6-10) one has
D X c (z, µ 0 ) = (4π 2 f B α s (2m q )α s (2m Q )m X m 2 Q C F ) 2ˆd 3P d 3 s ′ spin |Γ| 2 δ 3 (P + s ′ − p) P 0 p 0 s ′ 0 (P 0 + s ′ 0 − p 0 ) 2 .(14)
To perform the phase space integrations we start with the following integral
d 3P δ 3 (P + s ′ − p) P 0 p 0 (P 0 + s ′ 0 − p 0 ) 2 =ˆP 0 d 3P δ 3 (P + s ′ − p) p 0 (P 2 0 − (p 0 − s ′ 0 ) 2 ) 2 = zp 0 p 0 (m 2 X − (p − s ′ ) 2 ) 2 = z (m 2 X − 2m 2 Q + 2p · s ′ ) 2 . (15)
According to the Suzuki approach, in Eq. (14) instead of integrating over the transverse momentum of outgoing heavy quark Q, we simply replace the integration variable by its average value, i.e.
d 3 s ′ s ′ 0 g(z, s ′ T ) =ˆg(z, s ′ T ) ds ′ L d 2 s ′ T s ′ 0 = m 2 Q g(z, p 2 T ),(16)
where the average value p 2 T is a free parameter which can be specified phenomenologically. Putting all in Eq. (14), the FF of heavy tetraquark X reads
D X Q (z, µ 0 ) = N z × Σ spin ΓΓ (m 2 X − 2m 2 Q + 2p · s ′ ) 2 = N z × Σ spin ΓΓ [m 2 X − (m 2 Q + p 2 T )(1 + z − 1 1−z )] 2 ,(17)
where N = (16π 2 f B α s (2m q )α s (2m Q )m q m 4 Q C F ) 2 but it is related to the normalization condition:´dzD(z, µ 0 ) = 1 [27,28]. Using the traditional trace technique the squared amplitude ( |Γ| 2 ) reads where
Σ spin ΓΓ = G 2 z 3 (1 − z) z 6 p 2 T 3 + αz 4 p 2 T 2 + βz 2 p 2 T − γ ,(18)α = m 2 Q (11z 2 − 18z + 12) + 6m 2 q (2z 2 − z + 2) + 4m q m Q (3z 2 − 4z + 6), β = m 4 Q (19z 4 − 84z 3 + 168z 2 − 144z + 48) + 4m 2 Q m 2 q (6z 4 − 37z 3 + 102z 2 − 112z + 72) −8m 4 q (z 3 − 6z 2 + 6z − 6) − 16m Q m 3 q (3z 3 − 12z 2 + 14z − 12) + 8m q m 3 Q (3z 4 − 22z 3 + 54z 2 − 52z + 24), γ = −64m Q m 5 q (z 3 + 3z 2 − 10z + 6) + 8m 4 q m 2 Q (z 5 − 6z 4 + 42z 3 − 158z 2 + 236z − 120) + 16m 3 q m 3 Q (3z 5 − 24z 4 + 98z 3 − 196z 2 + 192z − 80) − 2m 2 q m 4 Q (6z 6 − 71z 5 + 422z 4 − 1232z 3 + 1872z 2 − 1424z + 480) − 4m q m 5 Q (3z 6 − 40z 5 + 182z 4 − 424z 3 + 544z 2 − 352z + 96) − 32m 6 q (z 3 − 3z + 2) − m 6 Q (z 2 − 2z + 4)(3z 2 − 8z + 4) 2 .(19)
In the Suzuki model, the fragmentation function depends on both the fragmentation variable z and the parameter p 2 T (the transverse momentum of produced hadron relative to the parent jet). In Ref. [32], it is shown that the choice of p 2 T = 1 GeV 2 is an extreme value for this quantity and any higher value will produce the peak even at lower-z regions. In this work we adopt this value for the transverse momentum of heavy tetraquark. The z-dependence of FFs is not yet calculable at each scale, however once they are computed at the initial scale µ 0 their µ evolution can be specified by the DGLAP evolution equations [19]. Here, we set the initial scale of fragmentation to µ 0 = m X + m Q which is the minimum value of the invariant mass of the fragmenting parton. Then, the FF presented in Eq. (17) should be regarded as a model for the heavy quark fragmentation Q into the heavy tetraquark X(QqQq) at the initial scale µ 0 . To present our numerical analysis, we adopt the following input parameters [34]:
m c = 1.67 GeV, m b = 4.78 GeV, f B = 0.25 GeV, α s (2m c ) = 0.26, α s (2m b ) = 0.18.(20)
Note that, in Ref. [34] two values are reported for the quark masses. The above masses show the pole masses, i.e. the renormalized quark masses in the on-shell renormalization scheme, which correspond to the values m c = 1.27 ± 0.02 GeV and m b = 4.18 ± 0.03 GeV for the M S masses (the renormalized quark mass in the modified minimal subtraction scheme [35] which is intimately related to the use of dimensional regularization [36]). The relation between the pole quark mass and the M S mass is known to three loops in QCD perturbation theory, details are given in Ref. [37]. In our analysis, to show the effect of variation of heavy quark masses on their corresponding FFs we will consider the charm and bottom quark masses as plotted. Assuming m u ≈ m d this result is also valid for the transitions c → X(cdcd), X(cucd), X(cdcū). In Fig. 3, the b-quark FF into the neutral hidden-bottom tetraquark including strangeness flavor, i.e. b → X(bsbs), is shown considering the mass uncertainty band. Since the b-quark is heavier than the c-quark, as we expected, the peak of b-quark FF shifted toward higher values of z. Besides the heavy tetraquark FFs, their first moment is also of phenomenological interest and subject to experimental determination. It corresponds to the average fraction of energy that the tetraquark receives from the fragmenting heavy quark and is defined as [24] z Q (µ) = 1
B Q (µ)ˆ1 0 zD X Q (z, µ)dz,(21)
where, B Q (µ) stands for the branching fraction which is defined as B Q (µ) =´1 0 D X Q (z, µ)dz, where Q = c, b. For the tetraquark X(cucū) the branching fraction and the average energy fraction are B c = 2 × 10 −3 and z c (µ 0 ) = 0.635, respectively, and the corresponding ones for the tetraquark X(bubū) read B b = 1.1 × 10 −4 and z b = 0.629. Considering the inclusive differential cross section given in Eq. (4), one can compute the inclusive cross section for the production of heavy tetraquark at LO as
σ LO (e + e − → X QqQq + jets) = 8πα 2 s q e 2 qˆ1 0 dzD X q (z, µ),(22)
By ignoring the contributions which are appeared at higher orders of perturbation, one has
σ LO (e + e − → X QqQq + jets) ≈ 8πα 2 s e 2 Q B Q (s).(23)
For example, the contributions of c → X bubū , b → X cucū or g → X bubū /X cucū are related to very complicated Feynman diagrams which are in higher orders in α s , because they do not occur directly. Then, their contributions are ignorable in comparison with the one occurs directly, i.e., b → X bubū and c → X cucū FFs (17). Finally, having the branching fraction B Q (µ) for the transition B → X at the center-of-mass energy µ = s one can compute the corresponding cross section at electron-positron annihilation.
III. CONCLUSION
Besides the standard mesons and baryons there exist hadrons made of more than three quarks/antiquarks or valence gluons. These new types of hadrons consist of exotic states including glueballs, hybrids, tetraquarks, pentaquarks, hexaquarks, etc. In last two decades a large number of these exotic states, especially tetraquarks and pentaquarks, have been observed in the particle factories. However, some of them are now well-established there are also some doubts on existence of some other members. Therefore, exact determinations of the nature, structure and quantum numbers of these exotic states need more experimental efforts. Specifically, there are many exotic states which are proposed in theory but their existence need to be confirmed by the experiments.
It should be noted that, most of heavy multiquark particles discovered in the experiment have hidden-charm or hidden-bottom quark structure, i.e., they contain cc or bb in their inner structures. The study of their production mechanism and using it as a probe to the structure of hadrons are among the most active research fields in particle and nuclear physics. In fact, the challenge is to understand the nonperturbative transition from high energy e + e − , photon-hadron, and hadron-hadron collisions to physical exotic states. It is well-known that the dominant mechanism to produce heavy hadrons at high transverse momentum is fragmentation; the production of a high energy parton followed by its fragmentation into the heavy bound states. In this work, using the Suzuki model we, for the first time, studied the fragmentation function of a heavy quark into an unpolarized S-wave heavy tetraquark at leading order perturbative QCD. To be specific, we analytically computed the FF of bottom and charm quark into the neutral hidden flavor tetraquarks (X|QqQq >) at the initial scale µ 0 = m X + m Q . Using the extracted FFs one can compute the production rates of heavy tetraquarks in hadron colliders.
To have more accurate FFs, one can think of other effects such as the Fermi motion of constituents, the tetraquark mass effect, NLO radiative corrections, etc. Although, including the effects such as the Fermi motion of constituents leads to very complicated integrals which should be solved numerically [33].
Figure 1 :
1The lowest order Feynman diagram contributing to the fragmentation of a heavy quark Q into a heavy tetraquark X(QqQq). The four momenta are also labeled.
Figure 2 :
2Fragmentation of c-quark into heavy tetraquark Xcucū at lowest-order of perturbative QCD. The uncertainty band due to the variation of charm mass is also shown.
Figure 3 :
31.27 ≤ m c ≤ 1.67 GeV and 4.18 ≤ m b ≤ 4.78 GeV. InFig. 2, our phenomenological prediction for the FFs of charm into the neutral hidden-charm state X(cucū) is shown at the starting scale µ 0 = m c + m X . The uncertainty band due to the variation of charm quark mass is also The b → X bsbs FF at leading order perturbative QCD along with the uncertainty band of bottom mass.
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| []
|
[
"Riccati observers for position and velocity bias estimation from either direction or range measurements",
"Riccati observers for position and velocity bias estimation from either direction or range measurements"
]
| [
"Tarek Hamel ",
"Claude Samson "
]
| []
| []
| This paper revisits the problems of estimating the position of an object moving in n (≥ 2)-dimensional Euclidean space using velocity measurements and either direction or range measurements of one or multiple source points. The proposed solutions exploit the Continuous Riccati Equation (CRE) to calculate observer gains yielding global exponential stability of zero estimation errors, even in the case where the measured velocity is biased by an unknown constant perturbation. These results are obtained under persistent excitation (p.e.) conditions depending on the number of source points and body motion that ensure both uniform observability and good conditioning of the CRE solutions. With respect to prior contributions on these subjects some of the proposed solutions are entirely novel while others are adapted from existing ones with the preoccupation of stating simpler and more explicit conditions under which uniform exponential stability is achieved. A complementary contribution, related to the delicate tuning of the observers gains, is the derivation of a lower-bound of the exponential rate of convergence specified as a function of the amount of persistent excitation. Simulation results illustrate the performance of the proposed observers. | null | [
"https://arxiv.org/pdf/1606.07735v1.pdf"
]
| 11,002,306 | 1606.07735 | 5f132c58de3f75cc09d01c052e3d91a5891bdaeb |
Riccati observers for position and velocity bias estimation from either direction or range measurements
Tarek Hamel
Claude Samson
Riccati observers for position and velocity bias estimation from either direction or range measurements
position estimationRiccati observerslinear time-varying systemspersistent excitationobservability
This paper revisits the problems of estimating the position of an object moving in n (≥ 2)-dimensional Euclidean space using velocity measurements and either direction or range measurements of one or multiple source points. The proposed solutions exploit the Continuous Riccati Equation (CRE) to calculate observer gains yielding global exponential stability of zero estimation errors, even in the case where the measured velocity is biased by an unknown constant perturbation. These results are obtained under persistent excitation (p.e.) conditions depending on the number of source points and body motion that ensure both uniform observability and good conditioning of the CRE solutions. With respect to prior contributions on these subjects some of the proposed solutions are entirely novel while others are adapted from existing ones with the preoccupation of stating simpler and more explicit conditions under which uniform exponential stability is achieved. A complementary contribution, related to the delicate tuning of the observers gains, is the derivation of a lower-bound of the exponential rate of convergence specified as a function of the amount of persistent excitation. Simulation results illustrate the performance of the proposed observers.
I. INTRODUCTION
The general problem of estimating the position, or the complete pose (position and orientation), of a body relatively to a certain spatial frame is central for a multitude of applications. This is common knowledge. Among all sensing modalities that can be used to acquire the necessary information, source points direction (or bearing) measurements has early motivated many studies, in particular for pose estimation when body and source points are motionless in the frame of interest, a problem referred to as the Perspective-n-Point (PnP) problem in the dedicated literature [1]. Proposed solutions may roughly be classified into two categories, namely non-iterative methods based on a finite set of measurements (one per source point) feeding polynomial equations that are either algebraically or numerically solved [2], and iterative methods involving ongoing measurements that feed gradientlike recursive algorithms [3], [4]. Such recursive algorithms are called observers in the Automatic Control community. Generically, at least three source points are needed to ensure that the number of body poses compatible with the measurements is finite [1]. It is commonly acknowledged that these two types of methods are complementary. Non-iterative T. Hamel is with I3S UNS-CNRS, Nice-Sophia Antipolis, France, [email protected].
C. Samson is with INRIA and I3S UNS-CNRS, Sophia Antipolis, France, [email protected], [email protected]. methods are of interest to work out an approximation of the body pose after elimination of non-physical solutions, whereas iterative methods, that are local by nature (since they may be stuck at local minima even in the case of a unique global minimum), allow for a more precise estimation in relation to their filtering properties [5]. The present paper focuses on the sole estimation of the body position. This corresponds to applications for which the body's attitude is either of lesser importance or is estimated by using other sensing modalities. In this case, iterative methods are all the more interesting that their domain of convergence can be global. The reason is that, without the compact group of rotations being involved, this simplified problem is amenable to exact linearisation and can be associated with globally convex cost functions, as shown further in the paper. Another advantage of iterative methods is that they are naturally suited to handle the non-static case, i.e. when either the body or the point source(s) move(s), by using on-line the extra data and information resulting from motion. In particular, the observation of a single point source may be sufficient in this case, provided that the body motion regularly grants a sufficient amount of "observability". This possibility has been studied recently in [6] where the problem is linearised by considering an augmented state vector. Another solution, not resorting on state augmentation, is proposed in [7]. The present paper offers a generalization of previous studies on this subject that encompasses the static and non-static cases with an arbitrary number of source points.
Global Navigation Satellite Systems (GNSS), and the American Global Positioning System (GPS) [8] in particular, have familiarized the larger public with the problem of body position estimation from source points distance (or range) measurements. In the static or quasi-static case, solutions to this problem may again be classified into non-iterative and iterative methods. Similarly to the direction measurements case, three point sources (satellites) are also required to obtain a finite number (equal to two) of theoretical solutions, with an extra source point (non-coplanar with the other points) needed to eliminate the non-physical solution and overcome the problem of desynchronized clocks resulting in constant range measurement bias. The resolution of this problem is also facilitated by the fact that constraint (output or measurement) equations can be made linear in the unknown position coordinates. Studies of the non-static case are much less numerous and more recent [9], [10]. To our knowledge, Batista and al. [11] were first to address this case by exploiting the possibility of linearising the estimation problem via state augmentation, even when the body velocity vector is biased by a constant vector. A similar idea is used in the present paper, but via lower-dimensional state augmentation. This yields simpler observers and reduced computational weight.
For five decades, Kalman filters for linear systems, and their extensions to non-linear systems known as Extended Kalman Filters (EKF), have consistently grown in popularity near engineers with various backgrounds (signal processing, artificial vision, robotics,...) to address a multitude of iterative state estimation problems involving additive "noise" upon the state and/or the measurements. The optimality of these filters in a stochastic framework under specific noise conditions and assumptions, and their direct applicability to Linear Time-Varying (LTV) systems, have undoubtedly contributed to this popularity. It is however important to keep in mind, or to recall, that the stability and robustness properties associated with them, i.e. features that supersede conditional stochastic optimality in practice, are not related to stochastic issues. They result from properties of the associated deterministic continuous-time (or discrete-time, depending on the chosen computational framework) Riccati equation that underlies a (locally) convex estimation error index (or Lyapunov function) and a way of forming recursive estimation algorithms that uniformly decrease this index exponentially (under adequate observability conditions). With this perspective, Kalman filters belong to the (slightly) larger set of Riccati observers that we intentionally derive here in a deterministic framework, knowing that a complementary stochastic interpretation may be useful to subsequently tune the Riccati equation parameters and observer gains. We also believe that, by contrast with standard Kalman filter derivations, this approach allows one to better comprehend how the system observability properties are related to the good conditioning of the Riccati equation solutions and to the observer's performance (the rate of convergence to zero of the estimation errors, in particular) via a Lyapunov analysis.
The research themes addressed in the present paper are not new, nor are the basic conceptual tools (Riccati equation, Lyapunov stability, uniform observability and persistent excitation,...) used to derive the propose observers. However, we believe that our approach to the problems and the resulting observer design synthesis are original. Also, by contrast with a majority of studies based on the application of Kalman filtering, uniform exponential stability of the observers is rigorously proved in association with explicit and simple observability conditions worked out from the corresponding observability Grammian condition. The connection between rate of convergence and amount of observability is also drawn out explicitly. Observers are derived for both direction measurements and range measurements, in n (≥ 2)dimensional Euclidean space so that both 2D and 3D cases (of particular practical interest) are covered, with an arbitrary number of source points. Concerning this latter aspect, the observers are designed by first considering a single source point, with stability and convergence of the observer relying on persistent excitation properties associated with the body motion. The solutions are then generalized to the case of multiple source points, with the augmentation of the number of these points reflecting on the gradual weakening of the body motion conditions needed to ensure uniform exponential stability. While measuring the body velocity is central to estimate the position, we also show how to modify the observers via state augmentation when velocity measurement are biased by a constant vector. Uniform exponential stability is then preserved under either the same observability conditions, when direction measurements are used, or slightly stronger ones, when range measurements are used with less than (n + 1) source points.
The paper is organized along six sections. Following the present introduction, Section II recalls basic observability concepts and central properties of the CRE, complemented with a few original technical results used for the design and analysis of the observers. Direction measurements and range measurements cases are treated in Sections III and IV respectively. Illustrative simulations results are presented in Section V, followed by a short section VI of concluding remarks. The proofs of several technical results are reported in the Appendix.
II. RECALLS
Although several of the definitions and results recalled in this section are well known, others are not. Our main intent here is to provide the reader with a self-contained overview of basic observability concepts and of state observers whose gains are calculated from solutions to the Continuous Riccati Equation (CRE). This overview is also an opportunity to recall natural Lyapunov functions associated with these observers for stability and convergence analysis.
Throughout the paper the following notation is used:
• A(t), B(t),
A. Observability definitions and conditions
Consider a generic linear time-varying (LTV) system
ẋ = A(t)x + B(t)u y = C(t)x(1)
with x ∈ R n the system state vector, u ∈ R s the system input vector, and y ∈ R m the system output vector. The following definitions and properties of observability associated with this system are borrowed from [12]. Definition 2.1 (instantaneous observability): System (1) is instantaneously observable if ∀t, x(t) can be calculated from the input u(t), the output y(t), and the time-derivatives
u (k) (t), y (k) (t), k ∈ N.
Lemma 2.2: Define the observation space at the timeinstant t as the space generated by Note that, with this definition, uniformity is related to time and not to the input. This definition of uniform observability, which we adopt here, is thus different from other definitions proposed in the literature, e.g. [13] or [14].
O(t) := N 0 (t) N 1 (t) . . . with N 0 = C, N k+1 = N k A +Ṅ k , k = 1, . . . Then System (1) is instantaneously observable if rank(O(t)) = n.
Theorem 2.4 (sufficient condition for uniform observability):
Sytem (1) is uniformly observable if there exist δ > 0, µ > 0 such that ∀t ≥ 0 W (t, t + δ) := 1 δ t+δ t Φ T (s, t)C T (s)C(s)Φ(s, t)ds ≥ µI d > 0(2)
with Φ(t, s) the transition matrix associated with A(t), i.e. such that d dt Φ(t, s) = A(t)Φ(t, s) with Φ(t, t) = I d . The matrix valued-function W (t, t + δ) is called the observability Grammian of System (1). A very useful result, derived in [15], gives a sufficient condition for uniform observability in terms of the properties of the matrices A(t) and C(t) and their time-derivatives Lemma 2.5: If there exists a matrix-valued function M (.) of dimension (p × n) (p ≥ 1) composed of row vectors of N 0 (.), N 1 (.),. . ., such that for some (strictly) positive numbers (δ,μ) and ∀t ≥ 0
1 δ t+δ t M T (s)M (s)ds ≥μI d > 0(3)
then the observability Grammian of System (1) satisfies the condition (2) (and this system is thus uniformly observable).
B. Riccati observers
We here call Riccati observer any observer of System (1) of the forṁ
x = A(t)x + B(t)u + K(t)(y − C(t)x) ;x(0) ∈ R n (4)
with the observer gain given by
K(t) = k(t)P (t)C T (t)Q(t) ; k(t) ≥ 0.5 (5)
where P (t) is the solution to the so-called Continuous Riccati Equation (CRE)
P = A(t)P + P A T (t) − P C T (t)Q(t)C(t)P + V (t) (6)
with P (0) any p.d. matrix and Q(t), V (t) p.s.d. matrices that have to be specified. Note that the optimal Kalman gain in the stochastic setting where the matrices V (t) and Q −1 (t) are p.d. matrices and interpreted as covariance matrices of additive noise on the system state and output is obtained by taking k(t) = 1.
Let us now quickly recall how the stability and convergence properties of a Riccati observer is directly related to the properties of the solution P (t) to the CRE. Define the estimation errorx := x −x, from (1) and (4) one obtains the error equationẋ
= (A(t) − K(t)C(t))x(7)
Assume (for the time being) that P (t), which is a symmetric matrix by construction, is well defined on R + and is p.d., so that its inverse is also well defined and p.d., and consider the candidate Lyapunov function
V(t) =x T (t)P −1 (t)x(t).
Then using the fact that the time-derivative ofṖ −1 satisfies the equatioṅ
P −1 = −P −1 A(t)−A T (t)P −1 +C T (t)Q(t)C(t)−P −1 V (t)P −1(8)
and using (5) and (7), one easily verifies that the timederivative of V(t) is given bẏ
V = −x T (2k(t) − 1)C T (t)Q(t)C(t) + P −1 V (t)P −1 x ≤ −x T P −1 V (t)P −1x ≤ − p 2 m p M vmV (≤ 0) (9) so that V(t) ≤ V(0)exp(− p 2 m p M v m t).
To conclude thatx = 0 is globally exponentially stable it thus suffices to choose V (t) > v m I d with v m > 0 and to show that P (t) i) is always well-defined, ii) that it is p.d., and -most importantly-iii) that it is well conditioned in the sense that p m is strictly positive and p M is finite so that the ratio P M p 2 m is bounded. Since this ratio essentially determines the exponential rate of convergence of the estimation errors to zero, it is of interest to know bounds of p m and p M in relation to the "amount" of observability. Such bounds are derived in Appendix B, with a lower bound of p M calculated from an expression derived in [16]. The central issue of boundedness and good conditioning of P (t) brings us to recall classical, and also point out less known, results concerning the CRE.
C. Properties of the Continuous Riccati Equation
The first results concerns the existence of the solutions to the CRE for t ∈ [0, +∞). Lemma 2.6: If P (0) is p.d. and Q(t) and V (t) are p.s.d, then P (t) is p.d. and well defined on [0, +∞). See the proof in Appendix A. Now, to ensure boundedness and good-conditioning of the solution P (t) to the CRE one has to impose other conditions upon the terms entering the equation. Sufficient conditions are pointed out in the next lemma.
Lemma 2.7: Define:
W V (t, t + δ) := 1 δ t+δ t Φ(t, s)V (s)Φ(t, s) T ds(10)
and
W Q (t, t + δ) := 1 δ t+δ t Φ T (s, t)C T (s)Q(s)C(s)Φ(s, t)ds (11) If there exist (strictly) positive numbers δ, µ v , and µ q such that W V (t, t + δ) ≥ µ v I d and W Q (t, t + δ) ≥ µ q I d , ∀t,
then the solution P (t) to the CRE (6) is bounded and wellconditioned in the sense that 0 < p m ≤ p M < ∞.
A proof of this result is given in [16] where lower and upper bounds of P (t) are also derived. The matrix Q(t) is in fact assumed p.d. because the inverse of Q is (for technical convenience) used in the proof. However, it is simple to verify that the proposed bounds for P (t) do not depend on the smallest eigenvalue of Q(t), so that these bounds are also valid when Q(t) is only p.s.d.
From now on, and by analogy with the previously defined observability Grammian W of System (1), W Q is called Riccati observability Grammian. It coincides with W when Q(t) = I d . Note that if Q(t) ≥ I d > 0 and the observability Grammian W satisfies the positivity condition (2), then the Riccati observability Grammian W Q satisfies a similar condition. This is just a consequence of that W Q (t, t + δ) ≥ λ min (Q(t))W (t, t + δ).
The above lemma calls for the following (well known) corollaries whose proofs are simple and omitted here for the sake of conciseness.
Corollary 2.8:
• If A and B are constant and such that the pair (A, B) is (Kalman) controllable, and if V (t) = BV (t)B T with V (t) ≥ I d > 0, ∀t, then the condition upon W V is satisfied. These conditions are themselves satisfied in the particular case where B = I d and V (t) ≥ I d > 0. • If A and C are constant and such that the pair (A, C) is (Kalman) observable, and if Q(t) ≥ I d > 0, ∀t, then the condition upon W Q is satisfied. It is however useful to keep in mind that the above conditions for the boundedness and good-conditioning of P (t) are only sufficient. For instance, when A is constant and Hurwitz stable, it is also sufficient to take Q(t) = 0, ∀t (so that the Riccati observability Grammian is identically equal to zero) and V constant. Indeed, it is simple to show that P (t) then converges to the p.d. solution to the Lyapunov equation
AP + P A T + V = 0.
A second corollary, that will be used further for the design of a position observer based on direction measurements, is as follows Corollary 2.9: Consider the projection matrix operator Π y(t) := I d − y(t)y T (t) with y(t) ∈ R n and such that |y(t)| = 1 (i.e. y(t) ∈ S n−1 ). If A(t) is the null matrix,
C(t) = Π y(t) and V (t) ≥ v I d > 0 or V (t) = k v Π y(t)
, with v and k v denoting positive numbers, and if the following persistent excitation (p.e.) condition is satisfied
∀t : 1 δ t+δ t Π y(s) ds ≥ I d > 0 , for some δ > 0 (12)
then the conditions on W V and W Q are also satisfied (and the solution P (t) to the CRE is thus bounded and well conditioned). If one chooses Q(t) = k q I d and V (t) = k v Π y(t) then the CRE writes aṡ
P = −k q P Π y(t) P + k v Π y(t) ; P (0) : p.d.
and
P (t) converges to (k v /k q ) 0.5 I d .
Note that this latter matrix is a solution to the CRE even when the condition (12) of persistent excitation is not satisfied. A technical extension of this corollary, also used further for the same estimation problem, but in the case where the velocity measurement is biased, follows Lemma 2.10: (12) is satisfied, then the Riccati observability Grammian W Q (t, t +δ) is positive for someδ > 0 and ∀t ≥ 0. See the proof in Appendix C. Let us just remark that the requirement of eigenvalues of A being all real is not fortuitous. Indeed, it is not difficult to find out examples for which the non-satisfaction of this condition forbids the Riccati observability Grammian from being positive. Such an example is
If 1) C(t) = Π y(t)C withC a constant matrix, 2) A is constant and such that the pair (A,C) is Kalman observable, 3) all eigenvalues of A are real, i.e. det(λI d − A) = 0 ⇒ λ ∈ R, 4) the p.e. conditionA = 0 1 −1 0 ,C = I d
The pair (A,C) is clearly observable, and det(λI d − A) = λ 2 +1 so that the eigenvalues of A are the imaginary numbers ±j. Choose
y(t) = [cos(t), − sin(t)] T so that t+2π t Π y(s) ds = t+2π t sin(s) 2 sin(2s)/2 sin(2s)/2 cos(s) 2 ds = πI d
This establishes that the p.e. property (12) is satisfied. However, using the fact that exp(At) = cos(t)I d + sin(t)A and setting b = [1, 0] T ∈ S 1 , one verifies that
Π y(s)C exp(A(s))b = Π y(s) y(s) = 0 , ∀s
This proves that the Riccati observability Grammian is never invertible in this case.
III. OBSERVERS FOR POSITION ESTIMATION FROM
DIRECTION MEASUREMENTS
The problem consists in estimating the position x of a body (or object) with respect to (w.r.t.) a fixed frame given its velocity u and the measurement of its direction x/|x|, knowing that the measured velocity may be biased by an initially unknown constant vector a. In practice, x will be a two-dimensional vector of coordinates (in the 2D, or planar, case) or a three-dimensional vector of coordinates (in the 3D, or spatial, case). For the sake of generality, we assume here that x ∈ R n , with n ≥ 2. The corresponding modelling equations areẋ
= u + ȧ a = 0 0 = Π y(t) x(13)
with y(t) := x(t)/|x(t)|. Let us distinguish two cases, depending on whether the velocity measurement is unbiased, i.e. a = 0, or is biased by an unknown constant vector a.
A. The unbiased velocity case
In this case the modelling equations can also be written asẋ
= Ax + u 0 = C(t)x(14)
with A = 0 n×n -the (n × n)-dimensional null matrixand C(t) = Π y(t) . This system can be associated with the following Riccati-like observeṙ
x = Ax + u + K(t)(0 − C(t)x)(15)
with the observer gain K(t) calculated as in relation (5) from the solution to the CRE (6). The resulting observer writes aṡ
x = u − K(t)Π y(t)x (16) with K(t) = k(t)P (t)Π y(t) Q(t) and P (t) the solution to the CREṖ = −P Π y(t) Q(t)Π y(t) P + V (t) Sinceẋ = (A − K(t)C(t))
x the Lyapunov analysis of section II-B applies, and global exponential stability of x = 0 is obtained provided that P (t) is bounded and well-conditioned. From Corollary 2.9 such is the case if V (t) is chosen positive (and larger than v I d ) or equal to k v Π y(t) , and -most importantly-if the p.e. condition (12) is satisfied. A loose interpretation of this condition is that the body must keep moving and not always in the direction of the source point. Note that, in the case where
V (t) = k v Π y(t) and Q(t) = k q I d , choosing the constant solution P = (k v /k q ) 0.5 I d simplifies the observer equation toẋ = u − k(t) k q k v Π y(t)
x , so that one recovers the solution proposed in [7].
B. The biased velocity case
In this more difficult case the velocity bias a has to be estimated as well. The modelling equations (13) can be written in the state form aṡ
X = AX +ū 0 = C(t)X(17)
with X := [x T , a T ] T the 2n-dimensional extended state vector,ū := [u T , 0 1×n ] T , and
A = 0 n×n I d 0 n×n 0 n×n , C = Π y(t)C ,C = I d 0 n×n Note that the pair (A,C) is Kalman-observable since [C T , (CA) T ] is equal to the (2n × 2n)-dimensional identity matrix. Consider now the CRĖ P = AP + P A T − P C(t) T Q(t)C(t)P + V (t)(18)
with P (0): a p.d. matrix. Provided that the p.e. condition (12) is satisfied, the solution P (t) to this CRE is bounded and well-conditioned, by application of Lemma 2.10 after noticing that all eigenvalues of A are equal to zero (and thus real). Consider now the following Riccati observeṙ
X = AX +ū + K(t)(0 − C(t)X) (19) withX := [x T ,â T ] T and K(t) = k(t)P (t)C(t) T Q(t) (k(t) ≥ 0.5).
This observer can also be written as
ẋ = u +â − k(t)P 11 (t)Π y(t) Q(t)ẋ a = −k(t)P 21 (t)Π y(t) Q(t)x(20)
with P ij (i, j ∈ {1, 2}) denoting the block components of P with adequate dimensions. Since the estimation error satisfies the equationẊ = (A − K(t)C(t))X, the Lyapunov analysis of section II-B applies and global exponential stability of X = 0 is obtained provided that the p.e. condition (12) is satisfied.
C. Extension to multiple direction measurements
We consider now the problem of estimating a vector x from l measurements y i = x−zi |x−zi| , i ∈ {1, . . . , l}. In the case where x represents the position of a body w.r.t. an inertial frame, and z i is the known vector of coordinates of a fixed source point, then y i is the unit vector measuring the direction between the object and this point. Setting X := [x T , a T ] T ,ū := [u T , 0 1×n ] T , and y = [(Π y1(t) z 1 ) T , . . . , (Π y l (t) z l ) T ] T , one obtains the systeṁ
X = AX +ū y = C(t)X(21)
with
A = 0 n×n I d 0 n×n 0 n×n C(t) = Π y1(t) 0 n×n . . . 0 n×n 0 n×n Π y2(t) . . . 0 n×n . . . . . . . . . 0 n×n 0 n×n . . . Π y l (t) C C = I d 0 n×n . . . I d 0 n×n with dim(C) = ln × 2n
It is simple to verify that the pair (A,C) is Kalmanobservable. Consider now the CRĖ
P = AP + P A T − P C(t) T Q(t)C(t)P + V (t)
with P (0): a p.d. matrix and
Q(t) = Q 11 (t) 0 n×n . . . 0 n×n 0 n×n Q 22 (t) . . . 0 n×n . . . . . . . . . 0 n×n 0 n×n . . . Q ll (t)
The solution P (t) to this equation is bounded and well conditioned provided that the corresponding Riccati observation Grammian W Q is positive. Using the fact that
C(t) T Q(t)C(t) = ∆(t) 0 n×n 0 n×n 0 n×n with ∆(t) := l i=1 Π yi(t) Q ii (t)Π yi(t)
, one verifies that this condition is satisfied if, for some δ > 0 and for all t > 0, 1 δ t+δ t∆ (s)ds, with∆(t) := l i=1 Π yi(t) , is greater than an arbitrarily small s.p.d. matrix. This p.e. condition clearly points out the interest of using multiple direction measurements in order to weaken, or even remove, conditions upon x and its time-variations. For instance, in the 3D case (n = 3), if l ≥ 2 then this p.e. condition is satisfied provided that the body is periodically not aligned with all the source points. If three or more source points are not aligned, then this condition is automatically satisfied independently of x and its time-variations. A Riccati observer associated with this system iṡ
X = AX +ū + K(t)(y − C(t)X) (22) withX := [x T ,â T ] T and K(t) = k(t)P (t)C(t) T Q(t) (k(t) ≥ 0.5).
One easily verifies that this observer can also be written as
ẋ = u +â − k(t)P 11 (t)( l i=1 Π yi(t) Q ii (t)(x − z i )) a = −k(t)P 21 (t)( l i=1 Π yi(t) Q ii (t)(x − z i ))(23)
From what precedes this observer globally exponentially stabilizesX = 0 if the previously evoked p.e. condition is satisfied.
Remarks:
• In the 3D-case, if l ≥ 2 and the matrix∆(t) is positive, and if the body moves with a constant unknown velocity, the above observer provides also an estimation of this velocity. To this aim it suffices to set u = 0 in the algorithm. The termâ is then an estimate of the body velocity that is equal to a in this case.
• In the unbiased case where a = 0 and the body velocity u is measured, the calculation ofâ is superfluous and the above observer reduces tȯ
x = u − k(t)P (t)( l i=1 Π yi(t) Q ii (t)(x − z i ))
with P (t) the solution to the CRĖ
P = −P ∆(t)P + V (t)
A particular solution to this latter equation, obtained by choosing V (t) = ∆(t), is P = I d .
IV. OBSERVERS FOR POSITION ESTIMATION FROM
RANGE MEASUREMENTS
The problem consists in estimating the position x of a body w.r.t. a fixed frame given its velocity u and the measurement of the distance (or range) |x|, knowing that the measured velocity may be biased by an initially unknown constant vector a. Again, for the sake of generality, we assume that x ∈ R n , with n ≥ 2. For the sake of simplifying forthcoming relations, and also to prevent the designed observer from being singular when |x| = 0, it is useful to formally set the system's output y equal to 0.5|x| 2 rather than |x|. The corresponding modelling equations arė
x = u + ȧ a = 0 y = 0.5|x| 2(24)
Let us again distinguish two cases, depending on whether the velocity measurement is unbiased, i.e. a = 0, or is biased by an unknown constant vector a.
A. The unbiased velocity case
In this case the modelling equations can be linearised by defining the (n + 1)-dimensional extended state vector X := [x T , y] T . Indeed, forming the time-derivative of X yields the LTV systemẊ = A(t)X +ū y = CX
withū := [u T , 0] T and
A(t) = 0 n×n 0 n×1 u T (t) 0 , C = [0 1×n 1]
A Riccati observer associated with this system iṡ
X = AX +ū + K(t)(y − CX) K(t) = k(t)P (t)C T q(t) P = A(t)P + P A T (t) − P C T q(t)CP + V (t)(26)
with P (0) a p.d. matrix, k(t) ≥ 0.5, q(t) ≥ > 0, and V (t) ≥ I d > 0. SettingX = [x T ,ŷ] T this observer can equivalently be written as
ẋ = u(t) + k(t)q(t)P 21 (t)(y(t) −ŷ(t)) y = u T (t)x + k(t)q(t)p 22 (t)(y(t) −ŷ(t))
This observer globally exponentially stabilizes the estimation errorX := X −X at zero if P (t) is bounded and wellconditioned. From what precedes such is the case if there exists δ > 0 and µ q > 0 such that the Riccati observability Grammian satisfies W Q (t, t + δ) ≥ µ q I d , ∀t ≥ 0. We claim that this latter condition is itself satisfied when the p.e. condition upon u(t) specified in the next lemma is satisfied. Remark: Another observer yielding the asymptotic stability ofX = 0 under the p.e. condition upon u(t) and when u(t) is uniformly continuous is
ẋ = u(t) + k 1 (y(t) −ŷ(t)) y = u T (t)x + k 2 (t)(y(t) −ŷ(t))
with k 1 > 0 and 0 ≤ k 2 ≤ k 2 (t) ≤ k 2 < ∞. This can be proved by considering the positive function V(t) := |x(t)| 2 /k 1 +ỹ(t) 2 whose time-derivative satisfiesV(t) = −2k 2 (t)ỹ(t) 2 (≤ 0). One deduces thatỹ(t) converges to zero and that |x(t)| is bounded and converges to some finite limit |x ∞ |. Then, by application of (extended) Barbalat's Lemma one deduces that the time-derivative ofỹ(t) converges to zero so that u T (t)x(t) also converges to zero. Therefore t+δ t |u T (s)x(s)| 2 ds converges to zero when t tends to infinity. The convergence ofỹ(t) to zero also implies that the time-derivative ofx(t) converges to zero. From there one finishes the proof by showing that the satisfaction of the p.e. condition is not compatible with |x ∞ | = 0.
An advantage of this second observer over a Riccati observer is that it involves less calculations. However, the proof of convergence of this observer, as sketched above, does not establish that the rate of convergence is exponential. This limitation epitomizes an important feature that goes with Riccati observer designs, namely the knowledge of an explicit Lyapunov function that allows for a more complete analysis of stability and convergence.
B. The biased velocity case
In this more difficult case the modelling equations can be linearised by defining the (2n + 3)-dimensional extended state vector X := [x T , a T , y, a T x, |a| 2 ] T . Indeed, forming the time-derivative of X yields the LTV systeṁ
X = A(t)X +ū y = CX(28)
withū := [u T , 0 1×(n+3) ] T and
A(t) = 0 n×n I n×n 0 n×1 0 n×1 0 n×1 0 n×n 0 n×n 0 n×1 0 n×1 0 n×1 u T (t) 0 1×n 0 1 0 0 1×n u T (t) 0 0 1 0 1×n 0 1×n 0 0 0 C = [0 1×n 0 1×n 1 0 0]
A Riccati observer associated with this sytem is given by (26), and global exponential stability of this observer follows if the system's observability Grammian satisfies (2). We claim that this latter condition is itself satisfied when the p.e. condition uponu(t) specified in the next lemma is satisfied. Lemma 4.2: If u(t) is twice differentiable with bounded first and second derivatives and ifu(t) satisfies the p.e. condition
∀t ≥ 0 : 1 δ t+δ tu (s)u T (s)ds ≥ µI d > 0(29)
for some δ > 0 and µ > 0, then the Riccati observer globally exponentially stabilizesX = 0. See the proof in Appendix E.
C. Extension to multiple range measurements
We consider now the problem of estimating a vector x from l measurements y i = 0.5|x − z i | 2 , i ∈ {1, . . . , l}. In the case where x represents the 3D position of a body w.r.t. an inertial frame, and z i is the known vector of coordinates of a fixed source point, then y i is half the squared distance between the body and this point.
1) Unbiased velocity case (a = 0): Define the (l × 1)dimensional constant vector ξ := [1, . . . , 1] T . Define the weighted output variable y 0 := l i=1 α i (y i − 0.5|z i | 2 ), with α = [α 1 , . . . , α l ] T denoting a l-dimensional vector of real numbers such that
l i=1 α i = 1. Sinceẋ = u and y 0 = 0.5|x| 2 − l i=1 α i z T i x, one hasẏ 0 = (x T − l i=1 α i z T i )u.
Define also the l-dimensional output vector y := [y 0 , (y 1 − y 0 − 0.5|z 1 | 2 ), . . . , (y l − y 0 − 0.5|z l | 2 )] T and the augmented state X :
= [x T , y 0 ] T . Since (y j − y 0 − 0.5|z j | 2 ) = l i=1 α i z T i x − α j z T j x one has y = CX with C := 0 1×n 1 D(α)Z T 0 l×1 D(α) := ξα T − I l×l : (l × l)-dimensional matrix ξ := [1, . . . , 1] T : (l × 1)-dimensional vector Z := [z 1 . . . z l ] : (n × l)-dimensional matrix
Note that the rank of the matrix D(α) is equal to (l − 1).
From the previous definitions one obtains the linear systeṁ
X = A(t)X +ū y = CX (30) with A(t) := O n×n O n×1 u(t) T 0 ,ū := I n×n − l i=1 α i z T i u
A Riccati observer associated with this system is of the form (26), except for the positive scalar q(t) involved in the CRE that is now replaced by a ((l + 1) × (l + 1))-dimensional p.d. matrix Q(t). (31) for some δ > 0 and µ > 0, then the above-mentioned Riccati observer globally exponentially stabilizesX = 0. The proof of this lemma is a straightforward adaptation of the proof of Lemma 4.1 given in Appendix D, after observing that the matrix M (t) involved in the proof is now
M (t) = O 1×n 1 D(α)Z T 0 l×1 u(t) T 0 .
We remark that the p.e. condition is automatically satisfied, independently of u(t), when l ≥ n + 1 and rank(D(α)Z T ) = n, i.e. when n vectors among the l vectors z j − l i=1 α i z i (j = 1, . . . , l) are independent. For instance, in the 3D case it is generically satisfied when the number of source points is equal to, or greater than, four. This result is coherent with the minimum number of four source points needed to geometrically determine the position of a motionless body with no ambiguity from a single set of multiple range measurements. Using four, or more, non coplanar source points provides redundancy that can be used to accelerate the rate of convergence and/or reduce the asymptotic variance ofX when the range measurements are corrupted by noise. More generally, Riccati observers performance depends on the tuning of the parameters involved in the CRE, namely k(t), Q(t), and V (t). In this respect the choice of parameters associated with the stochastic optimal Kalman filter can provide useful leads in complementation with the dependence, pointed out earlier, between the exponential convergence rate of the observer and the amount of persistent excitation. This tuning issue is important for practical purposes and deserves to be studied in its own right. However, it is out of the present paper's scope and is thus not pursued further here.
2) Biased velocity case (a is a priori unknown): Define
• X := [x T , a T , y 0 , a T x, |a| 2 ] T ; • y := [y 0 , (y 1 − y 0 − 0.5|z 1 | 2 ), . . . , (y l − y 0 − 0.5|z l | 2 )] T ,
i.e. the same output vector as in the unbiased case;
•ū := [u T , 0 1×n , −u T ( l i=1 α i z i ), 0, 0] T .
Forming the time-derivative of X yields a linear system alike (30) with the state matrix
A(t) = 0 n×n I n×n 0 n×1 0 n×1 0 n×1 0 n×n 0 n×n 0 n×1 0 n×1 0 n×1 u T (t) − l i=1 α i z T i 0 1 0 0 1×n u T (t) 0 0 1 0 1×n 0 1×n 0 0 0
and the output matrix
C := 0 1×n 0 1×n 1 0 0 D(α)Z T 0 1×n 0 l×1 0 l×1 0 l×1
A Riccati observer associated with this system is thus again given by (26), with the p.d. matrix Q(t) involved in the CRE chosen as in the unbiased case with multiple range measurements.
Lemma 4.4: If u(t) is twice differentiable with bounded first and second derivatives, and ifu(t) and the vectors z i (i = 1, . . . , l) satisfy the p.e. condition
∀t ≥ 0 : ZD(α) T D(α)Z T + 1 δ t+δ tu (s)u T (s)ds ≥ µI d > 0
(32) for some δ > 0 and µ > 0, then the Riccati observer globally exponentially stabilizesX = 0. The proof of this lemma is a simple adaptation of the proof of Lemma 4.2 given in Appendix E, with the matrix M (t) involved in the proof chosen as follows
M (t) = N 0 N 1 (t) N 2 (t)
with N 0 = C, N 1 (t) = CA(t), andN 2 (t) equal to the first line of N 2 (t) = N 1 (t)A(t) +Ṅ 1 (t). According to this lemma one finds again that the positivity of the matrix ZD(α) T D(α)Z T , which is generically ensured when the number of source points is greater than three, is sufficient to yield the exponential stabilization ofX = 0 independently of the input u(t). Nevertheless, the knowledge of the time-derivative of x, via the estimation of the bias a, is still required.
Remark: In the 3D-case, when the number of source points exceeds three and the matrix ZD(α) T D(α)Z T is positive, and when the body velocity is constant but unknown a priori, the observer provides also an estimate of this velocity. To this aim it suffices to set u = 0 in the algorithm. The termâ is then an estimate of the body velocity a.
V. SIMULATIONS
For these simulations in 3D-space we have considered three scenarios involving various body motions. Estimation of the body position is carried out from range and direction measurements with a minimal number of source points ensuring uniform observability. Since the conditions of observability are different in the two measurement cases, the number of source points may also be different. In all scenarios the body velocity measurement is corrupted by the constant bias a = (0.33, 0.66, 0.99) and initial state conditions are x(0) = (5, 0 4) ,x(0) = (4, 6, 12) and a(0) = (0, 0, 0) . Riccati observers are calculated with k(t) = 1, as for a Kalman filter, and the corresponding CRE are initialized with P (0) = 100I 6 , when using direction measurements, and P (0) = 100I 9 , when using range measurements. For each scenario, simulations are first carried out with noise-free measurements to validate theoretical exponential stability results, then with measurements corrupted by noise to illustrate the resulting (and inevitable) slight degradation of the observers following the transient phase when the estimation errors become small but no longer converge to zero. Concerning the body velocity u we have used a Gaussian zero mean additive noise with a standard deviation equal to 0.1m/s. As for the direction and range measurements, they are calculated from a body position corrupted by a Gaussian zero mean noise with standard deviation equal to 0.05m/s. For the matrix V (i.e. the state noise variance in the Kalman filtering terminology) involved in the CRE we have set V = 0.01diag{1, 1, 1, 0, 0, 0} + v I 6 , when using direction measurements, and V = 0.01diag{1, 1, 1, 0, 0, 0, 10, 0, 0} + v I 9 , when using range measurements, with the small number v set equal to 0.001 to ensure that the matrix is positive definite. As for the matrix Q (i.e. the inverse of the output noise variance in the Kalman filtering terminology) we have used Q ii = 1.5I 3 , ∀i = 1 . . . l (with l the number of source points) and Q = 1.5I l+1 respectively. In this case a single source point, again taken as the origin of the inertial frame, suffices to ensure uniform observability in the direction measurement case, whereas a second source point has to be used in the range measurement case to ensure the satisfaction of this property. Figures 2(a)-2(c) illustrate the performance of the two observers in the ideal noise-free case, and Figures 2(d)-2(e) show asymptotic estimation errors in the case of noisy measurements.
Scenario 3:
The body is motionless. Two source points are then needed in the direction measurement case to ensure uniform observability, whereas two other source points, non coplanar with them, are required in the range measurement case. Figures 3(a)-3(c) illustrate the performance of the two observers in the ideal noise-free case, and Figures 3(d)-3(e) show asymptotic estimation errors in the case of noisy measurements. By comparison with the previous two scenarios the estimation errors are smaller. This is coherent with the increased number of source points that yields less noisy information in the average.
VI. CONCLUDING REMARKS
In this paper, Riccati observers for the estimation of a body position from either direction or range measurements and from the knowledge of the body velocity have been reviewed. Even when the body velocity is biased by an unknown constant vector, these observers ensure global exponential stability of zero estimation errors under uniform observability conditions that have been worked out in relation to the number of source points and the body motion. Clearly the set of such observers extends without difficulty to the case where the available information comes from the combination of direction measurements (associated with certain source points) with range measurements (associated with other source points). A logical prolongation of this work is the derivation of Riccati observers for the estimation of the complete body pose (position and orientation). However, due to the specific structure of the group of rotations, exact linearisation of the problem is then no longer possible and globally convex cost functions do not exist. As a consequence Riccati observers for pose estimation, and corresponding Extended Kalman Filters (EKF), have to be derived from linear approximations of the system state and output equations. This also implies that only local exponential stability of zero estimation errors can be achieved. An important complementary issue, also in the prolongation of the present work, is the characterisation of uniform observability conditions under which this latter property is granted. We foresee several other possible extensions. Let us just mention visionbased robotic applications involving the control of the body position from estimates provided by Riccati observers, and a deterministic approach to Simultaneous Localication and Mapping (SLAM) that could usefully complement existing studies on the subject.
VII. ACKNOWLEDGMENTS
This work was supported by the ANR-ASTRID project SCAR "Sensory Control of Unmanned Aerial Vehicles", the ANR-Equipex project "Robotex". Recall that, as long as P (t) is defined and p.d., its trace is the sum of its eigenvalues. Accordingly, since the eigenvalues of P −1 (t) are the inverse of the ones of P (t), the trace of P −1 (t) is the sum of the inverse of the eigenvalues of P (t).
To prove that P (t) is well-defined for t ∈ [0, +∞) and is p.d. it suffices to show that neither the trace of P (t) nor the trace of P −1 (t), which are initially positive (since P (0) is p.d. by assumption), can tend to infinity in finite time. Indeed, this implies that none of the eigenvalues of P (t) can either reach zero or tend to infinity in finite time. To this aim, it suffices to show that neither tr(P (t) nor tr(P −1 (t)) can grow faster than exponentially, so that divergence in finite time is not possible.
Let us set x = tr(P ). In view of (6), and since tr(P (t)C(t) T Q(t)C(t)P (t)) ≥ 0, one haṡ
x ≤ tr(AP ) + tr(P A T ) + tr(V )
Let |A(t)| denote the spectral norm of A(t). By assumption it is bounded by some positive number k a . Similarly, tr(V (t)) is bounded by a positive number v. Since P is p.s.d., |tr(AP )| = |tr(P A T )| ≤ |A|tr(P ) and the previous inequality yieldsẋ ≤ 2k a x + v
This inequality in turn implies that x(t) ≤ (x(0) + v 2ka )exp(2k a t) − v 2ka , ∀t ≥ 0. Similar arguments applied to y = tr(P −1 ) yielḋ y ≤ |tr(P −1 A)| + |tr(A T P −1 ) + tr(C T QC) ≤ 2k a y +μ q withμ q denoting the supremum of tr(C(t) T Q(t)C(t)). Therefore, y(t) ≤ (y(0) +μ q 2ka )exp(2k a t) −μ q 2ka , ∀t ≥ 0. (end of proof) B. Determination of ultimate bounds for p m and p M 1) Ultimate lower bound of the smallest eigenvalue of P (t) when v m > 0: In practice the matrix V (t) is usually chosen strictly positive so that the assumption of positivity on v m is little restrictive. Set, as in the previous appendix,
y(t) = tr(P −1 (t)) = n i=1 1 λi(t) , with {λ i (t)} i=1.
..n the set of eigenvalues of P (t). The suprema of the spectral norm of A(t) and of tr(C(t) T Q(t)C(t)) are again denoted as k a and µ q respectively. From (8) one haṡ y ≤ 2k a y +μ q − tr(P −1 V P −1 )
with tr(P −1 V P −1 ) ≥ vm n y 2 . This inequality implies that y(t) is ultimately smaller than, or equal to, the largest (positive) root of the second degree equation vm n y 2 − 2k a y − µ q = 0. More precisely
lim sup t→∞ y(t) ≤ nk a v m 1 + (1 +μ q v m nk 2 a ) 0.5
Since 1/λ min (P (t)) ≤ y(t) the previous inequality yields
lim inf t→∞ λ min (P (t)) ≥ v m nk a 1 + (1 +μ q v m nk 2 a ) 0.5 −1 (33)
2) Ultimate upper bound of the largest eigenvalue of P (t) when W Q (t, t + δ) ≥ µI d > 0: Recall that W Q is the Riccati observability Grammian defined in (11). We use the expression of the upper bound of P (t) derived in [16]
P (t) ≤ δW −1 Q (t, t+δ)+δ 2 W −1 Q (t, t+δ)I2(t, t+δ)W −1 Q (t, t+δ)
with I 2 (t, t + δ) a positive matrix-valued function which, using the inequality |Φ(t, s)| ≤ exp(k a |t − s|)I d , is upper bounded by
exp(6kaδ)μ 2 q δ 3 v M 3 I d . Therefore, when W Q (t, t + δ) ≥ µI d > 0 one deduces that lim sup t→∞ λ max (P (t)) ≤ 1 µδ + 1 3 μ q µ 2 exp(6k a δ)δv M (34)
Relations (33) and (34) can in turn be used to estimate an ultimate lower bound of p 2 m p M v m , i.e. an estimate of the lower bound pointed out in (9) of the exponential rate of convergence associated with a Riccati observer.
C. Proof of lemma 2.10
For the sake of simplifying the reading of the proof by avoiding non-essential technicalities, we set Q = k q I d with k q > 0. Let us proceed by contradiction and assume that the lemma's conclusion is wrong, i.e.
∀ , ∀δ > 0, ∃t ≥ 0 : W Q (t, t +δ) < I d Consider a sequence { p } p∈N of positive numbers converging to zero, and an arbitrary positive numberδ. From the previous assertion there must exist a sequence of timeinstants {t p } p∈N and a sequence {x p } p∈N with x p ∈ S n−1 (i.e. |x p | = 1) such that ∀p ∈ N : x T p W Q (t p , t p +δ)x p < p .
Since S n−1 is a compact set there exists a sub-sequence of {x p } p∈N which converges to a limitx ∈ S n−1 . Therefore lim p→∞x T W Q (t p , t p +δ)x = 0
Using C = Π y(t)C and Φ(t, s) = exp(A(t − s)) in the definition (11) of W Q , the above equality is equivalent to lim p→∞ δ 0 |Π y(tp+s)C exp(As)x| 2 ds = 0 which in turn implies lim p→∞ δ δ−δ |Π y(tp+s)C exp(As)x| 2 ds = 0
provided thatδ ≥ δ. Consider now the following technical result whose proof is given at the end of the present appendix Lemma 0.1: Assume that the eigenvalues of the matrix A are all real, then, givenx ∈ S n−1 , there exist r ≥ 0, λ ∈ R, and z ∈ R m − {0} such thatC exp(At)x t r exp(λt) = z + η(t) with lim t→+∞ η(t) = 0. In view of this result, settingz = z/|z| ∈ S m−1 , and choosingδ large enough so that sup s∈[δ−δ,δ] |η(s)| < 2 |z| one deduces that 1 γ 2 |z| 2 δ δ−δ |Π y(tp+s)C exp(As)x| 2 ds ≥ δ δ−δ |Π y(tp+s) (z + η(s) |z| )| 2 ds ≥ tp+δ tp+δ−δ |Π y(s)z | 2 ds − δ 0 |Π y(tp+s) η(s) |z| )| 2 ds ≥ δ − δ /2 (= δ /2) with γ = inf s∈[δ−δ,δ] (s r exp(λs)) > 0. The p.e. condition (12) is used in the last inequality. Therefore δ δ−δ |Π y(tp+s)C exp(As)x| 2 ds ≥ γ 2 |z| 2 2 > 0
Since this latter inequality holds true for any t p , it contradicts (35) and the initial assumption according to which the result of the lemma is not true. It only remains to prove the technical Lemma 0.1. From Cayley-Hamilton's theorem, one has exp(At) = n−1
i=0 α i (t)A i with α i (t) = d k=1 ( l k −1
j=0 a ij t j )exp(λ k t), λ k a (real) eigenvalue of A, a ij ∈ R, d ≤ n the number of distinct eigenvalues, and l k the multiplicity of λ k . Thereforē the Kalman observability matrix whose rank is, by assumption, equal to n. This latter assumption in turn implies that the vectorz is different from zero, and thus that at least one of the z i components of this vector is different from zero. The previous sum can also be arranged as follows
Cexp(At)x =C n−1 i=0 α i (t)A ix = n−1 i=0 α i (t)CA ix = n−1 i=0 α i (t)z i withz = z 1 . . .n−1 i=0 α i (t)z i = k,j v k,j (t)z k,jzk,j ∈ R m
with v k,j (t) = t r k,j exp(λ k t), k ∈ [1, . . . , n], r k,j ∈ [0, . . . , n − 1]. We note that at least one of the vectors z k,j must be different from zero, due to the observability assumption and the full rank of O. Consider the largest (less negative, or most positive) root λ k for whichz k,j is different from zero, and the largest power r k,j that goes with such a vector. Denote this root as λ and this power as r, set v(t) := t r exp(λt), and denote the corresponding vectorz k,j as z ( = 0). The dominating coefficient in the development ofCexp(At)x, when t tends to infinity, is thus v(t) and one has lim t→∞C exp(At)x v(t) = z. This latter property can also be written asC exp(At)x v(t) = z + η(t) with lim t→∞ η(t) = 0.
D. Proof of lemma 4.1
Recalling that the positivity of the observability Grammian W yields the positivity of the Riccati observability Grammian W Q when Q(t) ≥ I d > 0, one only has to show -according to Lemma 2.5-the existence of an adequate matrix-valued function M (.) that satisfies (3) for some positive numbersδ andμ. with 0 ≤ |b 1 | ≤ 1. There are two possible cases: either b 1 = 0 or b 1 = 0. In the first case one obtains γ(t) = 1. In the second case, settingb 1 := b 1 /|b 1 | ∈ S n−1 , one obtains γ(t) = (1 − |b 1 | 2 ) + |b1| 2 δ t+δ tb T 1 u(s)u T (s)b 1 ds ≥ (1 − |b 1 | 2 ) + |b 1 | 2 µ ≥ inf(1, µ). Therefore γ(t) ≥μ with µ = inf(1, µ). Since the last inequality holds for any b ∈ S n , (3) holds true.
Definition 2.3 (uniform observability): Sytem (1) is uniformly observable if there exists τ > 0 such that, ∀t, x(t) can be calculated from the knowledge of the input u(.) and ouput y(.) on the time-interval [t, t + τ ].
Lemma 4. 1 :
1If u(t) satisfies the p.u T (s)ds ≥ µI d > 0 (27)for some δ > 0 and µ > 0, then the Riccati observer (26) globally exponentially stabilizesX = 0. See the proof in Appendix D.
Lemma 4 . 3 :
43If u(t) and the vectors z i (i = 1, . . . , l) satisfy the p.e. condition ∀t ≥ 0 : ZD(α) T D(α)Z T + 1 δ t+δ t u(s)u T (s)ds ≥ µI d > 0
Scenario 1 :
1The body moves along a Lissajous curve of equationx(t) = (20 cos t − 15, 20 sin t, −2 cos t + 6) , and a single source point located at the origin of the inertial frame is used for both direction and range measurements.
verifies that conditions for uniform observability are then satisfied in both cases. Figures 1(a)-1(c) illustrate the performance of the two observers in the ideal noise-free case. More precisely, Figure 1(a) shows the location of the source point and the trajectories followed by the body position x(t) and its estimatex(t), Figure 1(b) shows the convergence of the bias estimateâ to the velocity bias a and Figure 1(c) shows the evolution of the logarithms of the Lyapunov functions associated with the observers. The rate of exponential convergence to zero of the Lyapunov functions are given by the mean slopes of the curves. Asymptotic estimation errors in the case of noisy measurements are shown in Figures 1(d) and 1(e).
Scenario 2 :
2The body moves along a circular trajectory of equation x(t) = (20 cos t − 15, 20 sin t, 4) .
Fig. 1 .tFig. 2 .Fig. 2 .Fig. 2 .Fig. 3 .Fig. 3 .Fig. 3 .
1222333) Time evolution of velocity bias estimation errors in the case of noisy measurements Scenario (o bserver ba sed direct.) â1 → a1 = 0.33 a2 → a2 = 0.66 a3 → a3 = 0.99 (b) Time evolution of the velocity bias estimation Scenario Time evolution of velocity bias estimation errors in the case of noisy measurements Scenario ) Time evolution of position estimation errors in the case of noisy measurements Scenario ) Time evolution of the Lyapunov functions logarithms Scenario Time evolution of velocity bias estimation errors in the case of noisy measurements Scenario Time evolution of position estimation errors in the case of noisy measurements Scenario 3
For the system under consideration one has N 0= C = [0 1×n 1] and N 1 (t) = CA(t) = [u T (t) 0]. an arbitrary vector b ∈ S n . Then M (t)b = b 2 u T (t)b 1 with b = [b T 1 , b 2 ] T . Therefore |M (t)b| 2 = b T 1 u(t)u T (t)b 1 + b 2 2 . Define γ(t)
the unbiased case we show the existence of a matrixvalued function M (.) that satisfies (3) for some positive numbersδ andμ. For the system under consideration one has N 0 = C = [0 1×n 0 1×n 1 0 0], N 1 (t) = CA(t) = [u T (t) 0 1×n 0 1 0], and N 2 (t) = N 1 (t)A(t) +Ṅ 1 (t) = [u T (t) 2u T (t) 0 0 1]. Define M (t)
and consider an arbitrary vector b = [b T 1 , b T 2 , b T 3 ] T ∈ S 2n+2 , with b 1,2,3 sub-vectors of dimensions n, n, and 3 respec-|M (s)b| 2 ds and let us make a proof by contradiction by assuming that the condition(3)is not satisfied. In this case there exists a sequence {t p } and a vector b ∈ S 2n+2 such that lim p→+∞ γ(t p ) = 0. This in turn implies that b 3,1 = 0 and alsoandUsing the assumed boundedness ofu(t) the first of these two limits yields lim p→+∞ u T (t p + s)b 1 = −b 3,2 , ∀s ∈ (0, δ).Using now the assumed boundedness ofü(t) this in turn implies that lim p→+∞u T (t p + s)b 1 = 0, ∀s ∈ (0, δ). From (37) one deduces that lim p→+∞ (u T (t p + s)b 1 + 2u T (t p + s)b 2 + b 3,3 ) = 0, ∀s ∈ (0, δ) and, subsequently, that lim p→+∞ u T (t p + s)b 2 = −b 3,3 /2, ∀s ∈ (0, δ). Using the assumed boundedness ofü(t) this in turn implies that lim p→+∞u T (t p + s)b 2 = 0, ∀s ∈ (0, δ). If either b 1 or b 2 is different from zero one reaches a contradiction with (29). Therefore b 1 = b 2 = 0. But then, from what precedes, b 3,2 = b 3,3 = 0 so that b = 0. This is not possible since b ∈ S 2n+2 .
Review and analysis of solutions of the three point perspective pose estimation problem. R Haralick, C Lee, K Ottenberg, M Nlle, International Journal of Computer Vision. 133R. Haralick, C. Lee, K. Ottenberg, and M. Nlle, "Review and analysis of solutions of the three point perspective pose estimation problem," International Journal of Computer Vision, vol. 13, no. 3, pp. 331-356, 1994.
A novel parametrization of the perspective-three-point problem for a direct computation of absolute camera position and orientation. L Kneip, D Scaramuzza, R Siegwart, Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on. IEEEL. Kneip, D. Scaramuzza, and R. Siegwart, "A novel parametrization of the perspective-three-point problem for a direct computation of absolute camera position and orientation," in Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on. IEEE, 2011, pp. 2969-2976.
Pose estimation from corresponding point data. R Haralick, H Joo, C Lee, X Zhuang, V Vaidya, M Kim, IEEE transactions on Systems, Man and Cybernetics. 196R. Haralick, H. Joo, C. Lee, X. Zhuang, V. Vaidya, and M. Kim, "Pose estimation from corresponding point data," IEEE transactions on Systems, Man and Cybernetics, vol. 19, no. 6, pp. 1426-1446, 1989.
A kalman-filter-based methods for pose estimation in visual servoing. F Janabi-Sharifi, M Marey, IEEE transactions on Robotics. 265F. Janabi-Sharifi and M. Marey, "A kalman-filter-based methods for pose estimation in visual servoing," IEEE transactions on Robotics, vol. 26, no. 5, pp. 939-947, 2010.
Motion estimation via dynamic vision. S Soatto, R Frezza, P Perona, IEEE Transactions on Automatic Control. 413S. Soatto, R. Frezza, and P. Perona, "Motion estimation via dynamic vision," IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 393-414, 1996.
Globally exponentially stable filters for source localization and navigation aided by direction measurements. P Batista, C Silvestre, P Oliveira, Systems & Control Letters. 6211P. Batista, C. Silvestre, and P. Oliveira, "Globally exponentially stable filters for source localization and navigation aided by direction measurements," Systems & Control Letters, vol. 62, no. 11, pp. 1065- 1072, 2013.
Observer design for position and velocity bias estimation from a single direction output. F L Bras, T Hamel, R Mahony, C Samson, 54th IEEE International Conference on Decision and Control (CDC). F. L. Bras, T. Hamel, R. Mahony, and C. Samson, "Observer design for position and velocity bias estimation from a single direction output," in 54th IEEE International Conference on Decision and Control (CDC), 2015.
An introduction to the global positioning system and some geological applications. T Dixon, Reviews of gophysics. 292T. Dixon, "An introduction to the global positioning system and some geological applications," Reviews of gophysics, vol. 29, no. 2, pp. 249- 276, 1991.
Sensor-based long baseline navigation: observability analysis and filter design. P Batista, C Silvestre, P Oliveira, Asian J. Control. 16P. Batista, C. Silvestre, and P. Oliveira, "Sensor-based long baseline navigation: observability analysis and filter design," Asian J. Control, vol. 16, pp. 974-994, 2014.
Tightly coupled long baseline/ultra-short baseline integrated navigation system. International Journal of Systems Science. no. ahead-of-print--, "Tightly coupled long baseline/ultra-short baseline integrated navigation system," International Journal of Systems Science, no. ahead-of-print, pp. 1-19, 2014.
Position and velocity usbl/imu sensor-based navigation filter. M Morgado, P Batista, P Oliveira, C Silvestre, 18th IFAC World Congress. Milan, Italy13M. Morgado, P. Batista, P. Oliveira, and C. Silvestre, "Position and velocity usbl/imu sensor-based navigation filter," in 18th IFAC World Congress, Milan, Italy, 2011, pp. 13 642-13 647.
C.-T Chen, Linear System Theory and Design. CBS College Publishing2nd ed.C.-T. Chen, Linear System Theory and Design, 2nd ed. CBS College Publishing, 1984.
Observability and observers for nonlinear systems. J.-P Gauthier, J.-P Kupka, SIAM J. on Control Optim. 432J.-P. Gauthier and J.-P. Kupka, "Observability and observers for nonlinear systems," SIAM J. on Control Optim., vol. 4, no. 32, pp. 975-994, 1994.
An overview on observer tools for nonlinear systems. G Besançon, Besançon G.Springer-VerlagG. Besançon, An overview on observer tools for nonlinear systems. Besançon G., editor, Springer-Verlag, 2007.
Fusion de données visuo-inertielles pour l'estimation de pose et l'autocalibrage. G Scandaroli, Ph.D. dissertationG. Scandaroli, "Fusion de données visuo-inertielles pour l'estimation de pose et l'autocalibrage," Ph.D. dissertation, 2013.
On the boundedness of the solutions of the continuous riccati equation. M Pengov, E Richard, J.-C Vivalda, J. of Inequal. and Appli. 6M. Pengov, E. Richard, and J.-C. Vivalda, "On the boundedness of the solutions of the continuous riccati equation," J. of Inequal. and Appli., vol. 6, pp. 641-649, 2001.
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[
"NOTES ON VARIATION OF LEFSCHETZ STAR OPERATOR AND T -HODGE THEORY",
"NOTES ON VARIATION OF LEFSCHETZ STAR OPERATOR AND T -HODGE THEORY"
]
| [
"X U Wang "
]
| []
| []
| These notes were written to serve as an easy reference for[34]. All the results in this presentation are well-known (or quasi-well-known) theorems in Hodge theory. Our main purpose was to give a unified approach based on a variation formula of the Lefschetz star operator, following[32]. It fits quite well with Timorin's T -Hodge theory, i.e. the Hodge theory on the space of differential forms divided by T (i.e. forms like T ∧ u), where T is a finite wedge product of Kähler forms.Date: August 25, 2017. | null | [
"https://arxiv.org/pdf/1708.07332v1.pdf"
]
| 119,667,668 | 1708.07332 | 0773a05044984c974abdab1a40695a14b4eccf1e |
NOTES ON VARIATION OF LEFSCHETZ STAR OPERATOR AND T -HODGE THEORY
24 Aug 2017
X U Wang
NOTES ON VARIATION OF LEFSCHETZ STAR OPERATOR AND T -HODGE THEORY
24 Aug 2017
These notes were written to serve as an easy reference for[34]. All the results in this presentation are well-known (or quasi-well-known) theorems in Hodge theory. Our main purpose was to give a unified approach based on a variation formula of the Lefschetz star operator, following[32]. It fits quite well with Timorin's T -Hodge theory, i.e. the Hodge theory on the space of differential forms divided by T (i.e. forms like T ∧ u), where T is a finite wedge product of Kähler forms.Date: August 25, 2017.
1. Preliminaries 1.1. Primitivity: linear setting. Let V be an N -dimensional real vector space. Let ω be a bilinear form on V . We call ω a symplectic form if ω is non-degenerate and ω ∈ ∧ 2 V * , i.e. ω(u, v) = −ω(v, u), ∀ u, v ∈ V . Proposition 1.1. Assume that there is a symplectic form ω on V . Then N = 2n for some integer n and there exists a base, say {e * 1 , f * 1 ; · · · ; e * n , f * n }, of V * such that ω = n j=1 e * j ∧ f * j .
Proof. Since ω is non-degenerate, we know that N ≥ 2. If N = 2 and ω(e, f ) = 1 then
ω = e * ∧ f * ,
where {e * , f * } is the dual base of {e, f }. Assume that N ≥ 3, consider V ′ := {u ∈ V : ω(u, e) = ω(u, f ) = 0}.
Then for every u ∈ V , we have u ′ := u − ω(u, f )e + ω(u, e)f ∈ V ′ ,
and ae + bf ∈ V ′ iff a = b = 0, thus V = V ′ ⊕ Span{e, f }.
Since ω is non-degenerate, we know that for every v ∈ V ′ , there exists u ∈ V such that ω(u, v) = 0. Thus ω(u ′ , v) = ω(u, v) = 0, which implies that ω| V ′ is a symplectic form on V ′ . Thus the theorem follows by induction on N .
One may use ω to define a bilinear form, say ω −1 , on V * such that
ω −1 (f * j , e * k ) = −ω −1 (e * k , f * j ) = δ jk , ω −1 (f * j , f * k ) = ω −1 (e * j , e * k ) = 0. Let T ω : V → V * be the linear isomorphism defined by T ω (u)(v) = ω(v, u), ∀ u, v ∈ V.
Then we have T −1 ω = T ω −1 , thus the definition of ω −1 does not depend on the choice of bases in the above proposition. We shall also use ω −1 to denote the following bilinear form on ∧ p V * :
(1.1) ω −1 (µ, ν) := det(ω −1 (α i , β j )), µ = α 1 ∧ · · · ∧ α p , ν = β 1 ∧ · · · ∧ β p . Definition 1.1 (By Guillemin [16]). The symplectic star operator * s : ∧ p V * → ∧ 2n−p V * is defined by
(1.2) µ ∧ * s ν = ω −1 (µ, ν) ω n n! .
The following theorem is the key to decode the structure of * s . Theorem 1.2 (Hard Lefschetz theorem). For each 0 ≤ k ≤ n,
u → ω n−k ∧ u, u ∈ ∧ k V * ,
defines an isomorphism between ∧ k V * and ∧ 2n−k V * .
Proof. Notice that the theorem is true if n = 1 or k = 0, n. Now assume that it is true for n ≤ l, l ≥ 1. We need to prove that it is true for n = l + 1, 1 ≤ k ≤ l.
Put
ω ′ = l j=1 e * j ∧ f * j .
Then we have ω l+1−k = (ω ′ ) l+1−k + (l + 1 − k)(ω ′ ) l−k ∧ e * l+1 ∧ f * l+1 .
Let us write u ∈ ∧ k (V * ) as u = u 0 + u 1 ∧ e * l+1 + u 2 ∧ f * l+1 + u 3 ∧ e * l+1 ∧ f * l+1 , where each u j contains no e * l+1 or f * l+1 term. Then ω l+1−k ∧ u = 0 is equivalent to (ω ′ ) l+1−k ∧ u 0 = (ω ′ ) l+1−k ∧ u 1 = (ω ′ ) l+1−k ∧ u 2 = (ω ′ ) l+1−k ∧ u 3 + (l + 1− k)(ω ′ ) l−k ∧ u 0 = 0, which implies u 1 = u 2 = 0 by our theorem for n = l. Moreover, u 3 = 0 since
(ω ′ ) l+2−k ∧ u 3 = (ω ′ ) l+2−k ∧ u 3 + (l + 1 − k)(ω ′ ) l−k+1 ∧ u 0 = 0.
Thus (ω ′ ) l−k ∧ u 0 = 0, which implies u 0 = 0. Now we know that u → ω l+1−k ∧ u is an injection, thus an isomorphism since dim ∧ k V * = dim ∧ 2n−k V * .
The key notion in these notes is the following: Definition 1.2. We call u ∈ ∧ k V * a primitive form if k ≤ n and ω n−k+1 ∧ u = 0.
The following Lefschetz decomposition theorem follows directly from Theorem 1.2 (see the proof of Theorem 2.1 in the next section). Theorem 1.3 (Lefschetz decomposition formula). Every u ∈ ∧ k V * has a unique decomposition as follows:
(1.3) u = ω r ∧ u r , ω r := ω r r! ,
where each u r is a primitive (k − 2r)-form.
By the above theorem, it is enough to study the symplectic star operator on ω r ∧ u, where u is primitive.
Theorem 1.4. If u is a primitive k-form then * s (ω r ∧ u) = (−1) k+···+1 ω n−k−r ∧ u.
The above theorem implies * 2 s = 1. We shall use a symplectic analogy of the Berndtsson lemma (see Lemma 3.6.10 in [3]) to prove it. Definition 1.3. u ∈ ∧ k V * is said to be an elementary form if there exists a base, say
{e * 1 , f * 1 ; · · · ; e * n , f * n }, of V * such that ω = n j=1 e * j ∧ f * j , u = e * 1 ∧ · · · ∧ e * k .
Lemma 1.5 (Berndtsson lemma). The space of primitive forms is equal to the linear space spanned by elementary forms.
Proof.
Since
ω n−k+1 = j 1 <···<j n−k+1 e * j 1 ∧ f * j 1 ∧ · · · ∧ e * j n−k+1 ∧ f * j n−k+1 ,
we know that ω n−k+1 ∧ u = 0 if u is an elementary k-form. Thus every elementary form is primitive. Let us prove the other side by induction on n. Notice that the lemma is true if n = 1. Assume that it is true for n ≤ l, l ≥ 1. We whall prove that it is also true for n = l + 1. With the notation in the proof of Theorem 1.2, ω l−k+2 ∧ u = 0 is equivalent to
(ω ′ ) l+2−k ∧u 0 = (ω ′ ) l+2−k ∧u 1 = (ω ′ ) l+2−k ∧u 2 = (ω ′ ) l+2−k ∧u 3 +(l+2−k)(ω ′ ) l−k+1 ∧u 0 = 0,
which is equivalent to the ω ′ -primitivity of u 1 , u 2 , u 3 and (l + 2 − k)u 0 + ω ′ ∧ u 3 . Now it suffices to show that
u ′ := u 3 ∧ ((l + 2 − k)e * l+1 ∧ f * l+1 − ω ′ )
is a linear combination of elementary forms. Since u 3 is ω ′ -primitive, by the induction hypothesis, we can assume that
u 3 = e * 1 ∧ · · · ∧ e * k−2 . Thus u ′ = l j=k−1 e * 1 ∧ · · · ∧ e * k−2 ∧ (e * l+1 ∧ f * l+1 − e * j ∧ f * j ).
Now it suffices to show that if n = 2 then e * 1 ∧ f * 1 − e * 2 ∧ f * 2 is a linear combination of elementary forms. Notice that
e * 1 ∧ f * 1 − e * 2 ∧ f * 2 = (e * 1 + e * 2 ) ∧ (f * 1 − f * 2 ) + e * 1 ∧ f * 2 + f * 1 ∧ e * 2 . It is clear that e * 1 ∧ f * 2 and f * 1 ∧ e * 2 are elementary. (e * 1 + e * 2 ) ∧ (f * 1 − f * 2 ) is also elementary since we can write ω = (e * 1 + e * 2 ) ∧ f * 1 + e * 2 ∧ (f * 2 − f * 1 ). The proof is complete.
We shall also use the following Lemma from [16].
Lemma 1.6 (Guillemin Lemma). Assume that (V, ω) = (V 1 , ω (1) ) ⊕ (V 2 , ω (2) ). Then * s (u ∧ v) = (−1) k 1 k 2 * 1 s u ∧ * 2 s v, u ∈ ∧ k 1 V * 1 , v ∈ ∧ k 2 V * 2 , where * 1 s and * 2 s are symplectic star operators on V 1 and V 2 respectively. Proof. For every a ∈ ∧ k 1 V * 1 , b ∈ ∧ k 2 V * 2 , we have a ∧ b ∧ (−1) k 1 k 2 * 1 s u ∧ * 2 s v = ω −1 (a ∧ u, b ∧ v)ω n ,u = e * 1 ∧ · · · ∧ e * k . Consider V = Span{e * j , f * j } 1≤j≤k ⊕ Span{e * k+1 , f * k+1 } ⊕ · · · ⊕ Span{e * n , f * n } and write * s = * ≤k s ⊕ * k+1 s ⊕ · · · ⊕ * n s . Since * j s (1) = e * j ∧ f * j , * j s (e * j ∧ f * j ) = 1, ∀ k + 1 ≤ j ≤ n, by the Guillemin lemma, we have * s (e * k+1 ∧ f * k+1 ∧ · · · ∧ e * k+r ∧ f * k+r ∧ u) = e * k+r+1 ∧ f * k+r+1 ∧ · · · ∧ e * n ∧ f * n ∧ * ≤k s u, which implies * s (ω r ∧ u) = ω n−k−r ∧ * ≤k s u. Since * ≤k s = * 1 s ⊕ · · · ⊕ * k s and * j s e * j = −e * j , ∀ 1 ≤ j ≤ k, the Guillemin lemma gives * ≤k s u = (−1) k−1 (−e * 1 ) ∧ * ≤(k−1)
s (e * 2 ∧ · · · ∧ e * k ) = · · · = (−1) k+···+1 u, the proof is complete. We have:
Proposition 1.7. If u is a primitive k-form then Λ(L r u) = (n − k − r + 1)L r−1 u, B(L r u) = (k + 2r − n)L r u, for every 0 ≤ r ≤ n − k + 1. Proof. Put c = (−1) k+···+1 , then L * s (L r u) = cL(L n−k−r u) = (n − k − r + 1)cL n−k−r+1 u = (n − k − r + 1) * s (L r−1 u),
which gives the first identity. The second follows directly from the first. Now let us consider another structure on a linear space, which can be used to define an inner product structure on (V, ω). If J is an almost complex structure on V then
J(v)(u) := v(Ju), ∀ u ∈ V, v ∈ V * ,
defines an almost complex structure on V * . Definition 1.7. We call
J(v 1 ∧ · · · ∧ v k ) := J(v 1 ) ∧ · · · ∧ J(v k ),
the Weil operator on ⊕ 0≤k≤2n ∧ k V * .
Since the eigenvalues of J are ±i, its eigenvectors lie in C ⊗ V * . Put
E i := {u ∈ C ⊗ V * : J(u) = iu}, E −i := {u ∈ C ⊗ V * : J(u) = −iu}, we know that E i = {u − iJu : u ∈ V * }, E −i = {u + iJu : u ∈ V * }. and C ⊗ V * = E i ⊕ E −i . Put ∧ p,q V * := (∧ p E i ) ∧ (∧ q E −i ).
Then we have C ⊗ (∧ k V * ) = ∧ k (C ⊗ V * ) = ⊕ p+q=k ∧ p,q V * , and Ju = i p−q u, ∀ u ∈ ∧ p,q V * . We call ∧ p,q V * the space of (p, q)-forms.
Proposition 1.8. An almost complex structure J on (V, ω) is compatible with ω iff (α, β) := ω −1 (α, Jβ),
defines a Hermitian inner product structure on ∧ p,q V * , 0 ≤ p, q ≤ n.
Proof. Assume that J is compatible with ω. Then
T ω (Ju)(v) = ω(v, Ju) = −ω(Jv, u) = −J(T ω u)(v), u, v ∈ V, Thus T ω • J = −J • T ω . Now put a = T ω (u), b = J(T ω (v)) = −T ω (Jv).
Then
ω(v, Ju) = T ω (Ju)(v) = −(Ja)(v) = −a(Jv) = a(T ω −1 (b)), thus ω(v, Ju) = T ω −1 (b)(a) = ω −1 (a, b) = ω −1 (T ω u, J(T ω v)),
which gives the proposition.
Definition 1.8. The Hodge star operator * : ∧ p,q V * → ∧ n−q,n−p V * is defined by u ∧ * v = (u, v)ω n .
The above proposition gives
* = * s • J = J • * s . 1.2. Application in complex geometry. Let (X, ω) be an n-dimensional complex man- ifold with a Hermitian form (smooth positive (1, 1)-form) ω. Let (E, h E )
be a holomorphic vector bundle over X with a smooth Hermitian metric h E . Let us denote by V k the space of E-valued k-forms with compact support on X. The following theorem is a direct consequence of Theorem 1.2.
Theorem 1.9 (Hard Lefschetz theorem). For each 0 ≤ k ≤ n,
(1.4) u → ω n−k ∧ u, u ∈ V k ,
defines an isomorphism between V k and V 2n−k . Definition 1.9. We call an E-valued k-form, say u, on X a primitive form if k ≤ n and ω n−k+1 ∧ u ≡ 0. Now we have the following analogy of Theorem 1.3:
Theorem 1.10 (Lefschetz decomposition formula). Every E-valued k-form u on X has a unique decomposition as follows:
(1.5) u = ω r ∧ u r , ω r := ω r r! , where each u r is an E-valued primitive (k − 2r)-form.
Let {e α } be a local holomorphic frame of E, then
||u|| 2 := X h E (e α , e β )u α ∧ * ū β , u := u α ⊗ e α .
defines a Hermitian inner product structure on V k , we call it (ω, J, h E )-metric on V k .
Definition 1.10. The Hodge star operator on V k is defined by * u = ( * u α ) ⊗ e α , u := u α ⊗ e α . Put u α ⊗ e α , u β ⊗ e β h E := h E (e α , e β )u α ∧ū β .
Then we have
||u|| 2 = X {u, * u} h E .
The Hodge-Riemann bilinear relation is a direct consequence of Theorem 1.4 and * = J • * s .
Theorem 1.11 (Hodge-Riemann bilinear relation). If u is an E-valued primitive (p, q)- form then its (ω, J, h E )-norm satisfies (1.6) ||u|| 2 = X {u, ω n−k ∧ Iu} h E , Iu := (−1) k+···+1 i p−q u, k := p + q.
Lefschetz bundle
Definition 2.1. Let V = ⊕ 2n k=0 V k be a direct sum of complex vector bundles over a smooth manifold M . Let L be a smooth section of End(V ). We call (V, L) a Lefschetz bundle if L(V l ) ⊂ V l+2 , ∀ 0 ≤ l ≤ 2(n − 1), L(V 2n−1 ) = L(V 2n ) = 0, and each L k : V n−k → V n+k , 0 ≤ k ≤ n, is an isomorphism. Definition 2.2. Let (V, L) be a Lefschetz bundle. u ∈ V k is said to be primitive if k ≤ n and L n−k+1 u = 0.
Theorem 2.1. Let (V, L) be a Lefschetz bundle. Then every u ∈ V k has a unique decomposition as follows:
(2.1) u = L r u r , L r := L r r! .
where each u r is a primitive form in V k−2r .
Proof. We can assume that k ≤ n since we have the isomorphism L k : V n−k → V n+k . Notice that the theorem is trivial if k = 0, 1. Assume that 2 ≤ k ≤ n. The isomorphism
L n−k+2 : V k−2 → V 2n−k+2 ,
givesû ∈ V k−2 such that L n−k+2û = L n−k+1 u. Put u 0 = u − Lû, we know that u 0 is primitive and u = u 0 + Lû. Considerû instead of u, we haveû = u 1 + Lũ, where u 1 is primitive. By induction, we know that u can be written as
u = L r u r ,
where each u r ∈ V k−2r is primitive. For the uniqueness part, assume that
0 = j r=0 L r u r ,
where each u r ∈ V k−2r is primitive. Then we have
0 = L n−k+j j r=0 L r u r = L n−k+j L j u j ,
which gives u j = 0. By induction on j we know that all u r = 0.
Definition 2.3. We call the following C-linear map * s : V → V defined by * s (L r u) := (−1) k+···+1 L n−r−k u,
where u ∈ V k is primitive, the Lefschetz star operator on V .
Notice that * 2 s = 1. We know from the last section that the Lefschetz star operator is a generalization of the symplectic star operator.
Definition 2.4. Put Λ = * −1 s L * s , B := [L, Λ].
We call (L, Λ, B) the sl 2 -triple on (V, L) (Proposition 1.7 is also true for general Lefschetz bundle).
Variation of Lefschetz star operator
3.1. Main theorem. Our main theorem is a generalization of the main result in [32].
* s D * s (L r u) = D(L r u) + [Λ, θ](L r u), where u is a primitive k-form. Since [L, θ] = 0, we have D * s (L r u) = cD(L n−r−k u) = c(L n−r−k−1 θu + L n−r−k Du), c := (−1) k+···+1 .
Step 1 : Since u is primitive, we have L n−k+1 θu = θL n−k+1 u = 0, which implies that the primitive decomposition of θu contains at most three terms. Thus we can write
(3.1) θu = a + Lb + L 2 c, where a, b, c are primitive, which gives * s D * s (L r u) = −L r−1 a + M L r b − M (M + 1)L r+1 c + c * s L n−r−k Du, M := n − r − k.
Step 2 : Since
(3.2) θL r u = L r (a + Lb + L 2 c), Proposition 1.7 gives ΛθL r u = (M − 1)L r−1 a + (r + 1)M L r b + (r + 1)(r + 2)(M + 1)L r+1 c, and θΛL r u = (M + 1)θL r−1 u = (M + 1)L r−1 (a + Lb + L 2 c). Thus (3.3) [Λ, θ]L r u = −2L r−1 a + (M − r)L r b + (2r + 2)(M + 1)L r+1 c.
Step 3 : Put
A := * s D * s (L r u) − [Λ, θ]L r u, B := D(L r u). We have B = L r−1 θu + L r Du = L r−1 a + rL r b + r(r + 1)L r+1 c + L r Du.
Since the first two steps gives
A = L r−1 a + rL r b − (M + 1)(M + 2r + 2)L r+1 c + c * s L n−r−k Du, we have A − B = c * s L n−r−k Du − L r Du − [(M + 1)(M + 2r + 2) + r(r + 1)]L r+1 c.
Step 4 : Primitivity of u implies
0 = D(L n−k+1 u) = θL n−k u + L n−k+1 Du. Notice that θL n−k u = L n−k (a + Lb + L 2 c) = L n−k L 2 c. Thus L n−k+1 (Du + (n − k + 1)Lc) = 0, which implies the primitivity of (3.4) v := Du + (n − k + 1)Lc. Now we have c * s L n−r−k Du = c * s L n−r−k (v − (n − k + 1)Lc) = L r v + (n − k + 1)(M + 1)L r+1 c. and (3.5) L r Du = L r (v − (n − k + 1)Lc) = L r v − (n − k + 1)(r + 1)L r+1 c.
Thus c * s L n−r−k Du − L r Du can be written as
(3.6) [(n − k + 1)(M + 1) + (n − k + 1)(r + 1)]L r+1 c.
Step 5 : Since our formula is equivalent to A = B, by step 3 and 4, it is enough to prove (n − k + 1)(M + 1) + (n − k + 1)(r + 1) = (M + 1)(M + 2r + 2) + r(r + 1), which is true (recall that M = n − r − k).
Remark: If we write D = dt j ⊗ D j ,
where {t j } 1≤j≤m are smooth local coordinates. Then our main theorem is equivalent to On the other hand, notice that
(3.7) * −1 s D j * s u = D j u + [Λ, θ j ]u, θ j := [D j , L], for every 1 ≤ j ≤ m. 3.2. Corollary.− 1 2 [Λ, [Λ, θ]] = ΛθΛ − 1 2 (Λ 2 θ + θΛ 2 ). Now Λ 2 θL r u = Λ 2 L r (a + Lb + L 2 c).
thus Proposition 1.7 gives
Λ 2 θL r u = (M − 1)M L r−2 a + (r + 1)M (M + 1)L r−1 b +(r + 1)(r + 2)(M + 1)(M + 2)L r c.
By a similar argument, we also get The proof is complete.
θΛ 2 L r u = (M + 1)(M + 2)(L r−2 a + (r − 1)L r−1 b + r(r − 1)L r c),
The following proposition can be seen as a generalization of formula 1 in [26]. Since * 2 s = 1, the proposition follows.
T -Hodge theory
4.1. Timorin's theorem. Timorin's theorem [30] is a mixed linear version of the Hodge-Riemann bilinear relation. Let (V, ω, J) be a 2n-dimensional real vector space with compatible pair (ω, J). Let α 0 , α 1 , · · · , α n be J-compatible symplectic forms on V . Put T k := α k ∧ · · · ∧ α n , 0 ≤ k ≤ n, T n+1 := 1.
and
V k := C ⊗ ∧ k V * .
Timorin introduced the following definition in [30]. Proof. We claim that MHR-n follows MHL-n and usual Hodge-Riemann bilinear relation. Notice that MHL-n below implies that the space, say P k , of T k -primitive forms has constant dimension dim V k − dim V k−2 and Q is non-degenerate on P k , consider
α t j := (1 − t)α j + tω, 0 ≤ t ≤ 1,
then the positivity of Q at t = 0 follows from the positivity of Q at t = 1 (the usual Hodge-Riemann bilinear relation). Then MHR-(n-1) gives
Q j (u) := (−1) k+···+1 T k+2 | H j ∧ u| H j ∧ Ju| H j ≥ 0.
If u ∧ T k+1 = 0 then
0 = (−1) k+···+1 T k+1 ∧ u ∧ Ju = Q j (u) ∧ (iσ j ∧σ j ),
which implies each Q j (u) = 0. Thus u| H j = 0 for every 1 ≤ j ≤ n, which gives u∧α k+1 = 0, thus u = 0 since deg u ≤ n − 1.
4.2.
Hodge star operator on V T . Let (E, h E ) be a holomorphic vector bundle over an n-dimensional complex manifold (X, ω). Denote by V p,q the space of smooth E-valued (p, q)-forms with compact support on X. Put
V := ⊕ 0≤p,q≤n V p,q , V k := ⊕ p+q=k V p,q .
Fix 0 ≤ m ≤ n and smooth positive (1, 1)-forms α m+1 , · · · , α n on X. Consider
f T : u → T ∧ u, u ∈ V.
where T := α m+1 ∧ · · · ∧ α n , T := 1, if m = n. We call the Hodge theory on
Imf T := {T ∧ u : u ∈ V },
the T -Hodge theory. Put
V p,q T := f T (V p,q ), V T := ⊕ 0≤p,q≤n V p,q T , V k T := ⊕ p+q=k V p,q T . We have V T = ⊕ 2m k=0 V k T = ⊕ 0≤p,q≤m V p,q T , and L : u → ω ∧ u, u ∈ V T , maps V p,q T to V p+1,q+1
T . Timorin's mixed hard-Lefschetz theorem gives:
Theorem 4.3. For every 0 ≤ k ≤ m, L m−k : u → u ∧ ω m−k , defines an isomorphism from V k T to V 2m−k T .
Proof. By Timorin's theorem, we know that
u → T ∧ u, u ∈ V k
is injective. Thus f T defines an isomorphism from V k to V k T . Again by Timorin's theorem, we have the following isomorphism
A : u → u ∧ T ∧ ω m−k , from V k to V 2n−k . Thus L m−k = A • f −1
T is an isomorphism. Definition 4.2. We call u ∈ V k T a primitive k-form if k ≤ m and L m−k+1 u = 0. The proof of Theorem 2.1 implies:
Theorem 4.4. Every u ∈ V k
T has a unique decomposition as follows:
(4.1) u = L r u r , L r := L r r! .
where each u r is a primitive form in V k−2r T . Definition 4.3. We call the following C-linear map * s :
V T → V T defined by * s (L r u) := (−1) k+···+1 L m−r−k u,
where u ∈ V k T is primitive, the Lefschetz star operator on V T . In case m = n, the Lefschetz star operator above is just the symplectic star operator. Since J commutes with f T , the Weil-operator is also well defined on V T , we shall also denote it by J. Timorin's mixed Hodge-Riemann bilinear relation gives:
Theorem 4.5. Put (u, v) := X {f −1 T (u), * v} h E , u, v ∈ V k T , 0 ≤ k ≤ m, and (u, v) := X {u, f −1 T ( * v)} h E , u, v ∈ V k T , m ≤ k ≤ 2m.
Then (u, v) is a Hermitian inner product on V T .
Definition 4.6. Let us define (u, v) T such that
{f −1 T (u), * v} h E = (u, v) T ω m ∧ T, u, v ∈ V k T , 0 ≤ k ≤ m, and {u, f −1 T ( * v)} h E = (u, v) T ω m ∧ T, u, v ∈ V k T , m ≤ k ≤ 2m.E, h E ). Assume that T is d-closed, then Du ∈ V T if u ∈ V T . Let D * , ∂ * , (∂ E ) * be the adjoint of D, ∂, ∂ E : V T → V T ,
respectively. We shall use Theorem 3.1 and Proposition 3.4 to prove the following Tgeometry generalization of the Demailly-Griffiths-Kähler identity (see page 307 in [11]).
Theorem 4.6. If T is d-closed then [∂ * , L] = i(∂ E + [Λ, [∂ E , L]]) on V T .
Proof. Since * = * s • J = J • * s and * 2 s = 1, we have
∂ * = − * ∂ E * = (−1) k+1 i * s ∂ E * s = (−1) k+1 i n j=1 ( * −1 s ∂ E j * s )( * −1 s σ j * s ), on V k T , where σ j := dz j ∧. Thus [∂ * , L] = (−1) k+1 i n j=1 ( * −1 s ∂ E j * s )( * −1 s σ j * s )L − L( * −1 s ∂ E j * s )( * −1 s σ j * s ). Now Proposition 3.4 implies [ * −1 s σ j * s , L] = (−1) k+1 σ j , thus [∂ * , L] = (−1) k+1 i n j=1 ( * −1 s ∂ E j * s )((−1) k+1 σ j + L * −1 s σ j * s ) − L( * −1 s ∂ E j * s )( * −1 s σ j * s ) = i n j=1 ( * −1 s ∂ E j * s )σ j + (−1) k+1 [ * −1 s ∂ E j * s , L]( * −1 s σ j * s ). (3.8) gives [ * −1 s ∂ E j * s , L] = −θ j , where θ j := [∂ E j , L], thus our main theorem gives [∂ * , L] = i∂ E + i [Λ, θ j ]σ j + (−1) k θ j ( * −1 s σ j * s ).
Now it suffices to show
(4.2) [Λ, θ j ]σ j + (−1) k θ j ( * −1 s σ j * s ) = [Λ, [∂ E , L]].
Since
θ j σ j = [∂ E , L], we have (4.3) [Λ, θ j ]σ j = Λ[∂ E , L] − θ j Λσ j . Proposition 3.4 gives (4.4) (−1) k θ j ( * −1 s σ j * s ) = θ j Λσ j − θ j σ j Λ.
Thus the left hand side of (4.2) can be written as
(4.5) Λ[∂ E , L] − [∂ E , L]Λ,
which equals the right hand side of (4.2).
on V T , where D c := i∂ − i∂ E .
The above Kähler identity gives the following Bochner-Kodaira-Nakano identity in Tgeometry:
Theorem 4.8. Assume that both T and ω are d-closed. Then Proof. Recall that a differential operator of order l is said to be elliptic if σ l (D)(x, ξ) is invertible for every x ∈ M and every non-zero ξ ∈ T x M , where
✷ ∂ = ✷ ∂ E + [iΘ(E, h E ), Λ], on V T , where Θ(E, h E ) := D 2 , ✷ ∂ := ∂ * ∂ + ∂∂ * , ✷ ∂ E := ∂ E (∂ E ) * + (∂ E ) * ∂ E ; moreover ✷ D c = ✷ D = ✷ ∂ + ✷ ∂ E ,σ l (D)(x, ξ)u := lim t→∞ t −l e −itf D(e itf u)(x),
and f is a smooth function near x such that df (x) = ξ. We have
σ 2 (✷ ∂ )(x, ξ) = (∂f ) * (∂f ) + (∂f )(∂f ) * , σ 2 (✷ ∂ E )(x, ξ) = (∂f ) * (∂f ) + (∂f )(∂f ) * .
Theorem 4.8 gives
σ 2 (✷ ∂ )(x, ξ) = σ 2 (✷ ∂ E )(x, ξ).
Thus it suffices to prove that if u ∈ V k T , k = m, satisfies (∂f ) ∧ u = (∂f ) ∧ u = (∂f ) * u = (∂f ) * u = 0, then u(x) = 0. It is clear that we can assume that u ∈ V p,q T , p + q = k. Consider * u if k > m, one may assume further that k < m. Moreover, by a C-linear change of local coordinate, one may assume that ∂f = dz 1 , ∂f = dz 1 . Let
u = j r=0 ω r ∧ u r , be the Lefschetz decomposition of u. Then * u = (−1) k+···+1 i p−q j r=0 (−1) r ω m−k+r ∧ u r .
The lemma below gives
u r ∧ dz 1 = u r ∧ dz 1 = 0, ∀ 0 ≤ r ≤ j, hence f −1 T (u r ) ∧ dz 1 = f −1 T (u r ) ∧ dz 1 = 0
, since the degree of each u r is no bigger than m − 1. Thus each f −1 T (u r ) can be written as
dz 1 ∧ dz 1 ∧ v r .
Timorin's Hodge-Riemann bilinear relation implies that if f −1 T (u r ) = 0 then f −1 T (u r ) ∧ f −1 T (u r ) = 0. But obviously (dz 1 ∧dz 1 ) 2 = 0 gives f −1 T (u r )∧f −1 T (u r ) = 0. Thus f −1 T (u r ) = 0 and u = 0. The proof is complete.
Lemma 4.10. u r ∧ dz 1 = u r ∧ dz 1 = 0, ∀ 0 ≤ r ≤ j. Proof. dz 1 ∧ u = dz 1 ∧ * u = 0 gives A := j r=0 ω r ∧ (dz 1 ∧ u r ) = 0, B := j r=0 (−1) r ω m−k+r ∧ (dz 1 ∧ u r ) = 0.
The lemma is true if j = 0. Assume the lemma is true for 0 ≤ j ≤ l. We claim that it is true for j = l + 1. In fact, ω m−k ∧ A + (−1) l+1 C l m−k+l B = 0 gives
C := (ω m−k ∧ ω l+1 + C l m−k+l ω m−k+l+1 ) ∧ (dz 1 ∧ u l+1 ) + l−1 r=0 C r ω m−k+r ∧ (dz 1 ∧ u r ) = 0.
Since each u r is primitive, we know that
ω l ∧ C = ω r ∧ (ω m−k ∧ ω l+1 + C l m−k+l ω m−k+l+1 ) ∧ (dz 1 ∧ u l+1 ),
which gives dz 1 ∧ u l+1 = 0 by Timorin's hard Lefschetz theorem. Thus the lemma is also true for j = l + 1 by the induction hypothesis. Theorem 5.1. Let (X,ω) be an n-dimensional complete Kähler manifold with finite volume. Let α 1 , · · · , α n be smooth d-closed semi-positive (1, 1)-forms such that α j ≤ω on X for every 1 ≤ j ≤ n. Assume that n ≥ 2. Put T := α 3 ∧ · · · ∧ α n , T := 1, if n = 2.
Then X α 1 ∧ α 2 ∧ T 2 ≥ X α 2 1 ∧ T X α 2 2 ∧ T .
Remark: In case (X,ω) is compact Kähler, the above theorem is just the Khovanskii-Teissier inequality. The classical Alexandrov-Fenchel inequality follows from the above theorem with X = R n × (R/Z) n , see [34] for the proof.
Proof of Theorem 5.1. We shall follow the method in [34]. Consider α j + ǫω instead of ω, one may assume that
(5.1)ω C ≤ α j ≤ Cω,
for every 1 ≤ j ≤ n, where C is a fixed positive constant.
Step 1 : By Proposition 2.6 in [34], it suffices to show that
ψ : t → − log X ω 2 ∧ T, ω := tα 1 + (1 − t)α 2 ,
is convex on (0, 1).
Step 2 : Consider the trivial line bundle ker d := U × C over
U := {t + is : 0 < t < 1, s ∈ R},
with metric
h(1, 1)(t + is) := e −ψ(t) = X ω 2 ∧ T,
Then ψ is convex iff the curvature of (ker d, h) is positive.
Step 3 : Look at ker d as a holomorphic subbundle of
A := U × A, where A := {f ∈ C ∞ (X, C) : X |f | 2ω n < ∞}.
h extends to a metric on A as follows:
h(f, g)(t + is) := X fḡ ω 2 ∧ T = X {f, * g}, ∀ f, g ∈ A.
Thus the Chern curvature operator of (A, h) can be written as Step 4 : Denote by Θ K tt the Chern curvature operator of (ker d, h). By the subbundle curvature formula, we have
ψ tt = h(Θ K tt 1, 1) h(1, 1) = h(Θ A tt 1, 1) h(1, 1) − h((Λθ) ⊥ , (Λθ) ⊥ ) h(1, 1) , where Λθ = [Λ, θ]1 and (Λθ) ⊥ := (Λθ) − h(Λθ, 1) h(1, 1) .
is the L 2 -minimal solution of d(·) = d(Λθ).
Step 5 : Theorem 4.7 gives d(Λθ) = (d c ) * θ. If u is a smooth one-form with compact support on X then |((d c ) * θ, u)| 2 = |(θ, d c u)| 2 ≤ ||θ|| 2 ||d c u|| 2 , moreover, theorem 4.8 gives ||d c u|| 2 ≤ ||du|| 2 + ||d * u|| 2 , thus Hörmander's L 2 -theory (here we use (5.1) and the completeness of (X,ω), see the proof of Lemma 5.2 in [34] for the details) implies h((Λθ) ⊥ , (Λθ) ⊥ ) = ||(Λθ) ⊥ || 2 ≤ ||θ|| 2 = h(Θ A tt 1, 1), where the last identity follows from Θ A tt = [θ * , θ]. Thus ψ tt ≥ 0. 5.2. Dinh-Nguyên's theorem. Let (X, ω) be an n-dimensional compact Kähler manifold. Let α 1 , · · · , α n be smooth Kähler forms on X. Let A p,q be the space of smooth (p, q)-forms on X and A k be the space of real-valued smooth k-forms on X. We have the Dolbeault cohomology group (a C-vector space in fact) H p,q (X, C) := A p,q ∩ ker ∂ ∂A p,q−1 , and the de Rham cohomology group (an R-vector space)
H k (X, R) := A k ∩ ker d dA k−1 .
The following theorem depends on the theory of elliptic operators, see [17], and Theorem 4.8 (when (E, h E ) is trivial and T = 1).
Theorem 5.2 (Hodge-Dolbeault-de Rham theorem). Let (X, ω) be an n-dimensional compact Kähler manifold. Let ✷ ∂ be the ∂-Laplacian with respect to the (ω, J)-metric. Then each H p,q (X, C) is C-linear isomorphic to H p,q := A p,q ∩ ker ✷ ∂ which is finite dimensional. Let ✷ d be the d-Laplacian with respect to the (ω, J)-metric.
Then each H k (X, R) is R-linear isomorphic to H k := A k ∩ ker ✷ d . Moreover, H k + iH k = ⊕ p+q=k H p,q .
The above theorem implies that every class in H p,q (X, C) has a d-closed representative. Fix 0 ≤ m ≤ n, put T := α m+1 ∧ · · · ∧ α n , T := 1 if m = n.
Definition 5.2. We call a class [u] in H p,q (X, C), p + q = m, a T -primitive class if [u ∧ T ∧ ω] = 0 in H p+n−m+1,q+n−m+1 (X, C).
Theorem 5.3 (Dinh-Nguyên's theorem [12]). Assume that [u] ∈ H p,q (X, C), p + q = m, is a non-zero T -primitive class then
X (−1) m+···+1 u ∧ Ju ∧ T > 0, where u is a d-closed representative of [u], Ju = i p−q u.
Proof. The case m = 0 is trivial. Assume that m ≥ 1. Let u be a d-closed representative of [u]. Since [u] is T -primitive, there exists v ∈ A p+n−m+1,q+n−m such that
u ∧ T ∧ ω = ∂v.
Let us look at v as an element in V p+1,q T . Timorin's mixed hard Lefschetz theorem gives v ′ ∈ V p,q−1 T such that v ′ ∧ ω = v. By Theorem 4.9, ✷ ∂ are elliptic on V k T for every k = m. Thus the elliptic operator theory (see [17]
) gives v ′ = v ′ h + ✷ ∂ f ′ , v ′ h ∈ V m−1 T ∩ ker ✷ ∂ , f ′ ∈ V m−1 T ,
The Kähler identity in T -Hodge theory implies that ✷ ∂ commutes with L and Λ, thus
v h := v ′ h ∧ ω ∈ V m+1 T ∩ ker ✷ ∂ , and v = v h + ✷ ∂ f, f := ω ∧ f ′ . Since ∂v is ∂-closed, we have 0 = ∂∂v = ∂∂ * ∂∂f = 0.
Thus ||∂ * ∂∂f || 2 = (∂∂ * ∂∂f, ∂∂f ) = 0, which gives ∂ * ∂∂f = ∂∂ * ∂f = 0. The Kähler identity in T -Hodge theory implies [∂ * ∂∂, L] = 0.
Thus we have ω ∧ ∂∂ * ∂f ′ = 0, which gives ∂v = ω ∧ ∂v ′ = ω ∧ ∂∂∂ * f ′ . Let us write ∂ * f ′ = T ∧ g, thus (u + ∂∂g) ∧ T ∧ ω = 0.
Thus Timorin's mixed Hodge-Riemann bilinear relation gives
X (−1) m+···+1 (u + ∂∂g) ∧ J(u + ∂∂g) ∧ T ≥ 0,
where the equality holds iff u + ∂∂g ≡ 0. Stokes' theorem implies
X (−1) m+···+1 u ∧ Ju ∧ T > 0 = X (−1) m+···+1 (u + ∂∂g) ∧ J(u + ∂∂g) ∧ T,
thus the theorem follows. 5.3. Curvature of higher direct images. We shall use the following setup:
(1) π : X → B is a proper holomorphic submersion from a complex manifold X to another complex manifold B, each fiber X t := is an n-dimensional compact complex manifold;
(2) E is a holomorphic vector bundle over X , E t := E| Xt ;
(3) ω is a smooth (1, 1)-form on X that is positive on each fiber, ω t := ω| Xt ;
(4) h E is a smooth Hermitian metric on E, h Et := h E | Et .
For each t ∈ B, let us denote by A p,q (E t ) the space of smooth E t -valued (p, q)-forms on X t . Let us recall the following definition in [7]: Definition 5.3. Let V := {V t } t∈B be a family of C-vector spaces over B. Let Γ be a C ∞ (B)-submodule of the space of all sections of V . We call Γ a smooth quasi-vector bundle structure on V if each vector of the fiber V t extends to a section in Γ locally near t.
Consider
A p,q := {A p,q (E t )} t∈B . Denote by A p,q (E) the space of smooth E-valued (p, q)-forms on X . Let us define Γ p,q := {u : t → u t ∈ A p,q (E t ) : ∃ u ∈ A p,q (E), u| Xt = u t , ∀ t ∈ B}.
We call u above a smooth representative of u ∈ Γ p,q . We know that each Γ p,q defines a quasi-vector bundle structure on A p,q .
Definition 5.4. Let (V, Γ) = ⊕ 2n k=0 (V k , Γ k )
be a direct sum of quasi vector bundles over a smooth manifold B. Let L be a section of End(V ). We call (V, Γ, L) a Lefschetz quasi vector bundle if
L(Γ l ) ⊂ Γ l+2 , ∀ 0 ≤ l ≤ 2(n − 1), L(Γ 2n−1 ) = L(Γ 2n ) = 0,
and each L k : Γ n−k → Γ n+k , 0 ≤ k ≤ n, is an isomorphism.
Same as before, one may define the Lefschetz star operator and the sl 2 -triple on a general Lefschetz quasi vector bundle. Consider (A, Γ) := ⊕ 2n k=0 (A k , Γ k ), (A k , Γ k ) := ⊕ p+q=k (A p,q , Γ p,q ) and define L ∈ End(A) such that
Lu(t) = ω t ∧ u t , ∀ u ∈ Γ.
Then the hard Lefschetz theorem implies that (A, Γ, L) is a Lefschetz quasi vector bundle. One may also define the notion of connection on a general quasi vector bundle, see [7]. Thus our main theorem is still true for general Lefschetz quasi vector bundles. We shall use the following connection on (A, Γ).
Definition 5.5. The Lie-derivative connection, say ∇ A , on (A, Γ) is defined as follows:
∇ A u := dt j ⊗ [d E , δ V j ]u + dt j ⊗ [d E , δV j ]u, u ∈ Γ,
where d E := ∂ + ∂ E denotes the Chern connection on (E, h E ) and each V j is the horizontal lift of ∂/∂t j with respect to ω.
Our main theorem implies: For bidegree reason, the connection, say D A , on each (V p,q , Γ p,q ) induced by ∇ A satisfies Then ∇ A − D A = dt j ⊗ κ j + dt j ⊗ κj.
D A u := dt j ⊗ [∂ E , δ V j ]u + dt j ⊗ [∂, δV j ]u, u ∈ Γ p,q . Thus (∇ A − D A )u = dt j ⊗ [∂, δ V j ]u + dt j ⊗ [∂ E , δV j ]u.
Definition 5.6. We call κ j u := [∂, δ V j ]u, the non-cohomological Kodaira-Spencer action of κ j on u ∈ Γ.
The fiberwise Hodge star operator * equals * s • J, recall that J is the Weil-operator Ju = i p−q u, u ∈ Γ p,q .
Thus Theorem 5.4 implies:
Proposition 5.5. If dω ≡ 0 then D A * = * D A , κ j * = − * κ j and κj * = − * κj. Theorem 5.6. If dω ≡ 0 then D A defines a Chern connection on each (A p,q , Γ p,q ) and each κj is the adjoint of κ j .
Proof. First part: Since the metric on A p,q is defined by (u, v) = Xt {u, * v}, thus the above proposition implies that D A preserves the metric. The fact that the square of the (0, 1)-part of D A vanishes follows from the usual Lie derivative identity, see [7] for the details. Thus D A is a Chern connection.
Second part: Let u ∈ Γ p,q , v ∈ Γ p−1,q+1 , then for bidegree reason, we have 0 = ∂/∂t j (u, v) = (κ j u, v) + (u, * −1 κj * v), which gives (κ j u, v) = (u, κj v) by the above proposition.
Remark: One may also prove the above theorem by a direct computation without using the Hodge star operator, see [25]. For other related results on the Lie-derivative connection, see [6], [14], [20], [21], [23], [24], [28], [29], [31].
The curvature of the Lie-derivative connection is
(∇ A ) 2 = [[d E , δ V j ], [d E , δV k ]]dt j ∧ dt k .
For bidegree reason, we have
(5.2) (D A ) 2 = (∇ A ) 2 − [κ j , κk]dt j ∧ dt k .
One may get a curvature formula of the higher direct image bundles using the above formula and the sub-bundle-quotient-bundle curvature formula, see [7] for the details.
Definition 5.7. We call u ∈ Γ p,q a holomorphic section of A p,q if each [∂, δV j ]u = 0, on fibers, i.e. the (0, 1)-part of D A vanishes on u.
Theorem 5.6 and (5.2) give:
Proposition 5.7. If dω = 0 and u is a holomorphic section of A p,q then i∂∂||u|| 2 ≥ −i((∇ A ) 2 u, u) + i ((κ * k u, κ * j u) − (κ j u, κ k u))dt j ∧ dt k . We shall show how to use the above proposition in a future publication [33].
Definition 1 . 4 .
14We call {L, Λ, B} the sl 2 -triple on ⊕ 0≤k≤2n ∧ k V * , where Lu := ω ∧ u, Λ := * −1 s L * s , B := [L, Λ]. We have ω −1 (Lu, v) = ω −1 (u, Λv). Hence Λ is the adjoint of L. Put
L r := L r /r!, L 0 := 1, L −1 := 0.
Definition 1 . 5 .
15We call a linear map J : V → V an almost complex structure on V if J(Ju) = −u for every u ∈ V .
Definition 1 . 6 .
16An almost complex structure J on (V, ω) is said to be compatible with ω if ω(u, Jv) = ω(v, Ju), ∀ u, v ∈ V, and ω(u, Ju) > 0 if u is not zero.
Theorem 3 . 1 .
31Let D be a degree preserving connection on a Lefschetz bundle (V, L). Put θ := [D, L]. If [L, θ] = 0 then * −1 s D * s = D + [Λ, θ]. Proof. By the Lefschetz decompostion theorem and * 2 s = 1, it suffices to prove
Corollary 3. 2 .
2With the same assumption in Theorem 3.1, we have
Since Λ = * −1 s L * s and θ = [D, L] = DL − LD, we have * −1 s θ * s = [ * −1 s D * s , Λ]. Our main theorem and (3.9) give * −1 s θ * s = [D + [Λ, θ], Λ] = [D, Λ] + 2 * −1 s θ * s , which gives (3.8).
Proof. For a primitive k-form u, we have θ * s (L r u) := cL n−r−k θu. By the proof of our main theorem, we can write θu = a + Lb + L 2 c, where a, b, c are primitive. Thus * s θ * s (L r u) = −L r−2 a + (M + 1)L r−1 b − (M + 1)(M + 2)L r c, M := n − r − k.
andΛθΛL r u = (M + 1)(M L r−2 a + r(M + 1)L r−1 b + r(r + 1)(M + 2)L r c).Thus − 1 2 [Λ, [Λ, θ]](L r u) equals −L r−2 a + (M + 1)L r−1 b − (M + 1)(M + 2)L r c = * s θ * s (L r u).
Proposition 3. 4 .
4Let (V, L) be a Lefschetz bundle. Let σ be a smooth degree one section of End(V ). If [L, σ] = 0 then * −1 s σ * s = (−1) k [Λ, σ] on V k . Proof. For a primitive k-form u, we have σ * s (L r u) := cL n−r−k σu. Since σ is degree one, we can write σu = a + Lb, where a, b are primitive. Thus * s σ * s (L r u) = (−1) k+1 (L r−1 a − (M + 1)L r b), M := n − r − k. On the other hand, Proposition 1.7 gives ΛσL r u = M L r−1 a + (r + 1)(M + 1)L r b. and σΛL r u = (M + 1)(L r−1 a + rL r b).Thus (−1) k [Λ, σ](L r u) = * s σ * s (L r u).
Definition 4. 1 .
1We call u ∈ V k a T k -primitive form if k ≤ n and u ∧ T k = 0.
Theorem 4. 1 (
1Timorin's mixed Hodge-Riemann bilinear relation, MHR-n). Let u be a non-zero T k -primitive form. Then Q(u) := (−1) k+···+1 T k+1 ∧ u ∧ Ju > 0.
Theorem 4. 2 (
2Timorin's mixed hard-Lefschetz theorem, MHL-n). For every 0 ≤ k ≤ n,u → u ∧ T k+1 , defines an isomorphism from V k to V 2n−k .Proof. Since MHR-1 is true, it suffices to show MHR-(n-1) implies MHL-n. Assume that u ∈ V k , k ≤ n − 1. If u ∧ T k+1 = 0 then u| H ∧ T k+1 | H = 0, for every hyperplane H. Thus if u ∧ T k+1 = 0 then u| H is T k+1 | H -primitive for every H. Let us write α k+1 = n j=1 iσ j ∧σ j , H j := ker σ j .
Definition 4 . 4 .
44Put Λ = * −1 s L * s , B := [L, Λ]. We call (L, Λ, B) the sl 2 -triple on V T .
Definition 4 . 5 .
45We call * := * s • J the Hodge star operator on V T .
.
We call (u, v) T the pointwise Hermitian inner product of u, v in V T . Remark: If m = n then T = 1 and (u, v) T is just the pointwise (ω, J, h E )-inner product. Moreover, our Hermitian metric in Theorem 4.5 is compatible with the current norm on V Kähler identity in T -Hodge theory. Let D := ∂ + ∂ E , be the Chern connection on (
Theorem 4. 7 .
7Assume that both T and ω are d-closed, then D * = [Λ, D c ], (D c ) * = [D, Λ],
.
on V T , where ✷ D := DD * + D * D, ✷ D c := D c (D c ) * + (D c ) * D c .Remark: If 1 ≤ m < n then ✷ D is elliptic on V In general, we don't have (∂f ) * (∂f ) + (∂f )(∂f ) * = |∂f | 2 T , but still we have the following theorem: Theorem 4.9. ✷ ∂ , ✷ ∂ E are elliptic on V k T for every k = m.
.
Alexandrov-Fenchel inequality. Definition 5.1. A Hermitian manifold (X,ω) is said to be complete if there exists a smooth function, say φ : X → [0, ∞), such that φ −1 ([0, c]) is compact for every c > 0 and |dφ|ω(x) ≤ 1, ∀ x ∈ X.
Θ
A tt := [ * −1 (∂/∂t) * , ∂/∂t]. Our main theorem gives * −1 (∂/∂t) * = ∂/∂t + [Λ, θ], θ := [∂/∂t, ω] = α 1 − α 2 , thus we have Θ A tt = [[Λ, θ], ∂/∂t] = [θ * , θ], by (3.8).
Theorem 5 . 4 .
54If dω ≡ 0 then the Lie-derivative connection ∇ A commutes with the Lefschetz star operator * s on the Lefschetz quasi vector bundle (A, Γ, L). Proof. By our main theorem, it suffices to prove [∇ A , L] = 0, i.e. [[d E , δ V j ], L] = 0. Notice that [d E , L] = dω and [δ V j , L]| Xt = (V j ⌋ω)| Xt ≡ 0, since each V j is horizontal. By the Jacobi identity, dω = 0 gives [∇ A , L] = 0.
Put κ j u := [∂, δ V j ]u, κju := [∂ E , δV j ]u.
which gives the lemma. Now we are able to prove Theorem 1.4. Proof of Theorem 1.4. By the Berndtsson lemma, we can assume that
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. Department Of, Sciences, NO-7491 TRONDHEIMNORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY. NORWAY E-mail address: [email protected] OF MATHEMATICAL SCIENCES, NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY, NO-7491 TRONDHEIM, NORWAY E-mail address: [email protected]
| []
|
[
"Semileptonic B Decays",
"Semileptonic B Decays"
]
| [
"Vera G Lüth \nSLAC National Laboratory Accelerator\nStanford University\nUSA\n"
]
| [
"SLAC National Laboratory Accelerator\nStanford University\nUSA"
]
| [
"Flavor Physics and CP Violation (FPCP 2012)"
]
| The following is an overview of the measurements of the CKM matrix elements |V cb | and |V ub | that are based on detailed studies of semileptonic B decays by the BABAR and Belle Collaborations and major advances in QCD calculations. In addition, a new and improved measurement ofHere D ( * ) refers to a D or a D * meson and ℓ is either e or µ. The results, R(D) = 0.440 ± 0.058 ± 0.042 and R(D * ) = 0.332 ± 0.024 ± 0.018, exceed the Standard Model expectations by 2.0σ and 2.7σ, respectively. Taken together, they disagree with these expectations at the 3.4σ level. The excess of events cannot be explained by a charged Higgs boson in the type II two-Higgs-doublet model. | null | [
"https://arxiv.org/pdf/1209.4674v2.pdf"
]
| 119,198,263 | 1209.4674 | cc2e10bcdc16a6ff622e1deb5fde9114845487ae |
Semileptonic B Decays
29 Sep 2012. May 21-25, 2012
Vera G Lüth
SLAC National Laboratory Accelerator
Stanford University
USA
Semileptonic B Decays
Flavor Physics and CP Violation (FPCP 2012)
Hefei, China29 Sep 2012. May 21-25, 20121
The following is an overview of the measurements of the CKM matrix elements |V cb | and |V ub | that are based on detailed studies of semileptonic B decays by the BABAR and Belle Collaborations and major advances in QCD calculations. In addition, a new and improved measurement ofHere D ( * ) refers to a D or a D * meson and ℓ is either e or µ. The results, R(D) = 0.440 ± 0.058 ± 0.042 and R(D * ) = 0.332 ± 0.024 ± 0.018, exceed the Standard Model expectations by 2.0σ and 2.7σ, respectively. Taken together, they disagree with these expectations at the 3.4σ level. The excess of events cannot be explained by a charged Higgs boson in the type II two-Higgs-doublet model.
I. INTRODUCTION
Over the past decade, the vast samples of B mesons recorded at the B Factories at KEK and SLAC have allowed detailed studies of semileptonic B decays. In the Standard Model (SM), these decays proceed via first-order weak interaction and are mediated by the W boson. Decays involving electrons and muons are expected to be free of non-SM contributions and are therefore well suited for the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |V cb | and |V ub |. They are fundamental parameters of the SM and have to be determined experimentally. Decays involving the higher mass τ lepton provide additional information on SM processes. They are also sensitive to non-SM contributions, for instance, from the exchange of a charged Higgs boson.
This presentation will combine a summary of the current status of the measurements of the CKM matrix elements |V cb | and |V ub | from the Belle and BABAR experiments with the first report on the observation by BABAR of an excess of events beyond the SM expectations in B → D ( * ) τ − ν τ decays [1]. Here D ( * ) refers to the ground state charm mesons, D and D * .
II. |Vcb| AND |Vub|
There are two experimental methods to determine |V cb | and |V ub |. The first is based on the study of exclusive semileptonic B decays in which the hadron is a D, D * , D * * , π, or ρ meson. The second is based on inclusive decays of the form B → Xℓν, where X refers to either X c or X u , that is, to any allowed hadronic final state with charm or without charm, respectively. To extract |V cb | or |V ub | from the measured partial decay rates, both approaches depend on calculations of hadronic contributions to the matrix element. Since the two methods rely on different experimental techniques and involve different theoretical approximations, they result in largely independent measurements of |V cb | and |V ub |.
The tables summarizing the results from the B Fac-tories are taken from a recent report by the Belle and BABAR Collaborations [2]. They include updates of input parameter values and reflect the latest understanding of the theoretical uncertainties. The averages account for correlations among the various measurements. In particular, all theoretical uncertainties are considered to be correlated, as are the uncertainties on the modeling of B → X c ℓν and B → X u ℓν decays. Experimental uncertainties due to reconstruction efficiencies are fully correlated for measurements from the same experiment, and uncorrelated for different experiments. Statistical correlations are also taken into account, whenever available. The averaging procedure was developed by the Heavy Flavor Averaging Group (HFAG) [3].
A. |Vcb| from B → D ( * ) ℓ − νℓ Decays
The "exclusive" determination of |V cb | relies on studies of B → Dℓ − ν ℓ and B → D * ℓ − ν ℓ decays, where ℓ = e, µ. The differential rate for the decay B → Dℓν can be written as
dΓ(B → Dℓν) dw = G 2 F 48π 3 |V cb | 2 K D (w) η 2 EW G 2 (w),(1)
where K D (w) is a known phase-space factor and η EW = 1.0066 refers to the one-loop electroweak correction [4] defined relative to G F from muon decay.
In the limit of small lepton mass m ℓ , G(w) represents a single vector form factor that depends on the ratio of meson masses r = m D /m B and w = v B · v D , the product of the four-velocities of the D and the B mesons. w is related to the four-momentum trans-
fer q 2 = (P B − P D ) 2 = (P ℓ + P ν ) 2 , namely w = (m 2 B + m 2 D − q 2 )/(2m B m D ) = E D /m D .
The values of w are limited by kinematics, 1 ≤ w D ≤ 1.59.
The same ansatz for the differential rate also holds for B → D * ℓν decays, except that the phase-space factor K D * (w) and w differ numerically. The form factor G(w) is replaced by F (w, θ ℓ , θ V , χ), which depends also on three angles, θ ℓ and θ V , the helicity angles of the lepton and the D * , and χ is the angle between the ηEW G(w)|V cb | for B → Dℓν decays from tagged events [6], and for comparison (b) ηEW F(w)|V cb | for B → D * ℓν decays from untagged events [7]. decay planes of the D * and the W . F (w, θ ℓ , θ V , χ) contains a combination of three form factors (one vector and two axial vectors) related to the three helicity states of the charm meson. The axial vector form factor A 1 (w) dominates as w → 1, and therefore the decay rate is usually expressed in terms of the ratios R 1 (w) = V (w)/A 1 (w) and R 2 (w) = A 2 (w)/A 1 (w) .
In the limit of infinite b-and c-quark masses, heavy quark symmetry predicts a universal form factor F (w) with a normalization at zero-recoil, F (w = 1) = G(w = 1) and a dependence on w which, with constraints from analyticity and unitarity, can be express in terms of a single parameter ρ 2 D or ρ 2 D * [5]. The principal uncertainties for the determination of |V cb | stem from the form factors, both their shape and normalization. The form factor parameters are determined from fits to the differential decays rates. For B → Dℓ − ν ℓ decays, |V cb | G(1) and the slope ρ 2 D can be extracted from Γ(w) distribution. For B → D * ℓ − ν ℓ decays, |V cb | F (1), the slope ρ 2 D * , R 1 (w = 1), and R 2 (w = 1) are determined from fits to the decay distribution Γ(w, θ ℓ , θ V , χ). As an example, Figure 1 shows the extraction of the form factor slope and normalization from the efficiency-corrected decay rates for B → Dℓ − ν ℓ and B → D * ℓ − ν ℓ decays for two BABAR analyses.
The results of recent form factor measurements for B → Dℓ − ν ℓ and B → D * ℓ − ν ℓ decays from Belle and BABAR are listed in Table I. The branching fractions quoted for B 0 are based on B 0 and B + measurements, combined under the assumptions that isospin relations hold. The branching ratios are calculated using these form factor parameters, taking into account correlated systematic uncertainties. The errors are dominated by the uncertainties in the detector performance.
For B → D * ℓ − ν ℓ decays, only two of the four measurements exploit the angular dependence of the form factor F (w, θ ℓ , θ V , χ). The most precise measurement based on the full Belle data set [8] relies on a fit to the 1-dimensional distributions of the four variables. BABAR [7] enhances the sensitivity to R 1 (1) and R 2 (1) and also |V cb | by combining the results with a fit to the four-dimensional decay rate Γ(w, θ ℓ , θ V , χ) [9]. The results from the two experiments agree well. The average values R 1 (1) = 1.40 ± 0.03 ± 0.01 and R 2 (1) = 0.86 ± 0.02 ± 0.01, have a precision of 3%, and are used as input to measurements that are limited to the w dependence of the decay rate.
A precise determination of |V cb | requires corrections to G(1) and F (1) for finite quark masses and nonperturbative effects. Table II summarizes the latest results from lattice QCD (LQCD), heavy quark sum rules (HQSR), and HQE calculation. The LQCD predictions for the two decay modes are about 5% higher than the results from the other two QCD calculations.
While the results for the two decay modes agree well, |V cb | measured in B → D * ℓν decays is more precise and will be considered as the main result. The differences in the values for |V cb | underline the fact that the principal uncertainties stem from the form factor normalization.
B. |Vcb| from Inclusive B → Xcℓν Decays
At the parton level, the inclusive decay rate for b → cℓν can be calculated accurately: It is proportional to |V cb | 2 and depends on the quark masses m b and m c . To extract |V cb | from the measured B meson decay rate, the parton-level calculations have to be corrected for effects of strong interactions. These corrections are suppressed by powers of α s and Λ QCD /m b (Λ QCD refers to the perturbative QCD scale). Thus, the decay rate for inclusive semileptonic B decays can be expressed in terms of a Heavy Quark Expansion (HQE) in inverse powers of the b-quark mass and with a limited number of non-perturbative parameters. Due to confinement and non-perturbative effects, HQEs rely on the definition of the quark masses, which depends on the coupling to the SM Lagrangian.
Calculations of the decay rates for B → X c ℓν are available in the 1S mass scheme [19,20] which is derived from a perturbative expression for the mass of the Y (1S) state, and the kinetic mass scheme [21,22] which is derived from heavy quark sum rules and enters the non-relativistic expression for the kinetic energy of the b quark inside the B meson. In the following, we rely on calculations in the kinetic scheme for which the total decay rate for B → X c ℓν can be ν ℓ and B → D * ℓ − ν ℓ form factor parameters ηEWG(1)|V cb |, ηEWF(1)|V cb |, ρ 2 D and ρ 2 D * , and the branching fractions. The measurements have been rescaled to the end of year 2011 values of the common input parameters [3,10]. The stated errors correspond to the statistical and systematic uncertainties. [11] 40.8 ± 4.4 ± 5.0 1.12 ± 0.22 ± 0.14 2.07 ± 0.12 ± 0.52 BABAR DXlν [12] 43.4 ± 0.8 ± 2.1 1.20 ± 0.04 ± 0.06 2.18 ± 0.03 ± 0.13 BABAR tagged [6] 42.5 ± 1.9 ± 1.1 1.18 ± 0.09 ± 0.05 2.12 ± 0.10 ± 0.06 Average 42.7 ± 0.7 ± 1.5 1.19 ± 0.04 ± 0.04 2.14 ± 0.03 ± 0.10
B → Dℓ − ν ℓ ηEWG(1)|V cb | (10 −3 ) ρ 2 D B(B 0 → D − ℓ + ν) (%) BelleB → D * ℓ − ν ℓ ηEWF(1)|V cb | (10 −3 ) ρ 2 D * B(B 0 → D * − ℓ + ν) (%) BABAR D * − ℓ + ν [7] 34.1 ± 0.3 ± 1.B → Dℓ − ν ℓ Calculation ηEWG(1) |V cb | (10 −3 ) LQCD [14]
1.081 ± 0.018 ± 0.016 39.46 ± 1.54 ± 0.88 HQE [15] 1.047 ± 0.020 40.79 ± 1.58 ± 0.78 [16,17] 0.908 ± 0.005 ± 0.016 39.04 ± 0.55 ± 0.73 HQSR [18] 0.865 ± 0.020 40.93 ± 0.58 ± 0.95
B → D * ℓ − ν ℓ Calculation ηEWF(1) |V cb | (10 −3 ) LQCDexpressed to O(1/m 3 b ) in a simplified way as Γ cℓν ∼ = G 2 F m 5 b 192π 3 |V cb | 2 (1 + A ew )A pert (r, µ) × z 0 (r) + z 2 (r) µ 2 π m 2 b , µ 2 G m 2 b + z 3 (r) ρ 3 D m 3 b , ρ 3 LS m 3 b .(2)
A more detailed ansatz can be found in the literature [22]. The dependence on the charm quark mass Moment measurements are available from B → X c ℓν decays for the lepton energy spectrum E n ℓ with n = 1, 2, 3, the hadronic mass distribution, M 2n X with n = 1, 2, 3 These moments and the inclusive semileptonic decay rate ∆B are measured for different values of the minimum lepton energy E min ℓ in the range of 0.6 − 1.5 GeV.
The HFAG has developed a fit procedure based on the full O(α 2 s ) calculations of the moments in the kinetic scheme [24,25]. This fit combines moment measurements from the B Factories and determines |V cb |, the b-quark mass m b , and four higher order parameters in the OPE description of semileptonic decays. The only external input is the average B 0 and B + lifetime, (1.582 ± 0.007) ps [10]. Figure 2 shows the comparison of fitted HQE predictions in the kinetic scheme to some of the moments measured by the Belle Collaboration [26] as a function of the minimum lepton energy. Since the moments are derived from the same distribution, in particular those which differ only by the minimum lepton energy, are highly correlated, only about half of the measured data points are included in the fit. Similar fits have been performed by the BABAR collaboration, including also some mixed moments of different distributions, e.g., a combination of the hadronic mass and lepton energy [27]. The fit to selected 44 moments measured by Belle and BABAR has a χ 2 /N DF = 23.2/37, an indication that the experimental and theoretical uncertainties and estimated correlations among moments are not fully understood. Correlations between the fitted parameters are generally small.
To enhance the precision on m b , a precise constraint on the c-quark mass m c (3 GeV) = 0.998 ± 0.029 GeV, was introduced, derived from low-energy sum rules [28], one of several precise calculations of quark masses [29,30]. Fits in the kinetic scheme to the Belle and BABAR moments result in
|V cb | incl = (42.01 ± 0.47 f it ± 0.59 th ) × 10 −3 , m kin b (1 GeV) = (4.551 ± 0.025 f it ) GeV, µ 2kin π (1 GeV) = (0.499 ± 0.044 f it ) GeV 2 .(3)
The first error represents the combined experimen-tal and theoretical uncertainty of the fit, and the additional error on |V cb | reflects the estimated uncertainty of 1.4% for the expansion for the decay rate. The fitted semileptonic branching fraction is B(B → X c ℓν) = (10.51 ± 0.13)%. The result on |V cb | cited here agrees very well with the result of a fit to the same moments based on the 1S mass scheme [2].
C. |Vub| from B → πℓν Decays
For the determination of |V ub | from exclusive charmless decays, the most promising decays, both experimentally and theoretically, is B → πℓ − ν ℓ . Branching fractions for decays involving the pseudoscalar mesons η and η ′ and the vector mesons ρ and ω have been measured, albeit with considerable uncertainties. Thus, they currently provide only limited information on form factors and therefore on |V ub |.
As for B → Dℓ − ν ℓ decays, the differential decay rate for decays to low-mass charged leptons can be written as
dΓ(B 0 → π − ℓ + ν) dq 2 = G 2 F 24π 3 |p π | 3 |V ub | 2 |f + (q 2 )| 2 . (4)
Here f + (q 2 ) is the only form factor affecting the rate, because in the limit of small lepton masses m ℓ , the term proportional to the second form factor f 0 (q 2 ) can be neglected [31]. Since the rate depends on the third power of p π , the pion momentum in the B meson rest frame, it is suppressed at high q 2 . Among several parameterizations of the form factors, a model-independent approach based on the general properties of analyticity, unitarity and crossingsymmetry is preferred [32,33]. The stringent constraints on the form factor are expressed in the form of a rapidly converging series in the variable z(q 2 ,
q 2 0 ) = ( m 2 + − q 2 − m 2 + − q 2 0 ), with m ± = m B ± m π .
The simplest functional form by Bourreley, Caprini, and Lellouch (BCL) [34] is
f + (q 2 ) = 1 1 − q 2 /m 2 B * K k=0 b k (q 2 0 )z(q 2 , q 2 0 ) k .(5)
Here q 2 0 is a free parameter, chosen to optimize the convergence.
The principal goal of the studies of B → πℓ − ν ℓ decays is a precise measurement of the branching fraction and the determination of the q 2 dependence of the B → π form factor. The main experimental challenge is the reduction of the abundant background from continuum events and from B → X c ℓν decays. Also the isolation of the B → πℓν decays from the other B → X u ℓν decays, where X u is a charmless hadronic final state, is difficult due to very similar decay kinematics. Three analyses have been performed based on untagged event samples, two by the BABAR [35,36] and one by the Belle [37] Collaboration. The measured branching fractions show excellent agreements, the average, taking into account correlations, B(B 0 → π − ℓ + ν) = (1.44 ± 0.03 ± 0.05) −4 , is dominated by systematic uncertainties, primarily related to the reconstruction of the missing neutrino derived from the missing energy and momentum in the event, and the backgrounds from continuum events at low q 2 and from B → X u ℓν decays at high q 2 .
A few months ago the Belle collaboration [38] presented first results on exclusive charmless decays involving the pseudoscalar mesons π + , π 0 , η, and η ′ and the vector mesons, ρ + , ρ 0 , and ω. This analysis is based on the full data sample and benefits from a highly efficient selection of events tagged by the hadronic decay of one of the B mesons in the event. Figure 3 shows the missing mass distribution for a selected B 0 → π − ℓ + ν sample with a purity of about 65%. From a fit to this distribution, a signal of 468±28 B 0 → π − ℓ + ν decays has been extracted. Preliminary measurements of the dΓ/dq 2 distributions and branching factions are fully consistent with the untagged measurements (see Table III). While the statistical uncertainties are larger than for the untagged analyses, the systematic uncertainties are much reduced due to the kinematic reconstruction of the full event.
Currently, two principal methods are used to extract |V ub | from the measured differential decay rates. The more conventional method relates the measured partial branching fractions to ∆ζ(q 2 min , q 2 max ) = ∆Γ theory /|V ub | 2 , which is derived from QCD calculations integrated over specific q 2 ranges, and |V ub | 2 = ∆B(q 2 min , q 2 max )/∆ζ(q 2 min , q 2 max )/τ 0 , (6) where τ 0 is the B 0 lifetime. Table III lists the average partial branching fractions, the values of ∆ζ, and the |V ub | results relying on light cone sum rules (LCSR), and two sets of LQCD calculations.
More recently, |V ub | has been determined from a simultaneous fit to unquenched LQCD calculations [39] and the measured q 2 spectrum. The BCL parameterization is used as parameterization for f + (q 2 ) over the whole q 2 range to minimize the form factor model dependence. This method makes optimum use of the measured shape of the whole q 2 spectrum and normalization from LQCD which results in a reduced uncertainty on |V ub |. Figure 4 shows the combined fit to the FNAL/MILC lattice calculations and the data from the three untagged measurements. To avoid high correlations, only four of the twelve FNAL/MILC points have been included in the fit. This reduction of the theoretical input does not change the |V ub | result but leads to a better agreement of the fitted curve with the lattice points. The χ 2 probability of the fit is 2.2% (χ 2 /ndf = 58.9/31). The fit results for the parameters in the BCL parameterization are b 1 /b 0 = −0.82 ± 0.20 and b 2 /b 0 = −1.63 ± 0.62, and a value of f + (0)|V ub | = 0.945 ± 0.028 is obtained, which translates to f + (0) = 0.29 ± 0.03, in good agreement with the LCSR result, f + (0) = 0.28 ± 0.02. The |V ub | values obtained from fits to the individual untagged measurements agree with each other within about one standard deviation. The total uncertainty on |V ub | is about 9%; 3% from the branching fraction measurement, 4% from the shape of the q 2 spectrum determined with data, and 8% from the form-factor normalization obtained from LQCD. Table III summarizes various measurements of |V ub |, based on different form factor normalizations. In addition to the three untagged analysis, it also lists the preliminary results by Belle using the new tagging algorithm. All these results are fully consistent within the stated uncertainties.
D. |Vub| from Inclusive B → Xuℓν Decays
The total inclusive rate for B → X u ℓν decays can be expressed in terms of an OPE which has a similar structure as the one for B → X c ℓν decays, with nonperturbative corrections occuring at O(1/m 2 b ) and higher. Unfortunately, experimenters usually restrict the phase space to reduce the large background from Cabibbo-favored B → X c ℓν decays, and these restrictions spoil the HQE convergence. Perturbative and non-perturbative corrections are drastically enhanced and the rate becomes sensitive to the Fermi motion of the b quark inside the B meson, introducing terms that are not suppressed by powers of 1/m b . In practice, non-perturbative shape functions (SF) are introduced, which to leading order in 1/m b should be similar for b → u and b → s transitions. The form of the [40], HPQCD [41], FNAL/MILC [39]. The quoted errors on |V ub | are due to experimental uncertainties and theoretical uncertainties on ∆ζ. The last column shows the |V ub | results of the simultaneous fits to data and the FNAL/MILC prediction. Here the stated error represents the combined experimental and theoretical uncertainty. SF cannot be calculated from first principles, but has to be constrained by data. SF parameterizations are generally chosen such that their first and second moments are equal to Λ = m B − m b and µ 2 π , i.e., they are directly related to the non-perturbative HQE parameters and thus can be determined by fits to moments of B → X c ℓν and B → X s γ decays.
The extracted values of |V ub | are presented in Table IV for both untagged and tagged BB samples. The earlier untagged measurements placed cuts near the kinematic limit of the lepton spectrum. Thus they covered limited fractions of the total phase space and had sizable experimental and theoretical uncertainties. The most recent analyses by the Belle [46] and BABAR [42] Collaborations are based on their full data sample and use BB events tagged by the hadronic decays of the second B meson and thus cover up to 90% of the phase space. The most precise results are based on a fit to the two-dimensional q 2 versus M X (the mass of the X u system) distributions. An example of such a fit is shown in Figure 5. The average of the two partial branching fraction measurements, assuming full correlation of the uncertainty in the predicted signal spectrum, is ∆B(p * ℓ > 1 GeV) = (1.87 ± 0.10 ± 0.11) × 10 −3 . Here p * ℓ refers to the momentum of the charged lepton in the rest frame of the B meson. The systematic uncertainties are dominated by the simulation of the signal decays; in particular, they are sensitive to the shape function and the bquark mass.
There is a high degree of consistency among the Overview of |V ub | measurements based on inclusive B → Xuℓν decays analyzed in three untagged and 2 tagged data samples. The critical input parameters m b and µ 2 π depend on the different mass schemes and have been obtained from the OPE fits to B → Xcℓν hadronic mass moments in the kinetic mass scheme. For the BLNP and DGE calculations, they have been translated from the kinetic to the shape function and M S schemes, respectively. The additional uncertainties on m b and µ 2 π are due to these scheme translations. The first error is the experimental and the second reflects the uncertainties of the QCD calculations and the HQE parameters [3]. Table IV, we quote the unweighted arithmetic average of the tagged data samples as the overall result,
|V ub | incl = (4.42 ± 0.20 exp ± 0.15 th ) × 10 −3 .(7)
E. Summary on |Vcb| and |Vub| As a result of joint efforts by theorists and experimentalists, our understanding of semileptonic Bmeson decays has substantially advanced over the last decade.
Substantial progress has been made in the application of HQE calculations to extract |V cb | and m b from fits to measured moments from B → X c ℓν decays. The total error quoted on |V cb | is 1.8% and the introduction of a c-quark mass constraint, m c (3 GeV) = (0.998 ± 0.029) GeV, has reduced the overall uncertainty on m b to 25 MeV.
The measurement of |V cb | based on the exclusive decay B → D * ℓν ℓ has now a combined experimental and theoretical uncertainty of 2.3%. Values of |V cb | differ by about 5%, depending on the choice of the QCD calculation for the normalization of the form factors; lattice calculations lead to lower values of |V cb | than heavy flavor sum rules.
Consequently the comparison of the inclusive and exclusive determinations of |V cb | depends on the choice of the normalization of the form factors. For the LQCD calculations, the values of the inclusive and exclusive determination of |V cb | differ at the level of 2.5σ,
|V cb | excl = [39.04 (1 ± 0.014 exp ± 0.019 th )] × 10 −3 , |V cb | incl = [42.01 (1 ± 0.011 exp ± 0.014 th )] × 10 −3 .
On the other hand, based on heavy flavor sum rule calculations for the exclusive measurement, the value is Measurements of the differential decay rate as a function of q 2 for B → πℓν provide valuable information on the shape of the form factor, though with sizable errors due to large backgrounds. Results based on different QCD calculations agree within the stated theoretical uncertainties. While the traditional method of normalizing to QCD calculations in different ranges of q 2 results in uncertainties of +17% −10% , combined fits to LQCD predictions and the measured spectrum using a theoretically motivated ansatz [32][33][34] have resulted in a reduction of the theoretical uncertainties to about 8%.
The values of the inclusive and exclusive determinations of |V ub | are only marginally consistent, they differ at a level of 3σ, III. STUDY OF B → D ( * ) τ − ντ DECAYS So far, we have focused on semileptonic decays involving low-mass charged leptons, for instance, B → D ( * ) ℓ − ν ℓ decays which are well-understood SM processes. Decays involving the higher mass τ lepton provide an opportunity to search for contributions beyond the SM processes, for example, decays mediated by a charged Higgs boson of the Two Higgs Doublet Model (2HDM) of type II [47][48][49][50][51][52].
In the SM, the differential decay rate (integrated over angles) for B → D ( * ) τ − ν τ decays can be written in terms of helicity amplitudes as follows [53][54][55],
dΓ τ dq 2 = G 2 F |V cb | 2 |p|q 2 96π 3 m 2 B 1 − m 2 τ q 2 2 |H + | 2 + |H − | 2 +|H 0 | 2 1 + m 2 τ 2q 2 + 3 2 m 2 τ q 2 |H s | 2 ,(8)
where for simplicity, the q 2 dependence of the helicity amplitudes H n has been omitted. The amplitudes H ± only receive contributions from helicity λ D * = ± and therefore are absent for B → Dτ − ν τ decays. λ D * = 0 contribute to H 0 and H s . SM calculations [52,56], updated to account for recent FF measurements, predict for the ratios of decay rates,
R(D) SM = Γ(B → Dτ ν) Γ(B → Dℓν) = 0.297 ± 0.017, R(D * ) SM = Γ(B → D * τ ν) Γ(B → D * ℓν) = 0.252 ± 0.003.
These ratios are independent of |V cb | and to a large extent, insensitive to the parameterization of the hadronic matrix element. Previous measurements [57][58][59] have slightly exceeded these predictions, though due to sizable statistical uncertainties the significance of the measured excess was low.
The BABAR Collaboration reports results [60] of a major update of its earlier measurement [58] of the ratios R(D ( * ) ) for both charged and neutral B mesons. They choose to reconstruct only the leptonic decays τ − → ℓ − ν ℓ ν τ , so that the final states of the signal B → D ( * ) τ − ν τ and the normalization B → D ( * ) ℓ − ν ℓ decays contain the same particles, i.e., a charm meson D ( * ) and a charged lepton, e − or µ − . This leads to a cancelation of various experimental uncertainties in the ratios R(D ( * ) ).
The analysis relies on the reconstruction of the full BB final state. In addition to the semileptonic decay, the hadronic decay of the other B meson is fully reconstructed.Compared to the previous BABAR analysis [58], the efficiency of the B tagging algorithm and the reconstruction of the semileptonic decays has been increased by a factor of three, and the size of the data sample is doubled.
The events are divided into four subsamples identified by the charm meson from a semileptonic decay candidate, D 0 ℓ, D * 0 ℓ, D + ℓ, D * + ℓ. The missing mass m 2 miss = (p e + e − − p tag − p D ( * ) − p ℓ ) 2 separates the normalization decays with m 2 miss ∼ 0 (one neutrino) from signal decays with much larger m 2 miss (three neutrinos). The leptons in normalization decays have higher momenta than the secondary leptons from τ decays, and this feature is also utilized to separate the two decay modes. Decays to higher-mass, excited charm mesons, B → D * * ℓ/τ ν , enter the event selection, whenever the low momentum pion from D * * → D ( * ) π decays is undetected or incorrectly assigned to the B tag . These higher-mass states are poorly understood and their branching fractions are not well measured. Therefore the fit includes four control D ( * ) π 0 ℓ samples, enriched in B → D * * ℓ/τ ν decays, by adding a π 0 decay to the signal selection.
The background in these 8 samples is reduced by applying multivariate methods (BDT) that make use of variables describing the quality of the reconstruction, such as the mass of the reconstructed D ( * ) and ∆E = E tag − √ s/2, where E tag and √ s refer to the B tag and the center of mass energy, respectively. Candidates with one or more additional charged tracks are eliminated. The yields for semileptonic decay of the four signal B → D ( * ) τ − ν τ decays and four normalization B → D ( * ) ℓ − ν ℓ decays are extracted by an extended, unbinned maximum-likelihood fit to the twodimensional m 2 miss versus |p * ℓ | distributions. The fit is performed simultaneously to the four D ( * ) ℓ samples and the four D ( * ) π 0 ℓ samples. The distribution of each D ( * ) ℓ sample is described as the sum of eight contributions: Dτ ν, D * τ ν, Dℓν, D * ℓν, D * * ℓν, crossfeed between B 0 and B + due to misreconstruction of the B tag , and backgrounds from BB and continuum events. The yields and shapes of these backgrounds are taken from MC simulation and fixed in the fit. A large fraction of B → D * ℓν decays are reconstructed in the Dℓν samples. A total of 56 two-dimensional probability density functions (PDF) for the individual contributions to the samples are constructed from large Monte Carlo samples by using Gaussian Kernel Estimators (KEYS). The fitted distributions for Dℓ and D * ℓ samples are shown in Figure 6. The results for charged and neutral B mesons are combined, assuming isospin relations. In total, there are 489 ± 63 B → Dτ − ν τ compared to 2, 981 ± 65 B → Dℓ − ν ℓ decays, and 888 ± 63 B → D * τ − ν τ compared to 11, 953 ± 122 B → D * ℓ − ν ℓ decays.
The measured ratios, corrected for efficiencies and branching fractions, are R(D) = 0.440 ± 0.058 ± 0.042, R(D * ) = 0.332 ± 0.024 ± 0.017.
The principal contributions to the systematic errors are the uncertainties in the B → D * * ℓ/τ ν and continuum backgrounds and the limited size of the MC
ν τ D* ν τ D ν D*l ν Dl ν D**l Bkg. |p * ℓ | (GeV) |p * ℓ | (GeV) |p * ℓ | (GeV) |p * ℓ | (
The negative sign applies to B → Dτ − ν τ and the positive sign to B → D * τ − ν τ decays. Depending on the value of tan β/m H + , the ratio of two vacuum expectations values and the mass of the charged Higgs, this term would either enhance or decrease the ratios R(D ( * ) ) and affect the τ polarization. Figure 7 shows the impact a charged 2HDM type II Higgs boson [51,61] would have on the measured ratios R(D) and R(D * ) as a function tan β/m H + . This assessment was made by reweighting the simulated events to account for the changes in the matrix element, including the τ polarization.
The measured values of R(D) and R(D * ) match the predictions of this particular Higgs model for tanβ/m H + = (0.44 ± 0.02) GeV −1 and tanβ/m H + = (0.75 ± 0.04) GeV −1 , respectively. The R(D) and R(D * ) results together exclude the type II 2HDM charged Higgs boson at at 99.8% confidence level, or higher for larger values of tanβ/m H + . This conclusion is only valid for values of m H + greater than 15 GeV [48,51]. However, the region for m H + ≤ 15 GeV has already been excluded by B → X s γ measurements [62], and therefore, the type II 2HDM is excluded in the full tanβ-m H + parameter space. [60] with predictions that include a charged Higgs boson of type II 2HDM (dark grey, red). The SM corresponds to tanβ/m H + = 0.
IV. CONCLUSIONS AND OUTLOOK
While there has been tremendous progress, we have not achieved the precision of 1% for |V cb | or 5% on |V ub |, goals many of us had hoped to reach before the shutdown of Belle and BABAR experiments.
We are left with two puzzles:
• The puzzling difference, in the results of exclusive and inclusive measurements of |V ub | and to lesser extent of |V cb |, if we rely on non-lattice calculations, which challenge our current understanding of the experimental and theoretical techniques applied.
• The excess of events in B → D ( * ) τ − ν τ decays at the level of 3.4 standard deviations relative to the SM calculations, which might indicate non-SM contributions. This excess cannot be explained by contributions from a charged Higgs boson of the 2HDM of type II. However, it has been pointed out in recent publications [63,64] that this result can be accommodated in terms of other versions of the Two-Higgs Doublet Model.
To resolve these puzzles a major effort will be required. It will take much larger tagged data samples and a more detailed understanding of the detector performance and background composition to reduce experimental uncertainties. It will also require further progress in QCD calculations, based on lattice, heavy flavor sum rules, or other methods, to reduce the uncertainties of form factor predictions for exclusive decays, to improve the detailed prediction of inclusive processes, and to incorporate precision determinations of the heavy quark masses.
Measurements of the D * and τ polarization and forward-backward asymmetries as well as other kinematic distributions might be able to distinguish among various couplings of non-SM processes [65,66] and possibly lead to an explanation of the excess events in B → D ( * ) τ − ν τ decays.
FIG. 1 :
1BABAR measurements, corrected for the reconstruction efficiency, of the w dependence of the form factors, with fit results superimposed (solid line): (a)
m c is contained in the ratio r = m c /m b which enters the phase space factors z i (r). The most relevant scale for b → c transitions is the energy release m b −m c . The electroweak corrections are estimated to be 1 + A ew ≈ [1 + α/π ln(M Z /m b )] 2 ≈ 1.014 and the perturbative contributions are A pert ≈ 0.91 ± 0.01. The leading non-perturbative contributions arise at O(1/m 2 b ) and are parameterized in terms of µ 2 π (µ) and µ 2 G (µ), the expectation values of the kinetic and chro-momagnetic dimension-five operators. At O(1/m 3 b ), two additional parameters enter, ρ 3 D (µ) and ρ 3 LS (µ), the expectation values of the Darwin and spin-orbital (LS) dimension-six operators. These parameters, as well as the quark masses m b and m c , depend on the renormalization scale µ which separates shortdistance from long-distance QCD effects. For the kinetic scheme the chosen value is µ = 1 GeV [23]. Similar HQEs can be derived for moments of inclusive B → X c ℓν distributions; they also depend on α s , m b and m c and the same perturbative parameters. The leading terms are known to O(α s ) and O(α 2 s β 0 ) (with β 0 = (33 − 2n f )/3). Non-perturbative terms are included to O(1/m 3 b ), and corrections to O(α 2 s ) have recently been implemented.
FIG. 2 :
2Comparison of the measured moments and the fit to the HQE predictions (kinetic scheme) by the Belle Collaboration [26] as a function of the minimum lepton or photon energy: (a) branching fraction ∆B, (b,c) hadron mass (MX ), (d,e,f) and electron energy (Ee). The vertical bars represent the experimental uncertainties and the shaded bands show the theoretical uncertainties. Filled (open) circles mark data used (unused) in the fit.
FIG. 3 :
3Missing mass squared distributution from a tagged samples of B 0 → π − ℓ + ν decays by the Belle Collaboration[38].
FIG. 4 :
4Simultaneous fit of the BCL parameterization to the ∆B/∆q 2 distributions for B → πℓν decays and to four of the twelve points of the FNAL/MILC calculation (magenta, closed triangles) The FNAL/MILC prediction has been rescaled to the data according to the |V ub | value obtained in the fit.
FIG. 5 :
5BABAR[42]: Projections of measured distributions (data points) of (a) q 2 and (b) MX for inclusive B → Xuℓν decays, Upper row: comparison with the result of a χ 2 fit to the two-dimensional MX − q 2 distribution for the sum of two scaled MC contributions, signal (white) and background (grey). Lower row: corresponding spectra with equal bin size after background subtraction based on the fit. The data are not efficiency corrected.
|V cb | excl = [40.93 (1 ± 0.014 exp ± 0.023 th )] × 10 −3 and agrees very well with the inclusive measurement.For inclusive measurements of |V ub |, experimental and theoretical errors are comparable in size. The dominant experimental uncertainties are related to the limited size of the tagged samples, the signal simulation, and background subtraction. The theoretical uncertainties are dominated by the error on the bquark mass; a 20-30 MeV uncertainty in m b impacts |V ub | by 2-3%.
|V ub | excl = [3.23 (1 ± 0.05 exp ± 0.08 th )] × 10 −3 |V ub | incl = [4.42 (1 ± 0.045 exp ± 0.034 th )] × 10 −3 .
FIG. 6 :
6m 2 miss and |p * ℓ | projections of the isospinconstrained fit to the signal samples. The |p * ℓ | projections do not include the m 2 miss peak at m 2 miss < 1 GeV 2 , which excludes most of the normalization events[60].samples. The measurements exceed the SM calculations by 2.0σ and 2.7σ. The combination of these results, taking into account their correlation of -0.27, excludes the SM at the 3.4σ level.The charged Higgs boson H + would only impact the helicity amplitude H s ,
FIG. 7 :
7Comparison of the results of the BABAR analysis (light grey, blue)
TABLE I :
ISummary of the B Factory results for the B → Dℓ −
TABLE II :
IINormalization of the form factors for B → Dℓ − ν ℓ and B → D * ℓ − ν ℓ decays and the resulting values of |V cb | based on different QCD calculations.
TABLE III :
IIIOverview of |V ub | measurements based on B → πℓν decays (for 3 untagged and 1 tagged samples) for various q 2 regions and form factor calculations: LCSR
TABLE IV :
IV
Presented at Flavor Physics and CP Violation (FPCP 2012), Hefei, China,May 21-25, 2012
AcknowledgmentsThe author would like to thank the organizers of FPCP 2012, in particular Zheng-guo Zhao and his colleagues at USTC in Hefei for a very exciting conference. The summary of the results on |V cb | and |V ub | is heavily based on the Physics of the B Factories, a book that has been assembled by the BABAR and Belle Collaborations with the support of many theorists. I consider myself privileged to have been part of this effort. Last not least, I would like to acknowledge the contibutions of Manuel Franco Sevilla, who recently completed his PhD thesis on the B → D ( * ) τ − ν τ decays and the observation of an excess of events above Standard Model expectations.This work was supported by Department of Energy contract DE-AC03-76SF00515.
Throughout this report, charge-conjugate decay modes are implied. Throughout this report, charge-conjugate decay modes are implied.
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| []
|
[
"On the Quantum Boltzmann Equation near Maxwellian and Vacuum",
"On the Quantum Boltzmann Equation near Maxwellian and Vacuum"
]
| [
"Zhimeng Ouyang \nDepartment of Mathematics\nBrown University\n\n",
"Lei Wu \nDepartment of Mathematics\nLehigh University\n\n"
]
| [
"Department of Mathematics\nBrown University\n",
"Department of Mathematics\nLehigh University\n"
]
| []
| We consider the non-relativistic quantum Boltzmann equation for fermions and bosons. Using the nonlinear energy method and mild formulation, we justify the global well-posedness when the density function is near the global Maxwellian and vacuum. This work is a generalization and adaptation of the classical Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear operator Q in the quantum context. This is the first piece of a long-term project on the quantum kinetic equations.Motivation and Previous ResultsThe study of the quantum Boltzmann equation dates back to Uehling-Uhlenbeck [66]. Since then, there are many papers devoted to its theory and applications in physics, chemistry and engineering. Here, we briefly summarize the relevant literature regarding its mathematical theory.So far, the study mainly focuses on two types of problems: the derivation of quantum Boltzmann equations and homogeneous equations.The rigorous derivation of classical/quantum equation is an outstanding problem in kinetic theory (see Vallani [67]). The formal derivation of quantum Boltzmann equation from N -body Schrödinger equation can be found in Erdős-Salmhofer-Yau[22]and Spohn[61]. In a series of papers[8,9,10,12], Benedetto-Castella-Esposito-Pulvirenti partially derived the quantum Boltzmann equation both in the low-density limit and weak-coupling limit. The brief surveys of their results can be found in [11] and [59], and the references there are also very informative. Some recent progress can be found in Colangeli-Pezzotti-Pulvirenti [17] and Chen-Guo[16]. In summary, this problem is still largely open so far. | 10.1016/j.jde.2022.01.056 | [
"https://arxiv.org/pdf/2102.00657v2.pdf"
]
| 231,740,838 | 2102.00657 | afbb5d9f5d0dfeddc4f33681171a6b334bef6374 |
On the Quantum Boltzmann Equation near Maxwellian and Vacuum
6 Feb 2021
Zhimeng Ouyang
Department of Mathematics
Brown University
Lei Wu
Department of Mathematics
Lehigh University
On the Quantum Boltzmann Equation near Maxwellian and Vacuum
6 Feb 2021fermionsbosonsenergy methodstability
We consider the non-relativistic quantum Boltzmann equation for fermions and bosons. Using the nonlinear energy method and mild formulation, we justify the global well-posedness when the density function is near the global Maxwellian and vacuum. This work is a generalization and adaptation of the classical Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear operator Q in the quantum context. This is the first piece of a long-term project on the quantum kinetic equations.Motivation and Previous ResultsThe study of the quantum Boltzmann equation dates back to Uehling-Uhlenbeck [66]. Since then, there are many papers devoted to its theory and applications in physics, chemistry and engineering. Here, we briefly summarize the relevant literature regarding its mathematical theory.So far, the study mainly focuses on two types of problems: the derivation of quantum Boltzmann equations and homogeneous equations.The rigorous derivation of classical/quantum equation is an outstanding problem in kinetic theory (see Vallani [67]). The formal derivation of quantum Boltzmann equation from N -body Schrödinger equation can be found in Erdős-Salmhofer-Yau[22]and Spohn[61]. In a series of papers[8,9,10,12], Benedetto-Castella-Esposito-Pulvirenti partially derived the quantum Boltzmann equation both in the low-density limit and weak-coupling limit. The brief surveys of their results can be found in [11] and [59], and the references there are also very informative. Some recent progress can be found in Colangeli-Pezzotti-Pulvirenti [17] and Chen-Guo[16]. In summary, this problem is still largely open so far.
Introduction
Problem Setup
We consider the quantum Boltzmann equation in three dimensions with hard-sphere collisions: (1.1) where the collision operator
∂ t F + v · ∇ x F = Q[F, F ; F ],Q[F, F ; F ] := R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) (1.2) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) dωdu.
for q(ω, |v − u|) = ω · (v − u) (ω ∈ S 2 ). Here the unknown F (t, x, v) is the density function (for quantum particles), at time t ∈ R + , at the position x ∈ T 3 or R 3 , and having the velocity v ∈ R 3 . In the integral of (1.2), (u, v) denotes the pre-collision velocity, and (u ′ , v ′ ) the post-collision velocity with u ′ = u+ω ω ·(v −u) , v ′ = v −ω ω ·(v −u) . They satisfy the conservation of momentum u+v = u ′ +v ′ and energy |u| 2 + |v| 2 = |u ′ | 2 + |v ′ | 2 . θ = ±1 corresponds to the fermions (−) or bosons (+), respectively. The equation is equipped with initial data
F (0, x, v) = F 0 (x, v).
(1.
3)
The collision operator Q is essentially cubic, since the cancellation reveals that
Q[F, F ; F ] = R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) + θF (v) (1.4) − F (u)F (v) 1 + θF (u ′ ) + θF (v ′ ) dωdu.
For fermions (θ = −1), we require that 0 ≤ F 0 ≤ 1. For bosons (θ = 1), we require that F 0 ≥ 0. In this paper, we intend to study the global well-posedness and decay of the solution F when it is close to the global Maxwellian or the vacuum.
Background and Modelling
In this section, we briefly discuss various quantum kinetic equations. We refer to [61,11] for more detailed discussion.
Quantum kinetic theory concerns the dynamics of a large number of quantum particles, including fermions and bosons. The equation arising from first principles to describe N identical interacting particles is the N -body Schrödinger equation. Under proper scaling and in suitable regime, asymptotically the overall behavior of this particle system can be characterized by the quantum kinetic equations.
Model 1: Low density scaling: time ∼ ε −1 , volume ∼ ε −3 and N ∼ ε −2 .
• The particles are typically far apart and the difference between classical, Fermi, or Bose gas is irrelevant.
All three statistics should be described by the same equation.
• The interacting potential φ(x) is short-ranged.
• The mean free path and mean free time is of O(ε −1 ).
• In the classical regime, this model corresponds to dilute gas dynamics, which is described by the classical Boltzmann equation.
• In the quantum regime, this model is described by the quantum Boltzmann equation (1.5) where
∂ t F + v · ∇ x F = Q L [F, F ],Q L [F, F ] = R 3 S 2 B L (ω, v − u) F (u ′ )F (v ′ ) − F (u)F (v) dωdu. (1.6)
Here the collision kernel
B L (ω, v − u) ∼ q(ω, |v − u|) φ ω ω · (v − u) 2 + n≥3 B (n) L (ω, v − u),(1.7)
forφ(k) as the Fourier transform of φ(x), and higher-order Born approximation B (n) L (see [11]).
• The quantum Boltzmann equation has the same structure as the classical Boltzmann equation, but the collision kernel may be different.
Model 2: Weak coupling scaling: time ∼ ε −1 , volume ∼ ε −3 and N ∼ ε −3 .
• The gas is dense, but the coupling of neighboring particles is weak.
• There are two possible scaling for the potential φ(x).
-Scaling ε
1 2 φ(x). -Scaling φ(ε − 1 2 x).
• For classical particles, the first scaling gives rise to the classical Landau equation and the second scaling is equivalent to the low-density limit (which means that it will lead to the classical Boltzmann equation).
• However, for quantum particles, both scalings are genuine weak coupling. This is described by the quantum Boltzmann equation (or the so-called Uehling-Uhlenbeck equation [66])
∂ t F + v · ∇ x F = Q W [F, F ; F ],(1.8)
where
Q W [F, F ; F ] = R 3 S 2 B W (ω, v − u) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) (1.9) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) dωdu.
Here θ = 1 corresponds to Bose-Einstein statistics (for bosons), θ = −1 corresponds to Fermi-Dirac statistics (for fermions), and θ = 0 corresponds to Maxwell-Boltzmann statistics (for classical particles). The collision kernel
B W (ω, v − u) ∼ ω · (v − u) φ (v ′ − v) + θφ(v ′ − u) 2 .
(1.10)
Model 3: Mean field scaling: time ∼ ε −1 , volume ∼ ε −3 and N ∼ ε −3 .
• Particles are affected not only by its neighboring particles, but all the others.
• The potential φ(x) can be short-or long-ranged. The scaling is ε 3 φ(εx).
• In classical mechanics, this corresponds to the Vlasov equation, which is applicable to the long-ranged potential.
• In quantum mechanics, this is described by the quantum Vlasov equation
∂ t F + v · ∇ x F + R 3 R 3 F (y, u) ∇ x φ(y − x) · ∇ v F (x, v) dudy = 0.
( 1.11) In this paper, we focus on the weak coupling model with hard-sphere potential φ(x) = δ 0 (x), which yieldŝ φ(k) = 1. Hence, the collision kernel reduces to q(ω, |v − u|), which is exactly the same as the classical one.
Remark 1.2. Some comments regarding the models above:
1. We will discuss the more general models (e.g. inverse power laws) in the subsequent work. Note that a direct computation reveals that for φ(x) ∼ |x| −p with p ≥ 1, unlike the hard-sphere case, the quantum collision kernel is not the same as the classical hard/soft potential one. Hence, some of the work in the related literature using classical hard/soft potential may need reexamination.
2. The low density model with hard-sphere potential will roughly reduce to the classical Boltzmann equation (if we ignore the higher-order Born approximation terms), which is less appealing in the quantum context. Note that for general potential φ, the low density model is also of interest. There are much less work in this direction.
3. For classical particles, the Boltzmann equation describes dilute gas and the Landau equation describes dense gas. However, for quantum particles, both cases give rise to the quantum Boltzmann equations.
In other words, in the quantum regime, Boltzmann equation can be applied to more general scenarios. (1.12) where the collision operator
Similar to the classical Landau equation, quantum Landau equation can be derived by taking grazing collision limit in quantum Boltzmann equation. The typical quantum Landau equation is
∂ t F + v · ∇ x F = Q Lan [F, F ; F ],Q Lan [F, F ; F ] :=∇ v · R 3 Φ(v − u) F (u) 1 + θF (u) ∇ v F (v) (1.13) − F (v) 1 + θF (v) ∇ u F (u) du,
for a semi-positive definite matrix
Φ(v) = |v| γ+2 I − v ⊗ v |v| 2 with − 3 ≤ γ ≤ 1.
(1.14)
Just like in the classical Boltzmann theory, the homogeneous quantum equation
∂ t F = Q[F, F ; F ], (1.15)
is a good starting point to develop the whole theory. There are quite a few results in this direction. We refer to Lu [45], Lu-Wennberg [54] for fermions, and Lu [44,46,47], Lu-Zhang [55], Briant-Einav [14] for bosons. Their results basically follow from the moment-entropy approach and are based on L 1 theory. The theory has been developed to treat both the isotropic and anisotropic cases. One of the most interesting facts about the quantum Boltzmann equation is for bosons. Physicists have long predicted the existence of the so-called Bose-Einstein condensation and it was observed in 1995. Since then, the mathematical justification of such phenomenon has attracted a lot of attention. Mathematically, it means that for bosons, the solution to the quantum Boltzmann equation may have singularity (e.g. δfunction) at certain spatial point. The existence of equilibria with δ-function has been justified by minimizing the entropy functional by Escobedo-Mischler-Valle [23].
On one hand, it will be thrilling to justify the generation of such singular solutions when the temperature is sufficiently low (physical requirement). Escobedo-Velázquez [27] justifies that for some well-prepared initial data (e.g. smooth, radial, and sufficiently localized), the solution will become unbounded (i.e. blow up) within finite time. The work is done for the homogeneous equation with isotropic data. Such localization restriction for the initial data has been removed in Cai-Lu [15]. We refer to Escobedo-Mischler-Velázquez [24,25], Escobedo-Velázquez [26], Spohn [62], Lu [50,51], Bandyopadhyay-Velázquez [7] for more information.
On the other hand, studying the general well-posedness and regularity becomes much harder. We have to develop measure solutions for the Boltzmann equation. There are a lot of progress in this direction for the homogeneous equation (see Lu [52,53], Li-Lu [43]). In particular, the stability of Bose-Einstein distribution has also been studied in the sense of measure.
There are also some results regarding the dynamics of the excited states and interactions with the condensed states, see Arkeryd-Nouri [2,3,4] and Nguyen-Tran [58].
Compared with the homogeneous equation, there are much fewer results on the non-homogeneous equation, see Dolbeault [18], Zhang-Lu [68,69], Lu [48,49] and Arkeryd-Nouri [5]. In particular, all of the known results so far regarding the Bose-Einstein condensation concerns the homogeneous equation. Also, we refer to the nice introduction to the quantum Landau equation by Lemou [42]. The semi-classical limit from quantum Boltzmann equation to quantum Landau equation can be found in He-Lu-Pulvirenti [39].
As far as we are aware of, so far there are very limited study on the non-homogeneous quantum Boltzmann equation, and the Bose-Einstein condensation. In this paper, we plan to utilize the well-known nonlinear energy method (see Guo [32,31,36,34,33,35,37], Strain-Guo [63,64,65], Kim-Guo-Hwang [41]) and mild formulation (see Guo [30], Illner-Shinbrot [40], Duan-Yang-Zhu [21] and the references in [28]) to investigate the global well-posedness of solutions near the Maxwellian and the vacuum.
When we are preparing this work, we are aware of the recent preprint by Bae-Jang-Yun [6] on the relativistic quantum Boltzmann equation. They used the nonlinear energy method similar to ours. However, there are some key differences:
• They only consider the system in the periodic setting near the Maxwellian. On the other hand, we consider both the periodic and whole space settings near the Maxwellian and the whole space setting near the vacuum. This provides a more comprehensive picture of the solutions in these classical frameworks.
• More importantly, the nonlinear estimates are different. While in [6] they can bound the nonlinear term in the relativistic case without using L ∞ bounds in velocity, such an estimate is absent in the non-relativistic scenario. Therefore, we must take velocity derivatives and use Sobolev embedding, which introduces a lot of technical difficulties.
Main Results
Near the Maxwellian
For the case Ω = T 3 , let the multi-indices γ and β be γ = (γ 1 , γ 2 , γ 3 ), β = (β 1 , β 2 , β 3 ).
(1.16) Denote the differential operator by
∂ γ β = ∂ γ1 x1 ∂ γ2 x2 ∂ γ3 x3 ∂ β1 v1 ∂ β2 v2 ∂ β3 v3 . (1.17)
If each component of θ is not greater thanθ, we denote by θ ≤θ. 20) and
Let N ≥ 8. Denote |||f (t)||| = |γ|+|β|≤N ∂ γ β f (t) L 2 x,v , (1.18) |||f (t)||| ν = |γ|+|β|≤N ∂ γ β f (t) L ν x,v , (1.19) where f L ν x,v :∼ f 2, 1 2 is defined in Definition 3.6. Denote also E[f (t)] = |||f (t)||| 2 + t 0 |||f (s)||| 2 ν ds,(1.E[f 0 ] = |||f 0 ||| 2 . (1.21) Theorem 1.3 (Periodic Case). Let Ω = T 3 . Assume that the initial data F 0 (x, v) = µ(v)+ M 1 2 (v)f 0 (x, v) ≥ 0 with µ(v) and M(v) defined in (3.8) and (3.9), and f 0 (x, v) satisfying E[f 0 ] ≤ M 0 2 (1.22)
for some M 0 > 0, as well as the conservation laws (3.12) -(3.14). Then there exists a unique global solution
F (t, x, v) = µ(v) + M 1 2 (v)f (t, x, v) ≥ 0 to the quantum Boltzmann equation (1.1) such that E[f (t)] ≤ M 0 (1.23)
for any t ∈ [0, ∞). Also, the perturbation f (t, x, v) satisfies
|||f (t)||| ≤ Ce −kt |||f 0 ||| (1.24)
for some constant C, K > 0.
For the case Ω = R 3 , let the multi-indices γ and β be γ = (γ 0 , γ 1 , γ 2 , γ 3 ), β = (β 1 , β 2 , β 3 ).
(1.25)
Denote the differential operator by
∂ γ β = ∂ γ0 t ∂ γ1 x1 ∂ γ2 x2 ∂ γ3 x3 ∂ β1 v1 ∂ β2 v2 ∂ β3 v3 . (1.26)
If each component of θ is not greater thanθ, we denote by θ ≤θ.
Let N ≥ 8. Denote |||f (t)||| = |γ|+|β|≤N ∂ γ β f (t) L 2 x,v , (1.27) |||f (t)||| ν = |γ|+|β|≤N ∂ γ β f (t) L ν x,v .
(1.28)
Denote also
E[f (t)] = |||f (t)||| 2 + t 0 |||f (s)||| 2 ν ds,(1.29)
and (3.8) and (3.9), and f 0 (x, v) satisfying
E[f 0 ] = |||f 0 ||| 2 . (1.30) Theorem 1.4 (Whole Space Case). Let Ω = R 3 . Assume that the initial data F 0 (x, v) = µ(v)+M 1 2 (v)f 0 (x, v) ≥ 0 with µ(v) and M(v) defined inE[f 0 ] ≤ M 0 2 (1.31)
for some M 0 > 0, as well as the conservation laws (3.12) -(3.14). Then there exists a unique global solution
F (t, x, v) = µ(v) + M 1 2 (v)f (t, x, v) ≥ 0 to the quantum Boltzmann equation (1.1) such that E[f (t)] ≤ M 0 (1.32)
for any t ∈ [0, ∞).
Remark 1.5. In the whole space case, since the dissipation lacks the lowest-order term, we cannot easily obtain decay estimates. This loss of lowest-order terms is purely due to the absence of the Poincaré-type inequality. The optimal t − 3 4 decay may be obtained under other norms based on a different framework (see Glassey [28], Duan-Strain [20,19]). However, it is beyond the scope of our paper. Remark 1.6. For both Ω = T 3 and Ω = R 3 , the global solution satisfies the following positivity bounds:
• for fermions θ = −1, if 0 ≤ F 0 (x, v) ≤ 1, then 0 ≤ F (t, x, v) ≤ 1; • for bosons θ = 1, if F 0 (x, v) ≥ 0, then F (t, x, v) ≥ 0. See Theorem 3.
18. This is a byproduct of the standard iteration argument.
Near the Vacuum
Given β > 0, define Define also the solution set
S = F ∈ C 0 (R + × R 3 × R 3 ) : |F (t, x, v)| ≤ ce −β(|x| 2 +|v| 2 ) for some c > 0 ,(1.S R = F ∈ S : |||F ||| ≤ R . (1.35)
Theorem 1.7 (Whole Space Case). There exists an R 0 > 0 sufficiently small such that if |||F 0 ||| ≤ R0 2 , then the equation (1.1) has a unique mild solution F ∈ S R0 . Remark 1.8. In the near vacuum case, the global solution also satisfies the positivity bounds as in Remark 1.6. However, the proof is highly non-trivial. See Theorem 4.9 for bosons and Theorem 4.12 for fermions. Remark 1.9. We will skip the periodic case near the vacuum due to the following:
• The framework developed here highly relies on the dispersion properties of the transport operator ∂ t + v · ∇ x . However, such dispersive decay in time is absent in the periodic case for mild solution (smoothness may improve the decay). It is far beyond our methods discussed here.
• In the periodic case for the classical Boltzmann equation, as Mouhot [57] and Briant [13] proved, the solution will instantaneously fill the vacuum and be above a Maxwellian. This indicates that the near Maxwellian framework is more suitable for this case. We anticipate that the similar result will hold for the quantum Boltzmann equation.
• The general theory for solutions merely satisfying |F (t, x, v)| e −β|v| 2 is far from mature and there are very limited results on the global well-posedness. Remark 1.10. All of these results reveal that if the initial data is sufficiently close to the Maxwellian or vacuum, then the solution will remain finite and not blow up. Hence, for bosons, the Bose-Einstein condensation will not occur. In other words, the occurrence of Bose-Einstein condensation requires more delicate analysis. This is a sharp constrast with the result in Escobedo-Velázquez [27].
Notations
Throughout this paper, C > 0 denotes a constant that only depends on the domain Ω, but does not depend on the data. It is referred as universal and can change from one inequality to another. When we write C(z), it means a certain positive constant depending on the quantity z. We write a b to denote a ≤ Cb for some universal constant C > 0; we will use and ≃ in a similar standard way.
Difficulties and Ideas
The framework of studying the non-quantum Boltzmann equation near the Maxwellian is classical (see [28,32,31,35]). It is customary to linearize the solution F (t, x, v) around the global Maxwellianμ(v) = e −|v| 2 as F =μ +μ (1.36) for the linearized Boltzmann operator L and quadratic operator Γ. For the quantum Boltzmann equation, the first difficulty is that the naive perturbation F = µ + µ 1 2 f around the quantum Maxwellian
1 2 f , where the perturbation f (t, x, v) satisfies ∂ t f + v · ∇ x f + L[f ] = Γ[f, f ],µ(v) = 1 ρe |v| 2 − θ (1.37)
with ρ > 1 will result in a linearized Boltzmann operator without coercivity, which is devastrating for the energy method. To get around this, a crucial observation is that we need to redesign the perturbation as
F = µ + M 1 2 f with M(v) = µ(1 + θµ) = ρe |v| 2 (ρe |v| 2 − θ) 2 .
(1.38)
Then as Theorem 3.2 reveals, we have that L is a self-adjoint operator on L 2 v (R 3 ) and
R 3 f · L[f ] dv (I − P)[f ] L ν v ,(1.39)
where P[f ] is the orthogonal projection of f onto the five-dimensional null space of L, i.e.
Null space of L = M 1 2 (a + b · v + c |v| 2 ) : a, c ∈ R, b ∈ R 3 . (1.40)
Then this recovers the basic energy structure as in the classical Boltzmann equation. The most important distinction between the classical and quantum Boltzmann equation framework lies in the nonlinear estimates. Compared with [32, Lemma 2.3] using only L 2 v norms, Lemma 3.15 depends on sup v f . Thus in order to obtain L ∞ v estimate of f in the Sobolev-space framework, we resort to the high-regularity framework with derivatives in both space and velocity.
Intuitively, for the classical Boltzmann equation, the nonlinear operator Γ only involves integral for dudv of the form (we ignore the collision kernel)
f (u)f (v) or f (u ′ )f (v ′ ). (1.41)
These can be handled with the pre-post change-of-variable (u, v) ↔ (u ′ , v ′ ). However, for the quantum Boltzmann equation, due to the cubic nonlinearity, we must estimate the mixed-type integral
f (u)f (v ′ ) or f (u)f (u ′ ) or f (v)f (u ′ ) or f (v)f (v ′ ). (1.42)
Then the pre-post change-of-variable cannot resolve the difficulty. This kind of integrals have long been captured as an important ingredient to tackle long-range interactions and non-cutoff Boltzmann equations. So far, there are mainly two approaches to deal with them:
• The proof of Alexandre-Desvillettes-Villani-Wennberg [1, Lemma 1] justifies that under proper sense, du ′ du and dv ′ dv have positive lower bounds. With this in hand, we may bound f (u)f (v ′ ) and f (v)f (u ′ ). However, the difficulty lies in the remaining two integrals f (u)f (u ′ ) and f (v)f (v ′ ). It has been shown that du ′ dv and dv ′ du do not have positive lower bounds. In the literature, some authors claimed that it is possible to use the well-known cancellation lemma (see Alexandre-Desvillettes-Villani-Wennberg [1, Lemma 1]) to finish the job, but we cannot see the viability and it does not look hopeful. Actually, cancellation lemma is used to bound
f (u)(f (v) − f (v ′ )) and f (v)(f (u) − f (u ′ )), which can help handle f (u)f (v ′ ) and f (v)f (u ′ ), but not f (u)f (u ′ ) and f (v)f (v ′ )
, so the difficulty remains.
• In the analysis of non-cutoff Boltzmann equation, Gressman-Strain [29] introduces two Carleman-type representations. Roughly speaking, [29, Proposition A.1] makes the change-of-variables v → v ′ and u → u ′ with the help of σ integral, which is controllable (see [29,Section 3] At the end of the day, we arrive at the conclusion that we have to resort to L ∞ v estimate of f , which depends on high-regularity framework in the velocity variable and Sobolev embedding. This in turn creates a lot of technical difficulties. For example, we need to estimate the velocity derivatives of L and Γ operators. Such estimates have been done for classical hard-sphere case in [32, Lemma 2.1, Lemma 2.2]. However, the proof there highly depends on the explicit formula of L (see [28, Section 3.2, Section 3.3]), which we do not have. Then we have to use the fact that µ(v) ≃μ(v) and M(v) ≃μ(v) and the conservation laws to bound the derivatives term by term. In particular, we need to find the partially explicit formula as in [28, Section 3.2, Section 3.3] to complete the estimates.
For the quantum Boltzmann equation near the vacuum, we utilize the robust framework introduced in Illner-Shinbrot [40] (we refer to Glassey [28, Section 2] for clarity). While the global well-posedness is not too difficult to adapt, the positivity proof needs more thinking. The proof for the classical Boltzmann equation relies on a monotonicity argument to construct two approximate sequences from above and below. In particular, it highly depends on the monotonicity of the gain term F (u ′ )F (v ′ ). While this still holds for bosons with θ = 1, such a naive adaptation does not work for fermions θ = −1 since F (u ′ )F (v ′ ) 1 + θF (u) + θF (v) does not have the desired monotonicity. Our strategy is to redesign the artificial gain and loss terms to enforce the monotonicity. In particular, we also need to redesign the approximate sequences such that the convergence is preserved.
Organization of the Paper
Our paper is organized as follows: In Section 2, we record some preliminary results on the quantum Boltzmann collision operator Q, including the conservation laws, the entropy and H-Theorem, and the quantum Maxwellian. In Section 3, we study the near Maxwellian case through the nonlinear energy method, which comes with a careful derivation of the perturbation form, linear and nonlinear estimates, and the proof of local and global well-posedness. In Section 4, we study the near vacuum case via the mild formulation, proving the global well-posedness and positivity of solutions.
Properties of the Collision Operator
In this section, we present some basic results regarding the collision operator Q. They are mostly well-known for the classical Boltzmann equation and we derive them in the quantum context. We mainly adapt from Glassey [28].
Conservation Laws
Lemma 2.1. For all smooth functions F (v) and φ(v), small at infinity,
R 3 Q[F, F ; F ](v)φ(v) dv (2.1) = R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) φ(v) dωdudv = R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) φ(u) dωdudv = − R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) φ(v ′ ) dωdudv = − R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) φ(u ′ ) dωdudv.
Proof. The first equality is the definition of Q. Then switching the status of u and v, we find that the integral does not change, so the second equality naturally follows. The third equality comes from the pre-post collision substitution (u, v) → (u ′ , v ′ ). Based on [28, Lemma 1.4.1, Lemma 1.6.1], we may directly verify that ω · (v − u) = −ω · (v ′ − u ′ ), and the Jacobian J satisfies |J| = 1. Also, under this substitution, we know (u ′ , v ′ ) → (u, v). Hence, after renaming the variables, we get the third equality. Similarly, switching the status of u and v, we get the fourth equality.
R 3 Q[F, F ; F ](v)φ(v) dv (2.2) = 1 4 R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) φ(v) + φ(u) − φ(v ′ ) − φ(u ′ ) dωdudv.
Proof. Adding the four equalities in Lemma 2.1, this result naturally follows.
Corollary 2.3. For any smooth function F (v), small at infinity,
R 3 Q[F, F ; F ](v) dv = 0, R 3 Q[F, F ; F ](v)v i dv = 0, R 3 Q[F, F ; F ](v) |v| 2 dv = 0,(2.
3)
for i = 1, 2, 3.
Proof. Taking φ(v) = 1, v i , |v| 2 in Corollary 2.2, which satisfy φ(v)+φ(u)−φ(v ′ )−φ(u ′ ) = 0, we immediately obtain the results.
Remark 2.4. The test function φ(v) satisfying φ(v) + φ(u) − φ(v ′ ) − φ(u ′ ) = 0
is called a collisional invariant. Corollary 2.3 justifies that 1, v i , |v| 2 are collisional invariants.
Theorem 2.5. If F (t, x, v)
is a solution to the equation (1.1) which is suitable small at infinity, then we have for any t > 0 and for Ω = T 3 or R 3 ,
Ω R 3 F (t, x, v) dvdx = Ω R 3 F 0 (x, v) dvdx, (Conservation of Mass) (2.4) Ω R 3 F (t, x, v)v i dvdx = Ω R 3 F 0 (x, v)v i dvdx, (Conservation of Momentum) (2.5) Ω R 3 F (t, x, v) |v| 2 dvdx = Ω R 3 F 0 (x, v) |v| 2 dvdx. (Conservation of Energy) (2.6)
Proof. We multiply 1, v i , |v| 2 on both sides of the equation (1.1), and then integrate the resulting equation over (x, v) ∈ Ω × R 3 . Using Corollary 2.3 and integration by parts, we get d dt
Ω×R 3 F (t, x, v)φ(v) dvdx ≡ 0 for φ(v) = 1, v i , |v| 2 , which implies the desired result.
Remark 2.6. The above is the basic conservation laws for the quantum Boltzmann equation.
Entropy and the H-Theorem
Theorem 2.7 (Entropy: H-Theorem). If F (t, x, v) > 0 is a solution to the equation (1.1) which is suitable small at infinity (for θ = −1, we further require F (t, x, v) < 1 to ensure 1 + θF (t, x, v) > 0), then we have for any t > 0 and for Ω = T 3 or R 3 ,
d dt Ω R 3 F (t, x, v) ln 1 F (t, x, v) − θ 1 + θF (t, x, v) ln 1 1 + θF (t, x, v) dvdx ≥ 0. (2.7)
Proof. Denote
S[F ](t, x, v) := F (t, x, v) ln 1 F (t, x, v) − θ 1 + θF (t, x, v) ln 1 1 + θF (t, x, v)
.
(2.8)
We may directly compute
∂S[F ] ∂t = ∂ t F ln F 1 + θF . (2.9)
Hence, multiplying ln F 1 + θF on both sides of the equation (1.1) and integrating over (
x, v) ∈ Ω × R 3 , we obtain d dt Ω R 3 S[F ] dvdx = Ω R 3 ∂ t F ln F 1 + θF dvdx (2.10) = − Ω R 3 v · ∇ x F + Q[F, F ; F ] ln F 1 + θF dvdx = − Ω R 3 Q[F, F ; F ] ln F 1 + θF dvdx.
Based on Corollary 2.2 with φ = ln
F 1 + θF , we have R 3 Q[F, F ; F ] ln F 1 + θF dv (2.11) = 1 4 R 3 R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) ln F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) dωdudv = 1 4 R 3 R 3 S 2 q(ω, |v − u|)F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) (1 − A) ln(A) dωdudv, where A := F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v
Equilibrium: the Quantum Maxwellian
In this section, we study the equilibrium, which is independent of time and space.
µ(v) = 1 e −(a+b·v+c|v| 2 ) − θ , (2.14)
for some a, c ∈ R, b ∈ R 3 with c < 0. For bosons θ = 1, we also require a < 0.
Proof. An equilibrium means that it is invariant when time evolves. Suppose that the solution F (t, x, v) to the equation (1.1) is an equilibrium, then we must have
d dt Ω R 3 S[F ] dvdx ≡ 0, (2.15)
which, from the proof of Theorem 2.7, yields
Ω R 3 Q[F, F ; F ] ln F 1 + θF dvdx ≡ 0. (2.16)
We use the notation from the proof of Theorem 2.7. In the last line of (2.11), since the integrand is of one sign, we must have A = 1, i.e.
F (u)F (v) 1 + θF (u) 1 + θF (v) = F (u ′ )F (v ′ ) 1 + θF (u ′ ) 1 + θF (v ′ )
.
(2.17)
Hence, for φ = ln
F 1 + θF , we must have φ(v) + φ(u) = φ(v ′ ) + φ(u ′ ). (2.18)
Then based on [28, Lemma 1.7.2], for continuous φ, we must have
φ(v) = a + b · v + c |v| 2 , (2.19)
for some a, c ∈ R, b ∈ R 3 with c < 0 (to guarantee integrability). For bosons θ = 1, we also require a < 0 to avoid singularity of the denominator. Therefore, we know that if F is an equilibrium, then it must have the form
F (v) = 1 e −(a+b·v+c|v| 2 ) − θ . (2.20)
On the other hand, it is easy to verify that such F indeed satisfies the equation (
1 e −(a+b·v+c|v| 2 ) − 1 + dδ − b 2c , (2.21)
where a, c, d ∈ R and b ∈ R 3 . The presence of δ-function at finite time indicates the Bose-Einstein condensation.
For now on, we will consider the simplest equilibrium with b = 0 and c = −1, which is
µ(v) = 1 e |v| 2 −a − θ . (2.22)
When θ = 0 and a = 0 for the classical particles, this reduces to the standard Maxwellian µ(v) = e −|v| 2 . Actually, we allow any a ∈ R. However, for Fermion gas (θ = −1) or Boson gas (θ = 1), this Maxwellian is highly non-trivial. For fermions, the Maxwellian is well-defined for any a ∈ R. However, for bosons, we must require a < 0 to guarantee the positivity of F . If a = 0, the Maxwellian might contain a singularity at v = 0.
To handle all cases in a uniform fashion, we require a < 0 and denote
µ(v) := 1 ̺e |v| 2 − θ (2.23) with ̺ = e −a > 1.
Remark 2.12. Although it is easy to justify that for ̺ > 1,
R 3 1 ̺e |v| 2 − θ dv < ∞, (2.24)
it is almost impossible to evaluate this integral explicitly. This has serious consequences. For the classical Boltzmann equation with hard-sphere collision, the analysis heavily depends on the explicit computation of such type of integrals (see the beautiful arguments in [28, Section 3.2] for Gaussian functions). However, now we have to find other approaches to get around this difficulty.
Global Stability of the Maxwellian
Perturbation Formulation
The Quantum Boltzmann Operator
Recall the quantum Boltzmann collision operator
Q[F, F ; F ] := R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) 1 + θF (v) (3.1) − F (u)F (v) 1 + θF (u ′ ) 1 + θF (v ′ ) dωdu = R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) 1 + θF (u) + θF (v) − F (u)F (v) 1 + θF (u ′ ) + θF (v ′ ) dωdu.
We may further decompose
Q[F, F ; F ] = Q 1 [F, F ] + θQ 2 [F, F ; F ], (3.2) where Q 1 [F, F ] = R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) − F (u)F (v) dωdu, (3.3) Q 2 [F, F ; F ] = R 3 S 2 q(ω, |v − u|) F (u ′ )F (v ′ ) F (u) + F (v) − F (u)F (v) F (u ′ ) + F (v ′ ) dωdu. (3.4)
Here Q 1 is identical to the classical Boltzmann collision operator, and Q 2 is a trilinear form which contains the quantum effects. Note that Q 2 is not necessarily smaller than Q 1 (we should simply regard it as a correction), so the quantum effects will play a significant role. Denote the symmetrized operators
Q[F, G] := 1 2 R 3 S 2 q(ω, |v − u|) F (u ′ )G(v ′ ) + G(u ′ )F (v ′ ) − F (u)G(v) − G(u)F (v) dωdu,(3.5)
and
Q[F, G; H] := 1 2 R 3 S 2 q(ω, |v − u|) F (u ′ )G(v ′ ) H(u) + H(v) + G(u ′ )F (v ′ ) H(u) + H(v) (3.6) − F (u)G(v) H(u ′ ) + H(v ′ ) − G(u)F (v) H(u ′ ) + H(v ′ ) dωdu.
Obviously, we know
Q[F, F ] = Q 1 [F, F ], Q[F, F ; F ] = Q 2 [F, F ; F ]. (3.7)
Recall that the quantum Maxwellian is
µ(v) := 1 ̺e |v| 2 − θ (3.8) with ̺ > 1. Define also M(v) := µ(1 + θµ) = µ + θµ 2 = ̺e |v| 2 (̺e |v| 2 − θ) 2 .
(3.9)
We define the perturbation f (t, x, v) via
F (t, x, v) = µ(v) + M 1 2 (v)f (t, x, v), (3.10) with F 0 (x, v) = µ(v) + M 1 2 (v)f 0 (x, v), (3.11)
satisfying the conservation laws
Ω R 3 f (t, x, v)M 1 2 (v) dvdx = Ω R 3 f 0 (x, v)M 1 2 (v) dvdx = 0, (Mass) (3.12)a + b · v + c |v| 2 with a, c ∈ R and b ∈ R 3 .
(2) Self-adjointness: L is a self-adjoint (symmetric) operator, i.e. for any f (v), g(v) small at infinity,
we have L[f ], g = f, L[g] . (3) Null space: L[f ] = 0 if and only if f (v) = M 1 2 a + b · v + c |v| 2 with a, c ∈ R and b ∈ R 3 .
Proof.
(1) We may write each term explicitly:
− 2M − 1 2 Q µ, M 1 2 f (3.22) = − M − 1 2 (v) R 3 S 2 q(ω, |v − u|) µ(u ′ )M 1 2 (v ′ )f (v ′ ) + M 1 2 (u ′ )µ(v ′ )f (u ′ ) − µ(u)M 1 2 (v)f (v) − M 1 2 (u)µ(v)f (u) dωdu, − 2θM − 1 2 Q µ, M 1 2 f ; µ (3.23) = − θM − 1 2 (v) R 3 S 2 q(ω, |v − u|) µ(u ′ )M 1 2 (v ′ )f (v ′ ) µ(u) + µ(v) + M 1 2 (u ′ )µ(v ′ )f (u ′ ) µ(u) + µ(v) − µ(u)M 1 2 (v)f (v) µ(u ′ ) + µ(v ′ ) − M 1 2 (u)µ(v)f (u) µ(u ′ ) + µ(v ′ ) dωdu, and − θM − 1 2 Q µ, µ; M 1 2 f (3.24) = − 1 2 θM − 1 2 (v) R 3 S 2 q(ω, |v − u|) µ(u ′ )µ(v ′ ) M 1 2 (u)f (u) + M 1 2 (v)f (v) + µ(u ′ )µ(v ′ ) M 1 2 (u)f (u) + M 1 2 (v)f (v) − µ(u)µ(v) M 1 2 (u ′ )f (u ′ ) + M 1 2 (v ′ )f (v ′ ) − µ(u)µ(v) M 1 2 (u ′ )f (u ′ ) + M 1 2 (v ′ )f (v ′ ) dωdu = − θM − 1 2 (v) R 3 S 2 q(ω, |v − u|) µ(u ′ )µ(v ′ ) M 1 2 (u)f (u) + M 1 2 (v)f (v) − µ(u)µ(v) M 1 2 (u ′ )f (u ′ ) + M 1 2 (v ′ )f (v ′ ) dωdu.
Hence, summarizing all above, we have
L[f ] = M − 1 2 (v) R 3 S 2 q(ω, |v − u|) µ(v) + θµ(v)µ(u ′ ) + θµ(v)µ(v ′ ) − θµ(u ′ )µ(v ′ ) M 1 2 (u)f (u) (3.25) + µ(u) + θµ(u)µ(u ′ ) + θµ(u)µ(v ′ ) − θµ(u ′ )µ(v ′ ) M 1 2 (v)f (v) − µ(v ′ ) + θµ(v ′ )µ(u) + θµ(v ′ )µ(v) − θµ(u)µ(v) M 1 2 (u ′ )f (u ′ ) − µ(u ′ ) + θµ(u ′ )µ(u) + θµ(u ′ )µ(v) − θµ(u)µ(v) M 1 2 (v ′ )f (v ′ ) dωdu.
Direct computation reveals that
µ(v) + θµ(v)µ(u ′ ) + θµ(v)µ(v ′ ) − θµ(u ′ )µ(v ′ ) = µ(v)µ(u ′ )µ(v ′ )̺e |v| 2 ̺e |u| 2 − θ , (3.26) µ(u) + θµ(u)µ(u ′ ) + θµ(u)µ(v ′ ) − θµ(u ′ )µ(v ′ ) = µ(u)µ(u ′ )µ(v ′ )̺e |u| 2 ̺e |v| 2 − θ , (3.27) µ(v ′ ) + θµ(v ′ )µ(u) + θµ(v ′ )µ(v) − θµ(u)µ(v) = µ(v ′ )µ(u)µ(v)̺e |v ′ | 2 ̺e |u ′ | 2 − θ , (3.28) µ(u ′ ) + θµ(u ′ )µ(u) + θµ(u ′ )µ(v) − θµ(u)µ(v) = µ(u ′ )µ(u)µ(v)̺e |u ′ | 2 ̺e |v ′ | 2 − θ . (3.29)
Hence, we have
L[f ] = M − 1 2 (v) R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 (3.30) µ −1 (u)̺e −|u| 2 ̺e |u| 2 − θ M 1 2 (u)f (u) + µ −1 (v)̺e −|v| 2 ̺e |v| 2 − θ M 1 2 (v)f (v) − µ −1 (u ′ )̺e −|u ′ | 2 ̺e |u ′ | 2 − θ M 1 2 (u ′ )f (u ′ ) − µ −1 (v ′ )̺e −|v ′ | 2 ̺e |v ′ | 2 − θ M 1 2 (v ′ )f (v ′ ) dωdu = M − 1 2 (v) R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (u)f (u) + M − 1 2 (v)f (v) − M − 1 2 (u ′ )f (u ′ ) − M − 1 2 (v ′ )f (v ′ ) dωdu, due to |u| 2 + |v| 2 = |u ′ | 2 + |v ′ | 2 and µ −1 (v)̺e −|v| 2 ̺e |v| 2 − θ = M −1 (v).
Hence, using the similar techniques as in the proof of Theorem 2.7 (by symmetry of the "kernel" and change of variables), we have that
L[f ], f = 1 4 R 3 R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 (3.31) M − 1 2 (u)f (u) + M − 1 2 (v)f (v) − M − 1 2 (u ′ )f (u ′ ) − M − 1 2 (v ′ )f (v ′ ) 2 dωdudv.
Since the integrand is always nonnegative, we know
L[f ], f ≥ 0. (3.32)
In particular, if the equality holds, then we have
M − 1 2 (u)f (u) + M − 1 2 (v)f (v) − M − 1 2 (u ′ )f (u ′ ) − M − 1 2 (v ′ )f (v ′ ) = 0, (3.33)
which implies that
f (v) = M 1 2 a + b · v + c |v| 2 , (3.34)
for a, c ∈ R and b ∈ R 3 .
(2) Based on (3.30) and (3.31) in the proof of (1), it is clear that L is self-adjoint.
(
3) If f (v) = M 1 2 a + b · v + c |v| 2 ,a + b · v + c |v| 2 for some a, c ∈ R and b ∈ R 3 .
Now we know that the linear operator L given by (3.21) is a self-adjoint non-negative operator on
L 2 v (R 3 ). Its null space (kernel) is a five-dimensional subspace of L 2 v (R 3 ) spanned by M 1 2 , vM 1 2 , |v| 2 M 1 2
. We introduce the following notations: For the function
f (t, x, v) with fixed (t, x) , we define the projection of f in L 2 v (R 3 ) onto N(L) as Pf (t, x, v) := 4 i=0 f (t, x, · ), e i e i =: a f (t, x) + v · b f (t, x) + |v| 2 c f (t, x) M 1 2 , where a f (t, x) := f (t, x, · ), e 0 = R 3 f (t, x, · )M 1 2 dv , b i f (t, x) := f (t, x, · ), e i = R 3 f (t, x, · )v i M 1 2 dv , c f (t, x) := f (t, x, · ), e 4 = R 3 f (t, x, · )|v| 2 M 1 2 dv .
From (3.30), we may rewrite
L[f ] = νf − K[f ], (3.36) where ν(v) := R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M −1 (v) dωdu, (3.37) and K = K 2 − K 1 for K 1 [f ] := M − 1 2 (v) R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (u)f (u) dωdu, (3.38) (3.39) K 2 [f ] := M − 1 2 (v) R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (u ′ )f (u ′ ) + M − 1 2 (v ′ )f (v ′ ) dωdu. Lemma 3.4 (ν Estimate). ν(v) satisfies that ν 1 1 + |v| ≤ ν(v) ≤ ν 2 1 + |v| , (3.40)
for some ν 1 , ν 2 > 0.
Proof. We have the naive bounds for µ and M:
e −|v| 2 µ(v) e −|v| 2 , e −|v| 2 M(v) e −|v| 2 . (3.41) Hence, considering |u| 2 + |v| 2 = |u ′ | 2 + |v ′ | 2 , we may bound R 3 S 2 q(ω, |v − u|)e −|u| 2 dωdu ν(v) R 3 S 2 q(ω, |v − u|)e −|u| 2 dωdu, (3.42) which implies R 3 |v − u| e −|u| 2 dωdu ν(v) R 3 |v − u| e −|u| 2 dωdu. (3.43)
Then following the same proof as [28, Section 3.3.1], we obtain the desired result.
Lemma 3.5. We have |∂ β ν| 1 for |β| ≥ 1.
Proof. We first rearrange ν as
ν(v) = R 3 S 2 q(ω, |v − u|)µ(u) µ(v)M −1 (v) µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 dωdu (3.44) = R 3 S 2 A(u, v, ω)B(v)C(u, v, ω) dωdu.
We directly take v derivatives in ν, so it might hit one of A, B, C. Then we have
∇ v A ∼ ∇ v |v − u| µ(u) = v − u |v − u| µ(u), (3.45) ∇ v B = ∇ v 1 − ̺θe −|v| 2 = 2v̺θe −|v| 2 , (3.46) ∇ v C = ∇ v 1 ̺ 2 − ̺θ(e −|u ′ | 2 + e −|v ′ | 2 ) + θ 2 e −|u| 2 −|v| 2 (3.47) = 2̺θ u ′ · ∇ v u ′ e −|u ′ | 2 + v ′ · ∇ v v ′ e −|v ′ | 2 − 2vθ 2 e −|u| 2 −|v| 2 ̺ 2 − ̺θ(e −|u ′ | 2 + e −|v ′ | 2 ) + θ 2 e −|u| 2 −|v| 2 2 , where ∇ v u ′ = ω ⊗ ω and ∇ v v ′ = I − ω ⊗ ω. Then we have |∇ v A(u, v, ω)| µ(u), (3.48) |∇ v B(v)| e − 1 2 |v| 2 , (3.49) |∇ v C(u, v, ω)| e − 1 2 |u ′ | 2 + e − 1 2 |v ′ | 2 + e − 1 2 |u| 2 − 1 2 |v| 2 . (3.50)
Then using the substitution v − u → w (the integral is transformed to be with respect to dw), we know
A(w, v, ω) = w |w| µ(v − w). (3.51) Obviously, we know |∂ β µ(v − w)| e − 1 2 |v−w| 2 . B(v)
does not change and any v derivative will be controlled by e −|v| 2 . The structure of C(w, v, ω) will be preserved. In particular,
u ′ = v − w + ω(ω · w), v ′ = v − ω(ω · w). (3.52)
Hence, any v derivative will be controlled by e −|u ′ | 2 , e −|v ′ | 2 and e −|u| 2 −|v| 2 , and thus our result follows.
Definition 3.6 (ν Norms). We define
f L ν v := R 3 ν(v) |f (v)| 2 dv 1 2 ∼ |f | 2, 1 2 , (3.53) and f L ν x,v := Ω R 3 ν(v) |f (v)| 2 dvdx 1 2 ∼ f 2, 1 2 , (3.54) Lemma 3.7 (Compactness of K). K is a compact operator on L ν v (R 3 ) and on L 2 v (R 3 ). Proof. We may denote K i [f ] = R 3 k i (u, v)f (u) du (3.55)
for some kernel functions k i , i = 1, 2. We first consider K 1 . Obviously,
k 1 (u, v) = ̺ 2 e |u| 2 +|v| 2 M − 1 2 (u)M − 1 2 (v)µ(u)µ(v) S 2 q(ω, |v − u|)µ(u ′ )µ(v ′ ) dω. (3.56)
Based on a (3.41), we know
e − |u| 2 2 − |v| 2 2 M − 1 2 (u)M − 1 2 (v)µ(u)µ(v) e − |u| 2 2 − |v| 2 2 ,(3.57)
and
1 e |u| 2 +|v| 2 µ(u ′ )µ(v ′ ) 1. (3.58)
Thus, though we cannot derive an explicit formula for k 1 , we know it can be well-controlled by the correspondingk 1 for the classical Boltzmann equation, wherẽ
k 1 (u, v) = π |v − u| exp − |u| 2 2 − |v| 2 2 . (3.59)
For K 2 , k 2 is not easy to obtain. We first split
K 2 [f ] = R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (v)M − 1 2 (u ′ )f (u ′ ) dωdu (3.60) + R 3 S 2 q(ω, |v − u|)µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (v)M − 1 2 (v ′ )f (v ′ ) dωdu.
Following the argument as in [28, (3.34),(3.51)], we obtain that
K 2 [f ] = R R 2 2 |η − v| µ(η + V ⊥ )µ(v)µ(v + V ⊥ )µ(η)̺ 2 e |η+V ⊥ | 2 +|v| 2 M − 1 2 (v)M − 1 2 (η) dV ⊥ f (η) dη (3.61) where V = u − v, V = (V · ω)ω, V ⊥ = V − (V · ω)ω, η = v + V ,(3.62)
or equivalently
u = η + V ⊥ , u ′ = v + V ⊥ , v ′ = η. (3.63)
Therefore, we know
k 2 (v, η) = R 2 2 |η − v| µ(η + V ⊥ )µ(v)µ(v + V ⊥ )µ(η)̺ 2 e |η+V ⊥ | 2 +|v| 2 M − 1 2 (v)M − 1 2 (η) dV ⊥ . (3.64)
Unfortunately, due to the complexity of µ and M in the quantum Boltzmann equation, we can hardly further simplify k 2 as in [28, (3.45), (3.52)] to get an explicit formula. Hence, we turn to direct bounds. Note that in the original variables
e − 1 2 |u| 2 − 1 2 |u ′ | 2 µ(u)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |u| 2 +|v| 2 M − 1 2 (v)M − 1 2 (v ′ ) e − 1 2 |u| 2 − 1 2 |u ′ | 2 . (3.65)
Thus in the new variables, we know
(3.66) e − 1 2 |η+V ⊥ | 2 − 1 2 |v+V ⊥ | 2 µ(η + V ⊥ )µ(v)µ(v + V ⊥ )µ(η)̺ 2 e |η+V ⊥ | 2 +|v| 2 M − 1 2 (v)M − 1 2 (η) e − 1 2 |η+V ⊥ | 2 − 1 2 |v+V ⊥ | 2 .
Compared with [28, (3.35)], we know the upper bound and lower bound can be computed explicitly. In other words, though we cannot obtain explicitly formula of k 2 , we know that it can be well-controlled by the correspondingk 2 for the classical Boltzmann equation, where based on [28, (3.52)]
k 2 (u, v) = 2π |u − v| exp − 1 4 |u − v| 2 − 1 4 (|u| 2 − |v| 2 ) 2 |u − v| 2 .
(3.67)
Then similarly to the argument in [28, Section 3.5], we know K is compact.
Corollary 3.8. We have sup v∈R 3 R 3 k 1 (u, v) du < ∞, sup v∈R 3 R 3 k 1 (u, v) 2 du < ∞, (3.68) sup v∈R 3 R 3 k 2 (u, v) du < ∞, sup v∈R 3 R 3 k 2 (u, v) 2 du < ∞. (3.69) Proof. See [28, Section 3.3.2]. Corollary 3.9. K is bounded on L 2 v (R 3 ) and on L ν v (R 3 ). Proof. See [28, Section 3.5].
Corollary 3.10. Let α > 0 and k = k 2 − k 1 . Then we have
R 3 |k(u, v)| (1 + |u| 2 ) − α 2 du (1 + |u| 2 ) − α+1 2 (3.70) Proof. See [28, Lemma 3.3.1].
Lemma 3.11. Let |β| = k. Then we have
∂ β K[g] 2 L 2 v g 2 L 2 v + |α|=k ∂ α g 2 L 2 v . (3.71) Also, for any small η > 0, there exists C k,η > 0 such that for any g(v) ∈ H k v (R 3 ) and β ′ ≤ β, we have ∂ β ′ K[g] 2 L 2 v ≤ C k,η g 2 L 2 v + η |α|=k ∂ α g 2 L 2 v . (3.72)
Proof. Due to the standard interpolation estimate
∇ j v g L 2 v g L 2 v ∇ k v g L 2 v (3.73)
for 0 ≤ j ≤ k, it suffices to justify the L 2 boundedness of ∂ β K. Furthermore, it suffices to show that ∂ β K is compact.
We introduce substitution w = v − u. Then we regroup K 1 to obtain
K 1 [g] = M − 1 2 (v) R 3 S 2 |ω · w| µ(v − w)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |v−w| 2 +|v| 2 M − 1 2 (u)g(u) dωdw (3.74) = R 3 S 2 |ω · w| µ(v − w)M − 1 2 (v − w) ̺ 2 M − 1 2 (v)µ(v) µ(u ′ )µ(v ′ )e |v−w| 2 +|v| 2 g(u) dωdw = R 3 S 2 A(w, v, ω)B(v)C(w, v, ω)g(v − w) dωdw.
Then similar to the proof of Lemma 3.5, after taking v derivatives, we know
|∂ β1 A| e − 1 2 |v−w| 2 , (3.75) |∂ β2 B| e − 1 2 |v| 2 , (3.76) |∂ β3 C| e − 1 2 |u ′ | 2 − 1 2 |v ′ | 2 .
(3.77)
Hence, following the proof of Lemma 3.7, we know ∂ β K 1 is compact.
Similarly, for K 2 , using the substitution w = v − u and regrouping, we obtain (3.78)
K 2 [g] = M − 1 2 (v) R 3 S 2 |ω · w| µ(v − w)µ(v)µ(u ′ )µ(v ′ )̺ 2 e |v−w| 2 +|v| 2 M − 1 2 (u ′ )g(u ′ ) + M − 1 2 (v ′ )g(v ′ ) dωdw = R 3 S 2 |ω · w| µ 1 2 (v − w) ̺ 2 M − 1 2 (v)µ 1 2 (v) µ(u ′ )µ(v ′ )e |v−w| 2 +|v| 2 µ 1 2 (v − w)µ 1 2 (v) M − 1 2 (u ′ )g(u ′ ) + M − 1 2 (v ′ )g(v ′ ) dωdw = R 3 S 2 A(w, v, ω)B(v)C(w, v, ω)D(w, v, ω) dωdw.
Here, A, B, C can be handled as in K 1 case, so we focus on D. In particular,
∇ v µ 1 2 (v − w)µ 1 2 (v)M − 1 2 (u ′ ) (3.79) = µ − 1 2 (v − w)µ − 1 2 (v)M 1 2 (u ′ ) ∇ v M −1 (u ′ ) µ(v − w)µ(v) + ∇ v µ(v − w)µ(v) M −1 (u ′ ) .
Then direct computation reveals that
∇ v M −1 (u ′ ) |u ′ | e |u ′ | 2 , (3.80) ∇ v µ(v − w)µ(v) |v| + |v − w| e −|v| 2 −|v−w| 2 . (3.81)
Hence, we know
∇ v µ 1 2 (v − w)µ 1 2 (v)M − 1 2 (u ′ ) |u ′ | + |v| + |v − w| e − 1 2 |v ′ | 2 (3.82) |v ′ | + |v| + |v − w| e − 1 2 |v ′ | 2 1 + |v| + |v − w| e − 1 4 |v ′ | 2 .
Similar technique justifies that
∇ v µ 1 2 (v − w)µ 1 2 (v)M − 1 2 (v ′ ) 1 + |v| + |v − w| e − 1 4 |u ′ | 2 . (3.83)
The similar structure will be preserved when taking higher-order v derivatives. Therefore, we know
|∂ β4 D| 1 + |v| |β4| + |v − w| |β4| e − 1 4 |u ′ | 2 + e − 1 4 |v ′ | 2 |∂ β4 g(u ′ )| + |∂ β4 g(v ′ )| . (3.84) Here, 1 + |v| |β4| + |v − w| |β4|
can be handled by A and B. Summarizing all above, we know that v derivatives of K 2 will not change its fundamental structure, so we may follow the proof of Lemma 3.7 to show that ∂ β K 2 is compact. Since L 2 ν is a Hilbert space, based on the Eberlain-Shmulyan theorem, we have the weakly convergent sequence (up to extracting a subsequence with an abuse of notation) g n ⇀ g in L 2 ν . Therefore, by the weak semi-continuity, we have
g L ν v ≤ 1. (3.87) Notice that L[g n ], g n = g n 2 L ν v − K[g n ], g n = 1 − K[g n ], g n . (3.88)
Since K is a compact operator on L ν v , we know it maps weakly convergent sequence into strongly convergent sequence, i.e.
lim n→∞ K[g n ] − K[g] L ν v = 0. (3.89)
Hence, we naturally have
lim n→∞ K[g n ], g n − K[g], g = 0.
(3.90) Therefore, using (3.86), we may direct take limit n → ∞ in (3.88) to get
L[g], g = 1 − K[g], g = 0. (3.91)
On the other hand, the above equality may be written as
L[g], g = 1 − g 2 L ν v + g 2 L ν v − K[g], g = 0. (3.92)
Based on the weak semi-continuity, we just proved that the first term is non-negative. Also, the second term is actually L[g], g which is also non-negative due to Lemma 3.2 (1). Hence, both of them must be zero, i.e. L[g] = 0 and g 2 L ν v = 1. Then based on Lemma 3.2 (3), we know
g = M 1 2 a + b · v + c |v| 2 . (3.93)
Then our assumption P[g n ] = 0 implies that the limit P[g] = 0, which means a = c = 0 and b = 0. Therefore, we must have g = 0, which contradicts with g L ν v = 1.
Nonlinear Estimates
Recall that
Γ[f 1 , f 2 ; f 3 ] = M − 1 2 Q M 1 2 f 1 , M 1 2 f 2 + θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; µ + θM − 1 2 Q M 1 2 f 1 , µ; M 1 2 f 3 (3.94) + θM − 1 2 Q µ, M 1 2 f 2 ; M 1 2 f 3 + θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 f 3 . Lemma 3.13. For any f (v), we have P Γ[f, f ; f ] = 0. (3.95) Proof. Since Q µ + M 1 2 f, µ + M 1 2 f ; µ + M 1 2 f = Q[µ, µ; µ] + M 1 2 L[f ] + M 1 2 Γ[f, f ; f ] (3.96) = M 1 2 L[f ] + M 1 2 Γ[f, f ; f ],
and by direct computation in the same spirit as the proof of Corollary 2.3, and for fixed ω and u, the Jacobian of the transformation v → v ′ satisfies
P M − 1 2 Q[µ + M 1 2 f, µ + M 1 2 f ; µ + M 1 2 f ] = P L[f ] = 0,(3.dv ′ dv ≥ 1 8 . (3.100)
Proof. This is based on the proof of Alexandre-Desvillettes-Villani-Wennberg [1, Lemma 1].
Lemma 3.15. Let f i for i = 1, 2, 3, and g be smooth functions. Then we have
R 3 Γ[f 1 , f 2 ; f 3 ]g dv (3.101) g L 2 v f 1 L 2 v f 2 L 2 v f 3 L 2 v + g L ν v f a L 2 v f b L ν v + g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v |f 1 | sup v |f 2 | f 3 L ν v + g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v ν 1 2 f 1 f 2 L 2 v + f 1 L 2 v sup v ν 1 2 f 2 f 3 L 2 v + g L ν v min f 1 L ν v sup v |f 3 | , sup v ν 1 2 f 1 f 3 L 2 v + g L ν v min f 2 L ν v sup v |f 3 | , sup v ν 1 2 f 2 f 3 L 2 v .
Also, we have
R 3 Γ[f 1 , f 2 ; f 3 ]g dv sup v ν 3 g f a L 2 v f b L 2 v + g L ν v + sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 v ,(3.
102)
and
Γ[f 1 , f 2 ; f 3 ]g L 2 v sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 v + f a L 2 v f b L 2 v (3.103) + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 1 | sup v |f 2 | f 3 L 2 v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 2 | f 1 L 2 v f 3 L 2 v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 1 | f 2 L 2 v f 3 L 2 v .
Here (a, b) runs all combinations of {1, 2, 3}.
Proof. We look at formula (3.94) for the Γ operator and estimate term by term.
Step 1. Estimate of quadratic terms.
For M − 1 2 Q M 1 2 f 1 , M 1 2 f 2 and θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; µ , based on [32, Lemma 2.3], we have R 3 M − 1 2 Q M 1 2 f 1 , M 1 2 f 2 g dv f 1 L ν v f 2 L 2 v g L ν v , (3.104) R 3 M − 1 2 Q M 1 2 f 1 , M 1 2 f 2 g dv sup v ν 3 g f 1 L 2 v f 2 L 2 v , (3.105) M − 1 2 Q M 1 2 f 1 , M 1 2 f 2 g L 2 v sup v |νg| f 1 L 2 v f 2 L 2 v ,(3.106)
and
R 3 θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; µ g dv f 1 L ν v f 2 L 2 v g L ν v , (3.107) R 3 θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; µ g dv sup v ν 3 g f 1 L 2 v f 2 L 2 v , (3.108) θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; µ g L 2 v sup v |νg| f 1 L 2 v f 2 L 2 v . (3.109)
Step 2-1. Estimate of cubic terms for (3.101). We then focus on
θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 f 3 . Recalling (3.41), we have θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 f 3 (3.110) e 1 2 |v| 2 R 3 S 2 q(ω, |v − u|)e − 1 2 |u ′ | 2 − 1 2 |v ′ | 2 |f 1 (u ′ )f 2 (v ′ )| + |f 1 (v ′ )f 2 (u ′ )| e − 1 2 |u| 2 |f 3 (u)| + e − 1 2 |v| 2 |f 3 (v)| dωdu + e 1 2 |v| 2 R 3 S 2 q(ω, |v − u|)e − 1 2 |u| 2 − 1 2 |v| 2 |f 1 (u)f 2 (v)| + |f 1 (v)f 2 (u)| e − 1 2 |u ′ | 2 |f 3 (u ′ )| + e − 1 2 |v ′ | 2 |f 3 (v ′ )| dωdu R 3 S 2 q(ω, |v − u|)e − 1 2 |u| 2 |f 1 (u ′ )f 2 (v ′ )| + |f 1 (v ′ )f 2 (u ′ )| e − 1 2 |u| 2 |f 3 (u)| + e − 1 2 |v| 2 |f 3 (v)| dωdu + R 3 S 2 q(ω, |v − u|)e − 1 2 |u| 2 |f 1 (u)f 2 (v)| + |f 1 (v)f 2 (u)| e − 1 2 |u ′ | 2 |f 3 (u ′ )| + e − 1 2 |v ′ | 2 |f 3 (v ′ )| dωdu = J 1 + J 2 .
For both parts, we have the naive bound
q(ω, |v − u|)e − 1 2 |u| 2 ν(v). (3.111) Also, noticing that q(ω, |v − u|) = ω ω · (v − u) = |u − u ′ |, we have q(ω, |v − u|)e − 1 2 |u| 2 ν(u ′ ). (3.112) For J 1 , we split R 3 J 1 (v)g(v) dv (3.113) R 3 R 3 S 2 q(ω, |v − u|)e − 1 2 |u| 2 |f 1 (u ′ )f 2 (v ′ )| + |f 1 (v ′ )f 2 (u ′ )| e − 1 2 |u| 2 |f 3 (u)| |g(v)| dωdudv + R 3 R 3 S 2 q(ω, |v − u|)e − 1 2 |u| 2 |f 1 (u ′ )f 2 (v ′ )| + |f 1 (v ′ )f 2 (u ′ )| e − 1 2 |v| 2 |f 3 (v)| |g(v)| dωdudv = I 11 + I 12 .
We may directly use Cauchy's inequality to bound I 11 ,
I 11 R 3 R 3 S 2 f 2 1 (u ′ )f 2 2 (v ′ ) + f 2 1 (v ′ )f 2 2 (u ′ ) dωdudv 1 2 (3.114) × R 3 R 3 S 2 ν 2 (v)e −|u| 2 f 2 3 (u)g 2 (v) dωdudv 1 2 g L 2 v f 1 L 2 v f 2 L 2 v f 3 L 2 v .
The estimate of I 12 is a bit complicated due to g(v)f 3 (v) term. We may bound it in two different ways
I 12 R 3 R 3 S 2 ν(u ′ ) f 2 1 (u ′ )f 2 2 (v ′ ) + f 2 1 (v ′ )f 2 2 (u ′ ) dωdudv 1 2 (3.115) × R 3 R 3 S 2 |v − u| e − |u| 2 2 e −|v| 2 f 2 3 (v)g 2 (v) dωdudv 1 2 g L ν v f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | .
and
I 12 sup v |f 1 | sup v |f 2 | R 3 R 3 S 2 |v − u| g 2 (v)e − 1 2 |u| 2 e − 1 2 |v| 2 dωdudv 1 2 (3.116) × R 3 R 3 S 2 |v − u| f 2 3 (v)e − 1 2 |u| 2 e − 1 2 |v| 2 dωdudv 1 2 g L 2 v sup v |f 1 | sup v |f 2 | f 3 L 2 v .
Hence, we know
I 12 g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v |f 1 | sup v |f 2 | f 3 L 2 v . (3.117)
In total, we have
R 3 J 1 (v)g(v) dv g L 2 v f 1 L 2 v f 2 L 2 v f 3 L 2 v (3.118) + g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v |f 1 | sup v |f 2 | f 3 L 2 v .
For J 2 , we may use two different ways to bound it:
R 3 J 2 (v)g(v) dv sup v |f 3 | R 3 R 3 S 2 |v − u| g 2 (v)e − 1 2 |u| 2 dωdudv 1 2 (3.119) × R 3 R 3 S 2 ν(v)f 2 i (v)f 2 j (u) dωdudv 1 2 g L ν v f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | ,
and using the fact that |q(ω, |v − u|)| = |v − v ′ |,
R 3 J 2 (v)g(v) dv R 3 R 3 S 2 ν(v)g 2 (v)f 2 i (u) dωdudv 1 2 (3.120) × R 3 R 3 S 2 ν(u ′ )f 2 j (v)e −|u ′ | 2 f 2 3 (u ′ ) dωdudv 1 2 + R 3 R 3 S 2 ν(v)g 2 (v)f 2 i (u) dωdudv 1 2 × R 3 R 3 S 2 |v − v ′ | e − 1 2 |u| 2 f 2 j (v)e −|v ′ | 2 f 2 3 (v ′ ) dωdudv 1 2 = I 21 + I 22 .
Using Lemma 3.14 with substitution u → u ′ , we may obtain
I 21 g L ν v f i L 2 v f j L 2 v f 3 L 2 v .
(3.121)
Using Lemma 3.14 with substitution v → v ′ , we may obtain
I 22 sup v ν 1 2 f j g L ν v f i L 2 v f 3 L 2 v .
(3.122)
In total, we know
R 3 J 2 (v)g(v) dv g L ν v f 1 L 2 v f 2 L 2 v + sup v ν 1 2 f 1 f 2 L 2 v + f 1 L 2 v sup v ν 1 2 f 2 f 3 L 2 v .
Hence, we know
R 3 J 2 (v)g(v) dv g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , (3.123) f 1 L 2 v f 2 L 2 v + sup v ν 1 2 f 1 f 2 L 2 v + f 1 L 2 v sup v ν 1 2 f 2 f 3 L 2 v .
Summarizing all above, we know
R 3 θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 f 3 g dv (3.124) g L 2 v f 1 L 2 v f 2 L 2 v f 3 L 2 v + g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v |f 1 | sup v |f 2 | f 3 L ν v + g L ν v min f 1 L ν v f 2 L 2 v + f 1 L 2 v f 2 L ν v sup v |f 3 | , sup v ν 1 2 f 1 f 2 L 2 v + f 1 L 2 v sup v ν 1 2 f 2 f 3 L 2 v .
Step 2-2. Estimate of cubic terms for (3.102). On the other hand, similar to the estimates in Step 2-1, if we take supremum over v on g, we have
R 3 J 1 (v)g(v) dv g L ν v + sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 v ,(3.125)
and
R 3 J 2 (v)g(v) dv g L ν v + sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 v , Hence, we have R 3 θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 g dv g L ν v + sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 v . (3.126)
Step 2-3. Estimate of cubic terms for (3.103). Also, in a similar fashion, for any h ∈ L 2 v , we have
R 3 J 1 (v)g(v)h(v) dv (3.127) sup v |νg| h L 2 v f 1 L 2 v f 2 L 2 v f 3 L 2 v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 1 | sup v |f 2 | f 3 L 2 v , and R 3 J 2 (v)g(v)h(v) dv (3.128) sup v |νg| h L 2 v min sup v |f 2 | f 1 L 2 v f 3 L 2 v , sup v |f 3 | f 1 L 2 v f 2 L 2 v + min sup v |f 1 | f 2 L 2 v f 3 L 2 v , sup v |f 3 | f 1 L 2 v f 2 L 2 v .
Therefore, due to duality of L 2 , we have
θM − 1 2 Q M 1 2 f 1 , M 1 2 f 2 ; M 1 2 g L 2 v (3.129) sup v |νg| f 1 L 2 v f 2 L 2 v f 3 L 2 x,v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 1 | sup v |f 2 | f 3 L 2 v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 2 | f 1 L 2 v f 3 L 2 v + min sup v |f 3 | f 1 L 2 v f 2 L 2 v , sup v |f 1 | f 2 L 2 v f 3 L 2 v .
Step 3. More estimate of quadratic terms. By a similar argument, we can handle θM − 1 2 Q M
R 3 θM − 1 2 Q M 1 2 f 1 , µ; M 1 2 f 3 g dv (3.130) g L ν v f 1 L 2 v f 3 L ν v + g L ν v min f 1 L ν v sup v |f 3 | , sup v ν 1 2 f 1 f 3 L 2 v , and R 3 θM − 1 2 Q µ, M 1 2 f 2 ; M 1 2 f 3 g dv (3.131) g L ν v f 2 L 2 v f 3 L ν v + g L ν v min f 2 L ν v sup v |f 3 | , sup v ν 1 2 f 2 f 3 L 2 v .
In addition, for (3.102),
R 3 θM − 1 2 Q M 1 2 f 1 , µ; M 1 2 f 3 g dv g L ν v + sup v |νg| f 1 L 2 v f 3 L 2 v ,(3.132)
and
R 3 θM − 1 2 Q µ, M 1 2 f 2 ; M 1 2 f 3 g dv g L ν v + sup v |νg| f 2 L 2 v f 3 L 2 v . (3.133)
Also, for (3.103),
θM − 1 2 Q M 1 2 f 1 , µ; M 1 2 f 3 g L 2 v (3.134) sup v |νg| f 1 L 2 v f 3 L 2 v + sup v |νg| min f 1 L 2 v sup v |f 3 | , sup v |f 1 | f 3 L 2 v , and θM − 1 2 Q µ, M 1 2 f 2 ; M 1 2 f 3 g L 2 v (3.135) sup v |νg| f 2 L 2 v f 3 L 2 v + sup v |νg| min f 2 L 2 v sup v |f 3 | , sup v |f 2 | f 3 L 2 v . (3.136)
Remark 3.16. The nonlinear estimate for the quantum case is much more complicated than the classical Boltzmann version. In particular, unlike the classical Boltzmann equation, here we need sup v estimate via Sobolev embedding, which implies that we have to consider velocity derivatives.
Lemma 3.17. Let f i for i = 1, 2, 3, and g be smooth functions. Then we have
R 3 ∂ γ β Γ[f 1 , f 2 ; f 3 ]g dv (3.137) g L 2 v f ′ 1 L 2 v f ′ 2 L 2 v f ′ 3 L 2 v + g L ν v f ′ a L 2 v f ′ b L ν v + g L ν v min f ′ 1 L ν v f ′ 2 L 2 v + f ′ 1 L 2 v f ′ 2 L ν v sup v |f ′ 3 | , sup v |f ′ 1 | sup v |f ′ 2 | f ′ 3 L ν v + g L ν v min f ′ 1 L ν v f ′ 2 L 2 v + f ′ 1 L 2 v f ′ 2 L ν v sup v |f ′ 3 | , sup v ν 1 2 f ′ 1 f ′ 2 L 2 v + f ′ 1 L 2 v sup v ν 1 2 f ′ 2 f ′ 3 L 2 v + g L ν v min f ′ 1 L ν v sup v |f ′ 3 | , sup v ν 1 2 f ′ 1 f ′ 3 L 2 v + g L ν v min f ′ 2 L ν v sup v |f ′ 3 | , sup v ν 1 2 f ′ 2 f ′ 3 L 2 v .
Thus, we have
R 3 ∂ γ β Γ[f 1 , f 2 ; f 3 ]g dv sup v ν 3 g f ′ a L 2 v f ′ b L 2 v + g L ν v + sup v |νg| f ′ i L 2 v f ′ j L 2 v f ′ k L 2 v , and ∂ γ β Γ[f 1 , f 2 ; f 3 ]g L 2 v sup v |νg| f ′ 1 L 2 v f ′ 2 L 2 v f ′ 3 L 2 v + f ′ a L 2 v f ′ b L 2 v (3.138) + min sup v |f ′ 3 | f ′ 1 L 2 v f ′ 2 L 2 v , sup v |f ′ 1 | sup v |f ′ 2 | f ′ 3 L 2 v + min sup v |f ′ 3 | f ′ 1 L 2 v f ′ 2 L 2 v , sup v |f ′ 2 | f ′ 1 L 2 v f ′ 3 L 2 v + min sup v |f ′ 3 | f ′ 1 L 2 v f ′ 2 L 2 v , sup v |f ′ 1 | f ′ 2 L 2 v f ′ 3 L 2 v .
Here Proof. The proof is similar to that of Lemma 3.15 and Lemma 3.5. Note that the spatial and velocity derivatives will be distributed among the three arguments in Γ, but it will not change the fundamental structure here.
Local Solutions
• for fermions θ = −1, if 0 ≤ F 0 (x, v) ≤ 1, then 0 ≤ F (t, x, v) ≤ 1; • for bosons θ = 1, if F 0 (x, v) ≥ 0, then F (t, x, v) ≥ 0.
Proof. In the following, we mainly study Q[F, F ; F ], so we rearrange the terms as
Q[F, F ; F ] = Q p [F, F ; F ] − Q q [F, F ; F ],(3.R 3 S 2 q(ω, |v − u|)F (u ′ )G(v ′ ) 1 + θH(u) dωdu, (3.142) Q q [F, G; H] := R 3 S 2 q(ω, |v − u|) − θF (u ′ )G(v ′ )H(v) + F (u)H(v) 1 + θG(u ′ ) + θG(v ′ ) dωdu. (3.143) Q q [F, F ; F ]withQ p [F, G] = R 3 S 2 q(ω, |v − u|) − θF (u ′ )G(v ′ ) + F (u) 1 + θG(u ′ ) + θG(v ′ ) dωdu. (3.145)
Similar to the decomposition in (3.141), we may also decompose Γ as
Γ[f, g; h] = Γ p [f, g; h] − Γ q [f, g; h],(3.146)
where Γ p comes from the linearization of Q p , and Γ q from Q p .
Step 1. Boundedness. Define the iteration sequence via
∂ t F n+1 + v · ∇ x F n+1 + Q q [F n , F n ; F n+1 ] = Q p [F n , F n ; F n ], (3.147) with F n+1 (0, x, v) = F 0 (x, v). (3.148)
This is equivalent to the perturbation form
∂ t f n+1 + v · ∇ x f n+1 + νf n+1 = K[f n ] + Γ p [f n , f n ; f n ] − Γ q [f n , f n ; f n+1 ], (3.149) with f n+1 (0, x, v) = f 0 (x, v). (3.150)
The iteration starts with f 0 (t, x, v) = f 0 (x, v). Taking ∂ γ β on both sides of (3.149), we obtain
∂ t ∂ γ β f n+1 + v · ∇ x ∂ γ β f n+1 + ∂ γ β νf n+1 = ∂ γ β K[f n ] + ∂ γ β Γ p [f n , f n ; f n ] − ∂ γ β Γ q [f n , f n ; f n+1 ]. (3.151)
Multiplying ∂ γ β f n+1 on both sides of (3.151) and integrating over Ω × R 3 , we obtain 1 2
d dt ∂ γ β f n+1 2 L 2 x,v + Ω×R 3 ∂ γ β f n+1 · ∂ γ β νf n+1 (3.152) = Ω×R 3 ∂ γ β f n+1 · ∂ γ β K[f n ] + Ω×R 3 ∂ γ β f n+1 · ∂ γ β Γ p [f n , f n ; f n ] − Ω×R 3 ∂ γ β f n+1 · ∂ γ β Γ q [f n , f n ; f n+1 ].
Let η > 0 be sufficiently small and C η > 0. For the second term on the LHS of (3.152), using Lemma 3.4 and Lemma 3.5, we have
Ω×R 3 ∂ γ β f n+1 · ∂ γ β νf n+1 (3.153) Ω×R 3 ∂ γ β f n+1 · ν∂ γ β f n+1 − β ′ <β Ω×R 3 ∂ γ β f n+1 · ∂ γ β ′ νf n+1 ∂ γ β f n+1 2 L ν x,v − η ∂ γ β f n+1 2 L 2 x,v − C η β ′ <β ∂ γ β ′ f n+1 2 L 2 x,v ,
For the first term on the RHS of (3.152), using Lemma 3.11, we have
Ω×R 3 ∂ γ β f n+1 · ∂ γ β K[f n ] η ∂ γ β f n+1 2 L 2 x,v + C η β ′ ≤β ∂ γ β ′ f n 2 L 2 x,v . (3.154)
For the second term on the RHS of (3.152), using the first inequality in Lemma 3.17 with sup v falling on the term with the lowest v derivative, combining with Sobolev embedding theorem
H 2 v ֒→ L ∞ v , we obtain Ω×R 3 ∂ γ β f n+1 · ∂ γ β Γ p [f n , f n ; f n ] ∂ γ β f n+1 L ν x,v β ′ ≤β ∂ γ β ′ f n L ν x,v · β ′ ≤β ∂ γ β ′ f n 2 L 2 x,v ,(3.155)
and
Ω×R 3 ∂ γ β f n+1 · ∂ γ β Γ q [f n , f n ; f n+1 ] (3.156) ∂ γ β f n+1 L ν x,v β ′ ≤β ∂ γ β ′ f n+1 L ν x,v · β ′ ≤β ∂ γ β ′ f n 2 L 2 x,v + β ′ ≤β ∂ γ β ′ f n+1 L 2 x,v · β ′ ≤β ∂ γ β ′ f n L 2 x,v · β ′ ≤β ∂ γ β ′ f n L ν x,v .
Summarizing all above and running over |γ| + |β| ≤ N , we have Therefore, we know this iteration is uniformly bounded.
d dt f n+1 2 + f n+1 2 ν f n+1 2 + |||f n ||| ν |||f n ||| 2 f n+1 ν (3.157) + |||f n ||| 2 + |||f n ||| 2 f n+1 2 ν + |||f n ||| 2 + |||f n ||| ν |||f n ||| f n+1 ν f n+1 .
Then integrating over [0, t] and using the definition of E, with the help of Cauchy's inequality, we have
E[f n+1 (t)] E[f 0 ] + t sup s∈[0,t] E[f n+1 (s)] + t sup s∈[0,t] E[f n (s)] + sup s∈[0,t] E[f n (s)] sup s∈[0,t] E[f n+1 (s)].
Step 2. Contraction. On the other hand, taking the difference of the equations for f n+1 and f n , we obtain
∂ t f n+1 − f n + v · ∇ x f n+1 − f n + ν f n+1 − f n (3.161) = K[f n − f n−1 ] + Γ p [f n − f n−1 , f n ; f n ] + Γ p [f n−1 , f n − f n−1 ; f n−1 ] + Γ p [f n−1 , f n−1 ; f n − f n−1 ] − Γ q [f n − f n−1 , f n ; f n+1 ] − Γ q [f n−1 , f n − f n−1 ; f n+1 ] − Γ q [f n−1 , f n−1 ; f n+1 − f n ], with f n+1 − f n (0, x, v) = 0. (3.162)
Similarly to the above argument, we first take ∂ γ β , multiply ∂ γ β f n+1 − f n on both sides of (3.161), and integrate over Ω × R 3 . Using similar techniques as in proving boundedness, we obtain
sup s∈[0,T ] E f n+1 − f n (s) sup s∈[0,T ] E f n − f n−1 (s), (3.163)
where we use the nonlinear estimates
Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ p [f n − f n−1 , f n ; f n ] (3.164) sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E[f n ](s) 1 2 sup s∈[0,T ] E[f n ](s) 1 2 M sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 , Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ p [f n−1 , f n − f n−1 ; f n ] (3.165) sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E[f n−1 ](s) 1 2 sup s∈[0,T ] E[f n ](s) 1 2 M sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 , Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ p [f n−1 , f n−1 ; f n − f n−1 ] (3.166) sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E[f n−1 ](s) M sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E f n − f n−1 (s) 1 2 , and Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ q [f n − f n−1 , f n ; f n+1 ] (3.167) sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E[f n ](s) 1 2 sup s∈[0,T ] E[f n+1 ](s) 1 2 M sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E f n+1 − f n (s) 1 2 , Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ q [f n−1 , f n − f n−1 ; f n+1 ] (3.168) sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E f n+1 − f n (s) 1 2 sup s∈[0,T ] E[f n−1 ](s) 1 2 sup s∈[0,T ] E[f n+1 ](s) 1 2 M sup s∈[0,T ] E f n − f n−1 (s) 1 2 sup s∈[0,T ] E f n+1 − f n (s) 1 2 , Ω×R 3 ∂ γ β f n+1 − f n · ∂ γ β Γ q [f n−1 , f n−1 ; f n+1 − f n ] (3.169) sup s∈[0,T ] E f n+1 − f n (s) · sup s∈[0,T ] E[f n−1 ](s) M sup s∈[0,T ] E f n − f n−1 (s).
Hence, we know the iteration is a contraction, and thus defines a (uniform) Cauchy sequence. Therefore, we know that there exists a classical solution f . The uniqueness follows naturally from the contraction proof and the Gronwall's inequality. The inequality (3.157) also justifies the continuity of E[f (t)] with respect to t. The positivity of F follows from a standard induction. Our iteration is actually
∂ t F n+1 + v · ∇ x F n+1 +Q p [F n , F n ]F n+1 = Q q [F n , F n ; F n ]. (3.170)
We may verify that for θ = ±1, F n satisfies the positivity estimate
Q q [F n , F n ; F n ] ≥ 0. (3.171)
In addition, for θ = −1, F n satisfies the positivity estimate
(3.172) Q p [F n , F n ] = R 3 S 2 q(ω, |v − u|) − θF n (u ′ )F n (v ′ ) 1 + θF n (u) + F n (u) 1 + θF n (u ′ ) 1 + θF n (v ′ ) dωdu ≥ 0.
Hence, by solving the ODE for F n+1 , we may derive the positivity naturally.
Global Solutions for Ω = T 3
In this section, we will justify the global well-posedness and decay for Ω = T 3 case.
Positivity Estimate for
L Lemma 3.19. Assume f (t, x, v) satisfies (3.15) for t ∈ [0, T ] with T ≥ 1. Also, f (t, x, v) satisfies sup t∈[0,T ] E[f (t)] ≤ M . Then for 0 ≤ s < t ≤ T , we have |γ|≤N ∂ γ f (t) 2 L 2 x,v + t s ∂ γ f (τ ) 2 L ν x,v dτ ≤ e C(t−s) |γ|≤N ∂ γ f (s) 2 L 2 x,v , (3.173) and |γ|≤N t s ∂ γ f (τ ) 2 L ν x,v dτ ≥ 1 − e C(t−s) |γ|≤N ∂ γ f (s) 2 L 2 x,v . (3.174)
Proof. Similar to the proof of Theorem 3.18, applying ∂ γ on both sides of the equation (3.15), multiplying ∂ γ f , and integrating over Ω × R 3 , we have
1 2 d dt ∂ γ f 2 L 2 x,v + Ω×R 3 ∂ γ f · ν∂ γ f = Ω×R 3 ∂ γ f · K ∂ γ f + Ω×R 3 ∂ γ f · ∂ γ Γ[f, f ; f ].d dt |γ|≤N ∂ γ f 2 L 2 x,v + |γ|≤N ∂ γ f 2 L ν x,v |γ|≤N ∂ γ f 2 L 2 x,v + M |γ|≤N ∂ γ f 2 L 2 x,v . (3.176)
When M is small, we may absorb the last term into the LHS to obtain d dt
|γ|≤N ∂ γ f 2 L 2 x,v + |γ|≤N ∂ γ f 2 L ν x,v |γ|≤N ∂ γ f 2 L 2 x,v . (3.177)
Then by Gronwall's inequality, we obtain
|γ|≤N ∂ γ f (t) 2 L 2 x,v ≤ e C(t−s) |γ|≤N ∂ γ f (s) 2 L 2 x,v .
(3.178)
Then we integrate over τ ∈ [s, t] to obtain |γ|≤N t s ∂ γ f (τ ) 2 L ν x,v dτ ≤ e C(t−s) |γ|≤N ∂ γ f (s) 2 L 2 x,v . (3.179)
This justifies the first inequality in the lemma.
On the other hand, we may rearrange the terms in (3.175)
1 2 d dt ∂ γ f 2 L 2 x,v = − Ω×R 3 ∂ γ f · ν∂ γ f + Ω×R 3 ∂ γ f · K ∂ γ f + Ω×R 3 ∂ γ f · ∂ γ Γ[f, f ; f ]. (3.180)
Similar to the above argument, we have
d dt |γ|≤N ∂ γ f 2 L 2 x,v − |γ|≤N ∂ γ f 2 L ν x,v − |γ|≤N ∂ γ f 2 L 2 x,v − M |γ|≤N ∂ γ f 2 L 2 x,v , (3.181) which yields d dt |γ|≤N ∂ γ f 2 L 2 x,v − |γ|≤N ∂ γ f 2 L ν x,v , (3.182) Integrating over τ ∈ [s, t], we have |γ|≤N ∂ γ f (t) 2 L ν x,v |γ|≤N ∂ γ f (s) 2 L 2 x,v − |γ|≤N t s ∂ γ f (τ ) 2 L ν x,v dτ. (3.183)
Then by Gronwall's inequality, we obtain
|γ|≤N t s ∂ γ f (τ ) 2 L ν x,v dτ ≥ 1 − e C(t−s) |γ|≤N ∂ γ f (s) 2 L 2 x,v . (3.184)
This justifies the second inequality in the lemma.
|γ|≤N 1 0 T 3 R 3 L ∂ γ f (t) · ∂ γ f (t) dvdxdt ≥ δ M |γ|≤N 1 0 ∂ γ f (t) 2 L ν x,v dt. (3.185)
Proof. We prove by contradiction. If the result is not true, then there exists a sequence of solutions
f n (t, x, v) ∞ n=1 to (3.15) such that sup t∈[0,1] |γ|+|β|≤N ∂ γ β f n (t) 2 L 2 x,v ≤ M, (3.186) and 0 ≤ |γ|≤N 1 0 T 3 R 3 L ∂ γ f n (t) · ∂ γ f n (t) ≤ 1 n |γ|≤N 1 0 ∂ γ f n (t) 2 L ν x,v . (3.187)
We normalize
Z n (t, x, v) = f n (t, x, v) |γ|≤N 1 0 ∂ γ f n (t) 2 L ν x,v dt . (3.188)
Then noticing that L = νI − K, we know
0 ≤ 1 − |γ|≤N 1 0 T 3 R 3 K ∂ γ Z n (t) · ∂ γ Z n (t) ≤ 1 n . (3.189)
On the other hand, by Lemma 3.19, we know 3.190) and
|γ|≤N ∂ γ f n (t) 2 L 2 x,v |γ|≤N ∂ γ f n (0) 2 L 2 x,v ,(|γ|≤N 1 0 ∂ γ f n (t) 2 L ν x,v |γ|≤N ∂ γ f n (0) 2 L 2 x,v . (3.191)
Then based on the definition of Z n , we know that
sup t∈[0,1] |γ|≤N ∂ γ Z n (t) 2 L 2 x,v 1. (3.192)
Based on the equation (3.15), we know Z n satisfies
∂ t Z n + v · ∇ x Z n + L[Z n ] = Γ[f n , f n ; Z n ]. (3.193)
After applying ∂ γ on both sides, we obtain
∂ t ∂ γ Z n + v · ∇ x ∂ γ Z n + L ∂ γ Z n = ∂ γ Γ[f n , f n ; Z n ]. (3.194)
Also, we have the conservation laws (3.197) Due to the boundedness, we can extract weakly convergent subsequence in (3.198) where Z and f are the limit functions, respectively.
Ω R 3 Z n (t, x, v)M 1 2 (v) dvdx = 0, (Mass) (3.195) Ω R 3 Z n (t, x, v)M 1 2 (v)v i dvdx = 0, (Momentum) (3.196) Ω R 3 Z n (t, x, v)M 1 2 (v) |v| 2 dvdx = 0. (Energy)L 2 ([0, 1] × T 3 × R 3 ) to get ∂ γ Z n ⇀ ∂ γ Z, ∂ γ f n ⇀ ∂ γ f,
Step
1. K[∂ γ Z n ] → K[∂ γ Z] in L 2 ([0, 1] × T 3 × R 3 ) for |γ| ≤ N .
Based on Lemma 3.7, we know K is a bounded operator in L 2 (T 3 × R 3 ). Then for any ε > 0, we have
ε 0 K[∂ γ Z n ] 2 L 2 x,v + 1 1−ε K[∂ γ Z n ] 2 L 2 x,v ε. (3.199)
Hence, there is no time concentration in a neighborhood of t = 0 or t = 1. Then it suffices to consider
K[∂ γ Z n ] → K[∂ γ Z] in L 2 ([ε, 1 − ε] × T 3 × R 3 ). Notice that K[∂ γ Z n ] = R 3 k(u, v)∂ γ Z n (u)du.(k(u, v) = k m (u, v) + k(u, v) − k m (u, v) , (3.201) where k m (u, v) = k(u, v)1 {(u,v):|u−v|≥ 1 m and |v|≤m} ,(3.202)satisfying 1 0 R 3 (k − k m )∂ γ Z n (t)du L 2 x,v ≤ ε 1 0 ∂ γ Z n (t) L 2 x,v ε. (3.203)
Then naturally k m ∈ L 2 (R 3 × R 3 ). Based on the density lemma, we may find a smooth function κ ε (u, v) = κ 1 (u)κ 2 (v) with compact support satisfying
k m − κ ε L 2 u,v ε. (3.204)
Hence, we know
1 0 R 3 (k m − κ ε )∂ γ Z n (t)du L 2 x,v ≤ k m − κ ε L 2 u,v 1 0 ∂ γ Z n (t) L 2 x,v ε. (3.205)
Then it suffices to justify
R 3 κ 1 (u)∂ γ Z n (t)du → R 3 κ 1 (u)∂ γ Z(t)du (3.206)
in L 2 ([ε, 1 − ε] × T 3 ) since we can later multiply κ 2 (v) and integrating over v ∈ R 3 to complete the proof. Let χ(t, x) be a smooth cutoff function in (0, 1) × R 3 such that χ = 1 in [ε, 1 − ε] × T 3 . Multiplying κ 1 (v)χ(t, x) on both sides of (3.194) to obtain (3.207)
∂ t κ 1 χ∂ γ Z n + v · ∇ x κ 1 χ∂ γ Z n = −κ 1 χL ∂ γ Z n + κ 1 χ∂ γ Γ[f n , f n ; Z n ] + ∂ γ Z n ∂ t + v · ∇ x (κ 1 χ).
Due to compact support in (t, x, v) variables of κ 1 χ, we know
1 0 κ 1 χL ∂ γ Z n 2 L 2 x,v 1 0 ∂ γ Z n 2 L 2 x,v 1,(3.208)and 1 0 ∂ γ Z n ∂ t + v · ∇ x (κ 1 χ) 2 L 2 x,v 1 0 ∂ γ Z n 2 L 2 x,v 1. (3.209)
Then we are left with the nonlinear term. Based on the third inequality in Lemma 3.17 and our assumption of the lemma, we always put L 2 v,x on the term with highest-order derivative and use Sobolev embedding to handle the supremum in (x, v), to obtain
1 0 κ 1 χ∂ γ Γ[f n , f n ; Z n ] 2 L 2 x,v M 1 0 ∂ γ Z n 2 L 2 x,v + ∂ γ f n 2 L 2 x,v 1.
(3.210)
In total, we know that
∂ t κ 1 χ∂ γ Z n + v · ∇ x κ 1 χ∂ γ Z n ∈ L 2 ([0, 1] × R 3 × R 3 ). (3.211)
Then from the averaging lemma, we obtain
R 3 κ 1 χ∂ γ Z n (t, x, u)du ∈ H 1 4 ([0, 1] × R 3 ). (3.212)
Then by the compact embedding, we may extract a weakly convergent subsequence in H 1 4 , which is a strongly convergent subsequence in L 2 such that
R 3 κ 1 χ∂ γ Z n (t, x, u)du → R 3 κ 1 χ∂ γ Z(t, x, u)du.
(3.213)
Hence, our result naturally follows.
Step 2.
Z(t, x, v) = a(t, x)M 1 2 + b(t, x) · vM 1 2 + c(t, x) |v| 2 M 1 2 . From
Step 1, we know that
1 0 T 3 R 3 K ∂ γ Z n · ∂ γ Z n → 1 0 T 3 R 3 K ∂ γ Z · ∂ γ Z. (3.214)
Hence, taking limit n → ∞ in (3.189), we obtain
0 ≤ 1 − |γ|≤N 1 0 T 3 R 3 K ∂ γ Z · ∂ γ Z ≤ 0. (3.215)
Hence, we must have
|γ|≤N 1 0 T 3 R 3 K ∂ γ Z · ∂ γ Z = 1. (3.216)
On the other hand, the lower semi-continuity of ν-norm implies
|γ|≤N 1 0 ∂ γ Z(t) 2 L ν x,v ≤ 1. (3.217) Therefore, we know 0 ≤ |γ|≤N 1 0 T 3 R 3 L ∂ γ Z · ∂ γ Z (3.218) = |γ|≤N 1 0 ∂ γ Z(t) 2 L ν x,v − |γ|≤N 1 0 T 3 R 3 K ∂ γ Z · ∂ γ Z ≤ 1 − 1 = 0.
Therefore, we have
|γ|≤N 1 0 ∂ γ Z(t) 2 L ν x,v = 1,(3.219)
and |γ|≤N 1 0
T 3 R 3 L ∂ γ Z · ∂ γ Z = 0. (3.220)
In particular, the weak convergence and norm convergence imply strong convergence, i.e. ∂ γ Z n → ∂ γ Z in
L 2 ([0, 1] × R 3 × R 3 ).
Hence, we know Z belongs to the null space of L, i.e.
Z(t, x, v) = a(t, x)M 1 2 + b(t, x) · vM 1 2 + c(t, x) |v| 2 M 1 2 , (3.221)
where a, b, c are given by Z. In particular, the boundedness of Z implies
sup t∈[0,1] ∂ γ a(t) 2 L 2 x + ∂ γ b(t) 2 L 2 x + ∂ γ c(t) 2 L 2 x 1. (3.222)
Then taking limit n → ∞ in (3.194), we know that in the sense of distribution
∂ t ∂ γ Z + v · ∇ x ∂ γ Z = ∂ γ Γ[f, f ; Z]. (3.223)
Also, we have the conservation laws
Ω R 3 Z(t, x, v)M 1 2 (v) dvdx = 0, (Mass) (3.224) Ω R 3 Z(t, x, v)M 1 2 (v)v i dvdx = 0, (Momentum) (3.225) Ω R 3 Z(t, x, v)M 1 2 (v) |v| 2 dvdx = 0. (Energy) (3.226)
Step 3.
|γ|≤N 1 0 ∂ γ Z(t) 2 L ν x,v
M . If this is justified, then it contradicts (3.219) and we conclude our proof. Plugging (3.221) into (3.223), we obtain that
∇ x ∂ γ c · v |v| 2 M 1 2 + ∂ t ∂ γ c |v| 2 + v · ∇ x (v · ∂ γ ) M 1 2 (3.227) + ∂ t ∂ γ b + ∇ x ∂ γ a · vM 1 2 + ∂ t ∂ γ a M 1 2 = ∂ γ Γ[f, f ; Z].
Since v i |v| 2 M
∂ xi ∂ γ c = h γ ci , (3.229) ∂ t ∂ γ c + ∂ xi ∂ γ b i = h γ i , (3.230) ∂ xi ∂ γ b j + ∂ xj ∂ γ b i = h γ ij for i = j, (3.231) ∂ t ∂ γ b i + ∂ xi ∂ γ a = h γ bi , (3.232) ∂ t ∂ γ a = h γ a ,(3.233)
where h γ ci , h γ i , h γ ij , h γ bi and h γ a are linear combinations of (3.234)
R 3 ∂ γ Γ[f, f ; Z]v i |v| 2 M 1 2 , R 3 ∂ γ Γ[f, f ; Z]v i v j M 1 2 , R 3 ∂ γ Γ[f, f ; Z]v i M 1 2 , R 3 ∂ γ Γ[f, f ; Z]M 1 2 .
In particular, based on the second inequality of Lemma 3.17, we know sup
t∈[0,1] h γ ci L 2 x + h γ i L 2 x + h γ ij L 2 x + h γ bi L 2 x + h γ a L 2 x M.∆ x ∂ γ b i = i =j ∂ xj xj ∂ γ b i + ∂ xixi ∂ γ b i (3.237) = i =j − ∂ xixj ∂ γ b j + ∂ xj h γ ij + − ∂ t ∂ xi ∂ γ c + ∂ xi h γ i = i =j ∂ t ∂ xi ∂ γ c − ∂ xi h γ j + i =j ∂ xj h γ ij + − ∂ t ∂ xi ∂ γ c + ∂ xi h γ i = ∂ t ∂ xi ∂ γ c + i =j ∂ xj h γ ij − ∂ xi h γ j + ∂ xi h γ i = − ∂ xixi ∂ γ b i + ∂ xi h γ i + i =j ∂ xj h γ ij − ∂ xi h γ j + ∂ xi h γ i = −∂ xixi ∂ γ b i + i =j ∂ xj h γ ij − ∂ xi h γ j + 2∂ xi h γ i .
Then multiplying ∂ γ b i in the above equation and integrating by parts, we obtain
∇ x ∂ γ b i L 2 x h γ ij L 2 x + h γ i L 2 x M. (3.238)
We assume t > 1 2 and focus on [0, t] (otherwise, we can focus on [t, 1]). We integrate (3.232) over [0, t] to obtain that for 0 ≤ |γ| ≤ N − 1
∂ γ b i (t) − ∂ γ b i (0) + t 0 ∂ xi ∂ γ a(s)ds = t 0 h γ bi (s)ds. (3.239)
Then since ∂ t ∂ xi ∂ γ a = ∂ xi h γ a , we have
∂ xi ∂ γ a(s) = ∂ xi ∂ γ a(t) + s t ∂ xi h γ a (τ )dτ. (3.240)
Then plug this into the above equation, we have
∂ xi ∂ γ a(t) = − 1 t ∂ γ b i (t) − ∂ γ b i (0) − 1 t t 0 s t ∂ xi h γ a (τ )dτ ds + 1 t t 0 h γ bi (s)ds. (3.241)
Then taking x i derivative, we obtain
∂ xixi ∂ γ a(t) = − 1 t ∂ xi ∂ γ b i (t) − ∂ xi ∂ γ b i (0) − 1 t t 0 s t ∂ xixi h γ a (τ )dτ ds + 1 t t 0 ∂ xi h γ bi (s)ds. (3.242)
Then multiplying ∂ γ a in the above equation and integrating by parts, we obtain
∇ x ∂ γ a L 2 x sup t∈[0,1] b L 2 x + ∇ x h γ a L 2 x + h γ bi L 2 x M. (3.243)
In total, we have proved
0<|α|≤N 1 0 ∂ γ a 2 L 2 x + ∂ γ b 2 L 2 x + ∂ γ c 2 L 2 x M.2 L 2 x + b 2 L 2 x + c 2 L 2 x (3.246) 1 0 ∇ t,x a 2 L 2 x + ∇ t,x b 2 L 2 x + ∇ t,x c 2 L 2 x + t 0 R 3 a + t 0 R 3 b + t 0 R 3 c M + t 0 R 3 a + t 0 R 3 b + t 0 R 3 c .
The conservation law for Z implies
t 0 R 3 a = 0, t 0 R 3 b = 0, t 0 R 3 c = 0. (3.247) Hence, we have 1 0 a 2 L 2 x + b 2 L 2 x + c 2 L 2 x M. (3.248)
This concludes our proof.
Remark 3.21. This proof highly relies on Poincaré's inequality, so it cannot be naturally extended to Ω = R 3 case.
Lemma 3.22. Assume f (t, x, v) satisfies (3.15) for t ∈ [0, T ] with T ≥ 1.
Assume the initial data f 0 satisfies the conservation laws. Also,
f (t, x, v) satisfies sup t∈[0,T ] E[f (t)] ≤ M .
Then there exists a constant δ M ∈ (0, 1) such that for any t ′ ≥ 0 and a positive integer
n with t ′ + n ∈ [0, T ], |γ|≤N t ′ +n t ′ T 3 R 3 L ∂ γ f (t) · ∂ γ f (t)dvdxdt ≥ δ M |γ|≤N t ′ +n t ′ ∂ γ f (t) 2 L ν x,v dt. (3.249)
Proof. We apply Lemma 3.20 to each of the intervals [t ′ , t ′ + 1], [t ′ + 1, t ′ + 2], · · · , [t ′ + (n − 1), t ′ + n] and then sum them up.
Remark 3.23. It is not very easy to further extend the result to intervals with arbitrary length. In particular, if we take f 0 satisfying (I − P)[f 0 ] = 0, then for a short period of time, we know the results in Lemma 3.20 cannot be true. Hence, the lower bound of interval length is very important.
Global Well-Posedness and Time Decay
Theorem 3.24. There exists M 0 > 0 such that if Based on Theorem 3.18 for the local well-posedness, we know T > 0. For any t ∈ [0, T ], applying ∂ γ on both sides of the equation (3.15), multiplying ∂ γ f , and integrating over T 3 × R 3 , we have
E[f 0 ] ≤ M 0 2 ,(3.1 2 d dt ∂ γ f 2 L 2 x,v + T 3 ×R 3 ∂ γ f · L ∂ γ f = T 3 ×R 3 ∂ γ f · ∂ γ Γ[f, f ; f ]. (3.253)
We further integrate over time to obtain
∂ γ f (t) 2 L 2 x,v + t 0 T 3 ×R 3 ∂ γ f · L ∂ γ f = ∂ γ f 0 2 L 2 x,v + t 0 T 3 ×R 3 ∂ γ f · ∂ γ Γ[f, f ; f ]. (3.254)
For each t, we split t = t ′ + n, where t ′ ∈ [0, 1) and n is a positive integer. Then using Lemma 3.12, Lemma 3.22 and Lemma 3.17, and summing over |γ| ≤ N , we have
|γ|≤N ∂ γ f (t) 2 L 2 x,v + t t ′ |γ|≤N ∂ γ f 2 L ν x,v C M E[f 0 ] + C M sup s∈[0,t] E[f (s)] 2 ,
for some constant C M ≥ 1 depending on M . However, the second term in LHS still lacks the information on [0, t ′ ]. We fill this gap by adding the missing piece (the integral over [0, t ′ ]) on both sides
(3.255) |γ|≤N ∂ γ f (t) 2 L 2 x,v + t 0 |γ|≤N ∂ γ f 2 L ν x,v C M E[f 0 ] + C M sup s∈[0,t] E[f (s)] 2 + t ′ 0 |γ|≤N ∂ γ f 2 L ν x,v .
Then based on Lemma 3.19, we know for t ′ ∈ [0, 1)
t ′ 0 |γ|≤N ∂ γ f 2 L ν x,v |γ|≤N ∂ γ f 0 2 L 2 x,v . (3.256)
Hence, in total, we obtain
|γ|≤N ∂ γ f (t) 2 L 2 x,v + t 0 |γ|≤N ∂ γ f 2 L ν x,v ≤ C M E[f 0 ] + C M sup s∈[0,t] E[f (s)] 2 . (3.257)
Next, we consider the mixed derivative case. For any t ∈ [0, T ], applying ∂ γ β on both sides of the equation (3.15), multiplying ∂ γ β f , and integrating over T 3 × R 3 , we have
(3.258) 1 2 d dt ∂ γ β f 2 L 2 x,v + T 3 ×R 3 ∂ γ β f · L ∂ γ β f = − T 3 ×R 3 ∂ γ β f · ∂ β L[∂ γ f ] − L ∂ γ β f + T 3 ×R 3 ∂ γ β f · ∂ γ β Γ[f, f ; f ].
Note that the each term in
∂ β L ∂ γ f − L ∂ γ β f has ∂ γ β ′ f with 0 ≤ |β ′ | < |β|.
Hence, we may use a simple induction over |β| = 0, 1, 2, · · · , N to obtain
E[f (t)] ≤ C M E[f 0 ] + C M sup s∈[0,t] E[f (s)] 2 .
(3.259)
Note that we cannot directly absorb the last term into the LHS since we are not clear whether C M M < 1. We further choose M 0 such that
C M M 0 ≤ 1 2 . (3.260)
Also, we choose the initial data
E[f 0 ] ≤ ε M = M 0 4C M < M 0 2 < M 2 . (3.261) Denote T 0 = sup t ≥ 0 : sup s∈[0,t] E[f (s)] ≤ M 0 . (3.262)
For 0 < t < T 0 ≤ T , the bound (3.259) still holds. Hence, we have
E[f (t)] ≤ C M E[f 0 ] + C M sup s∈[0,t] E[f (s)] 2 ≤ M 0 4 + 1 2 sup s∈[0,t] E[f (s)]. (3.263) Taking supremum over t ∈ [0, T 0 ], we have sup t∈[0,T0] E[f (s)] ≤ M 0 2 < M 0 . (3.264)
Then by standard continuity argument, we know T 0 = ∞. for some constant C, K > 0.
Proof. We mainly use an argument similar to Hadžić-Guo [38] and Maslova [56]. Based on the proof of Theorem 3.24, we know for any s < t with |t − s| ≥ 1,
|||f (t)||| 2 + t s |||f (τ )||| 2 ν dτ ≤ C 0 |||f (s)||| 2 . (3.266) Since |||f (t)||| |||f (t)||| ν , we know |||f (t)||| 2 + t s |||f (τ )||| 2 dτ ≤ C 0 |||f (s)||| 2 . (3.267) Denote V (s) = ∞ s |||f (τ )||| 2 dτ. (3.268)
Then naturally
V (s) ≤ C 0 |||f (s)||| 2 ,(3.269)
and thus
V ′ (s) = −|||f (s)||| 2 ≤ − 1 C 0 V (s). (3.270) Therefore, we know V (s) ≤ V (0)e − 1 C 0 s . (3.271)
Then we integrate over s ∈ [t, 2t] for t ≥ 1 in (3.267), we have
t|||f (t)||| 2 ≤ 2t t |||f (τ )||| 2 dτ ≤ ∞ t |||f (τ )||| 2 dτ = V (t) (3.272)
Hence, we have
|||f (t)||| 2 ≤ V (0)e − 1 C 0 t .(3.273)
Since V (0) M 0 , our result naturally follows.
Global Solutions for Ω = R 3
In this section, we will prove the global well-posedness when Ω = R 3 . Denote a special dissipation rate
|||f ||| ν,0 = (I − P)f L ν x,v + 0<|γ|≤N ∂ γ β f L ν x,v . (3.274)
Note that this does not include Pf L ν
x,v , which has not time or spatial derivatives.
Positivity Estimate for L
Similar to Ω = T 3 case, we denote
f (t, x, v) = P[f ](t, x, v) + (I − P)[f ](t, x, v) (3.275) = a(t, x)M 1 2 + b(t, x) · vM 1 2 + c(t, x) |v| 2 M 1 2 + (I − P)[f ](t, x, v),
where a, b, c are given by f . Plugging (3.275) into (3.15) and compare the two sides with the basis
v i |v| 2 M 1 2 , v i v j M 1 2 , v i M 1 2 , M 1 2 (3.276)
we obtain the so-called macroscopic equations
∂ xi ∂ γ c = ℓ γ ci + h γ ci , (3.277) ∂ t ∂ γ c + ∂ xi ∂ γ b i = ℓ γ i + h γ i , (3.278) ∂ xi ∂ γ b j + ∂ xj ∂ γ b i = ℓ γ ij + h γ ij for i = j, (3.279) ∂ t ∂ γ b i + ∂ xi ∂ γ a = ℓ γ bi + h γ bi , (3.280) ∂ t ∂ γ a = ℓ γ a + h γ a ,(3.281)
where ℓ γ ci , ℓ γ i , ℓ γ ij , ℓ γ bi and ℓ γ a are the coefficients corresponding to the above basis for the linear term − ∂ t + v · ∇ x + (I − P) [
ℓ γ ci L 2 x + ℓ γ i L 2 x + ℓ γ ij L 2 x + ℓ γ bi L 2 x + ℓ γ a L 2 x |γ|≤N (I − P)[∂ γ f ] L 2 x . (3.282)
Proof. Assume the basis in (3.276) is {ε n (v)}. Then the coefficients ℓ γ ci , ℓ γ i , ℓ γ ij , ℓ γ bi and ℓ γ a are just linear combinations of
R 3 ∂ t + v · ∇ x + L (I − P)[∂ γ f ] · ε n .
(3.283)
Note that with Lemma 3.7 and Lemma 3.12,
R 3 ∂ t + v · ∇ x + L (I − P)[∂ γ f ] · ε n 2 L 2 x (3.284) R 3 |ε n | · R 3 ×R 3 |ε n | |(I − P)[∂ t ∂ γ f ]| 2 + |v| 2 |(I − P)[∇ x ∂ γ f ]| 2 + |(I − P)[L[∂ γ f ]]| 2 (I − P)[∂ t ∂ γ f ] 2 L 2 x + (I − P)[∇ x ∂ γ f ] 2 L 2 x + (I − P)[∂ γ f ] 2 L 2 x .
Hence, our result is obvious.
Remark 3.27. This lemma indicates that we must include ∂ t in the definition of γ.
Lemma 3.28. |γ|≤N h γ ci L 2 x + h γ i L 2 x + h γ ij L 2 x + h γ bi L 2 x + h γ a L 2 x |||f ||| 2 |||f ||| ν,0 . (3.285)
Proof. Similar to the above lemma, it suffices to bound
R 3 ∂ γ Γ[f, f ; f ] · ε n L 2 x . (3.286)
This is a bit delicate since |||f ||| ν,0 does not include the lowest order terms. For |γ| > 0, the derivative is distributed among the three arguments in Γ. Based on the second inequality in Lemma 3.15, we may assign L 2
x to the term with highest-order derivative to bound it by |||f ||| ν,0 . Then we assign L ∞
x for the other two terms, and the Sobolev embedding helps bound them by |||f ||| 2 . The more delicate case is |γ| = 0. We split f = P
+ Γ P[f ], P[f ]; P[f ] .
Since |||f ||| ν,0 includes (I − P)[f ] L 2
x , so the first three terms are good to go. We just need the estimates as in |γ| > 0 case. The difficult part is the last term
R 3 Γ P[f ], P[f ]; P[f ] · ε n L 2 x .
(3.288)
Since
P[f ] = a(t, x)M 1 2 + b(t, x) · vM 1 2 + c(t, x) |v| 2 M 1 2 , we have R 3 Γ P[f ], P[f ]; P[f ] · ε n L 2 x |a| 3 + |b| 3 + |c| 3 L 2 x a 3 L 6 x + b 3 L 6 x + c 3 L 6 x . (3.289)
Due to Sobolev inequality in R 3 , we have
a 3 L 6 x + b 3 L 6 x + c 3 L 6 x ∇ x a 3 L 2 x + ∇ x b 3 L 2 x + ∇ x c 3 L 2 x |||f ||| 2 |||f ||| ν,0 . (3.290)
Hence, our result naturally follows. such that
(3.291) 0<|γ|≤N T 3 R 3 L ∂ γ f (t) · ∂ γ f (t)dvdx ≥ δ M 0<|γ|≤N ∂ γ f (t) 2 L ν x,v − d dt R 3 a(∇ x · b) − |||f ||| 4 |||f ||| 2 ν,0 .
Proof. Due to Lemma 3.12, we know
0<|γ|≤N T 3 R 3 L ∂ γ f (t) · ∂ γ f (t)dvdx ≥ δ 0<|γ|≤N ∂ γ (I − P)[f ](t) 2 L ν x,v . (3.292)
Hence, it suffices to bound ∂ γ P[f ](t) L ν x,v .Similar the bound of Z in the proof of Lemma 3.20, and using the proof of Lemma 3.26 and Lemma 3.28, we know
∇ x ∂ γ c L 2 x ℓ γ ci L 2 x + h γ ci L 2 x 0<|γ|≤N (I − P)[∂ γ f ] L 2 x + |||f ||| 2 |||f ||| ν,0 ,(3.293)
and
∇ x ∂ γ b i L 2 x ℓ γ ij L 2 x + ℓ γ i L 2 x + h γ ij L 2 x + h γ i L 2 x (3.294) 0<|γ|≤N (I − P)[∂ γ f ] L 2 x + |||f ||| 2 |||f ||| ν,0 .
Also, from (3.230), we have
∂ t ∂ γ c L 2 x ∂ xi ∂ γ b i L 2 x + ℓ γ i L 2 x + h γ i L 2 x 0<|γ|≤N (I − P)[∂ γ f ] L 2 x + |||f ||| 2 |||f ||| ν,0 . (3.295)
The remaining term is for temporal derivative of b, which will be discussed later. For a, (3.233) implies
∂ t ∂ γ a L 2 x ℓ γ a L 2 x + h γ a L 2 x 0<|γ|≤N (I − P)[∂ γ f ] L 2 x + |||f ||| 2 |||f ||| ν,0 . (3.296)
The remaining term is for purely spatial derivative of a. Let γ = [0, γ 1 , γ 2 , γ 3 ]. For |γ| > 0, taking ∂ xi in (3.232) yields
∂ xixi ∂ γ a = −∂ t ∂ xi ∂ γ b i + ∂ xi (ℓ γ bi + h γ bi ). (3.297)
Multiplying ∂ γ a on both sides, integrating over R 3 , and integrating by parts, we have
∇ x ∂ γ a L 2 x ∂ t ∂ γ b i L 2 x + ℓ γ bi L 2 x + h γ bi L 2 x 0<|γ|≤N (I − P)[∂ γ f ] L 2 x + |||f ||| 2 |||f ||| ν,0 . (3.298)
For |γ| = 0, the same procedure implies
∇ x a 2 L 2 x R 3 a(∇ x · ∂ t b) + ℓ bi L 2 x + h bi L 2 x .
(3.299)
In particular, we know
R 3 a(∇ x · ∂ t b) = d dt R 3 a(∇ x · b) − R 3 ∂ t a(∇ x · b) ≤ d dt R 3 a(∇ x · b) + ∂ t a 2 L 2 x + ∇ x b 2 L 2 x .
(3.300)
In summary, we have
∇ x a 2 L 2 x d dt R 3 a(∇ x · b) + 0<|γ|≤N (I − P)[∂ γ f ] 2 L 2 x + |||f ||| 4 |||f ||| 2 ν,0 . (3.301)
Finally, we come to the purely temporal derivative of b. For γ = [γ 0 , 0, 0, 0] with |γ| ≥ 0 in (3.280), we have
∂ t ∂ γ b i L 2 x ∂ xi ∂ γ a L 2 x + ℓ bi L 2 x + h bi L 2 x . (3.302)
The RHS has been estimated as above. In particular, for |γ| = 0, we need to introduce d dt R 3 a(∇ x · b).
Global Well-Posedness
Denote
E[f (t)] = |||f (t)||| 2 + t 0 |||f (s)||| 2 ν,0 ds,(3.E[f (t)] ≤ M 0 , (3.306)
for any t ∈ [0, ∞).
Proof. Applying ∂ γ with |γ| > 0 to (3.15), multiplying ∂ γ f on both sides and integrating over R 3 × R 3 , we get
1 2 0<|γ|≤N ∂ γ f 2 L 2 x,v + 0<|γ|≤N R 3 ×R 3 L ∂ γ f · ∂ γ f = 0<|γ|≤N R 3 ×R 3 ∂ γ Γ[f, f ; f ] · ∂ γ f. (3.307)
Using Lemma 3.30 and similar techniques as in the proof of Lemma 3.28, we have
d dt 0<|γ|≤N ∂ γ f 2 L 2 x,v − R 3 a(∇ x · b) + 0<|γ|≤N ∂ γ f 2 L ν x,v |||f ||| 4 |||f ||| 2 ν,0 . (3.308)
For |γ| = 0, we have
1 2 f 2 L 2 x + R 3 ×R 3 L[f ] · f = R 3 ×R 3 Γ[f, f ; f ] · f. (3.309) Note that R 3 ×R 3 Γ[f, f ; f ] · f = R 3 ×R 3 Γ[f, f ; f ] · (I − P)[f ]. (3.310)
Using similar techniques as in the proof of Lemma 3.28, we have
f 2 L 2 x,v + (I − P)[f ] 2 L 2 x,v |||f ||| 2 |||f ||| 2 ν,0 . (3.311)
In total, we have
d dt C f L 2 x,v + 0<|γ|≤N ∂ γ f 2 L 2 x,v − R 3 a(∇ x · b) (3.312) + 0<|γ|≤N ∂ γ f 2 L ν x,v + (I − P)[f ] 2 L 2 x,v |||f ||| 2 |||f ||| 2 ν,0 .
In particular, we may choose C sufficiently large to kill R 3 a(∇ x · b). Hence, we have
d dt f L 2 x,v + 0<|γ|≤N ∂ γ f 2 L 2 x,v + 0<|γ|≤N ∂ γ f 2 L ν x,v + (I − P)[f ] 2 L 2 x,v |||f ||| 2 |||f ||| 2 ν,0 . (3.313)
This is for |β| = 0 case. When |β| > 0, we use similar induction as in the proof of Theorem 3.24 to obtain
d dt |||f ||| 2 + |||f ||| ν,0 |||f ||| 2 |||f ||| 2 ν,0 . (3.314)
Here note the key fact that
R 3 L[f ] · g = R 3 L (I − P)[f ] · (I − P)[g],(3.315)
and
R 3 Γ[f 1 , f 2 ; f 3 ] · g = R 3 Γ[f 1 , f 2 ; f 3 ] · (I − P)[g]. (3.316)
This helps handle the case when the velocity derivative hits ν, K or Γ. Since I − P part is included in the dissipation, we are good to go. Then by a similar argument as in the proof of Theorem 3.24, we obtain the global well-posedness.
Remark 3.32. This provide a different framework to justify global well-posedness. It also works for Ω = T 3 case. However, note that Theorem 3.24 is slightly better since there we do not need to take temporal derivatives.
Global Stability of the Vacuum
In this section, we focus on the global well-posedness and positivity of the mild solution near the vacuum.
Mild Formulation
As in the classical Boltzmann equation, we decompose the collision term
Q[F, F ; F ] =Q gain [F, F ; F ] − Q loss [F, F ; F ], (4.1) where Q gain [F, F ; F ] := R 3 S 2 q(ω, |v − u|)F (u ′ )F (v ′ ) 1 + θF (u) + θF (v) dωdu, (4.2) Q loss [F, F ; F ] := R 3 S 2 q(ω, |v − u|)F (u)F (v) 1 + θF (u ′ ) + θF (v ′ ) dωdu. (4.3)
In particular, we might write
Q loss [F, F ; F ](v) = F (v) · R[F, F ](v), (4.4) where R[F, F ] := R 3 S 2 q(ω, |v − u|)F (u) 1 + θF (u ′ ) + θF (v ′ ) dωdu. (4.5)
Given β > 0, define We name the weight function w(x, v) := e β(|x| 2 +|v| 2 ) . (4.8)
S = F ∈ C 0 (R + × R 3 × R 3 ) : there exists c > 0 such that |F (t, x, v)| ≤ ce −β(|x| 2 +|v| 2 ) ,(4.
We introduce the transported solution
F # (t, x, v) = F (t, x + tv, v). (4.9)
Then the quantum Boltzmann equation can be rewritten as
∂ t F # = Q # [F, F ; F ] = Q # gain [F, F ; F ] − Q # loss [F, F ; F ], (4.10) where Q # gain [F, F ; F ](t, x, v) = Q # gain [F, F ; F ](t, x + tv, v) (4.11) = R 3 S 2 q(ω, |v − u|)F (t, x + tv, u ′ )F (t, x + tv, v ′ ) 1 + θF (t, x + tv, u) + θF (t, x + tv, v) dωdu, = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u ′ ), u ′ F # t, x + t(v − v ′ ), v ′ 1 + θF # t, x + t(v − u), u + θF # t, x, v dωdu, Q # loss [F, F ; F ](t, x, v) = Q # loss [F, F ; F ](t, x + tv, v) (4.12) = R 3 S 2 q(ω, |v − u|)F (t, x + tv, u)F (t, x + tv, v) 1 + θF (t, x + tv, u ′ ) + θF (t, x + tv, v ′ ) dωdu, = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u), u F # t, x, v 1 + θF # t, x + t(v − u ′ ), u ′ + θF # t, x + t(v − v ′ ), v ′ dωdu.
Global Well-Posedness
The equation (4.10) can be written in the mild formulation
F # (t, x, v) = F 0 (x, v) + t 0 Q # [F, F ; F ](τ, x, v)dτ. (4.13)
We call the function F ∈ S satisfying the above a mild solution to (4.10). Hence, the key is to bound
t 0 Q # gain [F, F ; F ](τ, x, v)dτ and t 0 Q # loss [F, F ; F ](τ, x, v)dτ . Lemma 4.1. We have ∞ 0 e −β|x+τ (v−u)| 2 dτ ≤ β π 1 |v − u| .
(4.14)
Proof. This is [
|x + τ (u − v ′ )| 2 + x + τ (v − v ′ ) 2 = |x| 2 + |x + τ (v − u)| 2 . (4.15)
Proof. This is [28, (2.19)]. Note that the conservation laws of the classical and quantum Boltzmann equations are the same, i.e. u + v = u ′ + v ′ and |u|
2 + |v| 2 = |u ′ | 2 + |v ′ | 2 .
Lemma 4.3. For F # ∈ S and any t ≥ 0, we have
t 0 Q # loss [F, F ; F ](τ, x, v)dτ β −2 w −1 (x, v) F # 2 + F # 3 . (4.16)
Proof. We may further decompose
Q # loss [F, F ; F ](t, x, v) = Q # loss,1 [F, F ](t, x, v) + θQ # loss,2 [F, F ; F ](t, x, v) + θQ # loss,3 [F, F ; F ](t, x, v),(4.17)
where
Q # loss,1 [F, F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u), u F # t, x, v dωdu, Q # loss,2 [F, F ; F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u), u F # t, x, v F # t, x + t(v − u ′ ), u ′ dωdu, Q # loss,3 [F, F ; F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u), u F # t, x, v F # t, x + t(v − v ′ ), v ′ dωdu.
We first consider Q # loss,1 . Direct computation reveals
t 0 Q # loss,1 [F, F ](τ, x, v)dτ = t 0 F # τ, x, v dτ R 3 |v − u| F # τ, x + τ (v − u), u du (4.18) w −1 (x, v) F # t 0 R 3 |v − u| F # τ, x + τ (v − u), u dudτ w −1 (x, v) F # 2 t 0 R 3 |v − u| e −β|u| 2 e −β|x+τ (v−u)| 2 dudτ = w −1 (x, v) F # 2 R 3 |v − u| e −β|u| 2 du t 0 e −β|x+τ (v−u)| 2 dτ .
Based on Lemma 4.1, we know
t 0 e −|x+τ (v−u)| 2 dτ ≤ ∞ 0 e −|x+τ (v−u)| 2 dτ ≤ β π 1 |v − u| . (4.19)
Hence, we have
t 0 Q # loss,1 [F, F ](τ, x, v)dτ β − 1 2 w −1 (x, v) F # 2 R 3 e −β|u| 2 du β −2 w −1 (x, v) F # 2 .
(4.20)
Next, we turn to Q # loss,2 . We have
t 0 Q # loss,2 [F, F ; F ](τ, x, v)dτ (4.21) = t 0 F # τ, x, v dτ R 3 S 2 |ω · (v − u)| F # τ, x + τ (v − u), u F # τ, x + τ (v − u ′ ), u ′ dωdu w −1 (x, v) F # 3 t 0 R 3 S 2 |ω · (v − u)| e −β|u| 2 e −β|u ′ | 2 e −β|x+τ (v−u)| 2 e −β|x+τ (v−u ′ )| 2 dωdudτ .
Since e −β|u ′ | 2 ≤ 1 and e −β|x+τ (v−u ′ )| 2 ≤ 1, it reduces to Q # loss,1 case. Hence, we have
t 0 Q # loss,2 [F, F ; F ](τ, x, v)dτ β −2 w −1 (x, v) F # 3 . (4.22)
Similarly, we know
t 0 Q # loss,3 [F, F ; F ](τ, x, v)dτ β −2 w −1 (x, v) F # 3 . (4.23)
Lemma 4.4. For F # ∈ S and any t ≥ 0, we have
t 0 Q # gain [F, F ; F ](τ, x, v)dτ β −2 w −1 (x, v) F # 2 + F # 3 . (4.24)
Proof. We may further decompose
Q # gain [F, F ; F ](t, x, v) = Q # gain,1 [F, F ](t, x, v) + θQ # gain,2 [F, F ; F ](t, x, v) + θQ # gain,3 [F, F ; F ](t, x, v), (4.25) where Q # gain,1 [F, F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u ′ ), u ′ F # t, x + t(v − v ′ ), v ′ dωdu, Q # gain,2 [F, F ; F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u ′ ), u ′ F # t, x + t(v − v ′ ), v ′ F # t, x + t(v − u), u dωdu, Q # gain,3 [F, F ; F ](t, x, v) = R 3 S 2 q(ω, |v − u|)F # t, x + t(v − u ′ ), u ′ F # t, x + t(v − v ′ ), v ′ F # t, x, v dωdu. gain,1 . Direct computation reveals t 0 Q # gain,1 [F, F ](τ, x, v)dτ (4.26) = t 0 R 3 S 2 |v − u| F # τ, x + τ (v − u ′ ), u ′ F # τ, x + τ (v − v ′ ), v ′ dωdudτ F # 2 t 0 R 3 S 2 |v − u| e −β|u ′ | 2 e −β|v ′ | 2 e −β|x+τ (v−u ′ )| 2 e −β|x+τ (v−v ′ )| 2 dωdudτ .
Using Lemma 4.2 and the conservation laws for (u, v) and (u ′ , v ′ ), we have
t 0 Q # gain,1 [F, F ](τ, x, v)dτ F # 2 t 0 R 3 |v − u| e −β|u| 2 e −β|v| 2 e −β|x| 2 e −β|x+τ (v−u)| 2 dudτ (4.27) ≤ w −1 (x, v) F # 2 t 0 R 3 |v − u| e −β|u| 2 e −β|x+τ (v−u)| 2 dudτ .
Then it reduces to the Q # loss,1 case in Lemma 4.3, so we know
t 0 Q # gain,1 [F, F ](τ, x, v)dτ β −2 w −1 (x, v) F # 2 .
(4.28)
Next, we turn to Q # gain,2 . We have
t 0 Q # gain,2 [F, F ; F ](τ, x, v)dτ (4.29) F # 3 t 0 R 3 S 2 |ω · (v − u)| e −β|u| 2 e −β|u ′ | 2 e −β|v ′ | 2 e −β|x+τ (v−u)| 2 e −β|x+τ (v−u ′ )| 2 e −β|x+τ (v−v ′ )| 2 dωdudτ .
Using the same technique as in Q # gain,1 case, we have For Q # gain,3 , we directly get
t 0 Q # gain,3 [F, F ; F ](τ, x, v)dτ w −1 (x, v) F # t 0 Q # gain,1 [F, F ](τ, x, v)dτ (4.31) β −2 w −2 (x, v) F # 3 .
Define the operator
F [F # ](x, v) = F 0 (x, v) + t 0 Q # [F, F ; F ](τ, x, v)dτ.≤ w −1 (x, v)|||F 0 ||| + β −2 w −1 (x, v) F # 2 + F # 3 .
Therefore, we know
F [F # ] |||F 0 ||| + β −2 F # 2 + F # 3 (4.35) R 0 2 + β −2 (R 2 0 + R 3 0 ) ≤ R 0 .
Hence, F is a mapping from S R0 to S R0 . A similar argument justifies that this is a contraction. Hence, the solution exists uniquely.
Remark 4.6. This theorem justifies that for both fermions and bosons, in the space S, the solution remains small and smooth. Hence, if the initial data is in S and is sufficiently small, there is no possibility of Bose-Einstein condensation. This is significantly different from the result of homogeneous equation.
In the homogeneous equation, the lack of transport operator means that we lose the dispersion and cannot handle the time integral. This is exactly the key in non-homogeneous case. Suppose S T is the restriction of element F ∈ S to [0, T ] × R 3 × R 3 . Assume ℓ 0 (t, x, v) ≤ u 0 (t, x, v) with ℓ 0 , u 0 ∈ S. Define a sequence
Positivity of F for Bosons
θ = 1 Recall Q gain [F, G; H] = R 3 S 2 q(ω, |v − u|)F (u ′ )G(v ′ ) 1 + θH(u) + θH(v) dωdu,(4.∂ t ℓ # k+1 + ℓ # k+1 R # [u k , u k ] = Q # gain [ℓ k , ℓ k ; ℓ k ],(4.39)∂ t u # k+1 + u # k+1 R # [ℓ k , ℓ k ] = Q # gain [u k , u k ; u k ],(4.40)
with initial data ℓ k+1 = F 0 and u k+1 = F 0 . We would like to select some special starting point (ℓ 0 , u 0 ) and study the convergence property of (ℓ # k , u # k ). Lemma 4.7. If |||F 0 ||| and β −2 (R 0 + R 2 0 ) are sufficiently small, then there exists ℓ 0 , u 0 ∈ S T such that the Beginning Condition (BC) 0 ≤ ℓ 0 ≤ ℓ 1 ≤ u 1 ≤ u 0 (4.41)
holds for t ∈ [0, T ].
Proof. We take ℓ 0 = 0, which implies R # [ℓ 0 , ℓ 0 ] = 0 and Q # gain [ℓ 0 , ℓ 0 ; ℓ 0 ] = 0. Hence, we have ∂ t ℓ # 1 + ℓ # 1 R # [u 0 , u 0 ] = 0, (4.42)
∂ t u # 1 = Q # gain [u 0 , u 0 ; u 0 ]. (4.43) Due to the positivity of R # [u 0 , u 0 ] and Q # gain [u 0 , u 0 ; u 0 ], this naturally implies
0 ≤ ℓ # 1 ≤ F 0 ≤ u # 1 . (4.44)
Hence, it remains to show u # 1 ≤ u # 0 . This does not hold for arbitrary u 0 , so we need a delicate construction. Let
ψ(v) = sup x e β|x| 2 |F 0 (x, v)| .
(4.45)
Then we know ψ(v) e −β|v| 2 .
(4.46)
We know 47) or equivalently
u # 1 (t, x, v) = F 0 + t 0 Q # gain [u 0 , u 0 ; u 0 ](τ, x, v)dτ,(4.u 1 (t, x + tv, v) = F 0 + t 0 R 3 S 2 |ω · (v − u)| u 0 (τ, x + τ v, u ′ )u 0 (τ, x + τ v, v ′ ) (4.48) × 1 + θu 0 (τ, x + τ v, u) + θu 0 (τ, x + τ v, v) dωdudτ.
We will look for u 0 =ṽ(x − tv, v), Therefore, u 1 ≤ u 0 if and only if
t 0 R 3 S 2 |ω · (v − u)|ṽ(x + τ (v − u ′ ), u ′ )ṽ(x + τ (v − v ′ ), v ′ ) (4.49) × 1 + θṽ(x + τ (v − u), u) + θṽ(x, v) dωdudτ ≤ṽ(x, v) − F 0 (x, v).
Then we further requireṽ(x, v) = e −β|x| 2 w(v). Then we know
v(x + τ (v − u ′ ), u ′ )ṽ(x + τ (v − v ′ ), v ′ ) = w(u ′ )w(v ′ )e −β|x+τ (v−u ′ )| 2 e −β|x+τ (v−v ′ )| 2 (4.50) = w(u ′ )w(v ′ )e −β|x| 2 e −β|x+τ (v−u)| 2 .
Hence, u 1 ≤ u 0 if and only if
(4.51) t 0 R 3 S 2 |ω · (v − u)| w(u ′ )w(v ′ )e −β|x+τ (v−u)| 2 × 1 + θe −β|x+τ (v−u)| 2 w(u) + θw(v) dωdudτ ≤ w(v) − ψ(v).
Using Lemma 4.1, we know π β 1 + θw(v) Define operator Hence, when |||F 0 ||| and β −2 (R 0 + R 2 0 ) are sufficiently small, we can easily justify that T maps a small ball under W norm into the same ball and it is a contraction. Therefore, such w must exist.
R 3 S 2 w(u ′ )w(v ′ )dωdu + θ π 2β R 3 S 2 w(u ′ )w(v ′ )w(u)dωdu ≤ w(v) − ψ(v).T [w] = ψ(v) + π β 1 + θw(v) R 3 S 2 w(u ′ )w(v ′ )dωdu + θ π 2β R 3 S 2 w(u ′ )w(v ′ )w(u)dωdu.
Lemma 4.8. If ℓ 0 , u 0 ∈ S T such that the Beginning Condition (BC)
0 ≤ ℓ 0 ≤ ℓ 1 ≤ u 1 ≤ u 0 (4.57)
for t ∈ [0, T ], then the iterative sequence (ℓ k , u k ) are always well-defined for t ∈ [0, T ] and satisfies
ℓ k ≤ ℓ k+1 ≤ u k+1 ≤ u k . (4.58)
Proof. The sequence is naturally well-defined due to basic ODE theory. We will focus on the inequality. Rewrite the iteration into mild formulation Also, we assume
ℓ # k+1 (t) = F 0 e − t 0 R # [u k ,u k ] + t 0 e − t τ R # [u k ,u k ] Q # gain [ℓ k , ℓ k ; ℓ k ]dτ,(4.ℓ k−1 ≤ ℓ k ≤ u k ≤ u k−1 . (4.61) Then ℓ # k+1 (t) − ℓ # k (t) = F 0 e − t 0 R # [u k ,u k ] − e − t 0 R # [u k−1 ,u k−1 ]
(4.62)
+ t 0 e − t τ R # [u k ,u k ] − e − t τ R # [u k−1 ,u k−1 ] Q # gain [ℓ k , ℓ k ; ℓ k ]dτ + t 0 e − t τ R # [u k−1 ,u k−1 ] Q # gain [ℓ k , ℓ k ; ℓ k ] − Q # gain [ℓ k−1 , ℓ k−1 ; ℓ k−1 ] dτ.
Due to the monotonicity of R and Q gain , we know all the three terms on the RHS are nonnegative. Hence, we know ℓ # k+1 ≥ ℓ # k . Similarly, we have u # k+1 ≤ u # k . By induction, we know the desired inequality holds. Theorem 4.9. There exists a constant R 0 such that if |||F 0 ||| and β −2 (R 0 + R 2 0 ) are sufficiently small with F 0 ≥ 0, then the equation (4.10) for bosons has a unique mild solution F ∈ S R0 with F ≥ 0.
Proof. Let k → ∞ in the iteration, since we know ℓ k and u k are pointwise monotone with proper upper and lower bounds, dominated convergence theorem implies ℓ k → ℓ and u k → u satisfying
ℓ # − F 0 + t 0 ℓ # R # [u, u] = t 0 Q # gain [ℓ, ℓ; ℓ], (4.63) u # − F 0 + t 0 u # R # [ℓ, ℓ] = t 0 Q # gain [u, u; u].
(4.64)
Taking the difference, we have
u # − ℓ # = t 0 ℓ # R # [u, u] − t 0 u # R # [ℓ, ℓ] + t 0 Q # gain [u, u; u] − t 0 Q # gain [ℓ, ℓ; ℓ] .
(4.65)
Hence, we have
u # − ℓ # β −2 R 2 0 + R 3 0 u # − ℓ # ,(4.66)
which implies u # = ℓ # . They both converge to the solution F to the equation (4.10). Based on our construction, we know F is nonnegative. Since θ = −1, we have to define the iterative sequence in a different fashion
Positivity of F for Fermions
θ = −1 Recall Q gain [F, G; H] = R 3 S 2 q(ω, |v − u|)F (u ′ )G(v ′ ) 1 + θH(u) + θH(v) dωdu,(4.∂ t ℓ # k+1 + ℓ # k+1 R # [u k , ℓ k ] = Q # gain [ℓ k , ℓ k ; u k ], (4.70) ∂ t u # k+1 + u # k+1 R # [ℓ k , u k ] = Q # gain [u k , u k ; ℓ k ],(4.71)
with initial data ℓ k+1 = F 0 and u k+1 = F 0 . We would likee to select some special starting point (ℓ 0 , u 0 ) and study the convergence property of (ℓ # k , u # k ). Lemma 4.10. If |||F 0 ||| and β −2 (R 0 + R 2 0 ) are sufficiently small, then there exists ℓ 0 , u 0 ∈ S T such that the Beginning Condition (BC)
0 ≤ ℓ 0 ≤ ℓ 1 ≤ u 1 ≤ u 0 (4.72)
holds for t ∈ [0, T ].
Proof. We take ℓ 0 = 0, which implies R # [ℓ 0 , ℓ 0 ] = 0 and Q # gain [ℓ 0 , ℓ 0 ; ℓ 0 ] = 0. Hence, we have The rest of the proof follows from that of Lemma 4.7.
∂ t ℓ # 1 + ℓ # 1 R # [u 0 , 0] = 0,
Lemma 4.11. If ℓ 0 , u 0 ∈ S T such that the Beginning Condition (BC) 0 ≤ ℓ 0 ≤ ℓ 1 ≤ u 1 ≤ u 0 (4.76)
for t ∈ [0, T ], then the iterative sequence (ℓ k , u k ) are always well-defined for t ∈ [0, T ] and satisfies
ℓ k ≤ ℓ k+1 ≤ u k+1 ≤ u k . (4.77)
Proof. This follows naturally from that of Lemma 4.8 based on the monotonicity of R and Q gain .
Theorem 4.12. There exists a constant R 0 such that if |||F 0 ||| and β −2 (R 0 + R 2 0 ) are sufficiently small with F 0 ≥ 0, then the equation (4.10) for fermions has a unique mild solution F ∈ S R0 with F ≥ 0.
Proof. This follows from that of Theorem 4.9.
Remark 4.13. Since we consider the near vacuum case, F ≤ 1 is naturally true when R is small.
Remark 1 . 1 .
11When θ = 0, the cubic terms in Q vanish and the equation (1.1) reduces to the classical Boltzmann equation.
Corollary 2 . 2 .
22For all smooth functions F (v) and φ(v), small at infinity,
Theorem 2. 9 (
9Equilibrium). The equilibrium (a.k.a. global Maxwellian) of the equation (1.1) is
.
Actually, there exist solutions to the equation (1.1) in the form of (2.20) where a, b, c depend on time and space. We call such a solution the local Maxwellian.Remark 2.11. Based on Escobedo-Mischler-Valle [23] on miminization of the entropy functional, for bosons we actually allow the presence of δ-function. For any given mass, momentum and energy, there exists an equilibrium in the sense of distribution of the form
Lemma 3. 2 (
2Properties of L). (1) Non-negativity: For any f (v) small at infinity, we have L[f ], f ≥ 0, and the equality holds if and only if f (
inserting it into (3.30), we know L[f ] = 0. On the other hand, if L[f ] = 0, then using (3.31) (being zero and noting the non-negativity of the integrand), we have f (v
Definition 3 . 3 (
33Projection P onto the null space of L). Let N(L) := g ∈ L 2 v (R 3 ) : L[g] = 0 denote the null space of the linear operator L with a set of basis e 0
Theorem 3. 12 (
12Semi-Positivity of L). There exists a δ > 0 such thatL[g], g ≥ δ (I − P)[g]
Proof. We prove by contradiction. Assume that there exists a sequence of functions {g n } ∞ n=1 satisfying P[g n ] = 0, g n L ν v = 1 and L[g n ], g n ≤ 1 n .(3.86)
(a, b) runs all combinations of {1, 2, 3}. For s = 1, 2, 3, f ′ s := ∂ γs βs f s where |β 1 | + |β 2 | + |β 3 | = |β| and |γ 1 | + |γ 2 | + |γ 3 | = |γ|.
Theorem 3 .
318 (Local Well-posedness). There exists M > 0 and T (M ) > 0 such that if E[f 0 ] ≤ M 2 , (3.139) then there exists a unique solution f (t, x, v) to the quantum Boltzmann equation (3.18) such that E[f (t)] ≤ M, (3.140) for any t ∈ [0, T ]. Moreover, the energy E[f (t)] is continuous over t ∈ [0, T ]. Furthermore,
E
[f n (s)] ≤ M and E[f 0 ] ≤ M 2 . Then taking supremum over s ∈ [0, T ]for M sufficiently small and T (M ) sufficiently small, we know sup s∈[0,T ] E[f n+1 (s)] ≤ M. (3.160)
. 4 ,
4Lemma 3.7 and Lemma 3.15, and summing over |γ| ≤ N , we have
Lemma 3 . 20 .
320Assume f (t, x, v) satisfies (3.15) for t ∈ [0, T ] with T ≥ 1. Assume the initial data f 0 satisfies the conservation laws. Also, f (t, x, v) satisfies sup t∈[0,T ] E[f (t)] ≤ M . Then there exists a constant δ M ∈ (0, 1) such that
independent, in the sense of distribution, their coefficients on both sides of the equation should be equal, i.e. the so-called macroscopic equations
|γ| = 0 in (3.229)-(3.233), we know
250)then there exists a unique solution f (t, x, v) to the quantum Boltzmann equation(3.15) such thatE[f (t)] ≤ M 0 ,(3.251) for any t ∈ [0, ∞). Proof. We first choose the initial data E[f 0 ] ≤ M 2 . Denote T = sup t ≥ 0 : sup s∈[0,t] E[f (s)] ≤ M . (3.252)
Theorem 3 . 25 .
325Under the same assumption as in Theorem 3.24, the global solution f (t, x, v) satisfies |||f (t)||| ≤ Ce −kt |||f 0 |||, (3.265)
[f ] + (I − P)[f ] and get Γ[f, f ; f ] =Γ f, f ; (I − P)[f ] + Γ f, (I − P)[f ]; P[f ] + Γ (I − P)[f ], P[f ]; P[f ](3.287)
Remark 3 . 29 .
329The Sobolev inequality in R 3 plays a key role in the proof of this lemma. It does not hold in Ω = T 3 case.Lemma 3.30. Assume f (t, x, v) satisfies (3.15) for t ∈ [0, T ] with T ≥ 1. Assume the initial data f 0 satisfies the conservation laws. Also, f (t, x, v) satisfies sup t∈[0,T ] |||f (t)||| ≤ M . Then there exists a constant δ M ∈ (0, 1)
u| e −2β|u| 2 e −β|v| 2 e −β|x| 2 e −2β|x+τ (v−u)| 2 dudτ (4.30) β −2 w −1 (x, v) F # 3 .
S
R = {F ∈ S : |||F ||| ≤ R}.(4.33) Theorem 4.5. There exists a constant R 0 such that if |||F 0 ||| < R 0 2 , then the equation (4.10) has a unique mild solution F ∈ S R0 .
Q
loss [F, G; H](v) = F (v) · R[G, H](v), ω, |v − u|)G(u) 1 + θH(u ′ ) + θH(v ′ ) dωdu.(4.38)
the existence of solution w(v) ≥ 0, we introduce the space W = w ∈ C(R 3 ) : there exists c > 0 such that |w(v)| ≤ ce −β|v| 2 ,
≥ 0, then naturally T [w] ≥ 0. Then similar to the proof of Lemma 4.4 and Lemma 4.3, we have |||T [w]||| W |||ψ||| + β
e
− t τ R # [u k−1 ,u k−1 ] Q # gain [ℓ k−1 , ℓ k−1 ; ℓ k−1 ]dτ.(4.60)
Q
loss [F, G; H](v) = F (v) · R[G, H](v), ω, |v − u|)G(u) 1 + θH(u ′ ) + θH(v ′ ) dωdu.(4.69)
the positivity of R # [u 0 , 0] and Q # gain [u 0 , u 0 ; 0], this naturally implies0 ≤ ℓ # 1 ≤ F 0 ≤ u # 1 .(4.75)
Remark 2.8. We usually call S[F ] the entropy density.)
. Since the function (1 − A) ln(A) ≤ 0 for any A > 0, we
thus have
Ω R 3
Q[F, F ; F ] ln
F
1 + θF
dvdx ≤ 0.
(2.12)
Hence, we know
d
dt Ω R 3
S[F ] dvdx ≥ 0.
(2.13)
Lemma 3.14. For fixed ω and v, the Jacobian of the transformation u → u ′ satisfies97)
we must have
P Γ[f, f ; f ] = 0.
(3.98)
du ′
du
≥
1
8
,
(3.99)
contains all terms that depends on F (v), and Q p [F, F ; F ] contains all the other terms.(Here
the F, G, H is the not the same as (3.1)). We may further rewrite
Q q [F, G; H] = HQ p [F, G],
(3.144)
Theorem 3.31. There exists M 0 > 0 such that if then there exists a unique solution f (t, x, v) to the quantum Boltzmann equation (3.15) such that303)
and
E[f 0 ] = |||f 0 |||
2 .
(3.304)
E[f 0 ] ≤
M 0
2
,
(3.305)
28, Lemma 2.1.1].Lemma 4.2. We have
Proof. Using Lemma 4.4 and Lemma 4.3, we know for|||F 0 ||| ≤ F [F # ] ≤ |F 0 | + Q # gain [F, F ; F ](τ, x, v)dτ +R 0
2
, we have
t
0
t
0
Q #
loss [F, F ; F ](τ, x, v)dτ
(4.34)
AcknowledgementsWe would like to thank Ning Jiang for pointing out a mistake in the initial version of this paper. Also, we would like to thank Lingbing He and Xuguang Lu for suggesting some references. Lei Wu's research is supported in part by NSF grant DMS-1853002.
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| []
|
[
"On the Uniqueness of Supersymmetric Attractors",
"On the Uniqueness of Supersymmetric Attractors"
]
| [
"Taniya Mandal \nDepartment of Physics\nIndian Institute of Technology Madras\n600036ChennaiIndia\n",
"Prasanta K Tripathy \nDepartment of Physics\nIndian Institute of Technology Madras\n600036ChennaiIndia\n"
]
| [
"Department of Physics\nIndian Institute of Technology Madras\n600036ChennaiIndia",
"Department of Physics\nIndian Institute of Technology Madras\n600036ChennaiIndia"
]
| []
| In this paper we discuss the uniqueness of supersymmetric attractors in four dimensional N = 2 supergravity theories coupled to n vector multiplets. We prove that for a given charge configuration the supersymmetry preserving axion free attractors are unique. We generalise the analysis to axionic attractors and state the conditions for uniqueness explicitly. We consider the example of a two-parameter model and find all solutions to the supersymmetric attractor equations and discuss their uniqueness. | 10.1016/j.physletb.2015.07.070 | [
"https://arxiv.org/pdf/1506.06276v2.pdf"
]
| 119,200,503 | 1506.06276 | 2b6e2ff5568671e4342434d8b5ef5fe6075d2384 |
On the Uniqueness of Supersymmetric Attractors
23 Jun 2015
Taniya Mandal
Department of Physics
Indian Institute of Technology Madras
600036ChennaiIndia
Prasanta K Tripathy
Department of Physics
Indian Institute of Technology Madras
600036ChennaiIndia
On the Uniqueness of Supersymmetric Attractors
23 Jun 2015* email: [email protected] † email: [email protected] 1
In this paper we discuss the uniqueness of supersymmetric attractors in four dimensional N = 2 supergravity theories coupled to n vector multiplets. We prove that for a given charge configuration the supersymmetry preserving axion free attractors are unique. We generalise the analysis to axionic attractors and state the conditions for uniqueness explicitly. We consider the example of a two-parameter model and find all solutions to the supersymmetric attractor equations and discuss their uniqueness.
Introduction
Understanding the origin of black hole entropy has remained to be an important topic of research in gravity and string theory since the seminal work by Bekenstein [1] on this issue. One of the important developments in this area is the so called attractor mechanism, which states that, in a theory of gravity coupled to several scalar fields admitting a single centred extremal black hole, the scalar fields run into a fixed point at the horizon whose value depends only on the black hole charges [2][3][4][5]. There are several aspects of attractor mechanism which have been studied thoroughly [6,7]. Multiplicity of the attractors is one of the puzzling issues which remains to be understood better. Because of the presence of multiple basin of attractors, the near horizon geometry of the black hole is no longer uniquely determined by its charges and one needs to specify the area code in addition to the black hole charges.
The existence of multiple basin of attractors for a given set of charges has been first discussed in [8,9]. Area codes in the context of flux vacua and black hole attractors has been studied [10,11]. Subsequently, multiple supersymmetric attractors in five dimensional N = 2 supergravity theory has been discussed and explicit constructions in the simple case of a two parameter model has been carried out [12]. The analysis has been extended to four dimensional N = 2 supergravity [13] by using the know 4D − 5D correspondence of the attractor points [14]. Further, new multiple non-supersymmetric attractors which does not have obvious five dimensional embedding has been constructed [13]. Multiple attractors in a one parameter model in the presence of quantum corrections has already been studied [15].
The existence of multiple single centred supersymmetric attractors might at first sight appear to be in contradiction with the uniqueness results [16]. (For homogeneous moduli spaces, the solution is always unique up to a duality transformation [17]). However, as explained by Kallosh [18], this is not always the case, because the moduli space might in general possess several disconnected branches. The attractor solution in each of these branches remains unique. One might expect similar results in four dimensional N = 2 supergravity. However, though there exists multiple non-supersymmetric attractors and also multiple attractors with one of the attractor points being supersymmetric in these four dimensional supergravity theories there is no known example where both the attractor points are supersymmetric for these N = 2 supergravity theories in four dimensions [13]. This suggests that, unlike the five dimensional case, the supersymmetric attractors might be unique in these four dimensional supergravity theories. The present work aims to investigate this issue in detail.
The plan of this paper is as follows. In the following section, we will briefly overview the N = 2 supergravity theory. In §3 we will prove that the axion free attractors in four dimensions are unique. Subsequently, we will generalise this result for axionic attractors. This will be followed by an explicit construction of all supersymmetric attractors in a simple two-parameter model in §4. Finally, we will be summarise our results in §5.
Overview
The Lagrangian density for the bosonic part of the four dimensional N = 2 supergravity theory coupled t o n vector multiplet, is given by
L = − R 2 + g ab ∂ µ x a ∂ νxb h µν − µ ΛΣ F Λ µν F Σ λρ h µλ h νρ − ν ΛΣ F Λ µν * F Σ λρ h µλ h νρ . (2.1)
Here h µν is the space-time metric, R is the corresponding Ricci scalar, g ab is the metric on the vector multiplet moduli space parameterized by the corresponding n complex scalar fields x a and A Λ µ are the (n + 1) gauge fields with corresponding field strength F Λ µν . The gauge couplings µ ΛΣ , ν ΛΣ and the moduli space metric g ab are uniquely determined by the N = 2 prepotential F .
We are interested in static, spherically symmetric configurations. The line element corresponding to the space time metric h µν in this case is given by
ds 2 = e 2U dt 2 − e −2U γ mn dy m dy n .
(2.
2)
The wrap factor U depends only on the radial coordinate r. For extremal black holes, the metric of the spacial section γ mn must be identity. The equations of motion for these configurations simplifies and the system can now be described in terms of an effective one dimensional theory with a potential which is extremized at the horizon.
For the N = 2 Lagrangian (2.1), the effective black hole potential takes the form [4]:
V = e K g ab ∇ a W ∇ b W + |W | 2 . (2.3)
Here W and K are respectively the superpotential and the Kähler potential. The superpotential W is related to the central charge by Z = e K/2 W . In terms of the dyonic charges (q Λ , p Λ ) and the prepotential F , the expression for W is given by
W = n Λ=0 (q Λ X Λ − p Λ ∂ Λ F ) , (2.4)
The symplectic sections X Λ are related to the physical scalar fields by x a = X a /X 0 . The
Kähler potential is given in terms of F by the relation:
K = − log i n Λ=0 (X Λ ∂ Λ F − X Λ ∂ Λ F ) . (2.5)
The covariant derivative is defined as ∇ a W = ∂ a W +∂ a KW . For supersymmetric attractors ∇ a W = 0. In general, the attractor points are determined by ∂ a V = 0.
Throughout this paper, we will focus on the N = 2 prepotential which is of the form
F = D abc X a X b X c X 0 . (2.6)
The above prepotential appears as the leading term in the compactification of type IIA string theory on a Calabi-Yau manifold M in the large volume limit. In this case, D abc are the triple intersection numbers D abc = M α a ∧α b ∧α c , where the two forms α a form a basis of H 2 (M, Z). In this paper, we will use string theory terminologies to describe various charge configurations irrespective of whether the coefficients D abc are actually associated with a Calabi-Yau compactification or not.
In the following we will describe some of the well known supersymmetric attractor solutions. For this purpose we need explicit expressions for the Kähler and the superpotentials. The Kähler potential K corresponding to the N = 2 prepotential F has the following simple form
K = − log[−iD abc (x a −x a )(x b −x b )(x c −x c )] . (2.7)
(Now on we set the gauge X 0 = 1 without any loss of generality and express our formulae in terms of the physical scalar fields x a .) The superpotential depends on the specific charge configurations. In this paper we will mainly focus on D0 − D4 and D0 − D4 − D6
configurations. For the D0 − D4 configuration, the superpotential is given by
W = q 0 − 3p a D abc x b x c ,(2.8)
whereas for the D0 − D4 − D6 configuration, we have
W = q 0 − 3p a D abc x b x c + p 0 D abc x a x b x c . (2.9)
These configurations possess well known supersymmetric attractor solutions [19]. For the D0 − D4 configuration, we have
∇ a W = −6D ab x b − 3M a M W.
From here onwards we use the standard notations [20]
D ab = D abc p c , D a = D ab p b , D = D a p a , M ab = D abc (x c −x c ), M a = M ab (x b −x b ) and M = M a (x a −x a ). (Note that M a is
real where as M ab and M are pure imaginary.) Setting the ansatz, x a = p a t, we find
∇ a W = − 3D a 2tD (q 0 + t 2 D) ,
and hence,
x a = ip a q 0 D ,
for the supersymmetric D0 − D4 configuration. The entropy of the corresponding super-
symmetric black hole is S = 2π √ q 0 D.
The solution can be generalised in a straightforward manner upon adding D6 branes.
We find
∇ a W = −6D ab x b + 3p 0 D abc x b x c − 3M a M W .
Setting the ansatz x a = p a t, we find the supersymmetric configuration corresponds to [19]
t = 1 2D p 0 q 0 ± i 4q 0 D − (p 0 q 0 ) 2 .
(2.10)
The entropy for this configuration is
S = π 4q 0 D − (p 0 q 0 ) 2 .
The general solution
In this section, we will focus on the supersymmetric conditions more carefully and obtain the general solution without assuming any specific ansatz. We will first focus on the D0 − D4 configuration. Note that, in this case the superpotential contains only even powers of x a . Thus we can set the axionic parts of the scalar fields to zero: x a = ix a 2 . The supersymmetry condition now becomes
M ab p b + M a M (q 0 − 3 4 M b p b ) = 0 . (3.1)
Note that, for any configuration x a 2 satisfying the above equation, we have q 0 = − 1 4 M a p a . We can see this by multiplying by (x a −x a ) and simplifying the above equation. Thus, we can further simplify Eq.(3.1) by substituting 1 4 M a p a = −q 0 in it. We find
M ab p b + 4q 0 M a M = 0 . (3.2)
Assuming the matrix M ab to be invertible, we can rewrite the above equation as
p a = −8iq 0 x a 2 M . (3.3)
This is a cubic equation in We will now generalise this result in the presence of D6 branes. Note that in the presence of D6 branes it is no longer possible to set the axionic parts of the scalar fields to zero. We denote x a = x a 1 +ix a 2 and express the real and imaginary parts the supersymmetric condition ∇ a W = 0 as
4MM ab (p b − p 0 x b 1 ) = 3M a M b (p b − p 0 x b 1 ) − 4M a (q 0 − 3D bc x b 1 x c 1 + p 0 D bcd x b 1 x c 1 x d 1 ) ,(3.4) 8MD abc x b 1 (2p c − p 0 x c 1 ) − p 0 MM a = 12M a M bc x b 1 (2p c − p 0 x c 1 ) . (3.5)
For convenience we introduce ω a = p a − p 0 x a 1 . Expressing the above equations in terms of ω a and x a 2 , we find
4MM ab ω b = 3M a M b ω b − 4M a (p 0 ) 2 q 0 (p 0 ) 2 − 2D + 3D b ω b − D bcd ω b ω c ω d , (3.6) 8M p 0 (D a − D abc ω b ω c ) − p 0 MM a = 12M a p 0 M bc (p b p c − ω b ω c ) . (3.7)
We would like to find the most general solution of the above equations for the variables ω a , x a 2 . We first rewrite these equations in a simpler form so that it will be easier for us to solve them. Consider first (3. (3.8)
Using the above relation in (3.7) we obtain
4D a + (p 0 ) 2 M a = 4D abc ω b ω c . (3.9)
We can similarly simplify (3.6). Multiplication of (x a −x a ) on both sides of (3.6) provides
4 q 0 (p 0 ) 2 − 2D + 3D a ω a − D abc ω a ω b ω c + (p 0 ) 2 M a ω a = 0 . (3.10)
Putting back (3.10) in (3.6) we find
MM ab ω b = M a M b ω b . (3.11)
Introducing µ = ( 2iMaω a /M) the above equation can be rewritten as w a = µx a 2 . Substituting ω a = µx a 2 in (3.9) we get
D a = − 1 4 (p 0 2 + µ 2 )M a ,
which implies
x a 2 = 2i M ab D bc p c p 0 2 + µ 2 . (3.12)
Defining
I a b = 2i M ac D cb p 0 2 + µ 2 ,
we can rewrite Eqs.(3.12) along with ω a = µx a 2 as
w a = µ p 0 2 + µ 2 I a b p b , (3.13) x a 2 = 1 p 0 2 + µ 2 I a b p b .
(3.14)
It can be shown that the matrix I a b is involutory: I a b I b c = δ a c and it satisfies the relation
D abc I b e I c f = D aef . (3.15)
Using the explicit expressions for µ and after some simplifications, we can rewrite Eqs. (3.13) and (3.14) in terms of the variables x a 1 , x a 2 as
x a 1 = 1 p 0 p a − D − 1 2 q 0 p 0 2 D c I c d p d I a b p b , (3.16) x a 2 = 1 p 0 1 − D − 1 2 q 0 p 0 2 D c I c d p d 2 1/2 I a b p b . (3.17)
This is the most general solution for the supersymmetry conditions (3.6) and (3.7). Any involution I a b satisfying the relation (3.15) will give us a new supersymmetric attractor. The standard solution (2.10) can be recovered by setting I a b = δ a b . We will have multiple attractors if there exists nontrivial involutions satisfying (3.15) and if the moduli space metric as well as the gauge kinetic terms remain positive definite at more than one attractor points for the same charge configuration.
For supersymmetric black holes the entropy is given by S = πe K 0 |W 0 | 2 , where K 0 and W 0 are the values of the Kähler and superpotential at the attractor point respectively.
Substituting the value of K 0 and W 0 in the expression for entropy, we find
S = π p 0 4(D a I a b p b ) 2 − (2D − q 0 p 0 2 ) 2 .
(3.18)
An explicit example
In the previous section we have derived the most general expression for the supersymmetric D0 − D4 − D6 attractors. They are given in terms of the involution I a b satisfying the constraint (3.15). In general it is not possible to solve (3.15) for arbitrary number of vector multiplets. Here we will consider the simplest case of a two-parameter model where this condition can be solved exactly to obtain new supersymmetric attractors.
av − 2bu − cw = 0 , (4.2) bv − 2cu − dw = 0 . (4.3)
It is straightforward to solve the above set of equations. For L 2 − 4MN > 0 they admit a solution of the form:
u = L √ L 2 − 4MN , v = −2M √ L 2 − 4MN , w = 2N √ L 2 − 4MN
Thus we obtain a new D0 − D4 − D6 supersymmetric attractor in the two parameter case in addition to the standard solution (2.10). Using the above solution for the involutory matrix I a b we can obtain explicit expressions for the vector multiplet moduli x 1 = x 1 1 + ix 1 2 and x 2 = x 2 1 + ix 2 2 (for easy reading we denote χ = D a I a b p b in the following): We will first consider the moduli space metric g ab = ∂ a ∂bK. From the expression for it
x 1 1 = 1 p 0 p 1 − (D − 1 2 q 0 p 0 2 )(Lp 1 − 2Mp 2 ) χ √ L 2 − 4MN , x 1 2 = 1 p 0 1 − D − 1 2 q 0 p 0 2 χ 2 1/2 (Lp 1 − 2Mp 2 ) √ L 2 − 4MN , x 2 1 = 1 p 0 p 2 − (D − 1 2 q 0 p 0 2 )(2N p 1 − Lp 2 ) χ √ L 2 − 4MN , x 2 2 = 1 p 0 1 − D − 1 2 q 0 p 0 2 χ 2 1/2 (2N p 1 − Lp 2 ) √ L 2 − 4MN .
Kähler potential (2.7) it is straightforward to find
g ab = 3 M 2M ab − 3 M M a M b . (4.5)
At the attractor point (2.10) it takes the form
g ab = 9 q 0 4D − q 0 p 0 2 D a D b − 2 3 DD ab ,(4.g ab = 9p 0 2 χ 4 χ 2 − D − 1 2 q 0 p 0 2 2 D a D b − 2 3 χD abc I c d p d . (4.7)
For the two parameter model it is straightforward to diagonalise both the metrics. The explicit expressions for the eigenvalues are lengthy and we will not reproduce them here.
For our purpose it will be sufficient to consider the determinant of the metric. From (4.5)
we find
det g = (−1) n 3 M 2n det M a M b − 2M 3 M ab = (−1) n 3 M 2n − 2M 3 n−1 ǫ a 1 a 2 ···an M 1 M a 1 M 2a 2 · · · M nan + · · · +ǫ a 1 a 2 ···an M 1a 1 M 2a 2 · · · M (n−1)a n−1 M n M an + − 2M 3 n det M ab Note that ǫ a 1 a 2 ···an M 1 M a 1 M 2a 2 · · · M nan = ǫ a 1 a 2 ···an M 1 (x b 1 −x b 1 )M a 1 b 1 M 2a 2 · · · M nan = M 1 (x 1 − x 1 ) det(M ab ).
There are n such terms and adding them all we get M det(M ab ). Thus, the determinant of the moduli space metric is found to be −3 n 2 (n−1) det M ab M . Substituting the explicit solutions, we find, for (2.10), We can explicitly verify that the eigenvalues become positive in these respective regions of the charge lattice. We have numerically verified that the gauge kinetic terms can also simultaneously be made positive definite by suitable choice of charges.
det g = 18D 2 (N p 1 2 − Lp 1 p 2 + Mp 2 2 ) q 0 2 4D − q 0 p 0 2 2 ,(4.det g = − 18p 0 4 χ 2 4χ 2 − 2D − q 0 p 0 2 2 2 (N p 1 2 − Lp 1 p 2 + Mp 2 2 ) .
Summary
In this paper we have studied the uniqueness of supersymmetric attractors in N = 2 supergravity theories in four dimensions arising from type IIA compactification on a Calabi-Yau manifold. We have proved the uniqueness for D0 − D4 attractors. We found that the supersymmetry conditions admit more general solutions if we include D6 charges in addition.
These solutions are determined by involutions which satisfies certain constraints. For the two parameter model we can explicitly solve the constraint to find two independent solutions for the attractor equation. However, they exist in mutually exclusive domains of the charge lattice. Hence, the attractors are unique in the respective domains.
x a 2 .
2To solve it exactly, use the RHS of the above for p a in D = D abc p a p b p c to rewrite it as D = −64 q 0 3 M 2 . Solving this for M and substituting it in Eq.(3.3), we find x a = ip a q 0 /D as the most general axion free solution of the supersymmetric condition (3.1).
7). Multiplying (x a −x a ) on both side of of this equation and using the relation D a (x a −x a ) = M ab p a p b we find 4D a (x a −x a ) + (p 0 ) 2 M = 4M ab ω a ω b .
2 + vw = 1. To solve (3.15) for the two parameter case, we denote D 111 = a, D 112 = b, D 122 = c and D 222 = d. Further we use the notation L = ad − bc, M = c 2 − bd and N = b 2 −ac for convenience. Using u 2 + vw = 1 we find two linearly independent equations from the condition (3.15):
the above new configuration for the D0−D4−D6 attractors we would like to ask if it coexists with (2.10) for the same set of charges. Both the solutions are well defined for L 2 − 4MN > 0. However, this is not sufficient for the existence of the attractor solution and we need to make sure that both the moduli space metric and the gauge kinetic terms are positive definite.
above, we find that both the determinant are proportional to (N p 1 2 − Lp 1 p 2 + Mp 2 2 ) with the proportionality factor being positive for the first one where as negative for the second solution. Clearly, for a given set of charges, both the terms can't be made positive simultaneously. Thus the moduli space metric become positive definite in mutually exclusive regions of the charge lattice. The attractor solution becomes unique in each of these domains. For the attractor point (2.10), this domain is specified by (N p 1 2 − Lp 1 p 2 + Mp 2 2 ) > 0 where as for the solution (4.4) it is given by (N p 1 2 − Lp 1 p 2 + Mp 2 2 ) < 0.
6 )
6where as for the new solution Eqs.(3.16) and(3.17) we have
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| []
|
[
"Parallel Direction Method of Multipliers",
"Parallel Direction Method of Multipliers"
]
| [
"Huahua Wang [email protected] ",
"Arindam Banerjee [email protected] ",
"Zhi-Quan Luo ",
"\nDept of Computer Science and Engineering\nDept of Computer Science and Engineering\nUniversity of Minnesota\nTwin Cities\n",
"\nTwin Cities\nDepartment of Electrical and Computer Engineering\nTwin Cities\nUniversity of Minnesota\nUniversity of Minnesota\n\n"
]
| [
"Dept of Computer Science and Engineering\nDept of Computer Science and Engineering\nUniversity of Minnesota\nTwin Cities",
"Twin Cities\nDepartment of Electrical and Computer Engineering\nTwin Cities\nUniversity of Minnesota\nUniversity of Minnesota\n"
]
| []
| We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates primal and dual blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso. | null | [
"https://arxiv.org/pdf/1406.4064v6.pdf"
]
| 14,279,817 | 1406.4064 | 5778713be60baff602fa00e1a4fc12e8b4ee6b2c |
Parallel Direction Method of Multipliers
Sep 2014
Huahua Wang [email protected]
Arindam Banerjee [email protected]
Zhi-Quan Luo
Dept of Computer Science and Engineering
Dept of Computer Science and Engineering
University of Minnesota
Twin Cities
Twin Cities
Department of Electrical and Computer Engineering
Twin Cities
University of Minnesota
University of Minnesota
Parallel Direction Method of Multipliers
Sep 2014
We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates primal and dual blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso.
Introduction
In this paper, we consider the minimization of block-seperable convex functions subject to linear constraints, with a canonical form:
min {x j ∈X j } f (x) = J j=1 f j (x j ) , s.t. Ax = J j=1 A c j x j = a ,(1)
where the objective function f (x) is a sum of J block separable (nonsmooth) convex functions, A c j ∈ R m×n j is the j-th column block of A ∈ R m×n where n = j n j , x j ∈ R n j ×1 is the j-th block coordinate of x, X j is a local convex constraint of x j and a ∈ R m×1 . The canonical form can be extended to handle linear inequalities by introducing slack variables, i.e., writing Ax ≤ a as Ax + z = a, z ≥ 0.
A variety of machine learning problems can be cast into the linearly-constrained optimization problem (1). For example, in robust Principal Component Analysis (RPCA) [5], one attempts to recover a low rank matrix L and a sparse matrix S from an observation matrix M, i.e., the linear constraint is M = L + S. Further, in the stable version of RPCA [43], an noisy matrix Z is taken into consideration, and the linear constraint has three blocks, i.e., M = L + S + Z. The linear constraint with three blocks also appears in the latent variable Gaussian graphical model selection problem [6,23]. Problem (1) can also include composite minimization problems which solve a sum of a loss function and a set of nonsmooth regularization functions. Due to the increasing interest in structural sparsity [2], composite regularizers have become widely used, e.g., overlapping group lasso [42]. As the blocks are overlapping in this class of problems, it is difficult to apply block coordinate descent methods for large scale problem [24,27] which assume block-separable. By simply splitting blocks through introducing equality constraints, the composite minimization problem can also formulated as (1) [3].
A classical approach to solving (1) is to relax the linear constraints using the (augmented) Lagrangian [28,29], i.e.,
L ρ (x, y) = f (x) + y, Ax − a + ρ 2 Ax − a 2 2 .(2)
where ρ ≥ 0 is called the penalty parameter. We call x the primal variable and y the dual variable.
(2) usually leads to primal-dual algorithms which update the primal and dual variables alternatively. The dual update is simply dual gradient ascent where the dual gradient is the resiudal of equality constraint, i.e., Ax − a.
The primal update is to solve a minimization problem of (2) given y. The primal update determines the efficiency of this class of primal-dual algorithms and will be the focus of this paper. If ρ = 0, (2) decomposes into J independent subproblems provided f is separable. In this scenario, the primal-dual algorithm is called the dual ascent method [4,31], where the primal update is solved in a parallel block coordinate fashion. While the dual ascent method can achieve massive parallelism, a careful choice of stepsize and some strict conditions are required for convergence, particularly when f is nonsmooth. To achieve better numerical efficiency and convergence behavior compared to the dual ascent method, it is favorable to set ρ > 0 in the augmented Lagrangian (2). However, (2) is no longer separable since the augmentation term makes x coupled. A well-known primal-dual algorithm to solve (2) is the method of multipliers, which solves the primal update in one block. For large scale optimization problems, it is often difficult to solve the entire augmented Lagrangian efficiently. Considerable efforts have thus been devoted to solving the primal update of the method of multipliers efficiently. In [34], randomized block coordinate descent (RBCD) [24,27] is used to solve (2) exactly, but leading to a double-loop algorithm along with the dual step. More recent results show (2) can be solved inexactly by just sweeping the coordinates once using the alternating direction method of multipliers (ADMM) [14,3].
When J = 2, the constraint is of the form A c 1 x 1 + A c 2 x 2 = a. In this case, a well-known variant of the method of multipliers is the Alternating Direction Method of Multipliers (ADMM) [3], which solves the augmented Lagrangian seperately and alternatively. ADMM was first introduced in [14] and become popular in recent years due to its ease of applicability and superior empirical performance in a wide variety of applications, ranging from image processing [11,1,15] to applied statistics and machine learning [30,40,39,22,35,12,21,37]. For further understanding of ADMM with two blocks, we refer the readers to the comprehensive review by [3]. The proof of global convergence of ADMM with two blocks can be found in [13,3]. Recently, it has been shown that ADMM converges at a rate of O(1/T ) [35,18], where T is the number of iterations. For strongly convex functions, the dual objective of an accelerated version of ADMM can converge at a rate of O(1/T 2 ) [16]. For strongly convex functions, ADMM can achieve a linear convergence rate [10].
Encouraged by the success of ADMM with two blocks, ADMM has also been extended to solve the problem with multiple blocks [20,19,9,26,17,7]. The variants of ADMM can be mainly divided into two categories. One is Gauss-Seidel ADMM (GSADMM) [20,19], which solves (2) in a cyclic block coordinate manner. [20] established a linear convergence rate for MADMM under some fairly strict conditions: (1) A j has full column rank;
(2) f j has Lipschitz-continuous gradients; (3) certain local error bounds hold; (4) the step size needs to be sufficiently small. In [17], a back substitution step was added so that the convergence of ADMM for multiple blocks can be proved. In some cases, it has been shown that ADMM might not converge for multiple blocks [7]. In [19], a block successive upper bound minimization method of multipliers (BSUMM) is proposed to solve the problem (1). The convergence of BSUMM is established under conditions: (i) certain local error bounds hold; (ii) the step size is either sufficiently small or decreasing. However, in general, Gauss-Seidel ADMM with multiple blocks is not well understood and its iteration complexity is largely open. The other is Jacobi ADMM [38,9,26], which solves (2) in a parallel block coordinate fashion. In [38,26], (1) is solved by using two-block ADMM with splitting variables (sADMM). [9] considers a proximal Jacobian ADMM (PJADMM) by adding proximal terms. In addition to the two types of extensions, a randomized block coordinate variant of ADMM named RBSUMM was proposed in [19]. However, RBSUMM can only randomly update one block. Moreover, the convergence of RBSUMM is established under the same conditions as BSUMM and its iteration complexity is unknown. In [32], ADMM with stochastic dual coordinate ascent is proposed to solve online or stochastic ADMM [35,25,33] problem in the dual, which is not the focus of this paper.
In this paper, we propose a randomized block coordinate method named parallel direction method of multipliers (PDMM) which randomly picks up any number of blocks to update in parallel, behaving like randomized block coordinate descent [24,27]. Like the dual ascent method, PDMM solves the primal update in a parallel block coordinate fashion even with the augmentation term. Moreover, PDMM inherits the merits of the method of multipliers and can solve a fairly large class of problems, including nonsmooth functions. Technically, PDMM has three aspects which make it distinct from such state-of-the-art methods. First, if block coordinates of the primal x is solved exactly, PDMM uses a backward step on the dual update so that the dual variable makes conservative progress. Second, the sparsity of A and the number of blocks K to be updated are taken into consideration to determine the step size of the dual update. Third, PDMM can randomly choose arbitrary number of primal and dual blocks for update in parallel. Moreover, we show that sADMM and PJADMM are the two extreme cases of PDMM. The connection between sADMM and PJADMM through PDMM provides better understanding of dual backward step. PDMM can also be used to solve overlapping groups in a randomized block coordinate fashion. Interestingly, the corresponding problem for RBCD [24,27] with overlapping blocks is still an open problem. We establish the global convergence and O(1/T ) iteration complexity of PDMM with constant step size. Moreover, PDMM can also do randomzied dual block coordinate descent. We evaluate the performance of PDMM in two applications: robust principal component analysis and overlapping group lasso.
The rest of the paper is organized as follows. PDMM is proposed in Section 2. The convergence results are established in Section 3. In Section 4, we show PDMM can also do randomized dual block ascent. We evaluate the performance of PDMM in Section 5 and conclude the paper in Section 6. The proof of the convergence of PDMM is given in the Appendix.
Notations: Assume that A ∈ R m×n is divided into I × J blocks. Let A r i ∈ R m i ×n be the i-th row block of A, A c j ∈ R m×n j be the j-th column block of A, and A ij ∈ R m i ×n j be the ij-th block of A. Let y i ∈ R m i ×1 be the i-th block coordinate of y ∈ R m×1 . N (i) is a set of nonzero blocks A ij in the i-th row block A r i and d i = |N (i)| is the number of nonzero blocks. λ ij max is the largest eigenvalue of A T ij A ij . diag(x) denotes a diagonal matrix of vector x. I n is an identity matrix of size n × n. LetK i = min{d i , K} where K is the number of blocks randomly chosen by PDMM and T be the number of iterations. Table 1: Parameters (τ i , ν i ) of PDMM. K is the number of blocks randomly chosen from J blocks, and
K i = min{d i , K} where d i is the number of nonzero blocks A ij in the i-th row of A. K ν i τ i K = 1 0 1 2J−1 1 < K < J 1 − 1 K i K K i (2J−K) K = J 1 − 1 d i 1 d i
Parallel Direction Method of Multipliers
Consider a direct Jacobi version of ADMM which updates all blocks in parallel:
x t+1 j = argmin x j ∈X j L ρ (x j , x t k =j , y t ) ,(3)y t+1 = y t + τ ρ(Ax t+1 − a) .(4)
where τ is a shrinkage factor for the step size of the dual gradient ascent update. However, empirical results show that it is almost impossible to make the direct Jacobi updates (3)-(4) to converge even when τ is extremely small. [20,9] also noticed that the direct Jacobi updates may not converge.
To address the problem in (3) and (4), we propose a backward step on the dual update. Moreover, instead of updating all blocks, the blocks x j will be updated in a parallel randomized block coordinate fashion. We call the algorithm Parallel Direction Method of Multipliers (PDMM). PDMM first randomly select K blocks denoted by set I t at time t, then executes the following iterates:
x t+1 jt = argmin
x j t ∈X j t L ρ (x jt , x t k =jt ,ŷ t ) + η jt B φ j t (x jt , x t jt ) , j t ∈ I t ,(5)y t+1 i = y t i + τ i ρ(A i x t+1 − a i ) ,(6)y t+1 i = y t+1 i − ν i ρ(A i x t+1 − a i ) ,(7)
where τ i > 0, 0 ≤ ν i < 1, η jt ≥ 0, and B φ j t (x jt , x t jt ) is a Bregman divergence. Note x t+1 = (x t+1 jt , x t k =jt ) in (6) and (7). Table 1 shows how to choose τ i and ν i under different number of random blocks K and block sparsity of A. K is the number of blocks randomly chosen from J blocks, andK i = min{d i , K} where d i is the number of nonzero blocks A ij in the i-th row of A.
In the x jt -update (5), a Bregman divergence is addded so that exact PDMM and its inexact variants can be analyzed in an unified framework [36]. In particular, if η jt = 0, (5) is an exact update. If η jt > 0, by choosing a suitable Bregman divergence, (5) can be solved by various inexact updates, often yielding a closed-form for the x jt update (see Section 2.1).
Let r t = Ax t − a, then r t+1 = r t + jt∈It A c jt (x t+1 jt − x t jt ). (5) can be rewritten as
x t+1 jt = argmin x j t ∈X j t f jt (x jt ) + ŷ t , A c jt x jt + ρ 2 A c jt x jt + j =k A c jt x t jt − a 2 2 + η jt B φ j t (x, x t jt )
= argmin
x j t ∈X j t f jt (x jt ) + (A c jt ) T (ŷ t + ρr t ), x jt + ρ 2 A c jt (x jt − x t jt ) 2 2 + η jt B φ j t (x, x t jt ) . (8)
Therefore, we have the algorithm of PDMM as in Algorithm 1.
To better understand PDMM, we discuss the following three aspects which play roles in choosing τ i and ν i : the dual backward step (7), the sparsity of A and the choice of randomized blocks.
Algorithm 1 Parallel Diretion Method of Multipliers
1: Input: ρ, η j , τ i , ν i 2: Initialization: x 1 ,ŷ 1 = 0 3: if τ i , ν i are not defined, initialize τ i , ν i as given in Table 1 4: r 1 = Ax 1 − a = −a 5: for t = 1 to T do 6: randomly pick up j t block coordinates 7: x t+1 jt = argmin (3), which can be rewritten as:
x j t ∈X j t f jt (x jt ) + (A c jt ) T (ŷ t + ρr t ), x jt + ρ 2 A c jt (x jt − x t jt ) 2 2 + η jt B φ j t (x, x t jt ) 8: r t+1 = r t + jt∈It A c jt (x t+1 jt − x t jtx t+1 j = argmin x j ∈X j f j (x j ) + y t + ρ(Ax t − a), A c j x j + ρ 2 A c j (x j − x t j ) 2 2 .(9)
In the primal x j update, the quadratic penalty term implicitly adds full gradient ascent step to the dual variable, i.e., y t + ρ(Ax t − a), which we call implicit dual ascent. The implicit dual ascent along with the explicit dual ascent (4) may lead to too aggressive progress on the dual variable, particularly when the number of blocks is large. Based on this observation, we introduce an intermediate variableŷ t to replace y t in (9) so that the implicit dual ascent in (9) makes conservative progress, e.g.,ŷ t + ρ(Ax t − a) = y t + (1 − ν)ρ(Ax t − a) , where 0 < ν < 1.ŷ t is the result of a 'backward step' on the dual variable, i.e., y t = y t − νρ(Ax t − a). Moreover, one can show that τ and ν have also been implicitly used when using two-block ADMM with splitting variables (sADMM) to solve (1) [26,38]. Section 2.2 shows sADMM is a special case of PDMM. The connection helps in understanding the role of the two parameters τ i , ν i in PDMM. Interestingly, the step sizes τ i and ν i can be improved by considering the block sparsity of A and the number of random blocks K to be updated.
Sparsity of A: Assume A is divided into I × J blocks. While x j can be updated in parallel, the matrix multiplication Ax in the dual update (4) requires synchronization to gather messages from all block coordinates j t ∈ I t . For updating the i-th block of the dual y i , we need
A i x t+1 = jt∈It A ijt x t+1 jt + k /
∈It A ik x t k which aggregates "messages" from all x jt . If A ijt is a block of zeros, there is no "message" from x jt to y i . More precisely,
A i x t+1 = jt∈It∩N (i) A ijt x t+1 jt + k / ∈It A ik x t k where N (i)
denotes a set of nonzero blocks in the i-th row block A i . N (i) can be considered as the set of neighbors of the i-th dual block y i and d i = |N (i)| is the degree of the i-th dual block y i . If A is sparse, d i could be far smaller than J. According to Table 1, a low d i will lead to bigger step sizes τ i for the dual update and smaller step sizes for the dual backward step (7). Further, as shown in Section 2.3, when using PDMM with all blocks to solve composite minimization with overlapping blocks, PDMM can use τ i = 0.5 which is much larger than 1/J in sADMM.
Randomized Blocks: The number of blocks to be randomly chosen also has the effect on τ i , ν i . If randomly choosing one block (K = 1), then ν i = 0, τ i = 1 2J−1 . The dual backward step (7) vanishes. As K increases, ν i increases from 0 to 1 − 1 d i and τ i increases from 1 2J−1 to 1 d i . If updating all blocks (K = J),
τ i = 1 d i , ν i = 1 − 1 d i .
PDMM does not necessarily choose any K combination of J blocks. The J blocks can be randomly partitioned into J/K groups where each group has K blocks. Then PDMM randomly picks one group. A simple way is to permutate the J blocks and choose K blocks cyclically.
Inexact PDMM
If η jt > 0, there is an extra Bregman divergence term in (5), which can serve two purposes. First, choosing a suitable Bregman divergence can lead to a closed-form solution for (5). Second, if η jt is sufficiently large, the dual update can use a large step size (τ i = 1) and the backward step (7) can be removed (ν i = 0), leading to the same updates as PJADMM [9] (see Section 2.2).
Given a differentiable function ψ jt , its Bregman divergence is defiend as
B ψ j t (x jt , x t jt ) = ψ jt (x jt )−ψ jt (x t jt )− ∇ψ jt (x t jt ), x jt −x t jt ,(10)
where ∇ψ jt denotes the gradient of ψ jt . Rearranging the terms yields
ψ jt (x jt )−B ψ j t (x jt , x t jt ) = ψ jt (x t jt )+ ∇ψ jt (x t jt ), x jt −x t jt ,(11)
which is exactly the linearization of ψ jt (x jt ) at x t jt . Therefore, if solving (5) exactly becomes difficult due to some problematic terms, we can use the Bregman divergence to linearize these problematic terms so that (5) can be solved efficiently. More specifically, in (5), we can choose φ jt = ϕ jt − 1 η j t ψ jt assuming ψ jt is the problematic term. Using the linearity of Bregman divergence,
B φ j t (x jt , x t jt ) = B ϕ j t (x jt , x t jt ) − 1 η jt B ψ j t (x jt , x t jt ) .(12)
For instance, if f jt is a logistic function, solving (5) exactly requires an iterative algorithm. Setting ψ jt = f jt , ϕ jt = 1 2 · 2 2 in (12) and plugging into (5) yield
x t+1 jt = argmin
x j t ∈X j t ∇f jt (x t jt ), x jt + ŷ t , A jt x jt + ρ 2 A jt x jt + k =j A k x t k − a 2 2 + η jt x jt − x t jt 2 2 ,(13)
which has a closed-form solution. Similarly, if the quadratic penalty term ρ
2 A c jt x jt + k =j A c k x jt − a 2 2 is a problematic term, we can set ψ jt (x jt ) = ρ 2 A c jt x jt 2 2 , then B ψ j t (x jt , x t jt ) = ρ 2 A c jt (x jt − x t jt ) 2 2
can be used to linearize the quadratic penalty term.
In (12), the nonnegativeness of B φ j t implies that B ϕ j t ≥ 1
η j t B ψ j t .
This condition can be satisfied as long as ϕ jt is more convex than ψ jt . Technically, we assume that ϕ jt is σ/η jt -strongly convex and ψ jt has Lipschitz continuous gradient with constant σ, which has been shown in [36]. For instance, if
ψ jt (x jt ) = ρ 2 A c jt x jt 2 2 , σ = ρλ max (A c jt ) where λ max (A c jt ) denotes the largest eigenvalue of (A c jt ) T A c jt . If choosing ϕ jt = 1 2 · 2 2
, the condition is satisfied by setting η jt ≥ ρλ max (A c jt ).
Connections to Related Work
All blocks: There are also two other methods which update all blocks in parallel. If solving the primal updates exactly, two-block ADMM with splitting variables (sADMM) is considered in [26,38]. We show that sADMM is a special case of PDMM when setting τ i = 1 J and ν i = 1 − 1 J ( See Appendix B). If the primal updates are solved inexactly, [9] considers a proximal Jacobian ADMM (PJADMM) by adding proximal terms where the converge rate is improved to o(1/T ) given the sufficiently large proximal terms. We show that PJADMM [9] is also a special case of PDMM ( See Appendix C). sADMM and PJADMM are two extreme cases of PDMM. The connection between sADMM and PJADMM through PDMM can provide better understanding of the three methods and the role of dual backward step. If the primal update is solved exactly which makes sufficient progress, the dual update should take small step, e.g., sADMM. On the other hand, if the primal update takes small progress by adding proximal terms, the dual update can take full gradient step, e.g. PJADMM. While sADMM is a direct derivation of ADMM, PJADMM introduces more terms and parameters.
Randomized blocks: While PDMM can randomly update any number of blocks, RBUSMM [19] can only randomly update one block. The convergence of RBSUMM requires certain local error bounds to be hold and decreasing step size. Moreover, the iteration complexity of RBSUMM is still unknown. In contast, PDMM converges at a rate of O(1/T ) with the constant step size.
Randomized Overlapping Block Coordinate
Consider the composite minimization problem of a sum of a loss function ℓ(w) and composite regularizers g j (w j ):
min w ℓ(w) + L j=1 g j (w j ) ,(14)
which considers L overlapping groups
w j ∈ R b×1 . Let J = L + 1, x J = w. For 1 ≤ j ≤ L, denote x j = w j , then x j = U T j x J , where U j ∈ R b×L
is the columns of an identity matrix and extracts the coordinates of (14) can be written as: (15) can be solved by PDMM in a randomized block coordinate fashion. In A, for b rows block, there are only two nonzero blocks, i.e., d i = 2. Therefore, [3] can solve (15) where the equality constraint is
x J . Denote U = [U 1 , · · · , U L ] ∈ R n×(bL) and A = [I bL , −U T ] where bL denotes b × L. By letting f j (x j ) = g j (w j ) and f J (x J ) = ℓ(w),min x J j=1 f j (x j ) s.t. Ax = 0.(15)where x = [x 1 ; · · · ; x L ; x L+1 ] ∈ R b×J .τ i = K 2(2J−K) , ν i = 0.5. In particular, if K = J, τ i = ν i = 0.5. In contrast, sADMM uses τ i = 1/J ≪ 0.5, ν i = 1 − 1/J > 0.5 if J is larger. Remark 1 (a) ADMMx j = U T j x J . (b)
In this setting, Gauss-Seidel ADMM (GSADMM) and BSUMM [19] are the same as ADMM. BSUMM should converge with constant stepsize ρ (not necessarily sufficiently small), although the theory of BSUMM does not include this special case.
(c) Consensus optimization [3] has the same formulation as (15). Therefore, PDMM can also be used as a randomized consensus optimization algorithm.
Theoretical Results
We establish the convergence results for PDMM under fairly simple assumptions:
Assumption 1
(1) f j : R n j → R ∪ {+∞} are closed, proper, and convex.
(2) A KKT point of the Lagrangian (ρ = 0 in (2)) of Problem (1) exists.
Assumption 2 is the same as that required by ADMM [3,35]. Assume that {x * j , y * i } satisfies the KKT conditions of the Lagrangian (ρ = 0 in (2)), i.e.,
− A T j y * ∈ ∂f j (x * j ) ,(16)Ax * − a = 0.(17)
During iterations, (82) is satisfied if Ax t+1 = a. Let ∂f j be the subdifferential of f j . The optimality conditions for the x j update (5) is
−A c j [y t +(1−ν)ρ(Ax t − a)+A c j (x t+1 j −x t j )]−η j (∇φ j (x t+1 j )−∇φ j (x t j )) ∈ ∂f j (x t+1 j ) .(18)When Ax t+1 = a, y t+1 = y t . If A c j (x t+1 j − x t j ) = 0, then Ax t − a = 0. When η j ≥ 0, further assuming B φ j (x t+1
j , x t j ) = 0, (81) will be satisfied. Overall, the KKT conditions (81)-(82) are satisfied if the following optimality conditions are satisfied by the iterates:
Ax t+1 = a , A c j (x t+1 j − x t j ) = 0 ,(19)B φ j (x t+1 j , x t j ) = 0 .(20)
The above optimality conditions are sufficient for the KKT conditions. (85) are the optimality conditions for the exact PDMM. (86) is needed only when η j > 0.
Let z ij = A ij x j ∈ R m i ×1 , z r i = [z T i1 , · · · , z T iJ ] T ∈ R m i J×1 and z = [(z r 1 ) T , · · · , (z r I ) T ] T ∈ R Jm×1
. Define the residual of optimality conditions (85)-(86) as
R(x t+1 ) = ρ 2 z t+1 − z t 2 Pt + ρ 2 I i=1 β i A r i x t+1 − a i 2 2 + J j=1 η j B φ j (x t+1 j , x t j ) .(21)
where P t is some positive semi-definite matrix 1 and
β i = K JK i . If R(x t+1 ) → 0,(v t = (x t j , y t i ) and h(v * , v t ) as a distance from v t to a KKT point v * = (x * j , y * i ): h(v * , v t ) = K J I i=1 1 2τ i ρ y * i − y t−1 i 2 2 +L ρ (x t , y t ) + ρ 2 z * − z t 2 Q + J j=1 η j B φ j (x * j , x t j ) ,(22)
where Q is a positive semi-definite matrix 1
andL ρ (x t , y t ) with γ i = 2(J−K) K i (2J−K) + 1 d i − K JK i is L ρ (x t , y t ) = f (x t ) − f (x * ) + I i=1 y t i , A r i x t − a i + (γ i − τ i )ρ 2 A r i x t − a i 2 2 .(23)
The following Lemma shows that h(v * , v t ) ≥ 0.
1 See the definition in the Appendix A.
Lemma 1
Let v t = (x t j , y t i ) be generated by PDMM (5)- (7) and h(v * , v t ) be defined in (89). Setting (16)- (17) are satisfied.
ν i = 1 − 1 K i and τ i = K K i (2J−K) , we have h(v * , v t ) ≥ ρ 2 I i=1 ζ i A r i x t − a i 2 2 + ρ 2 z * − z t 2 Q + + J j=1 η j B φ j (x * j , x t j ) ≥ 0 . (24) where ζ i = J−K K i (2J−K) + 1 d i − K JK i ≥ 0. Moreover, if h(v * , v t ) = 0, then A r i x t = a i , z t = z * and B φ j (x * j , x t j ) = 0. Thus,
In PDMM, y t+1 depends on x t+1 , which in turn depends on I t . x t and y t are independent of I t . x t depends on the observed realizations of the random variable
ξ t−1 = {I 1 , · · · , I t−1 } .(25)
The following theorem shows that h(v * , v t ) decreases monotonically and thus establishes the global convergence of PDMM.
Theorem 1 (Global Convergence of PDMM)
Let v t = (x t j , y t i ) be generated by PDMM (5)-(7) and v * = (x * j , y * i ) be a KKT point satisfying (16)- (17).
Setting ν i = 1 − 1 K i and τ i = K K i (2J−K) , we have 0 ≤ E ξt h(v * , v t+1 ) ≤ E ξ t−1 h(v * , v t ) , E ξt R(x t+1 ) → 0 .(26)
The following theorem establishes the iteration complexity of PDMM in an ergodic sense.
Theorem 2 (The Rate of Convergence) Let (x t j , y t i ) be generated by PDMM (5)-(7). Letx T = T t=1 x t . Setting ν i = 1 − 1 K i and τ i = K K i (2J−K) , we have Ef (x T ) − f (x * ) ≤ J K I i=1 1 2β i ρ y * i 2 2 +L ρ (x 1 , y 1 ) + ρ 2 z * − z 1 2 Q + J j=1 η j B φ j (x * j , x 1 j ) T ,(27)E I i=1 β i A r ix T − a i 2 2 ≤ 2 ρ h(v * , v 0 ) T .(28)
where β i = K JK i and Q is a positive semi-definite matrix.
Extensions: PDMM with Randomized Dual Block Coordinate Ascent
In this section, we further show that PDMM can update the dual blocks randomly. The randomized dual block coordinate ascent (RDBCD) can further increase of dual step size τ i . More specifically, at time t + 1, PDMM randomly selects K primal blocks denoted by J t and K I dual blocks denoted by set I t , then executes the following iterates:
y t+1 i = y t+1 i − ν i ρ(A i x t+1 − a i ) ,(29)
x t+1 jt = argmin
x j t ∈X j t L ρ (x jt , x t k =jt ,ŷ t ) + η jt B φ j t (x jt , x t jt ) , j t ∈ J t ,(30)y t+1 it = y t it + τ it ρ(A it x t+1 − a it ) , i t ∈ I t ,(31)
where x = (x t+1 jt∈Jt , x k ∈Jt ), y = (y t+1 it∈It , y t k ∈∈It ), and τ i , ν i take the following values:
τ i = K K i [(2J − K) K I I + K(1 − K I I )] ≥ K K i (2J − K) , ν i = 1 − 1 K i .(32)
The dual step size τ i increases when using RDBCD. If I = J, K I = K = 1, τ i = I 3J−2 > 1 3 , which is far greater than 1 2J−1 in PDMM without RDBCD. In this setting, x t+1 depends on J t , and y t+1 depends on I t , J t . y t+1 depends on x t+1 , which in turn depends on the observed realizations of the random variable
ξ t = {(I 1 , J 1 ), · · · , (I t , J t )} .(33)
Define the current
iterate v t = (x t j , y t i ) and h(v * , v t ) as a distance from v t to a KKT point v * = (x * j , y * i ): h(v * , v t ) = K J I i=1 I 2K I τ i ρ y * i − y t−1 i 2 2 +L ρ (x t , y t ) + ρ 2 z * − z t 2 Q + η T B φ (x * , x t ) .(34)
The following Lemma shows that h(v * , v t ) ≥ 0. (34).
Lemma 2 Let h(v * , v t ) be defined inSetting ν i = 1 − 1 K i and τ i = K K i [(2J−K) K I I +K(1− K I I )]
, we have
h(v * , v t ) ≥ ρ 2 I i=1 ζ i A r i x t − a i 2 2 + ρ 2 z * − z t 2 Q + + J j=1 η j B φ j (x * j , x t j ) ≥ 0 .(35)
where ζ i = (16)- (17) are satisfied. The following theorem shows that h(v * , v t ) decreases monotonically and thus establishes the global convergence of PDMM.
+ 1 d i − K JK i ≥ 0. Moreover, if h(v * , v t ) = 0, then A r i x t = a i , z t = z * and B φ j (x * j , x t j ) = 0. Thus,
Theorem 3 (Global Convergence)
Let v t = (x t jt , y t it ) be generated by PDMM (29)- (31) and v * = (x * j , y * i ) be a KKT point satisfying (16)- (17).
Setting ν i = 1 − 1 K i and τ i = K K i [(2J−K) K I I +K(1− K I I )]
, we have
0 ≤ E ξt h(v * , v t+1 ) ≤ E ξ t−1 h(v * , v t ) , E ξt R(x t+1 ) → 0 .(36)
Theorem 4 (The Rate of Convergence)
Let (x t jt , y t it ) be generated by PDMM (29)- (31).
Letx T = T t=1 x t . Setting ν i = 1 − 1 K i and τ i = K K i [(2J−K) K I I +K(1− K I I )]
, we have
Ef (x T ) − f (x * ) ≤ I K I I i=1 1 2τ i ρ y 0 i 2 2 + J K 1 2β i ρ y * i 2 2 +L ρ (x 1 , y 1 ) + ρ 2 z * − z 1 2 Q + η T B φ (x * , x 1 ) T ,(37)E I i=1 β i A r ix T − a i 2 2 ≤ 2 ρ h(v * , v 0 ) T .(38)
where Table 1. Gauss-Seidel (GSADMM) is the fastest algorithm, although whether it converges or not is unknown. PDMM3 is faster than PDMM1 and PDMM2. For the two randomized one block coordinate methods, PDMM1 is faster than RBSUMM.
β i = K JK i
Experimental Results
In this section, we evaluate the performance of PDMM in solving robust principal component analysis (RPCA) and overlapping group lasso [42]. We compared PDMM with ADMM [3] or GSADMM (no theory guarantee), sADMM [26,38], and RBSUMM [19]. Note GSADMM includes BSUMM [19]. All experiments are implemented in Matlab and run sequentially. We run the experiments 10 times and report the average results. The stopping criterion is either residual norm(x-xold) norm(xold) + norm(y-yold) norm(yold) ≤ 10 −4 or the maximum number of iterations.
RPCA: RPCA is used to obtain a low rank and sparse decomposition of a given matrix A corrupted by noise [5,26]:
min 1 2 X 1 2 F + γ 2 X 2 1 + γ 3 X 3 * s.t. A = X 1 + X 2 + X 3 .(39)
where A ∈ R m×n , X 1 is a noise matrix, X 2 is a sparse matrix and X 3 is a low rank matrix. A = L + S + V is generated in the same way as [26] 2 . In this experiment, m = 1000, n = 5000 and the rank is 100.
The number appended to PDMM denotes the number of blocks (K) to be chosen in PDMM, e.g., PDMM1 randomly updates one block. Figure 1 compares the convegence results of PDMM with ADMM methods. In PDMM, ρ = 1 and τ i , ν i are chosen according to Table ( respectively. We choose the 'best' results for GSADMM (ρ = 1) and RBSUMM (ρ = 1, α = ρ 11 √ t+10 ) and sADMM (ρ = 1). PDMMs perform better than RBSUMM and sADMM. Note the public available code of sADMM 2 does not have dual update, i.e., τ i = 0. sADMM should be the same as PDMM3 if τ i = 1 3 . Since τ i = 0, sADMM is the slowest algorithm. Without tuning the parameters of PDMM, GSADMM converges faster than PDMM. Note PDMM can run in parallel but GSADMM only runs sequentially. PDMM3 is faster than two randomized version of PDMM since the costs of extra iterations in PDMM1 and PDMM2 have surpassed the savings at each iteration. For the two randomized one block coordinate methods, PDMM1 converges faster than RBSUMM in terms of both the number of iterations and runtime.
The effect of τ i , ν i : We tuned the parameter τ i , ν i in PDMMs. Three randomized methods ( RBSUMM, PDMM1 and PDMM2) choose the blocks cyclically instead of randomly. Table 2 compares the 'best' results of PDMM with other ADMM methods. In PDMM,
(τ i , ν i ) = {( 1 2 , 0), ( 1 3 , 1 2 ), ( 1 2 , 1 2 )}.
GSADMM converges with the smallest number of iterations, but PDMMs can converge faster than GSADMM in terms of runtime. Since GSADMM uses new iterates which increases computation compared to PDMM3, PDMM3 can be faster than GSADMM if the numbers of iterations are close. PDMM1 and PDMM2 can be faster than PDMM3. By simply updating one block, PDMM1 is the fastest algorithm and achieves the lowest residual.
Overlapping Group Lasso: We consider solving the overlapping group lasso problem [42]:
min w 1 2Lλ Aw − b 2 2 + g∈G d g w g 2 .(40)
where A ∈ R m×n , w ∈ R n×1 and w g ∈ R b×1 is the vector of overlapping group indexed by g. d g is some positive weight of group g ∈ G. As shown in Section 2.3, (40) can be rewritten as the form (15). The data is generated in a same way as [41,8]: the elements of A are sampled from normal distribution, b = Ax + ǫ with noise ǫ sampled from normal distribution, and x j = (−1) j exp(−(j − 1)/100). In this experiment, m = 5000, the number of groups is L = 100, and d g = 1 L , λ = L 5 in (40). The size of each group is 100 and the overlap is 10. The total number of blocks in PDMM and sADMM is J = 101. τ i , ν i in PDMM are computed according to Table (1).
In Figure 2, the first two figures plot the convergence of objective in terms of the number of iterations and time. PDMM uses all 101 blocks and is the fastest algorithm. ADMM is the same as GSADMM in this problem, but is slower than PDMM. Since sADMM does not consider the sparsity, it uses τ i = 1 J+1 , ν i = 1 − 1 J+1 , leading to slow convergence. The two accelerated methods, PA-APG [41] and S-APG [8], are slower than PDMM and ADMM.
The effect of K: The third figure shows PDMM with different number of blocks K. Although the complexity of each iteration is the lowest when K = 1, PDMM takes much more iterations than other cases and thus takes the longest time. As K increases, PDMM converges faster and faster. When K = 20, the runtime is already same as using all blocks. When K > 21, PDMM takes less time to converge than using all blocks. The runtime of PDMM decreases as K increases from 21 to 61. However, the speedup from 61 to 81 is negligable. We tried different set of parameters for RBSUMM ρ i 2 +1 i+t (0 ≤ i ≤ 5, ρ = 0.01, 0.1, 1) or sufficiently small step size, but did not see the convergence of the objective within 5000 iterations. Therefore, the results are not included here.
Conclusions
We proposed a randomized block coordinate variant of ADMM named Parallel Direction Method of Multipliers (PDMM) to solve the class of problem of minimizing block-separable convex functions subject to linear constraints. PDMM considers the sparsity and the number of blocks to be updated when setting the step size. We show two other Jacobian ADMM methods are two special cases of PDMM. We also use PDMM to solve overlapping block problems. The global convergence and the iteration complexity are established with constant step size. Experiments on robust PCA and overlapping group lasso show that PDMM is faster than existing methods.
A Convergence Analysis
A.1 Technical Preliminaries
We first define some notations will be used specifically in this section. Let
z ij = A ij x j ∈ R m i ×1 , z r i = [z T i1 , · · · , z T iJ ] T ∈ R m i J×1 and z = [(z r 1 ) T , · · · , (z r I ) T ] T ∈ R Jm×1 . Let W i ∈ R Jm i ×m i be a column vector of W ij ∈ R m i ×m i where W ij = I m i , if A ij = 0 , 0 otherwise .(41)
Define Q ∈ R Jm×Jm as a diagonal matrix of Q i ∈ R Jm i ×Jm i and
Q = diag([Q 1 , · · · , Q I ]) , Q i = diag(W i ) − 1 d i W i W T i .(42)
Therefore, for an optimal solution x * satisfying Ax * = a, we have
z t − z * 2 Q = I i=1 z t i − z * i 2 Q i = I i=1 z t i − z * i 2 diag(w i )− 1 d i w i w T i = I i=1 j∈N (i) z t ij − z * ij 2 2 − 1 d i w T i (z t i − z * i ) 2 2 = I i=1 z t i − z * i 2 2 − 1 d i A r i x t − a i 2 2 ,(43)
where the last equality uses w T i z * i = A r i x * = a i . In the following lemma, we prove that Q i is a positive semi-definite matrix. Thus, Q is also positive semi-definite.
Lemma 3 Q i is positive semi-definite.
Proof: As W ij is either an identity matrix or a zero matrix, W i has d i nonzero entries. Removing the zero entries from W i , we haveW i which only has d i nonzero entries. Then,
W i = I m i . . . I m i , diag(W i ) = I m i . . . I m i ,(44)diag(W i ) is an identity matrix. DefineQ i = diag(W i ) − 1 d iW iW T i . IfQ i is positive semi-definite, Q i is positive semi-definite.
Denote λ max W i as the largest eigenvalue ofW iW T i , which is equivalent to the largest eigenvalue of
W T iW i . SinceW T iW i = d i I m i , then λ max W i = d i . Then, for any v, v 2W iW T i ≤ λ max W i v 2 2 = d i v 2 2 .(45)
Thus,
v 2 Q i = v 2 diag(W i )− 1 d iW iW T i = v 2 2 − 1 d i v 2W iW T i ≥ 0 ,(46)
which completes the proof.
Let W t i ∈ R Jm i ×m i be a column vector of W ijt ∈ R m i ×m i where W ijt = I m i , if A ijt = 0 and j t ∈ I t , 0 otherwise .(47)
Define P t ∈ R Jm×Jm as a diagonal matrix of P t i ∈ R Jm i ×Jm i and
P t = diag[P t 1 , · · · , P t I ] , P t i = diag(W t i ) − 1 K i W t i (W t i ) T .(48)whereK i = min{K, d i } ≥ min{|I t ∩ N i |, d i }.
Using similar arguments in Lemma 3, we can show P t is positive semi-definite. Therefore,
z t+1 − z t 2 Pt = I i=1 z t+1 i − z t i 2 P t i = I i=1 z t+1 i − z t i 2 diag(w t i )− 1 K i w t i (w t i ) T = I i=1 jt∈It z t+1 ijt − z t ijt 2 2 − 1 K i (w t i ) T (z t+1 i − z t i ) 2 2 = I i=1 z t+1 i − z t i 2 2 − 1 K i A r i (x t+1 − x t ) 2 2 .(49)
In PDMM, an index set I t is randomly chosen. Conditioned on x t , x t+1 and y t+1 depend on I t . P t depends on I t . x t , y t are independent of I t . x t depends on a sequence of observed realization of random variable
ξ t−1 = {I 1 , I 2 , · · · , I t−1 } .(50)
As we do not assume that f jt is differentiable, we use the subgradient of f jt . In particular, if f jt is differentiable, the subgradient of f jt becomes the gradient, i.e., ∇f jt (x jt ). PDMM (5)-(7) has the following lemma.
Lemma 4 Let {x t
jt , y t i } be generated by PDMM (5)- (7). Assume τ i > 0 and ν i ≥ 0. We have
jt∈It f jt (x t+1 jt ) − f jt (x * jt ) ≤ − K J I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − jt∈It ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) + K J ŷ t + ρ(Ax t − a), Ax t − a + I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ 2 A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + jt∈It η jt (B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) − B φ j t (x t+1 jt , x t jt )) + ρ 2 I i=1 [(1 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i ] A r i x t − a i 2 2 − (1 − ν i − τ i + 1 d i ) A r i x t+1 − a i 2 2 + (1 − ν i − 1 K i ) A r i (x t+1 − x t ) 2 2 .
(51)
Proof: Let ∂f jt (x t+1 jt ) be the subdifferential of f jt at x t+1 jt . The optimality of the x jt update (5) is
0 ∈ ∂f jt (x t+1 jt ) + (A c jt ) T [ŷ t + ρ(A c jt x t+1 jt + k =jt A c k x t k − a)] + η jt (∇φ jt (x t+1 jt ) − ∇φ jt (x t jt )) ,(52)
Using (7) and rearranging the terms yield
− (A c jt ) T [ŷ t + ρ(Ax t − a) + ρA c jt (x t+1 jt − x t jt )] + η jt (∇φ jt (x t+1 jt ) − ∇φ jt (x t jt )) ∈ ∂f jt (x t+1 jt ) . (53)
Using the convexity of f jt , we have
f jt (x t+1 jt ) − f jt (x * jt ) ≤ − ŷ t + ρ(Ax t − a), A c jt (x t+1 jt − x * jt ) − ρ A c jt (x t+1 jt − x t jt ), A c jt (x t+1 jt − x * jt ) − η jt ∇φ jt (x t+1 jt ) − ∇φ jt (x t jt ), x t+1 jt − x * jt = − ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) − ŷ t + ρ(Ax t − a), A c jt (x t+1 jt − x t jt ) − ρ I i=1 A ijt (x t+1 jt − x t jt ), A ijt (x t+1 jt − x * jt ) + η jt B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) − B φ j t (x t+1 jt , x t jt ) .(54)
Summing over j t ∈ I t , we have
jt∈It f jt (x t+1 jt ) − f jt (x * jt ) ≤ − jt∈It ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) − ŷ t + ρ(Ax t − a), jt∈It A c jt (x t+1 jt − x t jt ) − ρ I i=1 jt∈It A ijt (x t+1 jt − x t jt ), A ijt (x t+1 jt − x * jt ) + jt∈It η jt B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) − B φ j t (x t+1 jt , x t jt ) = − jt∈It ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) + K J ŷ t + ρ(Ax t − a), Ax t − a − K J ŷ t + ρ(Ax t − a), Ax t − a − ŷ t + ρ(Ax t − a), A(x t+1 − x t ) H 1 + ρ 2 I i=1 jt∈It ( A ijt (x * jt − x t jt ) 2 2 − A ijt (x * jt − x t+1 jt ) 2 2 − A ijt (x t+1 jt − x t jt ) 2 2 ) H 2 + jt∈It η jt B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) − B φ j t (x t+1 jt , x t jt ) .(55)
H 1 in (55) can be rewritten as
H 1 = − ŷ t + ρ(Ax t − a), Ax t+1 − a + (1 − K J ) ŷ t + ρ(Ax t − a), Ax t − a .(56)
The first term of (56) is equivalent to
− ŷ t + ρ(Ax t − a), Ax t+1 − a = − I i=1 ŷ t i + ρ(A r i x t − a i ), A r i x t+1 − a i = − I i=1 y t i + (1 − ν i )ρ(A r i x t − a i ), A r i x t+1 − a i = − I i=1 y t+1 i − τ i ρ(A r i x t+1 − a i ), A r i x t+1 − a i + (1 − ν i )ρ A r i x t − a i , A r i x t+1 − a i = − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ A r i x t+1 − a i 2 2 − (1 − ν i )ρ 2 ( A r i (x t+1 − x t ) 2 2 − A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) = − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ 2 A r i x t+1 − a i 2 2 + I i=1 (1 − ν i )ρ 2 ( A r i (x t+1 − x t ) 2 2 − A r i x t − a i 2 2 ) − (1 − ν i − τ i )ρ 2 A r i x t+1 − a i 2 2 .(57)
The second term of (56) is equivalent to
(1 − K J ) ŷ t + ρ(Ax t − a), Ax t − a = (1 − K J ) I i=1 ŷ t i + ρ(A r i x t − a i ), A r i x t − a i = (1 − K J ) I i=1 y t i + (1 − ν i )ρ(A r i x t − a i ), A r i x t − a i = (1 − K J ) I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 + (1 − K J ) I i=1 (1 − ν i + τ i 2 )ρ A r i x t − a i 2 2 .
(58) (55) is equavilant to
H 2 inH 2 = ρ 2 I i=1 jt∈It ( z * ijt − z t ijt 2 2 − z * ijt − z t+1 ijt 2 2 − z t+1 ijt − z t ijt 2 2 ) = ρ 2 I i=1 ( z * i − z t i 2 2 − z * i − z t+1 i 2 2 − z t+1 i − z t i 2 2 ) = ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + ρ 2 I i=1 1 d i ( A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) − 1 K i A r i (x t+1 − x t ) 2 2 .(59)
where the last equality uses the definition of Q in (42) and P t (48), andK i = min{K, d i }. Combining the results of (56)-(59) gives
H 1 + H 2 = − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ 2 A r i x t+1 − a i 2 2 + I i=1 (1 − ν i )ρ 2 ( A r i (x t+1 − x t ) 2 2 − A r i x t − a i 2 2 ) − (1 − ν i − τ i )ρ 2 A r i x t+1 − a i 2 2 + (1 − K J ) I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 + (1 − K J ) I i=1 (1 − ν i + τ i 2 )ρ A r i x t − a i 2 2 + ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + ρ 2 I i=1 1 d i ( A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) − 1 K i A r i (x t+1 − x t ) 2 2 ) = − K J I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 + I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ 2 A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + ρ 2 I i=1 [(1 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i ] A r i x t − a i 2 2 − (1 − ν i − τ i + 1 d i ) A r i x t+1 − a i 2 2 + (1 − ν i − 1 K i ) A r i (x t+1 − x t ) 2 2 .(60)
Plugging back into (55) completes the proof.
Lemma 5
Let {x t jt , y t i } be generated by PDMM (5)- (7). Assume τ i > 0 and ν i ≥ 0. We have
jt∈It f jt (x t+1 jt ) − f jt (x * jt ) ≤ − K J I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − jt∈It ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) + K J ŷ t + ρ(Ax t − a), Ax t − a + I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − I i=1 y t+1 i , A r i x t+1 − a i − τ i ρ 2 A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − B φ (x * , x t+1 ) − B φ (x t+1 , x t )) + ρ 2 I i=1 γ i ( A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) − β i A r i x t+1 − a i 2 2 .(61)
where η T = [η 1 , · · · , η J ]. τ i > 0, ν i ≥ 0, γ i ≥ 0 and β i ≥ 0 satisfy the following conditions:
ν i ∈ (max{0, 1 − 2J K i (2J − K) }, 1 − 1 K i ] ,(62)τ i ≤ J 2J − K [ 4 K i − (4 − 2K J )(1 − ν i )] ≤ 2K K i (2J − K) ,(63)γ i = (3 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i − 2 K i ,(64)β i = 4 K i − (2 − K J )[2(1 − ν i ) + τ i ] .(65)
Proof: In (51), denote
H 3 = [(1 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i ] A r i x t − a i 2 2 − (1 − ν i − τ i + 1 d i ) A r i x t+1 − a i 2 2 ,(66)H 4 = (1 − ν i − 1 K i ) A r i (x t+1 − x t ) 2 2 .(67)
Our goal is to eliminate H 4 so that
H 3 + H 4 = γ i ( A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) − β i A r i x t+1 − a i 2 2 ,(68)
where γ i ≥ 0 and β i ≥ 0 . We want to choose a large τ i and a small ν i .
Assume 1 − ν i − 1 K i ≥ 0, i.e., ν i ≤ 1 − 1 K i , we have H 4 = (1 − ν i − 1 K i ) A r i (x t+1 − x t ) 2 2 ≤ 2(1 − ν i − 1 K i )( A r i x t − a i 2 2 + A r i x t+1 − a i 2 2 ) . (69)
Therefore, we have
H 3 + H 4 ≤ [(3 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i − 2 K i ] A r i x t − a i 2 2 + (1 − ν i + τ i − 1 d i − 2 K i ) A r i x t+1 − a i 2 2 = γ i ( A r i x t − a i 2 2 − A r i x t+1 − a i 2 2 ) − β i A r i x t+1 − a i 2 2 .(70)
where
γ i = (3 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i − 2 K i ≥ (3 − 2K J ) 1 K i + (1 − K J )τ i + 1 d i − 2 K i = (1 − K J ) 1 K i − K JK i + 1 d i + (1 − K J )τ i ≥ 0 .(71)
and
β i = −(1 − ν i + τ i − 1 d i − 2 K i + γ i ) = 4 K i − (2 − K J )[2(1 − ν i ) + τ i ] .(72)
We also want β i ≥ 0, which can be reduced to
τ i ≤ J 2J − K [ 4 K i − (4 − 2K J )(1 − ν i )] (73) ≤ J 2J − K [ 4 K i − (4 − 2K J ) 1 K i ] = 2K K i (2J − K)
.
It also requires the RHS of (73) to be positive, leading to
ν i > max{0, 1 − 2J K i (2J−K) }. Therefore, ν i ∈ (max{0, 1 − 2J K i (2J−K) }, 1 − 1 K i ]. Denote B φ = [B φ 1 , · · · , B φ J ] T as a column vector of the Bregman divergence on block coordi- nates of x. Using x t+1 = [x t+1 jt∈It , x t jt ∈It ] T , we have B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) = B φ (x * , x t ) − B φ (x * , x t+1 ), B φ j t (x t+1 jt , x t jt ) = B φ (x t+1 , x t ). Thus, jt∈It η jt B φ j t (x * jt , x t jt ) − B φ j t (x * jt , x t+1 jt ) − B φ j t (x t+1 jt , x t jt ) = η T (B φ (x * , x t ) − B φ (x * , x t+1 ) − B φ (x t+1 , x t )) .(74)
where η T = [η 1 , · · · , η J ].
Lemma 6
Let {x t jt , y t i } be generated by PDMM (5)- (7). Assume τ i > 0 and ν i ≥ 0 satisfy the conditions in Lemma 5. We have
f (x t ) − f (x * ) ≤ − I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 + J K L ρ (x t , y t ) − E ItLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E It A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − E It z * − z t+1 2 Q − E It z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − E It B φ (x * , x t+1 ) − E It B φ (x t+1 , x t )) .(75)
whereL ρ is defined as follows:
L ρ (x t , y t ) = f (x t ) − f (x * ) + I i=1 y t i , A r i x t − a i + (γ i − τ i )ρ 2 A r i x t − a i 2 2 .
(76) τ i , ν i , γ i , β i and η are defined in Lemma 5.
Proof:
Using x t+1 = [x t+1 jt∈It , x t jt ∈It ] T , we have f (x t+1 ) − f (x t ) = jt∈It f jt (x t+1 jt ) − f jt (x t jt ) = jt∈It [f jt (x t+1 jt ) − f jt (x * jt )] − jt∈It [f jt (x t jt ) − f jt (x * jt )] .(77)
Rearranging the terms and using Lemma 5 yield
jt∈It f jt (x t jt ) − f jt (x * jt ) = j∈It [f jt (x t+1 jt ) − f jt (x * jt )] + f (x t ) − f (x t+1 ) ≤ − K J I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 − jt∈It ŷ t + ρ(Ax t − a), A c jt (x t jt − x * jt ) + K J ŷ t + ρ(Ax t − a), Ax t − a +L ρ (x t , y t ) −L ρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − z * − z t+1 2 Q − z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − B φ (x * , x t+1 ) − B φ (x t+1 , x t )) ,(78)
whereL ρ (x t , y t ) is defined in (76). Conditioning on x t and taking expectation over I t , we have
K J [f (x t ) − f (x * )] ≤ − K J I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 +L ρ (x t , y t ) − E ItLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E It A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − E It z * − z t+1 2 Q − E It z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − E It B φ (x * , x t+1 ) − E It B φ (x t+1 , x t )) ,(79)
where we use
E It [− jt∈It ŷt + ρ(Ax t − a), A c jt (x t jt − x * jt ) ] = − K J ŷ t + ρ(Ax t − a), Ax t − a .(80)
Dividing both sides by K J and using the definition (76) complete the proof.
A.2 Theoretical Results
We establish the convergence results for PDMM under fairly simple assumptions:
Assumption 2
(1) f j : R n j → R ∪ {+∞} are closed, proper, and convex.
(2) A KKT point of the Lagrangian (ρ = 0 in (2)) of Problem (1) exists.
Assumption 2 is the same as that required by ADMM [3,35]. Let ∂f j be the subdifferential of f j . Assume that {x * j , y * i } satisfies the KKT conditions of the Lagrangian (ρ = 0 in (2)), i.e.,
− A T j y * ∈ ∂f j (x * j ) ,(81)Ax * − a = 0.(82)
During iterations, (82) is satisfied if Ax t+1 = a. The optimality conditions for the x j update (5) is
0 ∈ ∂f j (x t+1 j ) + A c j [ŷ t + ρ(A c j x t+1 j + k =j A c k x t k − a)] + η j (∇φ j (x t+1 j ) − ∇φ j (x t j )) ,(83)
which is equivalent to
−A c j [y t + (1 − ν)ρ(Ax t − a) + A c j (x t+1 j − x t j )] − η j (∇φ j (x t+1 j ) − ∇φ j (x t j )) ∈ ∂f j (x t+1 j ) . (84) When Ax t+1 = a, y t+1 = y t . If A c j (x t+1 j − x t j ) = 0, then Ax t − a = 0. When η j ≥ 0, further assuming B φ j (x t+1
j , x t j ) = 0, (81) will be satisfied. Overall, the KKT conditions (81)-(82) are satisfied if the following optimality conditions are satisfied by the iterates:
Ax t+1 = a , A c j (x t+1 j − x t j ) = 0 ,(85)B φ j (x t+1 j , x t j ) = 0 .(86)
The above optimality conditions are sufficient for the KKT conditions. (85) are the optimality conditions for the exact PDMM. (86) is needed only when η j > 0. In Lemma 5, setting the values of ν i , τ i , γ i , β i as follows:
ν i = 1 − 1 K i , τ i = K K i (2J − K) , γ i = 2(J − K) K i (2J − K) + 1 d i − K JK i , β i = K JK i .(87)
Define the residual of optimality conditions (85)-(86) as
R(x t+1 ) = ρ 2 z t+1 − z t 2 Pt + ρ 2 I i=1 β i A r i x t+1 − a i 2 2 + [η T B φ (x t+1 , x t )] .(88)
If R(x t+1 ) → 0, (85)-(86) will be satisfied and thus PDMM converges to the KKT point {x * , y * }.
Define the current iterate v t = (x t j , y t i ) and h(v * , v t ) as a distance from v t to a KKT point v * = (x * j , y * i ): h(v * , v t ) = K J I i=1 1 2τ i ρ y * i − y t−1 i 2 2 +L ρ (x t , y t ) + ρ 2 z * − z t 2 Q + η T B φ (x * , x t ) .(89)
The following Lemma shows that h(v * , v t ) ≥ 0.
Lemma 7 Let h(v * , v t ) be defined in (89). Setting ν i = 1 − 1 K i and τ i = K K i (2J−K) , we have h(v * , v t ) ≥ ρ 2 I i=1 ζ i A r i x t − a i 2 2 + ρ 2 z * − z t 2 Q + + J j=1 η j B φ j (x * j , x t j ) ≥ 0 . (90) where ζ i = J−K K i (2J−K) + 1 d i − K JK i ≥ 0. Moreover, if h(v * , v t ) = 0, then A r i x t = a i , z t = z * and B φ j (x * j , x t j ) = 0.
Thus, (81)-(82) are satisfied. Proof: Using the convexity of f and (81), we have
f (x * ) − f (x t ) ≤ − A T y * , x * − x t = I i=1 y * i , A r i x t − a i .(91)Thus,L ρ (x t , y t ) = f (x t ) − f (x * ) + I i=1 y t i , A r i x t − a i + (γ i − τ i )ρ 2 A r i x t − a i 2 2 ≥ I i=1 y t i − y * i , A r i x t − a i + (γ i − τ i )ρ 2 A r i x t − a i 2 2 = I i=1 y t−1 i − y * i , A i x t − a i + y t i − y t−1 i , A i x t − a i + (γ i − τ i )ρ 2 A r i x t − a i 2 2 ≥ I i=1 − K 2Jτ i ρ y t−1 i − y * i 2 2 − Jτ i ρ 2K A i x t − a i 2 2 + (γ i + τ i )ρ 2 A i x t − a i 2 2 = I i=1 − K 2Jτ i ρ y t−1 i − y * i 2 2 + [γ i + (1 − J K )τ i ] ρ 2 A i x t − a i 2 2 . (92) h(v * , v t ) is reduced to h(v * , v t ) ≥ ρ 2 I i=1 [γ i + (1 − J K )τ i ] A i x t − a i 2 2 + ρ 2 z * − z t 2 Q + η T B φ (x * , x t ) .(93)Setting 1 − ν i = 1 K i and τ i = K K i (2J−K) , we have γ i + (1 − J K )τ i = (3 − 2K J )(1 − ν i ) + (1 − K J )τ i + 1 d i − 2 K i + (1 − J K )τ i = (1 − K J ) 1 K i + (2 − K J − J K ) K K i (2J − K) + 1 d i − K JK i = (J − K) K i (2J − K) + 1 d i − K JK i ≥ 0 .(94)Therefore, h(v * , v t ) ≥ 0. Letting ζ i = J−K K i (2J−K) + 1 d i − K JK i
completes the proof.
The following theorem shows that h(v * , v t ) decreases monotonically and thus establishes the global convergence of PDMM.
Theorem 5 (Global Convergence of PDMM)
Let v t = (x t jt , y t i ) be generated by PDMM (5)- (7) and v * = (x * j , y * i ) be a KKT point satisfying (81)-(82). Setting ν i = 1 − 1
K i and τ i = K K i (2J−K) , we have 0 ≤ E ξt h(v * , v t+1 ) ≤ E ξ t−1 h(v * , v t ) , E ξt R(x t+1 ) → 0 .(95)
Proof: Adding (91) and (75) yields
0 ≤ I i=1 y * i − y t i , A r i x t − a i + τ i ρ 2 A r i x t − a i 2 2 + J K L ρ (x t , y t ) − E ItLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E It A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − E It z * − z t+1 2 Q − E It z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − E It B φ (x * , x t+1 ) − E It B φ (x t+1 , x t )) .(96)
Using (6), we have
y * i − y t i , A r i x t − a i + τ i ρ 2 A r i x t − a i 2 2 = 1 τ i ρ y * i − y t i , y t i − y t−1 i + τ i ρ 2 A r i x t − a i 2 2 = 1 2τ i ρ ( y * i − y t−1 i 2 2 − y * i − y t i 2 2 ) .(97)
Plugging back into (96) gives
0 ≤ I i=1 1 2τ i ρ ( y * i − y t−1 i 2 2 − y * i − y t i 2 2 ) + J K L ρ (x t , y t ) − E ItLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E It A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − E It z * − z t+1 2 Q − E It z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − E It B φ (x * , x t+1 ) − E It B φ (x t+1 , x t )) = J K h(v * , v t ) − E It h(v * , v t+1 ) − E It R(x t+1 ) .(98)
Taking expectaion over ξ t−1 , we have
0 ≤ J K E ξ t−1 h(v * , v t ) − E ξt h(v * , v t+1 ) − E ξt R(x t+1 ) .(99)
Since E ξt R(x t+1 ) ≥ 0, we have
E ξt h(v * , v t+1 ) ≤ E ξ t−1 h(v * , v t ) .(100)
Thus, E ξt h(v * , v t+1 ) converges monotonically.
Rearranging the terms in (99) yields
E ξt R(x t+1 ) ≤ E ξ t−1 h(v * , v t ) − E ξt h(v * , v t+1 ) .(101)
Summing over t gives
T −1 t=0 E ξt R(x t+1 ) ≤ h(v * , v 0 ) − E ξ T −1 h(v * , v T ) ≤ h(v * , v 0 ) .(102)
where the last inequality uses the Lemma 7. As T → ∞, E ξt R(x t+1 ) → 0, which completes the proof.
The following theorem establishes the iteration complexity of PDMM in an ergodic sense.
Theorem 6
Let (x t j , y t i ) be generated by PDMM (5)- (7). Letx T = T t=1 x t . Setting ν i = 1 − 1
K i and τ i = K K i (2J−K) , we have Ef (x T ) − f (x * ) ≤ I i=1 1 2τ i ρ y 0 i 2 2 + J K 1 2β i ρ y * i 2 2 +L ρ (x 1 , y 1 ) + ρ 2 z * − z 1 2 Q + η T B φ (x * , x 1 ) T ,(103)E I i=1 β i A r ix T − a i 2 2 ≤ 2 ρ h(v * , v 0 ) T . (104) where β i = K JK i .
Proof: Using (7), we have
− I i=1 y t i , A r i x t − a i − τ i ρ 2 A r i x t − a i 2 2 = − I i=1 1 τ i ρ y t i , y t i − y t−1 i − 1 2τ i ρ y t i − y t−1 i 2 2 = I i=1 1 2τ i ρ ( y t−1 i − y t i 2 2 ) .(105)
Plugging back into (75) yields
f (x t ) − f (x * ) ≤ I i=1 1 2τ i ρ ( y t−1 i 2 2 − y t i 2 2 ) + J K L ρ (x t , y t ) − E ItLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E It A r i x t+1 − a i 2 2 + ρ 2 ( z * − z t 2 Q − E It z * − z t+1 2 Q − E It z t+1 − z t 2 Pt ) + η T (B φ (x * , x t ) − E It B φ (x * , x t+1 ) − E It B φ (x t+1 , x t )) .(106)
Taking expectaion over ξ t−1 , we have
E ξ t−1 f (x t ) − f (x * ) ≤ I i=1 1 2τ i ρ (E ξ t−2 y t−1 i 2 2 − E ξ t−1 y t i 2 2 ) + J K E ξ t−1L ρ (x t , y t ) − E ξtLρ (x t+1 , y t+1 ) − ρ 2 I i=1 β i E ξt A r i x t+1 − a i 2 2 + ρ 2 (E ξ t−1 z * − z t 2 Q − E ξt z * − z t+1 2 Q − E ξt z t+1 − z t 2 Pt ) + η T (E ξ t−1 B φ (x * , x t ) − E ξt B φ (x * , x t+1 ) − E ξt B φ (x t+1 , x t )) .(107)
Summing over t, we have
T t=1 E ξ t−1 f (x t ) − f (x * ) ≤ I i=1 1 2τ i ρ ( y 0 i 2 2 − E ξ T −1 y T i 2 2 ) + J K L ρ (x 1 , y 1 ) − E ξ TL ρ (x T +1 , y T +1 ) − ρ 2 T t=1 I i=1 β i E ξt A r i x t+1 − a i 2 2 + ρ 2 ( z * − z 1 2 Q − E ξ T z * − z T +1 2 Q − E ξ T z T +1 − z T 2 Q ) + η T (B φ (x * , x 1 ) − E ξ T B φ (x * , x T +1 ) − E ξ T B φ (x T +1 , x T )) .(108)
Using (91), we havẽ
L ρ (x T +1 , y T +1 ) = f (x T +1 ) − f (x * ) + I i=1 [ y T +1 i , A i x T +1 − a i + (γ i − τ i )ρ 2 A i x T +1 − a i 2 2 ] ≥ − I i=1 y * i , A r i x T +1 − a i + 1 d i ) A r i (x t+1 − x t ) 2 2 + J j=1 η j B φ j (x * j , x t j ) − B φ j (x * j , x t+1 j ) − B φ j (x t+1 j , x t j ) .(124)
Proof: Let I t be all blocks, K = J. According the definition of P t in (42) and Q in (48), P t = Q. Therefore, (51) reduces to
f (x t+1 ) − f (x * ) ≤ I i=1 − y t+1 i , A r i x t+1 − a i + τ i ρ 2 A r i x t+1 − a i 2 2 + ρ 2 ( z t − z * 2 Q − z t+1 − z * 2 Q − z t+1 − z t 2 Q ) + J j=1 η j B φ j (x * j , x t j ) − B φ j (x * j , x t+1 j ) − B φ j (x t+1 j , x t j ) + ρ 2 I i=1 (ν i − 1 + 1 d i ) A r i x t − a i 2 2 − (1 − ν i − τ i + 1 d i ) A r i x t+1 − a i 2 2 + (1 − ν i − 1 d i ) A r i (x t+1 − x t ) 2 2 .(125)
Rearranging the terms completes the proof.
Corollary 2
Let {x t j , y t i } be generated by PDMM (5)- (7). Assume (1)τ i > 0 and ν i ≥ 0; (2) η j > 0; (3) φ j is α j -strongly convex. We have
f (x t+1 ) − f (x * ) ≤ I i=1 − y t+1 i , A r i x t+1 − a i + τ i ρ 2 A r i x t+1 − a i 2 2 + ρ 2 ( z t − z * 2 Q − z t+1 − z * 2 Q − z t+1 − z t 2 Q ) + J j=1 η j B φ j (x * j , x t j ) − B φ j (x * j , x t+1 j ) .(126)
ν i and τ i satisfy
ν i ∈ [1 − 1 d i − η j α j ρId i λ ij max , 1 − 1 d i ] and τ i ≤ 1 + 1 d i − ν i , where λ ij max is the largest eigenvalue of A T ij A ij .
In particular, if η j = (d i −1)ρIλ ij max α j , ν i = 0 and τ i ≤ 1 + 1 d i .
Proof: Assume η j > 0. We can choose larger τ i and smaller ν i than Lemma 5 by setting η j sufficiently large. Since φ j is α j -strongly convex,
B φ j (x t+1 j , x t j ) ≥ α j 2 x t+1 j − x t j 2 2 . We have J j=1 η j B φ j (x t+1 j , x t j ) ≥ I i=1 J j=1 η j α j 2I x t+1 j − x t j 2 2 ≥ I i=1 j∈N (i) η j α j 2Iλ ij max A ij (x t+1 j − x t j ) 2 2 . (127) A r i (x t+1 − x t ) 2 2 = j∈N (i) A ij (x t+1 j − x t j ) 2 2 ≤ d i j∈N (i) A ij (x t+1 j − x t j ) 2 2 ,(128)
where λ ij max is the largest eigenvalue of A T ij A ij . Plugging into (124) gives
Proof: Adding (91) and (126) together yields Plugging into (126) yields
f (x t+1 ) − f (x * ) ≤ I i=1 1 2τ i ρ ( y t i 2 2 − y t+1 i 2 2 ) + ρ 2 ( u t − u * 2 Q − u t+1 − u * 2 Q − u t+1 − u t 2 Q ) + J j=1 η j B φ j (x * j , x t j ) − B φ j (x * j , x t+1 j ) .(138)
Summing over t from 0 to T − 1, we have
T −1 t=0 f (x t+1 ) − f (x * ) ≤ I i=1 1 2τ i ρ ( y t i 2 2 − y t+1 i 2 2 ) + ρ 2 ( u 0 − u * 2 Q − u T − u * 2 Q ) + J j=1 η j B φ j (x * j , x t j ) − B φ j (x * j , x t+1 j ) .(139)
Applying the Jensen's inequality on the LHS and usingx T = T t=1 x t complete the proof.
If η j = (d i −1)ρIλ ij max α j , ν i = 0 and τ i = 1. Therefore, PDMM becomes PJADMM [9], where the convergence rate of PJADMM has been improved to o(1/T ).
Step: We attribute the failure of the Jacobi updates (3)-(4) to the following observation in
85)-(86) will be satisfied and thus PDMM converges to the KKT point {x * , y * }. Define the current iterate
Figure 1 :
1Comparison of the convergence of PDMM (with K blocks) with ADMM methods in RPCA. The values of τ i , ν i in PDMM is computed according to
Figure 2 :
21), i.e., (τ i , ν i ) = {( 1 5 , 0), ( Comparison of convergence of PDMM and other methods in overlapping group Lasso.
Table 2 :
2The 'best' results of PDMM with tuning parameters τ i , ν i in RPCA. PDMM1 randomly updates one block and is the fastest algorithm. PDMMs converges faster than other ADMM methods.time (s) iteration residual(×10 −5 ) objective (log)
PDMM1
118.83
40
3.60
8.07
PDMM2
137.46
34
5.51
8.07
PDMM3
147.82
31
6.54
8.07
GSADMM 163.09
28
6.84
8.07
RBSUMM 206.96
141
8.55
8.07
sADMM 2
731.51
139
9.73
8.07
http://www.stanford.edu/ boyd/papers/prox algs/matrix decomp.html
The following theorem shows that h(v * , v t ) decreases monotonically and thus establishes the global convergence of PDMM.Theorem 7 (Global Convergence of PDMM)Let v t = (x t j , y t i ) be generated by PDMM (5)-(7)and v * = (x * j , y * i ) be a KKT point satisfying (81)-(82). Assume τ i , ν i and γ i satisfy conditions in Lemma 2. Then v t converges to the KKT point v * monotonically, i.e.,
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| []
|
[
"Enhanced Purcell factor for nanoantennas supporting interfering resonances",
"Enhanced Purcell factor for nanoantennas supporting interfering resonances"
]
| [
"Rémi Colom \nZuse Institute Berlin\nTakustraße 714195BerlinGermany\n",
"Felix Binkowski \nZuse Institute Berlin\nTakustraße 714195BerlinGermany\n",
"Fridtjof Betz \nZuse Institute Berlin\nTakustraße 714195BerlinGermany\n",
"Yuri Kivshar \nNonlinear Physics Center\nResearch School of Physics\nAustralian National University\n2601CanberraACTAustralia\n",
"Sven Burger \nZuse Institute Berlin\nTakustraße 714195BerlinGermany\n\nJCMwave GmbH\nBolivarallee 2214050BerlinGermany\n"
]
| [
"Zuse Institute Berlin\nTakustraße 714195BerlinGermany",
"Zuse Institute Berlin\nTakustraße 714195BerlinGermany",
"Zuse Institute Berlin\nTakustraße 714195BerlinGermany",
"Nonlinear Physics Center\nResearch School of Physics\nAustralian National University\n2601CanberraACTAustralia",
"Zuse Institute Berlin\nTakustraße 714195BerlinGermany",
"JCMwave GmbH\nBolivarallee 2214050BerlinGermany"
]
| []
| We study the effect of coupled resonances and quasi-bound states in the continuum (quasi-BICs) on the Purcell factor in dielectric resonant nanoantennas. We analyze numerically interfering resonances in a nanodisk with and without a substrate when the modes are coupled to an emitter localized inside the nanodisk, and we quantify the modal contributions to the Purcell factor also reconstructing the radiation patterns of the resonant system. It is revealed that the Purcell effect can be boosted substantially for a strong coupling of resonances in the quasi-BIC regime. | 10.1103/physrevresearch.4.023189 | [
"https://arxiv.org/pdf/2111.03339v3.pdf"
]
| 243,832,684 | 2111.03339 | ce89d293c4fcc3069e34da598ce44cb828b3cf65 |
Enhanced Purcell factor for nanoantennas supporting interfering resonances
Rémi Colom
Zuse Institute Berlin
Takustraße 714195BerlinGermany
Felix Binkowski
Zuse Institute Berlin
Takustraße 714195BerlinGermany
Fridtjof Betz
Zuse Institute Berlin
Takustraße 714195BerlinGermany
Yuri Kivshar
Nonlinear Physics Center
Research School of Physics
Australian National University
2601CanberraACTAustralia
Sven Burger
Zuse Institute Berlin
Takustraße 714195BerlinGermany
JCMwave GmbH
Bolivarallee 2214050BerlinGermany
Enhanced Purcell factor for nanoantennas supporting interfering resonances
We study the effect of coupled resonances and quasi-bound states in the continuum (quasi-BICs) on the Purcell factor in dielectric resonant nanoantennas. We analyze numerically interfering resonances in a nanodisk with and without a substrate when the modes are coupled to an emitter localized inside the nanodisk, and we quantify the modal contributions to the Purcell factor also reconstructing the radiation patterns of the resonant system. It is revealed that the Purcell effect can be boosted substantially for a strong coupling of resonances in the quasi-BIC regime.
I. INTRODUCTION
Resonances play a central role in the control of lightmatter interactions in nanophotonics. Plasmonic resonances enable such a control via large near-field enhancements [1,2], which allows, e.g., for realizing plasmonic nanoantennas to tailor the radiation from quantum emitters [3,4]. Recently, the excitation of Mietype resonant modes [5,6] in high-refractive-index dielectric resonators has proven to be very useful for a wide range of applications, from the enhancement of nonlinear effects to a resonant control of the phase in metasurfaces [7,8]. One important figure of merit for measuring the effect of resonances on light-matter interactions is their quality factor (Q-factor), that quantifies the ability of a structure to trap light and to enhance the electromagnetic fields. Nanoresonators act as nanoantennas for strongly localized light sources, like quantum dots or defects in crystalline lattices, which can allow for the realization of efficient single-photon sources by enhancing the emission of light [9,10]. Such a control of the emission via the modification of the electromagnetic environment is a concept that dates back to the pioneering work of Purcell [11] performed in the microwave range followed by the experiments of Drexhage [12] that demonstrated the possibility of controlling the lifetime of fluorescent molecules in the visible range. This phenomenon is ubiquitous, and it has also been used to control the resonant scattering by dielectric nanorod antennas [13].
The figure of merit that quantifies the emission enhancement is called the Purcell factor [11], and it is proportional to the Q-factor. Optical nanoantennas were first realized with plasmonic materials [3,4], but recently dielectric resonators have been shown to allow for large enhancements of the Purcell factor via the excitation of both electric and magnetic optically-induced * Correspondence email address: [email protected]
Mie-type resonances [14,15]. The excitation of magnetic resonances presents the advantage of enhancing light emission also via the magnetic dipole transitions. This effect was first theoretically predicted [15][16][17] and confirmed later in experiments [18][19][20]. This is a very promising application for dielectric nanoantennas as the enhancement of light emission empowered by the magnetic dipole resonances is an emerging area of research [21,22]. The enhancement of the Purcell factor was used successfully to improve the emission of quantum dots in silicon nanoantennas [23] and also for metallic and hybrid nanoantennas [24]. Control of the emission can also be achieved dynamically [25]. Finally, nanoantennas can also be designed to enhance the performance of quantum emitters, providing promising platforms for the realization of single-photon sources [26].
Bound states in the continuum (BICs) appear as a special type of nonradiating modes associated with an infinite Q-factor [27]. Such states can originate from different physical mechanisms [28,29]. Symmetry protected BICs occur in photonic crystal slabs, and they result from the impossibility of these modes to couple to propagating fields outside the photonic crystal because of symmetry restrictions [27,30]. Further, the so-called accidental BICs appear from interferences between several resonances [27,28]. They are observed when a system parameter is varied continuously. This concept was introduced in quantum mechanics where the coupling between resonances is controlled by engineering the potential [31].
In optics, one of the first attempts to study BICs was made in the physics of photonic crystals [32]. While BICs can be realized in gratings or photonic crystals which are infinite in two directions, it is much more challenging to observe such BICs in compact structures and even more in subwavelength systems [27]. The existence of BICs, also called embedded eigenstates, was predicted theoretically in a coated nanosphere where the permittivity of the outer shell vanishes [33]. In more realistic configurations, it is still possible to take advantage of the coupling of resonances in nanostructures to arXiv:2111.03339v3 [physics.optics] 3 Jun 2022 increase the Q-factor of one resonance, even if it does not lead to accidental BICs with infinite Q-factors. In photonics, such an approach was suggested to enhance the Q-factors of the modes of optical micro-cavities [34] and coupled dielectric nanopillars [35]. It was shown recently that high-refractive index nanodisks supporting multiple resonances are a good platform to employ this approach [36][37][38]. Due to similarity of this approach with accidental BICs [31], the large Q-factors achieved through the interference of several resonances are called quasi-BICs. Quasi-BICs have been observed experimentally in AlGaAs nanodisks [39], and they have been used in various applications [28,40] including nonlinear optics [41][42][43] and lasing from a single nanoparticle [44]. Compared to photonic crystal cavities, ring resonators, and other setups [4], such compact nanostructures supporting quasi-BICs exhibit lower Q-factors and Purcell enhancements [45]. However, their relatively small device footprint allows these resonators to be used, e.g., as meta-atoms in metasurfaces [46].
Unlike BICs, which lead to a perfect confinement of light, quasi-BICs suffer from residual radiation losses. As a consequence, for a rigorous treatment of quasi-BICs it is important to use quasinormal modes (QNMs) and associated complex eigenfrequencies which generalizes modal approaches to dissipative and non-Hermitian systems [45,47,48]. The influence of quasi-BICs on light-matter interactions and, in particular, their coupling to a light source can be quantified by using QNM expansions. The QNM analysis of the coupling of an electromagnetic dipole source to an optical resonator, i.e., the modal expansion of the Purcell factor, has been carried out through several approaches [15,[49][50][51][52].
In this paper, we study quasi-BICs numerically. We consider dielectric nanoresonators either with or without a substrate and demonstrate that they can support interfering resonances with a strong coupling between a pair of modes. We choose to design the structure with the refractive index of GaAs. The motivations behind this choice come from the fact that including a dipole emitter into such a structure can be realized using modern nanofabrication methods allowing to include a quantum dot in a GaAs nanodisk [53]. We carry out numerical simulations with a localized source embedded into the resonator to demonstrate different physical regimes. Modal expansions of the Purcell factor and far-field patterns reveal a complex interplay between different modal contributions interfering destructively in the spectral vicinity of quasi-BICs, yielding a strong enhancement of the Purcell factor and single modal excitation when the parameters of the source and resonator are tuned to match the quasi-BIC conditions.
The major steps followed in this article are illustrated in Fig. 1. In Sec. II, we vary the aspect ratio D/H of a GaAs nanodisk to control the interference between the two modes of the nanodisk with or without a substrate. In particular, the strong coupling between these modes leads to the appearance of a high-Q mode: the quasi-BIC resonance. Sec. III considers the coupling of a dipole source with the nanodisk, leading to a complex electromagnetic response as seen in Fig. 1(b). Modal expansions are employed to analyze the role of the interfer-ence between the nanodisk modes for the coupling with the dipole. These expansions enable to identify how the constructive interference between two modal contributions leads to the enhancement of the dipole emission, as illustrated in Fig. 1(c). On the other hand, destructive interference leads to the inhibition of the dipole emission, as illustrated in Fig. 1(d). The modal analysis of the radiation pattern is carried out in Sec. IV. Finally, Sec. V concludes the paper.
II. QUASI-BICS IN ISOLATED NANODISKS
To understand the appearance of quasi-BIC states, first we review the theoretical approach employed to study the mode coupling [34,54,55]. A good insight in the physics of strong coupling for interfering resonances can be gained from a phenomenological model of mode coupling that involves the two modes with the uncoupled eigenfrequencies ω un,1 and ω un,2 . When these two eigenfrequencies are far apart in the complex plane, there is no coupling between them. However, when the eigenfrequencies get close to each other, the coupling has to be taken into account and modifies the trajectories of these eigenfrequencies when a parameter is varied. The eigenfrequencies of the coupled modes can be found as the eigenvalues of an effective two-mode Hamiltonian, and they are equal to
ω ± = ω un,1 + ω un,2 2 ± √ γ, where γ = ω un,1 − ω un,2 2 2 + v 2
with v being the coupling coefficient between the modes [54]. We are interested in the regime where these two resonances are close to each other, and therefore we assume that (ω un,1 ) = (ω un,2 ) and v is real as in Ref. [54].
As explained in [34,54], two regimes of the mode coupling may be realized depending on the relation between v and 1 2 (ω un,1 − ω un,2 ). When 2v < | (ω un,1 − ω un,2 ) |, the mode eigenvalues become ω ± = ω un,1 + ω un,2 2 ± i |γ|, and one observes that the coupling mostly alters the imaginary part of the eigenvalues resulting in an avoided crossing of the imaginary parts of the coupled eigenvalues and a crossing of their real parts. This behavior substrate (e, f). Field intensity maps |E| of the QNMs in an x − z-cross section through the 3D field distribution. (c), resp. (d), corresponds to the high-Q (resp. low-Q) mode of the isolated nanodisk [at the aspect ratio indicated by the black, resp. green, dot in (b)]. (g), resp. (h), corresponds to the high-Q (resp. low-Q) mode of the nanodisk with substrate [black, resp. green, dot in (f)].
(a) (e) (b) (f) (c) (g) (d) (h)
is a direct signature of the mode weak coupling. On the other hand, if 2v > | (ω un,1 − ω un,2 ) |, the mode eigenvalues are presented as
ω ± = ω un,1 + ω un,2 2 ± |γ|,
suggesting that the coupling of the eigenmodes mostly alters the real part of the eigenfrequencies yielding, this time, an avoided crossing of the real parts of the coupled eigenvalues and a crossing of the imaginary parts. A more detailed discussion on the coupling regimes between modes for a purely real or a purely imaginary coupling constant can be found in the Appendix B. In the following, we discuss how the mode coupling may result in the appearance of a hybridized quasi-BIC mode.
We consider a Gallium Arsenide (GaAs) nanodisk resonator with a height H = 1260 nm and varying diameter D in two different configurations: The nanodisk is just surrounded by air (case 1), and, the nanodisk is placed on a glass substrate and surrounded by a superspace of air (case 2). The optical properties of the system are investigated in the near-infrared wavelength range; the corresponding constant relative permittivities in our model are GaAs = 11.56, sub = 2.25, and air = 1.0. The time-harmonic optical fields are modeled using Maxwell's equations,
∇ × µ −1 0 ∇ × E(r, ω) − (r)ω 2 E(r, ω) = iωJ (r) ,(1)
where µ 0 is the vacuum permeability, (r) is the permittivity, and J (r) the source current density. For numerically solving Eq. (1), we use an adaptive, higher-order finite element method (FEM) [56]. For computing the eigenmodes E n of the system and their associated eigenfrequencies ω n , i.e., solutions to Eq. (1) where J = 0, the cylindrical symmetry of the system is taken into account. Only modes with an azimuthal quantum number equal to 1 or -1 are investigated because these are the only ones excited by a dipole located on the axis of rotation, which is the configuration we are investigating in the second part of this study. Furthermore, only the component of the polarization normal to the symmetry axis can couple to the modes of interest and therefore we restrict to a polarization with z = 0. Without loss of generality we chose a y-polarized dipole.
In order to find a quasi-BIC condition, the interference between two modes of the structure has to be tuned [34,36,57]. This is done by varying the geometry parameter, D, and computing eigenmodes E n and their associated eigenfrequencies ω n , where n is the mode index. Note that alternatively, a perturbation approach based on QNMs may be employed for finding the quasi-BICs [58]. Figure 2(a,b,e,f) shows how the normalized frequency, (ω n H/2c), and the Q-factor,
Q = − 1 2 (ω n ) (ω n ) ,
depend on the aspect ratio D/H. In Fig. 2(a,b), the case where the GaAs nanodisk is located in air is considered. It can be observed that the real part of the eigenfrequencies is showing a repulsion behavior at D/H = 0.909 and an almost coinciding peak reaching Q ≈ 800 is observed for the Q-factor of one of the modes while a minimum is seen for the other mode. As discussed above, this behavior is an indication of strong coupling between the two modes. The high-Q mode can thus be considered to be a quasi-BIC. Figures 2(e,f) show the results for the second case, where the nanodisk is put on a glass substrate. It can be observed that, for the investigated modes and parameter range, the real part of the eigenfrequencies shows a crossing at D/H = 0.933. We observe a peak of the Q-factor reaching Q ≈ 400 at D/H = 0.92. In fact, this peak is linked to the anti-crossing or level-repulsion occurring for the imaginary parts of the eigenfrequencies. This avoided crossing of the imaginary parts of the eigenfrequencies shows up in Fig. 2(f) at about D/H = 0.944. The qualitative analysis based on the effective Hamiltonian discussed above shows that this behavior is an indication of weak coupling between the two modes. The transition from strong to weak coupling when a substrate is added indicates that there must be an exceptional point, i.e., a condition for which the two coupled eigenvalues would become degenerated [59], when continuously varying the refractive index of the substrate from 1 to 1.5 [54,55,60,61]. To conclude the discussion on the avoided crossing of the eigenfrequencies, we show, in Figs. 2(c,d,g,h), the field patterns associated with both modes when the Q-factor is maximized. For the case without substrate, this occurs for D/H = 0.909 while, when the substrate is added, the maximum occurs for an aspect ratio of D/H = 0.92. This helps to understand the level repulsion observed since, in both cases, the modes have apparently very different field patterns: The high-Q mode field is concentrated in hot spots located at the top and bottom of the disk while, for the low-Q mode, it is concentrated at the center of the disk. This apparent difference in the localization of the modes certainly prevents their merging.
III. COUPLING OF A POINT SOURCE TO A NANORESONATOR
Now, we turn to the study of a dipole emitter coupled to the investigated nanoresonator considering the two cases, the nanoresonator with and without substrate. It is worth noting that the coupling of a dipole with a BIC in an array of nanoparticles have already been studied [62], but we will here focus on the coupling of a dipole with the quasi-BIC arising in an individual nanodisk. We consider Maxwell's equations, given by Eq. (1), with the current density J = jδ (r − r d ) that is a point source located at r d . The Purcell factor, which is used to quantify the enhancement of the emission, is defined as
Γ(ω) = −[ (E(ω, r d ) · j * (ω, r d ))]/[2Γ b (ω)],
where Γ b (ω) describes the emission of the dipole in a homogeneous medium of the permittivity GaAs . The interest of studying the Purcell factor and its modal analysis is twofold. On the one hand, one can see how a mode with a Q-factor as large as the one of the quasi-BIC can affect the dipole emission. On the other hand, looking at the modal analysis of the Purcell factor would allow to use it as a probe to study the interplay between several modes. This is particularly interesting for quasi-BICs since interferences between modes are at the origin of their formation.
To do so, we start by considering the Purcell factor for a dipole located at the maximum of the field amplitude of the high-Q mode. This position is on the symmetry axis of the nanodisk, about 30 nm below the top face. The consequences of the interplay between resonances at the origin of the quasi-BIC can be better understood by carrying out a modal analysis of the Purcell factor. Our method for deriving modal expansions relies on the use of Riesz projections [52,63]. The modal expansion of the Purcell factor reads as
Γ tot (ω) = 2 n=1 Γ n (ω) + Γ background (ω),(2)
where Γ n are the modal contributions to the Purcell factor that are computed using contour integrals around the eigenfrequencies. Here, we take into account only the two interfering modes, i.e., the modes which are also shown in Fig. 2. The modal Purcell factors Γ 1 and Γ 2 are contributions corresponding to these two modes. The term Γ background contains the contributions of all other poles as well as the nonresonant background [52,63]. Finally, Γ tot corresponds to the total expansion including both the modal and background contributions. The different black markers indicate the wavelengths at which the radiation patterns are computed in Fig. 4. Details about the modal expansions are provided in the Appendix A. First, we look at the coupling of the dipole to the nanoresonator with the geometry corresponding to the maximum of the Q-factor in Fig. 2. The results of the modal analysis of the Purcell factor are displayed in Fig. 3(a), for a nanodisk in air with an aspect ratio D/H = 0.909, and, in Fig. 3(c), for a nanodisk on a substrate with an aspect ratio D/H = 0.92. In both cases, the peak observed in the Purcell factor spectrum can be directly linked to the modal contribution corresponding to the high-Q mode. In the region around the peak, the contributions from the low-Q mode and the background are very small or even negligible. This demonstrates that, for a resonator supporting a quasi-BIC, an emitter may easily excite nearly exclusively this resonance. We note that the quasi-BIC allows to reach a high Purcell factor of Γ ≈ 40 in the case without substrate and Γ ≈ 20 in the case with substrate.
It is also worth looking at configurations where one can expect that the contributions to the Purcell factor from the two main modes would be of the same order of magnitude. This would allow us to investigate the interplay between modal contributions. This is the reason for showing, in Fig. 3(b), the Purcell factor for an aspect ratio of D/H = 0.933 for the disk without substrate corresponding to the crossing of the Q-factor of the two modes in Fig. 2(b). For the case with substrate, we consider the aspect ratio D/H = 0.944 as it corresponds to the avoided crossing of the imaginary parts of the eigenvalues as can be seen from the Q-factor trajectories in Fig. 2(f). This avoided crossing is caused by the interference of the interacting modes. Therefore, we expect that the interference will be seen in the contributions of the modal expansion. The Purcell factor again shows a distinct maximum, with a value of Γ ≈ 8 with substrate and Γ ≈ 10 without substrate. However, as expected, both modal contributions have the same order of magnitude. Also, the qualitative shape of both spectra of the modal Purcell factors are approximately mirror-symmetric to each other with respect to the resonance wavelength. This behavior yields the fact that, away from the resonance, the signs of the modal contributions of the two modes are opposite, leading to destructive interference in these spectral regions. This is the case in Fig. 3(b) for wavelengths below ∼ 1210 nm and above ∼ 1230 nm. For the case including a sub-strate in Fig. 3(d), we observe a similar behavior for wavelengths below ∼ 1225 nm and above ∼ 1245 nm. The destructive interference between modes has been used previously to qualitatively describe the appearance of quasi-BICs [40]. In the present study, we show that the effect can be quantified by using modal expansion techniques.
Note that the interplay between the modes at the optimal aspect ratio becomes visible when the position of the dipole is moved away from the hot spot of the high-Q mode. Corresponding simulation results can be found in the Appendix C.
IV. MODAL ANALYSIS OF RADIATION PATTERNS
It is well known that the coupling with nanostructures can alter the radiation pattern of a quantum emitter [3]. This ability to control the emission pattern of a dipole emitter with nanostructures has a great practical interest since it can improve the collection of the emitted field with an optical system. A modal analysis allows to understand how each mode but also the interferences between modes modifies the emission pattern. We will consider the far-field pattern of the energy flux density defined as s(r, ω) =
1 2 E * (r, ω) × 1 iωµ0 ∇ × E(r, ω) · n,
i.e., the projection of the Poynting vector on the normal vector n in the direction of field propagation. The modal expansion of s(r, ω) is computed using Riesz projections [63,64] leading to the expression s(r, ω) = 2 n=1 s n (r, ω) + s background (r, ω), where r is a point located in the farfield. We will in particular look at the dependency of the radiation pattern with θ in the x-z plane. In Fig. 4, the field patterns radiated by the dipole upwards towards the air are plotted for different wavelengths and for different aspect ratios. In Figs. 4(a-d), results are shown for the nanodisk without substrate for aspect ratios equal to 0.909 and 0.933 while Figs. 4(g-h) display results for the nanodisk on a substrate for aspect ratios equal to 0.92 and 0.944. Please note that the lower region of the plot shown in gray does not correspond to the field radiated downwards but to the negative modal contributions. Negative contributions are particularly important here, since, as for the Purcell factor, they are linked to the interferences between different modal contributions. Radiation pattern towards the substrate are actually shown in the supplemental material. In Figs. 4(a,e), we show the radiation pattern and its modal expansion at the aspect ratio and wavelength of the quasi-BIC. Just like for the Purcell factor, one mode has a much larger Q-factor than the other, it is not surprising to find that the radiation pattern can then be almost entirely understood from the contribution of the high-Q mode while the contributions from the low-Q mode is negligible compared to the contribution of the high-Q mode. The results of the modal expansion of the radiation pattern for the nanodisk without substrate with the aspect ratio equal to 0.933 are plotted in Figs. 4(b-d). These computations are made for the wavelengths on both sides of the main peak in Fig. 3(b), with 1204 nm, 1217 nm, and 1234 nm in Figs. 4(b, c, d), respectively.
(e) (f) (g) (h) (a) (b) (c) (d)
For λ = 1204 nm, we obtain a positive contribution for both the modes summing up to a radiation lobe between ∼ ±45 • . For λ = 1217 nm, the main contribution is from mode 2 leading to a quite directional emission between ∼ ±30 • . Eventually, for λ = 1234 nm, an interference between the two main modal contributions is observed with a positive contribution from mode 2 between ±30 • and a negative contribution of mode 1 in the same range. In Figs. 4(f-h), we show the re-sults of the modal expansions for the nanodisk on substrate with D/H = 0.944 for the wavelengths 1215 nm, 1237 nm, and 1247 nm, respectively. In Fig. 4(f), for λ = 1215 nm, the mode 1 has a positive contribution for angles between roughly 30 and -30 degrees while the mode 2 has a negative contribution in the same range of angles. Consequently, the total radiation pattern is suppressed, resulting from the interference between several modes. A very analogous behavior is observed at λ = 1247 nm in Fig. 4(h), however, in this case, the mode 1 has a negative contribution while the mode 2 has a positive contribution. There is, again, a strong interference between the two modes and the far-field pattern cannot be understood without taking this interference into account. Finally, in Fig. 4(g), for λ = 1237 nm corresponding to the peak of the Purcell factor in Fig. 3(b), we observe that the contribution from both modes of interest add up leading to a larger amplitude of the ra-diation by the dipole and to a confined far-field pattern.
V. CONCLUSIONS
We have numerically analyzed dielectric nanodisk resonators which support multiple resonances in overlapping frequency ranges. Using a finite-elementmethod-based framework, regimes where the resonators support quasi-BIC resonances have been investigated. The impact of the resonances on the Purcell factor describing the emission enhancement of a localized source has been shown in the quasi-BIC regime as well as in adjacent parameter regimes where several competing resonances are excited. The modal contributions to the Purcell factor have been computed using the Riesz projection method, and it has been shown that a single QNM causes the strongly enhanced dipole emission in the quasi-BIC situation. Further, we have investigated the modal, angular resolved far-field spectrum in on-resonance as well as off-resonance conditions. This demonstrated that modal interference strongly impacts both, far-field emission strength as well as angular resolved radiation patterns. It has been shown that micron-scale dielectric resonators supporting quasi-BICs allow for high Purcell enhancement as well as for highly directed emission of light. We expect that, apart from the gained insight in the complex interference behavior in multi-modal resonators, these findings will allow for the design of efficient and robust future photonic components, such as single-photon emitters for quantum technology applications.
Research Data: The source code and data for performing the numerical simulations and producing the resulting figures as reported in this article will be made available [65].
ACKNOWLEDGMENTS
The authors acknowledge funding from the German Research Foundation (DFG, Excellence Cluster MATH+, EXC-2046/1, project 390685689), the Helmholtz Association (Helmholtz Excellence Network SOLARMATH, project ExNet-0042-Phase-2-3), and the German Federal Ministry of Education and Research (BMBF Forschungscampus MODAL, project 05M20ZBM). Also, this project has received funding from the EMPIR program( European Metrology Programme for Innovation and Research) co-financed by the Participating States and by the European Union's Horizon 2020 research and innovation program (projects 20FUN05 SEQUME & 20FUN02 PO-LIGHT). Y.K. acknowledges support from the Australian Research Council (grants DP200101168 and DP210101292).
APPENDIX Appendix A. RIESZ PROJECTION PRINCIPLE AND RESULTS
Our approach for carrying out modal expansions relies on Riesz projections [52,63,64]. In a first step, the quantity of interest at real frequencies ω 0 is expressed as a contour integral using Cauchy's integral formula. To this end, it has to be analytically continued to the complex frequency plane. In a second step, the resonance expansion is obtained by deforming the contour around ω 0 until it encloses neighboring poles of the physical system and by the application of Cauchy's residue theorem. Each summand of the expansion corresponds to a contour integral.
Using the example of the Purcell factor Γ(
ω 0 ) = − 1 2 (E(ω 0 , r d ) · j * (ω 0 , r d )) /Γ b (ω 0 )
, whose expansion is shown in Fig. 3 of the main document, the first step yields
Γ(ω 0 ) = − 1 2Γ b (ω 0 ) C0 (E(z, r d ) · j * (z, r d )) z − ω 0 dz,
where C 0 is a contour around ω 0 . The second step results in the desired expansion of the Purcell factor,
Γ(ω 0 ) = n Γ n (ω 0 ) + Γ background (ω 0 ), with Γ n (ω 0 ) = − 1 2Γ b (ω 0 ) Cn (E(z, r d ) · j * (z, r d )) z − ω 0 dz and Γ background (ω 0 ) = − 1 2Γ b (ω 0 ) C background (E(z, r d ) · j * (z, r d )) z − ω 0 dz.
The contour C n is the contour around the nth pole and C background is the large outer contour. Please refer to Fig. 5 and note that, for quantities linear in the electric field, such as the Purcell factor in the given form, the complex conjugate poles located in the upper half of the complex plane can be ignored. The integrals are computed numerically using the trapezoidal rule for the used circular and ellipsoidal contours.
In Section IV, we expand the far-field pattern of the radiated flux which is quadratic in the electric field,
s (E(ω), E * (ω)) = 1 2 E * (ω) × 1
iωµ0 ∇ × E(ω) · n, and hence involves its complex conjugate. The method for expanding quadratic forms was developed in [63]. The application of Cauchy's residue theorem requires a holomorphic expression and therefore does not allow for complex conjugation. As the electric field is a real quantity in the time domain, we have E * (ω) = E(−ω) for real ω. With the analytic continuation to the complex plane E • (ω) associated with E(ω) as shown in Fig. 5. The expansion of s (E(ω), E • (ω)) consequently features resonant terms from poles in the lower and the upper part of the complex plane. Following this approach [63], one can derive the expansion of s (E(ω), E • (ω)),
of E(−ω), the holomorphic expression s (E(ω), E • (ω)) = 1 2 E • (ω) × 1 iωµ0 ∇ × E(ω) · n iss (E(ω 0 ), E • (ω 0 )) = − n 1 2iπ Cn s (E(z), E • (z)) z − ω 0 dz − n 1 2iπ C * n s (E(z), E • (z)) z − ω 0 dz + 1 2iπ C background s (E(z), E • (z)) z − ω 0 dz,
where C n is again the contour around the nth pole and C background is the large background contour. As mentioned above, we have to add the contours around poles in the upper part of the complex plane, which we denote by C * n . The expansion of the radiation pattern towards the air is discussed in Section IV. Here, for the sake of completeness, we show the expansion of the radiation towards the substrate at the same wavelengths and aspect ratios as in Fig. 4. Overall, the same behavior is observed for the flux radiated towards the substrate as it was for the flux radiated towards the air. In Fig. 6(a), one can see that the modal contribution from the high-Q mode dominates all the other contributions. Figures 6(b-d) show once more that the radiation pattern of the flux results from the interference of the two main modes. In Figs. 6(b) and (d), they interfere destructively and, in Fig. 6(c), constructively.
Appendix B. COUPLING OF RESONANCES
In the main text, we employed a phenomenological method to study the mode coupling. We used the following expressions for the coupled eigenfrequencies:
ω ± = ω un,1 + ω un,2 2 ± √ γ,(S1)
where γ = ω un,1 − ω un,2 2
2 + v 2 (S2)
with v being the coupling coefficient. While we then focused on explaining the crossings and avoided crossings of real and imaginary parts of the resonances as a consequence of different coupling regimes, these formulas can provide further insight into the couplings between resonances. Some additional results based on these formulas are provided in the following.
We will study the following uncoupled resonance frequencies:
ω un,1 = 1 − i0.01 ω un,2 = 1 + ∆ − i(0.01 + ∆ω i )(S3)
For this study, we keep a fixed value of ∆ω i = 0.0025 and study the trajectories of the coupled eigenvalues when ∆ is varied. The impact of the value of v on the coupling of resonances will be studied for two cases: in the first case v is real-valued and positive and in the second case v 2 is purely imaginary. We will also study the impact of the coupling of the resonances on their respective Q-factor. of γ and ∆ω i . We can start by reexpressing γ for ω un,1 and ω un,2 defined in Eq. (S3):
γ = ∆ − i∆ω i 2 2 + v 2 = ∆ 2 + 4v 2 − ∆ω 2 i 4 − i ∆ * ∆ω i 2 (S4)
One notices that the real part of γ cancels out for ∆ = ± ∆ω 2 i − 4v 2 while its imaginary part cancels out for ∆ = 0. Studying the roots of the real part of γ, three regimes of coupling can then be distinguished depending on the relative values of v 2 and ∆ω 2 i : v 2 > and v 2 < ∆ω 2 i 4 . In the example we study, ∆ω 2 i 4 = 1.5625 * 10 −6 . We then plot the trajectories of the real and imaginary parts of the coupled eigenvalues for v 2 larger, equal and smaller than 1.5625 * 10 −6 . The results are plotted in the following figures along with the variation of the Q-factor as a function of ∆. In Fig. 7 a)-c), we plot these trajectories for v 2 = 10 −4 and thus larger than 1.5625 * 10 −6 . The trajectories of the eigenvalues display a behavior typical for a strong coupling as discussed in the main text with an avoided crossing of the real parts and a crossing of the imaginary parts. On both sides of this crossing, the Q-factor of one mode increases while the Q-factor of the other one decreases.
The trajectories of the eigenvalues when v 2 = 1.5625 * 10 −6 are shown in Fig. 7 d)-f). It is clearly seen that the real part and the imaginary parts cross at ∆ = 0. The two eigenfrequencies are thus completely degenerated at ∆ = 0 which is linked to the existence of an exceptional point, a degeneracy existing in non-Hermitian systems. The behavior of the Q-factor is similar to the one observed in Fig. 7 c). Finally, the trajectories for v 2 < 1.5625 * 10 −6 are shown in Fig. 7 g)-i). In this case, there is a crossing of the real part and an avoided crossing of the imaginary part. ω + and ω − are actually swapped on one side and the other of ∆ = 0. This behavior might certainly be seen as a jump from one Riemann sheet to the other.
B.2. Coupling of resonance for v 2 purely imaginary
A similar analysis to the one done in the previous section can be performed for v 2 which is purely imaginary v 2 = iu with u being real-valued. γ can then be rewritten in the following way:
γ = ∆ − i∆ω i 2 2 + v 2 = ∆ 2 − ∆ω 2 i 4 − i ∆ * ∆ω i − 2u 2(S5)
This time, the roots of the real part of γ occur for ∆ = ±∆ω i while the imaginary part of γ vanishes for ∆ = 2u ∆ωi . The different regimes of coupling now depend on the relative values of ∆ω i and 2u ∆ωi or equivalently the relative values of u and Fig. 8 a)-c). These trajectories display a strong coupling behavior with an avoided crossing of the real parts of the coupled eigenvalues. This avoided crossing coincide this time with a peal of the imaginary part of one eigenvalue and a dip of the other one. As a consequence there is a peak of the Q-factor associated with one eigenvalue and a dip of the Q-factor of the other eigenvalue.
The trajectories of the eigenvalues for u = 3.12510 −6 which is equal to ∆ω 2 i 2 are shown in Fig. 8 d)-f). This reveals the existence of an degeneracy of the eigenvalue, i.e. an exceptional point, where both the real and imaginary parts of the two eigenvalues are identical.
Figure 1 .
1Principle of the enhanced and suppressed emission with interfering resonances. (a) Schematics of GaAs nanodisks with and without a substrate. The aspect ratio D/H is tuned to control the interference between the two main modes of the nanodisk. (b) Visualization of the electromagnetic field distribution resulting from a dipole emitter, represented by a white sphere, which is located below the top face of the nanodisk. Its frequency is chosen to excite the two modes of interest. (c) and (d): 2D cross-sections through the dominant two modal fields (left) and the total field distribution (right) visualizing the real part of the y field component. Red and blue colors correspond to negative and positive fields, respectively. The emitter position is indicated with a white circle. (c) When the two modal fields are excited in phase they interfere constructively, leading to enhancement of dipole emission. (d) Out-of-phase excitation of the two modal fields at a different dipole emission frequency, results in suppressed emission.
Figure 2 .
2Real parts of two eigenfrequencies of interest (a, e) and corresponding Q-factors (b, f) as function of nanodisk aspect ratio D/H. Avoided crossing of the real parts and local maximum and minimum of the Q-factors at D/H = 0.909 indicate strong coupling for the nanodisk without substrate (a, b). Crossing of the real parts of two eigenfrequencies at D/H = 0.93 and avoided crossing of the Q-factor curves at D/H = 0.944 leading to a peak at D/H = 0.92 indicate weak coupling for the nanodisk with
Figure 3 .
3Modal analysis of the wavelength (λ) dependent Purcell factor Γ for a y-polarized dipole located on the symmetry axis 20 nm and 27 nm below the top face of the nanodisk in the case without substrate (a,b) and with substrate (c,d), respectively. (a) Modal expansion for the aspect ratio D/H = 0.909 (maximum Q-factor in Fig. 2(b). The high-Q mode corresponds to the modal Purcell factor Γ2 (a black solid curve) and is solely responsible for the peak of the total Purcell factor Γtot (dashed red curve) at around 1205 nm. The contributions of the low-Q mode Γ1 (green solid curve) and of the background Γ background (dotted blue line) are negligible. (b) Modal expansion for D/H = 0.933 (crossing of Q-factors in Fig. 2(b). Both modal terms Γ1 and Γ2 are of the same order of magnitude and destructively interfere in regions where they are of different sign. The impact of Γ background is constant and negligible in resonant regions. (c) Modal expansion for D/H = 0.92 (peak of the high Q-factor inFig. 2(f). The high-Q mode corresponds to Γ2 and is responsible for the peak of Γtot at around 1220 nm. (d) Modal expansion for D/H = 0.944 (avoided crossing of Q-factors inFig. 2(f). The modal terms interfere, as in (b). The markers Γ optimized , Γ bd , Γresonance and Γ rd indicate the wavelengths for which far-field patterns are displayed inFig. 4.
Figure 4 .
4Modal decomposition of the θ-dependent, normalized radiation patterns towards the top for a dipole on the symmetry axis 20 nm and 27 nm below the top face of the nanodisk in the case without substrate (a-d) and with substrate (e-h), respectively. The green (black) solid curve corresponds to the angle-resolved, far-field modal energy flux s1 (s2) corresponding to the low-Q (high-Q) mode. The red dashed curve corresponds to the total energy flux stot. The upper half of each diagram shows positive contributions while the lower half in gray shows negative contributions. The dipole emission wavelengths correspond to the different Γ markers in Fig. 3 which are reproduced in the right bottom of each emission diagram. (a, e) show the on-resonant far field radiation for nanodisks supporting the quasi-BIC (D/H = 0.909 and λ = 1206 nm, resp. D/H = 0.92 and λ = 1219 nm) with clearly dominating contribution from the high-Q mode. (b-d), resp. (f-h) show the far field radiation for nanodisks with aspect ratios of D/H = 0.933, resp. D/H = 0.944 (i.e., at the avoided crossing, resp. crossing of the eigenfrequencies, cf., Figs. 2(a, e) for on-resonant sources (λ = 1217 nm/1237 nm in c/g) and off-resonant sources (λ = 1204 nm/1234 nm/1219 nm/1247 nm in b/d/f/h). While for on-resonant sources a single mode is predominantly contributing to the far field pattern (c, g), in off-resonant settings, the two relevant modes can interfere constructively (b) or destructively (d, f, h), as can be seen from the equal or different signs of the two dominant modal contributions in each case.
defined. The poles of E • (ω) are located in the upper part of the complex plane. They are the complex conjugates of the resonance poles
Figure 5 .
5Contours used for the modal expansions of Purcell factor and emission pattern of a nanodisk with different aspect ratios D/H, either isolated or placed on a substrate. While the emission pattern is based on a quadratic form and requires contours around the complex conjugate resonance frequencies ω * n , which are the poles of E • (ω), the circular contours in the upper half space can be ignored for the Purcell factor.
Figure 6 .
6Modal analysis of the radiation pattern towards the substrate for a dipole located on the symmetry axis 27 nm below the upper base of the nanodisk. The black markers refer to Fig. 3 where they mark the corresponding wavelengths. The optimized system (D/H = 0.92) shown in (a) illustrates the dominance of a single mode. For (b)-(d) the aspect ratio is D/H = 0.944. Here, the radiation pattern results from the interference between two dominant modes. In (b), the pattern is shown at a wavelength blue shifted from the maximal Purcell enhancement, in (c), at the maximum and, in (d), at a red shifted wavelength.
Figure 7 .
7Real parts Re(ω±), imaginary parts Im(ω±) and Q-factors Q± of the two coupled frequencies ω± as a function of the difference of the real parts of the uncoupled eigenfrequencies ∆ = Re(ωun,2 − ωun,1). The trajectories are given for three different couplings v 2 , with v being real and positive. For better readability a vertical line is shown at ∆ = 0.
B. 1 .Figure 8 .
18Coupling of resonance for v 2 real and positive Let us first study the coupling of resonances for a coupling coefficient v which is real and positive. A careful study of the behavior of the function γ in Eq. (S2) reveals 3 different behaviors depending on the relative values Real parts Re(ω±), imaginary parts Im(ω±) and Q-factors Q± of the two coupled frequencies ω± as a function of the difference of the real parts of the uncoupled eigenfrequencies ∆ = Re(ωun,2 − ωun,1). The trajectories are given for three different couplings v 2 = iu, with u being real and positive. For better readability a vertical line is shown at ∆ = 0.
Figure 9 .
9Purcell factor for a dipole located close to the hot spot of the low-Q mode, on the symmetry axis 700 nm above the bottom of the nanodisk. The contributions of both modes are similar and positive at this position leading to a total Purcell factor which is the superposition of the two main modes.
Appendix C. ADDITIONAL CALCULATIONS FOR THE EMISSION OF A DIPOLE EMITTERIn the main text we keep the position of the dipole emitter fixed and show how the coupling effects the modal contributions Γ n (ω) of the Purcell enhancement at the hot spot of the high-Q mode. Γ n (ω) depends on the electric field strength at the dipole position and if the quasi BIC condition is met, the contribution of the low-Q mode at this point is much smaller. For a slightly altered aspect ratio the field values become comparable and interference can be observed. Instead of varying the aspect ratio, one can also consider the enhancement at different dipole positions. For the sake of completeness inFig. 9we provide a corresponding example where the contributions of both modes are similar. For this example the dipole is positioned at the symmetry axis close to the center of the nanodisk 700 nm above the bottom face.
These trajectories reveal a weak coupling behavior with a crossing of the real part of the coupled eigenvalues and an anti-crossing of the imaginary part. Fig. 8 g)-i). These trajectories reveal a weak coupling behavior with a crossing of the real part of the coupled eigenvalues and an anti-crossing of the imaginary part.
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| []
|
[
"Canalization and symmetry in Boolean models for genetic regulatory networks Canalization and symmetry in Boolean models... 2",
"Canalization and symmetry in Boolean models for genetic regulatory networks Canalization and symmetry in Boolean models... 2"
]
| [
"C J Olson Reichhardt \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNM\n",
"Kevin E Bassler \nDepartment of Physics\nUniversity of Houston\n77204-5005HoustonTX\n\nTexas Center for Superconductivity\nUniversity of Houston\n77204HoustonTX\n"
]
| [
"Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNM",
"Department of Physics\nUniversity of Houston\n77204-5005HoustonTX",
"Texas Center for Superconductivity\nUniversity of Houston\n77204HoustonTX"
]
| []
| Canalization of genetic regulatory networks has been argued to be favored by evolutionary processes due to the stability that it can confer to phenotype expression. We explore whether a significant amount of canalization and partial canalization can arise in purely random networks in the absence of evolutionary pressures. We use a mapping of the Boolean functions in the Kauffman N-K model for genetic regulatory networks onto a k−dimensional Ising hypercube (where k = K) to show that the functions can be divided into different classes strictly due to geometrical constraints. The classes can be counted and their properties determined using results from group theory and isomer chemistry. We demonstrate that partially canalizing functions completely dominate all possible Boolean functions, particularly for higher k. This indicates that partial canalization is extremely common, even in randomly chosen networks, and has implications for how much information can be obtained in experiments on native state genetic regulatory networks. | 10.1088/1751-8113/40/16/006 | [
"https://arxiv.org/pdf/q-bio/0610011v2.pdf"
]
| 14,325,675 | q-bio/0610011 | 0e3a13c825422f5b9c623edd639c98c3e15243fa |
Canalization and symmetry in Boolean models for genetic regulatory networks Canalization and symmetry in Boolean models... 2
2 Mar 2007
C J Olson Reichhardt
Theoretical Division and Center for Nonlinear Studies
Los Alamos National Laboratory
87545Los AlamosNM
Kevin E Bassler
Department of Physics
University of Houston
77204-5005HoustonTX
Texas Center for Superconductivity
University of Houston
77204HoustonTX
Canalization and symmetry in Boolean models for genetic regulatory networks Canalization and symmetry in Boolean models... 2
2 Mar 2007
Canalization of genetic regulatory networks has been argued to be favored by evolutionary processes due to the stability that it can confer to phenotype expression. We explore whether a significant amount of canalization and partial canalization can arise in purely random networks in the absence of evolutionary pressures. We use a mapping of the Boolean functions in the Kauffman N-K model for genetic regulatory networks onto a k−dimensional Ising hypercube (where k = K) to show that the functions can be divided into different classes strictly due to geometrical constraints. The classes can be counted and their properties determined using results from group theory and isomer chemistry. We demonstrate that partially canalizing functions completely dominate all possible Boolean functions, particularly for higher k. This indicates that partial canalization is extremely common, even in randomly chosen networks, and has implications for how much information can be obtained in experiments on native state genetic regulatory networks.
Introduction
To preserve the identity of a species, biological organisms must be capable of maintaining relatively stable phenotype expression in the face of a variety of environmental factors and a certain level of genetic randomness. Experimental observations have shown that certain developmental traits appear to control the expression of other traits. Waddington [1] termed the control of one trait by another "canalization," a name derived from the analogy that the developmental pathway of the organism is like one particular canal in a floodplain, and the further development of the organism is completely constrained by that canal. Canalization produces robustness because it suppresses those changes in phenotype expression that would require development to deviate from the canalized pathway. For this reason it has been suggested that organisms evolve to be canalized.
The significance of canalization and how it might evolve remains a subject of debate [2]. Since canalization suppresses the expression of genetic variability, experimental detection of the existence of a canalized trait generally involves perturbing the organism out of the canalized state [3]. There is good evidence for the existence of canalization in a variety of organisms [4,5,6]; however, the microscopic mechanism for canalization is not well established. Presumably canalization is produced genetically by the complex interactions between genes known as the genetic regulatory network (GRN). As we shall see, however, a certain amount of canalization is expected to appear in GRNs even in the absence of an evolutionary preference for canalization. An open question is whether or not real GRNs contain more canalization than would be expected from a random graph, which could indicate that evolution favors canalization.
Genetic regulatory networks have been proposed as the mechanisms through which identical genetic information is expressed as different cell types within the same organism, and they can also control distinct stages in the life cycle of an individual cell. Depending on the conditions experienced by a given cell and the regulatory interactions between genes, at any moment a distinct subset of all possible genes are activated. The state or temporal pattern of expression produces particular cell types. Organisms with larger numbers of genes have a larger number of potentially realizable cell types. There has been a recent dramatic increase in the amount of experimental information available on the structure of genetic regulatory networks in a range of organisms, including E. coli [7], budding yeast S. cervisiae [8,9], Drosophila species [10], Xenopus [11], and the embryo of the sea urchin S. purpuratus [12]. In the simplest representation, the nodes of the network are genes and the links between genes describe their interactions. Generally, the interactions are directional, so that the expression of gene A may depend on, that is "listen to," the expression of gene B, but the expression of gene B doesn't necessarily depend directly on the expression of gene A. The connectivity of a gene indicates how many other genes it "listens to" when determining whether to be in an active or inactive state. Analysis of the connectivity of E. coli [13,14] and other GRNs shows a broad distribution of connectivity among the genes, with a significant amount Table 1. An example of a k = 3 Boolean function.
in 1 in 2 in 3 out 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1
of negative autoregulation. In the context of canalization, several questions arise. What types of structures in a GRN produce canalization of a trait? Do these structures arise randomly, or do they only appear because of a special evolutionary preference? How significant is canalization on the scale of the entire regulatory network?
The easiest way to approach such questions is through a simplified model for a genetic regulatory network such as the Kauffman N-K model [15], which represents the GRN as a random Boolean network. The N-K model has been studied extensively [16,17,18,19,20,21,22,23]. Certain features of real GRNs, including the ability of a single network to produce multiple cell types (which appear as multiple attractors for the network), are captured by the N-K model. In this model, each gene is represented by a single binary element which can be either on or off in the state 1 or 0. Every gene receives input from a fixed set of k other genes that are randomly chosen when the network is constructed, where k = K. Depending on the states of its input genes, a given gene determines whether to express the state 1 or 0 according to a randomly chosen Boolean function of k variables. An example of a Boolean function for k = 3 is given in table 1. The value of k may vary from gene to gene. The system evolves in discrete time steps and all genes update their states simultaneously. The entire network eventually settles into an attractor cycle which produces a specific series of network states as a function of time. The initial conditions of the states of the genes in the network determine which of the available attractors the network will reach. The different attractors are interpreted as representing different cell types expressed by a given set of genes.
A gene with connectivity k employs one of the 2 2 k possible Boolean functions to determine its response to its k inputs. Canalization occurs in a Boolean function if the output of the gene is fixed by a particular value of at least one of its input genes, regardless of the values of any other inputs to that gene. In this case the input that fixes the output of the regulated gene is a canalizing input. Note that one value of an input gene, say value 0, can be a canalizing input even if the other value 1 from the same input gene is not canalizing. Canalization also occurs in a Boolean function if particular values of two or more inputs together suffice to guarantee the next state of the regulated gene, regardless of the values of any other inputs to the gene. In this case, the inputs that together fix the output of the regulated gene are said to be collectively canalizing inputs. How canalizing a particular Boolean function is can be quantified by the set of numbers P n , n = 0, 1, . . . , k − 1, which are the fraction of sets of n individual input values that are canalizing or collectively canalizing. Note that Boolean functions with P 0 = 1 have a fixed output state regardless of their inputs. Boolean functions can also be characterized by their internal homogeneity p which is defined as the fraction of 1s or 0s, whichever is larger, output by the function due to all of the possible sets of input [24].
A consequence of canalization is that some of the interactions between genes may become irrelevant. As an extreme example, if the Boolean function of a particular gene has P 0 = 1, this gene will be insensitive to the state of the rest of the network and its interactions with its input genes are irrelevant. The number of canalizing functions as a function of k was derived recently in Ref. [25]. Although the behaviour of canalizing functions would certainly contribute to the stability of a network that is subjected to random perturbations, such an extreme behaviour has a detrimental effect on the ability of the network to respond to changing conditions. In contrast, other Boolean functions successfully maintain a degree of stability while retaining the ability to change. For these Boolean functions, which we will refer to as "partially canalizing," the gene may ignore one or more of its inputs under certain conditions. In some cases, the gene completely ignores n inputs at all times, so that its effective connectivity is k ef f = k − n. In other cases, if a particular input has the value 1, for example, the gene ignores the remaining inputs, but if that same input has the value 0, the gene listens to its other inputs. Here, the effective connectivity of the gene depends on the current state of the network. More complex categories are also possible, such as the nested canalizing functions proposed by Kauffman [19,26]. Since the fraction of canalizing functions drops rapidly with k, as shown in Ref. [27], it has been assumed that canalization plays a less important role at high k compared to small k [28]. The class of partially canalizing functions is considerably larger than the class of canalizing functions; however, is it large enough to dominate all classes of functions? As we will show below on mathematical grounds, the partially canalizing functions completely dominate the class of all possible Boolean functions as k increases, so that the emergence of canalization is essentially unavoidable in a complex network.
Results
Our approach is to examine the properties of individual Boolean functions and to determine the amount of canalization expected from a random sample of functions. Since the number of possible Boolean functions explodes combinatorially with k, we employ powerful techniques from group theory and isomer chemistry to classify the various functions and help obtain their properties. We provide results through k = 5 with these methods. The techniques can be applied readily to higher k, but become increasingly complicated. Therefore, to find results for larger k through k = 8 we employ For small enough k, the canalization properties of the functions can be obtained directly from inspection. When there are two inputs for each gene, k = 2, as shown in table 2, there are only 16 possible functions which fall into four classes: fixed (or completely canalizing) with P 0 = 1; sensitive to both inputs with P 0 = 0 and P 1 = 0; and the partially canalizing cases with P 0 = 0 and P 1 = 1/2: sensitive to only one input; sensitive to one or two inputs depending on the value of one input.
Inspection becomes a less viable option as k increases. In a simulation study of the evolution properties of the different Boolean functions, Bassler et al. [29] observed that functions with k = 3 inputs fell into 14 distinct classes. In their study all of the functions that were members of the same class evolved, on average, with equal probability. Upon examining representative functions from each class, they categorized the functions according to their canalization properties P n . The triple of numbers P 0 , P 1 , and P 2 possible for k = 3 was nearly enough, but not quite enough, to distinctly identify each class of function. Whether the function was symmetric about its midpoint also needed to be considered in defining the classes. These observations about the structures of the functions belonging to each class were essentially empirical. Class membership could be important for determining the properties of real networks since we expect that on average all functions in the same class will evolve with equal probability.
Here, we demonstrate that there is a fundamental geometric reason for the existence of distinct function classes. In the N-K model, a given function is normally represented by a Boolean string of numbers, such as 1001, of length 2 2 k . Comparing different functions by inspection amounts to comparing strings of numbers with each other. Instead of using this representation, we consider an alternative, equivalent representation of each function in the form of a unit k-dimensional Ising hypercube. Each axis of the kdimensional hypercube (or simply a k-hypercube) represents one of the k input variables. The coordinates on a given axis indicate the state of the corresponding input variable. Each vertex of the k-hypercube represents an output state of the gene. In figure 1 we illustrate the mapping of the input states onto a square and cube for k = 2 and k = 3, respectively. The output state of the gene corresponding to an input represented by a particular vertex can be indicated by colouring the vertex white or black to represent the values 0 or 1. It is important to note that this system obeys parity symmetry: replacing all 0's with 1's and vice versa results in the same canalization properties for the function.
With this hypercube mapping, it becomes clear that functions which belong to the same class have the geometric property that they can be rotated into each other by symmetry operations on the k-hypercube plus parity. In mathematical terms, the classes that were identified empirically in Ref. [29] are group orbits. We illustrate the mapping for the sixteen k = 2 functions in figure 2, where the rotational plus parity symmetries of the functions belonging to each of the four classes are obvious. In figure 3 we illustrate one representative cube for each of the 14 function classes in k = 3. The remaining functions in each class are obtained by applying all possible rotations plus parity to the cube. In the hypercube representation, the canalization properties of a Boolean function correspond to the fraction of homogeneous hypersurfaces. That is, for a Boolean function with k inputs P n is proportional to the fraction of the k − n dimensional hypersurfaces that have all vertices the same. For the two classes with P 1 = 1/2 in figure 2, 2 of the 4 one-dimensional sides are uniformly coloured.
We can now employ results from group theory to obtain information about the class structure of functions at all values of k, not merely those values of k which are small enough to permit direct inspection of all functions. The total number of classes for a given k can be obtained by an application of the orbit-counting theorem,
P G (x 1 , x 2 , ...) = 1 |G| g∈G |X g |(1)
Here, the symmetry group G of the set X contains |G| symmetry operators g, which together include all transformations of the hypercube onto itself. The set of elements in X that are left invariant by g is denoted as X g . In order to find X g , note that the mapping of all k-hypercube vertices onto themselves by a given symmetry operator g can be written as a permutation of the vertex numbers. As a result, each operator g can be expressed in terms of its cycle structure x b 1 1 x b 2 2 ...x bn n , where n = k. This notation indicates that g contains b 1 cycles of length 1, b 2 cycles of length 2, ... b n cycles of length n [31]. For example, the k = 2 permutation (14)(2)(3) has the cycle representation x 2 1 x 2 since it has 2 cycles of length 1 and a single cycle of length 2. To apply the orbit-counting theorem, we must first construct all of the operators of our group, sum the number of functions left invariant by each operator (the fixed points of that operator), and divide by the total number of operators.
The number of functions in, or size of, a class, is given by the number of elements in the group divided by the number of elements in the isometry group of the functions in the class. The isometry group of a class is the subgroup of the full group that describes the symmetry of a function in the class. Note that the particular isometry group will vary from function to function in the class, but the size of the isometry group will remain invariant.
First, consider the number of symmetry operations |G| in our group, which is the khypercube crossed with parity. The symmetry group for the k-hypercube is isomorphic to the hyperoctahedral group O n with n = k, which has n!2 n symmetry transformations [30]. As an example, there are eight operators for the k = 2 square. There is one operator with cycle structure (3) and (23)(1)(4). When these operators are combined with parity, which doubles the number of symmetry operators, we obtain Table 3. Cycle polynomials for k = 1 through 5 and the number of classes P G for each k.
k Cycle polynomial
P G 1 (1/2)(x 2 1 + x 2 ) 2 2 (1/8)(x 4 1 + 3x 2 2 + 2x 2 1 x 2 + 2x 4 ) 4 3 (1/48)(x 8 1 + 13x 4 2 + 8x 2 1 x 2 3 + 8x 2 x 6 + 6x 4 1 x 2 2 + 12x 2 4 ) 14 4 (1/384)(x 16 1 + 12x 8 1 x 4 2 + 51x 8 2 + 12x 4 1 x 6 2 + 32x 4 1 x 4 3 + 48x 2 1 x 2 x 3 4 + 84x 4 4 +96x 2 2 x 2 6 + 48x 2 8 ) 238 5 (1/3840)(x 32 1 + 384x 3 10 x 2 + 20x 16 1 x 8 2 + 60x 8 1 x 12 2 + 231x 16 2 + 80x 8 1 x 8 3 +320x 2 12 x 2 4 + 240x 4 1 x 2 2 x 6 4 + 240x 4 2 x 6 4 + 520x 8 4 + 384x 2 1 x 6 5 +160x 4 1 x 2 2 x 4 3 x 2 6 + 720x 4 2 x 4 6 + 480x 4 8 ) 698635
a total of |G| = 16 operators. For each operator without parity, the number of functions left invariant is equal to 2 Nc , where N c = k i=1 b i is the total number of cycles in the operator. Parity must be treated separately; no functions are left invariant by the parity operator with any k-hypercube operator containing at least one cycle of length 1. Thus there are 2 Np functions left invariant for the eight operators which include parity, where
N p = (1 − Θ(b 1 )) k i=1 b i
and Θ is the Heaviside step function. Applying the orbitcounting theorem produces the correct number of classes, but only if parity is included. In the case of k = 2 without parity, P G = (1/8)(2 4 + 2(2) + 3(2 2 ) + 2(2 3 )) = 6 classes. Including parity gives P G = (1/16)(2 4 + 2(2) + 3(2 2 ) + 2(2 3 ) + 2(2) + 3(2 2 )) = 4 classes, which is the correct result. We now face the task of identifying all operators g for the k-hypercube. This can be performed by simple inspection in k = 2 and k = 3, but it becomes more complicated to keep track of higher-dimensional rotation symmetries. Fortunately, this problem was solved in the middle of the last century, when Harrison [32] derived a formula that produces the complete cycle representation for all k in the group of interest to us, called the "Zyklenzeiger" in the notation of Ref. [32]. We have used this formula to obtain the cycle representations through k = 5, shown in table 3. The number of classes for each k, P G , is also listed in table 3. Clearly, obtaining the properties of the classes by inspection is not feasible by the time k = 5.
What we are interested in is not simply how many different classes are present, but also the size of each class, and the structure, particularly the canalization properties, of the functions belonging to them. For example, how many classes are there which have the same internal inhomogeneity p? To find this, we use an application of Pólya's theorem [33] which is frequently used in isomer chemistry [34]. In isomer chemistry, for a molecule composed of exactly two different types of atoms, the terms A and B can be used to represent the different types of atoms. In our case, A and B represent 0 and 1, such that either A = 0 and B = 1, or B = 0 and A = 1. Using the generating polynomial, substitute in a term of the form A a + B a for each x a . Divide the result by the total number of operators, including parity. Then, drop all terms in the result where the exponent of B exceeds that of A, as these terms are already accounted for by parity. Table 5. Class structure for k = 3.
Class type N h S c A 4 1 2 A 3 B 1 8 A 2 B 2 2 3Class type N h S c A 8 1 2 A 7 B 1 16 A 6 B 2 3 18.667 A 5 B 3 3 37.333 A 4 B 4 6 11.667
The multiplicity of each term indicates how many of the classes are of that form. For example, representatives of classes of the form A 2 B 2 are 1010 and 1100. This gives us the desired result of how many classes N h there are for each value of the internal homogeneity p. Since we also know that the total number N f (m, n) of functions of the form A m B n is simply N f (m, n) = (2 − δ m,n )(2 k )!/(m!n!), where m + n = 2 k , we can estimate the number of functions in each class by the average size of a class, S c = N f /N h . The class structure and average class size for k = 1 through 5 are listed in tables 4 through 7. We note that the actual size S c of each class is given by the number of operators that preserve the symmetry of that particular function class. Thus, as discussed earlier, the maximum class size S max c is equal to the total number of operators
S max c = k! 2 k+1 .(2)
S max c = 16 for k = 2, 96 for k = 3, 768 for k = 4, and 7680 for k = 5. This is consistent with the average class sizes which we obtain.
We also note that isomer chemistry provides a simple means for determining whether two randomly selected functions belong to the same class. Construct the adjacency matrix for the k-hypercube. Along the diagonal, place the values A or B corresponding to the colours of the vertices of one of the functions under consideration, and then find the determinant of the resulting matrix. Each function class has a unique determinant, so performing this procedure on both functions provides an immediate test of whether the two functions fall into the same class.
To show that canalization remains important even as k increases, we measured the average number of homogeneous d-dimensional sides present in a series of randomly generated functions for different k. We denote the number of d-dimensional homogeneous sides (which produce canalization) that a k-dimensional Boolean function has as C(d, k). The total number of d-dimensional sides is
N d (k) = 2 k−d k! (k − d)!d! .(3)
Note that the canalization properties P n discussed in the beginning of this paper are related to C(d, k) as P n = C(k − n, k)/N k−n (k). In figure 4 The mapping of Boolean functions onto k-hypercubes we have described here provides a means of constructing k + 1 functions recursively from pairs of k functions. A k + 1 function can be composed by stacking together two k functions. Depending on the symmetry properties of the two k functions chosen, there may be only one possible class of k + 1 functions that can be constructed from those k functions, or there may be several classes that depend on the relative orientation of the k functions when they are stacked together. This allows us to bound the amount of canalization present. When we assemble a k + 1 function out of two k functions, we must have
N d (k + 1) ≥ C(d, k + 1) ≥ 2 i=1 C i (d, k).(4)
The lower bound is obtained from the fact that the value of C(d, k + 1) must be at least as large as the sum of the values C i (d, k) of the two functions that have been combined. This is simply a consequence of the fact that homogeneous d-dimensional sides cannot be destroyed as a result of combining two functions. The new sides that are added when the functions are joined may or may not be homogeneous, depending on which two k functions are combined and how they are oriented with respect to each other. It is possible that none of the new sides would be homogeneous, in which case the lower limit of Eq. (4) would apply. The internal homogeneity p(k + 1) of the composite function is given simply by
p(k + 1) = 1 2 2 i=1 p(k).(5)
Discussion
Now that we have used the mapping of the functions onto k-hypercubes in order to obtain information about the class structures of the functions for several values of k, we can make some observations regarding how prevalent partial canalization is among all possible functions. Previous estimates of the fraction of canalizing functions indicated that canalization was of less and less importance as k increased. These estimates used a very narrow definition of canalization, however. Rather than counting the number of partially canalizing functions, consider the number of completely uncanalizing functions. These functions have the property that they are sensitive to all values of all inputs. There are exactly 2 such functions for each k, regardless of k. All of the remaining functions are at least partially canalizing. This means that partial canalization completely dominates the classes of functions, especially as k increases. The rampant occurrence of partial canalization has important implications for recent work on mapping of genetic regulatory networks. The experiments typically map only those connections between genes which are active in the native state of the organism. Here, "active" means that a change in one gene directly affects the second gene. This technique will not detect many of the partially canalizing interactions that could exist between genes. In the case where the partial canalization is of the form that a gene completely ignores one or more of its inputs, the actual value of k for that gene is larger than the apparent value of k. This could potentially impact the distributions of k that have been extracted from experimental measurements. A far more dangerous case is a partially canalizing interaction between genes in which a gene ignores one or more inputs when a canalizing input has a value of 1 (for example), but responds to the other inputs when that same canalizing input has a value of 0. If the gene ignores its inputs in the native state, the connection between that gene and its ignored inputs will not be detected experimentally. Suppose that the canalizing gene is identified as causing a disease state. Consultation of the experimentally determined genetic network map indicates that this gene does not appear to control anything else of importance. If, however, the canalizing gene is treated and switches to the state opposite from its canalizing value, the gene that received the canalizing input will suddenly start to respond to the values of its other inputs. This could result in unexpected side effects or worse effects. Thus, from a purely combinatorial point of view, it is important to consider all possible interactions between genes, and not merely those which are expressed in the native state.
The natural predominance of canalization as k increases suggests that the canalization observed experimentally could be due simply to the high fraction of the available Boolean functions which are canalizing, rather than evolutionary pressure to develop canalizing functions. It is, however, unclear how much canalization is present in real genetic regulatory networks, as discussed in Ref. [35]. It is possible that there is in fact a special evolutionary preference for canalization, which could result in real networks having even higher levels of canalization than would be expected from random selection at increasing k. In order to answer this question it would be necessary to measure the excess canalization, which is the difference between the P n s observed in real networks and that in random networks [36]. The existing experimental data on genetic regulatory networks is not extensive enough to determine whether an interaction between genes is canalizing or partially canalizing. As noted above, the difference between the two types of interactions can become important when the network is perturbed away from its native state. More experimental work is needed in order to determine the prevalence of canalization and/or partial canalization in actual genetic regulatory networks. The Boolean models can offer guidance in determining how likely it would be to observe any type of canalization in a random network.
Conclusion
In conclusion, we have used a mapping of the Boolean functions in the Kauffman model for genetic regulatory networks onto a k−hypercube to obtain information about the classes into which the functions can be divided. These classes arise due to geometrical constraints, and can be constructed by applying all possible rotations of the k−hypercube plus parity to each function. The classes can be counted and their properties determined using results from group theory and isomer chemistry. We emphasize that partially canalizing functions completely dominate all possible functions, particularly for higher k. This indicates that partial canalization should be extremely common, even in a randomly chosen network, and has implications for how much information can be obtained in experiments on native state genetic regulatory networks.
Figure 1 .
1Left: Mapping of the four possible input states for k = 2 onto the vertices of a square. Right: Mapping of the eight possible input states for k = 3 onto the vertices of a cube.
Figure 2 .
2Representation of the sixteen k = 2 functions on Ising squares. The functions are grouped into four classes. The members of each class are clearly related by symmetry operations on the square plus parity.
Figure 3 .
3A single representative Ising cube mapping of each of the 14 classes in k = 3.
x 4 1
4: (1)(2)(3)(4); two operators with cycle structure x 4 : (1243) and (3421); three operators with cycle structure x 2 2 : (12)(34), (13)(24), and (14)(23); and two operators with cycle structure x 2 1 x 2 : (14)(2)
Figure 4 .
4we plot the average fraction of homogeneous d dimensional sides, c d = C(d, k) /N d (k), for d = 1 through 4 and k = 2 through 8, obtained numerically. We sampled up to 1 × 10 8 functions generated with p = 0.5. For the case of d = 1, shown in figure 4(a), on average over 50% of the sides of the hypercube are uniformly coloured even for k = 8, indicating a significant amount of partial canalization. As d increases, c d drops considerably, as illustrated in figure 4(b-d) for d = 2, 3, and 4. Thus, the most prevalent type of partial canalization is that associated with homogeneous d = 1 sides. Average fraction c d of homogeneous d-dimensional sides in randomly selected Boolean functions versus k for (a) d = 1, (b) d = 2, (c) d = 3, and (d) d = 4.
Table 4 .
4Class structure for k = 2.
Table 6 .
6Class structure for k = 4.Class type N h
S c
A 16
1
2
A 15 B
1
16
A 14 B 2
4
60
A 13 B 3
6
186.667
A 12 B 4
19
191.58
A 11 B 5
27
323.56
A 10 B 6
50
320.32
A 9 B 7
56
408.57
A 8 B 8
74
173.9
Table 7. Class structure for k = 5.
Class type
N h
S c
A 32
1
2
A 31 B
1
64
A 30 B 2
5
198.4
A 29 B 3
10
992
A 28 B 4
47 1530.2
A 27 B 5
131 3074.4
A 26 B 6
472 3839.8
A 25 B 7
1326 5076.7
A 24 B 8
3779 5566.7
A 23 B 9
9013 6224.1
A 22 B 10
19963 6463.2
A 21 B 11
38073 6777.7
A 20 B 12
65664 6877.2
A 19 B 13
98804 7031.6
A 18 B 14
133576 7058.7
A 17 B 15
158658 7131.3
A 16 B 16
169112 3554.3
AcknowledgmentsWe thank Min Liu for discussions. This work was supported by the U.S. DoE under Contract No. W-7405-ENG-36 (CJOR), the LANL Laboratory Directed Research and Development program (CJOR), and the NSF through grant No. DMR-0427538 (KEB).
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"KKR TYPE BIJECTION FOR THE EXCEPTIONAL AFFINE ALGEBRA E MASATO OKADO AND NOBUMASA SANO",
"KKR TYPE BIJECTION FOR THE EXCEPTIONAL AFFINE ALGEBRA E MASATO OKADO AND NOBUMASA SANO"
]
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| For the exceptional affine type E(1) 6 we establish a statistic-preserving bijection between the highest weight paths consisting of the simplest Kirillov-Reshetikhin crystal and the rigged configurations. The algorithm only uses the structure of the crystal graph, hence could also be applied to other exceptional types. | 10.1090/conm/565/11181 | [
"https://arxiv.org/pdf/1105.1636v2.pdf"
]
| 119,121,681 | 1105.1636 | 3b1a53dc813fd303820c36b0f5717e26b8e8f024 |
KKR TYPE BIJECTION FOR THE EXCEPTIONAL AFFINE ALGEBRA E MASATO OKADO AND NOBUMASA SANO
10 Jun 2011
KKR TYPE BIJECTION FOR THE EXCEPTIONAL AFFINE ALGEBRA E MASATO OKADO AND NOBUMASA SANO
10 Jun 2011(1) 6
For the exceptional affine type E(1) 6 we establish a statistic-preserving bijection between the highest weight paths consisting of the simplest Kirillov-Reshetikhin crystal and the rigged configurations. The algorithm only uses the structure of the crystal graph, hence could also be applied to other exceptional types.
Introduction
In a pioneering work [15] Kerov, Kirillov and Reshetikhin introduced a new combinatorial object, called rigged configuration, through Bethe ansatz analysis of the Heisenberg spin chain, and constructed a bijection between rigged configurations and semistandard tableaux. One of the amazing properties of the rigged configuration is that it possesses a natural statistic and the statistic coincides with the charge by Lascoux and Schützenberger [21] on the tableau under the bijection. Subsequently, Nakayashiki and Yamada [25] studied the meaning of the charge in terms of Kashiwara's crystal bases. They considered the crystal base B l of the l-fold symmetric tensor representation of the n-dimensional irreducible U q ( sl n )-module. For the tensor product B l ⊗ B l ′ an integer-valued function H, called energy function, is defined via the q → 0 limit of the quantum R-matrix. Using this H they constructed a function D on the multiple tensor product B l1 ⊗ · · · ⊗ B lm . They then showed that under a certain bijection sending highest weight vectors or paths of B l1 ⊗ · · · ⊗ B lm to semistandard tableaux, the value of D agrees with the charge, thereby proving that the well-known Kostka polynomial is represented as a generating function of highest weight paths with statistic D. This generating function is denoted by X and the one of rigged configurations by M . The equality X = M was extended to the most general case for affine type A in [17]. See also [30] for review.
It did not take long before this kind of equality was conjectured to exist for other affine types. For the X side, crystal bases for some finite-dimensional modules, which are now called Kirillov-Reshetikhin (KR) modules, for quantum affine algebras have been discovered in [13]. For the M side, the existence of KR modules were conjectured and a formula to count the number of rigged configurations were presented in [16]. Introducing an appropriate q-analogue for the formula, the X = M conjecture [8,7] was presented. Imitating the one by KKR a bijection between rigged configurations and highest weight paths consisting of elements of KR crystals for other nonexceptional affine types was subsequently constructed in [28,29,31]. We note that these bijections have an important application for the analysis of the ultra-discrete integrable systems, also called box-ball systems [4,6,9]. In such systems rigged configurations give the complete set of the action and angle variables [19,20].
In this paper we consider the exceptional affine algebra of type E
6 . The KR crystal we deal with is the simplest one denoted in our notation by B 1,1 , whose crystal structure was revealed in [22,5]. We construct a map Φ from rigged configurations to highest weight elements of (B 1,1 ) ⊗L by executing a fundamental procedure δ repeatedly. We then show Φ is a statistic-preserving bijection (Theorem 3.2). It is worth mentioning that our procedure only uses the crystal graph structure of the KR crystal B 1,1 , hence similar constructions could be possible for other exceptional types.
We remark that recently Naoi [26] solved, with the help of the results in [3] and [23], the X = M conjecture for all untwisted affine types when the tensor product of KR crystals is of the form B r1,1 ⊗ · · · ⊗ B r l ,1 by showing both X and M are equal to the graded character of a Weyl module, a finite-dimensional current algebra representation defined in [2]. Hence his result includes ours as a special case. However, we think our direct method is also important, since it could also be used for more general cases by cutting larger KR crystals as in [17].
2. Quantum affine algebra and crystal 2.1. Affine algebra E (1) 6 . We consider in this paper the exceptional affine algebra E (1) 6 . The Dynkin diagram is depicted in Figure 1. Note that we follow [11] for the labeling of the Dynkin nodes. It is different from that in [1] or [5]. Let I be the index set of the Dynkin nodes, and let α i , α ∨ i , Λ i (i ∈ I) be simple roots, simple coroots, fundamental weights, respectively. Following the notation in [11] we denote the projection of Λ i onto the weight space of E 6 by Λ i (i ∈ I 0 ) and set P = i∈I0 ZΛ i , P
+ = i∈I0 Z ≥0 Λ i . Let (C ij ) i,j∈I stand for the Cartan matrix for E(1)
6 . For i, j ∈ I, i ∼ j means C ij = −1, namely, the nodes i and j are adjacent in the Dynkin diagram of E (1) 6 . 2.2. KR crystal. Let g be any affine algebra and U ′ q (g) the corresponding quantized enveloping algebra without the degree operator. Among finite-dimensional U ′ q (g)-modules there is a distinguished family called Kirillov-Reshetikhin (KR) modules [18,24,10]. One of the remarkable properties of KR modules is the existence of a crystal basis [14] called a KR crystal. It was conjectured in [8,7], and recently settled for all nonexceptional types in [27]. The KR crystal is indexed by (a, i) (a ∈ I 0 , i ∈ Z >0 ) and denoted by B a,i . For exceptional types the KR crystal is known to exist when the KR module is irreducible or the index a is adjacent to 0 [13]. Recently, the explicit crystal structure of all such cases of type E (1) 6 was clarified in [5].
The KR crystal we are interested in in this paper is an E
6 -crystal B 1,1 , whose crystal structure was clarified in [5]. The crystal structure of B 1,1 is depicted in Figure 2. Here vertices in the graph signify elements of B 1,1 and b
i −→ b ′ stands for f i b = b ′ or equivalently b = e i b ′ .
We adopt the original convention for the tensor product of crystals. Namely, if B 1 and B 2 are crystals, then for b 1 ⊗ b 2 ∈ B 1 ⊗ B 2 the action of e i is defined as
e i (b 1 ⊗ b 2 ) = e i b 1 ⊗ b 2 if ϕ i (b 1 ) ≥ ε i (b 2 ), b 1 ⊗ e i b 2 else,
where ε i (b) = max{k | e k i b = 0} and ϕ i (b) = max{k | f k i b = 0}. By glancing at Figure 2, one obtains the following lemma which will be used to prove our main theorem. Let B 0 be the subgraph obtained by ignoring the 0-arrows from B. A route is a sequence (β 1 , . . . , β l ) of arrows such that the sink of β j is the source of β j+1 for j = 1, . . . , l − 1.
Lemma 2.1. The graph B 0 has the following features.
(1) Suppose the initial arrow of a route R has the same color a as the terminal arrow and there is no intermidiate arrow of color a. Then there are exactly two arrows β i (i = 1, 2) of color b i such that b i ∼ a in R.
(2) Let R be a route starting from 1 , (a 1 , . . . , a l ) the colors from the initial arrow to the terminal one in R. Then we have
l−1 j=1
C aj a l = δ a l ,1 − 1.
(3) Let R be a route of two steps with colors (a, b) such that b ∼ a. Then there exists a route R ′ with colors (b, a) starting and terminating at the same vertices as R. (4) Let R be a route of colors (a 1 , . . . , a l ). Let v i be the source of the arrow of color a i (i = 1, . . . , l). Suppose a 1 ∼ a l and a i ∼ a l for any i = 2, . . . , l − 1.
Then there is an arrow of color a l starting from v i for any i = 2, . . . , l − 1.
Proof. (1) and (3) can be checked by direct observations. (2) and (4) are derived from (1) and (3).
In what follows in this paper we assume B = B 1,1 . The set of classically restricted paths in B ⊗L of weight λ ∈ P + is by definition One may check that the following are equivalent for b = b 1 ⊗ b 2 ⊗ · · · ⊗ b L ∈ B ⊗L and λ ∈ P + .
(1) b is a classically restricted path of weight λ ∈ P + .
(2) b 1 ⊗ · · · ⊗ b L−1 is a classically restricted path of weight λ − wt(b L ), and
ε i (b L ) ≤ λ − wt(b L ), α ∨ i for all i ∈ I 0 . The weight function wt : B → P is given by wt(b) = i∈I (ϕ i (b) − ε i (b))Λ i . The weight function wt : B ⊗L → P is defined by wt(b 1 ⊗ · · · ⊗ b L ) = L j=1 wt(b j ). Example 2.2. The element b = 1 · 2 · 3 · '&%$ !"# 16 · 2 · '&%$ !"# 24
of B ⊗6 is a classically restricted path of weight Λ 3 . The dot · signifies ⊗.
2.3.
One-dimensional sums. The energy function D : B ⊗L → Z gives the grading on B ⊗L . In our case where a path is an element of the tensor product of a single KR crystal it takes a simple form. Due to the existence of the universal R-matrix and the fact that B ⊗B is connected, by [12] there is a unique (up to global additive constant) function H : B ⊗ B → Z called the local energy function, such that
(2.2) H(e i (b ⊗ b ′ )) = H(b ⊗ b ′ ) + 1 if i = 0 and e 0 (b ⊗ b ′ ) = e 0 b ⊗ b ′ −1 if i = 0 and e 0 (b ⊗ b ′ ) = b ⊗ e 0 b ′ 0 otherwise.
We normalize H by the condition
(2.3) H( 1 ⊗ 1 ) = 0.
More specifically, the value of H is calculated as follows. Firstly, one knows the crystal graph of B 0 ⊗ B 0 decomposes into three connected components as
B 0 ⊗ B 0 = B(2Λ 1 ) ⊕ B(Λ 1 + Λ 2 ) ⊕ B(Λ 1 + Λ 5 ),
where B(λ) stands for the highest weight E 6 -crystal of highest weight λ and the highest weight vector of each component is given by
1 ⊗ 1 , 1 ⊗ 2 , 1 ⊗ '&%$ !"# 18.
H is constant on each component, and takes the value 0, −1, −2, respectively. One can confirm it from the fact that e 0 ( 1 ⊗ 1 ) = 1 ⊗ '&%$ !"# 17 and e 0 ( 1 ⊗ 2 ) = 1 ⊗ '&%$ !"# 22 belong to the second and third component.
With this H the energy function D is defined by
(2.4) D(b 1 ⊗ · · · ⊗ b L ) = L−1 j=1 (L − j) H(b j ⊗ b j+1 ). Define the one-dimensional sum X(λ, L; q) ∈ Z ≥0 [q −1 ] by (2.5) X(λ, L; q) = b∈P(λ,L) q D(b) .
Rigged configuration and the bijection
3.1. The fermionic formula. This subsection reviews the definition of the fermionic formula from [7,8]. We at first provide the definition that is valid for any simplylaced affine type g and datum L, and then restrict g and L to E i ) be another such matrix. Say that ν is an admissible configuration if it satisfies
(3.1) a∈I0 i∈Z>0 i m (a) i α a = a∈I0 i∈Z>0 i L (a) i Λ a − λ and (3.2) p (a) i ≥ 0 for all a ∈ I 0 and i ∈ Z >0 , where (3.3) p (a) i = j∈Z>0 L (a) j min(i, j) − b∈I0 (α a |α b ) min(i, j)m (b) j .
Write C(λ, L) for the set of admissible configurations for λ ∈ P + and L. Define the charge of a configuration ν by
c(ν) = 1 2 a,b∈I0 j,k∈Z>0 (α a |α b ) min(j, k)m (a) j m (b) k − a∈I0 j,k∈Z>0 min(j, k)L (a) j m (a) k . (3.4) Using (3.3) c(ν) is rewritten as (3.5) c(ν) = − 1 2 a∈I0,i∈Z>0 p (a) i m (a) i + a∈I0,j,k∈Z>0 min(j, k)L (a) j m (a) k .
The fermionic formula is then defined by
(3.6) M (λ, L; q) = ν∈C(λ,L) q c(ν) a∈I0 i∈Z>0 p (a) i + m (a) i m (a) i . We now set g = E (1) 6 and (3.7) L (a) i = Lδ a1 δ i1 (a ∈ I 0 , i ∈ Z >0 ).
The latter restriction corresponds to considering paths in (B 1,1 ) ⊗L . By abuse of notation we denote the fermionic formula under the restriction (3.7) by M (λ, L; q).
Then the X = M conjecture of [8,7] states in this particular case that
(3.8) X(λ, L; q) = M (λ, L; q).
3.2. Rigged configuration. The fermionic formula M (λ, L; q) can be interpreted using combinatorial objects called rigged configurations. These objects are a direct combinatorialization of the fermionic formula M (λ, L; q). Our goal is to prove (3.8) by defining a statistic-preserving bijection from rigged configurations to classically restricted paths. Let ν = (m (a) i ) a∈I0,i∈Z>0 be an admissible configuration. We identify ν with a sequence of partitions
{ν (a) } a∈I0 such that ν (a) = (1 m (a) 1 2 m (a) 2 · · · ). Let J = {J (a,i) } (a,i)∈I0×Z>0
be a double sequence of partitions. Then a rigged configuration is a pair (ν, J) subject to the restriction (3.1) and the requirement that J (a,i) be a partition contained in a m
(3.10) p (a) i = Lδ a1 − 2Q (a) i + b∼a Q (b) i , where b ∼ a stands for C ba = −1 as defined in §2.1.
The set of rigged configurations for fixed λ and L is denoted by RC(λ, L). Then (3.6) is equivalent to
M (λ, L; q) = (ν,J)∈RC(λ,L) q c(ν,J) where (3.11) c(ν, J) = c(ν) + |J|
with c(ν) as in (3.4) and |J| = (a,i)∈I0×Z>0 |J (a,i) |. The set RC(λ, L) with the restriction (3.7) is denoted by RC(λ, L). The partitions ν (1) , ν (2) , . . . , ν (6) are illustrated from left to right as Young diagrams. In ν (1) , 0 and 1 on the left signify p The bijection Φ is defined recursively as follows. For b ∈ B let P b (λ, L) be the set of paths in B ⊗L that have b as rightmost tensor factor. For L = 0 the bijection Φ sends the empty rigged configuration (the only element of the set RC(λ, L)) to the empty path (the only element of P(λ, L)). Otherwise assume that Φ has been defined for B ⊗(L−1) and define it for B ⊗L by the commutative diagram
(3.12) RC b (λ, L) Φ − −−− → P b (λ, L) δ RC(λ − wt(b), L − 1) Φ − −−− → P(λ − wt(b), L − 1)
where the right hand vertical map removes the rightmost tensor factor b. In short,
(3.13) Φ(ν, J) = Φ(δ(ν, J)) ⊗ γ(ν, J).
Here follows the main theorem of our paper.
The bijection
In this section, for (ν, J) ∈ RC(λ, L), an algorithm is given which defines b = γ(ν, J), the new smaller rigged configuration (ν,J) = δ(ν, J) such that (ν,J) ∈ RC(ρ, L − 1) where ρ = λ − wt(b), and the new vacancy numbers in terms of the old.
Illustrating a rigged configuration as in Example 3.1 we call a row in ν (a) singular if its rigging (number on the right) is equal to the corresponding vacancy number p We also use the notation ℓ
(4.1)m (a) i = m (a) i − ka k=1 (δ i,ℓ (a) k − δ i,ℓ (a) k −1 )
where k a is the maximum of k such that ℓ
i − p (1) i = −1 + 2χ(i ≥ ℓ (1) 1 ) − χ(i ≥ ℓ (2) 1 ) − χ(i ≥ ℓ (2) 2 ) + 2χ(i ≥ ℓ (1) 2 ) − χ(i ≥ ℓ (2) 3 ). Here we set ℓ (a) k = ∞ if k > k a .
This calculation is summarized in the following table.
• a = 1 [1, ℓ
1 ) [ℓ (1) 1 , ℓ (2) 1 ) [ℓ (2) 1 , ℓ (2) 2 ) [ℓ (2) 2 , ℓ (1) 2 ) [ℓ (1) 2 , ℓ (2) 3 ) [ℓ (2) 3 , ∞) -1 +1 0 -1 +1 0(1)
The first row signifies the range of i, namely, [1, ℓ
1 ) means 1 ≤ i < ℓ(1)
1 and the second rowp
(1) i − p (1) i
in this range. Similarly one obtains the following tables for other a.
• a = 2 [1, ℓ
1 ) [ℓ (1) 1 , ℓ (2) 1 ) [ℓ (2) 1 , ℓ(1)1 ) [ℓ (3) 1 , ℓ (3) 2 ) [ℓ (3) 2 , ℓ (2) 2 ) [ℓ (2) 2 , min(ℓ (1) 2 , ℓ(3)3 )) 0 -1 +1 0 -1 +1 [min, max) [max, ℓ(3)
3 ) [ℓ
3 , ℓ
4 ) [ℓ (3) 4 , ∞) 0 -1 +1 0(3)
In this table min, max without (·, ·) means the abbreviation of the previous parenthesis.
• a = 3 [1, ℓ(2)1 ) [ℓ (2) 1 , ℓ(3)1 ) [ℓ(3)
1 , min(ℓ
1 , ℓ
1 )) [min, max) [max, ℓ
2 ) 0 -1 +1 0 -1 [ℓ (3) 2 , min(ℓ (2) 2 , ℓ(3)
2 )) [min, max) [max, ℓ
3 ) [ℓ
3 , min(ℓ
2 , ℓ
3 )) +1 0 -1 +1
[min, max) [max, ℓ
4 ) [ℓ (3) 4 , ℓ(3)3 ) [ℓ (4) 3 , ∞) 0 -1 +1 0 • a = 4 [1, ℓ(4)1 ) [ℓ (3) 1 , ℓ(3)1 ) [ℓ(4)
1 , min(ℓ
1 , ℓ
2 )) [min, max) [max, ℓ
2 ) [ℓ (4) 2 , ℓ(4)3 ) 0 -1 +1 0 -1 +1 [ℓ (3) 3 , ℓ(3)4 ) [ℓ (3) 4 , ℓ(3)
3 ) [ℓ
3 , ℓ
2 ) [ℓ
(5) 2 , ∞) 0 -1 +1 0 • a = 5 [1, ℓ(4)1 ) [ℓ (4) 1 , ℓ(5)1 ) [ℓ (5) 1 , ℓ(4)
2 ) [ℓ (4) 2 , ℓ
3 ) [ℓ
3 , ℓ
2 ) [ℓ
(5) 2 , ∞) 0 -1 +1 0 -1 +1 • a = 6 [1, ℓ(3)1 ) [ℓ(3)
1 , ℓ
1 ) [ℓ(6)
1 , ℓ
2 ) [ℓ
2 , ℓ
3 ) [ℓ
3 , ℓ
2 ) [ℓ
2 , ℓ
4 ) [ℓ(3)4 , ∞) 0 -1 +1 0 -1 +1 0(3)
Example 4.1. The algorithm Φ for the rigged configuration in Example 3.1 is described at each step δ below.
? 0 0 0 0 0 ∅ ∅ ∅ ∅ δ 3 ? 0 0 ∅ ∅ ∅ ∅ ∅ δ 2 ? ∅ ∅ ∅ ∅ ∅ ∅ δ 1 ? ∅ ∅ ∅ ∅ ∅ ∅
Hence this rigged configuration corresponds to the path in Example 2.2 by Φ.
4.4.
Inverse algorithmδ. For a given rigged configuration (ν,J) and b ∈ B the inverse algorithmδ of δ is described as follows. From b ∈ B go back the arrow in the crystal graph B 0 . Let the maximal length of the singular row in ν (a) bel 0 . Repeat the following process for j = 1, 2, . . . until we arrive at 1 . Suppose the color of the arrow is a. Find the maximal integer i ≤l j−1 such that ν (a) has a singular row of length i and setl j = i, reset b to be the source of the arrow. If there are two arrows ending at b, compare the maximal integers and take the larger one. If the integers the same, either one can be taken. The output of the algorithm does not depend on the choices.
Proof of Theorem 3.2
Theorem 3.2 is proved in this section. The following notation is used. Let (ν, J) ∈ RC(λ, L), b = γ(ν, J) ∈ B, ρ = λ − wt(b), and (ν,J) = δ(ν, J). For (ν, J) ∈ RC(λ, L), define ∆(c(ν, J)) = c(ν, J) − c(δ(ν, J)). The following lemma is essentially the same as [28, Lemma 5.1].
Lemma 5.1. To prove that (3.14) holds, it suffices to show that it holds for L = 1, and that for L ≥ 2 with Φ(ν, J) = b 1 ⊗ · · · ⊗ b L , we have
(5.1) ∆(c(ν, J)) = −α(1)
1 , and
(5.2) H(b L−1 ⊗ b L ) =α (1) 1 − α (1) 1 where α (1) 1 andα (1)
1 are the lengths of the first columns in ν (1) andν (1) respectively, and δ(ν, J) = (ν,J).
There are five things that must be verified: We need several preliminary lemmas on the convexity and nonnegativity of the vacancy numbers p where λ a is defined by λ = a∈I0 λ a Λ a .
(I) ρ is dominant. (II) (ν,J) ∈ RC(ρ, L − 1). (III) b
Proof. This follows from the formula for the vacancy number (3.3) and the constraint (3.1).
Direct calculations show that
(5.3) − p (a) i−1 + 2p (a) i − p (a) i+1 = Lδ a1 δ i1 − 2m (a) i + b∼a m (b) i .
In particular these equations imply the convexity condition
(5.4) p (a) i ≥ 1 2 (p (a) i−1 + p (a) i+1 ) if m (a) i = 0.
Lemma 5.3. Let ν be a configuration. The following are equivalent:
(1) p (a) i ≥ 0 for all i ∈ Z >0 , a ∈ I 0 ; (2) p (a) i ≥ 0 for all i ∈ Z >0 , a ∈ I 0 such that m (a) i > 0.
Proof. This follows immediately from Lemma 5.2 and the convexity condition (5.4).
Proof of (I).
Here we show ρ = λ − wt(b) is dominant. Suppose not. Let λ = i∈I0 λ i Λ i . Since ε i (b), ϕ i (b) ≤ 1 for any i ∈ I 0 and b ∈ B, in order to make ρ not dominant there exists a ∈ I 0 such that λ a = 0 and ϕ a (b) = 1. (There may be at most two such a, but the proof is uniform.) Let R be the route taken by the algorithm δ. Although the arrow of color a sourcing from b is not taken by δ, we include it into R as a terminal arrow from notational reason. Let (a 1 , . . . , a l ) be colors of arrows in R. Let v j be the source of the arrow of color a j . Then a l = a, v l = b. Let ℓ j be the length of the singular row in ν (aj ) whose node is removed by δ.
Let ℓ be the largest part in ν (a) . We first show ℓ > 0. Suppose ℓ = 0. Then from (3.10) and Lemma 5.2 one gets
(5.5) 0 = Lδ a1 + c∼a Q (c) i for large i.
However, this is a contradiction since along the route R there has to be some c such that c ∼ a and a node in ν (c) was removed. There is only one exception: b = 1 and a = 1 case. This is also contradictory since the first term of the r.h.s. of (5.5) is positive. We can conclude ℓ > 0.
The convexity condition (5.4) implies p = 0 for all i > ℓ and c ∼ a. Set k = max{1 ≤ j < l | a j ∼ a}. Then from Lemma 2.1 (4) there is an arrow of color a sourcing from v j for any k < j < l, though by definition of a k and a l , all these arrows are not chosen by δ. In view of the fact that m (a k ) i = 0 for all i > ℓ and ν (a) has a singular row of length ℓ, one concludes that all length ℓ rows of ν (a) had been removed before a k . Thus we obtain (5.6) ♯{1 ≤ j < l | a j = a and ℓ j = ℓ} = m ℓ . Let l 1 = min{1 ≤ j < l | a j = a and ℓ j = ℓ}. Then the latter condition combined with Lemma 2.1 (1) and (5.6) imply that a node in each row of length ℓ in ν (c) (c ∼ a) should be entirely removed during the process of the algorithm between j = l 1 and j = l. Therefore length ℓ rows of ν (c) (c ∼ a) are not removed between j = 1 and j = l 1 − 1, which implies that ℓ j < ℓ for all j ≤ max{1 ≤ j < l 1 | a j ∼ a}. If m (a) ℓ−1 > 0, a node in all these rows should have been removed at the stage of j = l 1 during the algorithm since these rows are singular and after j = l 1 only length ℓ rows are removed. Hence
(5.8) ♯{1 ≤ j < l | a j = a and ℓ j = ℓ − 1} = m (a) ℓ−1 .
This equality is valid also when m ℓ−1 . The latter condition implies ℓ j < ℓ − 1 for all j ≤ max{1 ≤ j < l 2 | a j ∼ a} where l 2 = min{1 ≤ j < ℓ | a j = a and ℓ j ≥ ℓ − 1}, since from Lemma 2.1 (1) a node in all the rows of length ℓ − 1 in ν (c) (c ∼ a) should be removed between j = l 2 and j = l 1 . We continue this procedure until j = 1, where ♯{1 ≤ j < l | a j = a and ℓ j = 1} = m 1 . This equation implies that a node in all the rows of length 1 in ν (c) (c ∼ a) should be removed during the process j ≥ min{1 ≤ j < l | a j = a}. However, it is a contradiction, since there exists a j such that a j ∼ a and j < min{1 ≤ j < l | a j = a} by Lemma 2.1 (2). The proof is completed.
Proof of (II). To prove the admissibility of (ν,J) we need to show (i) There exists a singular row of length i in ν (a) such that ℓ j ≤ i < ℓ j ′ for some j < j ′ .
(5.12) 0 ≤J (a,i) max ≤p (a) i for all i ≥ 1, 1 ≤ a ≤ 6 whereJ(ii) m (a) ℓ j ′ −1 = 0, p (a)
ℓ j ′ −1 = 0, ℓ j < ℓ j ′ for some j < j ′ . In both cases ℓ j ′ corresponds to ν (a) and ℓ j to ν (c) such that c ∼ a and j is the maximum that is less than j ′ .
We show (i) and (ii) cannot occur. Firstly, suppose (i) occurs. Then, by Lemma 2.1 (4), a node of this singular row of length i should have been removed by δ, which is a contradiction. Suppose (ii) occurs. Let t be a maximal integer such that t < ℓ j ′ , m
i = 0 for all c ∼ a, t < i < ℓ j ′ .
Since ℓ j < ℓ j ′ this implies that ℓ j ≤ t. If t = 0, it contradicts ℓ j ≥ 1. Hence assume that t > 0. Since p (a) t = 0 and m (a) t > 0, there is a singular row of length t in ν (a) and therefore ℓ j ′ = t by Lemma 2.1 (4), which contradicts t < ℓ j ′ .
Proof of (III). Given (ν,J) ∈ P(ρ, L − 1) and b ∈ B, we want to show that one obtains the original (ν, J) ∈ P(λ, L) by the inverse procedure of δ. However, once one notices from the tables in §4.3 that if a node is removed from a row of length ℓ in ν (a) , then the differencep = +1 for all ℓ ≤ i < ℓ ′ where ℓ ′ is the length of the singular row in ν (c) such that c ∼ a removed by δ after ℓ, it is obvious that δ gives the inverse procedure of δ.
Proof of (IV). Let (ν,J) = δ(ν, J). Letm
C ab min(j, k)(m (a) j m (b) k −m (a) jm (b) k ) (5.13) + j (Lm (1) j − (L − 1)m (1) j ) + a ka k=1 (p (a) ℓ (a) k −p (a) ℓ (a) k −1 ). From (3.3) we obtain p (a) ℓ (a) k −p (a) ℓ (a) k −1 = δ a1 (1 + (L − 1)δ ℓ (a) k ,1 ) − b,j C ab χ(j ≥ ℓ (a) k )m (b) j + min(ℓ (a) k − 1, j) k b i=1 (δ j,ℓ (b) i − δ j,ℓ (b) i −1 ) .
Substituting (4.1) and the above into (5.13) one gets
∆(c(ν, J)) = k 1 − j m (1) j − V, where V = 1 2 a,b ka i=1 k b j=1 C ab (δ ℓ (a) i ℓ (b) j + χ(ℓ (a) i < ℓ (b)
j )).
Use another notation for ℓ (a)
i . Namely, let ℓ j (j = 1, . . . , ℓ) be the successive length of the singular rows by δ. V is calculated as
V = 1 2 ℓ i,j=1 C aiaj (δ ℓiℓj + 2χ(ℓ i < ℓ j )) = ℓ + i<j C aiaj = k 1 .
Here we have used Lemma 2.1 (2) in the last equality. This completes the proof.
Proof of (V). The proof is reduced to showing the following lemma.
Lemma 5.4. For (ν, J) ∈ RC(λ, L) with L ≥ 2 set γ(ν, J) = c, γ(δ(ν, J)) = b. Let ℓ (a) k
be the length of the singular row in ν (a) at the k-th time by the algorithm δ. Define the following subsets of B ⊗ B.
S 1 ={ 1 ⊗ j | j ≥ 18} ⊔ { 2 ⊗ j | j ≥ 23} ⊔ { 3 ⊗ j | j ≥ 25} ⊔ { 4 ⊗ j , 7 ⊗ j | j ≥ 26} ⊔ { i ⊗ '&%$ !"# 27 | i = 5, 8, 10, 13, 18}, S 2 ={ i ⊗ j |
i can be reached by following some (possibly zero) arrows from j }.
Then we have
(1) H(b ⊗ c) = −2 if b ⊗ c ∈ S 1 0 if b ⊗ c ∈ S 2 −1 otherwise.
(2) b ⊗ c belongs to S 1 if and only if ℓ Proof. Checking (1) reduces to a finite calculation that can be confirmed by computer.
To prove (2) letl (a) k be the length of the row in ν (a) at the k-th time by the second δ. We first show the condition ℓ
4 ≤l
1 , ℓ
3 ≤l
1 , ℓ
2 ≤l
1 ,l
1 = ∞. Let R andR be the routes taken by the first and second algorithms δ. Suppose ℓ (1) 1 = ℓ (1) 2 = 1. Then for all the arrows in R between the first one of color 1 and the second, the first δ removes a node from a row of length 1, namely, removes the row. In view of the table for a = 1 in §4.3 the length of the singular row after the first δ should be no less than ℓ (2) 3 . Hence we have ℓ 1 . Proceeding similarly we obtain (5.14). Suppose (5.14) next and assume ℓ 2 . However, it contradicts to the first inequality of (5.14). Therefore, we have ℓ (1) 1 = ℓ (1) 2 = 1. The fact that (5.14) is equivalent to b ⊗ c ∈ S 1 is checked as follows. Suppose for instance that b = 2 . This meansl
(1) 1 < ∞ andl(2)
1 = ∞. From the first inequality of (5.14) we have ℓ (2) 3 < ∞, which implies c = j for j ≥ 23. Other cases can be checked similarly.
We are left to show (3). From the assumption ℓ (1) 1 > 1, there are remaining singular rows after the first δ which could be removed by the second δ. Thus the "if" part is finished. To show the "only if" part, we assume ℓ
2010
Mathematics Subject Classification. Primary 17B37 82B23 05A19; Secondary 17B25 81R50 81R10 05E10 11B65. Date: May 20, 2011.
Figure 1 .
1Dynkin
Figure 2 .
2Crystal graph for B 1,1
(2. 1 )
1P(λ, L) = {b ∈ B ⊗L | wt(b) = λ and e i b = 0 for all i ∈ I 0 }.
case corresponding to paths we consider in this paper. Fix λ ∈ P + and a matrix L = (L (a) i ) a∈I0,i∈Z>0 of nonnegative integers, almost all zero. Let ν = (m (a)
of µ in the first i columns. Then setting Q (a) i = Q i (ν (a) ) the vacancy number (3.3) under the restriction (3.7) is rewritten as
we have c(ν) = −18, hence c(ν, J) = −14.3.3. The bijection from RCs to paths. We now describe the bijection Φ : RC(λ, L) → P(λ, L). Let (ν, J) ∈ RC(λ, L). We shall define a map γ : RC(λ, L) → B which associates to (ν, J) an element of B. Denote by RC b (λ, L) the elements of RC(λ, L) such that γ(ν, J) = b. We shall define a bijection δ : RC b (λ, L) → RC(λ − wt(b), L − 1). The disjoint union of these bijections then defines a bijection δ : RC(λ, L) → b∈B RC(λ − wt(b), L − 1).
Theorem 3. 2 .
2Φ : RC(λ, L) → P(λ, L) is a bijection such that(3.14) c(ν, J) = D(Φ(ν, J)) for all (ν, J) ∈ RC(λ, L).
.
Algorithm δ. Suppose you are at b = 1 in the crystal graph B 0 and set ℓ 0 = 1. Repeat the following process for j = 1, 2, . . . until stopped. From b proceed by one step through an arrow of color a. Find the minimal integer i ≥ ℓ j−1 such that ν (a) has a singular row of length i and set ℓ j = i, reset b to be the sink of the arrow. If there is no such integer, then set ℓ j = ∞ and stop. If there are two arrows sourcing from b, compare the minimal integers and take the smaller one. If the integers are the same, either one can be taken. The output of the algorithm does not depend on the choices by Lemma 2.1(3).
k
(= ℓ j ) if at the j-th step the arrow has color a and it is the k-th one having color a from the beginning.4.2. New configuration. The new configurationν =
3
3Change in vacancy numbers. Let A be a statement, then χ(A) = 1 if A is true and χ(A) = 0 if A is false. Then from (
can be appended to (ν,J) to give (ν, J). (IV) (5.1) in Lemma 5.1 holds. (V) (5.2) in Lemma 5.1 holds. Parts (I) and (II) show that δ is well-defined. Part (III) shows δ has an inverse. Part (IV) and (V) suffice to prove that Φ preserves statistics.
i = ℓ − 1 in (5.3). It yields (5.9) − p (a) ℓ−2 = Lδ a1 δ ℓ−1,1 − 2m
for the largest part ofJ (a,i) . In view of the definition of the algorithm δ in §4.1 and the tables ofp
§4.3, the condition (5.12) could only be violated when the following cases occur.
all t ≤ i ≤ ℓ j ′ . By (5.3) one finds that m (c)
be for (ν,J). Let ℓ (a) k (1 ≤ k ≤ k a ) be the length of the row a node of which is removed at the k-th time from ν (a) by the algorithm δ.
) b ⊗ c belongs to S 2 if and only if ℓ
the first δ there exists a singular row in ν (1) of length less than ℓ
<
In view of the table at §4.3, at a = 2∞ by the assumption b ⊗ c ∈ S 2 . Imitating the way to showl
k
′ . This procedure continues until we arrive at ℓ (a ′ ) k ′ = ∞. However, the previousl (a) k should be finite since the second δ can go further along the route taken by the first δ. This contradicts tõ ℓ
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| []
|
[
"Distributed Multi-Relay Selection in Accumulate-then-Forward Energy Harvesting Relay Networks",
"Distributed Multi-Relay Selection in Accumulate-then-Forward Energy Harvesting Relay Networks"
]
| [
"Yifan Gu ",
"He Chen ",
"Yonghui Li ",
"Ying-Chang Liang ",
"Branka Vucetic "
]
| []
| []
| This paper investigates a wireless-powered cooperative network (WPCN) consisting of one source-destination pair and multiple decode-and-forward (DF) relays. We develop an energy threshold based multi-relay selection (ETMRS) scheme for the considered WPCN. The proposed ETMRS scheme can be implemented in a fully distributed manner as the relays only need local information to switch between energy harvesting and information forwarding modes. By modeling the charging/discharging behaviours of the finite-capacity battery at each relay as a finitestate Markov Chain (MC), we derive an analytical expression for the system outage probability of the proposed ETMRS scheme over mixed Nakagami-m and Rayleigh fading channels. Based on the derived expression, the optimal energy thresholds for all the relays corresponding to the minimum system outage probability can be obtained via an exhaustive search. However, this approach becomes computationally prohibitive when the number of relays and the associated number of battery energy levels is large. To resolve this issue, we propose a heuristic approach to optimize the energy threshold for each relay. To gain some useful insights for practical relay design, we also derive the upper bound for system outage probability corresponding to the case that all relays are equipped with infinite-capacity batteries. Numerical results validate our theoretical analysis. It is shown that the proposed heuristic approach can achieve a near-optimal system performance and our ETMRS scheme outperforms the existing single-relay selection scheme and common energy threshold scheme. | 10.1109/tgcn.2017.2761872 | [
"https://export.arxiv.org/pdf/1602.00339v3.pdf"
]
| 3,987,086 | 1602.00339 | b5cb10f0cf99312e258b6f3d50985478734be690 |
Distributed Multi-Relay Selection in Accumulate-then-Forward Energy Harvesting Relay Networks
8 Oct 2017
Yifan Gu
He Chen
Yonghui Li
Ying-Chang Liang
Branka Vucetic
Distributed Multi-Relay Selection in Accumulate-then-Forward Energy Harvesting Relay Networks
8 Oct 20171Index Terms-Wireless energy harvestingcooperative com- municationsrelay selectionaccumulate-then-forwardMarkov Chainoutage probability
This paper investigates a wireless-powered cooperative network (WPCN) consisting of one source-destination pair and multiple decode-and-forward (DF) relays. We develop an energy threshold based multi-relay selection (ETMRS) scheme for the considered WPCN. The proposed ETMRS scheme can be implemented in a fully distributed manner as the relays only need local information to switch between energy harvesting and information forwarding modes. By modeling the charging/discharging behaviours of the finite-capacity battery at each relay as a finitestate Markov Chain (MC), we derive an analytical expression for the system outage probability of the proposed ETMRS scheme over mixed Nakagami-m and Rayleigh fading channels. Based on the derived expression, the optimal energy thresholds for all the relays corresponding to the minimum system outage probability can be obtained via an exhaustive search. However, this approach becomes computationally prohibitive when the number of relays and the associated number of battery energy levels is large. To resolve this issue, we propose a heuristic approach to optimize the energy threshold for each relay. To gain some useful insights for practical relay design, we also derive the upper bound for system outage probability corresponding to the case that all relays are equipped with infinite-capacity batteries. Numerical results validate our theoretical analysis. It is shown that the proposed heuristic approach can achieve a near-optimal system performance and our ETMRS scheme outperforms the existing single-relay selection scheme and common energy threshold scheme.
I. INTRODUCTION
The performance of many wireless communication networks in practice is largely confined by the energy constrained devices that require replenishment periodically. Recently, a novel radio-frequency (RF) energy transfer and harvesting technique has been proposed as a new viable and promising solution to prolong the lifetime of energy constrained wireless networks [1]. RF energy transfer and harvesting enables wireless devices to harvest energy from RF signals broadcast by ambient/dedicated energy transmitters to charge their batteries [2], [3]. This technique has opened a new research paradigm, termed wireless-powered communication ( become a hot research topic recently (see, e.g., [4], [5] and references therein).
The WPC technique has brought new research opportunities to cooperative communications, which have attracted an upsurge of interest during the past decade due to its various advantages [6]. In this paper, we refer to a cooperative communication network with wireless-powered 1 relay(s) as a wireless-powered cooperative network (WPCN). In [7], Nasir et. al first investigated a classical three-node WPCN consisting of one source-destination pair and one energy harvesting amplify-and-forward (AF) relay. Two practical relaying protocols, namely time switching-based relaying and power splitting-based relaying, were proposed and analyzed in [7]. Inspired by this seminal work, a plenty of works focusing on the design and/or analysis of WPCNs have published in open literature very recently (see [8]- [22] and references therein).
All aforementioned works on WPCNs assumed that the wireless-powered node(s) exhausts the harvested energy in the current time slot to perform information transmission/forwarding straight away. Equipping each wirelesspowered node with an energy storage (e.g., a rechargeable battery) such that they can accumulate the harvested energy and then perform information tranmission/forwarding in an appropriate time slot can improve the system performance significantly. The energy accumulation process of the classical three-node WPCNs with a single wireless-powered relay was modeled and the resulting network performance was analyzed in [23], [24] for finite and infinite storage scenario, respectively. [25] studied a multi-user network where all the users can harvest and accumulate energy from the base station simultaneously. Based on the considered system, the authors derived the transmission probability, the signal-to-interference ratio coverage probability and the overall success probability. It is also of great importance to investigate the network setup with multiple wireless-powered relays. Specifically, the relay selection problem was studied in [26] for a time-division and full-duplex block structure. Inspired by the max-max relay selection strategy, a new relay selection scheme is proposed that the relay with the best source-to-relay link is selected to receive information and store it in its buffer while the relay with the best relay-to-destination link is selected to transmit information. The corresponding outage probability and throughput were then analyzed. Inspired by the opportunistic relaying (OR) originally proposed in [27], a battery-aware relay selection (BARS) scheme was proposed and analyzed. In the BARS, the relays with accumulated energy exceeding a predetermined threshold will first form a subset, which need to feedback their channel state information (CSI) to the source. Then, the "best" relay among the subset with maximum endto-end signal-to-noise ratio (SNR) is selected by the source to forward its information, while other relays harvest energy in the first hop. [28] studied a WPCN with multiple randomly distributed relays. A distributed beamforming (DB) scheme was proposed that all relay nodes which are fully charged at the beginning of the transmission block form a forwarding subset. Among this subset of relays, the relays that are able to decode the source's signal create a virtual multiple antenna array and transmit source's signal to destination coherently.
In this paper we develop an efficient energy threshold based multi-relay selection (ETMRS) scheme with energy accumulation capability at each relay for WPCNs. In the proposed ETMRS, each relay can flexibly switch between energy harvesting and information forwarding modes in each transmission block. It should be noted that the proposed scheme is purely distributed and only the local battery status and local channel state information 2 (CSI) are required at each relay to perform mode selection, thus not involving extensive inter-relay information exchanges as in existing schemes. We also consider a practical and more general channel fading than [28], [31], where source-relay links and relay-destination links are assumed to experience independent but not necessarily identical distributed (i.n.i.d.) channel fading. Different from [28], we adopt a multiple-level battery model to characterize the charging/discharging behavior at relay batteries. In this case, the amount of harvested energy available at relays could be different. As such, to adequately exploit the relays, in our ETMRS scheme we set an individual energy threshold for each relay. This is in contrast to the existing schemes that considered independent and identical distributed channel fading and adopted a common energy threshold for all relays [28], [31]. The performance of the proposed ETMRS scheme can be further improved by jointly optimizing the energy thresholds of all relays. Moreover, compared with the DB scheme in which the weights in distributed beamforming for the relays within the forwarding set are obtained based on full CSI of the whole forwarding subset, the weight at each relay in the proposed ETMRS approach is calculated based only on the local CSI such that it can be implemented in a fully distributed manner. This can effectively reduce the network overhead and latency. Note that in our ERMRS scheme the relays that are selected to forward source's information to destination in each transmission block can be any combination of all relays, which depends on not only the instantaneous CSI but also the long-term evolution of all relay batteries. This makes the performance analysis and system design of the considered network a non-trivial task since we need to develop a systematic approach to characterize the probability of different relay combinations as well as the statistics of the 2 For more information about CSI acquisition for energy harvesting devices, interested readers could see [29] and [30]. corresponding end-to-end SNR as a sum of i.n.i.d random variables. Furthermore, the energy threshold of each relay that determines its long-term energy evolution should be jointly designed to boost the system performance.
Notation: Throughout this paper, we use f X (x) and F X (x) to denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable X. Γ (·) is the Gamma function [32,Eq. (8.310)], γ (·, ·) is the lower incomplete gamma function [32,Eq. (8.350.1)] and ⌈·⌉ is the ceiling function. We use (·) * and (·) T to represent the complex conjugate and the transpose of a matrix or vector, respectively. E [·] is the expectation operator and I denotes the identity matrix. P { A |B } is the conditional probability of A under a given condition B.
II. SYSTEM MODEL AND SCHEME DESIGN
A. System Model
As depicted in Fig. 1, in this paper we investigate a WPCN consisting of one source-destination pair and N decode-andforward (DF) relays, which are deployed to assist the source's information transmission. We assume that there is no direct link between source and destination due to obstacles or severe attenuation. Also, all nodes are equipped with single antenna and work in half-duplex mode. As in [28], we consider the scenario that all DF relays are wireless-powered devices and purely rely on the energy harvested from RF signals broadcast by source to perform information forwarding. Moreover, these relays are equipped with separate energy and information receivers [33]. As such, they can flexibly switch between energy harvesting (EH) mode and information forwarding (IF) mode at the beginning of each transmission block. We also assume that each relay is equipped with a finite-capacity rechargeable battery such that it can perform energy accumulation and scheduling across different transmission blocks. Specifically, they can accumulate the harvested energy to a certain amount before assisting the source's information transmission.
We denote by T the duration of each transmission block, which is further divided into two time slots with equal length T /2. During the first time slot, the source broadcasts its signal to all relays. At each relay operating in EH mode, the received signal is delivered to the energy receiver to convert to direct current and charge the battery. In contrast, the received signal at each relay in IF mode is connected to its information receiver to decode the information sent by the source. All relays that operate in IF mode and decode the source's information correclty form a decoding set. In the second time slot, all relays in the decoding set will jointly forward the source's information to destination by consuming part of the accumulated energy from their batteries. On the other hand, other relays outside the decoding set keep in silence during the second time slot 3 .
We hereafter use subscript-S and subscript-D to denote the source and destination respectively. We denote by R u , u ∈ {1, 2, · · · , N }, the u-th wireless-powered relay. Among existing fading models, Rician fading would be the most appropriate one to characterize the channel fading of S-R u links. This is mainly motivated by the fact that the up-to-date wireless energy transfer techniques could only be operated within a relatively short communication range such that the line-of-sight (LoS) path is very likely to exist in these links. However, the statistical functions (e.g., cumulative density function (CDF) and probability density function (PDF)) of Rician fading are very complicated, which would make the analysis extremely difficult [34]. Fortunately, the Rician distribution could be well approximated by the more tractable Nakagami-m fading model. Thus, in this paper we adopt an asymmetric scenario for the fading distributions of sourceto-relay links and relay-to-destination links. Specifically, the S-R u link is assumed to be subject to Nakagami-m fading with fading severity parameter m u and average power gain λ SR u , while the R u -D link suffers from Rayleigh fading with average power gain λ R u D as the distance between them may be much further. Besides, all channels in the considered system experience slow, independent, and frequency-flat fading such that instantaneous channel gains remain unchanged within each transmission block but change independently from one block to the other. It is worth mentioning that we do not require the channel fading parameters to be identical or non-identical for both hops. That is, we investigate a general independent but not necessarily identical fading model, which includes the independent and identical one as well as the independent and non-identical one for special cases. Without loss of generality, we consider a normalized transmission block (i.e., T = 1) hereafter.
Let P denote the source transmit power and x denote the transmitted symbol with E | x| 2 = 1. The received signal at the u-th relay during the first time slot is thus given by
y u = √ Ph u x + n,(1)
where h u is the channel coefficient between S and R u , and n denotes the additive white Gaussian noise (AWGN) with zero mean and variance N 0 at the receiver side. When the u-th relay works in EH mode, the received signal y u will be delivered to its energy receiver and converted to direct current to charge the battery. The amount of harvested energy at R u during the first time slot can thus be expressed asẼ
u = 1 2 ηPH u ,(2)
where 0 < η < 1 is the energy conversion efficiency and H u = |h u | 2 is the channel power gain between S and R u . Note that in (2), we ignore the amount of energy harvested from the noise since the noise power is normally very small and below the sensitivity of the energy receiver. On the other hand, if R u opts to decode information in the first time slot, it will harvest zero energy.
Let Φ denote the current decoding set. In the second time slot, all relays in the decoding set Φ will jointly forward the source's information to the destination by implementing the distributed beamforming technique [35]. Specifically, the transmitted signal at R u ∈ Φ is given by
x u = w u P u x,(3)
where w u is the weight of R u in distributed beamforming and P u is the transmit power of R u . In the considered WPCN, we assume that each relay only knows its local CSI of the second hop. In this case, the optimal weight for the u-th relay that maximizes the overall end-to-end SNR can be expressed as w u = g * u /|g u | [35], where g u is the complex channel coefficient between R u -D. We defineĝ u = |g u | for notation simplicity. The received signal at the destination can thus be expressed as
y d = u:R u ∈Φĝ u P u x + n.(4)
As a result, the conditional end-to-end SNR for a given decoding set Φ can be written as
γ Φ = u:R u ∈Φ √ P uĝu 2 N 0 .(5)
B. Energy Threshold Based Multi-Relay Selection
In this paper we develop an energy threshold based multi-relay selection (ETMRS) framework for the considered WPCN. In our ETMRS scheme, each relay R u determines its individual energy threshold, denoted byχ u . This energy threshold includes two parts: the first part is the energy consumption for circuit operation (e.g., information decoding), denoted byα for all the relays; the second part is the energy consumption for information forwarding, denoted byβ u for relay R u . Each relay decides to operate in EH mode or IF mode based on its own battery status at the beginning of each transmission block. Specifically, relay R u will perform the IF operation only when its accumulated energy is not less than its associated energy thresholdχ u . Otherwise, it will opt EH mode to further accumulate energy in its battery. Moreover, if R u works in the IF mode and falls in the decoding set, it will decode and forward the source signal to destination by consuming the amount of energyβ u from its battery in the second time slot. The conditional end-to-end SNR given in (5) can now be updated by substituting P u =β u /(1/2) = 2β u . That is,
γ Φ = u:R u ∈Φ 2β uĝu 2 N 0 .(6)
Note that we are investigating a distributed scheme such that each relay only requires local CSI and battery status to determine its operation modes. In this case, the optimal transmit power for each transmission block cannot be obtained as it requires global CSI and battery status. As a result, the fixed transmit power strategy is still preferable in our considered case and we can thus set fixed energy thresholds to each relay,χ u , u = 1, 2, · · · , N. Moreover, the energy thresholds for all the relays should be different in order to achieve the best system performance when the relays are in different locations. For the special case that all the relays are co-located in a cluster, the channels of each hop are now independent and identically distributed (i.i.d.). In this sense, all relays can adopt the same energy thresholdχ =χ 1 =χ 2 = · · · =χ N and consume the same power to forward information, denoted bỹ β =β 1 =β 2 = · · · =β N . We then have the conditional endto-end SNR of this i.i.d. case given by
⌢ γ Φ = 2β u:R u ∈Φĝ u 2 N 0 .(7)
III. PERFORMANCE ANALAYSIS
To analyze the performance of the proposed ETMRS scheme, in this section we first characterize the dynamic charging/discharging behaviors of the relay batteries. We consider a discrete-level and finite-capacity battery model. Thus, it is natural to use a finite-state Markov chain (MC) to model the dynamic behaviors of relays' batteries. From the MC model and the derived stationary distribution of the battery, we then derive an approximate analytical expression of the system outage probability for our ETMRS scheme. Based on the derived analytical expression, we subsequently discuss how to optimize the energy threshold of the relays to reduce the system outage probability.
A. Markov Chain of Relay Batteries
Thanks to the fact that our ETMRS scheme is a decentralized relay selection approach and the relays make decisions of their operation modes based only on their local CSI and battery status, we thus can evaluate the steady state distributions of all relays' batteries separately as they are independent to each other. In [23], the authors investigated a single relay WPCN and proposed an adaptive information forwarding scheme such that the relay forwards the source information only when its residual energy can guarantee an outage-free transmission in the second hop. The transition probabilities of the relay battery was summarized into eight general cases. Recall that our ETMRS scheme implements fixed transmit powers for multiple relays. By adequately using this feature, a compact mode-based method is used to evaluate the transition probabilities of the MC for each relay.
Let C denote the capacity of all relays' batteries and L denote the number of discrete levels excluding the empty level in each battery. Then, the i-th energy level of the relay battery can be expressed as ε i = iC/L, i ∈ {0, 1, 2 · · · L}. It is worth pointing out that as shown in [36], the adopted discrete battery model can tightly approximate its continuous counterpart when the number of energy levels (i.e., L) is sufficiently large, which will also be verified later in the simulation section. For each relay node, we define state S i as the relay residual energy in the battery being ε i . The transition probability T i, j u is defined as the probability of transition from state S i to state S j at the u-th relay. With the adopted discrete-level battery model, the amount of harvested energy can only be one of the discrete energy level. Thus, the discretized amount of harvested energy at the u-th relay during an EH operation is defined as
E u ∆ = ε j , j = arg max i ∈{0,1, ··· , L } ε i : ε i ≤Ẽ u .(8)
Moreover, for relays operating in the IF operation mode, the energy consumption for decoding operation α should also be discretized to one specific energy level of the battery with the definition give by
α ∆ = ε j , j = arg min i ∈{0,1, ··· , L } ε i : ε i ≥ α , ifα ≤ ε L ∞, ifα > ε L(9)
The energy consumption for information forwardingβ u and the energy threshold for each relayχ u should be chosen from one of the energy levels of the battery excluding the empty level. The descretized energy consumption for information forwarding β u and discretized energy threshold for each relay χ u can be defined as β u ∈ {ε 1 , ε 2 , · · · , ε L − α} and χ u = α+ β u ∈ {α + ε 1 , α + ε 2 , · · · , ε L }, respectively. Note that the system will work properly only when α < ε L , otherwise when α ≥ ε L , the fully charged battery even cannot support the circuit energy consumption of the relay and β u does not exist.
We now evaluate the state transition probabilities of the MC for each relay R u , u ∈ {1, 2, · · · , N }. Different from [23] that summarizes all the transition probabilities into eight general cases, we propose a compact mode-based approch which summarizes all the transition probabilities into the following two cases.
1) The relay R u operates in EH mode (S i to S j with 0 ≤ i < χ u ε 1 ≤ L and ∀ j): When the relay R u operates in the EH mode, it harvests energy from the source during the first time slot while remains in silence during the second time slot. Due to the fact that the relay battery is not discharged in the transition, the transition probability is none-zero only when the end state falls into the set S j ∈ {S i , S i+1 , · · · , S L }. Specifically, S j = S i indicates that the battery level of the relay remains unchanged and the harvested energy during the transitionẼ u is discretized to zero (i.e., E u = 0). S j = S L denotes the case that the battery is fully charged during the transition and the harvested energy should be larger than ε L−i . From the definition of discretization given in (8), the transition probabilities for R u operates in EH mode can be summarized in (10) on top of the next page.
Recall that the channels between source and relays are assumed to suffer from Nakagami-m fading. As such, the PDF and CDF of H u are given by [37, eq.2.21]
f H u (x) = b u mu Γ(m u ) x m u −1 exp(−b u x), F H u (x) = γ(m u ,b u x) Γ(m u ) , where b u = m u /λ SR u .
With the CDF, the transition probabilities for this case can now be expressed as
T i, j u = Pr ε j−i ≤Ẽ u < ε j−i+1 , if i ≤ j < L Pr Ẽ u ≥ ε L−i , if i ≤ j = L 0, if i > j = Pr 2 ( j − i) C ηPL ≤ H u < 2 ( j − i + 1) C ηPL , if i ≤ j < L Pr H u ≥ 2 (L − i) C ηPL , if i ≤ j = L 0, if i > j . (10) T i, j u = F H u 2( j − i + 1)C ηPL − F H u 2( j − i)C ηPL , if i ≤ j < L 1 − F H u 2(L − i)C ηPL , if i ≤ j = L 0, if i > j .(11)
2) The relay R u operates in IF mode (S i to S j with
χ u ε 1 ≤ i ≤ L and ∀ j):
In this case, the relay R u will try to decode the received signal and forward it to destination if the decoding is successful. In the first time slot, the relay consumes energy α to decode the received signal from S. If the decoding is unsuccessful, it remains in silence during the second time slot. On the other hand, if the relay R u decodes the information successfully, it forwards the decoded information to the destination by further consuming β u from its battery during the second time slot. We can now conclude that after the transition, the end state is none-zero only when
j = i − α ε 1 or j = i − χ u ε 1 .
Recall that ϕ u is the decoding indicator of relay R u and the transition probabilities for R u operating in IF mode can be summarized as
T i, j u = Pr {ϕ u = 0} , if j = i − α ε 1 1 − Pr {ϕ u = 0} , if i ≤ j = i − χ u ε 1 0, Otherwise .(12)
We now analyze the term Pr {ϕ u = 0} for the u-th relay. Let γ u = PH u N 0 denote the received SNR at relay R u . The channel capacity of the S-R u link is given by Θ u = 1 2 log 2 (1 + γ u ). According to the channel capacity, the term Pr {ϕ u = 0} can be evaluated as
Pr {ϕ u = 0} = Pr {Θ u < κ} = F H u vN 0 P ,(13)
where κ is the system transmission rate and v = 2 2κ − 1 is the SNR threshold for system outage. Based on the above analysis, the transition probabilities can be re-written as
T i, j u = F H u vN 0 P , if j = i − α ε 1 1 − F H u vN 0 P , if i ≤ j = i − χ u ε 1 0, Otherwise .(14)
We now define Z u = (T i, j u ) to denote the (L + 1) × (L + 1) state transition matrix of the MC for each relay R u , u ∈ {1, 2, · · · , N }. By using similar method in [23], we can easily verify that the MC transition matrix Z u derived from the above MC model is irreducible and row stochastic. Thus for each relay R u , there must exist a unique stationary distribution π π π u that satisfies the following equation π π π u = π u,0 , π u,1 , · · · , π u, L
T = (Z u ) T π π π u ,(15)
where π u,i , i ∈ {0, 1, · · · , L}, is the i-th component of π π π u representing the stationary distribution of the i-th energy level at relay R u . The battery stationary distribution of relay R u can be solved from (15) and expressed as [23] π π π u = (Z u )
T − I + B −1 b,(16)
where B i, j = 1, ∀i, j and b = (1, 1, · · · , 1) T . Moreover, when it comes to the i.i.d. channel model, all relays are equipped with an identical energy threshold and they have the same transition matrix. The corresponding identical stationary distribution of all relays can be similarly obtained as (16) and denoted by π π π 1 = π π π 2 = · · · = π π π N = π π π = (π 0 , π 1 , · · · , π L ) T .
B. System Outage Probability
With the above derived stationary distribution of the relay batteries, we now characterize the system outage probability of the proposed ETMRS scheme. Let O denote the outage event of the considered system employing our ETMRS scheme. According to the full probability theory, we can express the system outage probability as
P out = Pr {O} = Φ∈Λ Pr {Φ} Pr {O |Φ } ,(17)
where Λ = {R 1 , R 2 · · · , R N } denotes the set of all relays in the considered network and incorporates the decoding set Φ as its subset, Pr {Φ} denotes the probability that the current decoding set is Φ, and Pr {O |Φ } denotes the probability that system outage occurs under the decoding set Φ. In order to further expand (17), we define Φ k,n to denote the n-th k-subset of Λ (i.e., the n-th k-subset of Λ contains exactly k elements, k = 1, 2, · · · , N, n = 1, 2, · · · , N k ). Then the outage probability of the ETMRS scheme can be further expanded as
P out = Pr {∅} + N k=1 ( N k ) n=1 Pr Φ k,n Pr O Φ k,n ,(18)
by realizing that the system outage probability equals to one when the decoding set is an empty set ∅. The empty decoding set can be caused by two kinds of event: One is that none of the relays working in IF mode decodes the received signal from source correctly. The other is that all relays operate in EH mode and no relay performs IF. Based on the derived
Pr Φ k,n = u:R u ∈Φ k, n Pr {ζ u = ζ I , ϕ u = 1} u:R u Φ k, n Pr {ζ u = ζ I , ϕ u = 0} ∪ ζ u = ζ E = u:R u ∈Φ k, n (1 − Pr {ϕ u = 0}) L i=χ u /ε 1 π u,i u:R u Φ k, n Pr {ϕ u = 0} L i=χ u /ε 1 π u,i + χ u /ε 1 i=0 π u,i .(20)
stationary discrete distribution of relay batteries given in (16), we can calculate the first probability term in (18) as follows
Pr {∅} = u:R u ∈Λ Pr {ζ u = ζ I , ϕ u = 0} ∪ ζ u = ζ E = u:R u ∈Λ Pr {ϕ u = 0} L i=χ u /ε 1 π u,i + χ u /ε 1 i=0 π u,i .(19)
Similarly, the term Pr Φ k,n can be computed as (20) on top of the next page.
To evaluate the third probability term in (18), we first characterize the distribution of the conditional end-to-end SNR for a given decoding set. For notation simplicity, we use γ k,n to denote the received SNR at the destination when the decoding set is Φ n,k . Recall that the conditional end-to-end SNR for a certain decoding set is given in (6), which includes a weighted sum of Rayleigh random variables. However, to the best knowledge of authors, the exact distribution of a weighted sum of Rayleigh random variables does not exist in open literature. As a result, we cannot further characterize the exact distribution of γ k,n . Fortunately, with the aid of a tight approximation for the CDF of a weighted sum of Rayleigh random variables derived in [38], the CDF of γ k,n can be approximated as a gamma distribution and expressed as
F γ k, n (x) ≈ γ k, N 0 4 u:Ru ∈Φ k, n β u σ 2 u x Γ (k) ,(21)
where σ u = λ R u D /2 is the scale parameter of the Rayleigh fading channel between R u and D.
In order to further expand the summation term in (21), we use a similar method adopted in [39]- [41]. To this end, we define a set A = √ β u σ u : R u ∈ Λ, u ∈ {1, 2, · · · , N } with the same cardinality as the set Λ (i.e., |A| = |Λ| ). Similarly, let A k,n denote the n-th k-subset of A. The j-th element of the subset A k,n are denoted by φ k,n, j ∈ A k,n , j = 1, 2, · · · , k. To be more clear, we list the corresponding relationship between φ k,n, j and √ β u σ u in Table I. With the corresponding relation between φ k,n, j and √ β u σ u , (21) can now be expressed as
F γ k, n (x) ≈ γ k, a k,n x Γ (k) = 1 − exp(−a k,n x) k−1 i=0 a k,n x i i! ,(22)
where a k,n = √ β u σ u , u = 1, 2, · · · , N , FOR k = 1, · · · , N , n = 1, 2, · · · , N k AND j = 1, 2, · · · , k.
distribution of γ k,n , the third probability term in (18) can now be further expanded as
Pr O Φ k,n = Pr γ k,n < v = F γ k, n (v) ≈ 1 − exp(−a k,n v) k−1 i=0 a k,n v i i! ,(23)
where v = 2 2κ − 1 is the SNR threshold for system outage and κ is the system transmission rate. By substituting (19), (20) and (23) into (18), we have derived an approximate analytical expression of the outage probability for the proposed ETMRS scheme.
In terms of the i.i.d channel fading case where the relays have the same energy threshold, the number of different decoding sets reduces to N + 1 and the expression of system outage probability can be simplified to
⌢ P out = Pr {O} = Pr ⌢ ∅ + N k=1 Pr ⌢ Φ k Pr O ⌢ Φ k ,(24)
where Pr ⌢ ∅ is the probability that the decoding set is empty for the i.i.d. case, Pr ⌢ Φ k is the probability that the decoding set contains k relays, and Pr O ⌢ Φ k is the conditional outage probability when k relays falls in the decoding set. Similar to the analysis of the general case, the first and second probability terms in (24) can be expressed as
Pr ⌢ ∅ = Pr {ϕ = 0} L i=χ/ε 1 π i + χ/ε 1 i=0 π i N ,(25)Pr ⌢ Φ k = N k (1 − Pr {ϕ = 0}) L i=χ/ε 1 π i k × Pr {ϕ = 0} L i=χ/ε 1 π i + χ/ε 1 i=0 π i N−k ,(26)
where χ = α + β is the identical energy threshold for all the relays and Pr {ϕ = 0} is the outage probability of the relays for the special i.i.d. channel fading case, which can be derived from that for the general case given in (13) and written as
Pr {ϕ = 0} = γ m, mv N 0 λ S R P Γ(m)
, where m is the identical severity parameter of the Nakagami-m fading channels between source and relays and λ SR is the identical channel power gain of the first hop.
The third probability term in (24) can be deduced from (23) as
Pr O ⌢ Φ k ≈ 1 − exp(− N 0 v 4k βσ 2 ) k−1 i=0 N 0 v 4kβσ 2 i i! ,(27)
where σ = λ RD /2 is the identical scale parameter for the Rayleigh fading channels between relays and destination. Note that different from (23), the expression of the conditional system outage probability given in (27) does not require the parameters defined in Table I. The outage probability for the special i.i.d. channel fading case can be obtained by substituting (25), (26) and (27) into (24).
C. Energy Threshold Optimization
We first consider the special i.i.d. channel fading case where the associated energy thresholds for all the relays is identical, denoted by χ = α + β. When the energy used for information forwarding β grows, the conditional outage probability derived in (27) reduces and the overall system outage probability correspondingly decreases. On the other hand, increasing of β will lead to an increase of χ. According to (26), it decreases the probability of each relay working in IF mode and the number of relays falling into the decoding set. This will increase the overall system outage probability. In simple words, when the designed energy threshold χ for the relays is small, most of relays could fall into the decoding set but their transmit power is low. When the energy threshold is large, only a few relays fall into the decoding set but their associated transmit power is high. As χ = α + β ≥ α + ε 1 , we can now infer that for the special i.i.d. channel fading case, there should exist an optimal energy threshold χ ∈ {α + ε 1 , α + ε 2 , · · · , ε L } that minimize the system outage probability of the proposed ETMRS scheme. Similarly, for the general i.n.i.d. channel fading case where the relays may be located dispersively. There should also exist an optimal energy threshold set { χ 1 , χ 2 , · · · , χ N } , χ u ∈ {α + ε 1 , α + ε 2 , · · · , ε L } , ∀u = 1, 2, · · · , N, for all relays that minimize the overall system outage probability.
Due to the complexity of the adopted MC model, it is difficult to derive an analytical expression for the optimal energy threshold of each relay. However, for the special i.i.d. channel fading case, the optimal energy threshold can be easily achieved by performing a one-dimensional exhaustive search from all the possible energy levels with the derived analytical outage probability given in (24). The computation complexity of this search is given by O (L).
When it comes to the general i.n.i.d. case, the optimal set of energy thresholds can be found by a N-dimension exhaustive search from all the possible combinations of energy thresholds with the analytical expression derived in (18). The computation complexity is O L N , which grows exponentially with the number of relays N. Thus, finding the optimal set of energy thresholds becomes computationally prohibitive when the number of relays N and the level of batteries L are large. To overcome this problem, in the following subsection, we provide a heuristic approach to design the energy thresholds for the general i.n.i.d. case.
D. A Heuristic Approach for i.n.i.d Scenario
Intuitively, a higher energy threshold and forwarding transmit power should be set for those relays closer to the source node. This is because that the average amount of harvested energy is higher for those relays compared to the ones far away from the source node. On the other hand, relays closer to the source are relatively further away from the destination node such that the associated second hop channels are relatively weaker, a larger transmit power should be used to overcome the higher path loss. On the other hand, for the relays near to the destination node, smaller forwarding transmit power can be adopted due to their limited harvested energy and stronger second hop channels. Inspired by this fact, we set the energy consumption for information forwarding at each relay R u as
β u = zλ SR u /λ R u D ,(28)
where z is a scalar factor to adjust the overall transmit power of all the relays. For the considered discrete-battery model, the designed energy consumption for information forwarding β u should be discretized to one specific energy level of the battery. Note that the definition of discretization given below (9) no longer holds as we cannot simply choose β u ∈ {ε 1 , ε 2 , · · · , ε L − α}. We now define the discretized value ofβ u for the u-th relay as β u
∆ = ε j , j = arg min i ∈{0,1, ··· , L } ε i : ε i ≥β u , ifβ u ≤ ε L − α L − α ε 1 , ifβ u > ε L − α .(29)
Based on the above definitions, the discretized energy threshold for relay R u can be further expressed as
χ u = α + β u = α + min ε 1 zλ SR u λ R u D ε 1 , C − α .(30)
Similar to the analysis given in the previous subsection, we can deduce that there should exist an optimal value of z that minimizes the system outage probability. To find the optimal value of z, we first define λ max and λ min as the maximum and minimum value of the term λ SR u /λ R u D , among all relays, respectively. The optimal z should exist within the interval [ε 1 /λ max , ε L /λ min ], where z = ε 1 /λ max could make all relays choose ε 1 + α as their energy thresholds and z = ε L /λ min will force all relays to adopt energy level L as their energy thresholds. The optimal z thus can now be achieved by performing a one-dimension exhaustive search from this interval over the derived outage probability expression. In order to capture all the possible combinations, we search z with an increment of ε 1 /λ max each time and the computation complexity for the proposed heuristic approach can be expressed by O (Lλ max /λ min ). Note that the complexity of the heuristic approach is obviously much lower than that of the exhaustive search given by O L N .
IV. PERFORMANCE UPPER BOUND
As described in Section III, we adopt a practical finitecapacity battery model in this paper. With this model, it can be readily deduced that the system performance could be improved when we increase the battery capacity (i.e., C). This is because a larger battery capacity can reduce the energy loss caused by energy overflow (i.e., the battery cannot be charged when it is full) and thus the relays have more available energy to support their information forwarding operation. On the other hand, we can infer that the system performance improvement speed actually decreases as the battery capacity increases since the energy overflow happens more rarely when the battery capacity keeps increasing. A natural question that comes up here is "For a given network setup, how large of the battery capacity C will be sufficient?" This question is particularly important for the considered system as one of its potential applications will be low-cost and lower-power networks (e.g., wireless sensor networks), in which the network deployment cost should be kept as low as possible by carefully selecting the battery capacity. However, it is hard to find the answer of this question based on the derived expressions in the previous section due to their complexity. We are thus motivated to adopt an indirect way: We first derive performance upper bounds of the considered system and the sufficient battery capacity will then be obtained when a certain value of C can make the performance expressions derived in previous sections approach their corresponding upper bounds. In this sense, in this section we analyze the performance upper bounds of the considered system with infinite battery capacity (i.e., C → ∞ and implicitly L → ∞).
When the relay batteries are infinite, there will be no energy overflow. As such, we can implement the well-known flow conservation law to evaluate the system outage probability. Specifically, in our ETMRS scheme, the relay battery is charged if operated in EH mode and discharged if operated in IF mode. In this sense, for each relay, the total amount of harvested energy in a long term should equal to the total amount of energy consumed for information decoding and forwarding. Mathematically, we have the following formula for the u-th relay
p uĒu = (1 − p u ) [(1 − q u ) α + q u (α + β u )] ,(31)
where p u denotes the probability that relay R u performs EH operation, q u denotes the probability that the u-th relay falls in the decoding set,Ē u is the average amount of harvested energy during each EH operation, α is the circuit energy consumption for each decoding operation and β u is the energy consumption for information forwarding of R u adopted in the finite-capacity case. Since both C and L are infinity, the termĒ u can be obtained from (2) given bȳ
E u = 1 2 ηPλ SR u .(32)
With the fact that relay R u falls in the decoding set only when it performs IF operation and decodes the source's information correctly, we thus have
q u = (1 − p u ) (1 − Pr {ϕ u = 0}) ,(33)
where the term Pr {ϕ u = 0} has been derived in (13). By jointly considering (31)-(33), we can now obtain q u expressed as
q u = 1 1 1−Pr{ϕ u =0} + 2α+2β u (1−Pr{ϕ u =0}) η Pλ S Ru (1−Pr{ϕ u =0}) .(34)
The first and second probability terms in the system outage probability defined in (18) can now be evaluated by q u as
Pr {∅} = u:R u ∈Λ (1 − q u ) ,(35)Pr Φ k,n = u:R u ∈Φ k, n q u u:R u Φ k, n (1 − q u ) .(36)
The third probability term in (18) is independent of C and L, and only depends on the energy threshold of the relays. In this sense, the third probability term in (18) remains the same as the finite-capacity case, which has been derived in (23). We thus can express the upper bound of system outage probability for the proposed ETMRS scheme corresponding to infinite battery capacity as (37) on top of the next page.
When the special i.i.d. channel fading case is considered, similar to the above analysis, the system outage probability is upper bounded by (38) on top of the next page.
where q is the probability of each relay falling in the decoding set when the channels are i.i.d. and it can be easily derived based on (34) and expressed as
q = 1 1 1−Pr{ϕ=0} + 2α+2β(1−Pr{ϕ=0}) η Pλ S R (1−Pr{ϕ=0}) .(39)
Remark 1: By substituting (13) into (34), we can see that the probability for the u-th relay falling in the decoding set (i.e., q u ) is proportional to the term Pλ SR u . This is understandable since as the value of Pλ SR u increases, the u-th relay harvests more energy on average and is more likely to decode the source signal successfully in the first hop. Moreover, as expected, the value of q u is inversely proportional to that of the energy threshold β u . Furthermore, using the above performance upper bounds of system outage probability, we are able to judge whether a certain value of battery capacity C is sufficiently large for a given network setup via numerical results as shown in next section.
P ub out ≈ u:R u ∈Λ (1 − q u ) + N k=1 ( N k ) n=1 u:R u ∈Φ k, n q u u:R u Φ k, n (1 − q u ) 1 − exp(−a k,n v) k−1 i=0 a k,n v i i! . (37) ⌢ P ub out ≈ (1 − q) N + N k=1 N k q k (1 − q) N−k 1 − exp(− N 0 v 4k βσ 2 ) k−1 i=0 N 0 v 4kβσ 2 i i! ,(38)
V. SIMULATIONS AND DISCUSSIONS
In this section, we present some simulation and numerical results to validate and illustrate the above theoretical analysis. In order to capture the effect of path-loss, we use the model that λ XY = 10 −3 1+d ω XY , where λ XY is the average channel power gain between node X and Y, d XY denotes the distance between node X and Y, and ω ∈ [2,5] is the path-loss factor. Note that a 30 dB average signal power attenuation is assumed at a reference distance of 1 meter (m) in the above channel model [19]. For simplicity, we consider a linear topology that the relays are located on a straight line between the source and destination and denote by d SR d SR d SR = d SR 1 , d SR 2 , · · · , d SR N the set of distances between source and all relays. We use χ χ χ = { χ 1 , χ 2 , · · · , χ N } to represent the energy threshold set for all the relays. In all the following simulations, we set the distance between the source and destination d SD = 20m, the path-loss factor ω = 3, the severity parameter m u = 2, ∀u, the noise power N 0 = −90dBm, and the energy conversion efficiency η = 0.5.
We first compare the analytical system outage probability with its Monte Carlo simulation, which corresponds to the case that the charging of the relay batteries is continuous (i.e., L → ∞). To this end, we plot the system outage probability of our ETMRS scheme versus the source transmit power for different battery levels L in Fig. 2. We can see that the derived analytical expression of outage probability approaches the corresponding Monte Carlo simulation result as the discrete battery level L increases. Specifically, when L = 200, the analytical expression coincides well with the simulation result, which verifies the effectiveness of the adopted MC model and the correctness of our theoretical analysis presented in Sec. II-IV. Moreover, as expected, the performance of the ETMRS scheme with finite-capacity battery is bounded by the one with infinite-capacity battery. As the analytical results agree well with the simulation results and for the purpose of simplicity, in the following, we will only plot the analytical results of the ETMRS scheme.
Recall that we derived system performance upper bound in Section IV, which can be used to judge whether a certain value of C is sufficiently large via numerical results. To show this, we now plot the outage probability ratio of the proposed ETMRS scheme with finite battery capacity over its upper bound with infinite battery capacity versus relay battery capacity C in Fig. 3. For simplicity, we consider the special i.i.d. case where the relays are located in a cluster. The outage probability ratio is formally defined as
⌢ P ub out / ⌢ P out ∈ (0, 1],
where ⌢ P ub out derived in (38) is the system outage probability for infinite-capacity battery case and ⌢ P out given in (24) is the system outage probability for finite-capacity battery case. From Fig. 3, we can first observe that the outage probability ratio monotonically increases as the battery capacity C grows and gradually converges to 1 when the value of C is large enough. This indicates that the performance gap between the finite-capacity battery and infinite-capacity battery decays to 0 as C increases. However, the convergence speed varies for different network setups. Specifically, from Fig. 3(a), we can see that the outage probability ratio converges to 1 slower with either higher source transmit power or shorter distances between source and relays. This can be understood as follows. The amount of harvested energy at relays increases when the source transmit power increases or the distances between source and relays reduce. In this case, a larger battery capacity is required to avoid the potential energy overflow (i.e., the battery cannot be charged when it is full). From Fig. 3(b), we find out that as the number of relays N increases, the convergence speed is also reduced. This is understandable since energy overflow happens in a higher probability as N grows. Similarly, larger value of C is required to make the outage probability ratio close to 1. We can now summarize that larger capacity batteries should be equipped at energy harvesting relays for those network setups with higher source transmit power, shorter distances between source and relays or larger number of relays.
In Fig. 4, we plot the outage probability versus the energy threshold of the i.i.d. channel fading case for different source transmit power and relay location. Recall that the energy threshold χ is discretized to one of the L + 1 energy levels of the battery excluding the empty level. Thus, the outage probability is plotted in stair curve in the figure. First of all, Fig. 4 demonstrates that there exists an optimal energy threshold that minimizes the outage probability in all considered cases, which validates our deduction in Remark 1. Moreover, we can see from Fig. 4(a) that the higher the transmit power at the source, the larger the value of optimal energy threshold. This is because the relays can harvest more energy when the transmit power of the source increases and thus a larger energy threshold can be supported. We can also see from Fig. 4(a) that a smaller energy threshold should be chosen when the relays are located away from the source. This can be explained as two folds. Firstly, the harvested energy at the relays is limited when they are far away from the source. Secondly, the second hop channel condition becomes better when the relays are away from the source (i.e., close to the destination) and a small energy threshold is enough to avoid system outage. The number of the relays N also affects the optimal energy threshold. From Fig. 4(b) we can see that as the number N increases, the optimal energy threshold decreases. This is understandable since in cooperative relay networks, a decoding set with more relays can achieve the same outage probability with less transmit power. Note that similar results can be observed for the general i.n.i.d case, which are not provided here due to space limitation..
We then compare the system performance of the proposed ETMRS scheme with optimized energy threshold and the common energy threshold scheme in Fig. 5. The optimal energy threshold of the ETMRS is obtained by the exhaustive search based approach and the proposed heuristic one. As the optimal energy threshold for the i.i.d. case can be easily obtained via a one-dimension exhaustive search, we only consider the general i.n.i.d. case where the optimal energy thresholds for the relays could be different. In Fig. 5, we plot the system outage probability for these two approaches versus the source transmit power for two different relay topologies. The outage probability of exhaustive search approach is obtained via a N-dimension exhaustive search from all the possible combinations of energy thresholds. Due to the intense computation complexity mentioned before, we consider the case that each relay only has 20 energy levels. For the proposed heuristic approach, the computation complexity can be dramatically reduced and it is thus particularly suitable to those networks with large number of relays and energy levels. From Fig. 5, we can observe that the proposed heuristic approach with reduced complexity can achieve the near optimal system performance in both simulated scenarios. We also can see that our proposed scheme outperforms the common-energy strategy for all simulated cases. For the case that the relays are located closely, our proposed scheme and the one in [28] performs similarly. When the relays are located differently, our proposed scheme improves the performance significantly by using different energy thresholds for each relay.
Finally, we compare the proposed ETMRS scheme with the existing BARS scheme [31] in Fig. 6. As the BARS scheme proposed in [31] only considers a clustered topology, we thus compared BARS scheme with the special i.i.d. case of the proposed ETMRS scheme. In Fig. 6, we plot the optimal outage probabilities of the ETMRS and BARS schemes versus source transmit power for different transmission rate. It can be observed that the proposed ETMRS scheme can achieve a lower outage probability than the BARS scheme. And their performance gap is enlarged as the transmission rate grows.
VI. CONCLUSIONS
In this paper, we proposed an energy threshold based multirelay selection (ETMRS) scheme for accumulate-then-forward energy harvesting relay networks. We modeled the finitecapacity battery of the relays by a finite-state Markov Chain (MC) in order to evaluate their stationary distribution. We then derived an approximate analytical expression for system outage probability of the proposed ETMRS scheme over independent but not necessarily identical mixed Nakagami-m and Rayleigh fading channels. A heuristic approach was then designed to minimize the system outage probability, which was shown to achieve near-optimal performance with reduced computational complexity. Moreover, we derived upper bounds for the concerned system performance corresponding to the case that all relays are equipped with infinite-capacity batteries. Numerical simulations validated the accuracy of the analytical results, demonstrated the impact of various system parameters and provided some insights of practical relay battery design. Numerical results showed that larger capacity battery should be equipped at energy harvesting relays for those network setups with higher transmit power, shorter distance in the first hop or larger number of relays. Furthermore, the proposed ETMRS scheme can considerably outperform the existing single relay selection scheme and the common energy threshold scheme.
Fig. 1 .
1The considered WPCN with one source-destination pair and N wireless-powered relays.
the last equality in (22) holds according to [32, Eq. (8.352.6)] with integer k. Based on the
FiniteFig. 2 .
2−Capacity Battery, Analytical Eq. (18) Infinite−Capacity Battery, Analytical Eq. The outage probability of the proposed ETMRS scheme versus the source transmit power for different battery levels L, where transmission bit rate κ = 1, C = 2 × 10 −5 , α = 10 −7 , d S R d S R d S R = {5, 5.5, 6, 6, 6, 6, 6.5, 7} and χ χ χ = {3, 3, 3, 3, 3, 3, 4, 4} × 10 −6 .
of N and d S Ru
Fig. 3 .
3The ratio between the upper bound outage probability of the ETMRS scheme with infinite battery capacity and the outage probability with finite battery capacity versus battery capacity C for different source transmit power and relay topologies, where κ = 1, α = 10 −7 , χ u = 4×10 −6 , ∀u and L = 600.
of N and d S Ru
Fig. 4 .
4The outage probability of the proposed ETMRS scheme versus the identical energy threshold for different source transmit power, relay topologies where κ = 1, C = 2 × 10 −5 , α = 10 −7 and L = 200.
Fig. 5 .
5Outage probability of the proposed ETMRS scheme and the common energy threshold scheme for different relay locations, where κ = 1, C = 2 × 10 −5 , α = 10 −7 and L = 20.
Fig. 6 .
6The outage probability of the proposed ETMRS scheme and the existing BARS scheme with optimal settings versus source transmit power for different transmission bit rate, where N = 8, d S Ru = 3, ∀u, C = 2 × 10 −5 , α = 10 −7 and L = 200.
WPC), which has Y. Gu, H. Chen, Y. Li, Y.-C. Liang and B. Vucetic are with School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW 2006, Australia (email: [email protected], [email protected], [email protected], [email protected], [email protected]). The material in this paper was presented in part at the IEEE International Conference on Communications, Kuala Lumpur, Malaysia, in May 2016.
TABLE I THE
IRELATION BETWEEN φ k, n, j AND
Throughout this paper, we use the terms "wireless-powered" and "energy harvesting" interchangeably.
For simplicity, we consider that the relays out of the decoding set will not harvest energy in the second time slot since the amount of energy harvested from the forwarded signals would be negligible compared with the energy harvested from the source.
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| []
|
[
"Efficient probabilistic top-down and left-corner parsingt",
"Efficient probabilistic top-down and left-corner parsingt"
]
| [
"Brian Roark brian-roark@brown \nCognitive and Linguistic Sciences\nBrown University Providence\nBox 197802912RIUSA\n",
"Mark Johnson \nCognitive and Linguistic Sciences\nBrown University Providence\nBox 197802912RIUSA\n"
]
| [
"Cognitive and Linguistic Sciences\nBrown University Providence\nBox 197802912RIUSA",
"Cognitive and Linguistic Sciences\nBrown University Providence\nBox 197802912RIUSA"
]
| []
| This paper examines efficient predictive broadcoverage parsing without dynamic programming.In contrast to bottom-up methods, depth-first top-down parsing produces partial parses that are fully connected trees spanning the entire left context, from which any kind of non-local dependency or partial semantic interpretation can in principle be read. We contrast two predictive parsing approaches, topdown and left-corner parsing, and find both to be viable. In addition, we find that enhancement with non-local information not only improves parser accuracy, but also substantially improves the search efficiency. | 10.3115/1034678.1034743 | null | 6,217,983 | cs/0008017 | ffb0f378a5e2126ce2f139df5d0a85ff75e786f8 |
Efficient probabilistic top-down and left-corner parsingt
Brian Roark brian-roark@brown
Cognitive and Linguistic Sciences
Brown University Providence
Box 197802912RIUSA
Mark Johnson
Cognitive and Linguistic Sciences
Brown University Providence
Box 197802912RIUSA
Efficient probabilistic top-down and left-corner parsingt
This paper examines efficient predictive broadcoverage parsing without dynamic programming.In contrast to bottom-up methods, depth-first top-down parsing produces partial parses that are fully connected trees spanning the entire left context, from which any kind of non-local dependency or partial semantic interpretation can in principle be read. We contrast two predictive parsing approaches, topdown and left-corner parsing, and find both to be viable. In addition, we find that enhancement with non-local information not only improves parser accuracy, but also substantially improves the search efficiency.
Introduction
Strong empirical evidence has been presented over the past 15 years indicating that the human sentence processing mechanism makes online use of contextual information in the preceding discourse (Crain and Steedman, 1985;Altmann and Steedman, 1988;Britt, 1994) and in the visual environment (Tanenhaus et al., 1995). These results lend support to Mark Steedman's (1989) "intuition" that sentence interpretation takes place incrementally, and that partial interpretations are being built while the sentence is being perceived. This is a very commonly held view among psycholinguists today.
Many possible models of human sentence processing can be made consistent with the above view, but the general assumption that must underlie them all is that explicit relationships between lexical items in the sentence must be specified incrementally. Such a processing mecha-tThis material is based on work supported by the National Science Foundation under Grant No. SBR-9720368. nism stands in marked contrast to dynamic programming parsers, which delay construction of a constituent until all of its sub-constituents have been completed, and whose partial parses thus consist of disconnected tree fragments. For example, such parsers do not integrate a main verb into the same tree structure as its subject NP until the VP has been completely parsed, and in many cases this is the final step of the entire parsing process. Without explicit on-line integration, it would be difficult (though not impossible) to produce partial interpretations on-line. Similarly, it may be difficult to use non-local statistical dependencies (e.g. between subject and main verb) to actively guide such parsers.
Our predictive parser does not use dynamic programming, but rather maintains fully connected trees spanning the entire left context, which make explicit the relationships between constituents required for partial interpretation. The parser uses probabilistic best-first parsing methods to pursue the most likely analyses first, and a beam-search to avoid the nontermination problems typical of non-statistical top-down predictive parsers.
There are two main results. First, this approach works and, with appropriate attention to specific algorithmic details, is surprisingly efficient. Second, not just accuracy but also efficiency improves as the language model is made more accurate. This bodes well for future research into the use of other non-local (e.g. lexical and semantic) information to guide the parser.
In addition, we show that the improvement in accuracy associated with left-corner parsing over top-down is attributable to the non-local information supplied by the strategy, and can thus be obtained through other methods that utilize that same information.
Parser architecture
The parser proceeds incrementally from left to right, with one item of look-ahead. Nodes are expanded in a standard top-down, left-to-right fashion. The parser utilizes: (i) a probabilistic context-free grammar (PCFG), induced via standard relative frequency estimation from a corpus of parse trees; and (ii) look-ahead probabilities as described below. Multiple competing partial parses (or analyses) are held on a priority queue, which we will call the pending heap. They are ranked by a figure of merit (FOM), which will be discussed below. Each analysis has its own stack of nodes to be expanded, as well as a history, probability, and FOM. The highest ranked analysis is popped from the pending heap, and the category at the top of its stack is expanded. A category is expanded using every rule which could eventually reach the look-ahead terminal. For every such rule expansion, a new analysis is created 1 and pushed back onto the pending heap.
The FOM for an analysis is the product of the probabilities of all PCFG rules used in its derivation and what we call its look-ahead probability (LAP). The LAP approximates the product of the probabilities of the rules that will be required to link the analysis in its current state with the look-ahead terminal 2. That is, for a grammar G, a stack state [C1 ... C,] and a lookahead terminal item w:
(1) We recursively estimate this with two empirically observed conditional probabilities for every non-terminal Ci on the stack: /~(Ci 2+ w) and/~(Ci -~ e). The LAP approximation for a given stack state and look-ahead terminal is:
(2) PG([Ci . .. Ca] wot) P(Ci w) +
When the topmost stack category of an analysis matches the look-ahead terminal, the terminal is popped from the stack and the analysis 1We count each of these as a parser state (or rule expansion) considered, which can be used as a measure of efficiency.
2Since this is a non-lexicalized grammar, we are taking pre-terminal POS markers as our terminal items. is pushed onto a second priority queue, which we will call the success heap. Once there are "enough" analyses on the success heap, all those remaining on the pending heap are discarded. The success heap then becomes the pending heap, and the look-ahead is moved forward to the next item in the input string. When the end of the input string is reached, the analysis with the highest probability and an empty stack is returned as the parse. If no such parse is found, an error is returned.
The specifics of the beam-search dictate how many analyses on the success heap constitute "enough". One approach is to set a constant beam width, e.g. 10,000 analyses on the success heap, at which point the parser moves to the next item in the input. A problem with this approach is that parses towards the bottom of the success heap may be so unlikely relative to those at the top that they have little or no chance of becoming the most likely parse at the end of the day, causing wasted effort. An alternative approach is to dynamically vary the beam width by stipulating a factor, say 10 -5, and proceed until the best analysis on the pending heap has an FOM less than 10 -5 times the probability of the best analysis on the success heap. Sometimes, however, the number of analyses that fall within such a range can be enormous, creating nearly as large of a processing burden as the first approach. As a compromise between these two approaches, we stipulated a base beam factor a (usually 10-4), and the actual beam factor used was a •/~, where/3 is the number of analyses on the success heap. Thus, when f~ is small, the beam stays relatively wide, to include as many analyses as possible; but as /3 grows, the beam narrows. We found this to be a simple and successful compromise.
Of course, with a left recursive grammar, such a top-down parser may never terminate. If no analysis ever makes it to the success heap, then, however one defines the beam-search, a top-down depth-first search with a left-recursive grammar will never terminate. To avoid this, one must place an upper bound on the number of analyses allowed to be pushed onto the pending heap. If that bound is exceeded, the parse fails. With a left-corner strategy, which is not prey to left recursion, no such upper bound is necessary. Nijholt (1980) characterized parsing strategies in terms of announce points: the point at which a parent category is announced (identified) relative to its children, and the point at which the rule expanding the parent is identified. In pure top-down parsing, a parent category and the rule expanding it are announced before any of its children. In pure bottom-up parsing, they are identified after all of the children. Grammar transforms are one method for changing the announce points. In top-down parsing with an appropriately binaxized grammar, the paxent is identified before, but the rule expanding the parent after, all of the children. Left-corner parsers announce a parent category and its expanding rule after its leftmost child has been completed, but before any of the other children.
Delaying rule identification through binarization
Suppose that the category on the top of the stack is an NP and there is a determiner (DT) in the look-ahead. In such a situation, there is no information to distinguish between the rules NP ~ DT JJ NN andNP--+DT JJ NNS.
If the decision can be delayed, however, until such a time as the relevant pre-terminal is in the look-ahead, the parser can make a more informed decision. Grammar binaxization is one way to do this, by allowing the parser to use a rule like NP --+ DT NP-DT, where the new non-terminal NP-DT can expand into anything that follows a DT in an NP. The expansion of NP-DT occurs only after the next pre-terminal is in the look-ahead. Such a delay is essential for an efficient implementation of the kind of incremental parser that we are proposing.
There axe actually several ways to make a grammar binary, some of which are better than others for our parser. The first distinction that can be drawn is between what we will call left binaxization (LB) versus right binaxization (RB, see figure 1). In the former, the leftmost items on the righthand-side of each rule are grouped together; in the latter, the rightmost items on the righthand-side of the rule are grouped together. Notice that, for a top-down, left-to-right parser, RB is the appropriate transform, because it underspecifies the right siblings. With LB, a top-down parser must identify all of the siblings before reaching the leftmost item, which does not aid our purposes.
Within RB transforms, however, there is some variation, with respect to how long rule underspecification is maintained. One method is to have the final underspecified category rewrite as a binary rule (hereafter RB2, see figure lb). Another is to have the final underspecified category rewrite as a unary rule (RB1, figure lc). The last is to have the final underspecified category rewrite as a nullaxy rule (RB0, figure ld). Notice that the original motivation for RB, to delay specification until the relevant items are present in the look-ahead, is not served by RB2, because the second child must be specified without being present in the look-ahead. RB0 pushes the lookahead out to the first item in the string after the constituent being expanded, which can be useful in deciding between rules of unequal length, e.g. NP---+ DT NN and NP ~ DT NN NN. , 1993), and tested on section 23. For each transform tested, every tree in the training corpus was transformed before grammar induction, resulting in a transformed PCFG and lookahead probabilities estimated in the standard way. Each parse returned by the parser was detransformed for evaluation 3. The parser used in each trial was identical, with a base beam factor c~ = 10 -4. The performance is evaluated using these measures: (i) the percentage of candidate sentences for which a parse was found (coverage); (ii) the average number of states (i.e. rule expansions) considered per candidate sentence (efficiency); and (iii) the average labelled precision and recall of those sentences for which a parse was found (accuracy). We also used the same grammars with an exhaustive, bottom-up CKY parser, to ascertain both the accuracy and probability of the maximum likelihood parse (MLP). We can then additionally compare the parser's performance to the MLP's on those same sentences.
As expected, left binarization conferred no benefit to our parser. Right binarization, in contrast, improved performance across the board. RB0 provided a substantial improvement in coverage and accuracy over RB1, with something of a decrease in efficiency. This efficiency hit is partly attributable to the fact that the same tree has more nodes with RB0. Indeed, the efficiency improvement with right binarization over the standard grammar is even more interesting in light of the great increase in the size of the grammars.
3See Johnson (1998) for details of the transform/detransform paradigm.
It is worth noting at this point that, with the RB0 grammar, this parser is now a viable broadcoverage statistical parser, with good coverage, accuracy, and efficiency 4. Next we considered the left-corner parsing strategy.
3.2
Left-corner parsing
Left-corner (LC) parsing (Rosenkrantz and Lewis II, 1970) is a well-known strategy that uses both bottom-up evidence (from the left corner of a rule) and top-down prediction (of the rest of the rule). Rosenkrantz and Lewis
showed how to transform a context-free grammar into a grammar that, when used by a topdown parser, follows the same search path as an LC parser. These LC grammars allow us to use exactly the same predictive parser to evaluate top-down versus LC parsing. Naturally, an LC grammar performs best with our parser when right binarized, for the same reasons outlined above. We use transform composition to apply first one transform, then another to the output of the first. We denote this A o B where (A o B) (t) = B (A (t)). After applying the left-corner transform, we then binarize the resulting grammar 5, i.e. LC o RB. Another probabilistic LC parser investigated (Manning and Carpenter, 1997), which utilized an LC parsing architecture (not a transformed grammar), also got a performance boost 4The very efficient bottom-up statistical parser detailed in Charniak et al. (1998) measured efficiency in terms of total edges popped. An edge (or, in our case, a parser state) is considered when a probability is calculated for it, and we felt that this was a better efficiency measure than simply those popped. As a baseline, their parser considered an average of 2216 edges per sentence in section 22 of the WSJ corpus (p.c.).
5Given that the LC transform involves nullary productions, the use of RB0 is not needed, i.e. nullary productions need only be introduced from one source. Thus binarization with left corner is always to unary (RB1). Table 2 shows left-corner results over various conditions 6. Interestingly, options (a) and (d) encode the same information, leading to nearly identical performance 7. As stated before, right binarization moves the rule announce point from before to after all of the children. The LC transform is such that LC o RB also delays parent identification until after all of the children. The transform LC o RB o ANN moves the parent announce point back to the left corner by introducing unary rules at the left corner that simply identify the parent of the binarized rule. This allows us to test the effect of the position of the parent announce point on the performance of the parser. As we can see, however, the effect is slight, with similar performance on all measures.
(a) LB o LC (b) RB o LC (c) LC o LB (d) LC o RB
RB o LC performs with higher accuracy than the others when used with an exhaustive parser, but seems to require a massive beam in order to even approach performance at the MLP level. Manning and Carpenter (1997) used a beam width of 40,000 parses on the success heap at each input item, which must have resulted in an order of magnitude more rule expansions than what we have been considering up to now, and 6Option (c) is not the appropriate kind of binarization for our parser, as argued in the previous section, and so is omitted. 7The difference is due to the introduction of vacuous unary rules with RB. yet their average labelled precision and recall (.7875) still fell well below what we found to be the MLP accuracy (.7987) for the grammar. We are still investigating why this grammar functions so poorly when used by an incremental parser.
Non-local annotation
Johnson (1998) discusses the improvement of PCFG models via the annotation of non-local information onto non-terminal nodes in the trees of the training corpus. One simple example is to copy the parent node onto every nonterminal, e.g. the rule S ~ NP VP becomes S ~ NP~S VP~S. The idea here is that the distribution of rules of expansion of a particular non-terminal may differ depending on the nonterminal's parent. Indeed, it was shown that this additional information improves the MLP accuracy dramatically.
We looked at two kinds of non-local information annotation: parent (PA) and left-corner (LCA). Left-corner parsing gives improved accuracy over top-down or bottom-up parsing with the same grammar. Why? One reason may be that the ancestor category exerts the same kind of non-local influence upon the parser that the parent category does in parent annotation. To test this, we annotated the left-corner ancestor category onto every leftmost non-terminal category. The results of our annotation trials are shown in table 3.
There are two important points to notice from these results. First, with PA we get not only the previously reported improvement in accuracy, but additionally a fairly dramatic decrease in the number of parser states that must be visited to find a parse. That is, the non-local information not only improves the final product of the parse, but it guides the parser more quickly to the final product. The annotated grammar has 1.5 times as many rules, and would slow a bottom-up CKY parser proportionally. Yet our parser actually considers far fewer states en route to the more accurate parse. Second, LC-annotation gives nearly all of the accuracy gain of left-corner parsing s, in support of the hypothesis that the ancestor information was responsible for the observed accuracy improvement. This result suggests that if we can determine the information that is being annotated by the troublesome RB o LC transform, we may be able to get the accuracy improvement with a relatively narrow beam. Parentannotation before the LC transform gave us the best performance of all, with very few states considered on average, and excellent accuracy for a non-lexicalized grammar.
4
Accuracy/Efficiency tradeoff One point that deserves to be made is that there is something of an accuracy/efficiency tradeoff with regards to the base beam factor. The results given so far were at 10 -4 , which functions pretty well for the transforms we have investigated. Figures 2 and 3 show four performance measures for four of our transforms at base beam factors of 10 -3 , 10 -4 , 10 -5 , and 10 -6. There is a dramatically increasing efficiency burden as the beam widens, with varying degrees of payoff. With the top-down transforms (RB0 and PA o RB0), the ratio of the average probability to the MLP probability does improve substantially as the beam grows, yet with only marginal improvements in coverage and accuracy. Increasing the beam seems to do less with the left-corner transforms.
SThe rest could very well be within noise.
Conclusions and Future Research
We have examined several probabilistic predictive parser variations, and have shown the approach in general to be a viable one, both in terms of the quality of the parses, and the efficiency with which they are found. We have shown that the improvement of the grammars with non-local information not only results in better parses, but guides the parser to them much more efficiently, in contrast to dynamic programming methods. Finally, we have shown that the accuracy improvement that has been demonstrated with left-corner approaches can be attributed to the non-local information utilized by the method. This is relevant to the study of the human sentence processing mechanism insofar as it demonstrates that it is possible to have a model which makes explicit the syntactic relationships between items in the input incrementally, while still scaling up to broad-coverage.
Future research will include: • lexicalization of the parser • utilization of fully connected trees for additional syntactic and semantic processing • the use of syntactic predictions in the beam for language modeling • an examination of predictive parsing with a left-branching language (e.g. German) In addition, it may be of interest to the psycholinguistic community if we introduce a time variable into our model, and use it to compare such competing sentence processing models as race-based and competition-based parsing.
Figure 1 :
1Binaxized trees: (a) left binaxized (LB); (b) right binaxized to binary (RB2); (c) right binaxized to unary (RB1); (d) right binarized to nullaxy (
.Figure 2 :Figure 3 :
23.............. PAoLCoRB ~,~"~,.....,.,"'~, . ...... Changes in performance with beam factor variation M. Britt. 1994. The interaction of referential ambiguity and argument structure. Journal o/ Memory and Language, 33:251-283. E. Charniak, S. Goldwater, and M. Johnson. 1998. Edge-based best-first chart parsing. In Proceedings of the Sixth Workshop on Very Large Corpora, pages 127-133. S. Crain and M. Steedman. 1985. On not being led up the garden path: The use of context by the psychological parser. In D. Dowty, L. Karttunen, and A. Zwicky, editors, Natural Language Parsing. Cambridge University Press, Cambridge, UK. M. Johnson. 1998. PCFG models of linguistic tree representations. Computational Linguistics, 24:617-636. C. Manning and B. Carpenter. 1997. Probabilistic parsing using left corner language models. In Proceedings of the Fifth International Workshop on Parsing Technologies. Changes in performance with beam factor variation
Table 1
1summarizes some trials demonstrat-Binarization Rules in Percent of Avg. States Avg. Labelled Avg. MLP Ratio of Avg.Grammar
Sentences
Considered
Precision and Labelled
Prob to Avg.
Parsed*
Recall t
Prec/Rec t
MLP Prob t
None
14962
34.16
19270
.65521
.76427
.001721
LB
37955
33.99
96813
.65539
.76095
.001440
I~B1
29851
91.27
10140
.71616
.72712
.340858
RB0
41084
97.37
13868
.73207
.72327
.443705
Beam Factor = 10 -4
*Length ~ 40 (2245 sentences in F23 Avg. length --21.68)
tof those sentences parsed
Table 1 :
1The effect of different approaches to binarizationing the effect of different binarization ap-
proaches on parser performance. The gram-
mars were induced from sections 2-21 of the
Penn Wall St. Journal Treebank (Marcus et
al.
Table 2 :
2Left Corner Results through right binarization. This, however, is equivalent to RB o LC, which is a very different grammar from LC o RB. Given our two binarization orientations (LB and RB), there are four possible compositions of binarization and LC transforms:
Table 3 :
3Non-local annotation results
M.P. Marcus, B. Santorini, and M.A. Marcinkiewicz. 1993. Building a large annotated corpus of English: The Penn Treebank. Computational Linguistics, 19(2):313-330. A. Nijholt. 1980. Context-/tee Grammars: Covers, Normal Forms, and Parsing. Springer Verlag, Berlin. S.J. Rosenkrantz and P.M. Lewis II. 1970. Deterministic left corner parsing. In IEEE Conference Record of the 11th Annual Symposium on Switching and Automata, pages 139-152. M. Steedman. 1989. Grammar, interpretation, and processing from the lexicon. In W. Marslen-Wilson, editor, Lexical representation and process. MIT Press, Cambridge, MA. M. Tanenhaus, M. Spivey-Knowlton, K. Eberhard, and J. Sedivy. 1995. Integration of visual and linguistic information during spoken language comprehension. Science, 268:1632-1634.
Interaction with context during human sentence processing. G Altmann, M Steedman, Cognition. 30G. Altmann and M. Steedman. 1988. Interac- tion with context during human sentence pro- cessing. Cognition, 30:198-238.
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"ANALYSIS OF RING LASER GYROSCOPES INCLUDING LASER DYNAMICS A PREPRINT"
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"Virgilio \nINFN Sez. di Pisa\nPolo Fibonacci, Largo B Pontecorvo 3I-56127PisaItaly\n",
"Nicolò Beverini \nDipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly\n",
"Giorgio Carelli \nDipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly\n",
"Donatella Ciampini \nDipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly\n",
"Francesco Fuso \nDipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly\n",
"Enrico Maccioni \nDipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly\n"
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"INFN Sez. di Pisa\nPolo Fibonacci, Largo B Pontecorvo 3I-56127PisaItaly",
"INFN Sez. di Pisa\nPolo Fibonacci, Largo B Pontecorvo 3I-56127PisaItaly",
"Dipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly",
"Dipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly",
"Dipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly",
"Dipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly",
"Dipartimento di Fisica \"E. Fermi\"\nUniversità di Pisa\nLargo B Pontecorvo 3, I56127PisaItaly"
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| Inertial sensors stimulate very large interest, not only for their application but also for fundamental physics tests. Ring laser gyros, which measure angular rotation rate, are certainly among the most sensitive inertial sensors, with excellent dynamic range and bandwidth. Large area ring laser gyros are routinely able to measure fractions of prad/s, with high duty cycle and bandwidth, providing fast, direct and local measurement of relevant geodetic and geophysical signals. Improvements of a factor 10 − 100 would open the windows for general relativity tests, as the GINGER project, an Earth based experiment aiming at the Lense-Thirring test at 1% level. However, it is well known that the dynamics of the laser induces non-linearities, and those effects are more evident in small scale instruments. Sensitivity and accuracy improvements are always worthwhile, and in general there is demand for high sensitivity environmental study and development of inertial platforms, where small scale transportable instruments should be used. We discuss a novel technique to analyse the data, aiming at studying and removing those non-linearity. The analysis is applied to the two ring laser prototypes GP2 and GINGERINO, and angular rotation rate evaluated with the new and standard methods are compared. The improvement is evident, it shows that the back-scatter problem of the ring laser gyros is negligible with a proper analysis of the data, improving the performances of large scale ring laser gyros, but also indicating that small scale instruments with sensitivity of nrad/s are feasible. | 10.1140/epjc/s10052-019-7089-5 | [
"https://arxiv.org/pdf/1904.02533v1.pdf"
]
| 102,353,916 | 1904.02533 | 0709fb1faeb3df4c6531f980b5004f8e3c0ec3eb |
ANALYSIS OF RING LASER GYROSCOPES INCLUDING LASER DYNAMICS A PREPRINT
April 5, 2019 4 Apr 2019
Angela D V Di
INFN Sez. di Pisa
Polo Fibonacci, Largo B Pontecorvo 3I-56127PisaItaly
Virgilio
INFN Sez. di Pisa
Polo Fibonacci, Largo B Pontecorvo 3I-56127PisaItaly
Nicolò Beverini
Dipartimento di Fisica "E. Fermi"
Università di Pisa
Largo B Pontecorvo 3, I56127PisaItaly
Giorgio Carelli
Dipartimento di Fisica "E. Fermi"
Università di Pisa
Largo B Pontecorvo 3, I56127PisaItaly
Donatella Ciampini
Dipartimento di Fisica "E. Fermi"
Università di Pisa
Largo B Pontecorvo 3, I56127PisaItaly
Francesco Fuso
Dipartimento di Fisica "E. Fermi"
Università di Pisa
Largo B Pontecorvo 3, I56127PisaItaly
Enrico Maccioni
Dipartimento di Fisica "E. Fermi"
Università di Pisa
Largo B Pontecorvo 3, I56127PisaItaly
ANALYSIS OF RING LASER GYROSCOPES INCLUDING LASER DYNAMICS A PREPRINT
April 5, 2019 4 Apr 2019A PREPRINT -APRIL 5, 2019
Inertial sensors stimulate very large interest, not only for their application but also for fundamental physics tests. Ring laser gyros, which measure angular rotation rate, are certainly among the most sensitive inertial sensors, with excellent dynamic range and bandwidth. Large area ring laser gyros are routinely able to measure fractions of prad/s, with high duty cycle and bandwidth, providing fast, direct and local measurement of relevant geodetic and geophysical signals. Improvements of a factor 10 − 100 would open the windows for general relativity tests, as the GINGER project, an Earth based experiment aiming at the Lense-Thirring test at 1% level. However, it is well known that the dynamics of the laser induces non-linearities, and those effects are more evident in small scale instruments. Sensitivity and accuracy improvements are always worthwhile, and in general there is demand for high sensitivity environmental study and development of inertial platforms, where small scale transportable instruments should be used. We discuss a novel technique to analyse the data, aiming at studying and removing those non-linearity. The analysis is applied to the two ring laser prototypes GP2 and GINGERINO, and angular rotation rate evaluated with the new and standard methods are compared. The improvement is evident, it shows that the back-scatter problem of the ring laser gyros is negligible with a proper analysis of the data, improving the performances of large scale ring laser gyros, but also indicating that small scale instruments with sensitivity of nrad/s are feasible.
Introduction
Ring laser gyroscopes (RLGs) are inertial sensors based on the Sagnac effect [1,2,3]. They are largely utilised for inertial navigation, and applications in geodesy, geophysics and even for General Relativity tests are foreseen [4]. Since 2011 we are studying the feasibility of the Lense Thirring test at the level of 1% with an array of large frame RLGs [5,6,7]. For that purpose it is necessary to push the relative accuracy of the Earth rotation rate measurement in the range from 1 part in 10 9 up to 1 part in 10 12 . RLG consists of a laser with a cavity comprising of three or four mirrors, depending if the cavity is triangular or square, rigidly attached to a frame; large frame RLGs are utilised to measure the Earth rotation rate, being attached to the Earth crust. Because of the Sagnac effect, the two counterpropagating cavity modes have slightly different frequency, and the beat note of the two beams is proportional to the angular rotation rate felt by the ring cavity. Large frame RLGs are the most sensitive instruments for inertial angular rotation measurements. The Sagnac frequency of a RLG is in fact proportional to the angular rotation rate Ω which affects the apparatus:
f s = SΩ cos θ (1) S = 4 A λL
where A is the area of the ring cavity, L is its perimeter, λ the wavelength of the light, and θ is angle between the area versor of the ring and the orientation of Ω. For RLGs horizontally aligned (area versor vertical) θ is the colatitude angle, while for RLGs aligned at the maximum Sagnac signal θ = 0. Eq. 2 connects Ω with the scale factor S, which depends on the geometry, and λ, quantities than can be measured. Further to sensitivity, other advantages of such instruments rely on the broad bandwidth, which can span from kHz down to DC, and the very large dynamical range. In fact the same device can record microseismic events and high magnitude nearby earthquakes [8], owing to the fact that the signal is based on the measurement of the beat note. It has been proven that large size RLGs with state of the art mirrors can reach the relative precision of 3 parts in 10 9 in one day integration time, for the Earth rotation rate measurement [1]. If shot noise limited, the sensitivity scales with the second power of the size of the ring cavity.
The main limitation of RLG performances is given by the coupling between the two counter propagating laser modes. On each cavity mirror a little fraction of the two traveling waves is backscattered in the opposite direction. As a result of the interference between the reflected waves from each mirror, we have effective backscattering amplitude reflectivity r 1 and r 2 of the beam 1 over the beam 2 and of the beam 2 over the beam 1, respectively. We outline that the interference of the back reflected waves, and consequently the values of r 1,2 are very sensitive to any perturbation of the optical cavity geometry. This coupling produces for small rotational rate a pulling of the Sagnac frequency from the f s value given by eq. 2, and eventually a locking of the two laser frequencies when the f s value become lower than f lock = r 1,2 /π [9]. As a rough estimation, it is possible to evaluate for a square cavity f lock = cµλ πdL , where c is the velocity of light, d the diameter of the beam and µ the total scattered fractional amplitude at each reflection.
In order to ensure the functionality of small scale RLGs [10], mechanical dithering is usually implied to increase the bias between the two modes and avoid locking. Large frame rings utilise the Earth rotation rate as bias. Any improvement in the accurate evaluation of the backscatter noise, and in general of the systematics induced by the non linear dynamics of the lasing process, is advantageous for increasing the performance of both large and small frame RLGs. Presently there is large interest in this kind of device, large scale apparatus should further improve their sensitivity and accuracy for geodetic and fundamental physics application, and small scale and transportable devices at nrad/s sensitivity are required to improve the low frequency response of gravitational wave antennas, for the development of inertial platforms, and for seismology [11,12,13,14,15]. The problem of the reconstruction of signals is a general one, and sophisticated filters can be developed to this aim. In the past we have addressed this problem utilizing Kalman filters, with whom we have obtained good results, but which were rather time consuming. At present, we have the necessity to analyze a very large set of data and to set up mathematical tools for the analysis with the aim not only to evaluate the sensitivity, but also to precisely identify specific issues in the setup which are limiting the sensitivity. This paper presents a mathematical approach to measure the Sagnac frequency taking into account the laser dynamics. This issue has been addressed several times [9,16,17,18,19], but no general solution exists. Analytical solutions can be derived in the case the backscattered light is equal in the two counter propagating modes, or the ratio between the intensities of the two modes is constant, conditions which are not fulfilled in the actual generation of RLGs [17]. The discussion is composed of two main parts. The first one, after a short description of the RLG and the standard analysis approach, describes the general RLG dynamics and reconstructs the Sagnac frequency taking into account the laser dynamics in the general case through a single analytical formula containing the laser coefficients (Lamb coefficients), which can be separately evaluated based on experimental measurements. This formula can be further divided as linear sum of six contributions. One of the contributions, called ω s0 , is the dominant one, the others being small corrections. In the second part of the paper the implementation of ω s0 is discussed and applied to data of the RLG prototypes GP2 and GINGERINO. The other additional terms are not considered in the implementation, they will be subject of future work based on the data of GINGERINO. The appendix reports a short discussion about noise, and practical methods to identify portions of data which have to be discarded. Fig. 1 shows the typical lay-out of a square cavity RLG. The four mirrors are placed at the corner of the square ring, they are contained inside small vacuum chambers, which are connected by pipes. The whole setup is vacuum tight Figure 1: Typical scheme of RLG with a square ring cavity. and filled with Helium and an isotopic 50/50 mixture of 20 Ne and 22 Ne. In one of the side the laser discharge is located to generate the plasma required for laser operation (top side). Piezoelectric translators are utilised to translate the mirrors, allowing a control of the RLG perimeter length. The Sagnac beat note signal is observed at one corner (bottom-left) by superimposing the two output beams on a photodiode. Other two photodiodes monitor at one of the output corners (top-left) the laser output power of the two beams (P H 1 , P H 2 ). We will indicate them in the following as the mono-beam signals. Another photodiode (discharge monitor) records the fluorescence from the discharge, filtered around 633 nm by a narrow width interferometer filter, which provides a rough indication of the density of excited atoms. In the usual operations, the plasma discharge is electronically controlled in order to keep constant the output power of one of the two monobeam signal. All these signals are acquired by an ADC with a frequency rate of a few kHz, suitable to well reconstruct them from DC up to the Sagnac frequency.
Typical RLG and standard analysis method
In the standard analysis, the Sagnac angular frequency ω S is assumed to be equal to the instantaneous frequency ω m , reconstructed from the interferogram by means of the Hilbert transform or of the standard AR2 recursive algorithms based on autocorrelation. The backscatter noise is usually subtracted by fitting the quantity I S1 I S2 P H1P H2 cos 2 , where is the backscatter phase and P H 1,2 and I S1,S2 are the amplitudes of DC and ω S spectral components of mono-beams 1 and 2 , respectively [16].
Ring Laser dynamics, approximations and the stationary solution
The present analysis is dedicated to the evaluation of the Sagnac signal taking into account the dynamics of the ring laser. The most general description of the RLG is based on the model developed by Aronowitz following the more general Lamb theory [20,21]. The general equations arė
I 1 = c L (α 1 I 1 − β 1 I 2 1 − θ 12 I 1 I 2 + 2r 2 I 1 I 2 cos(ψ + ))(2)I 2 = c L (α 2 I 2 − β 2 I 2 2 − θ 21 I 1 I 2 + 2r 1 I 1 I 2 cos(ψ − )) (3) ψ =ω s − σ 1 + σ 2 − τ 12 I 2 + τ 21 I 1 + − c L (r 1 I 1 I 2 sin(ψ − ) + r 2 I 2 I 1 sin(ψ + ))(4)
where I 1 , I 2 are the intra-cavity laser intensities expressed in the dimensionless "Lamb units"; ψ andψ are the instantaneous phase difference and its time derivative; index 1 and 2 refers to the clockwise and counter-clockwise laser beam respectively. It is important to remind that all terms of eq. 2,3, and 4 are time depending. Here α 1,2 , σ 1,2 , β 1,2 , θ 12,21 , τ 12,21 are the Lamb parameters. In particular, α 1,2 , σ 1,2 are the threshold gain minus losses, β 1,2 is the self saturation, θ 12,21 , τ 12,21 describe cross-(mutual-)saturation. The Lamb theory involves a large number of parameters, however, the special mixture of two isotopes of Neon and the working point close to the laser threshold allow adoption of a simplified model [17,18,22]. In our present analysis, we assume β 1 = β 2 = β, and θ 21 = θ 12 = θ. This assumption is justified by the fact that our RLGs operate close to threshold in mono-mode regime (for operation near multi-mode regime, a further approximation is feasible). In the following θ will be neglected, owing to the mono-mode operation.
Without loss of generality we can define δ ns = σ 2 − σ 1 + τ 21 I 2 − τ 12 I 1 , which is usually referred to as null shift; it is generally accepted that δ ns is a small quantity to be neglected [9,17,20]. In the present analysis it will not be neglected: δ ns will be considered a perturbation ofψ, defining a new variableψ 0 ψ − δ ns (ψ being the frequency effectively measured by the interferogram, called also ω m ). Assuming that RLG is at the steady state [17], the solutions are the following:
I 1 (t) α 1 β + 2 √ α 1 α 2 r 2 ( Lωs sin(tωs+ ) c + α 1 cos(tω s + )) β(α 2 1 + L 2 ω 2 s c 2 ) + − 2cr 1 r 2 sin(2 ) βLω s I 2 (t) α 2 β + 2 √ α 1 α 2 r 1 (α 2 cos( − tω s ) − Lωs sin( −tωs) c ) β(α 2 1 + L 2 ω 2 s c 2 ) + + 2cr 2 r 1 sin(2 ) βLω s ψ 0 (t) c( α1 α2 r 1 cos( − tω s ) + α2 α1 r 2 cos(tω s + )) Lω s + + t(ω s − 2r 1 r 2 ( c L ) 2 cos(2 ) ω s )(5)
The validity of the above solutions has been previously tested with a Monte Carlo simulation and with the experimental data of the RLG G-Pisa, which was a 5.40 m perimeter RLG [18,23]. Here the validity of previous results is taken for granted and ω s is analytically expressed.
Assuming that the parameters are constant in the time interval between t and t + δt, we have ψ 0 (t + δt) − ψ 0 (t) = ω m δt−δ ns δt. From the above relation it is straightforward to deduce that, at the first order in δt:
(ω m − δ ns )δt (ω s + K(t) L − 2c 2 r 1 r 2 cos(2 ) L 2 ω s )δt. (6)
In the above equation ω s is the Sagnac angular frequency, the quantity we are looking for, and we have conveniently defined K(t):
K(t) = α 1 α 2 cr 1 sin( − tω s ) − α 2 α 1 cr 2 sin(tω s + ),(7)
K(t) contains oscillatory terms at the Sagnac frequency ω s . Considering that ω s is almost constant, for frequency much below the Sagnac frequency, it is possible to look for approximated solutions. Eq. 6 can be written as:
ω s = ω m 2 + 8c 2 r 1 r 2 cos(2 ) + (K − L(ω m + δ ns )) 2 4L 2 − K 2L + δ ns 2 (8)
where we dropped the time dependence in K. The occurrence of the oscillations of K at the Sagnac frequency makes the evaluation of ω s non trivial. The average value of K is very small for frequencies much below Sagnac frequency, since the average value of sinus and co-sinus oscillating at the Sagnac frequency goes to zero for frequency much below f s . ω s can be found with eq.8, provided that r 1 , r 2 , , δ ns , and the average value of K are available, which is in principle feasible utilising the mono-beam signals and the measured losses of the cavity and employing numerical recursive methods to evaluate K.
In the following eq. 8 will be decomposed in several pieces, which can be separately evaluated. In any case, when |K| Lω m and δ ns ω m , eq. 8 can be expanded in K and δ ns at first and second order, obtaining:
ω s ω s0 + ω ns1 + ω ns2 + ω K1 + ω K2 + ω nsK (9) ω s0 = ( 1 2 8c 2 r 1 r 2 cos(2 ) L 2 + ω 2 m + ω m 2 ) (10) ω ns1 = −δ ns × ( ω m 2 8c 2 r1r2 cos(2 ) L 2 + ω 2 m + 1 2 ) ω ns2 = δ 2 ns × 2c 2 r 1 r 2 cos(2 ) (8c 2 r 1 r 2 cos(2 ) + L 2 ω 2 ) 8c 2 r1r2 cos(2 ) L 2 + ω 2 m ω K1 = K × (− ω m 2L 8c 2 r1r2 cos(2 ) L 2 + ω 2 m − 1 2L ) ω K2 = K 2 × 2c 2 r 1 r 2 cos(2 ) 8c 2 r1r2 cos(2 ) L 2 + ω 2 m (8c 2 r 1 r 2 cos(2 ) + L 2 ω 2 m ) 2 ω nsK = δ ns K 2 √ 8c 2 r 1 r 2 cos 2 + L 2 ω m 2
Eq. 9 is composed of 6 terms, which can be independently evaluated, and analysed. Careful evaluation is necessary for ω ns1,2 , ω K1,2 , since the determination of the parameters β, σ 1 , σ 2 , τ 12 , and τ 21 , which are function of the beam area a, the output power, the mirrors transmission and the total losses µ, is required. The mathematical relationships to evaluate those terms can be found in previous papers [16,17,18]. The reconstruction of those terms will be addressed in future work, and applied to the analysis of the data of GINGERINO.
In the following the implementation of the first term ω s0 will be specialised for data acquired with large frame RLGs and compared with the standard analysis method. Backscatter noise is accounted for, and it has been checked that the standard method to subtract the backscatter noise can be derived from eq. 10 assuming 8c 2 r1r2 cos(2 ) L 2 ω 2 m and expanding at first order.
Application to the actual data
The analysis described in the following will take into account data streams at normal operation and far from transients of the laser as mode jumps and split modes. Appendix A describes methods to identify and eliminate those portions of data. As already said eq. 9 is valid for K Lω m ; referring to our smaller prototype G-Pisa (perimeter 5.40 m), and utilising published parameters [17], we obtain K ∼ 6 rad m/s, to be compared with ω m L ∼ 3600 rad m/s: consequently eq. 9 is valid. We underline that quoted values are conservative ones, since K depends on the mirror quality and the size of the ring. The prototype G-Pisa was smaller than GP2 and GINGERINO and equipped with less performing mirrors. Determining ω s0 requires in turn to evaluate r 1 and r 2 . Following previous works [17,18], it is possible to link such quantities with available measured data:
r 1 = I S2 ω m 2c √ P H1P H2 L(11)r 2 = I S1 ω m 2c √ P H1P H2 L(12)
with all symbols already defined. Similarly, the relative phase is found comparing the mono-beams signal at the Sagnac frequency. All above quantities are commonly used in the standard analysis [16,22,24]. Substituting and simplifying, it is straightforward to show that:
ω s0 = 1 2 (1 + ξ) 2I S1 I S2 ω 2 m cos(2 ) P H 1 P H 2 + ω 2 m + ω m 2(13)
The term ξ (ξ 1) has been added in order to take into account inaccuracies on the mono-beams signals. It is important to remark that the quantities P H 1 , P H 2 , and I S1 and I S2 refer to the laser power inside the optical cavity, while measured ones are obtained utilising the power transmitted outside the cavity. Since Eq. 13 exploits the ratios, in principle it is not affected by the measurement scheme, and the voltage output of the photodiodes can be used. However, it is necessary to consider the presence of noise in the mono-beams signals, which can be due to the inherent noise of the photodiodes or by the discharge fluorescence, which cannot be completely removed. The related noise affects the evaluation of ω s0 done with eq. 13. Therefore, in order to have the possibility to correct it with common statistical methods, the term ξ has been added. Expanding at first order in ξ, we obtain:
ω s0 = 1 2
2ω 2 m I S1 I S2 cos(2 )
I 1 I 2 + ω 2 m + ω m 2 + ω sξ (14) ω sξ = ξ × I S1 I S2 ω 2 m cos(2 ) 2I 1 I 2 2I S1 I S2 ω 2 m cos(2 ) I1I2 + ω 2 m(15)
It is straightforward to evaluate the term ω s0 , while the corrective one, ω sξ , must be evaluated by fitting the parameter ξ. Remarkably, the above relation does not contain any Lamb parameter of the laser and can therefore be determined without knowledge of such parameters.
Reconstruction of ω s0 for GP2 and GINGERINO
Data acquired by our prototypes GINGERINO and GP2 are utilised. GP2 is an apparatus with comparatively low quality mirrors and located in a noisy environment [25,26], while GINGERINO is located in a very quiet place [24,27], and is presently equipped with state of the art mirrors. We remark that GINGERINO is free running, the geometry is not controlled and long time operation and high duty cycle (> 90%) are possible since it is located in the underground Gran Sasso laboratory, which exhibits high natural thermal stability (typically ∼ 0.01 o C in one day). Fig. 2 shows the comparison between the Sagnac frequency from GINGERINO data reconstructed with the standard method (referred to as ω m ) and the one presented here. It is interesting to observe that the average value of the frequency ω s0 is higher, this is what we expect when the noise is dominated by backscatter. In such conditions, frequency is shifted upward ('pull') [9]. The average values are different for the two analysis methods; as far as GINGERINO is concerned, the difference is quite small, for example the analysis of 24 days in November 2018 gives a relative difference of 6 × 10 −5 , with < ω s0 > evaluated by the method presented here a bit larger than < ω m >. Since the absolute orientation of the RLG is unknown, in both cases the measured Sagnac frequency is compatible with the expected one assuming an inclination of ∼ Figure 2: Comparison of the old and new analysis of GIN-GERINO data. Blue trace: standard method with Hilbert transform; red trace: data corrected using eq. 15; green trace: data corrected after fitting for parameter ξ (eq. 15, in the fit ξ = 0.16).
Comparison of standard and new analysis
of phenomena with typical frequency below 20 Hz; we have checked that the two methods are equivalent in the high frequency band of interest. Fig. 3, showing the power spectral density (PSD) as a function of frequency, demonstrates that, for GINGERINO and above 200 mHz, the difference between the two methods is less than 0.1nrad/s in 1 second measurement. This comparison shows that the new analysis is not introducing extra noise above 200 mHz at this level of sensitivity. It has been also checked that the old method, which estimates and subtracts the backscattering effect through a linear fitting procedure, provides results distributed with width similar to ω s0 , and, as already said, slightly different mean value. We outline that systematics of the laser dynamics include non linear terms, which in principle cannot be eliminated with linear methods. Then, the standard method, being linear, cannot guarantee a full correction.
The systematics of RLG depends on the size and the mirror quality, large frame RLGs are usually closer to behave in an ideal manner. For reduced size RLG and when the mirrors are not top quality, deviations from the ideal case are more relevant. This is the case of our prototype GP2. Fig. 4 shows the histogram of the Sagnac frequency data of GP2 analysed with the two methods. The standard analysis leads to a broader distribution and the mean value is compatible with a mean rotation frequency 1 Hz higher than expected. GP2 is oriented at the maximum signal, so its response should be close to (and never higher than) the Earth rotation rate. The new method gives an averaged rotation rate Ω = 7.2922 × 10 −5 rad/s, in agreement with the Earth rotation rate Ω ⊕ = 7.292115 × 10 −5 rad/s. With the new analysis the average rotation rate is evaluated with a relative systematic error of 1 part in 10 −4 , while with 6 part in 10 −3 with the standard analysis: a factor 60 improvement in accuracy has been achieved. The present result is very similar to the one obtained in previous analysis based on the Kalman filter in term of accuracy and sensitivity. In both cases the best sensitivity was of the order of a few nrad/s with tens of seconds integration time [18].
It is puzzling to note that with the standard analysis method GP2 is showing higher than expected Sagnac frequency. A possible explanation is given within our mathematical approach to measure the Sagnac frequency. GP2 has been designed to test the geometry control developed for GIN-GER and based on diagonal length measurements; data with and without geometry control have been compared and it has been checked that, with adequate analysis, sensitivities are comparable [26]. Fig. 5 compares the fringe contrast (TOP) and the relative phase (BOTTOM) for GP2 data taken during the geometry control test. This was achieved by keeping constant the length of the two diagonals within 80 nm [26]. Fig. 5 shows that mode jumps occur in order to keep close to ± π 2 , this in support of the fact that GP2 has quite large backscatter light, and stable operation is favourite when is such that the two modes have an extra shift of 1 Hz, also called "dissipative coupling regime" [9]. It is not perfectly clear from the theory why this regime takes place, but it is a matter of fact that the coupling between the two modes decreases increasing the bias frequency, accordingly stable operation is favourite. Another consequence is that mode jumps occur also to keep the relative phase close to a certain range, not only to compensate changes of the perimeter. It has been checked that GINGERINO exhibits all values of .
Conclusion
Systematics induced by the non linear dynamics of the laser, mainly due to backscatter light, induces non linear terms in the output of high sensitivity RLGs, severely limiting the development of RLGs with sensitivity of the order of nrad/s level, which in principle should have a large range of applications. An analytical method, suitable to reconstruct the Sagnac frequency ω s taking into account the laser dynamics, has been developed in the general case in which the two backscattered beams are not equal, and the ratio between the power of the two counter-propagating modes is not constant. The application of this formula requires the knowledge of the fractions of backscattered waves, and the laser parameters α 1,2 , σ 1,2 , τ and β, all quantities which can be evaluated. In the present theory the term θ is not considered, this term takes into account the multimode operation, and can be neglected in the description of high sensitivity RLGs which operates mono-mode close to threshold. Expanding in series at first and second order it is possible to divide the general formula as the sum of six terms which can be separately evaluated. The analytical expansion for the whole set of 6 terms is reported. The term called ω s0 , which is the dominant one and does not contain any laser parameter, is evaluated in details and expressed as a function of the available measurements; this term has been evaluated for the two RLG prototypes GINGERINO and GP2, and compared with the standard analysis method. The advantage of the new approach is evident: not only the width of the distribution is reduced, but the reconstructed Sagnac frequency is more accurate and in better agreement with the expected value. In short, ω s0 eliminates the so called backscatter noise, which is the dominant systematics especially for small and medium size RLG. The GP2 prototype has more backscatter light, because it is smaller and has lower quality mirrors with respect to GIN-GERINO. In this case the standard method evaluates the Earth rotation rate with a relative systematic error of 6 part in 10 −3 , while in the new way 1 part in 10 −4 is obtained, a factor 60 improvement in accuracy, with a sensitivity in the range of 2 nrad/s with tens of seconds integration time. This work opens up the window for the development of high sensitivity transportable RLGs, for applications in seismology, as environmental monitors to improve the low frequency performance of the test mass suspension of the gravitational wave antennas, and for the development of inertial platforms in general. Further efforts will be devoted to analyse the data of GINGERINO using the full set of terms.
controlled, it happens that the RLG changes its operational points; accordingly the wavelength changes separately for both modes, or for one only, and mode jumps or split mode operations occur. Split mode operation occurs from time to time; in principle the split mode regime provides good measurement of ω s , but in this case ω m = ω s + 2πF SR (FSR, Free Spectral Range), and data acquisition at high rate and accurate knowledge of the perimeter are necessary. In the present analysis data affected by split mode operation have been disregarded. Mode jumps are very fast transients, affecting only few seconds of data acquisition. During these discontinuities the RLG is not at the stationary condition, so portions of data have to be discarded. The observation of the fringe contrast provides a very efficient tool to identify and eliminate those imperfections. Fig. 6 shows corresponding split mode and mode jumps. Sometime some instabilities in the operation are visible before the mode jump takes place. Fig. 7 shows the typical behaviour of the mode jump.
Figure 3 :
3Comparison of the power spectral density utilising the standard (blue data) and new analysis (red data). The two PSD are approximately equal above 200mHz, the main differences are at low frequency.
Figure 4 :
4Comparison of the histograms of the Sagnac frequency estimated with the standard method (blue) and by the new one (red). Clearly the new method leads to a narrower and more Gaussian-like distribution, with mean value 184.29 Hz.
Figure 5 :
5TOP: Fringe contrast during geometry control of GP2, a few features suggesting mode jumps are shown. BOTTOM: the relative phase between the two modes is shown. In correspondence of the jumps there is a rapid change in phase.
Figure 6 :
6TOP: typical fringe contrast, the mode jumps are evident, it is also clear that instabilities occur before the mode jumps, in the middle there is a split mode operation of the duration of 2.6 hours. BOTTOM: the corresponding Sagnac frequency.
Figure 7 :
7GINGERINO Sagnac frequency around a typical mode jump.
A Discussion about the noise Since we deal with high sensitivity measurements, it is important to estimate the noise injected in the evaluation of ω s0 . P H i and I Si (i, 1, 2) are utilised to evaluate ω s0 , and their noise will contribute to the total noise budget. In general all measurements of these quantities are limited by shot noise of the power collected by the photodiode δ i , i = 1, 2, and the total noise is the incoherent sum of the photodiode noise. The contribution of each term, δP H 1 and δI S1 gives:(I 2 and I S2 are obtained changing 1 with 2 in a symmetric way).Usually for top quality mirrors losses are minimised, but there are no requirements for the transmission. In order to minimise the contribution of the mono-beams to the total noise the optimal choice would be to have transmission of the same order of the losses, at least for one of the mirrors (one output only is enough since in order to evaluate it is necessary to observe the two mono-beams transmitted by the same mirror). Care is also necessary in order to avoid small spurious reflections from one of the windows of the vacuum chamber, and narrow band filters are necessary in order to reduce the spurious signal from the discharge fluorescence. In any case, especially the measurement of the two terms P H 1,2 could be a real limitation for the very low frequency measurements, since they are DC quantities affected by the well known 1/f noise of any electronic device.B The fringe contrast: a suitable tool to remove bad portions of dataIn large frame RLG attached to the Earth crust the Sagnac frequency is usually above 100 Hz, and it is determined by the Earth rotation rate, which is almost constant in time.The relative phase is slowly changing, since the cavity is rigid. In general, unless the geometry is electronically
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| []
|
[
"Discrete modelling of capillary mechanisms in multi-phase granular media",
"Discrete modelling of capillary mechanisms in multi-phase granular media"
]
| [
"L Scholtès ",
"B Chareyre ",
"F Nicot ",
"F Darve "
]
| []
| [
"CMES"
]
| A numerical study of multi-phase granular materials based upon micromechanical modelling is proposed. Discrete element simulations are used to investigate capillary induced effects on the friction properties of a granular assembly in the pendular regime. Capillary forces are described at the local scale through the Young-Laplace equation and are superimposed to the standard dry particle interaction usually well simulated through an elastic-plastic relationship. Both effects of the pressure difference between liquid and gas phases and of the surface tension at the interface are integrated into the interaction model. Hydraulic hysteresis is accounted for based on the possible mechanism of formation and breakage of capillary menisci at contacts. In order to upscale the interparticular model, triaxial loading paths are simulated on a granular assembly and the results interpreted through the Mohr-Coulomb criterion. The micro-mechanical approach is validated with a capillary cohesion induced at the macroscopic scale. It is shown that interparticular menisci contribute to the soil resistance by increasing normal forces at contacts. In addition, more than the capillary pressure level or the degree of saturation, our findings highlight the importance of the density number of liquid bonds on the overall behaviour of the material. | 10.3970/cmes.2009.052.297 | [
"https://arxiv.org/pdf/1203.1234v1.pdf"
]
| 119,024,924 | 1203.1234 | deefc626afea27c1a250be25255be926a836b0c0 |
Discrete modelling of capillary mechanisms in multi-phase granular media
2009
L Scholtès
B Chareyre
F Nicot
F Darve
Discrete modelling of capillary mechanisms in multi-phase granular media
CMES
112009Discrete Element Methodmicromechanicscapillaritymulti-phase materials
A numerical study of multi-phase granular materials based upon micromechanical modelling is proposed. Discrete element simulations are used to investigate capillary induced effects on the friction properties of a granular assembly in the pendular regime. Capillary forces are described at the local scale through the Young-Laplace equation and are superimposed to the standard dry particle interaction usually well simulated through an elastic-plastic relationship. Both effects of the pressure difference between liquid and gas phases and of the surface tension at the interface are integrated into the interaction model. Hydraulic hysteresis is accounted for based on the possible mechanism of formation and breakage of capillary menisci at contacts. In order to upscale the interparticular model, triaxial loading paths are simulated on a granular assembly and the results interpreted through the Mohr-Coulomb criterion. The micro-mechanical approach is validated with a capillary cohesion induced at the macroscopic scale. It is shown that interparticular menisci contribute to the soil resistance by increasing normal forces at contacts. In addition, more than the capillary pressure level or the degree of saturation, our findings highlight the importance of the density number of liquid bonds on the overall behaviour of the material.
Introduction
An outstanding feature of granular materials lies in the existence of different scales of interest. First, the microscopic scale corresponding to the scale of individual particles can be considered. The microscopic scale is intimately related to the interaction processes that occur between adjoining particles. Second, a mesoscopic scale can be identified, associated to a set of a few particles. The mesoscopic scale is relevant to describe local kinematics (Cambou, Chaze, and Dedecker (2000)), or to address the creation and deletion of force chains amongst several elements (Radjaï, Roux, and Moreau (1999)). Third, the macroscopic scale which is the scale of the specimen, generally denoted as the material scale.
As a matter of fact, the overall mechanical behaviour of granular materials is intimately related to the local properties that take place at the microscopic scale. In addition to dry contact interactions, the presence of a surrounding liquid can modify local deformation, attrition, and sliding between particles. The material is thus a tri-phasic medium where the skeleton, liquid and air interact depending on the thermodynamic equilibrium. According to the liquid content and to the void ratio of the medium, different saturation regimes can be observed. For small liquid contents, as long as the liquid tends to be concentrated as independent menisci between adjoining particles, this is the pendular regime. For increasing liquid content, menisci begin to merge. From this point, all liquid-gas configurations until the gaseous phase consists in gas bubbles within the liquid are classified as funicular configurations. Beyond, the medium is in a capillary state until it tends toward a fully saturated regime. For each case, the presence of both liquid and gas induces attractive forces that confer specific constitutive properties to the material. However, even though interparticular menisci have been dealt with a great attention in the pendular regime (Hotta, Takeda, and Ionya (1974); Lian, Thornton, and Adams (1993); Soulié, Cherblanc, Youssoufi, and Saix (2006)), their modelling for higher saturation regimes is still a great challenge to deal with due to complex thermodynamic arrangements between the three phases (Urso, Lawrence, and Adams (1999); Murase, Mochida, and Sugama (2004)). The present paper focuses on liquid bridges as defined in the pendular state as an attempt toward the understanding of induced complex macroscopic behaviours. The analysis is therefore limited to low saturation degrees where the pendular assumption can be assumed.
Assessing the constitutive behaviour of such multi-phasic materials at the macroscopic scale can be addressed through sophisticated phenomenological constitutive models, requiring specific assumptions and mathematical refinements (see Cambou (1998) for a general review). This line of thinking has prevailed during the past decades. A powerful alternative concerns the micromechanical approaches, in which the description of physical phenomena is dealt with at the microscopic scale. The overall behaviour is therefore derived thanks to homogeneous schemes, relating both microscopic and macroscopic scales as presented in Christoffersen, Mehrabadi, and Nemat-Nasser (1981); Cambou (1998); Kruyt and Rothenburg (2000); Nicot and Darve (2005) or Lu and Likos (2006) for example. Such methods are appealing since the local physics can be described by simple equations, requiring a small number of parameters with generally a clear physical meaning.
Following a micromechanical approach, discrete element methods (DEM) constitute a particularly relevant computational tool to track the constitutive response of granular masses along a variety of loading paths. Indeed, the rheological behaviour is obtained without any global hypotheses. Since the pioneering work of Cundall and Strack (1979), DEM approach has been popularized in the field of granular materials, and extension to multi-physical coupling issues is an emergent line of research. For example, hydro-mechanical modelling supports now a great source of efforts, both for unsaturated (Richefeu, Youssoufi, Peyrous, and Radjaï (2007); Shamy and Gröger (2008); ) and saturated flows (Han, Feng, and Owen (2007); Zeghal and Shamy (2008)). This paper is situated in this context, by showing how the local parameters characterizing wetted contact interactions within a granular medium can influence the overall response of the material. In the following, the numerical specimen is assumed to be homogeneous and constituted by a sufficient number of grains so as a constitutive relation can be derived from both homogeneous strain and stress tensors: the specimen is considered as a Representative Volume Element (RVE) of a granular medium 2 Micromechanical model DEM has been extensively used to study soil mechanics providing, for instance, some insights into shear strength and deformation properties of granular soils (see Iwashita and Oda (1988) and Ting, Corkum, Kauffman, and Greco (1989) for example). Recently, DEM has been enhanced to investigate unsaturated granulates features by considering the possible effects of capillary water between grains (Gili and Alonzo (2002) ;Jiang, Leroueil, and Konrad (2004); Richefeu, Youssoufi, and Radjaï (2006); Shamy and Gröger (2008) ;). The DEM is essentially a Lagrangian (mesh-free) technique where each particle of the material is a sphere identified by its own mass, radius and moment of inertia. For every time step of the computation, interaction forces between particles, and consequently resulting forces acting on each of them, are deduced from sphere positions through the interaction law. Newton's second law is then integrated through an explicit second-order finite difference scheme to compute new spheres' positions.
We present here a 3D discrete element model developed into the YADE-Open DEM platform (Kozicki and Donzé (2008)), where effects of capillarity have been implemented in both terms of force and water retention through the resolution of the Young-Laplace equation as presented hereafter. In the proposed model, each interaction can thus be described as the sum of a dry contact repulsive force and of an attractive capillary force resulting from the presence of liquid bridges between grains.
Contact friction model
The elastic-plastic model for contact friction is presented in Fig. 1(a). Although the formulation is quite simplistic compared to more realistic Hertz-Mindlin solutions, it has been proved to give accurate results (DiRenzo and DiMaio (2004)), with a significant computational efficiency.
Particles are considered to be rigid, but can overlap as shown in Fig. 1(b). This overlap accounts for the deformation induced by contacts forces. A linear elastic law provides the contact force as a function of the relative displacement between two interacting grains (see Fig. 2). A normal stiffness K n is defined to relate the normal force F n to the intergranular normal distance U n such as:
F n = K n U n , if U n 0 0, if U n > 0(1)
and a tangential stiffness K t allows to deduce the shear force F t induced by the incremental tangential relative displacement dU t :
dF t = −K t dU t(2)
K n and K t are dependent functions of the interacting particle radii R1 and R2 and of a characteristic modulus E of the material such as:
K n = 2. E.R1.R2 (R1+R2) K t = α.K n(3)
with α a fixed parameter. This definition results in a constant ratio between E and the effective bulk modulus of the packing, whatever the size of the particles.
Shear and normal forces are finally related by a slip Coulomb model such that F max t = −µF n , where µ is the contact friction coefficient defined as µ = tan(ϕ c ), with ϕ c the intergranular friction angle.
Capillary model
For simplicity, we assume that capillary water inside the sample is solely composed of interparticular independent menisci as defined in the pendular state (Fig. 3).
Their exact shape between spherical bodies is defined by the Young-Laplace equation, which relates the pressure difference ∆u = u g − u l across the gas-liquid interface to the radius of curvature C of the bridge surface and to the surface tension γ of the liquid phase:
∆u = γ C(4)
In the Cartesian coordinates of Fig. 3, C can be formulated as a function of the profile y(x) of the liquid-gas interface curve, the x axis coinciding with the axis of symmetry of the bridge, passing through the centers of the bonded spheres. Following the Young-Laplace equation (Eq. 4), the profile of the liquid bridge is thus related to the capillary pressure ∆u through the following non-linear differential equation:
∆u γ (1 + y 2 (x)) 3/2 + 1 + y 2 (x) y(x) − y (x) = 0 (5)
According to a recent study by Soulié, Cherblanc, Youssoufi, and Saix (2006), the corresponding meniscus volume V m and intergranular distance U n can be obtained by considering the x-coordinates (x c1 and x c2 ) of the three-phases contact lines defining the solid-liquid-gas interface such as:
V m = π x c2 x c1 y 2 (x)dx − 1 3 πR 3 1 (1 − acos(x c1 )) 2 (2 + acos(x c1 )) − 1 3 πR 3 2 (1 − acos(x c2 )) 2 (2 + acos(x c2 ))(6)
and
U n = R 2 (1 − acos(x c2 )) + x c2 + R 1 (1 − acos(x c1 )) − x c1(7)
The capillary force due to the liquid bridge can then be calculated at the profile apex y 0 according to the 'gorge method' (Hotta, Takeda, and Ionya (1974)) and consists of a contribution of both ∆u and γ:
F cap = 2πy 0 γ + πy 2 0 ∆u (8)
The relation between the capillary pressure and the configuration of the capillary doublet is thus described by a system of non-linear coupled equations (Eq. 5, Eq. 6, Eq. 7, Eq. 8) where local geometry (U n ) and meniscus volume (V m ) arise as a result of the solved system.
In order to account for capillarity in the YADE-Open DEM code, an interpolation scheme on a set of discrete solutions of the Young-Laplace equation has been developed so as to link directly ∆u to F cap and V m for a given grain-pair configuration (R 1 , R 2 , U n ). This numerical procedure results in a suction-controlled model where, at every time-step during the simulation, capillary forces and water volumes are computed based upon the local geometry and the imposed capillary pressure. Fig. 4 presents the evolution of the capillary force with the relative displacement between two interacting grains.
It is to be noted that capillary forces act exclusively in the axial direction of the interaction (the normal to the tangential contact plane), and that F cap is maximum Figure 4: Evolution of the capillary force with the relative displacement between interacting grains for a given capilary pressure ∆u: a meniscus can form for U n ≤ U creation and breaks for U n > U rupture .
for grains in contact (U n 0). U rupture is the debonding distance corresponding to the minimum U n value from which the Young-Laplace equation has no solution.
An hydraulic hysteresis can be accounted for by defining U creation as the distance from which a liquid bridge can form between two interacting grains. For instance, as a drying scenario results in a desaturation of the material, liquid bridges can exist at stretched contacts as long as the debonding distance U n is less than the critical separation distance U rupture (Fig. 5(a)). On the other hand, a wetting scenario only allows appearance of menisci between contacting or close grains ( Fig. 5(b)) such as it would occur during a capillary condensation when the relative humidity of the surrounding air is increased. The choice was made here to allow bridge appearance for strict contact (U creation = 0), thus neglecting the possible effect of adsorbed water. In the same way, once they break, menisci can reform when particles come strictly in contact.
Following these assumptions, wetting and drying scenarios can be taken into consideration for the initial state of the material, keeping in mind that, in our simulations, the contact angle θ (Fig. 3(b)) which defines the wetting of the material is set to 0 • for simplicity, assuming a perfect wetting of the material and occulting therefore its possible effect on local hysteresis mechanisms.
Interparticle behaviour
The interaction laws involve normal repulsion and Coulomb friction at contact as well as capillary adhesion as presented respectively in Fig. 2 Capillary forces are considered constant for the range of elastic contact deformation (U n 0), assuming a low deformability of particles during simulations. To pull two bonded spheres apart, an increasing external tensile force is needed. This force has to balanced the sum of the repulsive contact force F n and the adhesive capillary one F cap which tends to decrease with the interparticle distance U n . The tensile process is stable as long as U n is less than the debonding distance U rupture which will depend on both the doublet configuration (R 1 , R 2 ) and the capillary pressure ∆u. As presented in Fig. 6, the particle size ratio r = R 1 R 2 has a great influence on both the adhesive force and the liquid bridge volume. For instance, for a given ∆u value, the greater the ratio r is, the smaller the bridge volume V m is, resulting in less significant interacting distances U rupture before rupture as well as in diminishing force intensities.
Concerning now the influence of the capillary pressure (Fig. 7), it has to be noted that, for a given doublet configuration (r = 1), if higher values of ∆u result in smaller liquid bridge volumes, capillary force intensity at contact (U n = 0) is quite constant whatever ∆u. At the particle scale, capillary pressure, through its effect on menisci volumes, finally influences more the debonding distance than the resulting adhesive force.
To sum up, in addition to the great influence of particle size distribution on their intensity level, capillary effects between interacting grains are strongly driven by the amount of liquid involved in their behaviour. Such a complexity leads to numerous uncertainties when considering granular materials such as soils. The next section is therefore an attempt to provide some clarifications on capillarity consequences at the scale of a numerical granular assembly. as a function of the separation distance U * n = U n R 2 for r = R 1 R 2 = 1. Friction angle E (MPa) α ϕ c (deg.) 50 0.5 30 Figure 8: The discrete element model.
Macroscopical behaviour
In order to investigate macroscopic consequences of interparticle capillary menisci, numerical simulations were conducted on a polydisperse assembly composed of 10,000 spheres ( Fig. 8) with a uniform grain size distribution ranging from 0.035 mm to 0.07 mm. The specimen is contained inside a box made of six rigid frictionless boundary walls for which positions at each time steps are defined from the prescribed loading program. Tab. 1 provides a summary of the input parameters used in the simulations, referring to Eq. 1, Eq. 2 and Eq. 3.
The sample is prepared in a dry configuration by an isotropic-compaction technique which ensures the initial homogeneity of the packing ). The final porosity of the sample depends on the intergranular friction angle ϕ c defined during the compaction stage. The smaller ϕ c , the denser the specimen. Here, ϕ c was fixed to 1 • , leading to a dense specimen with a porosity of about 0.39. ϕ c was then set to 30 • for all the simulations. Starting from the stabilized isotropic configuration, capillary pressure ∆u is then switched on, letting menisci to be formed between all possible interacting grains following the chosen drying or wetting scenario (stretched or contacting menisci respectively). It is to be noted that the suction controlled scheme ensures menisci to be homogeneously dis-tributed inside the medium in accordance the thermodynamic equilibrium between both liquid and gas phases.
Numerical results
Several loading paths were applied to the specimen in order to assess both its hydraulic and mechanical properties.
Hydraulic properties
Determining hydraulic properties of unsaturated soils is of great interest because of the strong hydro-mechanical coupling consequences on the overall behaviour.
Herein, the proposed model allows to determine the water content of an assembly as a result of the summation from all menisci volumes contained inside the medium. Indeed, the degree of saturation Sr of the numerical sample is obtained by:
Sr = 100 * V l V v = 100 * ∑ N m m=1 V m V sample −V grains(9)
with V l the total liquid volume and V v the void volume resulting from the grains arrangement inside the cubic box. V v is equal to the difference between the total volume of the sample V sample minus the volume occupied by all the grains g forming the assembly (V grains = ∑ N g g=1 4 3 πR 3 g ). By varying the capillary pressure ∆u inside a given sample, one can therefore construct its soil water characteristic curve (SWCC).
However, due to the pendular assumption concerning liquid phase modelling (independent interparticular liquid bridges), the SWCC cannot be computed for high degrees of saturation Sr. Indeed, below a given capillary pressure, liquid bridges become big enough to allow their overlap with neighbours. As this overlap leads to menisci fusion associated with a complete fluid reorganisation which makes obsolete the pendular assumption, a numerical procedure has been developed in order to identify menisci superpositions through the definition of their geometrical configuration. As defined in Fig. 3, the wetting of a liquid bridge over the grain surface is defined by its filling angles δ 1 and δ 2 . It is therefore possible to determine if, for a given grain g of an assembly, neighbouring menisci meet through their wetted surfaces.
For instance (see Fig. 9), if the angle O i O g O j formed between three interacting grains g, i and j is smaller than the sum (δ gi + δ g j ) corresponding to the surface of the grain wetted by menisci m g,i , and m g, j linking g with i and g with j respectively, an overlap occurs and fusion can therefore process. Obviously, the occurence of menisci fusion strongly depends on the microstructural arrangement and on the porosity of the medium. The range of saturation that corresponds to the pendular regime cannot therefore be infered for all materials and has to be checked for each particular case. Fig. 10 presents the SWCC obtained for the predefined numerical assembly in the range of its pendular regime. For this particular case, our finding is that the independence of menisci is strictly ensured for Sr < 12 %, whereas overlappings are negligible until Sr = 15 %.
One can note that, through the possible local mechanisms of formation and breakage of menisci, an hydraulic hysteresis arises at the macroscopic level, depending on the wetting history of the material. This macroscopical hysteresis is directly linked to the liquid bridge density inside the sample. Indeed, due to the possibility of stretched liquid bridges between separated grains for a drying scenario, menisci are numerous inside the medium and the corresponding degrees of saturation are therefore larger for the simulated dried material than for the wetted one. Fig. 11 illustrates this point through the evolution of the average number of liquid bridge per particle K m during both the two scenarios.
It is remarkable that both curves present two distinct stages: -For ∆u > 3 MPa, menisci distribution is driven by the particle gradation.
Menisci progressively disappear with the increase in ∆u due to the persistence of capillary water between the smallest particles. For instance, this is the reason why clay needs higher capillary pressures than sand to be fully desaturated.
-For ∆u < 3 MPa, menisci distribution is driven by the capillary pressure according to the wetting history of the material. During the drying scenario, the increase of ∆u induces smallest debonding distances (Fig. 7) and therefore less stretched bridges inside the sample. During the wetting scenario, as menisci are only present at contacts, they keep constant in number whereas their respective volumes increase with the decrease of ∆u.
At the particle scale, the simulated hysteresis can be illustrated through Fig. 5 where one can see that the darkest particle has more neighbours (K m = 5) after a drying sequence, than after a wetting one (K m = 2). This phenomenon, also denoted as the ink-bottle effect, is probably the most significant hysteretic mechanisms involved in unsaturated soil mechanics. The consequences of this property on the mechanical behaviour of granular assemblies is investigated in section 3.2.
Stress-Strain response under triaxial loadings
To determine the effect of interparticular capillary menisci on the mechanical properties of a granular material, we performed a series of triaxial compression loadings (σ 2 = σ 3 ,ε 1 > 0) on the numerical assembly under 5, 10 and 20 kPa confining pressures.
As a reference, the response of the dry specimen is plotted in Fig. 12. The assembly presents a behaviour which is qualitatively close to that of a dense granular material with strain softening associated with dilatancy. As shown in Fig. 13, the discrete element model behaviour is well described through the Mohr-Coulomb criterion: In order to cover the entire pendular regime, numerical unsaturated triaxial tests were performed at different capillary pressures corresponding to degrees of saturation ranging from 0 to 10 % along the wetting path. Tab. 2 presents the correspondence between ∆u and initial values of Sr for the considered specimen. Fig. 14 displays the responses of the unsaturated samples compared to the dry one for triaxial compressions applied under a confining pressure of 10 kPa. For clarity reason, only four of the six tests are plotted. As expected, the shear strength is greater for unsaturated materials and depends on the degree of saturation. The more wetted the sample is, the higher the deviatoric strength is, with a trend for dilatancy more pronounced than in the dry case, suggesting a more interlocked structure. To sum up all simulated tests, the corresponding failure envelops as well as the evolution of the apparent cohesion c with Sr are plotted in Fig. 15.
τ = σ tan Φ + c(10)
As a first observation, one can see that the internal friction angle Φ is independent on the degree of saturation, whatever the water content level. This probably results from the assumption of null effect of capillarity on contact friction, but it is remarkable that such a local property arises at the scale of an assembly. Richefeu, Youssoufi, and Radjaï (2006) came to the same conclusion from laboratory experiments on glass beads assemblies with pure water, providing therefore a good prediction to the idealized model. However, real granular materials such as soil would certainly not highlight such an independence between local and global friction properties due to the complex combined effects of surface roughness and in situ liquid wetting features.
Second, it is clear that the cohesion varies significantly from dry to unsaturated states. c increases non-linearly till a maximum value for the higher degrees of saturation that corresponds to the upper limit of the pendular state. This is fairly typical based on synthesis of reported experimental data on sand (Lu, Bailin, and Tan (2007)) or glass beads (Richefeu, Youssoufi, and Radjaï (2006)). Nevertheless, this is quite remarkable in regards to the interparticle behaviour, since capillary force intensity at contact is not such dependent on capillary pressure (Fig. 7). For instance, computing the mean capillary force < F cap >=
∑ Nm i F i cap N m
in the specimen for all tested degrees of saturation (Fig. 16) indicates that capillary forces tends to be greater, on average, for higher capillary pressure levels (and therefore for smaller saturation degrees).
This paradox is clearly due to capillary pressure influence on the debonding distance of menisci as presented in Fig. 7. Indeed, strong internal rearrangements occur during the loading, leading to a redistribution of the liquid inside the microstructure which is not obvious at the macroscopic scale. How this redistribution occurs and how it influences the shear strength of the material is strongly linked to the elaborated properties of grains when they interact in an assembly. A micromechanical investigation is therefore needed in order to identify effects induced by structural evolution of the medium on the liquid bridge distribution. Figure 17: (a) Evolution of the average number of menisci per particle K m during a triaxial compression for several initial degree of saturation Sr, (b) K m values at the q-peak as a function of Sr. Fig. 17(a) shows the evolution of the average number of menisci per particle K m for different degrees of saturation during the unsaturated triaxial compressions presented before (Fig. 14).
Microscopic analysis
Due to particle rearrangements during the loading, liquid bridge distribution inside the material changes with deformations, depending on the defined capillary pressure level. For instance, despite the fact that the initial liquid bridge densities are identical for tests corresponding to degrees of saturation of 0.01, 2.5, 10 %, their evolutions during compression are totally different due to the local response of liquid bonds to loading. Typically, as a result of the local behaviour of menisci (see Fig. 7), liquid bridges tend to persist more with the medium dilatancy for low values of ∆u (higher Sr), due to larger debonding distance U rupture . As discussed in as well as in Scholtès, Hicher, Nicot, Chareyre, and Darve (2009), the microstructure strongly influences liquid distribution inside the medium. The initial bridge density number appears therefore not appropriate to accurately evaluate the shear strength of the material as it was suggested by Richefeu, Youssoufi, and Radjaï (2006), in the sense that failure does not occur for an isotropic configuration of the material, but after some deformations leading to significant changes inside the fabric due to the induced anisotropy. Cohesion, as well as the internal friction angle, is dependent on space direction, and has to be evaluated at the yield state. Indeed, by plotting K m values corresponding to the q − peak (Fig. 17(b)), one can recover the shear strength hierarchy induced by capillarity depending on the degree of saturation. Figure 18: (a) stress-strain relationships and (b) evolution of K m during a triaxial compression on a sample subjected to a drying and a wetting scenario respectively under the same capillary pressure ∆u=20 kPa.
It is important to note that, in the end, whatever the capillary pressure level or the degree of saturation, the pertinent parameter which determines the shear strength of a partially saturated granular material is the liquid bridge density inside the medium at the yield state. Obviously, the evolution of this density with deformations is totally driven by the capillary pressure level, and, thereafter, by the water content.
Another way to illustrate the primacy of the liquid bridge density on shear strength properties of a wet granular material is to compare its behaviour for both a drying and a wetting scenario. As suggested in section 3.1.1, a given sample can contain different numbers of capillary bonds for a defined ∆u value (Fig. 11) due to the possible hydraulic hysteresis. Fig. 18(a) presents the stress-strain relationships obtained for a numerical sample subjected to a triaxial loading under the same capillary pressure (∆u = 20 kPa), but with two different initial configurations resulting respectively from a drying and a wetting scenario. Due to numerous liquid bridges Fig. 18(b), the dried sample has a greater shear strength (q peak =30 kpa) than the wetted one (q peak =27 kpa), which results in respective cohesions of 5 and 6.5 kPa. It is remarkable that the ratio c Drying c Wetting = 1.3 is strongly correlated to the ratio between the corresponding values of K m at the q − peak: K Drying m K Wetting m = 7.4 5.5 = 1.34, confirming here the possibility to link K q−peak m to the overall cohesion of the material.
Another remarkable feature of the coupled hydro-mechanical process is that internal rearrangements caused by the loading tend to conceal the initial difference between the two specimens, leading to a common residual state for large deformations with the same number of liquid bridges on average. This confirms the strong influence of the microstructure on the liquid phase distribution. For instance, if the presence of fluids inside a polyphasic granular material implies some consequences on its overall behaviour, the coupling is also active in the other way. Microstructural rearrangements of the solid skeleton lead to strong modifications of the fluid phases and both aspects induce complex mechanisms at the macroscopic scale.
Conclusion
A micromechanical computational model for the analysis of wet granular soils in the pendular regime has been proposed. Capillary mechanisms are described at the contact scale based upon the capillary theory in both terms of interparticle adhesive force and water retention. Capillary menisci are distributed inside the medium according to the thermodynamic equilibrium between liquid and gas phases. Moreover, an hydraulic hysteresis is accounted for based on the possible mechanisms of formation and breakage of liquid bridges during wetting and drying phases.
Triaxial compression test simulations were performed on a granular assembly under several confining pressures for dry and partially saturated conditions in order to analyze effects of local capillary menisci on macroscale cohesion and friction properties. The results confirm that capillary-induced attractive forces and hydraulic hysteresis play a significant role in the shear strength of granular materials. By increasing normal forces at contact, capillary menisci contribute to the apparent cohesion of the material and enhances its stiffness. A remarkable aspect is that the shear strength of an assembly is basically controlled by the liquid bridge density and not by the capillary pressure level. A strong correlation is highlighted between the average number of liquid bridges at the yield state and the induced cohesion. On the other hand, the liquid bridge distribution inside the medium is controlled by the capillary pressure which leads to counter-intuitive response at the scale of an assembly. In a separate way to capillary forces, Coulomb cohesion increases with water content and tends to a maximum value for the upper limit of the pendular regime following the liquid bridge density. In additon, the results put in evidence the effect of the skeleton induced anisotropy on liquid phase properties which is generally occulted when considering hydro-mechanical modelling. Solid fabric evolution clearly has a strong influence on the liquid distribution.
The proposed model is able to simulate the macroscopic response of wet granular materials and revealed a number of outstanding micromechanical mechanisms and response patterns consistent with experimental data. The multi-scale approach presented here appears to be a pertinent complementary tool for the study of unsaturated soil mechanics. A fundamental extension of this work would be to extend the interaction model to higher degrees of saturation with, for example, a validation through tomographic imaging.
Figure 1 :
1Contact interaction: (a) model and (b) overlap.
Figure 2 :
2Contact friction law: (a) normal force and (b) tangential force.
Figure 3 :
3Illustration of a liquid bridge between two particles of unequal sizes: (a) global geometry, (b) details of the bridge.
and Fig. 4.
Figure 5 :
5Hydraulic hysteresis history accounted for in the code: assembly of particles resulting from (a) a drying scenario, (b) a wetting scenario.
Figure 6 :
6Effect of the particle size ratio r = R 1 R 2 on (a) the normalized capillary force F * cap = F cap 2πR 2 γ and (b) the normalized meniscus volume V * m = separation distance U * n = U n R 2 for ∆u = 1000 kPa.
Figure 7 :
7Effect of the capillary pressure ∆u on (a) the normalized capillary force F * cap = F cap 2πR 2 γ and (b) the normalized meniscus volume V * m =
Figure 9 :
9illustration of the numerical procedure for menisci fusion: (a) definition of the local geometry, (b) identification of merged menisci: m g, j , m g,k and m g,l overlap on grain g surface.
Figure 10 :
10Soil Water Characteristic Curve generated with the proposed discrete element model.
Figure 11 :
11Evolution of the average number of menisci per particle during both wetting and drying sequences.
Figure 12 :
12Simulation of drained triaxial compressions on the dry specimen under 5, 10 and 20 kPa confining pressures.
Figure 13 :
13Failure envelop for the dry sample. with a null apparent cohesion c and an internal friction angle Φ of about 28 • .
Figure 14 :
14Simulation of triaxial compressions at different capillary pressure levels under a 10 kPa confining pressure.
Figure 15 :
15(a) Morh-Coulomb failure envelops and (b) apparent cohesion c as a function of the degree of saturation Sr.
Figure 16 :
16Mean capillary force < F cap > as a function of the degree of saturation Sr.
Table 1 :
1Input parametersGlobal Modulus k t
k n
Table 2 :
2Capillary pressures and corresponding initial degrees of saturation of the
discrete element model
Capillary pressure ∆u [kPa] 5000 3000 80 50 30 20
Degree of saturation Sr [%] 0.001 0.01 1 2.5 5.5 10
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| []
|
[
"Optical and Infrared Emission from the AXPs and SGRs",
"Optical and Infrared Emission from the AXPs and SGRs"
]
| [
"U Ertan ",
"& Ş Ç Alışkan "
]
| []
| []
| We show that the irradiated accretion disk model can account for all the optical and infrared observations of the anomalous X-ray pulsars in the persistent state. Model fits do not constrain the outer disk radii, while placing an upper limit to the inner disk radii, and thus to the strength of the dipole component of the stellar magnetic field. While magnetar fields (B * > 10 14 G) in higher multipoles are compatible with the irradiated disk model, magnetic dipole components of magnetar strength are not consistent with optical data. | 10.1086/508347 | [
"https://arxiv.org/pdf/astro-ph/0608288v1.pdf"
]
| 119,058,495 | astro-ph/0608288 | 2b42be1f49c99f722f7a6aaba3b14002cc75a6b6 |
Optical and Infrared Emission from the AXPs and SGRs
14 Aug 2006
U Ertan
& Ş Ç Alışkan
Optical and Infrared Emission from the AXPs and SGRs
14 Aug 2006arXiv:astro-ph/0608288v1Subject headings: pulsars: individual (AXPs) -stars: neutron -X-rays: bursts -accretion, accretion disks
We show that the irradiated accretion disk model can account for all the optical and infrared observations of the anomalous X-ray pulsars in the persistent state. Model fits do not constrain the outer disk radii, while placing an upper limit to the inner disk radii, and thus to the strength of the dipole component of the stellar magnetic field. While magnetar fields (B * > 10 14 G) in higher multipoles are compatible with the irradiated disk model, magnetic dipole components of magnetar strength are not consistent with optical data.
Introduction
Anomalous X-ray Pulsars (AXPs) and Soft Gamma-ray Repeaters (SGRs) constitute a special class of neutron star systems (Mereghetti et al. 2002;Hurley 2000;Woods & Thompson 2004). They are identified mainly through their X-ray luminosities (L x ∼ 10 34 − 10 36 erg s −1 ), which are orders of magnitude higher than their rotational powersĖ rot = IΩΩ. Their spin periods are clustered to a very narrow range (5 − 12 s). All five known SGRs and two of the eight known AXPs show repetitive, short ( < ∼ 1s) super-Eddington bursts with luminosities up to 10 42 erg s −1 . Three giant flares with peak luminosities L p > 10 44 erg s −1 and durations of a few minutes were observed from three different SGRs (Mazets et al. 1979;Mazets et al. 1999;Hurley et al. 1999;Palmer et al. 2005).
The short time scales and the super-Eddington luminosities of the soft gamma-ray bursts strongly indicate a magnetar mechanism for these bursts. Magnetar models (Duncan & Thompson 1992;Thompson & Duncan 1995;Thompson & Duncan 1996) adopt strong magnetic fields with magnitude B * > 10 14 G on the stellar surface to explain the burst energetics. Are the burst energies stored in the dipole component or the higher multipoles of the magnetic field of the neutron star? In the current magnetar models, the dipole component of the magnetic field must be of magnetar strength to account for the spin down properties of the AXPs and SGRs. In these models, persistent X-ray luminosities are explained by the magnetic field decay, while the magnetic dipole torque is taken as the mechanism responsible for the spin down rates of the sources.
In the alternative fallback disk model (Chatterjee, Hernquist, & Narayan 2000;Alpar 2001), the source of the X-rays is accretion onto the neutron star, while the optical/IR light originates from the accretion disk. Rotational evolution of the neutron star is determined by the interaction between the disk and the magnetosphere of the neutron star (B * ∼ 10 12 − 10 13 G). Fallback disk models can account for the period clustering of AXPs and SGRs as the natural outcome of disk-magnetosphere interaction during their lifetimes (Alpar 2001;Ekşi & Alpar 2003). These models are consistent with magnetar fields on the neutron star provided that these fields are in higher multipole components of the magnetic field. As higher multipole fields rapidly decrease with increasing radial distance (as r −5 for quadrupole component), it is the dipole component of the magnetic field which determines the interaction and the angular momentum transfer between the disk and the neutron star. The strength of magnetic dipole field in order to explain the period clustering of the AXPs and SGRs over theirṀ history is B * ∼ 10 12 − 10 13 G (Alpar 2001;Ekşi & Alpar 2003).
Fallback disk models can also explain the enhancements observed in the persistent luminosities of SGRs and AXPs. The X-ray enhancement of the SGR 1900+14 following its giant flare can be explained by the relaxation of a disk which has been pushed back by a preceding burst (Ertan & Alpar 2003). The same model with similar disk parameters can also reproduce the correlated X-ray and IR enhancement of AXP 2259+58, which lasted for ∼ 1.5 years, if this is triggered by a burst, with a burst energy estimated to have remained under the detection limits (Ertan, Gögüş & Alpar 2006).
The suggestion of fallback disks has motivated observational searches for disk emission in the optical and IR bands, and resulted in various constraints on the models. Some of the AXPs were observed in more than one IR band (Hulleman et al. 2001;Israel et al. 2002;Wang & Chakrabarty 2002;Kaspi et al. 2003;Israel et al. 2003;Hulleman et al. 2004;Israel et al. 2004;Tam et al. 2004;Morii et al. 2005;Durant & van Kerkwijk 2006b). AXP 4U 0142+61 is the source with most extended observations, as it was also observed in the optical R and V bands (Hulleman et al. 2000;Hulleman et al. 2004;Dhillon et al. 2005), and recently in mid-IR bands with the SPITZER observatory (Wang et al. 2006). Discovery of modulation in the R band luminosity of 4U 0142+61 at the neutron star's rotation period P=8.7 s, with a pulsed fraction 27 % (Kern & Mertin 2002;Dhillon et al. 2005), is particularly significant. This fraction is much higher than the pulsed fraction of the X-ray luminosity of this source, indicating that the origin of the pulsed optical emission is not likely to be the reprocessed X-rays by the disk. Magnetospheric models for these pulsations can be built with either a dipole magnetar field or within a disk-star dynamo model (Cheng & Ruderman 1991), in which a magnetospheric pulsar activity is sustained by a stellar dipole field of ∼ 10 12 G and a disk protruding within the magnetosphere. Ertan & Cheng (2004) showed that this pulsed optical component of the AXP 4U 0142+61 can be explained by both types of magnetospheric models. Thus the presence of strong optical pulsations from the magnetosphere does not rule out the possibility of a fallback disk together with 10 12 − 10 13 G surface dipole magnetic field.
In the present work, we concentrate on the unpulsed optical/IR emission from the AXPs and SGRs in their persistent states, and test the expectations of the irradiated accretion disk model through the observations in different optical/IR energy bands (V, R, I, J, H, K, and K s ). The optical/IR emission expected from the irradiated fallback disks was first computed and discussed by Perna, Hernquist & Narayan (2000) and Hulleman, van Kerkwijk & Kulkarni (2000). Using similar irradiation strengths, Perna et al. (2000) and Hulleman et al. (2000) found similar optical fluxes that remain well beyond those indicated by the observations of AXP 4U 0142+61 and AXP 1E 2259+586. To explain this result, Perna et al. (2000) suggested that the inner disk regions could be cut by an advection dominated flow, while Hulleman et al. (2000) concluded that the then existing optical data of the AXP 4U 0142+61 (in I,R,V bands) can only be accounted for by an extremely small outer disk radius, around a few ×10 9 cm. In the present work, we show that the optical/IR data of the AXPs can be explained by the irradiated accretion disk model without any implausible constraints on the outer and inner disk radii. The main reason for the difference between our results and those of earlier works is that both Hulleman et al. (2000) and Perna et al. (2000) assumed a particular irradiation strength, while we keep it as a free parameter, to address the broad band full data set. This approach is supported by the observations of the low mass X-ray binaries (LMXBs) which indicate varying irradiation strengths. Furthermore, model fits are sensitive to the interstellar reddening parameter A V , which was estimated to be between 2.6 and 5.1 for 4U 0142+61 (Hulleman et al. 2004). For this source, we obtain the best model fit with A V = 3.5 which turned out to be consistent with the recent result A V = 3.5 ± 0.4 (Durant & van Kerkwijk 2006a). On the other hand, the test disk model with unreasonably small outer disk radius requires A V = 5.4 (Hulleman et al. 2000). In this model, optical emission comes from the outer disk, while in our model it is the inner regions of an extended disk that emits substantially through the optical bands (see § 3 for further discussion). We give the details of the disk model in § 2. We discuss our results in § 3, and summarize the conclusions in § 4.
Optical/IR Emission from the Irradiated Disk
Model fits to the X-ray and IR enhancement data (Kaspi et al. 2003) of AXP 1E 2259+586 favor the irradiated disk model, though they do not exclude the nonirradiated thin disk model (Ertan, Gögüş & Alpar 2006). We start by assuming that the AXP disks are irradiated and include the irradiation strength as a free parameter through our calculations.
When the disk is irradiated by the X-rays from the neutron star, both the intrinsic dissipation and the irradiation flux should be taken into account in calculations of the disk blackbody emission. A steady disk model is a good approximation for the present evolution of the AXP and SGR disks in their persistent states. For a steady thin disk, the intrinsic dissipation can be written as D = (3/8π)(GMṀ /R 3 ) (see e.g. Frank et al. 2002) whereṀ is the disk mass flow rate, M is the mass of the neutron star and R is the radial distance from the neutron star. In the absence of irradiation, the effective temperature T eff of the disk is proportional to R −3/4 for a givenṀ . For an irradiated disk, the irradiation flux can be written as F irr = σT 4 irr = (CṀ c 2 )/(4πR 2 ). (Shakura & Sunyaev 1973), where c is the speed of light. Iraadiation parameter C includes the effects of the conversion efficiency of the accretion into X-rays, disk geometry and the albedo of the disk face. Irradiation temperature T irr = (F irr /σ) 1/4 is proportional to R −1/2 . For small radii, dissipation is the dominant source of the disk emission. At a critical radius R c , the irradiation flux becomes equal to the dissipation rate, and beyond R c , the disk emission is supported mainly by reprocessed X-rays. Equating F irr to D, the critical radius is found to be R c = (3GM * )/(2Cc 2 ) ≃ (10 −4 /C)3×10 9 cm. The effective temperature profile of the disk can be obtained using σT 4 eff = D + F irr where σ is the Stefan-Boltzmann constant.
We adopt the observed magnitudes in the optical/IR bands, distances and the N H values given by Woods & Thompson (2004) and references therein and convert the magnitudes to energy flux values. We calculate A V values using N H = 1.79 × 10 21 A V (Predehl & Schmitt 1995). To find the model disk flux in a given observational band, we integrate the calculated blackbody emissions of all radial grids radiating in this band. For comparison with data, we calculate the model disk fluxes along the optical/IR bands V, R, H, I, J, K and K s . For all sources we set cos i = 1 where i is the angle between the disk normal and the line of sight of the observer. We equate the disk mass flow rateṀ to the accretion rate onto the neutron star, thus assuming the mass loss due to the propeller effect is negligible. We first adjusṫ M to obtain the observed X-ray flux. Next, using this value ofṀ and taking the strength of the magnetic dipole field B * = 10 12 G on the surface of the neutron star we calculate the Alfvén radius R A which we take to be the inner radius of the disk. Then, we look for a good fit to the overall available optical/IR data by adjusting the irradiation strength C within the uncertainties discussed in § 3.
Results and Discussion
Our results are summarized in Table 1. For each source, the first column gives the unabsorbed flux data obtained from the observed magnitudes and the estimated A V values (see Table 1) given in Woods & Thompson (2004), and the second column gives the model fluxes. For the AXP 4U 0142+61, the range of reddening quoted in earlier literature is 2.6 < A V < 5.1 (Hulleman et al. 2004). We obtain a good fit with A V = 3.5. Table 1 shows that the irradiated steady disk model is in agreement with all the AXPs observed in the optical and IR bands. The parameters of the model for each source are given in Table 2.
At present, AXP 4U 0142+61, which has been observed in five different optical/IR bands from K to V in the same X-ray luminosity regime, seems to be the best source to study the properties of AXPs in the persistent state. Earlier work by Hulleman et al. (2000) excluded the disk model for the AXP 4U 0142+61. They obtained an irradiation temperature profile by using a particular irradiation strength. The estimated optical flux for an extended disk with this irradiation efficiency remains above the optical data points of the AXP 4U 0142+61 (see Fig. 3 in Hulleman et al. 2000). Considering the possibility that the optical flux might originate from the outermost disk region, Hulleman et al. (2000) tried to fit the then observed three data points in the I, R and V bands to the Rayleigh-Jeans tail of a blackbody spectrum with the extinction parameter A V = 5.4. This placed an upper limit to the outer disk radius which is too small for a realistic disk. The key factor in the difference between the earlier results and our recent results is the irradiation efficiency, which we allow to vary in conjunction with A V , to provide the best fit to the current broad band data. We note that the irradiation efficiency indicated by the observations of the low mass X-ray binaries varies from source to source. Even for the same source, the ratio of the irradiation flux to the X-ray flux may change with accretion rate (de Jong et al. 1996;Dubus et al. 1999;Ertan & Alpar 2002). Taking these into account, we keep the irradiation efficiency as a free parameter for our model fits. With the parameters given in Table 2, the irradiated disk model can account for the optical/IR data of this source without setting any stringent constraints on the inner or outer disk radii. In our model, the optical luminosity is radiated from the inner disk, while longer wavelength IR emission comes from larger radii. A more detailed analysis of the AXP 4U 0142+61 with the new detections in the mid-IR SP IT ZER bands confirms the results here ). The irradiation parameter C obtained from our model fits turned out to be in the range (10 −4 < C < 10 −3 ) estimated from the observations of LMXBs and the disk stability analyses of the soft X-ray transients (de Jong et al. 1996;Dubus et al. 1999;Ertan & Alpar 2002). Within the critical radius R c given by Eq. 4, dissipation is the dominant heating mechanism. For the disk model of the AXP 4U 0142+61, R c ≃ 3 × 10 9 cm and R in = 1 × 10 9 cm. The innermost disk emitting mostly in the UV bands also contributes to the optical emission. The radial distance at which the disk blackbody temperatures peak at the optical bands (R,V) is about 10 10 cm. at this radial distance, about 35 % of of the optical radiation is due to dissipation, and the rets is due to irradiation. Peak temperatures of the IR bands from I to K s lie between R ∼ 2 × 10 10 cm and R ∼ 1.5 × 10 11 cm.
There are several uncertainties related to the inner disk emission characteristics of the AXPs, which are not possible to address by the irradiated thin disk model. Fistly, emission properties of the innermost disk boundary interacting with the magnetosphere are not very clear. Secondly, the contributions from the magnetospheric pulsed emission which is known to have a fraction about 27 % in the R band for 4U 0142+61, is likely to be radiated from the other IR and optical bands as well. Relative amplitudes of these pulsed contributions radiated from different optical/IR bands are not known at present. Finally, there could be some X-ray shielding effects depending on the details of the geometry of the innermost disk regions, which could also affect the optical/IR emission properties of these sources. For all the AXPs that were detected in the optical/IR bands, optical and IR flux values of our models remain within about 30 % of all the data points, which is a reasonable fit considering the uncertainties discussed above.
For AXP J1708-40, Durant and van Kerkwijk (2006b) recently found that the previously reported IR data in K s , H, and J bands are likely to be a background star. They found another object within the positional error cycle and argued that this second object is more likely to be the IR counterpart to the AXP J1708-40. For this source, we adopt the IR (K s , H, J) data set reported by Durant and van Kerkwijk (2006b).
For the persistent state of the AXP 1E 2259+586, we use the preenhancement data (Hulleman et al. 2001). This source was detected in K s band and there are upper limits for I and R bands. Our model flux values are three and ten times below the upper limits reported for I and R bands respectively. AXP 1E 1048-59 was detected in K s , H and I bands (Wang & Chakrabarty 2002). Observed X-ray flux from this source between December 2000 to January 2003 show a variation within a factor of 5 (Mereghetti et al. 2004). We use the X-ray flux obtained from the nearest X-ray observation to the date of the IR observations. AXP 1E 1841 was detected only in the K s band, and there is a high upper limit in the R band (Watcher et al. 2004). Model estimates in other optical/IR bands for this source (and the other AXPs) can be tested by future optical and IR observations.
Since there are no detections in short wavelength optical bands for the AXPs (except for AXP 4U 0142+61), model fits are not sensitive to the chosen inner disk radii. We equate the inner disk radii to the Alfvén radii (Table 2) corresponding to a magnetic field with magnitude B * = 10 12 G on the stellar surface and the accretion rates derived from the estimated X-ray luminosities (see Table 1 for references). For the AXP 4U 0142+61, optical data in R and V bands provide a constraint for the inner disk radius, and thereby for the strength of the magnetic dipole field of this source (see § 4).
Conclusion
We have shown that the optical, infrared and X-ray observations of the AXPs in their persistent states can be explained with irradiated disk models. Among the AXPs, 4U 0142+61 is currently the only source which provides an upper limit for the inner disk radius through its optical (R,V) data. For the best model fit for this source, which we have obtained with A V = 3.5, the model inner disk radius (∼ 10 9 cm) is around the Alfvén radius for the accretion rate, estimated from the X-ray luminosity, together with a dipole magnetic field strength B * ≃ 10 12 G on the neutron star surface. Nevertheless, it is possible to obtain reasonable fits by increasing the inner disk radius and decreasing the reddening accordingly. For A V = 2.6, the minimum value of the reddening in the range 2.6 < A V < 5.1 (Hulleman et al. 2004), we obtain the best fit with R in ≃ 8 × 10 9 cm which corresponds to the maximum reasonable dipole field strength B * ≃ 4 × 10 13 G on the pole and half of this on the equator of the neutron star. We note that these limits could be increased depending on the amount of possible mass loss due to propeller effect and/or on how much the inner disk radius penetrates inside the Alfvén radius (see for a detailed discussion for 4U 0142+61). On the other hand, even including these possibilities, very recent analysis concluding A V = 3.5 ± 0.4 (Durant & van Kerkwijk 2006a) implies surface dipole magnetic field strengths less than about 10 13 G. While the magnetar fields (B * > 10 14 G) in multipoles are compatible with this picture, optical (R, V) data excludes a hybrid model involving a disk surrounding a magnetar dipole field. In the latter case, inner disk regions emitting in the optical would be truncated by the magnetar dipole field. High magnetic fields in multipoles, on the other hand, decrease rapidly with increasing radial distance, and do not affect the disk magnetosphere interaction.
Uncertainties in the source distances, require modification of the modelṀ values, and thereby the disk temperature profiles. For the AXP 4U 0142+61, new fits with similar quality can be obtained by slightly modifying the model inner disk radius and the irradiation parameter C as long as such distance corrections remain within a factor of ∼ 3. In this case, surface dipole magnetic field should also be recalculated consistently with the newly obtaineḋ M and Alfvén radius, r A ∝Ṁ −2/7 B 4/7 * . For the other AXPs, which were observed only in the IR bands, model fits at present are not sensitive to the inner disk radius, and in the case of distance corrections, similar fits can be obtained only by changing the irradiation strength. This is because IR radiation is emitted from the irradiation dominated outer disk regions, and thereforeṀ corrections do not change the relative amplitudes of the fluxes in different IR bands. Future detections of these sources in the optical bands can constrain the inner disk radii together with the surface dipole field strengths.
On the other hand, existing IR data of the AXPs, including recent observations of 4U 0142+61 by SPITZER in 4.5 µm and 8 µm bands (Wang et al. 2006), do not put an upper limit for the extension of the outer disk radius R out . The lower limit for R out provided by the longest wavelength IR data of the AXP 4U 0142+61 is around 10 12 cm. Further observations in the longer wavelength infrared bands by SP IT ZER space telescope will provide valuable information about the structure and possibly the extension of the fallback disks around these systems. As a final remark, some AXPs and SGRs which are under the detection limits in some of the optical and IR bands could be observed in these bands if they exhibit phases of enhanced emission, as observed in the SGR 1900+14 and the AXP 1E2259+586.
We thank Ali Alpar, Ersin Gögüş and Yavuz Ekşi for valuable comments on the manuscript. We acknowledge support from the Astrophysics and Space Forum at Sabancı University. S.Ç . acknowledges support from the FP6 Marie Curie Reintegration Grant, INDAM. Note. -The data flux values were calculated by using the magnitudes and A V values given in the references below. For the AXP 4U 0142+61 plausible range for reddening is 2.6< A V < 5.1 (Hulleman et al. 2004); the data of this source here correspond to A V =3.5. REFERENCES: (J1708-40) Durant & van Kerkwijk 2006b, Rea et al. 2003; (1E 2259+586) Hulleman et al. 2001; (4U 0142+61) Hulleman et al. 2000, Patel et al. 2003, Morii et al. 2005(1E 1841-45) Wachter et al. 2004, Morii et al. 2003(1E 1048-59) Wang & Chakrabarty 2002, Mereghetti et al. 2004 Table 2: The parameters of the irradiated disk model which gives the optical/IR flux values seen in Table 1. For all the sources, we set cos i = 1 where i is the inclination angle between the disk normal and the line of sight of the observer, and we take the outer disk radius R out = 5 × 10 12 cm. See Section 3 for details.
1RXS J1708-40 1E 2259+58 4U 0142+61 1E 1841-045 1E 1048-59 R in (cm) 1.2 × 10 9 2.3 × 10 9 1.0 × 10 9 1.3 × 10 9 3.3 × 10 9 C 5.0 × 10 −4 1.6 × 10 −4 1.0 × 10 −4 7.2 × 10 −4 7.0 × 10 −4 d(kpc) 5 3 3 7 3 M (g s −1 ) 1.0 × 10 15 9.1 × 10 13 4.8 × 10 14 2.2 × 10 15 1.3 × 10 14
Table 1 .
110 −15 erg s −1 cm −2 ) (10 −15 erg s −1 cm −2 ) (10 −15 erg s −1 cm −2 ) (10 −15 erg s −1 cm −2 ) (10 −15 erg s −1 cm −2 )J1708-40
1E 2259+58
4U 0142+61
1E 1841-045
1E 1048-59
Flux
Flux
Flux
Flux
Flux
(Band
Data
Model
Data
Model
Data
Model
Data
Model
Data
Model
(A V = 7.8)
(A V = 6.1)
(A V = 3.5)
(A V = 8.4)
(A V = 5.6)
K s
49
44
3.7
3.6
14
14
68
68
29
22
K
54
4.5
18
18
84
27
H
51
57
4.8
19
19
89
22
28
J
50
53
4.4
14
18
83
33
26
I
56
<15
4.4
18
21
88
24
R
65
<42
4.5
19
25
< 3.8 × 10 5
100
26
V
48
3.0
28
20
76
18
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| []
|
[
"Typeset using REVT E X",
"Typeset using REVT E X"
]
| [
"N N Achasov *[email protected]†[email protected] \nLaboratory of Theoretical Physics\nSobolev Institute for Mathematics 630090\nNovosibirsk-90Russia\n",
"V V Gubin \nLaboratory of Theoretical Physics\nSobolev Institute for Mathematics 630090\nNovosibirsk-90Russia\n"
]
| [
"Laboratory of Theoretical Physics\nSobolev Institute for Mathematics 630090\nNovosibirsk-90Russia",
"Laboratory of Theoretical Physics\nSobolev Institute for Mathematics 630090\nNovosibirsk-90Russia"
]
| []
| Analysis of the nature of the φ → γπη and φ → γπ 0 π 0 decays.AbstractWe study interference patterns in the φ → (γa 0 + π 0 ρ) → γπη and φ → (γf 0 + π 0 ρ) → γπ 0 π 0 reactions. Taking into account the interference, we fit the experimental data and show that the background reaction does not distort the π 0 η spectrum in the decay φ → γπη everywhere over the energy region and does not distort the π 0 π 0 spectrum in the decay φ → γπ 0 π 0 in the wide region of the π 0 π 0 system invariant mass, m ππ > 670 MeV, or when the photon energy is less than 300 MeV. We discuss the details of the scalar meson production in the radiative decays and note that there are reasonable arguments in favor of the one-loop mechanism φ → K + K − → γa 0 and φ → K + K − → γf 0 . We discuss also distinctions between the four-quark, molecular, and two-quark models and argue that the Novosibirsk data give evidence in favor of the four-quark nature of the scalar a 0 (980) and f 0 (980) mesons. 13.40.Hq, 13.65.+i | 10.1103/physrevd.63.094007 | [
"https://arxiv.org/pdf/hep-ph/0101024v2.pdf"
]
| 119,058,916 | hep-ph/0101024 | ea0ee3686fdadafe3c3a66567b90b81128530b03 |
Typeset using REVT E X
11 Apr 2001
N N Achasov *[email protected]†[email protected]
Laboratory of Theoretical Physics
Sobolev Institute for Mathematics 630090
Novosibirsk-90Russia
V V Gubin
Laboratory of Theoretical Physics
Sobolev Institute for Mathematics 630090
Novosibirsk-90Russia
Typeset using REVT E X
11 Apr 2001(October 31, 2018)arXiv:hep-ph/0101024v2
Analysis of the nature of the φ → γπη and φ → γπ 0 π 0 decays.AbstractWe study interference patterns in the φ → (γa 0 + π 0 ρ) → γπη and φ → (γf 0 + π 0 ρ) → γπ 0 π 0 reactions. Taking into account the interference, we fit the experimental data and show that the background reaction does not distort the π 0 η spectrum in the decay φ → γπη everywhere over the energy region and does not distort the π 0 π 0 spectrum in the decay φ → γπ 0 π 0 in the wide region of the π 0 π 0 system invariant mass, m ππ > 670 MeV, or when the photon energy is less than 300 MeV. We discuss the details of the scalar meson production in the radiative decays and note that there are reasonable arguments in favor of the one-loop mechanism φ → K + K − → γa 0 and φ → K + K − → γf 0 . We discuss also distinctions between the four-quark, molecular, and two-quark models and argue that the Novosibirsk data give evidence in favor of the four-quark nature of the scalar a 0 (980) and f 0 (980) mesons. 13.40.Hq, 13.65.+i
I. INTRODUCTION
As was shown in a number of papers, see Refs. [1][2][3][4][5][6] and references therein, the study of the radiative decays φ → γa 0 → γπη and φ → γf 0 → γππ can shed light on the problem of the scalar a 0 (980) and f 0 (980) mesons. These decays have been studied not only theoretically but also experimentally. Present time data have already been obtained from Novosibirsk with the detectors SND [7][8][9][10] and CMD-2 [11], which give the following branching ratios : BR(φ → γπη) = (0.88±0.14±0.09)·10 −4 [9], BR(φ → γπ 0 π 0 ) = (1.221±0.098±0.061)·10 −4 [10] and BR(φ → γπη) = (0.9±0.24±0.1)·10 −4 , BR(φ → γπ 0 π 0 ) = (0.92±0.08±0.06)·10 −4 [11].
These data give evidence in favor of the four-quark (q 2q2 ) [1,[12][13][14][15][16] nature of the scalar a 0 (980) and f 0 (980) mesons. Note that the isovector a 0 (980) meson is produced in the radiative φ meson decay as intensively as the well-studied η ′ meson involving essentially strange quarks ss (≈ 66%), responsible for the decay.
As shown in Refs. [1,3,17], the background situation for studying the radiative decays φ → γa 0 → γπ 0 η and φ → γf 0 → γπ 0 π 0 is very good. For example, in the case of the decay φ → γa 0 → γπ 0 η, the process φ → π 0 ρ → γπ 0 η is the dominant background. The estimation for the soft, by strong interaction standard, photon energy, ω < 100 MeV, gives BR(φ → π 0 ρ 0 → γπ 0 η, ω < 100 MeV) ≈ 1.5 · 10 −6 . The influence of the background process is negligible, provided BR(φ → γa 0 → γπ 0 η, ω < 100 MeV) ≥ 10 −5 . In this paper, in Sec. II, we calculate the expression for the φ → γπ 0 η decay amplitude taking into account the interference between the φ → γa 0 → γπ 0 η and φ → π 0 ρ → γπ 0 η processes. We show that for the obtained experimental data the influence of the background processes is negligible everywhere over the photon energy region.
The situation with φ → γf 0 → γπ 0 π 0 decay is not much different. As was shown in [1,3,17] the dominant background is the φ → π 0 ρ 0 → γπ 0 π 0 process with BR(φ → π 0 ρ 0 → γπ 0 π 0 , ω < 100 MeV) ≈ 6.4 · 10 −7 . The influence of this background process is negligible, provided BR(φ → γf 0 → γπ 0 π 0 , ω < 100 MeV) ≥ 5 · 10 −6 .
The exact calculation of the interference patterns between the decays φ → γf 0 → γπ 0 π 0 and φ → ρ 0 π → γπ 0 π 0 , which we present in this paper in Sec. III, shows that the influence of the background in the decay φ → γπ 0 π 0 for the obtained experimental data is negligible in the wide region of the π 0 π 0 invariant mass, m ππ > 670 MeV, or in the photon energy region ω < 300 MeV.
In Sec. IV we discuss the mechanism of the scalar meson production in the radiative decays and show that experimental data obtained in Novosibirsk give the reasonable arguments in favor of the one-loop mechanism φ → K + K − → γa 0 and φ → K + K − → γf 0 of these decays . In the same place we discuss also distinctions between the four-quark, molecular, and two-quark models and explain why these data give evidence in favor of the four-quark nature of the scalar a 0 (980) and f 0 (980) mesons.
II. INTERFERENCE BETWEEN THE REACTIONS φ → γa 0 → γπ 0 η AND φ → π 0 ρ 0 → γπ 0 η.
As was shown in Refs. [1,3] the background process e + e − → φ → π 0 ρ 0 → γπ 0 η is dominant. The amplitudes of the processes e + e − → ρ 0 (ω) → ηρ 0 (ω) → γπ 0 η are much less than the amplitudes of the e + e − → ρ 0 (ω) → π 0 ω(ρ 0 ) → γπ 0 η processes. In its turn, the amplitudes of the e + e − → ρ 0 (ω) → π 0 ω(ρ 0 ) → γπ 0 η processes are much less than the amplitudes of the e + e − → φ → π 0 ρ 0 → γπ 0 η processes. The amplitude of the e + e − → φ → ηφ → γπ 0 η process is also much less than the amplitude of e + e − → φ → π 0 ρ 0 → γπ 0 η process.
The amplitude of the background process φ(p) → π 0 ρ 0 → γ(q)π 0 (k 1 )η(k 2 ) is
M B = g φρπ g ρηγ D ρ (p − k 1 ) φ α k 1µ p ν ǫ δ (p − k 1 ) ω q ǫ ǫ αβµν ǫ βδωǫ .(1)
For the amplitude of the signal φ → γa 0 → γπ 0 η we use the model suggested in Ref. [1], in which the one-loop mechanism of the decay φ → K + K − → γa 0 is considered:
M a = g(m) g a 0 K + K − g a 0 πη D a 0 (m) (φǫ) − (φq)(ǫp) (pq) ,(2)
where m 2 = (k 1 + k 2 ) 2 , φ α and ǫ µ are the polarization vectors of φ meson and photon, the function g(m) is determined in Refs. [1,3]. The mass spectrum is
Γ(φ → γπη) dm = dΓ a 0 (m) dm + dΓ back (m) dm ± dΓ int (m) dm ,(3)
where the mass spectrum for the signal is
dΓ a 0 (m) dm = 2 π m 2 Γ(φ → γa 0 (m))Γ(a 0 (m) → πη) |D a 0 (m)| 2 = 2|g(m)| 2 p ηπ (m 2 φ − m 2 ) 3(4π) 3 m 3 φ g a 0 K + K − g a 0 πη D a 0 (m) 2 .(4)
Accordingly, the mass spectrum for the background process e + e − → φ → π 0 ρ → γπ 0 η is
dΓ back (m) dm = (m 2 φ − m 2 )p πη 128π 3 m 3 φ 1 −1 dxA back (m, x) ,(5)
where
A back (m, x) = 1 3 |M B | 2 = = 1 24 m 4 η m 4 π + 2m 2 m 2 η m 2 πm ρ 2 − 2m 4 η m 2 πm ρ 2 − 2m 2 η m 4 πm ρ 2 + 2m 4m ρ 4 − 2m 2 m 2 ηm ρ 4 + 2m 4 ηm ρ 4 − 2m 2 m 2 πm ρ 4 + 4m 2 η m 2 πm ρ 4 + m 4 πm ρ 4 + 2m 2m ρ 6 − 2m 2 ηm ρ 6 − 2m 2 πm ρ 6 +m ρ 8 − 2m 4 η m 2 π m 2 φ − 2m 2 m 2 η m 2 φm ρ 2 + 2m 2 η m 2 π m 2 φm ρ 2 − 2m 2 m 2 φm ρ 4 + 2m 2 η m 2 φm ρ 4 − 2m 2 φm ρ 6 + m 4 η m 4 φ + m 4 φm ρ 4 × g φρπ g ρηγ D ρ (m ρ ) 2 ,(6)andm ρ 2 = m 2 η + (m 2 + m 2 η − m 2 π )(m 2 φ − m 2 ) 2m 2 − (m 2 φ − m 2 )x m p πη p πη = (m 2 − (m η − m π ) 2 )(m 2 − (m η + m π ) 2 )
2m .
The interference between the background process amplitude and the signal amplitude is written in the following way:
dΓ int (m) dm = (m 2 φ − m 2 )p πη 128π 3 m 3 φ 1 −1 dxA int (m, x) ,(8)
where
A int (m, x) = 2 3 Re M a M * B = 1 3 (m 2 − m 2 φ )m ρ 2 + m 2 φ (m ρ 2 − m 2 η ) 2 m 2 φ − m 2 × Re g(m)g a 0 K + K − g a 0 πη g φρπ g ρηγ D * ρ (m ρ )D a 0 (m) .(9)
The inverse propagator of a 0 meson, D a 0 (m), is presented in Refs. [1,3]. The inverse propagator of ρ meson has the following expression
D ρ (m) = m 2 ρ − m 2 − im 2 g 2 ρππ 48π 1 − 4m 2 π m 2 3/2 .(10)
We use the coupling constant g φK + K − = 4.68 ±0.05 obtained form the decay φ → K + K − [18], and the coupling constant g ρηγ = 0.572 ± 0.08 GeV −1 obtained from the decay ρ → ηγ [19], with the help of the following expressions
Γ(φ → K + K − ) = g 2 φK + K − 48π m φ 1 − 4m 2 K m 2 φ 3/2 , Γ(ρ → ηγ) = g 2 ρηγ 96πm 3 ρ m 2 ρ − m 2 η 3 . (11)
The coupling constant g φρπ = 0.811 ± 0.081 GeV −1 is obtained using the data on the decay φ → ρπ → π + π − π 0 [18] with the help of the formulas from the paper [20]. The fit of the experimental data from the SND detector [9], taking into account the relation g a 0 πη = 0.85g a 0 K + K − resulting from the q 2q2 model [1], chooses the constructive interference and gives m a 0 = 985.51 ± 0.8 MeV
g a 0 K + K − = 2.747 ± 0.428 GeV; g 2 a 0 K + K − 4π = 0.6 ± 0.015 GeV 2 χ 2 /dof = 3.1/6 .(12)
The total branching ratio, taking into account the interference, is BR(φ → (γa 0 +π 0 ρ) → γπη) = (0.79±0.2)·10 −4 , the branching ratio of the signal is BR(φ → γa 0 → γπη) = (0.75± 0.2)·10 −4 and the branching ratio of the background is BR(φ → ρ 0 π 0 → γπ 0 η) = 3.43·10 −6 . So, the integral part of the interference is negligible. The influence of the interference on the mass spectrum of the πη system is also negligible, see Fig. 1. The difference of the obtained parameters (12) from the parameters found in [9], which are m a 0 = 994± 33 8 MeV, g 2 a 0 K + K − /4π = 1.05± 0.36 0.25 GeV 2 , is due to the fact that in [9] a more refined fitting was performed considering the event distribution inside of the each bin. Notice that this difference is less than two standard deviations.
Let us specially emphasize that the value g 2 a 0 K + K − /4π = 0.6 ±0.015 GeV 2 obtained by us on no account points to the possibility of the KK molecule description [2] of the a 0 meson. In the KK molecule model, the imaginary part of the K + K − loop is dominant because the real part of the K + K − loop is suppressed by the wave function of the molecule [4], see also Sec. IV. Due to this fact, we have BR(φ → γa 0 → γπη) ≈ 1.5 · 10 −5 [4] in the KK molecule model at the same coupling constant and m a 0 = 985 MeV, which is almost by six times less than the experimental value BR(φ → γπη) = (0.88 ± 0.14 ± 0.09) · 10 −4 [9]. The divergence is by five standard deviations! Besides, in the case of molecule, the bump in the spectrum of the πη system is much narrower than the experimentally observed, [4], see also Sec. IV.
III. INTERFERENCE BETWEEN THE e + e − → γf 0 → γπ 0 π 0 AND e + e − → φ → π 0 ρ → γπ 0 π 0 REACTIONS
When analyzing the φ → γf 0 → γπ 0 π 0 decay, one should take into account the mixing of the f 0 meson with the isosinglet scalar states. The whole formalism of the mixing of two scalar f 0 and σ mesons was considered in Ref. [3]. In this paper, we consider only expressions in regard to the interference with the background reactions.
As was shown in Refs. [1,3], the dominant background is the e + e − → φ → π 0 ρ → γπ 0 π 0 reaction. The amplitude of the e + e − → ρ → π 0 ω → γπ 0 π 0 reaction is much less than the amplitude of the e + e − → φ → π 0 ρ → γπ 0 π 0 reaction. In its turn, the amplitude of the e + e − → ω → π 0 ρ → γπ 0 π 0 reaction is much less than the amplitude of the e + e − → ρ → π 0 ω → γπ 0 π 0 reaction. The amplitude of the background decay φ(p) → π 0 ρ → γ(q)π 0 (k 1 )π 0 (k 2 ) is written in the following way:
M back = g ρπ 0 φ g ρπ 0 γ φ α p ν ǫ δ q ǫ ǫ αβµν ǫ βδωǫ k 1µ k 2ω D ρ (q + k 2 ) + k 2µ k 1ω D ρ (q + k 1 )
.
The amplitude of the signal φ → γ(f 0 + σ) → γπ 0 π 0 takes into account the mixing of f 0 and σ mesons, see [3],
M f 0 = g(m)e iδ B (φǫ) − (φq)(ǫp) (pq) R,R ′ g RK + K − G −1 RR ′ g R ′ π 0 π 0 ,(14)
where R, R ′ = f 0 , σ. The matrix of propagators is defined in Ref. [3]. The phase of the signal amplitude is formed by the phase of the triangle diagram (φ → K + K − → γR) and by the phase of ππ scattering which in its turn is defined by the phase of the f 0 − σ complex, and by the phase of the elastic background of ππ scattering, δ B , see details in Refs. [6,3,13]. The mass spectrum of the process is
Γ(φ → γπ 0 π 0 ) dm = dΓ f 0 (m) dm + dΓ back (m) dm ± dΓ int (m) dm ,(15)
where the mass spectrum of the signal has the form
dΓ f 0 (m) dm = |g(m)| 2 m 2 − 4m 2 π (m 2 φ − m 2 ) 3(4π) 3 m 3 φ R,R ′ g RK + K − G −1 RR ′ g R ′ π 0 π 0 2 .(16)
The mass spectrum for the background process
e + e − → φ → π 0 ρ → γπ 0 π 0 is dΓ back (m) dm = 1 2 (m 2 φ − m 2 ) m 2 − 4m 2 π 256π 3 m 3 φ 1 −1 dxA back (m, x) ,(17)
where
A back (m, x) = 1 3 |M back | 2 = = 1 24 g 2 φρπ g 2 ρπγ m 8 π + 2m 2 m 4 πm 2 ρ − 4m 6 πm 2 ρ + 2m 4m 4 ρ − 4m 2 m 2 πm 4 ρ + 6m 4 πm 4 ρ + 2m 2m 6 ρ − 4m 2 πm 6 ρ +m 8 ρ − 2m 6 π m 2 φ − 2m 2 m 2 πm 2 ρ m 2 φ + 2m 4 πm 2 ρ m 2 φ − 2m 2m 4 ρ m 2 φ + 2m 2 πm 4 ρ m 2 φ − 2m 6 ρ m 2 φ + m 4 π m 4 φ +m 4 ρ m 4 φ 1 |D ρ (m ρ )| 2 + 1 |D ρ (m * ρ )| 2 + m 2 φ − m 2 m 2 − 2m 2 π + 2m 2 ρ − m 2 φ 2m 2 m 2 π + 2m 2 π m 2 φ − m 4 1 |D ρ (m * ρ )| 2 + 2Re 1 D ρ (m ρ )D * ρ (m * ρ m 8 π − m 6m 2 ρ + 2m 4 m 2 πm 2 ρ + 2m 2 m 4 πm 2 ρ − 4m 6 πm 2 ρ − 4m 2 m 2 πm 4 ρ + 6m 4 πm 4 ρ + 2m 2m 6 ρ − 4m 2 πm 6 ρ +m 8 ρ + m 2 m 4 π m 2 φ − 2m 6 π m 2 φ + 2m 4m 2 ρ m 2 φ − 4m 2 m 2 πm 2 ρ m 2 φ + 2m 4 πm 2 ρ m 2 φ − m 2m 4 ρ m 2 φ + 2m 2 πm 4 ρ m 2 φ − 2m 6 ρ m 2 φ − m 4 π m 4 φ − m 2m 2 ρ m 4 φ + 2m 2 πm 2 ρ m 4 φ +m 4 ρ m 4 φ(18)andm 2 ρ = m 2 π + (m 2 φ − m 2 ) 2 1 − x 1 − 4m 2 π m 2 m * 2 ρ = m 2 φ + 2m 2 π − m 2 −m ρ 2 .(19)
The interference between the amplitudes of the background process and the signal has the form
dΓ int (m) dm = 1 √ 2 m 2 − 4m 2 π 256π 3 m 3 φ 1 −1 dxA int (m, x) ,(20)
where
A int (m, x) = 2 3 Re M f M * back = 1 3 Re g(m)e δ b g φρπ g ρπ 0 γ ( R,R ′ g RK + K − G −1 RR ′ g R ′ π 0 π 0 ) (m 2 ρ − m 2 π ) 2 m 2 φ − (m 2 φ − m 2 ) 2m2 ρ D * ρ (m ρ ) + (m * 2 ρ − m 2 π ) 2 m 2 φ − (m 2 φ − m 2 ) 2m * 2 ρ D * ρ (m ρ * ) .(21)
The factor 1/2 in (17) and the factor 1/ √ 2 in (20) take into account the identity of pions. In (16), the identity of pions is taken into account by the definition of the coupling constant g Rπ 0 π 0 = g Rπ + π − / √ 2. For the fitting of the experimental data we use the model of ππ scattering considered in Ref. [3]. The phase of the elastic background of ππ scattering is taken in the form δ B = b m 2 − 4m 2 π . We fit simultaneously the phase of ππ scattering and the experimental data on the decay φ → γπ 0 π 0 .
The fit of the experimental data [10], obtained using the total statistics of SND detector, and the data on the ππ scattering phase [21][22][23][24][25], taking the value g ρπ 0 γ = 0.295±0.037 GeV −1 obtained from the data on the ρ → π 0 γ decay [19] with the help of the following expression:
Γ(ρ → π 0 γ) = g 2 ρπ 0 γ 96πm 3 ρ m 2 ρ − m 2 π 3 ,(22)
gives the constructive interference and the following parameters:
g f 0 K + K − = 4
.021 ± 0.011 GeV, g f 0 π 0 π 0 = 1.494 ± 0.021 GeV, m f 0 = 0.996 ± 0.0013 GeV, g σK + K − = 0, g σπ 0 π 0 = 2.58 ± 0.02 GeV, m σ = 1.505 ± 0.012 GeV, b = 75 ± 2.1 (1 • /GeV ), C = 0.622 ± 0.04 GeV 2 , g 2 f 0 K + K − /4π = 1.29 ± 0.017 GeV 2 . (23) The constant C takes into account effectively the contribution of multi particle intermediate states in the f 0 ↔ σ transition in G RR ′ matrix, see Ref. [3], and incorporates the subtraction constant for the R → (0 − 0 − ) → R ′ transition. We treat this constant as a free parameter.
The total branching ratio, with interference being taken into account, is BR(φ → (γf 0 + π 0 ρ) → γπ 0 π 0 ) = (1.26 ± 0.29) · 10 −4 , the branching ratio of the signal is BR(φ → γf 0 → γπ 0 π 0 ) = (1.01 ± 0.23) · 10 −4 , the branching ratio of the background is BR(φ → ρ 0 π 0 → γπ 0 π 0 ) = 0.18 · 10 −4 . The results of fitting are shown in Figs. 2 and 3. Note, that for our aim, the phase in the region m ππ < 1.1 GeV is important. The authors of Ref. [10] fit the data taking into account the background reaction φ → ρ 0 π 0 → γπ 0 π 0 . The parameters found [10] are m f 0 = 0.9698 ± 0.0045, g 2 f 0 K + K − /4π = 2.47± 0.73 0.51 GeV 2 and g 2 f 0 π + π − /4π = 0.54± 0.09 0.08 GeV 2 . They are different from the parameters found in our fitting. The difference is due to the fact that we perform the simultaneous fitting of the data on the decay φ → γπ 0 π 0 and the data on the S-wave phase of ππ scattering, taking into account the mixing of f 0 and σ mesons. In addition, in Ref. [10], the interference between the background and signal is found from the fitting meanwhile in our paper the interference is calculated. The branching ratio of the background BR(φ → ρ 0 π 0 → γπ 0 π 0 ) = 0.12 · 10 −4 used in Ref. [10] is taken from Ref. [17] in which the coupling constant g ρ 0 π 0 γ is less by 25% than resulting from the experiment. In our paper, the background is calculated on the basis of experiment and is accordingly larger, BR(φ → ρ 0 π 0 → γπ 0 π 0 ) = 0.18 · 10 −4 . Note that in Ref. [10], in contrast to us, the fitting is performed taking into account the event distribution inside each bin.
The fitting of the experimental data of the CMD-2 detector [11] and the data on the ππ scattering phase [21][22][23][24][25] gives the constructive interference and the following parameters: 17 GeV, g f 0 π 0 π 0 = 0.536 ± 0.03 GeV, m f 0 = 1.0019 ± 0.002 GeV, g σK + K − = 0, g σπ 0 π 0 = 2.61 ± 0.1 GeV, m σ = 1.585 ± 0.015 GeV, b = 70.7 ± 2.0 (1 • /GeV ), C = −0.593 ± 0.06 GeV 2 , g 2 f 0 K + K − /4π = 1.19 ± 0.03 GeV 2 . (24) The total branching ratio, taking into account the interference, is BR(φ → (γf 0 +π 0 ρ) → γπ 0 π 0 ) = (0.98 ± 0.21) · 10 −4 , the branching ratio of the signal is BR(φ → γf 0 → γπ 0 π 0 ) = (0.74 ± 0.2) · 10 −4 , the branching ratio of the background is BR(φ → ρ 0 π 0 → γπ 0 π 0 ) = 0.18 · 10 −4 . The results of fitting are shown in Figs. 4 and 5. The parameters found in [11], which are m f 0 = 0.969 ± 0.005, g 2 f 0 K + K − /4π = 1.49 ± 0.36 GeV 2 and g 2 f 0 π + π − /4π = 0.4 ± 0.06 GeV 2 , are different from the parameters found in our fitting. The difference is due to the fact that we perform the simultaneous fitting of the data on the decay φ → γπ 0 π 0 and the data on the S-wave phase of the ππ scattering, taking into account the mixing of f 0 and σ mesons and taking into account the background reaction φ → ρ 0 π 0 → γπ 0 π 0 .
g f 0 K + K − = 3.874 ± 0.
One can see from Figs. 2 and 4 that the influence of the background process on the spectrum of the φ → γπ 0 π 0 decay is negligible in the wide region of the π 0 π 0 invariant mass, m ππ > 670 MeV, or when photon energy less than 300 MeV. In the meantime, the difference from the experimental data is observed in the region m ππ < 670 MeV. We suppose this difference is due to the fact that in the experimental processing of the e + e − → γπ 0 π 0 events the background events e + e − → ωπ 0 → γπ 0 π 0 are excluded with the help of the invariant mass cutting and simulation, in so doing the part of the e + e − → φ → ρπ 0 → γπ 0 π 0 events is excluded as well.
It should be noted that the SND and CMD-2 data on the branching ratios of the φ → γπ 0 π 0 decay are quite consistent, in the meantime, the SND and CMD-2 data on the shapes of the spectra of the π 0 π 0 invariant mass are rather different. The CMD-2 shape is noticeably more narrow, compare Figs. 2 and 4. This difference reflects on the coupling constant g f 0 π 0 π 0 and the constant C, which are quite different, see Eqs. (23) and (24). In all probability, this difference will disappear when the CMD-2 group processes the total statistics.
IV. CONCLUSION.
The experimental data give evidence not only in favor of the four-quark model but in favor of the dynamical model suggested in Ref. [1], in which the discussed decays proceed through the kaon loop, φ → K + K − → γf 0 (a 0 ).
Indeed, according to the gauge invariance condition, the transition amplitude φ → γf 0 (a 0 ) is proportional to the electromagnetic tensor F µν (in our case to the electric field). Since there are no pole terms in our case, the function g(m) in (2) and (14) is proportional to the energy of photon ω = (m 2 φ − m 2 )/2m φ in the soft photon region. To describe the experimental spectra, the function |g(m)| 2 should be smooth (almost constant) in the range m ≤ 0.99 GeV, see Eqs. (4) and (16). Stopping the function ω 2 at ω 0 = 30 MeV, using the form-factor of the form 1/(1 + R 2 ω 2 ), requires R ≈ 100 GeV −1 . It seems to be incredible to explain the formation of such a huge radius in hadron physics. Based on the large, by hadron physics standard, R ≈ 10 GeV −1 , one can obtain an effective maximum of the mass spectra under discussion only near 900 MeV. In the meantime, the K + K − loop gives the natural description to this threshold effect, see Fig. 6. To demonstrate the threshold character of this effect we present Fig. 7 in which the function |g(m)| 2 is shown in the case of K + meson mass is 25 MeV less than in reality. One can see that in the region 950-1020 MeV the function |g(m)| 2 is suppressed by the ω 2 low. In the mass spectrum this suppression is increased by one more power of ω, see Eqs. (4) and (16), so that we cannot see the resonance in the region 980-995 MeV. The maximum in the spectrum is effectively shifted to the region 935-950 MeV. In truth this means that a 0 (980) and f 0 (980) resonances are seen in the radiative decays of φ meson owing to the K + K − intermediate state, otherwise the maxima in the spectra would be shifted to 900 MeV.
It is worth noting that the K + K − loop model is practically accepted by theorists, compare, for example, Ref. [26] with Ref. [27], true there is exception [28]. It was noted already in paper [1] that the imaginary part of the K + K − loop is calculated practically in a model independent way making use of the coupling constants g φK + K − and g a 0 (f 0 )K + K − due to the Low's theorem [29] for the photons with energy ω < 100 MeV which is soft by the standard of strong interaction. In the same paper it was noted that the real part of the loop (with accuracy up to 20% in the width of the φ → γf 0 (a 0 ) decay) is practically not different for the point-like particle and the compact hadron with form-factor which has the cutting radius in the momentum space about the mass of ρ meson (m ρ = 0.77 GeV). In contrast to the four-quark state which is the compact hadron [12], the bound KK state is the extended state with the spatial radius R ∼ 1/ √ m K ǫ, where ǫ is the binding energy. Corresponding form-factor in the momentum space has the radius of the order of √ m K ǫ ≈ 100 MeV for ǫ = 20 MeV, [30]. The more detailed calculation [2] gives for the radius in the momentum space the value p 0 = 140 MeV. As a result, the contribution of the virtual intermediate K + K − states in the K + K − loop is suppressed by the momentum distribution in the molecule, and the real part of the loop amplitude is negligible [4]. It leads to the branching ratio much less than the experimental one, as it was noted above. In addition, the spectrum is much narrower in the KK molecule case that contradicts to the experiment, see the behavior of the imaginary part contribution in Fig. 6 and in corresponding figures in [4].
Of course, the two-quark state is as compact as four-quark one. The question arises, why is the branching ratio in the two-quark model suppressed in comparison with the branching ratio in the four-quark model? There are two reasons. First, the coupling constant of two-quark states with the KK channel is noticeably less [3,13] and, second, there is the Okubo-Zweig-Iizuka (OZI) rule that is more important really. If the isovector a 0 (980) meson is the two-quark state, it has no strange quarks. Hence [1,3,15], the decay φ → γa 0 should be suppressed according to the OZI rule. On the intermediate state level, the OZI rule is formulated as compensation of the different intermediate states [31][32][33]. In our case these states are KK, KK * +KK * , K * K * and so on. Since, due to the kinematical reason, the real intermediate state is the only K + K − state, the compensation in the imaginary part is impossible and it destroys the OZI rule. The compensation should be in the real part of the amplitude only. As a result, the φ → γa 0 decay in the two-quark model is mainly due to the imaginary part of the amplitude and is much less intensive than in the four-quark model [1,3]. In addition, in the two-quark model, a 0 (980) meson should appear in the φ → γa 0 decay as a noticeably more narrow resonance than in other processes, see the behavior of the imaginary part contribution in Fig. 6.
As regards to the isoscalar f 0 (980) state, there are two possibilities in the two-quark model. If f 0 (980) meson does not contain the strange quarks the all above mentioned arguments about suppression of the φ → γa 0 decay and the spectrum shape are also valid for the φ → γf 0 decay. Generally speaking, there could be the strong OZI violation for the isoscalar qq states ( mixing of the uū, dd and ss states) with regard to the strong mixing of the quark and gluon degree of freedom which is due to the nonperturbative effects of QCD [34]. But, an almost exact degeneration of the masses of the isoscalar f 0 (980) and isovector a 0 (980) mesons excludes such a possibility. Note also, the experiment points directly to the weak coupling of f 0 (980) meson with gluons, B(J/ψ → γf 0 → γππ) < 1.4 · 10 −5 [35].
If f 0 (980) meson is close to the ss state [15,36] there is no suppression due to the the OZI rule. Nevertheless, if a 0 (980) and f 0 (980) mesons are the members of the same multiplet, the φ → γf 0 branching ratio, BR(φ → γπ 0 π 0 ) = (1/3)BR(φ → γππ) ≈ 1.8·10 −5 , is significantly less than that in the four-quark model, due to the relation between the coupling constants with the KK, πη and KK, ππ channels inherited in the two-quark model, see Refs. [1,3]. In addition, in this case there is no natural explanation of the a 0 and f 0 mass degeneration.
Only in the case when the nature of f 0 (980) meson in no way related to the nature of a 0 (980) meson (which, for example, is the four-quark state) the branching ratio experimentally observed φ → γf 0 could be explained by ss nature of f 0 (980) meson. But, from the theoretical point of view, such a possibility seems awful [15].
V. ACKNOWLEDGEMENT
We gratefully acknowledge discussions with V.P. Druzhinin, A.A. Kozhevnikov, G.N. Shestakov, and Z.K. Silagadze. This work was supported in part by INTAS-RFBR, grant IR-97-232.
FIG. 1 .
1Fitting of dBR(φ → γπη)/dm × 10 4 GeV −1 with the background is shown with the solid line, the signal contribution is shown with the dashed line.
FIG. 2 .
2Fitting of dBR(φ → γπ 0 π 0 )/dm × 10 4 GeV −1 with the background is shown with the solid line, the signal contribution is shown with the dashed line. The dotted line is the interference term. The data are from the SND detector.
FIG. 3 .
3Fitting of the phase δ 0 0 of ππ scattering.
FIG. 4 .FIG. 5 .
45Fitting of dBR(φ → γπ 0 π 0 )/dm × 10 4 GeV −1 with the background is shown with the solid line, the signal contribution is shown with the dashed line. The dotted line is the interference term. The data are from the CMD-2 detector. Fitting of the phase δ 0 0 of ππ scattering.
FIG. 6 .
6The function |g(m)| 2 is drawn with the solid line. The contribution of the imaginary part is drawn with the dashed line. The contribution of the real part is drawn with the dotted line.
FIG. 7 .
7The function |g(m)| 2 for m K = 469 MeV is drawn with the solid line. The contribution of the imaginary is drawn with the dashed line. The contribution of the real part is drawn with the dotted line.
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The authors of this paper use the amplitude of the φ → γ(f 0 + background) → γπ 0 π 0 decay which does not vanish when ω → 0, i.e. which does not satisfy the gauge invariance condition. This amplitude is not adequate to the physical problem since the mass spectrum under discussion should have the behavior ω 3 at ω → 0 and not ω as in hep-ph/0011191. With the same result one can study the electromagnetic formfactor of π meson. A V Anisovich, V V Anisovich, V A Nikonov, hep-ph/0011191in the e + e − → π + π − reaction near the threshold considering that the cross-section of the process is proportional to the momentum of π meson while it is proportional to the momentum in the third powerA.V. Anisovich, V.V. Anisovich, and V.A. Nikonov, hep-ph/0011191. The authors of this paper use the amplitude of the φ → γ(f 0 + background) → γπ 0 π 0 decay which does not vanish when ω → 0, i.e. which does not satisfy the gauge in- variance condition. This amplitude is not adequate to the physical problem since the mass spectrum under discussion should have the behavior ω 3 at ω → 0 and not ω as in hep-ph/0011191. With the same result one can study the electromagnetic form- factor of π meson in the e + e − → π + π − reaction near the threshold considering that the cross-section of the process is proportional to the momentum of π meson while it is proportional to the momentum in the third power.
Hereinafter the comment could not be included in Physical Review D (published 30 March 2001) for the temporal reasons. Hereinafter the comment could not be included in Physical Review D (published 30 March 2001) for the temporal reasons.
But the real trouble is that the calculation in hep-ph/0011191 is not gauge invariant. The calculation of the φ → qq → γf 0 amplitude requires a gauge invariant regularization ( for example, the substraction at ω = 0) in spite of the integral convergence. A text-book example of such a kind is the γγ → e + e − (or qq) → γγ scattering. The authors of the paper under discussion obtained that A φ→γf0 =. hep-ph/0011191To provide the spectrum behavior ω 3 at ω → 0 the authors, correcting some typos and undoing some references in hep-ph. our symbols) g(m)(g f0K + K − /e) = 0 at ω =. m 2 φ − m 2 )/2m φ = 0 (A φ→γf0 does not depend on m at all), seeTo provide the spectrum behavior ω 3 at ω → 0 the authors, correcting some typos and undoing some references in hep-ph/0011191 v4 27 Mar 2001, inserted a crazy common factor F thresh (ω) = 1 − exp {−(ω/36 MeV) 2 } in the φ → γ(f 0 + background) → γπ 0 π 0 amplitude without any ex- planations, see Eq. (39) in hep-ph/0011191. But the real trouble is that the calculation in hep-ph/0011191 is not gauge invariant. The calcula- tion of the φ → qq → γf 0 amplitude requires a gauge invariant regu- larization ( for example, the substraction at ω = 0) in spite of the in- tegral convergence. A text-book example of such a kind is the γγ → e + e − (or qq) → γγ scattering. The authors of the paper under discus- sion obtained that A φ→γf0 = (in our symbols) g(m)(g f0K + K − /e) = 0 at ω = (m 2 φ − m 2 )/2m φ = 0 (A φ→γf0 does not depend on m at all), see
This means that the authors created the false pole in the invariant amplitude free from kinematical singularites: eA φ→γf0 / m 2 φ − m 2 × (φ µ p ν − φ ν p µ ) (ǫ µ q ν − ǫ ν q µ ), compare with Eq. Eq, hep-ph/0011191So, once again, the calculation of hep-ph/0011191 is not adequate to the physical problemEq. (30) in hep-ph/0011191. This means that the authors created the false pole in the invariant amplitude free from kinematical singularites: eA φ→γf0 / m 2 φ − m 2 × (φ µ p ν − φ ν p µ ) (ǫ µ q ν − ǫ ν q µ ), compare with Eq. (9) in hep-ph/0011191. So, once again, the calculation of hep-ph/0011191 is not adequate to the physical problem!
. F E Low, Phys. Rev. 110574F.E. Low, Phys. Rev. 110, 574 (1958).
the potential in the momentum space was taken as the momentum distribution in the molecule instead of the wave function in the momentum space. ; V E Unfortunately, Eur Markushin, J Phys, 8389But the momentum distribution radius of the potential is 5-8 times as large as one of the wave function, that was the reason for the misleading conclusion on the possibility to explain the Novosibirsk reasults in the molecule caseUnfortunately, in the interesting paper V.E. Markushin, Eur. Phys.J., A8, 389 (2000), the potential in the momentum space was taken as the momentum distribution in the molecule instead of the wave function in the momentum space. But the momentum distribution radius of the potential is 5-8 times as large as one of the wave function, that was the reason for the misleading conclusion on the possibility to explain the Novosibirsk reasults in the molecule case.
. H Lipkin, Nucl. Phys. B. 291720H. Lipkin, Nucl. Phys. B 291, 720 (1987).
. P Geiger, N Isgur, Phys. Rev. D. 44799P. Geiger and N. Isgur, Phys. Rev. D 44, 799 (1991).
. N N Achasov, A A Kozhevnikov, Phys. Rev. D. 4927N.N. Achasov and A.A. Kozhevnikov, Phys. Rev. D 49, 27 (1994).
. A I Vainshtein, V I Zakharov, V A Novikov, M A Shifman, Fiz. Elem. Chastits At. Yadra. 13224Sov. J. Part. Nucl.A.I. Vainshtein, V.I. Zakharov, V.A. Novikov, M.A. Shifman, Fiz. Elem. Chastits At. Yadra 13, 542 (1982) [ Sov. J. Part. Nucl., 13, 224 (1982)].
G Eigen, Proc. of the XXIV Int. Conf. on High Energy Phys. of the XXIV Int. Conf. on High Energy PhysMunich; BerlinSpringer-Verlag4590G. Eigen, Proc. of the XXIV Int. Conf. on High Energy Phys., Munich, August 4-10, 1988 (Eds. R. Kotthaus and J.H. Kuhn) Session 4(Berlin: Springer-Verlag, 1988) p. 590.
. N A Törnqvist, Z. Phys. C. 68647N.A. Törnqvist, Z. Phys. C 68, 647 (1995).
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[
"Plasma solitons in gated two-dimensional electron systems: exactly solvable analytical model for the regime beyond weak non-linearity",
"Plasma solitons in gated two-dimensional electron systems: exactly solvable analytical model for the regime beyond weak non-linearity"
]
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"A A Zabolotnykh \nKotelnikov Institute of Radio-engineering and Electronics of the RAS\nMokhovaya 11-7125009MoscowRussia\n"
]
| [
"Kotelnikov Institute of Radio-engineering and Electronics of the RAS\nMokhovaya 11-7125009MoscowRussia"
]
| []
| We analytically study plasma solitary waves, or solitons, in a two-dimensional (2D) electron system (ES) placed in close proximity to and between two ideal metallic gates. As a rule, solitons are described using a perturbative approach applicable only in the weak non-linearity regime. In contrast, we analyze solitons considering a non-perturbative model. This framework enables an exact analytical description of the soliton shape. Moreover, it can be achieved in the regime beyond weak non-linearity -when the concentration deviation due to the soliton is of the order of the equilibrium concentration. We determine the conditions required for a soliton to exist and derive the relationship between its amplitude, width, and velocity. We believe that our results obtained for the given model can provide valuable insight into the physics of non-linear waves. | 10.1103/physrevb.105.l201403 | [
"https://arxiv.org/pdf/2202.04503v2.pdf"
]
| 246,680,045 | 2202.04503 | 869c2eea788d5cfd4c4447281e43bd1f275b53fa |
Plasma solitons in gated two-dimensional electron systems: exactly solvable analytical model for the regime beyond weak non-linearity
19 May 2022
A A Zabolotnykh
Kotelnikov Institute of Radio-engineering and Electronics of the RAS
Mokhovaya 11-7125009MoscowRussia
Plasma solitons in gated two-dimensional electron systems: exactly solvable analytical model for the regime beyond weak non-linearity
19 May 2022
We analytically study plasma solitary waves, or solitons, in a two-dimensional (2D) electron system (ES) placed in close proximity to and between two ideal metallic gates. As a rule, solitons are described using a perturbative approach applicable only in the weak non-linearity regime. In contrast, we analyze solitons considering a non-perturbative model. This framework enables an exact analytical description of the soliton shape. Moreover, it can be achieved in the regime beyond weak non-linearity -when the concentration deviation due to the soliton is of the order of the equilibrium concentration. We determine the conditions required for a soliton to exist and derive the relationship between its amplitude, width, and velocity. We believe that our results obtained for the given model can provide valuable insight into the physics of non-linear waves.
Studies of low-dimensional structures are at the heart of modern condensed matter physics. A significant part of the research is devoted to the electronic properties of matter, in particular, the collective charge density excitations -plasma waves, or plasmons. Plasmons are especially intriguing when considered in so-called gated two-dimensional (2D) electron systems (ESs), with a planar metal electrode (gate) in proximity and parallel to the 2DES. Remarkably, these gated plasmons make possible the detection of terahertz radiation [1][2][3][4][5][6][7][8], compression of light [9,10], study of relativistic effects [11], etc.
This Letter presents an analytical investigation of a particular type of non-linear electron-density waves in a gated 2DES -the solitary plasma waves, or solitons. Solitons are space-localized waves that preserve their shape as they propagate. Qualitatively, the waveprofile stability can be explained by the fact that the non-linearity of the wave propagation equations is compensated for by the dispersion effect, namely, by the dependence of phase velocity on the wave vector (in linear regime). To begin with, let us consider the dispersion law of gated plasmons. To be specific, we examine a 2DES screened by two ideal metal gates positioned above and below it, as illustrated on the inset of Fig. 1. Then, the plasmon spectrum can be expressed as follows [12,13] (we use CGS system of units throughout the Letter):
ω 2 = 2πe 2 n 0 mε q tanh qd,(1)
where ω and q are, respectively, the plasmon frequency and wave vector in the 2DES plane, n 0 is the equilibrium concentration of electrons, −e and m are the electron charge and effective mass, d and ε are the distance and dielectric permittivity between the 2DES and the gate. We note that in deriving (1), the 2DES is assumed to be "clean", with infinitely large electron relaxation time, and the electromagnetic retardation effects are neglected. The resultant spectrum is plotted in Fig. 1 in the solid blue line. Unlike the 3DES, the complexity of the analytical description of plasmons in a 2DES is associated with solving the Poisson equation, which provides the relation be- tween the potential and charge density in a plasma wave. The challenge arises from the non-local (integral) nature of the relationship since the charges are confined to the 2DES plane, whereas the potential occupies the whole 3D space. For example, to find the spectrum for linear waves (1), the Poisson equation is solved first using the Fourier transform in the coordinates along the 2DES plane. Then, the solution to the resultant differential equation is obtained in the z-coordinate -in the direction perpendicular to the 2DES plane (see the inset in Fig. 1). In the case of non-linear waves, however, the given technique becomes inefficient. Hence, describing these waves requires solving a non-linear integral equation, with the exact solution almost impossible to find. Therefore, in practice, obtaining an analytical description of non-linear waves in low-dimensional systems generally involves an expansion of the Poisson equation into a series governed by a small parameter [14][15][16][17][18].
Let us illustrate the expansion procedure on the example of plasmons in a gated 2DES. Considering a δ-thin in z-direction 2DES, we can use the Poisson equation to express the relation between the deviation of the 2D electron concentration n(x, t) from its equilibrium value and the potential ϕ(x, t) in the 2DES plane, at z = 0, as follows:
ϕ(x, t) = −e ε +∞ −∞ G(x − x ′ )n(x ′ , t)dx ′ ,(2)
where the Green's function of the Poisson equation G(x) and its Fourier transform G(q) are defined as
G(x) = −2 ln tanh π|x| 4d , G(q) = 2π q tanh(qd),(3)
and the spectrum in (1) is related to G(q) as ω 2 = e 2 n 0 q 2 G(q)/mε. Then, the above-mentioned simplification of integral equation (2) is achieved by expanding G(q) into a series with respect to the small parameter qd ≪ 1. Thus, the Green's function G(q) is replaced by its series expansion:
G e (q) = 2πd(1 − d 2 q 2 /3). Consequently, in x-space, G e (q) becomes G e (x) = 2πd δ(x) + δ ′′ (x)d 2 /3
, and the Poisson equation (2) takes the form of a simple differential equation [19]. The plasmon spectrum corresponding to G e (q) takes the form Fig. 1). Importantly, since the expansion is valid only for qd ≪ 1, this approach can be applied to non-linear waves and solitons, in particular, only in the regime of weak non-linearity, when the deviation of the electron concentration associated with the soliton n(x, t) is small compared to the equilibrium concentration n 0 : |n(x, t)| ≪ n 0 [20].
ω 2 = 2πe 2 n 0 q 2 d(1 − d 2 q 2 /3)/mε (gray dashed line in
In this Letter we analyze solitons in a gated 2DES using a different method that allows to obtain an exact analytical solution with no restrictions of weak nonlinearities |n| ≪ n 0 and qd ≪ 1, unlike the conventional approach of expanding G(q) into the q-series. We replace G(q) with the model function G m (q). This function has two key properties: it is proportional to 1/(α 2 + q 2 d 2 ), where α is the model constant, i.e. it has a simple pole; and it has the same coefficients for the q 0 and q 2 terms in the q-series as G(q) [21]. The function G m that meets the desired criteria can be formulated as follows:
G m (x) = πα exp −α|x| d , G m (q) = 2πd 1 + q 2 d 2 /α 2 ,(4)
and α = √ 3 [22]. The resultant plasmon spectrum, defined as ω 2 = 2πe 2 n 0 q 2 d/mε(1 + q 2 d 2 /α 2 ), is plotted (for α = √ 3) in the solid green line in Fig. 1. It is evident that the given model (4) describes the exact spectrum with sufficient accuracy up to the wavevector values of qd ≈ 3. Therefore, G m provides a much better approximation than G e , as it is applicable over a wider range of qd < ∼ 1 compared to the traditional expansion approach valid only for qd ≪ 1.
It is worth mentioning that overall, considering G m instead of G e is analogous to using the regularized rather than ordinary Korteweg-De Vries equation [23]. In hydrodynamics, for example, a similar simplification of the integral equations has been successfully applied to describe non-linear waves and solitons on the surface of a liquid, including their peaking and breaking [24].
In 2D plasma physics, a similar method of replacing the kernel in the Poisson equation was employed, most likely for the first time, to analyze the (linear) edge plasma oscillations in a half-plane 2DES [25]. Afterwards, it was widely used to describe plasmons in various bounded or inhomogeneous 2DESs, including conventional systems [26][27][28][29] with possible anisotropy [30,31], graphene [32,33], and topological insulators [34,35]. Now let us proceed to the analytical description of a 2D plasma soliton. At this point, besides the Poisson equation (2), we need to consider additional equations characterizing electron dynamics in a 2DES. In this Letter, we follow a standard approach of using the Euler equation for the drift velocity of electrons v(x, t) and the continuity equation for the deviation in the electron concentration n(x, t) from its equilibrium value n 0 :
∂ t v(x, t) + ∂ x v 2 (x, t) 2 = e m ∂ x ϕ(x, t), ∂ t n(x, t) + ∂ x [(n 0 + n(x, t))v(x, t)] = 0.(5)
We seek the solutions in the form of a wave traveling along the x-axis with the velocity u. Hence, we introduce the argument ξ = x−ut. After this substitution, Eqs. (5) can be integrated. Then, considering the result along with Eq. (2), we obtain the following set of equations:
− uv(ξ) + v 2 (ξ) 2 − e m ϕ(ξ) = 0, − un(ξ) + [n 0 + n(ξ)] v(ξ) = 0, ϕ(ξ) = −e ε +∞ −∞ G(ξ − ξ ′ )n(ξ ′ )dξ ′ ,(6)
where in the first two equations, the integration constants are set equal to zero since we seek the solitons with v, n, and ϕ vanishing as ξ → ±∞.
From the second equation in (6), the velocity can be expressed as v(ξ) = un(ξ) n 0 + n(ξ) .
Here, the denominator cannot equal zero or change its sign as the deviation −n(ξ) cannot exceed n 0 .
Next, we substitute v(ξ) from (7) into the first equation in (6) and, subsequently, n(ξ) into the Poisson equation. Thus, we arrive at a single expression for ϕ(x, t) that can be rewritten in terms of the dimensionless soliton potential Φ(ξ) = 2eϕ(ξ)/mu 2 as follows: At the same time, the condition of n 0 + n(ξ) > 0 implies the inequality Φ(ξ) > −1.
Φ(ξ) = 2e 2 n 0 mεu 2 +∞ −∞ G(ξ − ξ ′ ) 1 − 1 1 + Φ(ξ ′ ) dξ ′(8)
The derived non-linear integral equation (8) fully describes solitons in a gated 2DES. In principle, such and similar equations can be solved numerically [15,[36][37][38][39]. Nevertheless, in the following discussion, we focus on the analytical treatment of Eq. (8) to achieve a comprehensible and straightforward description of solitons.
As was mentioned above, to find the analytical solution, we substitute G m (ξ) from (4) for G(ξ) in (8), which greatly simplifies Eq. (8) since G m (ξ) is essentially the Green's function of the differential operator
L m = d 2πα 2 −∂ 2 ξ + α 2 d 2 .(9)
Hence, applying L m to Eq. (8) after the substitution yields the following differential equation:
Φ ′′ (X) − α 2 Φ(X) + 2 U 2 1 1 + Φ(X) − 1 = 0.
(10) Here we introduce the dimensionless argument X and the soliton velocity U defined as:
X = ξ d , U = u v p , where v p = 2πe 2 n 0 d mε .(11)
In this case, v p is the velocity of gated plasmons (1) considered in the long-wavelength limit qd ≪ 1 [40], and the double prime in Eq. (10) denotes the second derivative with respect to X. We note that Eq. (10) in effect represents a non-linear oscillator defined as Φ ′′ + V (Φ) = 0. Naturally, it can be integrated to obtain an equivalent of the energy conservation law. Thus, multiplying Eq. (10) by Φ ′ (X) and then integrating it results in the following relation:
Φ ′ (X) 2 2 + α 2 W (Φ) = E, where(12)W (Φ) = − Φ 2 2 + 2 U 2 2 + Φ − 2 √ 1 + Φ .(13)
Here, the "total energy" constant E is chosen so that W (0) = 0. It is important to emphasize that the model parameter α is included in the "potential energy" term α 2 W (Φ) simply as a scaling factor. The function W (Φ) itself and the features of the potential energy are entirely independent of α. In fact, the dimensionless soliton velocity U is the only critical factor that defines the shape of W (Φ). Therefore, it is natural to analyze the characteristic patterns of W (Φ) for different values of U , as indicated in Fig. 2. For solitons, Φ(X) and Φ ′ (X) tend to zero at X → ±∞. Consequently, both terms in the left-hand side of Eq. (12) vanish, which, in turn, leads to E = 0. From Fig. 2 it is clear that finite solutions Φ(X) with E = 0 exist only when the velocity satisfies the condition of 1 < U < 2. Although the investigation of the entire structure of non-linear waves is beyond the scope of this Letter, for a more consistent presentation, we provide a brief qualitative analysis of the cases for U < 1 and U > 2 as follows. When U < 1, there are finite solutions with E > 0, where Φ(X) oscillates about zero value. These solutions most likely correspond to non-linear plasma oscillations. In the limit of small amplitude (when E → 0), they take the standard form with the spectrum in (1). As the amplitude and E increase (assuming that U remains constant), when E becomes larger than E c (Fig. 2(a)), the solution ceases to be finite, and the oscillations break down, which is ordinary for non-linear oscillations. However, it should be noted that in the derivation of Eqs. (12) and (13), we rely on the condition of n, ϕ, and v vanishing at ξ → ∞, which is not satisfied for the oscillatory regime. Therefore, the analysis above is only qualitative. For U > 2, we have W (Φ = −1) < 0, and all waves with E near zero are likely to be unstable, breaking up to form shock waves, similar to the case of ion-acoustic waves in gaseous plasma [41]. Now, let us expound on the solitonic regime of 1 < U < 2 in particular. In this case, the amplitude of the soliton potential Φ max and the concentration n max can be determined on account of the vanishing potential energy. Hence, W (Φ max ) = 0 leads to −Φ max = 4(U − 1)/U 2 and
n max n 0 = 1 √ 1 + Φ max − 1 = 2 · U − 1 2 − U .(14)
Width, Velocity, Figure 3. The dimensionless soliton width ∆X = ∆ξ/d extracted from the concentration dependency n(X), plotted as a function of the velocity U = u/vp (11). Gray and green dashed lines denote the asymptotics (15) and (16), correspondingly. The width vanishes at u/vp = 2. The inset shows the soliton shape at u/vp = 1.7.
More significantly, Eqs. (12) and (13) can be integrated exactly to provide an explicit relationship between the potential Φ (along with the concentration n) and X = (x − ut)/d. As the resulting expression is rather cumbersome, it is included in the Supplementary Material [42]. The obtained equation allows us to calculate the soliton width as a function of velocity U , as shown in Fig. 3. We consider the two limiting cases to find a straightforward analytical expression for the soliton width extracted from the concentration dependency n(X). First, in the low-velocity regime of 0 < U − 1 ≪ 1, corresponding to the limit of weak non-linearity, the full soliton width ∆X at the half n max can be defined as
∆X = ∆ξ d = 2 √ 2 ln(1 + √ 2) α √ U − 1 ,(15)
which is in agreement with the previous studies of Korteweg-De Vries-like plasma solitons [14].
In the second limiting case of 0 < 2 − U ≪ 1, the full width can be expressed as
∆X = ∆ξ d = 8 3α · (2 − U ) 3/2 .(16)
Clearly, it follows from Eq. (16) that the width ∆X tends to zero as U → 2. In this case, it invalidates the given approach as it violates the assumption for the parameter qd ≈ d/∆ξ = (∆X) −1 to be of the order of unity or less required for valid substitution of G m for G. Therefore, we cannot conclude with certainty whether solitons with U > ∼ 2 exist. In the above analysis we assume that the electron relaxation time τ is infinitely large. The finite value of τ in the framework of our approach leads to the appearance of an additional term v(x, t)/τ in the left-hand side of the first equation in (5). That term leads to the damping of solitons (as well as usual plasma waves) over the length L, which can be estimated as L = u · τ . Clearly, as the soliton propagates, its amplitude decreases and its shape is not preserved. Therefore, in that sense, solitons do not exist at finite values of τ (see also discussion after Eq. (11) in Ref. [15]).
Last but not least, let us stress the significance of considering a gated 2DES. The critical issue is that for a soliton to be stable, its velocity must not coincide with the phase velocity of a linear wave. Otherwise, the soliton undergoes the decay with emission of linear waves [43]. In an ungated 2DES, linear plasmons have a square-root dispersion law ω ∝ √ q [44] (corresponding to the limit of qd ≫ 1 in the spectrum (1)), i.e., their phase velocities take on any value. Hence, stable solitons cannot exist in such a system (see also discussion in Sec. VI.B in Ref. [18]). However, the presence of the gates (or, at least, one gate) limits the range of possible velocities of linear plasmons by v p (11), allowing for the solitons with velocities of u > v p .
In summary, we consider solitary plasma waves in a 2DES with ideal metallic gates in its vicinity. In our investigation, we employ a non-perturbative model (4) that permits finding exactly and analytically the soliton velocity, amplitude, and shape. We overcome the drawback of conventional perturbative methods restricted by weak non-linearity conditions. In contrast, our technique enables describing the regime beyond weak non-linearity and dispersion limitations. In other words, it applies when the concentration deviation is of the order of the equilibrium concentration and the soliton width is comparable to the separation distance between the 2DES and the gate. We establish that solitons exist provided that their velocity u lies in a finite range. We find the lower limit of u to be determined by the maximum velocity of linear plasmons v p (11). As u reaches the higher limit 2v p , the soliton width (based on the concentration dependency) approaches zero, while the amplitude tends to infinity. We conclude that solitons do not exist beyond the specified range, for u > 2v p . Although we consider an ordinary 2DES with simple dynamics and constant effective mass of charges (5), we believe that the proposed approach can be useful in studies of non-linear waves in various advanced structures, for example, in van der Waals heterostructures, in which charge dynamics is much more complex.
ACKNOWLEDGMENTS
The author is grateful to Igor Zagorodnev and Vladimir Volkov for valuable discussions. The work was financially supported by the Russian Science Foundation (Project No. 21-72-00114).
Figure 1 .
1The exact spectrum of linear plasmons (1) (blue) compared against the approximations (gray and green) obtained, respectively, based on the expansion at qd ≪ 1, and the model Green's function of the Poisson equation(4). The inset shows the schematic view of the 2DES setup under consideration.
Figure 2 .
2The effective potential energy W (Φ) (13) plotted for different values of the dimensionless soliton velocity (11) U = 0.7 (a), U = 1.6 (b), and U = 2.4 (c). Solitons correspond to the finite solutions with zero energy that exist only for 1 < U < 2, as indicated by red line in plot (b).
Appendix A: Supplementary Material for "Plasma solitons in gated two-dimensional electron systems: exactly solvable analytical model for the regime beyond weak non-linearity"To derive the relationship between the potential Φ and X = (x − ut)/d, we can rewrite Eqs.(12)and(13)from the main text in the following form:where α is the model constant equal √ 3[21], and the integration limits are chosen so that Φ(X = 0) = Φ max . Considering √ Φ + 1 as a new function, the integral on the left-hand side of Eq. (A1) can be found exactly. Hence, Eq. (A1) can be written in a different form as follows:Thus, we have determined the direct relation between Φ and X. To find the deviation in the concentration n(X), we can substitute into Eq. (A2) the following relation between Φ and n:n(X) n 0 = 1where n 0 is the equilibrium concentration.
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| []
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[
"Spatially-temporal dynamics of a passively Q-switched Raman-active solid-state oscillator",
"Spatially-temporal dynamics of a passively Q-switched Raman-active solid-state oscillator"
]
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"V L Kalashnikov \nInstitut für Photonik\nGusshausstr. 27/387A-1040Wien, ViennaTUAustria\n",
"A M Malyarevich \nInstitute for Optical Materials and Technologies\n65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus\n",
"K V Yumashev \nInstitute for Optical Materials and Technologies\n65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus\n"
]
| [
"Institut für Photonik\nGusshausstr. 27/387A-1040Wien, ViennaTUAustria",
"Institute for Optical Materials and Technologies\n65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus",
"Institute for Optical Materials and Technologies\n65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus"
]
| []
| The spatially-temporal model of an all-solid-state passively Q-switched oscillator with an active medium providing the stimulated Raman scattering is presented. The model does not presume a Gaussian shape of the cylindrically symmetric modes at both fundamental and Stokes wavelengths. It is found, that the highly-nontrivial spatially-temporal dynamics can be regularized by the optimal choice of the oscillator parameters, viz. initial transmission of a saturable absorber, curvature of a spherical mirror, and output mirror transmission at the fundamental and Stokes wavelengths. As a result, the pulse can be substantially temporally squeezed and spatially broadened at both fundamental and Stokes wavelengths. | 10.1016/j.optcom.2009.12.032 | [
"https://arxiv.org/pdf/0909.3730v1.pdf"
]
| 118,592,544 | 0909.3730 | 015de1a03393cd77687952b91ea9606c5685f918 |
Spatially-temporal dynamics of a passively Q-switched Raman-active solid-state oscillator
21 Sep 2009
V L Kalashnikov
Institut für Photonik
Gusshausstr. 27/387A-1040Wien, ViennaTUAustria
A M Malyarevich
Institute for Optical Materials and Technologies
65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus
K V Yumashev
Institute for Optical Materials and Technologies
65 Nezavisimosti Ave., Bldg. 17220013MinskBelarus
Spatially-temporal dynamics of a passively Q-switched Raman-active solid-state oscillator
21 Sep 2009Q-switchingSolid-state lasersstimulated Raman scatteringspatially-temporal dynamics PACS: 4260Gd4255Ye4265Sf
The spatially-temporal model of an all-solid-state passively Q-switched oscillator with an active medium providing the stimulated Raman scattering is presented. The model does not presume a Gaussian shape of the cylindrically symmetric modes at both fundamental and Stokes wavelengths. It is found, that the highly-nontrivial spatially-temporal dynamics can be regularized by the optimal choice of the oscillator parameters, viz. initial transmission of a saturable absorber, curvature of a spherical mirror, and output mirror transmission at the fundamental and Stokes wavelengths. As a result, the pulse can be substantially temporally squeezed and spatially broadened at both fundamental and Stokes wavelengths.
Introduction
Solid-state Q-switched oscillators allowing nano-and sub-nanosecond pulsing find applications in a lot of areas including medicine, spectroscopy, environment monitoring, etc. Passive Q-switching based on the use of both semiconductor [1,2] and crystalline [3] saturable absorbers is of particular interest due to compactness, simplicity, high damage threshold, and diodepumping ability of an oscillator.
Among the active media allowing diode-pumped Q-switching, KY(WO 4 ) 2 (KYW) and KGd(WO 4 ) 2 (KGW) crystals doped by Yb 3+ and Nd 3+ ions are known as the materials providing an efficient stimulated Raman scattering (SRS) [4,5,6,7,8]. As a result, there is possible a high-efficient selffrequency shift and a simultaneous two-wavelengths pulsing (e.g., at 1.35 and 1.54 µm for a Nd:KGW active medium) directly from an oscillator.
Theoretical studies of a Q-switched oscillator with the intra-cavity SRS have been based on the well-established rate-equations approach [9,10,11]. The oscillator parameters providing a pulse width minimization [10] and an output energy maximization [11] at both fundamental and Stokes wavelengths have been defined. Simultaneously, it has been found that the spatial structure of a laser field is strongly affected by the SRS [9]. This means that the mode transformation has to be taken into account along with the temporal evolution of fields and populations inside an active medium and a saturable absorber. Such a model has been developed to a Gaussian mode approximation for both fundamental and Stokes fields [12]. This has allowed defining the optimal values of the Raman gain and the ratio of pump and laser beams, which provide a single pulse operation of oscillator with the most efficient SRS.
Nevertheless, a transient character of Q-switching can prevent from the CW mode formation and disturb substantially the spatial structure of a laser field. This requires to take into account the spatial dynamics on a par with the temporal one.
In this work we present the analysis of spatially-temporal dynamics of a passively Q-switched Raman-active oscillator. The analysis is based on a rate-equations approach but without imposing a limitation on the shape of cylindrically symmetric transverse distribution of a field. The results demonstrate that there exist both spatial extra-broadening and squeezing as well as transition between non-Gaussian and Gaussian spatial profiles in dependence on the oscillator parameters (viz. output mirror transmission at the fundamental and Stokes wavelengths, initial transmission of a saturable absorber and curvature of a spherical mirror). The SRS contribution controlled by the optimized sets of oscillator parameters allows the regularization of the spatially-temporal structure of a field and the substantial pulse shortening.
Numerical procedure
The oscillator under consideration consists of the flat (output) and spherical mirrors. The active medium (Nd 3+ :KGW) is placed at the resonator center, and the saturable absorber (V 3+ :YAG) is placed on the output mirror.
The model is based on a direct generalization of that presented in [10,12]. It is supposed that the time-dependent (t is the time, which is periodical with the cavity period T cav : t ∈ [0, T cav ]) fields a f,s (indexes f and s correspond to the fundamental and Stokes fields, respectively) have the radially-symmetric transverse spatial distributions (r ∈ ]0, R] is the radial coordinate).
Inside an active medium, the dynamics is modeled on basis of the splitstep Hankel's method. That is the active medium volume is considered to be divided into 10 transverse slices with the thickness ∆z (z is the longitudinal coordinate), in which the dynamics within the time-domain and the domain of spatial frequencies (ν ≡ c/2πR, c is the velocity of light, R = 0.5 ÷ 1.5 cm) [13] is evolved step-by-step:
a f (z, r, t) = ∆z 2T cav a f (z, r, t) g (r, t) − g s |a s (z, r, t)| 2 ȧ s (z, r, t) = ∆z 2T cav a s (z, r, t) g s |a f (z, r, t)| 2 g (r, t) = − σ g λ f hc g (r, t) |a f (z, r, t)| 2 ⊗ (1) ⊗ã f,s (z + ∆z, ν, t) =ã f,s (z, ν, t) exp −ik f,s ∆z + i 2 k f,s ∆zλ 2 f,s ν 2 .
We shall suppose, that the field intensities |a f,s | 2 are normalized to hc/λ s σ g T cav (h is the Planck constant; λ f =1.35 µm and λ s =1.54 µm are the fundamental and Stokes wavelengths, respectively; σ g =0.76×10 −19 cm 2 is the gain cross-section, T cav =0.8 ns), and the time is normalized to T cav . The gain coefficient is g and its initial value equals to the threshold one: (ln 1/T 2 0 + ln 1/ρ f + l)/2L g . In the last expression, the varied values T 0 and ρ f correspond, respectively, to the initial transmission of saturable absorber and the reflection of output mirror at the fundamental wavelength; l = 0.05 is the unsaturable net-loss coefficient. L g =5 cm is the active medium length. The stimulated Raman scattering inside an active medium is described by the coefficient g s and its dimensional value amounts to 6 cm/GW. It is assumed that there exists no anti-Stokes and higher-order Stokes scattering as well as that the gain is saturable by only fundamental field. Also, the initial gain distribution is assumed to be spatially homogeneous.
The field representationsã f,s within the spatial frequency domain are obtained by the means of the fast Hankel's transformation [13,14]. k f,s are the wave-numbers corresponding to the fundamental and Stokes fields, respectively.
A similar procedure describes the dynamics inside a saturable absorber:
a f (z, r, t) = − ∆z 2T cav a f (z, r, t) [n (r, t) − σ esa (N − n (r, t))] a s (z, r, t) = − ∆z 2T cav a s (z, r, t) n (r, t) n (r, t) = − σ f λ f hc n (r, t) |a f (z, r, t)| 2 − σ s λ f hc n (r, t) |a s (z, r, t)| 2 + N − n (r, t) T r ⊗ (2) ⊗ã f,s (z + ∆z, ν, t) =ã f,s (z, ν, t) exp −ik f,s ∆z + i 2 k f,s ∆zλ 2 f,s ν 2
Here, n is the loss coefficient and its initial value amounts to N = (ln 1/T 2 0 )/2L a (L a =0.1 cm is the saturable absorber thickness). The loss is saturable by both fundamental and Stokes fields (σ f = 95, σ s = 4, when the normalization to σ g is presumed). Excited-state absorption is taken into account for the fundamental field (σ esa =0.1). T r = 26 is the loss relaxation time (the normalization to T cav is presumed).
Free propagations between the active medium and the absorber as well as between the active medium and the spherical mirror are considered in the frequency domain:
a f,s (z + L, ν, t) =ã f,s (z, ν, t) exp −ik f,s L + i 2 k f,s Lλ 2 f,s ν 2 ,(3)
where L = 5 cm is the propagation length. The reflection from a spherical mirror obeys:
a f,s (z, r, t) = a f,s (z, r, t) exp ik f,s r 2 R M(4)
where R M is the variable radius of the mirror curvature. The loss on an output mirror as well as the net unsaturable loss are taken into account by means of the following mapping:
a f,s (z, r, t) = a f,s (z, r, t) exp [−0.5(l + ln(1/ρ f,s ))] ,(5)
where ρ s is the output mirror reflection at the Stokes wavelength.
In the simulations, the time window is approximated by 100 points and the window of transverse coordinate is approximated by 2000÷6000 points in dependence on the R value, which varies from 0.5 to 1.5 cm in order to exclude the boundary effects. The solution in the time domain is evaluated using a fourth-order Runge-Kutta method. The initial dimensionless fields have the Gaussian spatial profile with the size, which equals to that of the fundamental CW mode at λ f . The initial dimensionless amplitudes a f and a s amount to 10 −2 and 10 −10 , respectively.
Spatially-temporal structure of laser field from a Q-switched oscillator
The simulations based on the model described in the previous Section demonstrate that the spatially-temporal structure of laser field from a Qswitched oscillator depends non-trivially on the oscillator parameters, which effect on the field dynamics at both fundamental and Stokes wavelengths. intensities are averaged over the cavity period in Figs. 1, a; 2, 4, 5, 7, and 8. Both low initial transmission of a saturable absorber and comparatively small radius of curvature of a spherical mirror initiate the multiple pulse dynamics, which is clearly visible in Fig. 1, b (upper plot) showing the output intensity integrated over the transverse spatial coordinate. The greatest pulse is comparatively long (≈5 ns) and the narrow "mode" (the radius equals to ≈130 µm) has an elongate wing (lower plot). As will be shown, a non-Gaussian transverse shape is typical for the regime under consideration (see [9]).
When a field at the Stokes wavelength is locked inside an oscillator owing to non-zero ρ s , the dynamics changes. For the case presented in Fig. 1, the multipulsing disappears (Figs. 2, a and 3) and the pulse at the fundamental wavelength shortens substantially (≈1.2 ns, see Fig. 3, upper plot) [10]. The mode at the fundamental wavelength widens and becomes Gaussian (Fig. 3, lower plot; see also [9]). Simultaneously, the short pulse at the Stokes wavelength develops (Figs. 2, b and 3). Such a pulse is located on the tale of the fundamental pulse (Fig. 3, gray curve in upper plot) and has a broad trapezoidal transverse profile (the mode radius reaches ≈1.4 mm in Fig. 3, gray curve in lower plot).
Thus, the transverse distribution excesses substantially the CW mode size, which equals to ≈265 µm for the spherical mirror under consideration. Such a spatial extra-broadening can be attributed to the saturable absorber saturation, which results in an excitation of high-order spatial frequencies. A spatial extra-broadening is possible at the fundamental wavelength, as well. Fig. 4, a demonstrates such an extra-broadening (up to ≈0.9 cm). The field covers almost a whole active crystal radius. It should be noted, that the output CW mode radius equals to ≈440 µm for the spherical mirror under consideration. The Q-switching "mode" has a shape of a dilative ring, which rises and then disappears (after ≈12 ns) with a gain depletion. In the presence of SRS, such a "mode" is suppressed owing to an inhomogeneous "collapsing" spatially-temporal behavior at both fundamental and Stokes wavelengths (Fig. 4, b and [15]).
An effect of the SRS on the Q-switching dynamics allows controlling the temporal and spatial profiles of a pulse at both fundamental and Stokes wavelengths. An imbalance between the loss and gain saturation at the fundamental wavelength causing the multiple pulsing can be compensated by the nonlinear loss due to SRS. As a result, the well-shaped and substantially shortened [10] pulses with the broad spatial profiles at both wavelengths appear (Figs. 5,6).
When the oscillator parameters are not optimized, the spatially-temporal structure of the Stokes field can be complicated. For instance, the radial position of the SRS spike can depend on the time (Fig. 7, a, where the SRS spikes move off the optical axis with the pulse evolution; this effect has been observed experimentally in [16]). A similar situation is possible also at the fundamental wavelength (Fig. 4, a). Even through the SRS is well-synchronized (i.e. the SRS appears synchronously at the different radial positions), the mode profile is inhomogeneous, as a rule (Fig. 7, b). The oscillator parameters optimization allows regularizing the spatiallytemporal dynamics at both fundamental and Stokes wavelengths. In particular, the spatial profile of the Stokes field becomes homogeneous though non-Gaussian (Fig. 8). The numerical analysis reveals, that there exist some optimal sets of parameters (see Table 1), in particular the optimal curvature of a spherical mirror (≈1000 cm in the case under consideration), which provide the regular pulses within a broadest range of parameters (viz. ρ f , ρ s , and T 0 ). The initial transmission T 0 >0.5 results in a sufficiently strong fundamental field, that causes an efficient SRS. The reflectivity at the Stokes wavelengths has to be sufficiently high (>0.7) to provide an efficient conversion from the fundamental field to the Stokes one.
The resulting pulses at both wavelength are shortened in comparison with a single-wavelength regime and have the smooth and broad transverse spatial profiles. The last property allows using a larger volume of an active crystal. The "mode" shape is close to the Gaussian one for the fundamental field and to the trapezoidal one for the Stokes field.
Conclusion
The model of a Q-switched Nd:KGW oscillator with a V:YAG saturable absorber, which takes into account a transverse spatial field distribution at both fundamental and Stokes wavelengths, has been developed. The numerical analysis has demonstrated that an imbalance between the loss and gain saturation causes not only a multiple pulsing but an aberration of an oscillator "mode". When the initial transmission of a saturable absorber is sufficiently low, the spatial extra-broadening of a field develops so that a "mode" covers almost whole volume of an active crystal.
Manipulations with the beam size on a saturable absorber (by means of the change of a spherical mirror curvature), the initial transmission of an absorber, and the output mirror reflectivity at both fundamental and Stokes wavelengths allow controlling the spatially-temporal profiles of the output pulses. To obtain a single pulsing at two-wavelengths with a broad and homogeneous spatial distribution, the initial transmission of the saturable absorber has to be about of 0.5, the reflectivity of the output mirror at the fundamental wavelength has to approach 1 (it can range within 0.5 -1 for the optimal curvature of the spherical mirror, which is of ≈1 m in the case under consideration), and the reflectivity at the Stokes wavelength has to be about of 0.9. The SRS causes a temporal squeezing of the pulses at both wavelengths, whereas the spatial profiles are greatly stretched in comparison with the CW-mode size. When the oscillator parameters are optimized, the transverse spatial profiles are near-Gaussian at the fundamental wavelength and trapezoidal at the Stokes wavelength.
The obtained results are of interest for development of the passively Qswitched Raman-active oscillators producing the pulses with durations of about of few nanoseconds and possessing the smooth transverse spatial profiles covering a considerable part of an active crystal.
Figure 1 :
1a: Contour plot of the output intensity at the fundamental wavelength from a Q-switched oscillator with T 0 =0.3, ρ f =0.7, ρ s =0, R M =300 cm. b: Output intensity integrated over r (integral power; upper plot) or t (integral fluency; lower plot).
Fig. 1 ,
1a shows the contour plot of the output intensity at the fundamental wavelength in absence of an effective SRS in an oscillator (ρ s =0).
Figure 2 :
2Contour plots of the output intensity at the fundamental (a) and Stokes (b) wavelengths. T 0 =0.3, ρ f =0.7, ρ s =0.7, R M =300 cm.
Figure 3 :
3Output integral power (upper plot) and integral fluency (lower plot) at the fundamental (black curves) and Stokes (gray curves) wavelengths. Parameters correspond toFig. 2.
Figure 4 :
4Contour plots of the output intensity at the fundamental wavelength. T 0 =0.3, ρ f =0.5; ρ s =0 (a), 0.7 (b); R M =2000 cm.
Figure 5 :
5Contour plots of the output intensity at the fundamental (a, b) and Stokes (c) wavelengths. T 0 =0.5, ρ f =0.9; ρ s =0 (a), 0.9 (b, c); R M =1000 cm.
Figure 6 :
6Output integral power (upper plot) and integral fluency (lower plot) at the fundamental (black solid and dotted curves) and Stokes (gray curves) wavelengthes. ρ s =0 (black solid curves) and 0.9 (black dotted and gray curves). Another parameters correspond toFig. 5. Dotted and gray curves are vertically rescaled for convenience.
Figure 7 :
7Stokes intensity profiles for (a): R M =2000 cm, ρ f =0.7, ρ s =0.7, and (b):R M =1000 cm, ρ f =0.9, ρ s =0.5. T 0 =0.5.
Figure 8 :
8Stokes intensity profile for R M =1000 cm, ρ f =0.9, ρ s =0.9, and T 0 =0.5.
Table 1 :
1Oscillator parameters providing the regular spatial-temporal pulse profiles at both fundamental and Stokes wavelengths.
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| []
|
[
"SHRINKING RATES OF HORIZONTAL GAPS FOR GENERIC TRANSLATION SURFACES",
"SHRINKING RATES OF HORIZONTAL GAPS FOR GENERIC TRANSLATION SURFACES"
]
| [
"Jon Chaika ",
"Samantha Fairchild "
]
| []
| []
| A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R → ∞ for the difference in angle between two almost horizontal saddle connections.Remark 1.2. Note that the choice of a horizontal gap is a convenience. Apply a rotation to the full measure set in Theorem 1.1, and we obtain the same result in a different direction. Consider a countable subset D n = {θ n } n∈N ⊆ [0, 2π). Since a countable union of measure 0 subsets is still measure zero, we obtain a natural corollary that the smallest gap of any of the directions in D n has the same decay rate as given in Theorem 1.1.We first provide an explanation for the scaling factor of R 2 . Masur ([Mas88, Mas90]) showed |Λ ω (R)| has quadratic growth in the sense that for each ω there exist constants c 1 , c 2 so thatfor all large enough R. There are at most 4g − 4 saddle connections in the same direction, so |Θ ω (R)| also has quadratic growth. This result explains the scaling factor of R 2 in Theorem 1.1. The quadratic growth of saddle connections was subsequently built on in [Vee98, EM01, Vor05, EMM15, NRW20].For almost every translation surface, Theorem 1.1 gives partial information on how Θ ω (R) is distributed in [−π, π). For every translation surface,[Mas86]shows that Θ ω = R Θ ω (R) is dense in [−π, π). If we order the points θ 1 ≤ · · · ≤ θ ℓ(R) in Θ ω (R), then by density the adjacent differences θ j+1 − θ j → 0 as R → ∞ for every ω. Thus ζ ω (R) → 0 as R → ∞ for every ω. However, we cannot expect Theorem 1.1 to hold for every translation surface. Indeed when ω is a lattice surface (see [Mas22, Sections 5 and 7] for a definition), [AC12] showed for every unbounded ψ we have lim inf R→∞ ψ(R)R 2 ζ ω (R) = ∞.The results of Theorem 1.1 considers the behavior of a single gap. There is also substantial work done on studying the behavior of the family of gaps. Namely one can study the entire set Θ ω (R). The distribution of normalized gaps exists for almost every ω by [AC12]. In many cases, the distribution has also been computed [ACL15, BMMM + 21, KSW21, UW16, San21]. Considering the behavior of a single gap, which is the focus of the current paper, is orthogonal because the behavior of a single gap does not affect the distribution of gaps.1.1. Outline of proof. The proof follows the now standard strategy of relating a problem about the geometry of a translation surface ω to the orbit of ω under Teichmüller geodesic flow, g t = e t 0 0 e −t . In Section 2 we reduce our problem about gaps to a shrinking target problem for g t . The shrinking targets are sets A t obtained by relating Theorem 1.1 to whether or not g t ω ∈ A t for arbitrarily large t. To prove Theorem 1.1 (2) we use independence results for the sets g −t A t . A key tool to do this is the fact that the g t action is exponentially mixing (Section 3). | null | [
"https://arxiv.org/pdf/2207.03836v1.pdf"
]
| 250,407,988 | 2207.03836 | 7e7aede0d2e545de218faa17223c43c1e16d08e1 |
SHRINKING RATES OF HORIZONTAL GAPS FOR GENERIC TRANSLATION SURFACES
Jul 2022
Jon Chaika
Samantha Fairchild
SHRINKING RATES OF HORIZONTAL GAPS FOR GENERIC TRANSLATION SURFACES
8Jul 2022
A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R → ∞ for the difference in angle between two almost horizontal saddle connections.Remark 1.2. Note that the choice of a horizontal gap is a convenience. Apply a rotation to the full measure set in Theorem 1.1, and we obtain the same result in a different direction. Consider a countable subset D n = {θ n } n∈N ⊆ [0, 2π). Since a countable union of measure 0 subsets is still measure zero, we obtain a natural corollary that the smallest gap of any of the directions in D n has the same decay rate as given in Theorem 1.1.We first provide an explanation for the scaling factor of R 2 . Masur ([Mas88, Mas90]) showed |Λ ω (R)| has quadratic growth in the sense that for each ω there exist constants c 1 , c 2 so thatfor all large enough R. There are at most 4g − 4 saddle connections in the same direction, so |Θ ω (R)| also has quadratic growth. This result explains the scaling factor of R 2 in Theorem 1.1. The quadratic growth of saddle connections was subsequently built on in [Vee98, EM01, Vor05, EMM15, NRW20].For almost every translation surface, Theorem 1.1 gives partial information on how Θ ω (R) is distributed in [−π, π). For every translation surface,[Mas86]shows that Θ ω = R Θ ω (R) is dense in [−π, π). If we order the points θ 1 ≤ · · · ≤ θ ℓ(R) in Θ ω (R), then by density the adjacent differences θ j+1 − θ j → 0 as R → ∞ for every ω. Thus ζ ω (R) → 0 as R → ∞ for every ω. However, we cannot expect Theorem 1.1 to hold for every translation surface. Indeed when ω is a lattice surface (see [Mas22, Sections 5 and 7] for a definition), [AC12] showed for every unbounded ψ we have lim inf R→∞ ψ(R)R 2 ζ ω (R) = ∞.The results of Theorem 1.1 considers the behavior of a single gap. There is also substantial work done on studying the behavior of the family of gaps. Namely one can study the entire set Θ ω (R). The distribution of normalized gaps exists for almost every ω by [AC12]. In many cases, the distribution has also been computed [ACL15, BMMM + 21, KSW21, UW16, San21]. Considering the behavior of a single gap, which is the focus of the current paper, is orthogonal because the behavior of a single gap does not affect the distribution of gaps.1.1. Outline of proof. The proof follows the now standard strategy of relating a problem about the geometry of a translation surface ω to the orbit of ω under Teichmüller geodesic flow, g t = e t 0 0 e −t . In Section 2 we reduce our problem about gaps to a shrinking target problem for g t . The shrinking targets are sets A t obtained by relating Theorem 1.1 to whether or not g t ω ∈ A t for arbitrarily large t. To prove Theorem 1.1 (2) we use independence results for the sets g −t A t . A key tool to do this is the fact that the g t action is exponentially mixing (Section 3).
Introduction
Consider a finite collection of polygons in the plane, where all sides come in pairs of equal length with opposite orientations on the boundary of the polygons. Identifying these sides gives a compact finite type Riemann surface. The form dz on the plane endows this Riemann surface with a holomorphic 1-form ω. This structure is called a translation surface. Translation surfaces are stratified by their zeros and each connected component of each stratum supports a natural measure class called Masur-Veech measure. More background on translation surfaces can be found, for example in [Mas22,MS91]. A saddle connection γ is a geodesic starting and ending at the zeroes of ω with no zeroes in between. Associated to γ, we define the holonomy vector v γ = γ dω ∈ C. To understand the geometry of a typical surface, much effort has gone into understanding the asymptotic behavior of holonomy vectors of saddle connections of length at most R [Doz19, EM01, EMM15, EMZ03, NRW20, Mas88, Mas90, Vee98,Vor05], and more recently the asymptotic behavior of pairs of saddle connections [ACM19,AFM22]. We will consider Λ ω (R) = {v γ ∈ C ∩ B(0, R) : γ is a saddle connection} and Θ ω (R) = {arg(v γ ) : v γ ∈ Λ ω (R)}, where arg(v) ∈ [−π, π) is the angle v makes with the horizontal. The sets Λ ω (R) and Θ ω (R) are discrete subsets of C and [−π, π), respectively, so we can define
ζ ω (R) = min{φ ∈ Θ ω (R) : φ ≥ 0} − max{φ ∈ Θ ω (R) : φ < 0}.
The main result of this paper is: (2) If
Reductions
In this section we present a series of reductions:
(1) We use renormalization to relate gaps to a shrinking target problem in Section 2.1.
(2) In Section 2.3, we consider the convergence case Theorem 1.1(1) and prove it suffices to show that for almost every ω we have lim inf R→∞ ψ(R)R 2 ζ ω (R) > 0. (3) In Section 2.4, we consider the divergence case Theorem 1.1(2) and prove it suffices to show that for a positive measure set of ω we have lim inf R→∞ ψ(R)R 2 ζ ω (R) < ∞. (4) In Section 2.5 we state and prove a partial converse to the Borel-Cantelli lemma that we use as the framework for proving Theorem 1.1. (5) We prove Theorem 1.1 assuming Proposition 2.14.
2.1. Definition and measure of the shrinking target sets. Fix a stratum H, with complex dimension 2g + s − 1 with s the number of distinct singularities. We will sample ψ along a discrete set ψ(b k ) for k ∈ N where b = e ℓ 0 , and ℓ 0 ≥ 1 is as in Corollary 4.6, for 0 < δ < 1, and I = (− π 12 , π 12 ).
Definition 1 (Definition of the A's). Fix 0 < σ < 1, and let 0 ≤ c < 1. Define T ± c,σ,j to be the trapezoids with corners (c, 0), (1, 0), (1, ±σ ψ(b j ) ), (c, c ±σ ψ(b j ) ). Set H c,σ,j = {ω ∈ H : ω has a holonomy vector in T + c,σ,j and a holonomy vector in T − c,σ,j }.
Finally for k ∈ N, define A k (c, σ) = g log(b k ) H c,σ,k , where g t = e t 0 0 e −t .
Remark 2.1. When the choice of c or σ is clear, or arbitrary, then for clarity we will suppress the dependence and simply write A k instead of A k (c, σ).
We next obtain measure estimate for A k . Let µ denote the Masur-Veech measure on H. Since µ is SL(2, R) invariant, it suffices to understand µ(H c,σ,j ). Before obtaining the desired estimates, we quote the following result of Masur-Smillie (as quoted from [AC12])
Lemma 2.2. There is a constant M so that for all ǫ, κ > 0, the subset of H consisting of flat surfaces which have a saddle connection of length at most ǫ has measure at most M ǫ 2 . The subset of flat surfaces which have a saddle connection of length at most ǫ and another nonhomologous saddle connection of length at most κ has measure at most M ǫ 2 κ 2 .
Following [MS91, µ is defined on the flat structures of area 1 via a cone measure µ over flat structures with area at most 1. The cone measure µ is inherited from a measure on cohomology, defined via charts coming from the developing map. For ω ∈ H there exists r ω > 0 so that the ball of radius r ω about the preimage of ω in relative cohomology is sent injectively to the coordinate chart about ω. In a fixed coordinate chart we can write ω = (x 1 , x 2 , x 3 ) ∈ C × C × C 2g+s−3 and µ as Lebesgue measure on C 2g+s−1 . Let ω 0 = (x 0 1 , x 0 2 , x 0 3 ) have two horizontal holonomy vectors of length 1, and let r > 0 be the radius for the chart as above. One can find such a surface in every connected component of every stratum. Indeed, [Zor08] explicitly constructs surfaces, which are a single horizontal cylinder, in each connected component of every stratum of genus at least two. From such a surface one can vary the length of a pair of boundary horizontal saddle connections to obtain ω.
= 1 − 2 −n H mσ 2 ψ(b j ) 2 ≤ µ(H c H ,σ,j ) ≤ µ(H 0,σ,j ) ≤ M σ 2 ψ(b j ) 2 .
Proof. Since µ is defined via µ, it suffices to prove the statement for µ. Fix r > 0 as above for the surface ω 0 ∈ H. If necessary, shrink r so that r < 1 and the image of the chart is contained in a compact subset of H. Notice that for any c, since ψ(t) ≥ 1, T ± c,σ,j is always contained in the trapezoid T c,σ, * given by (c, cσ), (c, 0), (1, 0), (1, σ). So we can guarantee T c,σ,j is always contained in the chart whenever T c,σ, * ⊂ B C (1, r). This geometric condition can be satisfied for some 0 < σ H < r and n H ∈ N, where c H = 1 − 2 −n H and σ < σ H . Define the set
in C × C × C 2g+s−3 by H c H ,σ,j = T + c H ,σ,j × T − c H ,σ,j × B, where B = B C 2g+s−3 (x 0 3 , r). By our choice of r the measure µ on H is locally Lebesgue. Hence if m j is Lebesgue measure on C j , by symmetry of T ± c H ,σ,j , µ( H c H ,σ,j ) = m 1 (T + c H ,σ,j ) 2 m 2g+s−3 (B). Note that m 1 (T + c H ,σ,j ) = 1 2 σ ψ(b j ) + c H σ ψ(b j ) (1 − c H ) = σ ψ(b j ) (1 − c 2 H ) 2 . Hence m = 1−c 2 H 2 m 2g+s−3 (B) and µ(H 0,σ,j ) ≥ µ(H c H ,σ,j )
gives the desired bounds.
For the upper bound, flowing by geodesic flow which preserves measure, g
log √ σ/ψ(b k ) H 0,σ,k
has two non-homologous vectors of length at most 2σ ψ(b k ) , so by Lemma 2.2 we obtain the desired upper bound.
2.2.
A lemma on targets. For the following lemma and corollary, for generality we allow any b > 1. Note that b = e ℓ 0 as defined in Definition 1 satisfies b > 1 since we can choose ℓ 0 > 0.
Lemma 2.4. Let φ : [1, ∞) → [1, ∞) be non-decreasing. We have ∞ 1 (tφ(t)) −1 dt = ∞ if and only if ∞ j=1 φ(b j ) −1 = ∞ for any b > 1.
Proof. For ease of exposition we assume b = 2, the general case is similar, but requires using floor and ceiling functions. For each k ∈ N we have
2 k 1 2 k φ(2 k ) ≥ 2 k+1 2 k (tφ(t)) −1 dt ≥ 2 k 1 2 k+1 φ(2 k+1 )
.
It follows that ∞ j=1 φ(2 j ) −1 ≥ ∞ 2 (tφ(t)) −1 dt ≥ 1 2 ∞ j=2 φ(2 j ) −1 . Corollary 2.5. Let ψ : [1, ∞) → [1, ∞) be non-decreasing. We have ∞ 1 (tψ(t) 2 ) −1 dt = ∞ if and only if ∞ j=1 ψ(b j ) −2 = ∞ for any b > 1. 2.3. Convergence reduction.
Lemma 2.6. If whenever
∞ 1 1 tψ(t) 2 dt < ∞, µ {ω : lim inf R→∞ ψ(R)R 2 ζ ω (R) > 0} = 1, then whenever ∞ 1 1 tψ(t) 2 dt < ∞ we have µ ω : lim inf R→∞ ψ(R)R 2 ζ ω (R) = ∞ = 1.
Proof. We first construct a slightly smaller function ψ 0 (t) so that ∞ 1 1 tψ 0 (t) 2 dt < ∞ and lim t→∞ ψ(t) ψ 0 (t) = ∞.
To do this let n 1 = 1 and set n j = sup T > n j−1 :
T n j−1 1 tψ(t) 2 dt ≤ 2 −j ∞ 1 1 tψ(t) 2 dt.
For j ∈ N we piecewise define ψ 0 (t) = j −1 ψ(t) whenever n j ≤ t < n j+1 .
Then since j → ∞ as t → ∞, lim t→∞ ψ(t)/ψ 0 (t) = ∞. By our choice of n j , we also have ∞ 1 (tψ 0 (t) 2 ) −1 dt < ∞. By the assumption of Lemma 2.6 there is a full measure set of ω so that lim inf t→∞ ψ 0 (t)t 2 ζ ω (t) > 0. From this we have the desired result that for a full measure set of ω,
lim inf t→∞ ψ(t)t 2 ζ ω (t) ≥ lim inf t→∞ ψ(t) ψ 0 (t) lim inf t→∞ ψ 0 (t)t 2 ζ ω (t) = ∞. 2.4. Divergence Case.
Proposition 2.7. If for all σ > 0, µ(lim sup A k (0, σ)) > 0 then lim inf R→∞ ψ(R)R 2 ζ ω (R) = 0 for µ-a.e. ω.
Remark 2.8. Note that A k (0, σ) ⊆ A k (0, σ ′ ) whenever σ < σ ′ .
Thus the assumption of Proposition 2.7 is satisfied as long as µ(lim sup A k (0, σ)) > 0 for all σ small enough.
Proof. We first claim µ(lim sup A k ) = 1. We will show lim sup A k is invariant under forward geodesic flow g log(b t ) for t > 0. Let ω ∈ g log(b −t ) lim sup k→∞ g log(b k ) H 0,σ,k . For n ∈ N, there exists k ≥ n + t so that the monotonicity of ψ implies
ω ∈ g log(b −t+k ) H 0,σ,k ⊆ g log(b −t+k ) H 0,σ,−t+k .
Hence ω ∈ lim sup A k . By ergodicity of the geodesic flow and the assumption that µ(lim sup A k ) > 0, we conclude that µ(lim sup A k ) = 1.
We now translate lim sup A k by a small backwards geodesic flow to a set where we have the desired result. Namely we claim that for s 0 = log(2)
2 log(b) > 0, any ω ∈ g −s 0 log(b) lim sup A k satisfies lim inf R→∞ ψ(R)R 2 ζ ω (R) = 0.
First note that we are still working with a full measure set since µ is invariant under geodesic flow µ(g −s 0 log(b) lim sup A k ) = µ(lim sup A k ) = 1.
Let ω = g −s 0 log(b) ω for some ω ∈ lim sup A k . Since ω ∈ lim sup A k , for any m ∈ N we can find ρ m ≥ m so that ω ∈ g ρm log(b) H 0,σ,ρm . The choice of s 0 guarantees that the longest possible holonomy vectors in g −s 0 log(b) T ± 0,σ,b ρm are at most b ρm since the choice of s 0 is sufficient so that
b 2(ρm−s 0 ) + σ 2 b 2(ρm−s 0 ) ψ(b ρm ) 2 ≤ b 2ρm .
Thus the holonomy vectors detected by g −s 0 log(b) T ± 0,σ,b ρm have length at most b ρm and an upper bound on the angle around zero, giving the following upper bound
ζ g − log(b s ) ω (b ρm ) ≤ arctan σ ψ(b ρm )b 2ρm−2s 0 < σb 2s 0 ψ(b ρm )b 2ρm .
Since s 0 is fixed, we can take σ → 0 to obtain lim inf R→∞ ψ(R)R 2 ζ ω (R) = 0.
2.5. Axiomatic framework. We will first recall the Borel-Cantelli lemma, and then spend the remainder of the section stating and proving a partial converse.
Lemma 2.9 (Borel-Cantelli lemma). Suppose (A k ) ∞ k=1 are measurable sets with ∞ k=1 µ(A k ) < ∞. Then µ (∩ ∞ N =1 ∪ ∞ i=N A i ) = µ(lim sup A i ) = 0.
Proposition 2.10 (Exponential Decay Borel-Cantelli). Let C ≥ 1, 0 < δ < 1, and (A k ) ∞ k=1 be measurable sets. Suppose the following hold.
(1) ∞ k=1 µ(A k ) = ∞. (2) For all i ≤ j, µ(A i ) ≥ µ(A j ).
(3) For all i, for all j so that j > i + C log
1 µ(A i ) we have µ(A i ∩ A j ) ≤ Cµ(A i ) µ(A j ) + e − δ 4 |i−j| .
(4) For all i, for all j so that i < j
≤ i + C log 1 µ(A i ) we have that there exists measurable sets B i , C j so that (a) B i ⊂ A i and A j ⊂ C j , (b) µ(B i ) > 1 C µ(A i ), (c) µ(C j ) < Cµ(A j ) 1 2 , (d) µ(B i ∩ C j ) < Cµ(B i ) 2 −(j−i)(1−δ) + µ(C j ) 1+δ 2 .
Then ∩ ∞ N =1 ∪ ∞ i=N A i = lim sup A i has positive measure. Remark 2.11. One can observe that the decay of correlations in Assumption (3) is insufficient to handle the generality of Theorem 1.1. Indeed consider ψ(t) = log(t + 4) log(log(t + 4)). Since
µ(A k ) is proportional to ψ(b k ) −2 = 1 log(b k +4) log(log(b k +4)) < 1 k log(k) log(b) . Notice Assumption (3) gives no bounds on µ(A i ∩A j ) when i < j < C ′ log(i)+i. Define n k recur- sively by n 1 = 2 and n i+1 = ⌈n i + C ′ log(n i )⌉. Observe ∞ i=1 µ(A n i ) < ∞, so µ(lim sup A n k ) = 0 by the Borel-Cantelli lemma.
Thus we cannot draw any conclusions on µ(lim sup A k ) without Assumption (4).
We now prove Proposition 2.10. This proof is inspired by [Pet02, Theorem 2.1], which invokes the Chung-Erdős inequality.
Lemma 2.12 (Chung-Erdős inequality). Suppose(A k ) ∞ k=1 is a sequence of measurable sets with µ N k=1 A k > 0, then µ N k=1 A k ≥ N k=1 µ(A k ) 2 n j,k=1 µ(A j ∩ A k ) .
Proof of Lemma 2.12. Beginning with the numerator on the right hand side,
N k=1 µ(A k ) 2 = 1 { N k=1 1 A k >0} N k=1 1 A k dµ 2 .
By the Cauchy-Schwarz inequality
N k=1 µ(A k ) 2 ≤ µ N k=1 A k N k=1 1 A k 2 dµ = µ N k=1 A k N j,k=1 µ(A j ∩ A k ) .
Rearranging we obtain the desired inequality.
The next lemma will also be used to prove Proposition 2.10, which states the conditions on the sets B k ⊆ A k that we use to show lim sup(B k ) > 0, and thus lim sup(A k ) > 0.
Lemma 2.13. Under the assumptions of Proposition 2.10, there exists some C ≥ 1 depending only on C large enough so that if
m i = i + C + C log 1 µ(B i ) , the following hold. (a) ∞ k=1 µ(B k ) = ∞. (b) For all i ≤ j, µ(B i ) ≥ 1 C µ(B j ). (c) For all i and for all j so that j > m i , µ(B i ∩ B j ) ≤ Cµ(B i ) µ(B j ) + e − δ 4 |i−j| . (d) For all i and for all j with i < j < m i , µ(B i ∩ B j ) < Cµ(B i ) 2 −|i−j|(1−δ) + µ(B j ) 1+δ 4 .
Moreover there exists constants D, D ′ > 0 so that for any n > 0 and N ≥ n,
(2.1) N i,j=n µ(B i ∩ B j ) ≤ C D N k=n µ(B k ) + D ′ + N k=n µ(B k ) 2 .
Proof of Lemma 2.13. The first 4 parts use assumptions on A k in Proposition 2.10.
(a) Follows from Assumptions (1) and (4b).
(b) Follows from Assumptions (2) and (4b).
(c) Combine Assumption (3) with Assumptions (4a) and (4b).
(d) Combine Assumptions (4b) and (4c) with that fact that 1+δ 4 ∈ ( 1 4 , 1 2 ) and C ≥ 1 to obtain
µ(C j ) 1+δ 2 ≤ Cµ(A j ) 1+δ 4 ≤ C(Cµ(B j )) 1+δ 4 ≤ C 2 µ(B j ) 1+δ 4 .
Now we move to Eq. (2.1) where we want to find an upper bound for the denominator of Lemma 2.12 applied to the sets B k . First since C ≥ 1,
(2.2) N i,j=n µ(B i ∩ B j ) ≤ 2 N i=n j>i µ(B i ∩ B j ) + C N k=n µ(B k ).
We split the double sum on the right hand side of Eq. (2.2) into cases when j > m i and j ≤ m i .
In the first case when j > m i , part (c) implies
(2.3) 2 N i=n j> m i µ(B i ∩ B j ) ≤ 2 C N i=n j> m i µ(B i )µ(B j ) + µ(B i )e − δ 4 |i−j| .
In the first sum of Eq. (2.3), we re-expand the square so that
(2.4) 2 N i=n N j> m i µ(B i )µ(B j ) ≤ N i=n µ(B i ) 2 − N i=n µ(B i ) 2 ≤ N i=n µ(B i ) 2 .
In the second sum of Eq. (2.3) we making the change of variables k = j − i, the geometric series formula gives
(2.5) 2 N i=n N j> m i µ(B i )e − δ 4 |i−j| ≤ 2 N i=n µ(B i ) 1 1 − e − δ 4 .
Combining Eq. (2.3), Eq. (2.4), and Eq. (2.5), we obtain
(2.6) 2 N i=n j> m i µ(B i ∩ B j ) ≤ C N k=n µ(B k ) 2 + 2 1 − e − δ 4 N k=1 µ(B k ) .
We now consider the case for i < j ≤ m i . By (d)
(2.7) 2 N i=n i<j≤ m i µ(B i ∩ B j ) ≤ 2 C N i=n i<j≤ m i µ(B i )2 −|i−j|(1−δ) + µ(B i )µ(B j ) 1+δ 4 .
We again use a geometric series formula so that the first sum is bounded by
(2.8) 2 N i=n i<j≤ m i µ(B i )2 −|i−j|(1−δ) ≤ 2 1 − 2 −(1−δ) N i=n µ(B i ).
Note 5+δ 4 ∈ 5 4 , 6 4 so 5+δ 4 is a power bigger than 1 with µ(
B i ) ≤ 1. Therefore µ(B i ) 5+δ 4 ≤ µ(B i ). Combining this fact with (b), 2 N i=n i<j≤ m i µ(B i )µ(B j ) 1+δ 4 ≤ 2C N i=n µ(B i ) 5+δ 4 ( m i − i) ≤ 2C C N i=n µ(B i ) + 2C C N i=n µ(B i ) 5+δ 4 log 1 µ(B i ) . (2.9)
Now choose n 0 large enough so that for all i ≥ n 0 ,
log 1 µ(B i ) ≤ µ(B i ) −1−δ 4 . Then if n ≥ n 0 , N i=n µ(B i ) 5+δ 4 log 1 µ(B i ) ≤ N i=n µ(B i ). Otherwise if n ≤ n 0 N i=n µ(B i ) 5+δ 4 log 1 µ(B i ) ≤ n 0 −1 i=n µ(B i ) log 1 µ(B i ) + N i=n 0 µ(B i ) ≤ C ′ + N i=n µ(B i ).
where C ′ > 0 is the bound for the finite sum. Therefore there is a constant C ′ so that
(2.10) N i=n µ(B i ) 5+δ 4 log 1 µ(B i ) ≤ C ′ + N i=n µ(B i ).
Thus we conclude by combining Eq. (2.7), Eq. (2.8), Eq. (2.9), and Eq. (2.10) so that
(2.11) 2 N i=n i<j≤ m i µ(B i ∩ B j ) ≤ C 2CC ′ C + 2 1 − 2 −(1−δ) + 4C C N i=n µ(B i ) .
The proof of Eq. (2.1) is completed by combining Eq. (2.2), Eq. (2.6), and Eq. (2.11) where D ′ = 2C CC ′ and D = 1 + 2
1−e − δ 4 + 2 1−2 −(1−δ) + 4C C.
Proof of Proposition 2.10. It suffices show that that the measure of lim sup B k has positive measure. By the Chung-Erdős inequality (Lemma 2.12), Lemma 2.13 (a), and Eq. (2.1),
lim inf N →∞ µ N k=n B k ≥ lim inf N →∞ N k=n µ(B k ) 2 C D N k=n µ(B k ) + D ′ + N k=n µ(B k ) 2 = lim inf N →∞ 1 C D N k=n µ(B k ) + D ′ ( N k=n µ(B k )) 2 + 1 = 1 C Hence µ ( ∞ k=n B k ) ≥ 1 C . Notice ∞ k=n B k is a nested decreasing sequence of sets, so we conclude µ(lim sup B n ) = µ ∞ n=1 ∞ k=n B k = lim n→∞ µ ∞ k=n B k ≥ 1 C > 0.
2.6. Proof of Theorem 1.1. We now prove Theorem 1.1 conditional on the next proposition, whose proof is completed in Section 3 and Section 4.
Proposition 2.14. For some c > 0, there exist sets B k , C k so that along with the sets A k defined in Definition 1, the assumptions of Proposition 2.10 are satisfied.
Before outlining how to prove Proposition 2.14, we first show that it is sufficient to obtain Theorem 1.1.
Proof of Theorem 1.1. We first prove the convergence case. By Corollary 2.5, ∞ j=1 ψ(e j ) −2 < ∞, and thus lim j→∞ ψ(e j ) = ∞. Set
L k = {ω : ∃ R ∈ [e k , e k+1 ] so that R 2 ζ ω (R) < ψ(e k ) −1 }. For k large enough, if ζ ω (R) < ψ(e k ) −1
R 2 , then there are two saddle connections on ω with horizontal holonomy at most R and vertical holonomy of magnitude at most 2ψ(e k ) −1 R . It follows that for ω ∈ L k , g − log e k √ ψ(e k ) ω has two saddle connections of length at most ψ(e k ) −1/2 e √ 2.
By Lemma 2.2, there is some constant
C ′ so that µ(L k ) ≤ C ′ ψ(e k ) −2 so that ∞ k=1 µ(L k ) < ∞.
By the Borel-Cantelli lemma (Lemma 2.9), µ(lim sup L k ) = 0. Taking the complement, we have a full measure set of ω so that for all k ≥ k 0 , and for all R ∈ [e k , e k+1 ],
ψ(R)R 2 ζ ω (R) ≥ ψ(e k )R 2 ζ ω (R) ≥ 1. Therefore µ {ω : lim inf R→∞ ψ(R)R 2 ζ ω (R) > 0} ≥ µ {ω : lim inf R→∞ ψ(R)R 2 ζ ω (R) ≥ 1} = 1.
By Lemma 2.6, the convergence case of Theorem 1.1 is verified.
We now prove the divergence case. By Corollary 2.5, ∞ k=1 ψ(b k ) −2 = ∞. By Lemma 2.3 for any σ < σ H our set A k has measure proportional to ψ(b k ) −2 . By Proposition 2.10 and Proposition 2.14, for a positive measure set of ω we have g t ω ∈ A k for infinitely many k. Following Remark 2.8 we satisfy the assumptions of Proposition 2.7, which implies the divergence case of Theorem 1.1.
Proof outline of Proposition 2.14. We verify or state where each of the assumptions of Proposition 2.10 are verified.
(1) Assumption (1) follows by the measure bounds of Lemma 2.3, Corollary 2.5 and the fact that g t preserves measure:
∞ k=1 µ(A k ) ≥ ∞ k=1 m σ 2 ψ(b k ) 2 = ∞.
(2) Assumption (2) follows by Lemma 2. We begin by stating our key exponential mixing result:
3: i ≤ j implies ψ(b i ) ≤ ψ(b j ), so µ(A i ) = mσ 2 ψ(b i ) 2 ≥ mσ 2 ψ(b j ) 2 = µ(A j ).
Theorem 3.1 (Stated from [AEZ16] Theorem C.4, see [AGY06]). Fix H a connected component of the stratum and µ be Masur-Veech measure as above. There exists C > 0 and δ > 0 so that for all h 1 , h 2 Lipschitz and compactly supported, there exists a C K depending only on the shortest sytole of a surface in the compact set so that for all t ≥ 0
h 1 (h 2 • g t ) dµ − h 1 dµ h 2 dµ ≤ C(C K + h 1 ∞ + h 2 Lip )(C K + h 2 ∞ + h 2 Lip )e −δt .ǫ i,j = e − δ 4 |i−j| .
Then for ℓ ∈ {i, j}, define
ρ ℓ i,j (x 1 , x 2 , x 3 ) = f ℓ 1 (x 1 )f ℓ 2 (x 2 )f 3 (x 3 ) where f ℓ 1 (x 1 ) = min 1, 1 ǫ i,j dist x 1 , ∂T c H ,σ,ψ(b ℓ ) · 1 T c H ,σ,ψ(b ℓ ) , f ℓ 2 (x 2 ) = min 1, 1 ǫ i,j dist x 2 , ∂T c H ,σ,ψ(b ℓ ) 1 T c H ,σ,ψ(b ℓ ) , and f ℓ 3 (x 3 ) = min 1, 1 ǫ i,j dist (x 3 , ∂B(0, 1)) 1 B(0,1) . Lemma 3.3. The functions ρ ℓ i,j are 1 ǫ i,j -Lipschitz.
Proof. Note that f ℓ k for k = 1, 2, 3 are all 1 ǫ i,j -Lipschitz as the distance function being 1-Lipschitz implies
f ℓ k (x k ) − f ℓ k (y k ) ≤ 1 ǫ i,j dist x k , ∂T c H ,σ,ψ(b ℓ ) − dist y k , ∂T c H ,σ,ψ(b ℓ ) ≤ 1 ǫ i,j |x k − y k | .
Now we claim that the function ρ ℓ i,j is 1 ǫ i,j -Lipschitz with respect to the distance on H given by d H ((x 1 , x 2 , x 3 ) − (y 1 , y 2 , y 3 )) = |x 1 − y 1 | + |x 2 − y 2 | + |x 3 − y 3 |.
To see this, let (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) be fixed. We compute
|ρ(x 1 , x 2 , x 3 ) − ρ(y 1 , y 2 , y 3 )| ≤ |f 1 (x 1 )f 2 (x 2 )f 3 (x 3 ) − f 1 (y 1 )f 2 (x 2 )f 3 (x 3 )| + |f 1 (y 1 )f 2 (x 2 )f 3 (x 3 ) − f 1 (y 1 )f 2 (y 2 )f 3 (x 3 )| + |f 1 (y 1 )f 2 (y 2 )f 3 (x 3 ) − f 1 (y 1 )f 2 (y 2 )f 3 (y 3 )| ≤ 1 ǫ i,j (f 2 (x 2 )f 3 (x 3 )|x 1 − y 1 | + f 1 (y 1 )f 3 (x 3 )|x 2 − y 2 | + f 1 (y 1 )f 2 (y 2 )|x 3 − y 3 |) ≤ 1 ǫ i,j d H ((x 1 , x 2 , x 3 ) − (y 1 , y 2 , y 3 )). (Since f i ≤ 1)
We can now use the definition of the ρ ℓ i,j to state a corollary of Theorem 3.1. Corollary 3.4. Fix 0 < δ < 1. Then there exists a constant C so that for all j > m i ,
ρ i i,j (ρ j i,j • g ℓ 0 (j−i) ) ≤ Cµ(A i ) µ(A j ) + e − δ 4 |i−j| .
Proof. By definition ρ ℓ i,j ∞ = 1, and by Lemma 3.3 ρ ℓ i,j = 1 ǫ i,j = e δ 4 |i−j| . By Theorem 3.1, the fact that b ≥ e, and writing C K + 1 = C + K ,
ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) − ρ i i,j ρ j i,j ≤ C(C + K + e δ 4 |i−j| ) 2 e −δ|i−j| = C(C + K ) 2 e −δ|i−j| + 2CC + K e − 3δ 4 |i−j| + Ce − δ 2 |i−j| (Since e − δ 2 |i−j| ≥ e − 3 4 δ|i−j| ≥ e −δ|i−j| as e −δ|i−j| < 1) ≤ Ce − δ 2 |i−j| . By construction of ρ ℓ i,j , ρ ℓ i,j = µ(H c H ,σ,ℓ ) = µ(A ℓ ), so ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) ≤ ρ i i,j ρ j i,j + Ce − δ 2 |i−j| ≤ µ(A i )µ(A j ) + Ce − δ 2 |i−j| .
Since j > m i , we have e − δ 4 |i−j| < µ(A i ). Thus, using C > 1,
ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) ≤ Cµ(A i ) µ(A j ) + e − δ 4 |i−j| .
Next our goal is to get a relationship between µ(A i ∩ A j ) and ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )). Lemma 3.5. For all j > m i ,
µ(A i ∩ A j ) ≤ ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) + µ(E j ) + µ(E i )
where E ℓ = {ω : ρ ℓ i,j ∈ (0, 1)}.
Proof. We first make a general claim.
Claim If 0 ≤ g ≤ f ≤ 1 then f ≤ g + µ{f = g}.
To see this is true, we write
f = g + f − g = g + {f =g} f − g ≤ g + µ{f = g}
where the last inequality follows since f − g ≤ 1.
The proof now follows from the claim where f = 1 H c H ,σ,i ∩g ℓ 0 (j−i) )H c H ,σ,j and g = ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )), combined with the fact that f = g occurs when ρ i i,j ∈ (0, 1) or ρ j i,j ∈ (0, 1). So µ{f = g} ≤ µ(E i ) + µ(E j ).
3.1. Proof that Proposition 2.10 Assumption (3) holds. To finish the proof of Proposition 2.10 Assumption (3) we will relate ρ i i,j ρ j i,j to µ(E i ) + µ(E j ) from Lemma 3.5. In order to do so, we will show it suffices to add a technical assumption about the behavior of ψ.
Lemma 3.6. We may assume that (3.1)
∀i ∀j so that j > i+ 4 δ log 1 µ(A i ) , we in fact have e − δ 4 |i−j| ≤ min σ 4 7 4 1 ψ(b j ) 4 , r 2 , 2 −(n H +2) ,
where r is a fixed injectivity radius for a stratum H, and n H is chosen as in Lemma 2.3.
Lemma 3.7. Under the assumptions of Lemma 3.6 there exists a constant C > 1 so that
C ρ i i,j ρ j i,j > µ(E i ) + µ(E j ).
Indeed Lemma 3.7 is sufficient to conclude the proof of (3) as follows:
Proof of Proposition 2.14, part (3). Fix i and let j > m i . Then
µ(A i ∩ A j ) ≤ ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) + µ(A i ) + µ(A j ) (By Lemma 3.5) ≤ ρ i i,j (ρ j i,j • g ℓ 0 (j−i) )) + C ρ i i,j ρ j i,j (By Lemma 3.7) ≤ Cµ(A i ) µ(A j ) + e − δ 4 |i−j| + Cµ(A i )µ(A j ) (by Corollary 3.4, and since ρ ℓ i,j ≤ µ(H c H ,σ,ℓ ) = µ(A ℓ )) ≤ Cµ(A i ) µ(A j ) + e − δ 4 |i−j| (Setting C = 2C.)
Since Lemma 3.7 depends on the additional assumptions in Lemma 3.6, we will first prove Lemma 3.6 by replacing the function ψ with a function ψ which satisfies Eq. (3.1) and is still sufficient to prove the desired conclusion for ψ. The proof of Lemma 3.6 uses Lemma 3.8. After stating Lemma 3.8, we will then give a proof of Lemma 3.6, then Lemma 3.7, and the section concludes with the proof of Lemma 3.8.
Lemma 3.8. Let {a j } j∈N be a non-increasing sequence of positive numbers with ∞ j=1 a j = ∞. For any ρ, k, τ > 0 with τ < 1, there exists a constant C > 0 and a non-increasing sequence {c j } j∈N with c j = min 1 j , max{a j , 1 j 2 } , so that ∞ j=1 c j = ∞, j:c j >a j a j < ∞, and
(3.2) whenever i ≥ 3, for all j > max{i − C log(c i ), 9}, we have e −ρ(j−i) < τ c k j .
Proof of Lemma 3.6. We first note that from the proof of Lemma 2.3 that we can choose n H large enough so that we always have 2 −n H < 2r, and thus
min σ 4 7 4 1 ψ(b j ) 4 , r 2 , 2 −(n H +2) = min σ 4 7 4 1 ψ(b j ) 4 , 2 −(n H +2)
We next construct a modification to remove the constant upper bound. That is, we replace ψ by ψ so that e − δ 4 |i−j| ≤ σ 4 1 ψ(b j ) 4 , r 2 , 2 −(n H +2) . Namely using the constants from Lemma 2.3, define
ψ(b i ) = ψ(b i ) M σ 2 ψ(b i ) < 2 −(n H +2) M σ 2 2 (n H +2) otherwise.
We now have the constant upper bounds are trivially satisfied for ψ for all j > i + 4 δ log 1 µ(A i ) . Moreover for sets A j corresponding to ψ, A j ⊆ A j , and so µ(lim sup A j ) ≥ µ(lim sup A j ). Since our goal is to prove positive measure, we may now always assume the constant upper bounds hold.
We now construct ψ from ψ which satisfies Eq. (3.1) by defining ψ(b j ) = c −1/2 j where c j = min{1/j, max{a j , 1/j 2 }} is the sequence from Lemma 3.8 for a j = 1 ψ(b j ) 2 , ρ = δ 4 , k = 2, and τ = σ 4 7 4 < 1. Let A i be the set corresponding to ψ.
Notice that c i = 1 ψ(2 i ) 2 , so by Lemma 3.8 whenever i ≥ 3 and j > max{i − C log( ψ(2 i ) 2 ), 9}, we indeed have for i large enough that making C larger if necessary
e − δ 4 |i−j| < ( ψ(2 i )) −2·C δ 4 < mσ 2 ( ψ(2 i )) −2 ≤ µ( A i )
and moreover we have the desired inequality that
e − δ 4 |i−j| < σ 4 7 4 1 ψ(2 j ) 4 .
The restrictions on i ≥ 3 and j > 9 do not play a role in the conclusion for the limsup sets. For j ≥ n > 9, notice c j ≤ a j implies A j ⊆ A j , so
∞ j=n A j ⊇ j=n c j ≤a j A j ∪ j=n c j >a j A j ⊇ j=n c j ≤a j A j .
On the other hand since c j > a j exactly when c j = 1
j 2 , µ ∞ j=n A j ≤ µ ∞ j=n c j ≤a j A j + µ ∞ j=n c j >a j A j ≤ µ ∞ j=n c j ≤a j A j + ∞ j=n c j >a j mσ 2 j 2 .
Since the tail sum of convergent series goes to zero, for any ǫ we can choose n large enough so that j>n,c j >a j mσ 2
j 2 < ǫ, Hence, µ ∞ j=n A j ≥ µ j=n c j ≤a j A j ≥ µ ∞ j=n A j − ǫ.
Thus in the limit we conclude µ(lim sup A j ) ≥ µ(lim sup A j ).
With our preliminary work, Lemma 3.7 follows from geometric estimates on measures of balls and trapezoids.
Proof of Lemma 3.7. Since ρ j i,j ≤ ρ j i,j , we have
(3.3) ρ i i,j ρ j i,j ≥ ρ j i,j 2 ≥ µ{ρ j i,j = 1} 2 .
Now we want to get a lower bound for the measure of the set where ρ j i,j = 1. To do this, we need to find the area of the subset of the trapezoid T + c H ,σ,j where the ρ j i,j = 1. To do this, note the horizontal line y = ǫ i,j from c H + ǫ to 1 − ǫ give the height of the inner trapezoid. The line which is length ǫ i,j away from the line y = σ ψ(b j ) x is given by
y = σ ψ(b j ) x − ǫ 1 + σ 2 ψ(b j ) 2 .
Thus the four corners of the trapezoid where ρ j i,j = 1 are given by (c H + ǫ, ǫ), (1 − ǫ, ǫ),
1 − ǫ, σ ψ(b j ) (1 − ǫ) − ǫ 1 + σ 2 ψ(b j ) 2 , c H + ǫ, σ ψ(b j ) (c H + ǫ) − ǫ 1 + σ 2 ψ(b j ) 2 .
Hence the area of subset of T + c H ,σ,j where ρ j i,j = 1 is given by
(1 − c H − 2ǫ i,j ) 2 σ ψ(b j ) (1 + c H ) − 2ǫ i,j 1 + σ 2 ψ(b j ) 2 − 2ǫ i,j .
By symmetry, the area of T − c H ,σ,j where ρ j i,j = 1 is the same as the area for T + c H ,σ,j . Thus the total area is the product of the two trapezoids with the product of the ball where n + 4 = 2g + s − 1 and σ(n) is gives the volume of the n-ball. That is .1), ǫ i,j ≤ r 2 so d n is some constant depending only on n, and since 1 + c H ≥ 1)
µ({ρ j i,j = 1}) = (1 − c H − 2ǫ i,j ) 2 4 σ ψ(b j ) (1 + c H ) − 2ǫ i,j 1 + σ 2 ψ(b j ) 2 − 2ǫ i,j 2 · σ(n)(r − ǫ i,j ) n ≥ d n (1 − c H − 2ǫ i,j ) 2 σ ψ(b j ) − 2ǫ i,j 1 + σ 2 ψ(b j ) 2 − 2ǫ i,j 2 (by (3= d n 2 −n H − 2ǫ i,j 2 σ ψ(b j ) − 6ǫ i,j 2
(substituting 1 − c H = 2 −n H from Lemma 2.3 and and the trivial bounds
σ 2 /ψ(b j ) 2 < 1 < 3) ≥ d n 2 2(n H +1) σ ψ(b j ) − 6ǫ i,j 2 (assuming by (3.1) ǫ i,j < 2 −(n H +2) , which implies 2 −n H − 2ǫ i,j > 2 −(n H +1) ) ≥ d n 2 2(n H +1) ǫ 1 2 i,j .
(assuming by (3.1) that ǫ i,j ≤ σ 4 7 4 ψ(b j ) 4 , which implies 6ǫ i,j + ǫ
1 4 i,j ≤ 7ǫ 1 4 i,j ≤ σ ψ(b j ) )
Combining this fact with (3.3), we obtain
(3.4) ρ i i,j ρ j i,j ≥ d 2 n 2 4(n H +1) ǫ i,j .
Now from the other end we want an upper bound for µ{ρ i ∈ (0, 1)} + µ{ρ j ∈ (0, 1)} ≤ 2Cǫ i,j .
Given ℓ = i or ℓ = j, we have the area of the ǫ i,j -boundary of one of the trapezoids T ± c H ,σ,2 ℓ is given by
µ(∂ ǫ i,j T ± c H ,σ,2 ℓ ) = σ 2ψ(b ℓ ) (1 − c 2 H ) − 1 − c H 2 − ǫ i,j σ ψ(b ℓ ) (1 + c H ) − 2ǫ i,j 1 + σ 2 ψ(b ℓ ) 2
(taking area of T ± c H ,σ,ℓ less the area where ρ ℓ i,j = 1)
= ǫ i,j σ ψ(b ℓ ) (1 + c H ) + (1 − c H − 2ǫ i,j ) 1 + 1 + σ 2 ψ(b ℓ ) 2 ≤ ǫ i,j 1 + c H + (1 − c H )(1 +
√ 2) (assuming σ < ψ(b ℓ ) which is easy since σ < 1 is fixed and ψ ≥ 1 is non-decreasing)
≤ ǫ i,j C. (where C depends on c H ) Thus (3.5) µ({ρ i i,j ∈ (0, 1)}) + µ({ρ j i,j ∈ (0, 1)}) ≤ 2Cǫ i,j . Combining Equation 3.5 with Equation 3.4, (3.6) ρ i i,j ρ j i,j ≥ d 2 n 2 4(n H +1) ǫ i,j ≥ d 2 n 2 4(n H +1) (2C) µ{ρ i i,j ∈ (0, 1)} + µ{ρ j i,j ∈ (0, 1)} . Setting C = 2 4(n H +1) (2C) d 2 n
, we can assume C > 1 since d n is bounded above by a fixed constant, and we can make C larger if necessary.
Proof of Lemma 3.8. We first claim the following:
Claim: The sequence c j is non-increasing, has divergent sum and j:c j >a j a j < ∞ and j:c j =a j c j < ∞.
Proof of Claim. The maximum of two non-increasing sequences is non-increasing and so max{a j , 1 j 2 } is a non-increasing sequence (in j). Similarly the minimum of two non-increasing sequences is non-increasing and so c j is non-increasing.
If max{a j , 1 j 2 } = 1 j 2 , then c j = j −2 > a j . Otherwise c j = min{1/j, a j } ≤ a j .
If c j > a j then a j < 1 j 2 and so clearly c j >a j a j < 1 j 2 < ∞. On the other hand, since c j > a j is only possible when c j = j −2 ,
c j >a j c j = c j >a j j −2 < ∞. Now observe that (3.7) 2 k+1 −1 j=2 k c j ≥ min{ 1 2 , 2 k a 2 k+1 }.
Indeed we are estimating the sum from below by 2 k c 2 k+1 and considering the different possibilities of c 2 k+1 . Notice that 2 k 2 k a 2 k+1 ≥ j a j and so 2 k a 2 k+1 diverges and thus
k min{ 1 2 , 2 k a 2 k+1 } diverges. So j c j = k 2 k+1 −1 j=2 k c j diverges.
We have proved the Claim and now proceed with the remainder of the proof of Lemma 3.8. We now show if C > 4 k ρ ( 1 τ + 1) then Eq. (3.2) holds where τ = min{τ, 1} and k = max{ 1 2 , k}. Indeed it suffices to show that for all j > i + C log(i) we have that
(3.8) e −ρ(j−i) < τ j 2k .
Clearly smaller τ and larger 2k make (3.8) harder to satisfy. So from here we assume τ ≤ 1 and 2k ≥ 1, which motivated our choice of τ and k Under these assumptions, Equation
(3.8) is implied by j − i > 2k ρ log( j τ ).
If j ≤ 2i this follows from our condition on C. Indeed (2)).
j − i > 4 k ρ 1 τ + 1 log(i) > 4 k ρ 1 τ + 1 (log(j) − log
And so
j − i > 4 k ρ (log j 1 τ +1 − log(2 1 τ +1 )) > 2 k ρ log(j 1 τ +1 ) > 2k ρ log j τ .
Note that the second inequality uses that log(j 1 τ +1 ) > 2 log(2 1 τ +1 ) because j ≥ 9 > 2 2 and the third inequality uses that j 1 τ +1 > 1 τ j for all j ≥ 9 and τ > 0.
For the case when j > 2i, set f (x) = x − i and g(x) = 2k ρ log x τ . Note that f (2i) > g(2i) from the case where j ≤ 2i. Moreover, f ′ (x) = 1 > g ′ (x) = 2k xρτ for all x > 2k ρτ . Since i ≥ 3 and log(3) > 1,
j > i + C log(i) > 3 + 4k ρ ( 1 τ + 1) log(3) > 2k ρτ .
So for all j > 2i we have f ′ (j) > g ′ (j) and f (2i) > g(2i). Hence f (j) > g(j) for all j ≥ 2i as desired.
Verifying Proposition 2.10 Assumption (4)
We begin this section by defining the sets B i and C i required for Proposition 2.14, and then verify these sets satisfy Assumption (4) of Proposition 2.10.
Definition 3 (Definition of the B's). Set I def = (− π 12 , π 12 ). For k ∈ N define
B k = g log(b k ) g log ψ(b k ) σ θ∈I r θ W k
where we have the following definitions. We first pull back the set A k so that trapezoids in H c,σ,k makes a 45 degree angle so
W k = g log σ ψ(b k ) g −log(b k ) A k .
Then we restrict to a smaller subset of W k denoted W k so that r θ W k ⊂ W k for θ ∈ I. That is W k is the set of ω with two holonomy vectors v 1 and v 2 satisfying
c H 2σ ψ(b k ) ≤ |v 1 |, |v 2 | ≤ σ ψ(b k ) arg(v 1 ) ∈ π 12
, π 6 and arg(v 2 ) ∈ − π 6 , − π 12 .
In the graphic below, the gray shaded region corresponds to the region W k and the pink region corresponds to W k .
Definition 4 (Definition of the C's). Define
C k = g log(b k ) g log ψ(b k ) σ S(c H , σ, b k )
where S(c H , σ, t) = ω : ω has a holonomy vector v with |v| ∈ c H σ ψ(t)
, 2σ ψ(t) .
4.1.
Proof that (4) (a)-(c) hold. We now verify assumptions (4) (a), (b) and (c), where we set c = c H . Fix i and take j so that i < j ≤ i + C log 1 µ(A i ) .
(a) As constructed B i ⊆ A i and A j ⊆ C j . (b) Following the strategy of Lemma 2.3 where we compute the area of the sectors instead of trapezoids,
µ(W i ) ≥ σ 2 ψ(b i ) 2 π 24 (1 − 2c H ) 2 m 2g+s−3 (B).
Thus by Lemma 2.3,
µ(B i ) ≥ µ(W i ) ≥ µ(A i ) 1 m π 24 (1 − 2c H ) 2 m 2g+s−3 (B).
(c) Since the measure is invariant under geodesic flow and by Lemma 2.3,
µ(A j ) = µ(H c H ,σ,j ) ≥ m σ 2 ψ(b j ) 2 .
By Masur-Smillie Lemma 2.2,
µ(C j ) = µ(S(c H , σ, 2 j )) ≤ M σ ψ(b j )
.
Thus
µ(C j ) ≤ M √ m mσ 2 ψ(b j ) 2 = M √ m µ(A j ) 1 2 .
4.2. Construction and circle averages of logsmooth functions. The main goal of this section is to prove Corollary 4.6, which extends the statements of [Doz19] (giving averages over intervals) to include so-called logsmooth functions from [Ath06] (which gives averages over the full circle)
Definition 5. A complex K in ω is a closed subset of X whose boundary ∂K consists of a union of disjoint (in the interior) saddle connections such that if ∂K contains three saddle connections bounding a triangle, then the interior of that triangle is in K. Given a complex K the complexity of K is the number of saddle connections needed to triangulate K. For any δ > 0 and k ∈ N, if M is the complexity of ω,
α k (ω) = max K complexity k area(K)<2 k−M −1 1 |∂K| 1+δ .
If the set over which we take the maximum is empty, then we set α k (ω) = 0.
Definition 6. Given a function f on H and a point ω ∈ H, we let
Ave t (f )(ω) = 1 2π 2π 0 f (g t r θ ω) dθ.
Note that α 1 (ω) = 1 ℓ(ω) 1+δ where ℓ(ω) is the length of the shortest saddle connection. Since M is finite for all k large enough, α k (ω) = 0. For more information and intuition for the α k , see Section 5.3 of [Doz19]. From Proposition 5.3 of [Doz19], we have Proposition 4.1. Fix a stratum H, and 0 < δ < 1 2 . We can find a constant b such that for any interval I ⊆ S 1 , there exists a constant c I such that for all ω ∈ H and T ≥ 0,
I α k (g T r θ ω) dθ < c I e −(1−2δ)T j≥k α j (ω) + b|I|.
The strategy we will take is to extend this theorem to a function V δ which is a weighted average of the α k functions. We want to weight the average to have nice properties, so our goal is to recreate the following theorem for integrating over an interval I instead of [0, 2π).
(Ave) t (V (t) δ )(ω) = 2π 0 V (t) δ (g t r θ ω) dθ ≤ c 1 e −(1−δ)t V (t) δ (ω) + b t .
Moreover, V δ is logsmooth. That is
(4.1) V (t) δ (gω) ≤ c 3 V (t)
δ (ω) for all ω ∈ H and g ∈ V.
Finally, there exists a constant C δ,t so that
(4.2) V (t) δ (ω) V δ (ω) ∈ [C −1 δ,t , C δ,t ] where V δ = max{1, α 1 (ω)} = max{1, 1 ℓ(ω) 1+δ }.
We want to change Lemma 4.2 to restricting over an interval I. We now explicitly construct V δ using the following result. ). Fix H and 0 < δ < 1. There exists C > 0 so that for any t > 0, there exists constants b t and w t so that for any k and any ω ∈ H,
(4.3) Ave t (α k )(ω) ≤ Ce −t(1−δ) α k (ω) + w t j>k α j (ω) + b t .
Definition 7. Fix δ and t > 0. Define
λ (t) k = w t C + 1 k
where w t and C are the constants of 4.3. Define
V (t) δ (ω) = M k=0 λ (t) k α k (ω)
where M is the maximum complexity of ω.
Proof of Lemma 4.2. We first claim
(4.4) λ k Ce −(1−δ)t + w t k−1 j=0 λ j ≤ 2Cλ k e −(1−δ)t .
To see this holds, note that e −(1−δ)t ≥ 1. Thus since λ k ≥ 1, we have
1 − 1 λ k ≤ 1 ≤ w t C + 1 − 1 C w t e −(1−δ)t .
Simplifying and using the finite geometric series formula, this implies
k−1 j=0 λ j = λ k − 1 λ 1 − 1 ≤ λ k C w t e −(1−δ)t .
Multiplying by w t and adding λ k Ce −(1−δ)t to each side yields Equation 4.4.
We now want to prove that on average V δ shrinks over circles of radius t. To see this, we compute
Ave t (V δ )(ω) = n k=0 λ k Ave t (α k )(ω) ≤ n k=0 λ k Ce −t(1−δ) α k (ω) + w t j>k α j (ω) + b t (by 4.3) = n k=0 λ k Ce −t(1−δ) α k (ω) + w t n k=1 α k (ω) j = 0 k−1 λ j + b t (replacing b t = b t n j=1 λ j ) ≤ 2Ce −t(1−δ) λ 0 α 0 (ω) + n k=1 α k (ω) λ k Ce −t(1−δ) + w t k−1 j=0 λ j + b t (≤ Ce −t(1−δ) n k=0 λ k α k (ω) + b t = Ce −t(1−δ) V δ (ω) + b t .
The logsmoothness of the V δ follows from [EM01].
Now that we have defined V δ with the logsmooth property, we now proceed to extending the results of [Doz19] to include the V δ function.
Lemma 4.4. There exists a constant c 2 > 0 so that for any τ ≥ 0 and I ⊆ S 1 an interval, there exists t 0 (τ, |I|) ≥ 0 so that for any ω ∈ H and t > t 0 , we have
I V (τ ) δ (g t+τ r θ ω) dθ ≤ c 2 J Ave τ (V (τ ) δ )(g t r θ ω) dθ
where J ⊆ S 1 is an interval (that could depend on all other parameters) with |J| = |I|.
Proof. Note that this result would follow directly from linearity combined with Lemma 5.2 of [Doz19], except as stated in Lemma 5.2 the interval J could depend on α i . However following the proof exactly using linearity to replace each α i with V (τ ) δ , we take the interval 2I with the same center as I and twice the length. Then in the last 5 lines of the proof, we write 2I = J 1 ∪J 2 as a union of two intervals with |J 1 | = |J 2 | = |I|. Then
max j=1,2 J j Ave τ (V (τ ) δ (g t r θ ω) ≥ 1 2 2I Ave τ V (τ ) δ (g t r θ ω).
Now define J (which now depends on V τ δ instead of individual α k to be the interval on which the maximum is achieved, and the proof follows by linearity as desired. Let I ⊆ S 1 be an interval and by Lemma 4.4 let m be the smallest possible integer so that (m − 1)τ > t 0 (τ, |I|). That is m = 1 + t 0 (τ,|I|)
τ . There are constants c = c(τ, δ, |I|) > 0 and b τ = b(τ, δ) so that for all n ≥ m and for any ω ∈ H, (4.5)
I V (τ ) δ (g nτ r θ ω) dθ < ce −(1−δ)nτ V (τ ) δ (ω) + b τ |I|.
Proof. Let n ≥ m and ω ∈ H. Our goal is to construct the constants c and b τ to that Equation 4.5 holds. Indeed applying Lemma 4.4 followed by Lemma 4.2, we have (by our choice of τ the geometric sum is at most 2, so replacing b τ with 2c 2 b τ ,)
By the logsmooth property of V δ from Lemma 4.2, splitting into small steps, there exists some k(m, τ ) so that V
|I|
where we note m depends on τ and |I|, so c depends only on δ, τ and |I|.
Thus we obtain
I V (τ ) δ (g nτ r θ ω) dθ ≤ ce −τ n(1−δ) V (τ ) δ (ω) + b τ |I|.
Corollary 4.6. Fix a stratum H and 0 < δ < 1. There exists τ ≥ 0 so that for any interval I ⊆ S 1 , there exists constants c = c(τ, δ, |I|) > 0 and b τ = b(τ, δ) so that there exists an ℓ 0 so that for all ℓ ≥ ℓ 0 and for any ω ∈ H,
I V (τ ) δ (g ℓ r θ ω) dθ ≤ ce −(1−δ)ℓ V (τ ) δ (ω) + b τ |I|.
Proof. We choose τ to satisfy the assumption of Proposition 4.5. Choose ℓ ≥ mτ where m is defined in Proposition 4.5. Pick n 0 = min{n ∈ N : nτ > ℓ} and note n 0 − 1 ≥ m. Let r = n 0 τ − ℓ. Choose step sizes of r 0 so that r = kr 0 for some k ∈ N and g −r 0 ∈ V so we can apply (4.1).
Then from Proposition 4.5,
I V (τ ) δ (g ℓ r θ ω) dθ = I V (τ ) δ (g −kr 0 g n 0 τ r θ ω) dθ ≤ c k 3 I V (τ )
δ (g n 0 τ r θ ω) dθ (by (4.1)) ≤ c k 3 ce −(1−δ)n 0 τ V 4.3. Completion of the verification of (4) (d). To obtain an upper bound for µ(B i ∩C j ), by g t -invariance of µ, it suffices to find an upper bound for µ( B i ∩ C j ) where B i = θ∈I r θ W i = I ·W i and C j = g f (i,j) S(c H , σ, 2 j ) for f (i, j) = log b j−i ψ(b j ) ψ(b i ) .
We first use the fact that in S(c H , σ, 2 j ), the shortest possible saddle connection has length
ℓ(ω) ∈ c H σ ψ(b j ) , 2σ ψ(b j )
. Choose τ large enough to satisfy the assumption of Corollary 4.6. By (4.2),
C −1 δ,τ ψ(b j ) 2σ 1+δ 2 ≤ V (τ ) δ (ω) ≤ C δ,τ c H ψ(b j ) σ 1+δ 2 .
Thus by Markov's inequality,
µ( B i ∩ C j ) ≤ µ ω ∈ B i : V (τ ) δ (g −f (i,j) ω) ≥ C −1 δ,τ ψ(b j ) 2σ 1+δ 2 (4.6) ≤ C δ,τ ψ(b j ) 2σ − 1+δ 2 I·W i V (τ )
δ (g −f (i,j) ω) dµ(ω).
Disintegrating the measure µ = dθ d µ on SO(2) × (H/SO(2)) and increasing to a full SO(2) orbit (4.6) ≤ C δ,τ ψ(b j ) 2σ (since SO(2)W i is SO(2)-invariant)
Theorem 1. 1 .
1Let ψ : [1, ∞) → [1, ∞) be a nondecreasing function. In any connected component of a stratum of translation surfaces of genus at least
t) 2 dt < ∞, then for Masur-Veech almost every ω, lim inf R→∞ ψ(R)R 2 ζ ω (R) = ∞.
(t) 2 dt = ∞, then for Masur-Veech almost every ω, lim inf R→∞ ψ(R)R 2 ζ ω (R) = 0.1
Lemma 2 . 3 .
23Given a stratum H, there exists positive finite constants m = m(H), M = M (H), n H ∈ N, and σ H < 1 chosen so that for all 0 < σ < σ H and for c H
t) 2 dt = ∞, ψ satisfies the second assumption of Theorem 1.1. Motivated by Lemma 2.3 we assume
( 3 )
3Assumption (3) is proved in Section 3.1 using Corollary 3.4, Lemma 3.5, and Lemma 3.7. (4) Assumption (4) is proved as follows. The construction of the B k and C k sets along with proofs of Assumptions (4a), (4b), and (4c) are given in Section 4.1. The proof is completed by verifying Assumption (4d) in Section 4.3. 3. Verifying Proposition 2.10 Assumption (3): exponential decay of correlations for far away pairs
j ) 4 implies e − δ 4 |i−j| ≤ min σ 4 7 4
Lemma 4. 2 (
2Lemma 6.2[AG13], Proof in[Ath06]). Let V be a neighborhood of the identity in SL(2, R). Fix H a conncted stratum of H(α). For every 0 < δ < 1 there exists c 1 > 0 so that for all t > 0 there exists a function V(t) δ : H → [1, ∞) and a scalar b t satisfying the following properties. For all ω ∈ H,
Now we state Proposition 5.3 of[Doz19] for the V δ functions.
Proposition 4 . 5 .
45Fix a stratum H and 0 < δ < 1. Let c 1 and c 2 be the constants of Lemma 4.2 and Lemma 4.4, respectively. Choose τ ≥ 0 large enough so that c 1 c 2 e −(1−δ)τ < 1 2 .
δ
τ r θ ω) dθ + c 2 b τ |I| where the last equality follows from the fact that |J n−1 | = |I|. Now repeatedly applying this inequality for n − 1, n − 2, . . . , m with I replaced by J n−1 , J n−2 through J m which all have length |I|, we obtain I V (τ ) δ (g nτ r θ ω) ≤ c 1 c 2 e −((g (m−1)τ r θ ω) dθ + |I|b τ .
can write our constant c asc = (c 1 c 2 ) n−m+1 e −(−1δ)τ −m+1 c k(m,τ ) 3
δ
(ω) + b τ |I| . (since n 0 τ ≥ ℓ and r ≤ τ )Thus the final constants only depend on τ and not ℓ and we obtain the desired result.
δ
( ω) + b τ |I| d µ( ω) (by Corollary 4.6 and monotonicity of ψ,f (i, j) ≥ log(b j−i ) = ℓ 0 (j − i) > ℓ 0 ) = C δ,τ ψ(b j ) 2σ − 1+δ 2 ce −(1−δ)(f (i,j)) W i V (τ ) δ (ω) dµ(ω) + b τ |I|µ(W i ) .
In order to apply Theorem 3.1, we need to use bump functions to approximate the sets A i and A j . By g t -invariance of µ, µ(A i ∩ A j ) = µ(H c H ,σ,i ∩ g log(b j−i ) H c H ,σ,j ). So it suffices to define our bump function to approximate H c H ,σ,i and H c H ,σ,j .Remark 3.2. The metric used to define the Lipschitz norm in Theorem 3.1 is given in [AGY06,
Section 2.2.2]. The metric is uniformly comparable to the metric we use below, which uses period
coordinates. Indeed, the denominator in [AGY06, Equation (2.6)] is bounded away from zero
on compact sets.
Definition 2. For each i and each j > i + 4
δ log
1
µ(A i ) , define
by Equation 4.4 and replacing C with 2C)
Acknowledgements. We would like to thank Jayadev Athreya, Osama Khalil, and Howard Masur for useful discussions. The first author is supported by NSF grants DMS-2055354 and DMS-452762, the Sloan foundation, Poincaré chair, and Warnock chair. The second author is supported by the Deutsche Forschungsgemeinschaft (DFG) -Projektnummer 445466444.
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"Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation",
"Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation"
]
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"Chen-Huan Wu \nCollege of Physics and Electronic Engineering\nNorthwest Normal University\n730070LanzhouChina\n"
]
| [
"College of Physics and Electronic Engineering\nNorthwest Normal University\n730070LanzhouChina"
]
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| We investigate the properties of attractive polaron formed by a single impurity dressed with the particle-hole excitations in a three-dimensional (3D) doped (extrinsic) parabolic system. Base on the single particle-hole variational ansatz, we study the pair propagator, self-energy, and the non-self-consistent medium T -matrix. The non-self-consistent T -matrix discussed in this paper contains only the open channel since we don't consider the shift of center-of-mass due to the resonance (e.g., induced by the magnetic field). Besides, since we focus on the low-density regime of the majority particles, the effective Fermi wave vector is small. The scattering form factor is discussed in detail for the chiral case and compared to the non-chiral one. The effects of the bare coupling strength, which is momentum-cutoff-dependent, are also discussed. We found that the pair propagator and the related quantities, like the self-energy, spectral function, induced effective mass, and residue (spectral weight), all exhibit different features in the low-momentum regime and the high one, which also related to the polaronic instabilities as well as the many-body fluctuation and nonadiabatic/adiabatic dynamics. The pair-propagator and the energy relaxation time at finite temperature are also explored. | 10.1016/j.physb.2020.412127 | [
"https://arxiv.org/pdf/1908.10196v3.pdf"
]
| 201,646,135 | 1908.10196 | 22959d0d409b07a6816cac0adbd122d0ca9b2670 |
Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation
25 Aug 2019 August 28, 2019
Chen-Huan Wu
College of Physics and Electronic Engineering
Northwest Normal University
730070LanzhouChina
Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation
25 Aug 2019 August 28, 2019Attractive polaronnon-self-consistent medium T -matrixRetarded self- energyPair propagatorChiral systemLadder approximation
We investigate the properties of attractive polaron formed by a single impurity dressed with the particle-hole excitations in a three-dimensional (3D) doped (extrinsic) parabolic system. Base on the single particle-hole variational ansatz, we study the pair propagator, self-energy, and the non-self-consistent medium T -matrix. The non-self-consistent T -matrix discussed in this paper contains only the open channel since we don't consider the shift of center-of-mass due to the resonance (e.g., induced by the magnetic field). Besides, since we focus on the low-density regime of the majority particles, the effective Fermi wave vector is small. The scattering form factor is discussed in detail for the chiral case and compared to the non-chiral one. The effects of the bare coupling strength, which is momentum-cutoff-dependent, are also discussed. We found that the pair propagator and the related quantities, like the self-energy, spectral function, induced effective mass, and residue (spectral weight), all exhibit different features in the low-momentum regime and the high one, which also related to the polaronic instabilities as well as the many-body fluctuation and nonadiabatic/adiabatic dynamics. The pair-propagator and the energy relaxation time at finite temperature are also explored.
Introduction
The polaron formed in the cold-atom systems has been intensely studied where a lattice model is absent similar to the Frohlich model, but the polaron formed in a lattice model (atomic lattice or the crystal lattice) has been less studied, especially the topological [1,2] or chiral systems. In this paper, we focus on the properties of fermion polaron formed in a three-dimensional (3D) doped parabolic system in the crystal lattice. Considering the effect of impurities (in coherent background), the polaron as an excited quasiparticle in the population/spin-imbalanced Fermi gases, BEC, or the topological insulator [2,3,4], are important when the many-body effect is been taken into account. Unlike the gapless Dirac system where the electron-electron interaction usually gives rise to the renormalization of quasiparticle velocity [5,6], for systems with quadratic dispersion, the electron-electron interaction usually gives rise to the effective mass renormalization [7], which will be discussed in this paper, and we note that the effective mass approximation [8,9,10] is applicable for the polaron in small momentum regime. Besides, since the spin rotation is missing in our system in the presence of δ-type impurity field, the spin structure is fixed, and the interacting spins between impurity and majority particle are usually opposite in direction (due to the Pauli principle) which provides the opportunity to form the Cooper pair and the strongly bound dimer in superfluid. That also implies the low-temperature which with weaker spin relaxation is beneficial to the formation of polarons, that can also be seen from the finite temperature pair propagator as presented in this paper. We present in Appendix.A a brief discussion of the variational approach in mean-field approximation which is valid in the weak interacting regime with non-too-low density [11], for the polaron dressed by the partially polarized excitations-cloud, however, the mean-fields approximation sometimes overestimates the interaction effect [12] in the strongly interacting regime (with tightly bound dimers) where the self-trapping, soliton, and breather are harder to formed than that in the weakly bound pairs (e.g., the BCS superfluid state in the non-Fermi-liquid picture). The stable repulsive polarons are most likely to be found in the side away from the Feshbach resonance (for gases) where polaron energy is large and positive (with 1/k F a ≫ 1; a ψφ is the scattering length related to the impurity-majority interaction, which is produced by the attractive potential and k F is the fermi wave vector), while in strong scattering region (1/k F a 1), the tightly-bound molecule (within Fermi-liquid picture) could also be found which with binding energy E b = 2 /2a 2 m > 0 [9,13], as experimentally realized in Ref. [14,15]. However, in most cases the repulsive polaron is thermodynamically unstable in the strong interaction region even at low-temperature. While for the attractive polaron, whose eigenenergy is negative, it can be observed in the solid state systems [13,16,17,18,19,20] including the topological materials, through the, e.g., substrate-related polaronic effect. Besides, the effects of both the intrinsic and extrinsic (like the electric field-induced Rashba) spin-orbit coupling on the polaron have also been explored [16,17,18,19], and it is found that the spin-orbit coupling is also related to the polaron-molecule transition [21]. The quasiparticle residue Z (spectral weight), as an important property of the polarons, which can be measured by the Rabi oscillations [22], is finite even at zero-energy state for the parabolic system or normal Dirac semimetals, but it vanishes at zero-energy state for the Weyl semimetals which with higher dispersion [23,24] (i.e., the multi-Weyl semimetal).
In this paper, we will focus on the weak-interacting (short-range) regime in nonadiabatic configuration, where the quasiparticle spectral weight is away from zero and the polaron is welldefined (and thus long-lived). While the long-range Coulomb interaction, which will gives rise to the adiabatic effect and thus accelerates the electrons, is been ignored here due to the screening effect. For example, by including a Thomas-Fermi screening wave vector (k T F = me 2 /ǫ) into the Coulomb interaction (i.e., the static part of the particle-hole bubble), the interaction range will becomes shorter, and it satisfies a B k F ≪ 1 [25,26] (a B = k −1 T F ) at small density. In twobody problem (k F → 0), the feature of screened polaron will becomes more obvious in the non-relativistic limit, which with a slower motion compared to the intrinsic Dirac electron (relativistic), and that also gives rise to the nonadiabatic dynamics. Due to the nonadibatic character, the Matsubara Green's function is fully momentum-and frequency-dependent in renormalization group flow, which is different from the bare Coulomb interaction in the clean Dirac system with the instantaneous nature. The vertex function as well as the pair propagator is also fully momentum-and frequency-dependent.
For the attractive fermionic polaron formed through the attractive interaction between the impurity with the electron-hole pairs excited in a doped parabolic system in weak coupling regime, its polaronic effects are investigated by using the non self-consistent T -matrix approach (consist of the undressed propagators, and thus the particle/energy conserving is broken) within the ladder approximation. The accuracy of non-self-consistent method used here has been verified [27,28]. Note that, however, the energy of polaron obtained here will shows deviation from the DFT result (for unit cell) [29] since the Coulomb energy are ignored, also, the nonself-consistent treatment will results in deviations from the real total energy [28]. We consider the low-doping case of the parabolic system so that the interband electron-hole excitation can be created by the impurity, and not be suppressed by the phase space restriction. Thus the system discussed in this paper (no matter the chiral or nonchiral one) will exhibits a little Fermi liquid charater, which can be seen from the power-law behavior in frequency of the self-energy imaginary part. Our theory can also be applied to the massive (doped) Dirac or Weyl systems where the band structure in low-energy regime is gapped.
In Sec.2, we introduce the model we studying. The Hamiltonian of the nonchiral or chiral impurity model are presented. In Sec.3, we present the expressions of the non-self-consistent T -matrix, part-propagator, and self-energy. The interacting Hamiltonian are presented, where we can see that the polaron problem can be related to the Schrodinger-type eigenvalue problem subjected to a wave functional constraint, which is similar to Ref. [29,30]. Besides, the validity for non-self-consistent T -matrix in studying the many-body effect are also discussed in this section. In Sec.4, we present the main results of this paper, where the single-channel BCS model as well as the variational wave function are introduced, and the pair propagator-related quantities (including the self-energy and spectral function) are derived for the chiral and nonchiral cases for zero-temperature case. The induced effective masses and the residue are also calculated for different coupling strength. The scattering form factor here is different from the one with single-particle propagation. The polaronic stability and the many-body fluctuation are also discussed. In Sec.5, we discuss the pair propagator and the relaxation time at finite temperature which exhibit some differences compared to the zero-temperature case.
Theories
Different to the non-Fermi-liquid picture in the Dirac/Weyl semimetal, the impurities of the Fermi gases are mobile as widely studied [13,14,31], while for the immobile one the Kondo effect as well as the indirect exchange interactions are considered. For such mobile impurity, the δ-type impurity field (within Born approximation) is valid in measuring the effect of impurity-interaction. However, that also cause the vanishing of the spin rotation as well as the intrinsic spin current during the impurity-majority scattering (collision) process. Then, the fixed spin structure guarantees that there exists only the singlet pairing between the impurity and majority particles, otherwise the triplet pairing exists as discussed in Ref. [21]. In Fermi gases or the dilute BEC, the interactions between impurity and majority component, and that between the majority particles needed to taken into account, such problems are usually dealed by the non-self-consistent many-body T -matrix (i.e., the ladder approximation). Here we consider the interactions between the fermions (bath) and the single fermionic impurity. The difference compared to the normal solid state system [32,33,34] is that we are taking the mobile impurity into account and note that we still focus on the single impurity problem since the Fermi polaron is well defined (with symmetry and easy-to-identify spectral function) in the single-impurity-limit [13]. At first, we write the microscopic euclidean action as
S = dτ d 3 r{ψ † [∂ τ + H 0 (k)]ψ + 1 2 α=x,y,z (∂ α φ) 2 + 1 2 g ψψ ψ † ψψ † ψ + g ψφ ψ † ψφ † φ},(1)
where ψ and φ denote the majority and impurity field, respectively. g ψψ and g ψφ are the intraspecies (only around the impurity) and interspecies coupling, respectively, for the mobile particles. For parabolic chiral system (doped), the noninteracting Hamiltonian of the impurity reads
H 0 = |p| 2 2mp · σ σ σ + v z p z σ z − µ, = p 2 x 2m σ x + 2 p x p y 2m σ y − p 2 y 2m σ x + v z p z σ z − µ.(2)
Note that here p can be replaced by other momenta to represent the other particles (like the majority component). While for the nonchiral parabolicn system, the Hamiltonian can be obatin by removing the termp · σ σ σ in above equation or replacing it by a (spin) Pauli matrix σ z . We assume the longitudinal term involving p z is small, and thus the eigenenergy can be approximated as p 2 2m − µ. Due to the chirality (as can be seen, the momentum is locked with the Pauli matrices acting on the band space), the eigenvectors can be written as [24] (after ignoring the p z -term)
|p = e ip·r √ S 1 √ 2 1 λe i2φλ ,(3)
where φ = arctan py px is the polar angle of p. λ = ±1 correspond to the electron and hole states, respectively.
In this paper, we only consider the interspecies interaction, g ψφ , which describes the bare attractive contact interaction strength. The contact potential (with Guassian broadening) which with only the short-range attractive/repulsive interaction is used here since the size of pair is much larger than the range of potential, thus the long-range Coulomb repulsion is reduced here. While for relativistic particles in, e.g., the 3D Dirac/Weyl system, the mass term is necessary to form the polaron. We here use the coupling constants of the impurity-fermion sea mixture system,
g −1 ψψ = m ψ 4π 2 a ψψ , g b ψφ = [ m 2π 2 a ψφ − d 3 k (2π) 3 2 m k 2 ] −1 , where m = m ψ m φ /(m ψ + m φ )
is the renormalized mass, superscript b denotes the bare coupling. Λ is the momentum cutoff. a ψφ is the impurity-majority scattering length. The g ψψ here follows the general definition of the background coupling constant which related to the background scattering length a ψψ . Indeed, the interaction effect of the polaron system requires the investigation of the effective mass in contrast to the semimetal system, especially in the strong interacting regiem with the obvious renormalization effect and the Fermi-liquid feature.
T -matrix and the self-energy
The ladder approximation (non-perturbative) is applicable not only for the imbalanced Fermi gases or nuclear physics, but also for the solid state systems with finite effective mass. Further, for large-species Fermion system, the ladder approximation is similar to the leading order 1/Nexpansion. In thermodynamic limit with N → ∞, the self-energy which describes the pairing fluctuation becomes zero [35] which implies that the interaction between impurity and majority particles vanishes (i.e., without the polaron). In the strong interacting case, the self-energy effect as well as the resummation of ladder diagrams (for the forward scattering) are important to be considered. The non-self consistent T -matrix, which does not contains the self-consistency of the Green's function, describes the fluctuations in s-wave cooper channel. Firstly, we can write the T -matrix between the single impurity and the majority component as
T (p + q, ω + Ω) = [ m 2π 2 a ψφ + Π(p + q, ω + Ω)] −1 ,(4)
where q is the momentum of the majority particle, p is the momentum of the impurity. Note that for the T -matrix here, we only consider the closed channel scattering (i.e., the bare case), and without consider the interchanging as well as the spin/valley degrees of freedom. The term (p + q) can be treated as the center-of-mass momentum. Ω is the Fermionic frequency since we assume the zero-temperature limit, similarly, ω ≈ ε p↓ = 2 (p 2 )
2m φ −µ ↓ = 2 (p 2 )
2m φ −Re Σ(p = 0, ω = 0) is the Bosonic frequency where Σ(p, ω) is the impurity self-energy as stated below. Here µ ↑ = µ ↓ due to the spin-imbalance. Note that this T -matrix is non-self-consistent, which with the bare impurity propagator and the majority propagator as diagrammatically shown by the Bethe-Salpeter equation (see, e.g., Ref. [36]). While the self-consistent T -matrix requires the dressed impurity propagator which containing the impurity self-energy effect, and it is more suitable to apply when take into account an infinite number of the majority particles, in which case its statistical properties emerge including the imbalance between the two majority species [37]. The Bethe-Salpeter equation about the non-self-consistent many-body T -matrix reads
T (p + q, ; p + q − k ′ ) = V 0 (p + q, ; p + q − k ′ ) + k V 0 (p + q, ; k)G φ 0 (p + q − k)G ψ 0 ( + k)T (p + q − k, + k; p + q − k − k ′ ) (5)
where V 0 are the bare impurity-majority interactions, specially, V 0 (p + q, ; k) is the interaction induced by the polarization operator (consist of the two bare Green's functions; see Appendix. B). k, k ′ are the relative momentum. G ψ 0 and G φ 0 are the bare Fermionic and Bosonic Green's function, respectively, as presented below. The symbol can be omitted, but we retain it here for the integrity of the above equation. In the absence of the center-of-mass momentum (p + q = 0), the Bethe-Salpeter equation reduced to the Lippmann-Schwinger equation
T (k 1 , k 2 ; ω) = V 0 (k 1 , k 2 ) + k 3 V 0 (k 1 , k 3 ) 1 ω + i0 − 2ε k 3 T (k 3 , k 2 , ω).(6)
The impurity-majority pair propagator (unrenormalized) in non-self consistent T -matrix approximation reads
Π(p + q, ω + Ω) = d 3 k (2π) 3 dν 2π G ψ 0 (−Ω + ν, k − q)G φ 0 (ω + Ω − ν, p + q − k),(7)
where
G ψ 0 (−Ω + ν, k − q) = [iν − iΩ − k 2 2m ψ + µ ↑ ] −1
is the noninteracting (in the absence of a condensate and the long-range Coulomb interaction) majority particle (Fermion) propagator and
G φ 0 (ω + Ω − ν, p + q − k) = [iω + iΩ − iν − (p+q−k) 2 2m φ + µ ↓ ] −1
is the bare impurity propagator (not the scalar-field one). In this majority particle propagator we ignore the perturbation from the single impurity to the Fermi sea. We at first discuss the self-energy of polaron in the general Fermi sea, which is usually referred to as the polaronic binding energy [12] or the molecule binding energy [30,13]. This self-energy can be obtained by the following impurity-majority interaction Hamiltonian
H int = g ψφ |Ψ ψ | 2 0 0 g ψφ |Ψ φ | 2 ,(8)
and the related the interaction energy
ε ψφ = n F d 3 RΨ(R)[ − 2 ∇ 2 R 2 m + U ψφ ]Ψ † (R),(9)where Ψ(R) = (Ψ ψ (R), Ψ φ (R)) is the normalized wave function. n F = d 3 q (2π) 3 dΩ
2π G F (q, Ω)e i0 + Ω is the numerical density of the Fermions, where G F (q, Ω) is the dressed (full) Green's function unlike the above ones, and gives the actural Fermion dispersion. There exists the constrains
4πn F Λ 0 dRR 2 Ψ(R)Ψ † (R) =I, |Ψ(R ≥ Λ r )| =1,(10)
where I is the identity matrix. Base on the many-body scattering theory at low-temperature, where we consider only the s-wave scattering, the T -matrix in Fermi (or Bose) gases is usually self-consistent, i.e., it's a two-channel T -matrix [38,39] while in our model, the T -matrix contains only the open channel (i.e., the bare one) which is non-self-consistent.
To study the many-body effect, the non-self-consistent T -matrix is similar but not exactly like the leading order 1/N expansion, since it ignores the dynamical screening effect. For this reason, the non-self-consistent T -matrix is more like the leading-order loop expansion within GV approximation rather than the leading order 1/N expansion within GW approximation. It's also found that the static screening (GV) to the Coulomb interaction can be a good approximation for the dynamical screening in the low doping regime, especially for carriers frequency of the order of binding energy [40]. The related studies are also reported in Refs. [35,27,41]. Besides, the validity of the non-self-consistent T -matrix in studing the BCS-BEC crossover has also been verified [42,43]. In our model, the self-energy about the interaction between mobile impurity and the bath reads
Σ(p, ω) = q<k F N F (Ω) m 2π 2 a ψφ + Π(p + q, ω + Ω) ,(11)
and the numerator can be replaced by θ(k F − q) at zero-temperature limit, where θ stands the step function. Note that this self-energy expression describes only the region around the single impurity (the attractive polaron) which is small but not localized (since the impurity is mobile). It's different to the Fermi gases that, the self-energy of the impurity does not contain the condensate density as well as the condensate-related spin fluctuation and the pair propagator contains the chiral factor F λλ ′ (λ, λ ′ = ±1) (the wave function overlap) which suppresses the backscattering and is absent in the 2D electron gas. The chiral factor here is indeed observble in the polarons formed in the Dirac system [44]. While for 2D electron gas, F λλ ′ = 1 and contains only the intraband contribution, except at a quantum Hall setup with strong magnetic filed as report in Ref. [45]. We can also see that, in the narrow gap limit, the pair propagator reduced to the well known dynamical polarization, and T = Π −1 . In the surface of Dirac system, since away from the condensed phase, the condensate density vanishes but the related pairing fluctuations remain as long as g ψφ = 0, i.e., the pairing instability exists (especially when the spin-orbit coupling turns on [21]) even in the case of g ψψ = 0.
Pair propagator and related quantities at zero-temperature limit
To describe the polaronic dynamics, we use the following BCS-type many-body Hamiltonian
H = k ε k↑ c † k↑ c k↑ + p ε p↓ c † p↓ c p↓ + 1 N k,p,q g q p − q|p k + q|k c † p−q↓ c † k+q↑ c k↑ c p↓ ,(12)
where N = S/s 0 is the total number of the unit cell where S is the total area and s 0 is the area of the unit cell. Within one-particle-hole approximation, which is valid according to the Monte Carlo calculation and the experimental results due to the destructive interference in the presence of more than one particle-hole part, the variational wave function reads [46,47]
|ψ = ψ 0 c † p↓ |0 ↑ + k>k F ,q<k F ψ kq c † p+q−k,↓ c † k,↑ c q,↑ |0 ↑ ,(13)
where |0 ↑ = Π k<k F c † k↑ |vac is the ground state of the majority particles [48] and |vac is the vacuum electron state. We focus on the coherence case, where the masses of impurity and the majority particle are comparable, and thus decoherence effect [49] is weak while the nonadiabatic dynamics is dominating. The first term in the right-hand-side of above equation describes the free impurity which assumed is that totally delocalized. k is the momentum of a majorityparticle scattered out of fermi surface, and q is the momentum of a majority-particle before scattering. Through the normalization condition ψ|ψ = 1, we have, after minimizing the total energy,
ψ kq =ψ 0 T (p + q, ω + Ω) ω − ε p+q−k,↓ − ε k,↑ + ε q,↑ , ψ 0 = 1 1 + k>k F ,q<k F ( ψ kq ψ 0 ) 2 .(14)
At zero-temperature limit, the above renormalized pair propagator can be written as
Π(p + q, ω + Ω) = − d 3 k (2π) 3 1 − N F (ε k↑ ) −ω − i0 − Ω + ε k↑ + ε p+q−k↓ F λλ ′ = − d 3 k (2π) 3 N F (ε k↑ ) − 1 ω + i0 + Ω − ε k↑ − ε p+q−k↓ F λλ ′ = − 4π (2π) 3 Λ k F −k 2 θ(k − k F ) ω + i0 + ε q↑ − ε k↑ − ε p+q−k↓ F λλ ′ dk,(15)
where N F is the Fermi-distribution function and it appears only in the presence of nonzero center-of-mass momentum. Note that instead of using a term 2 m k 2 to make the pair propagator convergent even in the ultraviolet limit [38,50] (renormalized pair propagator), we here use the momentum cutoff Λ similar to the literatures [9,26], and the value of momentum cutoff is setted as 3 eV which is the same as the graphene-like systems [51]. Interestingly, it is also found that, for open channel T -matrix (which is what we focus on throughout this paper), the open channel shift due to the medium effect (i.e., the integral for open channel propagator over the scattering wave vector [39,10]) just equals to this term ( k 2 m k 2 ) in the vacuum limit (with zero center-of-mass momentum and zero impurity frequency). Base the above expression of the pair propagator, we can obtain that, the polaron self-energy increases with the increasing mass term or coupling parameter g ψφ , however, there is an exception: when the intrinsic spinorbit coupling (not the extrinsic one) is presented, then the increase of g ψφ will reduces the self-energy since it will greatly reduces the mass [16]. As we can see, although the chiral factor F λλ ′ is contained, it has F λλ ′ = 1 for the non-chiral systems (like the 2D electron gas) or the systems which are dominated by the backscattering (like the bilayer Dirac system [52]).
Different to the retarded polarization function (density-density correlation) in one-loop approximation which only describes the scattering of one kind of particle (like the electron) due to the interaction (like the Coulomb interaction), the pair propagator describes both the scatterings of the impurity and the majority particle (electron-hole pair here). The second term in above trial wave function corresponds to the excited state where the impurity scattered from state with intrinsic momentum p to the final state with momentum p + q − k, i.e., the scattering wave vector is q − k; while the majority particle (electron-hole pair) scattered from the initial state with momentum 0 (does not exis yet until created by the impurity) to the final state with momentum k − q, i.e., the scattering wave vector is k − q. Note that here we assume the scattering wave vectors are small and thus the inter valley scattering can be ignored. We can see that the scattering wave vectors for impurity and majority particles are opposite in direction. Thus for such a pair propagator, the spinor wave function overlap (form factor) according to Eq.(3) reads
F λλ ′ = (p + q − k)|p k − q|0 = 1 2 (e iφ p+q−k λ ′ λ ′ e −iφ p+q−k λ ′ ) e −iφpλ λe iφpλ ,(16)
where the θ is the angle between the initial impurity wave vector p and the scatterig wave vector (q − k). Appearly such form factor is a little different to the one which is widely seen in the chiral solid system [24,53]. Through this form factor. we can clearly see the difference between the pair propagator and the one loop diagrams where the two propagators describe the same particle beforce and after scattering, respectively. For intraband scattering (λλ ′ = 1),
F λλ ′ = cos(φ p+q−k − φ p ) = p + (q − k)cosθ p 2 + (q − k) 2 + 2p(q − k)cosθ ≈ 1 − sin 2 θ 2p 2 (q − k) 2 .(17)
where θ is the angle between initial momentum p and the scattering one q − k. For interband scattering (λλ ′ = −1),
F λλ ′ = isin(φ p+q−k − φ p ) ≈ i 1 − [1 − sin 2 θ 2p 2 (q − k) 2 ] 2 .(18)
Next we only focus on the nonchiral and intraband chiral cases. For the polarons formed in parabolic systems with nonrelativistic interacting particles, the renormalized interacting strength can be represented in another representation
g −1 ψφ = 1 + g b Λ k=k F 1 2ε k g b .(19)
where the bare coupling g b is tunable and it tends to zero when let Λ → ∞.
Obviously, as showed in Fig.1(a), the above scattering form factor (Eq.(16)) corresponds to the pair propagator of Π 11 , while for the particle-hole propagator up to second order[10] Π 22 as showed in the Fig.1(b), we will see that the value of its form factor is the same as the one corresponds to Π 11 , i.e.,
F λλ ′ = (p + q − k)|p k − q|0 = F ′ λλ ′ = (p + q − k)|p 0|k − q .(20)
Note that we assume the direction of initial wave vector of the majority particle is along the finial one, i.e., k − q. But for the case that the direction of initial wave vector of the majority particle is the same with the impurity before scattering, i.e., p, and the scattering wave vector q − k ≫ p is mainly along the x-direction in the momentum space, then the form factor F λλ ′ (F ′ λλ ′ ) equals zero, which means that for chiral system the pair propagator vanishes, which is the case for helical system like Bi 2 Se 3 [3].
In Fig.2-3 and Fig.4-5, we show the pair propagator and the corresponding self-energy, respectively, for nonchiral and chiral cases. By comparing the chiral case to the nonchiral one, we can see that, when the angle θ (between the initial impurity wave vector and the scattering wave vector) is nonzero, both the pair propagator and self-energy diverges away from the nonchiral ones, and such divergence locates mainly in the low-momentum region. Thus the chiral effect leads to the instability to both the pair propagator and self-energy in small-p region. for low initial momentum of impurity. In Fig.4-5, we can also see that, for stronger bare coupling |g b |, the polaron has a lower self-energy. Since we consider the polaron formed in weak-coupling regime, the final value of self-energy obtained here at large momentum agrees with the result of perturbation theory: Σ(p, ω) ∝ g b n (the total particle-number density n is setted as 1 here). That is also in agreement with the mean-field result for the homogeneous condensate [27]. Note that during the calculation of pair propagator and self-energy, the direction of p is unfixed, and thus we don't integrate over the all possible angle θ (between p and q − k), as obviously can be seen from the figures. The suppression of marginal fermi liquid character can be seen from the self-energy as a function of ω as shown in lower-panel of Fig.5, where we can see that the imaginary part of the self-energy diverges away from the intrinsic case |ImΣ| ∼ ω [7] at large value of ω. Through the imaginary part of the lower-panel of Fig.5, we can also see that the marginal fermi liquid character is suppressed while the fermi liquid character is rised with the increasing g b . The increase of g b also gives rise to the intraband single-particle excitation.
We note that, in Fermi-liquid picture with strong screening effect, the Fermion interaction is dominated by the short-ranged one, i.e., the Hubbard interaction, then the strong spin fluctuation as well as the particle-hole fluctuation are possible to build the bipolaron [54]. The quasiparticle properties can also be detected by the spectral function, which measures the propability of exciting or removing a (quasi)particle at a certain momentum. Next we consider the particle spectral function containing the many-body effect,
A(p, Ω) = − 1 π |ImΣ(p, ω)| (Ω − ReΣ(p, ω) − 2 p 2 2m φ + µ ↓ ) 2 + (ImΣ(p, ω)) 2 .(21)
From Fig.6-7, we can see that, except for zero frequency, the spectral functions exhibit symmetry feature, and the peaks decrease with the increase of |g b |. From the intensity plot of spectral function (Fig.8), we can see that the spectral function exhibits similar dispersion with the parabolic impurity (before scattering) in low-momentum region, but exhibits linear dispersion in the large-p region. Also, we find that the width of spectral function becomes very narrow when close to zero-momentum. Further, by comparing to the dash-line in the center inset of Fig.8, we can easily see that the dispersion of original impurity ( p 2 2m ↓ − µ ↓ ) is been lowered by the polaronic effect (negative interaction energy), and, in the mean time, the effective mass is also increased due to the decrease of dispersion slope (compared to the original one).
At zero energy limit, the quasiparticle residue of the usual Dirac Fermions remains finite (here we assuming a noninteracting initial state) and thus the coefficients ψ 0 and ψ kq remain finite too, while for the multi-Weyl semimetal, the residue vanishes at zero-energy and then ψ 0 = ψ kq = 0. Foe zero momentum (q = 0) with the lowest dispersion, the impurity self-energy becomes,
Σ(p, ω) = 1 S µ ↑ 1 m 2π 2 a ψφ + Π(ω + Ω, p) 1 ω + i0 + µ ↑ ,(22)
with Π(ω + Ω, p) = Π(ω + Ω, p, q = 0), and S is the volume of the space where all the chemical potential around the polaron are taken into account. Now the center of mass is just p. That's also agree with the results of the Fröhlich polaron model [19,55] in long-wavelength limit which with the strong electron-phonon coupling as well as the observable optical excitations [20]. For simplicity, we further set µ ↑ = 0 and the Fermi frequency Ω = 1 (which is possible in the zero-temperature limit), then the polaron self-energy becomes Σ(p, ω) = T (p, ω)G 0 ψ (0, 0) where G 0 ψ (0, 0) = 1 The self-energy is shown in Fig.7(a). Then by substituting the above self-energy to the Eq.(22), we obtain the corresponding particle spectral function as shown in the Fig.7(b), where we set ω > 0 (ω > µ ↑ ) to make sure the spectral function here describes only the particle states. Through Fig.7(b), the total density of states can be obtained by integrating over the p-axis, while the occupation probability [56] can be obtained by integrating over the ω-axis.
In Fig.9, we show the effective masses induced by the polaronic effect and the quasiparticle residue, which read
∆m * =( ∂ 2 Σ(p, ω) ∂p 2 ) −1 , Z = 1 1 − Re∂ ω Σ(p, ω) ω=E(p) ,(23)
where E(p) is determined self-consistently by the equation
E(p) = p 2 2m ↓ − µ ↓ + ReΣ(E(p), p),(24)
after the expansion coefficients of the polaron trial wave function are obtained by performing the variational minimization. In Fig.9, we plot the induced effective mass and the quasiparticle residue as a function of momentum. In small momentum region (p < 4), both the induced effective mass and residue exhibit unusual behavior (due to the instabilities of the slow polaron with strong nonadiabatic dynamics). In this region, the interaction may very strong and thus the residue could be very low, and the induced effective mass may even becomes negative. In the large momentum region (p > 4), the polaron becomes relatively stable, then the power law behavior of ∆m * and the logarithmic behavior of Z can be seen. We can also see that, for stronger attractive interaction (i.e., for larger |g b |), the polaron has a higher effective mass (m * = m ↓ + ∆m * ), which is consistent with the result of Ref. [10,14], and the residue asymptotically approaches one more slowly. From the effective masses, we can easily see that the positive interaction (repulsive polaron) induces stronger instability compares to the negative one in the low-momentum region (especially for ω = 0). In stable regime, the induced effective masses ∆m * increase more and more fast with the increasing initial impurity momentum p. That is in consistent with the result of Ref. [27], which, in addition, shows that the polaron is possible to changes to molecule if the p keeps increasing. In large p region, the many-body fluctuation is supressed, and the interaction effect is faded (while the adiabatic effect is enhanced). When the residue reaches one, the impurity will acts like a free particle. Note that here the coupling strength is mainly controlled by the parameter g b , but also affected by the value of initial momentum p as shown in the figure.
Pair propagator and relaxation time at finite temperature
For the case of finite temperature, we introduce the fermionic Matsubara frequencies read Ω = (2n + 1)πT , ν = (2n ′ + 1)πT (n, n ′ are integer numbers). which are discrete variables. At finite temperature, the pair propagator can be written as
Π(p + q, ω + Ω) = d 3 k (2π) 3 [ n ′ T V G ψ 0 (ν, k)G φ 0 (ω + Ω − ν, p + q − k)],(25)
where we consider the single band model and regard the Green's functions as the only eigenvalue of the matrix. Here we define
G ψ 0 (ν, k) = 1 iν − k 2 2m ψ + µ ↑ , G φ 0 (ω + Ω − ν, p + q − k) = 1 iω + iΩ − iν − (p+q−k) 2 2m φ + µ ↓ .(26)
The summation over Matsubara frequencies (iν) can be calculated as
ν G ψ 0 (ν, k)G φ 0 (ω + Ω − ν, p + q − k) = ∞ n ′ =−∞ 1 i(2n ′ + 1)πT − a 1 −i(2n ′ + 1)πT − b = tanh b 2T + tanh a 2T 2T (a + b) ,(27)
where
a ≡ ε k↑ = k 2 2m ψ − µ ↑ , b ≡ −iω − iΩ + (p+q−k) 2 2m φ
− µ ↓ , and the above result can also be rewritten as For small a and b, i.e., in the limit of small energy and small frequency, we approximate tanh(x) ≈ x − x 3 3 + O(x 5 ), then the pair propagator (nonchiral) becomes
ν G ψ 0 (ν, k)G φ 0 (ω + Ω − ν, p + q − k) = e a+b T − 1 T (a + b)(e a/T + 1)(e b/T + 1) = N F (a)N F (b) N B (a + b)T (a + b) ,(28)Π(p + q, ω + Ω) =4π 1 (2π) 3 Λ k F k 2 dk tanh b 2T + tanh a 2T 2T (a + b) =4π 1 (2π) 3 Λ k F k 2 dk b 2T − 1 3 ( b 2T ) 3 + a 2T − 1 3 ( a 2T ) 3 2T (a + b) =4π 1 (2π) 3 Λ k F k 2 dk 1 4T 2 − 1 3 b 3 + a 3 (2T ) 4 1 a + b =4π 1 (2π) 3 Λ k F k 2 dk 1 4T 2 − 1 3 1 2T 4 (a 2 − ab + b 2 ) = 1 8π 2 T 2 (Λ − k F ) 3 3 − F ,(29)
where Fig.10, the momentum-or energy-dependence of the polarization function decreases with the increase of temperature. At high-enough temperature, both the imaginary and real pat of the polarization function become constant, and thus we can suspect that the induced effective mass will become infinite (selftrapped polaron) at high enough temperature while the residue will equals to one in the same case.
F = 1 6π 2 1 (2T ) 4 Λ k F k 2 (a 2 − ab + b 2 )dk = 1 6π 2 1 (2T ) 4 k 3 420m 2 ψ m 2 φ [35m 2 ψ (4c 2 m 2 φ + 2cm φ ((p + q) 2 − 2dm φ ) + ((p + q) 2 − 2dm φ ) 2 ) + 21k 2 m ψ (2cm φ (m ψ − 2m φ ) + 2dm φ (m φ − 2m ψ ) + (6m ψ − m φ )(p + q) 2 ) − 105km 2 ψ (p + q)(cm φ − 2dm φ + (p + q) 2 ) + 15k 4 (m 2 ψ − m φ m ψ + m 2 φ ) − 35k 3 m ψ (2m ψ − m φ )(p + q)] Λ k F . (30) where we define c ≡ µ ↑ , d ≡ iω + iΩ + µ ↓ . As shown in the
The ladder approximation (by summing over the ladder diagrams which correpond to the forward scattering) results in accurate results of the pair propagator and self-energy, and it also agrees with the Quantum Monte-Carlo calculation as well as the experimemtal results. The single-channel T -matrix (which contains the pair propagator) introduces the tuneable s-wave scattering length to the manipulation of the behavior of a single impurity embedded to a fermi sea, which describes the scattering between a pair of atoms with up and down spins, respectively, and within the center of mass frame with energy ε = ω −(p + q)/2(m ψ + m φ ) + µ ↑ + µ ↓ . At finite temperature and for the configuration that the number density of the bose impurity is much lower than the fermions (without the effect of three-atom loss (the Efimov trimers) [31,38,37]), the pair propagator can also be written as [50,31] Π(p + q, ω + Ω) =
k 1 − N F (ε k↑ ) − N F (ε p+q−k↓ ) ω + i0 − ε p+q−k↓ − ε k↑ + ε q↑ ,(31)
where p and q correspond to the momentum of impurity and hole respectively, ω and Ω correspond to the frequency of impurity and hole respectively. For pairing mechanism, this expression is definitely important, e.g., for the pairing instability [57,37,58,59] and the resonantly enhanced correlation, and its real part and imaginary part are easy to obtained by firstly replacing the imaginary frequencies in denominator with the analytical continuation and then using the Dirac identity (for retarded functions) lim η→0
1 x±iη = P ( 1 x ) ∓ iπδ(x).
We can see that the factor F λλ ′ is in fact related to the angle between the wave vectors of polaron (coherently dressed by the particle-hole excitations of majority part) and the electron with momentum k. And this term is unnecessary in the three (or two)-dimensional electron (or hole) gases, it is nonzero only when the eigenstates at different wave vectors have overlap (corresponds to the two statistical functions in the numerator), which for three-dimensional system (consider the longitudinal wave vector k z ) reads
Ψ + = e iθ cos θ ⊥ 2 sin θ ⊥ 2 , Ψ − = e iθ sin θ ⊥ 2 −cos θ ⊥ 2 ,(32)
where the indices ± denote the sign of band energy (i.e., the conduction band and valence band), and θ = atank y /k x , θ ⊥ = atan k 2 x + k 2 y /k z . Then the overlap factor reads F λλ ′ =
1±(cosθ ⊥ cosθ ′ ⊥ −sinθ ⊥ sinθ ′ ⊥ sin b) 2 where θ ′ ⊥ = atan k ′2 x + k ′2 y /k ′ z .
That is clearly different from the ones appear in two-dimensional system [60]. For the calculation in main text, we use the twodimensional chiral factor F λλ ′ = 1±cos b 2 = 1 2 (1 ± k+qcos a √ k 2 +q 2 +2kqcos a ) due to the nature of weakchirality of the system we discussed. For another case of three-dimensional system, at longwavelength limit (k z → 0) and with isotropic dispersion, such approximation is also applicable as shown in, e.g., Ref. [61]. When the vertex correction is not taken into account, for the case of inversed frequency Π(p+q, −ω −Ω), we can use the identity Π(p+q, −ω −Ω) = Π * (p+q, ω +Ω), i.e., Re Π(p + q, −ω − Ω) = ReΠ(p + q, ω + Ω), Im Π(p + q, −ω − Ω) = −ImΠ(p + q, ω + Ω) Further, when the chirality (from the Weyl system) appears, the causality relations are studied in Ref. [62]. Then the self-energy at finite temperature can be obtained as
Σ(p, ω) = T V k F 0 d 3 q (2π) 3 n T (p + q, ω + Ω)G 0 ψ (q, Ω),(33)
where G 0 ψ (q, Ω) can also be replaced by G ψ (q, Ω) which contains the self-energy term when consider the self-energy effect as done in Ref. [35] with strong scattering strength. In addition, we discuss the case when consider the ladder vertex correction, where summation over Matsubara frequency can be done by using the method of coutour integral (in optical limit) [63],
T V n ′ G ψ (ν)G φ (ω + Ω − ν)Γ(ν, ω + Ω − ν) = − C dz 2πi G ψ (z)G φ (ω + Ω − z)Γ(z, ω + Ω − z),(34)
where Γ(ν, ω + Ω − ν) denotes the vertex function.
At finite temperature where the s-wave scattering is still dominating, the low-energy excitations induced by quantum fluctuation has a more significant effect on the properties of polaron compared to the thermal excitations especially for the case of small-chemical potential, like the particle-hole parts (especially at low dimension [13,64]) or the phonon-like (Frøhlich type) excitations. For chiral system at finite temperature, the transport relaxation time (in ladder diagram) of impurity due to the scattering by the electron-hole pair contains a (1 − cosθ) term, which supresses the forward scattering (θ = 0) and exists as long as the elastic scattering is involved in the scattering event. This term cannot be found in the quasiparticle relaxation time in single bubble diagram, and it together with the chiral term determines the scattering cross section [52]. While for the gapless Dirac system, like the intrinsic graphene, both the forward and backward scattering are supressed, as calculated by literatures [52,65,66]. The Boltzmann transport theory gives the following inversed relaxation time (in second order Born approximation)
1 τ p = 2π k,q (1 − cosθ)W p+q−k,p ,(35)
where p ′ = p + q − k is the wave vector of impurity after scattering, and θ = φ p+q−k − φ p is the angle between wave vectors before and after scattering.
W p+q−k,p =(1 − N F (ε k−q↑ ))δ(ω − ε p+q−k↓ − ε k↑ + ε q↑ )|g b (k − q)| 2 | p + q − k|p | 2(36)
is the transition rate and can be approximated as the imaginary part of self-energy ImΣ(p, ω) and the from factor can be found in Eq. (16). Then the relaxation time can be obtained as
1 τ p = 2π k,q (1 − cos(φ p+q−k − φ p ))(1 − N F (ε k−q↑ )) δ(ω − ε p+q−k↓ − ε k↑ + ε q↑ )|g b (k − q)| 2 cos 2 (φ p+q−k − φ p ) ≈ 2π (2π) 3 Λ k F k 2 dk π 0 sinΦdΦ 2π 0 dφ p+q−k (1 − N F (ε k−q↑ )) δ(ω − ε p+q−k↓ − ε k↑ + ε q↑ )|g b (k − q)| 2 cos 2 (φ p+q−k − φ p ) = 4π (2π) 3 Λ k F k 2 dk 2π 0 dφ p+q−k (1 − N F (ε k−q↑ )) δ(ω − ε p+q−k↓ − ε k↑ + ε q↑ )|g b (k − q)| 2 cos 2 (φ p+q−k − φ p ),(37)
where we assume the case of satic hole (q = 0) for simplicity. Also, we assume the scattering wave vector is larger than the initial impurity wave vector (k > p) so that the integral of φ p+q−k can over the whole range. Unlike the above pair propagator or self-energy, the variation of scattering angle is integrated (with the momentum k), and thus the transition rate here is scattering angle-independent. g b (k − q) is the scattering wave vector-dependent bare interaction vertex (kernel). Note that in case of electron-phonon interaction, where both the electron-phonon polaron and the electron-ion polaron are produced with the emergent electronic screening and lattice screening [29], the interaction vertex here will dependent on both the scattering wave vector and the initial wave vector (like p) [67]. While for the case of multi-impurity (Eq.(22)), the above relaxation time should be rewritten as 1
τ p = 2π q,k (1 − cosθ)W p+q−k,p 1 − N F (p + q − k) 1 − N F (p) ,(38)
which can be obtained through the following relation in large-impurity momentum regime,
1 − N F (ε k↑ ) − N F (ε p+q−k↓ ) = (1 − N F (ε k−q↑ )) 1 − N F (p + q − k) 1 − N F (p) .(39)
Summary
In this paper, we discuss the polaron formed in a doped chiral/nonchiral parabolic system. The method reported here can also be applied to the Dirac systems with finite effective mass. In the numerical simulations, we studied the effect of bare coupling as well as the instability of the pair propgator and spectral function with a small-renormalized effective mass and chemical potential. The many-body effect is also analyzed through the study of spectral function. Although the T -matrix approximation here takes into account the pairing interaction with the leading instability even in the presence of weak intraspecies interactions (the p-wave interaction), it is indeed a nonperturbation theory which is evidened by the absence of the self-consistency (i.e., the Coulomb induced exchange self-energy is the Hartree-Fock type and in lack of the dynamical dielectric function), thus the energy is unconserving here and the quasiparticle weigth is lower than the one in random phase approximation (RPA) theory (with dynamical screening) or the partial self-consistent theory (with static screened interaction or dielectric function) [28]. Recently, the formation and properties of the attractive polaron formed in a two-dimensional semi-Dirac system is reported in Ref. [36,71], where the anisotropic effective masses distribution takes an important role, and we approximate the dispersion as the parabolic anisotropic one similar to the plasmon-polaron formed by the phosphorene locates on polar substrates [72]. Besides, the p-wave scattering of the polaron system is also been studied in topological superfluid and the weak-coupled BEC recently [73,4]. The attractive polaron as a quasiparticle can be observed experimently through the momentum-resolved photoemission spectroscopy [13]. For Bose-Hubbard model in superfluid phase, the Bose polaron with a spin impurity can be created by the off-resonance laser and microscope objective [75,76]. Specially, at half-filling (µ = 0) where the electron density equals 1 and the on-site Hubbard U is much larger than the mobility of impurity, the spin impurity is localized and in this case the non-self-consistent T -matrix approximation has high accuracy due to the weak dynamical screening effect from the carriers. In one-dimensional geometry, this experiment also provides a platform to explore the other polaronic physics like the propagation velocity affected by the self-trapping effect, which implies that the polaronic effect can emergent also in the superconductors or the Mott insulators [64]. The self-trapping effect will becomes more obvious at finite temperature due to the emergent electron-phonon coupling [71]. While at low-temperature limit (e.g., < 1µK) the magnetic or electric trapping can be applied to the molecule cloud or the hyperfine states (can be treated as the species as we discuss in above) of the alkali atoms, to design the quantum memory setups in the quantum circuit [77]. For solid state like the Dirac system, in the presence of, e.g., the separable s-wave potential [78], the s-wave scattering as well as the elastic scattering can be treated as dominating at low-temperature limit, and the two-body Lippmann-Schwinger equation is still valid in obtaining the coupling parameters and the T -matrix. In fact for impurity and the particle-hole part (excited by the quantum fluctuation) with energies similar to the same (gapped) Dirac cone, the scattering can be treated as the intravalley one, which can help us to deal with the multichannel problem.
7 Appendix: Variational approach in mean-field approximation for isotropic lattice For Bose polaron in BEC, the method of mean-field approximation is valid in the presence of the weak on-site Boson-Boson or Boson-Fermion (impurity) interaction, i.e., the dilute BEC, and certainly, the physical parameters like the lattice parameter or the interaction strength can be controlled by the Feshbach technique, and the strong-interaction regime can also reached by this method. Here we use the variational approach base on Gaussian variational ansatz and the Lagrangian optimization. The variational approach can be generalized by the differential of the matrix element [12] ∂
Ψ|H − E|Ψ ∂(iψ * ) = 0,(40)
where Ψ is the trial wave function including the interaction effect, H is the effective Hamiltonian of the discussing system, E is the Lagrange multiplier which gives the local minimal energy, ψ is the real components. We make the mean-field approximation to the Grassmann field which written as c j at site j, then the Lagrangian reads
L = j i ∂H M F ∂(ic * j ) c * j − H M F ,(41)
where H M F is the mean-field Hamiltonian, the Grassmann field is treated as a dynamical Gaussian profile as
c j = √ 2 r √ π exp[− (j − c) 2 r 2 + ik(j − c)],(42)
where c and k is the coordinate and momentum of the center of wave package, respectively, r is the width of wave package. Then base on the Euler-Lagrangian relation
∂L ∂c − d dt ∂L ∂ċ = −mc = 0,(43)
we can obtain c = sink e − 1 2r 2 t,
m = 1 2 cosk e − 1 2r 2 ,(44)
In the absence of the external potential, the effective Hamiltonian is independent of c,
∂H M F ∂c =0, ∂H M F ∂ċ = − A r 2 − cosk e −1 2r 2 1 r 3 r 3 sink e −1 2r 2 ,(45)
where the effective coupling parameter A = U/(4J √ π) as a ratio between the on-site interaction U and the tunneling strength J, the mean-field Hamiltonian here reads H M F = A r − cose −1 2r 2 . Here we note that the critical value of effective coupling parameter A for the self-trapping, soliton, and breather are not continued during the BEC-BCS crossover, unlike the attractive self-energy beyong the Hartree-Fock approximation [12]. In the strong interacting case, the electron may become self-trapped and with localized wave package characterized by a diverging effective mass m. q-k=0.5 q-k=1 q-k=2 q-k=2.5 The rows from top to bottom correspond to the Bosonic frequency (impurity) ω = 0, 1, 2, respectively. The momentum cutoff Λ is setted as 3 eV (which is large enough for the system we discuss) and the chemical potential is setted as 0.1 eV. The blue circle and the red lines correspond to the case of g b = −0.5 for nonchiral and chiral cases, respectively, and the dash lines correspond to the case of g b = −0.8. We can see that the real part of the self-energy is always negative, which agrees with the attractive feature of the polaron.
where N F (x) = 1/(e x/T − 1) is the Bose distribution function and N F (x) = 1/(e x/T + 1) is the Fermi distribution function.
Figure 1 :
1Pair propagators Π 11 (a) and particle-hole-like propagator Π 22 (b). (c) Polar plot of the scattering form factor F λλ ′ (q − k, ∆φ) (Eq.(16)) for different values of scattering wave vector (q − k). The impurity is setted as p = 1.
Fig. 2 FigFigure 2 :
22Real part (left) and imaginary part (right) of the pair propagator in chiral and nonchiral case with θ = 0 as a function of the impurity momentum p. The rows from top to bottom correspond to the Bosonic frequency (impurity) ω = 0, 1, 2, respectively. The momentum cutoff Λ is setted as 1 and the chemical potential is setted as 0.1. The masses of impurity, electron, and hole are setted as the same for realize the coherent and nonadiabatic configuration.
Figure 3 :
3The same asFig.1but for θ = π/4.
FigFigure 4 :
4Real part (left) and imaginary part (right) of the self-energy in chiral and nonchiral case with θ = 0 as a function of the impurity momentum p.
Figure 5 :
5The same asFig.4but for θ = π/4. The lower panel shows the self-energy as a function of frequency ω, which is presented to show the divergence from the marginal fermi liquid character.
Fig. 6 Fig
6Fig.6 Fig.7
Figure 6 :Figure 7 :
67The moemntum-and frequency-dependent polaron spectral function in chiral and nonchiral case with θ = 0. The same asFig.5but for θ = π/4.
Fig. 8 Figure 8 :
88Intensity plot of spectral function in ω − p plane. For comparation, the inset in the center shows the dispersion of the noninteracting impurity as a function of momentum p.
FigFigure 9 :
9Induced effective masses by the polaronic effect for attractive polaron with negative g b (a) and repulsive polaron with positive g b (b). (c) is the quasiparticle residue Z as a function of impurity momentum for different bare coupling constant g b in nonchiral doped parabolic system. For zero coupling g b = 0, the induced effective mass vanishes and the residue equals one. ∆m * (0) denotes the induced effective mass in static case (p = 0).
Fig. 10
10Fig.10
Figure 10 :
10Pair propagator at finite temperature as a function of the impurity momentum (left panel) and frequency (right panel). Related parameters are setted the same asFig.2.
Also here the existence of factor 1−N F (p+q−k)implies the scattering is inelastic[68,69,70]in multi-impurity case. The impurity scattering angle θ here is defined as the angle between p and p + q − k, which can be written as θ = arccos(n p+q−k /n p ) where n p is the direction projection, and it is related to the scattering wave vector by q − k = 2p F sin(θ/2)
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| []
|
[]
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"E Poretti [email protected] \nOsservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly\n",
"C Koen \nSouth African Astronomical Observatory\nPO Box 97935Cape TownObservatorySouth Africa\n",
"M Bossi \nOsservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly\n",
"E Rodríguez \nInstituto de Astrofisica de Andalucia\n3004, 18080GranadaC.S.I.C., ApdoSpain\n",
"S Martín \nOsservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly\n\nInstituto de Astrofisica de Andalucia\n3004, 18080GranadaC.S.I.C., ApdoSpain\n",
"K Krisciunas \nCerro Tololo Inter-American Observatory\nCasilla 603, La SerenaChile\n",
"M C Akan \nScience Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey\n",
"R Crowe \nDepartment of Physics and Astronomy\nUniversity of Hawaii − Hilo\n200 West Kawili Street96720-4091HiloHawaiiUSA\n",
"M Wilcox \nDepartment of Physics and Astronomy\nUniversity of Hawaii − Hilo\n200 West Kawili Street96720-4091HiloHawaiiUSA\n",
"C Ibanoglu \nScience Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey\n",
"S Evren \nScience Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey\n"
]
| [
"Osservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly",
"South African Astronomical Observatory\nPO Box 97935Cape TownObservatorySouth Africa",
"Osservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly",
"Instituto de Astrofisica de Andalucia\n3004, 18080GranadaC.S.I.C., ApdoSpain",
"Osservatorio Astronomico di Brera\nVia Bianchi 46I-23807MerateItaly",
"Instituto de Astrofisica de Andalucia\n3004, 18080GranadaC.S.I.C., ApdoSpain",
"Cerro Tololo Inter-American Observatory\nCasilla 603, La SerenaChile",
"Science Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey",
"Department of Physics and Astronomy\nUniversity of Hawaii − Hilo\n200 West Kawili Street96720-4091HiloHawaiiUSA",
"Department of Physics and Astronomy\nUniversity of Hawaii − Hilo\n200 West Kawili Street96720-4091HiloHawaiiUSA",
"Science Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey",
"Science Faculty\nDept of Astronomy and Space Sciences\nEge University\n35100Bornova, IzmirTurkey"
]
| []
| We discuss new photometric data collected on the γ Dor variables HD 224945 and HD 224638. Multiperiodicity was detected in both stars, thanks to the clear spectral window of a multisite campaign that involved five observatories. HD 224945 shows the shortest period among the γ Dor stars, i.e., 0.3330 d. The pulsation behaviour is very different: HD 224945 displays a set of frequencies spread over an interval much wider than that of HD 224638. We clearly found evidence for amplitude variations in the excited modes by comparing data from different years. HD 224945 and HD 224638 are among the best examples of γ Dor stars that show multimode pulsations, which make them very interesting from an asteroseismological point of view. | 10.1051/0004-6361:20020068 | [
"https://arxiv.org/pdf/astro-ph/0201200v1.pdf"
]
| 18,041,378 | astro-ph/0201200 | dc47c288dbc44fbf161ba7a98990f509fb497369 |
14 Jan 2002
E Poretti [email protected]
Osservatorio Astronomico di Brera
Via Bianchi 46I-23807MerateItaly
C Koen
South African Astronomical Observatory
PO Box 97935Cape TownObservatorySouth Africa
M Bossi
Osservatorio Astronomico di Brera
Via Bianchi 46I-23807MerateItaly
E Rodríguez
Instituto de Astrofisica de Andalucia
3004, 18080GranadaC.S.I.C., ApdoSpain
S Martín
Osservatorio Astronomico di Brera
Via Bianchi 46I-23807MerateItaly
Instituto de Astrofisica de Andalucia
3004, 18080GranadaC.S.I.C., ApdoSpain
K Krisciunas
Cerro Tololo Inter-American Observatory
Casilla 603, La SerenaChile
M C Akan
Science Faculty
Dept of Astronomy and Space Sciences
Ege University
35100Bornova, IzmirTurkey
R Crowe
Department of Physics and Astronomy
University of Hawaii − Hilo
200 West Kawili Street96720-4091HiloHawaiiUSA
M Wilcox
Department of Physics and Astronomy
University of Hawaii − Hilo
200 West Kawili Street96720-4091HiloHawaiiUSA
C Ibanoglu
Science Faculty
Dept of Astronomy and Space Sciences
Ege University
35100Bornova, IzmirTurkey
S Evren
Science Faculty
Dept of Astronomy and Space Sciences
Ege University
35100Bornova, IzmirTurkey
14 Jan 2002Received date; Accepted DateAstronomy & Astrophysics manuscript no. (will be inserted by hand later) The multiperiodicity of the γ Doradus stars HD 224945 and HD 224638 as detected from a multisite campaign ⋆Methods: data analysis -Stars: oscillations -stars: variables:general -techniques: photometric
We discuss new photometric data collected on the γ Dor variables HD 224945 and HD 224638. Multiperiodicity was detected in both stars, thanks to the clear spectral window of a multisite campaign that involved five observatories. HD 224945 shows the shortest period among the γ Dor stars, i.e., 0.3330 d. The pulsation behaviour is very different: HD 224945 displays a set of frequencies spread over an interval much wider than that of HD 224638. We clearly found evidence for amplitude variations in the excited modes by comparing data from different years. HD 224945 and HD 224638 are among the best examples of γ Dor stars that show multimode pulsations, which make them very interesting from an asteroseismological point of view.
Introduction
The variability of HD 224638≡BT Psc (V =7.5, F1 V) and HD 224945≡BU Psc (V =6.93, F0 V) was announced by Mantegazza & Poretti (1991), as a by-product of the monitoring of the δ Sct star HD 224639≡BH Psc. Both stars had been used as comparison stars in the first observing run devoted to BH Psc. They increased the number of known F -type stars located close to the low-temperature edge of the Cepheid instability strip which exhibit small amplitude variability on time scales of several hours, usually longer than the length of a night of observation and therefore easily detectable only when used as comparison stars for short-period variables. At that time, the debate on the nature of these variations was divided between spot activity (observed periods as rotational periods) and pulsation (nonradial g-modes). Mantegazza et al. (1994, hereinafter Paper I) tried to explain the complicated light behaviour of HD 224639 and HD 224945 in the simplest way possible, by means of periodicities shaped as double-Send offprint requests to: E. Poretti ⋆ Based on observations partially collected at ESO-La Silla (Proposals 54.E-018 and 56.E-0308) or triple-wave curves. Balona et al. (1994) reported on the multiperiodicity of γ Dor, giving decisive evidence in favour of pulsation. Also, as a result of combined photometry and line profile variations for 9 Aurigae (Krisciunas et al. 1995) and γ Dor (Balona et al. 1996), plus the preliminary results on HD 224945 , the hypothesis of variability caused by g-modes achieved a wide consensus. The properties of this new class of variable stars were delineated step-by-step through observational efforts and have been summarized by Kaye et al. (1999), while the problem of the driving mechanism and the excitation of g-modes constitutes the target of continuing theoretical investigations. As a result, γ Dor variables are now considered intermediate between A − F pulsators and solar-type stars, finding a special place in the main programmes of asteroseismologic missions such as corot and mons.
On the basis of this progress, it is worth investigating in more detail the pulsational behaviour of HD 224639 and HD 224945. Here we present the results of a multisite campaign carried out in October 1995.
Observations
HD 224945 and HD 224638 are located very close to the celestial equator and therefore can be monitored from both the northern and southern hemispheres. As comparison stars we used the same two stars as in the 1991 campaign (HD 225086 and HD 200), since they proved to be stable within a few mmag. Five telescopes were used for this multisite campaign: Earlier observations were performed in 1994 at European Southern Observatory, using the ESO 50-cm telescope equipped with the same instrumentation used in 1995. On that occasion the observer was M. Bossi; the results obtained on BH Psc are reported by Mantegazza et al. (1996). The measurements were distributed over 13 nights, with a total useful observing time of 64 hours and a baseline of 14.9 d.
Data reduction and analysis
The original measurements were provisionally reduced by the observers themselves to check the data quality and for comparison purposes. The original raw data were then re-reduced by means of the same algorithm; this homogenous procedure allowed us to avoid the introduction of local drifts caused by routines using different interpolation formulae and/or different methods of applying extinction corrections. Since the weather was not particularly favourable during the campaign, the rejection of some parts of nights characterized by unfavourable weather conditions was decided on the basis of a uniform criterion. We also decided to use the code which allows us to calculate instantaneous values of the extinction coefficients (Poretti & Zerbi 1993) since in some cases changes were expected to occur. Table 1 lists some details regarding observing runs at each site. As can be noted, the mean values of the magnitude differences between the two comparison stars show the presence of systematic shifts (up to 0.03 mag). This is not surprising, taking into account that different photomultipliers and filters were used. The systematic shifts between HD 225086 and HD 200 cannot be used to correct magnitude differences between HD 225086 and the two program stars since they have different colours from HD 200; uncertainties of the colour transformations are larger than the precision we need. Hence, to apply this correction we exploited the very useful circumstance that the spread in longitude between different observing sites is not larger than the length of the night. There were always some overlapping segments of the light curve, which allowed us to determine the amounts of the corrections in a straightforward way.
It should also be noted that the Turkish measurements show a scatter larger than all the others. At first, all the Turkish data were considered, but in the last step of the analysis (once it was established that rapid variations are not present in the light curves) those measurements were averaged in groups of 4-5. Thus, the final set of data contains the original measurements from all the observatories except for the averaged values from Turkey. The standard deviations of the ∆B and ∆V values between the comparison stars are 3.7 and 3.8 mmag, respectively. Figure 1 shows a part of the B light curves of HD 224945 and HD 224638. The fits derived in the next sections are also shown. The original measurements can be requested from the authors.
Since data in the final form circulated amongst all the participants, several co-authors analysed the time series independently and by different period search algorithms (least-squares, clean, dft, ...) and it was particularly satisfying to see that they detected the same terms, even if, owing to the complexity of the light variations, the terms did not always stand out in a clear way.
We present here the analysis carried out by using the least-squares method (Vaniĉek 1971) used by the Merate group in the analysis of δ Sct star light curves. This method has the advantage of not using any data prewhitening since only the frequency values previously found are considered as input values (known constituents; k.c.'s); their amplitude and phase values are recalculated as unknowns in the new searches. That means that after the detection of the f 1 term, only the frequency value f 1 was considered as established (i.e. a k.c.) and in the second search the unknowns were ∆m o , A 1 , φ 1 , f 2 , A 2 , φ 2 . The ordinates of the power spectra show the reduction factor
Red.Factor = 1 − σ 2 fin σ 2 in(1)
Moreover, we can present the term detection step-bystep. The frequency values were refined after each new detection. We can also obtain useful evidence concerning the amplitude by applying a least-squares fit to the datasets. The interpolating formula we used is
∆m(t) = ∆m o + N i A i cos[2πf i (t − T o ) + φ i ](2)
Since the 1995 observations are concentrated in an interval of 15 d only, the frequency resolution is 1.5/∆T=0.10 cd −1 . However, the absence of relevant aliases makes the detection of the excited terms very simple. We note that the frequencies mentioned in the text and figures are given to better accuracies that the formal frequency resolution, because they were determined by a least-squares procedure. The formal least-squares standard errors on the frequencies no doubt understimate the true uncertainties -aside from problems pointed out by e.g. Montgomery & O'Donoghue (1999), it is clear that we have not always resolved close frequencies, and that some unidentified signals may remain in the data.
4. Frequency analysis of the HD 224945 data Table 1. Summary of the photometric B and V data collected in the 1995 multisite campaign. N is the total number of measurements, n the number of nights. t 1 and t N are the times of the first and last measurement, respectively. Magnitude differences are calculated with respect to HD 225086. Fig. 2. In each panel the horizontal line indicates the level for S/N = 4.0, i.e., the limit usually accepted for significance (Kuschnig et al. 1997).
The top panel shows the spectrum obtained without any k.c.: the peak at 3.00 cd −1 stands out clearly, and the alias structure mimics very well the spectral window (see also Fig. 1 in Poretti et al. 1996). In the discussion of the 1991 data (Paper I), the alias at 2.00 cd −1 was erroneously preferred. The reality of this term close to a multiple of 1 cd −1 was discussed in Paper I. We note that it is also detected in the new campaign data for HD 224945, but in neither the HD 224638, nor the HD 200, time series; its physical presence in the HD 224945 data is therefore certain. When introducing the 3.00 cd −1 frequency as k.c., the power spectrum shows an almost flat pattern, indicating that the amplitude of the remaining terms is smaller. However, two peaks reach the acceptance level: the first is located at 1.16 cd −1 (second panel), the second at 2.84 cd −1 (third panel). After introducing these three terms as k.c.'s, the detection of further terms becomes delicate. The highest peak in the fourth panel is at 2.42 cd −1 and after that the power spectrum does not show a really dominant peak (bottom panel).
The analysis of the V measurements yields the same results, except with a slight difference in the frequency of the 2.84 cd −1 term. A value close to 2.77 cd −1 is preferred. Since the peak at 2.84 cd −1 is broad (see third panel of Fig. 2), the presence of another undetectable term combined with the noise distribution could be responsible for the difference between the two values. In the least-squares solution we will consider the 2.84 cd −1 term as the only well-established term.
These independent terms describe a different scenario from that in Paper I. In that case the single-site observations make the power spectrum very complicated, with a great uncertainty between a peak and its ±1 cd −1 aliases. This uncertainty led us to propose a solution based on two periodicities each having a triple-wave shape, as it was considered the simplest. Thanks to the multisite observations, we now know that many pulsational modes are simultaneously excited.
The 1994 and 1991 observations
The frequency analysis of the B data collected in the 1994 season is much more complicated than that of the multisite campaign. In spite of that, we could detect the 3.00 and the 2.84 cd −1 terms. When introducing these two terms as k.c.'s, the highest peak is at 2.27 cd −1 . This peak is significantly different than the 2.42 cd −1 detected in the 1995 multisite campaign. The search for a fourth component is not easy since we get numerous peaks: however, we could recognize the 1.16 cd −1 term and its aliases. We considered it as a real mode on the basis of the results of the 1995 campaign. After that, the new search showed evidence of a complicated structure centered at 1.66 cd −1 . We have no way of selecting in a reliable way the true term amongst this peak and the aliased ones. Moreover, the S/N of this term is less than 3.0. The V data allowed us to identify the same terms, supporting the presence of the new term at 2.27 cd −1 .
In the 1991 dataset the detection order of the terms is 3.00, 2.84 and 2.31 cd −1 : the only difference with respect to the previous least-squares search (see the beginning of § 3.2 in Paper I) is the identification of the first term as 3.00 cd −1 instead of 2.00 cd −1 . No other term can be detected after considering this triplet as k.c.'s; in particular there is no evidence of the presence of the 1.16 cd −1 term. It should be emphasized that it would be very difficult to solve the 1994 and 1991 light curves without having the way shown by the results of the 1995 campaign. In turn, the similar structures observed in the three independent datasets corroborate the proposed identification of the excited frequencies, even if some of them are at the limit of the acceptance level.
Frequency refinement and least-squares fitting
The much longer time interval covered by the 1991 data (the two runs span 14 and 10 days, respectively, and they are separated by a gap of 10 days) allowed us to use these data to refine the frequency values. First, we calculated the best solution using all the 1991 data. Then we calculated the amplitudes for each 1991 subset keeping fixed frequencies and phases.
Before applying these frequency values to other datasets, we verified the correspondence between some terms. We considered the 2.27 cd −1 term detected in the 1994 data as the same detected at 2.31 cd −1 in the 1991 campaign. However, we note that the least-squares solution yields a better fit when considering different values for different seasons. On the other hand, we cannot consider the 2.42 cd −1 term detected in 1995 as coincident with the 2.31 cd −1 term. We suggest that there are many terms excited (simultaneously or not) in the narrow range 2.2-2.5 cd −1 and their detection is not an easy task. We considered the following five terms as independent modes detected in the three campaigns: 3.003, 2.836, 2.313, 2.424 and 1.160 cd −1 . We also note the almost 2:1 ratio between the 2.313 and 1.160 cd −1 terms: the similar amplitude does not support a monoperiodic contribution to the light curve as it should be very asymmetric. The resonance effect is more plausible, even if it should be noted that the 1.160 cd −1 term was not observed in the 1991 dataset.
Using the four frequencies above, we calculated the amplitudes from the 1994 and 1995 datasets. Tab. 2 summarizes the results. The formal errors of the amplitudes are about 0.4 mmag. Looking at the differences between the amplitudes of the two 1991 subsets we can see that they are marginally significant, being in the interval 0.7-0.9 mmag. This very close similarity demonstrates that solutions obtained from short runs are self-consistent. The dense time coverage in each dataset ensures the reliability of the amplitude determination. Therefore, we can look at the differences in the amplitudes between the 1991, 1994 and 1995 datasets with greater confidence.
We also analyzed in frequency the data obtained by merging the 1991, 1994 and 1995 datasets. For each detected term, it is not possible to select the true value owing to the numerous aliases separated by integer values of ±1 cy −1 ; therefore, in the case of HD 224945 we cannot propose a full, comprehensive solution and evaluate in a reliable way the phase coherence of the pulsation. The amplitude variability is a further complication. It should be noted that an attempt at such an analysis detected two peaks at 2.27 and 2.31 cd −1 , suggesting that these two peaks are related to two independent terms (see the above comparison of the 1994 and 1991 results).
The sum of the squared amplitudes are the same in 1991 and 1994 (61 and 58 mmag 2 , respectively), while it is lower in 1995 (50 mmag 2 ). However, the amplitude of a mode can change dramatically: the 2.836 cd −1 term halved its B amplitude from 1991 to 1995 while the 2.313 cd −1 term disappeared. As a matter of fact, in the 1991 dataset there are three terms having the same amplitude, while in the 1995 ones the 3.003 cd −1 term is dominating. The amplitude variability looks like a wellestablished fact for this γ Dor star.
Frequency analysis of the HD 224638 data
The 1995 campaign
The analysis of the most recent B and V measurements of HD 224638 identified the same terms as before, i.e., 1.627, 1.368, 1.697, 1.565 and 1.145 cd −1 (Fig. 3). Also in this case the multisite observations allowed us to identify the terms without any ambiguity. Differently than the case of HD 224945, there are several very close frequencies: 1.627 (top panel), 1.697 (third panel), and 1.565 cd −1 (fourth panel). The separations between the two side peaks and the central one are 0.07 and 0.06 cd −1 ; these values are comparable to the half width at the half maximum, i.e. Table 2. HD 224945: least-squares fit of the B and V measurements performed in the 1991, 1994 and 1995 observing seasons. The term detected at 2.27 cd −1 in the 1994 season is considered the same as that detected at 2.31 cd −1 in the other seasons, while that at 2.42 cd −1 is considered an independent frequency. Formal errors on the frequencies are calculated with respect to the 1991 data (when applicable) or to the 1995 data.
Amplitude B [mmag] Amplitude V [mmag]
Freq . 1991 1991 1991 1994 1995 1994 1995 1/∆T =0.07 cd −1 . Indeed, in the power spectra the peaks are not well resolved, but they appear to be double and/or enlarged. In particular, the double peaks appearing in the third and fourth panels of Fig. 3 should be noted. This reveals the presence of overlapping peaks. Two peaks flanking a central one at the limit of the frequency resolution can be generated by a single term modulated in amplitude; however, the amplitudes of the side peaks are comparable to that of the central one and this fact supports the reality of the triplet. The analysis of the 1991 dataset definitely confirms the hypothesis of the three independent modes (see next subsection). Note that the amplitudes of these terms are larger than those detected in the HD 224945 data, giving a higher S/N value. The other two terms (1.368, second panel, and 1.145 cd −1 , fifth panel) are more separated. The power spectrum obtained by introducing the five frequencies as k.c.'s is not homogenously flat (bottom panel). The noise in the region 0.0-2.0 cd −1 is 1.34 mmag, while in the region 4.0-6.0 cd −1 it is only 0.65 mmag. However, it should be noted that the general patterns of the residual B and V power spectra are slightly different. This fact suggests that noise dominates over pulsation and the latter is really undetectable owing to the very small amplitude of the involved terms.
The 1994 and 1991 seasons
Once again, the light curves of the 1991 and 1994 seasons can only be understood if extra alias-free frequency information is available. The frequency analysis of the 1991 B data provided evidence for the terms 1.697, 1.627, 1.368 and 1.565 cd −1 . The first three terms were already reported in Paper I, where two periodicities having each the shape of a double-wave were proposed. The 1.565 cd −1 (detected also in the 1995 dataset) is a new term obtained by pushing one further step in the analysis. As a result, we also detect here the close triplet observed in the 1995 campaign. Since the frequency resolution in the 1991 dataset is much better than in the 1995 one, this confirmation strengthens our confidence in the reality of the triplet. The peak at 1.145 cd −1 does not appear very clearly in the power spectrum, but it has been considered for the least-squares solution.
The first four terms are also detected in the 1994 B and V datasets, but residual signal is left, especially in the region 0.5-1.0 cd −1 . The 1.145 cd −1 term shows an amplitude much smaller than the other four terms. As a Table 3. HD 224638: least-squares fit of the B and V measurements performed in the 1991, 1994 and 1995 observing seasons. Formal errors on frequencies are calculated with respect to the 1991 data.
Amplitude B [mmag] Amplitude V [mmag]
Freq . 1991 1991 1991 1994 1995 1994 1995 matter of fact, in this season it seems that many terms having amplitudes at the mmag level are excited.
Frequency refinement and least-squares fitting
We followed the same procedure used for HD 224945, refining the frequency values by means of the 1991 dataset and then calculating the amplitudes for the two 1991 subsets and the other datasets. The frequency analysis already gave some hint about the amplitude variability of some terms, since for example the 1.145 cd −1 peak was not clearly detected in all datasets. This is confirmed by the large amplitude variability, which can be discerned from Tab. 3. Considering the 1991 subsets we can see differences of up to 2.3 mmag, but note the +3.7, +2.9 and -3.9 mmag differences between the amplitudes of the 1.627, 1.368 and 1.565 cd −1 terms in 1995 and 1994 datasets, which look extremely large. The term ranking is also largely changing. Note that the largest amplitude of the 1.565 cd −1 term was in 1994, whereas usually this term was below the first three in 1991 and 1995. The amplitude is also greatly changing. The sum of the squared amplitudes is higher in the 1995 data (311 mmag 2 ) and lower in the 1994 dataset (250 mmag 2 ), with a variation of 25%. This difference supports the intrinsic variability of the mode amplitudes. Note for example that in the 1991 data there are three terms having a large amplitude, in 1994 data there are four and in 1995 data two of them clearly have the largest amplitude, considerably greater than the others. Hence, the amplitude changes look more conspicuous in HD 224638 than in HD 224945.
In the case of HD 224638 we can merge the 1991, 1994 and 1995 datasets knowing that all the datasets show the same frequency content. Indeed, the frequency analysis of the whole set yields the same frequencies listed in Tab. 3. The detection of the three terms at 1.565, 1.627 and 1.697 cd −1 supports the identification as three independent modes proposed in Sect. 5.1. Unfortunately, we cannot give refined values for the frequencies, owing to the cy −1 aliases. However, we performed some tests assuming a constant value for the frequency and then calculated the amplitudes and phases of each term for each observing season. Besides the verification of the amplitude variability, we found that the same frequency displays similar phase values from one season to the next. This fact supports an intrinsic variability of the mode amplitudes, rather than a beating phenomenon between two very close terms.
Physical properties
The properties of γ Dor stars have been reviewed by Zerbi (2000); driving mechanisms have been proposed by Guzik et al. (2000), Wu (2002) and Löffler (2002), but the origin of the pulsation in γ Dor stars still remains an open problem.
HD 224945 and HD 224638 have very similar physical properties (Zerbi 2000): M V =2.98, L/L ⊙ =5.5, T eff =7200 K, R/R ⊙ =1.51 and M/M ⊙ =1.52 for HD 224638, M V =3.07, L/L ⊙ =5.1, T eff =7250 K, R/R ⊙ =1.43 and M/M ⊙ =1.51 for HD 224945. Also the difference in metallicity is not significant, [Me/H]=-0.15 vs. -0.30, respectively. The two stars occupy very similar positions also in the colour-magnitude diagram, being located on the ZAMS, in the middle of the domain of γ Dor stars and on the low temperature edge of the δ Sct region (Handler 1999). Therefore, two very similar stars display different pulsational modes, since HD 224945 is characterized by four frequencies spread over a large interval, while HD 224638 displays a much more closely spaced set (it is similar to HR 2740; see Poretti et al. 1997). Moreover, the residual power spectrum of HD 224638 has a very low noise level above 4.0 cd −1 , while that of HD 224945 has a higher level, suggesting some signal contribution.
The large spread in the frequency values observed for HD 224945 is another piece of evidence that the cause of light variability in these stars is pulsation and not stellar activity. Differential rotation is not able to match a spread in frequencies as large as 0.7 cd −1 (considering the three terms always observed, i.e., not considering the even more distant frequency at 1.16 cd −1 ).
We note that multiperiodicity is not observed in all γ Dor stars. Some display only a single photometric period (i.e., a single low-order mode). A relevant example is HD 207223≡HR 8330 (Aerts & Kaye 2001), which is also monoperiodic from a spectroscopic point of view (i.e. it does not show any high-order modes). Another example is HD 164615 (Zerbi et al. 1997), one of the first known γ Dor stars. It is probably a monoperiodic variable which shows amplitude modulation.
Finally, we note that HD 224945 exhibits the shortest known period (i.e. 0.3330 d) of the γ Dor stars. Guzik et al. (2000) considered 0.48 d as the lower limit (probably the old value of the same term, i.e. 2.00 cd −1 , Paper I). Therefore, the period value cannot be used to separate pand g-modes, as δ Sct stars can display such 'long-period' p-modes. A careful evaluation of the physical parameters is necessary. Indeed, in HD 224638 and HD 224945 the fundamental radial mode is shorter than 0.07 d, confirming that p-and g-modes are well separated in γ Dor stars.
Conclusions
The solution of the light curves of HD 224638 and HD 224945 has been deduced only on the basis of a multisite campaign, since the true peaks could not be recognized if a ±1 cd −1 effect is present in the spectral window. We did our best to identify the correct excited modes, but the more important characteristic of these two stars is the strong multiperiodicity itself, independent of the exact frequency values. In our opinion, HD 224638 and HD 224945 provide two of the best examples of features typical of multiperiodicity among γ Dor variables: different sets of frequency content, amplitude variations, disappearing terms, close doublets of frequencies. Also, after detecting five or six terms, the rms scatter is larger than the observational error. Therefore, residual signal is hidden in the noise. Such extreme multiperiodicity can be clearly ascribed to pulsation, as stellar activity is not able to generate it.
The differences in the frequency and amplitude ranges of HD 224638 and HD 224945 (clearly visible in Fig. 1) remind us of the unpredictable frequency content of δ Sct stars, where the selection mechanism among all the possible modes seems to be different from one star to the next (Poretti 2000). The amplitude variability of the excited modes is another point of similarity between γ Dor and δ Sct stars. Other than multiperiodicity, HD 224945 and HD 224638 provide the best examples of amplitude variability among γ Dor stars, following the three campaigns carried out on these stars. This observational evidence suggests strategies for asteroseismological space missions: a long observing run (or two separate runs) may be very helpful in detecting more terms (excited at different levels at different times) or in studying damping effects.
Fig. 1 .
1Light curves (B light) of HD 224945 and HD 224638 obtained during the 1995 multisite campaign; measurements from JD 2450005 to 2450014 are shown. Note the different timescales and amplitudes between the two curves.
Fig. 2 .
2Power spectra of the B measurements of HD 224945 obtained during the 1995 multisite campaign. Note the different scale of the top panel.
Fig. 3 .
3Power spectra of the B measurements of HD 224638 obtained during the 1995 multisite campaign. Note the different scales of the panels.
The 50-cm telescope located in Sutherland, South African Astronomical Observatory, was equipped with a photon-counting photometer and B and V filters.The observer was C. Koen. 4. The 61-cm telescope located on the Mauna Kea (Hawaii) was equipped with a photon-counting photometer and standard B and V filters. The observers were K. Krisciunas, R. Crowe and M. Wilcox. 5. The 90-cm telescope located at Sierra Nevada Observatory (Spain). The observers were E. Rodríguez and S. Martín. Photometry was performed in the uvby system, but only v and y measurements are discussed here, as the more compatible with the B and V ones, respectively.1. The European Southern Observatory 50-cm telescope
located in La Silla (Chile), equipped with a photon-
counting photometer (EMI 9789 QB photomultiplier)
and B and V filters. The observer was E. Poretti.
2. The 48-cm telescope, located at Ege University
Observatory, was equipped with a solid-state pho-
tometer and B and V filters. The observer team was
led by M. C. Akan.
3.
We collected 442 V and 417 B measurements of HD 224945, considering the Turkish measurements as binned points. The results of the frequency analysis of the B data are shown inSite
Check star (HD 200)
HD 224945
HD 224638
∆V
N
s.d.
t1
tN
n
N
t1
tN
n
N
ESO
0.1472 198 0.0029
4.605 14.597 10 205
4.512 14.572 10 219
Ege Obs.
0.1180 117 0.0102
3.250 12.317
5
36
3.252 12.311
5
33
SAAO
0.1294
41 0.0055
1.427 14.479
6
36
1.432 14.482
6
44
Mauna Kea
0.1371
39 0.0051
7.824 14.003
6
37
7.840 14.023
6
38
OSN
0.1304 126 0.0032
7.364 19.524
9 128
7.362 19.527
9 133
∆B
N
s.d.
t1
tN
n
N
t1
tN
n
N
ESO
0.2569 198 0.0029
4.604 14.597 10 203
4.512 14.572 10 217
Ege Obs.
0.2672 125 0.0091
3.264 12.312
5
32
3.251 12.322
5
29
SAAO
0.2458
41 0.0060
1.436 14.479
6
32
1.432 14.482
6
44
Mauna Kea
0.2554
20 0.0041
10.751 14.002
3
21
10.743 14.021
3
21
OSN
0.2449 126 0.0029
7.364 19.524
9 129
7.362 19.527
9 133
4.1. The 1995 campaign
Acknowledgements. ER and SM acknowledge the partial support by the Junta de Andalucia and by the Direccion General de Investigacion (DGI) under project AYA2000-1559. SM also acknowledges the financial support by the Osservatorio Astronomico di Brera and by the Agenzia Spaziale Italiana (ASI Contract I/R/037/01). Thanks are due to an anonymous referee and to M. Breger for useful comments and suggestions on the first version of the manuscript.
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| []
|
[
"Neutron Stars in Supernova Remnants",
"Neutron Stars in Supernova Remnants"
]
| [
"Franco Pacini \nArcetri Astrophysical Observatory and Department of Astronomy and Space Sciences\nUniversity of Florence\nLargo E. Fermi 5I-50125FirenzeItaly\n"
]
| [
"Arcetri Astrophysical Observatory and Department of Astronomy and Space Sciences\nUniversity of Florence\nLargo E. Fermi 5I-50125FirenzeItaly"
]
| [
"Highly Energetic Physical Processes and Mechanisms for Emission from Astrophysical Plasmas, IAU Symposium 195 ASP Conference Series"
]
| In the following, we shall briefly summarize some facts and ideas concerning the presence of neutron stars in Supernova remnants. While sources similar to the Crab Nebula require the presence of a central energetic object, shell-type remnants such as Cas A are compatible with the presence of neutron stars releasing a weak relativistic wind. | 10.1017/s0074180900162813 | [
"https://arxiv.org/pdf/astro-ph/9911418v1.pdf"
]
| 18,044,694 | astro-ph/9911418 | da61662f5978f0e80fa38dc16e0d837d67ead71a |
Neutron Stars in Supernova Remnants
1999
Franco Pacini
Arcetri Astrophysical Observatory and Department of Astronomy and Space Sciences
University of Florence
Largo E. Fermi 5I-50125FirenzeItaly
Neutron Stars in Supernova Remnants
Highly Energetic Physical Processes and Mechanisms for Emission from Astrophysical Plasmas, IAU Symposium 195 ASP Conference Series
1951999
In the following, we shall briefly summarize some facts and ideas concerning the presence of neutron stars in Supernova remnants. While sources similar to the Crab Nebula require the presence of a central energetic object, shell-type remnants such as Cas A are compatible with the presence of neutron stars releasing a weak relativistic wind.
Supernova remnants are usually classified into two extreme categories: shelltype and filled-center (plerions). In the case of shell remnants, the edges of the source appear bright, the interior rather faint. The typical radio spectrum is steep (S ν ∝ ν α with α ≃ −0.5) and is due to synchrotron radiation from relativistic electrons produced by shocks in the region where the expanding debris interact with the circumstellar/interstellar medium. Cas A is the prototype of shell-type remnants.
On the opposite end, the Crab Nebula has been assumed to be the typical plerion, where a central neutron star continuously converts its rotational energy into a magnetized relativistic wind. This wind expands and produces a centerfilled nebular emission which has a rather flat radio spectrum (α ≃ −0.2). It is not surprising that several remnants show both characteristics (internal emission and bright limbs) since some plerions expand into a relatively dense medium (composite remnants). For more details we refer the reader to a broad review by Frail (1998) and to a full coverage of the subject in the Proceedings of the Arcetri -Elba Workshop "Relationship between Neutron Stars and Supernova Remnants" (Bandiera et al. 1998). We also refer to two accompanying papers in these Proceedings which deal with the Crab Nebula (Amato 1999) and with plerions in general (Bandiera 1999). Among the 215 catalogued Supernova remnants (Green 1996) 85% are of the shell-type.
In the past, it was widely held that plerions contain a neutron star while shell-type remnants do not, either because the explosion blows apart the entire star or because the central object becomes a black hole. The basis for this belief was the lack of evidence for an internal radio pulsar and/or of a source of relativistic wind. On the other hand, already in the early work on pulsars, it had been suggested that magnetic fields of neutron stars can reach values up to, say, 10 14 − 10 15 gauss (Woltjer 1968). The initial loss of rotational energy would then be very fast and the central neutron star would soon become unable to produce a strong relativistic wind.
Pacini
In this framework the possible presence of a neutron star hidden in Cas A (with period P ≥ 0.7 sec B > 10 14 gauss) was discussed by Cavaliere and Pacini (1970).
It is also worth recalling that the estimate for the rate of core collapse Supernovae (around one every 30-50 years) is about a factor two larger than the same estimate for the birthrate of radiopulsars (about one every 100 years). Despite the uncertainty of these estimates, this suggests the existence of a large fraction of neutron stars which do not appear as radiopulsars (Helfand andBecker 1984, Helfand 1998).
Indeed, in recent years the observational evidence for the presence of neutron stars in Supernova remnants has changed, largely because of observations in the X and γ-rays bands. According to Frail (1998), at least 19 Supernova remnants contain neutron stars which manifest themselves in different ways. Seven of them are classical radiopulsars (only one third of the total!); 3 are X-ray binaries; 2 are slow X-ray pulsars; 2 are soft gamma-ray repeaters; 5 are radio quiet neutron stars (detected because of their thermal X-ray emission). Although this list is not updated, the picture has not changed substantially in its implications.
The reasons which lead to different manifestations of neutron stars in Supernova remnants are not fully understood. However, the binary nature of the system (i.e. the role of accretion) or the initial period of the neutron star and/or the magnetic field strength are likely to play a dominant role in determining the evolution. These factors may not be independent. In particular the initial rotation of neutron stars could be related to the strength of the coupling between the collapsing core and the stellar envelope (Tsuruta andCameron 1966, Pacini 1983). Strong magnetic fields would lead to slow initial rotation. Even if this coupling is not important, the wind produced by a newly born, strongly magnetized, neutron star could damp the rotation immediately or shortly after the explosion. As a consequence, some young remnants could contain a slow neutron star, unable to support a strong wind. The evidence (from the slowing down rate) for the presence of neutron stars with periods of several seconds and fields up to and above 10 14 gauss in some remnants suggests that this occurs rather frequently. This could lead to the dominance of the stellar magnetic energy ≃ B 2 R 3 over the rotational energy ≃ M R 2 Ω 2 , a so-called "magnetar". Magnetic fields would then become the main energy source, which could be released, e.g. through flares. This possibility, already envisaged by Woltjer (1968), has been revived in more recent times as a possible explanation of the soft gammaray repeaters (Duncan and Thompson 1992). We stress however that the standard determination of the field value is based upon the energy loss in a dipole field: any deviation from the dipolar geometry inside the speed of light cylinder (or, even worse, the existence of alternative slowing down mechanisms) would invalidate the determination of the strength on the stellar surface.
In any case it is now evident that the lack of a radiopulsar or of the plerionic component cannot be taken as proof for the non existence of a neutron star inside Supernova remnants. In this case only observations can tell us whether the central object is a slow rotator with a strong magnetic field or, at the opposite end, a fast but weakly magnetized neutron star (or any other combination of rotation and field strength unable to provide a strong pulsar wind).
Finally, we note that SN 1987A does not show, in its light curve, evidence for a pulsar energy input larger than 2 × 10 36 ergs s −1 (Danziger, private com-
Neutron Stars in Supernova Remnants
3 munication). This lends additional support to the idea that some remnants may contain neutron stars less energetic than those usually present in plerions.
Note added (October 1999). After I.A.U. Symposium 1995 took place, a central point source (neutron star?) in Cas A has indeed been found by the AXAF -Chandra X-ray satellite, shortly after its launch (Tananbaum et al., 1999). Observations in the various bands are currently underway. These will be able to provide more information about the nature and possible periodicity of this source, as well as about the correctness of the points discussed above.
Acknowledgments. I am indebted to E. Amato, R. Bandiera, M. Salvati, L. Woltjer for many discussions on the subject of this talk. This work was partly supported by the Italian Space Agency.
. E Amato, these proceedingsAmato, E. 1999 (these proceedings)
R Bandiera, E Masini, F Pacini, M Salvati, L Woltjer, Proc. of the Workshop "The relationship between neutron stars and Supernova remnants. of the Workshop "The relationship between neutron stars and Supernova remnants69n. 4.Bandiera, R., Masini, E., Pacini, F., Salvati, M., Woltjer, L. 1998 Proc. of the Workshop "The relationship between neutron stars and Supernova remnants", Mem. SAIt, vol. 69, n. 4.
. R Bandiera, these proceedingsBandiera, R. 1999 (these proceedings)
The many faces of neutron stars. A Cavaliere, F Pacini, L21, R C Duncan, C Thompson, L9, D A Frail, Proc. of the NATO ASI. of the NATO ASIKluwer Academic Publishers159ApJCavaliere, A., Pacini, F. 1970, ApJ, 159, L21. Duncan, R.C., Thompson, C. 1992, ApJ, 392, L9. Frail, D.A. 1998, in Proc. of the NATO ASI "The many faces of neutron stars", Kluwer Academic Publishers, pp. 179-194.
A catalogue of galactic Supernova remnants, Mullard Radio Astronomy Observatory. D A Green, Cambridge, UKGreen, D.A. 1996, A catalogue of galactic Supernova remnants, Mullard Radio Astronomy Observatory, Cambridge, UK.
D J Helfand, R H Becker, D J Helfand, Proc. of the Workshop "The relationship between neutron stars and Supernova remnants. of the Workshop "The relationship between neutron stars and Supernova remnants307NatureHelfand, D.J., Becker, R.H. 1984, Nature, 307, 215. Helfand, D.J. 1998, in Proc. of the Workshop "The relationship between neutron stars and Supernova remnants", Mem. SAIt, vol. 69, n. 4, pp. 791-800.
. F Pacini, L11, B Tananbaum, A&A. 1267246Pacini, F. 1983, A&A, 126, L11. Tananbaum, B. et al. 1999, IAUC No. 7246.
. S Tsuruta, A G W Cameron, Nature. 211179Woltjer, LApJTsuruta, S., Cameron, A.G.W. 1966, Nature, 211, 356. Woltjer, L. 1968, ApJ, 152, L179.
| []
|
[
"Particle-like wave packets in complex scattering systems",
"Particle-like wave packets in complex scattering systems"
]
| [
"Benoît Gérardin \nUMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance\n",
"Jérôme Laurent \nUMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance\n",
"Philipp Ambichl \nInstitute for Theoretical Physics\nVienna University of Technology (TU Wien)\nWiedner Hauptstraße 8-10/136A-1040ViennaAustria\n",
"Claire Prada \nUMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance\n",
"Stefan Rotter \nInstitute for Theoretical Physics\nVienna University of Technology (TU Wien)\nWiedner Hauptstraße 8-10/136A-1040ViennaAustria\n",
"Alexandre Aubry \nUMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance\n"
]
| [
"UMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance",
"UMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance",
"Institute for Theoretical Physics\nVienna University of Technology (TU Wien)\nWiedner Hauptstraße 8-10/136A-1040ViennaAustria",
"UMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance",
"Institute for Theoretical Physics\nVienna University of Technology (TU Wien)\nWiedner Hauptstraße 8-10/136A-1040ViennaAustria",
"UMR 7587\nESPCI Paris\nPSL Research University\nCNRS\nInstitut Langevin\n1 rue JussieuF-75005ParisFrance"
]
| []
| A wave packet undergoes a strong spatial and temporal dispersion while propagating through a complex medium. This wave scattering is often seen as a nightmare in wave physics whether it be for focusing, imaging or communication purposes. Controlling wave propagation through complex systems is thus of fundamental interest in many areas, ranging from optics or acoustics to medical imaging or telecommunications. Here, we study the propagation of elastic waves in a cavity and a disordered waveguide by means of laser interferometry. From the direct experimental access to the time-delay matrix of these systems, we demonstrate the existence of particle-like wave packets that remain focused in time and space throughout their complex trajectory. Due to their limited dispersion, their selective excitation will be crucially relevant for all applications involving selective wave focusing and efficient information transfer through complex media. arXiv:1602.05812v2 [physics.optics] | 10.1121/1.4950178 | [
"https://arxiv.org/pdf/1602.05812v2.pdf"
]
| 5,620,561 | 1602.05812 | 4d69bfcdd52a8a319abb4ac8b4983ea2d9fafa8b |
Particle-like wave packets in complex scattering systems
Benoît Gérardin
UMR 7587
ESPCI Paris
PSL Research University
CNRS
Institut Langevin
1 rue JussieuF-75005ParisFrance
Jérôme Laurent
UMR 7587
ESPCI Paris
PSL Research University
CNRS
Institut Langevin
1 rue JussieuF-75005ParisFrance
Philipp Ambichl
Institute for Theoretical Physics
Vienna University of Technology (TU Wien)
Wiedner Hauptstraße 8-10/136A-1040ViennaAustria
Claire Prada
UMR 7587
ESPCI Paris
PSL Research University
CNRS
Institut Langevin
1 rue JussieuF-75005ParisFrance
Stefan Rotter
Institute for Theoretical Physics
Vienna University of Technology (TU Wien)
Wiedner Hauptstraße 8-10/136A-1040ViennaAustria
Alexandre Aubry
UMR 7587
ESPCI Paris
PSL Research University
CNRS
Institut Langevin
1 rue JussieuF-75005ParisFrance
Particle-like wave packets in complex scattering systems
(Dated: July 27, 2016)
A wave packet undergoes a strong spatial and temporal dispersion while propagating through a complex medium. This wave scattering is often seen as a nightmare in wave physics whether it be for focusing, imaging or communication purposes. Controlling wave propagation through complex systems is thus of fundamental interest in many areas, ranging from optics or acoustics to medical imaging or telecommunications. Here, we study the propagation of elastic waves in a cavity and a disordered waveguide by means of laser interferometry. From the direct experimental access to the time-delay matrix of these systems, we demonstrate the existence of particle-like wave packets that remain focused in time and space throughout their complex trajectory. Due to their limited dispersion, their selective excitation will be crucially relevant for all applications involving selective wave focusing and efficient information transfer through complex media. arXiv:1602.05812v2 [physics.optics]
I. INTRODUCTION
Waves propagating in complex media typically undergo diffraction and multiple scattering at all the inhomogeneities they encounter. As a consequence, a wave packet suffers from strong temporal and spatial dispersion while propagating through a scattering medium. Eventually, the incident wave is converted into a diffuse halo that gives rise to a complicated interference pattern (speckle) at the output of the medium. Albeit complex, this wave-field remains, however, deterministic. By actively shaping the wave-field at the input, one can manipulate the interference between all the scattering paths that the wave can follow. On the one hand, this insight has given rise to spectacular focusing schemes in which scattering enables -rather than impedes -wave focusing and pulse compression [1][2][3][4][5][6][7][8][9]. On the other hand, it can lead to an optimized control of wave transport [10][11][12][13]. A designed wave-front can, e.g., be completely transmitted/reflected at will [14,15] as a result of a multiple scattering interference that is intrinsically narrowband [16]. Here we will aim at the more challenging goal to generate states that are fully transmitted/reflected, yet very robust in a broadband spectral range. As we will demonstrate explicitly, this goal can be reached by way of highly collimated scattering states that are concentrated along individual particle-like bouncing patterns inside the medium [17]. By avoiding the multi-path interference associated with conventional scattering states, these wave beams also avoid the frequency sensitivity associated with this interference. As we shall see, particle-like scattering states give rise, in the time domain, to wave packets that remain focused in time and space throughout their trajectory within the medium. This crucial feature makes these states uniquely suited for many applications in a variety of fields, ranging from high intensity focused ultrasound [18,19] or underwater acoustics [20,21] to endoscopic microscopy [22][23][24], fibre optics [25][26][27][28][29] or telecommunications [30][31][32]. * [email protected]
The key aspect of our experimental study is to demonstrate that these particle-like wave packets can be created just based on the information stored in the scattering matrix [17]. This highly dimension S-matrix relates any arbitrary wave-field at the input to the output of the scattering medium, and in principle, allows the reconstruction or prediction of either. It fully describes wave propagation across a scattering medium and can meanwhile be routinely measured not only in acoustics [33,34], but also in microwave technology [13,35] and optics [6,12]. The sub-blocks of the scattering matrix contain the complex-valued transmission (t, t ) and reflection (r, r ) matrices with a certain number N of input and output channels,
S = r t t r .(1)
To describe the statistical properties of S for wave transport through complex media, random matrix theory (RMT) has been very successful [36]. One of the striking result of RMT is the universal bimodal distribution followed by the transmission eigenvalues T of tt † [37] through diffusive media [36,38,39] or chaotic cavities [40,41]. In contradiction with a classical diffusion or chaotic picture, a substantial fraction of propagation channels are found to be essentially closed (T ∼ 0) or open (T ∼ 1). Going beyond such a statistical approach, Rotter et al. [17] recently showed how a system-specific combination of fully open or fully closed channels may lead to scattering states that follow the particlelike bouncing pattern of a classical trajectory throughout the entire scattering process. Such particle-like scattering states with transmission close to 1 or 0 are eigenstates of the Wigner-Smith time-delay matrix:
Q = − i 2π S † ∂ f S,(2)
where ∂ f denotes the derivative with respect to the frequency f . Originally introduced by Wigner in nuclear scattering theory [42] and extended by Smith to multichannel scattering problems [43], the Q-matrix generally describes the time that the incoming wave accumulates due to the scattering process: each eigenvalue yields the time delay of the associated scattering state. Compared to a mere study of the S-matrix, the Q-matrix provides an elegant and powerful tool to harness the dispersion properties of a complex medium. In this article, we show, in particular, how a time-delay eigenstate can be engineered to be "particle-like" not only in its stationary wave function patterns [17], but also in the sense that a non-dispersive wave packet can be propagated along the corresponding particle-like bouncing pattern. The associated eigenvalue of Q then corresponds to the propagation-time of this wave packet. Our experimental setup consists of an elastic cavity and a disordered elastic wave guide at ultrasound frequencies [see Fig. 1]. In a first step we measure the entries of the Sand Qmatrices over a large bandwidth using laser-ultrasonic techniques. The eigenvalues of the transmission matrix are shown to follow the expected bimodal distributions in both configurations. The wave-fields associated with the open/closed channels are monitored within each system in the time domain by laser interferometry. Not surprisingly, they are shown to be strongly dispersive as they combine various path trajectories and thus many interfering scattering phases. To reduce this dispersion and to lift the degeneracy among the open/closed channels, we consider the eigenstates of the Q-matrix that have a well-defined time-delay, corresponding to a wave that follows a single path trajectory. In transmission, a one-to-one association is found between time-delay eigenstates and raypath trajectories. The corresponding wave functions are imaged in the time domain by laser interferometry. The synthesized wave packets are shown to follow particle-like trajectories along which the temporal spreading of the incident pulse is minimal. In reflection, the Q-matrix yields the collimated wave-fronts that focus selectively on each scatterer of a multitarget medium. Contrary to alternative approaches based on time-reversal techniques [44][45][46][47], the discrimination between several targets is not based on their reflectivity but on their position. The eigenvalues of Q directly yield the time-of-flight of the pulsed echoes reflected by each scatterer.
II. EXPERIMENTAL RESULTS
A. Revealing the open and closed channels in a cavity
The waves we excite and measure are flexural waves in a duralumin plate of dimension 500 × 40 × 0.5 mm 3 [see Fig. 1].The frequency range of interest spans from 0.23 to 0.37 MHz (∆f = 0.14 MHz) with a corresponding average wavelength λ of 3.5 mm. We thus have access to N = 2W/λ ∼ 22 independent channels, W being the width of the elastic plate. Two complex scattering systems are built from the homogeneous plate: (i) a regular cavity formed by cutting the plate over 20 mm on both sides of the plate [see Fig. 1(a)] and (ii) a scattering slab obtained by drilling several circular holes in the plate [see Fig. 1(b)]. The thickness L of each system is 45 mm and 52 mm, respectively.
The S-matrix is measured for each system with the laserultrasonic set-up described in Fig. 1, following the procedure explained in the Appendix A. Transmission and reflection matrices are expressed in the basis of the modes of the homoge- neous plate [15]. These eigenmodes and their eigenfrequencies have been determined theoretically using the thin elastic plate theory [48,49]. They are renormalized such that each of them carries unit energy flux across the plate section [15]. Figure 2(a) displays an example of an S-matrix recorded at the central frequency f 0 = 0.30 MHz for the cavity. Most of the energy emerges along the main diagonal and two subdiagonals [50] of the reflection/transmission matrices. These reflection and transmission matrix elements correspond to specular reflection of each mode on the cavity boundaries and to the ballistic transmission of the incident wave-front, respectively.
We first focus on the statistics of the transmission eigenvalues T l computed from the measured t-matrix (see Appendix B). Their distribution, ρ(T ) is estimated by averaging the corresponding histograms over the frequency bandwidth. Figure 2(b) shows the comparison between the distribution measured in the rectangular cavity and the bimodal law ρ b which is theoretically expected in the chaotic regime [40,41],
ρ b (T ) = 1 π T (1 − T )(3)
Even though our system is not chaotic, but exhibits regular dynamics, a good agreement is found between the measured eigenvalue distribution and the RMT predictions, confirming previous numerical studies [51]. A similar bimodal distribution of transmission eigenvalues is obtained in the disordered plate, as shown in the Supplemental Material [52]. Whereas the eigenvalues T l of tt † yield the transmission coefficients of each eigenchannel, the corresponding eigenvectors provide the combination of incident modes that allow to excite this specific channel. Hence, the wave-field associated with each eigenchannel can be measured by propagating the corresponding eigenvector. To that aim, the whole system is scanned with the interferometric optical probe. A set of impulse responses is measured between the line of sources at the input and a grid of points that maps the medium. The wave function associated with a scattering state is then deduced by a coherent superposition of these responses weighted by the amplitude and phase of the eigenvector at the input (see Appendix C). Hence all the wave functions displayed in this article are only composed of experimentally measured data and do not imply any theoretical calculation or numerical simulation.
Figures 2(c) and 2(d) display the wave-field associated with the two most open eigenchannels (T l ∼ 1) of the cavity. Although such open channels allow a full transmission of the incident energy, they do not show a clear correspondence with a particular path trajectory. The same observation holds in the disordered plate [52]. As a consequence, the associated scattering state undergoes multiple scattering when passing through the cavity. Figures 3(a) and 3(b) illustrate this dispersion by displaying the output temporal signal associated with the two open channels shown in Figs. 2(c) and 2(d) (see Appendix B). Both signals contain several peaks occurring at altogether three different times of flight. As we will see further, each peak is associated with a particular path trajectory and can be addressed independently by means of the Wigner-Smith time-delay matrix.
B. Addressing particle-like scattering states in a cavity
The Wigner-Smith time-delay matrix Q is now investigated to generate coherent scattering states from the set of open channels. Since Q is Hermitian when derived from a unitary S-matrix [Eq. (2)], the time-delay eigenstates q in m form an orthogonal and complete set of states, to each of which a real proper delay time τ m can be assigned, such that Qq in m = τ m q in m . In general, q in m is a 2N -dimensional eigenvector which implies an injection from both the left and the right leads of the system. However, among this set of timedelay eigenstates, a subset features an incoming flux from only one lead that also exits through just one of the leads. These are exactly the desired states that belong to the subspace of open or closed channels and display trajectory-like wave function patterns.
As was shown by Rotter et al. [17], the above arguments can be translated into a straightforward operational procedure (see Appendix B), which we apply here to identify the particle-like scattering states among the time-delay eigenstates of the measured Q-matrix. The litmus test for this procedure in the present context will be to show that the three ] can now be individually addressed through an associated particle-like state. The results we obtain for the cavity geometry [ Fig. 1(a)] fully confirm our first successful implementation of particle-like scattering states: The propagation of the states we obtain from our procedure yields monochromatic wave states that are clearly concentrated on individual bouncing patterns [see Fig. 4]. Whereas Fig. 4(a) corresponds to the direct path between the input and output leads, Figs. 4(b) and 4(c) display a more complex trajectory with two and four reflections on the boundaries of the cavity, respectively. The associated time-delays τ m do correspond to the run-times of a particle that would follow the same trajectory at the group velocity v g ∼ 2.6 mm/µs of the flexural wave [53]. Their transmission coefficients |t m | are equal to 0.90, 0.95 and 0.85, respectively, meaning that they are almost fully transmitted through the cavity.
The time trace associated with each particle-like scattering state is computed from the frequency-dependent t-matrix (see Appendix B). The result is displayed in Figs. 3(c), 3(d) and 3(e). Unlike the open transmission channels studied above, each particle-like state gives rise to a single pulse that arrives at the output temporally unscattered at time τ = τ m . Fig. 2(d) consists of a linear combination of the paths displayed in Fig.4. This association is also confirmed by explicitly analyzing the vectorial decomposition of the particle-like state in terms of the transmission eigenchannel basis.
The frequency dependence of the particle-like scattering states is investigated in the Supplemental Material [52]. They are shown to be stable over the frequency ranges f = 0.2−0.6 MHz [ Fig. 4(a) Fig. 4(c)]. The corresponding bandwidths are at least one order of magnitude larger than the frequency correlation width of the transmission matrix coefficients which is equal to 0.02 MHz [52]. This proves the robustness of particle-like states over a broadband spectral range. Given this non-dispersive feature, they turn out to be perfect candidates also for the formation of minimally dispersive wave packets in the time domain. To check this conjecture, we investigate here the spatio-temporal wave functions of these states over the aforementioned bandwidths (see Appendix B). The propagation of the particle-like wave-packets through the cavity can be visualized in the three first movies of the Supplemental Material [52]. Figure 5 displays successive snapshots of the wave-packet synthesized from the particle-like scattering state displayed in Fig. 4(c). Quite remarkably, the spatio-temporal focus of the incident wave-packet is maintained throughout its trajectory despite the multiple scattering events it undergoes in the cavity.
], f = 0.3 − 0.6 MHz [Fig. 4(b)] and f = 0.2 − 0.4 MHz [
C. Lifting the degeneracy of particle-like scattering states
In a next step, we investigate particle-like scattering states in the disordered wave guide [ Fig. 1(b)]. The corresponding Q-matrix is measured at the central frequency f 0 = 0.3 MHz (see Appendix A). Figures 6(a) and 6(b) display the monochromatic wave functions associated with two fully transmitted time-delay eigenstates. Whereas Fig. 6(a) displays the typical features of a particle-like scattering state that winds its way through the scatterers, the time-delay eigenstate of Fig. 6(b) is clearly associated with two scattering paths of identical length. We thus encounter here a degeneracy in the time-delay eigenvalues that needs to be lifted by an additional criterion, such as by considering well-defined subspaces of the measured S-matrix [54]. In this instance, the two ray paths can be discriminated by their different angles of incidence. Correspondingly, we consider two subspaces S of the original S-matrix by keeping either positive or negative angles of incidence from the left lead (see Appendix D). The corresponding time-delay matrices lead to two distinct particlelike scattering states displayed in Figs. 6(c) and 6(d). The two scattering paths that were previously mixed in the original time-delay eigenstate [ Fig. 6(b)] are now clearly separated. The frequency stability of these states is investigated in the Supplemental Material [52]. They are shown to be stable over the frequency ranges f = 0. Fig. 6(d)]. The corresponding particlelike wave packets are shown in the two last movies of Supplemental Material [52] where the high quality of their focus in space and time is immediately apparent.
D. Revealing time-delay eigenstates in reflection.
Time-delay eigenstates can also result from a suitable combination of closed channels. Figures 6(e) and 6(f) display two such closed channels derived from the S-matrix of the disordered slab. The closed channels combine multiple re-flections from the holes of the scattering layer. However, the closed channel displayed in Fig. 6(e) mixes the contributions from the scatterers labeled 1 and 2. Also the closed channel displayed in Fig. 6(f) is associated with reflections from altogether three scatterers (2, 3 and 4). A simple eigenvalue decomposition of rr † or tt † does not allow a discrimination between the scatterers. On the contrary, the analysis of the Q-matrix allows a one-to-one correspondence with each scatterer based on a time-of-flight discrimination. Figures 6(g) and 6(h) actually display the wave functions associated with two reflected time-delay eigenstates. Each of these eigenstates is associated with a reflection from a single scatterer (2 and 3, respectively). The corresponding time delays τ m given in the caption of Fig. 6 are directly related to the depth z of each scatterer, such that τ m ∼ 2z/v g .
III. DISCUSSION
The first point we would like to emphasize is the relevance of the time-delay matrix for selective focusing and imaging in multi-target media. The state-of-the-art approach is the DORT method (French acronym for Decomposition of the Time Reversal Operator). Initially developed for ultrasound [44,45] and more recently extended to optics [46,47], this widely used approach takes advantage of the reflection matrix r to focus iteratively by time reversal processing on each scatterer of a multi-target medium. Mathematically, the time-reversal invariants can be deduced from the eigenvalue decomposition of the time reversal operator rr † or, equivalently, from the singular value decomposition of r. On the one hand, the eigenvectors of r should, in principle, allow selective focusing and imaging of each scatterer. On the other hand, the associated eigenvalue directly yields the scatterer reflectivity. However, a one-to-one association between each eigenstate of r and each scatterer only exists under a single-scattering approximation and if the scatterers exhibit sufficiently different reflectivities. Figures 6(e) and 6(f) illustrate this limit. Because the holes display similar scattering cross-sections in the disordered slab, closed channels are associated with several scatterers at once. On the contrary, the time-delay matrix allows a discrimination between scatterers based on the time-of-flight of the reflected echoes [see Figs. 6(g) and 6(h)]. Moreover, unlike the DORT method, a time-delay analysis also allows to discriminate the single scattering paths from multiple scattering events, the latter ones corresponding to longer time-of-flights. Hence, the time-delay matrix provides an alternative and promising route for selective focusing and imaging in multi-target media.
A second relevant point to discuss is the nature of transmitted time-delay eigenstates in other complex systems. Recently, Carpenter et al. [27] and Xiong et al. [28] investigated the group delay operator, i/(2π)t −1 ∂ f t, in a multimode optical fiber. The eigenstates of this operator are known as the principal modes in fiber optics [25,26]. For a principal mode input, a pulse that is sufficiently narrow-band reaches the output temporally un-distorted, although it may have been strongly scattered and dispersed along the length of propagation. On the contrary, for a particle-like scattering state input, the focus of the pulse is not only retrieved at the output of the scattering medium but maintained throughout its entire propagation through the system. This crucial difference provides particle-like states with a much broader frequency stability than principal modes, which translates into the possibility to send much shorter pulses through these particle-like scattering channels. Last but not least, we also emphasize that even though particle-like states are unlikely to occur in diffusive scattering media, the time-delay eigenstates are still very relevant also in such a strongly disordered context. Consider here, e.g., that the eigenstates with the longest time delay can be of interest for energy storage, coherent absorption [55] or lasing [56,57] purposes. From a more fundamental point-ofview, the trace of the Qmatrix directly provides the density of states of the scattering medium [58], a quantity that turns out to be entirely independent of the mean free path in a disordered system [59].
Finally, we would like to stress the impact of our study on other fields of wave physics and its extension to more complex geometries. For this experimental proof-of-concept, a measurement of the wave-field inside the medium was required in order to image the wave functions and prove their particle-like feature. However, such a sophisticated protocol is not needed to physically address particle-like wave packets. A simple measurement of the scattering matrix (or a subpart of it) at neighboring frequencies [27] yields the time-delay matrix from which particle-like state inputs can be extracted. Such a measurement can be routinely performed through 3D scattering media whether it be in optics [6,12], in the microwave regime [13,35] or in acoustics [33,34]. As to the generation of particle-like wave packets, multi-element technology is a powerful tool for the coherent control of acoustic waves and electromagnetic waves [60]. Moreover, recent progress in optical manipulation techniques now allows for a precise spatial and temporal control of light at the input of a complex medium [60]. Hence there is no obstacle for the experimental implementation of particle-like wave packets in other fields of wave physics. At last, we would like to stress the fact that in our experimental implementation we are in the limit of only a few participating modes with a wavelength that is comparable to the spatial scales of the system. In this limit the implementation of particle-like states is truly nontrivial since interference and diffraction dominates the scattering process as a whole. When transferring the concept to the optical domain one may easily reach the geometric optics limit where the wavelength is much shorter than most spatial scales of the system and particle-like states may in fact be much easier to implement in corresponding complex optical media.
IV. CONCLUSION
In summary, we experimentally implemented particle-like scattering states in complex scattering systems. Based on an experimentally determined time-delay matrix, we have demonstrated the existence of wave packets that follow particle-like bouncing patterns in transmission through or in reflection from a complex scattering landscape. Strikingly, these wave packets have been shown to remain focused in time and space throughout their trajectory within the medium. We are convinced that the superior properties of these states in terms of frequency stability and spatial focus will make them very attractive for many applications of wave physics, rang-ing from focusing to imaging or communication purposes. In transmission, the efficiency of these states in terms of information transfer as well as their focused in-and output profile will be relevant. In reflection, selective focusing based on a time-of-flight discrimination will be a powerful tool to overcome aberration and multiple scattering in detection and imaging problems.
V. ACKNOWLEDGMENTS
The authors wish to thank A. Derode for fruitful discussions and advices. The authors are grateful for funding provided by LABEX WIFI (Laboratory of Excellence within the French Program Investments for the Future, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*). B.G. acknowledges financial support from the French "Direction Générale de l'Armement"(DGA). P.A. and S.R. are supported by the Austrian Science Fund (FWF) through Projects NextLite F49-10 and I 1142-N27 (GePartWave).
Appendix A: Experimental procedure
The first step of the experiment consists in measuring the impulse responses between two arrays of points placed on the left and right sides of the disordered slab [see Fig. 1]. These two arrays are placed 5 mm away from the disordered slab. The array pitch is 0.8 mm (i.e < λ/2) which guarantees a satisfying spatial sampling of the wave field. Flexural waves are generated in the thermoelastic regime by a pumped diode Nd:YAG laser (THALES Diva II) providing pulses having a 20 ns duration and 2.5 mJ of energy. The out-of-plane component of the local vibration of the plate is measured with a heterodyne interferometer. This probe is sensitive to the phase shift along the path of the optical probe beam. The calibration factor for mechanical displacement normal to the surface (100 mV/nm) was constant over the detection bandwidth (100 kHz -400 kHz). Signals detected by the optical probe were fed into a digital sampling oscilloscope and transferred to a computer. The impulse responses between each point of the same array (left and right) form the timedependent reflection matrices (r and r , respectively). The set of impulse responses between the two arrays yield the timedependent transmission matrices t (from left to right) and t (from right to left). From these four matrices, one can build the S-matrix in a point-to-point basis [Eq. (1)]. A discrete Fourier transform (DFT) of S is then performed over a time range ∆t = 120 µs that excludes the echoes due to reflections on the ends of the plate and ensures that most of the energy has escaped from the sample when the measurement is stopped. The next step of the experimental procedure consists in decomposing the S-matrices in the basis of the flexural modes of the homogeneous plate. These eigenmodes and their eigenfrequencies have been determined theoretically using the thin elastic plate theory [15,48,49]. They are normalized such that each of them carries unit energy flux across the plate section. Theoretically, energy conservation would imply that S is unitary. In other words, its eigenvalues should be distributed along the unit circle in the complex plane. However, as shown in a previous work [15], this unitarity is not retrieved experimentally because of experimental noise. A dispersion of the eigenvalues s i of the S−matrix is observed around the unit circle. We compensate for this undesirable effect by considering a normalized scattering matrix with the same eigenspaces but with normalized eigenvalues [15]. The Q−matrix is then deduced from S using Eq. (2). The frequency-derivative of S at f = f 0 is estimated from the centered finite difference,
∂ f S (f 0 ) = S (f 0 + δf ) − S (f 0 − δf ) 2δf ,
with δf = 3 kHz.
t(f 0 ) = l T l (f 0 )u l (f 0 )v † l (f 0 )
with T l (f 0 ) the intensity transmission coefficient associated with the l th scattering eigenstate at the central frequency.
The frequency-dependent amplitude transmission coefficient t l (f ) of this eigenstate can be obtained from the set of transmission matrices t (f ) measured over the whole frequency bandwidth, such that
t l (f ) = u † l (f 0 )t (f ) v l (f 0 ) An inverse DFT of t l (f )Q(f 0 ) = m τ m q in m (f 0 ) q in m (f 0 ) †
The time-delay eigenvector q in m is a 2N -dimensional column vector that can be decomposed as
q in (f 0 ) = q in m,L (f 0 ) q in m,R (f 0 )
where q in m,L (f 0 ) and q in m,R (f 0 ) contain the complex coefficients of q in m (f 0 ) in the basis of the N incoming modes in the left and right leads, respectively. Among the set of timedelay eigenstates, particle-like scattering states injected from the left lead should fulfill the following condition [17]:
||q in m,L (f 0 )|| 2 >> ||q in m,R (f 0 )|| 2 0.
A particle-like scattering state is thus associated with a Ndimensional input eigenvector q in m,L (f 0 ). The corresponding output eigenvector q out m,R (f 0 ) and transmission coefficient t (q) m (f 0 ) can be deduced from the t-matrix:
t(f 0 )q in m,L (f 0 ) = t (q) m (f 0 ) q out m,R (f 0 ).
The frequency-dependent amplitude transmission coefficient t (q) m (f ) of this time-delay eigenstate can be obtained from the set of transmission matrices t (f ) measured over the whole frequency bandwidth, such that
t (q) m (f ) = q out m,L (f 0 ) † t (f ) q in m,R (f 0 ).
An inverse DFT of t Impulse responses are measured between the line of sources (denoted by the index i) and a grid of points (denoted by the index j) that maps the medium, following the same procedure as the one described above. The grid pitch is 1.3 mm.
This set of impulse responses forms a transmission matrix k(τ ) = [k ji (τ )]. A discrete Fourier transform (DFT) of k(τ ) yields a set of frequency-dependent transmission matrices k(f ). The lines of k(f ) are then decomposed in the basis of the plate modes. The monochromatic wave-field Ψ(f ) = [ψ j (f )] associated with a transmission/reflection or time-delay eigenchannel is provided by the product between the matrix k(f ) and the corresponding eigenvector (v l (f 0 ) or q in m,L (f 0 ), respectively). The time-dependent wave-field Ψ(τ ) = [ψ j (τ )] is deduced by an inverse DFT over a frequency bandwidth of our choice. Note that a Hann window function is priorly applied to k(f ) to limit side lobes in the time-domain.
Appendix D: Unmixing degenerated time-delay eigenstates.
Depending on the geometry of the scattering medium, the different scattering paths involved in a degenerated time-delay eigenstate can be discriminated either in the real space or in the spatial frequency domain [54]. Here, the time-delay eigenstate displayed in Fig. 6(b) shows two scattering paths with opposite angles of incidence. Hence, they can be discriminated by analyzing each block of the S-matrix in the spatial frequency domain. The left lead of each block is decomposed over the positive or negative angles of incidence. This subspace of the S-matrix, refered to as S , is then used to compute a reduced time-delay matrix Q [54] such that
Q = − i 2π S −1 ∂ f S .
Note that the transpose conjugate operation of Eq. (2) is here replaced by an inversion of S because of its non-unitarity [54]. Depending on the sign of the angle of incidence chosen for the left lead, the reduced matrix Q provides the timedelay eigenstates displayed in Figs. 6(c) and 6(d).
Supplemental Material
This document provides further information on the scattering matrix measured in the disordered wave guide and the stability of particle-like states in the frequency domain. The captions of the Supplementary Movies are also listed at the end of the document. Figure S1(a) displays an example of S−matrix recorded in the disordered wave guide depicted in Fig. 1(a) of the accompanying paper. Despite a moderate level of disorder, the S−matrix exhibits an overall random appearance. Nevertheless, a residual ballistic wave-front slightly emerges along the diagonal of the transmission matrices. The statistics of the transmission eigenvalues T l computed from the t−matrix is now investigated. As for the regular cavity, their distribution ρ (T ) is estimated by averaging their histograms over the frequency bandwidth. Fig. S1(b) displays the estimator of this distribution. Most of the transmission coefficients are found to be either close to zero or one, accounting respectively for closed and open channels. Figures S1(c) and S1(d) display the wave-fields associated with two open channels at the central frequency (see Methods). As in the cavity, the open channels combine multiple path trajectories. As a consequence, they undergo a strong spatial and temporal dispersion while propagating through the scattering slab. As shown in Fig.6 . These values should be compared to the frequency correlation width δf c = 0.02 MHz of the transmission matrix coefficients. We see that the spectral correlation width δf m of particle-like states is 8, 8.5 and 6 times larger than δf c . This illustrates the stability of the cavity particle-like states in the frequency domain and accounts for the spatio-temporal focusing of particle-like wave packets in the time domain (see Supplementary Movies 1, 2 and 3). The same analysis can be performed for the disordered wave guide. As an example, we investigate the frequency stability of the particle-like states displayed in Figs.6(c) and 6(d). Figures S3(a) and S3(b) display the frequency dependence of the corresponding transmission coefficients t Fig.S3(b)]. The corresponding spectral correlation functions C m (δf ) are displayed in Figs.S3(c) and S3(d). We measure a frequency correlation width δf m of 0.13 and 0.12 MHz. Contrastingly, the transmission matrix elements exhibit a frequency correlation width δf c = 0.025 MHz. δf m is thus 5 times higher than δf c . This demonstrates the frequency stability of these two particle-like states and accounts for the spatio-temporal focusing of the corresponding particle-like wave packets in the time domain (see Supplementary Movies 4 and 5).
I. SCATTERING MATRIX ANALYSIS IN THE DISORDERED WAVEGUIDE
III. CAPTIONS OF THE SUPPLEMENTARY MOVIES
Movie 1: Particle-like wave-packet in the cavity synthesized from the scattering state displayed in Fig. 4(a) over the frequency range f = 0.2 − 0.6 MHz. Movie 2: Particle-like wave-packet in the cavity synthesized from the scattering state displayed in Fig. 4(b) over the frequency range f = 0.3 − 0.6 MHz. Movie 3: Particle-like wave-packet in the cavity synthesized
FIG. 1 .
1The two systems under investigation consist (a) of a regular cavity and (b) of a disordered slab machined in an elastic plate. In both configurations, the S-matrix is measured in the time-domain between two arrays of points placed on the left and right sides of the system (see Appendix A). Flexural waves are generated on each point by a pulsed laser via thermo-elastic conversion over a focal spot of 1 mm 2 . The normal component of the plate vibration is measured with an interferometric optical probe. The laser source and the probe are both mounted on 2D translation stages.
FIG. 2 .
2(a), Real part of the S-matrix measured in the cavity [Fig. 1(a)] at f0 = 0.30 MHz. The black lines delimit transmission and reflection matrices as depicted in Eq. (1). (b), Transmission eigenvalue histogram, ρ (T ), averaged over the frequency bandwidth f = 0.23 − 0.37 MHz. The distribution is compared to the bimodal law ρ b (T ) [red continuous line, Eq. (3)]. (c)-(d), Absolute values of the monochromatic wave-fields (f0 = 0.30 MHz) associated with the two most open channels.
FIG. 3 .
3The time trace of a scattering or time-delay eigenstates is computed from the set of t-matrices measured over the frequency range f = 0.23 − 0.37 MHz (see Appendix B). The output intensity is displayed versus time for, (a)-(b), the two first open channels displayed in Figs. 2(c) and 2(d) and for, (c)-(e), the three time-delay eigenstates displayed in Fig. 4. For each time trace, the different echoes are labeled with a number 1, 2 or 3 that corresponds respectively, to the direct, double or quadruple scattering paths highlighted in Fig. 4.
FIG. 4 .
4Absolute value of the wave-field associated with the three particle-like scattering states derived from the matrix Q. The corresponding time delays, (a), τm = 20 µs, (b), τm = 30 µs and, (c), τm = 59 µs, nicely match with the run-times of the corresponding classical trajectories shown in the inset (top left). different time traces that are identifiable in the open transmission channels [see Figs. 3(a) and 3(b)
Figure 3 also shows that each temporal peak in the time trace of the open channels can be attributed to a particular path trajectory. We may thus conclude that the open channel displayed in Fig. 2(c) is mainly associated with the double and quadruple scattering paths displayed in Figs. 4(b) and 4(c). The open channel displayed in
2 − 0.4 MHz [Fig. 6(c)] and f = 0.2 − 0.5 MHz [
FIG. 5 .FIG. 6 .
56A spatio-temporal wave-packet is synthesized from the particle-like scattering state displayed inFig. 4(c) over the frequency bandwidth 0.2 − 0.4 MHz. The different subsets (a)-(h) display successive snapshots of its propagation across the cavity versus time. Absolute value of the wave-field associated with several time-delay/scattering eigenstates of the disordered system at the central frequency f0. (a), Transmitted particle-like scattering state (τm = 29 µs, |tm| = 0.84). (b) Degenerate time-delay eigenstate mixing two particle-like trajectories (τm = 26 µs, |tm| = 0.91). (c)-(d) Unmixing of the time-delay eigenstate displayed in (b) by considering different subspaces of the S-matrix (see Appendix D). Their transmission coefficients |tm| are equal to 0.94 and 0.98, respectively. (e)-(f) Closed channels deduced from the S-matrix (T l ∼ 0) showing simultaneous reflections from the scatterers of the disordered wave guide. (g) Reflected time-delay eigenstate showing selective focusing on the scatterer n o 2 (time delay τm = 11 µs, reflection coefficient |rm| = 0.94). (h) Reflected time-delay eigenstate showing selective focusing on the scatterer n o 3 (time delay τm = 30 µs, reflection coefficient |rm| = 0.88).
B: Revealing transmission/time-delay eigenchannels and their temporal/spectral features The transmission and time-delay eigenchannels are derived from the matrices S and Q measured at the central frequency f 0 . The transmission matrix t(f 0 ) (from the left to the right lead) is extracted from S(f 0 ) [Eq. (1)]. The output and input transmission eigenvectors, u l and v l , are derived from the singular value decomposition of t(f 0 ):
finally yields the time-dependent amplitude transmission coefficient t l (τ ) of the l th scattering eigenstate measured at the central frequency f 0 . The time traces displayed in Figs. 3(a) and 3(b) correspond to the square norm of this quantity. The time-delay eigenchannels at the central frequency are derived from the eigenvalue decomposition of Q(f 0 )
m
(f ) finally yields the time-dependent amplitude transmission coefficient t(q) m (τ ) of the m th timedelay eigenstate measured at the central frequency f 0 . The time traces displayed in Figs. 3(c), 3(d) and 3(e) correspond to the square norm of this quantity. Appendix C: Imaging spatio-temporal wave functions of transmission/reflection and time-delay eigenchannels
FIG. S1. (a), Amplitude of the S−matrix measured in the scattering slab at f0 = 0.30 MHz. The black lines delimit transmission and reflection matrices. (b), Transmission eigenvalue histograms, ρ (T ), averaged over the frequency bandwidth. (c)-(d), Absolute value of the monochromatic wave-fields associated with two open channels at the central frequency.
. S2. (a)-(b)-(c), Frequency dependence of the transmission coefficients t(q) m (f ) for the three cavity particle-like states displayed in Fig. 4(a)-(b)-(c), respectively (blue line: real part, red line: imaginary part). The black vertical dashed lines delimit the bandwidth of each state. (d)-(e)-(f), Spectral correlation function |Cm(δf )| (blue line) of the transmission coefficient t
m
(f ) displayed in (a)-(b)-(c), respectively. Cm(δf ) is computed by considering the central frequency f0 of each state. Each curve is compared with the spectral correlation function C(δf ) of the transmission matrix elements (black dashed line). Figures S2(a) and S2(c) display the frequency dependence of the transmission coefficients t
m
(f ) for the three cavity particle-like states displayed inFig. 4. The frequency evolution of t(q) m (f ) allows to delimit the frequency range over which each particle-like state remains stable: f = 0.2 − 0.6 MHz [Fig. S2(a)], f = 0.3 − 0.6 MHz [Fig. S2(b)] and f = 0.2 − 0.4 MHz [Fig. S2(c)]. To be more quantitative, we have estimated the spectral correlation function of each state,C m (δf ) = t (q) m (f 0 )t (q) * m (f 0 + δf ),with f 0 the associated central frequency. The result is displayed in Figs. S2(d), S2(e) and S2(f), and compared to the mean spectral correlation function,C(δf ) = t ij (f )t * ij (f + δf ) ,of the measured transmission matrix coefficients t ij (f ). The symbol . here denotes an average over the frequency bandwidth. The FWHM of C m (δf ) yields the spectral correlation width δf m of each state displayed in Fig. 4. We find δf m = 0.16 MHz [Fig. S2(d)], 0.17 MHz [Fig. S2(e)] and 0.12 MHz [Fig. S2(f)]
. S3. (a)-(b), Frequency dependence of the transmission coefficients t q m (f ) for the two particle-like states in the disordered wave guide, displayed in Figs. 6(c) and 6(d), respectively (blue line: real part, red line: imaginary part). The black vertical dashed lines delimit the bandwidth of each state. (c)-(d), Spectral correlation function, |Cm(δ f )|, for the transmission coefficient t q m (f ) displayed in (a)-(b), respectively. Cm(δf ) is computed by considering the central frequency f0 of each state. Each curve is compared with the spectral correlation function |C(δf )| of the transmission matrix elements (black dashed line).
Each state remains stable over a frequency range f = 0.2−0.4 MHz [Fig.S3(a)] and f = 0.2 − 0.5 MHz [
of the accompanying paper, the study of the time-delay matrix allows to lift this degeneracy.II. STABILITY OF THE PARTICLE-LIKE STATES IN THE FREQUENCY DOMAINδf [MHz]
spectral correlatoin
function
(d)
transmission
coe cient
(a)
δf [MHz]
(e)
δf [MHz]
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Particle-like wave-packet in the disordered wave guide synthesized from the scattering state displayed in Fig. 6(c) over the frequency range f = 0.2 − 0.4 MHz. Movie 5: Particle-like wave-packet in the disordered wave guide synthesized from the scattering state displayed in Fig. MHz. Movie. 4d) over the frequency range f = 0.2 − 0.5 MHzMHz. Movie 4: Particle-like wave-packet in the disordered wave guide synthesized from the scattering state displayed in Fig. 6(c) over the frequency range f = 0.2 − 0.4 MHz. Movie 5: Particle-like wave-packet in the disordered wave guide synthesized from the scattering state displayed in Fig. 6(d) over the frequency range f = 0.2 − 0.5 MHz.
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[
"Combined hapto-visual and auditory rendering of cultural heritage objects",
"Combined hapto-visual and auditory rendering of cultural heritage objects"
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"Praseedha Krishnan \nDepartment of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia\n",
"Aniyath \nDepartment of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia\n",
"Sreeni Kamalalayam Gopalan \nDepartment of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia\n",
"Priyadarshini Kumari \nDepartment of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia\n",
"Subhasis Chaudhuri \nDepartment of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia\n"
]
| [
"Department of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia",
"Department of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia",
"Department of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia",
"Department of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia",
"Department of Electrical Engineering\nVision and Image Processing Lab\nIIT Bombay\nIndia"
]
| []
| In this work, we develop a multi-modal rendering framework comprising of hapto-visual and auditory data. The prime focus is to haptically render point cloud data representing virtual 3-D models of cultural significance and also to handle their affine transformations. Cultural heritage objects could potentially be very large and one may be required to render the object at various scales of details. Further, surface effects such as texture and friction are incorporated in order to provide a realistic haptic perception to the users. Moreover, the proposed framework includes an appropriate sound synthesis to bring out the acoustic properties of the object. It also includes a graphical user interface with varied options such as choosing the desired orientation of 3-D objects and selecting the desired level of spatial resolution adaptively at runtime. A fast, point proxy-based haptic rendering technique is proposed with proxy update loop running 100 times faster than the required haptic update frequency of 1 kHz. The surface properties are integrated in the system by applying a bilateral filter on the depth data of the virtual 3-D models. Position dependent sound synthesis is incorporated with the incorporation of appropriate audio clips. | 10.1007/978-3-319-16631-5_36 | [
"https://arxiv.org/pdf/2010.02015v1.pdf"
]
| 14,449,653 | 2010.02015 | a43432e58bd5014aff5e1a2a32cd18682a3cbbb2 |
Combined hapto-visual and auditory rendering of cultural heritage objects
Praseedha Krishnan
Department of Electrical Engineering
Vision and Image Processing Lab
IIT Bombay
India
Aniyath
Department of Electrical Engineering
Vision and Image Processing Lab
IIT Bombay
India
Sreeni Kamalalayam Gopalan
Department of Electrical Engineering
Vision and Image Processing Lab
IIT Bombay
India
Priyadarshini Kumari
Department of Electrical Engineering
Vision and Image Processing Lab
IIT Bombay
India
Subhasis Chaudhuri
Department of Electrical Engineering
Vision and Image Processing Lab
IIT Bombay
India
Combined hapto-visual and auditory rendering of cultural heritage objects
In this work, we develop a multi-modal rendering framework comprising of hapto-visual and auditory data. The prime focus is to haptically render point cloud data representing virtual 3-D models of cultural significance and also to handle their affine transformations. Cultural heritage objects could potentially be very large and one may be required to render the object at various scales of details. Further, surface effects such as texture and friction are incorporated in order to provide a realistic haptic perception to the users. Moreover, the proposed framework includes an appropriate sound synthesis to bring out the acoustic properties of the object. It also includes a graphical user interface with varied options such as choosing the desired orientation of 3-D objects and selecting the desired level of spatial resolution adaptively at runtime. A fast, point proxy-based haptic rendering technique is proposed with proxy update loop running 100 times faster than the required haptic update frequency of 1 kHz. The surface properties are integrated in the system by applying a bilateral filter on the depth data of the virtual 3-D models. Position dependent sound synthesis is incorporated with the incorporation of appropriate audio clips.
Introduction
We interact with our physical world mostly through visual, auditory and tactile sensations. Haptics is an emerging research field which tries to emulate the latter sensation in the virtual world. Incorporation of haptics in the virtual environment provides a better immersive experience to the users, especially to the visually impaired persons. Further, a combined hapto-visual rendering enhances the realism of the haptic interaction even for the sighted users. Many authors have also suggested the inclusion of surface properties like texture and friction in the haptic domain for realistic haptic perception. However, in most cases of prior art, the rendering of surface properties such as texture and friction is not realistic enough to provide sufficient immersion to the users. This is more so as common haptic interface devices like Novint Falcon and SensAble Phantom are kinaesthetic and not tactile.
Haptic rendering methods like god object rendering algorithm [17] work well in the case of polygon based representation of 3-D models. On the other hand, it fails to properly render 3-D objects represented using point cloud data. Moreover, 3-D objects may appear at various different scales and orientations, and the user needs to experience the object at different levels of details. A polygon-based rendering scheme is not suitable to implement variations in scale and orientation. This is because recomputing the mesh structure during the interaction is not feasible as it is time consuming. The authors in [18] did speed up the multi-level hapto-visual rendering using a Monge surface. However, their focus is mainly on fast rendering of a single valued function at different levels of details and cannot handle any rotation of the space and does not provide any surface friction. This paper aims at enhancing the virtual immersion during haptic interaction with point cloud 3-D models by incorporating surface properties such as texture and friction. The coefficient of friction is computed from the texture component extracted from the scanned depth data. But for these properties, the virtual environment would often feel slippery since the direction of the force vector is always perpendicular to the surface and thus would not be sufficiently realistic. The new rendering framework can also handle affine transformations such as rotation, translation, expansion and geometric contraction of 3-D objects as illustrated in Fig. 1. Initially, we generate depth data at each point of the model at any desired orientation by reading the contents of depthbuffer in OpenGL using the inbuilt GLUT command glReadPixels() and create a Monge surface from it. This surface is then hapto-visually rendered at different scales adaptively at run time. We propose the use of a bilateral filter to extract the local surface texture and compute the dynamic friction as a function of the local texture. We show that the user's experience can be enriched by allowing the user to interact with the object at multiple resolutions. Audio rendering is also incorporated in the proposed system for enhanced virtual experience by playing appropriate au-dio files based on the position of the haptic probe inside the virtual environment. We also implement a graphical user interface for easier accessibility.
The organization of the paper is as follows. Section 2 provides a review of the related literature. Section 3 explains the proposed method. Haptic and graphic rendering techniques and additional functionalities are discussed in section 4. Section 5 illustrates the results. The conclusions are summarized in section 6.
Literature Review
The basic haptic rendering technique is polygon based in which virtual objects are represented using polygonal meshes. In polygon based rendering, each time the haptic interface point (HIP) penetrates the object, the rendering algorithm finds the closest point on the mesh defined surface and computes the penetration depth of HIP inside the object. If x is the vector representing the penetration depth, the reaction force is calculated as F = −kx, where k is the stiffness constant of the surface. This method has problems in determining the appropriate force direction while rendering thin objects. Authors in [24], [16] independently proposed the concept of god-object algorithm and proxy algorithm, respectively, to solve this problem. The god-object rendering algorithm includes another point in addition to the HIP, called "god-object or proxy". In free space the proxy and the HIP are collocated. However, as the HIP penetrates the virtual object, the proxy is constrained to lie on the surface of the virtual object [7]. But in the case of virtual objects represented using point cloud data, the proxy would slip into the object. It is called the "fall-through" problem of the god-object. This can be avoided by using a spherical proxy instead of a point. However, an increase in radius of the proxy impairs the haptic interaction by smoothing out the surface texture.
Many authors have also developed algorithms to render point cloud data directly. Lee et al. have proposed a rendering method with point cloud data which estimates the distance from HIP to the closest point on the moving least square surface defined by the given point set [9]. El-Far et al. used axis aligned bounding boxes to fill the voids in the point cloud and then rendered with a god object rendering technique [12]. Leeper et al. described a constraint based approach of rendering point cloud based data where the points are replaced by spheres or surface patches of approximate size [10]. Another proxy based technique of rendering a dense 3-D point cloud data was proposed in [19], where the surface normal is estimated locally from the point cloud. Most of the methods are unable to handle scale changes during rendering and variable density of point cloud.
In order to enrich the experience of virtual world, many authors have incorporated surface properties like texture and friction in the haptic domain. Adi et al. [1] introduced the technique of rendering the tactile cues from visual information using wavelet transforms. It was more realistic than primitive haptic texture rendering methods implemented using sine waves [2] and Fourier series [21]. But this technique was found to be less stable than the existing meth-ods. Some authors, as in [15], have proposed data driven approaches to realistic haptic texture rendering. However, these methods are computationally expensive for real-time applications. Similarly, Richard et al. presented a friction rendering model in haptics using modified Karnopp model [14]. Hayward et al. [6] developed a discrete implementation of friction exhibiting four friction regions: sticking, creeping, oscillating, and sliding. Harwin et al. [5] proposed the friction cone algorithm for providing friction in haptic environments which also has a few shortcomings as suggested in [11]. Hence, still much needs to be explored in this domain in order to augment the user's virtual experience.
Research in ecological acoustics imply that auditory feedback can effectively convey information about a number of object attributes such as its shape, size and material [3]. The effectiveness of auditory sensations was studied by Lederman et al. [8] who have showed that sound plays a dominant role when a probe is used to interact with a surface as compared to the case of direct contact with the bare fingers. We make use of this fact in order to improve the virtual immersion of the user interacting with the virtual environment using a haptic probe. Simultaneous audio rendering is required in our application where we try to render ancient cultural monuments where the supporting pillars have very interesting acoustic (read musical) properties.
Proposed Proxy Updation Algorithm
The proposed system incorporates a proxy based algorithm to render a point cloud data. It is to be noted that the proxy is a point and not a sphere as it is popular in point cloud rendering techniques, thus allowing us to perceive surface textures. Let us first assume that one has the depth buffer data corresponding to 3-D models available. Issues related to transformation of the haptic space and scaling will be discussed in section 4.
In order to render the object in the haptic domain, we need to find the collision of HIP with the bounding surface and hence the penetration depth of HIP into the surface. The proposed algorithm tries to translate the proxy over the object surface in short steps during the haptic interaction such that each time it finds the most suitable proxy position which would provide the minimum distance between HIP and proxy and simultaneously applies the reaction force normal to the surface at the point of contact. Fig. 3(a) illustrates the proxy movement during collision with an arbitrary object represented by a Monge surface. The Monge surface corresponding to a 3-D object is shown as a curve in 1−D. HIP is shown with a yellow circle penetrated inside the object and the proxy is also shown with a yellow dot constrained to the surface. The vector v n represents the normal at the initial proxy position with n = vn |vn| representing the unit normal and v h is the vector from proxy to the HIP and is given by v h = X h − X p , where X h and X p represent the position vectors of HIP and proxy, respectively. The tangential vector in the plane of v n and v h is evaluated which provides a fast approximation of the direction for the proxy to move so that the distance between proxy and HIP can be minimized. When the proxy is moved continuously along the tangential direction on the curve, proxy will finally come to rest at a point where the angle between v n and v h is 180 degrees. The tangent vector v t can be computed from n and v h using equation 1.
z = f (x) x proxy position xp (xp, f (xp)) (xp, zp) f (xp)v t = v h − (v n .v h )n.(1)
We use the following proxy update equation to translate the proxy along the tangent plane.
X (k+1) p = X (k) p + ρv k t .(2)
The parameter ρ < 1 and is arbitrarily chosen. As the value of ρ increases, the proxy quickly converges, but it does not move close to the surface during the convergence. On the other hand if ρ is very small, the proxy point moves close to the object surface, but needs a little more time to converge. As a matter of fact, the value of ρ relates to the frictional force on the surface and should depend on the material property as explained in section 3.2. A small value of ρ signifies larger surface friction. Hence the proposed method provides an easy way of including dynamic friction during rendering.
Once the proxy moves in the direction of tangential vector, it may deviate from the the boundary of the object. This deviation of the proxy from the boundary is avoided by projecting the proxy along n onto the surface before updating its position along the tangential direction. The proxy position update is performed within 1 ms of time, so that the user's interaction with the object through the haptic device is unhindered and is carried out at 1 kHz. Since the updated proxy location need not be on the chosen lattice for depth representation, the surface needs to be locally interpolated. In case of 2 − D depth data, we project the proxy onto the X-Y plane and the corresponding depth value is obtained by interpolating the neighbourhood depth values to form a continuous function z = f (x, y) as illustrated in Fig. 3
(b) for a 1 − D function z = f (x).
Since the available points are sampled quite densely, bilinear interpolation is sufficient to find the bounding surface as shown in Fig. 3(b). In order to check the collision of HIP with the surface, we compare z p with the depth at the projected point z = f (x, y) for a given proxy position (x p , z p ). If f (x p ) > z p , the proxy has touched the surface, otherwise it is free to move towards the HIP.
Recovery of surface texture
Any real or virtual surface can be represented as a sum of a general shape or envelope of the surface and the minute surface variations called texture. The latter provides a realistic feel to an otherwise slippery envelope. Hence the texture component of the surface can be extracted by subtracting the envelope (low frequency component) from the surface as illustrated in Fig. 3. The figure shows how a textured Monge surface can be represented as a combination of a macro profile (general geometry or shape) and micro profile (texture). In this work, a bilateral filter is used for the purpose of envelope subtraction. Bilateral filtering [20] provides simple and non-iterative edge-preserving data smoothing. The bilateral filter takes a weighted sum of the depth map at a local neighbourhood at the lattice; the weights depend on both the spatial distance and similarity in depth values. In this way, edges are preserved well while "noise" is averaged out. The bilateral filtered output f b (x, y) of the depth data f (x, y) is obtained from pixels (x,ỹ) in the neighbourhood as shown in the following equation.
f b (x, y) = 1 W (x, y) x ỹ G σ S (x −x, y −ỹ)G σ R (f (x, y) − f (x,ỹ))f (x,ỹ) (3)
where G(x, y) is a Gaussian kernel and σ R and σ S represent the spread in amplitude values and spatial distances, respectively. The term W (x, y) is a normalization factor. Although this is a non-linear filter, computationally efficient algorithms exist to obtain the filtered output [20]. The bilateral filtered output f b (x, y) of the 3-D object is subtracted from the depth data f (x, y) so as to obtain the texture component alone. Hence Fig. 3.1(a) shows the depth image of a 3-D model and Fig. 3.1(b) shows the extracted texture from the depth image using a bilateral filter. As explained earlier, bilateral filter provides edge-preserving smoothing and hence its output is the smoothed depth data without texture details. Haptic rendering techniques that do not consider the surface friction induce the feeling of a very smooth and slippery surface which is not the case in practice. Adding friction along with the texture details to the model provides a more realistic feeling of the surface. Let f be the reaction force on the haptic device as shown in Fig. 5. The component of the applied force normal to the surface f n is given by |f | cos β. Similarly the tangential force on the surface is |f | sin β. The magnitude of the retarding force on the proxy is proportional to the normal force f n . If µ s denotes the static friction coefficient, proxy is in static contact with the surface as long as |f t | < µ s |f n |. During haptic interaction, dynamic friction exerts a retarding force on the proxy while moving on the surface of the object. µ d which denotes the coefficient of dynamic friction is made proportional to the resultant curvature of texture component at each point of the rendered surface since friction depends on the surface property of the material. Since the curvature encodes the unevenness of a surface very well, we use µ d proportional to the curvature. Now, the magnitude of the retarding force is given by µ d |f n | and is in a direction opposite to f t . The curvature is computed as the resultant of mean and Gaussian curvatures whose magnitudes are given by the following equations. The physical significance of mean curvature(H) is the first variation of the surface area and Gaussian curvature(K) represents the local convexity. Hence the use of resultant curvature as a representative of µ d can be justified as it represents the amount of "bending" at each point on the surface [23].
h(x, y) = f (x, y) − f b (x, y).(4)
Incorporation of surface friction
H = h xx (1 + h 2 y ) + h yy (1 + h 2 x ) − 2h xy h x h y 2(1 + h 2 x + h 2 y ) 3 2 (5) K = h xx h yy − h 2 xy (1 + h 2 x + h 2 y ) 2(6)
where the parameters h x , h y are first partial derivatives of the texture component h(x, y) of the surface w.r.t x and y axis. Similarly h xx , h yy are second partial derivatives of h(x, y) w.r.t x and y. The dynamic friction coefficient µ d is computed as follows:
µ d = 1 R √ H 2 + K 2(7)
where R is the radius of the largest inscribable sphere within the haptic work space. Since R is much larger than the radius of curvature in the micro texture, usually µ d < 1.
The resultant force f r on the proxy is given by
f t − µ d f n . f r = f t (1 − µ d cot β) if |f t | ≥ µ s |f n | = 0 otherwise(8)
Equation 8 provides a direct description of the frictional force. For proxy based haptic rendering, such an equation can easily be incorporated while defining the proxy movement. Comparing equations (8) with (2), we observe that f r is proportional to v t . Hence the parameter ρ with friction is given by ρ f = ρ(1 − µ d cot β). Then the proxy update equation is given by:
X (k+1) p = X (k) p + ρ(1−µ d cot β k )v (k) t(9)
where k denotes the iteration number.
Multimodal Rendering
Haptic rendering involves generating force feedback in order to provide the sensation of touch to the users. Any haptic rendering algorithm would include the following two steps:
1. detection of collision of the HIP with the object.
computation of force feedback if a collision is detected.
If z p < f (x p , y p ) in Fig. 3(b) then the proxy has touched the object and a force needs to be fed back to the user through the haptic device. Subsequently as explained in section 2, the reaction force is computed as F = −kx where k is the Hooke's constant, and x is the current penetration depth given by x = X h − X p , where X h is the HIP position and X p is the proxy position.
For a combined hapto-visual rendering, the 3-D object surface is displayed as a simple quad mesh formed out of the depth values in OpenGL. We have opted for the mesh-based graphical display in order to give a better perception to the viewer since using point cloud data for graphic display would result in gaps in the visually rendered image. We have used the stereoscopic display technique for creating the effect of depth in the image by presenting two offset images on the screen corresponding to two different points of projection. This provides a 3-D perception to the user. Anaglyphic glasses can be used to view such displays. In practice, 3-D objects come at various physical scales and orientations. In a virtual environment, one should be able to experience objects of all sizes at different scales to get a sense of overall structure to finer details from the same data set. Hence, we have implemented adaptive scaling in both graphical and haptic domains. In order to scale the surface we resize depth data of resolution N × N depending on the level user selects, with N × N as the finest level. If we load the level N × N into the haptic space the full object can be rendered visually as well as haptically. Users can select the level as well as the region of interest at run time either using buttons in the haptic device or using keyboard functions. Additionally, we have developed a graphical user interface for easy acessibility. Fig. 6 illustrates the selection of scale and orientation by the user. Fig. 6(b) shows the rotated version of Fig. 6(a). Fig. 6(c) illustrates how user can select different levels of details and the corresponding textural component is shown.
Rendering at Different Scales and Orientations
The user can select the desired orientation before the depth scan using either the mouse controls or using the graphical user interface. Further, depending on the scale selected by the user, the corresponding depth data is dynamically loaded into the active haptic space and an appropriate haptic force is rendered. As only a limited subset of data is loaded, the rendering is very fast. In general, at higher levels of resolution, the user should be able to view finer details. The haptic force also varies accordingly. Hence in order to incorporate realistic haptic and graphic perception, we need to appropriately scale the depth values at each level of depth map. Further, trying to map a large physical dimension over a small haptic work space (typically about 4 inch cube of active space) leads to a lot of unwanted vibrations (something similar in concept to aliasing) during rendering. Hence the depth values need to be smoothed before being down-sampled and mapped into the haptic work space. Multi-scale data generation for procuring different levels of details is performed based on [4]. Sounds are incorporated in the rendering framework by playing appropriate audio files based on the position of haptic probe inside the virtual environment. A demonstration of this technique is given in the attached video using the 3-D model of musical pillars at Vitthala temple, Hampi which is an early 14 th century world heritage site. The pillars in this temple have musical columns which produces distinct sounds when struck. The temple consists of 56 pillars which are monolithic sculptures each having granite stone columns of height 10 feet as shown in Fig. 4.2(a). Each major pillar is surrounded by 7 minor pillars that can reverberate at 7 primary notes of Indian classical music. We have used the recorded audio files of these pillars (data courtesy: www.daiict.com) in our work. A sample audio file is shown in Fig. 4.2(b). A single wavelength of the sound file was extracted from the above audio file and was played back whenever the haptic probe touched the virtual pillar. During audio rendering, one could synthesize various types of sounds synthetically. However, we provide the real data from the actual heritage site to provide a real feel of the musical pillars. During haptic rendering whenever a collision takes place for the first time with a specific pillar, the corresponding note is played, the volume of which is made proportional to the rendered force.
Audio Rendering
Results
The proposed method was implemented in visual C++ in a Windows XP platform with an Intel i5 CPU @ 2.66 GHZ with 2 GB RAM. For obtaining texture details, the depth data obtained from OpenGL depthbuffer is fed to a bilateral filter using OpenCV inbuilt functions and the output from the filter is subtracted from the original depth image. We have experimented with various models of 3-D objects and a few of them are displayed below. Fig. 8 shows the model of Ganesh, visually rendered in OpenGL. For haptic rendering we use HAPI library. The blue ball represents the position of the proxy constrained to lie on the surface. The discrete position in the model is displayed in a fixed 200 × 200 haptic space. The size and spatial resolution of the model depend on two factors: the active space of the haptic device used to render the model, and the resolution at which the model should be displayed. We use a 3-DOF haptic device, NOVINT FALCON with a 4 inch cube of active space. While interacting with the object haptically, the average proxy update time was found to be 0.0056 ms which is much faster than the required upper bound of 1 ms, and hence the user has a very smooth haptic experience. The average time required for dynamic data generation and loading it into the haptic space depends on the resolution of input depth data and it was observed to be around 0.5 s and 0.02 s, respectively, for depth data with a resolution of 800 × 800. We also carry out the rendering at finer levels of details by successively zooming into the object. In above cases, each figure consists of two parts where the left part is the reference for the users to select the part of the object they wish to explore haptically as shown in Fig. 6 and Fig. 8. The right part of the figure corresponds to the selected region at the appropriate resolution for haptic rendering. Fig. 8(b) shows the scaled up version of Fig. 8(a). It is quite clear from Fig. 8(c) that the users are able to feel even minute details of the sculpture and have visual perception of closeness in depth. Hence they can have a more realistic experience. Furthermore, the audio-playback feature with respect to the musical pillars augmented the user experience. However, we observe that there is a small time lag between the haptic and audio rendering due to data access time to open the stored audio files. For a small number of musical notes, these audio clips can be stored in RAM to circumvent this problem.
Validation of proposed method
Validation of result is not quite easy during haptic rendering. Authors in [18] propose a validation technique using a standard 3-D sphere model. But, since rendering of friction in our work involves computation of resultant curvature, the use of a spherical model (having a fixed curvature) is not justified. We demonstrate a validation technique using a known sinusoidal surface. Fig. 9 shows the z-component of proxy as a function of time while haptically interacting with the depth data. The green line and the blue line in the figure show the zcomponent of the proxy point with and without texture and friction, respectively. We observe that the proxy position converges, but with a time-lag in the former case, as expected due to surface effects of texture and friction. The shapes of both the curves are quite similar except for the time lag. Fig. 10(a) shows the reaction force versus time during haptic interaction with a known sinusoidal surface without incorporating the surface effects. In free space the HIP and proxy positions are almost the same and hence the reaction force on the haptic device is zero for initial few seconds as illustrated in figure. As the HIP penetrates the object, the proxy stays on the surface according to the iteration method discussed in section 3. The proxy point moves continuously during interaction, whenever there is a change in HIP position and appropriate reaction force is fed back to the haptic device. The force-time graph in Fig. 10(b) illustrates how the net reaction force fed back to the haptic device is effectively delayed by constraining the proxy movement which gives the perception of friction to the user. In Fig. 11, we show the actual set up of our rendering framework. A user wearing the anaglyphic glasses watches the stereoscopic visual rendering of an object and at the same time haptically interacts with the object through the Falcon device. This provides an excellent hapto-visual immersion of the subject into the virtual object. However, for the visually impaired users, the selection of scale and the location for rendering cannot be based on the small navigation window on the screen. For such subjects, we use the buttons available on the haptic device for the user to explore the object at different scales and locations. We also conducted a user survey on the proposed virtual set-up in order to understand how realistically the users can perceive the shape and surface prop- Fig. 11. Illustration of hapto-visual immersion of a subject for a 3-D object. On the left, the user wearing anaglyphic glasses is holding the FALCON haptic device while interacting with the virtual 3-D object displayed on the screen. erties of the virtual models. All subjects were made to sit at a desk with the virtual environment displayed in front of them. Then they were asked to explore the virtual system by grasping the haptic device. After an exploration time of 1 minute, each user was asked to rate the realism of the virtual surface in terms points from 1-10 (with 10 being the highest rating). A total of 12 subjects (8 males and 4 females) volunteered for user study. All the users were in the age group of 18 to 40. The survey resulted in an average user rating of 7.23 (out of 10) for the realism of virtual environment. We propose to conduct richer user study in future including distribution and statistical meaning of user ratings.
Conclusions
In this work we have proposed a new technique for multi-modal rendering of 3-D objects represented as a dense depth map data. Rendering of surface properties like texture and friction is found to enrich the user's experience in the virtual world. We include scalability, rotation, translation and stereoscopic display of 3-D models as additional features to enhance the realism in experience. Presently, we have integrated audio rendering with respect to a single spatially segmented 3-D object i.e., the musical pillars at Hampi, India. In future, we propose to include continuous audio rendering in a more generalized framework by annotating the acoustic property at each point in the 3-D model. We conducted experiments with several 3-D models. We also conducted an user survey on a few subjects and observed that hapto-visual and auditory rendering of virtual 3-D models using the proposed method greatly augmented the user's experience. and his team for providing us with the audio signals of musical pillars ate Hampi which is the input to our proposed rendering framework.
Fig. 1 .
1Block diagram illustrating the proposed system. LoD stands for level of details.
Fig. 2 .
2(a) Illustration of tangent vector evaluation using the current proxy and HIP positions. (b) Illustration of surface approximation from depth values.
Fig. 3 .
3Envelope subtraction from a textured surface.
Fig. 4 .
4Illustration of texture retrieval from depth data using a bilateral filter:(a) Depth image corresponding to 3-D model of Buddha (b) Extracted texture details (Data Courtesy: www.archibaseplanet.com).
Fig. 5 .
5Illustration of calculation of the resultant vector for the proxy movement while incorporating surface friction.
Fig. 6 .
6Illustration of user defined selection of orientation (before depth scan): 3-D model of Buddha (a) before rotation (b) after rotation. (c) Illustration of user defined selection of level of details (at run-time) for another model(Data Courtesy:www.archibaseplanet.com).
Fig. 7 .
7(a) Front view of Vitthala temple, Hampi(courtesy:[13]). (b) Plot of an audio file recorded from the musical pillar. (courtesy:[13]).
Fig. 8 .
83-D model of a statue with texture: at (a) lowest level of details (b) at double the resolution and (c) at the finest resolution. (Data Courtesy: www.archibaseplanet.com).
Fig. 9 .Fig. 10 .
910Plot of z-position of proxy as a function of time to illustrate the effect of surface friction. Force-vs-time graph for a particular interaction with the depth data (a) without surface effects and (b) with surface effects.
Acknowledgement. The authors would like to thank DST for the grant provided on the Indian Digital Heritage Project and MCIT for the grant on perception engineering. The authors would also like to thank Prof. Manjunath Joshi
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| []
|
[
"Nonlinear excitation of photonikos and plasmons by high-power, short pulse lasers",
"Nonlinear excitation of photonikos and plasmons by high-power, short pulse lasers"
]
| [
"Levan N Tsintsadze \nGraduate School of Pure and Applied Sciences\nDepartment of Plasma Physics, E.Andronikashvii Institute of Physics\nUniversity of Tsukuba\nTsukuba, TbilisiJapan, Georgia\n",
"Hitoshi Hojo \nGraduate School of Pure and Applied Sciences\nDepartment of Plasma Physics, E.Andronikashvii Institute of Physics\nUniversity of Tsukuba\nTsukuba, TbilisiJapan, Georgia\n"
]
| [
"Graduate School of Pure and Applied Sciences\nDepartment of Plasma Physics, E.Andronikashvii Institute of Physics\nUniversity of Tsukuba\nTsukuba, TbilisiJapan, Georgia",
"Graduate School of Pure and Applied Sciences\nDepartment of Plasma Physics, E.Andronikashvii Institute of Physics\nUniversity of Tsukuba\nTsukuba, TbilisiJapan, Georgia"
]
| []
| Modulational excitation of longitudinal photons (photonikos) and electron Langmuir waves, as well as ion sound waves by an incoherent strong and superstrong radiation (high-power short pulse lasers, non-thermal equilibrium cosmic field radiation, etc.) in plasmas are investigated. A simultaneous generation of photonikos and plasmons are demonstrated. Furthermore, the kinetic instability is considered when a low frequency photonikos are generated alone. Growth rates of these new modes are obtained. | 10.1103/physreve.78.065401 | [
"https://arxiv.org/pdf/0810.5180v1.pdf"
]
| 28,006,531 | 0810.5180 | 64bc2677a9e530a311d85762ea157ec2c4527bd0 |
Nonlinear excitation of photonikos and plasmons by high-power, short pulse lasers
29 Oct 2008 (Dated: October 29, 2008)
Levan N Tsintsadze
Graduate School of Pure and Applied Sciences
Department of Plasma Physics, E.Andronikashvii Institute of Physics
University of Tsukuba
Tsukuba, TbilisiJapan, Georgia
Hitoshi Hojo
Graduate School of Pure and Applied Sciences
Department of Plasma Physics, E.Andronikashvii Institute of Physics
University of Tsukuba
Tsukuba, TbilisiJapan, Georgia
Nonlinear excitation of photonikos and plasmons by high-power, short pulse lasers
29 Oct 2008 (Dated: October 29, 2008)
Modulational excitation of longitudinal photons (photonikos) and electron Langmuir waves, as well as ion sound waves by an incoherent strong and superstrong radiation (high-power short pulse lasers, non-thermal equilibrium cosmic field radiation, etc.) in plasmas are investigated. A simultaneous generation of photonikos and plasmons are demonstrated. Furthermore, the kinetic instability is considered when a low frequency photonikos are generated alone. Growth rates of these new modes are obtained.
Electromagnetic (EM) radiation in plasma is a fundamental physical system which has played a crucial role in opening up new frontiers in physics, such as the fast ignition in laser fusion [1], plasma based high-energy particle acceleration [2], [3], the electron-positron and the neutrino-antineutrino pair production [4], [5], [6], [7], nonlinear optics [8], optically induced nuclear fission [9], etc. One of the most salient phenomenon in the above mentioned problems of laboratory plasmas, as well as astrophysical and space plasmas are the relativistic parametric and modulational instabilities. Which gained prominence largely due to its importance in connection with the strongly nonlinear structures in plasmas, the deposition of EM energy through different modes into the plasma, heating and/or acceleration of plasma particles to very high energies. The relativistic parametric instabilities has a long history starting with the work of Tsintsadze [10]. Since then a great number of theoretical and simulation results were reported [11]. Recent progress in development of high-power, short pulse lasers has renewed the interest in this phenomenon [12], [13], [14], [15]. The above treatments where restricted to the case of monochromatic EM waves. For ultrashort pulses the bandwith of coherent wave is increasingly broad. Even if the bandwith may be initially narrow, its spectrum may eventually broaden, either as a result of several kinds of instability processes, or as the result of other nonlinear wave-wave interaction processes.
In order to study the interaction of spectrally broad relativistically intense EM waves with a plasma, in previous papers [16], [17] starting from the fully relativistic equations, we have derived a general kinetic equation for the photon gas incorporating two forces of distinct nature. One force appears due to the redistribution of electrons in space, ∇n e , and time, ∂ne ∂t . The other force arises by the variation of the shape of wavepacket. In other words, this force originates from the alteration of the average kinetic energy of the electron oscillating in a rapidly varying field of EM waves, and is proportional to ∇ 1 γ and ∂ ∂t 1 γ , where γ is the relativistic Lorentz factor. In the field of superstrong femtosecond pulses, it is expected that the character of the nonlinear response of medium would radically change. At high intensities the motion of free electrons near the focal volume would be extremely relativistic. Thus, the relativistic nonlinear effect, which is basically associated with the increase in the electron mass, will tend to determine the dynamics of EM pulses. Currently, lasers produce pulses whose intensity approaches 10 22 W/cm 2 [18]. At these intensities, the highly nonlinear and complicated relativistic dynamics of the laser-plasma system gives rise to a number of interesting phenomena, for example the Bose-Einstein condensation (BEC) and a new intermediate state of the photon gas [19]. It was shown in Refs. [20] that the behavior of photons in a plasma is radically different from the one in a vacuum. Namely, plasma particles perform oscillatory motion in the field of EM waves affecting the radiation field. Photons acquire the rest mass, m γ , and become one of the Bosons in plasmas and posses all characteristics of nonzero rest mass, i.e., we may say that the photon is the elementary particle of the optical field. This difference leads to certain novel phenomenon, such as photonikos [21], [22], [23], which originates from the decay process. Namely, the photon passing through the photon bunch absorbs and emits photonikos, with frequencies Ω = ∓(ω−ω′) and wavevectors q = ∓( k− k′). In Refs. [19], [21] we have shown that under certain conditions the photon-photon interactions dominate the photon-plasma particle interactions. So that under such conditions the variation of the plasma density can be neglected in comparison with the variation of the photon density. In recent 2 dimensional fully relativistic particle-in-cell simulations (Fourier space code -EPIC3D-AP) [24], which is based on potential form, the simultaneous emission of photonikos and plasmons by the laser pulse were observed, as well as the low-frequency photons that can be associated with BEC photons were seen. Thus, it is of interest to examine these new modes.
In this letter, we study analytically the nonlinear excitation of photonikos and plasmons by high-power, short pulse lasers. As we will see in the following there are cases, when both photonikos and plasmons are simultaneously generated, that confirm the results of recent simulations [24]. For our purpose, we employ the general dispersion relation obtained in Refs. [16], [17], which reads
ε 1 + ω 2 Le 2γ 2 • dk (2π) 3 P + • − P − • qkc 2 − Ωω(k) + (1 + δε i )δε e q 2 c 2 2γ 2 • dk (2π) 3 P + • − P − • qkc 2 − Ωω(k) = 0 ,(1)
where
ε = 1 + δε e + δε i , δε α = 4πe 2 q 2 (q∂f •α /∂p) Ω − qv α dp, ω 2 Lα = 4πe 2 n 0e m 0α γ 0 ,(2)γ 0 = 1 + Q 0 = 1 + 2 dk (2π) 3 P 0 and P 0 = e 2 A(k)A * (k) (m 0e c 2 ) 2
is the spectral f unction ,
Eq.(1) rewrite as
ε 1 + 1 A + (1 + δε i )δε e q 2 c 2 ω 2 Le = 0 ,(3)
where
A = ω 2 Le 2γ 2 • dk (2π) 3 P + • − P − • qkc 2 − Ωω(k) = ω 2 Le 2γ 2 • dk (2π) 3 P 0 1 Ω − qu + q 2 c 2 /2ω − 1 Ω − qu − q 2 c 2 /2ω − (4) ıπ δ(Ω − qu + q 2 c 2 2ω ) − δ(Ω − qu − q 2 c 2 2ω
) .
If A ≫ 1, then Eq.(3) reduces to Eq.(20) of Ref. [16].
For the photon gas we use the spectral Gaussian distribution function
P 0 = Q 0 (2πσ 2 k ) −3/2 exp − (k − k 0 ) 2 2σ 2 k ,(5)
where
Q 0 = e 2 | A(k 0 ) | 2 (m 0e c 2 ) 2 = γ 2 0 − 1 .
Note that if there is no variation of the plasma density δn α = 0, δε α = 0, then we have the equation that was studied in Ref. [21]. We now rewrite Eq.(3) taking into account poles in the integrals
Ω − qu = 0 ,(6)Ω − qv α = 0 ,(7)
where u = kc 2 ω(k) .
Using the well known relation
lim ε→0 1 x + ıε = ℘ 1 x − ıπδ(x) ,(8)
and recalling [21] A = ReA + iImA = A 0 + ıA 1 , (9) δε α = δε ′ α + ıδε ′′ α , and Ω = Ω ′ + ıΩ ′′ and assuming that A 0 ≫| A 1 |, and | δε ′ α |≫| δε ′′ α |, we rewrite Eq.(3) as
ε 1 + 1 A 0 − ı A 1 A 2 0 + (1 + δε i )δε e q 2 c 2 ω 2 Le = 0 .(10)
We first consider the excitation of the longitudinal photons and electron Langmuir waves (Ω ′ ∼ ω Le , and Ω ≫ ω Li , or δε i = 0). In this case
ε = 1 + δε ′ e + ıδε ′′ e = 1 − ω 2 Le (1 + 3q 2 r 2 De ) Ω 2 − ı 4π 2 e 2 m 0e q 2 ∂f 0e ∂p x vx=Ω/q ,(11)1 + 1 A 0 = 1 q 2 V 2 E (Ω − qu) 2 − q 2 (V 2 s − V 2 E ) − α 2 q 4 ,(12)A 1 = − π 2 ω 3 (σ k c) 3 V 2 E c 2 (Ω − qu) qc exp − 3 2 (Ω − qu) 2 q 2 V 2 s ,(13)
where
V 2 E = c 2 2 ω Le ω 2 γ 2 0 − 1 γ 2 0 , V s = c √ 3 σ k c ω(k 0 ) , α = c 2 2ω(k 0 ) ,(14)
and r De is the Debay length for electrons. We now neglect the small imaginary term in Eq.(10) and examine the following dispersion relation
ε ′ e 1 + 1 A 0 + q 2 c 2 ω 2 Le δε ′ e = 0 ,(15)
or more explicitly
Ω 2 − ω 2 Le (1 + 3q 2 r 2 De ) (Ω − qu) 2 − q 2 U 2 − α 2 q 4 = q 4 V 2 E c 2 (1 + 3q 2 r 2 De ) ,(16)
where
U 2 = V 2 s − V 2 E , u(k 0 ) = k 0 c 2 ω(k 0 ) , qu = qucosΘ .
The maximum growth rate we obtain if the density energy of photons compensate the kinetic "thermal" energy of photons and diffraction term, i.e.,
V 2 s + α 2 q 2 = V 2 E(17)
and from Eq. (16) for Ω = ω Le (1 + 3 2 q 2 r 2 De ) + δ ≈ qu + δ, we obtain
δ 3 = 1 2ω Le q 4 V 2 E c 2 (1 + 3q 2 r 2 De ) 1/2 ,(18)
or
Imδ = √ 3 2 qc 2ω Le V 2 E c 2 (1 + 3q 2 r 2 De ) 1/2 1/3 qc .(19)
If the relation (17) does not hold, then Eq. (16) has the unstable solution such as Ω = ω Le 1 + 3q 2 r 2 De + δ and Ω = qu − q U 2 + α 2 q 2 + δ .
From Eq.(16) follows for the imaginary part of δ Imδ = qV E 2ω Le qc qu ω Le 1 + 3q 2 r 2
De − 1 −1/2 .(21)
Note that qu > ω Le 1 + 3q 2 r 2 De always, as follows from relation (20). Eqs. (19) and (21) demonstrate the modulational excitation of photonikos and plasmons simultaneously.
We now consider the kinetic instability for the case when q 2 v 2 tre < Ω ′2 < ω 2 Le , which means that in this case the low frequency photonikos are generated alone. From Eq.(10) a simple calculation gives for the imaginary part of Ω(q) the following expression
Ω ′′ = − π 8 (Ω ′ − qu) ω Le Ω ′ ω Le qc 3 V E c 2 ω σ k c 3 exp − 3 2 (Ω ′ − qu) 2 2q 2 V 2 s + Ω ′ Ω ′ qv tre 3 exp − Ω ′2 2q 2 v 2 tre .(22)
From Eq. (22) follows that Ω ′′ changes sing and becomes positive if the Cherenkov condition is satisfied
u > Ω ′ qcosΘ 1 + Ω ′ ω Le qc ω Le 3 c V E 2 σ k c ω 3 exp − Ω ′2 2q 2 v 2 tre .(23)
So that this instability leads to the excitation of photonikos. Here we have supposed that | Ω ′ − qu |< qV s . We next study the range of frequencies
q 2 v tri < Ω ′ < q 2 v tre(24)
and the case when δn i = 0. In this case
ε = ε ′ + ıδε ′′ = 1 + 1 q 2 r 2 De + ı π 2 ω 2 Le Ω (qv tre ) 3 − ω 2 pi Ω 2 .(25)
First, we neglect the small imaginary terms in Eqs. (25) and (4), and use Eq.(10) to obtain the following dispersion relation
(Ω 2 − Ω 2 s ) (Ω − qu) 2 − q 2 (U 2 + α 2 q 2 ) + q 4 c 2 V 2 E (Ω 2 − ω 2 pi ) ω 2 Le (1 + q 2 r 2 De ) = 0 ,(26)
where Ω s = qvs √ 1+q 2 r 2
De is the ion sound frequency.
If the length of waves is shorter than the Debay length, i.e., λ ≪ r De , then Ω s → ω pi , and Eq.(26) reduces to
(Ω − qu) 2 − q 2 (V 2 s + α 2 q 2 ) + q 2 V 2 E 1 + c 2 v 2 tre = 0 .(27)
This relation demonstrates that the ions play no role in the instabilities. Whereas, in the presence of the hot electrons, the growth rate becomes large, because of the coupling term (the last term in Eq. (26)), as c 2 /v 2 tre , and the imaginary part of the frequency is
ImΩ = Ω ′′ ≃ qV E c v tre .
In the opposite case, i.e., q 2 r 2 De ≪ 1, we have the modulational excitation of photonikos and ion sound waves simultaneously. That is the photon flux triggers the both waves. To show this, we first consider the case when the condition (17) is satisfied. In this case for the coincide roots Ω = Ω s + δ = qu + δ, where Ω s ≫| δ |, we obtain from Eq.(26)
Imδ = √ 3 2 qc m e 2m i V 2 E cv s 1/3 .(28)
Next, if the relation (17) does not hold, then Eq.(26) has the unstable solution such as Ω = Ω s + δ and Ω = qu − q U 2 + α 2 q 2 + δ
and we get the growth rate from Eq. (26) as
Imδ = qc 2 V E v tre 1 (1 + q 2 r 2 De ) 1/2 1 (qu/Ω s − 1) 1/2 .(30)
Which is true for the Cherenkov condition u > Ωs qcosΘ = vs cosΘ . Furthermore, the equation (30) is valid when V E ≪ v tre , since Ω s = qv s > Imδ.
To summarize, we have shown a simultaneous nonlinear excitation of photonikos and plasmons in the interaction of relativistically intense nonmonochromatic radiation bunches with a nonmagnetized plasma. Cases without and with ion dynamics are discussed. The generation of only low frequency photonikos is also confirmed. The growth rates of these new modes, which have no counterpart in the case of monochromatic EM waves, are obtained. These investigations may play an essential role in advanced fusion concepts and advanced accelerators, as well as for the description of extremely complex phenomena that appear only in energetic astrophysical systems and in experiments modeling the high energy density astrophysics in the laboratory. It was argued in Ref. [25] that the m γ would lead to a catastrophic emission of longitudinal photons. This gas may play the decisive role in the expansion of the Universe. In addition in the fireball model of Gamma-ray bursts, the afterglow may be due to the decay process discussed in this and the previous papers [19], [21]. A cursory examination of burst profiles indicates that some are chaotic and spiky with large fluctuations on all time scales, while others show rather simple structures with few peaks. However, some bursts are seen with both characteristics present within the same burst [26].
* Also at Department of Plasma Physics, E.Andronikashvii Institute of Physics, Tbilisi, Georgia
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| []
|
[
"Effects of Li doping on H-diffusion in MgH 2 : a first-principles study",
"Effects of Li doping on H-diffusion in MgH 2 : a first-principles study"
]
| [
"Wenmei Ming \nDepartment of Materials Science and Engineering\nUniversity of Utah\n84112Salt Lake CityUTUSA\n",
"Zhigang Zak Fang \nDepartment of Metallurgical Engineering\nUniversity of Utah\n135 South, Room 4121460, 84112-0114East, Salt Lake City, Utah\n",
"Feng Liu \nDepartment of Materials Science and Engineering\nUniversity of Utah\n84112Salt Lake CityUTUSA\n"
]
| [
"Department of Materials Science and Engineering\nUniversity of Utah\n84112Salt Lake CityUTUSA",
"Department of Metallurgical Engineering\nUniversity of Utah\n135 South, Room 4121460, 84112-0114East, Salt Lake City, Utah",
"Department of Materials Science and Engineering\nUniversity of Utah\n84112Salt Lake CityUTUSA"
]
| []
| The effects of Li doping in MgH 2 on H-diffusion process are investigated, using first-principles calculations. We have identified two key effects: (1) The concentration of H vacancy in the +1 charge state (V +1 H ) can increase by several orders of magnitude upon Li doping, which significantly increases the vacancy mediated H diffusion rate. It is caused by the preferred charge states of substitutional Li in the −1 state (Li −1 Mg ) and of interstitial Li in the +1 state (Li +1 i ), which indirectly reduce the formation energy of V +1 H by up to 0.39 eV depending on the position of Fermi energy. (2) The interaction between V +1 H and Li −1 Mg is found to be attractive with a binding energy of 0.55 eV, which immobilizes the V +1 H next to Li −1 Mg at high Li doping concentration. As a result, the competition between these two effects leads to large enhancement of H diffusion at low Li doping concentration due to the increased H-vacancy concentration, but only limited enhancement at high Li concentration due to the immobilization of H vacancies by too many Li. | 10.1063/1.4853055 | [
"https://arxiv.org/pdf/1312.1620v1.pdf"
]
| 18,679,503 | 1312.1620 | 2c573a3139cd79bf66c07478aee462e7a1b7f166 |
Effects of Li doping on H-diffusion in MgH 2 : a first-principles study
5 Dec 2013
Wenmei Ming
Department of Materials Science and Engineering
University of Utah
84112Salt Lake CityUTUSA
Zhigang Zak Fang
Department of Metallurgical Engineering
University of Utah
135 South, Room 4121460, 84112-0114East, Salt Lake City, Utah
Feng Liu
Department of Materials Science and Engineering
University of Utah
84112Salt Lake CityUTUSA
Effects of Li doping on H-diffusion in MgH 2 : a first-principles study
5 Dec 2013
The effects of Li doping in MgH 2 on H-diffusion process are investigated, using first-principles calculations. We have identified two key effects: (1) The concentration of H vacancy in the +1 charge state (V +1 H ) can increase by several orders of magnitude upon Li doping, which significantly increases the vacancy mediated H diffusion rate. It is caused by the preferred charge states of substitutional Li in the −1 state (Li −1 Mg ) and of interstitial Li in the +1 state (Li +1 i ), which indirectly reduce the formation energy of V +1 H by up to 0.39 eV depending on the position of Fermi energy. (2) The interaction between V +1 H and Li −1 Mg is found to be attractive with a binding energy of 0.55 eV, which immobilizes the V +1 H next to Li −1 Mg at high Li doping concentration. As a result, the competition between these two effects leads to large enhancement of H diffusion at low Li doping concentration due to the increased H-vacancy concentration, but only limited enhancement at high Li concentration due to the immobilization of H vacancies by too many Li.
I. INTRODUCTION
Light metal hydride MgH 2 is one of the most promising hydrogen-storage materials for on-board clean-fuel application, because it has both high gravimetric (7.7wt%) and volumetric densities (6.7× 10 22 H/cm 3 ) 1 . However, their dehydrogenation process is too slow to be practically useful, and for bulk materials as high as about 300 K above room temperature is required to obtain an equilibrium H 2 pressure of 1 bar 1-4 . Such poor dehydrogenation kinetics is primarily due to the strong ionic bonding between Mg and H and large enthalpy of formation of MgH 2 (∼ 75KJ/molH 2 ), as evidenced by both experiments 5-7 and first-principles calculations 8,9 . Various attempts have been made to help facilitate dehydrogenation process. For example, to improve the kinetics, ball milling processing 10,11 has been used to shorten the diffusion length, doping of transition metals [10][11][12] were adopted to reduce the strength of H-Mg bond, and applying tensile stress was tried to weaken the Mg-H stability 2, 13 .
On the other hand, doping has been known to enhance H diffusion in metal hydrides, which is usually mediated by H vacancy, by inducing a higher concentration of Hvacancy [14][15][16][17][18][19] . For example, Van de Walle et. al recognized that in certain charged state, Zr(Ti) can enhance the dehydrogenation kinetics of NaAlH 4 14 , because the formation energy of H-vacancy is decreased upon doping. In particular, when the H-vacancy is charged, its formation energy depends on the position of Fermi energy; and conversely, selective doping of the hydride with impurities that take different charged states will tune the Fermi energy with respect to the dopant-free system. And the shift of Fermi energy can result in a decrease of H-vacancy formation energy depending on the sign of the H-vacancy charge state. Consequently, the a) Corresponding author. E-mail: [email protected] concentration of H-vacancy will increase to enhance the vacancy-mediated H diffusion.
In this work, we investigated the effects of Li doping on H diffusion in MgH 2 . One important reason that we chose Li is because it is a lighter metal than Mg, so that it will not degrade the high H gravimetric density. We focused on the effects of charge state of Li impurity and H-vacancy, as recognized before in other systems 14 , but also went beyond the previous works by taking into account the effects of interaction between the charged impurities and defects. In many previous studies of the charged-impurity-enhanced H diffusion [14][15][16][17][18][19] , it implicitly assumed no interaction between the dopant and defect. This might be true in the limit of low doping concentration and weak defect-dopant interaction, but unlikely at high doping concentration. Especially, if there is an attractive impurity-defect interaction, such as the binding between the Li-dopant and H-vacancy in MgH 2 as shown by Smith et al. 20 , the impurity may immobilize the H-vacancy, counteracting the enhancement effect of H-vacancy on H diffusion. Therefore, by taking into account the binding between Li and H-vacancy and its dependence on the charge states of Li and H-vacancy, we have systematically studied the effects of Li doping on H diffusion in MgH 2 as a function of Li concentration. We have determined the favored charge states of Li by calculating its formation energy as a function of Fermi energy, the equilibrium concentration of H vacancies by calculating the H vacancy formation energy as a function of Li doping concentration, and the percentage of immobilized H-vacancies by calculating the binding energies between H-vacancy and Li-dopant. We have also calculated the diffusion barrier of H-vacancy in the presence of Li dopant.
II. CALCULATION DETAILS
Our first principles calculations based on density functional theory (DFT) were conducted using projector augment wave pseudopotential (PAW) 21 with the generalized gradient approximation (GGA) 22 to the exchange-correlation functional, as implemented in VASP package 23 . Supercell technique was used to calculate the formation energy of defects and dopants, interaction energy and diffusion barrier. We used a supercell comprised of 3×3×4 primitive MgH 2 rutile unit cells with the dimensions of 13.481 × 13.481 × 12.012Å
3 . 400 eV energy cutoff and 2×2×2 k-mesh were used for wavefuntion expansion and k-space integration, respectively. All the structures were relaxed in terms of internal atomic coordinates using conjugate gradient method until the force exerted on each atom was smaller than 0.005eV /Å 3 . The charged system was simulated by adding to or removing from the system electrons with a compensating uniform opposite charge background. Diffusion barrier was calculated using the nudged elastic band method 24 .
The formation energy E f (X q ) of defect or dopant (X) with charge q was computed according to Ref. 25 :
∆E f (X q ) = E tot (X q ) − E tot (bulk) − i n i µ i +q(E F + E ν + ∆V ). (1)
Where E tot (bulk) and E tot (X q ) are the total energies of supercell for pure MgH 2 and for MgH 2 containing defect or dopant (X q ), respectively. E ν is chosen to be the valence band maximum (VBM) energy. E F is the Fermi energy with respect to E ν . ∆V is additional electrostatic energy alignment due to different energy references between the defect-containing structure and defect-free structure. i denotes H-defect or dopant Li and n i is the number of species i in the supercell. µ i is the chemical potential of species i. In low concentration limit, the equilibrium defect concentration can be related to the formation energy using:
C = N exp(−∆E f /k B T )(2)
N is the number of sites that can be occupied by defect, k B is Boltzman constant and T is temperature in K. For the chemical potential µ i , the externally added dopant Li is assumed to have its bulk chemical potential E Li (bulk). The chemical potential of H, µ H is in be-
tween 1 2 E(H 2 ) + 1 2 ∆H f (M gH 2 ) (
III. RESULTS AND DISCUSSION
First we calculated the formation energy of native defects: H-vacancy (V H with charge -1, 0, +1) and interstitial H (H i with charge -1, 0, +1). The preferred defects are V +1 H and V −1 H in H-poor condition, and V +1 H and H −1 i in H-rich condition, respectively. Charge neutrality condition requires Fermi-energy to be 2.85 eV and 2.65 eV for the H-poor condition and H-poor condition, respectively. These results are in good agreement with those for MgH 2 in Ref. 17 . In Table I we give an estimate of the concentration for the favored H-defects from equation (2). In order to study how the formation energies of the Hrelated defects are affected by Li doping, we then calculated the formation energy for both substitutional Li configuration (Li Mg ) and interstitial Li configuration (Li i ) in the (-1, 0, +1) charge states. As shown in Figure 1 for both H-poor and H-rich conditions, Li −1 Mg is more stable than Li 0
Mg and Li +1 Mg in almost the whole range of Fermi energy in the gap except very close to the VBM. While Li +1 i is more stable than other two charge states in almost the whole range of Fermi energy in the gap except very close to conduction band minimum (CBM). This indicates that the defect level remains close to the VBM and CBM for Li Mg and Li i , respectively (see Ref. 26 for similar behavior of native defects in anatase TiO 2 ). Under the charge-neutrality condition, the Fermi energy of the Li-doped system (vertical solid lines) is shifted to the left by 0.25 eV (Fig. 1(a)) and 0.39 eV ( Fig. 1(b)) with respect to the Fermi energy of the undoped system (vertical dashed lines) for the H-poor and H-rich condition, respectively, assuming the concentration of dopant Li is much higher than that of H-defects so that the Li +1 Table I, at 400K the concentration of V +1 H in the Li-doped system is 1.40×10 3 and 8.12×10 4 times larger than that in the undoped system under the H-poor and H-rich condition, respectively. On the contrary, the concentration of H −1 i in the Li-doped system is ∼10 5 and ∼ 5×10 7 times lower than that in the undoped system under the H-poor and H-rich condition, respectively.
A previous calculation 18 showed that in the undoped We note that we have neglected entropy contribution in our analysis. Usually, this is a good approximation because the contribution due to the entropy difference is much smaller than the contribution due to the total energy difference. Of course, more accurate results can be obtained by calculating the phonon spectra of all the MgH 2 systems and H 2 . On the other hand, for the MgH 2 system we consider, it has been shown that even though H has a low mass, but the vibrational entropies for H in the lattice and in the H 2 reservoir are rather similar and hence the net entropy difference is small 14 . Also we have used a relatively large supercell dimension so that the added defect charge density in the supercell is very low. Consequently, the interaction energy between the charged defects in the neighboring cells is expected to be sufficiently small, not to affect our conclusion.
The results above suggest the dominating defect and dopant species to be V + H , Li −1 Mg and Li +1 i . However, we didn't consider the interaction between V + H and Li −1 Mg . Next, we calculated the attractive interaction energy between V + H and Li −1 Mg as a function of their separation as shown in Fig. 2. We didn't consider the interaction between V + H and Li +1 i because it is repulsive. Two key features are found in Fig. 2(b): (1) V + H prefers to sit in one of the six nearest-neighbor H-sites (site 1 and site 2 in Fig. 2(a)) of Li with binding energy of 0.50-0.55 eV; (2) Once beyond the nearest-neighbor H-site, their attraction decays rapidly to be insignificant. Based on this observation, we propose a nearest-neighbor interaction model to determine how many V + H being trapped by Li −1
Mg as a function of Li doping concentration. We assume that the interaction energy is ∆E b = -0.55 eV when V + H is in any of the six nearest-neighbor sites and negligible otherwise. Following the Boltzmann distribution 27 we have
R trapped = 3n exp[−∆E b /k B T ] [2N − 3n] + 3n exp[−∆E b /k B T ] .(3)
Where R trapped is the ratio of the number of trapped V + H to the total number of V + H , n is the number of doped Li and N is the number of Mg sites. The number of substitutional and interstitial Li are taken to the same under the charge-neutrality condition, as shown in Fig. 1. Fig. 2(c) shows the calculated R trapped as a function of Li doping concentration. We see that even in the low concentration (for example, n N = 1 × 10 −4 ), the trapping ratio R trapped is very close to one, indicating that almost all the V + H next to Li are immobilized due to their attractive interaction. This also indicates that H vacancy prefers to stay next to Li −1 Mg , because its formation energy is effectively decreased by 0.55 eV.
Furthermore, we studied kinetically how H-vacancy diffusion is affected by the presence of Li Mg through the calculation of diffusion barriers. In Fig. 3, we show the barriers for the H-vacancy diffusing from the nearestneighbor sites of Li (sites 1 and 2 in Fig. 3(a) ) to its closest H site (sites 4 and 5) (path 1) and from the next nearest-neighbor site (site 3) to its closet H site (site 5) (path 2). For the path 1, the diffusion barrier is found to increase by 0.15 eV compared to that in the undoped MgH 2 . For the path 2, the diffusion barrier is found only ∼30 meV higher than that in the undoped MgH 2 . This strong site dependence of H-vacancy diffusion barrier is consistent with the fast decay of the attractive interaction between V H and Li Mg as shown in Fig. 2(b). The 0.15 eV increase of diffusion barrier, together with the large V H trapping ratio suggest that H vacancies will mostly be immobilized in the vicinity of Li dopants, inhibiting the V H mediated H diffusion.
IV. CONCLUSIONS
In conclusion, we have investigated the effects of Lidoping in MgH 2 on the H-vacancy meditated H-diffusion, using DFT calculations. The formation energy calculation shows that the Li dopant favors two charged configurations of Li −1 Mg and Li +1 i . The charge neutrality condition requires the Fermi energy be shifted towards the VBM by 0.25 eV and 0.39 eV upon Li doping under the H-poor and H-rich conditions, respectively, which decreases the formation energy of V + H by the same amount. This leads to an increase of V + H concentration by up to about 5 orders of magnitude at T=400 K. Furthermore, the calculations of interaction energy between V +1 H and Li −1 Mg as well as diffusion barrier of H vacancy in the presence of Li show that almost all the H-vacancy next to Li are immobilized. Therefore, the H-diffusion is enhanced by Li doping in MgH 2 only at the low Li doping concentration but not at the high concentration.
V. ACKNOWLEDGEMENT
This work was supported by NSF MRSEC (Grant No. DMR-1121252) and DOE-BES (Grant No. DE-FG02-04ER46148). We thank the CHPC at University of Utah and NERSC for providing the computing resources.
FIG. 1 .
1Formation energy of Li-dopant in MgH2:(a) H-poor condition. (b) H-rich condition. The vertical solid and dashed lines indicate the Fermi energy in MgH2 with and without Li, respectively. EF ermi=0 eV corresponds to the VBM and EF ermi=3.8 eV corresponds to the CBM.
Mg are the dominant charged dopants to maintain the charge-neutrality condiction. The Fermi energy in both situations is deep inside the band gap. Thus, the thermally excited free carriers in both valence and conduction band are negligible. The consequence of the shift of Fermi energy is that the formation energy of V +1 H is reduced by 0.25 eV and 0.39 eV under the H-poor and H-rich condition, respectively, according to Eq. (1). And the opposite effect happens to V −1 H and H −1 i : their formation energy is increased by 0.25 eV and 0.39 eV, respectively. As shown in
MgH 2 the diffusion barrier of V +1 H is 0.25 eV smaller than that of V −1 H under the H-poor condition, and the diffusion barrier of V +1 H is 0.36 eV higher than that of H −1 i under the H-rich condition. This means that without Li doping, the V +1 H is the dominant diffusing species under the H-poor condition, while the H −1 i is the dominant diffusing species under the H-rich condition. Our calculations show that upon Li doping, the formation energy of V +1 H is decreased by 0.25 eV under that H-poor condition, while that of H −1 iis increased by 0.39 eV under H-rich condition. Because the H-related defect diffusion is determined by the activation barrier, which is the sum of the diffusion barrier and the formation energy. The V +1 H remains the dominant diffusing species under the Hpoor condition because its formation energy is decreased, leading to a lower activation barrier. In contrast, the H −1 i becomes the less favorable diffusing species under the Hrich condition because its formation energy is increased, leading to a higher activation barrier. Consequently, the Li doping makes the V +1 H the dominant diffusion species in the whole range of H chemical potential.
FIG. 2 .
2(color online) (a) The structure of Li-dopant plus H-vacancy with H-vacancy at different positions labeled with number and distance from Li; (b)Interaction energy between V + H and Li −1 M g as a function of their separation distance (in Angstrom); (c) Ratio of the trapped V + H with Li −1 M g to the number of V + H , T=400 K. Green balls are Mg atoms, white balls are H atoms and orange ball is Li dopant.
. 3. (color online) (a) Illustration of different diffusion path for VH to diffuse away further way from Li site. The arrow indicates diffusion direction; (b) VH mediated H-diffusion barrier change with the presence of Li. The purple ball indicates the position of VH in our calculation. The arrow indicates the diffusion path
, Mg and H 2 . ∆H f (M gH 2 ) is enthalpy of formation of MgH 2 , E(H 2 ) is the energy of hydrogen molecule at 0 K. Similarily, the chemical potential of Mg is in between E(bulk Mg) and E(bulk Mg)+∆H f (MgH 2 ). We specifically considered two extreme cases: H poor condition and H rich condition.H-poor condition) and
1
2 E(H 2 ) (H-rich condition), considering thermodynamic
equilibrium between MgH 2
TABLE I .
IThe formation energy(∆ E f ) and concentration(C)
of relevant H-defects without(a) and with(b) dopant Li, at
T=400 K.
H-poor
H-rich
V +
H
V −
H
V +
H
H −
i
∆ E f (eV) a 1.225
1.225
1.358
1.358
C(/cm 3 ) a 2.5×10 7 2.5×10 7
5.3×10 5 5.3×10 5
∆ E f (eV) b 0.975
1.475
0.968
1.748
C(/cm 3 ) b 3.5×10 10 1.772×10 4 4.13×10 10
6.5
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| []
|
[
"Littlewood-Paley-Stein functions for Schrödinger operators",
"Littlewood-Paley-Stein functions for Schrödinger operators",
"Dedicated ; to the memory of Abdelghani Bellouquid (2/2/1966-8/31/2015)"
]
| [
"El Maati Ouhabaz "
]
| []
| []
| We study the boundedness on L p (R d ) of the vertical Littlewood-Paley-Stein functions for Schrödinger operators −∆ + V with nonnegative potentials V . These functions are proved to be bounded on L p for all p ∈ (1, 2). The situation for p > 2 is different. We prove for a class of potentials that the boundedness on L p , for some p > d, holds if and only if V = 0.Mathematics Subject Classification: 42B25, 47F05 | null | [
"https://arxiv.org/pdf/1705.06794v1.pdf"
]
| 119,288,123 | 1705.06794 | 4406e25a3f237663d5e1ef60eae5c291b8fcda1f |
Littlewood-Paley-Stein functions for Schrödinger operators
18 May 2017
El Maati Ouhabaz
Littlewood-Paley-Stein functions for Schrödinger operators
Dedicated ; to the memory of Abdelghani Bellouquid (2/2/1966-8/31/2015)
18 May 2017Schrödinger operatorsLittlewood-Paley-Stein functionsFunc- tional calculus
We study the boundedness on L p (R d ) of the vertical Littlewood-Paley-Stein functions for Schrödinger operators −∆ + V with nonnegative potentials V . These functions are proved to be bounded on L p for all p ∈ (1, 2). The situation for p > 2 is different. We prove for a class of potentials that the boundedness on L p , for some p > d, holds if and only if V = 0.Mathematics Subject Classification: 42B25, 47F05
Introduction
Let L := −∆ + V be a Schrödinger operator with a non-negative potential V . It is the self-adjoint operator associated with the form
a(u, v) := R d ∇u.∇vdx + R d V uvdx with domain D(a) = {u ∈ W 1,2 (R d ), R d V |u| 2 dx < ∞}.
We denote by (e −tL ) t≥0 the semigroup generated by (minus) L on L 2 (R d ).
Since V is nonnegative, it follows from the Trotter product formula that
0 ≤ e −tL f ≤ e t∆ f(1)
for all t ≥ 0 and 0 ≤ f ∈ L 2 (R d ) (all the inequalities are in the a.e. sense). It follows immediately from (1) that the semigroup (e −tL ) t≥0 is sub-Markovian and hence extends to a contraction C 0 -semigroup on L p (R d ) for all p ∈ [1, ∞). We shall also denote by (e −tL ) t≥0 the corresponding semigroup on L p (R d ).
The domination property (1) implies in particular that the corresponding heat kernel of L is pointwise bounded by the Gaussian heat kernel. As a consequence, L has a bounded holomorphic functional calculus on L p (R d ) and even Hörmander type functional calculus (see [6]). This implies the boundedness on L p (R d ) for all p ∈ (1, ∞) of the horizontal Littlewood-Paley-Stein functions:
g L (f )(x) := ∞ 0 t| √ Le −t √ L f (x)| 2 dt 1/2 and h L (f )(x) := ∞ 0 t|Le −tL f (x)| 2 dt 1/2 .
Indeed, these functions are of the form (up to a constant)
S L f (x) = ∞ 0 |ψ(tL)f (x)| 2 dt t 1/2 with ψ(z) = √ ze − √ z for g L and ψ(z) = ze −z for h L .
The boundedness of the holomorphic functional calculus implies the boundedness of S L (see [8]). Thus, g L and h L are bounded on L p (R d ) for all p ∈ (1, ∞) and this holds for every nonnegative potential V ∈ L 1 loc (R d ). Now we define the so-called vertical Littlewood-Paley-Stein functions
G L (f )(x) := ∞ 0 t|∇e −t √ L f (x)| 2 + t| √ V e −t √ L f (x)| 2 dt 1/2 and H L (f )(x) := ∞ 0 |∇e −tL f (x)| 2 + | √ V e −tL f (x)| 2 dt 1/2 .
Note that usually, these two functions are defined without the additional terms t|
√ V e −t √ L f (x)| 2 and | √ V e −tL f (x)| 2 .
The functions G L and H L are very different from g L and h L as we shall see in the last section of this paper. If V = 0 and hence L = −∆ it is a very well known fact that G L and H L are bounded on L p (R d ) for all p ∈ (1, ∞). The Littlewood-Paley-Stein functions are crucial in the study of non-tangential limits of Fatou type and the boundedness of Riesz transforms. We refer to [14]- [16]. For Schrödinger operators, boundedness results on L p (R d ) are proved in [10] for potentials V which satisfy |∇V |
V + ∆V V ∈ L ∞ (R d )
. This is a rather restrictive condition. For elliptic operators in divergence form (without a potential) boundedness results on L p (R d ) for certain values of p are proved in [2]. For the setting of Riemannian manifolds we refer to [4] and [5]. Again the last two papers do not deal with Schrödinger operators.
In this note we prove that G L and H L are bounded on
L p (R d ) for all p ∈ (1, 2] for every nonnegative potential V ∈ L 1 loc (R d ). That is R d ∞ 0 t|∇e −t √ L f (x)| 2 + t| √ V e −t √ L f (x)| 2 dt p/2 dx ≤ C R d |f (x)| p dx and similarly, R d ∞ 0 |∇e −tL f (x)| 2 + | √ V e −tL f (x)| 2 dt p/2 dx ≤ C R d |f (x)| p dx for all f ∈ L p (R d ).
Our arguments of the proof are borrowed from the paper [4] which we adapt to our case in order to take into account the terms with √ V in the definitions of G L and H L . Second we consider the case p > 2 and d ≥ 3. For a wide class of potentials, we prove that if G L (or H L ) is bounded on L p (R d ) for some p > d then V = 0. Here we use some ideas from [7] which deals with the Riesz transform on Riemannian manifolds. In this latter result we could replace
G L by ∞ 0 t|∇e −t √ L f (x)| 2 dt 1/2
and the conclusion remains valid. Many questions of harmonic analysis have been studied for Schrödinger operators. For example, spectral multipliers and Bochner Riesz means [6] and [12] and Riesz transforms [12], [1], [13] and [3]. However little seems to be available in the literature concerning the associated Littlewood-Paley-Stein functions G L and H L . Another reason which motivates the present paper is to understand the Littlewood-Paley-Stein functions for the Hodge de-Rham Laplacian on differential forms. Indeed, Bochner's formula allows to write the Hodge de-Rham Laplacian on 1-differential forms as a Schrödinger operator (with a vector-valued potential). Hence, understanding the Littlewood-Paley-Stein functions for Schrödinger operators L could be a first step in order to consider the Hodge de-Rham Laplacian. Note however that unlike the present case, if the manifold has a negative Ricci curvature part, then the semigroup of the Hodge de-Rham Laplacian does not necessarily act on all L p spaces. Hence the arguments presented in this paper have to be changed considerably. We shall address this problem in a forthcoming paper.
Boundedness on
L p , 1 < p ≤ 2 Recall that L = −∆ + V on L 2 (R d ). We have Theorem 2.1. For every 0 ≤ V ∈ L 1 loc (R d ), G L and H L are bounded on L p (R d ) for all p ∈ (1, 2].
Proof. By the subordination formula
e −t √ L = 1 √ π ∞ 0 e − t 2
4s L e −s s −1/2 ds it follows easily that there exists a positive constant C such that
G L (f )(x) ≤ CH L f (x) (2) for all f ∈ L 1 (R d ) ∩ L ∞ (R d )
and a.e. x ∈ R d . See e.g. [4]. Therefore it is enough to prove boundedness of H L on L p (R d ).
In order to do so, we may consider only nonnegative functions f ∈ L p (R d ).
Indeed, for a general f we write f = f + − f − and since
|∇e −tL (f + − f − )| 2 ≤ 2(|∇e −tL f + | 2 + |∇e −tL f − | 2 ) and | √ V e −tL (f + − f − )| 2 ≤ 2(| √ V e −tL f + | 2 + | √ V e −tL f − | 2 )
we see that it is enough to prove
H L (f + ) p + H L (f − ) p ≤ C p ( f + p + f − p ),
which in turn will imply H L (f ) p ≤ 2C p f p . Now we follow similar arguments as in [4]. Fix a non-trivial 0 ≤ f ∈ L 1 (R d ) ∩ L ∞ (R d ) and set u(t, x) = e −tL f (x). Note that the semigroup (e −tL ) t≥0 is irreducible (see [12], Chapter 4) which means that for each t > 0, u(t, x) > 0 (a.e.). Observe that
( ∂ ∂t + L)u p = (1 − p)V u p − p(p − 1)u p−2 |∇u| 2 .
This implies
p|∇u| 2 + V |u| 2 = − u 2−p p − 1 ( ∂ ∂t + L)u p .(3)
Hence, there exists a positive constant c p such that
(H L (f )(x)) 2 ≤ −c p ∞ 0 u(t, x) 2−p ( ∂ ∂t + L)u(t, x) p dt ≤ c p sup t>0 u(t, x) 2−p J(x) where J(x) = − ∞ 0 ( ∂ ∂t + L)u(t, x) p dt.
The previous estimate uses the fact that ( ∂ ∂t + L)u(t, x) p ≤ 0 which follows from (3). Since the semigroup (e −tL ) t≥0 is sub-Markovian it follows that
sup t>0 e −tL f (x) p ≤ C f p .(4)
The latter estimate is true for all p ∈ (1, ∞), see [15] (p. 73). Therefore, by Hölder's inequality
R d |H L (f )(x)| p dx ≤ c p f p 2 (2−p) p R d J(x)dx p/2 .(5)
On the other hand,
R d J(x)dx = − R d ∞ 0 ( ∂ ∂t + L)u(t, x) p dtdx = f p p − ∞ 0 R d Lu(t, x) p dxdt = f p p − ∞ 0 R d V u(t, x) p dxdt ≤ f p p .
Inserting this in (5) gives
R d |H L (f )(x)| p dx ≤ c p f p p
which proves the theorem since this estimates extends by density to all f ∈ L p (R d ).
Boundedness on L p , p > 2
We assume throughout this section that d ≥ 3. We start with the following result.
Proposition 3.1. Let 0 ≤ V ∈ L 1 loc (R d ). If G L (or H L ) is bounded on L p (R d ) then there exists a constant C > 0 such that ∇e −tL f p ≤ C √ t f p (6)
for all t > 0 and all f ∈ L p (R d ).
Proof. Remember that by (2), if H L is bounded on L p (R d ) then the same holds for G L . Suppose that G L is bounded on L p (R d ). We prove that
∇f p ≤ C L 1/2 f p + Lf 1/2 p f 1/2 p .(7)
The inequality here holds for f in the domain of L, seen as an operator on L p (R d ). 1 In order to do this we follow some arguments from [5]. Set P t := e −t √ L and fix f ∈ L 2 (R d ). By integration by parts,
∇P t f 2 2 = (−∆P t f, P t f ) ≤ (LP t f, P t f ) = L 1/2 P t f 2 2 .
In particular,
∇P t f 2 ≤ C t f 2 → 0 as t → +∞.
The same arguments show that t ∇L 1/2 P t f 2 → 0 as t → +∞. Therefore,
|∇f | 2 = − ∞ 0 d dt |∇P t f | 2 dt = − t d dt |∇P t f | 2 ∞ 0 + ∞ 0 d 2 dt 2 |∇P t f | 2 t dt ≤ ∞ 0 d 2 dt 2 |∇P t f | 2 t dt = 2 ∞ 0 (|∇L 1/2 P t f | 2 + ∇LP t f.∇P t f )t dt =: I 1 + I 2 .
Using the fact that G L is bounded on L p (R d ) it follows that
I 1 p/2 ≤ G L (L 1/2 f ) 2 p ≤ C L 1/2 f 2 p .(8)
By the Cauchy-Schwartz inequality,
|I 2 | ≤ ∞ 0 (|∇LP t f | 2 tdt 1/2 ∞ 0 (|∇P t f | 2 tdt 1/2 ≤ G L (Lf )G L (f ).
Integrating gives
I 2 p/2 p/2 ≤ R d |G L (Lf )| p 1/2 R d |G L (f )| p 1/2 ≤ C Lf p/2 p f p/2 p . (9)
Combining (8) and (9) gives (7) for f ∈ D(L)∩L 2 (R d ). In order to obtain (7) for all f ∈ D(L) we take a sequence f n ∈ L 2 (R d )∩L p (R d ) which converges in the L p -norm to f . We apply (7) to e −tL f n (for t > 0) and then let n → +∞ and t → 0. For f ∈ L p (R d ) we apply (7) to e −tL f and we note that
L 1/2 e −tL f p ≤ C √ t f p and Le −tL f p ≤ C t f p .
Both assertions here follow from the analyticity of the semigroup on L p (R d ) (see [12], Chap. 7). This proves the proposition.
Remark. In the proof we did not use the boundedness of the function G L but only its gradient part, i.e. boundedness on L p (R d ) of the Littlewood-Paley-Stein function:
G(f )(x) = ∞ 0 t|∇e −t √ L f (x)| 2 dt 1/2 .(10)
In the next result we shall need the assumption that there exists ϕ ∈ L ∞ (R d ), ϕ > 0 such that
Lϕ = 0.(11)
The meaning of (11) is e −tL ϕ = ϕ for all t ≥ 0. Note that (11) is satisfied for a wide class of potentials. This is the case for
example if V ∈ L d/2−ε (R d ) ∩ L d/2+ε (R d )
for some ε > 0, see [9]. See also [11] for more results in this direction. Suppose now that V is as in the theorem and G L is bounded on L p (R d ) for some p > d.
Let k t (x, y) be the heat kernel of L, i.e.,
e −tL f (x) = R d k t (x, y)f (y)dy for all f ∈ L 2 (R d ).
As mentioned in the introduction, due to the positivity of V ,
k t (x, y) ≤ 1 (4πt) d/2 e − |x−y| 2 4t .(12)
On the other hand, using the Sobolev inequality (for p > d)
|f (x) − f (x ′ )| ≤ C|x − x ′ | 1− d p ∇f p we have |k t (x, y) − k t (x ′ , y)| ≤ C|x − x ′ | 1− d p ∇k t (., y) p .
Using (12), Proposition 3.1 and the fact that
k t (x, y) = e − t 2 L k t 2 (., y)(x), we have |k t (x, y) − k t (x ′ , y)| ≤ C|x − x ′ | 1− d p t − 1 2 t − d 2 (1− 1 p ) .(13)
Thus, using again (12) we obtain
|k t (x, y) − k t (x ′ , y)| = |k t (x, y) − k t (x ′ , y)| 1/2 |k t (x, y) − k t (x ′ , y)| 1/2 ≤ C|x − x ′ | 1 2 − d 2p t − d 2 + d 4p − 1 4 e − |x−y| 2 8t + e − |x ′ −y| 2 8t . Hence, for x, x ′ ∈ R d |ϕ(x) − ϕ(x ′ )| = |e −tL ϕ(x) − e −tL ϕ(x ′ )| = | R d [k t (x, y) − k t (x ′ , y)]ϕ(y)dy ≤ ϕ ∞ R d |k t (x, y) − k t (x ′ , y)|dy ≤ C|x − x ′ | 1 2 − d 2p t d 4p − 1 4 .
Letting t → ∞, the RHS converges to 0 since p > d. This implies that ϕ = c > 0 is constant. The equality 0 = Lϕ = Lc = V c and hence V = 0.
Remark. 1. The above proof is inspired from [7] in which it is proved that the boundedness of the Riesz transform ∇L −1/2 on L p (R d ) for some p > d implies that V = 0. 2. According to a previous remark, we could replace in the last theorem the boundedness of G L by the boundedness of G defined by (10).
Theorem 3. 2 .
2Suppose that there exists 0 < ϕ ∈ L ∞ (R d ) which satisfies(11).Then G L (or H L ) is bounded on L p (R d ) for some p > d if and only if V = 0Proof. If V = 0 then L = −∆ and it is known that the Littlewood-Paley-Stein function G L (and also H L ) is bounded on L p (R d ) for all p ∈ (1, ∞).
Since the semigroup e −tL is sub-Markovian, it acts on L p (R d ) and hence the generator of this semigroup in L p (R d ) is well defined. This is the operator L we consider on L p (R d ).
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"Lower bound for monotone Boolean convolution",
"Lower bound for monotone Boolean convolution"
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"Mike Paterson "
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| Any monotone Boolean circuit computing the n-dimensional Boolean convolution requires at least n 2 and-gates. This matches the obvious upper bound. The previous best bound for this problem was Ω(n 4/3 ), obtained by Norbert Blum in 1981. More generally, exact bounds are given for all semi-disjoint bilinear forms. | null | [
"https://arxiv.org/pdf/1708.03523v3.pdf"
]
| 9,213,012 | 1708.03523 | 5823d0d684dbaa28d9b5fbf5cc67b8ae2b63f5d0 |
Lower bound for monotone Boolean convolution
11 Aug 2017 August 14, 2017
Mike Paterson
Lower bound for monotone Boolean convolution
11 Aug 2017 August 14, 2017Boolean vector convolutionmonotone Boolean circuit complexitysemi-disjoint bilinear forms
Any monotone Boolean circuit computing the n-dimensional Boolean convolution requires at least n 2 and-gates. This matches the obvious upper bound. The previous best bound for this problem was Ω(n 4/3 ), obtained by Norbert Blum in 1981. More generally, exact bounds are given for all semi-disjoint bilinear forms.
Introduction
We consider the monotone circuit complexity of Boolean convolution, i.e., the number of logical gates needed in a Boolean circuit which has only and-gates and or-gates to compute the convolution of two Boolean vectors. The Boolean convolution of vectors x 1 , . . . , x n and y 1 , . . . , y n is f 1 , . . . , f 2n−1 , where
f k = i+j−1=k x i ∧ y j .
This shows explicitly that at most n 2 ∧-operations (and's) and (n−1) 2 ∨-operations (or's) are needed to compute the Boolean convolution of two length-n vectors. The wrapped Boolean convolution is just the same but gives n functions defined as above but with i+j−1=k (mod n) . The naive circuit for this uses n 2 ∧'s and n(n − 1) ∨'s.
Substantially fewer operations are needed if the restriction to monotone circuits is removed. With negation added, arithmetic operations can be implemented allowing transform techniques which can compute convolution in O(n log 2 n log log n) Boolean operations. While it is generally believed that n 2 ∧'s are required for convolution with monotone circuits, the best lower bound known to date for the ∧-complexity is Ω(n 4/3 ), due to Blum [1] over thirty years ago. Very recently Lingas [4] has given a bound of Ω(n 2−ǫ ), but conditional on a depth bound for conjunctions. There are larger lower bounds known for the ∨-complexity, e.g., n 3/2 due to Weiss [7], and, Ω(n 2 / log 6 n) due to Grinchuk and Sergeev [2,3].
The problem seems similar to finding the Boolean complexity of matrix multiplication. The naive algorithm for multiplying two n × n Boolean matrices requires n 3 ∧'s for monotone circuits, whereas, with negations permitted, fast algorithms for matrix multiplication over rings can be implemented yielding circuits with onlyÕ(n ω ) operations where ω < 2.373. In this case however it was shown over forty years ago ( [6,5]) that exactly n 3 ∧'s are necessary in monotone Boolean circuits.
Here we give lower bounds for conjunctive complexity matching exactly the obvious upper bounds for monotone Boolean convolution (both wrapped and unwrapped). More generally, exact results are also shown for all semi-disjoint bilinear forms.
Preliminaries
Let X and Y be disjoint sets of variables and Z = X ∪ Y .
Definition 1. A set F of functions is a semi-disjoint bilinear form over X and Y if
(1) each prime implicant of functions in F has the form x ∧ y where x ∈ X and y ∈ Y , (2) for each function f in F and each z in Z there is at most one prime implicant of f containing z, and
(3) the sets of prime implicants of functions in F are disjoint.
We will often omit ∧ symbols and represent conjunctions by juxtaposition.
Definition 2. The domain of F , dom(F ) is the set of pairs (i, j) such that x i y j is a prime implicant of a function in F . We define the domain size of F to be D(F ) = |dom(F )|.
To consider Boolean circuits, we will use the equivalent formulation of Boolean chains, i.e., sequences of Boolean functions where each function is the conjunction or disjunction of two previous functions in the sequence. More formally, we define a sequence s −q , . . . , s 0 , s 1 , . . . s T , where the inputs, x 0 , . . . , x I−1 , y 0 , . . . , y J−1 , appear as s −q , . . . , s 0 and each function s k , for 1 ≤ k ≤ T , is either s i ∧ s j or s i ∨ s j for some i, j < k. A Boolean chain computes a set of functions F if each f ∈ F occurs in the chain.
The ∧-complexity of a chain is the number of ∧ operations used in the chain, and similarly for the ∨-complexity. The complexity (∧-or ∨-) of a set F of Boolean functions is the minimal such complexity of a chain which computes F .
Motivating examples
It would simplify our investigations into the ∧-complexity of semi-disjoint bilinear forms if we could ignore implicants of degree three or more. Our first example shows that this would lead to the loss of an exact lower bound. Example 1. Computing w, given by
w = x 1 (y 2 ∨ y 3 ∨ y 4 ) ∨ y 1 (x 2 ∨ x 3 ∨ x 4 ) requires two conjunctions. But now w(x 2 ∨ y 2 ) = x 1 y 2 ∨ x 2 y 1 ∨ (x 1 x 2 y 3 ∨ x 1 x 2 y 4 ∨ x 3 y 1 y 2 ∨ x 4 y 1 y 2 )
and we note that the terms in parenthesis on the right are all of degree three. Similarly we have w(x 3 ∨ y 3 ) = x 1 y 3 ∨ x 3 y 1 ∨ (higher order terms) and
w(x 4 ∨ y 4 ) = x 1 y 4 ∨ x 4 y 1 ∨ (higher order terms).
Hence, were we to ignore higher order terms we could use a total of only 5 conjunctions to evaluate a semi-disjoint bilinear form with domain size 6.
We seek to define an appropriate measure of progress along a Boolean chain towards the final goal. We will define measure i,j (G) to represent the contribution of a set of functions G towards the computation of the prime implicant x i y j in the bilinear form F . Initially each measure i,j is to be zero, and finally each will be 1, i.e.,
measure i,j (∅) = 0 and measure i,j (G) = 1 for all (i, j) ∈ dom(F ) if G ⊇ F.
A disjunction should not increase any measure and we will want to show that the total progress made by one conjunction is at most 1. For hints towards an appropriate progress measure, the following examples from a convolution computation are instructive.
Example 2. Given w 1 = (x 1 ∨x 2 )y 2 and w 2 = x 1 (y 1 ∨y 2 ), what should measure i,j ({w 1 , w 2 }) be? We note that
w 3 = (w 1 ∨ x 2 )(w 2 ∨ y 1 ) = (x 1 y 2 ∨ x 2 )(x 1 y 2 ∨ y 1 ) = x 1 y 2 ∨ x 2 y 1 .
This suggests that the conjunction producing w 3 increases measure 2,1 from zero to one, and so w 1 and w 2 should already jointly provide measure of one for x 1 y 2 , i.e., measure 1,2 ({w 1 , w 2 }) = 1. Hence by symmetry we should regard the computation of w 1 as providing measure 1/2 to each of x 1 y 2 and x 2 y 2 .
Example 3. What should measure 1,k (x 1 y k ∨ x k−1 ∨ x k ) be?
Suppose we take this measure to be α, and similarly
measure i,k+1−i (x i y k+1−i ∨ x k−1 ∨ x k ) = α for 1 < i ≤ k − 2. The "or" of all these terms is v 1 = v ∨ x k−1 ∨ x k , where v = x 1 y k ∨ · · · ∨ x k−2 y 3 . We expect the total measure, measure(v 1 ) = i,j measure i,j (v 1 ) to be (k − 2)α.
We define v 2 , v 3 , and v 4 similarly, so that
v 1 = v ∨ x k−1 ∨ x k , v 2 = v ∨ y 2 ∨ x k , v 3 = v ∨ x k−1 ∨ y 1 , and v 4 = v ∨ y 2 ∨ y 1 .
Then measure({v 1 , v 2 , v 3 , v 4 }) is at most 4(k−2)α. However with three more conjunctions the term v 1 v 2 v 3 v 4 = x 1 y k ∨ · · · ∨ x k y 1 is generated which shows that 4(k − 2)α + 3 ≥ k, i.e., α ≥ k−3 4(k−2) . This suggests that an appropriate choice of α would be 1/4. More generally, we expect that
measure i,j (x i y j ∨ z 1 ∨ · · · ∨ z k ) ≥ 2 −k ,
where x i , y j ∈ {z 1 , . . . , z k } ⊂ Z. We choose to take this measure to be 2 −k .
Notation. We will identify a Boolean function f with the set f −1 (1), i.e., the set of arguments which f maps to 1. Then f ∧ g = f ∩ g and f ∨ g = f ∪ g. Set inclusion corresponds with implication: f ⊆ g is equivalent to f ∧ g = f or f ∨ g = g or f → g.
Definitions
For a semi-disjoint bilinear set of functions F = f 1 , . . . , f K and (i, j) ∈ dom(F ), we define h(i, j) = k where f k is the unique function in F with prime implicant x i y j . For example, in the notation we are using for Boolean convolution, we have h(i, j) = i + j − 1.
Definition 3. For any Boolean function g, any semi-disjoint bilinear set of functions F and any (i, j) ∈ dom(F ), we can partition the prime implicants, P I(g), of g into the "harmless" ones H i,j (g) and the "bad" ones
B i,j (g), where H i,j (g) = {p ∈ P I(g) : p ⊆ f h(i,j) } and B i,j (g) = {p ∈ P I(g) : p ⊆ f h(i,j) }.
We can further partition B i,j (g) into those prime implicants which are dependent on x i , those that are dependent on y j , and those that are independent of x i and y j . Any prime implicants involving both x i and y j are clearly in H i,j (g). The (i, j)-decomposition of g is the 4-tuple (h, a, b, c) of functions such that g = h ∨ x i a ∨ y j b ∨ c, corresponding to the partition described, and where a, b, c are independent of x i and y j .
Let G be a set of Boolean functions and F a semi-disjoint bilinear set of functions. We need to measure the progress of G towards the computation of each prime implicant x i y j of F . It is natural that such a progress measure should be dependent only on functions in G which have x i y j as a prime implicant. What is less natural but technically convenient is that our definitions for measuring the progress of G depend only on the conjunction of these functions.
We will write J and J for the conjunction and disjunction, respectively, of a set J of functions.
Definition 4. The (i, j)-support of a set of functions G, denoted S i,j (G)
, is the conjunction of the subset of functions that have x i y j as a prime implicant,
S i,j (G) = {g ∈ G : x i y j ∈ P I(g)}.
Note that this support either has x i y j as a prime implicant or is trivial, i.e., 1. Suppose the (i, j)-support of G is nontrivial and (h, a, b, c) is its (i, j)-decomposition, then h has x i y j as a prime implicant and contains only implicants of f h(i,j) . The other terms a, b, c contain non-implicants of f h(i,j) , which we regard as "pollutants". At the end of the computation we must have eliminated these pollutants, i.e., a∨b∨c = 0. Our (i, j)-measure is designed to quantify progress in pollutant elimination.
In this paper, a projection maps some subset of the variables to 0 and leaves the rest unchanged.
Definition 5. Attenuation for a monotone Boolean function g is the random projection which, independently for each variable z, maps z to 0 with probability 1/2 and leaves z unchanged otherwise. Definition 6. The vacuity, vac(g), of a monotone Boolean function g is the probability that the attenuation of g is 0, i.e., the zero function corresponding to the empty set ∅.
For examples, if z 1 , z 2 , . . . are distinct variables,
• vac(0) = 1, • vac(1) = 0, • vac(z 1 ) = 1/2, • vac(z 1 ∨ . . . ∨ z k ) = 2 −k , • vac(z 1 . . . z k ) = 1 − 2 −k , • vac(z 1 (z 2 ∨ z 3 )) = 5/8, and • vac(z 1 (z 2 ∨ · · · ∨ z k )) = 1/2 + 2 −k .
Vacuity is used to give a measure of progress for a prime implicant.
(i) if U ⊆ V then vac(U) ≥ vac(V ); and (ii) vac(U) + vac(V ) = vac(U ∨ V ) + vac(U ∧ V ). Proof. For (i), since any projection π is monotone increasing, if U ⊆ V and π(V ) = ∅ then π(U) = ∅. Therefore vac(U) ≥ vac(V ).
For (ii), we see that for any projection π:
(a) if π(U) = 0 and π(V ) = 0 then π(U ∨ V ) = 0 and π(U ∧ V ) = 0, (b) if π(U) = 0 and π(V ) = 0 (or vice versa) then π(U ∨ V ) = 0 and π(U ∧ V ) = 0, and (c) if π(U) = 0 and π(V ) = 0 then π(U ∨ V ) = 0 and π(U ∧ V ) = 0.
In each case π makes an equal contribution to the probability on each side of the equation.
Corollary 1. For all Boolean functions U, V and W ,
(i) vac(UV ) ≤ vac(U) + vac(V ); (ii) vac(U ∨ V W ) = vac((U ∨ V )(U ∨ W )) ≤ vac(U ∨ V ) + vac(U ∨ W ). Definition 7. Let p i,j (g) = a ∨ b ∨ c where (h, a, b, c) is the (i, j)-decomposition of the function g. It is convenient to abbreviate vac(p i,j (g)) as vac i,j (g). Recall that a ∨ b ∨ c does not depend on x i or y j . Definitions 8. For a set of Boolean functions G, measure i,j (G) = vac i,j (S i,j (G)), i.e., measure i,j (G) = vac(p i,j (S i,j (G))).
For a set G of Boolean functions, the total measure of G is
measure(G) = (i,j)∈dom(F ) measure i,j (G).
Revisiting Example 1, we see that
measure 1,2 (w(x 2 ∨ y 2 )) = vac(x 2 (y 3 ∨ y 4 ) ∨ y 1 (x 3 ∨ x 4 )) = 5 8 · 5 8 = 25/64.
So the conjunction with x 2 ∨ y 2 increases measure 1,2 , and similarly measure 2,1 , by 1/4 from vac(y 3 ∨ y 4 ∨ y 1 (x 2 ∨ x 3 ∨ x 4 )) = 1 2 · 1 2 · 9 16 = 9/64 to 25/64. This conjunction therefore increases the total measure by 1/2.
First results
Suppose G is a set of Boolean functions and
g † = g ′ •g ′′ is a new function where {g ′ , g ′′ } ⊆ G and • ∈ {∧, ∨}. Let G † = G ∪ {g † }. Suppose (i, j) ∈ dom(F ).
We begin by showing some elementary properties of our progress measures.
Lemma 2. For all i, j, p i,j is a monotonic increasing function, i.e., if g 1 ⊆ g 2 then
p i,j (g 1 ) ⊆ p i,j (g 2 ).
Proof. Monotonicity follows from the observations that if (h, a,
b, c) is the (i, j)-decomposition of g then ax i ∨ by j ∨ c = g \ f h(i,j) and p i,j (g) = a ∨ b ∨ c is obtained from ax i ∨ by j ∨ c by setting x i = y j = 1.
Lemma 3. The following inequalities hold.
(i) If • = ∨, then measure i,j (G † ) = measure i,j (G). (ii) measure i,j (G † ) ≥ measure i,j (G).
Proof.
For (i), since S i,j (G † ) = S i,j (G), the result is clear . For (ii),
G † ⊇ G =⇒ S i,j (G † ) ⊆ S i,j (G) (immediate from Definition 4) =⇒ p i,j (S i,j (G † )) ⊆ p i,j (S i,j (G)) (by Lemma 2) =⇒ vac(p i,j (S i,j (G † ))) ≥ vac(p i,j (S i,j (G))) (by Lemma 1(i)) ⇐⇒ measure i,j (G † ) ≥ measure i,j (G)
The next lemmas establish limits on the amount of progress which can be made with a single conjunction.
measure i,j (G 1 ∪ G 2 ) + measure i,j (G 1 ∩ G 2 ) ≤ measure i,j (G 1 ) + measure i,j (G 2 ). Proof. We observe that S i,j (G 1 ∪ G 2 ) = S i,j (G 1 ) ∧ S i,j (G 2 ) and S i,j (G 1 ∩ G 2 ) ⊇ S i,j (G 1 ) ∨ S i,j (G 2 ).
Since p i,j is monotonic increasing (Lemma 2),
measure i,j (G 1 ∪ G 2 ) + measure i,j (G 1 ∩ G 2 ) = vac(p i,j (S i,j (G 1 ∪ G 2 ))) + vac(p i,j (S i,j (G 1 ∩ G 2 ))) ≤ vac(p i,j (S i,j (G 1 ) ∧ S i,j (G 2 ))) + vac(p i,j (S i,j (G 1 ) ∨ S i,j (G 2 ))) = vac(p i,j (S i,j (G 1 ))) + vac(p i,j (S i,j (G 2 ))) (by Lemma 1(ii)) = measure i,j (G 1 ) + measure i,j (G 2 ).
Lemma 5. The progress for measure i,j from G to G † is at most the progress seen by considering the last operation in isolation, i.e.,
measure i,j (G † ) − measure i,j (G) ≤ measure i,j ({g ′ , g ′′ , g † }) − measure i,j ({g ′ , g ′′ }).
Proof. Since measure i,j is supermodular (Lemma 4),
measure i,j (G † ) + measure i,j ({g ′ , g ′′ }) = measure i,j (G ∪ {g ′ , g ′′ , g † }) + measure i,j (G ∩ {g ′ , g ′′ , g † }) ≤ measure i,j (G) + measure i,j ({g ′ , g ′′ , g † })
.
When g † = g ′ ∨ g ′′ , no progress is made since S i,j (G † ) = S i,j (G)
. So now we need only analyse the single operation g † = g ′ ∧ g ′′ . Lemma 6. In the following cases, no progress is made with respect to measure i,j , i.e., measure i,j ({g ′ , g ′′ }) = measure i,j ({g ′ , g ′′ , g † }), where g † = g ′ ∧ g ′′ :
(i) x i y j ∈ P I(g † );
(ii) x i y j ∈ P I(g ′ ) and x i y j ∈ P I(g ′′ );
(iii) x i y j ∈ P I(g ′ ) and x i ∈ P I(g ′′ ) and y j ∈ P I(g ′′ );
(iv) x i y j ∈ P I(g ′′ ) and x i ∈ P I(g ′ ) and y j ∈ P I(g ′ ).
Proof. Cases (i) and (ii) are obvious since S i,j ({g ′ , g ′′ }) = S i,j ({g ′ , g ′′ , g † }).
For Case (iii), let g ′ = h ′ ∨x i y j ∨x i a ′ ∨y j b ′ ∨c ′ and g ′′ = h ′′ ∨x i ∨y j ∨c ′′ , corresponding to their (i, j)-decompositions (h ′ ∨ x i y j , a ′ , b ′ , c ′ ) and (h ′′ , 1, 1, c ′′ ). Then
g † = g ′ ∧ g ′′ has (i, j)-decomposition (h * ∨ x i y j , a ′ ∨ c ′ , b ′ ∨ c ′ , c * ) where c * ⊆ c ′ c ′′ and h * ⊆ f h(i,j) . Hence p i,j (g ′ ∧ g ′′ ) = a ′ ∨ c ′ ∨ b ′ ∨ c ′ ∨ c * = a ′ ∨ b ′ ∨ c ′ = p i,j (g ′ ), so measure i,j ({g ′ , g ′′ }) = measure i,j ({g ′ , g ′′ , g † }). Case (iv) is similar.
There are six remaining cases where measure i,j may be improved. These are characterised by the significant occurrences of x i and y j given below. A precise description is given by the decompositions shown in the subsequent case analysis.
The six cases
XY: x i ∈ P I(g ′ ) and y j ∈ P I(g ′′ );
QY: x i y j ∈ P I(g ′ ) and y j ∈ P I(g ′′ );
XQ: x i ∈ P I(g ′ ) and x i y j ∈ P I(g ′′ );
YX: y j ∈ P I(g ′ ) and x i ∈ P I(g ′′ );
YQ: y j ∈ P I(g ′ ) and x i y j ∈ P I(g ′′ );
QX: x i y j ∈ P I(g ′ ) and x i ∈ P I(g ′′ ).
Let
g ′ = X ′ ∨ Y ′ ∨ u,v m ′ u,v x u y v ∨ c ′ and g ′′ = X ′′ ∨ Y ′′ ∨ u,v m ′′ u,v x u y v ∨ c ′′ , where X ′ ⊆ X, X ′′ ⊆ X, Y ′ ⊆ Y , Y ′′ ⊆ Y
, m ′ and m ′′ are (0, 1)-valued matrices, and c ′ and c ′′ contain no terms linear both in x's and y's. We define the integers
n ′ X = |X ′ |, n ′ Y = |Y ′ |, n ′′ X = |X ′′ |, n ′′ Y = |Y ′′ |
, and also, for each i and j, the linear functions
R ′ j = u m ′ u,j x u , S ′ i = v m ′ i,v y v , R ′′ j = u m ′′ u,j x u , S ′′ i = v m ′′ i,v y v ,
and the corresponding integers
r ′ j = u m ′ u,j , s ′ i = v m ′ i,v , r ′′ j = u m ′′ u,j and s ′′ i = v m ′′ i,v .
It will be convenient in the following to abbreviate, e.g., X ′ by X ′ for a subset X ′ of variables, where this presents no ambiguity. Lemma 7. In Case XY, x i ∈ X ′ and y j ∈ Y ′′ , and
measure i,j (g † ) ≤ 2 −(n ′ X − 1 + n ′′ Y − 1 + max(n ′′ X , r ′ j ) + max(n ′ Y , s ′′ i ) ≤ 2 −(n ′ X − 1 + n ′′ Y − 1 + n ′′ X + n ′ Y ) .
Proof. We have
g ′ ⊇ x i ∨ (X ′ \ x i ) ∨ Y ′ ∨ y j R ′ j , g ′′ ⊇ y j ∨ X ′′ ∨ (Y ′′ \ y j ) ∨ x i S ′′ i , and g † = g ′ ∧ g ′′ ⊇ x i y j ∨ x i (X ′′ ∨ (Y ′′ \ y j ) ∨ S ′′ i ) ∨ y j ((X ′ \ x i ) ∨ Y ′ ∨ R ′ j ).
Hence
measure i,j (g † ) ≤ vac((X ′ \ x i ) ∨ (Y ′′ \ y j ) ∨ X ′′ ∨ R ′ j ∨ Y ′ ∨ S ′′ i ) = 2 −(n ′ X − 1 + n ′′ Y − 1 + |X ′′ ∪ R ′ j | + |Y ′ ∪ S ′′ i |) ,
since the six sets of variables involved are disjoint except possibly for the two unions indicated. The inequalities of the lemma follow immediately.
Lemma 8. The contribution to the total measure under Case XY, is at most
{(i,j)|x i ∈X ′ &y j ∈Y ′′ } 2 −(n ′ X − 1 + n ′′ Y − 1 + max(n ′′ X , r ′ j ) + max(n ′ Y , s ′′ i )) ≤ n ′ X 2 n ′ X −1 n ′′ Y 2 n ′′ Y −1 2 −(n ′′ X + n ′ Y ) ≤ 2 −(n ′′ X + n ′ Y ) .
Proof. The first expression is immediate from Lemma 7. The next inequality replaces the max's by their first argument, removing dependency on i and j. The final inequality follows since n/2 n−1 ≤ 1 for all n ≥ 0.
The analysis of Case QY is more complicated since we need to bound the difference, measure i,j (g † ) − measure i,j (g ′ ), whereas in Case XY measure i,j (g ′ ) = measure i,j (g ′′ ) = 0.
Lemma 9. In Case QY, suppose g ′ = h ′ ∨ x i a ′ ∨ y j b ′ ∨ c ′ and g ′′ = h ′′ ∨ x i a ′′ ∨ y j ∨ c ′′ corresponding to their (i, j)-decompositions, where x i y j ∈ h ′ , then the (i, j)-progress made in this step is measure i,j (g † ) − measure i,j (g ′ ) ≤ vac i,j (a ′ (a ′′ ∨ c ′′ ) ∨ b ′ ∨ c ′ ) − vac i,j (a ′ ∨ b ′ ∨ c ′ ). Proof. We have g ′′ ⊇ x i a ′′ ∨ y j ∨ c ′′ , so g † = g ′ ∧ g ′′ ⊇ x i y j ∨ x i a ′ (a ′′ ∨ c ′′ ) ∨ y j (b ′ ∨ c ′ ).
Hence
measure i,j (g † ) ≤ vac i,j (a ′ (a ′′ ∨ c ′′ ) ∨ b ′ ∨ c ′ ) and measure i,j (g ′ ) = vac i,j (a ′ ∨ b ′ ∨ c ′ ).
Here is a cautionary example showing why we cannot simply use an inequality analogous to
vac(UV ∨ W ) − vac(U ∨ W ) ≤ vac(V ∨ W )
from Corollary 1(ii) to get an upper bound of vac(a ′′ ∨ c ′′ ∨ b ′ ∨ c ′ ) on the progress made in QY.
Example 4. We denote by e the unique index such that h(e, e) = h(i, j), and assume i and j are disjoint from 1, 2, 3, 1, 2, 3. Suppose
g ′ = x i y j ∨ x i a ′ where a ′ = x 1 x 2 ∨ x 2 x 3 ∨ x 3 x 1 , g ′′ = x i a ′′ ∨ y j where a ′′ = y 1 ∨ y 2 ∨ y 3 , and so g † = g ′ g ′′ = x i y j ∨ x i a ′ a ′′ .
For this example, vac(a ′′ ∨c ′′ ∨b ′ ∨c ′ ) = vac(a ′′ ) = 1/8. Note that a ′ is the majority function of x 1 , x 2 , x 3 , and so vac i,j (g ′ ) = vac(a ′ ) = 1/2, and hence vac(g ′ g ′′ ) = 1/2 + 1/2 · 1/8 = 1/2 + 1/16. However, We can however obtain a weaker inequality provided that V , say, is linear, i.e., a disjunction of variables. We first need the following monotonicity property.
a ′ a ′′ ⊆ f h(i,j) ∨ x 1 x 2 y 3 ∨ x 2 x 3 y 1 ∨ x 3 x 1 y 2 so vac i,j (g ′ g ′′ ) = vac(x 1 x 2 y 3 ∨ x 2 x 3 y 1 ∨ x 3 x
Lemma 10. For any
(i, j) ∈ dom(F ), vac i,j (UV ∨ W ) − vac i,j (U ∨ W ) is monotone decreasing in V and W .
(Note that Example 4 shows that this expression is neither monotone decreasing nor increasing in U. With V = a ′′ and W = 0, setting U to 0, a ′ , 1 gives values of 0, 13/64, 1/8 respectively.)
Proof. If V ⊇ V 0 and W ⊇ W 0 then for any projection π, if π(UV ) ≡ 0 and π(W ) ≡ 0 and π(U) ≡ 0 then π(UV 0 ) ≡ 0 and π(W 0 ) ≡ 0 and π(U) ≡ 0.
Hence vac i,j (UV 0 ∨ W 0 ) − vac i,j (U ∨ W 0 ) ≥ vac i,j (UV ∨ W ) − vac i,j (U ∨ W ).
Technical crux
For expressing our upper bound in Case QY, we need the function q(k) = (3 k + 1)/2 2k+1 .
Lemma 11. For any (i, j) ∈ dom(F ), if V is the disjunction of k variables then
vac i,j (U ∧ V ) − vac i,j (U) ≤ q(k).
Proof. Within this proof it is convenient to define the mateẑ of a variable z ∈ Z as the unique variable such that zẑ ⊆ f h(i,j) , i.e., zẑ ≡ 0. For example,ŷ 2 is x 2 (in the notation of Example 4), and for all z,ẑ = z.
Now vac i,j (U ∧ V ) − vac i,j (U)
is the proportion of projections π such that π(UV ) ≡ 0 and π(U) ≡ 0. Suppose that V = z 1 ∨ · · · ∨ z k . We begin by observing that we need only consider functions U which depend only onẑ 1 , . . . ,ẑ k . Suppose that U depends on the variable z, and U = U 0 ∨ U 1 z where U 0 and U 1 are independent of z. Then
UV = U 0 V ∨ U 1 zV . Case (i): z ∈ {z 1 , . . . , z k }. Then zV = z and so UV = U 0 V ∨ U 1 z. Hence vac i,j (UV ) − vac i,j (U) = vac i,j (U 0 V ∨ U 1 z) − vac i,j (U 0 ∨ U 1 z) ≤ vac i,j (U 0 V ) − vac i,j (U 0 )
by Lemma 10. Case (ii): z ∈ {z 1 , . . . , z k ,ẑ 1 , . . . ,ẑ k }. We can assume that U 1 is independent ofẑ since any tẑ ∈ P I(U 1 ) yields the term tẑz ≡ 0 in U 1 z, and so, for any projection π such that π(z) = z, π(U 1 zV ) ≡ 0 if and only if π(U 1 V ) ≡ 0. Then
vac i,j ((U 0 ∨ U 1 z)V ) − vac i,j (U 0 ∨ U 1 z) = 1 2 (vac i,j (U 0 V ) − vac i,j (U 0 ) + vac i,j ((U 0 ∨ U 1 )V ) − vac i,j (U 0 ∨ U 1 )),
i.e., the average of the terms when the projection of z is zero or nonzero.
In each case we see that vac i,j (UV ) − vac i,j (U) ≤ vac i,j (U ′ V ) − vac i,j (U ′ ) for some U ′ derived from U by eliminating the variable z. Hence it is sufficient for the proof to assume that U depends only onẑ 1 , . . . ,ẑ k . We denote {1, . . . , k} by I k and define Boolean variables b S for S ⊆ I k by the disjunctive normal form
U = S⊆I k i∈Sẑ i b S . By definition, if b S = 1 then b S ′ = 0 for any S ′ ⊂ S.
For any projection π, we define
S π = {i | π(z i ) = z i } andŜ π = {i | π(ẑ i ) =ẑ i }.
Consider a projection π such that π(U) ≡ 0 but π(UV ) ≡ 0. Then for some S ⊆Ŝ π , π( i∈Sẑ i b S ) ≡ 0. Hence {ẑ i |i ∈ S} cannot contain a pair of mates and b S = 1.
Since π(UV ) ≡ 0, i∈Sẑ i i∈Sπ z i ≡ 0, which implies S π ⊆ S. So S π ⊆ S ⊆Ŝ π , and hence S π ⊆Ŝ π . The probability that |S π | = r is 2 −k k r , and with the extra condition that {ẑ i |i ∈ S π } does not contain a pair of mates (since S π ⊆ S), this probability is at most 2 −k k r . The probability when |S π | = r thatŜ π ⊇ S π is 2 −r . Hence
vac i,j (U ∧ V ) − vac i,j (U) ≤ 2 −k 0≤r≤k k r 2 −r = 2 −k (1 + 1/2) k = (3/4) k .
We improve this bound using the following observation. Consider a pair of dual projections π 1 and π 2 , i.e., such that S π 1 = I k \Ŝ π 2 and S π 2 = I k \Ŝ π 1 . If both π 1 and π 2 contribute to vac i,
j (U ∧ V ) − vac i,j (U) then (a) S⊆Ŝπ 1 b S = 1, since π 1 (U) = S⊆Ŝπ 1 i∈Sẑ i b S ≡ 0, and (b) S ⊇Sπ 2 b S = 0, since π 2 (UV ) = S⊆Ŝπ 2 i∈Sẑ i b S i∈Sπ 2 z i ≡ 0.
From (a), there exists some S 1 such that S 1 ⊆Ŝ π 1 = (I k \ S π 2 ) and b S 1 = 1. From
(b), b S 1 = 1 implies that S 1 ⊇ S π 2 . Hence S π 2 ⊆ S 1 ⊆ (I k \ S π 2 )
, which implies that S π 2 = S 1 = ∅, and S π 1 = ∅ by a symmetric argument. Therefore, unless S π 1 = ∅ and S π 2 = ∅, we find π 1 and π 2 cannot both contribute to vac i,j (U ∧ V ) − vac i,j (U), and so the two projections together contribute at most 2 −2k to vac i,j (U ∧ V ) − vac i,j (U) instead of each contributing 2 −2k .
So we can successfully match dual pairs (π 1 , π 2 ) of projections except when S π 1 = ∅ and S π 2 = ∅. The number of such pairs is (3 k − 1)/2 which yields q(k) = (3 k + 1)/2 2k+1 .
Lemma 12. The function q has the following monotonicity properties:
(i) for all k ≥ 1, q(k − 1) ≥ q(k) ,
(ii) for all k ≥ 1, k q(k − 1) ≥ (k + 1)q(k) , (iii) for all k ≥ 1 and all t ≥ 0, k q(k − 1 + t) ≤ 1 .
Proof. For (i),
(3 k−1 + 1)/2 2k−1 ≥ (3 k + 1)/2 2k+1 ⇐⇒ 4(3 k−1 + 1) ≥ 3 k + 1 ⇐⇒ 3 k−1 + 3 ≥ 0. For (ii), k(3 k−1 + 1)/2 2k−1 ≥ (k + 1)(3 k + 1)/2 2k+1 ⇐⇒ 4k(3 k−1 + 1) ≥ (k + 1)(3 k + 1) ⇐⇒ 4k − (k + 1) ≥ (3(k + 1) − 4k)3 k−1 ⇐⇒ 3k − 1 ≥ (3 − k)3 k−1 .
The last inequality holds for k ≥ 1.
Inequality (iii) follows from inequalities (i) and (ii), and checking the value for k = 1 and t = 0, i.e., 1q(0) = 1.
Lemma 13. For any (i, j) ∈ dom(F ), if V and W are sets of variables, where |W | = m and |V \ W | = k then
vac i,j (UV ∨ W ) − vac i,j (U ∨ W ) ≤ (1/2) m q(k).
Proof. We need to estimate the probability of a projection π such that π((UV ∨ W ) ≡ 0 but π(U ∨ W ) ≡ 0, i.e., π(W ) ≡ 0 and π(UV ) ≡ 0 but π(U) ≡ 0 and π(V ) ≡ 0. We can regard any such π as the composition of a projection π 1 over the variables of W , a projection π 2 over the variables of |V \ W | and a projection π 3 over the remaining variables. The required result is reached with probability (1/2) m for π 1 , probability at most q(k) for π 2 and probability at most 1 for π 3 .
We use the monus (limited subtraction) notation, i.e., r . − s = max(r − s, 0).
Lemma 14. In Case QY, x i y j ∈ P I(g ′ ) and y j ∈ Y ′′ , and
measure i,j (g † ) − measure i,j (g ′ ) ≤ (1/2) m q(k), where m = r ′ j − 1 + n ′ X + n ′ Y and k = (n ′′ Y − 1 + (n ′′ X . − (r ′ j − 1)) + (s ′′ i . − n ′ Y ). Proof. If g ′ = h ′ ∨ x i a ′ ∨ y j b ′ ∨ c ′ and g ′′ = h ′′ ∨ x i a ′′ ∨ y j ∨ c ′′ then since b ′ ⊇ (R ′ j \ x i ), c ′ ⊇ X ′ Y ′ , a ′′ ⊇ S ′′ i and c ′′ ⊇ X ′′ (Y ′′ \ y j )
, from Lemmas 9 and 10 we have
measure i,j (g † ) − measure i,j (g ′ ) = vac i,j (a ′ (a ′′ ∨ c ′′ ) ∨ b ′ ∨ c ′ ) − vac i,j (a ′ ∨ b ′ ∨ c ′ ) ≤ vac i,j (a ′ V ∨ W ) − vac i,j (a ′ ∨ W ), where V = (Y ′′ \ y j ) ∪ X ′′ ∪ S ′′ i and W = (R ′ j \ x i ) ∪ X ′ ∪ Y ′ . Now |W | = r ′ j − 1 + n ′ X + n ′ Y and |V \ W | = |(Y ′′ \ y j ) ∪ (X ′′ \(R ′ j \ x i )) ∪ (S ′′ i \ Y ′ )| ≥ n ′′ Y − 1 + (n ′′ X . − (r ′ j − 1)) + (s ′′ i . − n ′ Y ).
By Lemmas 13 and 12(i),
vac i,j (a ′ V ∨ W ) − vac i,j (a ′ ∨ W ) ≤ (1/2) |W | q(|V \ W |),
which establishes the result.
Lemma 15. The contribution to the total measure under Case QY, is at most
{(i,j)|x i ∈R ′ j &y j ∈Y ′′ } 2 −(r ′ j − 1 + n ′ X + n ′ Y ) q n ′′ Y − 1 + (n ′′ X . − (r ′ j − 1)) + (s ′′ i . − n ′ Y ) ≤ {j|y j ∈Y ′′ } r ′ j 2 −(r ′ j − 1 + n ′ X + n ′ Y ) q n ′′ Y − 1 + (n ′′ X . − (r ′ j − 1)) ≤ 2 −(n ′ X + n ′ Y ) .
(d) n ′ X , n ′′ X > 0 and n ′ Y = n ′′ Y = 0;
C XY + C QY + C XQ + C Y X + C Y Q + C QX ≤ 0 + 0 + 1 2 + 0 + 0 + 1 2 = 1.
(e) n ′ X , n ′′ Y > 0 and n ′ Y = n ′′ X = 0;
C XY + C QY + C XQ + C Y X + C Y Q + C QX ≤ 1 + 1 2 + 1 2 + 0 + 0 + 0 = 2.
(f) n ′ X > 0 and n ′ Y = n ′′ X = n ′′ Y = 0;
C XY + C QY + C XQ + C Y X + C Y Q + C QX ≤ 0 + 0 + 1 + 0 + 0 + 0 = 1.
(g) n ′ X = n ′ Y = n ′′ X = n ′′ Y = 0;
C XY + C QY + C XQ + C Y X + C Y Q + C QX ≤ 0 + 0 + 0 + 0 + 0 + 0 = 0.
Main results
The results so far already give a new lower bound for semi-disjoint bilinear forms. Now we refine Theorem 1 to give the exact bound, i.e., increasing D(F )/2 to D(F ). We use the stronger bounds given in Lemmas 8 and 15 to show that the contribution to the total measure is at most 1 per step.
The case analysis is long and complicated. (We have used Mathematica for this, but plan eventually to present a readable analysis.) As a preliminary step we use simpler, weaker bounds to eliminate most cases. Define − (r ′ j − 1)) ,
Lemma 1 .
1The function vac is (i) monotone decreasing and (ii) modular, i.e., for all Boolean functions U and V ,
Lemma 4 .
4For all (i, j) ∈ dom(F ), measure i.j is supermodular, i.e., for all sets of functions G 1 , G 2 ,
an increase in measure i,j of 13/64, which exceeds vac(a ′′ ) = 1/8. (The value 45/64 can be shown by considering the three cases: (i) at most one x is 1, (ii) exactly two x's are 1 and (iii) all three x's are 1. The corresponding contributions to vac are 1/2, 3/8·1/2 and 1/8 · 1/8.)
Theorem 1 .
1The ∧-complexity for a semi-disjoint bilinear form F is at least D(f )/2, i.e., half the number of prime implicants in F . Proof. For some Boolean chain computing F , suppose that conjunctions are used only at steps t 1 , . . . , t k . Let M s = measure({g t | t ≤ t s }). Now M 0 = 0 and, by Lemma 16, M s −M s−1 ≤ 2 for all s > 0. Hence M s ≤ 2s for all s. So D(F ) = measure(F ) = M k ≤ 2k implies that k ≥ D(F )/2. Corollary 2. Boolean convolution (wrapped or unwrapped) for n-vectors requires at least n 2 /2 conjunctions.
j)|x i ∈X ′ &y j ∈Y ′′ } 2 −(n ′ X − 1 + n ′′ Y − 1 + n ′′ X + n ′ Y ) , and C ⋆ QY = {(i,j)|x i ∈R ′ j &y j ∈Y ′′ } 2 −(r ′ j − 1 + n ′ X + n ′ Y ) q n ′′ Y − 1 + (n ′′ X .
Concluding remarksWe have used a classic approach, giving an explicit "measure of progress" for each successive Boolean operation. We believe that our technique of attenuation defining the vacuity probability measure is novel in this context. It will be interesting to see if the technique has wider uses.We have dealt only with bounds for the conjunctive complexity for bilinear Boolean forms. The problem of giving an exact bound for the corresponding disjunctive complexity is open.I am grateful to Andrzej Lingas for bringing the problem to my attention again and giving very helpful feedback on a previous draft.Proof. The first expression is immediate from Lemma 14. The next inequality results by dropping the term (s ′′ i . − n ′ Y ) and summing over i. The final inequality uses r/2 r−1 ≤ 1 and Lemma 12(iii).Bounds for the remaining cases, (XQ,YX,YQ,QX), follow by symmetry. We denote the upper bounds for contributions to the total measure as given in Lemmas 8 and 15 by C XY , C QY , C XQ , C Y X , C Y Q , and C QX for the six cases indicated. For example,Then C = C XY +C QY +C XQ +C Y X +C Y Q +C QX is an upper bound for the total progress in one step.Lemma 16. The total contribution made in one step of the Boolean chain is at most 2.Proof. Lemmas 8 and 15 together with symmetry show thatWe bound the total contribution, C = C XY + C QY + C XQ + C Y X + C Y Q + C QX , by considering the following seven cases. Other cases all follow by symmetry.(a) n ′ X , n ′ Y , n ′′ X , n ′′ Y > 0;(b) n ′ X , n ′ Y , n ′′ X > 0 and n ′′ Y = 0;(c) n ′ X , n ′ Y > 0 and n ′′ X = n ′′ Y = 0;C XY + C QY + C XQ + C Y X + C Y Q + C QX ≤ 0 + 0 + 1 + 0 + 1 + 0 = 2.We first useas an upper bound on the progress per step. Symmetry permits us to assume without loss of generality thatAn analysis based on Mathematica shows that C ⋆ < 1 unless (n ′ X , n ′ Y , n ′′ X , n ′′ Y ) = (i) (4, 0, 0, * ), The better bound C resolves these cases, showing that C ≤ 1 and that the only values for which C = 1 are (n ′ X , n ′ Y , n ′′ X , n ′′ Y ) = (1, 0, 0, 1), (1, 1, 1, 1), (2, 0, 0, 1) or (2, 0, 0, 2).Lemma 17. The total contribution made in one step of the Boolean chain is at most 1.Proof. This is shown by the case analysis outlined above.Main Theorem. The ∧-complexity for a semi-disjoint bilinear form F is at least D(f ), i.e., the number of prime implicants in F .Proof. The proof is as in Theorem 1 but uses the tighter bound in Lemma 17 instead of that in Lemma 16.Main Corollary. Boolean convolution (wrapped or unwrapped) for n-vectors requires n 2 conjunctions.
. X Q Similarly For, Y X , Y Q , Q X By, Referencessimilarly for XQ, Y X, Y Q, and QX. By monotonicity, References
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