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Joint Sparse Recovery with Semi-Supervised MUSIC Zaidao Wen, Biao Hou, Member, IEEE, Licheng Jiao, Senior Member, IEEEDecember 30, 2023 ==========================================================================Discrete multiple signal classification (MUSIC) with its low computational cost and mild condition requirement becomes a significant non-iterative algorithm for joint sparse recovery (JSR). However, it fails in rank defective problem caused by coherent or limited amount of multiple measurement vectors (MMVs). In this paper, we provide a novel sight to address this problem by interpreting JSR as a binary classification problem with respect to atoms. Meanwhile, MUSIC essentially constructs a supervised classifier based on the labeled MMVs so that its performance will heavily depend on the quality and quantity of these training samples. From this viewpoint, we develop a semi-supervised MUSIC (SS-MUSIC) in the spirit of machine learning, which declares that the insufficient supervised information in the training samples can be compensated from those unlabeled atoms. Instead of constructing a classifier in a fully supervised manner, we iteratively refine a semi-supervised classifier byexploiting the labeled MMVs and some reliable unlabeled atoms simultaneously. Through this way, the required conditions and iterations can be greatly relaxed and reduced. Numerical experimental results demonstrate that SS-MUSIC can achieve much better recovery performances than other MUSIC extended algorithms as well as some typical greedy algorithms for JSR in terms of iterations and recovery probability. The code is available on <https://github.com/wzdammy/semi_supervised_MUSIC>.MUSIC, greedy pursuit, multiple measurement vectors, joint sparse recovery, semi-supervised classification. § INTRODUCTION The emerging theory of compressed sensing (CS) supplies a paradigm for recovering an unknown sparse signal from some compressed linear measurements and it has been devoted to many applications in signal processing as well as machine learning(ML) <cit.>. This theory primarily addresses the recovery problem of a signal 𝐱∈ℝ^n from its single measurement vector (SMV) 𝐲∈ℝ^m such that 𝐲=𝐀𝐱, where 𝐀∈ℝ^m× n models the linear measurement matrix with m≪ n. Practically, we may encounter the problem of simultaneously recovering a group of N sparse signals 𝐗=[𝐱_1,…,𝐱_N]∈ℝ^n× N from their multiple measurement vectors (MMVs) 𝐘in many tasks, e.g., multivariate regression <cit.>, classification <cit.>, direction of arrival estimation <cit.>, etc. When these underlying signals share some particular sparse patterns, it will enable to reduce the condition for successful recovery. One of the most prevalent patterns expressed as joint sparse suggests that these signals will share the same support so that 𝐗 will contain only a few non-zero rows. In this scenario, if the row-sparsity, the number of non-zero rows of 𝐗 is equal to K, the problem of joint sparse recovery (JSR) from a common 𝐀 can be formulated asmin_𝐗𝐘-𝐀𝐗_F^2,s.t. 𝐗_row,0≤ K,where ·_F is the Frobenius norm (F-norm) and 𝐗_row,0 counts the non-zero rows in 𝐗. Unfortunately, (<ref>) is generally a combinatorial non-convex optimization problem due to 𝐗_row,0. To solve this problem, two strategies have been developed in optimization field, namely convex relaxation with a mixed norm and greedy methods <cit.>. Focusing on the greedy algorithm, the central issue becomes to iteratively estimate a certain amount of atoms according to the correlations with the residual matrix so as to mostly decrease the value of objective function (<ref>). Once a support set is determined, the recovery problem will be reduced to a standard overdetermined linear problem solved with the least square.As a consequence, numerous greedy JSR algorithms have been extended from SMV to MMVs <cit.>, yielding the orthogonal matching pursuit (OMP) for MMVs (OMP-MMV) <cit.> or the so-called simultaneously OMP (SOMP) <cit.>, simultaneously compressive sampling matching pursuit (SCoSaMP) <cit.>, rank aware order recursive matching pursuit (RA-ORMP) <cit.> etc. Another significant algorithm referred to as discrete multiple signal classification (MUSIC) takes a different viewpoint in the field of signal processing <cit.>. It reveals that each measurement vector and the correct atoms should reside in the same subspace if rank(𝐘)≐ r=K. Under a mild condition, those atoms can be straightforward determined by singular value decomposition (SVD) without iterative process, which achieves far more remarkable performance than those greedy optimization algorithms in terms of complexity and required conditions. When r<K caused by limited amount of MMVs or information loss due to their correlations, MUSIC will however yield a failing estimation in this rank defective case. To overcome this drawback, some MUSIC extended algorithms have been developed for rank defective problem, such as iMUSIC, compressive MUSIC (CS-MUSIC) and subspace-augmented MUSIC (SA-MUSIC) <cit.>. iMUSIC, an initial version of SA-MUSIC, involves an iterative atom refinement procedure in MUSIC so that some falsely determined atoms could be gradually refined during iterations. However, some operations in atoms refinement are not optimal so that it can only achieve a marginal improvement than MUSIC and some conventional greedy approaches in noiseless case. Later, two almost equivalent algorithms of SA-MUSIC and CS-MUSIC provide a two-stages framework, which indicates that if any K-r atoms could be correctly estimated in the first stage with any an off the shelf algorithm, the rest r atoms will be simply determined by applying MUSIC on an augmented subspace <cit.>. It follows that the required conditions and iterations for such a combinational framework will actually depend on the algorithm in the first stage, which is usually suboptimal compared to MUSIC. How to fully exploit the advantages of MUSIC to relax the condition and reduce the iterations become two important issues. In recent years, the field of ML attracts much more attentions because of some significant progresses in both theory and industry. If we revisit the support estimation from the perspective of ML, it can be regarded as a binary classification task with respect to atoms and MUSIC actually constructs a nearest subspace classifier (NSC) according to the positive labeled training samples 𝐘 in a fully supervised way <cit.>. Therefore, its discriminative ability will be naturally affected by the quality and quantity of these training samples. Following this novel viewpoint, we are motivated to address the rank defective problem by means of the strategy in ML.In this paper, we present a novel semi-supervised MUSIC (SS-MUSIC) for JSR, in which both the labeled MMVs and some reliable unlabeled atoms are iteratively exploited for classifier construction <cit.>. Through this way, the inadequate supervised information in rank defective MMVs can be additionally compensated from those unlabeled data so as to increase the discrimination of the classifier. As a consequence, SS-MUSIC will successfully classify all atoms within K-r iterations as long as at least one positive atom can be newly determined and preserved in each iteration. The simulation results clearly demonstrate the superiorities of SS-MUSIC, compared with the other MUSIC extended frameworks as well as some state-of-the-art greedy algorithms. The rest paper is organised as follows. Sec. <ref> proposes our algorithm in detail. Numerical experiments are conducted in Sec. <ref> and Sec. <ref> concludes this paper.§ SEMI-SUPERVISED MUSICIn this section, we will formally reformulate the JSR problem and MUSIC from the viewpoint of ML in the first place. Then a novel SS-MUSIC framework is developed and compared with the other algorithms in order to demonstrate its superiorities.§.§ Reformulation of JSR and MUSICLet 𝐘∈ℝ^m× N contain N labeled noiseless training samples drawn from the positive class. Given n unlabeled atoms {𝐚_i}_i=1^n, the central task for JSR is classifying these atoms into two classes by assigning a proper label l_i∈{0,1} to 𝐚_i such that 𝐘=∑_iδ(l_i)𝐚_i𝐗^i, where 𝐗^i is the i-th row vector in 𝐗, δ stands for the indicator function as δ(l_i=0)=0 for negative label and δ(l_i=1)=1 for positive one. Additionally, we have a prior knowledge that the amount of the positive atoms will be K. Since each 𝐲_i will reside in the subspace 𝒮_𝐀(𝐥_+) spanned by those positive labeled atoms, we can measure the sum of Euclidean distance from each 𝐲_i to 𝒮_𝐀(𝐥_+) to evaluate the fitness of a label configuration 𝐥∈{0,1}^n, which is defined as following. ℓ(𝐥)=∑_i=1^N dis(𝐲_i,𝒮_𝐀(𝐥_+))=∑_i=1^N𝐲_i-𝒫_𝒮_𝐀(𝐥_+)(𝐲_i)_2^2where 𝐀(𝐥_+) stands for the subset of atoms with positive labels and 𝒫_𝒮 is the orthogonal projection operator onto subspace 𝒮 here and after. If 𝐔 is the orthogonal basis of subspace 𝒮 computed from truncated SVD or principal component analysis (PCA) <cit.>, 𝒫_𝒮=𝐔𝐔^T. It follows that if the classification is correct, ℓ(𝐥) will reach its lower bound, namely zero in a noiseless situation. In practical situation with noisy MMVs, we can exploit a threshold ϵ related to signal-to-noise ratio (SNR) to indicate the fitness of 𝐥, namely ℓ(𝐥)≤ϵ. Considering this task, one of the most prevalent strategies in ML is supervised classification, which focuses on constructing a classifier based on the training samples. Following this way, the novel MUSIC algorithm essentially constructs a NSC with 𝐘 and each query atom will be classified by measuring dis(𝐚_i,𝒮_𝐘) <cit.>. Then K atoms with the closest distance will be classified into the positive class. Since this classifier will be frequently exploited in the following paper, it will be denoted by 𝐥𝒲(𝒬|𝒮_𝒞,K), where 𝒬 contains the query testing samples, 𝒮_𝒞 is the subspace spanned by samples in set 𝒞 and K controls the number of positive labels in the output of label configuration 𝐥.§.§ Algorithm PresentationIt has been indicated that when rank(𝐘)=K, such a supervised classifier will produce a perfect classification result if any K+1 atoms are linearly independent <cit.>. However, when the number of training samples is limited or they are coherent, rank(𝐘)<K. In this case, supervised information in training data will be insufficient to construct a discriminative NSC so that the performance will be degraded. To overcome this deficiency, we will consider the strategy of semi-supervised classification (SSC) to develop a novel SS-MUSIC framework <cit.>, whose central idea is to simultaneously make use of the labeled and some reliable unlabeled samples to construct a semi-supervised classifier. Then the information required for classification will be compensated from unlabeled data. For better understanding the difference between MUSIC and SS-MUSIC, two frameworks will be illustrated in Figs.<ref>, where the notation 𝒯 and 𝒯 containing unlabeled atoms will represent the candidate and actual training set for semi-supervised classifier construction, respectively. we will address the two central issues of constructing 𝒯 and 𝒯 to explain the framework in Fig. <ref> as following.We will start from a label configuration 𝐥^t obtained in t-th iteration, t≥ 0. If it is not fitted according to (<ref>), it implies that the current training set cannot provide sufficient or correct discriminative information for classifier construction, i.e., 𝒯^t will contain some outlying atoms so as to bias the classifier or the number of involved atoms is inadequate. To address this issue, except for those atoms in current positive class, we will reappraise the confidence of each atom in the negative class so that some with high confidences will be also involved in 𝒯^t+1. We suggest that if an atom is much similar to 𝐘 measured in a feature domain, a high confidence of being involved will be encouraged. To avoid the redundant information, the high confident atoms should have the ability of providing extra information compared with 𝐀_𝐥_+^t. To meet these two requirements, we will firstly project 𝐘 onto the orthogonal complement subspace of 𝐀(𝐥_+^t) as 𝐘=Φ(𝐘) with a feature extractor Φ in order to eliminate the information of 𝐀(𝐥_+^t). Then NSC will be exploited to select K-r atoms 𝐀(𝐥_-^t) that is nearest to 𝒮_𝐘. Finally, the candidate set will be constructed as 𝒯^t+1{𝐀(𝐥_+^t),𝐀(𝐥_-^t)}. The above procedures are denoted by purple flows in Fig. <ref>.After we obtain 𝒯^t+1 containing 2K-r candidates, K-r representative and reliable atoms will be further refined to update 𝒯^t+1 and construct the semi-supervised classifier. This task will be simply interpreted as the following overcomplete variables selection problem <cit.>.𝐗^*min_𝐗𝐘-𝐀_𝒯𝐗_F^2whose solution is given by 𝐗^*=𝐀_𝒯^†𝐘 and 𝐀_𝒯^† stands for the pseudo-inverse of 𝐀_𝒯. Then the atoms corresponding to the first K-r largest (𝐗^*)^j_2 will be selected into 𝒯^t+1. Next, atoms in 𝒯^t+1 and the labeled MMVs will be simultaneously used to construct a semi-supervised classifier as𝐥^t+1𝒲(𝐀|𝒮_𝒯^t+1+𝐘,K). Since K-r atoms are already devoted to classifier construction, their labels will be consequently positive and we only need to assign the rest r positive labels to other atoms in 𝐀.The complete SS-MUSIC is summarised in following Algorithm <ref>. §.§ Discussion and ComparisonTo demonstrate the superiorities of SS-MUSIC to make it more convinced, some discussions and comparisons with other algorithms will be carried out in this subsection, in spite of their distinct motivations. In the first place, let us focus on the iterations. According to the principle of SA-MUSIC or CS-MUSIC, once 𝒯^t+1 has contained the K-r atoms belonging to the positive class, the subsequent 𝒲(𝐀|𝒮_𝒯^t+1+𝐘,K) will generate the fitted label configuration. It follows that if one more positive atom could be newly involved and preserved in 𝒯 in each iteration, the upper bound on iterations will be K-r. In fact, we will empirically show in the next section that the actual iterations will be much fewer than K-r. On the contrary, CS-MUSIC and SA-MUSIC respectively exploit M-OMP and OSMP to determine K-r atoms iteratively in the first stage so that their iterations will be always K-r. Since iMUSIC also adopts M-OMP for initial K-r atoms estimation, the lower bound on iterations will be K-r. Accordingly, SS-MUSIC requires fewer iterations than these MUSIC extended algorithms while its computational complexity will be still comparable with that of iMUSIC. Now let us compare the required conditions for each algorithm. For SA-MUSIC and CS-MUSIC, their required conditions mainly come from the first stage that should guarantee the correctness of selecting one atom in each iteration. On the contrary, SS-MUSIC will only require at least one correct atom to be selected and preserved in 𝒯, which will be reasonably much relaxed than that of SA-MUSIC and CS-MUSIC. This can be also concluded from the proof of the generalized OMP which selects a set of atoms in one iteration to relax the condition of OMP in SMV problem <cit.>. Additionally, SS-MUSIC involving an atom refinement process will further relax the conditions, which is similar to iMUSIC. Nevertheless, SS-MUSIC is different from iMUSIC in following implementations. 1). iMUSIC utilizes 𝒲(𝐀(𝐥_-)|𝒮_𝒯^t+𝐘,K_0) to construct 𝒯, where K_0 is the number of selected atoms controlled by the condition number of 𝐀_𝒯. On the contrary, SS-MUSIC selects K-r atoms based on 𝒲(𝐀(𝐥_-)|𝒮_𝐘,K-r) in a different feature subspace. 2). Eq. (<ref>) in iMUSIC is different and the resulted K atoms will be directly served as the label configuration in this iteration. In SS-MUSIC, those resulted K-r atoms will be subsequently utilized to build a semi-supervised classifier 𝒲(𝐀|𝒮_𝒯^t+1+𝐘,K) to obtain the label configuration. Comparing SS-MUSIC with some conventional greedy optimization algorithms in CS, it will be analogous to SCoSaMP which estimates 2K atoms and makes a refinement in each iteration. Nevertheless, SCoSaMP requires a more strict condition and more iterations for exact recovery. Ambat and Hari presented a general iterative framework for SMV problem <cit.>, in which they introduce a regularization procedure on both atoms and measurement vector to remove the effect of the previous estimated atoms. However, they aim at estimating complete K atoms in each iteration while SS-MUSIC focuses on K-r in the spirit of SSC. § EMPIRICAL PERFORMANCE In this section, we consider the following experimental setting to evaluate the performance for rank defective JSR problem. 𝐗∈ℝ^100× N is drawn from the standard Gaussian distribution and K>N rows in general position are randomly retained with other rows setting as zeros. In this case, rank(𝐘)=r=N. 𝐀∈ℝ^m× 100 is also chosen as the standard Gaussian random matrix with each atom normalized. T_max=100. In the first part, the phase transition of SS-MUSIC will be evaluated in Fig. <ref>, where m and K will vary from 1 to 100, respectively and N=20 and 1000 independent recovery experiments are conducted for each pair of (m,K). We can observe from the transition map that when m>K, SS-MUSIC will perfectly recover the signals with high probabilities for most pairs of (m,K). When K<20, it becomes the full rank JSR problem, in which case SS-MUSIC will be the MUSIC sharing condition of m=K+1. When K>20, we however observe that the required m for recovery will not greatly increase to reach the high probability performance, which empirically demonstrates a mild condition for 𝐀. To evaluate the required iterations, we choose the recovery results from pair (35,30) and (40,30) and count the iterations for successfully recovery. For those failure trials, the iterations will be denoted by 101 as T_max=100. We plot the histograms of the iterations in Fig. <ref>. For comparison, the results for iMUSIC will be also illustrated in Fig. <ref>. It should be declared that the standard iMUSIC leverages the condition number to control the amount of involved atoms for refinement, but it will involve another parameter. Instead, a fixed number is selected which is the same with SS-MUSIC, namely K-r. We can see from the results in Fig. <ref> that SS-MUSIC can achieve perfect performances in both cases and the required iterations can reach the lower bound 2 in the most experiments among 1000 trials. On the contrary, 300 trials for iMUSIC in the case of (35,30) are failed and the iterations in the rest experiments are 12.In the next experiments, SS-MUSIC will be compared with other JSR algorithms to demonstrate its superiority in terms of required condition. For this purpose, we will vary one parameter of m, K and N to evaluate the recovery probabilities of each algorithm with the other two fixed, respectively. In the first place, the noiseless situation is considered, yielding the results shown in Figs. <ref>-<ref> and the corresponding settings are also illustrated in the figures. From the results in Fig. <ref>, we can see that SS-MUSIC outperforms all competitive algorithms when we vary m, followed by RA-ORMP, SA-MUSIC and CS-MUSIC. It is worth noting that when m=31=K+1 reaches its lower bound, SS-MUSIC can still achieve over 90% probability of exact recovery, which empirically verifies a milder condition of 𝐀 for SS-MUSIC. Similar conclusions can be also derived from the performances in Figs. <ref> and <ref> and we will not discuss for the sake of space limitation. Finally, we will evaluate their performances in noisy situation, where the measurement noise distributed from Gaussian will be considered for simplicity, namely 𝐘_noise=𝐘+𝐄 and 𝐄 stands for the measurement noise matrix. Before starting, since SS-MUSIC is developed for noiseless MMVs, some operations should be modified, where the rank and the subspace of 𝐘 should be estimated from 𝐘_noise in the first stage. For simplicity and fair comparison, we adopt an efficient method proposed in SA-MUSIC to address this issue. Then we will exploit the estimated rank and basis of subspace 𝐘 to construct the semi-supervised classifier in SS-MUSIC. The comparison results are illustrated in Figs. <ref>-<ref>. We can conclude that SS-MUSIC can still preserve its remarkable recovery performance to outperform the other algorithms in the most cases. It can be also observed that iMUSIC will also achieve the better performances than other algorithms due to its refinement procedure. § CONCLUSION This paper develops a novel SS-MUSIC for rank defective JSR, which brings the strategy of ML into the optimization problem to shed a new light on this direction. We show that simultaneously exploiting the labeled MMVs and some unlabeled atoms can significantly improve the performance in terms of required iterations and conditions. In our future work, we will develop a robust framework to address the noisy JSR problem straightforwardly, in which more ML strategies will be considered and involved to avoid estimating r and subspace of 𝐘 in the first place.IEEEtran | http://arxiv.org/abs/1705.09446v1 | {
"authors": [
"Zaidao Wen",
"Biao Hou",
"Licheng Jiao"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170526063747",
"title": "Joint Sparse Recovery With Semisupervised MUSIC"
} |
Gossip in a Smartphone Peer-to-Peer Network Calvin NewportGeorgetown UniversityWashington, [email protected]===================================================================================== In this paper, we study the fundamental problem of gossip in the mobile telephone model: a recently introduced variation of the classical telephone model modified to better describe the local peer-to-peer communication services implemented in many popular smartphone operating systems. In more detail, the mobile telephone model differs from the classical telephone model in three ways: (1) each device can participate in at most one connection per round; (2) the network topology can undergo a parameterized rate of change; and (3) devices can advertise a parameterized number of bits about their state to their neighbors in each round before connection attempts are initiated.We begin by describing and analyzing new randomized gossip algorithms in this model under the harsh assumption of a network topology that can change completely in every round. We prove a significant time complexity gap between the case where nodes can advertise 0 bits to their neighbors in each round, and the case where nodes can advertise 1 bit. For the latter assumption, we present two solutions: the first depends on a shared randomness source, while the second eliminates this assumption using a pseudorandomness generator we prove to exist with a novel generalization of a classicalresult from the study of two-party communication complexity. We then turn our attention to the easier case where the topology graph is stable, and describe and analyze a new gossip algorithm that provides a substantial performance improvement for many parameters. We conclude by studying a relaxed version of gossip in which it is only necessary for nodes to each learn a specified fraction of the messages in the system. We prove that our existing algorithms for dynamic network topologies and a single advertising bit solve this relaxed version up to a polynomial factor faster (in network size) for many parameters. These are the first known gossip results for the mobile telephone model, and they significantly expand our understanding of how to communicate and coordinate in this increasingly relevant setting.§ INTRODUCTIONThis paper describes and analyzes new gossip algorithms in the mobile telephone model: an abstraction that captures the local device-to-device communication capabilities available in most smartphone operating systems; e.g., as implemented by services such as Bluetooth LE <cit.>, WiFi Direct <cit.>,and Apple's Multipeer Connectivity framework <cit.>.Motivation. Smartphones are a ubiquitous communication platform: there arecurrently over 3.9 billion smartphone subscriptions worldwide <cit.>. Most smartphone communication leverages one-hop radio links to cell towers or WiFi access points. In recent years, however, the major smartphone operating systems have included increasingly stable and useful support for local peer-to-peer communication that allows a device to talk directly to a nearby device (using local radio broadcast) while avoiding cellular and WiFi infrastructure.The ability to create these local links, combined with the ubiquity of smartphones, enables scenarios in which large groups of nearby smartphone users run applicationsthat create peer-to-peer meshes supporting infrastructure-free networking. There are many possible motivations for these smartphone peer-to-peer networks. For example, they can support communication in settings where network infrastructure is censored (e.g., government protests), overwhelmed (e.g., a large festival or march), or unavailable (e.g., after a disaster or at a remote event). In addition, in developing countries, cellular data minutes are often bought in blocks and carefully conserved—increasing interest in networking operations that do not require cellular infrastructure. To further validate the potential usefulness of smartphone peer-to-peer networks, consider the FireChat application, which implements group chat using smartphone peer-to-peer services. In the few years since its initial release, it has been widely adopted in over 120 countries and has been used successfully in multiple government protests, festivals (e.g., at Burning Man, which is held far from cell towers), and disaster scenarios <cit.>.Developing useful applications for this smartphone peer-to-peer setting requires distributed algorithms that can provide global reliability and efficiency guarantees on top of an unpredictable collection of local links. As detailed below, the models that describe this emerging setting are sufficiently different from existing models that new algorithms and analysis techniques are required. This paper addresses this need by describing and analyzing new gossip algorithms for this important setting.The Mobile Telephone Model. The mobile telephone model studied in this paper was introduced in recent work <cit.>. It is a variant of the classical telephone peer-to-peer model(e.g., <cit.>) modified to better describe the capabilities and constraints of existing smartphone peer-to-peer services. The details of the mobile telephone model are inspired, in particular, by the current specifications of Apple'sMultipeer Connectivity framework <cit.>: a peer-to-peer service available in every iOS version since iOS 7 that allows nodes to advertise services, discover nearby advertisers, and attempt to connect to nearby advertisers, using only local radio broadcast. (The definition of the classical telephone model, and differences between the classical telephone and mobile telephone model, are detailed and discussed below in the related work section.) In more detail, the mobile telephone model abstracts the basic scan-and-connect dynamics of the Multipeer framework as follows. Time proceeds in synchronous rounds. In each round, a connected graph describes the underlying network topology for that round. At the beginning of each round, each device (also called a node in the following) learns its neighbors in the topology graph (e.g., as the result of a scan).Each device can then attempt to initiate a connection with a neighbor. Each node can support at most one connection—so if multiple nodes attempt to connect with the same target, only one connection will succeed. If two nodes connect, they can perform a bounded amount of reliable communication before the round ends.We parameterize this model with a tag length b≥ 0. At the beginning of each round, each node can choose a tag consisting of b bits to advertise. When performing a scan, each node learns both the ids and chosen tags of its neighbors (where b=0 means there are no tags). These tags can change from round to round. In our previous study of rumor spreading with parameter b=1 <cit.>, for example, at the beginning of a given round, each node that already knows the rumor advertises a 1 with its tag, while other nodes advertise a 0. This simplified the rumor spreading task by enabling nodes that know the rumor to only attempt to connect to nodes that do not. This capability of nodes to use tags to deliver limited information to their neighbors is motivated by the ability of devices to choose and change their service advertisements in the Multipeer framework.We also parameterize the model with a stability factor τ≥ 1. The underlying network topology must stay stable for at least τ rounds between changes. For τ=1, for example, the network topology can change completely in every round, while for τ=∞, the topology never changes. There exist finer-grained approaches for capturing intermediate levels of stability (e.g., T-interval connectivity <cit.>), but in this paper we study only the two extreme cases of fully dynamic and fully stable topologies, so our simpler stability factor definition is sufficient. The need to model topology changes is motivated by the inherently mobile nature of the smartphone setting. Results. In this paper, we describe and analyze new algorithms for the gossip problem in the mobile telephone model with respect to different model parameter and algorithm assumptions. This problem assumes a subset of nodes start with messages (also called tokens). The goal is to spread these messages to the entire network. Gossip is fundamental in distributed computing and is considered particularly important for ad hoc networks such as the smartphone meshes studied in this paper(c.f., the introductory discussion in <cit.>).Below (and in Figure <ref>) we state and discuss our main results. In the following, let n>1 be the network size and k, 1≤ k ≤ n, be the number of tokens in the system. For a given topology graph,we use α to describe its vertex expansion (see the model discussion below) and Δ to describe its maximum degree.[If the topology is dynamic, then α is defined as the minimum expansion over all rounds, andΔ is defined as the largest maximum degree over all rounds.] We assume the topologies are connected. All round complexity results hold with high probability in n (i.e., probability at least 1-1/n). We start by considering the difficult setting where b=0 and τ=1; i.e., nodes cannot use tags and the network topology graph can change completely in each round. In Section <ref>, we describe and analyze a natural strategy for this setting called BlindMatch, which has nodes select neighbors with uniform randomness to send connection attempts.[This is essentially the well-known PUSH-PULL strategy from the classical telephone model with the key exception that in our model if a node receives multiple connection attempts, only one succeeds. As discussed in the related work and Section <ref>, this well-motivated model change requires new analysis techniques to understand information propagation.] We prove that BlindMatch solves gossip in O((1/α)kΔ^2log^2n) rounds. This bound might seem pessimistic at first glance, but it is known that disseminating even a single message in the mobile telephone model with this strategy can take Ω(Δ^2/√(α)) rounds in somenetworks <cit.>.Indeed, this lower bound holds even for the easier assumption that τ=∞. Accordingly, we do not consider b=0 and τ=∞ as a distinct case in this paper. (To provide intuition for why Ω(Δ^2) rounds are sometimes necessary, consider two stars centered on u and v, respectively, where each star has around Δ points and u and v are connected by an edge. Assume u starts with a gossip message. For v to receive this message two events must happen: (1) u selects v for a connection; and (2) v accepts u's connection from all incoming connections in that round. The first event occurs with probability ≈ 1/Δ, and because v can expect a constant fraction of its neighbors to send it connection attempts in any given round, the second event also occurs with probability ≈ 1/Δ.) Our BlindMatch result provides the benchmark against which we attempt to improve with the algorithms that follow.In Section <ref>, we consider the case where b=1 and τ≥ 1; i.e., the network can still change completely in each round, but now nodes can advertise a single bit to their neighbors. We begin by describing and analyzing an algorithm called SharedBit. This algorithm assumes a shared randomness source which is used to implement (essentially) a random hash function that allows nodes to hash their current set of known messages to a single bit to be used as their one-bit advertising tag. The key guarantee of this function is that nodes with the same sets advertise the same bit, and nodes with different sets have a constant probability of advertising different bits. This helps nodes seek out productive connections with neighbors (e.g., connections in which at least one node learns something new). We prove that SharedBit solves gossip in O(kn) rounds.We next seek to eliminate the shared randomness assumption. To do so, we describe SimSharedBit which solves gossip in O(kn + (1/α)Δ^1/τlog^6n) rounds, without assuming a shared randomness source. Notice, because α≥ 2/n and Δ≤ n, this solution is always within log factors of the SharedBit for large k, and for small k it is still comparable for many values of α, Δ, and/or τ.The SimSharedBit algorithm depends on a novel generalization of Newman's Theorem <cit.>—a well-known result onpublic randomness simulation from the study of two-party communication complexity.We prove that there exists an appropriate pseudorandom number generator that can provide sufficient randomness for the SharedBit strategy. We then elect a leader in O((1/α)Δ^1/τlog^6n) rounds using an algorithm from <cit.>, and use this leader to disseminate a small generator seed. We note that our generalization of Newman's Theorem is potentially of standalone interest as the techniques we introduced can be used to study pseudorandomness in many different graph algorithm settings.In Section <ref>, we consider the impact of topology changes on gossip time. In particular, we consider the case where b=1 and τ=∞; i.e., the network topology is stable. We describe and analyze CrowdedBin, an algorithm that solves gossip in O((1/α)klog^6n) rounds. This algorithm matches or outperforms the O(kn) round complexity of SharedBit for all α values (ignoringlog factors). For well-connected networks (e.g., constant α), it performs almost a factor of n faster. These results hint that large increases to stability are more valuable to gossip algorithms than large increases to tag length (for most of our solutions, increasing b beyond 1 only improves performance by at most logarithmic factors). The benefit of stable network topologies is that nodes can transmit larger amounts of information about their current state to their neighbors by using their single bit advertisement tag over multiple rounds. CrowdedBin leverages this capability to help nodes efficiently converge on an accurate estimate of k—which is not known in advance. This process depends on nodes testing guesses by throwing their tokens into a number of bins corresponding to the current guess, and then seeking/spreading evidence of crowding (as established by a new balls-in-bins algorithm described in Section <ref>).Once all nodes learn an appropriate guess of k, CrowdedBin deploys an efficient parallel rumor spreading strategy to efficiently disseminate the k tokens.Finally,we consider the ϵ-gossip problem, which is parameterized with a fraction ϵ, 0 < ϵ <1, assumes that k=n, and relaxes the gossip problem to require only that every node receives at least nϵ of the n total tokens. This variation is useful for settings where it is sufficient for nodes to learn enough rumors to complete the task at hand; e.g., when an algorithm requires responses from only a majority quorum of nodes.In Section <ref>, we re-analyze the SharedBit gossip algorithm from Section <ref>. Deploying a novel argument based on finding productive “coalitions" of nodes, we show that SharedBit solves ϵ-gossip in O(n√(ΔlogΔ)/(1-ϵ)α)rounds. Recall that SharedBit solves regular gossip in O(n^2) rounds under the k=n assumption. Therefore, when ϵ is a constant fraction and the network is well-connected (α is large), SharedBit solves ϵ-gossip up to a (sub-linear) polynomial factor faster than the standard gossip problem.Related Work.The mobile telephone model used in this paper was firstintroduced in a study of rumor spreading by Ghaffari and Newport <cit.>. We also recently studied leader election in this same model <cit.>. As noted, the mobile telephone modelis a variation of the classical telephone model (first introduced by Frieze and Grimmett <cit.>) adapted to better describe smartphone peer-to-peer networks. The mobile model differs from the classical model in two ways: (1) the classical model implicitly fixes b=0 and (typically) τ = ∞; and (2) the classical model allows nodesto accept an unbounded number of incoming connections.It is important to emphasize that most of the well-known bounds in the classical model depend on this assumption of unbounded connections, and removing this assumption requires new analysis techniques; c.f., the discussion in <cit.>. We note that work by Daum et al. <cit.> (which preceded <cit.>) also pointed out the dependence of existing telephone model bounds on unbounded concurrent connections. A fundamental problem in peer-to-peer networks is rumor spreading, in which a single message must be disseminated from a designated source to all nodes (this is equivalent to gossip with k=1). This problem is well-understood in the classical telephone model, where spreading times are often expressed with respect tospectral properties of the network topology graph such asgraph conductance (e.g., <cit.>)and vertex expansion (e.g., <cit.>). This existing work established that efficient rumor spreading is possible with respect to both graph properties in the classical model.In <cit.>, we studied this problem in the mobile telephone model. We proved that efficient rumor spreading with respect to conductance is not possible in the mobile telephone model, but efficient spreading with respect to vertex expansion is possible. We then proved that for b=1 and τ≥ 1, a simple random spreading strategy solves the problem in O((1/α)Δ^1/τpolylog(n)) rounds—matching the tight Θ((1/α)log^2n) result from the classical telephone model within log factors for τ≥logΔ. In <cit.>, we built on these results to solve leader election in similar asymptotic time.Though gossip is well-studied in peer-to-peer models (see <cit.> for a good overview), little is known about how to tackle the problem in the mobile telephone model, where concurrent connections are now bounded but nodes can leverage advertising tags.[It might betempting to simply run k parallel instances of the rumor spreading strategy from <cit.> to gossip k messages, but this approach fails for three reasons: (1) our model allows only O(1) tokens to be sent per connection per round; (2) each of the k instances requires its own advertising tag bit, whereas all of our new gossip results focus on the case where b≤1; and (3) nodes do not know k in advance. Accordingly, most results presented in this paper require substantial technical novelty.] Finally, we note that there are application similarities between gossip in the mobile telephone model and existing reliable multicast solutions for mobile ad hoc (e.g., <cit.>) and delay-tolerant (e.g., <cit.>) networks. These existing solutions, however, tend to be empirically evaluated and depend on the ability to predict information about link behavior (e.g., predicted link duration or an advance schedule of when given links will be present). § MODEL AND PROBLEMWe describe a smartphone peer-to-peer network usingthe mobile telephone model. As elaborated in the introduction, the basic properties of this model—including its scan-and-connect behavior, dynamic topologies,and the nodes' ability to advertise a bounded tag—are inspired in particular by the behavior of the Apple Multipeer Connectivity framework for smartphone peer-to-peer networking. In more detail, we assume executions proceed in synchronous rounds labeled 1,2,.... We assume all nodes start in the same round. We describe a peer-to-peer network topology in each round r as an undirected connected graphG_r=(V,E_r) that can change from round to round,constrained by the stability factor (see below). We call the sequence of graphs G_1, G_2,... that describe the evolving topology a dynamic graph. We assume the definition of the dynamic graph is fixed at the beginning of the execution.We assume a computational process (also called a node in the following) is assigned to each vertex in V, and use n=|V| to indicate the network size. At the beginning of each round r, we assume each node u learns its neighbor set N(u) in G_r. Node u can then select at most one node from N(u) and send a connection proposal. A node that sends a proposal cannot also receive a proposal. If a node v does not send a proposal, and at least one neighbor sends a proposal to v, then v can accept an incoming proposal. There are different ways to model how v selects a proposal to accept. In this paper, for simplicity, we assume v accepts an incoming proposal selected with uniform randomness from the incoming proposals. If node v accepts a proposal from node u, the two nodes are connected and can perform a bounded amount of interactive communication to conclude the round. We leave the specific bound on communication per connection as a problem parameter. Model Parameters. We parameterize the mobile telephone model with two integers, a tag length b≥ 0 and a stability factor τ≥ 1.We allow each node to select a tag containing b bits to advertise at the beginning of each round. That is, if node u chooses tag b_u at the beginning of a round, all neighbors of ulearn b_u before making their connection decisions in this round. A node can change its tag from round to round. We also allow for the possibility of the network topology changing between rounds. We bound the allowable changes with a stability factor τ≥ 1. For a given τ,the dynamic graph describing the changing topologymust satisfy the property that at least τ rounds must pass between any changes to the topology. For τ=1, the graph can change arbitrarily in every round. We use the convention of stating τ=∞ to indicate the graph never changes.Vertex Expansion and Maximum Degree. Several of our results express time complexity bounds with respect to the vertex expansion α of the dynamic graph describing the network topology. To define α, we first review a standard definition of vertex expansion for a fixed static unconnected graph G=(V,E).For a given S ⊆ V,define the boundary of S, indicated ∂ S, as follows:∂ S = { v∈ V ∖ S : N(v) ∩ S ≠∅}: that is, ∂ S is the setof nodes not in S that are directly connected to S by an edge in E. Next define α(S) = |∂ S|/|S|. As in <cit.>, we define the vertex expansion α(G) of our static graph G = (V,E)as follows: α(G) = min_S ⊂ V, 0 < |S| ≤ n/2α(S). Notice that despite the possibility of α(S) >1 for some S, we always have α(G) ≤ 1. We define the vertex expansion α of a dynamic graph G_1, G_2..., to be the minimum vertex expansion over all of the dynamic graph's constituent static graphs (i.e., α = min{α(G_i) : i ≥ 1}).Similarly, we define the maximum degree Δ of a dynamic graph to be the maximum degree over all of the dynamic graph's constituent static graphs. The Gossip Problem. The gossip problem assumes each node is provided an upper bound[For the sake of concision, the results described in the introduction and Figure <ref> make the standard assumption that N is a polynomial upper bound on n, allowing us to replace N with n within logarithmic factors inside asymptotic notation. In the formal theorem statements for these results, however, we avoid this simplification and leave N in place where used—enabling a slightly finer-grained understanding of the impact of the looseness of network size estimation on our complexity guarantees.] N ≥ n on the network size and a unique ID (UID) from [N]. The problem assumes some subset of nodes begins with a gossip message to spread (which we also call a token). We use k to describe the size of this subset and assume that k is not known to the nodes in advance. A given node can start the execution with multiple tokens, but no token starts at more than one node.We treat gossip tokens as comparable black boxesthat can only be communicated between nodes through connections (e.g., a node cannot transmit a gossip token to a neighbor by spelling it out bit by bit using its advertising tags). If a node begins an execution with a token or has received the token through a connection, we say that the node owns, knows or has learned that token. We assume that a pair of connected nodes can exchangeat most O(1) tokens and O(N) additional bits during a one round connection.Solving the Gossip Problem. The gossip problem requires all nodes to learn all k tokens, Formally, we say a distributed algorithm solves the gossip problem in f(n,k,α, b,τ) rounds, if with probability at least 1-1/n,all nodes know all k tokensby round f(n,k,α,b,τ) when executed in a network of size n, with k tokens, vertex expansion α, tag length b, and stability factor τ. We omit parameters when not relevant to the bound. Probability Preliminaries. The analyses that follow leverage the following well-known probability results: For p ∈ [0, 1], we have (1-p) ≤ e^-p and (1+p) ≥ 2^p.Let Y = ∑_i=1^t X_i be the sum of t > 0 i.i.d. random indicator variables X_1, X_2,..., X_t, and let μ = E(Y). Fix some fraction δ, 0 < δ < 1. It follows:(X ≤ (1-δ)μ) ≤ e^- δ^2 μ/2.Let Y = ∑_i=1^t X_i be the sum of t > 0 i.i.d. random indicator variables X_1, X_2,..., X_t, and let μ = E(Y). Fix some value δ>1. It follows:(X ≥ (1+δ)μ) ≤ e^- δμ/3.Let X_1, X_2, ..., X_t, be t ≥ 1 i.i.d. random indicator variables. Let μ = E(X_i) and fix some δ > 0. It follows: (1/t∑_i=1^t X_i ≥μ + δ) ≤ e^- 2 δ^2 t. Let X be a nonnegative random variable and a>0 be a real number. It follows: (X ≥ a) ≤E(X)/a. Useful Notation. For each round r≥ 1 and node u∈ V, we define T_u(r) to describe the set of tokens that u has received by the beginning of round r. It follows, therefore, that if u starts with a token, then T_u(1) contains only this token, and it is otherwise empty. We assume each token is labeled by the UID from [N] of the node where it originated.To simplify notation, we will sometimes refer to the tokens by these id labels. We also use the notation b_u(r), for each node u∈ V and round r ≥ 1, to describe thetag used by u in r. § TOKEN TRANSFER SUBROUTINE An obstacle to solving gossip in the mobile telephone model is deciding which tokens to exchange between two connected nodes. In more detail, once two nodes u and v with respective token sets T_u and T_v connect,even if they know T_u ≠ T_v,they must still identify at least one token t∉ T_u ∩ T_v to transfer for this round of gossip to be useful. Complicating this task is the model restriction that u and v can only exchange O(N) bits before deciding which tokens (if any) to transfer. This is not (nearly) enough bits to encode a full token set (a simple counting argument establishes that every coding scheme will require Ω(N) bits for some sets). Therefore, a more efficient routine is needed to implement this useful token transfer.Here we describe a transfer subroutine that solves this problem and is used by multiple gossip algorithms described in this paper.This routine, which we call Transfer(ϵ), for an error bound ϵ, 0 < ϵ < 1, is a straightforward application of an existing algorithmic tool from the literature on two-party communication complexity. It guarantees the following: if Transfer(ϵ) is called by two connected nodes u and v, with respective token sets T_u and T_v, and T_u ≠ T_v, then with probability at least 1-ϵ the smallest token t (by a predetermined token ordering) that is not in T_u ∩ T_v, will be transferred by the node that knows t to the node that does not. This routine requires u and v to exchange only O(log^2N·log(logN/ϵ)) controls bits in additionto token t. It also assumes some fixed ordering on tokens. Equality Testing. We use one of the many known existing solutionsto the set equality (EQ) problem from the study of two-party communication complexity. In our setting with u and v (described) above, these existing solutions provide u and v a way to test the equality of T_u and T_v, and they offer the following guarantee: if T_u = T_v, then u and v will correctly determine their sets are equal with probability 1, else if T_u ≠ T_v then u and v will erroneously determine their sets are equal with probability no more than 1/2. These existing solutions assume only private randomness and require u and v to exchange no more than O(logN) bits. A nice property of most such solutions is that each trial is independent. Therefore, if u and v repeat this test c times, for some integer c≥ 1, then the error probability drops exponentially fast with c to 2^-c. Let us fix one such equality testing routine and call it EQTest(c), where parameter c≥ 1 determines how many trials to execute in testing the equality. The Transfer Subroutine. We now deploy EQTest(ϵ'), for ϵ' = ⌈log(logN/ϵ)⌉, as a subroutine to implementthe Transfer(ϵ) routine. In particular, recall that for a given u and v, we can understand T_u and T_v to both be subsets of the values in [N] (as each node in the network can label each token with its UID from [N] at the beginning of the execution). Our goal is to identify the smallest location value in [N]that is in T_u ∪ T_v but not in T_u ∩ T_v. To do so, we can implement a binary search over the interval [N], using EQTest(ϵ') to test the equality of the interval in question between u and v.In more detail:Transfer(ϵ):a1; b Nwhile a≠ b result EQTest(ϵ') executedonT_u∩ [a,⌊ b/2 ⌋] and T_v ∩ [a,⌊ b/2 ⌋] if result = notequal then b ⌊ b/2 ⌋ else a ⌊ b/2 ⌋ +1transfer token a to the other node if you know token a The above logic implements a basic binary search over the interval [N] to identify the smallest value in this interval that is in exactly one of the two sets T_u and T_v. If every call to EQTest succeeds then the search succeeds and Transfer behaves correctly. There are at most logN calls to EQTest, each of which fails with probability 2^-ϵ'≤ϵ/logN. Therefore, by a union bound, the probability that at least one of the logN calls to EQTest fails is less than ϵ, as claimed. From a communication complexity perspective, each call to EQTest(ϵ') requires O(logN·ϵ') = O(logN·log(logN/ϵ)) bits, and we make logN such calls. Therefore, the total communication complexity is in O(log^2N·log(logN/ϵ)), as claimed. § GOSSIP WITH B=0 AND Τ≥ 1 Here we consider the most difficult case for gossip in our model:nodes cannot advertise any information to their neighbors (b=0),and the network topology graph can change arbitrarily in every round (τ = 1).We will study the straightforward strategy in which nodes randomly select neighborsfor attempted connections and then use the token transfer routine to select tokens to exchange during successful connections.We will show this strategy solves gossipin O((1/α)kΔ^2log^2N) roundswhen executed with k tokens in a network graph with expansion α and maximum degree Δ.This result might seem pessimistically large at first glance,but as shown in <cit.>, there are networks in which simple blind connection strategies like thoseimplemented heredo require Ω(Δ^2/√(α)) rounds to spread even a single message.The BlindMatch Gossip Algorithm. At the beginning of each round r≥ 1, each node u∈ V flips a fair coin to decide whether to be a sender or a receiver in r. If u decides to be a sender, it selects a neighbor uniformly from among its neighbors in this round and sends it a connection proposal. If u decides to be a receiver it waits to receive proposals.If two nodes u and v connect, they execute the token transfer subroutinewhich attempts to transfer the smallest token in (T_u(r) ∪ T_v(r)) ∖ (T_u(r) ∩ T_v(r)), assuming such a token exists. Analysis.We now prove the below theorem concerning about the performance of the BlindMatch algorithm. The proof adapts our recent analysis of leader election strategies in the mobile telephone model under the assumption that b=0 <cit.>. The main contribution of this section, therefore, is less technical than it is the establishment of a baseline against which to compare the other results studied in this paper. The BlindMatch gossip algorithm solves the gossip problem in O((1/α)kΔ^2log^2N) rounds when executed with tag length b=0 in a network with stability τ≥ 1. In <cit.>, we study a leader election algorithm called BlindGossip that essentially matches the behavior of BlindMatch. As in BlindMatch, this algorithm has each node in each round flip a coin to decide whether or not to send or receive, and senders choose a neighbor uniformly to send a connection proposal. If two nodes connect, they transfer the smallest UIDs they have seen so far in the execution. In <cit.>,we prove that this strategy will disseminate the smallest UID in the network to all nodes in the network in O((1/α)Δ^2log^2N) rounds, with high probability in N.This existing analysis follows the progress of the smallest token in the network showing that after this many rounds it will have spread to all nodes.In BlindMatch, by contrast, a connected pair executes the transfer routine to attempt to transfer the smallest token known by one but not both of the connected nodes. It follows, therefore, that under the assumption that the transfer routine works correctly every time it is called,BlindMatch will spread the smallest token in the network to all nodes in the time stated above. Once this has been accomplished, however, we can turn our attention to the second smallest token (once all nodes know the smallest token, the transfer routine will always transfer the second smallest when a node that knows the second smallest is connected to a node that does not). After the above number of rounds, the second smallest token will also have spread. We repeat this process for all k tokens to get the finalO((1/α)kΔ^2log^2N) timeclaimed above.§ GOSSIP WITH B=1 AND Τ≥ 1 Here we describe and analyze two gossip algorithm that now assume b=1. The first, called SharedBit, assumes shared randomness, while the second, SimSharedBit, does not. Both solutions offer a substantial time complexityimprovement over the BlindMatch algorithm for many graph parameters. Discussion: Shared Randomness. For the sake of clarity, we begin by making a strong assumption that we will subsequently eliminate: the nodes have access to a shared randomness source. In more detail, we assume at the beginning of the execution a bit string r̂ of length T=O(N^3logN) is selected with uniform randomness from the space R of all bit strings of this length. All nodes can access r̂. This shared random string simplifies the description and analysis of an efficient gossip algorithm for the assumptions tackled in this section.In particular, the key challenge for gossip in this setting isgenerating useful 1-bit advertising tags in each round. We would like nodes with the same token set to generate the same bit (so they will know not to attempt to connect to each other), while pairs of nearby nodes with different token sets to have a reasonable probability of generating different bits (so they will know a connection would prove useful). Shared randomness enables this property as each node can associate the same fresh random bit for each token in a given round, and the bit advertised for a given set can simply consist of the sum of the bits associated with tokens in the set (mod 2). Discussion: Eliminating the Shared Randomness Assumption. The assumption of shared randomness might be unrealistic in some settings. With this in mind, we will then proceed to show how to eliminate this assumption by simulating public randomness using a much smaller number of private random bits that disseminate quickly throughout the network.The core strategy of this simulation borrows and expands key ideas from the proof of Newman's Theorem (e.g., <cit.>)—a well-known result on public randomness simulation from the study of two-party communication complexity. Our result is existential in the sense that it establishes that there existsan efficient simulation of our shared randomness that works well enough. An equivalent formulation of this result in the language of pseudorandomness is that there exists a pseudorandom number generator that can generate the needed number of bits with a seed sufficiently small to fit in our message size bound. §.§ Shared RandomnessHere we describe and analyze the SharedBit gossip algorithm.The SharedBit Gossip Algorithm. Let r̂ be a shared random string of length cN^3(⌈logN⌉ + 1) bits. We assume nodes partition r̂ into cN^2 groups each consisting of N bundles (one for each id that might show up in the network) that each contain ⌈logN⌉ + 1 bits. We label these groups 1,2,...,cN^2, and label the bundles within a given group 1,2,...,N.At the beginning of each round r ≤ cN^2, node u must decide which bit to advertise to its neighbors (i.e., what value to select for b_u(r)). If T_u(r) is empty, then u advertises 0 (i.e., b_u(r)=0). Otherwise, node u calculates its advertisement by first extracting a shared bit from r̂ to assign to each t∈ T_u(r). In particular, for each such t∈ T_u(r),u sets its bit, indicated t.bit,to be the first bit in bundle t of group r from r̂. Node u then calculates the bit b_u(r) to advertise in this round as follows: b_u(r) = (∑_t∈ T_u(r) t.bit)2. If b_u(r) = 0 then u will receive connection proposals in this round. If b_u(r)=1 and u has at least one neighbor advertising 0, then u will choose one these neighbors with uniform randomness and send it a connection proposal. To make this random choice, u uses the random bits in positions 2 to ⌈logN + 1 ⌉ in the the bundle corresponding to its id in group r of r̂.[The reason we have u use shared random bits to select the receiver of its proposal is because it will simplify our subsequent effort to eliminate shared randomness for this algorithm. There are many straightforward ways a node can use (up to) logN bits to uniformly select a value from a set containing no more than N values.]If two nodes u and v connect in round r, they will deploy the token transfer subroutine, with parameter ϵ = n^-c_t, for some sufficiently large constant c_t ≥ 1 we fix in the analysis. This routine will identify and transfer the smallest token in (T_u(r) ∪ T_v(r)) ∖ (T_u(r) ∩ T_v(r)), without sending more than N bits in the interaction (the bound enforced by our model). Recall, this transfer subroutine is probabilistic and succeeds in identifyinga token to transfer with probability at least 1-ϵ.Once the algorithm proceeds past round cN^2 it can terminate or fall back to a simpler behavior (such as our algorithm for b=0), or recycle back to the beginning of the shared string. Analysis. Our goal is toprove the following theorem regarding the SharedBit gossip algorithm: The SharedBit gossip algorithm solves the gossip problem in O(kn) rounds when executed with shared randomness and tag length b=1, in a network with stability τ≥ 1.To setup our analysis, recall that we define T_u(r) for node u and round r≥ 1, to be the set of tokens u knows at the beginning of round r, and use b_u(r) to indicate the bit advertised by u in round r. Also recall that cN^2 is the maximum number of rounds for which the shared string r̂ contains bits (our below analysis will specify the needed lower bound on constant c≥ 1 ), and that t.bit, for a given token t and a fixed round,describes the shared random bit extracted from r̂ and assigned to t in this round.We begin withthe following lemma, which bounds the probabilistic behavior of the advertising tags generated using a given shared r̂. Fix two nodes u,v∈ V, u≠ v, and a round r, 1 ≤ r ≤ cN^2. Fix a r-1 round execution of SharedBit, and let p=(b_u(r) ≠ b_v(r)) be the probability (defined over the random selection of the relevant bits in r̂) that u and v generatedifferent advertising bits in round r. If T_u(r) = T_v(r) then p=0, else if T_u(r) ≠ T_v(r), then p = 1/2. If T_u(r) = T_v(r) then by definition of the algorithm b_u(r) = b_v(r). We turn our attention, therefore, to the remaining case where T_u(r) ≠ T_v(r). In the following, for a given non-empty token set T,define:adv_r(T) =(∑_t∈ T t.bit)2. And for the case of an empty set, we define by default adv_r(∅) = 0.Fix T'_u(r) = T_u(r) ∖ T_v(r) and T'_v(r) = T_v(r) ∖ T_u(r). Let T'_u,v(r) = T_u(r) ∩ T_v(r). It follows: b_u(r)=adv_r(T'_u(r)) + adv_r(T'_u,v(r)) 2 b_v(r)=adv_r(T'_v(r)) + adv_r(T'_u,v(r)) 2Given the above observation, we note that b_u(r) = b_v(r) if and only if adv_r(T'_u(r)) = adv_r(T'_v(r)). By definition, T'_u(r) and T'_v(r) have no values in common and at least one of these sets is non-empty. The bits used in these sums are all therefore pairwise independent and generated uniformly. The probability that both these sums are equal is exactly 1/2, and therefore so is the complementary probability of inequality. We next define the following useful potential function that captures the amount of information spreading still required in the network to solve gossip after a given round: ∀ r≥ 1: ϕ(r) = ∑_u∈ V( k-|T_u(r)| ).Notice that this function is non-increasing (as nodes never unlearn a token), and once the function evaluates to 0, there is no more information to spread and therefore gossip is solved. We now leverage the definition of potential function ϕ from above to define what it means for a round to be good with respect to making progress with the gossip problem: We say a given round r ≥1 is good if and only if one of the following two properties is true: (1) ϕ(r) = 0; or (2) ϕ(r+1) < ϕ(r). The following result leverages Lemma <ref> to formalize the key propertythat each round of our algorithm has a reasonable probability of being good by our above definition.For every round r, 1≤ r ≤ cN^2, the probability that round r is good is at least 1/4. There are two cases depending on the value of ϕ(r). If ϕ(r) = 0, then by definition this round is good. Else if ϕ(r) > 0, we must consider the probability that at least one node learns a new token in this round. To do so, fix some token t that is not known by all n nodes at the beginning of r (such a token must exist by the assumption that ϕ(r) >0). Let S be the nodes that know t.Because we assume the network topology is connected in each round, there must be an edge during round r between a node u∈ S and a node v∈ V∖ S.Because t∈ T_u(r) and t∉ T_v(r), we know T_u(r) ≠ T_v(r). By Lemma <ref>, the probability that b_u(r) ≠ b_v(r) is 1/2. Assume this event occurs. Also assume b_u(r)=1 and b_v(r)=0 (the opposite case is symmetric).By the definition of the algorithm, u will attempt to send a proposal in this round and it has at least one neighbor to choose from to receive this proposal. Let v' be the neighbor u chooses. Whether or not v'=v, we know that v' advertised 0 in this round. By Lemma <ref>, it follows that v' has a different token set than u in this round. Indeed, this must be true of v' and any node that sends it a proposal in this round.Now that we have established that v' receives at least one proposal, we know v' will form a connection this round. As we just noted, this connection will be with a node u' such that T_u'(r)≠ T_v'(r). Therefore, with high probability in n, the transfer subroutine will successfully identify a missing token to transfer between u' and v'—reducing ϕ.We have just shown that for r to be good in the case where ϕ(r) > 0, it is sufficient that the following two events occur: (1) b_u(r) ≠ b_v(r); and (2) the transfer subroutine between u' and v' succeeds. The first occurs with probability 1/2, and the second with high probability, which is at least 1/2 for n>1 (which must be true if ϕ(r) >0). Both events occur, therefore, with probability at least 1/4—as required. We can now leverage Lemma <ref> to prove Theorem <ref>. The key argument in the following is that ϕ(1) ≤ kn, therefore kn good rounds are sufficient to solve the gossip problem. With high probability, T=Θ(kn) total rounds is sufficient to achieve this goal—assuming that r̂ is long enough to supply random bits for T rounds. To assure this holds we fix the constant c in the definition of r̂ to be at least the constant identified in the analysis below for the definition of T (which turns out to be 32).Formalizing this intuition, however, requires some care in dealing with potential dependencies between different rounds with respect to their goodness.The potential function ϕ measures the number of missing values over the n total nodes. Each node can miss at most k values. Therefore: ϕ(1) ≤ kn. Because ϕ is non-increasing, it is sufficient to ask how many rounds are required to ensure kn good rounds with high probability. Here we show that 32kn rounds are more than sufficient.If we fix the constant c used in the definition of r̂ to 32, therefore, it follows that r̂ is sufficiently long to supply random bits for all 32kn rounds needed for high probability termination.Continuing with the proof, let X_r, for each round r≥ 1, be the random indicator variable that evaluates to 1 if and only if round r is good. Let Y_t, for some round count t ≥ 1, be defined as: Y_t = ∑_r=1^t X_r. The Y_t variable, in other words, measures the number of good rounds in the first t rounds. By Lemma <ref>, we know E(Y_t) ≥ t/4. Therefore, in expectation, 4kn rounds are sufficient to achieve kn good rounds. To achieve high probability, however, we cannot simply concentrate on this expectation as there may be dependencies between different X variables (e.g., the outcome in one round might increase the probability that the next is good).BecauseLemma <ref> establishes a lower bound on this probability that holds regardless of the execution history, we can deploy astochastic dominance argument to achieve our needed result. In more detail, let X̂_r, for each r ≥ 0, be the trivial random indicator variable that evaluates to 1 with independent probability 1/4. Let Ŷ_t = ∑_r=1^t X̂_r. Clearly, E(Ŷ_t) = t/4. Because the X̂ variables are pairwise independent, we can concentrate on this expectation. For example, fix t= 32kn. Applying the Chernoff bound from Section <ref> (Theroem <ref>) with δ = 1/2 and μ = E(Ŷ_t) = t/4 = 8kn, it follows: (Ŷ_t ≤ 4kn) ≤ e^- 8kn/8≤ e^-n < 1/n. That is, for this particular value of t∈Θ(kn), the probability that Ŷ_t is less than kn is small in n. We now note that for each r≥ 1, X_r stochastically dominates X̂_r. It follows that our above bound on Ŷ_t holds for Y_t as well—which is sufficient to conclude the proof.§.§ Eliminating the Shared Randomness AssumptionHere we discuss how to remove the assumption of shared randomness. In more detail, we describe SimSharedBit, a variation of SharedBit that does not use shared randomness. We emphasize that this new algorithm is existential instead of constructive. Formally, it depends on a small set of bit strings, called R', that we prove exists but do not explicitly construct. Accordingly, our maintheorem statement below references the existence of a string set R' for which SimSharedBit isan efficient solution.The SimSharedBit algorithm adds an additive cost of Õ(Δ^1/τ/α) rounds to the existing time complexity of SharedBit. For most combinations of Δ, τ, and α, and k,this additive cost is swamped by the O(kn) time complexity of SharedBit. For the worst-case values of these parameters, this extra cost can make SimSharedBit up to a factor of n slower than SharedBit(e.g., when k=1, α = 1/n, Δ = n-1, and τ=1).Strategy Summary. The high-level strategy for SimSharedBit is to first elect a leader that disseminates a seed string that can be used to generate sufficient randomness to run SharedBit. Notice, the number of shared bits required by SharedBit is much too large to be efficiently disseminated (our model restricts connections to deliver N bits per round, while SharedBit requires Ω(N^3) shared bits). The seed selected and disseminated by the leader, by contrast, is small enough to be fully transmitted over a connection in a single round. To prove that there exists a randomness generator that can extract sufficient randomness for our purpose from seeds of this small size, we adapt the technical details of Newman's Theorem ( e.g., <cit.>) from the simpler world of two-party communication to the more complicated world of n parties on a distributed and changing network topology. In more detail,we prove the existence of a multiset R', containing only poly(N) bit strings of the length required for SharedBit, that is sufficiently random to guarantee that if a leader chooses r̂ uniformly from R', the SharedBit algorithm using shared randomness r̂ is still likely to solve gossip efficiently. Because R' contains only poly(N) strings, the leader can identify the string it selected using only polylog(N) bits (this selection is the seed it disseminates)—enabling efficient dissemination of this information. The existential nature of SimSharedBit is entirely encapsulated in the existence of this set R'.Below we begin by describing the guarantees of the leader election primitive we will leverage in the SimSharedBit algorithm. We then describe the operation of SimSharedBit before proceeding with its analysis.Leader Election. To elect a leader we can deploy the BitConvergence leader algorithm described in our recent study of leader election in the mobile telephone model <cit.>. When run in a network with expansion α, stability factor τ≥ 1, and maximum degree Δ, this algorithm guarantees with high probability in N to solve leader election in O((1/α)Δ^1/τN) rounds. We emphasize that the algorithm does not require advance knowledge of α, Δ, or τ—its time complexity adapts to the network in which it is executed.To provide slightly more detail about this algorithm,in each round, each node identifies a single identifier to be its candidate leader for that round. To “solve leader election" means that eventually all candidate leaders in the network have permanently stabilized to the same identifier. As noted in <cit.>, a trivial extension to the algorithm allows each node to also generate a payload consisting of polylog(N) bits that follows its identifier. Each node now maintains a variable for its current candidate leader and a variable for that candidate's payload. We will leverage this payload in SimSharedBit to carry a pointer to a r̂ value from R'. Finally, we note that BitConvergence also maintains the useful property thatthe eventual leader will be the node with the smallest identifier of all participating nodes. This simplifies our analysis.The SimSharedBit Gossip Algorithm. We are now ready to describe the SimSharedBit gossip algorithm. This newgossip algorithm interleaves the BitConvergence leader election algorithm described above with the logic from SharedBit gossip. In more detail, we will prove below the existence of a multiset R', containing poly(N) bit strings, that is “sufficiently random" (a concept we will formalize soon) that it is sufficient for the nodes in the network to agree on a shared string r̂ sampled from R', instead of from the space of all possible strings of the needed length.In more detail, at the beginning of the execution, each node selects its own string from R' with uniform randomness. Assume we have fixed in advance a deterministic unique labeling of the poly(N) strings in R' with the values 1,2,...,| R'|. Each node can therefore refer to the string it selected with its label. Following the standard conventions of pseudoranomness, we call this label the seed for the string. Notice, each seed can be described with only polylog(N) bits. We take advantage of this small size by having each node run the leader election algorithm summarized above with this string stored in its payload. Therefore, once we elect a leader, all nodes also know its seed.To interleave gossip and leader election we will treat even and odd rounds differently. In even rounds, nodes execute the BitConvergence leader election algorithm described above, using their seed as their payload. In odd rounds, nodes execute the SharedBit gossip algorithm. In each odd round,each node uses as the shared string r̂ whatever string from R' is pointed to by the seed in their current candidate leader's payload. In defining R' below, we will fix the length of strings in this set to be slightly longer than the strings used by SharedBit, so as to capture the extra rounds required for the network to converge on a single string (the rounds before this point are potentially wasted with respect to making gossip progress).Proving the Existence of a Sufficiently Random R'. To prove SimSharedBit solves gossip efficiently with high probability, we must prove that a shared string sampled uniformly from R' is sufficiently random that the SharedBit logic executed in odd rounds will still solve gossip with high probability.To do so, we begin by establishing some preliminary assumptions and definitions. First, we note that the string r̂ used by SharedBit consists of t_SB = cN^2 groups consisting of N bundles that in turn each contain t_b =(⌈logN⌉ + 1) bits. The algorithm consumes bits from one group per round, and the analysis of SharedBit requires at most t_SB rounds worth of shared randomness to terminate with high probability.For SimSharedBit, we will need to extend this length to account for the early rounds in the execution when leader election has not yet converged, and therefore we cannot yet guarantee useful progress for the gossip logic executing in the odd rounds. For the worst case values of α, τ, and n, BitConvergence requires no more than t_BC = O(N^2N) rounds to converge. Therefore weextend the length of shared bit strings to consist of t_SSB = t_SB + t_BC = O(N^2N) groups. This ensures that after leader election converges we still have at least the full t_SB rounds of randomness needed for the analysis of SharedBit to apply. At the risk of slightly overloading previous notation,we will use R = {0,1}^t_SSB· N· t_b to refer to the set of all bit strings of length t_SSB· N· t_b—the maximum size shared string needed to give nodes time to converge to a leader and then subsequently solve gossip with the leader's shared string. The shared strings used in SimSharedBit come from R.Next, for a given network size n>1, let 𝔾(n) be the set containing every t_SB-round dynamic graph defined over n nodes. That is, if we run our algorithm for t_SB rounds in a network of size n, it will be executed in some dynamic graph G∈𝔾(n). Let A(n) be the set containing every assignment oftoken sets to the n nodes in a network of size n. We define “assignment" to capture two key pieces of information: (1) which nodes in the network started with a token; and (2) which of these tokens does each node know at the moment. Formally, a given A∈ A(n) can be described as a function from [n] to 2^n.[This function maps each of the n nodes to some subset of [1,n] indicating the tokens that node knows. The set of nodes that started with a token according to this assignment is the set of nodes that have a token show up somewhere in the assignment function's range.]For each network size n ∈ [2,N], round ℓ∈ [1,t_BC], dynamic graph G∈𝔾(n),token assignment A∈ A(n), and shared bit string r̂∈ R: let Z(n,ℓ, G, A, r̂) be the random indicator variable that evaluates to 0 if SharedBit solves gossip when run in a network of size n,starting with token assignment A, andexecuting for t_SB rounds in dynamic graph G, using the shared random bits from groups ℓ to ℓ + t_SB in r̂. It otherwise evaluates to 1. (In the evaluation of Z, assume that the probabilistic token transfer subroutine used by SharedBitalways works correctly.) Notice, we are using 0 to indicate a positive outcome (gossip works),and a 1 to indicate a negative outcome (gossip failed).In other words, Z(n,ℓ, G, A, r̂) answers the following question (with 0 indicating yes) : If we assume we are in a network of size n, and that leader election converges to a single leader at round ℓ, and this leader points toward shared string r̂, and that at this point the tokens in the network are spread according to A: will the SharedBit logic solve gossip sometime inthe next t_SB rounds, using the corresponding bits from r̂, assuming the graph evolves as G during this round interval?Our analysis of SharedBit tell us that if we select r̂ uniformly from R, with high probability: Z(n,ℓ, G, A, r̂) = 0. Our goal is to prove that there exists a multiset R', made up of values from R, such that R' only contains poly(N) strings, and yet if we select r̂ uniformly from R', the probability Z(n,ℓ, G, A, r̂) = 0 remains high. In particular, if ϵ is an upper bound on the small failure probability of SharedBit gossip when run in a setting with shared randomness, then we show the probability that Z evaluates to 1 when drawing r̂ from our multiset R' is at most only a constant factor larger. We formalize this goal with the following lemma. We emphasize that this setup (analyzing the probability that Z evaluates to 1 with our reduced R') comes from the proof of Newman's Theorem. We are generalizing this approach, however, to account for multiple nodes operating on a dynamic graph starting from an arbitrary round within a larger interval, with an arbitrary distribution of gossip tokens:There exists a multiset R' of size N^Θ(1) containing values from R, such that for every n ∈ [2,N],ℓ∈ [1,t_BC], G∈𝔾(n) and A∈ A(n), it follows:_r̂ R'(Z(n,ℓ, G, A, r̂) = 1) <2ϵ, where ϵ=N^-c (for some constant c ≥ 1) is an upper bound on the failure probability of SharedBit gossip when executed with shared randomness. Fix some network size n∈ [2,N], leader election termination round ℓ∈ [1,t_BC], G∈𝔾(n) and A∈ A(n). Consider an experiment in which we uniformly select t values r_1,r_2,...,r_t from R (with replacement), where t > 0 is a value defined with respect to N that we fix below. Let X_i be the random indicator variable defined as X_i = Z(n,ℓ, G, A, r_i).That is, X_i = 0 if SharedBit solves gossip using the relevant bits in r_i in Gstarting with assignment A.By Theorem <ref> and our definition of t_SB (which captures the worst case time complexity from this theorem),we know X_i = 0 with probability at least 1-ϵ.Therefore: E(X_i) = 0·(X_i=0) + 1·(X_i=1) ≤ϵ.Note that these random variables X_1, X_2, ..., X_t are i.i.d. as they are eachdetermined by a random string selected with uniform and independent randomness with replacement from a common set.It follows that we can apply a Chernoff-Hoeffding bound (Theorem <ref> from Section <ref>) to X_1, X_2, ..., X_t to prove that their average value is unlikely to deviate too much from the expected average. In more detail, let μ = E(X_i). This bound tells us that for any δ > 0:(1/t∑_i=1^t X_i ≥μ + δ) ≤ e^- 2 δ^2 t. Fix δ = ϵ and t = N^β/ϵ^2, for a constant β≥ 1 we will define below. We say for our fixed choice of n, ℓ, G and A, that a given selection of t strings from R is bad if 1/t∑_i=1^t X_i ≥ p = 2ϵ.For our fixed values of δ and ϵ, and our above bound, we know our random choice of strings is bad with probability no more than e^-2 N^β < 2^-N^β. Put another way,for a fixed network size, leader election termination round, dynamic graph and token assignment, we are very unlikely to have made a bad selection of strings. Now we consider other values for our parameters. We know there are no more than N choices for n and c'N^2N choices for ℓ, for some constant c' ≥ 1. For a given n, we can bound 𝔾(n) as |𝔾(n)| < (2^n^2)^t_SB = 2^n^2· t_SB≤ 2^N^γ, for some small constant γ≈ 4. And to bound A(n), we note: | A(n)| ≤ (2^n)^n ≤ 2^n^2≤ 2^N^2. The total number of combinations of n, ℓ, G and A values, therefore, is upper bounded by:N· (c'N^2N)· 2^N^γ· 2^N^2 ≤ c' · 2^logN^3+ log(N) + N^γ + N^2≤2^N^γ· c” for some constant c”≥ 1. Given this upper bound value, we fix the constant β used in the definition of t to be some constant strictly greater than c”·γ (say, ⌈ c”·γ + 1⌉).We now apply the probabilistic method to prove the existence of a selection of t values from R that is not bad for any of the possible combinations of network sizes, leader election termination points, graphs and token assignments. To do, note that the probability of a given selection being bad for a fixed set of parameters was shown above to be less than 2^-N^β. By applying a union bound over the less than 2^N^c”·γ combinations of parameters, the probability that there exists at least one such combination for which our selection is bad is less than: (2^N^c”·γ)·(2^-N^β) < 1.It follows that there exists at least one collection of t values from R that is not bad for every combination of the relevant parameters. Let us call this multiset of t values R'. The definition of being not bad for a given graph and assignment is that:1/t∑_i=1^t X_i ≤ 2ϵ. It follows that ∑_i=1^t X_i ≤ 2tϵ. From this it follows that at most a 2ϵ fraction of the X_i values evaluate to 1. Therefore, if we uniformly sample a string r_i from R', the probability that X_i = 0 is at least 1-2ϵ, as required by the lemma statement.To conclude the proof, we must show that | R'| = t is in poly(N). We earlier fixed: t=N^β/ϵ^2, where β = Θ(1) and ϵ = N^-c for a constant c≥ 1. It follows that t=N^β+2c = N^Θ(1).We now leverage Lemma <ref> to prove our main theorem concerning SimSharedBit:There exists a bit string multiset R' of size N^Θ(1), such that the SimSharedBit gossip algorithm using this R' as its source of simulated shared bit strings solves the gossip problem in O(kn + (1/α)Δ^1/τlog^6N) rounds when executed with tag length b=1 in a networkwith stability τ≥ 1. Fix the multiset R' proved to exist in Lemma <ref>. We now study the performance of SimSharedBit using this multiset as the source of shared random strings selected by leader candidates.First, we note that by Theorem <ref>, we know that SharedBit gossip solves gossip in O(kn) rounds with high probability. In <cit.>, we proved that BitConvergence leader election solves leader election inO( (1/α)Δ^1/τlog^6N) rounds with high probability. In Section <ref>, we proved that the transfer routine succeeds with high probability. By a union bound, we can therefore assume that with (slightly less) high probability the transfer routine works every time it is called in a poly(N) round execution.Let ϵ be the smallest of these three small failure probabilities. In a given execution of SimSharedBit, it follows (by a union bound) that the probability that the transfer routine fails at least once, or BitConvergence fails to elect a leader in the provided time bound, is less than 2ϵ.Assume neither of these two bad events occur. We now study the probability that SimSharedBit, running with a r̂ selected uniformly by the node with the smallest ID from the R', starting from the round right after leader election succeeds, and runnings on the given dynamic graph for the execution. By Lemma <ref>, the probability that SimSharedBit fails to solve gossip is also less than 2ϵ.A final union bound on these two failure probabilities establishes that the probability SimSharedBit gossip fails is less than 4ϵ, and therefore it succeeds with probability at last 1-4ϵ. So long as we set the constant factors in the time complexity of SharedBit, BitConvergence, and the transfer routine, to ensure that ϵ≤1/4N,SimSharedBit succeeds with high probability.§ GOSSIP WITH B=1 AND Τ = ∞Here we describe and analyze a gossip algorithm that requires only Õ(k/α) rounds when executed with b=1 and a stable network (where Õ hides N factors). Because Ω(k) is a trivial lower bound for gossip k messages in our model, this algorithm is optimal for larger α. Recall that for τ≥ 1 our best solution required O(kn) rounds. This algorithm matches this time for the worst-case α values but then improves over it as α increases. For constant α, this algorithm performs a factor of n faster (ignoring log factors). These results indicate that network stability is valuable from a gossip algorithm perspective. Notice, for the sake of presentation clarity, the algorithm analysis that follows does not attempt to optimize the polylogarithmic factors multiplied to the leading k/α term.Discussion: Crowded Bins We call this algorithm CrowdedBin gossip. This name comes from a core behavior in the algorithm in which nodes toss their tokens into a fixed number of bins corresponding to their current estimate k̂ of k (the number of tokens in the network). Nodes do not know k in advance. Determining this value is crucial to enabling efficient parallel dissemination of their tokens. Leveraging a new balls-in-bins analysis, we upper bound the number of tokens in any given bin if the estimate k̂ is sufficiently large. The nodes therefore search for crowded bins as evidence that they need a larger estimate of k. This mechanism provides a way to check that a current guess k̂ is too small while only paying a time complexity price relative to k̂ (as there are only k̂ bins required to check for crowding). Because the sequence of guesses we try are geometrically increasing, the cost of checking estimates smaller than k will sum up to Õ(k).Discussion: Spreading Bits versus Spreading Tokens. We also emphasize that the CrowdedBin algorithm makes a clear distinction between propagating information using the advertising bits and propagating the tokens themselves (which are treated as black boxes, potentially large in size, that require a pairwise connection for transfer). Combining the stability of the network with each node's ability to advertise a bit to all its neighbors in each round, nodes first attempt to stabilize to a consistent and accurate estimate of k, and a consistent set of tags describing the network's tokens. Once stabilized, this information can then support the efficient spreadingof the tokens, link by link, to the whole network. The PPUSH Rumor Spreading Strategy. The CrowdedBin algorithm uses a simple rumor spreading strategy called PPUSH as a subroutine to help spread tokens once the network has stabilized. This algorithm was introduced in our earlier study of rumor spreading in the mobile telephone model <cit.>. PPUSH assumes a subset of nodes start with a common rumor m, and the goal is to spread m to all nodes. It requires b ≥ 1.In more detail, the strategy PPUSH works as follows: (1) at the beginning of each round, if a nodes knows m (i.e., it is informed), it advertises bit 1, otherwise if it does not know m (i.e., it is uninformed), it advertises bit 0; (2) each informed node that has at least one uninformed neighbor in this round,chooses an uninformed neighbor with uniform randomness and attempts to form a connection to spread the rumor.In <cit.>, we proved the following key result about the performance of PPUSH: With high probability in N: PPUSH succeeds in spreading the rumor to all nodes in O(log^4N/α) rounds when executed in the mobile telephone model with b≥ 1, τ = ∞, and a topology graph with expansion α.We will leverage this theorem in our analysis of our gossip algorithm. We also use the following useful property proved in <cit.> which relates network diameter to expansion:[The actual result we proved in <cit.> is that it is always possible to spread a rumor in O(logn/α) rounds in the mobile telephone model in a graph with expansion α. The rumor spreading time in a given network can never be smaller than the network diameter, which provides a trivial lower bound on the problem.]Fix a connected graph with n nodes, expansion α, and diameter D. It follows that D = O(logn/α).§.§ The CrowdedBin Gossip AlgorithmWe divide our description of this analysis into several named parts to clarify its presentation. In the following, we assume each node u∈ V identifies itself with a tag t_uchosen uniformly from the space {1,2,...,N^β}, where β≥ 2 is constant we fix in our analysis. Let ℓ = βlogN be the number of bits needed to describe a tag. To simplify notation, we assume in the following that N is a power of 2.Parallelizing Instances. Nodes do not know in advance the value of k (the number of tokens in the system). They consider logN estimates of k: k_1, k_2,...,k_logN, where each k_i = 2^i. The nodes run in parallel a separate gossip instance for each estimate. We use the notation instance i to refer to the instance corresponding to estimate k_i. In order to run logN instances in parallel, each node uses logN rounds to simulate one round each of the logN instances. That is, nodes divide rounds into simulation groups consisting of logN rounds. Round j of simulation group i is used to simulate round i of instance j.Instance Schedules.Each instance i groups its rounds into blocks containing ℓ + logN rounds each. It then groups these blocks into bins containing γlogN blocks each, where γ > 1 is a constant we fix in our analysis below. Finally, it groups the bins into phases consisting of k_i bins each. In other words, the schedule for instance i is made up of phases, where each phase has k_i bins, which are each made up of γlogN blocks, which each contain ℓ+logN rounds: adding to a total of γ(β+1) k_i log^2Ntotal rounds per phase. Initialization. Each node u∈ V that begins an execution of the CrowdedBin algorithm with a gossip token, independently selects a bin for its token for each of the logN instances. That is, for each instance i, u selects a bin b_u(i) with uniform independent randomness from {1,2,...,k_i}. Each node u also maintains, for each instance i, and each bin j for this instance, a set T_u(i,j) containing the tags it has seen so far for tokens in bin j in instance i. For each instance i, if node u has a token it initializes T_u(i, b_u(i))= {t_u} (i.e., it places its own tag in the bin it selected for that instance). Node u also maintains a set Q_u containing the tokens it has received so far, where each token in Q_u is also labeled with its tag. Finally, each node u maintains a variable est_u, initialized to 1, which describes the current instance node u is participating in. Participation. Each node will only participate in a single instance at a time, and it will only participate in complete phases of an instance. In more detail, if some instance i starts a new phase in round r, and some node u has est_u = i at the start of round r, node u is now committed to participate in this full phase of instance i. As we will detail, its estimate cannot change again until this phase completes.To participate in a phase of instance i, node u does the following. First, for each bin j, 1 ≤ j ≤ k_i, u orders the tags in T_u(i,j) (if any) in increasing order. It will use the first ℓ rounds of the first block to spell out the smallest such tag, bit by bit, using its advertising bits (here the assumption that b≥ 1 is needed). It will then use the first ℓ rounds of the second block to spell out the second smallest tag, and so on. There are γlogN total blocks in this bin. If u knows more than this many tags for this bin, it transmits only the first γlogN. Node u transmits all 0's during the blocks in this bin for which it has no tags to advertise (here is where we use the assumption that the smallest possible tag is 1—preventing a block of all 0's from being mistaken for a tag.)During the rounds dedicated to bin j, node u also collects the bits advertised by its neighbors in each block. If it learns of a tag t_v that is not currently in T_u(i,j), it will put it aside and then add it to this set once the rounds dedicated to bin j in this phase conclude. We have only so far described what node u does during the first ℓ rounds for each block in our fixed instance j. During the remaining logN rounds in these blocks, u will attempt to disseminatethe actual tokens corresponding to the tags advertised (here we emphasize the difference between spelling out the bits of a tag using advertising bits and actually transmitting a token, which requires two nodes to form a connection). In more detail, u executes the PPUSH rumor spreading strategy discussed above during the last logN rounds of each block in the current bin. In more detail, for a given block h in this bin, if u advertised tag t in the first ℓ rounds of this block, and u actually has the token corresponding to tag t in Q_u, it executes PPUSH in the remaining rounds of this block using thistoken as the rumor and advertising 1 (i.e., it runs PPUSH with the status of an already informed node). Otherwise, node u runs PPUSH advertising 0 (i.e., it runs the PPUSH as an uniformed node).Increasing Size Estimates. A core behavior in this algorithm is how nodes upgrade their current estimate of the value k (stored in est_u for each node u). As described above, each node initializes their estimate to 1. As described below, these estimates can only grow during an execution. We call an increase in this estimate at a given node an upgrade. There are two events that trigger an upgrade at a given node u.The first event is that node u sees “activity" on an instance i' > est_u, where est_u is its current estimate. The term “activity" in this context means seeing a 1-bit advertised in an instance i' round.If this event occurs, then u knows that some other node has already increased its estimate beyond est_u, so u should upgrade its estimate as well.The second event is that node u fills a bin in its current estimate. That is, there is some bin j such that |T(est_u, j)| ≥γlogN. We call this event a crowded bin, and u can use this as evidence that est_u does not have enough bins for the number of tags in the system and therefore est_u is too small of an estimate for k. If this event occurs, u will increase est_u by 1 (unless est_u is already at its maximum value in which case it will remain unchanged.).Recall, as specified above, that if a node u increases its estimate est_u to a new value, it will complete the phase of whatever instance it was participating in before switching to the new estimate moving forward. This restriction simplifies the analysis that follows. §.§ Analysis In the following analysis, let D be the diameter of the fixed underlying topology graph. Some of intermediate results below will reference D. Our final result, however, will be expressed only with respect to α to maintain comparability to earlier results defined for non-stable networks in which D is not well-defined.At the beginning of an execution each node randomly assigns a tag from {1,2,...,N^β} to its token, and then randomly assigns the token to a bin in each of the logN instances. We call the global collection of these assignments for a given execution a configuration. Fix a configuration. We call a given instance i of this configuration, 1≤ i ≤logN, crowded, if the configuration has an instance i bin with at least γlogN unique tags assigned to it. The target instance for our fixed configuration is the smallest instance i that is not crowded. If every instance is crowded, then we say the target instance is undefined.We begin our analysis by defining what it means for a configuration to be good with respect to these terms:A configuration is good if and only if it satisfies the following two properties: (1) every token is assigned a unique tag; and (2) the target instance i is defined, and k_i ≤ 2k. A direct corollary of the above definition is that if a configuration is good, and i is the target, then k_i > k/(γlogN). We now bound the probability that the nodes generate a good configuration. We will show that increasing the constant β, used to define the space {1,2,...,N^β} from which tags are drawn, and the constant γ, used to define the number of blocks per bin,increases the high probability that a configuration is good. To make this argument we begin by proving a non-standard balls-in-bins argument that will prove useful to our specific algorithm's behavior.Fix some constant γ≥ 9. Assume k balls, 1 ≤ k ≤ N, are thrown into k' ≥ k bins with independent and uniform randomness. The probability that at least one bin has at least γlogN balls,is less than 1/N^(γ/3) -2. Label the balls 1,2,...,k and the bins 1,2,...,k'. Let b_1 be bin in which ball 1 is thrown. We now calculate the expected number of other balls to land in b_1. To do so, for each ball i > 1,let X_1 be the random indicator variable that evaluates to 1 if i lands in b_1 and otherwise evaluates to 0. Let Y_b_1 = ∑_1< i≤ k X_i be the total number of additional balls to land in b_1. By linearity of expectation and the observation that E(X_i) =1/k' ≤ 1/k, it follows that μ = E(Y_b_1) < 1.By definition of the process, X_i and X_j are independent for i ≠ j. We can therefore apply an upper bound form of a Chernoff Bound(Theorem <ref>) to concentrate near this expectation. In particular, define δ = (γlogN - 2)/μ. Notice, δ > (γlogN -2) > 1. We can therefore apply Theorem <ref> to Y=Y_b_1, and our above definitions of δ and μ. It follows that: (Y_b_1≥ (1+δ)μ)≤ exp{- (γlogN -2)/3} =exp{ -((γ/3)logN - 2/3) } =exp{ -((γ/3)lnNloge - 2/3) } <e^2/3/e^(γ/3)lnN < 2/N^γ/3≤ 1/N^γ/3 - 1 Notice, (1+δ)μ = μ + (γlogN - 2), and μ = 1/k' ∈ (0,1).Therefore, we can interpret the above bound saying that the probability that b_1 has at least γlogN -1 extra balls is less than 1/N^γ/3 -1. When we add in ball 1, which by definition is also in b_1, we get that the probability that b_1 has at least γlogN ballsis also less than 1/N^γ/3 -1. By symmetry, the same result holds for b_2 through b_k as well. There are dependencies between the outcomes in different bins, but we can dispatch this issue by applying a union bound over the k≤ N occupied bins, which provdes that the probability at least one bins has more than γlogN balls is less than N/N^(γ/3) - 1 = 1/N^(γ/3) - 2. Fix some constant c≥ 1. For a tag space constant β≥ c+3, and a bin size constant γ≥ 3c + 9, the nodes generate a good configuration with probability at least 1-1/N^c. There are two parts to the definition of good. The first requires each tag to be unique. The probability that there is at least one collision among the tag chocies, given that no more than N tags are drawn from N^β options, can be loosely upper bounded as 1/N^β -2. If we define β = c+3 then this failure probability is less than 1/N^c+1.The second part of the definition requires that the target instance is defined and it is not too large compared to the actual number of tokens, k. Let î = argmin_1 ≤ i ≤logN{k ≤ k_i}. That is, k_î is the smallest estimate of k considered by our algorithm that is at least as large as k. Because our estimates grow by a factor of 2, we know that k_î < 2k. If we can show that k_î is not crowded, therefore, it will follow that the target instance i for this configuration is defined,and i ≤î: which is sufficient to satisfy the second part of the definition of good. To make this argument, we can treat the selection of bins for each token in instance î as a balls in bins problem. We therefore apply Lemma <ref> to k and k'=k_î, which tells us that for any constant γ≥ 9, the probability that instance î crowded is less than 1/N^(γ/3) -2. If we set out bin size constant γ≥ 3c+9, this probability is less than 1/N^c+1.Pulling together the pieces, for β≥ c+3 and γ≥ 3c + 9, a union bound provides that the probability that we fail to satisfy at least one of the two parts of the definition of good is less than 2/N^c+1≤ 1/N^c, satisfying the lemma statement. Now that we have established that good configurations are likely, we establish the below lemma about these configurations that follows directly from the definition of good and the mechanism by which our algorithm updates estimates: In an execution with a good configuration with target instance i, no node ever sets its local estimate to a value larger than i. That is, for all u and all rounds, est_u ≤ i.We now continue our analysis by bounding the time required for all nodes to reach the target instance. We do so with two arguments: the first concerning the rounds required for nodes to learn of a larger estimate existing in the system, and the second concerning the rounds required for the largest estimate to increase if it is still less than the target. For the following results, recall that D is the network diameter.Fix an execution with a good configuration with target instance i. Assume that at the beginning of round r of this execution the largest estimate in the system is i_max≤ i. By round r' = r + O(D k_i_maxlog^3N) either: the largest estimate in the system is larger than i_max, or all nodes have estimate i_max.Fix a node u that has est_u=i_max at the beginning of round r. If u maintains that estimate at the beginning of its next instance i_max phase, then during that phase it will advertise at least one 1-bit (as it has at least its own tag in one of the bins for this instance). It follows that all u's neighbors in the underlying topology will learn that u has est_u = i_max and will upgrade their estimate to i_max, if their estimate is currently less than this value. We can then repeat this argument for u's neighbors, then their neighbors, and so on until either: at least one node adopts a larger estimate than i_max (which might impede the application of this logic), or all nodes adopt i_max. If the first event occurs, we satisfy the lemma statement. If the first event does not occur, the second event will occur after at most diameter D+1 instance i_max phases (the extra phase upper bounds the rounds required between round r and the start of the next instance i_max phase). The number of rounds to complete an instance i_max phase can be calculated as: k_i_max bins times γlogN blocks per bin times ℓ + logN = O(logN) instance i_max rounds per block times logN real rounds for each instance i_max rounds. This product evaluates to O(k_i_maxlog^3N) rounds per instance. Therefore, O(D k_i_maxlog^3N) rounds are sufficient to guarantee the lemma statement holds. Fix an execution with a good configuration with target instance i. Assume that at the beginning of round r of this execution the largest estimate in the system is i_max < i. By round r' = r + O(D k_i_maxlog^3N) the largest estimate in the system is larger than i_max.We start by applying Lemma <ref> to i_max and round r. This establishes that by round r' = r + O(D k_i_maxlog^3N) rounds either all nodes have estimate i_max, or at least one node has an estimate larger than i_max. If the latter is true than the lemma is satisfied directly at round r'.Moving forward, therefore, assume all nodes have the same estimate i_max by round r'.By assumption, i_max < i. It follows that instance i_max has at least one crowded bin. Call this bin j.Let T_j be the tags of the γlogN smallest tokens assigned to bin j in instance i_max in this configuration. Because nodes spell out tags from order of smallest to largest,we know that any node that knows tags from T_j, will assign each of these tags a block in any execution of instance i_max.It follows, therefore, that in each execution of an i_max phase, if all nodes start that phase with an estimate of i_max, then each of these tags in T_j will spread another hop. Applying the same argument as in the proof of Lemma <ref>, after at most D executions of i_max phases, either at least one node has increased its estimate to a value larger than i_max, or the tokens in T_j will have spread to all nodes in the network. If the latter event happens, then, by the definition of the algorithm, all nodes will have discovered a crowded bin in instance i_max and will increment their estimate. Either way, the lemma is satisfied. Therefore, by round r' + O(D k_i_maxlog^3N) = r+ O(D k_i_maxlog^3N), the conditions of the lemma is satisfied—as required. The following key result leverages Lemmas <ref> and <ref> to bound the total rounds required for all nodes to permanently stabilize their estimates to the target instance. Fix an execution with a good configuration with target instance i. By round r = O(Dk_ilog^3N), every node has estimate i. That is, for every node u, est_u=i by round r.By the definition of our algorithm, estimates never decrease. By Lemma <ref>, no node will ever adopt an estimate greater than i. Combined, it follows that we can keep applying Lemma <ref> to increase the largest estimate until the largest estimate reaches i. We can then apply a single instance of Lemma <ref> to ensure all nodes have this estimate—at which point the lemma will be permanently satisfied.To bound the time required for these applications of the above lemmas, we leverage our observation that the largest estimate can only increase. It follows that in the worst case we apply Lemma <ref> exactly once for each of the estimates leading up to the target i. Because these estimates form a geometric sequence (e.g., 2,4,8,...), the total rounds needed for these applications of Lemma <ref> is upper bounded by: O(D k_1log^3N) +O(D k_2log^3N) + ... +O(D k_ilog^3N) = O( (Dlog^3N)(k_1 + k_2 + ... + k_i ) ) =O(Dk_ilog^3N) The final application of Lemma <ref> to spread estimate i to all remaining nodes once it exists in the system adds only a single aan additional O(Dk_ilog^3N) rounds. The lemma statement follows. The preceding arguments bound the rounds required for useful information to propagate through the network via the nodes' advertising bits. We now conclude our proof by turning our attention to the rounds required forthe actual tokens (which must be passed one at a time through pairwise connections) to spread. We will tackle this problem by picking up where Lemma <ref> left off: a point at which the system is prepared for the PPUSH instances executing in the second half of blocks to make consistent progress. We will apply our bound on PPUSH from Theorem <ref> to establish the time required for this final propagation. We will then leverage Theorem <ref> to replace the network diameter in our complexity with an upper bound expressed with respect to the network size and expansion. The CrowdedBin gossip algorithm solves the gossip problem in O((1/α)klog^6N) rounds when executed with tag length b=1 in a network with stability τ = ∞.Assume for now that the configuration is good and i its target instance. Let round r = O(Dk_ilog^3N) be the round specified by Lemma <ref> for the network to converge its estimate. That is, every node has the same estimate i by round r. By definition, no bin is crowded for instance i in a good configuration. It follows that every tag for every bin in this instance will be spread inevery round by the nodes that know that tag in that round. Following the same propagation arguments used in Lemmas <ref> and <ref>, after at most D more phases of instance i, all nodes will know all tags.This requiresat most O(Dk_ilog^3N) rounds. Therefore by some round r' = O(Dk_ilog^3N), the system will have reached a stable state in which every node has the same estimate i and knows the tag for every token in the system. This information will never again change so we can turn ourattention for the rounds required to finish propagating the actual tokens after this point of stabilization.To bound this token propagation time, fix an arbitrary token t with tag q in instance i. Because we assume the system has stabilized, every node has q assigned to the same block of the same bin in their instance i phase. It follows that if we append together the last logN rounds from these blocks (i.e., the rounds in which nodes run PPUSH for the tag described in the first ℓ rounds of the block), we obtain a proper execution of PPUSH rumor spreading for token t during these rounds. That is, every time we come to the last logN rounds of q's block, all nodes are running PPUSH for rumor t, picking up where they left off in the previous instance.Applying Theorem <ref> from above, it follows that with high probability in N, O(log^4N/α) rounds are sufficient for t to spread to all nodes after stabilization. Each phase provides logN rounds of PPUSH, so O(log^3N/α) phases are sufficient after stabilization. The key observation is that each execution of instance i services all k rumors after stabilization, as each rumor has its own fixed bin in the instance i phase. Therefore, O(log^3N/α) phases are sufficient to spread all k rumors in parallel. A union bound establishes that all k ≤ N instances succeed with a slightly reduced high probability.From a probability perspective, we know from Lemma <ref> that the configuration is good with high probability. We just argued above that if the configuration is good, then with an additional high probability the tokens will all spread in the stated time, once the system stabilizes. We can increase both high probabilities to the desired exponent by increasing the constant β and γ used in the definition of crowded bins, and the constant factor in the time bound for PPUSH. A union bound then shows that both good events occur with high probability.From round cost perspective, we established that the time to stabilization isat most O(Dk_ilog^3N) rounds, while the time to complete propagation after stabilization is at most O(log^3N/α) instance i phases, which each require O(k_ilog^3N) rounds. The final time complexity is then in: O(Dk_ilog^3N + (k_ilog^6N)/α).By the definition of a good configuration, we know k_i ≤ 2k, and by Theorem <ref>, we know D=O(logN/α). We can therefore simplify this complexity to O((klog^6N)/α) rounds, as required. § Ε-GOSSIP WITH B=1 AND Τ≥ 1 In this section we consider ϵ-Gossip: a relaxed version of the gossip problem that is parameterized with some ϵ, 0 < ϵ < 1 (e.g., as also studied in <cit.>). In more detail, the problem assumes all n nodes start with a token. To solve ϵ-gossip there must be a subset S of the n nodes in the system, where |S| ≥ϵ n and for every u,v∈ S, u knows v's token and v knows u's token. Our goal here is to prove that for reasonably well-connected graphs and constant ϵ, almost solving gossip can be significantly faster than fully solving gossip. In particular, we prove that our SharedBit algorithm from before solves ϵ-gossip in O(n√(ΔlogΔ)/(1-ϵ)α) rounds. Given that Δ≤ n,this is faster than the O(n^2) required by SharedBit (for k=n) when ϵ is a constant fraction and α = ω(logΔ/(√(ΔlogΔ))). Preliminaries. We restrict our attention in this analysis to the case where ϵ≥ 1/2. We can then handle smaller values for this fraction by applying the below analysis for ϵ=1/2: a value that (more than) solves the problems for the smaller fraction, and at a cost of at most an extra constant factor in the time complexity (i.e., when we replace (1-ϵ) in the denominator with (1-1/2), where ϵ < 1/2 is the actual value we are analyzing, the stated bound is less than a factor of two larger than what we would get with the smaller ϵ).A key tool in our analysis is a set that describes the frequency of different token sets owned by nodes in the network at the beginning of a given round. To do so, let T be the set of tokens in the network. The definition of ϵ-gossip requires that |T|=n. For each token subset S⊆ T and round r ≥ 1, we define: count(S,r) = |{ u∈ V| T_u(r) = S }|,where T_u(r) is defined the same as in our above SharedBit analysis (i.e., the set of tokens u knows at the beginning of round r). Therefore, count(S,r) equals the number of nodes with token set S at the beginning of r. We now use the definition of count to define, for each round r ≥ 1, the following multiset: F(r) = { (S, q) | (S⊆ T)∧ (q=count(S,r)) ∧ (q ≥ 1)} This multiset contains all the token sets that appear at least once in the network at the beginning of round r,along with their frequency of occurrence. Finally, we also make use of the following potential function ϕ, which was first defined in Section <ref> to analyze SharedBit gossip: ∀ r≥ 1: ϕ(r) = ∑_u∈ V( n-|T_u(r)| ). Our analysis will also leverage two useful lemmas from our earlier study of rumor spreading in the mobile telephone model <cit.>. The first lemma is graph theoretic, and accordingly requires two definitions concerning graph properties. First, for a given graph G=(V,E) and node set S ⊂ V, we define B_G(S) to be the bipartite graph containing all (and only) the edges from E that connect a node in S to a node in V∖ S, with a vertex set consisting of these endpoints. Second, for a given graph H, let ν(H) the edge independence number of H, which describes the size of a maximum matching on H. We now proceed with our lemma:Fix a graph G=(V,E) with |V| = n and vertex expansion α. Fix some S⊂ V such that |S| ≤ n/2. It follows that ν(B_G(S)) ≥ |S| · (α/4). The second lemma adapted from <cit.> is algorithmic in that bounds the performance of a simple randomized strategy for approximating a maximum matching in a bipartite graph: Fix a network topology graph G=(V,E) with maximum degree Δ.Fix some subset C⊂ V. Assume there is a matching M of size m≥ 1 defined over B_G(C). Assume each node in C randomly chooses a neighbor in B_G(C) to send a connection proposal. With constant probability, at least Ω(m/√(ΔlogΔ)) nodes from V∖ Cthat are endpoints in M will receive a connection proposal from a node in C.Analysis. Our main strategy is to attempt to identify for each round a coalition of nodes such that: (1) the size of the coalition is within a target range (ϵ/2) n to ϵ n; and (2)no node in the coalition has the same token set as a node outside the coalition. If we can find such a coalition, the graph property result captured in Lemma <ref> tells us that there are many edges between coalition and non-coalition nodes (where the definition of “many" depends on α and ϵ). We can then show that a reasonable fraction of these edges will connect and therefore reduce ϕ. We begin this argument by leveraging the above definitions to prove that either we can find such a coalition or we have already solved the problem.Fix a round r ≥ 1. One of the following must be true about this round: (1) ϵ-gossip is solved by the beginning of round r; or (2) there exists a C ⊂ F(r) such that: (ϵ/2)n ≤∑_(S,q)∈ C q ≤ nϵ.Let q_max = max{ q : (*,q)∈ F(r)} (i.e., the number of nodes that own the set owned by the most nodes in r). We consider three cases for q_max and show that all three satisfy our lemma.The first case is that q_max > nϵ. In this case, we have identified a token set S that is owned by more than nϵ nodes. Let V_S be the set of nodes that own S at the beginning of r. Because every node starts with its own token in its token set, and no token ever leaves a token set, we know for each u∈ V_S, u's token is in S. It follows that every node in V_S knows the token of every other node in this set—meaning we have solved ϵ-gossip and therefore satisfy option (1) from the lemma statement.The second case is that(ϵ/2)n ≤ q_max≤ nϵ. In this case, we can set C={(S,q_max) }, where S is the set we identified owned by q_max nodes (if more than 1, choose one arbitrarily),and directly satisfy option (2) from the lemma statement.The third and final case is that q_max < (ϵ/2)n. In this case, we can apply the following simple greedy strategy for defining C: keep adding pairs from F(r) to C in decreasing order ofq values until ∑_(S,q)∈ C q first grows larger than (ϵ/2)n. By our case assumption, every q value in F is less than (ϵ/2)n. Therefore, the step of the greedy strategy that first pushes us over the (ϵ/2)n threshold must increase this sum to fall within our target range of (ϵ/2)n and ϵ n. That is, the greedy strategy described above will always terminate having identified a set C that satisfies option (2) from the lemma statement. Repeatedly applying Lemma <ref> will provide that in each round either we are done with the ϵ-gossip problem or we have a large coalition that is likely to generate lots of progress toward solving the problem. We are now ready to pull together our pieces to prove our main theorem. The main technical contribution of the below proof is arguing that a large coalition likely generates lots of new token transfers.This claim will pull from Lemmas <ref> and <ref> from above, as well as Lemma <ref> from the SharedBit analysis in Section <ref>. Fix some ϵ, 0 < ϵ < 1. The SharedBit gossip algorithm solves the ϵ-gossip problem in O(n√(ΔlogΔ)/(1-ϵ)α) rounds when executed with shared randomness with tag length b=1 in a network with stability τ≥ 1. Fix some ϵ that satisfies the theorem statement. Assume w.l.o.g. that ϵ≥ 1/2 (as argued at the beginning of thisanalysis, if ϵ is smaller, we can apply our analysis for ϵ=1/2 which more than solves the problem at the cost of only an extra constant factor in the stated time complexity). We begin by focusing on a single round, then extend the argument to the full execution. In particular, fix a round r, 1≤ r ≤ cN^2 (i.e., a round for which we still havebits in the shared string r̂ used by SharedBit). Let G_r = (V,E) be the network topology graph in this round. Assume ϵ-gossip has not finished by the beginning of this round. By Lemma <ref>, there exists a C⊂ F(r) such that:(ϵ/2)n ≤∑_(S,q)∈ C q ≤ nϵ.Let V_C be the set of nodes that start round r with one of the token sets in C. By our above assumption: (ϵ/2)n ≤ |V_C|≤ nϵ. Let q = min{ |V_C|, |V ∖ V_C|}. It follows that q ≤ n/2. By Lemma <ref>, therefore, there exists a matching M of size m≥ (α/4)q in B_G_r(V_C) (the bipartite subgraph of G_r that keeps only edges from E with one endpoint in V_C and one endpoint in V∖ V_C). For each edge e∈ M, we define e.c to be the endpoint from e in V_C and e.v to be the endpoint from e in V ∖ V_C.We say an edge e∈ M is wasted if both endpoints in e advertise the same bit; i.e., b_e.c(r) = b_e.v(r). By the definition of the coalition used in Lemma <ref>, it follows for each e∈ M it must be the case that T_e.c(r) ≠ T_e.v(r). We can therefore apply Lemma <ref> which provides that the probability they advertise different bits is 1/2.The probability that e is wasted is therefore also 1/2.To argue more precisely about wasted edges we define some random variables. For each e∈ M, let X_e be the random indicator variable that evaluates to 1 if e is wasted and otherwise evaluates to 0. Let Y= ∑_e∈ M X_e. By linearity of expectation and our above argument about the probability of wastefulness, it follows: E(Y) = m/2.We now want to bound the probability that the actual number of wasted edges is not too much larger than E(Y). We cannot apply a Chernoff-style bound as there might be dependency between the outcomes of different edges in M (as they may share tokens, and therefore share random bits used to determine their tag). To sidestep these issues, we apply Markov's Inequality (Theorem <ref> in Section <ref>) to derive the following: (Y ≥ (3/2)· E(Y)) ≤E(Y)/(3/2)· E(Y) = 2/3. Notice that (3/2)· E(Y) = (3/4)· m. We can therefore reword this result to say that with probability at least 1/3, at least m/4 edges in M are not wasted. For clarity, we will subsequently refer to an edge from M that is not wasted as an edge that is primed (as in the edge is primed for the possibility of its endpoints connecting in a manner that helps spread tokens).Moving forward in this analysis, assume this event occurs, and therefore at least m/4 edges in M are primed. Let M̂⊆ M be this set of primed edges. (Notice, because m≥ 1 and the size of M̂ must be a whole number, we know M̂ is non-empty under this assumption.)We want to now apply Lemma <ref> to the connections described by M̂. To do so, let Ĉ be the endpoints in M̂ that advertise a 1 in this round. Let Ĝ be the topology graph G_r for this round modified such that we remove every node that is not in Ĉ, but neighbors Ĉ and also advertises a 1 (along with their incident edges).We emphasize two properties of this modification: (1) by definition, no node in M̂ is removed by this step; (2) it is correct to say that nodes in Ĉ will choose a neighbor from Ĝ uniformly to send a connection proposal, because the SharedBit algorithm only has nodes that advertise a 1 choose among neighbors that advertise a 0, and we only removed neighbors from Ĉ nodes that also advertised a 1.We can therefore apply Lemma <ref> with G=Ĝ, C=Ĉ, and M=M̂. It follows that with constant probability, at least Ω(|M̂|/√(ΔlogΔ)) = Ω(m√(ΔlogΔ)) nodes in M̂ receive a connection proposal fromtheir neighbor in this matching. Each such node u will subsequently connect with some node v in this round (though not necessarily its neighbor in M̂). By Lemma <ref>, however, T_u(r) ≠ T_v(r) (as each advertised different bits in r), so each of these connections reduces ϕ by at least 1.Combining our probabilistic events from above, it follows that with constant probability, ϕ(r+1) - ϕ(r) ≥δ∈Ω(α q/√(ΔlogΔ)), where, as defined above, q=min{|V_C|, |V ∖ V_C|}. Let us call a round in which this event occurs a good round. To bound the number of good rounds until ϕ reduces to 0 (and the ϵ-gossip problem is solved, regardless of ϵ), we must first lower bound the size of δ. To do so, we first note that |V ∖ V_C| ≥ (1-ϵ)n. It follows that in the case where q=|V ∖ V_C|, we know q ≥ (1-ϵ)n. On the other hand, if q= |V_C|, we can apply our assumption that ϵ≥ 1/2 (see the beginning of this proof)to conclude that q ≥ (1/3)(1-ϵ)n. Combined: (1/3)(1-ϵ)(n) provides a general lower bound on q for all rounds.We now know that in a good round r: ϕ(r+1) - ϕ(r) ≥δ∈Ω(α (1-ϵ)n/√(ΔlogΔ)). Because ϕ(1) ≤ n^2 and ϕ can only decrease, it follows thatn^2/δ = t_good∈ O(n √(ΔlogΔ)/α(1-ϵ)) good rounds are sufficient to conclude gossip. As established above,the probability of a given round being good is lower bounded by a constant, regardless of the execution history preceding that round. For each round r, let X_r be the random indicator variable that evaluates to 1 if and only if r is good. We know Pr(X_r = 1) ≥ p, for the constant probability mentioned above. Therefore, in expectation, t_good/p ∈Θ(t_good) rounds are sufficient to achieve t_good good rounds. To obtain a high probability result we cannot directly apply a Chernoff bound to these indicator variables as they are not necessarily independent.Each X_r, however, stochastically dominates the trivial random variable X̂_r that evaluates to 1 with probability p. We can then apply a concentration result to the expectation calculated on the X̂ variables to determine that Θ(t_good) rounds are sufficient, with high probability in n.Pulling together the pieces, by Lemma <ref>, for each round r, either we have solved ϵ-gossip or we can find a coalition that provides us a constant probability of r being a good round. With high probability, the latter can occur at most O(t_good)= O(n√(ΔlogΔ)/(1-ϵ)α) times before we still solve the problem. The following corollary follows directly from our analysis inSection <ref> concerning the elimination of the shared randomness assumption when solving gossip with SharedBit. Fix some ϵ, 0< ϵ < 1. There exists a bit string multiset R', such that the SimSharedBit gossip algorithm using this R' solves the ϵ-gossip problem in O(n√(ΔlogΔ)/(1-ϵ)α + (1/α)Δ^1/τlog^6N)= Õ( n√(ΔlogΔ)/(1-ϵ)α) rounds when executed with tag length b=1 in a network with stability τ≥ 1. plain | http://arxiv.org/abs/1705.09609v1 | {
"authors": [
"Calvin Newport"
],
"categories": [
"cs.DS",
"cs.DC"
],
"primary_category": "cs.DS",
"published": "20170526151039",
"title": "Gossip in a Smartphone Peer-to-Peer Network"
} |
Optimization of Measurement Device Independent Scarani-Acìn-Ribordy-Gisin protocol C. Tannous[Tel.: (33) 2.98.01.62.28,E-mail: [email protected]] and J. Langlois Version December 30, 2023 ======================================================================================== New types of machine learning hardware in development and entering the market hold the promise of revolutionizing deep learning in a manner as profound as GPUs. However, existing software frameworks and training algorithms for deep learning have yet to evolve to fully leverage thecapability of the new wave of silicon. We already see the limitations of existing algorithms for models that exploit structured input via complex and instance-dependent control flow, which prohibits minibatching. We present an asynchronousmodel-parallel (AMP) training algorithm that isspecifically motivated by training on networks of interconnecteddevices.Through an implementation on multi-core CPUs, we show that AMP training converges to the same accuracy as conventional synchronous training algorithms in a similar number of epochs, but utilizesthe available hardware more efficiently even for small minibatch sizes, resulting in significantly shorter overall training times. Our framework opens thedoor for scaling up a new class of deep learning models that cannot be efficiently trained today. § INTRODUCTION A new category of neuralnetworks is emerging whose common trait is their ability to react in dynamic and unique ways to properties of their input. Networks like tree-structured recursive neural networks <cit.> and graph neural networks (GNNs) <cit.> defy the modern GPU-driven paradigm of minibatch-based data management. Instead, these networks take a tree or a graph as input and carry out a computation thatdepends on their individual structures. We refer to this new class of models with dynamic control flow asdynamic neural networks.Modern neural network frameworks are certainly capable of expressing dynamic networks. TensorFlow <cit.>introduces , , and other higher order functional abstractions, whileChainer <cit.>, DyNet <cit.>, and PyTorch <cit.> dynamically construct the computation graph using the control flow of the host language.However, training these networks with existing software frameworks and hardware can be painfully slow because these networks require highly irregular, non-uniform computation that depends on individual instances.This makes batching impractical or impossible, thus causing the cost of matrix-vector product to be dominated by the cost of loading the weights from DRAM – typically orders of magnitude slower than the peak compute on both CPUs and GPUs[ For example, the TitanX GPU performs 10^13 FLOPS but only 10^11 floats/s can be brought into the chip due to memory bandwidth (480 GB/s).]. Moreover, these frameworks are not typically optimized with single instance batch size in mind. Dynamically unfolding thecomputation graph, for example, is a concern when there are not enough instances to amortize the cost for it. [Recently proposed TensorFlow Fold <cit.> mitigates these issues with dynamicbatching. (Section <ref>)] With limited batching, we show in this paper that a way to scale up dynamic models is by exploiting an extreme form of model parallelism, amenable to distributed execution on a cluster of interconnected compute devices. By model parallelism, we not only mean computing disjoint parts of the computational graph in parallel, but also computing sequential operations in the graph in a pipeline-parallel fashion <cit.>.Conventional (pipeline) model parallelism, however, can only maximize device utilization if we can keep the pipeline full at all times.Unfortunately, as we show in Figure <ref>, keeping a conventional pipeline full is at odds with convergence speed due to a decreased parameter update frequency; compare Figure <ref> (a) and (b). This is analogous to the trade-off we face in batching. To overcome this problem, we propose asynchronous model-parallel (AMP) training, where we allow asynchronous gradient updates to occur, whenever enough gradients have been accumulated; see Figure <ref> (c). With this design we aim for both high device utilization and update frequency.In this setting, however, model parameters may be updated between the forward and the backward computation of an instance, introducing gradient “staleness”.Despite staleness, we show that AMP training can converge fast with good hardware utilization. Specifically, our contributions are: * We present the AMPNet framework for efficient distributed training of dynamic networks.* We present an intermediate representation (IR) with explicit constructs for branching and joining control flow that supports AMP training. Unlike previous work that considers static computation graphs for static control flow (e.g., Caffe), and dynamic computation graphs for dynamic control flow (e.g., Chainer), our IR encodes a static computation graph to execute dynamic control flow[Our IR bears similarity to TensorFlow but we discuss differences in Section <ref>.]. As a consequence, training becomes easy to distribute and parallelize. Further, IR nodes can process forward and backward messages from multiple instances at the same time and seamlessly support simultaneous training and inference. * We show that, thanks to explicit control flow constructs, our IR can readily encode replicas, a form of data parallelism (see Sec. <ref>). In addition, our IR includes operators for data aggregation which recover forms of batching. These features can further improveefficiency, even on CPUs.* We show that AMP training converges to similar accuracies as synchronous algorithms but often significantly faster. (Sec. <ref>) For example on the QM9 dataset <cit.> our implementation of gated graph sequence neural network (GGSNN) <cit.> on a 16 core CPU runs 9x faster than a (manually optimized) TensorFlow CPU implementation and 2.1x faster than a TensorFlow GPU implementation on the TitanX GPU, because it can better exploit sparsity. Though we do not aim to compete across-the-board with mature frameworks such as TensorFlow, our evaluation proves that AMPNet is particularly beneficial for dynamic networks. In summary, our work demonstrates the benefits of AMP training and gives a novel way to design and deploy neural network libraries with dynamic control flow.Together these contributions open up new ways to scale up dynamic networks on interconnected compute devices. Inspired by the increasing investment and innovation in custom silicon for machine learning (i.e., FPGAs <cit.> and ASICs <cit.>), we perform a simple calculation on the QM9 dataset that shows that AMPNet on a network of 1 TFLOPS devices can be 10x faster than our CPU runtime requiring only 1.2 Gb/s network bandwidth (Sec. <ref>). § NEURAL NETWORKS WITH COMPLEX AND DYNAMIC CONTROL FLOWBelow we highlight three models with dynamic control flow, which will be studied in depth in this paper:Variable-length RNNs iterate over the tokens of variable-length sequences. Pseudo-code for a simple RNN is given in Figure <ref>. The linear layer can be substituted with a more sophisticated unit such as a gated recurrent unit <cit.>. Though each instance has a different length, it is possible to add padding to enable batching. However this may lead to limited speedup due to variability in sequence lengths.Tree-structured neural networks are powerful models used for parsing of natural language and images, semantic representation, and sentiment analysis <cit.>. They require evaluation of (potentially multiple) trees withshared parameters but different topology for each instance. Even if one only needs to evaluate a single computational tree per instance as in <cit.>, the tree is instance-specific and batching requires nontrivial planning <cit.>. A simple form of tree neural network performs a bottom up traversal of the instance, starting from an embedding of the leaves. At each level the values from the child nodes are concatenated and sent through a specialized unit (e.g. LSTM). The result is then propagated further up the tree.Backpropagation over the tree structure is known as backpropagation through structure <cit.>. Graph neural networks <cit.> combine both the temporal recurrence in variable length RNN and recurrence over the structure in tree RNN. GNNs can be seen as performing aggregation/distribution operations over a general graph structure with shared parameters. Apart from the models above, there exist many recently proposed models with flexible control flow (e.g. hierarchical memory networks <cit.>, neural programmer interpreters <cit.>, adaptive computation networks <cit.>, and networks with stochastic depth <cit.>), to which our framework can be applied.§ ASYNCHRONOUS MODEL-PARALLEL TRAININGThe basic idea behind AMP training is to distribute a computation graph across compute nodes and communicate activations.For training, the nodes of the computation graph exchange forward or backward messages. Parameterized computations (e.g. fully-connected layers) can individually accumulate gradients computed from backwards messages. Once the number of accumulated gradients since the last update exceeds a threshold min_update_frequency, a local update is applied and the accumulator gets cleared.The local parameter update occurs without further communication or synchronization with other parameterized computations. The staleness of a gradient can be measured by the number of updates between the forward and backward computation that produces the gradient. Small min_update_frequency may increase gradient staleness. On the other hand, large min_update_frequency can reduce the variance of the gradient but can result in very infrequent updates and also slow down convergence. In addition,controls the maximum number of active instances that are in-flight at any point in time. By setting = 1 we restrict to single-instance processing.[Note this is usually, but not always, equivalent to synchronous training. For example, a single instance can be comprised of a stream of messages (e.g. tree nodes in a tree RNN) and depending on the model some updates may occur asynchronously, even if all the messages in-flight belong to a single instance.] More in-flight messages generally increase hardware utilization, but may also increase gradient staleness. We have implemented an AMPNet runtime for multi-core CPUs, the details of which are given in Appendix A. Section <ref> demonstrates the effects of these parameters.§ A STATIC INTERMEDIATE REPRESENTATION FOR DYNAMIC CONTROL FLOW Overview Motivated by the need to distribute dynamic networks on networks of interconnected devices and apply AMP training, we have designed a static graph-like intermediate representation (IR) that can serve as a target of compilation for high-level libraries for dynamic networks (e.g. TensorFlow or our own frontend), and can itself admit multiple backends (e.g. the multi-core CPU runtime that we consider in detail in this paper, or a network of accelerators). The key feature of our IR is that it is a static graph, but can execute dynamic and instance-dependent control flow decisions.A neural network model is specified by (i) an IR graph, and (ii) a specialized controller loop that pumps instances and other data – e.g. initial hidden states – and is responsible for throttling asynchrony.Each IR node can receive and process either forward messages (from its predecessor in the IR graph) or backward messages (from its successors). During training, forward propagation is carried out by passing forward messages through the IR graph. Each message consists of a payload and a state. The payload is typically a tensor, whereas the state is typically model-specific and is used to keep track of algorithm and control flow information. For example, in a variable-length RNN the state contains the instance identifier, the current position in the sequence, and the total sequence length for the instance. The final loss layer initiates the backward propagation through the IR graph. An invariant of our IR is that for every forward message that is generated by a node with a specific state, this node will eventually receive a backward message with the same state. Depending on(Section <ref>) multiple forward or backwardmessages can be in-flight, from one or more instances.In the rest of this section we discuss the most important IR nodes along with their operational semantics, and show how they are used in the example models from the previous section.Payload transformations Parameterized payload transform (PPT) nodes can be used to encode, for instance, fully connected layers. They apply a transform in the forward pass, but also record the activation in order to use it to compute gradients in the backward pass. An activation is recorded by keying on the state of the message, and hence this state must include all necessary information to allow the node to process multiple messages from potentially different instances without conflating the activations. We require specifications of the forward and the backward transformation, the operation to produce a new gradient, as well as the state keying function to be used. A PPT node may decide to independently apply accumulated gradients to update its parameters. For transformations that do not involve parameters (e.g. ReLUs) our IR offers a simpler non-parameterized payload transform.Loops, state, and control flow A condition node (f) is parameterized by a function f that queries the state (but not the payload) of the incoming message and, based on the response, routes the input to one of the successor nodes. A join node () propagates the messages it receives from each of its ancestor nodes but records the origin so that in the backward pass it can backpropagate them to the correct origin. Like PPT nodes, anode must be parameterized over the keying function on the state of the incoming message. An invertible state update node (f f^-1) is parameterized by two functions f and f^-1 that operate on the state of a message, and satisfy f^-1(f(x)) = x.Figure <ref> shows how to encode an RNN. The controller pumps sequence tokens into a lookup table – just a PPT node, where the parameter is the embedding table and is also being learned. The controller also pumps labels to the loss layer (dashed boxes are compound graphs whose details we omit), and an initial hidden state h_0 for every sequence. Message states contain the sequence time-step. Following the embedding, messages are concatenated ( node, see next paragraph) with the hidden state, and the result goes into a linear node followed by a ReLU activation. Thenode increments the time-step, and the conditional node tests whether the end of the sequence has been reached. Depending on the answer it either propagates the hidden state back to , or pushes the hidden state to the final linear and loss layers. In backward mode, the gradient is propagated inside the body of the loop, passes through the(which decrements the time-step), and reaches thenode. Thenode will (based on information from the forward phase) either backpropagate to thenode, or to the controller. Hence the loop is executed in both the forward and backward direction. Aggregation and disaggregationOur IR offers several constructs for aggregation and disagreggation; the most important ones are outlined below, and their behavior is summarized in Figure <ref>. , , andperform concatenation, partition, and broadcast of incoming messages as their names suggest.can group together several incomingmessages based on their state. The outputmessage contains a tensor composed of the input payloads, whereas the state is afunction of the incoming states. In forward mode (and also ) must key on this new state to cache thestates of the original messages, so as to restorethose in the backward phase.is a symmetric version of . creates a sequence of outgoing messages per incoming message, with replicated payload and new statesgiven by a state generation function that is a parameter of the node. The node keys on the outgoing states and cachesthe incoming state and number of expected messages, so as to sum all the gradients and restorethe original state in backward mode. Figure <ref> describes a GNN that combines aggregation on the structure of a graph instancewith an outer iteration. The iteration controls in effect the locality of information propagation. The controller pumps data, as before, to a lookup table and labels for this instance to the loss layer. The lookup table emits payloads that are matrices where each row corresponds to the embedding of an instance node, and states that contain the current iteration counter, the instance id, and a reference to the graph structure. The messages are broadcast and ungrouped so that each outgoing message corresponds to each node of the graph instance. Next, each message is goes through anode that replicates the payload for each outgoing edge and creates states that record the incoming node, outgoing node, and type of that edge, resulting in a stream of messages,one for each edge in the graph. Next, all edges are grouped by edge type and each group is sent to a designated linear layer. Each group is then dismantled back and edges are re-grouped by their target node. Each group is passed through a non-parameterized payload transformation that sums together all payloads. The result is a stream of messages where each message contains an aggregated value for a graph node. Finally, we group back all these aggregated values and send the result to the RNNCell for another outer iteration. We note that this constitutes a form of batching – the information about all nodes is batched together before been sent to the RNNCell.§ INTERACTION WITH DATA PARALLELISM AND REPLICAS Pipeline-style parallelism can often be augmented with forms of data parallelism. Consider the RNN in Fig. <ref>. The only heavy operation (Linear-1) in the body of the loop is going to act as a bottleneck for computation. One solution is to split the linear layer into smaller tiles and compute them in parallel.This is expressible in our IR but the linear operation needs to be large enough to benefit from tiling in this way.Another approach is to replicate the linear layer in full. Fortunately this requires only minimal new machinery – we can replicate the linear layer and place the replicas insideandnodes as in Figure <ref>. Different instances or messages from the same instance but with different position in the sequence can be processed in an (pipeline-)parallel fashion using 3 replicas in this case. To enable parameters to be shared among the replicas, we have implemented infrequent end-of-epoch replica synchronization (averaging) to keep the communication cost negligible, as well as a message-passing protocol asynchronous trigger of whole-replica group synchronization, but found that infrequent synchronization was sufficient for fast convergence.§ EXPERIMENTSWe evaluate AMPNet using the dynamic models introduced in Section <ref>. For completeness, we additionally consider a simple multi-layer perceptron (MLP) as an example of a static network with instances that are easy to batch. For each model we select a dataset and compare the throughput and convergence profile of AMPNet against traditional training schemes implemented in TensorFlow. MLP: MNIST As preliminary task, we train a 4-layer perceptron with ReLUs on MNIST <cit.>. We choose 784-dimensional hidden units, and we affinitize the 3 linear operations on individual workers (or threads; see Appendix A). Both AMP runtime and TensorFlow use SGD with learning rate 0.1 and batch size of 100. RNN: List reduction dataset As a starting point for experiments on networks with complex control flow we use a synthetic dataset solved by a vanilla RNN. Specifically, we train an RNN to perform reduction operations on variable length lists of digits. Each training instance is a sequence of at most 10 tokens: The first token indicates which of 4 reduction operations [The operations considered in our toy dataset act on a list L and are expressed in python syntax as: , , and .]is to be performed, and the remaining tokens represent the list of digits. The output is the result of the calculation rounded modulo 10.The dataset consists of 10^5 training and 10^4 validation instances.We present this task as a classification problem to a vanilla RNN with ReLU activation and a hidden dimension of 128. All parameterized operations are affinitized on individual workers. We bucket training instances into batches of 100 sequences (in the baseline and in AMPNet). Tree-LSTM: Stanford Sentiment Treebank As a non-synthetic problem, we consider a real-world sentiment classification dataset <cit.> consisting of binarized constituency parse trees of English sentences with sentiment labels at each node. Following Tai et al. <cit.>, we use 8,544 trees for training, 1,101 trees for validation, and 2,210 trees for testing.We use a Tree LSTM for this classification task based on the TensorFlow Fold <cit.> benchmark model. Both the AMP and Fold models are trained following <cit.> with the additional architectural modifications proposed by <cit.>. Furthermore, we split our Tree-LSTM cell into Leaf LSTM and Branch LSTM cells. This does not affect the expressiveness of the model because the LSTM cell receives either zero input (on branch) or zero hidden states (on leaves); i.e., the two cells do not share weights except for the bias parameters, which are learned independently in our implementation.We compare the time to reach 82 %fine grained (5 classes) accuracy (averaged over all the nodes) on the validation set.GNN: Facebook bAbI 15 & QM9 datasets We verify our GNN implementation using a toy logic deduction benchmark (bAbI task 15 <cit.>) and study a real-world application for GNNs: prediction of organic molecule properties from structural formulae in the QM9 dataset <cit.>. GNNs have previously been applied to these tasks in <cit.> and <cit.> respectively.For the bAbI 15 dataset we inflate each graphs from the default 8 nodes to 54 nodes to increase the computational load, but we preserve the two-hop complexity of the deduction task. The architecture of the model follows <cit.> with a hidden dimension of 5, and 2 propagation steps.For the QM9 dataset we concentrate on prediction of the norm of a molecule's dipole moment using a regression layer build on the propagation model from <cit.> (corresponding to the simplest setting in <cit.>). We use a hidden dimension of 100 and 4 propagation steps, initializing the graph nodes (atoms) following <cit.>. The molecules contain up to 29 atoms and in a TensorFlow baseline we bucket molecules into batches of 20 with atom counts differing by at most 1 within a batch. Following <cit.>, we report regression accuracies in multiples of a target accuracy from the chemistry community.ResultsOn MNIST, Table <ref> shows 3x speedup from synchrony (=1) to asynchrony (=4). This is almost ideal as the first three linear layers are the heaviest operations. As we can see in the fourth column of the table, mild asynchrony has negligible effect on the convergence while greatly improving throughput and time to convergence.The list reduction dataset demonstrates the power of replicas. As there is only one heavy operation (Linear-1, Figure <ref>), the speedup from asynchrony is mild (1.3x). However we get 2.5x and 3.5x speedup for 2 and 4 replicas, respectively, which is nearly ideal. Again, the # of epochs to convergence is not affected by increasing . The slowdown in convergence for 4 replicas is due to the increased effective minibatch size – also commonly observed in data parallel training.Next the sentiment tree-RNN dataset shows that our runtime is competitive without batching to TensorFlow Fold <cit.> using dynamic batching of batch size 100. It is worth mentioning that our runtime allows us to specify differentparameter for each parameterized operation. We set this parameter to 1000 for the embedding layer, which is initialized by Glove vectors, and 50 for all other layers. This greatly reduced gradient staleness in the embedding layer.Finally bAbI 15 (54 nodes) and QM9 datasets demonstrates the importance of sparsity. Note that the TensorFlow implementation of GGSNN <cit.> implements the message propagation and aggregation over the input graph as a dense NH× NH matrix multiplication where N is the number of nodes and H is the hidden state dimension. Since each input graph has a unique connectivity, this matrix needs to be constructed for each instance. By contrast, we handle this by message passing and branching as we described in Section <ref>. As a result we get roughly9x speedup on QM9 against TensorFlow implementation on CPUs with the same number of threads. Our runtime was also faster than a GPU TensorFlow implementation by 2.1x. AMPNet and TensorFlow implementation were comparable on the small bAbI 15 (54 nodes) dataset.r0.48 < g r a p h i c s > Performance of an 8-replica RNN model on the as a function of asynchrony hyperparameters. Solid gray lines show constant convergence time trajectories. muf stands for min_update_frequency. Asynchrony The degree of asynchrony is controlled by hyperparameters min_update_frequency andmax_active_keys. In Fig. <ref> we use an 8-replica RNN model on the list reduction dataset to investigate how these parameters affect the data and time required to converge to 96% validation accuracy. We find, in analogy with minibatch size in traditional systems, that min_update_frequency must neither be too large nor too small. Increasing max_active_keys (increasing asynchrony) monotonically increases performance when the number of keys is similar to the number of individually affinitized heavy operations in the model 8 in this case). Increasing max_active_keys significantly beyond this point produces diminishing returns.§ RELATED WORK Chainer <cit.>, DyNet <cit.>, and PyTorch <cit.> belong to a new class of deep learning frameworks that define the computation graph dynamically per-instance by executing the control flow of the host language (e.g. Python) that can limit cross-instance parallelism and has a cost that is difficult to hide when the minibatch size is small<cit.>. By contrast our IR graph is static so it is easier to distribute, optimize, and pipeline-parallelize across instances.Theano <cit.> and TensorFlow (TF)<cit.> provide powerful abstractions for conditional execution ( in Theano andin TF) and loops ( and , respectively); TF also provides higher-order functions, such as , , , and .The main difference between AMPNet and the above frameworks is that AMPNet is streaming and asynchronous whereas Theano is non-streaming and synchronous. Although not designed for streaming, TF can support streaming programmatically as it exposes first-class queues, as well as data prefetching with so called input pipelines. In our IR, all the queuing is implicit and stream-based execution is the default. TF additionally does support static description of dynamic control flow and state update, but we depart from the classic dataflow architecture that TF follows <cit.>: First, instead of having nodes that represent mutable reference cells, we encapsulate the state with which a message should be processed through the graph in the message itself. Second, because we encapsulate algorithmic state in the messages, we do not introduce the notion of control dependencies (which can be used to impose a specific execution order on TF operations). Our choices complicate algorithmic state management from a programming point of view and make the task of designing a high-level compiler non-trivial, but allow every node to run asynchronously and independently without a scheduler and without the need for control messages: For example, nodes that dynamically take a control flow path or split the data simply consult the state of the incoming message, instead of having to accept additional control inputs. For “small” states (e.g. nested loop counters or edge and node ids) this might be preferable than out-of-band signaling. Our IR can implement loops by simply using state-update, conditional, and phi nodes, because the state accompanies the payload throughout its lifetime, whereas TF introduces specialized operators from timely dataflow <cit.> to achieve the same effect.TensorFlow Fold (TFF) <cit.> is a recent extension of TensorFlow that attempts to increase batching for TF dynamic networks and is an interesting alternative to our asynchronous execution. TFF unrolls and merges together (by depth) the computation graphs of several instances, resulting in a batch-like execution. TFF effectiveness greatly depends on the model – for example, it would not batch well for random permutations of a sequence of operations, whereas our IR would very succinctly express and achieve pipeline parallelism through our control-flow IR nodes. Asynchronous data parallel training <cit.> is another popular approach to scale out optimization by removing synchronization, orthogonal to and combinable with model-parallel training. For example, convolutional layers are more amenable to data-parallel training than fully connected layers, because the weights are smaller than the activations. Moreover, when control flow differs per data instance, data parallelism is one way to get an effective minibatch size > 1, which may improve convergence by reducing variance. The impact of staleness on convergence <cit.> and optimization dynamics <cit.> have been studied for data parallelism. It would be interesting to extend those results to our setting.<cit.>, like us, aim to to train different parts of a model in a decoupled or asynchronous manner. More precisely, their goal is to approximate a gradient with a synthetic gradient computed by a small neural network that is locally attached to each layer. Hence, the local gradient calculation becomes independent of other layers (except for the training of the gradient predictor network) and allows asynchronous parameter updates. This would be especially useful if the evaluation of the local network is cheaper than the computation of the real gradient; for example, if the computation of the real gradient required communication of forward/backward messages between devices. § CONCLUSION AND OUTLOOK We have presented an asynchronous model-parallel SGD algorithm for distributed neural network training. We have described an IR and multi-core CPU runtime for models with irregular and/or instance-dependent control flow.Looking forward, we aim to deploy our system on specialized hardware. To give an idea of performant FPGA implementations of AMPNet, we perform a simple estimate of the peak throughput on the QM9 dataset running on a network of 1 TFLOPS FPGAs (see Appendix C for details). Our calculation shows that we achieve 6k graphs/s (10x compared to our CPU runtime) on the QM9 dataset with 200 hidden dimensions and 30 nodes per graph on average. This only requires a very reasonable 1.2 Gb/s network bandwidth.Equally importantly, we plan to build a compiler that automatically deduces the information to be placed in the states and generates state keying functions from a higher-level description of the models.By unlocking scalable distributed training of dynamic models, we hope to enable exploration of this class of models that are currently only on the horizon but may become more mainstream in the future. § ACKNOWLEDGEMENTSWe would like to thank Eric Chung, Doug Burger, and the Catapult team for continued discussions and feedback from the early stage of our work. We would also like to thank Krzysztof Jozwik for discussions on FPGAs, Stavros Volos for discussions on various memory architectures, Miguel Castro for discussions on data parallelism vs. model parallelism, John Langford for a discussion on asynchrony and reproducibility, and Frank Seide for discussions on dynamic networks. abbrvnat § AMPNET RUNTIME IMPLEMENTATION We have implemented an AMPNet runtime for multi-core CPUs. Our runtime spawns multiple workers each associated with a hardware thread and hosting one or more IR nodes – in a more general setting each worker corresponds to a compute device. To remain faithful to a distributed environment communication is only through message passing. Each worker is equipped with a multiple-producer single-consumer queue that can accept messages for any IR node hosted on that worker.The main worker loop periodically offloads messages from the concurrent queue to a worker-local priority queue that assigns higher priority to backward messages. Backward prioritization is designed for situations when multiple IR nodes with a dependency on the IR graph end up hosted on the same worker. As a consequence, backpropagation can complete faster and new instances can be pumped in by the controller. We dequeue the top message and invoke the forward or backward method of the target IR node. These methods may update internal IR node state (such as cache the state of the incoming message and wait for more messages) or post new forward or backward messages.How to update the parameters using the gradients is a configuration option that selects amongst a range of optimization algorithms. We have implemented runtime configuration options for selecting several well-known schemes such as (momentum-)SGD and Adam <cit.>, and for controlling the training hyper-parameters.§ DETAILS OF THE EXPERIMENTAL RESULTSWe providemore details of the experiment and analysis in this section. All experiments were carried out on machines with 16 cores and 112 GB of RAM. The validation curves were averaged over at least 20 independent runs. The time/epoch to reach a target accuracy was calculated as median of the time an algorithm takes to reach the target accuracy over the repetitions. We found this approach to be more reliable than reporting the time/epoch when the averaged accuracy reaches the target. Table <ref> show both the training and validation throughputs we obtained with AMPNet and our TensorFlow baselines. §.§ MNISTFigure <ref> shows the validation accuracy vs. time, validation accuracy vs. epochs, and throughputs of synchronous and asynchronous versions of AMPNet as well as TensorFlow. The throughput greatly increases from synchronous (=1) to asynchronous (=4) while the speed of convergence (middle panel) is hardly affected for mild amount of asynchrony. Taking higher =8 increase throughput only very little (because there is no more work) and seems to rather make the convergence more unstable. This is due to the fact that our current scheduler is greedy and pumps in a forward message whenever the first layer is unoccupied, which leads to large gradient staleness. Clearly a better scheduling will remove this sensitivity.§.§ List reduction datasetSimilarly Figure <ref> shows the validation accuracy vs. time and the number of epochs, and throughputs of the methods we discussed in the main text on the list reduction dataset. We first notice that increasing the asynchrony from synchronous (=1) to =4 and =16 affects the convergence very little at least in average. However, there is also very little speedup without introducing replicas as we discussed in the main text. Increasing the number of replicas increases the throughput almost linearly from 15k sequences/s (synchronous) to 30k sequences/s (2 replicas) and over 60k sequences/s (4 replicas). Convergence is almost unaffected for 2 replicas. This was rather surprising because the parameters of the replicas are only synchronized after each epoch as we described in Sec. <ref>. A slight slow-down in convergence can be noticed for 4 replicas. Since even =16 has almost no effect on the convergence without replicas, this is not due to asynchrony. We also tried to synchronize more frequently but this did not help. Thus we believe that the slow-down is due to the increase in the effective minibatch size resulting in reduced number of updates per epoch, which is commonlyobserved in data parallel training.§.§ Sentiment Tree Bank datasetFigure <ref> shows the averaged fine grained validation accuracy for the tree RNN model with different on the Stanford Sentiment Tree Bank dataset. Interestingly although TensorFlow Fold achieves higher throughput, AMPNet converges faster (in terms of the number of epochs). This speedup is mainly due to the fact that we are not batching and updating whenever we have accumulated 50 gradients (except for the lookup table node that updates every 1000 gradients); 50 gradients correspond to roughly 2 trees. The reason for the lower throughput compared to TensorFlow Fold is that we are only grouping the leaf operations and not the branch operations. Grouping the branch operations is possible by extending our IR nodes and we are actively working on it.Figure <ref> shows the same information for fixed =16 and different . We can see that as we increasefrom the originally used 50 to larger values, the peak of the validation accuracy shifts later and lower becoming closer to the curve obtained by TensorFlow Fold. This is consistent with the parallels betweenand minibatch size we drew in Section <ref>. Theparameter has marginal influence on the training throughput. §.§ bAbI 15 (54 nodes) and QM9 datasetFigures <ref> and <ref> show that GGSNN can tolerate relatively large =16. In particular, on the more challenging QM9 dataset taking =16 increased the throughput significantly from 152 graphs/s (synchronous) to 640 graphs/s.We sample 100 fresh samples for every epoch. § THROUGHPUT CALCULATION FOR THE GGSNN MODEL FOR QM9Suppose that the hidden dimension H is sufficiently wide so that the speed of matrix-vector product dominates the throughput of the system compared to element-wise operations, such as sigmoid and tanh; we takeH=200 in the calculation below.In an idealized scenario, pipeline parallel execution of the network consists of roughly 3 stages per time step. In the first stage, all four H× H linear nodes corresponding to different edge types execute in parallel. In the second stage, the two 2H× H linear nodes (#9 and #12) inside the GRU cell corresponding to update and reset gates execute in parallel (see Fig. <ref>,). Finally, the last 2H× H linear node in the GRU cell immediately before the Tanh node executes. We would need at least 7 devices that executes these linear nodes in a pipelined parallel fashion. The memory requirement for each device is 4 times the size of the H× H or 2H× H weight matrix, which consists of the parameter, gradient buffer, and two slots for the statistics that need to be accumulated in the Adam optimizer. This would be 1.2MB for H=200 and float32.The throughput of training this model is either limited by the speed of the GRU block or that of the linear nodes corresponding to edges. The number of operations in the forward and backward passes per time step can thus be estimated asfwdop = 2·max(2NH^2, EH^2/C), bwdop = 6·max(2NH^2, EH^2/C),where N and E are the average number of nodes and edges per instance, respectively and C is the number of edge types, which is 4 in this task. We assume that the backward operation is 3 times more expensive than the forward operation because it requires matrix transpose, matrix multiplication, and gradient accumulation.Moreover, in an idealized scenario, we can expect that each neural network node alternates between forward and backward (we thank Vivek Seshadri for pointing this out).Thus we can estimate the throughput of training this model on a network of 1 TFLOPS devices (e.g., Arria 10) asthroughput (samples/s) = 0.5·10^12/(fwdop+bwdop)· 4,where the last 4 is the number of propagation time steps and 0.5 accounts for all the other operations and communication overheadwe ignored in the calculation. For H=200, N=E=30 and C=4 we obtainthroughput (samples/s) = 0.5·10^12/64·NH^2≃ 6.5· 10^3 (samples/s).The network bandwidth required in this scenario is network bandwidth (bits/s) = 32·throughput·max(N,E) · H = 1.2·10^9 (bits/s). | http://arxiv.org/abs/1705.09786v3 | {
"authors": [
"Alexander L. Gaunt",
"Matthew A. Johnson",
"Maik Riechert",
"Daniel Tarlow",
"Ryota Tomioka",
"Dimitrios Vytiniotis",
"Sam Webster"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.DC",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170527081040",
"title": "AMPNet: Asynchronous Model-Parallel Training for Dynamic Neural Networks"
} |
http://arxiv.org/abs/1705.09128v1 | {
"authors": [
"S. Stalin",
"M. Senthilvelan",
"M. Lakshmanan"
],
"categories": [
"nlin.SI",
"nlin.PS"
],
"primary_category": "nlin.SI",
"published": "20170525112427",
"title": "Nonstandard bilinearization of $\\cal{PT}$-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions"
} |
|
@nat@width>@nat@width kframe@end@of@kframe @end@of@kframe ##1totalleftmargin -##1---totalleftmargin -totalleftmargin@ setminipage@end@of@kframe -.5in -.5in 1in -.3in -.8in =4 theoremTheorem lemmaLemma corollaryCorollary propositionProposition definition exampleExample remarkRemarkupquote.sty #1 1 Tractable Post-Selection Maximum Likelihood Inference for the Lasso Amit Meir[To whom correspondence should be addressed: [email protected]] Department of Statistics University of WashingtonMathias Drton Department of StatisticsUniversity of Washington December 30, 2023 ==================================================================================================================================================================================================== Applying standard statistical methods after model selection may yield inefficient estimators and hypothesis tests that fail to achieve nominal type-I error rates.The main issue is the fact that the post-selection distribution of the data differs from the original distribution.In particular, the observed data is constrained to lie in a subset of the original sample space that is determined by the selected model. This often makes the post-selection likelihood of the observed data intractable and maximum likelihood inference difficult.In this work, we get around the intractable likelihood by generating noisy unbiased estimates of the post-selection score function and using them in a stochastic ascent algorithm that yields correct post-selection maximum likelihood estimates.We apply the proposed technique to the problem of estimating linear models selected by the lasso.In an asymptotic analysis the resulting estimates are shown to be consistent for the selected parameters and to have a limiting truncated normal distribution.Confidence intervals constructed based on the asymptotic distribution obtain close to nominal coverage rates in all simulation settings considered, and the point estimates are shown to be superior to the lasso estimates when the true model is sparse. Keywords: Stochastic Optimization; Model Selection; Selective Inference; Linear Regression 1.45 § INTRODUCTION §.§ Inference After Model SelectionConsider the linear regression model y = β + ε, where y ∈ℝ^n is a response vector, ∈ℝ^n× p is a matrix of covariate values and ε∈ℝ^n is a noise vector.When the number of available covariates p is large, it is often desirable or even necessary to specify a more succinct model for the data. This is commonly done by selecting a subset of the columns ofto serve as predictors for y. Here, we focus on model selection with the lasso <cit.>, which uses an ℓ_1 penalty to estimate a sparse coefficient vector. A well known, yet not as well understood problem, is the problem of performing inference after a model has been selected. In particular, it is known thatconfidence intervals for parameters in selected models often do not achieve target nominal coverage rates, hypothesis tests tend to suffer from an inflated type-I error rate and point estimates are often biased. A simple Gaussian example serves well to illustrate the issues that may arise when using the same data for selection and inference. Let Y_1,…,Y_n∼ f i.i.d., with _f(Y_i) = μ and _f(Y_i) = 1. Furthermore, suppose that estimation of μ is of interest only if a statistical test provides evidence that it is nonzero.Specifically, suppose that at a 5%-level, we reject H_0: μ=0 if |y̅|>1.96/√(n).In this setting, if |μ|<1.96/√(n), the uncorrected estimator μ̂=y̅ will overestimate the magnitude of μ whenever we choose to estimate it. An example of early work emphasizing the fact that data-driven model selection may invalidate standard inferential methods is the article by <cit.>, with its aptly chosen title `validity, reliability and baloney'.Subsequently, this problem has been studied in the context of regression modeling. In particular, it has been shown that it is impossible to uniformly approximate the post-selection distribution of linear regression coefficient estimates <cit.>.The field of post-selection (or selective) inference is concerned with developing statistical methods that account for model selection in inference.The majority of work in selective inference is concerned with constructing confidence intervals and performing tests after model selection; see for example <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.The particular case of model selection with ℓ_1 penalization is treated by <cit.> and <cit.>. <cit.> consider the general problem of testing after model selection.Estimation after model selection is in the focus of the work of <cit.>, <cit.>, and <cit.>.In order to reconcile the aforementioned impossibility results with the recent advances in post-selection inference, we must clearly define the targets of inference. §.§ Targets of Inference In the context of variable selection in regression, let ℳ := 𝒫({1,…,p}) be the set of models under consideration, defined as the power set of the indices of the columns of the design matrix .Further, let S : ℝ^n→ℳ be a model selection procedure that selects a model M ∈ℳ based on the observed data y ∈ℝ^n.When discussing estimation after model selection in linear regression, one may consider two different targets for inference.The first are the `true' parameter values in correct models where all variables with non-zero coefficient are present. An alternative target for estimation is the vector of regression coefficients in the selected modelβ_0(y)= (_M^T_M)^-1^T_M (Y).In (<ref>), M=S(y) is the selected model, and _M is the sub-matrix ofmade up of the columns indexed by M.These two targets of estimation coincide when the selected model is true, meaning that it contains all variables that have a non-zero regression coefficient.Indeed, if the observed value y is such that S(y) = M for a model M that contains all covariates with non-zero coefficients, then (y) =_Mβ_0^M and β_0^M = β_0(y). Here β_0^M is the vector of non-zero true coefficients padded with zeros to make it a vector of length |M|.<cit.> and <cit.> study the behavior of least squares coefficients as estimators of the true regression coefficients in a sequential testing setting. In contrast, works such as <cit.> and <cit.> consider inference with respect to the regression coefficients in the selected model. In this work, we follow the latter point of view, taking the stance that a true model does not necessarily exist or, even if one exists, may be difficult to identify. Thus, the interest is in the parameters of the model the researchers have decided to investigate.§.§ Conditioning on SelectionA data-driven model selection procedure tends to choose models that are especially suited for the observed data rather than the data-generating distribution. In linear regression this would often be in the form of inclusion of variables that are correlated with the dependent variable only due to random variation. A promising approach for correcting for this bias towards the observed data is to condition on the selection of a model. Consider once again the univariate normal example, simplified via sufficiency to a single observation. Let Y ∼ N(μ, 1) and assume that we are interested in estimating μ if and only if |Y|>c for some constant c>0. Standard inferential techniques assume that we observe values from the distribution Y ∼ N(μ, 1). However, when inference is preceded by testing we never observe any values -c < Y < c and the post-selection distribution of the observed value is not normal but truncated normal.Thus, the conditional post-selection maximum likelihood estimator (MLE) is:μ̂= max_μ f(y|{|Y|>c}) = max_μf_μ(y)/P(|Y|>c) I_{|Y|>c}. The right-hand panel of Figure <ref> plots the post-selection MLE (as a function of y) for the two-sided case described above.Since this MLE is an even function we show the graph only for y>0. The left-hand panel describes the post-selection MLE for the one-sided case where we estimate μ if y>c. In the two-sided case the estimator is an adaptive shrinkage estimator that shrinks the observed value towards zero when it is close to the threshold and keeps it as it is when its magnitude is far away from the threshold. More generally, let Y ∼ f_θ follow a distribution from an exponential family with sufficient statistic T(Y)∈ℝ^p. The likelihood of T(y) given that model M has been selected isℒ_M(θ) = P(M|T(y)) f(T(y))/P(M) I_M,where we use the shorthand P(M|T(y)) := P(S(Y) = M | T(Y) = T(y)) for the conditional probability of selecting model M given T(y). Similarly, P(M) := P(S(Y) = M) is the unconditional probability of selecting M, f(T(y)) is the unconditional density function of T(y), and I_M=I_{S(y)=M} is the indicator function for the selection event.The main obstacle in performing post-selection maximum likelihood inference is the computation of the probability of model selection P(M), which is typically a p dimensional integral. Such integrals are difficult to compute when p is large, and much of the work in the field of post-selection inference has been concerned with getting around the computation of these integrals. For example, <cit.> propose to condition on the signs of the selected variables as well as some additional information contained in the sub-space orthogonal to the quantity of interest in order to obtain a tractable post-selection likelihood. <cit.> approximate P(M) with a barrier function.Conditioning on information beyond the selection of the model of interest, while having the benefit of providing tractable solutions to the post-selection inference problem, may drastically change the form of the likelihood. Consider once again the post-selection estimators for the univariate normal problem (Figure <ref>). Suppose that we observe y>0. Then the right-hand panel plots the conditional estimator for the scenario where two-sided testing is performed. On the left-hand side we plot the conditional estimator for μ as a function of y when we condition on the two-sided selection event as well as the sign of y. Indeed, since our observed value is positive we condition on {|Y| > c, Y >0} = {Y > c}. This second estimator is close to the observed value y when y is far from the threshold but approaches negative infinity as y→ c, see Appendix C for details. Thus, even in the univariate normal case, conditioning on the sign of y in two-sided testing, may drastically alter the resulting conditional estimator. §.§ OutlineIn this work, instead of working with the intractable post-selection likelihood, we base our inference on the post-selection score function which can be approximated efficiently even in multivariate problems. The following lemma describes the post-selection score function for exponential family distributions.Suppose the observation y is drawn from a distribution f_θ that belongs to an exponential family with natural parameter θ and sufficient statistic T(y).If the model selection procedure S(y) satisfies P(S(Y) = M | T(y)) ∈{0,1} for a given model M, then the conditional (post-selection) score function is given by:∂/∂θlogℒ(θ)= T(y) - _θ(T(Y) | M).This result follows directly from the fact that the conditional distribution of an exponential family distribution is also an exponential family distribution as long as P(M|T(y)) ∈{0,1}. See <cit.> for details. In the specific setup we consider subsequently, the conditional distribution of T(Y) given M is a multivariate truncated normal distribution. While it is then difficult to compute (T(Y)|M), we are able to sample efficiently from the multivariate truncated normal distribution using a Gibbs sampler <cit.>. The main idea behind the method we propose is to use the samples from the truncated multivariate normal distribution as noisy estimates of (T(Y)|M) and take small incremental steps in the direction of the estimated score function, resulting in a fast stochastic gradient ascent algorithm.Our framework has similarities with the contrastive divergence method of <cit.>.The rest of the article is structured as follows. In Section 2 we present the proposed inference method in detail and apply it to selective inference on the mean vector of a multivariate normal distribution. In Section 3 we describe how the proposed framework can be adapted for post-selection inference in a linear regression model that was chosen by the lasso. In Section <ref> we formulate conditions under which the conditional MLE is consistent.A simulation study in Section 5 demonstrates that the proposed approach yields improved point estimates for the regression coefficients, and that our confidence intervals, despite lacking a rigorous theoretical justification, achieve close to nominal coverage rates. Finally, in Section 6 we conclude with a discussion.§ INFERENCE FOR SELECTED NORMAL MEANSBefore considering the Lasso, we first discuss the simpler problem of selectively estimating the means of a multivariate normal distribution. Let Y ∼ N(μ, ) with mean vector μ∈ℝ^p and a known covariance matrix .Observing y, we select the modelM = {j∈{1,…,p} :y_j ≤ l_j or y_j≥ u_j },where l_1,…,l_p,u_1,…,u_p ∈ [-∞, ∞] are predetermined constants with l_1 < u_1,…,l_p < u_p. We then perform inference for the coordinates μ_j with j ∈ M (or possibly inference for a function of these coordinates).This seemingly simple problem has garnered much attention. For the univariate case of p=1, <cit.> propose a method for constructing valid confidence intervals, and <cit.> compute the post-selection MLE for μ. For p≫1, <cit.> develop a recipe for constructing valid confidence intervals for the selected means or linear functions thereof. <cit.> discuss ML estimation when = σ^2.To the best of our knowledge, the method we propose below is the first to address the computation of the conditional MLE when p≫1 and the covariance matrixis of general structure.Conditionally on selection, the distribution of y is truncated multivariate normal, as the jth coordinate of y is constrained to lie in the interval (l_j,u_j) if j∉ M or in its complement if j∈ M. In Section <ref> we describe the Gibbs sampler we use to sample from a truncated multivariate normal distribution, in Section <ref>we describe how such samples can be used to compute the post-selection estimator and in Section <ref> we propose a method for constructing confidence intervals based on the conditional MLE and samples obtained from the truncated normal distribution. §.§ Sampling from a Truncated Normal DistributionSampling from the truncated multivariate normal distribution is a well studied problem <cit.>. We choose to use the Gibbs sampler of <cit.>, as it is especially suited to our needs and simple to implement. Assume we wish to generate a draw from the univariate truncated normal distribution constrained to lie in the interval [l,u]⊆[-∞,∞]. This distribution has CDFΦ(y ; μ,σ^2,l,u) := Φ(y;μ,σ^2) - Φ(l;μ,σ^2)/Φ(u;μ,σ^2) - Φ(l;μ,σ^2),where Φ(y;μ,σ^2) denotes the CDF of the (untruncated) univariate normal distribution with mean μ and variance σ^2. A simple method for sampling from the truncated normal distribution samples a uniform random variable U∼ U(0,1) and setsy = Φ^-1(U;μ,σ^2,l,u) = Φ^-1(U(Φ(u) - Φ(l)) - Φ(l) ;μ,σ^2). Next, consider sampling from the truncated normal constrained to the set (-∞, l]∪[u,∞).In this case, we may first sample a region within which to include y and then sample from a truncated univariate normal distribution constrained to the selected region using the formula given in (<ref>).Given this preparation, we may implement a Gibbs sampler for a truncated multivariate normal distribution as follows.Let y∼ N(μ,), and let f(y|M) be the conditional distribution of y given the selection event.While the marginal distributions of f(y|M) are not truncated normal, the full conditional distribution f(y_j | M, y_-j) for a single coordinate y_j is truncated normal with parametersμ_j,-j = μ_j + _j,-j_-j,-j (y_-j - μ_-j), σ^2_j,-j = _j,j - _j,-j^-1_-j,-j_-j,j.The Gibbs sampler repeatedly iterates over all coordinates of y and draws a value for y_j conditional on M and y_-j.So at the tth iteration we sampleY^t_j ∼ f(y_j | M, y_1^t,…,y_j-1^t,y_j+1^t-1,…,y_p^t-1), j=1,…,p.The support of the truncated normal distribution is determined by whether or not j ∈ M.§.§ A Stochastic Gradient Ascent AlgorithmThe Gibbs sampler described above can be used to closely approximate (Y|M) but computation of the likelihood ℒ_M(μ) remains intractable. However, for optimization of the likelihood, we can simply take steps of decreasing size in the direction of the evaluated gradientμ^i = μ^i-1 +γ_i^-1(y - y^i(μ^i-1)),where y is the observed data, y^i(μ^i-1) is a sample from the truncated multivariate normal distribution taken at μ^i-1 and the step size γ_i satisfies:∑_i=1^∞γ_i = ∞,∑_i=1^∞γ_i^2 < ∞.We emphasize that while it is technically possible to compute an MLE for the entire mean vector of the observed random variable, it is not necessarily desirable. To see why, consider once again the left-hand panel ofFigure <ref> where the estimator tends to -∞ as the observed value approaches the threshold. Such erratic behavior may arise when we estimate the coordinates of μ which were not selected, based on observations that are constrained to lie in a convex set, resulting in poor estimates also for the selected coordinates. We plot the conditional log-likelihood for a two-dimensional normal model in Figure <ref>.In such a low-dimensional case, the likelihood function can be computed using routines from the `mvtnorm' R package <cit.>.Our plot is for a setting where we observe y = (1.45, 1.8) with _ij = 0.5^I{i≠ j}, and only the first coordinate of μ was selected based on the thresholds l_1 = l_2 = -1.65, u_1 = u_2 = 1.65.The point y is marked in the figure as an `O', and the log-likelihood is maximized at the point marked with `C', which is μ̂= (5.4, 2.5).We see that instead of performing shrinkage on the observed selected coordinate, the selected coordinate was estimated to be far larger than the observed value. In order to mitigate this behavior, we propose using a plug-in estimator for the coordinates outside of M. Particularly, we limit ourselves to taking steps of the formμ_j^i =μ^i-1 +γ_i^-1_j,.(y - y^i(μ^i-1))if j ∈ M, y_jif j∉ M,where ^-1_j,. is the jth row of ^-1.In other words, we impute the unselected coordinates of μ with the corresponding observed values of y, and maximize the likelihood only with respect to the selected coordinates of μ. These plug-in estimates for the coordinates of μ which were not selected are consistent, as we show in Section <ref>.The plug-in conditional MLE for Example <ref> is shown as a `P' in Figure <ref>.It is approximately μ̂= (1.45, 0.8).Next, we give a convergence statement for the proposed algorithm. Since our gradient steps are based on y^i(μ^i-1), a noisy estimate of _μ^i-1(Y|M), the resulting algorithm fits into the stochastic optimization framework of <cit.>. In short, the theory for stochastic optimization guarantees that taking steps in the form of (<ref>) leads to convergence to the MLE as long as the variance of the gradient steps can be bounded. Let Y ∼ N(μ,), and let M be defined as in (<ref>). Then for all j∈ M:E_μ(Y^i_j(μ) - _μ(Y_j | M))^2 ≤()/P(⋂_j∉ M{l_j < y_j < u_j}) ∏_j∈ MΦ(l_j; u_j, σ^2_j, -j).The algorithm described in (<ref>) converges to the Z-estimator given by the root of the functionψ(μ)_j =_j, .^-1(y_j - _μ(Y_j|M) ) if j∈ M, y_j - μ_jif j ∉ M.A precise description of the optimization algorithm is given in Algorithm 1 in the appendix. Figure <ref> shows typical optimization paths for Algorithm 1 as well as the stochastic gradient method for the Lasso described inSection <ref>. §.§ Conditional Confidence Intervals In the absence of model selection, the MLE is typically asymptotically normal, and it is common practice to construct Wald confidence intervals based on this limiting distribution:μ̂^naive = y, ^naive_j = (μ̂^naive_j - z_j,1-α/2, μ̂^naive_j - z_j,α/2),where z_j,α denotes the (1-α) quantile of the asymptotic normal distribution for the jth coordinate.The post-selection setting is more complicated, however, because wecan no longer rely the asymptotic normality of the estimators. Instead, we propose to construct confidence intervals based on the second order Taylor expansion of the conditional likelihood. In order to describe our proposed approximation to the distribution of the conditional MLE, we extend the normal means problem to the setting of an n-sample.So assume that instead of observing a single vector y∈ℝ^p, we have a set of observations y_1,…,y_n∈ℝ^p and perform model selection and inference based on y̅_n = n^-1∑_i=1^n y_i. Our confidence intervals are based on the approximation√(n)(μ̂^M_n - μ^M_0) ≈√(n)Var_μ^M_0(√(n)^-1Y̅_n|M)^-1^-1(y̅_n - _μ^M_0(Y̅_n|M)). Based on this approximation, we construct confidence intervals_j = (μ̂^M_j,n - _j, 1-α/2 / √(n),μ̂^M_j,n - _j, α/2 / √(n)).Here,stands for the conditional distribution given selection ofVar_μ̂^M(√(n)^-1Y̅|M)^-1^-1√(n)(y̅_n - _μ̂^M(Y̅_n|M)).We estimate the quantiles _j, 1-α/2 and _j, α/2 using empirical quantiles of samples from the truncated normal distribution.While we are unable to provide theoretical justification for these confidence intervals, a comprehensive simulation study reveals that they obtain coverage rates that are significantly better than those of the naive confidence intervals, and are surprisingly close to the desired level (Section <ref>).Figure <ref> shows point estimates and confidence intervals for selected means in a normal means problems.The figure was generated by sampling Y∼ N(μ,Σ) with μ_1,…,μ_20∼ N(0,4) i.i.d., μ_21=…=μ_100 =0 and Σ_i,j = 0.3 I_i ≠ j + 1 I_i = j.The applied selection rule was S(y)={j: |y_j| > 1.65}.The plotted estimates are the conditional estimates computed using the algorithm defined by (<ref>) along with the 95% confidence intervals described in (<ref>). In addition, we plot the estimates and confidence intervals described in (<ref>) which we termnaive. These were not adjusted for selection.As we had seen in the univariate case, the conditional estimator acts as an adaptive shrinkage estimator. When the observed value is far away from the threshold, then no shrinkage is performed and when it is relatively close to the threshold then it is shrunk towards zero.§ MAXIMUM LIKELIHOOD ESTIMATION FOR THE LASSOIn this section we demonstrate how the ideas from the previous section can be adapted for computing the post-selection MLE in linear regression models selected by the Lasso. The Lasso estimator minimizes the squared error loss augmented by an ℓ_1 penalty,β̂_Lasso = min_β1/2y - β^2_2 + λβ_1with λ≥ 0 being a tuning parameter.Model selection results from the fact that the ℓ_1 penalty may shrink a subset of the regression coefficients to zero.As in <cit.>, we perform inference on the non-zero regression coefficients in the Lasso solution, that is, the selection procedure is S(y) = {j:β̂_Lasso,j≠ 0 }.Given selection of a model M, we are interested in estimating the unconditional mean of the regression coefficients β = (^T_M_M)^-1^T_M(Y).We begin by describing the Lasso selection event (Section <ref>) and then give a Metropolis-Hastings sampler for the post-selection distribution of the least-squares estimates (Section <ref>).In Section <ref>, we describe a practical stochastic ascent algorithm for estimation after model selection with the Lasso. §.§ The Lasso Selection EventLet M⊆{1,…,p} be a given model.In order to develop a sampling algorithm for a normal distribution truncated to the event that S(, y):= {j:β̂_Lasso,j≠ 0 }=M, we invoke the work of <cit.> who provide a useful characterization of this Lasso selection event.Let s ∈{-1, 1}^|M| be the vector of signs of β̂_Lasso over the active set.We will consider two setsA_1(M,s) := {_1(M,s) y < u_1(M,s) }, A_0(M,s) := {l_0(M,s) < _0(M)y < u_0(M,s)},where in the first event _1(M, s) = -(s) (^T_M _M)^-1_M^T,u_1(M, s) = -λ(s)(_M^T_M)^-1 s,and in the second event_0(M)= 1/λ^T_-M (I - _M(_M^T_M)^-1_M^T) , l_0(M, s) = -1 - _-M^T_M(^T_M_M)^-1 s,u_0(M, s) = 1 - _-M^T_M(^T_M_M)^-1 s.Here, _M is the submatrix of the design matrixmade up of the columns indexed by the selected model M and the columns in the submatrix _-M correspond to variables which were not selected. It can be shown that {S(, Y) = M and sign vector equal to s} =A_0(M,s) ∩ A_1(M,s).Suppose that Y ∼ (β, σ^2), then conditional score function for a model selected by the Lasso is given byσ^2∂/∂βlogℒ(β | M)= _M^Ty- (^T_MY|M) = ^T_My - ∑_s P(M, s) (^T_M Y |A_1(M,s))/∑_s P(M,s),where for a given set of signs P(M, s) = P(A_0(M,s))× P(A_1(M,s)).As in the normal means problem, parameters related to the set of variables excluded from the model play a role in the conditional likelihood. In the normal means problem we advocated excluding those from the optimization of the conditional likelihood.For the Lasso, we similarly must compute a conditional expectation which is a function of _0(M) (Y).We again advocate for avoiding conditional likelihood-based estimation of this quantity.In computational experiments we observed that the value of _0(M) (Y) tends to be very small and rather well approximated by a vector of zeros. For more on this and some numerical examples see Appendix B. In the next subsection, we devise an algorithm for sampling from the post-selection distribution of the regression coefficients selected by the Lasso without conditioning on the sign vector s.The sampler will operate by updating the two quantitiesη := (^T_M _M)^-1^T_M y, ξ := 1/λ_-M(I - _M(^T_M _M)^-1^T_M) y. §.§ Sampling from the Lasso Post-Selection Distribution With a view towards Gibbs sampling, we examine the region where a single regression coefficient may lie given the signs of all other coefficients.Let j∈ M be an arbitrary index. Denote by s^+j and s^-j vectors of signs where the signs for all coordinates but j are held constant and the jth coordinates are set to either 1 or -1, respectively.A necessary condition for the selection of M is that η_j ≤λ(^T_M_M)^-1_j,.s^-j or η_j ≥λ(^T_M_M)^-1_j,. s^+j. Ideally, we would be able to implement a Gibbs sampler that allows for the change of signs as we have done in Section <ref> by settingl_j = λ(^T_M_M)^-1_j,.s^-j, u_j = λ(^T_M_M)^-1_j,. s^+j.However, an important way in which the Lasso selection event differs from the one described in Section <ref> is that when a single coordinate of s is changed, the thresholds for all other variables change. Thus, in order for a single coordinate of η to change its sign, all other variables must be in positions that allow for that. In order to explore the entire sample space (and sign combinations) we propose a delayed rejection Metropolis-Hastings algorithm <cit.>. The algorithm works by attempting to take a Gibbs step for each selected variable in turn.If the proposed Gibbs step for the jth variable satisfies the constraints induced by the selection event then the proposal is accepted. Otherwise, we keep the proposal for the jth variable and make a global proposal for all selected variables keeping their signs fixed. We use the notation: η∼ N_p(β, _1),β = (_M^T_M)^-1_M^T(Y), _1 = σ^2 (^T_M _M)^-1,ξ∼ N(0, _0),_0 = σ^2_0(M) _0(M)^T. At some arbitrary iteration t, oursampler first makes the drawξ^t∼ f(ξ |M, η^t-1).This sampling task is quite simple in the sense that ξ |M, η has a multivariate normal distribution constrained to a convex set. Next, we make a proposal for each selected variable. For the jth selected variable we sample:r_j ∼ f(η_j | {η_j < l_j}∪{u_j < η_j},η_1^t,…,η_j-1^t,η_j+1^t-1,…,η^t-1_p),where l_j and u_j are as defined in (<ref>). If the sign of r_j differs from the sign of η_j^t-1, then we must verify that ξ^t from (<ref>) satisfies the constraints imposed by the new set of signs. If the constraints described in (<ref>)are not satisfied, then the proposal is rejected.If the proposalyields a point that satisfies both (<ref>) and(<ref>) then no further adjustment is necessary and the acceptance probability is 1 because the proposal is full conditional <cit.>. On the other hand, if the proposed point is not in the set from (<ref>),thena sign change has been performed andwe must update the values for other coordinates. Denote by TN(a, b, μ, σ^2) a univariate normal distribution with mean μ and variance σ^2 constrained to the interval (a,b). For all variables k ≠ j we sample a proposal from the following distribution:r_k ∼TN(a_k, b_k, η_k^t, σ^2_k,-k),where a_k=u_k and b_k=∞ if s_k^t=1, and a_k=-∞ and b_k=l_k if s_k^t=-1. Note that in (<ref>) l_k and u_k must be recomputed according to the proposed sign change. The Metropolis-Hastings algorithm in its entirety is described in Algorithm <ref> in the Appendix. The following Lemma describes the transitions of the proposed sampler.For the jth variable at the tth iteration define: r_1^→ =(η_1^t,…,η^t_j-1,r_j,η^t-1_j+1,…,η^t-1_p, ξ^t ), r_2^→ = (r_1,…,r_j-1,r_j,r_j+1,…,r_p, ξ^t ), r_1^← =(r_1,…,r_j-1,η^t-1_j,r_j+1,…,r_p, ξ^t ), r_2^← = (η_1^t,…,η^t_j-1,η_j^t-1,η_j+1^t-1,…,η^t-1_p, ξ^t ).Here, r_2^← represents the current state of the sampler after the Gibbs step from (<ref>). If ξ^t from (<ref>) is not in the set from(<ref>),thenthe proposal for r_j is rejected and the sampler stays in state r_2^←.If ξ^t is in (<ref>) and r_1^→ is in the set from (<ref>) then the sampler moves to r^→_1.Otherwise, if r_1^← is in the set from (<ref>) then the sampler stays in state r_2^←. Finally if neither r^→_1 nor r^←_1 are in the set from (<ref>) then thesampler either moves to r_2^→ or stays put at r_2^←.In this case, the move to r_2^→ occurs with probabilityp^t_j = min(φ(r_2^→ ; β, )/φ(r^←_2; β, )q(r^→_2,r^←_2)/q(r^←_2,r^→_2) , 1),whereq(x, y) =f(y_j | {Y_j < l_j}∪{u_j < Y_j},x_-j) ∏_k≠ jφ(y_k ; x_k, σ^2_k,-k)/P(Y_k ∈ (a_k, b_k); x_k, σ^2_k,-k).§.§ A Stochastic Ascent Algorithm for the Lasso We now propose an algorithm for computing the post-selection MLE when the model is selected via Lasso. We begin by defining the gradient ascent step, which uses samples from the post-selection distribution of the refitted regression coefficients.We give a convergence statement for the resulting algorithm, and we discuss practical implementation for which we address variance estimation and imposing sign constraints.Let M=S(y) be the Lasso-selected model.Given a sample η^i ∼ f_β̂^i-1(η|M) from the post-selection distribution of the least squares estimator, we take steps of the form:β̂^i = β̂^i-1 + γ_i( _M^Ty - (^T_M _M)η^i),where the γ_i satisfy the conditions from (<ref>).In Theorem <ref> we give a convergence statement for the algorithm defined by (<ref>). As in Theorem <ref>, the main challenge is bounding the variance of the stochastic gradient steps. Let η follow the conditional distribution of η∼ N(β, ) given the Lasso selection S(y)=M. Then there exists a constant A such that for all β∈ℝ^p:_β(^T_M_M)η - _M^T_β(Y|M)^2_2 ≤ A.Furthermore, the sequence (β̂^i) from (<ref>) converges, and its limit β̂^∞ := lim_i→∞β̂^i satisfies ψ(β̂^∞) := _M^Ty - _β̂^∞(^T_M Y|M) = 0. Before we exemplify the behavior of the proposed algorithm we first discuss some technicalities. The sampling algorithm proposed in the previous section assumes knowledge of the residual standard error σ, a quantity that in practice must be estimated from the data. We find that the cross-validated Lasso variance estimate recommended by <cit.> works well for our purposes.As in the univariate normal case, the post-selection estimator for the Lasso performs adaptive shrinkage on the refitted regression coefficients.However, the asymmetry between the thresholds dictated by different sign setsmay cause the sign of the conditional coefficient estimate to be different than the one inferred by the Lasso. Empirically we have found some benefit for constraining the signs of the estimated coefficients to those of the refitted least-squares coefficient estimates.We illustrate the proposed method via simulated data that are generated as follows.We form a matrix of covariates by sampling n rows independently from N_p(0,) with _i,j = ρ^|i-j|.We then generate a coefficient vector β by sampling k coordinates from the 𝐿𝑎𝑝𝑙𝑎𝑐𝑒(1)distribution and setting the rest to zero. Next, we sample a response vector Y ∼ N(μ, σ^2), where μ = β and σ^2 is chosen to obtain a certain signal-to-noise ratio defined as := (μ)/σ^2.We set n = 400, p = 1000, k = 5, ρ = 0.3 and = 0.2.Given a simulated dataset we select a model using the Lasso as implemented in the R package `glmnet' <cit.>.Following common practice and the default of the package, the tuning parameter λ is selected via cross-validation.Strictly speaking, this yields another post-selection problem.In Figure <ref> we plot three types of estimates for the regression coefficients selected by the Lasso. The conditional estimator proposed here, the refitted least-squares estimates and the Lasso estimates. In addition to the point estimates, we also plot three types of confidence intervals. The first are the Conditional-Wald confidence intervals analogous to the ones described in Section <ref>.They are given by:_j = (β̂^M_j,n - _j, 1-α/2 /√(n),β̂^M_j,n - _j, α/2 /√(n)),=^Dσ^2_β̂_n^M(n^-0.5^T_MY|M)^-1 n^-0.5(^T_My - _β̂^M_n(^T_M Y|M)).The second intervals are the Refitted-Wald confidence intervals obtained from fitting a linear regression model to the selected covariates without accounting for selection. Finally, we also include the intervals of <cit.> as implemented in the R package `selectiveInference' <cit.>.We term these Polyhedral confidence intervals. In Figure <ref>, black circles mark the conditional estimates, triangles the refitted least squares estimates, squares the lasso estimates and plus signs the true coefficient values. The conditional estimator tends to lie between the refitted and the lasso estimates.When the refitted estimate is far from zero the conditional estimator applies very little shrinkage, and when the refitted estimator is closer to zero the conditional estimator is shrunk towards the lasso estimate.The conditional confidence intervals also exhibit a behavior that depends on the estimated magnitude of the regression coefficients.When the conditional estimator is far from zero the size of the confidence intervals is similar to the size of the refitted confidence interval. When the conditional estimator is shrunk towards zero, its variance tends to be the smallest. The confidence intervals are the widest when the conditional estimator is just in-between the lasso and refitted estimates. The Polyhedral confidence intervals tend to be the largest in most cases.Section <ref> gives a more thorough examination of these estimates and confidence intervals. § ASYMPTOTICS FOR CONDITIONAL ESTIMATORSWe now present asymptotic distribution theory that supports the estimation method proposed in the previous sections.Such theory is complicated by the fact that model selection induces dependence between the previously i.i.d. observations.In Section <ref> we first give a consistency result for naive unconditional estimates, which in particular justifies our plug-in likelihood method for the normal means problem.We then outline conditions under which the conditional MLE is consistent for the parameters of interest in a general exponential family setting.In Section <ref> we adapt the theory to the Lasso post-selection estimator. We remark that theory on the efficiency of conditional estimators can be found in <cit.>.Proofs for this section are deferred to the appendix. §.§ Theory for exponential families Suppose we have an i.i.d. sequence of observations (Y_i)_i=1^∞ drawn from a distribution f^*.As a base model for the distribution of each observation y_i, consider a regular exponential family {p_θ:θ∈Θ} with sufficient statistic T ∈ℝ^p and natural parameter θ.So, Θ⊂ℝ^p. For the sample y_1,…,y_n, define T̅_n := n^-1∑_i=1^n T(y_i). Now, let ℳ be a countable set of submodels, which we denote by M = {p_θ^M : θ^M ∈Θ^M} with parameter space Θ_M⊂Θ.We consider a model selection procedure S_n : ℝ^p→ℳ that selects a model M as a function of T̅_n.Based on the true distribution f^* the sample is taken from, the selection procedure S_n induces a distribution P_n(M):=P(S_n(T̅_n) = M) over ℳ.We emphasize that f^* need not belong to any model in ℳ nor the base family {p_θ:θ∈Θ}. In the normal means problem, p_θ is a normal distribution with mean vector θ.The sufficient statistic is T(y)=^-1y, whereis the known covariance matrix. Each model M∈ℳ corresponds to a set of mean vectors with a subset of coordinates equal to zero.The selection procedure S_n is based on comparing the coordinates of T̅_n to predetermined thresholds l_j and u_j, recall (<ref>). In an asymptotic setting l_j and u_j willoften scale with the sample size to obtain a pre-specified type-I error rate.We consider estimation of a parameter θ^M_0 of a fixed model M, which represents the model selected in the data analysis.If the data-generating distribution f^* belongs to M, then f^*=p_θ^M_0 for a parameter value θ^M_0 ∈Θ^M and consistency can be understood as referring to the true data-generating distribution.If f^*∉M, then the parameter in question corresponds to the distribution in M that minimizes the KL-divergence from f^*, soθ^M_0 := inf_θ^M∈Θ^M -_f^*[log p_θ^M(Y) - log f^*(Y) ] = sup_θ^M∈Θ^M_f^*[ℓ_θ^M(Y)].Note that even under model misspecification we have _f^*(T̅_n) = _θ_0^M(T̅_n) because θ^M_0 is the solution to the expectation of the score equation. The post-selection setting is unusual in the sense that we are only interested in a specific model M if S_n(T̅_n) = M.Hence, it only makes sense to analyze the asymptotic properties of an estimator of θ^M_0 if model M is selected infinitely often as n→∞.This justifies our subsequent focus on conditions that involve the probability of selecting M.Our first result applies in particular to the normal means problem and is concerned with the post-selection consistency of the unconditional/naive MLE for θ^M_0. Let M be a fixed model with P_n(M)^-1e^-δ n = o(1) for all δ>0.Let θ̃_n^M=(θ̃^M_n,j)_j=1^p be an estimator that unconditionally is unbiased for θ^M_0.Suppose there is a constant C∈(0,∞) such that for all 1≤ j≤ p and n≥ 1 the distribution of √(n)(θ̃^M_n,j-θ^M_0,j) is sub-Gaussian for parameter C.Then θ̃^M_n is post-selection consistent, that is,lim_n→∞ P(θ̃^M_n - θ^M_0_∞ > ε | S_n(T̅_n) = M ) =0 ∀ε > 0.Next, we turn to the conditional MLE.Let ℓ_θ^M(y_i) be the log-likelihood of y_i as a function of θ^M, and let P_n,θ^M(M) be the probability of {S_n(T̅_n) = M} where y_1,…,y_n is an i.i.d. sample from p_θ^M.Then the conditional MLE isθ̂^M_n = max_θ^M ∈Θ^M( 1/n∑_i=1^nℓ_θ^M(y_i) ) - 1/nlog P_n,θ^M(M).We now give conditions for its post-selection consistency. Suppose the fixed model M satisfies P_n(M)^-1 = o(n), lim_n→∞inf_θ^M P_n,θ^M(M)e^n =∞.Furthermore, suppose that for a sufficiently small ball U⊂Θ centered at θ^M_0sup_θ^M∈ U(θ^M_0) P_n,θ^M(M)^-1 = o(n).Then the conditional MLE is post-selection consistent for θ^M_0, that is,lim_n→∞ P(θ̂^M_n - θ^M_0_∞ > ε | S_n(T̅_n) = M ) = 0∀ε > 0.Condition (<ref>) concerns the model-based selection probability and ensures that the conditional MLE exists with probability 1 as n→∞. Both the plug-in likelihood for the selected means problem and the Lasso likelihood satisfy this condition.We note that this condition excludes examples such as the singly truncated univariate normal distribution, where the probability that an MLE does not exist is positive <cit.>. Condition (<ref>) concerns the true probability of selecting the considered model M, which is required to not decrease too fast. Condition (<ref>) serves to ensure that the conditional score function is well behaved in the neighborhood of the estimand.§.§ Theory for the LassoIn this section we describe how the theory from the previous section applies to inference in linear regression after model selection with the Lasso. Suppose that we observe an independent sequence of observations(Y_i)_i=1^∞∼ N(μ_i,σ^2).Each observation Y_i is accompanied by a vector of covariatesX_i∈ℝ^p which we consider fixed, or equivalently,conditioned upon. The sufficient statistic for the linear regression model is given by T_n(, y)= ^Tyand the model selection function S_n(,y) is the Lasso, which selects a model:S_n(,y) = {j:β̂_Lasso,j≠ 0 }. For a selected model M, the conditional MLE for the regression coefficients is given by:β̂^M_n = max_βf(_1 y)/P_β(M),where P_β(M) = ∑_s P_β(_1(M,s)) × P_n(_0(M, s)). Notice that in our objective function the probabilities for not selecting the null-set are not a function of the parameters over which the likelihood is maximized. Instead, they are defined as a function of the sample size n and are determined by the imputed value for _0(M)μ. In practice we set _0(M)μ = 0. This imputation method can be justified by the fact that a model is unlikely to be selected infinitely often if lim_n→∞_0(M)μ≠ 0.For good behavior of the conditional MLE we made assumptions regarding the probabilities of selecting models of interest. Many previous works have investigated the properties that a data generating distribution must fulfill in order for the Lasso to identify a correct model with high probability. See for example <cit.>, and <cit.>. While we do not limit our attention to the selection of the correct model, this line of study sheds light on the conditions that any model M∈ℳ must satisfy in orderto be selected with sufficiently high probability. In the following we assume that the number of covariates p_n = p is kept fixed while the sample size n grows to infinity. We touch on high-dimensional settings briefly at the end of the section. The set of models for which we are able to guarantee convergence depends on the scaling of the ℓ_1 penalization parameter. We consider two types of scalings:λ_n ∝√(n), lim_n→∞λ_n/√(n) = ∞, lim_n→∞λ_n/n = 0. We begin by discussing the case where the ℓ_1 penalization parameter scales as in (<ref>). In this setting, the model selection probabilities can be bounded in a satisfactory manner as long as the expected projection of the model residuals on the linear subspace spanned by the inactive variables is not too large. Suppose that λ_n scales as in (<ref>) and that y follows a normal distribution as defined in (<ref>).Suppose further that for an arbitrary model of interest M∈ℳ there is a matrixand a vector β_0^M such that following holds:1/n^T →, (^T_M _M)^-1_M^Tμ→β_0^M,_0(M)μ→ 0,a.s.Then there exists an asymptotic lower bound for the probability of selecting M:lim_n→∞ P_n(M) ≥lim_n→∞inf_β^M P_n,β^M(M) = c > 0.Next, we discuss the setting where λ_n grows faster than √(n). Here we must impose stronger conditions on the selected model becausethe probability of selecting a model which contains covariates with zero coefficient values may decrease to zero at an exponential rate. Furthermore, we make assumptions similar to the Irrepresentable Conditions of <cit.> on the selected model in order to make sure that the model selection conditions corresponding to the variables not included in the model are satisfied with high probability. We emphasize that we do not assume that the Irrepresentability Conditions hold in order to satisfy the selection of a true model, rather, we make these assumptions in order to identify models (correct or not) for which we can guarantee the consistency of our estimators. Suppose that λ_n scales as in (<ref>) and that conditions (<ref>) and (<ref>) hold. Furthermore, assume that:1/n^T_M_M→_M,a.s.,|β^M_0j| > 0,∀ j∈ M,and thatlim_n→∞sup |^T_-M_M(^T_MX_M)^-1s| ≤ν < 1, ∀s∈{0,1}^|M|,for some constant ν, where 1 is a vector of ones and the inequality holds element wise. Under these conditions the following limits hold:lim_n→∞inf_β^M P_n,β^M(M)e^n = ∞, lim_n→∞inf_β^M ∈ U(β^M_0) P_n,β^M(M) = 1.The linear regression model trivially satisfies the modeling assumptions we made in the previous section. Thus, under the conditions given in the lemmas stated in this section, the conditional MLE for a model selected by the Lasso can be guaranteed to be well behaved.Fix a model M∈ℳ and suppose that the conditions of either Lemma <ref> or Lemma <ref> are satisfied. Then the conditional MLE (<ref>) is consistent for β^M_0.The Lasso is often used in cases where the number of covariates p is much larger than n. In order to make asymptotic analysis relevant to such cases it is common to assume that p grows with the sample size. While the theory developed here does not explicitly treat such a high-dimensional setting, none of our assumptions prevent us from allowing the model selection function S_n to consider a growing number of covariates as n grows. Specifically, if we assume that the ℓ_1 penalty scales at the rate ofλ_n = O( √(nlog p_n)) as prescribed e.g. by <cit.>, then our theory applies as long as the assumptions of Lemma <ref> are satisfied and log p _n = o(n). While we made a simplifying normality assumption, we expect that for fixed dimension p, non-normal errors can be addressed using conditions similar to those outlined by <cit.>. For theory for selective inference with non-normal errors in the high-dimensional case, see the work of <cit.>. § SIMULATION STUDYIn order to more thoroughly assess the performance of the proposed post-selection estimator for the Lasso, we perform a simulation study, which we pattern after that in <cit.>.We consider prediction and coefficient estimation using Lasso, our conditional estimator and refitted Lasso.We note already thatwhile some existing theoretical works outline conditions under which the refitted Lasso should outperform the Lasso in prediction and estimation <cit.>, this does not occur in any of our simulation settings. For confidence intervals we compare our Wald confidence intervals to the confidence intervals of <cit.> which we term Polyhedral. We find that both selection adjusted methods achieve close to nominal coverage rates. We generate artificial data for our simulations in a similar manner as we have done for Example <ref> in Section <ref>. We vary the sample size n= 100, 200, 400, 800, signal-to-noise ratio = 0.2, 0.8, and the sparsity level k = 2, 5, 10. For each combination of parameter values we generate data and fit models 400 times.We keep the amount of dependence fixed at ρ = 0.5 and the number of candidate covariates fixed at p = 400. In Figure <ref> we plot the log relative estimation error of the refitted-Lasso estimates and the conditional estimates compared to the Lasso as defined by:1/|M|(∑_j∈ Mlog_2(β̂_j - β_j) - log_2(β̂_Lasso_j - β_j)).This measure of error gives equal weights to all regression coefficients regardless of their absolute magnitude. In all simulation settings the refitted least-squares estimates are significantly less accurate than the Lasso or the conditional estimates. The conditional estimates tend to be more accurate than the Lasso estimates in all simulation settings. The conditional estimate tends to do better when there are at least some large regression coefficients in the true model. In Figure <ref> we present the relative prediction error of the refitted least-squares Lasso estimates and the conditional estimates, as defined by:log_2β̂- μ^2_2 - log_2β̂_Lasso - μ^2_2.Here, the Lasso provides more accurate predictions when the true model has more non-zero coefficients and the conditional estimator tends to be more accurate when the true model is sparse. In Figure <ref> we plot the coverage rates obtained by the Conditional-Wald confidence intervals proposed here, the Polyhedral confidence intervals and the refitted `naive' confidence intervals. Bothof the selective methods obtain close to nominal coverage rates.The coverage rates of the refitted confidence intervals which were not adjusted for selection were far below the nominal levels in all simulation settings. While the two types of selection adjusted confidence intervals seem to be roughly on par with respect to their coverage rate, they tend to differ in their size. For Figure <ref> wegenerate the additional datasets with a smaller number of candidate covariates p = 200, a larger range of sample sizes- n = 40, 75, 150, 300, 600, 1250, 2500, 5000, 10000, a signal-to-noise ratio of = 0.2 and k = 10 non-zero regression coefficients. We face some difficulty in assessing the average size of the Polyhedral confidence intervals, as these sometimes have an infinite length. a measure for the length of a typical confidence interval, we take the median confidence interval length in each simulation instance. In Figure <ref> we plot boxplots describing the distribution of the log relative size of the selection adjusted confidence intervals to that of the unadjusted refitted confidence intervals which tend to be the shortest. We find that as the sample size increases, the sizes of the Conditional-Wald confidence intervals are roughly twice the size the unadjusted confidence intervals, while the typical size of a Polyhedral interval is about twice the size of the Conditional-Wald confidence interval.§ CONCLUSIONIn this work we presented a computational framework which enables, for the first time, the computation of correct maximum likelihood estimates after model selection with a possibly large number of covariates. We applied the proposed framework to the computation of maximum likelihood estimates of selected multivariate normal means and regression models selected via the lasso. Our methods take the arguably most ubiquitous approach to data analysis, that of computing maximum likelihood estimates and constructing Wald-like confidence intervals. Furthermore, we do not involve conditioning on information additional to the identity of the selected model. A practice which, as shown by <cit.>, may lead to a loss in efficiency.We experimented with the proposed estimators and confidence intervals in a comprehensive simulation study. The proposed conditional confidence intervals were shown to achieve conservative coverage rates and the point estimates were shown to be preferable to the refitted-least squares coefficients estimates in all simulation settings, and preferable to the Lasso coefficient estimates when there are large signals in the data. While in this work we focused on inference in the linear regression method, our framework and theory are directly applicable to any exponential family distribution. Specifically, it is immediately applicable to estimation of parameters of selected generalized linear models using the normal approximations proposed by <cit.>. §.§ Supplementary Material Proofs of theorems can be found in Appendix A. Some numerical examples for different plug-in methods for the Lasso MLE are in Appendix B. Analysis for maximum likelihood inference after one-sided testing is in Appendix C. Pseudo-codefor the algorithms used in the paper is in Appendix D. A software package and example scripts can be found at: https://github.com/ammeir2/selectiveMLE.apalike§ PROOF OF THEOREMS §.§ Proof of Theorem <ref>In their work on the convergence of stochastic gradient methods, <cit.> formulate a general stochastic gradient method as an iterative optimization method consisting of steps of the form:x_t+1 = x_t + γ_t(s_t + w_t),where γ_t satisfies the condition from (<ref>), s_t is a deterministic quantity related to the true gradient and w_t is a noise component. They outline conditions regarding s_t and w_t that ensure the convergence of the ascent algorithm to an optimum of a function f(x) which possesses a gradient ∇ f(x). The conditions require that there exist positive scalars c_1 and c_2 such that for all t:c_1∇ f(x_t)^2≤∇ f(x_t)^T s_t,s_t≤ c_2(1 + ∇ f(x_t)),and that[ w_t | ℱ_t ]= 0, [ w_t ^2 | ℱ_t ]≤ A(1 + ∇ f(x_t)),where ℱ_t is the filtration at time t, representing all historical information available at time t regarding the sequence (w_t,s_t)_i=1^∞.In our case, the function of interest is the conditional log-likelihood f(x) = l(μ) := logℒ(μ), where the coordinates of μ which were not selected are imputed with the corresponding observed coordinates of y. The conditions regarding the deterministic component in (<ref>) hold as s_t = ∇ l(μ | M), is the gradient itself.In Theorem <ref> we assumed that we are able to take independent draws from the truncated multivariate normal distribution, meaning that[ w_t| ℱ_t ] = [y^t - ∇ l(μ | M)] = 0.In practice, we should make sure that we run the Markov chain for a sufficiently large number of iterations between gradient updates in order for (<ref>) to hold in good approximation.The remaining issue is to bound the variance of w_t.The first step is finding an upper bound for the variance of w_t as a function of μ. In the following, we denote by f(y) the unconditional density of y, by f(y_j) the marginal (unconditional) density of y_j and by f(y_-j |y_j) the conditional distribution of y_-j given y_j. Since the mean minimizes an expected squared deviation we have[(y_j - (y_j | M))^2 |M ]≤[(y_j - μ_j)^2 |M ] =∫(y_j - μ_j)^2 f(y | M) dy=∫_M(y_j - μ_j)^2f(y)/P(M)dy .Let C(y_j) = ∫_M f(y_-j|y_j)dy_-j, which satisfies 0≤ C(y_j) ≤ 1.Then∫_M(y_j - μ_j)^2f(y)/P(M)dy=∫_M(y_j - μ_j)^2C(y_j)/P(M) f(y_j)dy_j ≤∫_M(y_j - μ_j)^21/P(M) f(y_j)dy_j ≤∫_ℝ(y_j - μ_j)^21/P(M) f(y_j)dy_j= σ^2_j/P(M). The next step in bounding the variance of w_t is bounding P(M) from below. The difficulty with finding a lower bound P(M) is that one may make it arbitrarily small by varying the coordinates of μ for the non-selected coordinates. This is the motivation behind setting them to the observed values and only estimating the selected coordinates, resulting in the Z-estimator described in (<ref>).Assume without loss of generality that the first k coordinates of μ were not selected and that the last p - k + 1 were selected. We writeP(M) = ∫_M f(y) dy = ∫ _M f(y_1 | y_2,…,y_p)×…× f(y_p) dy.We begin with the integration with respect to y_1:∫_M f(y_1 |y_2,…,y_p) dy_1= 1 - Φ(u_1; μ_1,-1, σ^2_1, -1) + Φ(l_1 ; μ_1,-1, σ^2_1, -1).Now, denote by m_j = (l_j + u_j) / 2 the mid-point between l_j and u_j. We have 1 - Φ(u_1; μ_1,-1, σ^2_1, -1) + Φ(l_1 ; μ_1,-1, σ^2_1, -1) ≥1 - Φ(u_1; m_1, σ^2_1, -1) + Φ(l_1 ; m_1, σ^2_1, -1)≥ Φ(l_1 ; m_1, σ^2_1, -1) ≥ Φ(l_1 ; u_1, σ^2_1, -1).We can apply a similar lower bound to all selected coordinates to obtain:P(M)≥∏_j∈ MΦ(l_j; u_j, σ^2_j, -j)∫_M f(y_p-k+1|y_p-k+2,…,y_p)×…× f(y_p) dy_p-k+1… dy_p=P(j∉ S(y)∀ j∉ M) ∏_j∈ MΦ(l_j; u_j, σ^2_j, -j).Taking (<ref>) and (<ref>) together, we obtain the desired bound:(y_j) ≤()/P(⋂_j∉ M{j∉ M}) ∏_j∈ MΦ(l_j; u_j, σ^2_j, -j).The proof of Theorem <ref> follows in a similar fashion. §.§ Proof of Lemma <ref>The proposal vectors defined in the lemma are given by: r_1^→ =(η_1^t,…,η^t_j-1,r_j,η^t-1_j+1,…,η^t-1_p, ξ^t ), r_2^→ = (r_1,…,r_j-1,r_j,r_j+1,…,r_p, ξ^t ), r_1^← =(r_1,…,r_j-1,η^t-1_j,r_j+1,…,r_p, ξ^t ), r_2^← = (η_1^t,…,η^t_j-1,η_j^t-1,η_j+1^t-1,…,η^t-1_p, ξ^t ).The proposed algorithm for sampling η | M, ξ is a two-step Delayed Rejection Metropolis-Hastings sampler.In our case the first step is to propose a sample from the full conditional distribution of η_j given η_-j.We denote the first proposal by r_1^→. Note that at this stage only the jth coordinate has been changed. The acceptance probability for this step is given by:α(r_2^←, r_1^→) =f(r_1,j^→|r_1,-j^→)/f(r_2,j^←|r_2,-j^←)f(r_2,j^←|r_2,-j^←)/f(r_1,j^→|r_1,-j^→) I{S_n(X, r^→_1) = M}= I{S_n(, r^→_1) = M}.That is, the acceptance probability of the first proposal is either 1 or 0 depending on whether the proposal satisfies conditions (<ref>) and (<ref>). If the first proposal is not accepted and (<ref>) is satsifeid, then we make a second proposal r^→_2. The acceptance probability for the second proposal as defined by <cit.> is given by:α(r_2^←,r_1^←,r_2^←) =f(r^→_2) q_1(r^→_2, r^←_1) q_2(r^→_2, r^←_1, r^←_2)(1 - α(r_2^→, r_1^←))/f(r^←_2) q_1(r^←_2, r^→_1) q_2(r^←_2, r^→_1, r^→_2)(1 - α(r_2^←, r_1^→)),where q_1(x,y) is the density of the first proposal and q_2(x,z,y) is the density of the second proposal.We only make a second proposal if α(r_2^←, r_1^→) =0 and therefore the ratio is always zero if r_1^← is a legal value. If both r^←_1 and r^→_1 are illegal then α(r_2^←,r_1^←,r_2^←) is non-zero and the proposal densities are given by:q_1(x,y) = f(y_j | {y_j < l_j}∪{u_j < y_j},x_-j), q_2(x,z,y) = ∏_k≠ jφ(y_k ; x_k, σ^2_k,-k)/P(y_k ∈ (a_k, b_k) ; x_k, σ^2_k,-k).Put together, we get:q(x, y) := q_1(x,y)q_2(x,z,y),which yields the desired result.§.§ Proof of Theorem <ref>Under the assumptions of Theorem <ref> we show that the unadjusted MLE is consistent even in the presence of model selection, in the sense that:lim_n→∞ P(θ̂^M_n - θ^M_0_∞≥ε | M)= 0.We prove this result by showing that it holds for a model M∈ℳ that satisfies the conditions of the theorem. Assume without loss of generality that θ^M ∈Θ^M ⊆ℝ^p. In the following we will use the shorthand I_n(M) = I_{ S_n(y) = M}. The results follows from the fact that as long as the probability of model selection can be bounded from below, then the selection thresholds cannot be too far a way from the true parameters. lim_n→∞ P(θ̂^M_n - θ^M_0_∞≥ε | M) = lim_n→∞P_n(M | {θ̂^M_n - θ^M_0_∞≥ε}) P(θ̂^M_n - θ^M_0_1 ≥ε)/P_n(M)≤lim_n→∞P(θ̂^M_n - θ^M_0_∞≥ε)/P_n(M)=lim_n→∞P(⋃_j=1^p{|θ̂^M_nj - θ^M_0j| ≥ε})/P_n(M)≤lim_n→∞∑_j=1^pP(|θ̂^M_nj - θ^M_0j| ≥ε)/P_n(M)= lim_n→∞∑_j=1^pP( |√(n)(θ̂^M_nj - θ^M_0j)| ≥√(n)ε)/P_n(M)≤^(*)lim_n→∞∑_j=1^p2e^-nε^2/2σ_Mj/P_n(M)=^(**) 0 ,where σ^2_Mj is the jth diagonal element of ^M and (*) holds by subgaussian concentration. The equality (**) holds by our assumption regarding the rate at which P_n(M) is allowed to tend to zero.§.§ Proof of Theorem <ref>Before we prove the theorem, we first state and and prove a couple of Lemmas that will come in handy in the proof of Theorem <ref>. Lemma <ref> to follow states that the conditional MLE is consistent for θ^M_0 even when used in the non-conditional setting (when the model to be estimated is pre-determined). Set a family of distributions M and assume that no data-driven model selection has been performed.Then under the conditions of Theorem <ref> the conditional MLE is consistent for θ^M_0, that is,P(θ̂^M_n - θ^M_0_∞ > ε) → 0.Consider once again the conditional MLEθ̂^M_n= max_θ^M G_n^M = max_θ^M1/n∑_i=1^n[ ℓ_θ^M(y_i)-1/nlog P_n,θ^M(M) ]:= ℓ̅_n(θ^M) - 1/nlog P_n,θ^M(M).where ℓ_θ^M(y_i) is the unconditional log-likelihood of y_i. We are evaluating the properties of the conditional estimator in the unconditional setting where M is designated for inference before the data are observed. In this setting, the conditional MLE can be considered an M-estimator obtained from performing inference under a misspecified likelihood. We now show that θ̂_n^M is consistent for the θ^M_0.We havesup_θ^MG_n^M(θ^M) ≥G_n^M(θ_0^M),which implies thatℓ̅_n(θ̂^M_n) ≥ℓ̅_n(θ_0^M) - 1/nlog P_n,θ^M_0(M) + 1/nlog P_n,θ̂^M_n(M).Equation (<ref>) together with assumption (<ref>) givesℓ̅_n(θ̃^M_n) ≥ℓ̅_n(θ_0^M) - o(1).Thus, the conditions for consistency as given by <cit.> (Theorem 5.14 p. 48) are satisfied. The implication of (<ref>) is that in the unconditional setting the conditional M-estimator is a consistent estimator.Next, we show that the difference between the conditional expectation of the sufficient statistic T̅_n converges to the unconditional expectation. This result will assist us later in proving a law-of-large number type statement for T̅_n under the conditional distribution. Under the assumptions of Theorem <ref>, for all δ < 1/2,n^δ_θ^M_0(T̅_n) -_θ^M_0(T̅_n|M) → 0 .According to Lemma <ref>, if y_i ∼ f_θ^M with f_θ^M an exponential family distribution and P_n,θ^M(M|T̅_n) ∈{0, 1} then the first derivative of the conditional log-likelihood is ∂/∂θ^M G_n(θ^M)= 1/n∑_i=1^n T(y_i) - _θ^M(T(y_i) | M) := T̅_n - _θ^M(T̅_n |M).At the maximizer of G_n(θ^M), for any δ < 1/2, we have:n^δ[ T̅_n - _θ̂^M_n(T̅_n |M) ] = 0 ,which implies thatn^δ(T̅_n - _θ^M_0(T̅_n) ) + n^δ(_θ^M_0(T̅_n) -_θ̂^M_n(T̅_n |M)) = 0 .Since n^δ(T̅_n - _θ^M_0(T̅_n) )=o_p(1) by law of large numbers, we obtain thatn^δ(_θ^M_0(T̅) -_θ̂^M_n(T̅ |M)) = o_p(1). Finally in order to prove the desired results we must show thatE_θ^M_0(T̅ |M) - E_θ̂^M_n(T̅ |M) → 0.It is clear that since θ^M_n →θ^M_0, a fixed continuous function of θ̂^M_n will converge as the sample size grows. However,E_θ̂^M_n(T̅ |M) is a function of both θ̂^M_n and n, and we must make sure that it does not vary too much with n in order for the desired convergence to hold. Define t = a^TT̅_n. Byassumption (<ref>) we have that for some sufficiently large n:sup_θ^M: θ^M - θ^M_0 < 1/√(n)|_θ^M_0(t |M) - _θ^M(t|M)|≤sup_θ^M: θ^M - θ^M_0 < 1/√(n)_θ^M(t)/P_n,θ^M(M)1/√(n). Because y is of an exponential distribution and t is an average we can bound the unconditional variance in the neighborhood of θ^M_0. For a sufficiently small ε > 0 there exists a constant C > 0 such that, sup_θ^M: θ^M - θ^M_0≤ε_θ^M(t) < C/n because _θ^M(t) is a continuous function and the supremum is taken over a compact set. Thus, by the √(n) consistency of θ̂^M_n for θ^M_0, the difference satisfies n^δ |_θ^M_0(t |M) - _θ̂_n^M(t|M)| = o(1) for any vector a as well as for T̅_n itself and the claim follows.We are now ready to prove Theorem <ref>. The first step in the proof is showing that T̅_n converges in probability conditionally on M. This result is a simple consequence of Markov's inequality and our assumption that P_n(M)^-1 = o(n). Set an arbitrary vector a ∈ℝ^p and define t = a^TT̅_n. By Markov's inequality,P_n(|t - _n(t|M)| > ε|M)≤_n(t|M)/ε^2≤O(n^-1)/ε ^2 P_n(M) = o(1).To see why (<ref>) holds, write:_n(t|M) = ∫(t - (t))^2/P_n(M) I{S_n(T̅_n) = M}f(t)d(t) - [(t) - _n(t|M)]^2≤a^T( T(y_i)) a/nP_n(M).By the fact that (<ref>) holds for any arbitrary vector a, together with Lemma <ref>, we can determine that conditionally on M, T̅_n →_p (T̅_n). By our assumption thatthe log-likelihood l_θ^M(y) is a continuous mapping of T(y), assumption (<ref>) and Lemma <ref>, conditionally on the selection of M we have:1/n∑_i=1^nℓ_θ^M(y_i) - 1/nlog P_n,θ^M(M) →_p[ℓ_θ^M(y_i)].The rest of the proof follows in a similar manner to the proof of Lemma <ref> where the law of large numbers in the proof of Theorem 5.14 in <cit.> is replaced by (<ref>) and our assumption that ℓ̅_n(θ^M) is a continuous function of T̅_n. §.§ Proof of Lemma <ref>In the context of this proof we use the following notation:A_0(M,s) := { l_o(M, s) ≤_0(M, s) y < u_0 (M, s)},A_1(M,s) := {_1(M, s) y < u_1 (M, s)} .For ease of exposition, we make a simplifying assumption that lim_n→∞λ_n/n^1/2 = λ^*. We begin by bounding the probability of not selecting the null-set. By our assumption that n^-1^T converges, we have that the thresholds l_0(M,s) and u_0(M,s) also convergence for all candidate models and sign permutations. Furthermore, by our assumption regrading the rate in which λ_n grows and the expectation of _0(M)y, _0(M) y →^D N(0, (_0)),where,(_0) = lim_n→∞σ^2/λ_n^2^T_-M(I - _M(^T_M _M)^-1^T_M)_-M.Thus, lim_n→∞ P_n(_0(M,s)) = c_0(M,s) > 0, ∀ M, s.Since the probability of _0(M,s) can be bounded in a uniform manner, we can setc_0(M) := min_s c_0(M,s),and obtain a lower bound for the probability of selecting M by boundingP_n(M) ≥ c_0(M) P_n( ∪_s_1(M,s) ) := c_0(M) P_n(_1(M) ). We bound P_n(_1(M)) next. Recall that the threshold a regression coefficient must cross is given byu_1(M,s) = -λ_ndiag(s) (^T_M _M)^-1 s.This threshold is a bit unwieldy, as it depends on the signs of the active set and an exact realization of _M. Since we are interested in asymptotic behavior of random quantities, it will be sufficient to work with the limiting value of the threshold:u^*_1(M,s) = lim_n→∞√(n) u_1(M,s) = -λ^*diag(s) _M^-1 s,Now, in order to eliminate the dependence on the signs of the active set define:u^*_1(M) := sup_ssup_j|(λ^*diag(s) _M^-1 s)_j|,and define an event:_1 := {√(n)|η_j| > u^*_1(M), ∀ j ∈ M}.In Ã_1 we replaced all coordinate thresholds with the largest threshold, and so it is clear that:limsup_n→∞P_n,β^M(Ã_1)/P_n,β^M(A_1)≤ 1.Furthermore, we have the lower boundP_n,β^M (Ã_1) ≥∏_j ∈ M( Φ(-u^*(M) ; 0, σ^2_j, -j) + 1 -Φ(u^*(M) ; 0, σ^2_j, -j)) , ∀β^M ∈ℝ^|M|,where σ^2_j, -j := Var(√(n)η_j | η_-j). See the proof of Theorem <ref> for details on how this bound is derived. The rest follows by our normality assumption and the fact that (<ref>) holds for all β^M including β^M_0. §.§ Proof of Lemma <ref>We begin by treating the probability of satisfying the conditions for not selecting the variables not in the model. Using the same notations as in the proof of Lemma <ref>, the following limit holds:(A_0) → 0,and consequently, by assumption (<ref>):lim_n→∞P_n(_0(M,s)) = 1, ∀ s. Next, we treat the probabilities of satisfying the conditions for selecting the variables included in the model. As before, we make a simplifying assumption that there exists a constant 0 < δ <0.5 such that:λ_n/n^0.5 + δ = λ^*,In the fast scaling case, a lower bound on P_n,β^M(Ã_1) no longer exists because the threshold u^*(M) grows with the sample size. However, we can show that a satisfactory bound exists at β^M_0. Since in this setting λ_n grows faster than √(n), we redefine the limit of the selection threshold:u^*_1(M,s) = lim_n→∞√(n)/n^δ u_1(M,s) = -λ^*diag(s) _M^-1 s.We can redefine u^*_1(M) in an analogous manner. Now, we rewrite the bound (<ref>) at the point β^M = β^M_0 and with u^*_1(M) properly scaled asP_n,β^M_0 (Ã_1) ≥∏_j ∈ M( Φ(- u^*(M) n^δ; √(n)β^M_0, σ^2_j, -j) + 1 -Φ( u^*(M) n^δ; √(n)β^M_0, σ^2_j, -j))With no loss of generality assume that β^M_0 < 0 to obtain the desired bound:lim_n→∞ P_n,β^M_0 (Ã_1) ≥lim_n→∞∏_j ∈ MΦ(- u^*(M) n^δ; √(n)β^M_0, σ^2_j, -j) = 1,where the limit holds because δ < 0.5. A similar result holds in a small neighborhood U of β^M_0 because the probability of selection is continuous in β^M.In order to bound the infimum of P_n,β^M(A_1), we again start from (<ref>) to get:P_n, β^M (Ã_1)≥∏_j ∈ M( Φ(- u^*(M) n^δ;0, σ^2_j, -j) + 1 -Φ( u^*(M) n^δ; 0, σ^2_j, -j))≥∏_j ∈ MΦ(- u^*(M) n^δ; 0, σ^2_j, -j)≥ C (n^δ u^*_1(M) / σ_j,-j/1 +u^*_1(M)^2 n^2δ / σ^2_j, -j)^|M|∏_j ∈ M e^- u_1^*(M)^2 n^2δ/2σ^2_j, -j= O(e^-n^2δ/n^δ|M| / 2).The lemma follows by our assumption that δ < 0.5. In (<ref>) we used the inequality:Φ(t;0, σ^2) ≥ C t/σ/1 + t^2 / σ^2 e^-t^2/2σ^2. § NUMERICAL EXAMPLES FOR THE LASSO MLEIn Section <ref> we discuss the conditions that must hold in order for a specific model to be selected by the Lasso and propose to estimate the mean vector _0(M)E(y) by 0. Here, we propose some alternatives and seek to demonstrate that the proposed method is a reasonable one. We generate data using the same process as described in Example <ref> with parameter values ρ = 0.5, n = p = 100, k = 3 and = 0.5. We selected a model with two active parameters of positive sign with observed values of 0.17 and 0.13. In order tocompute the conditional log-likelihood for this example we must decide on appropriate estimates for E(_0(M)y). We present results for three options. The first is to use the observed value, _0y as an estimate for its expectation, we term this method `plug-in'. The second is to work under the assumption that E(_0y) ≈ 0, estimating the expectation with a vector of zeros, we term this method `zero'. A third option is to simply assume that P(l < A_0y <u )≈ 1 for all signs sets, we term this method `none'. Finally, we also compute the likelihood under the truth, setting E(_0 y) = _0 E(y).We draw the contour plots for the two-dimensional log-likelihoods as a function of the selected regression coefficients in Figure <ref>. While the contour plots are visually similar, the values of the log-likelihoods differ slightly. For the `none' and `zero' methods the log-likelihood was maximized at 0.14, 0.02 at a log-likelihood value of 14.2. This is similar to the log-likelihood computed under the true expectation, where the maximum was also obtained at 0.14, 0.02 and at a slightly different value of 14.3. Finally, for the plug-in method the maximum was obtained at 0.13, 0.02 with a value of 16.9. Thus, for this example, the maximum likelihood estimates computed using the different imputation methods yielded results that are essentially equivalent. In this example the true probability of P(l_0 < _0y < u_0) was close to 1 for all sign permutations. In a second example we generate data using parameter values ρ = 0.8, n = 100, p = 500, k = 5 and = 0.2. Here we selected a model with four variables where the observed refitted regression coefficients estimates were 0.13, 0.17, 0.21 and 0.15. For all estimation methods the maximum of the log-likelihood was obtained atapproximately 0, -0.05, 0.1, 0. The values of the log-likelihood function at its maximum was 15.9 when no imputation was used, 19.9 for plugin imputation, 16.1 for the zero imputation and 16.7 when the true parameter value was used to compute the log-likelihood. The contour of the log-likelihood function are plotted in Figure <ref> for the second and third variables, keeping the values of the first and last coefficients fixed at zero.§ DESCRIPTION OF ALGORITHMS1em1em | http://arxiv.org/abs/1705.09417v2 | {
"authors": [
"Amit Meir",
"Mathias Drton"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170526024559",
"title": "Tractable Post-Selection Maximum Likelihood Inference for the Lasso"
} |
End-to-end Global to Local CNN Learning for Hand Pose Recovery in Depth Data[This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.] 16cmMeysam Madadi^1,2, Sergio Escalera^1,3, Xavier Baró^1,4, Jordi Gonzàlez^1,2^1 Computer Vision Center, Edifici O, Campus UAB, 08193 Bellaterra (Barcelona), Catalonia Spain^2 Dept. of Computer Science, Univ. Autònoma de Barcelona (UAB), 08193 Bellaterra, Catalonia Spain^3 Dept. Mathematics and Informatics, Universitat de Barcelona, Catalonia, Spain^4 Universitat Oberta de Catalunya, Catalonia, Spain ================================================================================================================================================================================================================================================================================================================================================================================================================================================= Individuals on social media may reveal themselves to be in various states of crisis (e.g. suicide, self-harm, abuse, or eating disorders). Detecting crisis from social media text automatically and accurately can have profound consequences.However, detecting a general state of crisis without explaining why has limited applications. An explanation in this context is a coherent, concise subset of the text that rationalizes the crisis detection. We explore several methods to detect and explain crisis using a combination of neural and non-neural techniques. We evaluate these techniques on a unique data set obtained from Koko, an anonymous emotional support network available through various messaging applications. We annotate a small subset of the samples labeled with crisis with corresponding explanations. Our best technique significantly outperforms the baseline for detection and explanation. § INTRODUCTIONApproximately one person dies by suicide every 40 seconds <cit.>. It accounts for approximately 1.5 % of all deaths, and is the second leading cause of death among young adults <cit.>. There are indications that for each adult who dies of suicide there may have been more than 20 others attempting suicide <cit.>. Closely tied to suicide are self-harm, eating disorders, and physical abuse. 13 to 23% of adolescents engage in self-injury at some point <cit.>. In the United States, about 7 million females and 1 million males suffer from eating disorders annually <cit.> and an average of 20 people are physically abused by intimate partners every minute <cit.>. Self-harm victims are more likely to die by suicide by an order of magnitude <cit.>. Additionally, eating disorders and physical abuse increase the risk of suicide <cit.>.We identify all of these phenomena (suicide, self-harm, eating disorders, physical abuse), with the term crisis. Someone who is in crisis is likely in need of some form of immediate support, be itintervention, therapy, or emergency. Roughly a third of people who think about suicide make a plan; 72% of those who report making a suicide plan actually make an attempt <cit.>. Accurate, automatic detection of someone in crisis in social media, messaging applications, and voice assistants has profound consequences. A crisis classifier can enable positive outcomes by enabling human outreach at earlier stages, and rescues at later stages. In many ways, however, it can still fall short of a human detector, by way of lacking an explanation or rationale of why classifier detected crisis. The factors that explain why someone is in crisis can range from suicidal ideation to eating disorders, from self-harm to sexual abuse.In crisis situations, triage can improve if the detection system can explain why the person is crisis. Someone who is about to die by suicide via overdose should receive a different response than someone who is considering anorexia due to self-image issues. Population level surveillance, diagnostics, and statistics are much improved due to factor based explanation. Finally, in human-in-the-loop crisis systems, the human responder can better sift through information if the factors of crisis were visually highlighted through automated means <cit.>. With the rise of complex neural models in classification tasks,we've seen gains in accuracy at the cost of transparency and interpretability <cit.>.In this work, we present models that we validate both for their raw accuracy and for the quality of their explanations. Validating a model's explanations, in addition to its detection performance, can improve interpretability in the model. In summation, automatically generating explanations for crisis detection scales and pays off in many ways. While we evaluate our models' explanations against human reference explanations, it is not practical to collect enough explanations to train these models on such data. Collecting an explanation requires an annotator to write free text or to highlight text for every case of crisis, sometimes more than once for a post (e.g. for a sexual abuse victim considering suicide), while merely identifying crisis is a simple binary decision task that can be performed much more quickly and cheaply. In this paper, we explore the problem of generating explanations for crisis without explicit supervision using modern representation learning techniques. We demonstrate our success comparing our proposed models with a variety of explanatory methods and models on a rationale-labeled test set. We evaluate the generated explanations through ROUGE metrics against human-produced references. In addition, we show detection performance that outperforms prior methods. § RELATED WORKDetecting Crisis <cit.> identify 125 Twitter users who publicly state their suicide attempt online on a specific date and have tweets preceding the suicide attempt. They artificially balance these tweets using data from random users who are assumed to be neurotypical, acknowledging that this data will be contaminated with users who also ideate and attempt suicide. They train simple, linear classifiers that show promise in detecting these suicidal users and discuss the difficulties of realizing this technology, highlighting privacy and intervention concerns. In our work, we attempt detection and explanation on phenomena that includes but is not limited to suicide on a dataset that is significantly larger and not artificially balanced. However, we do not incorporate the record of suicide attempt as signal when labeling. <cit.> operate on a filtered subset of mental health records to determine whether a mention of suicide is affirmed or negated. They do classification on mental health records,which are filtered by thekeyword. The goal of their work was the development of improved information retrieval systems for clinicians and researchers, with a specific focus on suicide risk assessment. Thus, the domain is constrained. Their work also differs from ours significantly in its technical execution. Rather than use neural network classifiers, they use probabilistic context free grammars to execute negation detection. This task is quite different than ours, both in dataset and approach, and is most likely not applicable to open-domain social media text. They also do not aim to or need to provide explainable detections, as mentions of suicide are clearly present in all of their data and negation detection is sufficient 'rationale' for affirming or negating that mention. <cit.> annotate Twitter data for suicide-risk with the level of distress as the label and achieve high inter-annotator agreement.They use a combination of specialized keyword search and LIWC sadness scores <cit.>to filter 2.5 million tweets down to 2000 in order to make the annotation task tractable. Our source dataset, which we introduce in the next section, has a significantly higher base rate of crisis; thus, no filtering is necessary.They train SVM classifiers on bag of n-grams to detect distress on different subsets of annotations, but do not explore neural classifiers, nor unsupervised explanations of detections. <cit.> and <cit.> also detect distress on small datasets using simple classifiers. <cit.>annotate 200 samples for distress level and discretize counts related to bag of word, part of speech, sentence complexity and sentiment word features to train a variety of multiclass classifiers. <cit.> annotated nearly 2000 tweets for different levels of suicidality and used word counts as features, filtered by document frequency. In our work, we compare neural techniques against linear models trained on word frequency counts both for detection and explanation as a baseline. Due to the relatively large amount of data in our training set, we do not use any custom features for the baseline. <cit.> detect depression in Twitter data in two stages: 1) detecting evidence of depression at all and 2) classify the specific depressive symptom if depression was detected.This is a kind of explanation in that it directly detects one of three symptoms of depression (fatigue, disturbed sleep, depressed mood). However, their data is explicitly annotated for these sub-factors, whereas our data is not. 1,656 tweets in their dataset were annotated with specific depressive symptoms.Interpretable Neural Networks In the past few years, neural attention mechanisms over words <cit.> have led to improvements in performance and interpretability in a range of tasks, such as translation (Bahdanau) and natural language inference <cit.>. These models induce a soft alignment between two sequences with the primary aim of using it to remove an information bottleneck, but this alignment can be also be used quite effectively to visualize which inputs drive model behavior.<cit.> present a more direct method for training interpretable text classifiers. Their model is also trained end-to-end, but instead of inducing a soft weighting, it extracts a set of short subspans of each input text that are meant to serve as sufficient evidence for the final model decision.In another work with similar goals, <cit.> introduce a model agnostic framework for intepretability, LIME, that learns an interpretable model over a given sample input that is locally faithful to the original trained model. § METHODSOur training set consists of N examples {X^i, Y^i}_i=1^N where the input X^i is a sequence of tokens w_1, w_2, ..., w_T, and the output Y^i is a binary indicator of crisis. §.§ Word Embeddings Each token in the input is mapped to an embedding. We use reference GloVe embeddings trained on Twitter data <cit.>. We used the 200 dimensional embeddings for all our experiments, so each word w_t is mapped to x_t ∈ℝ^200. We denote the full embedded sequence as x_1:T. §.§ Recurrent Neural NetworksA recurrent neural network (RNN) extends on a traditional neural network to recursively encode a sequence of vectors, x_1:T, into a sequence of hidden states. The hidden state of the RNN at t-1 is fed back into the RNN for the next time step.h_t = f(x_t, h_t-1;Θ)This allows the network to construct a representation incrementally as it reads the input sequence. In particular, we encode the sequence using a gated recurrent unit <cit.> RNN. The GRU employs an update gate z_t and reset gate r_t that are used to compute the next hidden state h_t h_t = (1 - z_t)h_t-1+ z_t h̃_tz_t = σ(W_zx_t + U_zh_t-1)h̃_t = tanh(Wx_t + U(r_t ⊙h_t-1))r_t = σ(W_rx_t + U_rh_t-1) We use a bidirectional RNN (running one model in each direction)and concatenate the hidden states of each model for each word to obtain a contextual word representation h^bi_t.§.§ Attention over WordsWith attention, a scoring function scores the relevance of each contextual word representation h^bi_t. We employ the unconditional attention mechanism used to do document classification employed by <cit.>. u_t = tanh(W_wh^bi_t + b_w) α_t = expu_t/∑_texpu_td = ∑_tα_t h_t The attention mechanism serves two purposes. d acts as a contextual document representation which can be fed into a downstream model component for detection. In addition, the score vectoru_1:N, can be utilized to seed our explanation, which will be expanded on in a following section. Optionally for detection, we encode the document by using the last hidden state of a single forward GRU, without the reverse GRU and attention mechanism. Both encoding schemes are evaluated in our experiments. §.§ Training Objective The final document encoding of each sample, d, is fed into a sigmoid layer with one node to detect the probability of crisis.We minimize the logistic loss objective during training. l(y,p) = -ylog(p) - (1 - y)log(1-p)where y is the true value and p is the output of the logistic output layer.§.§ Seeding the ExplanationsOur next goal is to generate explanations given the inputs and outputs for our trained model. We do this by building a subset of words which `seed' the explanation generation function. The explanation generation function is fixed while testing across all seeding techniques, thus allowing extensibility by modularizing the seed function using a relevant model. The seed function is meant to give a set of tokens from the input that most influenced the prediction, thus sewing the initial stitches of the explanation. For the task of detecting crisis, descriptive content words, such as adjectives, nouns, and verbs, are desirable compared to stop words or punctuation. We test three techniques of seeding words for a given input: (1) Using the magnitudes of activated coefficients in a logistic regression model. This acts as our baseline. (2) Using the distribution of attention from our neural model. (3) Using LIME, which can generate words that led to a prediction for any model. Each seed function is passed in the number of seed words to return, k. This allows us to maintain similar output behavior for all three techniques; it also allows us to extend the seed functions to more complex models.We will now detail how seeding works for each of these mechanisms.* Logistic Coefficients: Logistic regression is a linear model that learns a vector of weights for a fixed set of features to detect in binary classification. As a baseline, we train a logistic regression model on unigrams to learn a vector of weights for each word in the vocabulary. For our seed function, we find the k most highly-weighted activated features according to the model. A feature is activated if the word occurs in the given input. * Neural Attention: In this setting, we select seeds by sorting the words by their attention weights u. In order to get human interpretable scores for attention, we introduced a configurable dial to control how attention was distributed over the input by introducing an L2 penalty on the output of the attention. * LIME: The LIME API contains aparameter in thefunction. Each explanation will then result in learning an interpretable model, which can be used to then seed the explanation.The LIME API is applied to both models, the baseline logistic and the neural model. §.§ Explanation Generation Algorithm We use a novel algorithm for producing explanations that depends on seeds from a separately-developed seeding module.The algorithm acts on the input text and the k explanation seeds. It works as follows. First, the sentence of importance is identified by taking the sentence with the most seeds. The identified sentence is then parsed with a dependency parser <cit.>, and traversed from the root to find the highest seed in the sentence.If the highest seed token is not a verb and not the head of the entire sentence, we then traverse to the seed's head node. Subsequently, the subtree phrase of the highest seed is used for the explanation. Since the parse is projective, the subtree is necessarily a contiguous sequence of tokens.§ EXPERIMENTS §.§ Training Data Koko has an anonymous peer-to-peer therapy network based on an clinical trial at MIT <cit.>, which is made available through chatbots on a variety of social media platforms including Facebook, Kik, and Tumblr. They provided us with our training data through a research partnership. The posts on the platform generally come from users who are experiencing negative thoughts and need some form of emotional support. Each post is on average 3.1 sentences long with a standard deviation of 1.7 sentences. The training set is roughly 106,000 binary labeled posts (crisis or not).Their data was annotated for crisis by crowdworkers.During annotation, annotations were given clear instructions on what consists of crisis, examples, and common mistakes and helpful tips. These instructions were revised over multiple iterations of small batches of data to improve inter-annotator agreement. Using a minimum of three labelers per sample, they achieve over 95% inter-annotator agreement.Because the platform is a support network, the rates of depression and other mental disorders are high: the annotation task identified roughly 20% of the training data as crisis. This is in contrast to previous work using Twitter data, where multiple layers of filtering are required to get a reasonable sample of distress <cit.>. Our dataset requires no filtering and estimates the natural distribution of the platform. §.§ Explanation dataWe have select a set of 1242 labeled posts as our test set. Of these, 200 are labeled crisis.We annotate the 200 crisis samples with their corresponding explanations. An explanation is a phrase or clause in the post that most strongly identifies the rationale behind the crisis label. When selecting the explanation, we aim for them to be accurate, coherent, and concise.§.§ Model Configuration and TrainingWe tokenize the data using Spacy <cit.>. We do not fine-tune the pretrained GloVe embeddings, but rather learn a simple embedding transformation matrix that intervenes between the embeddings and the RNNs. We use 200 dimensional embeddings and 100 dimensional forward and backward GRUs (yielding 200 dimensional contextual representations). We apply an L2 penalty on the attention output using λ = .0001. We pad each input to 150 words. We train using RMSprop with a learning rate of .001 and a batch size of 256. We add dropout with a drop rate of 0.1 in the final layer before detecting to reduce overfitting. We determined the dropout rate, batch size, and input length empirically through random hyperparameter search and determined λ for the attention penalty using human evaluation. We use the best model from 20 epochs of training selected using a validation sample of 10% of the source data (excluding the test data).§ RESULTS AND DISCUSSION §.§ Detection EvaluationThe neural models significantly outperform the logistic model in detection accuracy (Table 1), with the best neural model achieving a .80 F1 on the crisis detections, compared to .66 for the logistic model. The neural attention model achieves a .76 F1 score, which is still significantly better than the linear baseline. The best model does not have an attention penultimate layer, bur rather a single feedforward GRU layer.§.§ Attention VisualizationWe first validate that the attention mechanism yields distributions that meet our expectations. This is done by visualizing the attention using a heat map, with each normalized attention weight aligned with the corresponding token in the input. Initially, we found that the attention distribution had a very low entropy, placing the bulk of the probability in a single token of the input. We penalized low entropy outputs using an L2 penalty, controlled by a λ parameter.We did not further tune it to boost explanation evaluation scores, though we expect this could improve performance. Figure 1 demonstrates the attention output for two crisis samples. For the first sample, we see that attention is focused around the final clause, and is not concentrated entirely on one word. As one would expect, “i cut myself” fetches the highest weight in the attention distribution. The second visualization shows singular attention on the word `suicide', thus placing markedly less importance on the rest of the input. This differentiation between background information and crisis signal provides a reassuring signal that the model is using reasonable features. §.§ Qualitative Explanation Results Interestingly, all of the techniques resulted in several high quality explanations. We surveyed about 20 samples and for each one, at least one of the seeding functions contained the correct explanation. Surprisingly, the logistic baseline performed quite well in this capacity. In Table 2,we show an example where all of the techniques got the identical result. This is likely due to the predictive power of the phrase `kill myself'. In many cases, the generated explanation contained more text than is necessary to accurately capture the gold explanation. The second example (Table 2) shows this in the neural+attention technique. This may suggest room for improvement in the explanation generation technique. The third example shows a difficult case in which the majority of the text is background information and only the last word of the input is included in the gold explanation.We see that both neural models and logistic+LIME are successful in capturing roughly the correct explanation. §.§ Quantitative Explanation Results We evaluate the generated explanations using ROUGE-1 and ROUGE-2 <cit.>, which measure the overlapping units (unigrams and bigrams respectively) of the generated text and reference texts. In Table 3 and 4, the average ROUGE-1 and ROUGE-2 scores for the generated explanations are listed for each model and seed strategy.By a large margin, the neural classifier[We use the RNN with attention in this result. The forward RNNin conjunction with LIME showed nearly identical ROUGE performance.] in conjunction with the LIME seed function outperformed the rest of the models. In ROUGE-2 evaluation, it beats the next best average F1 score by a margin of 10 points and in ROUGE-1 evaluation, it beats the next best average F1 by 12 points. Since LIME directly determines which input most influences the prediction, while attention does so onlyindirectly, this result makes sense. However, the LIME seeding function is the slowest approach we consider, taking up to a minute to generate a explanation. The neural attention seeding is negligible in contrast to this.In Table 3, the ROUGE Metrics show similar performance for the baseline logistic model and the neural model. However, in Table 1, we see that detection output is much better for the neural models. This suggests that though the logistic regression is quite reasonable in ranking features by weights, it fails to capture subtleties and dependencies in a sequence that an RNN captures. Thus, neural+attentionis a better choice between the two. The logistic+LIME outperforms the baseline by 5 points in precision for ROUGE-1 and around 3.5 points in precision for ROUGE-2. This exemplifies the efficacy of LIME, which is tuned for the individual example, rather than the model coefficients, which are tuned for the training data.§ CONCLUSION In this paper, we present and compare explanation-oriented methods for the detection of crisis in social media text. We introduce a modular approach to generating explanations and make use of neural techniques that significantly outperform our baseline. The best models presented are both effective at detection and produce explanations similar to those produced by human annotators. We find this exciting for two reasons:Within the domain of crisis identification, successes in explanation help to build the trust in trained models that is necessary to deploy them in such a sensitive context. Looking beyond this, we expect that our techniques may generalize to text classification more broadly. In the future experiments, we hope to explorehuman evaluation of the generated explanations as an indicator of trust in the model, to investigate compression-based approaches to explanation <cit.>, and to consider richer architectures for text classification.§ ACKNOWLEDGMENTSWe thank the anonymous reviewers and Kareem Kouddous for their feedback. Bowman acknowledges support from a Google Faculty Research Award and gifts from Tencent Holdings and NVIDIA Corporation. We thank Koko for contributing a unique dataset for this research. acl_natbib | http://arxiv.org/abs/1705.09585v1 | {
"authors": [
"Rohan Kshirsagar",
"Robert Morris",
"Sam Bowman"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170526134454",
"title": "Detecting and Explaining Crisis"
} |
Department of Physics, The University of Texas, Austin, Texas 78712-0264, USAFor the spin Hall effect arising from strong band-structure spin-orbit coupling, a semiclassical Boltzmann theory reasonably addressing the intriguing disorder effect called side-jump is still absent. In this paper we describe such a theory of which the key ingredient is the spin-current-counterpart of the semiclassical side-jump velocity (introduced in the context of the anomalous Hall effect). Applying this theory to spin Hall effects in a two-dimensional electron gas with giant Rashba spin-orbit coupling, we find largely enhanced spin Hall angle in the presence of magnetic impurities when only the lower Rashba band is partially occupied.72.10.-d, 72.10.Bg, 72.25.-b Semiclassical Boltzmann theory of spin Hall effects in giant Rashba systems Cong Xiao December 30, 2023 ===========================================================================§ INTRODUCTION It is now generally accepted that three mechanisms – intrinsic, side-jump and skew scattering – contribute to both the spin Hall effect (SHE) and anomalous Hall effect (AHE) <cit.>. Among the three mechanisms, the side-jump mechanism is of special interest because it originates from scattering but can, in some simple cases <cit.>, be independent of both the disorder density and scattering strength. In particular, when the SHE or AHE arises from strong spin-orbit coupling in the band-structure, the side-jump belongs to the category of disorder-induced interband-coherence effect which has recently been an important topic in condensed matter physics <cit.> .In investigating transport phenomena in solids, the semiclassical Boltzmann approach is appealing due to its conceptual intuition <cit.>. In the study of SHE-AHE, how to incorporate side-jump effects into the semiclassical formalism is an attracting theoretical issue <cit.>. In the study of AHE, the renewed semiclassical theory addressing this issue has proven useful in obtaining physical pictures <cit.>. In such a theory, the quantum mechanical information on side-jump is coded in the expressions of gauge-invariant classical concepts such as the coordinate-shift and side-jump velocity <cit.>. On the other hand, in the field of SHE when the spin is not conserved due to strong spin-orbit coupling in the band structure, such as in a Rashba two-dimensional electron gas (2DEG), a semiclassical description to side-jump SHE is still absent <cit.>. Although the modified Boltzmann equation <cit.> developed in studying the AHE can be directly applied to SHE, the spin–current-counterpart of the side-jump velocity in this case has not been addressed before.In the present paper we formulate a semiclassical Boltzmann framework of SHE when the spin is not conserved due to strong band-structure spin-orbit coupling. This semiclassical theory takes into account interband-coherence effects induced by both the dc uniform electric field and weak static disorder. We work out the spin-current-counterpart of the side-jump velocity based on scattering-induced modifications of conduction-electron states. When the electric field turns on, this quantity contributes one part of the side-jump SHE.As applications we consider the SHE in a 2DEG with giant Rashba spin-orbit coupling and short-range impurities. We focus on the enhancement of spin Hall angle when the Fermi energy is tuned down towards and below the band crossing point in giant Rashba 2DEGs with magnetic disorder. The spin Hall angle which measures the generation efficiency of the transverse spin current from the longitudinal electric current is the figure of merit of the SHE. Giant Rashba spin-orbit coupling energy comparable to or even larger than the Fermi energy is possible in the polar semiconductor BiTeX (X=Cl, Br and I) family and related surfaces and interfaces <cit.>. Thus these systems are promising to realize efficient conversion of charge current into spin current.The paper is organized as follows. In Sec. II we outline the semiclassical formulation of SHE. Section III introduces the Rashba model and calculates the SHE. Section IV concludes the paper.§ SEMICLASSICAL FORMULATION Considering the linear response of the spin current polarized in one particular direction (z direction is chosen in the following) to a weak dc uniform electric field 𝐄 in non-degenerate multiband electron systems in the weak disorder regime, one has the semiclassical formula𝐣^z=∑_lf_l𝐣_l^z,where 𝐣_l^z is the amount of spin current carried by the conduction-electron state denoted by l, f_l is the semiclassical distribution function.The conduction-electron state may be modified by the electric field and static impurity scattering, 𝐣_l^z can thus deviate from the customary pure-band value <cit.>: (𝐣_l ^z)^0≡⟨ l|^z|l⟩ with ^z the spin-current operator. Here |l⟩=|η𝐤⟩=|𝐤⟩|u_𝐤^η⟩ is the Bloch state, η is the band index, 𝐤 is the crystal momentum, |𝐤⟩ and |u_𝐤^η⟩ are the plane-wave and periodic parts of |η𝐤⟩, respectively. Following the recipe based on the quantum-mechanical perturbation theory for the electric-field modified Bloch state and the Lippmann-Schwinger equation for the scattering modified conduction-electron state in Ref. Xiao2017SOT, in the weak disorder regime nontrivial corrections caused by interband-coherence effects to (𝐣_l^z) ^0 read:𝐣_l^z=(𝐣_l^z)^0+δ ^in𝐣_l^z+δ^ex𝐣_l^z.The intrinsic correction δ^in𝐣_l^z=2Re⟨ l|^z|δ^𝐄l⟩ arises from the interband-virtual-transition (electron charge e) |δ^𝐄l⟩=-iħ e𝐄·∑_η^'≠η|η^'𝐤⟩⟨ u_𝐤^η^'|𝐯̂|u_𝐤^η⟩/(ϵ_𝐤^η -ϵ_𝐤^η^')^2 induced by the electric field <cit.>, with ϵ_l≡ϵ_𝐤^η the energy of Bloch state |η𝐤⟩ and 𝐯̂ the velocity operator. Thusδ^in𝐣_l^z=ħ e∑_η^'≠η2Im⟨η𝐤|^z |η^'𝐤⟩⟨ u_𝐤^η^'|𝐯̂· 𝐄|u_𝐤^η⟩/(ϵ _𝐤^η-ϵ_𝐤^η^')^2is an electric-field-induced interband-coherence effect.The extrinsic correction δ^ex𝐣_l^z originates from the interband-coherence during the elastic electron-impurity scattering process. The scattering-induced modification to conduction-electron states is captured by the Lippmann-Schwinger equation describing the scattering state |l^s⟩=|l⟩+(ϵ_l-Ĥ_0+iϵ) ^-1T̂|l⟩ with the T-matrix T̂|l⟩=V̂ |l^s⟩ related to the disorder potential V̂ and disorder-free Hamiltonian Ĥ_0. |δ l^s⟩≡|l^s⟩-|l⟩ denotes the scattering-induced modification to the Bloch state. Thus δ^ex𝐣_l^z is related to the values of 2Re⟨⟨ l|^z|δ l^s⟩⟩ _c and ⟨⟨δ l^s |^z|δ l^s⟩⟩ _c in the lowest nonzero order in the disorder potential. Here ⟨ ..⟩ _c denotes the average over disorder configurations and we assume that the statistical average of the disorder potential is zero (nonzero value only shifts the origin of total energy) ⟨ V⟩ _c=0. Only the terms containing interband matrix elements of ^z represent the disorder-induced interband-coherence effects, therefore <cit.>δ^ex𝐣_l^z =∑_η^''≠η^'∑_η^'𝐤^'×⟨⟨η𝐤|V̂|η^'𝐤^'⟩⟨η^''𝐤^'|V̂|η𝐤⟩⟩ _c⟨η^'𝐤^'| ^z|η^''𝐤^'⟩/( ϵ_𝐤^η-ϵ_𝐤^'^η^'-i0^+)(ϵ_𝐤^η-ϵ_𝐤 ^'^η^''+i0^+)+2Re∑_η^'≠η∑_η^''𝐤^''×⟨⟨η^'𝐤|V̂|η^''𝐤^''⟩⟨η^''𝐤^''|V̂|η𝐤⟩⟩ _c⟨η𝐤|^z|η^'𝐤⟩/(ϵ_𝐤^η-ϵ_𝐤^η^'+i0^+)( ϵ_𝐤^η-ϵ_𝐤^'' ^η^''+i0^+).It has been shown <cit.> that the side-jump velocity 𝐯_l^sj which is an important ingredient in the semiclassical theory of AHE <cit.> can also be obtained in this way (δ ^ex𝐯_l=𝐯_l^sj) and thus shares the same origin. δ^ex𝐣_l^z can therefore be deemed as the spin–current-counterpart of the side-jump velocity in the case of band-structure spin-orbit coupling.The properly modified steady-state linearized Boltzmann equation in the presence of weak static disorder has been proposed as <cit.>:e𝐄·𝐯_l^0∂ f^0/∂ϵ_l=-∑_l^'ω_l,l^'[f_l-f_l^' -∂ f^0/∂ϵ_le𝐄·δ𝐫_l',l],where 𝐯_l^0=∂ϵ_l/ħ∂𝐤 is the band velocity, f^0 is the Fermi distribution function, δ𝐫_l',l is the coordinate-shift in the scattering process (l→ l^') <cit.> and ω_l,l^' the semiclassical scattering rate (l^'→ l). Up to the linear order of the electric field one has the decomposition <cit.>f_l=f_l^0+g_l^n+g_l^a,with g_l^n the normal part of the out-of-equilibrium distribution function satisfying the Boltzmann equation in the absence of δ𝐫_l',l and g_l^a the anomalous distribution function related to δ𝐫_l',l. It is now clear <cit.> that δ r_l',l is a disorder-induced interband-coherence effect and so is g_l^a.Given that the semiclassical formulation is relevant in the weak disorder regime, Eq. (<ref>) reduces to <cit.>𝐣^z=∑_lf_l(𝐣_l^z)^0+∑ _lg_l^2sδ^ex𝐣_l^z+∑_lf_l^0δ ^in𝐣_l^z,up to the zeroth order of total impurity density and scattering strength. g_l^2s represents the value of g_l^n in the lowest Born order <cit.>. In higher Born orders, some additional contributions to g_l^n appear and are responsible for the transverse transport due to the breakdown of the principle of microscopic detailed balance. The analysis of these higher-Born-order contributions under the non-crossing approximation has been detailed in Ref. Xiao2017AHE. Here we only mention that there is an interband-coherence scattering effect called “intrinsic-skew-scattering induced side-jump” appearing in the third Born order under the Gaussian disorder. Below we set 𝐣 ^z,in=∑_lf_l^0δ^in𝐣_l^z which is just the intrinsic contribution to the spin current independent of the disorder <cit.>, and 𝐣^z,sj=∑_lg_l^2sδ ^ex𝐣_l^z because it is related to the spin–current-counterpart of the side-jump velocity. In general case of SHE induced by strong band-structure spin-orbit coupling, 𝐣^z,sj is just one part of the side-jump SHE arising from disorder-induced interband-coherence effects. Other two semiclassical contributions to the side-jump SHE (from the anomalous distribution function g_l^a and the intrinsic-skew-scattering induced side-jump) <cit.> and the skew scattering SHE arising from non-Gaussian disorder are all included in the first term of Eq. (<ref>) <cit.>.To be more clear we can consider the case of randomly distributed scalar pointlike Gaussian disorder with density n_0 and average strength V_0. Then g_l^2s∼ n_0^-1V_0^-2, δ^exj_l^z∼ n_0V_0^2, g_l^a∼ n_0^0V_0^0, and the third-Born-order contribution to g_l^n behaves as ∼ n_0^0V_0^0 (thus is called the intrinsic-skew-scattering <cit.>). In this case the side-jump SHE may consist of three semiclassical contributions in the zeroth order of both the impurity density and scattering strength: j^z,sj, ∑_lg_l^a(𝐣_l^z)^0 and the intrinsic-skew-scattering induced side-jump.§ MODEL CALCULATION§.§ Model The model Hamiltonian of a Rashba 2DEG is H_0=ħ^2𝐤 ^2/2m+α_Rσ̂·(𝐤×ẑ), where 𝐤 is the 2D wavevector, m is the effective mass, σ̂ is the vector of Pauli matrices, α_R the Rashba coefficient. The internal eigenstates read |u_𝐤^η⟩=1/√(2)[1,-iηexp( iϕ)]^T, where η=± label the two bands ϵ_k^η=ħ^2k^2/(2m)+ηα_Rk, and tanϕ=k_y/k_x.For ϵ>0 the corresponding wave number in η band is given as k_η(ϵ)=-η k_R+k_0(ϵ). Here k_R=mα_R/ħ^2=1/2(k_-( ϵ)-k_+(ϵ)) measures the momentum splitting of two Rashba bands, whereas k_0(ϵ) ≡α_R^-1√(ϵ_R^2+2ϵ_Rϵ)=1/2∑_ηk_η(ϵ). The density of states of η band takes the form D_η(ϵ)=D_0k_η(ϵ)/k_0(ϵ), with D_0=m/2πħ^2.For 0>ϵ>-ϵ_R/2, the iso-energy surface slices the spectrum two rings of radii k_-ν(ϵ)=k_R+(-1) ^ν-1k_0(ϵ), where ν=1,2 denote the two monotonic segments (Fig. 1), k_0(ϵ)≡α _R^-1√(ϵ_R^2+2ϵ_Rϵ)=1/2( k_-1(ϵ)-k_-2(ϵ)). The density of states of -ν branch reads D_-ν(ϵ) =D_0k_-ν(ϵ)/k_0(ϵ).The conventional definition of the spin current as an anti-commutator of velocity and spin is employed: ^z=ħ/21/2{σ̂_z,𝐯̂}=ħ/2ħ𝐤/mσ̂_z. It is purely off-diagonal in band-index space in this model: (𝐣_l^z)^0=0, thus the SHE in Eq. (<ref>) is determined only by𝐣^z=𝐣^z,in+𝐣^z,sj.The Boltzmann equation can be conveniently solved by using variables l=(ϵ,η,ϕ) for ϵ>0 and l=( ϵ,-ν,ϕ) for 0>ϵ>-ϵ_R/2. Correspondingly, ∑_l=∑_η(ν)∫ dϵ D_η(-ν)(ϵ)∫dϕ/2π for ϵ>0 (0>ϵ>-ϵ_R/2). If ϵ>0, in the lowest Born order the energy-integrated elastic scattering rate is ω_η,η^'^ϕ,ϕ^'( ϵ)=∫ dϵ^'D_η^'( ϵ^')ω_l',l^2s. Whereas if 0>ϵ >-ϵ_R/2 there exists elastic scattering between the two branches ν=1 and 2, and one has ω_-ν,-ν^'^ϕ,ϕ^'(ϵ)=D_-ν^'(ϵ)∫ dϵ_l^'ω_l',l^2s. Assuming isotropic disorder potential, transport-time type solutions to g_l^2s exist <cit.>. For ϵ>0 we haveg_η^2s(ϵ)=(-∂_ϵ f^0)e𝐄·𝐯_η(ϵ,ϕ) τ_η^tr(ϵ),where the transport time τ_η^tr(ϵ) is determined by1=∑_η^'∫dϕ^'/2πω_η,η ^'^ϕ,ϕ^'[τ_η^tr-cos( ϕ^'-ϕ)τ_η^'^tr].For 0>ϵ>-ϵ_R/2 we haveg_-ν^2s(ϵ)=(-∂_ϵ f^0)e𝐄·𝐯_-ν(ϵ,ϕ) τ_-ν^tr(ϵ),with the transport time τ_-ν^tr(ϵ) decided by1=∑_ν^'∫dϕ^'/2πω_-ν,-ν^'^ϕ,ϕ^'[τ_-ν^tr-(-1) ^ν^'-νcos(ϕ^'-ϕ)τ_-ν^'^tr]. §.§ Calculations We consider the impurity potential is produced by randomly distributed short-range scatters at 𝐑_i, i.e., V(𝐫) =∑_i,μV_μ^iσ_μδ(𝐫-𝐑 _i) with μ=0,1,2,3 and σ_0 the unity matrix in spin space <cit.>. Here the short-range potential is approximated by the delta-potential. We assume Gaussian disorder approximation and isotropic magnetic scattering <cit.>. n_0 and n_m are the concentrations of nonmagnetic and magnetic impurities, respectively. V_0 and V_m are the average strengths for the nonmagnetic and magnetic scattering, respectively. The external electric field is applied in x direction.§.§.§ Nonmagnetic impurities When ϵ>0, straightforward calculation leads to the spin-current-counterpart of the side-jump velocityδ^ex(𝐣_l^z)_y^nm=-1/τ_0η/k_Rħ/8cosϕ,with τ_0=(2π n_0V_0^2D_0/ħ)^-1. The transport time reads <cit.> τ_η^tr( ϵ)=τ_0D_η(ϵ)/D_0, and then the side-jump spin Hall current isj_y^z,sj=∑_lg_l^2sδ^ex(𝐣_l^z) _y^nm=e/8πE_x,which completely cancels out the intrinsic spin Hall current j_y ^z,in=-e/8πE_x. This just reproduces the well-known <cit.> vanishing spin Hall current j_y^z=0 in the semiclassical Boltzmann theory for the first time.When 0>ϵ_F>-ϵ_R/2, the intrinsic spin Hall current is j_y^z,in=k_0(ϵ_F)/k_R-e/8πE_x. Meanwhile the spin-current-counterpart of the side-jump velocity readsδ^ex(𝐣_l^z)_y^nm=1/τ_01/k_0(ϵ)ħ/8cosϕ.and thus the side-jump spin Hall currentj_y^z,sj=∑_lg_l^2sδ^ex(𝐣_l^z) _y^nm=k_0(ϵ_F)/k_Re/8πE_xagain cancels out the intrinsic one. This also coincides with the zero SHE obtained by the Kubo formula <cit.>.§.§.§ Magnetic impurities For isotropic delta-like magnetic impurity potential, since the contributions from V_x^i and V_y^i cancel out in Eq. (<ref>), the spin–current-counterpart of the side-jump velocity is given byδ^ex(𝐣_l^z)_y^m=-1/3n_mV_m^2/n_0V_0^2δ^ex(𝐣_l^z) _y^nm.The transport time is given byτ_η^tr(ϵ)/τ_m=8k_0( ϵ)-k_η(ϵ)/7k_0( ϵ)for ϵ>0, andτ_-ν^tr(ϵ)/τ_m=k_0( ϵ)/k_R8k_R-k_-ν(ϵ) /7k_Rfor 0>ϵ>-ϵ_R/2, with τ_m=(2π/ħn_mV_m^2D_0)^-1.When both Rashba bands are partially occupied, the side-jump spin Hall currentj_y^z,sj=∑_lg_l^2sδ^ex(𝐣_l^z) _y^m=1/7-e/8πE_x=1/7j_y^z,inenhances the total spin Hall current to j_y^z=j_y^z,in+j_y ^z,sj=8/7j_y^z,in. This j_y^z is the same as the weak-disorder-limit value of that obtained by Kubo diagrammatic calculations <cit.>. The longitudinal electric current is j_x =e^2/πħ^2τ_m3ϵ_R+7ϵ_F/7E_x, the spin Hall angle is thereforeα_sH=ej_y^z/(ħ/2)/j_x =-2ħ/τ_mϵ_R1/3+7ϵ_F/ϵ_R.When only the lower Rashba band is partially occupied, the side-jump and the total spin Hall currents arej_y^z,sj=k_0(ϵ_F)/7k_R-e/8πE_x=1/7j_y^z,inand j_y^z=8/7j_y^z,in, respectively. The longitudinal electric current is j_x=e^2/πħ^2τ_m3ϵ_R-ϵ_F/7k_0^2(ϵ_F) /k_R^2E_x, thusα_sH=-2ħ/τ_mϵ_R1/( 3-ϵ_F/ϵ_R)√(1+2ϵ_F/ϵ_R).Although ħ/τ_mϵ_R is a small quantity in giant Rashba systems, the factor √(1+2ϵ_F/ϵ_R) can be very small leading to a large spin Hall angle when ϵ_F is located close to the band bottom of the lower Rashba band. For instance, if ħ/τ_mϵ_R=0.02, 1+2ϵ_F/ϵ_R=0.1 leads to α_sH≃-4%, which is quite large <cit.>. Smaller τ_m and smaller √(1+2ϵ_F/ϵ_R) may lead to larger α_sH. However, the quantitative analysis of this possibility is beyond the scope of the semiclassical theory which is valid only in the weak disorder regime. From the above equation, α_sH goes to infinity as ϵ_F goes to the band bottom of the lower Rashba band. But this low carrier density limit is actually beyond the Boltzmann regime, and more rigorous microscopic treatments are called for.§.§.§ Both nonmagnetic and magnetic impurities The coexistence of nonmagnetic and magnetic impurities may be the more realistic case <cit.>. Only main results will be given in this case. Since there is no mixing between the nonmagnetic and magnetic scattering as pointed out by Inoue et al. <cit.>, the spin-current-counterpart of the side-jump velocity is δ^ex( 𝐣_l^z)_y=[1-1/3τ_0/τ_m]δ^ex(𝐣_l^z)_y^nm. The total spin Hall current reads j_y^z=8/3τ_0/τ_m/1+7/3τ_0/τ_mj_y^z,in, depending on the relative weight of different types of scattering <cit.>. The side-jump effect vanishes when τ_0=3τ_m, the same condition as that for the vanishing of the ladder vertex correction in the Kubo diagrammatic calculation in the spin-σ_z basis for the case ϵ_F>0 <cit.>.For Fermi energies above and below the band crossing point, the spin Hall angles areα_sH=-ħ/τ_Sϵ_R2/3τ _0/τ_m/1+τ_0/τ_m+(1+7/3τ_0/τ_m)ϵ_F/ϵ_Randα_sH=-ħ/τ_Sϵ_R2/3τ _0/τ_m/1+τ_0/τ_m+[1-1/3τ_0/τ_m]ϵ_F/ϵ_R1/√(1+2ϵ_F/ϵ_R),respectively. Here we define τ_S^-1≡τ_0^-1+τ_m^-1. Tuning the ratio τ_0/τ_m one can find that α_sH changes monotonically and continuously from the scalar-disorder-dominated case to the magnetic-disorder-dominated regime.§ DISCUSSION AND SUMMARY Before concluding this paper, we comment on some important issues not mentioned in above sections.First, the simple form of the semiclassical Boltzmann equation (<ref>) is exactly valid only for isotropic bands and isotropic scattering <cit.>. In our opinion, in the presence of anisotropy a more generic and complicated form of the Boltzmann equation may be necessary, we refer the readers to Ref. Luttinger1957 for detailed discussions.Second, the recently highlighted “coherent skew scattering” under the Gaussian disorder beyond the non-crossing approximation <cit.> is also included in the first term of Eq. (<ref>). This additional contribution is also in the zeroth order of both the impurity density and scattering strength in the weak disorder limit in the presence of only one type of disorder, like the side-jump contribution, but is not an interband-coherence scattering effect <cit.>. Thus how to place this contribution into the classification of AHE-SHE mechanisms suggested in Refs. Nagaosa2010,Sinova2015 is still an open question. Therefore, in presenting our theory we avoid this issue. Fortunately, in the Rashba model considered in Sec. III the first term of Eq. (<ref>) vanishes. Besides, we should remind the interested readers that this so-called “coherent skew scattering” has actually already been proposed sixty years ago by Kohn and Luttinger <cit.>. We will provide a comprehensive description of a semiclassical Boltzmann theory going beyond the non-crossing approximation in a future publication.Finally, in the presence of spin-orbit coupling the electron spin is not conserved thus the spin current is not uniquely defined. The conventionally defined spin current adopted in this study is not a conserved transport current. A physically attracting definition of the conserved spin current has been suggested by Shi et al. <cit.> by introducing the torque dipole moment. However, disorder effects on the torque dipole spin current <cit.> in the Bloch representation are hard to deal with under the uniform external electric field in the Boltzmann theory. We reserve these for future studies.In summary, we have formulated a semiclassical Boltzmann framework of spin Hall effects induced by strong band-structure spin-orbit coupling in non-degenerate multiband electron systems in the weak disorder regime. We worked out the absent ingredient in previous semiclassical theories, i.e., the spin–current-counterpart of the semiclassical side-jump velocity. This gauge-invariant quantity arises from the interband-coherence during the elastic electron-impurity scattering, and contributes one part of the side-jump spin Hall effect.Applying this theory to a 2DEG with giant Rashba spin-orbit coupling, we showed an enhanced spin Hall angle when only the lower Rashba band is partially occupied in the presence of magnetic impurities. We note that this energy regime below the band crossing point in Rashba systems and similar systems is of intense theoretical interest also from the standpoint of enhanced efficiency of spin-orbit torque and of Edelstein effect <cit.>, as well as enhanced thermoelectric conversion efficiency <cit.>. The author is indebted to Qian Niu, Dingping Li and Zhongshui Ma for insightful discussions.99Sinova2015J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015).Nagaosa2010N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).Kovalev2010A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Anomalous Hall Effect in Disordered Multiband Metals, Phys. Rev. Lett. 105, 036601 (2010).Culcer2010D. Culcer, E. M. Hankiewicz, G. Vignale, and R. Winkler, Side jumps in the spin Hall effect: Construction of the Boltzmann collision integral, Phys. Rev. B 81, 125332 (2010).Yang2011S. A. Yang, H. Pan, Y. Yao, and Q. Niu, Scattering universality classes of side jump in the anomalous Hall effect, Phys. Rev. B 83, 125122 (2011).SinitsynN. A. Sinitsyn, Semiclassical theories of the anomalous Hall effect, J. Phys.: Condens. Matter 20, 023201 (2008).Hou2015D. Hou, G. Su, Y. Tian, X. Jin, S. A. Yang, and Q. Niu, Multivariable Scaling for the Anomalous Hall Effect,Phys. Rev. Lett. 114, 217203 (2015).Culcer2017D. Culcer, A. Sekine, and A. H. MacDonald, Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects, Phys. Rev. B 96, 035106 (2017).Xiao2017SOTC. Xiao and Q. Niu, Semiclassical theory of spin-orbit torques in disordered multiband electron systems, preprint, arXiv: 1705.07947, to be published in Phys. Rev. B.Ziman1960J. M. Ziman, Electrons and Phonons (Clarendon, Oxford, 1960).Zhang2005S. Zhang and Z. Yang, Intrinsic Spin and Orbital Angular Momentum Hall Effect, Phys. Rev. Lett. 94, 066602 (2005).noteZhang2005For instance, one can verify that the semiclassical framework described in Ref. Zhang2005 cannot produce any extrinsic contribution to the SHE of the conventional spin current polarized in z direction in a disordered Rashba 2DEG. (Equations (2) – (7) in that paper).Eremeev2012S. V. Eremeev, I. A. Nechaev, Yu. M. Koroteev, P. M. Echenique, and E. V. Chulkov, Ideal two-dimensional electron systems with a giant Rashba-type spin splitting in real materials: Surfaces of Bismuth tellurohalides, Phys. Rev. Lett. 108, 246802 (2012).Sakano2013M. Sakano, M. S. Bahramy, A. Katayama, T. Shimojima, H. Murakawa, Y. Kaneko, W. Malaeb, S. Shin, K. Ono, H. Kumigashira, R. Arita, N. Nagaosa, H. Y. Hwang, Y. Tokura, and K. Ishizaka, Strongly spin-orbit coupled two dimensional electron gas emerging near the surface of polar semiconductors, Phys. Rev. Lett. 110, 107204 (2013).Wu2014L. Wu, J. Yang, S. Wang, P. Wei, J. Yang, W. Zhang, and L. Chen, Thermopower enhancement in quantum wells with the Rashba effect, Appl. Phys. Lett. 105, 202115 (2014).note-SO scatteringIn the present paper we do not consider side-jump induced by spin-orbit scattering. A semiclassical treatment of this case can be found in P. M. Levy, H. Yang, M. Chshiev, and A. Fert, Spin Hall effect induced by Bi impurities in Cu: Skew scattering and side-jump, Phys. Rev. B 88, 214432 (2013).Xiao2017AHEC. Xiao, D. Li, and Z. Ma, Role of band-index-dependent transport relaxation times in anomalous Hall effect, Phys. Rev. B 95, 035426 (2017).note-sjThe reason why the side-jump AHE-SHE induced by band-structure spin-orbit coupling is defined as the sum of these three semiclassical contributions was detailed in Ref. Nagaosa2010. Simply speaking, there are at least two motivations: one is the equivalence described in Ref. note-vertex, the other is that all these three contributions belong to the disorder-induced interband-coherence effect (see Refs. Nagaosa2010,Kovalev2010,Hou2015,Xiao2017SOT).Xiao2016PRBC. Xiao, D. Li, and Z. Ma, Unconventional thermoelectric behaviors and enhancement of figure of merit in Rashba spintronic systems, Phys. Rev. B 93, 075150 (2016).Lu2013H.-Z. Lu and S.-Q. Shen, Extrinsic anomalous Hall conductivity of a topologically nontrivial conduction band, Phys. Rev. B 88, 081304(R) (2013).Inoue2006J.-I. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G. E.W. Bauer, and L.W. Molenkamp, Vertex Corrections to the Anomalous Hall Effect in Spin-Polarized Two-Dimensional Electron Gases with a Rashba Spin-Orbit Interaction, Phys. Rev. Lett. 97, 046604 (2006).Grimaldi2006C. Grimaldi, E. Cappelluti, and F. Marsiglio, Off-Fermi surface cancellation effects in spin-Hall conductivity of a two-dimensional Rashba electron gas, Phys. Rev. B 73, 081303(R) (2006).Wang2007P. Wang, Y.-Q. Li, and X. Zhao, Nonvanishing spin Hall currents in the presence of magnetic impurities, Phys. Rev. B 75, 075326 (2007).Ebert2015K Chadova, S. Wimmer, H. Ebert, and D. Kodderitzsch, Tailoring of the extrinsic spin Hall effect in disordered metal alloys, Phys. Rev. B 92, 235142 (2015).Fert2011A. Fert and P. M. Levy, Spin Hall Effect Induced by Resonant Scattering on Impurities in Metals, Phys. Rev. Lett. 106, 157208 (2011).note-vertexIn the context of AHE induced by band-structure spin-orbit coupling, it has been established that the disorder-induced interband-coherence contribution (side-jump) calculated in the semiclassical Boltzmann theory is equivalent to the ladder vertex correction in the weak disorder limit to the bubble of the anomalous Hall conductivity in the non-chiral basis (σ_z basis for the Rashba model). The present calculations suggest that this equivalence is also valid for the spin Hall conductivity of the conventionally defined spin current. In fact, this equivalence has been employed in the statement of Ref. Sinova2015.Luttinger1957W. Kohn and J. M. Luttinger, Quantum Theory of Electrical Transport Phenomena, Phys. Rev. 108, 590 (1957).Ado2015I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Anomalous Hall effect with massive Dirac fermions, Europhys. Lett. 111, 37004 (2015).Luttinger1958J. M. Luttinger, Theory of the Hall Effect in Ferromagnetic Substances, Phys. Rev. 112, 739 (1958).Shi2006J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper definition of spin current in spin–orbit coupled systems, Phys. Rev Lett. 96. 076604 (2006).Nagaosa2006N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa, Spin Hall effect of a conserved current: Conditions for a nonzero spin Hall current, Phys. Rev. B 73, 113305 (2006).Murakami2012K. Tsutsui and S. Murakami, Spin-torque efficiency enhanced by Rashba spin splitting in three dimensions, Phys. Rev. B 86, 115201 (2012).Xiao2016FOPC. Xiao, D. Li, and Z. Ma, Thermoelectric response of spin polarization in Rashba spintronic systems, Front. Phys. 11, 117201 (2016).Zhang2016N. Zhang, Y. Wang, J. Berakdar and C. Jia, Giant spin-orbit torque and spin current generation in carriers at oxide interfaces, New J. Phys. 18, 093034 (2016). | http://arxiv.org/abs/1705.09399v3 | {
"authors": [
"Cong Xiao"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170525235541",
"title": "Semiclassical Boltzmann theory of spin Hall effects in giant Rashba systems"
} |
We study almost prime solutions of systems of Diophantine equations inthe Birch setting. Previous work shows that there exist integer solutions of size B with each component havingno prime divisors below B^1/u, where u=c_0n^3/2, n is the number of variables and c_0 is a constant depending on the degree and the number of equations. We improve the polynomial growthn^3/2 to the logarithmic (log n) (loglog n)^-1.Our main new ingredients are the generalisation of the Brüdern–Fouvry vector sieve in any dimension and theincorporation of smooth weightsinto the Davenport–Birch version of the circle method. Sarnak's saturation problem for complete intersections ]Sarnak's saturation problem for complete intersections Universiteit UtrechtMathematisch InstituutBudapestlaan 6Utrecht3584 CD [email protected] Max Planck Institute for MathematicsVivatsgasse 7Bonn 53111Germany [email protected] [ E. Sofos December 30, 2023 ===================== § INTRODUCTION Let f_1,…, f_R∈[x_1,…, x_n] be forms of degree d and write =(f_1,…, f_R).We consider the affine variety defined byV_ : f_i(x_1,…, x_n)=0, 1≤ i≤ R. We are interested inSarnak's saturation problem, that is to finda Zariski-dense setofinteger zeros (x_1,…,x_n) ∈ V_() where each x_i is either a prime or has a small number of prime divisors. Recent work of Cook and Magyar <cit.> is concerned with finding prime solutions to the Diophantine system ()= for ∈^R, i.e. solutions for which every variable x_i is a prime number.They succeed in establishing a local toglobal principle, including an asymptotic formula, via the circle method if the Birch rank 𝔅(f̱), that will be defined at the beginning of <ref>), satisfies 𝔅(f̱)≥χ(R,d) for somefunctionχ(R,d) which only depends on the degree d and the number of polynomials R.However, the value of χ(R,d), as it would result from the current proof in <cit.>, is expected to be tower exponential in d and R.For systems of quadratic forms one hasχ(R,2)≤ 2^2^CR^2. For more general systems we do not have any explicit upper bounds on this function. It is therefore natural to ask whether one can find an explicit condition which ensures the existence of a Zariski dense set of integer solutions with all coordinates being almost prime; this is usually referred to as Sarnak's saturation problem. Let Ω(m) denote the number of prime factors of m counted with multiplicity. Almost primes have zero density in the integers owing to the generalised prime number theorem: for each fixed integer k ≥ 1we have 1/x#{m ∈ℕ∩ [1,x]: Ω(m)≤ k }∼(loglog x)^k-1/(k-1)! log x,asx →∞ .The fact that one seeks solutions inthin subsets of integers places problems of this type in a higher level of difficulty than studying the number of all integer solutionsin expanding regions.Yamagishi <cit.>showed the existence of infinitely manyinteger solutions in the case R=1 and for large n, with every solutionhaving exactly 2 prime factors. This corresponds to taking k=2 in the asymptotic above.In this paper we are interestedin a harder question than that of almost primes,namely in finding solutions within sets that have asymptotically zero densitycompared to the set of almost primes.Let P^-(m) denote the least prime divisor of a positive integer m≠ 1and define P^-(1):=1. Integers m with P^-(m)≥ m^1/u for some u>1 are almost primes, however their density is arbitrarily smaller in comparison.Indeed, by Buchstab's theorem <cit.> one has the followingfor all fixedk∈_≥ 2 and u ∈_>1,#{m ∈ℕ∩ [1,x]: P^-(m)≥ x^1/u}/#{m ∈ℕ∩ [1,x]: Ω(m)≤ k}∼(k-1)!u w(u)/(loglog x)^k-1≪_k,u1/(loglog x)^k-1,asx →∞ ,where w(u) is the Buchstab function. Progress on the saturation problem within this thinner set of solutions was recently made by Magyar and Titichetrakun <cit.>.They managed to treat systems of equations where the number of variables is the same as in Birch's work <cit.>, i.e. assuming that the Birch rank exceedsR(R+1)(d-1)2^d-1.They proved lower bounds of the correct order of magnitude regarding the number of integer solutionswith each coordinate x_i satisfying P^-(|x_i|)≥ |x_i|^1/u, where u is any constant in the rangeu≥2^8n^3/2d(d+1)R^2(R+1)(R+2) .The ultimate goal of showing thatall variables x_i can simultaneously beprimecorresponds to the value u>2-ϵ for some ϵ>0, hence any result decreasing the admissible value for u in (<ref>) is an equivalent reformulation of progress towards this goal. Our aim in this paperis to decrease the admissible value for u when the degree and the number of equations is fixed so as to haveat most logarithmic growth in terms of n rather thanpolynomial.§.§ Summary of our results In order to prove quantitative or qualitative results for the system of equations (<ref>) one typically needs n to besufficiently large in terms of d and R and the singular locus of V_. Thus, for example, the Hasse principle is known for non-singular cubic hypersurfaces when n≥ 9 (Hooley <cit.>), for non-singular quartics when n≥ 40 (Hanselmann <cit.>) and for non-singular quintics in at least n≥ 101 variables (Browning and Prendiville <cit.>). One may expect that the dependence of (<ref>)on n should decrease when n increases; we are not able to provide a bound that is independent of nbut we shall provide a bound that depends logarithmically on n rather than polynomially. For thiswe shall use the vector sieve of Brüdern and Fouvry to show thatfor fixed d,R one can improve (<ref>) to u≫log n/loglog n .This constitutes a major improvement over (<ref>) and it applies to almost all situations in the Birch setting, see Theorem <ref>. This is the main result in this paper.As an additional result we shall provide an improvement in all situations in the Birch setting, however, this will not be of logarithmic nature. Namely, using the Rosser–Iwaniec sievewe shall prove that one can takeu≫ nin all of the remaining cases, see Theorem <ref>,while, in some situations covered by Theorem <ref> but not by Theorem <ref> we shall show via the weighted sievethat there are many integer zeros(x_1,…,x_n) where the total number of prime factors of|x_1⋯ x_n| is ≪ n log n, while at the same time every prime factor of each |x_i| is at least |x_i|^α for some 0<α<1 independent of x̱, see Theorem <ref>.§.§ The vector sieve in arbitrary dimension The vector sieve was brought into light by Brüdern and Fouvry <cit.> to show thatfor all sufficiently largepositive integers N satisfying N≡ 4 24 the Lagrange equationN=x_1^2+x_2^2+x_3^2+x_4^2has many solutions x̱∈^4 with each x_ibeing indivisible by any prime of size at most N^1/u withu≥ 68.86. Problems of type Waring–Goldbach become less hard the more variables are available and the expectation is that one can take each x_i to be a prime forN as above- this is still open while the case of representations by 5 squares of primes was settled by Hua <cit.>. The vector sieve was later used to make improvementson the admissible value for u in Lagrange's equation by Heath-Brown and Tolev <cit.>, Tolev <cit.> and Cai <cit.>, as well as in other sieving problems (<cit.>, <cit.>, <cit.>).The main idea of the vector sieve is to use a combinatorialinequality that replaces the usual lower bound sieveby a linear combination of products of sieving functions each of dimension 1, one of the advantages being an improvement over the admissible value for u. There are other applications of the vector sieve in the literature but to our knowledge it has not been applied for sieves ofarbitrarily large sieve dimension (the reader is referred to the book ofFriedlander and Iwaniec <cit.> for the terminology). Let us now proceed to the statement of our main theorem. Denoting the p-adic unitsby _p^×we will always make the assumption thatf̱=0̱ has non-singular solutions in(0,1)^nand in(_p^×)^nfor every primep.We shall define the quantity K=K() in (<ref>) using the notion of the Birch rank 𝔅(f̱). LetΥ:= d𝔅()/(d-1)2^d-1(d-1/R) +R,as well as θ':= min{1/ρ, ϵ_1,1-dR/ϵ_1,2+ϵ_1,3, ϵ_2,1-dR/ϵ_2,2+ϵ_2,3, ϵ_3,1-dR/ϵ_3,2+ϵ_3,3} ,whereρ:=4R(R+1)dł(1+d/2R(d-1)+1+3Rd/3R(d-1)+1)̊andthe vectors (ϵ_i,1,ϵ_i,2,ϵ_i,3) are defined as thecolumns of the following matrix,ϵ:= [ Rd+1/2K(K-R(R+1)(d-1))/4R(R+1)d+Rd; Rd+1/2K d(K-R^2(d-1))/2R(d-1)+R+K+2dK/d-1-Rd;00 max{0,K-R(R+1)(d-1)/4R(R+1)d-R-K+Rd} ]. Here follows the main result of our paper. There exists a positive absolute constant c_0 such that wheneverthe forms f_1,…,f_R∈[x_1,…,x_n] of degree d≥ 2 satisfy (<ref>) and𝔅(f̱) > max{ 2^d-1(d-1) R(R+1), 2^d-1 (d-1)R^2 +(R+1) (Υ+1), 2^d-1 (d^2-1)R^2 } ,then we have for all large enough B≥ 1, #{x̱∈ ((0,B]∩)^n:f̱()=0̱, P^-(x_1⋯ x_n)> B^θ' loglog n/c_0log n}≫B^n-Rd/(log B)^n ,where the constant θ' satisfies θ'≫_d,R 1.This provides a lower boundlog P^-(x_1⋯ x_n)/log B in terms of nthat vanishes logarithmically slow as n→+∞, which constitutesa large improvement overthe previously best known result that gave a polynomial decay <cit.>. The proof of Theorem <ref> will be given in <ref>. A crucial input for the sieving argumentswill be a generalversion of Birch's theorem that we shall prove in <ref>, see Theorem <ref>. Note that a similar result for one quadratic form is proved in work of Browning and Loughran <cit.>, whereas our result aims atgeneral complete intersections. More importantly,Theorem <ref> allows situations wherecongruence conditionsare imposed to every integer coordinate with a different moduli for every coordinate, while in their result one is only allowed to consider the same moduli for every coordinate. This extra feature will be of central importance for the vector sieve. An inspection of the argument at the end of <ref> shows thatwe can take c_0=3in Theorem <ref> when the number of variables n issufficiently large. For s∈ℝ_>2 let 0<f(s)≤ 1≤ F(s)be the sieve functions associated to the linear Rosser–Iwaniec sieve, defined for example in <cit.>, which satisfy F(s),f(s)=1+O(s^-s). One can improve the lower bound forlog P^-(x_1⋯ x_n)/log B given by Theorem <ref> by replacing the term c_0log n/loglog n by any value s>2 that satisfies F(s)^n<(1+1/n-1)f(s).A special case of Theorem <ref> isthe case of non-singular hypersurfaces. There exists a positive absolute constant c_1 such thatwhenever f is an integer non-singular formof degree d≥ 5in more than 2^d-1(d^2-1) variables that fulfils (<ref>) then the following estimate holds for all large enough B≥ 1, #{x̱∈ ((0,B]∩)^n:f()=0, P^-(x_1⋯ x_n)> B^c_1 loglog n/d log n}≫B^n-d/(log B)^n .Our results require a few more variables than in the Birch setting,which for non-singular hypersurfaces requires n>2^d(d-1). The reason for this is rooted to the way that the vector sieve works: in introducing n linear sieving functions in place of a single n-dimensional lower bound sievethe technique requires that we have a good control onthe independency of the events that a large prime p divides several coordinates of an integer zero, this is related to the function δ that will be studied in <ref>. The Birch assumption𝔅()>2^d-1(d-1)R(R+1)does not always allow a good bound for δ, however a slightly stronger geometricassumption will be shown to be sufficient via a version ofWeyl's inequality that is uniform in the coefficients of the underlying polynomials. It must be noted that thework ofYamagishi <cit.>only applies tosmooth hypersurfacesin n>8^d (4d-2) variables, which ought to be comparedwith the assumptionn>2^d-1(d^2-1)of Corollary <ref> §.§ Applications to the saturation problem One further advantage of Theorem <ref> is that it allows the use of any smooth weight with compact support. We can therefore establish a version of Theorem <ref> where one counts solutions near an arbitrary non-singular point in V_(). This allows to settle Sarnak's problem for the complete intersections under consideration. To phrase our resultwe first need the following definition. Eachx∈^n-1() can be written uniquely up to sign in the form x=[±x̱], where x̱=(x_1,…,x_n)∈^n and (x_1,…,x_n)=1. We can then define the function Ļ:^n-1()→_≥ 0 throughĻ(x):=max_1≤ i ≤ nx_i≠ 0max{log |x_i|/log p: pis a primedividing x_i } .Thus Ļ(x)≤ u holds for some x=[± (x_1,…,x_n)]∈^n-1() and u∈_≥ 0 if and only if x_i≠ 0 ⇒P^-(|x_i|) ≥|x_i|^1/u .Assume thatX⊂^n-1 is a variety defined over . Thelevel of saturation of Xis the infimum of all real non-negative numbers usuch that{x∈ X():Ļ(x)≤ u }is Zariski dense in X.Note that in this definitionthe level of saturation is allowed to be infinite, for example if X() is not Zariski dense. Recalling the definition of the number of prime divisorsΩ_^n-1()(x) of a rational point x∈^n-1() in the paragraph before <cit.>, we observe that if ∏_ix_i≠ 0 then Ω_^n-1()(x) ≤ n Ļ(x) .Therefore, according to <cit.>, if X has a finite level of saturationthen it has a finite saturation number. Therefore one could perceive Definition <ref> asa refinement of the standard notion of saturation.There exists a positive absolute constant c_0 such that wheneverthe forms f_1,…,f_R∈[x_1,…,x_n] of degree d≥ 2 satisfy (<ref>) and𝔅(f̱) > max{ 2^d-1(d-1) R(R+1), 2^d-1 (d-1)R^2 +(R+1) (Υ+1), 2^d-1 (d^2-1)R^2 }andthe complete intersection in ^n-1 that is defined throughV_f̱: f_1=f_2=⋯=f_R=0is geometrically irreducible then V_f̱ has finitelevel of saturation. In addition, thelevel of saturation is at mostc_0log n/θ'loglog n ,where the constant θ' satisfies θ'≫_d,R 1.§.§ Results via the Rosser–Iwaniec sieveWe next provide an almost prime result that covers all situations in the Birch setting, thus completing the treatment of the cases not covered by Theorem <ref>. This will provide a lower bound for log P^-(x_1⋯ x_n)/log B that is worse than the one in Theorem <ref> but still better than (<ref>); this is due to the strength of the level of distribution result implied by Theorem <ref>. For any forms f_1,…,f_R∈[x_1,…,x_n] of degree d≥ 2 satisfying (<ref>) and K>R(R+1)(d-1) we have for all large enough B≥ 1, #{x̱∈ ((0,B]∩)^n:f̱()=0̱, P^-(x_1⋯ x_n)> B^θ'/3.75 n}≫B^n-Rd/(log B)^n ,where θ' is given in (<ref>) and satisfies θ'≫_d,R 1. §.§ Results via the weighted sieve Theorem <ref> supplies apolynomially fast convergence to zero for log P^-(x_1⋯ x_n)/log B with respect to n. This is slightly undesired, thus we shall provide a complementary resultthat furnishes many integer zeros satisfying a bound of similar quality for log P^-(x_1⋯ x_n)/log B with the additional desired property thatx_1⋯ x_n has few prime factors.This will be implemented via the weighted sieve. We choose to include this result here because along the proof we shall provide a potentially useful reformulation of the weighted sieve given in the book of Diamond and Halberstam <cit.>. This reformulation allowsthe incorporation of further weights and will be given in Theorem <ref>.Defineu”:=(n-Rd)max{ (2ϵ_i,2-1)/ϵ_i,1-Rd:1≤ i ≤ 3 }, u:= max{u”, 1/θ', 2(n-Rd) ρ}, v := n c_n-1/θ'-1/u ,where c_n is a sequence that satisfies lim_n→+∞c_n=2.44…. We furthermore let r_0:= nu/n-Rd -1 + n(1+u/vc_n) logv/u -n (1-u/v) .For any forms f_1,…,f_R∈[x_1,…,x_n] of degree d≥ 2 satisfying (<ref>) and𝔅(f̱) > max{(d-1)R(R+1)2^d-1, (d^2-1)R2^d-1 , (d-1)R^2 2^d-1 +2(R+1) } we have for all r_1>r_0 and all large enough B≥ 1, #{x̱∈ ((0,B]∩)^n:f̱()=0̱, P^-(x_1⋯ x_n) >B^1/v, Ω(x_1⋯ x_n)≤ r_1 }≫B^n-Rd/(log B)^n ,where v=O_d,R(n) and r_0=O_d,R(n logn).A simpleconsequence of Theorem <ref> is that it provides many integer zeroswith Ω(x_1⋯ x_n) ≪n log n/loglog n ,which constitutes an asymptotic saving compared to the estimate Ω(x_1⋯ x_n) ≪ n log nsupplied by Theorem <ref>. This is surely surprising to those familiar withthe weighted sieve and its applications to higher dimensional sieve problems. The reason that the vector sieve gives a better saturation number here is the strong level of distribution supplied byTheorem <ref>, which is a resultof using smooth weights. Indeed, Theorem <ref> allows to estimate asymptotically the number ofinteger solutions of ()=0̱ subject to divisibility conditions of the form k_i|x_i for in a region having the shape ∈ B[-1,1]^nand vectors ∈^n of size||≤ B^1/s, where s>1 depends on d and R but not on n. Such a level of distribution is usually not availablein other problems related to the weighted sieve.We shall reserve the symbolν(m) for the counting function of distinct prime factors of a positive integer m. For vectorsx̱∈^n, n ∈, we shall reserve the symbols |x̱| and|x̱|_1for the supremum and the ℓ^1 norm respectively. For vectors ,∈^n we shall abbreviate the simultaneous conditions k_i|x_i by |. Similarly we write ≤ or < or ||≤ for the simultaneous conditions k_i≤ x_i (resp. k_i < x_i and |k_i|≤ x_i) for 1≤ i≤ n.We shall furthermore find it convenient to introduce the notation:=k_1⋯ k_n ,as well as ⟨⟩ = (k_1x_1,…, k_nx_n). For q ∈, z ∈ℂwe shall writee_q(z):=e^2 π i z/qande(z):=e^2 π i z .The letter ϵ will refer to an arbitrarily small positive fixed constant and to ease the notation we shall not record the dependence of the implied constant in the ≪ and O(·) notation. The letter w will be reserved to denote certain weight functions that will be considered constant throughout our work, thuswe shall not record the dependence of the implied constant in the ≪ and O(·) notation. Throughout our work the forms are considered to be constant, thus each implied constant in the ≪ and O(·) notationwill depend on the coefficients of ,d,n,z_0 and W, where the constants z_0,W are functions ofwhose meaning will become clear in due course. Any extra dependencies will be specified by the use of a subscript. Acknowledgements: We would like to thank Prof. T. D. Browning and Dr. S. Yamagishi for their comments on an earlier version of this paper, as well as the anonymous referee for numerous helpful comments that have clarified the exposition considerably. The first author is supported by a NWO grant 016.Veni.173.016. § A VERSION OF BIRCH'S THEOREM WITH LOPSIDED BOXES AND SMOOTH WEIGHTSIn our applications of sieve methods it will be important tobe able to count integer zeros of (x̱)=0̱such that each integer coordinate x_i is divisible by a fixed integer k_i≤ |x_i|. A change of variables makes clear that a version of Birch's theorem with lopsided boxes and with uniformity of the error term in the coefficients of the polynomials is sufficient.One can do this without smooth weights howeverthe resulting error terms will give a weak level of distribution for our sieve applications. We shall instead use smooth weightsand as a result we shall later be ableto take k_i much closer to the size of x_i.We now proceed to describe the version of Birch's theorem that we shall need. Assume that we are given any finite collection of polynomials g_i∈[x_1,…,x_n], 1≤ i ≤ R ,denote the homogeneous part of g_i by g_i^♮ and assume that there existsd ∈_≥ 2 such that 1≤ i ≤ R ⇒(g_i^♮)=d .The Birch rank, denoted by 𝔅(g̱^♮), is defined as the codimension of the affine variety inℂ^n which is given by rank((∂ g_i^♮()/∂ x_j)_1≤ i≤ R,1≤ j≤ n)<R. We setK:=2^-(d-1)𝔅(g̱^♮).Let us fix any smooth compactly supported weight function w: →_≥ 0 with the property (w) ⊂ [-2,2]. For=(P_1,…,P_n) ∈ (_≥ 1)^nwe denoteP̱ :=∏_i=1^n P_i, P_max:= max_1≤ i≤ n P_i and P_min :=min_1≤ i≤ n P_iand fix an element ∈[-1,1]^n. Our aim is to find an asymptotic formula for the counting functionN_w():= ∑_∈^n ()=0∏_i=1^n w(y_i/P_i-z_i).Birch's influential work <cit.> treated the case where w is replaced by the characteristic function of a finite interval andK>R(R+1)(d-1), P_min=P_max.For ourapplications of sieving methods a result that is uniform in the size of each P_i as well as the coefficients of each g_i is required. For h ∈ℂ[x_1,…,x_n] we denote by ẖ the maximum of the absolute values of its coefficientsand for h_1,…,h_R ∈ℂ[x_1,…,x_n] we letẖ:=max{h_i:1≤ i ≤ R} . Let g_i,w,ẕ,P_i be as above, assume that K>R(R+1)(d-1) and P_max/P_min<‖‖^-1/2R(d-1)+1‖^♮‖^-3R/3R(d-1)+1 P_max^1/4R(R+1)d . Then one has for each ϵ>0,N_w()- J_w ≪ P̱(P_max/P_min)^R P_max^-Rd-1/2 + P̱^1+(P_max/P_min)^KP_max^- K + ‖^♮‖^2K/d-1-R‖‖^ K-R^2(d-1)/2R(d-1)P̱^1+(P_max/P_min)^R+KP_max^-Rd-K-R(R+1)(d-1)/4R(R+1)d ,where the implied constant depends at most on ϵ>0. Hereand J_ are the usual circle method singular series and singular integral and are defined in (<ref>) and (<ref>) respectively.Our sole aim in this section is to establish Theorem <ref>. All implied constants may depend on n,R,d but not on the coefficients of the polynomials g_i(), 1≤ i≤ n. We start by introducing the exponential sumS_w():= ∑_∈^n∏_i=1^n w(y_i/P_i-z_i)e(· ()),where we use the vector notation ·()=∑_i=1^R _i g_i(). By orthogonality we now haveN_w()=∫_[0,1]^R S_w().We shall follow Birch's approach <cit.> to approximate N_w(). Our first step is to produce a Weyl type inequality for S_w().Recall that g_i^♮() are homogeneous polynomials of degree d, which can be written asg_i^♮()= d! ∑_1≤ j_1,…, j_d≤ ng_j_1,…, j_d^(i) y_j_1… y_j_d,with symmetric coefficients g_j_1,…, j_d (i.e. such that g_j_1,…, j_d=g_(j_1),…, (j_d) for a permutationof the indices). We associate its multilinear formsΦ_i(^(1),…, ^(d)) = d! ∑_1≤ j_1,…, j_d≤ n g_j_1,…, j_d^(i)y_j_1^(1)… y_j_d^(d),and setΦ (^(1),…, ^(d)):=∑_i=1^R _i Φ_i(^(1),…, ^(d)).With the notation above we have|S_w()|^2^d-1/P̱^2^d-1≪P̱^-d∑_-2 < ^(1) < 2…∑_-2< ^(d-1)< 2∏_i=1^n min{P_i, ‖Φ (^(1),…, ^(d-1),^(i))‖^-1}For w(x) a weight function and h∈ we introduce the notationw_h(x)=w(x+h)w(x).Moreover, for h_1,…, h_m∈, we iteratively definew_h_1,…, h_m= w_h_1,…, h_m-1(x+h_m)w_h_1,…, h_m-1(x).The same Weyl differencing process as in the proof of Lemma 3.3 (in particular equation (3.5)) in <cit.> or in Lemma 2.1 in <cit.> leads to |S_w()|^2^d-1/P̱^2^d-1≪P̱^-d∑_-2 < ^(1) < 2…∑_-2< ^(d-1)< 2 |S_w(^(1),…, ^(d-1),)|,where S_w (^(1),…, ^(d-1),)=∑_∈^n{∏_i=1^n w_h_i^(1)/P_i,… ,h_i^(d-1)/P_i(y_i/P_i-z_i) }e(∑_i=1^R _i Φ_i(^(1),…, ^(d-1),) + c(^(1),…, ^(d-1))),with integers c(^(1),…, ^(d-1)) independent of . Hence|S_w(^(1),…, ^(d-1),)|=|∑_∈^n{∏_i=1^n w_h_i^(1)/P_i,… ,h_i^(d-1)/P_i(y_i/P_i-z_i) }e(Φ (^(1),…, ^(d-1),) )|.The estimate S_w(^(1),…, ^(d-1),)≪∏_i=1^n min{P_i, ‖Φ (^(1),…, ^(d-1),^(i))‖^-1}can then be obtained via partial summation.We define the counting functionM(,):=♯{-2≤^(i)≤ 2, 1≤ i≤ d-1: ‖Φ (^(1),…, ^(d-1),^(j))‖ < P_j^-1∀ 1≤ j≤ n}. As Lemma 3.2 is deduced from Lemma 3.1 in <cit.> we obtain the following lemma.One has|S_w()|^2^d-1≪P̱^2^d-1-d+1+M(,).Next we need a version of Lemma 12.6 in <cit.> which is modified for lopsided boxes.Let L_1,…, L_n be symmetric linear forms given by L_i=_i1 u_1+… + _in u_n for 1≤ i≤ n, i.e. such that _ij=_ji for 1≤ i,j≤ n. Let a_1,…, a_n>1 be real numbers. We denote by N(Z) the number of integers solutions u_1,…, u_2n of the system of inequalities|u_i|< a_iZ, 1≤ i≤ n, |L_i- u_n+i|<a_i^-1Z, 1≤ i≤ n.Then for 0< Z_1≤ Z_2≤ 1 we haveN(Z_2)/N(Z_1)≪(Z_2/Z_1)^n. Letbe the 2n-dimensional lattice defined byx_i =a_i^-1u_i, 1≤ i≤ nx_n+i = a_i(_i1 u_1+… +_inu_n+u_n+i), 1≤ i≤ n.As in the proof of Lemma 12.6 in <cit.> we note that the inequalities describing N(Z) are equivalent to|x_i| < Z,1≤ i≤ 2n,for a point (x_1,…, x_2n) in the lattice . We identify the latticewith its matrix=([a_1^-1 … 0 0 … 0; ⋮ ⋮ ⋮ ⋮; 0 …a_n^-1 0 … 0;a_1_11 …a_1_1n a_1 … 0; ⋮ ⋮ ⋮ ⋮; a_n _n1 … a_n _nn 0 … a_n ])and we find that the adjoint lattice is given byM=(^t)^-1=([ a_1 … 0 -a_1_11 … -a_1_n1; ⋮ ⋮ ⋮ ⋮; 0 … a_n -a_n_1n … -a_n_nn; 0 … 0a_1^-1 … 0; ⋮ ⋮ ⋮ ⋮; 0 … 0 0 …a_n^-1 ]).Since _ij=_ji for all 1≤ i,j≤ n the two latticesand M can be transformed into one another by interchanging the order of x_1,…, x_2n and u_1,…, u_2n and changing signs at some variables. Hence they have the same successive minima. Now the proof of Lemma 12.6 in <cit.> applies to our situation andan identical argument concludes our proof.We now apply Lemma <ref> to the counting function M(,). Let 0<<1 and set Z=P_max^-1. We then obtain the following bound,|S_w()|^2^d-1≪P̱ ^2^d-1-d+1+/Z^(d-1)n♯,whereis defined by{ (^(1),…, ^(d-1))∈^(d-1)n: |^(i)|≤ Z, ‖Φ(^(1),…, ^(d-1),_j)‖ < Z^d-1P_j^-1, ∀ 1≤ j≤ n }.We are now in a position to obtain a form of Weyl's inequality for S_w() (for a Weyl's inequality in a similar setting see for example Lemma 4.3 in <cit.>). Let V^* be the affine variety defined by(∂ g_i^♮()/∂ x_j)_1≤ i≤ R, 1≤ j≤ n<R,and recall thatK=n- V^*/2^d-1.Assume that 0< <1. Then one has either(i)S_w() ≪P̱^1+(P_max/P_min)^KP_max^- K,or(ii) there are integers 1≤ q≤‖^♮‖^RP_max^R(d-1), and 0≤ a_1,…, a_R < q with (,q)=1 and|q_i-a_i|≤‖^♮‖^R-1 P_min^-1P_max^-(d-1)+R(d-1), 1≤ i≤ R.First assume that P_max^ -1P_min≥ 1. We start with the bound|S_w()|^2^d-1≪P̱^2^d-1-d+1+P_max^(1-)(d-1)n♯.Consider the affine variety ⊂^n(d-1) given by:(Φ_i(^(1),…, ^(d-1),_j))_1≤ i≤ R, 1≤ j≤ n<R.We set:={(^(1),…, ^(d-1))∈^n(d-1)∩: |^(i)|≤ P_max^-1, ∀ 1≤ i≤ d-1}.Now we distinguish two cases. (i) Assume that ⊂. Then we bound the cardinality ofby dimension bounds. We dissect the region given by the conditions that |^(i)|≤ P_max^-1 into boxes where all the side length are equal (at the boundaries we allow for overlapping boxes which will result in slight overcounting) and of size P_max^ -1P_min. The number of such boxes is bounded by≪(∏_i=1^n P_i/P_min)^d-1.On each of the boxes we apply a linear transformation to move the box to the origin. Then we apply Theorem 3.1 in <cit.>. Note that this bound is independent of the coefficients of the variety (only depending on the dimension and degree) and hence uniform in the shift. We obtain♯≪(∏_i=1^n P_i/P_min)^d-1 (P_max^ -1P_min)^.By <cit.> we have ≤ V^* + (d-2)n, hence we obtain the bound♯≪(∏_i=1^n P_i/P_min)^d-1 (P_max^ -1P_min)^ V^*+(d-2)n.Together with our assumption ⊂ we obtain|S_w()|^2^d-1 ≪P̱^2^d-1+P_max^(-1)( V^*+(d-2)n)P_max^-(d-1)n(-1)P_min^-n+ V^*≪P̱^2^d-1+ P_max^(1-)(n- V^*)P_min^-n+ V^*≪P̱^2^d-1+P_max^- (n- V^*)(P_max/P_min)^n- V^*.This estimate gives option (i) in the statement of our lemma. Next we assume that ∖≠∅. Let (^(1),…, ^(d-1)) be such a point in the difference set, i.e. (Φ_i(^(1),…, ^(d-1),_j))_1≤ i≤ R, 1≤ j≤ n=R.With no loss of generality we assume that the leading R× R minor is of full rank, and setq:= | (Φ_i(^(1),…, ^(d-1),_j))_1≤ i,j≤ R| .Note thatq≪‖^♮‖^R P_max^R(d-1).Moreover, we have the system of equations∑_i=1^R _i Φ_i(^(1),…, ^(d-1),_j) = a_j+_j,1≤ j≤ R,with a_1,…, a_R integers and |_j|≪ P_max^(-1)(d-1)P_j^-1, 1≤ j≤ n.We now obtain (after changingbyforarbitrarily small) as in the proof of <cit.> an approximation 1≤ a_1,…, a_R≤ q to the real numbers _i of the quality|q_i-a_i|≤‖^♮‖^R-1 P_min^-1P_max^-(d-1)+R(d-1), 1≤ i≤ R.Note that alternative (i) in Lemma <ref> trivially holds if P_max^-1P_min≤ 1.Next we come to the definition of the major arcs. Let 0<<1 and assume thatP_max^-1P_min≥ 1.For q∈ and 1≤ a_1,…, a_R≤ q we define the major arc_,q():={∈ [0,1]^R: |q_i-a_i|≤‖^♮‖^R-1 P_min^-1P_max^-(d-1)+R(d-1),1≤ i≤ R}.Moreover we define the major arcs () as the union()=⋃_1≤ q≤‖^♮‖^R P_max^R(d-1)⋃_1≤ a_1,…, a_R≤ q (,q)=1_,q()and set ():=[0,1]^R∖(). A short calculation gives the following bound for the measure of the major arcs (). Assume that 0<<1 such that (<ref>) holds. Then one has(())≪‖^♮‖^R^2 P_min^-RP_max^-R(d-1)+R(R+1)(d-1). We are now ready to provide an L^1-bound for the exponential sum S_w() over the minor arcs, which is a modification of Lemma 4.4 in <cit.> and proved in the very same way. Let 0<<1 such that (<ref>) holds. Assume thatK>R(R+1)(d-1).Then one has∫_()|S_w()|≪P̱^1+(P_max/P_min)^KP_max^- K+P^1+‖^♮‖^R^2(P_max/P_min)^R+KP_max^-Rd -(K-R(R+1)(d-1)) +,for > 0 arbitrarily small.For technical convenience we introduce the slightly larger major arcs'_,q():={∈ [0,1]^R: |q_i-a_i|≤ q‖^♮‖^R-1 P_min^-1P_max^-(d-1)+R(d-1),1≤ i≤ R},and'()=⋃_1≤ q≤‖^♮‖^R P_max^R(d-1)⋃_1≤ a_1,…, a_R≤ q (,q)=1'_,q(). We record that the major arcs '_,q() are disjoint forsmall enough and that('())≪‖^♮‖^2R^2P_min^-RP_max^-R(d-1)+(2R^2+R)(d-1). A minor modification of the proof of Lemma 4.1 in <cit.> gives the following result. Assume that‖^♮‖^3R-1 P_min^-1P_max^-(d-1)+3R(d-1) <1.Then for 1≤ q≤‖^♮‖^R P_max^R(d-1) and 1≤ a_1,…, a_R≤ q, (,q)=1 the major arcs '_,q() are disjoint. We now come to the major arc approximation of S_w(). Let q∈ and 1≤ a_1,…, a_R≤ q. We define the exponential sumS_,q:=∑_ qe(/q·())and the integralI_w():=∫_^n e(·()) ∏_i=1^n w (u_i/P_i-z_i).Let q∈ and 0≤ a_1,…, a_R <q. Write =/q+. Assume that q<P_minP_max^- and|| q P_max^d-1‖‖ < P_max^-.Then one has the following approximation for any real N ≥ 1,S_w()=q^-nS_,qI_w() + O_N(P̱P_max^-N).We recall the definition of the exponential sum S_w() asS_w()=∑_∈^n∏_i=1^n w(x_i/P_i-z_i)e(·()).We split the summation variablesinto residue classes modulo q and obtainS_w()=∑_ qe(/q·())∑_∈^n∏_i=1^n w(y_i+w_iq/P_i-z_i)e(·(+q)).We now consider the inner sum for a fixed vectormodulo q. Letψ():=∏_i=1^n w(y_i+w_iq/P_i-z_i)e(·(+q)).We apply Euler–Maclaurin's summation formula (see Theorem B.5 in <cit.>)of order κ̃ into each coordinate direction. If we choose κ̃ large enough depending only on , n and N we obtain∑_∈^nψ()= ∫_∈^nψ()+ O_N(P̱ P_max^-N).Note that all the boundary terms in Euler–Maclaurin's summation formula vanish due to the smooth weight function w. Since N was arbitrary we find after even enlarging κ̃ thatS_w()=S_,q∫_∈^nψ()+ O_N(P̱P_max^-N).A variable substitution now gives the statement of the lemma.Next we consider the singular integral. Note that in contrast to most approaches we defined the integral I_w() with the inhomogeneous polynomials () instead of taking their homogenizations. We now replace () by ^♮() in I_w() which will simplify the discussion of absolute convergence. DefineI_w^♮()=∫_^n e(·^♮()) ∏_i=1^n w(u_i/P_i-z_i). Assume that || ≤ 1. Then one hasI_w()-I_w^♮() ≪P̱ ||‖‖P_max^d-1. The proof of the lemma follows from directly comparing the integrands of the two integrals.Under the assumptions of Lemma <ref> we observe thatS_w()=q^-nS_,qI_w^♮() +O_N(P̱ P_max^-N)+O(P̱ || ‖‖P_max^d-1).We define the truncated singular series(Q):=∑_q≤ Q q^-n S_,q,and the truncated singular integralJ_w(Q):=∫_|| ≤ QI_w^♮(). With these definitions we can write the major arc contribution in the following way. Assume|| ≤ P_max and that (<ref>) holds, as well asmax{‖^♮‖^R P_max^R(d-1)P_min^-1,‖‖‖^♮‖^2R-1 P_min^-1 P_max^2R(d-1)}< P_max^-.Then one has∫_'() S_w()= (‖^♮‖^R P_max^R(d-1)) J_w (‖^♮‖^R-1P_min^-1P_max^-(d-1)+R(d-1)) + O_N(P̱ P_max^-N) +O(‖^♮‖^2R^2+R‖‖P̱ P_min^-R-1P_max^-R(d-1)+(2R^2+2R)(d-1)),for any real N≥ 1.By Lemma <ref> the major arcs are disjoint thus the proof follows from Lemma <ref>.Next we aim to complete the singular series. We recall Lemma 2.2 from <cit.> (see also <cit.>). For any >0 one has|S_,q|≪‖^♮‖^K/(d-1)q^n-K/R(d-1)+. We shall soon see thatthe truncated singular series (Q) is converging for Q→∞, thus we shall set= lim_Q→∞(Q).Lemma <ref> gives the following speed of convergence. Assume that K>R(d-1). Thenis absolutely convergent. Moreover one has-(Q)≪‖^♮‖^K/(d-1) Q^1-K/R(d-1)+,for any >0 and ||≪‖^♮‖^K/(d-1).In preparation for the proof of the absolute convergence of the singular integral, we note the following lemma, which is a consequence of Lemma <ref>. Assume that ||^3 ‖^♮‖^2 P_min^2P_max^2(d-1)<1. Then one hasS_w() ≪P̱^1+(P_max/P_min)^K (||‖^♮‖^-R+1P_minP_max^d-1)^-K/R(d-1),for any positive . Assume that P_i≥ 1 for 1≤ i≤ n and that || ≤ 1.ThenI_w^♮()≪P̱min{1,P̱^(P_max/P_min)^K(1+1/R(d-1))(P_max^d|| ‖^♮‖^-R+1)^-K/R(d-1)}.The proof of Lemma <ref> is relatively standard (see Lemma 5.2 in <cit.>), with the exception that we compare the oscillatory integral I_w^♮() with parameters P_1,…, P_n to an exponential sum with box length B_1,…, B_n such that B_i/B_max=P_i/P_max for all 1≤ i≤ n.We shall show that the truncated singular integral J_w(Q) converges for Q→∞, we will therefore letJ_w:=lim_Q→∞ J_w(Q),and call it the singular integral.In the following we will always assume that 1≤ P_i for 1≤ i≤ n and that || ≤ 1. As a consequence of Lemma <ref> we obtain the following result.Assume that K>R^2(d-1). Then J_w is absolutely convergent andJ_w -J_w(Q)≪P̱^1+(P_max/P_min)^K(1+1/R(d-1))(P_max^d ‖^♮‖^-R+1)^-K/R(d-1)Q^-K/R(d-1)+R.Moreover, we haveJ_w ≪P̱^1+(P_max/P_min)^R^2(d-1)+RP_max^-Rd‖^♮‖^R(R-1).We can now complete both the singular series and singular integral in our major arc analysis. According to Lemma <ref> and Lemma <ref> we obtain the following result. Assume that equations(<ref>), (<ref>) and (<ref>)hold and || ≤ P_max, as well as K>R^2(d-1). Then the following holds for any real N≥ 1,∫_'() S_w()=J_w+O(‖^♮‖^2R^2+R‖‖P̱ P_min^-R-1P_max^-R(d-1)+(2R^2+2R)(d-1))+O_N( P̱ P_max^-N+ ‖^♮‖^K/(d-1)+R^2-RP̱^1+(P_max/P_min)^R+K P_max^-Rd-K+R^2(d-1)). Theorem <ref> is now a consequence of the major arc analysis in Lemma <ref> in combination with the minor arc analysis from Lemma <ref>. For this, we choosebyP_max^ = ‖‖^-1/2R(d-1)+1‖^♯‖^-3R/3R(d-1)+1 P_max^1/4R(R+1)d.Then we clearly have 0<<1 and equation (<ref>) reduces to the assumption (<ref>). Moreover, one quickly sees that with this choice ofboth of the conditions (<ref>) and (<ref>) are satisfied. It remains to understand that the error terms in Lemma <ref> and Lemma <ref> are both majorised by the error term in Theorem <ref>. We bound the first error term in Lemma <ref> by‖^♯‖^2R^2+R‖‖P̱(P_max/P_min)^R P_max^-Rd-1P_max^(2R^2+2R)d≪P̱(P_max/P_min)^R P_max^-Rd-1/2.Note that the last error term in Lemma <ref> as well as the second error term in Lemma <ref> are bounded by ‖^♮‖^2K/d-1-R‖‖^ K-R^2(d-1)/2R(d-1)P̱^1+(P_max/P_min)^R+KP_max^-Rd-K-R(R+1)(d-1)/4R(R+1)d.The first error term in Lemma <ref> is also present in the statement of Theorem <ref>.Lastly let us remark that it is a well-known fact that the singular series factorises as =∏_pσ_p(), where for any prime p we haveσ_p():= lim_l→∞ p^-l(n-R)♯{ 1≤≤ p^l : p^l| (x̱) } .§ LOCAL DENSITIES Throughout this section we will have R formsof degree d> 1,f_1,…,f_R ∈[x_1,…,x_n]and we will always assume that the Birch rank satisfies𝔅()>2^d-1(d-1)R(R+1) .For a prime p and a vector =(j_1,…,j_n) ∈ (_≥ 0)^n we shall be concerned with bounding the quantitiesδ():= lim_l→∞ p^-l(n-R)♯{ 1≤ x_1,…,x_n ≤ p^l : p^l| (p^j_1x_1,…,p^j_nx_n) } ,these estimates will be applied later towards the proof of Theorems <ref>, <ref>and <ref>. We suppress the letter p from the notation for δ to make the presentation easier to follow. The formswill be considered constant, however the prime p and the vectorwill not, thus we shall require uniformity of our bounds with respect to p and . For later applications we only have to consider all big enough primes p>z_0, where z_0 is a constant depending at most on the coefficients ofand n,d,R. This constant will be enlarged, if needed, with no further comment. Let us emphasize that the entities δ() encode the probability of the eventsp^j_1|x_1,…,p^j_n|x_nas x̱∈^n sweeps through the zeros of =0̱, therefore, they are intimately connected with certain closed subvarieties of f̱=0̱. This is manifested even in the most simple of situations: for a primitive integer zero of x_1x_2=x_3^2 and a prime p|x_3 we always have p^2|x_1 or p^2|x_2 as a result of the subvariety x_1x_2=x_3^2, x_3=0 being reducible. We shall give geometric conditions that prevent δ()to attain large values for general systems =0̱. For every ∈{0,1}^n we define the system f̱^=0̱ of R formsin n-|j̱|_1 variables viaf_ξ^()=f_ξ(x_1,…, x_n)|_x_i=0j_i=1, ξ∈∩ [1,R].We later need a lower bound for the Birch rank of the new systems, as for example obtained in <cit.>. As there is a slight oversight in the proof of <cit.>, we give here the statement and proof of the corrected lemma where the quantity R in <cit.> is replaced by R+1. One has𝔅(^)≥𝔅() - (R+1)|j̱|_1.It is important to note here that we view ^ as a system of R equations in n-||_1 variables. Let V^*⊂_^n-1 be the projective variety given by (∂ f_ξ/∂ x_i)_ξ,i<R,and note that this is well-defined as all of the polynomials f_ξ are homogeneous. Then the Birch rank ofis given by𝔅()=n- (V^*)-1.Similarly, let V^*,⊂_^n-||_1-1 be the projective variety given by (∂ f_ξ^/∂ x_i)_ξ,i<R,such that we have𝔅(^)=n-||_1- (V^*,)-1.The variety V^*, naturally embeds into the linear subspace of _^n-1 given by x_i=0 for j_i=1. We write ι(V^*,) for this embedding.Then we observe that ι(V^*,) ∩⋂_1≤ξ≤ R⋂_1≤ i≤ n j_i=1{∂ f_ξ/∂ x_i=0}⊂V^*.Hence we obtain(V^*,) - R||_1≤ (V^* ) .Finally, this implies𝔅(^)≥ n-||_1-( (V^* ) + R||_1)-1 = 𝔅() - (R+1)|j̱|_1.This is a convenient place to introduce thehelpful notationΘ() := 𝔅(f̱^)/R (d-1)2^d-1and Θ(0̱) will be denoted by Θ. For non-negative integers j_1,…, j_n, any prime pand a vector x̱ we use the notationp^j̱|x̱ p^j_i|x_i, ∀ 1≤ i≤ n .This enables us to introduce the densities _p(p^j̱|x̱)= lim_l→∞ p^-l(n-R)♯{1≤ x_1,…,x_n ≤ p^l: p^l| (),p^j|x}and from the definition of δ we infer thatδ(j̱)/p^|j̱|_1 =_p(p^j̱|x̱).Let t,d be integers with 2≤ d <t. Thenfor each ∈ (/p^t-d)^R with p∤ and anyvector polynomial g̱∈[x̱]^R with max_1≤ i≤ R(g_i) ≤ d-1 we have∑_p^t-1 e_p^tł(p^d a̱·f̱() +pa̱·g̱())̊≪_ p^(t-1)(n-t-d/t-1Θ +ϵ) ,where the implied constant is independent of p,t, g̱ and a. We shall use <cit.> with P=p^t-1 and α=p^-t+da̱;in doing so we observe that lower degree polynomials leave the strength of the bounds in <cit.> unaffected. Recall that the constant K in <cit.> is given via 𝔅()/2^d-1. Our aim is to acquire a constant η>0, as large as possible, such thatα∉M̧(η), where M̧(η) is given in <cit.>. This would then implythat the sum in our lemma is ≪p^ (t-1) (n- 𝔅()η/2^d-1 + ) .The assumption α∈M̧(η) provides non-negativeintegersq',a_1',…,a_R' fulfilling (a_1',…,a_R',q')=1, 1≤ q' ≤ p^(t-1)R(d-1)ηand such that for all i=1,…,Rthe succeeding inequality is valid,2|q'a_i -a_i' p^t-d |≤ p^ t-d + (t-1)(-d+R(d-1)η) .As explained in <cit.>, we need to assume 2R(d-1)η <d in orderto ensure thatthe major arcs are disjoint. It is straightforward to infer that this condition is metupon choosing η := η()= t-d/(t-1)R(d-1)-for any small enough >0. Furthermore, this choice of η makes the exponent of p in (<ref>) non-positive, thus giving birth to the equalities q'a_i=a_i' p^t-d for all i. In particular, we obtain p^t-d=q'≤ p^(t-1)R(d-1)η, thus t-d≤(t-1)R(d-1)η, which constitutes a violation to the the definition of η.For ∈{0,1}^n, c ∈ and any prime p define E(p^c;):=♯{p^c: ()≡0p^c, p^j_i| x_i∀ i }.This quantity is intimately related to the geometry of ^=0̱ and we begin by using it to approximate δ(j̱). Let ∈{0,1}^nand assume that Θ>R. Then there is some z_0>0, such that for p>z_0 and each sufficiently small >0, we haveδ()=p^d(R-n)+||_1E (p^d;) +O(p^ -Θ +R(d+1) +) ,where the implied constant depends at most on f̱. For t≥ 1, j̱∈{0,1}^n and any a̱∈^R webring into play the entitiesW_,p^t(p^j̱|x̱) :=∑_p^tp^j̱|x̱ e_p^t(·()) and G(;p^t) := p^-tn_p^t W_,p^t(p^j̱|x̱) ,where the summation _a̱q is over vectorsa̱∈ (/q)^R with (a̱,q)=1. We have δ() p^-||_1 =lim_l→∞p^-l(n-R)∑_p^l1/p^Rl∑_p^lp^j̱|x̱ e_p^l(·())= lim_l→∞∑_p^lp^-ln∑_p^lp^j̱|x̱ e_p^l(·())= lim_l→∞(∑_t=1^l G(;p^t) + p^-ln♯{p^l: p^|})= p^-||_1+ lim_l→∞∑_t=1^l G(;p^t),whenceδ()=1 + p^|j̱|_1∑_t=1^∞ G(;p^t).Observe that for each form F ∈[x̱], any prime p and any fixed integer vectorthere exists an integerpolynomial F_∈[]of degree strictly smaller than (F), such thatF(+p)= p^(F)F() +F() +pF_() .Hence, if t≥ d+1, this allows us to rewrite the exponential sum W_,p^t(p^j̱|x̱) as∑_∈ (∩ [1,p])^n p^j̱|y̱∑_∈ (∩ [1,p^t-1])^n e_p^t(·(+p))=∑_∈ (∩ [1,p])^n p^j̱|y̱ e(p^-t·()) ∑_∈ (∩ [1,p^t-1])^ne(p^d-t·()+ p^-t+1·_()),where the polynomials _() have degree strictly smaller than d in .Invoking Lemma <ref> endows us with the following bound for the inner sum over ẖ,≪ p^(t-1)(n+)-(t-d)Θ,where the implicit constant is independent of p, t, and .Hence, for t>d we deduce that W_,p^t(p^|) ≪ p^t(n+)-|j̱|_1-(t-d)Θ ,thereby procuring the validity of ∑_t=d+1^∞ |G(;p^t)| ≪ p^-|j̱|_1+dΘ∑_t=d+1^∞ p^-t(Θ-R-) .Our assumption R<Θ shows that for each 0< < (Θ-R)/2the sum over t has the valuep^-(d+1)(Θ-R-)/1-p^-(Θ-R-)and increasing the value of z_0 to ensure thatz_0^(Θ-R)/2≥ 2 shows that ∑_t=d+1^∞ |G(;p^t)| ≪ p^-|j̱|_1 -Θ+R(d+1) +(d+1). To control the contribution of the terms with t ≤ dwe note thatp^-|j̱|_1+∑_t=1^d G(;p^t) = p^d(R-n)♯{p^d: ()≡0p^d, p^j̱|x̱},thus concluding our proof.Observe that, at least when ||_1 is relatively small, the quantity E(p^d;) regards the number of zeros p of a variety in sufficiently many variables; thus the estimates of Birch yield the required estimation of E(p^d;).Let ∈{0,1}^n and assume thatΘ() >R is fulfilled. Then for all >0 and primes p>z_0 we have E(p^d;)=p^d(n-R)-|j̱|_1+O_(p^d(n-R)-|j̱|_1-(Θ()-R)+),with an implicit constant that is independent of p.We initiate our argument by slicing the counting function E(p^d;) along the variables which are divisible by p.Let I={1≤ i≤ n: j_i =1}and for '=(x_i)_i∈ I∈ (/p^d)^|I| we defineE(p^d;;'):=♯{ x_i p^d, i∉ I:()≡0p^d}.We rewrite this counting function with exponential sums as follows,E(p^d;;') = p^d(n-|j̱|_1)-dR+ p^-dR∑_t=1^dp^(n-|j̱|_1)(d-t)_p^t∑_x_ip^ti ∉ Ie_p^t(·()).Note that the degree d part of the polynomial () when viewed as a polynomial in the variablesx_i,i ∉ I, is ^().We now apply <cit.>, the strength of which is unaffected by lower degree polynomials, to obtain for any >0 and uniformly for all p>z_0,∑_x_ip^ti ∉ Ie_p^t(·())≪_p^ t(n-|j̱|_1 - Θ()) +.We use this to estimate E(p^d;;') as follows,E(p^d;;')-p^d(n-|j̱|_1-R) ≪_ p^d(n-|j̱|_1-R)+∑_t=1^d p^t(R-Θ())≪_ p^d(n-|j̱|_1-R)-(Θ()-R-). We can now evaluate E(p^d;) as∑_x_ip^d, i∈ Ip| x_iE(p^d;;') =p^dn-|j̱|_1-Rd+O_ł(p^d(n-R)-|j̱|_1-(Θ()-R)+)̊ ,which concludes our proof. Tying Lemmas <ref> and <ref> together providesthe succeeding estimate. Assume that ∈{0,1}^n, min{Θ,Θ()}>R and that p is a prime in the range p>z_0. Then the following holdsfor each >0with an implied constant depending only onand ,δ() =1+Oł( p^R-min{Θ-dR,Θ()}+)̊ . Utilising (<ref>) to find lower bounds forΘ() gives the following consequence of Corollary <ref>. Assume that for some j̱∈{0,1}^nwe have 𝔅(f̱) > max{ (d-1)R^2 2^d-1 +(R+1) |j̱|_1 , (d^2-1)R^2 2^d-1} . Then there exists λ>0such thatfor all large enough primes p>z_0=z_0(), we have δ() =1+O(p^-λ) ,with an implied constant depending only on .We can see that the boundδ()≪ 1 fails when ||_1 approaches n hence the assumption Θ()>R of Corollary <ref> is no longer applicable. Indeed, a moment's thought reveals thatδ(1,…,1)=p^dRσ_p and that wheneverh_i≥ j_i for all 1≤ i≤ n then δ()≥δ() p^||_1-||_1. The bound σ_p ≫ 1, valid with an implied constant independent of p when p is sufficiently large, reveals that for such p we have n-dR/2<||_1≤ n ⇒δ() ≫ p^dR/2with an implied constant independent of p. Therefore we need to provide (necessarily weaker) bounds for the densities δ() which are however valid through the whole range 1≤ ||_1≤ n. The crucial import will be bounds for the exponential sums in Birch's workwith the additional property that the dependence on the coefficients of theunderlying forms is explicitly recorded.Assume that Θ>R. Then there exists a large z_0=z_0() such thatforeach ∈{0,1}^n, >0 and prime p>z_0the followingholds with an implicit constant depending at most onand ,δ()≪ p^dR Θ+R-Θ+.We start by rewriting W_,p^t(p^j̱|x̱) =p^-||_1∑_p^te_p^t(·(p^j_1x_1,…, p^j_nx_n))and considering (p^j_1x_1,…, p^j_nx_n) as a system of homogeneous polynomials in the variables x_1,…, x_n. Note that the maximum of the coefficients is bounded by C_1p^d for a positive constantC_1=C_1() that is independent of p. Moreover, the Birch rank ofthe system ()=0̱ equals the Birch rank of the system(p^j_1x_1,…, p^j_nx_n)=0̱. Alluding to the estimate <cit.> supplies us with the boundW_,p^t(p^j̱|x̱) p^||_1≪_ p^ dRΘ +t(n-Θ +),which, onceinjected into (<ref>), offers the validity ofδ()-1 ≪p^dRΘ∑_t=1^∞ p^t(R- Θ +) .Enlarging z_0 and 1/ϵ if needed, ensures the convergence of the sum over t to a value that is ≪_z_0p^R-Θ+ϵ, independently of p. For a prime p and a vector ∈ (_≥ 0)^n we define ϖ(p^j_1,…,p^j_n):=δ()/σ_p() .The standard estimate _p=1+O(p^-1- (f̱) ) holds for some (f̱)>0. Alluding to Lemma <ref> supplies us with the following corollary. Assume that 𝔅(f̱) > R^2 (d-1)2^d-1and recall the definition of Υ in (<ref>). Then the followingbound holds uniformlyfor each ∈{0,1}^n and p> z_0,ϖ(p^)≪ p^Υ.§ PROOF OF THEOREMS <REF> AND <REF>§.§ Preparations Owing to (<ref>), there exists positive integers z_0=z_0(f̱), m=m(f̱)such that if we let W:=∏_p≤ z_0p^m,then there exists y̱∈ (∩ [1,W])^n fulfilling the following,(y_1⋯ y_n,W)=1and p≤ z_0⇒σ_p((+Wx̱))>0.DefineA̧:={x̱∈^n: f̱(x̱)=0̱,≡W} .Let us now choose a non-singular point ζ∈ V_f̱() (whose existence is guaranteed by (<ref>)) andwe let η∈ (0,min_i{min{ζ_i/2,(1-ζ_i)/2}}) be arbitrary. DefiningB̧_η :={x̱∈^n: |x̱-ζ/ 2|ζ|| <η},we see that for any such η, one hasB̧_η⊂ (0,1)^n. Now we choose any smooth functionw:→_≥ 0of compact support in [-η/2,η/2]and such thatif|t|≤η/4 then w(t)>0. Lettingw_0:=sup{w(t):t∈} we have 1_{0<t≤ B}(t) ≥w_0^-1 w( t/B -ζ_i/(2|ζ|) ) and therefore for every x̱∈^n, ∏_i=1^n 1_{0<x_i≤ B}(x̱) ≥w_0^-n∏_i=1^n w( x_i/B -ζ_i/2|ζ|) .§.§ A level of distribution result Let us now take the opportunity to record a level of distribution result that will be the main input in the forthcomingsieving arguments. For ∈^n with (,W)=1 and each k_i being square-free let w:→_≥ 0 be a smooth weight as above.We let N_w(B;) :=∑_x̱∈A̧k_i| x_i∏_i=1^n w( x_i/B -ζ_i/2|ζ|) .Recall the definition of the matrixϵ in (<ref>). Our result will involve an error termrelated to the following function, defined for m̱∈^n and B≥ 1,E(B;m̱) := ∑_i=1^3 B^-ϵ_i,1 |m̱|^ϵ_i,2min{m_j}^ϵ_i,3 .Furthermore, extend the functionϖ defined in (<ref>) to ^n by letting for ∈^n,ϖ(ḵ) :=∏_p| k_1⋯ k_nϖ( p^ν_p(k_1),…,p^ν_p(k_n))and if(k_1⋯ k_n,W)=1 wedefineτ∈ (∩ [0,W))^n via⟨τ⟩≡y̱W. Finally, welet (,W):=∏_p|Wσ_p(f̱(τ+Ws̱)) ∏_p∤ Wσ_p()and𝒥_w(,W) :=1/W^n∫_^R∫_^n e(γ·(u̱)) ∏_i=1^nw ( u_i -ζ_i/2|ζ|) ̱̣uγ.Assume 𝔅()>2^d-1R(R+1)(d-1) and that∈^n satisfies(k_1⋯ k_n,W)=1and ||≤ B^1/ρ (log B)^-1 ,where B ∈ℝ_≥ 1and the constant ρ was defined in (<ref>). Then for each >0 we haveN_w(B;)= 𝒥_w(,W) (,W) ϖ()/ B^n-Rd +O( B^n+/ḵ E(B;ḵ) ) .Defining g̱(s̱):=f̱(⟨(τ +Ws̱)⟩ ) givesN_w(B;)= ∑_s̱∈^n g̱(s̱)=0̱∏_i=1^n w(s_i/B/k_iW - ( ζ_i/2|ζ|-τ_i/B/k_i) ) .We shall apply Theorem <ref> at this point; before doing so we need to verify that | ζ_i/2|ζ|-τ_i/B/k_i| ≤ 1 and thatcondition (<ref>) is met. The former is easy to verify due to |τ|≤ W≪ 1 and ρ>1, which implies thatB/k_i ≥ B^1-1/ρ (log B)→ +∞. Regarding (<ref>), the obvious equality ^♮(s̱)=W^d(⟨s̱⟩) presents us with max{^♮,}≪ ||^d, thus the growth condition on || in our lemma is sufficient. The last issue to be commented regards thereal densities. The real density provided by the application of Theorem <ref> is∫_^R∫_^n e( W^d β·(⟨s̱⟩)) ∏_i=1^nw(s_i/B/Wk_i -(ζ_i/2|ζ| -τ_i/B/k_i)) ̱̣sβ .Note that the proof of Theorem <ref> in fact shows that the real density can also be replaced by its inhomogeneous version,∫_^R∫_^n e( β·(⟨τ⟩ +W⟨s̱⟩)) ∏_i=1^nw(s_i/B/Wk_i -(ζ_i/2|ζ| -τ_i/B/k_i)) ̱̣sβ .For this we note that the major arc analysis initially camein itsinhomogeneous form, namely having(⟨τ⟩ +W⟨s̱⟩) in the exponential.Moreover, by shifting the center of the weight functions, one sees that Lemma <ref> still applies to the inhomogeneous form and then everything stays exactly the same with regard to the error terms.To continuethe proof of our lemma we performthe linear change of variables s_i↦ u_i andβ_i ↦γ_igiven byk_i(τ_i+Ws_i)=B u_i, B^d β_i=γ_i. This leads tothe following expression for the real density in our lemma,B^n-Rd/W^n∫_^R∫_^n e(γ·(u̱)) ∏_i=1^nw ( u_i -ζ_i/2|ζ|) ̱̣uγ ,which equals 𝒥_w(,W)^-1B^n-Rd. The most noteworthyproperty of Lemma <ref> is related to the presence of ^-1 in the error term; this allows to drastically improve the level of distribution in the forthcomingapplications. §.§ Using the Rosser–Iwaniec sieve By (<ref>) we have the followingwheneverz satisfies z_0<z<B,∑_x̱∈ (∩ [-B,B])^n()=0̱, P^-()>z1 ≥w_0^-n∑_x̱∈A̧ P^-()>z∏_i=1^n w( x_i/B -ζ_i/2|ζ|). Let us now bring into play a lower bound sieve sequence λ_k^- of dimension n.Recall the definition of θ' in (<ref>). We shall make use of the terminology in <cit.>; in doing so we shall call the support of λ^- by D:=B^δ, for some constant δ∈ (0,θ'). Using (1∗μ)(l)≥ (1∗λ^-)(l) for l=(P(z_0,z),x̱) yields∑_x̱∈ (∩ [-B,B])^n()=0̱, P^-()>z1 ≥w_0^-n∑_k|P(z_0,z) k≤ B^δλ_k^- ∑_x̱∈A̧k| ∏_i=1^n w( x_i/B -ζ_i/2|ζ|) .The proof of <cit.> can be directly adapted in the setting of arbitrary dimension, thus providingthe equality of the inner sum overtoμ(k)∑_k∈^n p|⇔ p|kμ() N_w(B;) ,where here and throughout the rest of the paper we will use the notationμ():=μ(k_1)⋯μ(k_n) .A moment's thought reveals that the succeeding function is multiplicative,g(k):= 1_(k,W)=1(k)μ(k) ∑_∈^np|⇔ p|kμ() ϖ(ḵ) ^-1 ,a notation which allows to assort our conclusions so far in the following form,∑_x̱∈ (∩ [-B,B])^n ()=0̱,P^-()>z1 ≫ B^n-Rd∑_k|P(z_0,z) k≤ B^δλ_k^-g(k) +O(B^n+∑_k≤ B^δ |μ(k)| ∑_ḵ∈ℕ^np|⇔ p|k |μ()| /ḵ E(B;ḵ)) .In bounding the error term we will be confronted with sums of the formb_k:= |μ(k)| ∑_ḵ∈ℕ^np|⇔ p|k |μ()| /ḵ |ḵ|^α_1min{k_i}^α_2 ,where α_i≥ 0. Each ḵ making a contribution to b_ksatisfies||≤ k ≤ḵ, therefore b_k ≪ |μ(k)| k^α_1+α_2-1+ .We deduce that for each 1≤ j ≤ 3, the quantityB^-ϵ_j,1∑_k≤ B^δ k^ϵ_j,2+ϵ_j,3-1≪ B^-ϵ_j,1+δ(ϵ_j,2+ϵ_j,3)becomes ≪ B^-Rd-ϵ' for some ϵ'>0 due to δ<θ'.Therefore, we can see thatfor eachδ∈ (0,θ') and >0there exists η=η(,δ)>0 such that B^n+∑_k≤ B^δ |μ(k)| ∑_k∈^n p|⇔ p|k |μ()| /ḵ E(B;ḵ)≪ B^n-Rd-η .This leads to the conclusion that subject to the assertion ∑_k|P(z_0,z) k≤ B^δ λ_k^-g(k) ≫ (log B)^-nwe can establish Theorem <ref> due to∑_x̱∈ (∩ [-B,B])^n ()=0̱,P^-()>z1 ≫B^n-Rd/(log B)^n .To prove (<ref>) we shall use <cit.>. To this end, for any polynomials h_i ∈[x_1,…,x_n] we abbreviate σ_p(p| ẖ(x̱)) := lim_l→+∞ p^-(n-R)l#{1≤≤ p^l : p^l|f̱(x̱),p| ẖ(x̱) } .For each prime p>z_0we haveg(p) σ_p = σ_p(p|x_1⋯ x_n). The definition (<ref>) furnishesg(p) σ_p = ∑_m=1^n (-1)^m-1/p^m∑_∈{0,1}^n ||_1=m δ() ,thus, letting N_(p^l) := #{ 1≤ x_1,…,x_l≤ p^l: f̱((p^j_ix_i))≡0̱p^l}, we conclude thatg(p) lim_l→+∞ N_0̱(p^l) /p^(n-R)l = lim_l→+∞∑_m=1^n (-1)^m-1/p^m∑_∈{0,1}^n ||_1=mN_(p^l) /p^(n-R)l . Obviously, ifj_i=1 and y_i≡ x_i p^l-1 thenp^j_iy_i≡ p^j_ix_i p^l. Therefore we may split the interval [1,p^l]into p subintervals of length p^l-1 to obtainN_(p^l) =p^||_1#{ j_i=1⇒ 1≤ x_i ≤ p^l-1, j_i=0⇒ 1≤ x_i ≤ p^l : f̱((p^j_ix_i))≡0̱p^l} .One can see that this entity equals #{x̱≤ p^l : f̱(x̱)≡0̱p^l, j_i=1⇒ p|x_i }, hence, combining this with (<ref>)yields the desired result.There exists _0∈ (0,1) such that one hasg(p)= n/p + O(p^-1-_0).For a prime p and t∈ let M(p^t):=♯{1≤≤ p^t: p^t|() , p∤ x_1·…· x_n}. Then Lemmas 11-12in <cit.> imply that there exists a positive _0>0 such that(1-1/p)^-nlim_t→∞ p^-t(n-R)M(p^t) = 1+O(p^-1-_0).We observe that lim_t→∞ p^-t(n-R)M(p^t)= _p - _p(p|x_1… x_n), thus Lemma <ref> reveals that g(p)= _p(p|x_1… x_n)/_p= 1 - _p^-1lim_t→∞ p^-t(n-R)M(p^t) = 1-(1-1/p)^n1/_p+O(_p^-1p^-1-_0).The work of Birch <cit.> establishes the existence of a positive ϵ_1 such that _p=1+O(p^-1-_1). This is sufficient for our lemma. Enlarging z_0 if necessary,ensures that for all primes p we have0≤ g(p) <1 andg(p)≤n/p+O(p^-1-ϵ_0) .This means that one can take κ=n in <cit.>, hence our sieve problem has dimensionn. By <cit.>, the sieving limit β fulfilsβ≤3.75n, thus <cit.>, in combination with Lemma <ref>, guarantees the veracity of (<ref>) under the conditionlog D/log z> 3.75 n .This concludes the proof of Theorem <ref>. §.§ Using the weighted sieve In the last section we saw that sieving out small prime divisors of x_1⋯ x_n for integer zeros of (x̱)=0̱ gives rise to a sieve of dimension n. When the dimension of the sieve increases then the weighted sieve gives better results for the number of prime divisors in our sequence. We would like to use the weighted sieve in the form given in the Cambridge Tract of Diamond and Halberstam <cit.>, however we shall need a more flexible version of their work; one that allows the use of smooth weights. This will follow from a generalisation of the weighted sieve that will be given in <ref>. This generalisation permits the use of any suitable non-negative function rather than just a smooth weight as well as sieving in multisets. §.§.§ The weighted sieve with smooth weights We assume that M̧ is any set equipped with two functions π:M̧→, h:M̧→ such that h(M̧)⊂ [0,1], h ≠ 0, #M̧<∞ .For convenience of presentation we shall prefer the notationm=π(m). We also assume that there exists a set of primesP̧, a constant X∈_>0 and a multiplicative function ω:→_≥ 0such that, when lettingr_M̧,h(k) := ∑_m ∈M̧ b|mh(m) -ω(b)/bX , (b ∈),there exist constants τ∈ (0,1],κ∈, A_1≥ 1 and A_2≥ 1 such that∑_1≤ b ≤ X^τ (log X)^-A_1μ(b)^24^ν(b) |r_M̧,h(b)| ≤A_2 X/(log X)^κ+1 ,where the function ωenjoys the following properties for some constants κ≥ 1,A>1,0≤ω(p)<p(p ∈P̧), ω(p)=0 (p∉P̧) ∏_w_1≤ p <w(1-ω(p)/p)^-1≤(log w/log w_1)^κ(1+A/log w_1), 2≤ w_1 < w.We furthermore demand that m ∈M̧,p| m⇒ p ∈P̧,and that that there exists a constant μ_0>0 such that max{ |m|: m ∈M̧}≤ X^τμ_0 .Lastly, we shall say that the property 𝐐(u,v) holds for two real positive numbers u<v if𝐐(u,v): ∑_X^1/v≤ p ≤ X^1/up ∈P̧∑_m ∈M̧ p^2|mh(m) ≪X/log X∏_ p∈P̧p<X^1/v(1-ω(p)/p) .Before stating the main theorem in this section recall the definition of f=f_κ,F=F_κ and β_κ in <cit.> through certain differential equations. The inequality β_κ < ν_κ is proved for κ≥ 200 in <cit.>; here ν_κis the Ankeni-Onishi sieving limit<cit.> that satisfies ν_κ∼ c κ as κ→+∞, wherec=2/elog 2exp( ∫_0^2 e^u-1/udu) =2.445….In particular there exists an absolute positive constant c_0 such that β_κ≤ c_0 κ for allκ≥ 1. Assume that κ≥ 1,M̧,X,ω,μ_0 are as above, that each one of the conditions (<ref>)-(<ref>) holds, that r is a natural number satisfying r>N(u,v;κ,μ_0,τ), whereN(u,v;κ,μ_0,τ) := τμ_0 u-1+κ/f_κ(τ v)∫_u^v F_κ(v(τ-1/s)) (1-u/s) ds/sand u,v satisfy𝐐(𝐮,𝐯), τ v>β_κ as well as 1/τ<u<v. Then we have #{ m ∈M̧, P^-(m)≥ X^1/v, Ω(m)≤ r }≫ X ∏_ p∈P̧p<X^1/v(1-ω(p)/p) .The proof is merely a careful recast of the proof of Theorem 11.1 in <cit.>. In place of the function defined in <cit.> we shall use the following function that combinesthe classical weights related to the weighted sieve in addition to the new weight h,W_h(M̧,P̧,z,y,λ) :=∑_ m ∈M̧ (m,P(z))=1 h(m) {λ- ∑_p ∈P̧,p|mz≤ p < y(1-log p/log y) } ,whereP(z):=∏{p:p∈P̧,p<z}. A statement analogous to <cit.> can be verified once the entitiesS(A̧,P̧,X^1/v) and S(A̧_p,P̧,X^1/v) are replaced by∑_m ∈M̧ (m,P(X^1/v))=1h(m)and∑_m ∈M̧, p|m (m,P(X^1/v))=1h(m)respectively. The rest of the arguments in <cit.> are carried automatically to our setting since, once the level of distribution result (<ref>) is applied, all information regarding M̧ and h is absorbed intoX. The only point of departure is the use of various sieve estimates from previous chaptersof the book. These sieve estimates boil down to the use of the Fundamental lemma of sieve theoryand the Selberg sieve, both of which can be adapted to our setting. This is due to the non-negativity of the function h, which allows various combinatorial inequalities to be adapted once multiplied by h. One example of this is in the case of an upper bound sieve, say λ^+: opening up the convolution in the right side of (1∗μ) ≤ (1∗λ^+) gives∑_m ∈M̧ (m,P(z))=1 h(m) ≤∑_k|P(z)λ_k^+ ∑_m ∈M̧k|mh(m) ,and one can now use (<ref>) to absorb M̧ and h in X for the rest of the argument.For the proof of the presenttheorem it remains to adapt the arguments in <cit.>. First, the contribution towards ∑_mh(m)of thosem ∈M̧ such that m is divisible by the square of a prime p∈P̧ in the range X^1/v≤ p ≤X^1/u can be safely ignoreddue to condition (<ref>). An inspection of <cit.> reveals that condition 𝐐_0 in <cit.> is used in the proof of <cit.> only to deal with this particular sum over primes in P̧∩ [X^1/v,X^1/u]. We are thus free to focus our attention exclusively on the contribution of the elements of the setM̧':={m'∈M̧:there is no prime p ∈P̧∩ [X^1/v,X^1/u]such thatp^2|m'} .The last inequality in <cit.> becomes∑_X^1/v≤ p <X^1/u p∈P̧,p|m'(1-u log p/log X) ≥Ω(m') -u log |m'|/log X ,which, when multiplied by h(m'), gives, as in <cit.>,W_h(M̧',P̧,z,y,λ) ≤ (r+1)∑_m'∈M̧', Ω(m')≤ r(m',P(X^1/v))=1h(m')for the choice of λ and r made in <cit.>. The property h(M̧)⊂ [0,1]shows that#{m∈M̧:P^-(m)≥ X^1/v, Ω(m)≤ r} ≥#{m'∈M̧': P^-(m')≥ X^1/v, Ω(m')≤ r}≥∑_m'∈M̧, Ω(m')≤ r(m',P(X^1/v))=1h(m') ≥1/r+1 W_h(M̧',P̧,z,y,λ),which allowsthe rest of the proof of <cit.> to be adapted to our case. Finally, the choice of the constants v and r given in our theoremis borrowed from the inequalities succeeding <cit.>.The setting of Theorem <ref> includes that of <cit.>; indeed, one can choose (M̧,π,h) =(A̧, id, 1). In most cases it is easy to verify𝐐(u,v) for all u,v>0, however this is not the case for the problem of prime factors ofx_1⋯ x_n for integer solutions x̱=(x_1,…,x_n) of general Diophantine equations, since, as explained in <ref>, quite often a prime could divide two coordinates of x̱.A table of estimates for β_κ for 1≤κ≤ 10 is given in <cit.>. Furthermore, <cit.> contains estimates for r that are slightly weaker but simpler than that of <cit.>. For example, the choice ξ=β_κ in <cit.> shows that, as long as𝐐(2β_κ-1/τβ_κ,2β_κ-1/τ) holds, then the conclusion of Theorem <ref> remains valid withv=(2β_κ-1)/τ andfor all natural numbers r satisfyingr≥μ_0-1+(μ_0-κ)(1-1/β_κ)+(κ+1)logβ_κ .In fact <cit.> with ξ=β_κ shows thatif𝐐(u,v) holds for some u>1/τ and any v>u, thenlettingv':=β_κ-1/τ-1/uwe deduce that the conclusion of Theorem <ref> still holds with any r satisfyingr≥τμ_0 u-1 + (κ+u/v'β_κ) logv'/u -κ(1-u/v') .To prove Theorem <ref> we take M̧:={x̱∈^n: f̱(x̱)=0̱,≡W,|x̱|≤ B }, π():=,andwe let h():=∏_i=1^nw( x_i/B -ζ_i/2|ζ|).Then for P̧ being the set of all primes p>z_0, g as in (<ref>), θ' as in (<ref>) and any 0<ϵ < θ' we can verify all conditions (<ref>)-(<ref>) with X:=𝒥_w(,W)𝔖(,W) B^n-Rd, ω(b):=b g(b), κ:=n, τ:= θ'-ϵ , μ_0=n/n-Rd1+ϵ/θ'-ϵwith an argument that is identical to that in <ref>. It remains to check condition 𝐐(u,v) and for this we note that in our setting, the sum in (<ref>) is at most ∑_X^1/v< p ≤ X^1/u∑_∈^n =p^2∑_x̱∈A̧k_i|x_i∏_i=1^n w( x_i/B -ζ_i/2|ζ|) .Invoking Lemma <ref> we see that, if u> 2(n-Rd) ρ, where ρ is defined in (<ref>), this is ≪ B^n-Rd( ∑_X^1/v≤ p ≤ X^1/u p^-2∑_∈^n =p^2ϖ() ) +B^n+( ∑_X^1/v <p ≤ X^1/up^-2∑_∈^n =p^2 E(B;ḵ) ) .Assuming max{ (d^2-1)R^2 2^d-1 , (d-1)R^2 2^d-1 +2(R+1) } < 𝔅(f̱), we obtain viaCorollary <ref> that the first sum overabove is ≪ 1, thus, when v> 0, the first term contributes≪ B^n-Rd-(n-Rd)/v≪ B^n-Rd/ (log B)^n .It remains to verify that there exists ϵ'>0 such that ∑_X^1/v <p ≤ X^1/up^-2∑_∈^n =p^2 E(B;ḵ)≪ B^-ϵ'-Rd .For this we note thateach ϵ_i,2 is at least 1/2 owing to K≥max{Rd,R^2(d-1)}and K≥ 1. Thus the error term above becomes≪∑_i=1^3 B^-ϵ_i,1∑_X^1/v <p ≤ X^1/up^-2+2ϵ_i,2≪∑_i=1^3 B^-ϵ_i,1+(n-Rd)/u(2ϵ_i,2-1) .Therefore, if u>max{(n-Rd) (2ϵ_i,2-1)/ϵ_i,1-Rd:1≤ i ≤ 3 }then (<ref>) holds. Now defineu_0:=(1+ϵ) max{u_1,1/(θ'-ϵ), 2(n-Rd)ρ}, whereu_1:= max{(n-Rd) (2ϵ_i,2-1)/ϵ_i,1-Rd:1≤ i ≤ 3 }. Then applying (<ref>) with u=u_0 and v':=(n c_n-1)/(θ'-ϵ-1/u_0), allows to take r≥n/n-Rd (1+ϵ) u_0-1 + n(1+u_0/v'c_n) logv'/u_0 -n (1-u_0/v') ,wherec_n:=β_n/n satisfies lim_n→+∞c_n=2.44…. Lettingϵ>0 be arbitrarily close to zeroconcludes the proof of the lower bound claimed in Theorem <ref>. This is becausethe quantitiesu”,u,v introduced in (<ref>) and (<ref>) are such that for fixed f̱,n,d,R we have lim_ϵ→ 0 (u_1,u_0,v)= (u”,u,v) .To complete the proof ofTheorem <ref> it remains to verify the estimates regardingv and r_0, where r_0 is defined in (<ref>). It is easy to see that u”/(n-Rd) is a function of K that is bounded away from 0 and +∞, while a similar remark applies to ρ and θ'.This implies that u≪_d,R n and noting thatu<v, one hasr_0≪_d,Ru+n logv/u≪_d,R n(1+logv/u) , where the implied constant is independent of K and n. The identityn c_n-1=τv-v/ushows that v/u≪ n+v≪ n+n/θ'-1/u≪ n, thereforer_0=O_d,R(n log n), with an implied constant depending at most on d and R.§ MULTIDIMENSIONAL VECTOR SIEVE The next lemmaconstitutesa generalisation of the vector sieve of Brüdern and Fouvry <cit.> to arbitrarily many variables. Let n∈ and assume that we are given2 sequencesλ_i^-,λ_i^+, (i=1,…,n) such that for each m ∈ and 1≤ i ≤ nwe have (1∗λ_i^-)(m) ≤(1∗μ)(m) ≤(1∗λ_i^+)(m) .Then the following inequality holds for each m̱∈^n,∏_i=1^n (1∗μ)(m_i) ≥∑_i=1^n (1∗λ_i^-)(m_i) ∏_1≤ j ≤ n j ≠ i(1∗λ_j^+)(m_j) -(n-1) ∏_i=1^n (1∗λ_i^+)(m_i)In light of (<ref>) it is sufficient to verify ∏_i=1^n (1∗μ)(m_i) ≥-(n-1) ∏_i=1^n (1∗λ_i^+)(m_i) + ∑_i=1^n (1∗μ)(m_i) ∏_1≤ j ≤ n j ≠ i(1∗λ_j^+)(m_j) .If m_i=1 for all i=1,…,n then (1∗λ_i^+)(m_i)≥ 1, thus the entities x_i:=1/(1∗λ_i^+)(m_i) fulfill0<x_i≤ 1. The inequality (<ref>) becomesx_1⋯ x_n ≥ -n+1+ (x_1+⋯+x_n). LettingA_i=1-x_i the last inequalitybecomes(1-A_1)⋯ (1-A_n) ≥ 1-(A_1+⋯+A_n), which is the Weierstrass product inequality, see <cit.>. In the remaining case where there exists i with m_i≠ 1 we can assume that(1∗λ_i^+)(m_i)≠ 0 for each such i, for otherwise both sides of (<ref>) vanish. We may now introduce for each 1≤ i ≤ n the variables x_i:=1/(1∗λ_i^+)(m_i); then (<ref>) becomesn-1≥∑_1≤ i ≤ nm_i=1x_i.The proof is concluded upon observing thatthe condition m_i=1 impliesx_i≤ 1. Our aim now becomes to prove a version of the Fundamental Lemma of sieve theoryin the context of prime divisors of coordinates of integer zeros in varieties. The exact form is given in Proposition <ref> and the rest of this section is devoted to its proof. The quantity under consideration is the weighted density of vectors ∈A̧ with||≤ B such thatx̱ does not haveprime divisors in the range p≤ z_1 for any z_1 withz_0<z_1≤ B. We prefer to keep the choice of z_1 unspecified in this section and we shall only need the value z_1= (log B)^A for A>0 independent of B in <ref>.For ḵ∈^nand y_1,y_2 ∈ with y_1<y_2 we define μ(ḵ):=∏_i=1^n μ(k_i) and P(y_1,y_2):=∏_y_1<p ≤ y_2p . For a smoothfunction w:→_≥ 0that is as in subsection <ref>,any z_1>z_0 and any ḻ∈^n we letG(B,z_1;ḻ) := ∑_x̱∈A̧, l_i|x_i,p|x_1⋯ x_n ⇒ p>z_1 ∏_i=1^nw( x_i/B -ζ_i/2|ζ|). We are interested inestimatingG(B,z_1;ḻ) whenever ḻ∈^n fulfillsl_i|P(z_1,z), where z is any constant satisfying z>z_1. This is analogous to <cit.> and we shall also begin by proving the upper bound. We shall use the upper and lower bound sieves, λ^+ and λ^-,as defined at the bottom of <cit.>. Assume that λ^+ is an upper bound sievesupported in [1,D_1] and note thatthe condition x̱≡y̱W ensures that p∤for all p≤ z_0. Recalling definition (<ref>) we see that whenever l_i|P(z_1,z) then G(B,z_1;ḻ) ≤∑_ḵ∈^n k_i |P(z_0,z_1) N_w(B;(k_1l_1,…,k_nl_n)) ∏_i=1^n λ^+_k_i .Note that all ḵ and ḻ above must satisfy(,)=1= ( ,∏_p≤ z_0p), μ(k_i)^2=1=μ(l_i)^2 .Recall definition (<ref>) and assume that |ḻ| ≤B^1/ρ/D_1log B.Then Lemma <ref> shows thatif K>R(R+1)(d-1) and (<ref>) holds thenN_w(B; (k_1l_1,…,k_nl_n) ) = ϖ()/ X_ +O( B^n+ϵ/ḻE(B; (k_1l_1,…,k_nl_n) )/ḵ) ,whereX_:= 𝔖() 𝒥_w(,W) ϖ()/ḻ B^n-Rd . We may now set (D_1,z_1)=∑_ḵ∈^n k_i |P(z_0,z_1)ϖ()/∏_i=1^nλ^+_k_ito obtainG(B,z_1;ḻ) ≤(D_1,z_1) X_ +O(B^n+ϵ/ḻ∑_|| ≤ D_1p|⇒ z_0<p ≤ z_1μ(ḵ)^2/ḵ E(B; (k_1l_1,…,k_nl_n) ) ). §.§ Bounds for ϖ. One has to be careful whenadapting the approach <cit.>to homogeneous equations. The reason is that in the case of Lagrange's equation there exists a multiplicative function ϖ satisfyingϖ(m̱)≤∏_i=1^n ϖ(m_i) and such that for all large primes p one has ϖ(p)≤ 2, see <cit.>. It is easy to see that bounds of this quality fail to holdratherspectacularly for systems offorms =0̱ as in Theorem <ref>. Indeed,ϖ(p,…, p) =_p^-1lim_l→∞p^-l(n-R)♯{p^l: (p)≡ 0 p^l}= p^Rd_p^-1lim_l→∞p^-(l-d)(n-R)♯{p^l-d: ()≡ 0 p^l-d}= p^Rd. To confront this issue our first task is to control the contributiontowards Σ(D_1,z_1) of integer vectors such that there exists i<j with k_ij:=(k_i,k_j)attaining a large value. Define^*(D_1,z_1)=∑_ḵ∈^nk_i |P(z_0,z_1) max k_ij≤ϖ()/∏_i=1^nλ^+_k_iand recall the definition of Υ in (<ref>).Assuming max{ (d-1)R^2 2^d-1 +(R+1) (Υ+1), (d^2-1)R^2 2^d-1}<𝔅(f̱), one has (D_1,z_1)-^*(D_1,z_1)≪^-1+ (log z_1)^n.The quantity under investigation is ≪∑_1≤ l_1<l_2≤ n(l_1,l_2), where (l_1,l_2):=∑_ >μ()^2 ∑_k_i|P(z_0,z_1)|k_l_1,|k_l_2ϖ()/ .We may now use the multiplicative properties of ϖ to deduce that(l_1,l_2) ≪∑_> ( ∏_z_0<p≤ z_1p|∑_∈{0,1}^nj_l_1=j_l_2=1ϖ(p^)/p^||_1) ( ∏_z_0<p≤ z_1p∤∑_∈{0,1}^nϖ(p^)/p^||_1) . Fix η∈ (0,1/4) and let usdenotes_0:=Υ+1+η. By Corollary <ref> we obtain ∑_||_1≥ s_0ϖ(p^)/p^||_1≪ p^Υ∑_s_0≤ s ≤ nns p^-s≪p^-1-η .The assumptions of ourlemma allow us to apply Corollary <ref> whenever ||_1≤ s_0. Thus it supplies us with some λ>0 such that ϖ(p^)=1+O(p^-λ), which yields ∑_∈{0,1}^nϖ(p^)/p^||_1 = 1+n/p +O(p^-1-) and ∑_∈{0,1}^nj_l_1=j_l_2=1ϖ(p^)/p^||_1= p^-2 +O(p^-2-),for some ϵ>0. Assorting all related estimates we obtain for square-freethat∏_z_0<p≤ z_1p|∑_∈{0,1}^nj_l_1=j_l_2=1ϖ(p^)/p^||_1≪^-2+,and ∏_z_0<p≤ z_1p∤∑_∈{0,1}^nϖ(p^)/p^||_1 ≪∏_z_0< p≤ z_1(1+n/p+O(p^-1-)) ≪ (log z_1)^n.These estimates prove that(l_1,l_2)≪(log z_1)^n ∑_>^-2+, which is sufficient. For any square-free integer m andindex 1≤ i ≤ n define ϖ_i(m):=ϖ(1,…,1,m,1,…,1) ,where m appears in the i-th position. For >0 define the multiplicative functionϕ_(m):=∏_p|m p>z_0(1+p^-) .Note that if assumptions of Corollary <ref> hold for ||_1=1 then there exists =(f̱)>0 such that σ_p(p^e_i|)=1/p+O(p^-1-). Enlarging z_0 and replacingby a smaller positive constant if needed yields the following result. Assume that 𝔅()>max{ R^2 2^d-1(d^2-1), R^2 2^d-1 (d-1) +(R+1) }. Then there exists =(f̱)>0 such that for all square-free integers m,max_1≤ i ≤ nϖ_i(m) ≤ϕ_(m). Observe that for all ∈^n with μ()^2=1 the expressionϖ()/∏_i=1^nϖ_i(d_i)is a function of the vector ((d_i,d_j))_1≤ i<j ≤ n. To see this, it is enough to consider the case whenis divisible by a single prime, say p. We need to show thatif, ∈{0,1}^n and i≠ j ⇒min(k_i,k_j) = min(h_i,h_j)then ϖ(p^)/∏_i=1^nϖ_i(p^k_i) = ϖ(p^)/∏_i=1^nϖ_i(p^h_i) .Obviously this holds in the case that = and we can therefore assume that ≠. A little thought reveals that in this case (<ref>) guarantees that there exist l,m,i ≠ j such that(,) equals one of the following,(e_l,0), (0̱,e_m), (e_i,e_j) .For any such instance we can verify that both sides of (<ref>) equal 1, hence our claim holds. We have proved that there exists a function g :^n2→_≥ 0such that μ()^2=1 ⇒ϖ ()= g((d_i,j)) ∏_i=1^n ϖ_i(d_i) .The function ϖ_i(d_i) keeps track of the probability that d_i|x_i and the function g((d_i,j))takes values close to 1 when the events d_i|x_iare independent (in a suitable sense) but can obtain larger values in general. Defining S((u_i,j)):=∑_ḵ∈^n k_i|P(z_0,z_1) (k_i,k_j)=u_i,j∏_i=1^nλ^+_k_iϖ_i(k_i)/k_ienables us to write^*(D_1,z_1)= ∑_u_i,j≤Δ 1≤ i<j ≤ ng((u_i,j))S((u_i,j)) .We may now usethe expression (μ∗ 1)((k_i/u_i,j,k_j/u_i,j)) to detect the condition (k_i,k_j)=u_i,j, thus inferringS((u_i,j))= ∑_(l_i,j) ∈^n2 1≤ i ≠j ≤ nu_i,jl_i,j|P(z_0,z_1)μ(ḻ) ∏_i=1^n ł(∑_k∈ k|P(z_0,z_1) ξ_i|kλ^+_kϖ_i(k)/k)̊ ,whereξ_i:= radł( ∏_1≤ j ≤ n j≠ iu_i,jl_i,j)̊and rad stands for the radical of a positive integer. Under the assumptions of Lemma <ref> we thus obtain the following estimate for all square-free integers δ,| ∑_k∈ k|P(z_0,z_1) δ|kλ^+_k ϖ_i(k)/k| ≤ϕ_(δ)/δ∏_z_0<p≤ z_1(1+p^-1+p^-1-) ≪ϕ_(δ)/δlog z_1 .Note that the succeeding inequality holdsfor all divisors m' of m, ϕ_(m)/m≤ϕ_(m')/m'.Letting ξ_i^* be the radical of ∏_j≠ il_i,j and using the last inequalities with δ=m=ξ_i and m'=ξ_i^*allows us to truncate the sum in (<ref>) to the range l_i,j≤Δ^B_1,where B_1>0 is a constant that will be chosen in due course. The contribution of l_1,2>Δ^B_1 is ≪ (log z_1)^n ∑_l_i,j≤ D_1,l_i,j|P(z_0,z_1) l_1,2>Δ^B_1μ()^2/ξ^*∏_i=1^n ϕ_(ξ_i^*) ,where D_1 is the support of λ^+. We may now use the inequalityϕ_(ξ_i^*)≤∏_j≠ iϕ_(l_i,j)to obtain∏_i=1^n ϕ_(ξ_i^*) ≤∏_1≤ i≠ j≤ nϕ_(l_i,j)^2 .Hence the last sum is ≤∑_l_1,2>Δ^B_1μ()^2/ξ_1^*⋯ξ_n^*∏_1≤ i≠ j≤ nϕ_(l_i,j)^2 .This is really a summation over the variables l_1,2,…,l_n-1,n because each expression ξ_i^* is a function of some of these variables. We first perform a summation over l_n-1,n. Recalling that ξ_i^*=rad(∏_j≠ il_i,j)we see that only ξ_n-1^* and ξ_n^* depend on l_n-1,n, since they satisfyξ_n^*=[l_n-1,n,ξ_n^**], ξ_n-1^*=[l_n-1,n,ξ_n-1^**],where both ξ_n-1^** and ξ_n^** are defined asξ_n-1^* and ξ_n^* but with the variable l_n-1,n missing, i.e.ξ_n-1^**:=rad(∏_j≠ n-1,nl_n-1,j), ξ_n^**:=rad(∏_j≠ n-1,nl_n,j) .Hence the sum over l_n-1,n is ∑_l_n-1,nμ(l_n-1,n )^2ϕ_(l_n-1,n)^2/[l_n-1,n,ξ_n-1^**][l_n-1,n,ξ_n^**] ,which equals1/ξ_n-1^**ξ_n^**∑_l_n-1,nμ(l_n-1,n)^2ϕ_(l_n-1,n)^2/l_n-1,n^2(ξ_n-1^**,l_n-1,n) (ξ_n^**,l_n-1,n) .The last sum is ≤∏_p|ξ_n-1^**ξ_n^**(2+2p^-+p^-2) ∏_p (1+p^-2(1+p^-)^2) ≪τ(ξ_n-1^**)^A τ(ξ_n^**)^A ,where A=3. Of course we can bound any ξ_k^** by the product of all available variables except l_n-1,n, i.e. ∏_{i,j}≠{n-1,n}l_i,j, thus we obtain ≪1/ξ_n-1^**ξ_n^**∏_{i,j}≠{n-1,n}τ(l_i,j)^2A .The process above is the first step of a finite induction that eliminates all variables l_i,j, beginning from l_n-1,n and terminating with l_1,2. At each step expressions of the form ∑_l_1,2>Δ^B_1μ()^2/ξ_1'⋯ξ_n'_1≤ i≠ j≤ nτ(l_i,j)^Aare bounded by ≪∑_l_1,2>Δ^B_1μ()^2/ξ_1^”⋯ξ_n^”_1≤ i≠ j≤ nτ(l_i,j)^100 A ,where the notation ξ', means that some of the variables l_i,j have been eliminated, the notation ξ^”, that one further variable has been eliminated and the constant A' depends at most on A and f̱. At the last step of the induction we will arrive at the expression∑_l_1,2>Δ^B_1μ(l_1,2)^2/l_1,2^2τ(l_1,2)^C ,where C=C(f̱). Obviously this is≪Δ^-B_1/2.The arguments above show thatS((u_i,j))= ∑_(l_i,j) ∈^n2 l_i,j≤^B_1u_i,jl_i,j|P(z_0,z_1)μ(ḻ) ∏_i=1^n ł(∑_k∈ k|P(z_0,z_1) ξ_i|kλ^+_kϖ_i(k)/k)̊+O((log z_1)^n ^-B_1/2),where the implied constant is independent of the u_i,j. We now aim to use a consequence of the linear case of the Rosser–Iwaniec sieve (in fact the linear case wassettled first by Jurkat and Richert <cit.>) that is given in <cit.>.We shall find it convenient to use the error term appearing in <cit.>, this will lead to replace the terme^√(L-s)(log D)^-1/3 in <cit.> and <cit.> bye^√(L) Q(s)(log D)^-1/3where, as stated in <cit.>, the function Q(s) satisfiesQ(s)<exp{-s log s+s loglog 3s+O(s)} , s≥ 3 . The constant L in our case will depend at most on the coefficients of f̱, which is considered constant throughout our paper-thus we can assume thatthe terms above are≪_ s^-s (log D)^-1/3, with an implied constant depending at most on . Let us choose the set of primes :={p : p>z_0}.Moreover, we observe that ϖ_i(k) is a multiplicative function for all 1≤ i≤ n.We define the modified multiplicative function ϖ_i(k) byϖ_i(p):={[ϖ_i(p) p>z_0; 0 p≤ z_0. ].So far we can only assume that ϖ_i(p)≤ 1+p^-, whereas in <cit.> they work with the stronger statement thatϖ(p)≤ 1+1/(p-1). However, we still get the bound present in <cit.> for a uniform L.For this we observe thatlog∏_w_1<p≤ w_2(1-ϖ_i(p)/p)^-1 ≤∑_w_1<p≤ w_2log(1-1/p-C/p^1+)^-1≤∑_w_1<p≤ w_2(1/p+C/p^1+)+O( w_1^-1) ≤loglog w_2-loglog w_1 +O(1/log w_1).by Mertens' theorem. This leads to the bound ∏_w_1<p≤ w_2(1-ϖ_i(p)/p)^-1≤(log w_2/log w_1)(1+L/log w_1),for a uniform constant L=L(). We can now directly apply <cit.> to the inner sums appearing in (<ref>).Introduce the constant s_0 through s_0:=(log D_1)/(log z_1), which we demand that it fullfills s_0≥ 3, and set U_i(z_1,ξ_i):=μ(ξ_i) ∏_p|ξ_i p>z_0ϖ_i(p)/p-ϖ_i(p)∏_z_0<p≤ z_1(1-ϖ(p)/p).This provides us with ∑_k∈ k|P(z_0,z_1) ξ_i|kλ^+_kϖ_i(k)/k = U_i(z_1,ξ_i)+O(τ(ξ_i) s_0^-s_0).Owing to the apparent bounds 0≤ϖ_i(p)< p/2, valid for p>z_0 (as long as z_0 is enlarged)we deduce that |U_i(z_1,ξ_i)|≤ 1 for all 1≤ i≤ n and divisors ξ_i|P(z_0,z_1). We use this approximation in (<ref>), to obtainS((u_i,j))-∑_ l_i,j≤^B_1u_i,jl_i,j|P(z_0,z_1)μ(ḻ)∏_i=1^n U_i(z_1,ξ_i)≪ (log z_1)^n ^-B_1/2+^B_1n2+1/100(s_0^-s_0+ s_0^-s_0 (log D_1)^-1/3).Assume that the assumptions in Lemma <ref> are satisfied. Together with equation (<ref>) we now obtain(D_1,z_1)= ^MT(D_1,z_1)+ ^ET(D_1,z_1),with a main term given by^MT(D_1,z_1)=∑_u_i,j≤Δ 1≤ i<j ≤ ng((u_i,j))∑_(l_i,j) ∈^n2 l_i,j≤^B_1u_i,jl_i,j|P(z_0,z_1)μ(ḻ)∏_i=1^n U_i(z_1,ξ_i),and an error term satisfying^ET(D_1,z_1) ≪(log z_1)^n/^1- +^ C+ n2 -B_1/2/(log z_1)^-n+ ^C+(B_1+1)n2+1/100 s_0^-s_0,where C=C()>0 is such that |g((u_i,j))| ≪max{u_i,j}^C .We will assume that such a C exists for the moment,this will be proved later in Lemma <ref>. Therefore we maychoose B_1>0 large enoughso thatC+ n2 -B_1/2<-1. We can then obtain(D_1,z_1)-^MT(D_1,z_1) ≪(log z_1)^n/^1-ϵ +^c s_0^-s_0 ,where c=c()>0. Note that here we implicitly assume that s_0≥ 3, thuslog D_1/log z_1≥ 3.Assume that 𝔅()>max{ R^2 2^d-1(d^2-1), R^2 2^d-1 (d-1) +(R+1) }. Let ∈^n2 be such that μ^2()=1 and such that p| implies that p>z_0. Then, for z_0 sufficiently large one hasg((u_i,j))≪(∏_i≠ ju_i,j)^d𝔅()/(d-1)2^d-1(d-1/R)+R+.First we recall thatg((u_i,j))∏_i=1^n ϖ_i(u_i)= ϖ(),where we have u_i,j=(u_i,u_j). For bounding g((u_i,j)) we may make the following assumption: if p is a prime with p|u_i for some 1≤ i≤ n, then there is a 1≤ j≤ n, j≠ i such that p|u_i,j. Otherwise we could replace in (<ref>) the vectorwith a vectorwhere u_k=u_k for k≠ i and u_k=u_k/p for k=i. In particular, we may assume thatu_i≤∏_j≠ i u_ij,for every 1≤ i≤ n.Next we observe that∏_i=1^n ϖ_i(u_i) = ∏_i=1^n ∏_p|u_iϖ_i(p).We recall the identity ϖ_i(p)=p(p^_i|)_p^-1. By Corollary <ref> we have(p^_i|)=1/p+O(p^-1-).Therefore we obtain∏_i=1^n ϖ_i(u_i)=∏_i=1^n ∏_p|u_i(1+O(p^-1-))^-1(1+O(p^-)),and ∏_i=1^n ϖ_i(u_i)^-1≪_μ (u_1⋯ u_n)^μ, for any μ>0.By Corollary <ref> we haveϖ(p^)≪ p^d𝔅()/(d-1)2^d-1 (d-1/R) +R .Injecting these bounds into (<ref>) yieldsg((u_i,j))≪( ∏_p|p)^d𝔅()/(d-1)2^d-1(d-1/R)+R+≪(∏_i≠ ju_i,j)^d𝔅()/(d-1)2^d-1(d-1/R)+R+,thus concluding the proof.As in <cit.>, we now observe that ^MT(D_1,z_1) is independent of D_1. We set D_2:= max (D_1,3^z_1)and with equation (<ref>) applied to D_2 instead of D_1, we obtain that(D_1,z_1)-(D_2,z_1) ≪(log z_1)^n/^1-ϵ +^c s_0^-s_0 .For this choice of D_2 we have ^+_d=μ(d) for d|P(z_0,z_1) (note that with the change of D_1 to D_2 also the sieve weightschange). Hence we can compute (D_2,z_1) as(D_2,z_1)=∑_ḏ∈^n d_i |P(z_0,z_1)ϖ()/∏_i=1^nμ(d_i)=∏_z_0<p≤ z_1(1-g(p)/p),with g(p) defined as in (<ref>). Injecting our estimates for Σ(D_1,z_1) into (<ref>) yields the upper bound in the next result. Assuming l_i|P(z_1,z), |ḻ|D_1log B ≤ B^1/ρ and that 𝔅(f̱) exceedsmax{ 2^d-1(d-1) R(R+1), 2^d-1 (d-1)R^2 +(R+1) (Υ+1), 2^d-1 (d^2-1)R^2 }we haveG(B,z_1;ḻ) = X_∏_z_0<p≤ z_1(1-g(p)/p)+O(ϖ() B^n-Rd/ḻ((log z_1)^n/^1-ϵ +^c s_0^-s_0)) + O(B^n+ϵ/ḻ∑_|| ≤ D_1p|⇒ z_0<p ≤ z_1μ(ḵ)^2 E(B; (k_1l_1,…,k_nl_n) )/ḵ). The lower bound can be procured upon writingG(B,z_1;ḻ) = ∑_x̱∈A̧ |( ∏_i=1^n w( x_i/B -ζ_i/2|ζ|) ) ( ∏_i=1^n (1∗μ) ((P(z_0,z_1),x_i)) )and using Lemma <ref> to obtainG(B,z_1;ḻ) ≥∑_i=0^nc_i M_i,where for1≤ i ≤ n we definec_i:=1 and M_i:= ∑_k_i |P(z_0,z_1)( λ_k_i^- ∏_j≠ iλ_k_j^+ ) N_w(B;(k_1l_1,…,k_nl_n)),in addition toc_0:=-(n-1)and M_0:= ∑_k_i |P(z_0,z_1)(∏_i=1^nλ_k_i^+) N_w(B;(k_1l_1,…,k_nl_n)) .The treatment of each individual M_i, (i≠ 0), is identical to the treatment of M_0 earlier in this section. The only difference arises at the last step (the calculation of the Euler products in the main term). Here the coefficients c_i satisfy∑_0≤ i ≤ nc_i=1, thus completing the proof of Proposition <ref>. § PROOF OF THEOREMS <REF> AND <REF>Recall the definition of the set A̧ in (<ref>). Our aim is to find a large function z=z(B)≤ B such thatS(B,z) := #{x̱∈A̧:||≤ B, p|x_1⋯ x_n ⇒ p>z }≫B^n-Rd/(log B)^n .By (<ref>) we haveS(B,z)≥ w_0^-nS_ζ(B,z) , where S_ζ(B,z) := ∑_x̱∈A̧p|x_1⋯ x_n ⇒ p>z ∏_i=1^nw( x_i/B -ζ_i/2|ζ|) .One may now write the sum overas ∑_x̱∈A̧( ∏_i=1^nw( x_i/B -ζ_i/2|ζ|) ) ( ∏_i=1^n (1∗μ) ((P(z),x_i)) ) .For a parameter D let λ^± be a sieve sequence supported in [1,D].Letting β(ḻ) := ∑_i=1^n λ_l_i^- ( ∏_1≤ j ≤ n j ≠ i λ_l_j^+ ) -(n-1) ∏_i=1^n λ_l_i^+, ḻ∈^n,alluding to Lemma <ref> and recalling (<ref>) allows us to infer thatfor any z>z_1 we have S_ζ(B,z) ≥ ∑_ḻ∈^nl_i | P(z_1,z)β(ḻ) G(B,z_1;ḻ) .Define the entitiesΣ(D,z_1,z):=∑_ḻ| P(z_1,z)β(ḻ)ϖ()/, B_1:=∑_| P(z_1,z)ϖ() /and B_2:= B^dR+ϵ∑_|| ≤ D | P(z_1,z) 1/ḻ∑_||≤ D_1 | P(z_0,z_1)E(B; (k_1l_1,…,k_nl_n) )/ḵ.Proposition <ref> now leads toS_ζ(B,z)/𝔖() 𝒥_w(,W) B^n-Rd≥Σ(D,z_1,z) ∏_z_0<p≤ z_1(1-g(p)/p) + Oł(((log z_1)^n/^1-ϵ+^c/s_0^s_0)B_1+B_2)̊ .Letting m_i:=k_il_i and taking advantage of the coprimality of k_i,l_ishows that B_2 ≤B^dR+ϵ∑_|m̱|≤ DD_1 m̱| P(z_0,z)E(B;m̱)/m̱.Recalling the definition of the matrix ϵ given in (<ref>), shows that, under the conditionD D_1≤B^1/ρ/log B ,the sum over m̱ is≪∑_i=1^3 B^-ϵ_i,1∑_1≤ m_1≤ D D_1 m_1^ϵ_i,2-1∑_1≤ m_2≤ m_1 m_2^-1…∑_1≤ m_n-1≤ m_n-2 m_n-1^-1∑_1≤ m_n≤ m_n-1 m_n^ϵ_i,3-1 .Since each ϵ_i,j is non-negativewe can use the estimate∑_1≤ m ≤ zm^λ-1≪_λ z^λlog z, valid for each fixed λ≥ 0to deduce that for every ϵ>0 one has B_2 ≪B^dR+ϵ∑_i=1^3 B^-ϵ_i,1 (D D_1)^ϵ_i,2+ϵ_i,3 .Our remaining task will be to give a lower bound for Σ and an upper bound for B_1. We begin by studying the contribution to Σ(D,z_1,z) of vectorswith δ:=(l_i_1,l_i_2)≠ 1; this task is similar to the one in Lemma <ref> and we adapt its assumptions in what follows. Each such δ is a product of primes p>z_1, therefore this contribution is ≪∑_δ>z_1μ(δ)^2 ∑_ |P(z_1,z)| l_i_1,|l_i_2ϖ()/ .As in the proof of Lemma <ref> we find that this is ≪ł(log z/log z_1)̊^n∑_δ>z_1δ^-2+≪z_1^-1+ (log z)^n .Note that if l_i,l_j are coprime for all i≠ j guarantees thatϖ()=∏_i=1^n ϖ_i(l_i). This givesΣ(D,z_1,z)= ∑_ḻ| P(z_1,z)i≠ j ⇒(l_i,l_j)=1β(ḻ)/∏_i=1^n ϖ_i(l_i) +O(z_1^-1+ (log z)^n).The same argument can also be used to establish∑_ḻ| P(z_1,z)β(ḻ)/∏_i=1^n ϖ_i(l_i) = ∑_ḻ| P(z_1,z)i≠ j ⇒(l_i,l_j)=1β(ḻ)/∏_i=1^n ϖ_i(l_i) +O(z_1^-1+ (log z)^n).LettingΨ^±_i:=∑_l|P(z_1,z)λ^±_l ϖ_i(l)/lshows that the sum on the left equalsΨ:= ∑_i=1^n (Ψ_i^- ∏_1≤ j ≤ n j ≠ i Ψ_j^+) -(n-1) ∏_i=1^n Ψ_i^+ ,thus providingΣ(D,z_1,z)=Ψ +O(z_1^-1+ (log z)^n). Under the assumptions of Lemma <ref> we can similarly show that the contribution of with (l_i_1,l_i_2)≠ 1 to B_1 is ≪∑_δ>z_1μ(δ)^2 ∑_| P(z_1,z) δ|l_i-1, δ| l_i_2ϖ() /≪B^n-Rd (z_1^-1+ (log z)^n) .Therefore B_1 ≪ z_1^-1+ (log z)^n +∑_| P(z_1,z)i≠ j ⇒(l_i,l_j)=1∏_i=1^nϖ_i(l_i)/l_iand the last sum is ≤∏_i=1^n∑_l|P(z_1,z)ϖ_i(l)/l≤∏_i=1^n ∏_z_1<p≤ zł(1+1/p+O(p^-1-))̊≪ (log z)^n,hence B_1 ≪ (log z)^n. We therefore find via (<ref>) the following lower boundS_ζ(B,z)/𝔖() 𝒥_w(,W) B^n-Rd ≥Ψ∏_z_0<p≤ z_1(1-g(p)/p) +O(B^dR+ϵ∑_i=1^3 B^-ϵ_i,1(D D_1)^ϵ_i,2+ϵ_i,3)+O( (log z)^n/z_1^1-(log z_1)^n +((log z_1)^n/^1-ϵ+^c/s_0^s_0)(log z)^n ), where a use of ∏_z_0<p≤ z_1(1-g(p)/p) ≪ (log z_1)^-nhas been made; this can be inferred from the estimateg(p)=n/p+O(p^-1-). Let us now fix any θ>0 which satisfiesθ< θ', where θ' was defined in (<ref>). Then there exists a small positive θ_1 such that ifD:=B^θ and D_1:=B^θ_1 then B^dR+ϵ∑_i=1^3 B^-ϵ_i,1 (D D_1)^ϵ_i,2+ϵ_i,3≪ B^-δ ,for some δ>0 that is independent of B. Choosing Δ=z_1=(log B)^2n+1 shows that s_0=log D_1/log z_1 =θ_1 log B/(2n+1) loglog B→∞ ,hence one can verify that (log z)^n/z_1^1-(log z_1)^n +((log z_1)^n/^1-ϵ+^c/s_0^s_0)(log z)^n ≪1/(log B)^n loglog BandS_ζ(B,z)/𝔖() 𝒥_w(,W) B^n-Rd≥Ψ∏_z_0<p≤ z_1ł(1-g(p)/p)̊+ Oł((log B)^-n(loglog B)^-1)̊ .The last product is ≫ (log z_1)^-n, thus it remains to show thatΨ≫ (log z_1/log z)^n. Let s:=log D/log z and assume that s>2. Using the inequalities stated in <cit.> one deduces that whens=O_n(1) with an implied constant depending at most on n, then Ψ≥ (Ψ_n(s)+O_n((log B)^-1/3)) ∏_i=1^n ∏_z_1<p≤ z(1-ϖ_i(p)/p) , where Ψ_n(s):= n f(s)-(n-1) F(s)^n. Here f(s) and F(s) denote the well-knownfunctions associated to the linear sieve, their definition can be found in <cit.>, for example. Further information on f and Fis located in <cit.>. In light of thelast lower bound for Ψ, it is sufficient to find the smallest possiblevalue for s such that Ψ_n(s)>0. This is equivalent toF(s)^n/f(s)<1+1/n-1 .It is a standard fact that when s>2 then 0<f(s)≤ 1 ≤ F(s). Therefore if s remains constant and independent of n then one cannot prove (<ref>) for large n, this forces us to take s as a function of n that tends to infinity. At this point we have to employ asymptotic approximations forf(s) and F(s), these can be found in <cit.>. They are given byF(s),f(s)=1±exp{-s log s-sloglog s+s+O(s loglog s/log s)}and one sees that if s≥3(log n)(loglog n)^-1 then slog s ≥ 3 log n+3 (log 3) (log n)/loglog n -3 (log n) (logloglog n)/loglog n .Therefore, for all large enough n, say n≥ n_0 for some positiveabsolute constant n_0, we obtainF(s)^n/f(s)<( 1+1/n^5/2)^n+1and the inequality 1+n^-5/2<(1+(n-1)^-1)^1/(n+1), valid for all n≥ 2, makes (<ref>) available.In the case that 1≤ n<n_0 one can immediately infer from the approximations to F(s) and f(s) that if s→+∞ then (<ref>) is automatically satisfied. This gives a constant σ_0 that depends at most on n_0 (and is therefore absolute) such that (<ref>) is valid whenever s≥σ_0. Hence there exists a positive absolute constant σ_0 such that if s≥3 log n/loglog n+σ_0then, alluding to (<ref>),Theorem <ref> holds with any constant c_0>3+σ and withP^-(x_1⋯ x_n) exceeding the sieving parameter z=D^1/s=B^θ/s. The arguments in the present section have so farproved that S_ζ(B,B^θ/s) ≫ B^n-Rd (log B)^-n.This is sufficient forTheorem <ref> because to show that a subset of V_f̱() is Zariski dense in an absolutely irreducible variety V_f̱, it is sufficient to choose an arbitrary neighbourhood in the real analytic topology of a non-singular point ζ∈ V_f̱() and show that any real point in the neighbourhood (on the variety) can be approximated by a rational point.In our case theneighbourhood is given by B B̧_η (where B̧_η was defined in (<ref>)). amsalpha | http://arxiv.org/abs/1705.09133v3 | {
"authors": [
"Damaris Schindler",
"Efthymios Sofos"
],
"categories": [
"math.NT",
"11D72, 11N36, 11P55"
],
"primary_category": "math.NT",
"published": "20170525113922",
"title": "Sarnak's saturation problem for complete intersections"
} |
^1 Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea ^2 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea ^3 National Institute of Advanced Industrial Science and Technologh (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan ^4 Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France ^5 Neutron Group, National Synchrotron Radiation Research Center, Hsinchu 30077, Taiwan ^6 Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan ^7 Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic ^8Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France [email protected] * March 2018YFeO_3 and LaFeO_3 are members of the rare-earth orthoferrites family with Pbnm space group. Using inelastic neutron scattering, the low-energy spin excitations have been measured around magnetic Brillouin zone center. Splitting of magnon branches and finite magnon gaps (∼2 meV) are observed for both compounds, where the Dzyaloshinsky-Moriya interactions account for most of this gap with some additional contribution from single-ion anisotropy. We also make comparisons with multiferroic BiFeO_3 (R3c space group), in which similar behavior was observed. By taking into account all relevant local Dzyaloshinsky-Moriya interactions, our analysis allows for the precise determination of all experimentally observed parameters in the spin-Hamiltonian. We find that different properties of the Pbnm and R3c space group lead to the stabilization of a spin cycloid structure in the latter case but not in the former, which explains the difference in the levels of complexity of magnon band structures for the respective compounds. Keywords: Ferrites, Multiferroics, Inelastic neutron scattering, Dzyaloshinskii-Moriya interaction, weak ferromagnetism§ INTRODUCTIONMagneto-electric (ME) multiferroic materials, in which both magnetic and ferroelectric ordering coexist, have attracted much attention due to the tunable magnetic properties via electric field or vice versa. Such materials also present the possibility of various applications in recording device technology or spintronics <cit.>. While searching for appropriate candidates is far from trivial, one may consider compounds with weak ferromagnetism (wFM) where the reversal of wFM by 180^∘ using electric field has been predicted theoretically <cit.>. In many cases, the microscopic mechanism of wFM is either Dzyaloshinsky-Moriya (DM) interaction or single-ion anisotropy (SIA) <cit.>. In this regard, accurately measuring the values of such quantities in real materials is of considerable importance for future applications.The rare-earth orthoferrites RFeO_3 are one of most promising model systems in this regard. The Fe^3+ ions in all of the RFeO_3 family undergo an antiferromagnetic transition with T_N ranging from 623 K in R=Lu to 738 K in R=La. These high transition temperatures are due to a strong nearest-neighbor exchange interaction (J > 4 meV) along the Fe-O-Fe bond and the large magnetic moment of Fe^3+ (S = 5/2). Most perovskites of ABO_3-type exhibit a cubic Pm3̅m structure at high temperature, and a structural transition occurs upon cooling which lowers the symmetry via tilting of edge-shared BO_6 octahedra. RFeO_3 adopts the Pbnm space group at this structural transition, the most frequent structure among the perovskites. Such octahedra tilting to Pbnm symmetry can be described by Glazer notation: a^-a^-c^+ <cit.>. Since this structure does not break space inversion symmetry (i.e. Pbnm is centrosymmetric), no net polarization in RFeO_3 is expected. In the case of RFeO_3, the tilting of FeO_6 octahedra is the origin of local DM interaction in this compound (see figure <ref>). Competition between DM and exchange interactions results in canting magnetic moments <cit.>. Below T_N, all RFeO_3 adopt a canted antiferromagnetic ground state Γ_4(G_a, A_b, F_c) with basic G-type antiferromagnetism along the a-axis, weak antiferromagnetism along the b-axis, and weak ferromagnetism along the c-axis as shown in figure <ref>(a). Such weak canted magnetic moments were extensively studied both theoretically and experimentally <cit.>. A. S. Moskvin and E. V. Sinitsyn derived a simple formula connecting the canting of magnetic moment and the crystal properties (unit cell parameter, position of oxygen and the bond length), deducing a relation between the A_y and F_z <cit.>. This theoretical prediction was confirmed for several orthoferrites by the polarized neutron diffraction <cit.>. For YFeO_3, calculated value of A_y/F_z = 1.1 is consistent with the experimental results within errorbars. It is worth noting that in case of RFeO_3 with magnetic rare-earth ions, there is a magnetic ordering of R^3+ at low temperature and a spin reorientation transition of Fe^3+ at intermediate temperatures due to the interaction between R^3+ and Fe^3+ ions. Such additional interactions between the two magnetic ions sometimes induces multiferroicity below the spin reorientation transition temperature, and often results in the rotation of Fe^3+ ions by exchange-striction mechanism <cit.>.BiFeO_3 is the only example that is well-established to exhibit multiferroicity above room temperature. BiFeO_3 shares several characteristics with RFeO_3: it has the similar exchange interaction and the very high antiferromagnetic transition temperature T_N at ∼650 K <cit.>. However, there are also clear contrasts between these two materials such as the distinct rotation of FeO_6 octahedra, much of which is due to the lone pair of Bi breaking the inversion symmetry for BiFeO_3 unlike the other centrosymmetric RFeO_3. BiFeO_3 has the non-centrosymmetric space group R3c coming from the Glazer tilting a^-a^-a^-. BiFeO_3 exhibits a large polarization with a ferroelectric transition at T_c=1100 K <cit.>. Below T_N, an incommensurate spin cycloidal magnetic structure develops along the [1 1 0] direction with an extremely long period of 620 Å and is superimposed on the simple G-type antiferromagnetism <cit.>. It was also reported to have a negative magnetostrictive magnetoelectric coupling at T_N <cit.>. Small angle neutron scattering (SANS) experiments revealed a spin density wave (SDW) fluctuation, which is perpendicular to the spin cycloid <cit.>. The local wFM moment made by this fluctuation is cancelled out over the whole cycloid, giving no wFM in bulk BiFeO_3. The spin-Hamiltonian of BiFeO_3 has been extremely well studied both theoretically and experimentally throughout many studies <cit.>. Recent study on the magnetic excitation spectra over the full Brillouin zone using inelastic neutron scattering (INS) measurements determined the values for the two exchange interactions and the Dzyaloshinskii-Moriya (DM) interaction <cit.>. Subsequently, a detailed examination was done on the low-energy region with the observation of the unique island-like feature at 1 meV. Separately, this can also be identified as the peak-and-valley feature in the constant Q-cut graph at the magnetic zone center <cit.>. By employing the full spin Hamiltonian in spin wave calculations, it was further determined that this feature originates from the interplay of the DM interaction and the easy-axis anisotropy <cit.>.The rare-earth orthoferrites have also been previously characterized in the literature, including studies on the spin waves of RFeO_3 with INS <cit.> and Raman spectroscopy <cit.>. Much of the focus in the INS studies was concentrated on the high energy transfer region of the excitation spectra to determine the structural and magnetic interaction strengths. For LaFeO_3, only powder INS spectra was reported that confirmed Heisenberg type nearest-neighbor exchange interactions between Fe^3+ ions <cit.>. For YFeO_3, a recent INS study successfully measured the overall shape of magnon dispersion up to ∼70 meV and deduced the best fit parameters including the nearest- and next nearest-exchange interactions J_1 and J_2, DM interactions, and SIA <cit.>. In addition, the low-energy transfer region at the Brillouin zone center was examined by Raman spectroscopy. These Raman measurements for YFeO_3 determined the magnon peaks around ∼1.4 and 2.2 meV at the Γ point <cit.>. Using these data, they also determined the parameters of the spin Hamiltonian of YFeO_3. However, the model Hamiltonian used in the above studies needs to be improved as it does not capture all the salient details of Pbnm symmetry, in particular the local DM vectors and their relation with the canted ferromagnetic moment. We note that local DM vectors are present even for centrosymmetric space group like Pbnm of RFeO_3, and it is rather poorly understood how this local DM vectors affect the spin waves.To understand the differences between these two compounds and the role of local DM vectors, it is necessary to quantitatively determine their full spin Hamiltonian. In this work, we have carried out comprehensive studies on the low-energy magnon excitations of YFeO_3 and LaFeO_3 since this is where effects of DM interaction and SIA are expected to manifest most strongly. We also collected new data of the low-energy spin waves of BiFeO_3 focusing on higher momentum resolution. Note that we have purposely selected the nonmagnetic rare-earth YFeO_3 and LaFeO_3 orthoferrites in order to focus directly on the magnetism of Fe^3+. Based on the allowed form of the DM interactions in the Pbnm symmetry, we have quantified the parameters of the full spin Hamiltonian for YFeO_3 and LaFeO_3, and reproduced two characteristic features observed in the low-energy magnetic excitation spectra: (1) a finite spin wave gap and (2) splitting of two magnon branches at the zone center. Also, two additional shoulders in constant energy cuts of BiFeO_3 have been identified, demonstrating the more complex nature of the magnon branches in comparison to the other orthoferrites.§ EXPERIMENTAL DETAILSSingle crystals of YFeO_3 and LaFeO_3 with masses of 1.52 and 1.41 g respectively were grown with floating zone furnaces. INS experiments were performed ulitizing the cold-neutron triple axis spectrometer SIKA <cit.> at the Australian Nuclear Science and Technology Organisation (ANSTO). Samples were mounted with their orthorhombic b^*-axis vertical, such that the wave vectors of the observed spin waves were all confined to the a^*-c^* plane. Based on the reflection conditions of magnetic Bragg peaks for the Pbnm space group, all constant-Q energy scans were carried out along the [H 0 0] direction centered on Q = (1 0 1). The final neutron energy was fixed at 5 meV giving a full width at half maximum (FWHM) energy resolution of 0.106 meV at the elastic position. A beam collimator configuration of 40'-40'-60'-40' was used to obtain optimized beam intensity and resolution. A cooled polycrystalline berylium filter was installed to remove the higher-order contamination of the scattered beam. Data were collected at 300 K without a cryostat, and then at 1.5 K with an orange cryostat.For BiFeO_3, the INS experiments were done with two cold-neutron triple axis spectrometers: 4F2 at Laboratoire Leon Brillouin (LLB) and ThALES at Institute Laue-Langevin (ILL). The data obtained at LLB have already been presented by Jeong et. al. <cit.> and reproduced here for comparison and subsequent discussion. In all measurements with 4F2, eight co-aligned single crystals of total mass 1.6 g with 3^∘ mosaicity were used. To achieve better momentum resolution, one single crystal with mass of 0.58 g was used in ThALES experiment. Similar with RFeO_3, BiFeO_3 samples were aligned in the a^*-c^* plane. Using 4F2, energy scans along the [H 0 0] direction centered on Q = (1 0 -1) at T = 16 and 270 K with fixed k_f = 1.2Å^-1. In additional measurements with the ThALES instrument, we have measured the constant-energy (E = 3 meV) cut along the [H 0 0] direction centered on Q = (1 0 -1) at T = 270 K. § RESULTS AND DISCUSSIONFigure <ref> shows scans in energy transfer at various Q points along the [H 0 0] direction centered at Q = (1 0 1) of YFeO_3 and LaFeO_3 for T = 300 and 1.5 K. After corrected for the Bose factor, the measured neutron intensities are proportional to the dynamic susceptibility Im[χ(Q,ω)]. For both compounds, the defining features in the constant-Q cuts are as follows: (1) a finite spin wave gap of E ∼1 meV (YFeO_3) and 2 meV (LaFeO_3) and (2) two distinct peaks directly above the gap, although the valleys between the two peaks are quite small. The two peaks are, as expected, most distinguishable at Q = (1 0 1), signifying that the magnon branches are split at the magnetic Brillouin zone center. Figure <ref> denotes the constant-energy transfer graphs of YFeO_3 and LaFeO_3 for T = 300 K. One can see that the magnetic signals at low energy are separated as two peaks as the energy transfer increases, implying the V-shaped dispersion of the magnetic excitation of YFeO_3 and LaFeO_3.In order to fully explain the low-energy magnetic excitations, we employ a minimal spin Hamiltonian of RFeO_3 to model the experimental data:H=J_c∑_alongcS_i·S_j +J_ab∑_ab planeS_i·S_j +J'∑_⟨⟨ ij ⟩⟩S_i·S_j +∑_⟨ ij ⟩D_ij·S_i×S_j +K_a∑_i(S_i^x)^2+K_c∑_i(S_i^z)^2, where J_c and J_ab represent the nearest-neighbor exchange constants along the c-axis and the ab plane, respectively. In the previous INS study on YFeO_3 <cit.>, these J_c and J_ab were set as same value J_1. However, we note that the difference between J_c and J_ab can reach up to 10 % due to Bloch's rule <cit.>, especially in the case of YFeO_3. J' denotes the exchange constant along the next-nearest neighbor bonds (see figure <ref>(a)). The fourth term represents the DM interactions defined on the Fe(i)-O-Fe(j) bonds with the antisymmetric relation: (D_ij=-D_ji). Transition ions having a 3d^5 configuration such as Fe^3+ lead to A_1g orbital symmetry. Therefore, we may assume that the DM interaction of ferrites can be given by a microscopically derived form (D_ij∝x̂_i ×x̂_j) <cit.>, where x̂_i is the unit vector connecting i-th Fe atom and oxygen atom between i-th and j-th Fe atoms. This means that in the Pbnm structure all DM interactions between two adjacent iron atoms may be characterized by five parameters: α_ab, β_ab, γ_ab, α_c, β_c <cit.>, as shown in figure <ref>(b). The density functional theory (DFT) calculation on LaFeO_3 <cit.> shows good agreement with the DM vectors obtained from our structural analysis, supporting this assumption. Normalized values of the local DM vectors of YFeO_3 and LaFeO_3 are shown in table <ref>. We note that the in-plane DM vectors defined in different basal planes, e.g. D_41 and D_32, are different along the b-axis. The result of combining all contributions of adjacent ions is that every Fe^3+ ion feels a different DM interaction, therefore global DM interactions cannot be as expected defined in this space group. This is an assumption contrary to those used in previous studies on YFeO_3 <cit.>. The last two terms of equation <ref> denote the easy-axis (K_a, K_c 0) SIA terms to stabilize the G-type antiferromagnetic order along the a-axis and the wFM along the c-axis, respectively. With respect to the spin wave theory, SIA is the origin of the spin wave gap at the Brillouin zone center. It is worth noting that the most generalized form of the spin Hamiltonian also includes the symmetric anisotropic exchange interaction, i.e. two-ion anisotropy (TIA). Such TIA terms are formulated as the form ∑_ijS_iΩ_ijS_j, where Ω_ij denotes 3 × 3 symmetric matrix. This is characterized by eight different parameters related to its Pbnm symmetry. This anisotropy, however, seems to be small with the order of D^2/J <cit.>, and would add unnecessarily too many parameters to our model Hamiltonian. The TIA mostly affects the spin wave gap at the zone center, like the SIA. In that sense, this TIA can be neglected and therefore will not be discussed further in this study.After combining all contributions from the oxygen environments, the four-sublattice magnetic ground state Γ_4(G_a, A_b, F_c) of the RFeO_3 can be stabilized <cit.>. In spherical coordinates, the four spins can be defined using two spin canting angles θ and ϕ, which are related to the weak ferro- and antiferro- magnetic moment, respectively. S_1=S( -cosθ cosϕ, -cosθ sinϕ, sinθ) S_2=S( cosθ cosϕ, cosθ sinϕ, sinθ) S_3=S( -cosθ cosϕ, cosθ sinϕ, sinθ)S_4=S( cosθ cosϕ, -cosθ sinϕ, sinθ) Since the spin cantings are very small (∼0.5^∘) for RFeO_3, we can ignore terms higher than second order with respect to the spin-orbit coupling λ_SO to obtain the relationship between spin canting angles and the spin Hamiltonian parameters from the ground state energy <cit.>:θ = 2β_ab+β_c/4J_ab+2J_c+K_c-K_a, ϕ = -2γ_ab/4J_ab-8J'-K_a Using this Hamiltonian of RFeO_3, we tried to find the best fit parameters that reproduce the experimental result well. First, an initial set of parameters was chosen under several constraining conditions. As the Hamiltonian contains many parameters: J_c, J_ab, J', D_ab, D_c, K_a and K_c, utilizing all the reasonable initial and constraining conditions is important for determining a reliable set of best fit parameters. Therefore, starting with the previously reported exchange coupling constants J_1 = 4.77 and J_2 = 0.21 meV derived from high energy INS experiment <cit.>, J_c, J_ab and J' were refined. Since only the J_1 value of LaFeO_3 has been previously reported <cit.>, we made the assumption that the J_1/J_2 ratio of LaFeO_3 is similar with that of YFeO_3. This assumption combined with the ratio of T_N of both compounds, yields J_c = J_ab = 5.47 and J' = 0.24 meV for LaFeO_3. We also used in our analysis the canting angle θ as derived from polarized neutron diffraction results <cit.> and magnetization measurements along the c-axis <cit.>. To obtain the consistency between the spin canting angles and the spin Hamiltonian parameters, equation <ref> was used as one of the constraint conditions.Secondly, with the chosen initial parameters fitting was performed by a bounded non-linear least squares fit to the experimental data set. Due to the presence of the constraint condition (equation <ref>), fmincon programming solver implemented in MATLAB was used. During the non-linear fit, the theoretical magnon dispersion curve and dynamic structure factor S(Q, ω) have been calculated. We note that the derivation of the analytic form of the dispersion is not easy as the size of Hamiltonian matrix is 8 × 8. We used SpinW software package <cit.> to diagonalize the spin Hamiltonian in the Holstein-Primakoff approximation. Since the neutron intensity obtained from the triple axis spectrometer is convoluted with the instrumental 4D resolution ellipsoid in the momentum-energy space, the theoretically derived dynamic structure factor should also be convoluted with the resolution ellipsoid for direct comparison with experimental data. The total INS intensity measured by the triple axis spectrometer is given by <cit.>:I(Q_0,ω_0)≈ R_0∫ d^3Qdω S(Q,ω)× exp[-1/2Δ℘^iM_ij(Q_0, ω_0)Δ℘^j],where Q_0=k_i-k_f represents the momentum transfer to the sample, ħω = E_i-E_f is the energy transfer, Δ℘≡(Q-Q_0,ħ(ω-ω_0)), and M is a 4 × 4 matrix defining a 4-dimensional resolution ellipsoid. Based on the geometry of the SIKA beamline and information of the sample, M matrices were calculated via a Cooper-Nathans method in the Reslib library <cit.>. Uniformly sampled 41 × 41 × 41 q-points within the ellipsoid were used for a convolution function in the Reslib library. Finally, the convoluted intensity I(Q,ω) was compared with the experimentally obtained Im[χ(Q,ω)] until we get satisfactory convergence of the paramteter. Throughout the above process, the set of parameters that best explain the data was determined. In figure <ref>, <ref>(d) and <ref>(e), the overall V-shapes of the spin-dispersions are modelled accurately by calculations for both compounds. The splitting of magnon branches at the zone center are not as noticeable in the INS data (figure <ref>(a) and <ref>(b)). But nevertheless it is fully consistent with theoretical dispersion curves. The constant-Q cuts in figure <ref>(b)(d) show this consistency more clearly, especially given the tendency for the convoluted I(Q,ω) to have slightly higher energies due to instrumental resolutions than the calculated energies of the two low-lying magnon branches. For example, the two measured peak positions at Q = (1 0 1) for YFeO_3 are at ∼1.7 and 2.4 meV, whereas the theoretically calculated magnon energies are at ∼1.2 and 2.42 meV. We note that the calculated energies of magnon branches at the magnetic zone center are consistent with Raman data (∼1.4 and 2.2 meV) <cit.>.The best fit parameters are given in table <ref> together with values for T_N and the spin canting angles. In our work, the values obtained for the DM interactions for YFeO_3 are quite different compared to those of Hahn et. al. <cit.>. We point out two possibilities for this discrepancy: * The spin Hamiltonian used in <cit.> doesn't include DM interaction along the c-axis. Since the magnitude of D_ab and D_c is similar in RFeO_3, they should be considered together. * The canting angles θ and ϕ of YFeO_3 used in <cit.> are much less than the known values (∼0.5^∘). Underestimation of the DM vectors is therefore inevitable since they are proportional to the canting angles (equation <ref>).The ratio between DM interaction and exchange interaction, D/J, is a criterion that indicates the competition between them. A rough estimate for the spin canting angle is given by tan^-1(D/J), and so one can find an approximate value for D/J from equation <ref>. For LaFeO_3 the value we obtain for D/J is ∼0.026, which is larger than the values obtained from DFT calculations (∼0.018 in reference <cit.>, 0.021 in reference <cit.>). It is also noteworthy that the canting angles of YFeO_3 and LaFeO_3 are remarkably similar, which is quite unexpected because they have significantly different values for their respective FeO_6 octahedra rotation angles. In case of YFeO_3, the ratio between canting angles θ/ϕ∼ 1.137 is consistent with previous theoretical and experimental results <cit.>. Having said that, the low-energy magnetic excitations of YFeO_3 and LaFeO_3 have several common features with that of BiFeO_3 (see figure <ref>(c)(f) and figure <ref>) such as the shoulder-like signal seen below the modes dispersing from the zone center. This feature has been shown to be the result of competition between the three different terms in the Hamiltonian: exchange interaction, DM interaction and SIA. Of course, there is room for this feature to manifest itself in several ways depending on the details. In Pbnm, centrosymmetricity and local DM vector constrain RFeO_3 to have the commensurate 4-sublattice magnetic structure, resulting in the simple V-shape dispersion curves with two of four magnon branches as shown in figure <ref>. In contrast, all local DM interactions in R3c can be effectively expressed as a global DM interaction along two directions, [1 1 0] and [0 0 1]. Thus, a spin cycloid structure can be stabilized. Furthermore, SDW fluctuations and anharmonicity add more complexity to the structure, making the magnon branches to become more complex. All of these effects combined lead to the distinct behavior of Im[χ(Q,ω)] above 4 meV for RFeO_3 and BiFeO_3. Since the magnon branches showing up in the INS susceptibility of RFeO_3 are nearly doubly degenerate: the degeneracy being broken by the DM interaction, there are only two peaks shown in the energy scan. In contrast, as BiFeO_3 has many branches of magnons, the scattering intensity remains high above 4 meV at the zone center. INS measurements on BiFeO_3 with substantially improved momentum resolution allowed for the observation of the individual magnon branches, as shown in the upper part of figure <ref>(c). The two peaks at E = 3 meV agree with theoretically calculated magnon dispersion, verifying the 4-fold nature of the magnon dispersions. In YFeO_3, LaFeO_3 and BiFeO_3, some care is necessary in choosing the proper relative strengths of the DM interaction and SIA in order to model the spin wave spectra correctly. The SIA in BiFeO_3 is not only affected by the DM interaction, but it is also influented by various properties such as ferroelectric distortion and A-site lone-pair effect. This complicated nature of the SIA in BiFeO_3 has been explained by new DFT-based calculations (see reference <cit.>). The mixing of such parameters yields a temperature dependence of several properties of BiFeO_3, e.g. static properties such as the cycloid periodicity and FE distortion as well as the dynamical properties such as the spin wave spectrum. Therefore, the spin Hamiltonian parameters of BiFeO_3 are also expected to vary as a function of temperature, which has indeed been observed <cit.>. However, both YFeO_3 and LaFeO_3 do not show any clear temperature dependence of spin wave spectrum (based on our results collected at T = 300 K and 1.5 K). This implies that the aforementioned static and dynamic properties, and therefore the spin-Hamiltonian parameters of YFeO_3 and LaFeO_3 remain largely unchanged between 1.5 and 300 K. § CONCLUSIONThe low-energy magnon spectra of YFeO_3, LaFeO_3 and BiFeO_3 were studied by our INS experiments. Several features of the magnetic excitation spectra have been explained by the full spin Hamiltonian, which includes the DM interaction and SIA. Best fit parameters of spin Hamiltonian were obtained for YFeO_3 and LaFeO_3. With the careful quantitative examination of the magnon behavior in these three compounds, we have shown how the relationships between the DM interaction, J, and SIA serves as the underlying mechanism driving the spin dynamics. Our study provides a guide for future work on other perovskite systems, in particular with regard to the delicate balance among DM, J and SIA. The values of the magnon mode splitting in most of the other RFeO_3 compounds is currently available in the literature <cit.>. Exploiting the relations between these parameters will play a key role in any future implementation of technological applications which utilize Fe^3+-based perovskites.We thank Joosung Oh, Y. Noda and D. T. Adroja for fruitful discussions. This work was supported by the Institute of Basic Science in Korea (Grant No. IBS-R009-G1). The work of M.K. and V.S. was supported within the program of Large Infrastructures for Research, Experimental Development and Innovation (project No. LM2015050) and project LTT17019 financed by the Ministry of Education, Youth and Sports, Czech Republic.§ REFERENCES iopart-num | http://arxiv.org/abs/1705.09441v3 | {
"authors": [
"Kisoo Park",
"Hasung Sim",
"Jonathan C. Leiner",
"Yoshiyuki Yoshida",
"Jaehong Jeong",
"Shin-ichiro Yano",
"Jason Gardner",
"Philippe Bourges",
"Milan Klicpera",
"Vladimír Sechovský",
"Martin Boehm",
"Je-Geun Park"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170526061324",
"title": "Low-energy spin dynamics of orthoferrites AFeO$_3$ (A = Y, La, Bi)"
} |
^1 Department of Physics and Institude of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, China^2 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China^3 School of Physical Sciences, University of Chinese Academy of Sciences, Beingjing 100049, ChinaGiven the recent progress in dark matter direction detection experiments, we examine a light bino-higgsino dark matter (DM) scenario (M_1<100 GeV and μ<300 GeV) in natural supersymmetry with the electroweak fine tuning measure Δ_EW<30. By imposing various constraints, we note that: (i) For sign(μ/M_1)=+1, the parameter space allowed by the DM relic density and collider bounds can almost be excluded by the very recent spin-independent (SI) scattering cross section limits from the XENON1T (2017) experiment. (ii) For sign(μ/M_1)=-1, the SI limits can be evaded due to the cancelation effects in the hχ̃^0_1χ̃^0_1 coupling, while rather stringent constraints come from the PandaX-II (2016) spin-dependent (SD) scattering cross section limits, which can exclude the higgsino mass |μ| and the LSP mass m_χ̃^0_1 up to about 230 GeV and 37 GeV, respectively. Furthermore, the surviving parameter space will be fully covered by the projected XENON1T experiment or the future trilepton searches at the HL-LHC. Status and prospects of light bino-higgsino dark matter in natural SUSY Murat Abdughani^2,3, Lei Wu^1 [Email: [email protected]] , Jin Min Yang^2,3 December 30, 2023 =================================================================================§ INTRODUCTIONScrutinizing the mechanism for stabilizing the electroweak scale becomes more impending after the Higgs discovery at the LHC <cit.>. Besides, there is overwhelming evidence for the existence of dark matter from cosmological observations. Identifying the nature of dark matter is one of the challenges in particle physics and cosmology.The weak scale supersymmetry is widely regarded as one of the most appealing new physics models at the TeV scale. It can successfully solve the naturalness problem in the Standard Model (SM) and also provide a compelling cold dark matter candidate. Among various supersymmetric models, the natural supersymmetry is a well motivated framework (see examples <cit.>), which usually indicates the light higgsinos in the spectrum <cit.>. If unification of gaugino mass parameters is further assumed, the current LHC bound on the gluino (m_g̃≳ 2 TeV <cit.>) would imply correspondingly heavy winos and binos, resulting in a higgsino-like lightest supersymmetric particle (LSP). However, the thermal abundance of light higgsino-like LSP is typically lower than the observed value of the dark matter in the universe, due to the large higgsino-higgsino annihilation rate. These considerations motivate us to explore the phenomenology of neutralino dark matter in natural SUSY by giving up the gaugino mass unification assumption. One of the possibilities is to allow for the light bino in natural SUSY. Such a mixed bino-higgsino neutralino dark matter can solve the above mentioned problems of a pure higgsino LSP without worsening the naturalness in natural SUSY. The studies of bino-higgsino dark matter have also been carried out in<cit.>.In this work, we will confront the light bino-higgsino dark matter scenario in natural SUSY with the recent direct detection data. In particular, we focus on the light dark matter regime (m_χ̃^0_1<100 GeV) and attempt to address the lower limit of the mass of LSP that saturates the dark matter relic abundance. In natural SUSY, a small μ parameter leads to a certain bino-higgsino mixing, so that the spin-independent/dependent neutralino LSP-nucleon scattering cross sections can be enhanced. We will utilize the recent XENON1T <cit.> and PandaX-II <cit.> limits to examine our parameter space. Since the couplings of the LSP with the SM particles depend on the relative sign (sign(μ/M_1)) between the mass parameters μ and M_1, we will include both of sign(μ/M_1)=± 1 in our study and show its impact on the exclusion limits for our scenario. Besides, we explore the potential to probe such a scenario by searching for the trilepton events at 14 TeV LHC.The structure of this paper is organized as follows. In Section <ref>, we will discuss the light bino-higgsino neutralino parameter space in natural SUSY. In Section <ref>, we will perform the parameter scan and discuss our numerical results. Finally, we draw our conclusions in Section <ref>.§ LIGHT BINO-HIGGISINO NEUTRALINO IN NATURAL SUSYIn the MSSM, the minimization of the tree-level Higgs potential leads to the following equation <cit.>M^2_Z/2 = m^2_H_d-m^2_H_utan^2β/tan^2β-1-μ^2,where m^2_H_u,d denote the soft SUSY breaking masses of the Higgs fields at the weak scale, respectively. It should be noted that the radiative EWSB condition usually imposes a non-trivial relation between the relevant soft mass parameters at the high scale in a UV model, such as mSUGRA. However, the scenario we studied in our work is the so-called low energy phenomenological MSSM, in which a successful EWSB is always assumed and in this case the above mentioned strong correlation between parameters from radiative EWSB condition in UV models is not applicable. Using the electroweak fine tuning measure Δ_EW <cit.>, one can see that the higgsino mass parameter μ should be of the order of ≲ 300 GeV to satisfy the requirement of Δ_EW < 30 <cit.>. The light higgsinos have been searched for through chargino pair production in the LEP-2 experiment <cit.>, which indicates μ≳ 100 GeV. We will use this LEP-2 limit as a lower bound for the higgsino mass. However, the relic abundance of thermally produced pure higgsino LSP falls well below dark matter measurements, unless its mass is in the TeV range. In order to provide the required relic density, several alternative ways have been proposed, such as the multi-component dark matter that introducing the axion <cit.>. On the other hand, without fully saturating the relic density (under-abundance), the higgsino-like neutralino dark matter in radiatively-driven natural supersymmetry with Δ_EW < 30 <cit.> or natural mini-landscape <cit.> has been confronted with various (in-)direct detections and is also expected to be accessible via Xenon1T experiment. In our study, we achieve the correct dark matter relic density by allowing the light bino to mix with the higgsinos. The two neutral higgsinos (H̃_u^0 and H̃_d^0) and the two neutral guaginos (B̃ and W̃^0) are combined to form four mass eigenstates called neutralinos. In the gauge-eigenstate basis (B̃, W̃^0, H̃_d, H̃_u), the neutralino mass matrix takes the form:M_χ̃^0 = ( M_1 0-c_β s_W m_Z s_β s_W m_Z 0 M_2c_β c_W m_Z-s_β s_W m_Z -c_β s_W m_Zc_β c_W m_Z0 -μs_β s_W m_Z-s_β s_W m_Z-μ 0 )where s_β = sinβ, c_β = cosβ, s_W = sinθ_W, c_W = cosθ_W, M_1 and M_2 are the soft-breaking mass parameters for bino and wino, respectively. M_χ̃^0 can be diagonalized by a 4× 4 unitary matrix N. In the limit of M_1<μ≪ M_2, the lightest neutralino is bino-like (with some higgsino mixture), while the second and third neutralinos are higgsino-like. The LSP can interact with nuclei via exchange of squarks and Higgs bosons (spin-independent) and via exchange of Z boson and squarks (spin-dependent). Given the strong LHC bounds on the squarks and non-SM Higgs bosons, one can neglect their contributions to the scattering cross section. Then, the couplings of the LSP with the Higgs boson can be written byC_hχ̃^0_1χ̃^0_1 ≈-√(2)g_1 N_11^2 M_Z s_W/μ M_1/μ + sin2β/1-(M_1/μ)^2.where N_11 denotes the bino component of the lightest neutralino mass eigenstate. It can be seen that the SI scattering cross section depends on the relative sign of M_1 and μ. When sign(M_1/μ)<0, the coupling C_hχ̃^0_1χ̃^0_1 can be suppressed and even vanish if M_1/μ =-sin2β so that the strong LUX SI limits can be escaped. For the SD scattering cross section, it should be noted that the coupling Zχ̃^0_1χ̃^0_1 can appear via the higgsino component in the LSP. The pure bino/wino LSP will not have interactions with Z boson, while the pure higgsino LSP can only have the non-zero coupling Zχ̃^0_1χ̃^0_2. Another blind spot in SD scattering can happen in the limit of tanβ=1, where the left-right parity is restored and the parity-violating Z coupling will vanish <cit.>. However, a low value of tanβ is disfavored by the observed Higgs mass in the MSSM.§ PARAMETER SCAN AND NUMERICAL RESULTSIn our numerical calculations, we vary the relevant parameters in the ranges of100 GeV≤ |μ| ≤ 300 GeV,30 GeV≤ |M_1| ≤ 100 GeV,10 ≤ tanβ≤ 50.We scan the values of M_1 up to 100 GeV since we are interested in light DM region and attempt to address the lower limit of the LSP mass. For higher upper values of μ and M_1, a heavy mixed higgsino-bino LSP may also produce the right DM relic abundance <cit.>, while the result for lower bound of LSP mass obtained in the following calculation will not change. The stop and gluino can contribute to the naturalness at loop level, which are expected to be m_t̃_1≲ 2.5 TeV and m_g̃≲ 3-4 TeV for Δ_EW <30 <cit.>. By recasting the LHC Run-2 with ∼ 15 fb ^-1 of data, it is found that the lower bounds of stop mass and gluino mass are about 800 GeV <cit.> and 1.5 TeV<cit.> in natural SUSY, respectively. Given the irrelevance of the third generation parameters for our neutralino dark matter, we fix the third generation squark soft masses as M_Q̃_3L=3 TeV, M_t̃_3R=M_b̃_3R=1 TeV and vary the stop trilinear parameters in the range |A_t|<2 TeV for simplicity. The physical stop mass m_t̃_1 has to be less than 2.5 TeV to satisfy Δ_EW <30. We also require that each sample can guarantee the correct Higgs mass and the vacuum stability <cit.>. The first two generation squark and all slepton soft masses are assumed to be 3 TeV. Other trilinear parameters are fixed as A_f=0. We also decouple the wino and gluino by setting M_2,3 = 2 TeV. We impose the following constraints in our scan:(1) The light CP-even Higgs boson masses of our samples should be within the range of 122–128 GeV. The package<cit.> is used to calculate the Higgs mass. (2) The samples have to be consistent with the Higgs data from LEP, Tevatron and LHC. We use the package<cit.> and<cit.> to implement the constraints. (3) The relic density of neutralino dark matter Ω_χ̃h^2 is computed by<cit.>. Including 10% theoretical uncertainty, we require our samples to satisfy the observed value 0.1186± 0.0020 <cit.> within 2σ range. (4) If m_χ̃^0_1<m_h/2, the SM Higgs boson can decay to χ̃^0_1χ̃^0_1 invisibly. We require the branching ratio Br(h →χ̃^0_1χ̃^0_1)<24%, which has been recently given by CMS collaboration at 95% C.L. <cit.>. (5) The invisible width of the Z boson is required less than 0.5 MeV to satisfy the LEP limit. (6) The LEP searches for χ̃^0_1χ̃^0_2,3 associated production gives an upper limit, σ(e^+e^- →χ̃^0_1χ̃^0_2,3× Br(χ̃^0_2,3→χ̃^0_1Z^*)<100 fb.In Fig. <ref>, we show the samples satisfying the dark matter relic density for sign(μ)=±1. Since a bino-like LSPhas rather small couplings with the SM particles, a certain portion of higgsino components is required to meet the observed relic density. Otherwise, the universe will be overclosed. Therefore, except for the two resonance regions m_χ̃^0_1≃ m_Z/2 and m_h/2, the higgsino mass parameter μ is expected to be as low as possible in our scan ranges. It should be noted that the difference of sign(μ/M_1)=± 1 in calculating the relic abundance mainly happens around and after the Higgs resonance region, in which more samples are allowed for sign(μ/M_1)=-1. This is because that the negative sign of μ/M_1 can reduce the coupling of the LSP with the Higgs boson and the suppress the enhanced annihilation cross section of χ̃^0_1χ̃^0_1 by the Higgs resonant effect. When m_χ̃^0_1 > m_h/2, the LSP for sign(μ/M_1)=± 1 is still bino-like so that the relic density easily exceeds the observed value. But if M_1 is close to μ, the LSP for sign(μ/M_1)=-1 can have sizable higgsino components, which allows samples in the lower right corner on the left panel of Fig. <ref>. However, such a region will be excluded by the dark matter direct detections as shown in the following. In Fig. <ref>, we present the spin-independent/dependent neutralino LSP-nucleon scattering cross sections, which are calculated by using<cit.>. All samples satisfying the constraints (1-6). The neutron and proton form factors are taken as f^p_d ≈ 0.132 and f^n_d ≈ 0.140. It can be seen that the very recent SI cross section limits from XENON1T experiment can almost exclude the whole parameter space of sign(μ/M_1)=+1. While for sign(μ/M_1)=-1, a large portion of our samples can escape the SI limits since the hχ̃^0_1χ̃^0_1 coupling is suppressed by the cancelation effect in Eq. (<ref>).On the other hand, the SD cross section is largely determined by Z-boson exchange and is sensitive to the higgsino asymmetry, σ_SD∝ |N^2_13-N^2_14|^2. The relic density constraint requires a large higgsino asymmetry so that the SD cross section is enhanced. Therefore, a strong bound on such a scenario comes from the PandaX-II (2016) SD neutralino LSP-neutron scattering cross section limits, which can rule out about 70% of our samples and exclude the higgsino mass |μ| and the LSP mass m_χ̃^0_1 up to about 230 GeV and 37 GeV, respectively. Such lower limits will not changed even if we extend the scan ranges of M_1 and μ to larger values. The current SD neutralino LSP-proton limits from PandaX and PICO are still weak. Both of sign(μ)=± 1 scenarios can be completely covered by the projected XENON1T experiment in the future. Besides the direct detections, the neutralino annihilation in the Sun to neutrinos can also be enhanced by the higgsino component in the LSP. The null results from the neutrino telescopes, such as IceCube, have produced a strong bound on the SD neutralino LSP-proton scattering cross sections and has excluded a sizable portion of the parameter space for sign(μ)=-1. Next, we discuss the LHC potential of probing the current parameter space of our scenario allowed by the constraints (1-6) and the above direct/indirect detections.In Fig. <ref>, we plot the decay branching ratios of χ̃_2^0 and χ̃_3^0. For sign(μ)=-1, we can see that the neutralinos χ̃^0_2,3 mainly decay to χ̃^0_1 Z. When Br(χ̃^0_2 →χ̃^0_1 Z) increases, Br(χ̃^0_3 →χ̃^0_1 Z) decreases because of the goldstone theorem <cit.>. A similar correlation can be seen in the decay channel χ̃^0_2,3→χ̃^0_1 h. But for sign(μ)=+1, the neutralino χ̃^0_2 still dominantly decay to χ̃^0_1 Z, while the neutralino χ̃^0_3 preferently decay to χ̃^0_1 h. This indicates that the samples with negative sign of μ/M_1 will produce more trilepton events through the process pp →χ̃^0_2,3(→ Zχ̃^0_1)χ̃^±_1(→ W^±χ̃^0_1) than those with positive sign of μ/M_1, and can be more easily excluded by the null results of searching for electroweakinos at the LHC.Given the above decay modes, we first recast the LHC searches for the electroweakinos listed in Table <ref> with<cit.>. We generate the parton level signal events by<cit.> and perform the shower and hadronization procedure by<cit.>. The fast detector simulation are carried out with the tuned<cit.>. We implement the jet clustering by<cit.> with the anti-k_t algorithm <cit.>. We use<cit.> to calculate the QCD corrected cross sections of the electroweakino pair productions at the LHC. Then, we estimate the exclusion limit by evaluating the ratio r = max(N_S,i/S^95%_obs,i), where N_S,i is the event number of signal for i-th signal region and S^95%_obs,i is the corresponding 95% C.L. observed upper limit. A sample is excluded at 95% C.L. if r > 1. After checking all surviving samples, we find that the LHC data in Tab. <ref> can not further exclude the parameter space because of the strong direct detection bound on higgsino mass parameter μ>230 GeV. In Fig. <ref>, we show the prospect of testing our surviving samples through searching for electroweakino pair production in the trilepton final states at 14 TeV LHC with the luminosity L=3000 fb^-1. Such an analysis <cit.> has been implemented inpackage. In order to reduce the Monte Carlo fluctuations, we generate 200,000 events for each signal point. In Fig. <ref>, we can see that all red triangles allowed by the constraints (1)-(6) and the XENON1T (2017) and PandaX (2016) experiments can be excluded by the HL-LHC at 95% C.L.. Therefore, we conclude that our light bino-higgsino neutralino dark matter scenario will be fully tested by either future XENON1T or HL-LHC experiments.§ CONCLUSIONIn this work, we examined light bino-higgsino neutralino dark matter in natural SUSY by imposing various constraints from the LEP, dark matter and LHC experiments. We found that the relative sign between the mass parameters μ and M_1 can significantly affect the dark matter and LHC phenomenology of our scenario. For sign(μ/M_1)=+1, the very recent SI limits from the Xenon1T (2017) experiment can almost exclude the whole parameter space allowed by the relic density and collider bounds. But for sign(μ/M_1)=-1, the SI limits can be avoided due to the cancelation effects in hχ̃^0_1χ̃^0_1 coupling. 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"authors": [
"Murat Abdughani",
"Lei Wu",
"Jin Min Yang"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170525132359",
"title": "Status and prospects of light bino-higgsino dark matter in natural SUSY"
} |
http://arxiv.org/abs/1705.09806v1 | {
"authors": [
"Satyajit Chowdhury",
"Subir Biswas",
"Nikhil Chakrabarti",
"Rabindranath Pal"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170527111833",
"title": "Experimental Observation of Electron-Acoustic Wave Propagation in Laboratory Plasma"
} |
|
Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations Hadi Ravanbakhsh and Sriram Sankaranarayanan Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80302Email: [email protected] 6 January 2017; accepted 30 May 2017 ============================================================================================================================================================================================ We present a technique for learning control Lyapunov (potential) functions, which are used in turn tosynthesize controllers for nonlinear dynamical systems.The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.§ INTRODUCTION We propose a novel learning from demonstration scheme for inferring control Lyapunov functions (potential functions) for stabilizing nonlinear dynamical systems to reference states/ trajectories.Control Lyapunov functions (CLFs) have wide applications to motion planning problems in robotics. They extend the classic notion of Lyapunov functions to systems involving control inputs <cit.>. Finding a CLF also leads us to an associated feedback control law that can be used to solve the stabilization problem. Additionally, they can be extended for feedback motion planning using extensions to time-varying or sequential CLFs <cit.>. Likewise, they have been investigated in the robotics community in many forms including artificial potential functions to solve path planning problems involving obstacles <cit.>. However, synthesizing CLFs for nonlinear systems remains a challenge <cit.>. Standard approaches to finding CLFs include the use of dynamic programming, wherein the value function satisfies the conditions of aCLF <cit.>, or using non-convex bilinear matrix inequalities (BMI) <cit.>. BMIs can be solved using alternating minimization methods <cit.>. However, these approaches often get stuck in local minima and exhibit poor convergence guarantees <cit.>. In this article, we investigate the problem of learning a CLF using a black-box demonstrator that can be queried with a given system state, and responds by demonstrating control inputs to stabilize the system starting from that state.However, our framework uses just the control input at the query state.Such a demonstrator can be realized using an expensive nonlinear model predictive controller (MPC) that uses a local optimization scheme, or even a human operator under certain assumptions [We do not handle noisy or erroneous demonstrations in this paper].The framework has a Learner which selects a candidate CLF and a Verifier that tests whether this CLF is valid. If the CLF is invalid, the Verifier returns a state at which the current candidate fails. The Learner queries the demonstrator to obtain a control input corresponding to this state. It subsequently eliminates the current candidate along with a set of related functions from further consideration. The framework continues to exhaust the space of candidate CLFs until no CLFs remain or a valid CLF is found in this process.We prove the process can converge in finitely many steps provided the Learner chooses the candidate function appropriately at each step. We also provide efficient SDP-based approximations to the verification problem that can be used to drive the framework. Finally, we test this approach on a variety of examples, by solving stabilization problems for nonlinear dynamical systems. We show that our approach can successfully find simple CLFs using finite horizon nonlinear MPC schemes with appropriately chosen cost functions to serve as demonstrators. In these instances, the CLFs yield control laws that are computationally inexpensive, andguaranteed against the original dynamical model. §.§ Illustrative Example: Tora SystemFigure <ref>(a) shows a mechanical system consisting of a cart attached to a wall using a spring. The position of the cart x is controlled by an arm with a weight that can be moved back and forth by applying a force u, as shown. The goal is to stabilize the cart to x=0, with its velocity, angle, and angular velocity ẋ = θ=θ̇ = 0. We refer the reader to the Jankovic et al. <cit.> for a derivation of the dynamics shown below in terms of state variables (x_1,…,x_4) andcontrol input u_1, after a suitable change of basis transformation:ẋ_̇1̇ = x_2, ẋ_̇2̇ = -x_1 + ϵsin(x_3), ẋ_̇3̇ = x_4, ẋ_̇4̇ = u_1 .We approximate sin(x_3) using a degree 3 approximation which is quite accurate over the range x_3 ∈ [-2,2]. The equilibrium x = ẋ = θ = θ̇ = 0 now corresponds to x_1 = x_2 = x_3 = x_4 = 0.The state space is taken to be S: [-1,1] × [-1,1] × [-2,2] × [-1,1], the control inputu_1 ∈ [-1.5, 1.5].MPC Scheme: A first approach to doing so uses a nonlinear model-predictive control (MPC) scheme using the time horizon = 30, time step τ = 1 and a simple cost function∑_t=[0,τ,...,](||(t)||_2^2 + ||(t)||_2^2) +||()||_2^2.Given the model in (<ref>), such a control is implemented using a first order numerical gradient descent method to minimize the cost function over a finite time horizon.The convergence was informally confirmed by observing hundreds of such simulations from different initial conditions for the system. However, the MPC scheme is expensive, requiring repeated solutions to (constrained) nonlinear optimization problems in real-time. Furthermore, the closed loop lacks formal guarantees despite the high confidence gained from numerous simulations. Learning a Control Lyapunov Function: The approach in this paper uses the MPC scheme as a “demonstrator”, and attempts to learn a simpler control law through a control Lyapunov function. The key idea depicted in Fig. <ref> is to pose queries to the MPC at finitely many witness states W = {_1, …, _j } and use the corresponding instantaneous control inputs _1, …, _j, respectively.The learner attempts to find a candidate function V() that is positive definite and which decreases at each witness state _j through the control input _j.This function is fed to the verifier, which checks whether V() is indeed a CLF, or discover a state _j+1 at which the condition fails. This new state is added to the witness set and the process is iterated.The procedure described in this paper synthesizes the control Lyapunovfunction after 60 iterations of the learning loop and synthesizes the CLF V() below: ([ 1.22 x_2^2+ 0.31 x_2x_3 + 0.44 x_3^2- 0.28 x_4x_2 + 0.8 x_4x_3; + 1.69 x_4^2+ 0.069 x_1x_2- 0.68 x_1x_3 - 1.85 x_4 x_1+ 1.60 x_1^2 ]) This function yields a simple associated control law that can be implemented and guarantees the stabilization of the model (<ref>). Figure <ref>(b) shows a closed loop trajectory of this control law vs control law extracted by MPC. The advantage of this law is that its calculation is much simpler, and furthermore, the control is formally guaranteed, at least for the model of the system. Contributions: In this paper, we instantiate the learning scheme sketched above, and show that under suitable assumptions terminates in finitely many iterations to either yield a control Lyapunov function V() that is guaranteed to be valid, or show that Lyapunov function of a specific form does not exist. We demonstrate this scheme and its scalability on several interesting vehicle dynamics taken from the literature to solve stabilization to state and trajectory stabilization problems.§ BACKGROUND In this section, we briefly describe control Lyapunov(potential) functions. A state feedback control system Ψ(X, U, , ) consists of a plantand a controllerover state space X ⊆^n and input space U ⊆^m.The planthas a state ∈ X and input ∈ U. The vector field for the plant is defined by a smooth function f: X × U →^n. Throughout the paper, we consider control affine systems that are possibly nonlinear in , but affine in , of the form = f(, ) = f_0() + ∑_i=1^m f_i() u_i,where f_i: X →^n. The controllermeasures the state of the plant (∈ X) and provides feedback ∈ U. The controller is defined by a continuous feedback function :X → U.< g r a p h i c s >For a given feedback function , the execution trace of the system Ψ is defined as (.) : ^+ → X, which maps time to state. Formally, given (0) = _0, (.) is defined according to (t) = f_0((t)) + ∑_i=1^m f_i((t))(((t)))_i, where (.) is the right derivative of (.). In this article, we study stabilization of nonlinear systems using Lyapunov functions (or potential function) inside a compact set S. More precisely, we consider a compact and connected set S ⊂ X. Without loss of generality, the originis the state we seek to stabilize to. Furthermore, ∈ int(S). We restrict the set S to be a basic semi-algebraic set defined by a conjunction of polynomial inequalities. Likewise, the control inputs U are restricted to a polytope. Our approach relies on control Lyapunov functions. A control Lyapunov function (CLF) <cit.> V, is a continuous function that respects the following conditions:[ V() = 0; (∀∈ S ∖{0}) V() > 0; (∀∈ S ∖{0})(∃∈ U)∇ V · f(, ) < 0.; ]The last condition ensures that the value of V can be decreased everywhere by choosing a proper feedback u. LetV^β = { | V() β},∈{ =, ≤, <, ≥, > } .Let β^* be maximum β s.t. V^≤β⊆ S. Once a CLF is obtained, it guarantees that the system initialized toany state belonging to V^< β^*, can be stabilized to the origin (Fig. <ref>).A control Lyapunov function guarantees that there is a control strategy, which stabilizes the system. Given a control affine system Ψ, where U : ^m and a polynomial control Lyapunov function V satisfying Eq. (<ref>), there is a feedback functionfor which if _0 ∈ V^< β^*, then: * (∀ t ≥ 0)(t) ∈ S * (∀ϵ > 0)(∃ T ≥ 0)(T) -< ϵ . 1) Using results from Artstein <cit.>, there exists a feedback function ^* s.t. while ∈ S, then dV/dt = ∇ V · f(, ) < 0. Assuming (0) = _0 ∈ V^<β^*⊂ S, then initially V((0)) < β^*. Now, assume the state reaches ∂ S at time t_2. By continuity of V, there is a time 0 < t_1 ≤ t_2 s.t. V((t_1)) = β^* and (∀ t ∈ [0, t_1])(t) ∈ S. Therefore V((t_1)) = (V((0)) + ∫_0^t_1dV/dt dt) < V((0)). This means V((t_1)) < β^*, which is a contradiction. Therefore, the state never reaches ∂ S and remains in int(S) forever. 2) V would be a Lyapunov function for the closed loop system when the control unit is replaced with the feedback function ^* and using standard results in Lyapunov theory (∀ϵ > 0)(∃ T ≥ 0)||(T) - 0|| < ϵ.Given, a control Lyapunov function, it is possible to then obtain a feedback law in a closed form that stabilizes the system. Sontag <cit.> provides a method for extracting a continuous feedback functionfor control affine systems from a control Lyapunov function. This can be extended to systems with constraints on the control inputs <cit.>.Also, feedback synthesis for periodic time/event triggered switching is possible <cit.>.We have thus far considered the problem of stabilizing to a fixed equilibrium state. However, given this primitive, we can extend the CLF approach to related problems of (a) Reach-while-stay: reaching a given target set of states T starting from an initial set I, while staying inside a safe setS using Lyapunov-barrier functions <cit.>;(b) Trajectory stabilization: stabilizing to a trajectory (t) rather than to a fixed state using time-varying Lyapunov functions; or similarly, (c) Feedback motion planning whichaddresses the robustness of a plan using funnels <cit.> ; (d) Obstacles: problems that involve reaching while avoiding an obstacle region in thestate-space using artificial potential functions <cit.>. We will focus our exposition on the basic formulation for stability (Eq. (<ref>)) while demonstrating extensions to some of other applications mentioned above through numerical examples.§ ALGORITHMIC LEARNING FRAMEWORK Finding CLFs is known to be a hard problem, requiring the solution to BMIs <cit.> or hard polynomialconstraints<cit.>.A standard approach to discovering such functions is to choose a set of basis functions g_1, …, g_r and search of a function of the formV_() = ∑_j=1^r c_j g_j(), where ∈^r is vector of unknowns. One possible choice of basis functions involves monomials g_j(): ^α_j wherein |α_j|_1 ≤ D_L for some degree bound D_L for the learning concept (CLF). Then, the problem is reduced to findings.t. V_ satisfies Eq. (<ref>).We now present the algorithmic learning framework. Let us fix a control affine systemover a state-space X, control inputs U given by (<ref>). Let ^* = be the equilibrium we wish to stabilize the system to, while remaining inside S ⊂ X. Next, we assume a demonstrator as a function : S ↦ U that given a state ∈ S, provides us an appropriate feedback () ∈ U for the state , such thatis presumed to be a valid function that stabilizes the system.Our definition of a demonstrator is general enough to allow offline MPC, sample based methods <cit.>, human operator demonstrations <cit.>, or even demonstrations that rely on opaque models such as neural networks. Also, we assume that the demonstrator is presumed correct. However, the approach can work even if the demonstrator may fail on some input states. Finally, a faulty demonstrator may, at the worst, lead our technique to fail without finding a CLF. In particular, such a demonstrator will not cause our technique to synthesize an incorrect CLF.The CLF learning problem has the following inputs: * A dynamical systemin the form (<ref>),* A safe set S, * A “black-box” demonstrator function : S ↦ U that presumably stabilizes the system, and * A candidate space for CLFs of the form ∑_j=1^r c_j g_j() given by basis functions (): g_1(),…, g_r() and a compact set C ∋ (c_1, …, c_r). We represent the coefficients (c_1, …, c_r) collectively as . The output can be Success: a function V_c(): ^t ·() that is a CLF; or Failure: no function could be discovered by our procedure. §.§ Algorithmic Learning Framework The algorithmic learning framework is shown in Fig. <ref>, and implements two modules (a) Learner and (b) Verifier that interact with each other and the demonstrator.The framework works iteratively until termination. At the j^th iteration, the learner maintains a (witness) setW_j: { (_1, _1) ,…, (_j, _j) }⊆ S × U. W_j is a finite set of pairs of states _i and correspondingdemonstrated feedback _i. Corresponding to W_j, C_j ⊆ C is defined as set of candidatecoefficientsfor functions V_(): ^t () with∈ C_j.Formally, C_j is a set of all candidatess.t. V_ satisfies the CLFcondition (<ref>) for every point in the finite set W_j:C_j : {∈ C | ⋀_(_i, _i) ∈ W_j[V_(_i) > 0; ∇ V_c · f(_i, _i) < 0 ] .} . The flowchart for the overall procedure is shown in Fig. <ref>.To begin with, W_0 : ∅ and C_0: C.Each iteration works as follows: * The learner samples a value _j ∈ C_j and outputs the corresponding function V_j(): _j^t ·(). If C_j = ∅ then no sample is found and the algorithm fails. * The verifier checks if V_j is a CLF by checking the conditions in (<ref>). If V_j satisfies the conditions, then the algorithm stops to declare success. Otherwise the verifier selects a(counterexample) state _j+1∈ S for which the CLF condition fails. Assume without loss of generality that _j+1≠.* Failing verification, the demonstrator is called to choose a suitable control _j+1 corresponding to _j+1.* The new set W_j+1 := W_j ∪{ (_j+1, _j+1) }. Furthermore, C_j+1: C_j ∩{ | [V_(_j+1) > 0; ∇ V_· f(_j+1, _j+1) < 0 ]} . We now prove some core properties that guarantee the correctness of the proposed scheme. We assume that the learner and the verifier are implemented without any approximations/relaxations (as will be subsequently presented).The algorithmic learning framework as described above has the following property: * _j ∉C_j+1. I.e, the candidate found at the j^th step is eliminated from further consideration.* If the algorithm succeeds at iteration j, then the output function V_j() is a valid CLF for stabilization.* The algorithm declares failure at iteration j if and only if no linear combination of the basis functions is a CLF compatible with the demonstrator. 1) Suppose that _j ∈ C_j+1. Then, _j satisfies the following conditions (Eq. (<ref>)): V_j(_j+1) > 0∇ V_j· f(_j+1, _j+1) < 0. However, the verifier guarantees that _j is a counterexample for Eq. (<ref>)). I.e. V_j(_j+1) ≤ 0∇ V_j· f(_j+1, _j+1) ≥ 0, which is a contradiction. Therefore, _j ∉C_j+1. 2) The algorithm declares success if the verifier could not find a counterexample. In other words, V_j satisfies conditions of Eq. (<ref>) and therefore a CLF. 3) A CLF V is compatible with a Demonstratoriff (∀∈ S ∖{})∇ V· f(, ()) < 0. The algorithm declares failure if C_j = ∅ and Eq. (<ref>) is equivalent to C_j: C^1_j∩ C^2_j where C^1_j : {∈ C_0 | _(_i, _i)∈ W_j∇ V_· f(_i, _i) < 0 } C^2_j : {∈ C_0 | _(_i, _i)∈ W_j V_(_i) > 0 } Suppose there exists a ∈ C_0 (a linear combination of the basis function) that yields a CLF V_. If V_ is compatible with , then (∀ (_i, _i = (_i)) ∈ W_j)∇ V_· f(_i, _i) < 0. Therefore, ∈ C^1_j. But we know that ∉C_j = ∅. As a result ∉C^2_j. I.e. (∃ (_i, _i) ∈ W_j)V_(_i) ≤ 0, and it is concluded that V_ is not a CLF, which is a contradiction.Proofs are available in <cit.>. Inverse results <cit.> suggest polynomial basis for Lyapunov functions are expressive enough for verification of exponentially stable, smooth nonlinear systems. This, justifies using polynomial basis for CLF. Assuming (i) the demonstrator functionis smooth, (ii) the closed loop system with controlleris exponentially stable, then there exists a polynomial CLF, compatible with . Under assumption (i) and (ii), one can show that a polynomial Lyapunov function V (not control Lyapunov function) exists for the closed loop system Ψ(X, U,, ) <cit.>. I.e. V() = 0 (∀∈ S ∖{})V() > 0 (∀∈ S ∖{}) ∇ V· f(, ()) < 0 V is also a CLF since it satisfies Eq. (<ref>). Moreover, V is compatible with(Eq. (<ref>)).Lemma <ref> guarantees success of the learning procedure if the set of basis functions is rich enough. We now present implementations of each of the modules involved, starting with the learner.§.§ Learner: Finding a CandidateThe findCandidate function simply samples a point from the setC_j defined in Eq. (<ref>).Note that V_(_j) : ^t ·(_j) is linear inand therefore ∇ V_.f(_j, _j) is linear inas well. The initial space of all candidates C is assumed to be a hyper-rectangular open box.At each iteration, the candidate _j ∈ C_j is chosen.Suppose the algorithm does not terminate at this iteration.According to Eq. (<ref>), the new set C_j+1 is obtained as C_j+1: C_j ∩ H_j, wherein H_j is defined by two linear inequalities H_j1 H_j2 (H_j1: _j1^t< b_j1, H_j2: _j2^t< b_j2). Let C_j represent the topological closure of the set C_j.For each j ≥ 0, C_j is a convex polyhedron. We prove by induction. Initially C_0 is an open box and as a result C_0 is a convex polyhedron. Now, assume C_j is a polyhedron. Recall that C_j+1 is defined as C_j+1: C_j ∩{ | [ ∑_i=1^r (c_ig_i(_j+1)) > 0; ∑_i=1^r (c_i∇ g_i · f(_j+1, _j+1)) < 0 ]} . Notice that f and g_i are fixed (∀ i) and _j+1 and _j+1 are given. Therefore, C_j+1 is intersection of an open polyhedron (C_j) and two half-space which yields an other polyhedron. And since the inequalities in equation above are strict, C_j+1 is an open polyhedron.Thus the problem of implementingis that of checking emptiness of a polyhedron with some strict inequality constraints. This is readily solved using a slight modification of standard linear programming algorithms using infinitesimals for strict inequalities.There exists a halfspaceH_jk: _jk^t _j ≥ b_jk that passes through _j, and C_j+1⊆ C_j ∩ H_jk. By definition _j ∈ C_j. By Theorem <ref>, it is guaranteed that _j ∉C_j+1. Since C_j+1 : C_j ∩{ | a_j1^t _j < b_j1 a_j2^t _j < b_j2}, we can conclude _j ∉{ | a_j1^t _j < b_j1 a_j2^t _j < b_j2}, I.e.a_j1^t _j ≥ b_j1 a_j2^t _j ≥ b_j2. Thus one of the two disjuncts holds for k ∈1,2. Further, we note thatC_j+1⊆ C_j∩{ | a_jk^t _j < b_jk}.Choosing a Candidate _j: The choice of _j ∈ C_j governs the overall convergence of the algorithm. Figure <ref> demonstrates the importance of this choice by showing the candidate _j, the hyperplane H_jk and the new region C_j+1.We wish to choose _j s.t. Vol(C_j+1) ≤αVol(C_j), for a fixed α, independent of H_jk.Let _j be chosen as the center of the maximum volume ellipsoid (MVE) of C_j. Then,Vol(C_j+1) ≤(1-1/r) Vol(C_j). This leads us to ascheme that guarantees termination of the overall iterative scheme in finitely many steps under a robustness assumption. A candidate ∈ C is δ-robust (δ > 0), ifffor each ∈_δ(), V_:^t ·() is a CLF, where _δ() is a ball ofradius δ around . Let Vol(C) < γΔ^r for γ > 0 the volume of the unit r-ball, and Δ > 0. Furthermore, the procedure terminates whenever Vol(C_j) < γδ^r following the robustness assumption above. If at each step _j is chosen as the center of the maximum volume ellipsoid in C_j, the learning loop terminates in at mostr (log(Δ) - log(δ))/- log(1 - 1/r)= O(r^2) . Initially, Vol(C_0) < γΔ^r. Then by Theorem <ref> Vol(C_j) ≤ (1 - 1/r)^jVol(C_0) < (1 - 1/r)^j γΔ^r log(Vol(C_j)) - log(γΔ^r) < jlog(1-1/r) Suppose the algorithm does not terminate after r(log(Δ)-log(δ))/-log(1-1/r) iterations. Then log(Vol(C_j)) - log(γΔ^r) < r(log(Δ)-log(δ))/-log(1-1/r) log(1-1/r) and log(Vol(C_j)) - log(γΔ^r) < r(log(δ) - log(Δ)) log(Vol(C_j)) - log(γΔ^r) < log(γδ^r) - log(γΔ^r) log(Vol(C_j)) < log(γδ^r) And it is concluded that Vol(C_j) < γδ^r, which is the termination condition. This is a contradiction and therefore, the algorithm terminates in at most r(log(Δ)-log(δ))/-log(1-1/r) iterations. And asymptotically -log(1 - 1/r) is Ω(1/r) (can be shown using Taylor expansion as r →∞) and therefore, the maximum number of iterations would be O(r^2).The maximum volume ellipsoid itself can be computed by solving a convex optimization problem<cit.>. §.§ Implementing the Verifier The verifier given a candidate V_j(): _j^t ·() checks the CLF conditions in Eq. (<ref>), split into two separate checks:(A) Check if V_j() is a positive definite polynomial over ∈ S. Assuming that () =0 for the basis, this reduces to:(∃ ∈ S ∖{}) V_j() ≤ 0. (B) Check if the Lie derivative of V_j can be made negative for each ∈ S by a choice ∈ U:(∃∈ S∖{} )(∀∈ U) (∇ V_j) · f(, ) ≥ 0.This problem seems harderdue to the presence of a quantifier alternation. Let U be a polyhedral set defined by U: {∈^m |A ≥}. Recall that f(, ) is control affine function f_0() + ∑_i=1^m f_i() u_i.Eq. (<ref>) holds for some ∈ S iff[(∃ ∈ S∖{},) ≥, ^t ≥ -∇ V_j.f_0();A_i^t =∇ V_j.f_i() (i ∈{1 … m}). ] Considering a fixed V and , then (∀∈ U)∇ V · f(, ) = ∇ V · f_0() + ∑_i=1^m∇ V · f_i() u_i ≥ 0, which is equivalent to: (∄) A ≥∇ V · f_0() + ∑_i=1^m∇ V · f_i() u_i < 0. This yields a set of linear inequalities (w.r.t. ). Using Farkas lemma, it is concluded (∃≥ 0)A_i^t= ∇ V · f_i() (i ∈{1...m}), ^t ≥ -∇ V · f_0().In other words, assuming that the dynamics and chosen bases are polynomials, the verification problem reduces to checking if a given semi-algebraic set defined by polynomial inequalities has a solution.Failing the polynomial assumption, the problem of verification is in general undecidable. However, it can be approximated by techniques such as δ-decision procedures proposed by Gao et al <cit.>. Solvers like dReal can thus be directly used for the verification problem. While dReal does a good job in adaptive space decomposition, in our experience, they do not scale reliably. Nevertheless,these solvers allow us to conveniently implement a verifier for small but hard problems involving rational and trigonometric functions.The verification problem for polynomial systems and polynomial CLFs through a polynomial basis function () is NP-hard, in general. Exact approaches using semi-algebraic geometry and the branch-and-bound solvers (including the dReal approach cited above) can tackle this problem precisely. However, scalability is still an issue.We sketch a relaxation using SDP solvers <cit.>. Let us fix a basis of monomial terms of degree up to D_V, ^t: [1 x_1… ^D_V], wherein D_V is chosen as at least half of the maximum degree inamong all monomials in g_j() and ∇ g_j · f(): D_V ≥1/2max( ⋃_j( {(g_j) }∪{(∇ g_j · f ) }) ). Let Z(): ^t. Thus, each polynomial of degree up to 2D_V may now be written as a trace inner product p() =P, Z() = 𝗍𝗋𝖺𝖼𝖾( P Z() ), wherein the matrix P is constant. Let S be the semi-algebraic set defined as S: {∈^n | r_1() ≤ 0, …, r_k() ≤ 0 } ,for polynomials r_1, …, r_k. The constraint in (<ref>) is equivalent to solving the following optimization problem over [𝗆𝖺𝗑_I,Z() ; 𝗌.𝗍. R_i, Z()≤ 0, i ∈{ 1,…, k };𝒱_j, Z()≤ 0,;]where V_j() (r_i()) is written as 𝒱_j, Z() (R_i, Z()). and those in (<ref>) are written as[𝗆𝖺𝗑_, I,Z() ; 𝗌.𝗍. R_i, Z()≤ 0, i ∈{ 1,…, k };F_ji, Z() = A_i^t, i ∈{1,…, m};-F_j0, Z()≤^t,≥ 0,;]wherein the components ∇ V_j · f_i()defining the Lie derivatives of V_j are now written in terms of Z() as F_ji,Z().The SDP relaxation is used to solve these problems and provide an upper bound of the solution <cit.>. The result of the relaxation treats Z() as a matrix variable Z that will satisfy Z ≽ 0.Notice that each optimization problem is feasible simply by setting Z andto be zero. However, if the optimal solution of both problems is Z = 0 in the SDP relaxation, then we will conclude that the given candidate is a CLF. If the relaxed optimization problems inEqs. (<ref>) and (<ref>) yield a zero solution,then the given candidate V_j() is in fact a CLF. Suppose that V_j is not a CLF. Then it does satisfies either Eq. (<ref>) or Eq. (<ref>). I.e. (∃^* ≠) s.t. V_j(^*) ≤ 0 or (∃ ≥)A_i^t =∇ V_j.f_i(^*) (i ∈{1 … m}) ,^t ≥ - ∇ V_j.f_0(^*).Then Z(^*) is a non-zero solution for Eq. (<ref>) or Eq. (<ref>). Z(^*) can also yield a non-zero relaxed matrix Z^* which satisfies the relaxed optimization problem.On the other hand, For any non-zero Z ≽ 0, I,Z is positive. Therefore, since the optimization returns a zero as the solution, it means the relaxed optimization does not have a non-zero solution (since I,Z is being maximized). Which is a contradiction as it is shown non-zero solution exists when V_j is not a CLF. Therefore, V_j is a CLF.However, the converse is not true. It is possible for Z ≻ 0 to be optimal for either relaxed condition, but no∈^n corresponds to the solution. This happens because the relaxation drops two key constraints to convexify the conditions: (1) Z has to be a rank one matrix written as Z: ^t and (2) there is a ∈^n such thatis the matrix of monomials corresponding to . To deal with this, we adapt our learning framework towork with witnesses W_j: { (Z_i, _i) }_i=1^j replacing states _i by matrices Z_i.* Each basis function g_j() inis now written instead as G_j, Z. The candidates are therefore, ∑_j=1^rc_jG_j, Z.Likewise, we write the components of its Liederivative ∇ g_j · f_i in terms of Z. * The learner maintains the set W as { (Z_j, _j) }, wherein Z_j is the feasible solution returned by the SDP solver while solving Eqs. (<ref>) and (<ref>). In other words, the CLF conditions are, in fact, taken to be the relaxed conditions. * We use a suitable projection operator π mapping each Z to a state : π(Z), such that the demonstrator receives . In practice, since the vector of monomials used to define Z fromincludes the terms 1, x_1, …, x_n, the projection operator simply selects a few entries from Z corresponding to the variables. Other more sophisticated projections are also possible.The relaxed framework thus lifts counterexamples to work over matrices Z_j. However, the candidate space begins with C and is refined each step as before. I.e, the relaxed framework continues to satisfy Lemmas <ref>, <ref>, Theorem <ref> and Theorem <ref> with the definition of (control) Lyapunov function changed to relaxed conditions. § EXPERIMENTSIn this section, we describe numerical results on some example benchmark systems. The algorithmic framework is implemented using quadratic template forms for the CLFs with the tool Globoptipoly used toimplement the verifier <cit.>. The demonstrator is implemented using a nonlinear MPC implemented using a gradient descent algorithm. For each benchmark, we tuned the time horizon, discretization stepand the cost function until the control objectives were satisfied by the MPC over hundreds of simulations starting from randomly selected initial states. Most of the benchmarks consider a reach-while-stay problem, wherein the goal is to reach target set T, starting initial set I, while remaining in the safe set S.We also illustrate an example involving a trajectory stabilization problem. All the computations are performed on a Mac Book Pro with 2.9 GHz Intel Core i7 processor and 16GB of RAM. The reported CLFs are rounded to 2 decimal points.The summary of results is provided in Table. <ref>.Bicycle Model: This system is two-wheeled mobile robot modeled with five states [x, y, v, θ, γ] and two control inputs <cit.>.The goal is to stabilize the carto a target velocity v^*=5, as shown in Fig. <ref>. We drop the variable x (since it is immaterial to our stabilization problem) and obtain a model with four state variables:[ [ẏ; v̇; θ̇; σ̇ ]] = [ [ vsin(θ); u_1;vσ; u_2 ]], [U: [-10, 10]×[-10, 10]; S: [-2, 2]×[3, 7]×[-1, 1]^2; I: _0.4();T: _0.1(), ]where σ = tan(γ) (see Fig. <ref>).sin function is approximated with a polynomial of degree 1. The method finds the following CLF:V =+ 0.42 y^2 + 0.59 yθ + 2.57 θ^2 + 0.79 yσ + 4.64 σθ+ 4.06 σ^2 - 0.38 vy + 1.46 vθ + 1.18 vσ + 2.39 v^2 .Inverted Pendulum on a Cart: This example has applications in balancing two-wheeled robots <cit.> (cf. <cit.> for list of such robots). The system has four state variables [x, ẋ, θ, θ̇] and one input u. The dynamics, after partial linearization, have the following form <cit.>: [ [ ; θ̈ ]] = [ [ 4u + 4(M+m)g tan(θ) - 3mgsin(θ)cos(θ)/4(M+m)-3mcos^2(θ);- 3 u cos(θ)/l ]],where m = 0.21, M=0.815, g=9.8 and l=0.305. The trigonometric and rational functions are approximated with polynomials of degree 3. The sets are S: [-1, 1]^4, U:[-20, 20], I:_0.1(), T: _0.1(). We obtain the following CLF:V =+ 11.64 θ̇^2 + 45.93 θ̇θ + 85.47 θ^2 + 12.15 xθ̇+ 36.57 xθ+ 6.44 x^2 + 15.07 θ̇ẋ + 33.06 ẋθ + 8.98 ẋx + 6.09 x^2. Forward Flight for Caltech Ducted Fan: The Caltech ducted fan models the aerodynamics of a single wing of a thrust vectored, fixed wing aircraft <cit.>.This problem is to design forward flight control in which the angle of attack needs to be changed. The model of the system is carefully calibrated through wind tunnel experiments. The system has 4 states [v, γ, θ, q] and two control inputs: u and δ_u. The dynamics are:[ [ m v̇; m v γ̇; θ̇; J q̇ ]] = [ [ -D(v, α) - W sin(γ) + u cos(α + δ_u);L(v, α) - W cos(γ) + u sin(α + δ_u);q; M(v, α) - u l_T sin(δ_u) ]],where α = θ - γ, and D, L, and M are polynomials in vand α. For full list of parameters, see <cit.>. The goal is to reach ^*: [6, 0, 0.1771, 0] as the steady state. We perform a translation so that the ^* is the origin in the new coordinate system.U:[0, 13.5] × [-0.45, 0.45]S:[3, 9]×[-0.75, 0.75]×[-0.75, 0.75]×[-2, 2]] I: {[v, γ, θ, q] |(4v)^2 + (10γ)^2 + (10θ)^2 + (10q)^2 < 4^2 } T: {[v, γ, θ, q] |(4v)^2 + (10γ)^2 + (10θ)^2 + (10q)^2 < 0.5^2 } . First, we approximate v^-1, sin and cos with polynomials of degree 1, 3 and 3 respectively, to get polynomial dynamics. However, the system is not affine in control. We replace u and δ_u input variables with u_1 = u sin(δ_u) and u_2 = u cos(δ_u) and U is under-approximated with a polytope. These changes yield a polynomial control affine dynamics, which fits the description of our model. The procedure finds the following CLF:V = + 3.21 q^2 + 2.18 qθ + 3.90 θ^2 + 0.40 qv - 0.15 vθ + 0.56 v^2 + 1.78 qγ - 1.42 γθ - 0.11 vγ + 3.90 γ^2.Hover Mode for Caltech Ducted Fan: This problem is again taken from <cit.>. The goal is to keep the ducted fan in a hover mode. The systemhas 6 variables x, y, θ, ẋ, ẏ, θ̇ and two control inputs u_1, u_2. The dynamics are[ [m ẍ;m ÿ; J θ̈ ]] = [ [-d_cẋ + u_1 cos(θ) - u_2 sin(θ); -d_cẏ + u_2 cos(θ) + u_1 sin(θ) - mg;r u_1 ]],where m = 11.2, g = 0.28, J = 0.0462, r = 0.156 and d_c = 0.1. The sets are:S: [-1,1]×[-1,1]×[-0.7,0.7]×[-1, 1]^3 U: [-10, 10]×[0, 10], I:_0.25(), T:_0.05(). The sin and cos are approximated with polynomials of degree 2 and the procedure finds a quadratic CLF:V =1.63 θ̇^2 - 1.09 θ̇ẏ + 15.00 ẏ^2 - 0.02 θ̇y + 1.54 yẏ+ 1.25 y^2 + 1.74 θθ̇ + 0.79 ẏθ + 0.08 yθ + 4.86 θ^2 - 4.93 θ̇ẋ+ 0.57 ẋẏ + 0.05 yẋ - 8.26 ẋθ + 12.58 ẋ^2 - 0.44 θ̇x - 0.38 ẏx - 0.20 yx- 4.27 xθ + 3.86 xẋ + 2.14 x^2.Unicycle: Now, we consider a simpler system, namely the unicycle model <cit.>, which is known not to havea polynomial CLF. However, for trajectory tracking problem, one can provide a time varyingCLF <cit.> for a finite time horizon control (using funnels).The unicycle model has the dynamics: ẋ_̇1̇ = u_1, ẋ_̇2̇ = u_2,ẋ_̇3̇ = x_1 u_2 - x_2 u_1.We consider a planning problem: starting near [π/2, -1, -1], the goal is to reach near [0, 2, 0], and by near we mean within distance 1. In the first step, a feasible trajectory (t) is generated as shown in Fig. <ref>.Then (t) is approximated with piecewise polynomials. More precisely, trajectory consistsof two segments. The first segment brings the car to the origin and the second segment moves the car to the destination. Each segment is approximated using polynomials in t with degree up to 3.The time varying CLF V is a function of boththe stateand time t. Choosing a template for V, our method is applied to this problem and we are able to find a strategy to implement the plan with guarantees. A level-set of the funnel (CLF) is shown in Fig. <ref>. § RELATED WORKSynthesis of Lyapunov Functions from Data: The problem of synthesizing Lyapunov functions for a control system by observing the states of the system in simulation has been investigated in the past by Topcu et al to learn Lyapunov functions along with the resulting basin of attraction <cit.>. Whereas the original problem is bilinear, the use of simulation data makes it easier to postulate states that belong to the region of attraction and therefore find Lyapunov functions that belong to this region by solving LMIs in each case. The application of this idea to larger black-box systems is demonstrated by Kapinski et al <cit.>. Our approach focuses on controller synthesis through learning a control Lyapunov function in a bid to replace an existing controller. In doing so, we do not attempt to prove that the original demonstrator is necessarily correct but find a control Lyapunov function by assuming that the demonstrator is able to stabilize the system for the initial conditions we query on. Another important contribution lies in our analysis of the convergence of the learning with a bound on the maximum number of queries needed. In fact, these results can also be applied to the Lyapunov function synthesis approaches mentioned earlier. Counter-Example Guided Inductive Synthesis (CEGIS): Our approach of alternating between a learning module that proposes a candidate and a verification module that checks the proposed candidate is identical to the CEGIS framework originally proposed by Solar-Lezama et al. <cit.>. As such, the CEGIS approach does not include a demonstrator that can be queried. The extension of this approach Oracle-guided inductive synthesis <cit.>, generalizes CEGIS using an oracle that serves a similar role as a demonstrator in this paper. Jha et al.prove bounds on the number of queries for discrete concept classes using results on exact concept learning in discrete spaces <cit.>.The CEGIS procedure has been used for the synthesis of CLFs recently by authors <cit.>, combining it with SDP solvers for verifying CLFs and a robust version for switched systems. The key difference here lies in the use of the demonstrator module that simplifies the learning module. In the absence of a demonstrator module, the problem of finding a candidate reduces to solving linear constraints with disjunctions, an NP-hard problem <cit.>. Likewise, the convergence results are quite weak <cit.>. In the setting of this paper, however, the use of a MPC scheme as a demonstrator allows us to use faster LP solvers and provide convergence guarantees. Learning from Demonstration: The idea of learning from demonstration has had a long history in robotics <cit.>. The overall framework uses a demonstrator that can in fact be a human operator <cit.> or a complex MPC-based control law <cit.>.The approaches differ on the nature of the interaction between the learner and the demonstrator; as well as how the policy is inferred. Our approach stands out in many ways: (a) We represent our policies by CLFs which are polynomial. On one hand, these are much less powerful than approaches that use neural networks <cit.>, for instance. However, the advantage lies in our ability to solve verification problems to ensure that the resulting policy learned through the CLF is correct with respect to the underlying dynamical model. (b) Our framework is adversarial. The choice of the counterexample to query the demonstrator comes from a failed attempt to validate the current candidate. (c) Finally, we use simple yet powerful ideas from convex optimization to place bounds on the number of queries, paralleling someresults on concept learning in discrete spaces <cit.>.Control Lyapunov functions were originally introduced by Artstein and the construction of a feedback law given a CLF was first given by Sontag <cit.>. As such, the problem of learning CLFs is well known to be hard, involving bilinear matrix inequalities (BMIs) <cit.>. An equivalent approach involves solving bilinear problems simultaneously for a control law and a Lyapunov function certifying it <cit.>. BMIs are well known to be NP-hard, and hard to solve for a feasible solution <cit.>. The common approach is to perform an alternating minimization by fixing one set of bilinear variables while minimizing in the other. Such an approach has poor guarantees in practice, often “getting stuck” on a saddle point that does not allow the technique to make progress in finding a feasible solution. To combat this, Majumdar et al. (ibid) use LQR controllers and their associated Lyapunov functions for the linearization of the dynamics as good initial seed solutions <cit.>. In contrast, our approach simply assumes a demonstrator in the form of a MPC controller that can be used to resolve the bilinearity. Furthermore, our approach does not encounter the local saddle point problem.The use of the learning framework with a demonstrator distinguishes the approach in this paper from recently developed ideas based on formal synthesis <cit.>. These techniques focus on a given dynamical system and a specification of the correctness in temporal logic to solve the problem of controller design to ensure that the resulting trajectories of the closed loop satisfy the temporal specifications. Majority of the approaches are based on discretization of the state-space into cells to compute a discrete abstraction of the overall system <cit.>. A smaller set of approaches synthesize Lyapunov-like functions by solving nonlinear constraints either through branch-and-bound techniques or by sum-of-square (SOS) relaxation techniques <cit.>. In this paper, we use the Lyapunov function approach to synthesizing controllers. An alternative is to use occupation measures <cit.>. These methods formulate an infinite dimensional problem to maximize the region of attraction and obtain a corresponding control law. This is relaxed to a sequence of finite dimensional SDPs <cit.>. Note however that the approach computes an over approximation of the finite time backward reachable set from the target and a corresponding control. Our framework here instead seeks an under approximation that yields a guaranteed controller. § CONCLUSION AND FUTURE WORKWe have proposed an algorithmic learning framework for synthesizing CLFs using a demonstrator and demonstrated our approach on challenging numerical examples with 4-8 state variables.As future work, we are considering many directions including the extensions to noisy/erroneous demonstrations, using output feedback (rather than full state feedback) synthesis and allowing disturbances in our framework. We are also working on integrating our control framework with RRT-based path planning and implementing it on board robotic vehicles. § ACKNOWLEDGMENTSThis work was funded in part by NSF under award numbers SHF 1527075 and CPS 1646556. All opinions expressed are those of the authors and not necessarily of the NSF.plainnat | http://arxiv.org/abs/1705.09619v4 | {
"authors": [
"Hadi Ravanbakhsh",
"Sriram Sankaranarayanan"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170526152309",
"title": "Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations"
} |
APS/123-CTPCorresponding author at: Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China. E-mail: [email protected]^1 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China ^2 Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China ^3 Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China The rarefied effect of gas flow in microchannel is significant and cannot be well described by traditional hydrodynamic models. It has been know that discrete Boltzmann model (DBM) has the potential to investigate flows in a relatively wider range of Knudsen number because of its intrinsic kinetic nature inherited from Boltzmann equation.It is crucial to have a proper kinetic boundary condition for DBM to capture the velocity slip and the flow characteristics in the Knudsen layer.In this paper, we present a DBM combined with Maxwell-type boundary condition model for slip flow. The tangential momentum accommodation coefficient is introduced to implement a gas-surface interaction model. Both the velocity slip and the Knudsen layer under various Knudsen numbers and accommodation coefficients can be well described. Two kinds of slip flows, including Couette flow and Poiseuille flow, are simulated to verify the model. To dynamically compare results from different models, the relation between the definition of Knudsen number in hard sphere model and that in BGK model is clarified.PACS numbers51.10.+y, 47.11.-j, 47.45.-n. Key words Rarefied gas, Discrete Boltzmann method, Maxwell-type boundary, slip flow. Discrete Boltzmann method with Maxwell-type boundary condition for slip flow Zhihua Chen^2 December 30, 2023 ============================================================================§ INTRODUCTIONIn recent years, the development of natural science and engineering technology has moved towards miniaturization. One of the most typical examples is Micro-Electro- Mechanical System (MEMS)<cit.>. It is extremely important to investigate the underlying physics of unconventional phenomena at the micro-scale. Those unconventional phenomena cannot be explained by the traditional macro-model and has become a key bottleneck limiting the further development of MEMS. Among these unconventional physical problems, the gas flow and heat transfer characteristics at the mico-scale are especially critical.Due to the reduction of the geometric scale, the mean free path of gas molecules may be comparable to the length scale of the device. The Knudsen number (Kn), a dimensionless parameter used to measure the degree of rarefaction of the flow and defined as the ratio of the mean free path of molecules to characteristic length of the device,may be larger than 0.001 and reaches slip-flow regime(0.001 < Kn < 0.1 ) or even transition-flow regime (0.1 < Kn < 10). As we know, Navier-Stokes (NS) equations are applicable to continuum flow (Kn < 0.001) where the continuum hypothesis is acceptable. The slip boundary models based on kinetic theory should be adopted to describe the slip-flow regime. While in the transition-flow region, the NS equations totally fail to calculate viscous stress and heat flow accurately and higher order equations such as Burnett equation<cit.> and Grad's 13-moment equations<cit.> are needed.In practical, gas flow in a microchannel may encounter continuum, slip and transition regimes simultaneously. Traditionally macro-scale models cannot apply to such a broad range of Knudsen numbers by only one set of equations. Besides, the numerical solutions of higher order macro-equations are difficult to obtain because of the numerical stability problem <cit.>. It is commonly accepted that Direct Simulation Monte Carlo (DSMC)<cit.> is a accurate method for rarefied gas flow which has been verified by experimental data. However, the computation cost in numerical simulations is too expensive for low speed gas flow. To reduce the huge ratio of the noise to the useful information, extremely large sample size is needed. Although, the information preservation (IP) method was presented to treat this problem<cit.>, the contradiction between the noise problem and sample size has not been well solved.It has been know that rarefied gas dynamics are represented properly by the Boltzmann equation due to its kinetic nature. That is continuum, slip and transition regimes can be described by one equation. Unfortunately, the Boltzmann equation is a 6-dimensional problem and the computational cost is formidable to solve such an equation by numerical method directly. In order to alleviate the heavy computational burden of directly solving Boltzmann equation, a variety of Boltzmann equation-based methods, such as the unified gas-kinetic scheme (UGKS)<cit.>, the discrete velocity method (DVM)<cit.>, the discrete unified gas-kinetic scheme (DUGKS)<cit.>, the lattice Boltzmann method (LBM)<cit.>, have been presented and well developed. Recently, Discrete Boltzmann Method (DBM) has also been developed and widely used in various complex flow simulations<cit.>, such as multiphase flows<cit.>, flow instability<cit.>, combustion and detonation<cit.>, etc. From the viewpoint of numerical algorithm, similar to finite-different LBM, the velocity space is substituted by a limited number of particle velocities in DBM. However, the discrete distribution function in DBM satisfy more moment relations which make it fully compatible with the macroscopic hydrodynamic equations including energy equation. The macroscopic quantities, including density, momentum, and energy are calculated from the same set of discrete distribution functions. From the viewpoint of physical modeling, beyond the traditional macroscopic description, the DBM presents two sets of physical quantities so that the nonequilibrium behaviors can have a more complete and precise description. One set includes the dynamical comparisons of nonconserved kinetic moments of distribution function and those of its corresponding equilibrium distribution function. The other includes the viscous stress and heat flux. The former describes the specific nonequilibrium flow state, the latter describes the influence of current state on system evolution. The study on the former helps understanding the latter and the nonlinear constitutive relations<cit.>. The new observations brought by DBM have been used to understand the mechanisms for formation and effects of shock wave, phase transition, energy transformation and entropy increase in various complex flows<cit.>, to study the influence of compressibility on Rayleigh-Taylor instability<cit.>. In a recent study, it is interesting to find that the maximum value point of the thermodynamic nonequilibrium strength can be used to divide the two stages, spinnodal decomposition and domain growth, of liquid-vapor separation. Some of the new observations brought by DBM, for example, the nonequilibrium fine structures of shock waves, have been confirmed and supplemented by the results of molecular dynamics<cit.>. It should be pointed out that the molecular dynamics simulations can also gives microscopic view of points to the origin of the slip near boundary, such as the non-isotropic strong molecular evaporation flux from the liquid<cit.>, which might help to develop more physically reasonable mesoscopic models for slip-flow regime.In order to extend DBM to the micro-fluid, it is critical to develop a physically reasonable kinetic boundary condition. Many efforts have been made to devise mesoscopic boundary condition for LBM to capture the slip phenomenon<cit.>. However, the previous works are most suitable for two-dimensional (2D) models with a very small number of particle velocities and can not directly applicable to the DBM. On the other hand, those boundary conditions fail to capture flow characteristics in the Knudsen layer so the effective viscosity or effective relaxation time approach needs to be adopted<cit.>. Besides, the results of LBM and DBM should be verified by the results of continuous Botlzmann equation. In 2009, Watari<cit.> gave a general diffuse reflection boundary for his thermal LB model<cit.> and investigated the velocity slip and temperature jump in the slip-flow regime. Then, in his sequent work<cit.>, he compared the relationship between accuracy and number of particle velocities in velocity slip. Two types of 2D models, octagon family and D2Q9 model, are used. It was found that D2Q9 model fails to represent a relaxation process in the Knudsen layer and the accuracy of the octagon family is improved with the increase in the number of particle velocities. However, all the boundary conditions were set as diffuse reflection wall and the tangential momentum accommodation coefficient (TMAC) was not taken into account.Because of the dependence of the mean free path on microscopic details of molecular interaction, especially the collision frequency, the Knudsen number may have different values in various interaction models for the same macroscopic properties.In this paper, we first clarify the definitions of Knudsen number and the connection between the hard sphere model and BGK model for three-dimensional (3D) condition so that the results obtained from various models can be compared dynamically. Then a general Maxwell-type boundary condition for DBM is represented and accommodation coefficient is introduced to implement a gas-surface interaction model. Two kinds of gas flows, Couette flow and Poiseuille flow, in a microchannel are simulated. In the section of Couette flow, the relation between the analysis solutions based on hard sphere and BGK model are verified. The simulation results with various Knudsen numbers and accommodation coefficients are compare with analytical ones based on linear Boltzmann equation in the literature not only on the velocity slip but on the Knudsen profiles. While in the section of Poiseuille flow, the simulation results are compare with analytical solution based on Navier-Stokes equation and the second order slip boundary condition. § MODELS AND METHODS §.§ Definition of Knudsen number The Knudsen number is defined as the ratio of the free path of molecules (λ) to the characteristic length (L),Kn=λ/L.Throughout the paper, we consider the characteristic length L as unit, so the Knudsen number Kn is equal to the value of λ.For the hard sphere collision model, the molecules are considered as hard spheres with diameter d, the mean free path of molecules λ_HS can be calculated byλ_HS=1/√(2)nπ d^2,where n is the number density of molecules<cit.>. According to Chapman and Enskog<cit.>, the viscosity coefficient μ of hard sphere molecules can be expressed byμ= 5/16√(m k T/π)/d^2,where k is the Boltzmann constant, m is the molecular mass, and T is the temperature. It should be note that gas constant R can be expressed by R=k/m. Combining the state equation of ideal gas (p=ρ R T), we have the following relationship between λ_HS and macroscopic quantities:λ _HS = 4/5μ/p√(8RT/π). For the BGK model, the mean free path of molecules λ _BGK is defined asλ _BGK = τc,where τ is the reciprocal of collision frequency and called relaxation time,c is the average thermal speed <cit.>. The definition of Kn in DBM is in accordance with the definition here.According to the kinetic theory of gas molecules, c is expressed byc = √(8RT/π)in 3D case. From the Chapman-Enskog expansion, we know that τ has the following relation with macroscopic quantities:μ= τ p .Consequently, it hasλ _BGK = μ/p√(8RT/π).The comparison of Eq.(<ref>) and Eq.(<ref>) yields the relationship between viscosity-based mean free path λ_HS and λ_BGK,λ _HS = 4/5λ _BGK. §.§ Discrete Boltzmann ModelThe 3D discrete Boltzmann model taking into account the effect of the external force was presented based on the thermal model represented by Watari<cit.>. The evolution of the discrete distribution function f_ki for the velocity particle 𝐯_ki is given as∂f_ki/∂ t + v_kiα∂f_ki/∂r_α - a_α(v_kiα - u_α)/Tf_ki^eq =- 1/τ(f_ki - f_ki^eq),where the variable t is the time, r_α is the spatial coordinate and τ is the relaxation-time constant. a_α and u_α denote the macroscopic acceleration and velocity, respectively, in the r_α direction. T denotes the temperature. f_ki^eq is the local equilibrium distribution function. The subscript k indicates a group of the velocity particles whose speed is c_k and i indicates the direction of the particles. The subscript α indicates an x, y, or z component.To recover the NS equations, the local equilibrium distribution function should retain up to the fourth order terms of flow velocity. The discrete local equilibrium distribution f_ki^eq containing the fourth rank tensor is written asf_ki^eq = ρF_k[ (1 - u^2/2T + u^4/8T^2) + 1/T(1 - u^2/2T)v_kiξu_ξ.+ 1/2T^2(1 - u^2/2T)v_kiξv_kiηu_ξu_η 40pt+ 1/6T^3v_kiξv_kiηv_kiτu_ξu_ηu_τ 52pt .+1/24T^4v_kiξv_kiηv_kiτv_kiχu_ξu_ηu_τu_χ], 8pt The velocity particles 𝐯_ki consist of a rest particle and 32 moving particles. Each moving particle has four speeds and can be obtained from the unit vectors in Table <ref> multiplied by difference c_k. The speeds c_k of moving particle is determined according to the method presented by Watariin Ref<cit.>.The F_k in Eq.(<ref>) is the weighting coefficient for the particle velocity v_ki and is determined by c_k using the following equations:F_0 = 1 - 32(F_1 + F_2 + F_3 + F_4), F_1 = 1/c_1^2(c_1^2 - c_2^2)(c_1^2 - c_3^2)(c_1^2 - c_4^2) 50pt ×[ 945/32T^4- 105/32(c_2^2 + c_3^2 + c_4^2)T^3.36pt . + 15/32(c_2^2c_3^2 + c_3^2c_4^2 + c_4^2c_2^2)T^2 - 3/32c_2^2c_3^2c_4^2T ], F_2 = 1/c_2^2(c_2^2 - c_3^2)(c_2^2 - c_4^2)(c_2^2 - c_1^2) 50pt ×[ 945/32T^4- 105/32(c_3^2 + c_4^2 + c_1^2)T^3.36pt .+ 15/32(c_3^2c_4^2 + c_4^2c_1^2 + c_1^2c_3^2)T^2- 3/32c_3^2c_4^2c_1^2T ], F_3 = 1/c_3^2(c_3^2 - c_4^2)(c_3^2 - c_1^2)(c_3^2 - c_2^2) 50pt ×[ 945/32T^4 - 105/32(c_4^2 + c_1^2 + c_2^2)T^3.36pt .+ 15/32(c_4^2c_1^2 + c_1^2c_2^2 + c_2^2c_4^2)T^2- 3/32c_4^2c_1^2c_2^2T ], F_4 = 1/c_4^2(c_4^2 - c_1^2)(c_4^2 - c_2^2)(c_4^2 - c_3^2) 50pt ×[ 945/32T^4- 105/32(c_1^2 + c_2^2 + c_3^2)T^3.36pt .+ 15/32(c_1^2c_2^2 + c_2^2c_3^2 + c_3^2c_1^2)T^2- 3/32c_1^2c_2^2c_3^2T ] . §.§ Boundary condition models To solve the evolution equation (<ref>), finite-difference method is adopted. The spatial derivative is solved by the second-order upwind scheme and time derivative is solved by the first-order forward scheme. Then the evolution equation (<ref>) can be rewritten asf_ki^t + Δ t = f_ki^t - v_kiα∂f_ki/∂r_αΔ t - 1/τ(f_ki - f_ki^eq)Δ t + a_α(v_kiα - u_α)/Tf_ki^eqΔ t.40ptThe derivation at position I (see Fig.<ref>)is calculated by∂f_ki/∂r_α = {[3f_ki,I - 4f_ki,I - 1 + f_ki,I - 2/2Δr_α 14ptif 5ptv_kiα≥ 0,; 3f_ki,I - 4f_ki,I + 1 + f_ki,I + 2/ - 2Δr_α 14ptif 5ptv_kiα < 0 .; ].For v_kiα≥ 0, the evolution equation (<ref>) with Eq.(<ref>) is applied from I=3 up to the right wall. However, at the node I=2, the second-order upwind scheme in Eq.(<ref>) is not applicable. The first-order upwind scheme,∂f_ki/∂r_α = f_ki,2 - f_ki,1/Δr_α, is applied there. For v_kiα < 0, the evolution equation (<ref>) with Eq.(<ref>) is applied from the left wall to the node I=N-2. Likewise, at the node I=N-1, the first-order upwind scheme,∂f_ki/∂r_α = f_ki,N - 1 - f_ki,N/ - Δr_α, is applied.To solve the value of the distribution function on the left wall for v_kiα > 0and on the right wall for v_kiα < 0, boundary condition models are required.§.§.§ Diffuse reflection boundaryThe complete diffuse reflection model assumes that the molecules leaving the surface with a local equilibrium Maxwellian distribution irrespective of the shape of the distribution of the incident velocity. It can be expressed asf_ki,N = f_ki^eq(ρ _w^R,u_w^R,T_w^R),8ptv_kiα < 0, f_ki,1 = f_ki^eq(ρ _w^L,u_w^L,T_w^L),8ptv_kiα > 0. The equilibrium distribution functions, f_ki^eq(ρ _w^R,u_w^R,T_w^R) and f_ki^eq(ρ _w^L,u_w^L,T_w^L),are determined from the wall conditions including the velocities and the surface temperatures. Using the zero-mass flow normal to the wall<cit.>, the density ρ _w^R and ρ _w^L can be respectively calculated by the following two equations:∑_c_kiα > 0f_ki,Nc_kiα+ ρ _w^R∑_c_kiα < 0f_ki^eq(1.0,v_w^R,e_w^R)c_kiα= 0, ∑_c_kiα < 0f_ki,1c_kiα+ ρ _w^L∑_c_kiα > 0f_ki^eq(1.0,v_w^L,e_w^L)c_kiα= 0. As a result, the distribution function on the left wall (f_ki,1) for v_kiα > 0 and on the right wall (f_ki,N) for v_kiα < 0 are solved under the diffuse reflection boundary condition.§.§.§ Specular reflection boundaryThe specular reflection model assumes that the incident molecules reflect on the wall surface as the elastic spheres reflect on the entirely elastic surface. The component of the relative velocity normal to the surfaces reverses its direction while the components parallel to the surface remain unchanged. As an example, the direction normal to the wall surface parallels to the x axis, then the molecules leave the surface with a distribution asf_ki,N(v_kix,v_kiy,v_kiz) = f_ki,N( - v_kix,v_kiy,v_kiz), 4ptv_kix< 0, f_ki,1(v_kix,v_kiy,v_kiz) = f_ki,1( - v_kix,v_kiy,v_kiz), 4ptv_kix> 0. Since the distribution function f_ki,N( - v_kix,v_kiy,v_kiz) for v_kix < 0and f_ki,1( - v_kix,v_kiy,v_kiz) for v_kix > 0 can be solved by Eq.(<ref>) with Eq.(<ref>), the distribution function on the right wall (f_ki,N) for v_kiα < 0 and on the left wall (f_ki,1) for v_kiα > 0 are easy calculated from Eqs.(<ref>) and (<ref>).§.§.§ Maxwell-type boundaryIn practice, the real reflection of molecules on the body surfaces cannot be described properly by complete diffuse reflection or pure specular reflection. So the Maxwell-type reflection model which is composed of the two reflection modes is needed. The TMAC, α is introduced to measure the proportion of complete diffuse reflection<cit.>. The α portion of the incident molecules reflect diffusely and the other (1-α) portion reflect specularly. The value of TMAC is used to characterize the degree to which the reflected molecules has adjusted to the tangential momentum of the surface,α =τ_i-τ_r/τ_i-τ_w,where τ_i and τ_r are the tangential components of the momentum fluxes of the incident and reflected molecules, respectively. τ_w is the tangential momentum fluxes of the molecules in the wall. α=1 corresponds to the case of complete tangential momentum accommodation and the molecules reflect with the Maxwellian distribution under wall condtion, u_w and T_w. α=0 corresponds to the the case when the incident molecules are entirely not adjusted to the conditions of the surface, τ_r=τ_i.Under this boundary condition, the distribution function on the right wall, (f_ki,N), for v_kiα < 0 and on the left wall, (f_ki,1), for v_kiα > 0 are solved by the following equations, respectively,f_ki,N(v_kix,v_kiy,v_kiz) = αf^eq(ρ _w^R,u_w^R,T_w^R) 50pt+ (1 - α )f_ki,N( - v_kix,v_kiy,v_kiz),8ptv_kiα < 0 ,20pt f_ki,1(v_kix,v_kiy,v_kiz) = αf^eq(ρ _w^L,u_w^L,T_w^L)50pt+ (1 - α )f_ki,1( - v_kix,v_kiy,v_kiz),8ptv_kiα > 0 .20pt § SIMULATION RESTLTS §.§ Couette flowConsider a gas flow between two parallel walls, one at x=-Land the other at x=L. The two plates are kept at uniform temperature T_0 and move with velocity (0,-v,0) and velocity (0,v,0), respectively. Velocity slip becomes more significant with the decrease of the distance between the two plates or with the increase of the mean free path of the molecules, more exactly, with the increase of Knudsen number.The typical velocity profile between parallel plates in the slip-flow regime is depicted in Fig.<ref>. Only right half of the profile is shown because of its antisymmetry. The gas flow away from the wall can be described by NS equations, and the corresponding flow area is referred to as the NS flow area. The flow near the wall possesses pronounced non-equilibrium characteristics,and the corresponding flow layer is known as the Knudsen layer whose thickness is of the order of the mean free path. In Fig.<ref>, the linear portion A-B, whose gradient is dv/dx, corresponds to the NS flow area and the portion B-D corresponds to the Knudsen layer. The line B-C is extended from the line A-B and the point C is the cross point of the extended line with the right wall.The slip velocity v_slip is defined as the difference between the velocity of the right wall (the value the point W) and the velocity value of the point C. Considering the complete diffuse reflection, Sone gave the relation between v_slip and the mean free path λ in Ref.<cit.>.For the hard sphere model,v_slip = 1.2540√(π)/2λ_HSdv/dx.For the BGK model,v_slip = 1.0162√(π)/2λ_BGKdv/dx.The relationship between λ_HS and λ_BGK deduced in Sec.<ref> is verified by Eq.(<ref>) and Eq.(<ref>) since 1.2540 λ_HS≈ 1.0162 λ_BGK.Knudsen profile Δ v is defined as the difference between the curves, B-D and B-C.Sone<cit.> gave also the relation between Δ v and λ,Δ v = Y_0(η )√(π)/2λdv/dx,by introducing the so-called Knudsen layer function, Y_0(η), where η is a coordinate transformed from x through the following conversion:η= x - L/√(π)/2λ. The Knudsen layer function for hard sphere model, Y_0^HS(η), and for BGK model, Y_0^BGK(η), are both shown in Fig.<ref>. The correction of the function Y_0^HS(η) according to the relation in Eq.(<ref>) is also plotted. It can be seen that, the profile of the corrected function is in excellent agreement with the profile Y_0^BGK(η). As a consequence, the Eq.(<ref>) is revalidated. In addition, the results based on hard sphere model can be compared with those from BGK model under same macro conditions by using the relation of Eq.(<ref>). Considering the Maxwell-type boundary condition, Onishi<cit.> gave the expression of slip velocity and Knudsen layer function Y_0^(α)(η) under various TMACs asv_slip = k_s √(π)/2λdv/dx, Y_0^(α )(η ) = ∑_i = 0^N A_iJ_i(η ),whereJ_n(η ) = ∫_0^∞x^nexp ( - x^2 - η/x)dx,A_i and k_s is the coefficient calculated by refined moment methods<cit.>. According to Onishi<cit.>, the solutions of N=7 are good approximations with high and sufficient accuracy to the exact ones.The coefficients for partial values of α are listed in Table <ref>. It should be noted that Y_0^(α )(η ) is in complete agreement with Y_0(η ) shown in Fig.<ref> when α=1 . The DBM simulation results for the Couette flow with different Knudsen numbers under complete diffusion boundary condition are shown in Fig.<ref>. Results for different Knudsen numbers are obtained by changing the relaxation-time constant τ according to Eq.(<ref>). Figure <ref>(a) shows that the phenomena of velocity slip become more significant with the increase of Kn. Comparison of the values of slip velocity normalized by dv/dx between the DBM results and Sone's results is shown in Fig.<ref>(b).The two kinds of resultsare in excellent agreement with each other. The DBM accurately capture the velocity slip. Besides, the Knudsen layer is also well described by DBM. As shown in Fig.<ref>, comparison of normalized Knudsen profiles calculated from DBM are also in excellent agreement with Sones results. The Knudsen profiles Δ v in Fig.<ref> are normalized by √(π)/2λdv/dx.Taking the TMAC (α) into consideration, the Maxwell-type boundary condition is adopted in the following simulation. The DBM simulation results with several different values of α are shown in Fig.<ref>. From Fig.<ref>(a), we can see that the phenomena of velocity slip are more significant with the decrease of α. The values of velocity slip for different values of α are compared with those given by Eq.(<ref>). Figure <ref>(b) shows good agreement of DBM simulation results with those of Onishi<cit.>.The results of Y_0^α (η)for various values of α are compared in Fig.<ref>. The DBM results also show good agreement with those of Eq.(<ref>).§.§ Poiseuille flowPressure driven gas flow, known as Poiseuille flow,in a microchannel is also very common in MEMS. In the slip-flow regime, the Navier-Stokes equations with slip boundary condition are applicable. Accurate second-order slip coefficients are of significantly since they directly determines the accuracy of the results given by Navier-Stokes equations<cit.>. The first second-order slip model was presented by Cercignani. Using the BGK approximation he obtainedu|_wall=1.016θ∂ u/∂ y|_wall-0.7667θ ^2∂ u^2/∂ y^2|_wall,where θ=μ /p√(2RT). It can be seen, the first-order coefficient is same with Eq.(<ref>). Subsequently, Hadjiconstantinou<cit.> improved the model for a hard sphere gas by considering Knudsen layer effects. In his article, the viscosity-based mean free path, λ=μ /p√(π RT/2), was used. Then he obtained the following slip velocityu|_wall=1.1466λ∂ u/∂ y|_wall-0.31λ ^2∂ u^2/∂ y^2|_wall.However, in our DBM model, viscosity-based mean free path is defined as λ=μ /p√(8RT/π), so the first-order and second-order coefficients should be rescaled by π/4. The slip velocity formulate should beu|_wall=0.9004λ∂ u/∂ y|_wall-0.1912λ ^2∂ u^2/∂ y^2|_wall.Considering the Maxwell-type boundary, the fully-developed velocity profile can be expressed byu(y) =- dp/dxH^2/2μ[- (y/H)^2 + y/H. 55pt . + 0.9004 2 - σ/σKn + 0.3824 Kn^2 ] .10ptwhere dp/dx is the pressure gradient in the streamwise direction. H is the width of the micro-channel. Nondimensionalize the two sides of Eq.(<ref>) by the mean channel velocity u givesU(y) = u(y)/u̅ =- (y/H)^2 + y/H + 0.9004 2 - σ/σKn + 0.3824 Kn^2/1/6 + 0.9004 2 - σ/σKn + 0.3824 Kn^2.It is clear that the pressure gradient and the viscosity coefficient vanish. Firstly, complete diffuse boundary condition is adopted.The simulation results by DBM for different Knudsen numbers are shown in Fig.<ref>. It can be found that the velocity slip is significant with the increase of Knudsen number. The nondimensional velocity has a higher maximum value for a smaller Knudsen number. The velocity profile described by Eq.(<ref>) with α=1 is also plotted for comparison. The simulation results show good agreement with the expression of Eq.(<ref>).Considering the behaviours of velocity slip under various TMACs. Then the Maxwell-type boundary condition is used. It can be seen from Fig.<ref> that the velocity slip is more significant with the decrease of TMAC and the nondimensional velocity has the highest maximum value when the complete diffuse reflection occurs. It concluded that the effect of Knudsen and TMAC on velocity is in the opposite direction. The numerical results are in well agreement with Eq.(<ref>) for different values of α which verify the accuracy of the Maxwell-type boundary condition. § CONCLUSIONA discrete Boltzmann method with Maxwell-type boundary condition for slip flow is presented. The definition of Knudsen number is clarified for DBM. The relation between the Knudsen number based on hard sphere model and that based on BGK model is given. Two kinds of gas flows, including Couette flow and Poiseuille flow, are simulated to verify and validate the new model. The results show that the DBM with Maxwell-type can reasonably capture both the velocity slip and the flow characteristics in Kundsen layer undervarious Knudsen numbers and tangential momentum accommodation coefficients.§ ACKNOWLEDGEThe authors would like to thank Drs. Wei Jiang, Hongwei Zhu, Wei Wang and Ge Zhang for fruitful discussions on discrete Boltzmann modeling of slip flows. We acknowledge support of National Natural Science Foundation of China [under grant nos. 11475028 and 11772064], Science Challenge Project (under Grant No. JCKY2016212A501 and TZ2016002), and Science Foundation of Laboratory of Computational Physics.*00Gad2001FlowM. Gad-El-Hak, Mcanique and Industries 2, 313 (2001). Zhang2005Gas Y. Zhang, R. Qin, Y. Sun, R. Barber, and D. Emerson, Journal of Statistical Physics 121, 257 (2005). Bao2005Linear F. Bao and J. 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"authors": [
"Yudong Zhang",
"Aiguo Xu",
"Guangcai Zhang",
"Zhihua Chen"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170526113056",
"title": "Discrete Boltzmann method with Maxwell-type boundary condition for slip flow"
} |
2120193104288 Entanglement properties of the time periodic Kitaev Chain Aditi Mitra December 30, 2023 ========================================================= Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G=(V,E), ageodesic triangle △(x,y,z) with x, y, z∈ V is the union P(x,y) ∪ P(x,z) ∪ P(y,z) of three shortest paths connecting these vertices. A geodesic triangle △(x,y,z) is called δ-slim if for any vertex u∈ V on any side P(x,y) the distance from u to P(x,z) ∪ P(y,z) is at most δ, i.e. each path is contained in the union of the δ-neighborhoods of two others. A graphG is called δ-slim, ifall geodesic triangles in G are δ-slim. The smallest value δ for which Gis δ-slim is calledthe slimness of G.In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter Δ(G) of a layering partition of G, (2) graphs with tree-length λ, (3) graphs with tree-breadth ρ,(4) k-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we showthat the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraphis at most 1.§ INTRODUCTIONRecently, there has been a surge of empirical works measuring and analyzing geometric characteristics of real-world networks, namely the hyperbolicity (sometimes called also the global negative curvature) of the network (see, e.g., <cit.>). Hyperbolicity measures the local deviation of a metric from a tree metric. It has been shown that a number of data networks, including Internet application networks, web networks, collaboration networks, social networks, and others, have small hyperbolicity. It has been confirmed (see <cit.>) that the property, observed in real-world networks (see <cit.>),in which traffic between vertices (nodes) tends to go through a relatively small core of the network, as if the shortest path between them is curved inwards, is indeed due to global negative curvature of the network.Fortunately, graphs and general geodesic metric spaces with small hyperbolicities have many algorithmic advantages. They allow efficient approximate solutions for a number of optimization problems. For example, Krauthgamer and Lee <cit.> presented a PTAS for the Traveling Salesman Problem when the set of cities lie in a hyperbolic metric space. Chepoi and Estellon <cit.> established a relationship between the minimum number of balls of radius r+2δ covering a finite subset S of a δ-hyperbolic geodesic space and the size of the maximum r-packing of S and showed how to compute such coverings and packings in polynomial time. Edwards et al. <cit.> provided a quasilinear time algorithm for the p-center problem with an additive error less than or equal to 3 times the input graph's hyperbolicity. Chepoi et al. <cit.> gave efficient algorithms for quick and accurate estimations of diameters and radii of δ-hyperbolic geodesic spaces and graphs. Additionally, Chepoi et al. <cit.> showed that every n-vertex δ-hyperbolic graph has an additive O(δlog n)-spanner with at most O(δ n) edges and enjoys an O(δlog n)-additive routing labeling scheme with O(δlog^2n)-bit labels and O(logδ) time routing protocol.Efficient embeddings of hyperbolic graphs into hyperbolic spaces are provided by Verbeek and Suri <cit.>. Effect of hyperbolicity parameter on cuts and expansions in graphs with some algorithmic implications is considered by DasGupta et al. <cit.>. In case of graphs (and general geodesic metric spaces), there exist several "equivalent" definitions of δ-hyperbolicity involving different but comparable values of δ <cit.>. In this paper, we are interested in two of them, in Gromov's4-point condition and in Rips'condition involving geodesic triangles. Let G=(V,E) be a graph, d(·,·) be the shortest path metric defined on V, and δ≥ 0.The Gromov product of y,z∈ V with respect to w is defined to be (y|z)_w=1/2(d(y,w)+d(z,w)-d(y,z)). Let δ≥ 0. A G=(V,E)is said to be δ- hyperbolic <cit.> if (x|y)_w≥min{ (x|z)_w, (y|z)_w}-δ for all w,x,y,z∈ V. Equivalently, G=(V,E)is δ-hyperbolic iffor any four vertices u,v,x,y of V, the two larger of the three distance sums d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x) differ by at most 2δ≥ 0.A graph G=(V,E)is said to be δ-hyperbolic<cit.> if for any four vertices u,v,x,y of V, the two larger of the three distance sums d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x) differ by at most 2δ. The smallest value δ for which Gis δ-hyperbolic is called the hyperbolicity of G and denoted by hb(G).A geodesic triangle △(x,y,z) with x, y, z∈ V is the union P(x,y) ∪ P(x,z) ∪ P(y,z) of three shortest paths connecting these vertices. A geodesic triangle △(x,y,z) is called δ-slim if for any vertex u∈ V on any side P(x,y) the distance from u to P(x,z) ∪ P(y,z) is at most δ, i.e.each path is contained in the union of the δ-neighborhoods of two others (see Figure <ref>). We say that a graph G isδ-slim, ifall geodesic triangles in G are δ-slim. The smallest value δ for which Gis δ-slim is calledthe slimness of G and denoted by sl(G).It is known that for every graph G,hb(G) ≤ 2sl(G)+ 1/2 <cit.> and sl(G) ≤ 3hb(G)+ 1/2 <cit.> and these inequalities are sharp for general graphs. It is clear from the definition that the hyperbolicity hb(G) of an n-vertex graph G can be computed in at most O(n^4) time. It is less obvious, however, that the slimness sl(G) of G can also be computed in at most O(n^4) time <cit.>.Motivated by the abundance of real-world networks with small hyperbolicities and by many algorithmic advantages that small hyperbolicity provides, a few researchers started investigating possible bounds on the hyperbolicity hb(G) of special classes of graphs (see <cit.>).(1/2)-Hyperbolic graphs were characterized in <cit.> and then later another characterization with more direct algorithmic applications was proposed in <cit.>. The hyperbolicity of chordal graphs was investigated in <cit.>. It was shown that the hyperbolicity of a chordal graph is at most 1. Furthermore,chordal graphs with hyperbolicity at most 1/2 were characterized by two forbidden isometric subgraphs <cit.>. Later, k-chordal graphs were studied in <cit.>. It was proven that every k-chordal graph (k≥ 4) has hyperbolicity at most ⌊k/2⌋/2 <cit.>. Additionally, all 4-chordal (5-chordal) graphs with hyperbolicity at most 1/2 were characterized by three (five, respectively) forbidden isometric subgraphs <cit.>. The hyperbolicity of graphs with tree-length λ was considered in <cit.>. It was shown that it is at most λ.Recently, in <cit.>, also graphs with bounded cluster-diameter Δ(G) (the minimum cluster-diameter over all layering partitions of G; see Section <ref> for more details) were considered. It was shown that their hyperbolicity is at most Δ(G).Much less is known about the slimness of particular classes of graphs. One can get straightforward bounds using the general inequality sl(G) ≤ 3hb(G)+ 1/2 and known bounds on hb(G), but the bounds on sl(G) obtained this way are usually far from being sharp.In <cit.>, a non-trivial result was obtained by Diestel and Müller. It was shown that the slimness of graphs with tree-length λ is at most ⌊3/2λ⌋, and the result is sharp for every λ≥ 1. Recently, Bermudo et al. <cit.> proved that the slimness of a k-chordal graph is at most k/4+1.In this paper, using the layering partition technique, in an unified way we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter Δ(G) of a layering partition of G, (2) graphs with tree-length λ, (3) graphs with tree-breadth ρ,(4) k-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we showthat the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraphis at most 1. Table <ref> summarizes our results for slimness and known results for hyperbolicity in special graph classes. §.§ Basic notions and notationsAll graphs appearing here are connected, finite, unweighted, undirected, loopless and without multiple edges.For a graph G=(V,E), we use n and |V| interchangeably to denote the number of vertices in G. Also, we use m and |E| to denote the number of edges. A path P of length k in a graph G is a sequence of vertices (v_0,v_1,…,v_k) such that v_i is adjacent to v_i+1 for each i, 0≤ i< k.The distance d_G(u,v) between vertices u and v is the length of a shortest path connecting u and v in G. The distance d_G(v,M) between a vertex v and a set M⊆ V is defined by d_G(v,M)=min{d_G(v,u): u∈ M}. The ball B_r(s,G) of a graph G centered at a vertex s ∈ V and with radius r is the set of all vertices with distance no more than r from s (i.e., B_r(s,G)={v∈ V: d_G(v,s) ≤ r }). We omit the graph name G and writeB_r(s) if the context is about only one graph. For any two vertices u, v of G, I(u,v)= { z∈ V:d(u,v)=d(u,z)+d(z,v) } is the (metric) interval between u and v, i.e., all vertices that lay on shortest paths between u and v. A subgraph H of a graph G is called isometric if for every two vertices u,v of H, d_G(u,v)=d_H(u,v) holds. The eccentricity of a vertex v, denoted by ecc_G(v), is the largest distance from that vertex v to any other vertex, i.e., ecc_G(v)=max_u ∈ V d_G(v,u).The radius rad(G) of a graph G is the minimum eccentricity of a vertex in G, i.e., rad(G)=min_v ∈ Vmax_u∈ Vd_G(v,u). Definitions of other graph parameters considered, as well as notions and notation local to a section, are given in appropriate sections.§ SLIMNESS AND OTHER TREE-LIKENESS PARAMETERS In this section, we establish a relation between the slimness and the cluster-diameter of a layering partition of a graph. As a corollary, we get a recent result of Diestel and Müller <cit.> on a relation between the slimness and the tree-length of a graph. §.§ Slimness and cluster-diameter of a layering partition Layering partition is a graph decomposition procedure introduced in <cit.> and used in <cit.>for embedding graph metrics into trees.A layering of a graph G=(V, E) with respect to a start vertex s is the decomposition of V into r+1 layers (spheres), where r=ecc_G(s) L^i(s)={u∈V:d_G(s,u)=i},i=0,1,…,r. A layering partition ℒ𝒫(G,s)={L^i_1,…,L^i_p_i:i=0,1,…,r}. of G is a partition of each layer L^i(s) into clusters L^i_1,…,L^i_p_i such that two vertices u,v ∈ L^i(s) belong to the same cluster L^i_j if and only if they can be connected by a path outside the ball B_i-1(s) of radius i-1 centered at s. Here, p_i is the number of clusters in layer i. See Figure <ref> for an illustration. A layering partition of a graph can be constructed in O(n+m) time (see <cit.>).A layering tree Γ(G,s) of a graph G with respect to a layering partition ℒ𝒫(G,s)is the graph whose nodes are the clusters of ℒ𝒫(G,s) and where two nodes C=L_j^i and C'=L_j'^i' are adjacent in Γ(G,s) if and only if there exist a vertex u ∈ C and a vertex v∈ C' such that uv ∈ E. It was shown in <cit.> that the graph Γ(G,s) is always a tree and, given a start vertex s,it can be constructed in O(n+m) time <cit.>. Note that, for a fixed start vertex s∈ V, the layering partition ℒ𝒫(G,s) of G and its tree Γ(G,s) are unique.The cluster-diameter Δ_s(G) of the layering partition ℒ𝒫(G,s) with respect to vertex s is the largest diameter of a cluster in ℒ𝒫(G,s), i.e., Δ_s(G)=max_C ∈ℒ𝒫(G,s) max_u,v∈Cd_G(u,v). The cluster-diameter Δ(G) of a graph G is the minimum cluster-diameter over all layering partitions of G, i.e., Δ(G)=min_s ∈VΔ_s(G). Let alsoΔ(G) denote the maximum cluster-diameter over all layering partitions of G, i.e., Δ(G)=max_s ∈VΔ_s(G). Finding the cluster-diameter Δ_s(G) for a given layering partition ℒ𝒫(G,s) of a graph G requires O(n m) time [The parameters Δ(G) and Δ(G)can also be computed in total O(n m) time for any graph G.] (we need to know the distance matrix of G), although the construction of the layering partition ℒ𝒫(G,s) itself, for a given vertex s, takes only O(n+m) time.It is not hard to show that, for any graph G and any two of its vertices s,q, Δ_q(G)≤ 3 Δ_s(G). Thus, the choice of the start vertex for constructing a layering partition of G is not that important. Let s be an arbitrary vertex of G. For every vertex q of G, Δ_q(G)≤ 3Δ_s(G). In particular, Δ(G)≤ 3 Δ(G) for every graph G. Let ℒ𝒫(G,s) be a layering partition and let Γ(G,s) be the corresponding layering tree. Consider an arbitrary vertex q and let x,y betwo vertices of a cluster of ℒ𝒫(G,q) such that d_G(x,y)=Δ_q(G). If both x,y belong to the same cluster in ℒ𝒫(G,s), thend_G(x,y)≤Δ_s(G). Thus, assume that x,y are in different nodes of Γ(G,s). Let X,Y,Q be the nodes of Γ(G,s) containing vertices x,y,q, respectively. Consider the median node M of nodes X,Y,Qin tree Γ(G,s) (see Figure <ref> for an illustration), i.e., the unique node common to three paths of tree Γ(G,s) connecting corresponding nodes from{X,Y,Q}. We will show that d_G(y,M) is at most Δ_s(G) (the same holds for d_G(x,M)) and since cluster M hasdiameter at most Δ_s(G) in G, we will get that d_G(x,y) is at most 3Δ_s(G). By the definition of layering partition ℒ𝒫(G,q), there exists a path P(x,y) in G (not necessarily shortest) between x and ysuch that no w∈ P(x,y) with d_G(q,w) < d_G(q,y) exists. Let w be a vertex of M∩ P(x,y) and y' be a vertex from M laying on a shortest path of G between y and q (such vertices necessarily existby the construction of tree Γ(G,s) and choice of M; M separates in G y from x and q, or it contains some of those vertices). We know d_G(q,y) ≤ d_G(q,w). Hence, d_G(q,y')+d_G(y,y')=d_G(q,y) ≤ d_G(q,w) ≤ d_G(q,y')+d_G(y',w), i.e., d_G(y,y') ≤ d_G(y',w). As d_G(y',w)≤Δ_s(G), we get d_G(y,M)≤ d_G(y,y')≤ d_G(y',w)≤Δ_s(G).To see that the result is sharp, consider a graph in Figure <ref>. For that graph and its vertices q and s,Δ_q(G)=6 and Δ_s(G)=2 hold. Furthermore, subdividing every edge k-1 times yields a graph G for which Δ_q(G)=6k and Δ_s(G)=2k for every integer k≥ 1.The following theorem establishes a relationship between the slimness and the cluster-diameter of a layering partition of a graph. For every graph G,sl(G)≤⌊Δ(G)/2⌋. Consider any geodesic triangle △=△ (x,y,z) of G formed by shortest paths P(x,y), P(x,z) and P(z,y) connecting corresponding vertices. Assume that c is an arbitrary vertex of P(x,y) and ℓ≥ 0 is the maximum integer such that B_ℓ(c) does not intersect P(x,z)∪ P(z,y). Let ℒ𝒫(G,c) be a layering partition of G starting at vertex c. Consider two vertices a and b in P(x,y) such that d_G(c,a)=d_G(c,b)=ℓ, a≠ b. Let also a' and b' be vertices in P(x,y) such that d_G(c,a')=d_G(c,b')=ℓ+1, a is adjacent to a' and b is adjacent to b' (see Figure <ref> for an illustration). Since B_ℓ+1(c) intersects P(x,z)∪ P(z,y) but B_ℓ(c) does not, vertices a',b' are in the same cluster of ℒ𝒫(G,c). Hence, d_G(a',b')≤Δ_c(G)≤Δ(G). On the other hand, d_G(a',b')=1+d_G(a,b)+1=2ℓ+2. Hence, d_G(c,P(x,z)∪ P(z,y))=ℓ+1= d_G(a',b')/2≤Δ(G)/2. To see that the result is sharp, consider the graph in Figure <ref>.In a geodesic triangle △ (x,y,z) of G, formed by shortest paths P(x,y)=(x,9,5,2,c,1,3,6,y), P(x,z)=(x,15,11,14,z) and P(z,y)=(z,13,10,12,y), vertex c∈ P(x, y) is at distance 4 from P(x, z) ∪ P(y, z). Hence, the slimness of G is at least 4. In a layering partition starting at c, all three vertices x,y,z belong to the same cluster. Therefore, Δ_c(G)=8 and, since diam(G)=8 as well, Δ(G)=8.That is, sl(G)=4=Δ(G)/2. Also, subdividing every edge k-1 times yields a graph G for which sl(G)=4k=Δ(G)/2 for every integer k≥ 1.Combining Proposition <ref> with Theorem <ref>, one gets the following corollary. For every graph G and every vertex s of G, sl(G)≤⌊3/2Δ_s(G)⌋. To see that the inequalityin Corollary <ref> is sharp, consider again the graph in Figure <ref>. We have Δ_s(G)=2 and sl(G)≥ 3 (check geodesic triangle △ (x,y,z), formed by shortest paths P(x,z) (the north-west one), P(y,z) (the north-west one) and P(x,y) (the south-east one), and vertex q in P(x,y)). Furthermore, subdividing every edge k-1 times yields a graph G for which Δ_s(G)=2k and sl(G)≥ 3k for every integer k≥ 1. Hence, sl(G)=3k=3/2Δ_s(G). §.§ Slimness and tree-lengthOur next graph parameter is based on the notion of tree-decomposition introduced by Robertson and Seymour in their work on graph minors <cit.>.A tree-decomposition of a graph G=(V,E) is apair ({X_i: i∈ I},T=(I,F)) where {X_i: i∈ I} is a collection of subsets of V, called bags, and T is a tree. The nodes of T are the bags {X_i: i∈ I} satisfying the following three conditions: 1) ⋃_i∈ IX_i=V;2) for each edge uv∈ E, there is a bag X_i such that u,v ∈ X_i;3) for all i,j,k ∈ I, if j is on the path from i to k in T, then X_i ∩ X_k⊆ X_j.For simplicity we denote a tree-decomposition ({X_i: i∈ I},T=(I,F)) of a graph G by 𝒯(G).The width of a tree-decomposition 𝒯(G)=({X_i: i∈ I},T=(I,F)) is max_i∈ I|X_i|-1. The tree-width of a graph G, denoted by tw(G), is the minimum width over all tree-decompositions 𝒯(G) of G <cit.>.The length of a tree-decomposition 𝒯(G) of a graph G is λ:=max_i∈ Imax_u,v∈ X_id_G(u,v) (i.e., each bag X_i has diameter at most λ in G). Thetree-length of G, denoted by tl(G), is the minimum lengthover all tree-decompositions of G <cit.>. The chordal graphsare exactly the graphs with tree-length 1. Note that these two graph parameters are not directly related to each other on general graphs. For instance, a clique on n vertices has tree-length 1 and tree-width n-1, whereas a cycle on 3n vertices has tree-width 2 and tree-length n. However, if one involves also the size ℓ(G) of a largest isometric cycle in G, then a relation is possible: tl(G)≤⌊ℓ(G)/2⌋(tw(G)-1) <cit.> (see also <cit.> and <cit.> for similar but slightly weaker bounds: tl(G)≤ℓ(G)(tw(G)+1) <cit.>; tl(G)≤ℓ(G)(tw(G)-2) <cit.>). Furthermore, the tree-width of a planar graph G is bounded by O(tl(G)) <cit.>.The breadth of a tree-decomposition 𝒯(G) of a graph G is the minimum integer r such that for every i∈ I there is a vertex v_i∈ V with X_i⊆ B_r(v_i,G) (i.e., each bag X_i can be covered by a ball B_r(v_i,G) of radius at most r in G).The tree-breadth of G, denoted by tb(G), is the minimum breadth over all tree-decompositions of G <cit.>. Evidently, for any graph G, 1≤ tb(G)≤ tl(G)≤ 2 tb(G) holds.Unfortunately, while graphs with tree-length 1 (as they are exactly the chordal graphs) can be recognized in linear time, the problem of determining whether a given graph has tree-length at most λ is NP-complete for every fixed λ >1 (see <cit.>). Judging from this result, it was conceivable that the problem of determining whether a given graph has tree-breadth at most ρ is NP-complete, too. Indeed, it was confirmed recently that the problem is NP-complete for every ρ≥ 1 <cit.>. On the positive side, both parameters tl(G) and tb(G) can be easily approximated within a factor of 3 using a layering partition of G <cit.>.The following proposition establishes a relationship between the tree-length and the cluster-diameter of a layering partition of a graph.<cit.> For every graph G and any vertex s, Δ_s(G)/3 ≤ tl(G) ≤Δ_s(G)+1.Thus, the cluster-diameter Δ_s(G) of a layering partition provides easily computable bounds for the hard to compute parameter tl(G).Diestel and Müller <cit.> proved that every graph Gof tree-length tl(G) has slimness at most ⌊3/2tl(G)⌋. Using Theorem <ref> and Proposition <ref> by Dourisboure and Gavoille <cit.>, we obtain that resultas a simple corollary.<cit.> For every graph G, sl(G)≤⌊3/2tl(G)⌋. Diestel and Müller in <cit.> showed also that the bound in Corollary <ref> is sharp; for every integer k there is a graph G with tree-length k such that sl(G)=⌊3/2k⌋. To show that the bound of ⌊3/2tl(G)⌋ is sharp, consider a path-decomposition of adhesion 2 into many copies of K^4 and with disjoint adhesion sets. Subdividing every edge tl(G) - 1 times yields a graph G of tree-length at most tl(G): the tree-decomposition witnessing this is a path-decomposition whose parts are the 4-vertex-sets from the K^4 s together with pendent parts each consisting of a subdivided edge. It is easy to find in G three vertices x, y, z with shortest paths P(x, y), P(y, z) and P(x, z) between them such that each subdivided K^4 (other than the first and the last) meets P(x, z) in exactly one subdivided edge P and P(x, y) ∪ P(y, z) in exactly one subdivided edge Q disjoint from P (Figure <ref>). For each subdivided K^4 except the leftmost and the rightmost one, the vertex closest to the mid-point of P ⊆ P(x, z) has distance exactly ⌊3/2tl(G)⌋ not only from Q but from all of P(x, y) ∪ P(y, z). As for every graph G, tl(G)≤ 2tb(G), we have also the following inequality. For every graph G, sl(G)≤ 3 tb(G).The inequalityin Corollary <ref> is also sharp. Consider the graph G in Figure <ref>. The tree-breadth of this graph is at most 2 as witnessed by the shown tree-decomposition. The slimness of G is at least 6. Indeed, consider a geodesic triangle △ (x,y,v), formed by two horizontal shortest paths P(x,v) and P(y,v) and a shortest path P(x,y) containing vertex z. Then, vertex z of P(x,y) is at distance 6 from P(x,v)∪ P(y,v). Thus, 6≤ sl(G)≤ 3 tb(G)≤ 6, i.e., sl(G)= 3 tb(G)= 6. Clearly, this graph can be modified to get a graph G such that tb(G)= k and sl(G)= 3k (contract some edges to get k=1; subdivide some edges to get k≥ 3).§ GRAPHS WITH (SMALL) CHORDALITYA graph G is k-chordal if every induced cycle of G has length at most k. The parameter k is usually called the chordality of G. When k=3, G is called a chordal graph.Using Theorem <ref>, we can also obtain a sharp bound on the slimness through the chordality of a graph. In <cit.>, Chepoi and Dragan proved that for every k-chordal graph G and every its vertex s, Δ_s(G)≤ k/2+2. Combining this with Theorem <ref>, we obtain a result of Bermudo et al. <cit.> as a simple corollary. <cit.> For every k-chordal graph G, sl(G)≤⌊ k/4⌋+1. To see that the result is sharp, consider again the graph from Figure <ref>. We know that sl(G)=4. The chordality of G is 12 as witnessed by an induced cycle (c,2,5,9,11,14,z,13,10,6,3,1) of length 12. Considering (half) rectilinear grids with longer or shorter sides, we obtain for every integer k≥ 1 a graph G with chordality 4k and slimness k+1.From Corollary <ref> we know that every chordal graph has slimness at most 1 and every k-chordal graph with k≤ 7has slimness at most 2.In this section, we also show that the graphs with slimness equal 0 are exactly the block graphs and an interesting subclass of 5-chordal graphs, namely the class of AT-free graphs, has slimness at most 1. Furthermore, we characterize those 4-chordal graphs for which every induced subgraph has slimness at most 1. As a consequence, we get that every house-hole-domino–free graph (HHD-free graph) has slimness at most 1.§.§ AT-free graphs and block graphs An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph G is an AT-free graph if it does not contain any asteroidal triples. The class of AT-free graphs contains many intersection families of graphs, including permutation graphs, trapezoid graphs, co-comparability graphs and interval graphs. Every AT-free graph has slimness at most 1.Consider any geodesic triangle △=△ (x,y,z) of G formed by shortest paths P(x,y), P(x,z) and P(z,y) connecting corresponding vertices. Assume there is a vertex u inP(x,y) such that B_1(u) does not intersect P(x,z)∪ P(z,y). Then, x,y,u form an independent set in G and there is a path in P(x,z)∪ P(z,y) that connects vertices x and y and avoids the neighborhood of u. Furthermore, as P(x,y) is a shortest path, its subpath P(x,u) (subpath P(y,u)) avoids the neighborhood of y (of x, respectively). Hence, vertices x,u,y form an asteroidal triple in G, contradicting with G being an AT-free graph.A block graph is a connected graph whose blocks (i.e., maximal biconnected subgraphs) are cliques.A diamond K_4-e is acomplete graph on 4 vertices minus one edge. We will need the following characterization of block graphs.<cit.> G is a block graph if and only if G has neither diamonds nor cycles C_k of length k≥ 4 as isometric subgraphs. sl(G)=0 if and only if G is a block graph. Let G be a block graph. Since hb(G)=0 for a block graph, it follows immediately from the inequality sl(G) ≤ 3hb(G)+ 1/2 and the integrality of the slimness that sl(G)=0. Conversely, if slimness of a graph is 0 then, for every geodesic triangle △(x,y,z), P(x,y)⊆ P(x,z)∪ P(z,y) must hold. So, a graph G with slimness 0 cannot have a diamond or any cycles C_k of length k≥ 4 as an isometric subgraph. Hence, by Proposition<ref>, G is a block graph. §.§ 4-chordal graphs We know that all 4-chordal graphs have slimness at most 2. In what follows, we characterize those 4-chordal graphs for which every induced subgraph has slimness at most 1. As a corollary, we get that every house-hole-domino–free graph (HHD-free graph) has slimness at most 1. A chord in a cycle is an edge connecting two non-consecutive vertices of the cycle. Let C_k denote an induced cycle of length k. The following two lemmata will be frequently used in what follows.(Cycle Lemma for 4-chordal graphs) <cit.> Let C be a cycle of length at least 5 in a 4-chordalgraph G=(V,E). Then, for each edge xy∈ C there are vertices w_1, w_2 in C such that xw_1∈ E, yw_2∈ E, and d_G(w_1, w_2)≤ 1, i.e., each edge of a cycle is contained in a triangle or a 4-cycle. A building B(w|uv) is a chain of k C_4s (k≥ 0) ending with a C_3 depicted in Figure <ref>.(building) In a 4-chordal graph G=(V,E), if an edge uv∈ E is equidistant from a vertex x, that is, d_G(u,x)=d_G(v,x), then G contains abuilding B(w|uv) as an isometric subgraph(see Figure <ref>), where w∈ I(x,u)∩ I(x,v). We proceed by induction on k=d_G(x,u)=d_G(x,v). Let P(x,u), P(x,v) be two shortest paths connecting u and v with x. Note that the union of P(x,u), P(x,v) and the edge uv contains a cycle. Applying the Cycle Lemma to this cycle and its edge uv, by distance requirements, we obtain either a vertex w adjacent to both u and v and at distance k-1 from x, or two adjacent vertices u' and v' such that uu',vv' ∈ E, uv',vu' ∉ E, and d(u',x)=d(v',x)=k-1. In the former case, w,u,v form a triangle and we are done. In the latter case, by the inductionhypothesis, there exists in G an isometricbuilding B(w|u'v') with w∈ I(x,u')∩ I(x,v'). Adding to B(w|u'v') vertices u,v and edges u'u,uv,vv' we obtain an isometricbuilding B(w|uv) with w∈ I(x,u)∩ I(x,v).Abuilding with k=1 is known as a house. For every 4-chordal graph Gthe following two statements are equivalent. * G and each its induced subgraph has slimness at most 1. * None of the following graphs (see Figure <ref>) is an induced subgraph of G. It is easy to verify that each graph in Figure <ref> has slimness 2. In allgraphs consider △(x,y,z) and vertex a∈ P(x,y). For the other direction, consider three vertices x,y,z in a 4-chordal graph G and three shortest paths P(x,y), P(x,z) and P(y,z) connecting appropriate pairs.Assume, by way of contradiction, that there is a vertex a on P(x,y) which is not adjacent to any vertex on P(x,z)∪P(y,z) (i.e., the slimness of G is greater than 1). Assume also that the triple x,y,z for which such a vertex a exists is chosen with the minimum sum d(x,y)+d(y,z)+d(z,x). Let b be a neighbor of a on P(x,y) which is closer to y (see Figure <ref>). By the Cycle Lemma applied to C:=P(x,y)∪ P(x,z)∪ P(y,z) and edge ab, there must exist vertices t and c in C such that ac∈ E, bt∈ E and d(t,c)≤ 1. Since a is not adjacent to any vertex from P(x,z)∪P(y,z), vertex c must belong to P(x,y). As P(x,y) is a shortest path, t cannot belong to P(x,y), implying d(t,c)= 1 and t∈ P(x,z)∪ P(y,z). See Figure <ref> for an illustration. By symmetry, in what follows, we will assume that t is on P(x,z). To facilitate our further discussion, we introducea few more vertices on paths P(x,y), P(x,z) and P(y,z)(see Figure <ref>). Since paths P(x,z) and P(x,y) are shortest, d(x,t)≤ d(x,c)+1 and d(x,t)+1≥ d(x,c)+2 must hold. Therefore, d(x,t)=d(x,c)+1=d(x,a). From d(x,t)=d(x,a), we also get d(x,w)=d(x,c) (see Figure <ref>). If vertices w and c are adjacent then, by Lemma <ref>, there is an isometric building B(x'|wc) in G, where x'∈ I(x,w)∩ I(x,c). If vertices w and c are not adjacent then, by the Cycle Lemma applied to C:=P(x,c) ∪ P(x,t) ∪tc and edge tc, there must exist a vertex g∈ P(x,c)∪ P(x,t) at distance d(x,c)-1 from x which formsa C_4 with w,t,c (see Figure <ref>). Note that c≠ x as a is not adjacent to any vertex of P(x,z).This completes the construction of left parts (left to the vertex a) in allforbidden subgraphs from Figure <ref>. If t=z then, by symmetry, we get the same construction on the right side of a and, therefore, get either one of the graphs (a)-(b) from Figure <ref> as an induced subgraph of G or a graph that has a graph (d) from Figure <ref> as an induced subgraph. So, in what follows, we will assume that t≠ z. Also, in what follows, if P(u,v) is a shortest path of G and w is a vertex ofP(u,v), then by P(u,w) we denote a subpath of P(u,v) connecting u and w. We will distinguish between two cases, whether vertices c and b have a common neighbor on path P(y,z) or not. Case 1: Vertices c and b have also a common neighbor t' on path P(y,z). First, notice that vertices w and t' cannot be adjacent by the minimality of the sum d(x,y)+d(y,z)+d(z,x) (in case t'w∈ E, consider geodesic triangle△(x,y,t'), formed by shortest paths P(x,y),P(y,t') and P(x,w)∪{t'}, and vertex a from P(x,y), where P(y,t') and P(x,w) are subpath of shortest paths P(y,z) and P(x,y), respectively). Additionally, by distance requirements, t' is not adjacent to any other vertices from P(x,w)∪ P(x,c)∖{c} (otherwise, there is a path from x to b via t' which is shorter than P(x,b)). With similar to earlier arguments, we get the neededconstruction on the right side of a in allforbidden subgraphs (e)-(j) from Figure <ref>. Furthermore, as P(x,z), P(y,z) are shortest paths and d(x,t)=d(x,c)+1, d(y,b)+1=d(y,t'), necessarily, d(z,t)≤ d(z,t') and d(z,t')≤ d(z,t), i.e., d(z,t')=(z,t). If vertices t and t' are adjacent then, by Lemma <ref>, there is an isometric building B(z'|tt') in G, where z'∈ I(z,t)∩ I(z,t'). If vertices t and t' are not adjacent (see Figure <ref>) then, by the Cycle Lemma applied to C:=P(z,t) ∪ P(z,t') ∪{ct,ct'} and edge tc, there must exist a vertex g∈ P(z,t)∪ P(z,t') at distance d(z,t)-1 from z which formsa C_4 with t,c,t'. This completes the construction of upper parts (upper to the vertex a) in allforbidden subgraphs (e)-(j) from Figure <ref>. Case 2: Vertices c and b have no common neighbors on path P(y,z). In what follows, we can assume that b is not adjacent to any vertex of path P(f,z)⊂ P(t,z) (see Figure <ref>). If b is adjacent to f, then vertices f,s,t,b induce a C_4, when bs∉ E, or a diamond, otherwise. Hence, G contains one of the graphs (a)-(b) from Figure <ref> as an induced subgraph. Furthermore, since P(t,z) is a shortest path, b cannot have neighbors in P(e,z)⊂ P(t,z) (otherwise, there is a path from t to z via b which is shorter than P(t,z)). Consider now the neighbor u of b on path P(b,y)⊂ P(x,y) (see Figure <ref>). We distinguish between three subcases. Case 2.1: Vertex u is adjacent to a vertex of path P(t,z). Since P(x,y) and P(x,z) are shortest paths, u is not adjacent to t (otherwise, d(c,u)=2<3) and can be adjacent only to s, f or e (otherwise, there is a path from t to z via u which is shorter than P(t,z)). If u is adjacent to s, then vertices u,s,t,b induce a C_4, when bs∉ E, or a diamond, otherwise. Hence, G contains one of the graphs (a)-(b) from Figure <ref> as an induced subgraph. If u is adjacent to f but not to s, then vertices u,f,s,t,b form a 5-cycle. This cycle must have a chord. The only possible chord is bs (see Figure <ref>).Hence, G contains one of the graphs (c)-(d) from Figure <ref> as an induced subgraph. If u is adjacent to e but not to s,f, then vertices u,e,f,s,t,b form a 6-cycle. This cycle must have chords and the only possible chord is bs. Hence, we get a forbidden C_5 or C_6 in G.We conclude that u has no neighbors in P(t,z)⊂ P(x,z). Case 2.2: Vertex u has no neighbors in P(t,z) but has a neighbor inP(y,z). Let k be a neighbor of u in P(y,z) closest to z. Hence, u has no other neighbors in P(k,z)⊂ P(y,z) except k. By the distance requirements, k is not adjacent to any vertex from P(x,w)∪P(x,c)(otherwise, there is a path from x to u via k which is shorter than P(x,u)). Assume k has a neighbor in P(t,z) and pick such a neighbor v which is closest to t. Consider a cycle Z formed by vertices b,u,k and subpath P(t,v) of P(t,z). In this cycle of length at least 4, the only possible chords are bs and bk. To avoid induced cycles of length at least 5, v must coincide with t,s or f. If v=t, then vertices u,k,t,b induce a C_4, when bk∉ E, or a diamond, otherwise. Hence, G contains one of the graphs (a)-(b) from Figure <ref> as an induced subgraph. If v=s, then vertices u,k,s,t,b form a 5-cycle. This cycle must have chords. The only possible chords are bk and bs.Hence, G contains one of the graphs (a)-(d) from Figure <ref> as an induced subgraph. If v=f, then vertices u,k,f,s,t,b form a 6-cycle (see Figure <ref>). This cycle must have chords and the only possible chords are bs and bk. Hence, again G contains one of the graphs (c)-(d) from Figure <ref> as an induced subgraph. Thus, k cannot have any neighbors in P(t,z). Recall also that u has no neighbors in P(t,z)∪ P(k,z)∖{k}. Applying the Cycle Lemma to edge bu, we get bk∈ E or bq∈ E, whereq is the neighbor of k in P(k,z). Denote by v the neighbor of b in P(k,z) closest to z. Note that v is not adjacent to c as c and b do not have common neighbors in P(y,z). By the minimality of the sum d(x,y)+d(y,z)+d(z,x), v is not adjacent to w (in case vw∈ E, consider geodesic triangle △(x,y,v), formed by shortest paths P(x,y),P(y,v) and P(x,w)∪{v}, and vertex a from P(x,y)). Additionally, by distance requirements, v is not adjacent to any other vertices from P(x,w)∪ P(x,c)(otherwise, there is a path from x to b via v which is shorter than P(x,b)). As P(k,z) is a shortest path, d(v,k)≤ 3, when bk∉ E, and d(v,k)≤ 2, when bk∈ E.Applying the Cycle Lemma to edge bv, weget vt∈ E, or vs∈ E, or vf∈ E (and hence, bs∈ E, otherwise, we get a C_5 in G), or G contains an induced C_4 formed by vertices b,s,v',v or vertices t,b,v,v', where v' is the neighbor of v in P(v,z). If v is adjacent to t, then vertices t,b with some vertices of path P(v,k)⊂ P(y,z) and possibly with u (e.g., when v=q) induce a C_4 or a house or a diamond in G (we leave these small technical details to the reader; one needs to analyze a subgraph on at most 7 vertices). Note that in this case v≠ k as k has no neighbors in P(t,z).So, G contains one of the graphs (a)-(d) from Figure <ref> as an induced subgraph. Now, we assume that tv∉ E. If v is adjacent to s, then vertices v,s,t,b induce a C_4, when bs∉ E, or a diamond, otherwise(see Figure <ref>). Hence, G contains one of the graphs (a)-(b) from Figure <ref> as an induced subgraph. If v is adjacent to f but not to s, then vertices v,f,s,b form an inducedC_4. Hence, G contains one of the graphs (c)-(d) from Figure <ref> as an induced subgraph. So, we can assume that tv,vs,vf∉ E. If vertices b,s,v',v or vertices t,b,v,v' form an induced C_4 in G, then G contains one of the graphs (a)-(d) from Figure <ref> as an induced subgraph. In this case, one needs to notice only that v' is not adjacent to any vertex from P(x,w)∪ P(x,c). If v' is adjacent to a vertex w' from P(x,w)∪ P(x,c)∖{c,w}, then we get a contradiction with distance requirements (subpath P(x,b) of a shortest path P(x,y) is a shortest path) or with the minimality of the sum d(x,y)+d(y,z)+d(z,x) (consider geodesic triangle △(x,y,v'), formed by shortest paths P(x,y),P(y,v') and P(x,w')∪{v'}, and vertex a from P(x,y)). If v' is adjacent to c then G has an induced C_5 formed by c,a,b,v,v', which is impossible.If v' is adjacent to w then G has an induced C_6 formed by c,a,b,v,v',w, when wc∈ E, or an induced C_7 formed by c,a,b,v,v',w,g, otherwise (vertex g can be seen in Figure <ref>). Both induced cycles are forbidden in G. Case 2.3: Vertex u has no neighbors in P(y,z)∪ P(t,z). In this case, we have a contradiction to our assumption that the triple x,y,z certifying sl(G)>1 has the smallest sum d(x,y)+d(y,z)+d(z,x). For triple c,z,y with shortest paths P(c,y)⊂ P(x,y), P(z,y) and P(c,z)={c}∪ P(t,z), u∈ P(c,y) has no neighbors in P(c,z)∪ P(z,y) and d(x,y)+d(y,z)+d(z,x)> d(c,y)+d(y,z)+d(z,c). Here, P(c,z)={c}∪ P(t,z) is a shortest path since d(x,w)=d(x,c) and hence d(c,z)=d(w,z)=1+d(t,z). Ahole is an induced cycle of length at least 5.A domino is an induced cycle on 6 vertices with one additional chord dividing it into two cycles of length 4 each. Ahouse-hole-domino–free graph (HHD-free graph) is a graph not containing holes, houses and dominoes as induced subgraphs. As 4-chordalgraphs do not contain any induced holes and the graphs shown in Figure <ref> have houses or dominoes as induced subgraphs, we have. For every HHD-free graph G, sl(G)≤ 1.We would like to thank anonymous reviewers for many useful suggestions and comments. * abbrvnat | http://arxiv.org/abs/1705.09797v4 | {
"authors": [
"Feodor F. Dragan",
"Abdulhakeem Mohammed"
],
"categories": [
"cs.DM",
"math.CO"
],
"primary_category": "cs.DM",
"published": "20170527094444",
"title": "Slimness of graphs"
} |
myamericanaddress,myitalianaddress]D. Bravo-Berguño RiccardoAddress]Riccardo Mereu myamericanaddress]R.B. Vogelaar RiccardoAddress]F. Inzoli[myamericanaddress]Physics Department, Virginia Tech, 24061 Blacksburg, VA (USA) [myitalianaddress]INFN Sezione Milano, Via Celoria 16, 20133 Milano (Italy) - (+39) 3394636849 - [email protected] [RiccardoAddress]Department of Energy - Politecnico di Milano, via Lambruschini 4, 20156 - Milano, Italy The strategy to install Borexino's Thermal Monitoring and Management System (BTMMS) successfully stabilized the thermal environment inside the Borexino neutrino observatory, which is understood to be a necessary step to improve and minimize radioactive background contamination inside the active volume of the detector, allowing for it to achieve better sensitivity in the regions of interest. Two-dimensional numerical simulations to achieve a proper understanding of Borexino's fluid-dynamics were developed and optimized for different regions and periods of interest, focusing on the most critical effects that were identified as influencing background concentrations. Literature experimental case studies were reproduced to benchmark the method and settings, and a Borexino-specific benchmark was constructed in order to validate the model's thermal transport. Finally, fully-convective models were implemented to understand general and specific fluid motions impacting the active detector volume.Neutrino detector Computational Fluid Dynamics Thermal control Radiopurity Background stabilityNatural convection § INTRODUCTIONThe Borexino liquid scintillator detector is devoted to performing high-precision neutrino observations. In particular, it is optimized to study the low energy part of the solar neutrino spectrum in the sub-MeV region, having the precision measurement of the ^7Be solar neutrinos as its design objective. Borexino has succeeded in performing high-precision measurements of all the major components of the solar neutrino spectrum (first direct detections of pp<cit.>, pep<cit.>, ^7Be<cit.>, and lowest (3 MeV) threshold observation of ^8B<cit.>), as well as in reaching the best available limit in the subdominant CNO solar neutrino rate<cit.>, with just the DAQ time of 767 days comprising its first dataset Phase 1 from 2007-10. Geoneutrinos have also been detected by Borexino with high significance (5.9σ<cit.>) thanks to the extremely clean ν channel. Results on searches for new particles, (anti)neutrino sources and rare processes like <cit.>, <cit.>, <cit.>, <cit.>, <cit.> are expected to gain even more relevance during the SOX phase of the experiment, where a ν_e generator will be placed in close proximity to the detector, in order to probe for anomalous oscillatory behaviors and unambiguously cover the allowed phase space for light sterile neutrinos<cit.>.These results were possible thanks to the unprecedented, extremely radio-pure conditions reached in the active section of the detector –achieved thanks to a combination of ultra-clean construction and fluid-handling techniques as well as dedicated scintillator purification campaigns<cit.>. Detailed detector response determination was made possible thanks to very successful internal calibration campaigns<cit.> which did not disturb the uniquely radio-pure environment. Moreover, results with even higher precision are under development thanks to the Phase 2 dataset, started in late 2011, which offers greatly enlarged statistics with improved background conditions, being parsed with new analysis techniques. Reduced background levels in the Phase 2 dataset have raised the need for increased stability in their spatio-temporal distribution inside the detector, due to the low statistics available for determining their rate, particularly for some background species. The liquid nature of the scintillator in use by Borexino means the best strategy to ensure background stability is to minimize fluid mixing, namely by means of external environmental control and stabilization. It is assumed the extremely dilute concentrations of background radioisotopes are carried by the fluid movement in ideally point-like, non-interacting particulates –thereby establishing a direct correlation between fluid dynamics and background migration, which is only attenuated through the corresponding radioisotopes' lifetimes.Section 2 of this paper will detail the recent background situation in Borexino and the correlation existent with temperature trends within the detector, pointing toward the strategy chosen to attempt an improvement in Fiducial Volume (FV) background levels. Section 3 will discuss the benchmarking strategy for the Computational Fluid Dynamics (CFD) convective simulations that would confer confidence on the simulative approach used, as well as the empirical LTPS-data-based thermal benchmarking performed for the Borexino geometry. Section 3 will instead focus on the Inner Detector models developed in order to understand fluid movement inside this closed, near-equilibrium system. It will also showcase the evolution toward a more focused model with greater detail around Borexino's active section, the Inner Volume (IV). Finally, Section 4 will discuss the conclusions reached through these models, and the perspectives on new studies building upon the present work.§ BACKGROUND STABILITY AND CFDBorexino, located in the Hall C of the Gran Sasso National Laboratories' (LNGS) underground facilities (3,800 m w.e.), measures solar neutrinos via their interactions with a 278 tonnes target of organic liquid scintillator. For more details about its structure, design and signal/background issues relevant to the present discussion, as well as the Thermal Monitoring and Management System developed and installed during 2014-16, refer to <cit.>.The installation of the Latitudinal Temperature Probe System (LTPS) multi-sensor hardware, especially the internal Phase I, offered an unprecedented chance at utilizing abundant data from evenly distributed points on both sides of the detector for applications beyond trivial temperature monitoring, such as the thermal transport constant from the Outer to the Inner detector, liquid stratification, 180^∘-resolution thermal asymmetries... Conductive 2D simulations were also developed with the aim of establishing a first layer of information about which heat transfer processes were conductive-dominated, in particular the cooldown effect dominated by the bottom heat sink in contact with the local aquifer temperature, the cooling constant and behavior it would cause, the extent of the Thermal Insulation System (TIS) insulating power and expectable boundary temperature trends, and the influence of structures as temperature bridges, including the conductive heat transmission expectable from the Active Gradient Stabilization System (AGSS) heating circuit. This would provide information needed to validate the simulative approach and move toward more detailed simulations.However, as is obvious, these conductive cases would not account for convection or fluid movement, imposing a clear limitation in simulating thermal trends away from stable stratifications or solid structures, as well as understanding where backgrounds would be led by fluid-dynamical currents.Considering the internal LTPS dataset, we can neglect the detailed modeling of the water convection in the WT: a convective model of the full detector, even in 2D, would either be too coarse for the expected convective speeds to surface from under the numerical noise, or take up too much computing time (i.e. the simulated time of interest, which is on the order of months, would approach the real time, negating its usefulness as a predictive tool, and delaying too much the results' availability). § CFD METHODOLOGYIn order to validate the CFD approach <cit.> in reproducing the main characteristics and fundamental phenomena inside Borexino, such as closed system, Newtonian water-like fluids, prevalent natural convection, vertical temperature difference (∼10^∘C) and thermal stratification, several benchmarking cases have been modeled and related results compared with available experimental data. §.§ Design parameters The choice of dimensions and Δ T for the following benchmark models needed to be motivated to at least lie close to, or ideally overlap, Borexino's regime of interest. The determination of the Rayleigh number for Borexino offers the simplest, most rigorous way of relating seemingly dissimilar geometries to the detector case. The definition of the Rayleigh number is very dependent on the model geometry, and in non-standard ones (such as Borexino's spheric geometry with distributed, gradual temperature differences) may be somewhat arbitrary if not keeping a close watch on the phenomenon under study. The Rayleigh number (Ra) is a dimensionless parameter defined, in general, as: Ra = β [K^-1] Δ T [K] g [m/s^2] L^3 [m^3]/ν ^2 [m^4/s^2] Pr where β is the thermal expansion coefficient of the fluid, Δ T is the temperature difference in the characteristic lengthscale of the system, g is the gravitational field acting on the system, L is the characteristic lengthscale for natural convection in the system, ν is the kinematic viscosity of the fluid and Pr is the Prandtl number, which is itself defined as the quotient between the momentum diffusivity and the thermal diffusivity. In practice, the Prandtl number is only dependent on the fluid's nature and state. The quotient multiplying Pr is referred to as the Grashof number Gr, which is a measure between the buoyancy and viscosity forces on a fluid.As can be inferred from this definition, the Rayleigh number is most dependent on the characteristic lengthscale L for convection in the considered system. Ra is therefore a way of relating buoyancy-driven fluid flows coming from different fluid natures, conditions and system geometries –and therefore, contains information about the convective/conductive dynamics of a fluid flow, irrespective of the fluid. This is in contrast to the Grashof number, which depends upon the fluid under consideration.If we consider the typical overall gradient of Borexino's IV to be ∼5^∘C, over the 8.5 m between the top and bottom poles of the vessel, we get ∼1.7 m/^∘C: that is, ∼17 cm separating each 0.1^∘C isotherm. Considering this is our L, Ra ∼𝒪(10^7-10^8) (with Pr=7.78 for PC, β _PC^10C∼10^-3 K^-1, and ν_PC^10C∼7·10^-7 m^2/s). We consider the 𝒪(0.1)^∘C temperature differences routinely happening in short timescales in Borexino, which may be causing the internal stirring concerning us. If the overall gradient was very large, the isotherms would be very close together, and a given Δ T seeping in from the outside would show up at a smaller lengthscale than if the overall gradient was smaller, and the isotherms were farther apart from each other –in which case the isotherm displacement to match the boundary condition would occur over larger lengthscales.We are of course assuming a linearly-stratified fluid, which is not the real case in Borexino (which exhibits a laxer stratification on the top than on the bottom). Therefore, we should keep in mind the order-of-magnitude Rayleigh number estimate above would be approximately 1-2 order(s) of magnitude larger, locally, on the top, and smaller on the bottom. Consequently, we can estimate Borexino's Rayleigh range as Ra ϵ [𝒪(10^6),𝒪(10^9)]. §.§ Governing EquationsThe finite volume commercial solver ANSYS-Fluent is used for modeling the flow field via mass, momentum and energy conservation equations for incompressible Newtonian fluids with constant viscosity and density. Governing equations of mass, momentum and energy are numerically treated using the Direct Numerical Simulation (DNS) approach for laminar flows, as reported in Eq. <ref>, <ref> and <ref>, respectively. ∂ρ/∂ t+∇·(ρ𝐮)=0∂ρ𝐮/∂ t+∇·(ρ𝐮𝐮)=-∇ p+∇·τ̅̅̅+ρ̅𝐠∂ρ E/∂ t+∇·(𝐮(ρ E+p))=∇·(k∇ T+(τ̅̅̅·𝐮)) where u, p and E represent the velocity vector, static pressure and energy, respectively, ρ̅𝐠 the gravitational body force, κ the thermal conductivity and τ̅̅̅ the stress tensor <cit.>. For the natural-convection flows the fluid density is modeled as a function of temperature. the Boussinesq model used here treats density as a constant value in all solved equations, except for the buoyancy term in the momentum equation, where it is approximated as: ρ̅𝐠=(ρ-ρ_0)𝐠≃ -ρ_0β(T-T_0)𝐠 where β=-1/ρ(∂ρ/∂ T)_p is the thermal expansion coefficient. §.§ Geometrical modeling The numerical domain has been defined for each benchmarking case and Borexino reproducing the real geometry with a two-dimension full-scale model. The computational mesh used to discretize the domain is a structured Cartesian or unstructured polygonal/polyhedral grid, depending on the geometry. Specific refinements near the walls are applied to take into account the viscous and thermal boundary layer.The mesh size (Δ x) is defined for each modeled geometry and it is based on a preliminary mesh sensitivity analysis. This permitted to quantify the influence of different grid sizes selecting the largest acceptable size, with computational grids ranging from 𝒪(10^4) to 𝒪(10^5) elements. §.§ Numerical modelingAll the domain modeled here are closed and boundary conditions are imposed considering no-slip conditions for the fluid dynamics and adiabatic or fixed external temperature for heat transfer conditions.The solver used for performing the transient simulations is based on the coupling pressure-velocity PISO algorithm. It is able to guarantee the convergence at each time step, through inner loops, using a restrictive Courant-Friedrichs-Lewy (CFL) condition (CFL≤1) to maintain the necessary accuracy. The time-step size Δ t is defined considering the physical Δ T_p and numerical Fourier stability analysis Δ T_Fo natural convection constrains: Δ t_p = τ/4≈L/4√(β g L Δ T)Δ t_Fo = Fo(Δ x)^2/α with τ the time constant, L the characteristic length, Fo the Fourier number (limited to Fo=0.1), Δ x the grid size and α=k/ρ c_p the numerical diffusivity. Based on fluids properties, operational conditions and geometrical characteristics the maximum time-step size has been defined for each simulation presented below.The same discretization schemes have been used for all simulations, as reported in Table <ref>.§ BENCHMARKING VALIDATION §.§ Phenomenological benchmarksThe level of currents that represent a concern inside Borexino's IV is derived from the background's half-life (τ_1/2^^210Po∼ 138 days) and the IV's dimensions (4.25 m nominal radius, or ∼1 m from the vessel where the ^210Pb progenitor sits and provides an "inexhaustible" ^210Po source for our purposes). This means that currents under 𝒪(10^-7) m/s would be so slow that more than half of the detached polonium will decay away in the trip, even under directly radial motion. Therefore, the level of admissible numerically-induced systematic uncertainties should not exceed this magnitude and ideally be ≤𝒪(10^-8) m/s.Simple scenarios involving a cylindrical 2D geometry were implemented to characterize the basic phenomena at work in a well-studied scenario with a regular square mesh that ideally avoids the creation of preferential directions. This rectangular mesh grid employed features an average cell size of ∼3 cm (11 cm^2). Initial conditions for all cases were set as a linearly-stratified temperature gradient of [10,18]^∘C according to: T(h)=T_2 + (T_1-T_2) h-h_0/H where H is the cylinder's height (13.7 m, to keep it within Borexino's dimensions, along with its width of 11.2 m) and T_1 (T_2) is the top (bottom) temperature.A scenario where no motion would be expected established the level of irreducible numerical noise for the model at <3.5·10^-7, with an undefined, random pattern across the model.Sudden temperature variations on the bottom (raising T) and/or top (reduced T) surfaces, keeping the walls with an adiabatic boundary condition, showed regional effects circumscribed to those areas, which extended only until the height of the corresponding interior isotherm was reached. Threshold for convection triggering recirculation cell formation was Δ T >0.1^∘C. Equivalent dynamics were found for both top and bottom. Largest achieved currents were 𝒪(dm/s). This result proves the inherently-safe principle of operation for the Active Gradient Stabilization System based on heat application on Borexino's top dome<cit.>.In contrast, the application of non-adiabatic boundary conditions to all boundary surfaces, including the walls, prompted the generation of a global convective mode spanning the whole cylinder geometry, showing varied characteristics depending on the Δ T, but organized around robust currents of rising/falling fluid along the walls, and weaker recirculation currents along the central axis, with the addition of meandering horizontal currents or recirculation cells for large Δ T. Uniform, height-weighted or delayed (through the addition of varying thicknesses of insulation on the model's boundaries) Δ Ts were employed to characterize the different behaviors –nevertheless, an important final conclusion is that there is no "allowable" threshold on the amount of temperature difference that would not induce a global circulation pattern, if Δ T/Δ t is small. Some currents enter the realm of the resolution limit for the model (𝒪(10^-9-10^-8 m/s), and would not be relevant in Borexino's case, but the organized convective structure remains in place. Literature models and conditions Reproduction of experimental results from selected literature examples were identified as the proper benchmarking strategy. The selected benchmarks <cit.> are characterized by two-dimensional geometry, with circular cylindrical section (or spherical, but it is identical in the two-dimensional case) and the presence of a concentric annulus fully contained inside, with different ratios between the inner and outer radii, as shown in Table <ref>. The inner annulus' outer surface is the one at high temperature and the exterior's inner one at low temperature for all cases presented here. The analysis focuses on the fluid behavior in the space between and it is initially considered at a constant, volume-weighted, mean temperature defined by: T_m = (r_av^3-r_i^3)T_i + (r_o^3-r_av^3) T_o/r_o^3-r_i^3 where r_av is the average radius (r_o+r_i)/2 and r_i (T_i), r_o (T_o) are the inner and outer radii (temperatures), respectively<cit.>.The numerical geometry for the benchmarking cases is a 2D model of the real geometry, with a structured mesh including a local refinement close to the walls. In all the geometries the minimum cell size (Δ x)_min ranges from 210^-4 to 510^-5 m and the number of cells from 2.310^4 to 3.6810^5 for coarse and fine mesh, respectively. Numerical methods and algorithms are those described in Section <ref>, with a time-step sizeΔ t ranging from 2 to 9 s for fine and coarse mesh, respectivelyFor the first cases, taken from<cit.>, the Grashof numbers given in the reference were converted to the dimensionless Rayleigh number and this was taken as reference to calculate the inner/outer surface temperatures, as well as the fluid's parameters according to Equation <ref>. This involved an amount of (informed) arbitrariness, since the literature reference did not indicate the absolute temperatures they worked with. For that reason, ranges around Borexino's 10-20^∘C were chosen when possible. The fluid employed was water, since the reference parameters are much better constrained than for PC/benzene at the small Δ Ts involved. Sometimes, owing to the low Ra used, the temperature difference for a high-viscosity fluid such as water would be too small (<𝒪(10^-3 ^∘C)), and air was chosen instead. Dimensions were kept as in the reference.ResultsA summary of the achieved results, based on the main relevant features in each of the benchmark cases, is included in Table <ref>.We can confront the average Nusselt number Nu from the inner cylinder's exchanged power (59.654 W). The Rayleigh number is the adimensional number to compare natural convection in different cases while the Nusselt number compares the heat exchange behavior of different cases. From the reference, we know that: Nu_conv = h_i D_i/κ where h_i is the local heat transfer coefficient in the inner cylinder, D_i is its diameter and κ is the thermal conductivity. We also know that Nu = κ_eq· Nu_cond, and since Nu_cond = 2 / ln(D_o/D_i)=2.09, we can calculate the Nu from the κ_eq=7.88 provided in the reference's Table 1 (for our Ra=2.51·10^6). Indeed, the total average Nu is then Nu=7.88·2.09=16.47.We shall compare this adimensional number to the one obtained through the data from the simulation, because the Nusselt number offers a way to find a correspondence between very different cases, from the point of view of geometry and fluids, with regard to heat exchange. It is noted the heat transfer coefficient cannot be the same as in the reference case because we use water instead of air for reference fluid, among other model considerations. However, the fact that the model is not precisely equal to that in the paper increases confidence in the validity of the modelization. In any case, from the numerical point of view, we have a different κ_eq, and we want to obtain the Nu shown in Equation <ref>, so we need h_i, defined as: h_i = Q/π D_i Z (T_i-T_o) where Q is the total power exchanged and Z is the depth of the cylinder (20.8 cm). The Δ T is 9.26^∘C. With this data, we have h_i=276. This yields a Nu=16.431 (with a thermal conductivity κ=0.6 W/(m·K)), which is in very good agreement with the Nu found with the reference data.We can also compare the h for different points along the inner/outer cylinder's surface to establish a comparison with Figure 8 in the reference, which shows the local heat transfer coefficient versus the angular position considered, from the total surface heat flux (W/m^2) shown in Figure <ref>. Values were converted to Nusselt number in order to establish a one-to-one comparison irrespective of differences in the employed fluid.From the values reported, we can readily see the extremal point for the inner cylinder at 0^∘ (top) tends to zero, as is the case for the simulated model. This is expectable since it will be in this region where the "chimney" structure will rise, at a very similar temperature to the imposed inner cylinder one. This tendency starts at a steep decline ∼4 on the κ_eq (i.e. 4·2.09=8.36) on the reference plot, which is in reasonable agreement with the value we get from the ∼1000 W/m^2 from the FLUENT plot, from the formula h_i = q_w_i/ Δ T and converting that to a Nusselt number by dividing by the thermal conductivity κ=0.6 W/(m·K), into Nu=6.4. On the other extremal point at 180^∘, which in the reference Figure 8 lies at κ_eq∼11 for the inner cylinder, here we find a value of ∼3.7·10^3W/m^2. This translates to a Nusselt of Nu=23.7, in even better agreement with the plot-provided value of Nu=11·2.09≈23.For the outer cylinder, we can do similar calculations to compare the extremal point at 0^∘ between the κ_eq∼31 in the reference plot (Nu≈64.8) and the 4500 W/m^2 from the surface heat flux plot (Nu=74.9), or the plateau value at ∼60^∘ of around 11 in the reference plot (Nu ≈23) and the ∼1500 W/m^2 in the surface heat flux plot (Nu=24.97). The curves also follow similar trends, as can be appreciated in Figure <ref>.In conclusion, the benchmarking showed good reproducibility of the thermal environment (when available to compare in the references) as well as the large- and medium-scale features present in each of the cases. Furthermore, while some small-scale features were not well-reproduced in some of the lowest Rayleigh number cases (and thus further from Borexino's regime), the general fluid flow pattern was faithfully reproduced in practically all regions of all cases. §.§ Borexino thermal benchmarkIt is advisable to benchmark not only the general reproducibility of results just described, but also the model behavior in our particular cases of interest. Although we have no way of directly measuring fluid-dynamic effects inside the SSS, apart from the limited inference obtained from the background movement analyses, we do have a good thermal transport probing system: the Phase I LTPS sensors. Indeed, we were able to empirically measure the time constant for the thermal inertia between the inside and outside of the SSS from the Phase I.a and I.b sensors<cit.>. We can now employ the temperatures registered on the outside (i.e. in the water around the SSS, through the Phase I.b probes) and study their transmission toward the inside of the Sphere. The most interesting feature here is that the simulated transmitted temperatures for the inside of the SSS can be confronted with the internal recorded temperatures (i.e. in the buffer just inside the SSS, through the Phase I.a probes), and therefore establish the level of fidelity on thermal transport that the CFD strategy can offer.The WaterRing geometry has a 0.5-m-thick water volume around a PC-filled SSS segmented by spherical, rigid unsupported vessels. The water volume around the Sphere is a ring truncated at the poles, to avoid complications with interpolating the temperatures at higher/lower regions than the approximate measuring heights of the temperature probes. A separate initialization was used for the water and SSS' interior, using the following linear interpolation: T_N/S(t,y) = 1/h_1^i-h_0^i( T_N/S^i+1(t) (y-h_0^i) + T_N/S^i(t) (h_1^i-y) ) where h^i_0/1 is the interpolation domain's upper(lower) height limit, which obviously coincides with the lower(upper) limit for the next i+1 (previous i-1) domain; y is the vertical coordinate in any of the model's internal points, and T_N/S^i are the recorded (North or South side) temperatures used as anchors on the square domains' corners, taken from the historical, time-varying LTPS Phase I(.a or .b) dataset. Care was exercised in order to keep the vertical coordinate of the LTPS probes the same as the domains' vertical limits, even though the horizontal position of the anchor point may vary slightly (𝒪(10cm)) to ensure smooth physical interpolation within the internal region of interest. The fact that this model is closed and enables fluid movement requires careful handling of the simulating conditions, in particular to the iterative timing, mesh geometry and iteration divergence probability. To ensure an appropriate numerical modeling the mesh for the external part of the WaterRing benchmark has been meshed using a radial Cartesian grid, with local refinement close to the boundaries representing the internal buffers. The internal part, representing the IV, has been meshed using both Cartesian and polygonal meshes in order to check accuracy and independence of numerical results from it. Such analysis showed a dependence for the Cartesian grid, with preferential direction of the fluid inside the IV, and the total independence with the use of the polygonal mesh. Based on such analysis and the guidelines carried out from the benchmarking cases reported in Section<ref> and <ref> the typical cell size (Δ x) is around 310^-2 m, the number of cells 1.210^5 and the time-step sizeΔ t equal to 9s. This enabled for the best compromise between computational efficiency and expediency, and appropriate low numerical noise levels as described above. A custom-made software tool took care of checking, at each iteration, at which point the simulated time was, comparing it with the listed times in the recorded data from the LTPS probes. Once this simulated time reached or exceeded a given time limit (set at 1800 s, since that is the standard time delay between data acquisitions by the LTPS sensors), the appropriate historical temperatures were updated as imposed boundary conditions on the water ring's surface. Provisions were implemented to ensure good data would always be available: in case of dropouts in DAQ, the imposed temperature would be kept at the last available time. This can cause a slight upset once new data is available, but the need to select relatively small periods for computation efficiency meant the dropout periods were short and few –furthermore, these jumps were verified not to cause large enough deviations on the boundary conditions to motivate divergences or unphysical effects on the numerical solver. Conditions inside the SSS were left free too evolve when t≠0, including on its boundary. This smoothed out the ring-to-SSS initialization differences in a few numerical iterations.This model's benchmarking power was realized by placing "tallying spots" in the nominal positions where the Phase I.a sensors would be. Simulated temperatures on these points, 1 m away from the imposed boundary condition, could be checked against the historical recorded data, to verify good thermal transport behavior across this instrumented region, representative of the whole Inner Detector. These cases focused on ∼1-month periods during the Transient and Insulated periods described in <cit.>. The earliest possible date for this model would be April 10th, 2015, since the Phase I.b sensors entered operation then. This roughly coincides with the end of the Transient period. Results Figure <ref> shows the residuals ("true" historical temperature minus simulated temperature at the same time and position) for the 14 positions of the LTPS Phase I.a probes. Good agreement can be seen, with a smooth exponential trend toward stable errors, up to ∼ <2.25^∘C –although equilibrium errors are no bigger than ∼0.15^∘C. Discretization and temperature interpolation account for this level of errors. The overall trend shows a remarkable agreement between recorded and simulated data, as shown in Figure <ref>. Even more remarkably, the simulation shows a situation whereupon the temperature initialization profile that sent temperatures to slightly incorrect values, due to the interpolation strategy followed, is corrected by the time evolution profile and allows the behavior to follow the recorded data profile within 2 days of simulated time. It should be noted the 1-1.3 m/day thermal inertia in the detector<cit.> accounts for at least part of this delay.A temporal phase shift is evident when focusing on the sharpest available features in the recorded temperatures and their simulated counterpart, as depicted, for example, in Figure <ref>. This shift is constant and the features (if the phase shift is cancelled out manually) are seen to line up almost perfectly, albeit with a certain –small– decrease in slope change. The cause for this effect is still under investigation, but is considered not to negatively impact the overall reliability of the thermal transport benchmarking power of this model, given it represents a small and constant shift, and is moreover shown not to cause feature broadening.The Water Ring models are seen to provide a powerful benchmark for thermal transport, quite faithfully (< ±0.2^∘C, and much better in some cases) replicating the temperature evolution in the OB's LTPS Phase I.a probes positions when the boundary condition represented by the Phase I.b sensors is imposed ∼1 m away, in a different medium (water) and having to pass through the SSS structural element. Therefore, at least as far as thermal transport capabilities, the implemented FLUENT models are a useful tool to understand, replicate and foresee the thermal environment in the detector. Further, it is reasonable to believe that this extrapolation will hold, for the same geometry (and possibly for similar ones), at other points in the model. It is not, however, a benchmark for fluid transport: there is no "ground truth" data from Borexino, since no "tracer" is available –other than the same backgrounds we are trying to study through this research.§ FLUID DYNAMICS IN THE ROI (SSS→IV)The WaterRing benchmarking model was adapted to include just the Inner Detector, neglecting the water around it, and imposing the Phase I.a buffer probes readings on the SSS boundary, assuming a constant temperature from the real position of the LTPS sensors to the SSS surface, at the same vertical position. An increase of node density (with several cycles of mesh adaptation to improve performance and reliability) and a finer timestep of 4.5 simulated seconds per iteration was thus possible. CFD-derived temperatures were recorded at several points on the Inner Vessel in order to impose them as boundary conditions later on. Effectively, the benchmarking power of the Water Ring models was employed as a confidence anchor to ensure the temperatures at the vessel boundary, where no probe is available, would be accurate. These measures should allow for minimization of systematic model-dependent errors, which could not be shown to be uncorrelated enough with the Water Ring mesh geometries. §.§ IV-onlySetup The IV is modeled as a perfect circle of nominal Inner Vessel radius of 4.25 m. No attempt at modeling the actual vessel shape is made, although the deviations are small enough (≤±20 cm, or ∼0.2%) for this to be a good approximation. The polygonal mesh approach is used, with a cell size of ∼5 cm^2 (𝒪(10^5) cells. No internal structures or localized mesh tightening is employed away from the model's boundary. Initialization is performed picking the simulated temperatures a few centimeters outside the vessel in the Water Ring models. This small distance away from the vessel is chosen so as to avoid boundary layer effects that may locally shift the isotherms in a way that would falsify the most realistic temperature mapping in the bulk of the IV. As such, these temperatures were also used as input for a time-evolution script, in order to impose time-varying boundary conditions on the model's outer wall. Boundary conditions A perfectly-stratified model was first run in order to characterize the level of unphysical currents induced by the numerical iterative process, yielding a background level of ∼𝒪(10^-5) m/s. It is noted part of these currents, despite having a physical origin (especially in the horizontal direction), are mesh-enhanced. The absence of boundary currents along the vessel is notable, in sharp contrast with the model with time-changing realistic temperatures imposed as boundary conditions. The model's intrinsic numerical noise level, if the mesh can be regularized, can be much higher (𝒪(10^-8-10^-9) m/s), although the circular geometry of this model prevents a mesh that is completely regular all over the geometry. Trial runs with a so-called "quad" mesh, with four approximately-checkerboard patterns that converge in a central rectangular mesh, akin but not equal to the very first meshes employed in these simulations, yielded these order-of-magnitude currents, but induced localized unphysical instabilities in the transition areas that would be unassumable in a realistic case. A three-dimensional model may be able to sidestep these geometrical instabilities, but the large computational time required left this potential for numerical noise reduction as a future perspective to be developed.Five time-dependent models were then run, using the temperatures derived from as many other Water Ring cases. These were chosen from the Phase I.b dataset in order to characterize the model in the most diverse array of thermal profiles as possible, from mid-2015 to the latest data available in late winter 2017. This time span comprises the end of the "Transient" period<cit.> and four regimes during the "Insulated" period, including the phase with the warm, stable top, and the later phase with a relatively-rapidly cooling top. The simulated time periods were between 2 and 8 weeks long. §.§ ResultsHorizontal currents are the dominant feature in all of the convective models considered with Borexino geometries and data-based temperature fields. Indeed, as expectable from the Rayleigh number calculation in Section <ref>, the large energy gap between the top and bottom, separating the stably-stratified fluid layers, precludes bulk, organized motion in the IV. The spherical geometry, in contrast to the cylindrical symmetry from the cases in Section <ref>, will have a role on this –but asymmetries between both sides are the main driving force behind this preponderance of horizontal currents in place of vertical ones in realistic Borexino scenarios. Indeed, these asymmetries favor elongated recirculation cells that transport fluid from one side of the Sphere to the other one, while leaving the stratification in place, when they favor the breaking of organized vertical movement into minimal-energy transport along slightly-inclined isotherms.This is better observed through iso-stream-function views, which show the discretized local fluid-carrying capacity of the flow (see Figure <ref>), and allow for a better identification of the general fluid behavior, rather than studying the current velocities directly. These may skew the attention toward small high-speed regions (which can be very localized and have no global importance, or even be numerically-induced by mesh irregularities): while these regions may propel fluid, and with it, background radioisotopes, at the highest velocities, they need not account for most of the background injection into the FV –indeed, often they are not correlated. Current velocities may be useful for determining organized global movement (i.e. in case there was a vertical global trend, as in the cylindrical benchmarks), of which we see no evidence in these cases. Tracking virtual massless particles "attached to the fluid" through the pathline Fluent utility provides confirmation of this fact.Furthermore, a common characteristic to all models is a strong (up to 5 orders of magnitude) surface current along the vessel internal periphery (see Figure <ref>), although in general it does not span the whole surface with the same direction, owing to the inhomogeneous vertical separation between isotherms. It is worthwhile to note that this feature was not present in any area of the stably-stratified control model, indicating it is a feature uniquely derived from the boundary conditions and geometry, but definitely not numerically-induced. This feature was hypothesized as an explanation for the introduction of background components (particularly ^210Po) from the less-radiopure vessel nylon into the FV's scintillator, although its detachment mechanism was the subject of much speculation and remained unexplained before numerical simulations were carried out.Interestingly, this rules out the idea of a "chimney" effect picking up radioimpurities from the vessel and propelling them up or down the vertical axis of the vessel. For similar temperature fields, and changes, as the ones studied here for the 2015-17 timespan, the dominant mixing mechanism is instead mainly horizontal. Extended discussion of background stability and transport with fluid movement will not be further discussed here, being out of the scope of the current CFD work. Instead, it will be reported in an upcoming publication detailing the techniques used for low-statistics background tracking through data selection and localization, and its correlation with CFD scenarios under development.Horizontal velocities between 𝒪(10^-5) and 𝒪(10^-7) m/s were seen to be the largest in magnitude in all models, including the control scenario with ideally stratified bulk with adiabatic boundary conditions. Even though these currents are seen to be mesh-enhanced locally, thanks to the aforementioned control stratified adiabatic case, large scale features are thought to be physical, since a markedly different horizontal current distribution is seen in the realistic scenarios, and much less horizontal organization develops in the stratified case, as well as with slower currents.No large-scale organized vertical motion was seen to exist, and areas of vertical velocity larger than mesh cell hotspots (still, no larger than a few tens of centimeters) are typically between 𝒪(10^-7) and 𝒪(10^-9) m/s. This discussion excludes the boundary layer in close proximity to the vessel boundary, where current magnitudes are much larger (up to ∼10^-3 m/s in some cases).§ CONCLUSIONS AND PROSPECTSThe current work provides valuable insights into the determination of non-turbulent convective dynamics in a closed, pseudo-stably stratified system such as Borexino, showing excellent benchmarking reproducibility in the Rayleigh number ranges of 10^5-10^7, and appropriate large-scale reproducibility down to 𝒪(10^3). Attainment of global, organized convective modes is seen to exist in cylindrical geometries with no threshold in the time of Δ T application, its lateral symmetry or its magnitude, as long as this Δ T is applied on the lateral walls. Conversely, only local convection will appear, up to the vertical distance the isotherms will be displaced, when applying the Δ T to just the terminal caps. In a spherical geometry such as the Borexino detector, good thermal reproducibility was reached by comparing the large historical dataset of recorded temperatures to the temperature field obtained in CFD runs at the same positions. This allowed for the study of the smallest unsegmented region of interest possible, the Inner Volume of the detector, where the understanding of small currents causing the mixing of the scintillator it contains is critical to further improve radioactive background levels that may allow the detector data to be of even higher quality. Horizontal movement caused by lateral asymmetrical imbalances in the boundary conditions is seen to be the main driving factor in bringing fluid from the periphery to the center of the volume, while no global forced convective fields are seen to develop. Observed horizontal currents are seen to be of the order and span that would be of concern for background transport (> 𝒪(10^-7), but this is not so for vertical currents. Carrying capacity maps ("iso-stream-function plots") clearly mark the regions where most of the fluid motion occurs, offering a powerful tool to understanding past behavior in the detector, as well as to engineer minimal-mixing temperature profiles for future directives toward establishing an ultra-low level of scintillator mixing –and, with it, unprecedented levels of radioactive background presence in Borexino's Fiducial Volume.The present results may not only inform the particular case of the Borexino neutrino observatory's internal facilities, but also expand the limited modeling of non-turbulent fluid mixing in pseudo-steady-state closed systems near equilibrium, subject to small asymmetrical perturbations in their temperature field, such as liquid reservoirs (water, liquified gas, petroleum-derived, deep cryogenics...), which share equivalent geometries and conditions.§ ACKNOWLEDGEMENTS The Borexino program is made possible by funding from INFN (Italy), NSF (USA), BMBF, DFG, HGF and MPG (Germany), RFBR (Grants 16-02-01026 A, 15-02-02117 A, 16-29-13014 ofi-m, 17-02-00305 A) (Russia), and NCN Poland (Grant No. UMO-2013/10/E/ST2/00180). We acknowledge the generous hospitality and support of the Laboratori Nazionali del Gran Sasso (Italy). All numerical simulations reported in this study are performed in the HPC system of the interdepartmental laboratory 'CFDHub' of Politecnico di Milano.§ REFERENCES | http://arxiv.org/abs/1705.09658v1 | {
"authors": [
"David Bravo-Berguño",
"Riccardo Mereu",
"Robert Bruce Vogelaar",
"Fabio Inzoli"
],
"categories": [
"physics.ins-det",
"hep-ex"
],
"primary_category": "physics.ins-det",
"published": "20170526150059",
"title": "Fluid-dynamics in the Borexino Neutrino Detector: behavior of a pseudo-stably-stratified, near-equilibrium closed system under asymmetrical, changing boundary conditions"
} |
Automatic Response Assessment in Regions of Language Cortex in Epilepsy Patients Using ECoG-basedFunctional Mapping and Machine Learning Harish RaviPrakash Center for Research in Computer VisionCollege of Engineering and Computer ScienceUniversity of Central FloridaOrlando, Florida 32826Milena KorostenskajaFunctional Brain Mappingand Brain Computer Interface Lab Florida Hospital for Children Orlando, Florida - 32803Ki LeeFunctional Brain Mappingand Brain Computer Interface Lab Florida Hospital for Children Orlando, Florida - 32803James BaumgartnerFunctional Brain Mappingand Brain Computer Interface Lab Florida Hospital for Children Orlando, Florida - 32803Eduardo CastilloMEG Lab Florida Hospital for ChildrenOrlando, Florida 32803Ulas BagciCenter for Research in Computer Vision College of Engineering and Computer Science University of Central Florida Orlando, Florida 32826 December 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= We study causal inference in a multi-environment setting, in which the functional relations for producing the variables from their direct causes remain the same across environments, while the distribution of exogenous noises may vary. We introduce the idea of using the invariance of the functional relations of the variables to their causes across a set of environments. We define a notion of completeness for a causal inference algorithm in this setting and prove the existence of such algorithm by proposing the baseline algorithm. Additionally, we present an alternate algorithm that has significantly improved computational and sample complexity compared to the baseline algorithm. The experiment results show that the proposed algorithm outperforms the other existing algorithms.§ INTRODUCTION Causal inference is a fundamental problemin machine learning with applications in several fields such as biology, economics, epidemiology, computer science, etc.When performing interventions in the system is not possible (observation-only setting), the main approach to identifying direction of influences and learning the causal structure is to perform statistical tests based on the conditional dependency of the variables on the data <cit.>. In this case, a “complete” conditional independence based algorithm allows learning the causal structure to the extent possible, where by complete we mean that the algorithm is capable of distinguishing all the orientations up to the Markov equivalence. Such algorithms perform a conditional independence test along with the Meek rules[Recursive application of Meek rules identifies the orientation of additional edges to obtain the Markov equivalence class.] introduced in <cit.>. IC <cit.> and PC <cit.> algorithms are two well known examples. Within the framework of structural equation models (SEMs) <cit.>, by adding assumptions to the model such as non-Gaussianity <cit.>, nonlinearity <cit.> or equal noise variances <cit.>, it is even possible to identify the exact causal structure. When the experimenter is capable of intervening in the system to see the effect of varying one variable on the other variables in the system (interventional setting), the causal structure could be exactly learned. In this setting, the most common identification procedure assumes that the variables whose distributions have varied are the descendants of the intervened variable and hence the causal structure is reconstructed by performing interventions in different variables in the system <cit.>.We take a different approach from the traditional interventional setting by considering a multi-environment setting, in which the functional relations for producing the variables from their parents remain the same across environments, while the distribution of exogenous noises may vary. This is different from the customary interventional setting, because in our model, the experimenter does not have any control on the location of the changes in the system, and as will be seen in Figure <ref>(a), this may prevent the ordinary interventional approaches from working. The multi-environment setting was also studied in <cit.> and <cit.>; we will put our work into perspective in relationship to these in the related work below.We focus on the linear SEM with additive noise as the underlying data generating model (see Section <ref> for details). Note that this model is one of the most problematic models in the literature of causal inference, and if the noises have Gaussian distribution, for many structures, none of the existing observational approaches can identify the underlying causal structure uniquely[As noted in <cit.>, “nonlinearities can play a role similar to that of non-Gaussianity”, and both lead to exact structure recovery.].The main idea in our proposed approach is to utilize the change of the regression coefficients, resulting from the changes across the environments to distinguish causes from the effects. r0.4< g r a p h i c s >Simple examples of identifiable structures using the proposed approach. Our approach is able to identify causal structures previously not identifiable using the existing approaches.Figure <ref> shows two simple examples to illustrate this point. In this figure, a directed edge form variable X_i to X_j implies that X_i is a direct cause of X_j, and change of an exogenous noise across environments is denoted by the flash sign. Consider the structure inFigure <ref>(a), with equations X_1=N_1, and X_2=aX_1+N_2, where N_1∼𝒩(0,σ_1^2) and N_2∼𝒩(0,σ_2^2) are independent mean-zero Gaussian exogenous noises. Suppose we are interested in finding out which variable is the cause and which is the effect. We are given two environments across which the exogenous noise of both X_1 and X_2 are varied.Denoting the regression coefficient resulting from regressing X_i on X_j by β_X_j(X_i), in this case, we have β_X_2(X_1)=Cov(X_1X_2)/Cov(X_2)=aσ_1^2/a^2σ_1^2+σ_2^2, andβ_X_1(X_2)=Cov(X_1X_2)/Cov(X_1)=a. Therefore, except for pathological cases for values for the variance of the exogenous noises in two environments, the regression coefficient resulting from regressing the cause variable on the effect variable varies between the two environments, while the regression coefficient from regressing the effect variable on the cause variable remains the same. Hence, the cause is distinguishable from the effect. Note that structures X_1→ X_2 and X_2→ X_1 are in the same Markov equivalence class and hence, not distinguishable using merely conditional independence tests. Also since the exogenous noises of both variables have changed, commonly used interventional tests are also not capable of distinguishing between these two structures <cit.>. Moreover, as it will be shortly explained (see related work), because the exogenous noise of the target variable has changed, the invariant prediction method <cit.>, cannot discern the correct structure either.As another example, consider the structure in Figure <ref>(b). Suppose the exogenous noise of X_1 is varied across the two environments. Similar to the previous example, it can be shown that β_X_2(X_1) varies across the two environments while β_X_1(X_2) remains the same. This implies that the edge between X_1 and X_2 is from the former to the later. Similarly, β_X_3(X_2) varies across the two environments while β_X_2(X_3) remains the same. This implies that X_2 is the parent of X_3. Therefore, the structure in Figure <ref>(b) is distinguishable using the proposed identification approach. Note that the invariant prediction method cannot identify the relation between X_2 and X_3, and conditional independence tests are also not able to distinguish this structure.Related Work.The best known algorithms for causal inference in the observational setup are IC <cit.> and PC <cit.> algorithms. Such purely observational approaches reconstruct the causal graph up to Markov equivalence classes. Thus, directions of some edges may remain unresolved. There are studies which attempt to identify the exact causal structure by restricting the model class <cit.>. Most of such work consider SEM with independent noise. LiNGAM method <cit.> is a potent approach capable of structure learning in linear SEM model with additive noise[ There are extensions to LiNGAM beyond linear model <cit.>.], as long as the distribution of the noise is not Gaussian. Authors of <cit.> showed that nonlinearities can play a role similar to that of non-Gaussianity.In interventional approach for causal structure learning, the experimenter picks specific variables and attempts to learn their relation with other variables, by observing the effect of perturbing that variables on the distribution of others. In recent work, bounds on the required number of interventions for complete discovery of causal relationships as well as passive and adaptive algorithms forminimize the number of experiments were derived <cit.><cit.><cit.> <cit.>. In this work we assume that the functional relations of the variables to their direct causes across a set of environments are invariant. Similar assumptions have been considered in other work <cit.>. Specifically, <cit.> which studies finding causal relation between two variables related to each other by an invertible function, assumes that “ the distribution of the cause and the function mapping cause to effect are independent since they correspond to independent mechanisms of nature”.There is little work on multi-environment setup <cit.>. In <cit.>, the authors analyze the classes of structures that are equivalent relative to a stream of distributions and present algorithms that output graphical representations of these equivalence classes. They assume that changing the distribution of a variable, varies the marginal distribution of all its descendants. Naturally this also assumes that they have access to enough samples to test each variable for marginal distribution change. This approach cannot identify the causal relations among variables which are affected by environment changes in the same way. The most closely related work to our approach is the invariant prediction method <cit.>, which utilizes different environments to estimate the set of predictors of a target variable. In that work, it is assumed that the exogenous noise of the target variable does not vary among the environments. In fact, the method crucially relies on this assumption as it adds variables to the estimated predictors set only if they are necessary to keep the distribution of the target variable's noise fixed.Besides high computational complexity, invariant prediction framework may result ina set which does not contain all the parents of the target variable. Additionally, the optimal predictor set (output of the algorithm) is not necessarily unique. We will show that in many cases our proposed approach can overcome both these issues. Recently, the authors of <cit.> considered the setting in which changes in the mechanism of variables prevents ordinary conditional independence based algorithms from discovering the correct structure. The authors have modeled these changes as multiple environments and proposed a general solution for a non-parametric model which first detects the variables whose mechanism changed and then finds causal relations among variables using conditional independence tests. Due to the generality of the model, this method requires a high number of samples. Contribution. We propose a novel causal structure learning framework, which is capable of uniquely identifying structures which were not identifiable using existing methods. The main contribution of this work is to introduce the idea of using the invariance of the functional relations of the variables to their direct causes across a set of environments. This would imply the invariance of coefficients in the special case of linear SEM, in distinguishing the causes from the effects. We define a notion of completeness for a causal inference algorithm in this setting and prove the existence of such algorithm by proposing the baseline algorithm (Section 3). This algorithm first finds the set of variables for which distributions of noises have varied across the two environments, and then uses this information to identify the causal structure. Additionally, we present an alternate algorithm (Section <ref>) which has significantly improved computational and sample complexity compared to the baseline algorithm. § REGRESSION-BASED CAUSAL STRUCTURE LEARNINGConsider a directed graph G=(V,E) with vertex set V and set of directed edges E. G is a DAG if it is a finite graph with no directed cycles. A DAG G is called causal if its vertices represent random variables V={X_1, ...,X_n} and a directed edges (X_i,X_j) indicates that variable X_i is a direct cause of variable X_j.We consider a linear SEM <cit.>as the underlying data generating model. In such a model the value of each variable X_j ∈ V is determined by a linear combination of the values of its causal parents PA(X_j) plus an additive exogenous noise N_j, where N_j's are jointly independent as followsX_j = ∑_X_i∈PA(X_j)b_jiX_i+N_j,∀j∈{1,⋯,p},which could be represented by a single matrix equation 𝐗=𝐁𝐗+𝐍. Further, we can write𝐗=𝐀𝐍,where 𝐀=(𝐈-𝐁)^-1.This implies that each variable X∈ V can be written as a linear combination of the exogenous noises in the system. We assume that in our model, all variables are observable. Also, for the ease of representation, we focus on zero-mean Gaussian exogenous noise; otherwise, the results could be easily extended to any arbitrary distribution for the exogenous noise in the system. The following definitions will be used throughout the paper.Graph union of a set 𝒢 of mixed graphs[A mixed graph contains both directed and undirected edges.] over a skeleton, is a mixed graph with the same skeleton as the members of 𝒢 which contains directed edge (X,Y), if ∃ G∈𝒢 such that (X,Y)∈ E(G) and ∄ G'∈𝒢 such that (Y,X)∈ E(G'). The rest of the edges remain undirected.Causal DAGs G_1 and G_2 over V are Markov equivalent if every distribution that is compatible with one of the graphs is also compatible with the other. Markov equivalence is an equivalence relationship over the set of all graphs over V <cit.>.The graph union of all DAGs in the Markov equivalence class of a DAG G is called the essential graph of G and is denoted by Ess(G).We consider a multi-environment setting consisting of M environments ℰ={E_1,...,E_M}. The structure of the causal DAG and the functional relations for producing the variables from their parents (the matrix 𝐁), remains the same across all environments, the exogenous noises may vary though. For a pair of environments E_i,E_j∈ℰ, let I_ij be the set of variables whose exogenous noise changed between the two environments. Given I_ij, for any DAG G consistent with the essential graph[DAG G is consistent with mixed graph M, if G does not contain edge (X,Y) while M contains (Y,X).] obtained from the conditional independence test, define the regression invariance set as followsR(G,I_ij){(X,S):X∈ V,S⊆ V\{X},β^(i)_S(X)=β^(j)_S(X)},where β^(i)_S(X) and β^(j)_S(X) are the regression coefficients of regressing variable X on S in environments E_i and E_j, respectively. In words, for all variables X∈ V, R(G,I_ij) contains all subsets S⊆ V\{X} that if we regress X on S, the regression coefficients do not change across E_i and E_j. Given I, the set of variables whose exogenous noise has changed between two environments, DAGs G_1 and G_2 are called I-distinguishable if R(G_1,I)≠ R(G_2,I). We make the following assumption on the distributions of the exogenous noises. The purpose of this assumption is to rule out pathological cases for values of the variance of the exogenous noises in two environments which make special regression relations. For instance, in Example <ref>, β_X_2^(1) (X_1)=β_X_2^(2)(X_1) only if σ_1^2 σ̃_2^2=σ_2^2 σ̃_1^2 where σ_i^2 and σ̃_i^2 are the variances of the exogenous noise of X_i in the environments E_1 and E_2, respectively. Note that this special relation between σ_1^2, σ̃_1^2, σ_2^2, and σ̃_2^2 has Lebesgue measure zero in the set of all possible values for the variances.[Regression Stability Assumption]For a given set I and structure G, perturbing the variance of the distributions of the exogenous noises by a small value ϵ does not change the regression invariance set R(G,I). We give the following examples as applications of our approach.Consider DAGs G_1:X_1→ X_2 and G_2:X_1← X_2. For I={X_1}, I={X_2} or I={X_1,X_2}, calculating the regression coefficients as explained in Section <ref>, we see that (X_1,{X_2})∉R(G_1,I) but (X_1,{X_2})∈ R(G_2,I). Hence G_1 and G_2 are I-distinguishable. As mentioned in Section <ref>, structures G_1 and G_2 are not distinguishable using the ordinary conditional independence tests. Also, in the case of I={X_1,X_2}, the invariant prediction approach and the ordinary interventional tests - in which the experimenter expects that a change in the distribution of the effect would not perturb the marginal distribution of the cause variable - are not capable of distinguishing the two structures either. r0.4< g r a p h i c s >DAG related to Example 3.Consider the DAG G in Figure <ref>(b) with I={X_1}. Consider an alternative DAG G' in which compared to G the directed edge (X_1,X_2) is replaced by (X_2,X_1), and DAG G” in which compared to G the directed edge (X_2,X_3) is replaced by (X_3,X_2). Since (X_2,{X_1})∈ R(G,I) while this pair is not in R(G',I), and (X_2,{X_3})∉R(G,I) while this pair belongs to R(G”,I), the structure of G is also distinguishable using the proposed identification approach. Note that G is not distinguishable using conditional independence tests. Also, the invariant prediction method cannot identify the relation between X_2 and X_3, since it can keep the variance of the noise of X_3 fixed by setting the predictor set as {X_2} or {X_1}, which have empty intersection.Consider the structure in Figure <ref>(a) with I={X_2}. Among the six possible triangle DAGs, all of them are I-distinguishable from this structure and hence, with two environments differing in the exogenous noise of X_2, this triangle DAG could be identified. Note that all the triangle DAGs are in the same Markov equivalent class and hence, using the information of one environment alone, observation only setting cannot lead to identification. For I={X_1}, the structure in Figure <ref>(b) is not I-distinguishable from a triangle DAG in which the direction of the edge (X_2,X_3) is flipped. These two DAGs are also not distinguishable using usual intervention analysis and the invariant prediction method.Let the structure G^* be the ground truth DAG structure. Define 𝒢_I{G:R(G,I)=R(G^*,I)}, which is the set of all DAGs which are not I-distinguishable from G^*. Using this set, we form the mixed graph M_I over V, as the graph union of members of 𝒢_I. An algorithm 𝒜:(Ess(G),R)→ M which gets an essential graph and a regression invariance set as the input and returns a mixed graph, is regression invariance complete if𝒜(Ess(G^*),R(G^*,I))=M_I. for any directed graph G^* and set I. In other words, we say an algorithm 𝒜 is regression invariance complete if given the correct essential graph and regression invariance set, it is able to return the appropriatemixed graph.In Section <ref> we will introduce a structure learning algorithm which is complete in the sense of Definition <ref>. § EXISTENCE OF COMPLETE ALGORITHMS In this section we show the existence of complete algorithm for learning the causal structure among a set of variables V whose dynamics satisfy the SEM in (<ref>) in the sense of Definition <ref>. The pseudo-code of the algorithm is presented in Algorithm <ref>. r0.5 0.5Suppose G^* is the ground truth structure. The algorithm first performs a conditional independence test followed by applying Meek rules to obtain the essential graph Ess(G^*). For each pair of environments {E_i,E_j}∈ℰ, first the algorithm calculates the regression coefficients β^(i)_S(Y) and β^(j)_S(Y), for all Y∈ V and S⊆ V\{Y}, and forms the regression invariance set R_ij, which contains the pairs (Y,S) for which the regression coefficients did not change between E_i and E_j. Next, using the function ChangeFinder(·), we discover the set I_ij which is the set of variables whose exogenous noises have varied between the two environments E_i and E_j. Then using the function ConsistantFinder(·), we find 𝒢_ij which is the set of all possible DAGs, G that is consistent with Ess(G^*) and R(G,I_ij)=R_ij. After taking the union of graphs in 𝒢_ij, we form the graph M_ij which is the mixed graph containing all causal relations distinguishable from the given regression information between the two environments. Clearly, since we are searching over all DAGs, the baseline algorithm is complete in the sense of Definition <ref>.After obtaining M_ij for all pairs of environments, the algorithm forms a mixed graph M_ℰ by taking graph union of M_ij's. We perform the Meek rules on M_ℰ to find all extra orientations and output M̂. Obtaining the set R_ij: In this part, for a given significance level α, we will show how the set R_ij can be obtained correctly with probability at least 1-α. For given Y∈ V and S⊆ V\{Y} in the environments E_i and E_j, we define the null hypothesis H_0,Y,S^ij as follows:H_0,Y,S^ij: ∃β∈ℝ^|S|β^(i)_S(Y)=ββ^(j)_S(Y)=β.Let β̂^(i)_S(Y) and β̂^(j)_S(Y) be the estimations of β^(i)_S(Y) and β^(j)_S(Y), respectively, obtained using the ordinary least squares estimator computed from observational data.If the null hypothesis is true, then(β̂^(i)_S(Y)-β̂^(j)_S(Y))^T (s_i^2Σ_i^-1+s_j^2Σ_j^-1)^-1 (β̂^(i)_S(Y)-β̂^(j)_S(Y))/p ∼ F(p,n-p),where s^2_i and s_j^2 are unbiased estimates of variance of Y^(i)-X_S^(i)β^(i)_S(Y) and Y^(j)-X_S^(j)β^(j)_S(Y), respectively(see Appendix <ref> for details). Furthermore, we have Σ_i=(X^(i)_S)^T X^(i)_S and Σ_j=(X^(j)_S)^T X^(j)_S. We reject the null hypothesis H_0,Y,S^ij if the p-value of (<ref>) is less than α/(p× (2^p-1-1)). By testing all null hypotheses H_0,Y,S^ij for any Y∈ V and S⊆ V\{Y}, we can obtain the set R_ij correctly with probability at least 1-α.Function ChangeFinder(·): We use Lemma <ref> to find the set I_ij with probability at least 1-2α.Given environments E_i and E_j, for a variable Y∈ V, if 𝔼{(Y^(i)-X_S^(i)β^(i)_S(Y))^2}≠𝔼{(Y^(j)-X_S^(j)β_S^(j)(Y))^2} for all S⊆ N(Y) such that (Y,S)∈ R_ij, where N(Y) is the set of neighbors of Y, then the variance of exogenous noise N_Y is changed between the two environments. Otherwise, the variance of N_Y is fixed.See Appendix <ref> for the proof. Based on Lemma <ref>, we try to find a set S⊆ N(Y), (Y,S)∈ R_ij such that the variance of residual Y-X_Sβ_S(Y) remains fixed between two environments. To do so, we check whether the variance of exogenous noise N_Y is changed between two environments E_i and E_j by testing the following null hypothesis for any set S⊆ N(Y), (Y,S)∈ R_ij:H̅_0,Y,S^ij: ∃σ∈ℝ𝔼{(Y^(i)-X^(i)_Sβ_S^(i)(Y))^2}=σ^2 𝔼{(Y^(j)-X_S^(j)β_S^(j)(Y))^2}=σ^2. In order to test the above null hypothesis, we can compute the variance of residuals Y^(i)-X_S^(i)β̂_S^(i) and Y^(j)-X_S^(j)β̂_S^(j) and test whether these variances are equal using an F-test. If the p-value for the set S is less than α/(p× (2^Δ-1)), then we will reject the null hypothesis H̅_0,Y,S^ij where Δ is the maximum degree of the causal graph. Ifwe reject all hypothesis tests H̅_0,Y,S^ij for any S∈ N(Y), (Y,S)∈ R_ij, then we will add Y to set I_ij. Function ConsistentFinder(·): Let D_st be the set of all directed paths from variable X_s to variable X_t. For any d∈ D_st, we define the weight of directed path d ∈ D_st as w_d:= Π_(u,v)∈ d b_vu where b_vu are coefficients in (<ref>). By this definition, it can be seen that the entry (t,s) of matrix 𝐀 in (<ref>) is equal to [𝐀]_ts=∑_d∈ D_st w_d. Thus, the entries of matrix 𝐀 are multivariate polynomials of entries of 𝐁. Furthermore,β_S^(i)(Y)= 𝔼{ X^(i)_S (X^(i)_S)^T }^-1𝔼{X^(i)_S Y^(i)}= (𝐀_S Λ_i 𝐀_S^T)^-1𝐀_S Λ_i 𝐀^T_Y,where 𝐀_S and 𝐀_Y are the rows corresponding to set S and Y in matrix 𝐀, respectively and matrix Λ_i is a diagonal matrix where [Λ_i]_kk= 𝔼{(N^(i)_k)^2}.From the above discussion, we know that the entries of matrix 𝐀 are multivariate polynomials of entries of 𝐁. Equation (<ref>) implies that the entries of vector β_S^(i)(Y) are rational functions of entries in 𝐁 and Λ_i. Therefore, the entries of Jacobian matrix of β_S^(i)(Y) with respect to the diagonal entries of Λ_i are also rational expression of these parameters. In function ConsistentFinder(.), we select any directed graph G consistent withEss(G^*) and set b_vu=0 if (u,v)∉G. In order to check whether G is in 𝒢_ij, we initially set R(G,I_ij)=∅. Then, we compute the Jacobian matrix of β_S^(i)(Y) parametrically for any Y∈ V and S∈ V\{Y}. As noted above, the entries of Jacobian matrix can be obtained as rational expressions of entries in 𝐁 and Λ_i. If all columns of Jacobian matrix corresponding to the elements of I_ij are zero, then we add (Y,S) to set R(G,I_ij) (since β^(i)_S(Y) is not changing by varying the variances of exogenous noises in I_ij). After checking all Y∈ V and S∈ V\{Y}, we consider the graph G in 𝒢_ij if R(G,I_ij)=R_ij. § LRE ALGORITHM The baseline algorithm of Section <ref> is presented to prove the existence of complete algorithms but it is not practical due to its high computational and sample complexity. In this section we present the Local Regression Examiner (LRE) algorithm, which is an alternative much more efficient algorithm for learning the causal structure among a set of variables V. The pseudo-code of the algorithm is presented in Algorithm <ref>. We make use of the following result in this algorithm. Consider adjacent variables X, Y∈ V in causal structure G. For a pair of environments E_i and E_j, if (X,{Y})∈ R(G,I_ij), but (Y,{X})∉R(G,I_ij), then X is the parent of Y.See Appendix <ref> for the proof. LRE algorithm consists of three stages. In the first stage, similar to the baseline algorithm, it performs a complete conditional independence test to obtain the essential graph. Then for each variable X∈ V, it forms the set of X's discovered parents, PA(X), and discovered children, CH(X), and leaves the remaining neighbors as unknown in UK(X). In the second stage, the goal is that for each variable Y∈ V, we find Y's relation with its neighbors in UK(Y), based on the invariance of its regression on its neighbors across each pair of environments. To do so, for each pair of environments, after fixing a target variable Y and for each of its neighbors in UK(X), the regression coefficients of X on Y and Y on X are calculated. We will face one of the following cases: * If neither is changing, we do not make any decisions about the relationship of X and Y. This case is similar to having only one environment, similar to the setup in <cit.>. * If one is changing and the other is fixed, Lemma <ref> implies that the variable which fixes the coefficient as the regressor is the parent. * If both are changing, we look for an auxiliary set S among Y's neighbors with minimum number of elements, for which β^(i)_S∪{X}(Y)=β^(j)_S∪{X}(Y). If no such S is found, it implies that X is a child of Y. Otherwise, if S and X are both required in the regressors set to fix the coefficient,we set {X}∪ S as parents of Y; otherwise, if X is not required in the regressors set to fix the coefficient, although we still set S as parents of Y, we do not make any decisions regarding the relation of X and Y (Example <ref> when I={X_1}, is an instance of this case). After adding the discovered relationships to the initial mixed graph, in the third stage, we perform the Meek rules on resulting mixed graph to find all extra possible orientations and output M̂.Analysis of the Refined Algorithm.We can use the hypothesis testing in (<ref>) to test whether two vectors β_S^(i)(Y) and β_S^(j)(Y) are equal for any Y∈ V and S⊆ N(Y). If the p-value for the set S is less than α/(p× (2^Δ-1)), then we will reject the null hypothesis H_0,Y,S^ij. By doing so, the output of the algorithm will be correct with probability at least 1-α. Regarding the computational complexity, since for each pair of environments, in the worse case we perform Δ (2^Δ-1) hypothesis tests for each variable Y∈ V, and considering that we have M2 pairs of environments, the computational complexity of LRE algorithm is in the order of M2 pΔ(2^Δ-1). Therefore, the bottleneck in the complexity of LRE is having to perform a complete conditional independence test in its first stage.§ EXPERIMENTS We evaluate the performance of LRE algorithm by testing it on both synthetic and real data. As seen in the pseudo-code in Algorithm <ref>, LRE has three stages where in the first stage, a complete conditional independence is performed. In order to have acceptable time complexity, in our simulations, we used the PC algorithm[We use the pcalg package <cit.> to run the PC algorithm on a set of random variables.] <cit.>, which is known to have a complexity of order O(p^Δ) when applied to agraph of order p with degree bound Δ.Synthetic Data. We generated 100 DAGs of order p=10 by first selecting a causal order for variables and then connecting each pair of variables with probability 0.25. We generated data from a linear Gaussian SEM with coefficients drawn uniformly at random from [0.1,2], and the variance of each exogenous noise was drawn uniformly at random from [0.1,4]. For each variable of each structure, 10^5 samples were generated. In our simulation, we only consider a scenario in which we have two environments E_1 and E_2, where in the second environment, the exogenous noise of |I_12| variables were varied. The perturbed variables were chosen uniformly at random.Figure <ref> shows the error ratio and undirected edges (UD) ratio, for stage 1, which corresponds to the PC algorithm, and for the final output of LRE algorithm. Define a link to be any directed or undirected edge. The error ratio is calculated as follows: Error ratio(|miss-detected links|+|extra detected links|+|wrongly oriented edges|)/p2. For the UD ratio, we count the number of undirected edges only among correctly detected links, i.e., UD ratio(|correctly detected undirected edges|)/(|correctly detected directed edges|+|correctly detected undirected edges|). As seen in Figure <ref>, only one change in the second environment (i.e., |I_12|=1), reduces the UD ratio by 8 percent compared to the PC algorithm. Also, the main source of error in LRE algorithm results from the application of the PC algorithm. We also compared the error ratio of LRE algorithm with the Invariant Prediction (IP) <cit.> and LiNGAM <cit.> (since there is no undirected edges in the output of IP and LiNGAM, the UD ratio of both would be zero). For LiNGAM, we combined the data from two environments as the input. Therefore, the distribution of the exogenous noise of variables in I_12 is not Guassian anymore.As it can be seen in Figure <ref>(a), the error ratio of IP increases as the size of I_12 increases. This is mainly due to the fact thatin IP approach it is assumed that the distribution of exogenous noise of the target variable should not change, which may be violated by increasing |I_12|. The result of simulations shows that the error ratio of LiNGAM is approximately twice of those of LRE and PC.Real Data. We considered dataset of educational attainment of teenagers <cit.>. The dataset was collected from 4739 pupils from about 1100 US high school with 13 attributes including gender, race, base year composite test score, family income, whether the parent attended college, and county unemployment rate. We split the dataset into two parts where the first part includes data from all pupils who live closer than 10 miles to some 4-year college. In our experiment, we tried to identify the potential causes that influence the years of education the pupils received. We ran LRE algorithm on the two parts of data as two environments with a significance level of 0.01 and obtained the following attributes as a possible set of parents of the target variable: base year composite test score, whether father was a college graduate, race, and whether school was in urban area. The IP method <cit.> also showed that the first two attributes have significant effects on the target variable.abbrv§ DERIVATION OF EQUATION (<REF>) The null hypothesis H_0,Y,S^ij can be written in the following form: 𝐂[β_S^(i)(Y);β_S^(j)(Y)]=0 where 𝐂 is a |S|× (2|S|) matrix such that nonzero entries of 𝐂 are [𝐂]_k,k=1, [𝐂]_k,k+|S|=-1, for all 1≤ k≤ |S|. Thus, the following statistic (β̂^(i)_S(Y)-β̂^(j)_S(Y))^T (𝐂Σ̂𝐂^T)^-1 (β̂^(i)_S(Y)-β̂^(j)_S(Y))/phas a F(p,n-p) distribution <cit.> where Σ̂=[s_i^2 Σ_i^-1,0_|S|× |S|;0_|S|× |S|,s_j^2 Σ_j^-1]. Since 𝐂Σ̂𝐂^T=s_i^2 Σ_i^-1+s_j^2 Σ_j^-1, the statistic in (<ref>) has the same F(p,n-p) distribution. § PROOF OF LEMMA <REF>For any set S⊆ N(Y) and (Y,S)∈ R_ij, using representation (<ref>), we have:Y^(i)=∑_X_k∈AN(Y)\{Y} c_k N^(i)_k +N^(i)_Y, X_S^(i)β_S^(i)(Y)=∑_X_k ∈AN(Y)\{Y} b_k N^(i)_k+∑_X_k ∈AN(S_CH)\AN(Y) b'_k N^(i)_k + b_Y N^(i)_Y,where S_CH:=S∩CH(Y) and the ancestral set AN(X) of a variable X consists of X and all the ancestors of nodes in X. Moreover, coefficients b_k's and c_k's are functions of 𝐁 and β_S(Y) which are fixed in two environments. ThereforeY^(i)-X^(i)_Sβ_S^(i)(Y) = ∑_X_k ∈AN(Y)\{Y} (c_k-b_k) N^(i)_k -∑_X_k ∈AN(S_CH)\AN(Y) b'_k N^(i)_k +(1-b_Y) N^(i)_Y,If the variance of N_Y is not changed, then clearly for the choice of S=PA(Y), the second summation vanishes, and in the first summation c_k=b_k. Therefore, the variance of residual remains unvaried. Otherwise, if the variance of N_Y varies, then its change may cancel out only for specific values of the variances of other exogenous noises which according to a similar reasoning as the one in Assumption <ref>, we ignore it.§ PROOF OF LEMMA <REF> Suppose X is the parent of Y. Consider environments E_i, E_j∈ℰ. It suffices to show that if β_Y^(i)(X)=β_Y^(j)(X), then β_X^(i)(Y)=β_X^(j)(Y). Using representation (<ref>), X and Y can be expressed as followsX=∑_X_k∈AN(X)a_kN_kY=∑_X_k∈AN(X)b_kN_k+∑_X_k∈AN(Y)\AN(X)c_kN_k.Hence we have 𝔼[X^2]=∑_X_k∈AN(X)a^2_k var(N_k)𝔼[Y^2]=∑_X_k∈AN(X)b^2_kvar(N_k)+∑_X_k∈AN(Y)\AN(X)c^2_kvar(N_k)𝔼[XY]=∑_X_k∈AN(X)a_kb_k var(N_k)Thereforeβ_X(Y)=∑_X_k∈AN(X)a_kb_k var(N_k)/∑_X_k∈AN(X)a^2_k var(N_k)β_Y(X)=∑_X_k∈AN(X)a_kb_k var(N_k)/∑_X_k∈AN(X)b^2_kvar(N_k)+∑_X_k∈AN(Y)\AN(X)c^2_kvar(N_k)in the expression for β_Y(X), the first summation contains the same exogenous noises as the numerator while the second summation contains terms related to the variance of other orthogonal exogenous noises. Therefore, by Assumption <ref>, β_Y^(i)(X)=β_Y^(j)(X) only if for all X_k∈AN(Y), var(N_k) remains unchanged. In this case, we will also have β_X^(i)(Y)=β_X^(j)(Y). Note that β_X(Y) can always remain unchanged of the exogenous noise of variables in AN(X) affect Y only through X. If structures G_1 and G_2 differ in the direction of the edge between adjacent variables X_i and X_j, it suffices to test the identifiably condition for X=X_i, and S⊆ N(X_i) and X=X_j, and S⊆ N(X_j).Equipped with Theorem <ref>, in Section <ref> we propose and algorithm which which focuses on a target variable Y and based on the information obtained from the available environments finds subsets of PA_Y and CH_Y. We in special cases this algorithm return the complete PA_Y and CH_Y sets.§ ALGORITHM T is the set of variables whose marginals is changing.The function rel-finder finds T_PA⊆CH(Y)∩ T and T_CH⊆PA(Y)∩ T.B_S is coefficient vector of the linear regression of Y over S.β_X: Coefficient of linear regression of Y on X in π. For each pair of environments, the algorithm comprises of two stages and 2 tests in each stage. In the first stage the regressand is the target variable, Y and in the second stage, the regressand is its neighbor X which is under study. 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(1995) Template-based algorithms for connectionist rule extraction. In G. Tesauro, D.S. Touretzky and T.K. Leen (eds.), Advances in Neural Information Processing Systems 7, pp. 609–616. Cambridge, MA: MIT Press.[2] Bower, J.M. & Beeman, D. (1995) The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System.New York: TELOS/Springer–Verlag.[3] Hasselmo, M.E., Schnell, E. & Barkai, E. (1995) Dynamics of learning and recall at excitatory recurrent synapses and cholinergic modulation in rat hippocampal region CA3. Journal of Neuroscience 15(7):5249-5262. | http://arxiv.org/abs/1705.09644v1 | {
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Abnormality Detection and Localization in Chest X-Rays using Deep Convolutional Neural Networks Mohammad Tariqul Islam^1, Md Abdul Aowal^1, Ahmed Tahseen Minhaz^1, Khalid Ashraf^2 ^1Semion, House 167, Road 3, Mohakhali DOHS, Dhaka, Bangladesh.^2Semion, 1811 Francisco St., St 2,Berkeley, CA 94703, USA.{mhdtariqul, aowal.eee, tahseenminhaz92}@gmail.com, {khalid}@semion.ai December 30, 2023 ==================================================================================================================================================================================================================================================================================================Chest X-Rays (CXRs) are widely used for diagnosing abnormalities in the heart and lung area. Automatically detecting these abnormalities with high accuracy could greatly enhance real world diagnosis processes. Lack of standard publicly available dataset and benchmark studies, however, makes it difficult to compare and establish the best detection methods. In order to overcome these difficulties, we have used the publicly available Indiana chest X-Ray dataset, JSRT dataset and Shenzhen Dataset and studied the performance of known deep convolutional network (DCN) architectures on different abnormalities. We employed heat maps obtained from occlusion sensitivity as a measure of localization in the CXRs. We find that the same DCN architecture doesn't perform well across all abnormalities. Shallow features or earlier layers consistently provide higher detection accuracy compared to deep features. We have also found ensemble models to improve classification significantly compared to single model. Combining these insight, we report the highest accuracy on chest X-Ray abnormality detection on this dataset. We find that in the cardiomegaly classification task, where comparison could be made, the deep learning method improves the accuracy by a staggering 17 percentage point compared to rule based methods. We applied the techniques developed along the way to the problem of tuberculosis detection on a different dataset and achieved the highest accuracy on that task. Our localization experiments using these trained classifiers show that for spatially spread out abnormalities like cardiomegaly and pulmonary edema, the network can localize the abnormalities successfully most of the time. One remarkable result of the cardiomegaly localization is that the heart and its surrounding region is most responsible for cardiomegaly detection, in contrast to the rule based models where the ratio of heart and lung area is used as the measure. We believe that through deep learning based classification and localization, we will discover many more interesting features in medical image diagnosis that are not considered traditionally. § INTRODUCTIONMedical X-rays are one of the first choices for diagnosis due to its “ability of revealing some unsuspected pathologic alterations, its non-invasive characteristics, radiation dose and economic considerations" <cit.>. X-Rays are mostly used as a preliminary diagnosis tool. There are many benefits of developing computer aided detection (CAD) tools for X-Ray analysis. First of all, CAD tools help the radiologist to make a quantitative and well informed decision. As the data volume increases, it will become increasingly difficult for the radiologists to go through all the X-Rays that are taken maintaining the same level of efficiency. Automation and augmentation is severely needed to help radiologists maintain the quality of diagnosis. Over the past decade, a number of research groups have focused on developing CAD tools to extract useful information from X-Rays. Historically, these CAD tools depended on rule based methods to extract useful features and draw inference based on them. The features are often useful for the doctor to gain quantitative insight about an X-Ray, while inference helps them to connect those abnormal features to certain disease diagnosis. However, the accuracy of these CAD tools has not achieved a significantly high level to work as independent inference tool. Thus CAD tools in X-Ray analysis are left as mostly providing easy visualization functionality.In recent time, deep learning has achieved superhuman performance on a number of image based classification<cit.>. This success in recognizing objects in natural images has spurred a renewed interest in applying deep learning to medical images as well. A number of reports recently have emerged where indeed superhuman accuracies were obtained in a number of abnormality detection tasks. This success of classifying abnormalities in images have not translated to other radiological modalities mainly because of the absence of large standard datasets. Creation of high quality and orders of magnitude larger dataset will certainly drive the field forward. In this work, we report DCN based classification and localization on the publicly available datasets for chest X-Rays. Our contributions are the following:* We show a 17 percentage point improvement in accuracy over rule based methods for Cardiomegaly detection using ensemble of deep convolutional networks.* Multiple random train/test data split achieve robust accuracy results when the number of training examples are low.* Shallow features or earlier layers perform better than deep features for classification accuracy. * Ensemble of DCN models performs better than single models. However, mix of rule based and DCN ensemble model degraded accuracy.* Sensitivity based localization provides correct localization for spatially spread out diseases.* Results of 20 different abnormalities which we believe will serve as a benchmark for other studies to be compared against.* Direct application of the methods developed in the paper on the Shenzen dataset achieve the highest accuracyfor tuberculosis detection. The paper is organized as follows. In section <ref> we overview of the related work. In section <ref>, we describe the dataset, analysis method, evaluation figure of merits and the localization method used. Then in section <ref>, we present our results on single and ensemble models and critique various issues discussed above. In section <ref>, we describe the localization results and discuss their performance. The cardiomegaly detection and tuberculosis detection are discussed in detail along with comparison sections <ref> and <ref>. Finally we conclude summarizing our results in section <ref>. The conclusions in the paper are derived by analyzing two representative abnormalities i.e. cardiomegaly and pulmonary edema. The classification results for other abnormalities are given in the supplementary materials.§ RELATED WORKS Local binary pattern (LBP) features were employed in segmented images to classify normal vs. pathology on CXRs in <cit.> for early detection purposes. The dataset used in the study was private and contained 48 images total. In <cit.>, image registration technique was used to localize the heart and lung region and then computed radiographic index like cardiothoracic ratio (CTR), cardiothoracic area ratio (CTAR) to classify cardiomegaly from the X-ray images. In <cit.> lung segmentation was performed using 247 images from JSRT, 138 images from Montgomery and 397 images from the India dataset with segmentation accuracies of 95.4%, 94.1%, and 91.7% respectively.Jaeger et. al <cit.> segmented lungs using graph cut method and used large features sets both from the domain of object detection and content based image retrieval for early screening of tuberculosis (TB) and made the databases public. Additionally, few other works on TB screening has been conducted using the public datasets <cit.> and using additional data along with the public datasets <cit.>. They achieved near human performance in detecting TB.Gabor filter features were extracted from histogram equalized CXRs in <cit.> in order to detect pulmonary edema using 40 pulmonary edema and 40 normal images and achieved 97% accuracy. The dataset is private hence the accuracy cannot be compared. In an attempt to identify multiple pathologies in a single CXR, bag of visual words is constructed from local features which are fed to probabilistic latent semantic analysis (PLSA) pipeline <cit.>. They used the ImageClef dataset and clustered various types of X-Rays present in the dataset. However, they didn't detect any abnormality in the paper.In a view to classifying abnormalities in the CXRs, a cascade of convolutional neural network (CNN) and recurrent neural network (RNN) are employed <cit.> on the Indiana dataset chest X-Rays. However, no test accuracy was given nor any comparison with previous results was discussed. Hence it was impossible to determine the robustness of the results.Usage of pre-trained Decaf model in a binary classifier scheme of normal vs. pathology, cardiomegaly, mediastinum and right pleural effusion have been attempted <cit.>. This work was reported on a private dataset, and hence no comparison can be made. §.§ Deep Learning on Medical Image Analysis A detailed survey of deep learning in medical image analysis can be found in <cit.>. Localization of cancer cells is demonstrated in <cit.>. Using inception network, human level diabetic retinopathy detection is shown in <cit.>. Using a multiclass approach, inception network is used in <cit.>, to obtain human level skin cancer detection.§ EXPERIMENTS§.§ DatasetsThe three publicly available datasets for our studies in this paper are:* Indiana Dataset <cit.>: Set consists of 7284 CXRs, both frontal and lateral images with disease annotations, such as cardiomegaly, pulmonary edema, opacity or pleural effusion. Indiana Set is collected from various hospitals affiliated with the Indiana University School of Medicine. The set is publicly available through Open-i SM, which is a multimodal (image + text) biomedical literature search engine developed by U.S. National Library of Medicine. A typical example of a normal CXR (left) and a CXR with cardiomegaly abnormality (right) is shown in Fig. <ref>. Visually, it can be observed that the heart in the cardiomegaly example is quite big compared to that of the normal CXR.* JSRT Dataset <cit.>: Set compiled by the Japanese Society of Radiological Technology (JSRT). The set contains 247 chest X-rays, among which 154 have lung nodules (100 malignant cases, 54 benign cases), and 93 have no nodules. All X-ray images have a size of 2048×2048 pixels and a gray-scale color depth of 12 bit. The pixel spacing in vertical and horizontal directions is 0.175 mm. The JSRT set is publicly available and has gold standard masks <cit.> for performance evaluation. * Shenzhen Dataset <cit.>: This set is compiled at Shenzhen No.3 People’s Hospital, Guangdong Medical College, Shenzhen, China. The recorded frontal CXRs are classified into two categories: normal and tuberculosis (TB). In a one month period, 326 normal cases and 336 cases with tuberculosis have been recorded from the outpatient clinics comprising a total of 662 CXRs in the dataset. The clinical reading of each of the CXRs is also provided. §.§ Deep Convolution Network Models As described in section <ref>, deep convolutional networks (DCN) have achieved significantly higher accuracy than previous methods in disease detection in various diagnostic modalities. In many cases, these accuracies have surpassed human detection capabilities. Here, we explore the performance of various DCNs for heart disease detection on chest X-Rays. We use binary classification of Cardiomegaly and Pulmonary Atelectasis against normal chest X-Rays as representative examples. Results for other diseases are given in the supplementary materials. We explored several DCN models, e.g, AlexNet <cit.>, VGG-Net <cit.> and ResNet <cit.>. These models vary in the number of convolution layers used and achieve higher classification accuracy as the number of convolution layers is increased. Specifically, ResNet and its variants have achieved superhuman performance on the celebrated ImageNet dataset. In the experiments we have extracted features from one of the layers of the DCN. We have frozen all the layers upto this layer and added a binary classifier layer to detect the abnormality. The second fully connected layer has been selected for feature extraction in AlexNet, VGG-16 and VGG-19 networks. The features from the ResNet-50, ResNet-101 and ResNet-152 are extracted from the ,andlayers respectively. All the DCN models have been implemented in Tensorflow and have been finetuned using Adam optimizer <cit.> with learning rate 0.001. The weights of the networks AlexNet, and VGG were obtained from the respective project pages, while weights of the ResNet models were obtained from MatConvNet Pre-train Library [http://www.vlfeat.org/matconvnet/pretrained/]. §.§ Evaluation Metrics The quality of detection was evaluated in terms for four measures: accuracy, area under receiver operating characteristics (ROC) curve (AUC), sensitivity and specificity. The accuracy is the ratio of number of correctly classified samples to total samples. Unless otherwise stated, classifier threshold is set to 0.50 in the reported values of accuracy, sensitivity and specificity. ROC curve is the graphical plot of true positive rate (TPR) vs false positive rate (FPR) of a binary classifier when classifier threshold is varied from 0 to 1. The number of pathological samples that are correctly identified as pathological sample by the classifier is called true positive (TP). The number of pathological samples that are incorrectly classified as normal by the classifier is called false negative (FN). The number of normal samples that are correctly classified as normal is called true negative (TN), and in a similar fashion, the number of normal samples that are incorrectly identified as pathological samples is called false positive (FP). True positive rate (TPR) is the proportion of pathological samples that are correctly identified as pathological sample, given asTPR=sensitivity=TP/TP+FNTPR is also called sensitivity which is called such as this measure shows the degree to which does not miss a pathological sample. False positive rate (FPR) is proportion of normal samples that are incorrectly identified as pathological samples, given as,FPR=1-specificity=FP/FP+TNThe measure specificity shows the degree to which the classifier correctly identifies normal samples as normal. The objective of a classifier to attain high sensitivity as well as specificity so that the classifier attains low diagnosis error. §.§ Localization SchemeThe sensitivity of softmax score to occlusion of a certain region in the chest X-Ray was used to find which region in the image is responsible for the classification decision. We followed the localization using occlusion sensitivity described in <cit.>. In this experiment, a patch of square size is occluded in the CXRs and is observed whether the classifier can detect pathology in the presence of the occlusion. If the region corresponding pathology is occluded then the classifier should no longer detect the pathology with higher probability and thus this drop in probability indicates that the pathology is located at the location of the occlusion. This occluded region is slid through the whole CXR and thus a probability map of the pathology corresponding to the CXR is obtained. The regions where the probabilities are below a certain threshold indicates that the pathology is likely to be occupying that region. Thus, the pathology in the CXR can be localized.The overall classification scheme and localization scheme is visualized in Fig. <ref>. In summary, the classification scheme (top) is ensemble of different types of DCNS and the localization (bottom) is obtained from the overlapping occlusions.§ RESULTS §.§ Classification §.§.§ Classification using single models Our first experiment use single model with DCNs fine-tuned from a model trained on ImageNet. Detection of cardiomegaly is done only for the frontal CXR images from the Indiana Dataset. It contains 332 frontal CXRs with cardiomegaly. In order to balance the binary classification, 332 normal frontal CXRs have been selected randomly from the database. Of these images, 282 of each class have been selected for training and 50 of each class for testing. In addition to training the DCNs, we also performed rule based features for cardiomegaly detection. Overall, we ran experiments with the following characteristics: (1) The NNs are fine-tuned on the Indiana dataset, (2) The NNs are fine-tuned using dropout technique <cit.>, (3) The fusion of NN feature and rule based features, and (4) The fusion of NN feature and rule based feature trained using dropout technique. The results are summarized in tables <ref>-<ref>. In table <ref>, the results obtained by fine-tuning the DCNs are shown. We find that deeper models like VGG-19 and ResNet improve the classification accuracy significantly. For example, the accuracy of Cardiomegaly detection improves by 6 percentage point from that using AlexNet when VGG-16 and ResNet-101 are used. In order to understand the robustness of theseresults, we further calculate the sensitivity, specificity, sensitivity vs 1-specificity curve and derive the area under curve (AUC) metric for classification using different networks. We find that although ResNet-101 gives the highest specificity and VGG-16 gives the highest sensitivity, VGG-19 gives an overall better performance with the highest AUC of 0.94. The AUC calculated using VGG-19 is at least one percentage point higher than the other networks considered here.Adding dropout improves the classification accuracy of the shallower networks but degrades the performance of deep models. We find that VGG-16 and AlexNet achieve the highest accuracy and AUC respectively when dropout is used as shown in table <ref>. On the contrary, the accuracy of deeper models like ResNet-101 and VGG-19 drops by about 4 percentage points.For all these experiments, we found that taking features from earlier layers compared to later layers improve accuracy by 2 to 4 percentage points. Shallow DCN features are often useful for detecting small objects in images <cit.>. Our findings are similar for chest X-Ray abnormality classification as well. As an example, we are showing the performance obtained by taking features from different layers of ResNet-152 model. The candidate layers are chosen from the 4th, 5th and final stage of the network based on what type of operations they perform. The chosen layers and their corresponding operations are listed in Table <ref>. The notation of the layers is based on the pre-trained model obtained from MatConvNet Pre-train Library. We trained five models to detect cardiomegaly using features from each of the layers and the average performance of these features in terms of accuracy, AUC, sensitivity, and specificity for Cardiomegaly detection are shown in Fig. <ref>. It can be observed that the performance of the final pooling layer () is degraded compared to the other layers in terms of accuracy, sensitivity and specificity. In particular features from residual connections (, ) and ReLU (, ) are considerably better with features fromproviding highest accuracy. Similar observations are made for other ResNet variants, VGG nets and AlexNet.In addition to the DCN features, we experimented with DCN and rule based feature fusion for single model classification. The rule based features that were used in the study are 1D-cardio-thoracic ratio (CTR), 2D-cardio-thoracic ratio and cardio-thoracic area ratio (CTAR) <cit.>. 1D-CTR is the ratio between the maximum transverse cardiac diameter and the maximum thoracic diameter measured between the inner margins of ribs, which is formulated as,1D-CTR = Maximum transverse cardiac diameter/Maximum thoracic diameter The 2D-CTR is the ratio between the perimeter of the heart region to the perimeter of the entire thoracic region and formulated as2D-CTR = Perimeter of Heart/Perimeter of Thoracic Region while CTAR, the ratio between area of the heart region to the sum of the area of the left and right lung region, is formulated asCTAR=Area of Heart/Area of Left Lung+Area of Right Lung. In the experiments involving rule based features, we concatenated the features with the features extracted from a DCN and trained a fully connected layer to detect cardiomegaly. However, the results degraded and hence are not shown here.Observation from these single model classification results is that different figure of merit is maximized by different DCNs. We wanted to explore if this is expected or due to some limitation of the data or training process itself. Hence, rather than taking a single train-test split of the data, we randomly split the train-test data and trained nine different model for each architecture. Then we calculated the mean and standard deviation for the figure of merits of interest. The results can be seen in table <ref>. We find that after averaging the nine random train-test sample results, a clear trend emerges where a single model, ResNet-152 in this case, achieves the highest accuracy, AUC and sensitivity. The mean specificity for ResNet-152, in this case is close to the highest number, however, the max specificity is indeed highest for ResNet-152.Having around 600 images for training a network is not sufficient. We wanted to see how does the mean accuracy and the standard deviation vary as we change the number of training examples. Since averaging over multiple train-test splits gave a robust classification accuracy and other figures of merit, we used this classification process to identify the deviation of the result as a function of the number of training images. The results are shown in Fig. <ref>. As expected, for both accuracy and AUC, the mean is lower and deviation is higher for less than 50 training example per category. As the number of example increases, the mean increases and the deviation decreases coming to a saturation at about 200 images.To check if the same model gives the highest accuracy for different abnormalities, we model pulmonary edema using the same averaging process described above. Our dataset for the detection of pulmonary edema contains available 45 frontal CXR images with pulmonary edema and randomly chosen 45 normal frontal CXRs from the Indiana Dataset. We partitioned the dataset in train and test set such that 30 of each class have been selected for training and 15 of each class for testing. We have run our program with 15 different seeds and reported the overall performance metrics in the table <ref> as (mean ± s.d.). We find that whereas ResNet-152 gave the highest accuracy for cardiomegaly detection, for pulmonary edema detection ResNet-50 gives the highest accuracy, highest AUC and highest sensitivity. ResNet-152 has a slightly higher specificity. This shows that there is no single model appropriate for all abnormalities, rather the suitable network varies for different abnormalities. This observation is consistent with the conclusions drawn in <cit.>. In this case, ResNet-152 which gave the highest accuracy for cardiomegaly detection achieves almost one percentage point reduced accuracy compared to ResNet-50. §.§.§ Classification using ensemble of modelsWe trained four different instances of each of the DCNs, i.e, AlexNet, VGG-16, VGG-19, ResNet-50, ResNet-101 and ResNet-152, to detect cardiomegaly. Thus a total of 24networks were trained on the same training data. There are a number of ways to perform ensemble on the trained model. The methods include linear averaging, bagging, boosting, stacked regression <cit.> etc. Since, the number of images in the training dataset is only 564, which is far less than the number of trainable parameters in the classifiers, the individual classifiers always overfit the training set. In this situation, if bagging, boosting and/or stacked regression are employed to build the ensemble model, it will result in a completely biased model. Thus, the ensemble models were obtained by using simple linear averaging of the probabilities given by the individual models. The performance of the ensembles was measured using 50 cardiomegaly and 50 normal images for all the possible combinations of the trained individual models. The performance of these combinations is shown in Fig. <ref> using boxplots. The horizontal red bars indicate the 50 percentile values and the spread of the blue boxes indicate the 25 and 75 percentile values. The black stars indicate extreme points in the data. It can be observed from the figure that, combinations of 7 to 10 models can achieve higher accuracy, however they have the largest spread. On the other hand, as number of models in the ensemble increases, the accuracy of the ensemble model converges to a certain value which for this experiment was 92%. The ROC curves of one instance AlexNet, VGG-19, ResNet-152 and one ensemble model, that is linear average of 6 different types of DCNs, are shown in Fig. <ref>. The curves are obtained using 50 cardiomegaly and 50 normal images. The AUC obtained for each model are 0.8624, 0.8888, 0.8896 and 0.9728, respectively. We can understand from the AUC values that, the separation between the pathology class and the normal class increases when an ensemble of multiple DCNs are performed. For the ensemble model to be used as a screening tool with high sensitivity, the operating point on the curve is set to achieve 98% sensitivity. The specificity obtained at this point is 82%. The second operating point is set for high specificity of 98% and the sensitivity at this point is 86%. §.§ Localization For any diagnostic task, it is desirable to gain intuitive understanding of why a certain classification decision is made rather than being a black box method. In other words, it is desirable to distinguish features that contributed most to certain abnormality in the entire chest X-Ray. There are various ways of achieving this goal <cit.>. The method used in <cit.> is the simplest, where a patch is occluded in the image to measure its impact on the eventual classification confidence score. We have used this method to find the regions in the image responsible for a certain abnormality detection. As a representative example, we have used cardiomegaly and pulmonary edema which occur in heart and lung areas respectively. The localization scheme described in section <ref> is followed with a patch size of 40×40 pixels taking lowest 20% values of probabilities. Instead of gray level occlusion as in <cit.> we found that black level occlusion works better for CXRs. This is due to the fact that the CXRs themselves are mostly gray level and occlusion of the same level does not hide much information compared to the neighborhood.§.§.§ Cardiomegaly LocalizationThe localization of abnormalities in cardiomegaly examples are shown in Fig. <ref>. Here, 20% of the image area is shown which has the highest sensitivity. It can be observed from the figures that the network is indeed most sensitive to the region where the heart is larger than a normal heart. We have performed this experiment on 50 cardiomegaly and 50 normal images and found this localization to be consistent for most examples. There is not much functional difference between a normal and cardiomegaly example other than the fact that the heart in cardiomegaly is larger than a normal heart. Given the fact that the normal images could also have various size of heart depending on the age or physical attributes of a patient, we found this level of localization sensitivity to be remarkable. Also interesting is the fact that the standard rule based features like CTR and CTAR take into account the relative size of heart and lung to determine if there is cardiomegaly present or not. In the DCN localization experiment, we see counter-intuitively that most of the signals contributing to the softmax score are coming from the heart only. This means that there are characteristic features in the shape of the heart and its surrounding regions that alone is sufficient to detect cardiomegaly. The lung and its relative size are probably less important features when trying to detect cardiomegaly. This observation is counterintuitive and needs to be explored further in future work.§.§.§ Pulmonary Edema LocalizationIn order to test the effectiveness of the localization procedure inareas other than the heart region, we chose pulmonary edema which occurs in the lung region. Also, pulmonary edema is detected by the net like white structure in the lung area. No anatomical shape change is associated with the abnormality. We have found that the localization is obtained best when the ROIs of lungs are taken to compute the map. Following the scheme in section <ref>, localization experiment on pulmonary edema is performed as shown in Fig. <ref>. It has been observed that the classifier is not sensitive to the fine features like septal or Kerley B lines. The localization is mainly obtained in the lung region where excess fluid is observed. Some localization regions are outside the lung region which occurs primarily for the fact that, even though the occlusion center is outside the lung, it occludes lung region and thus the probability drop occurs.§.§ Comparison between Rule based and DCN based cardiomegaly detectionA comparison between rule based and DCN based cardiomegaly detection is shown in table <ref>. State-of-the-art method by Candemir et al. <cit.> reported an accuracy of 76.5% while classifying between 250 cardiomegaly and 250 normal images. They employed 1D-CTR, 2D-CTR, and CTAR computed from segmented CXRs as features. A brief discussion about the rule based approach is given in the supplementary materials in section <ref>. In verifying that claim in the paper, we reproduced those results and achieved an accuracy of 75.6% on the same train-test set split on which the DCNs are trained. It can be observed from the table that the results are similar to that obtained by <cit.>. However, it is evident from the table that all DCN based approaches outperform the rule based method. As stated earlier, DCNs were fine tuned on a sample of 560 images and validated on 100 images. Among the independent DCN models, VGG-19 model achieves the highest accuracy of 92% and highest AUC of 0.9408 for detecting cardiomegaly. The ensemble model, which is linear average of the six individual DCN models, shows the best accuracy of 93% and AUC of 0.9728. The accuracy is 17 percentage point higher thanthat reported in Candemir's paper and the AUC is 18percentage points higher than our implementation of the paper. Similarly, a 17 percentage higher sensitivity and 16 percentage point higher specificity from the Candemir's paper is reported. This quantum of improvement in accuracy, AUC, sensitivity and specificity makes a strong case for use of deep learning based detection techniques in real world application of medical image analysis. §.§ Tuberculosis Detection In this section we evaluate the effectiveness of the network design and DCN pipelines for a different dataset and abnormality. We use the Shenzen dataset as it is often used for reporting accuracy on tuberculosis detection. Detailed study of tuberculosis detection will be provided in a future publication. But the a comparison among several TB classification methods and proposed DCN based methods along with their ensemble using Shenzhen Dataset is shown in table <ref>. Previously, Jaeger et. al <cit.> extracted several features from lung segmented CXRs and employed various classification methods to benchmark the features. The results reported in the table is obtained using low-level content-based image retrieval based features and linear logistic regression based classification. Hwang et. al <cit.> trained three different DCNs on three different train/test split on a large private KIT dataset and tested the ensemble of the model on Shenzhen dataset. It is to be noted that, both the KIT and Shenzhen dataset were obtained using Digital Radiography. Lopes and Valiati <cit.> employed bags of features and ensemble method using features from ResNet, VGG and GoogLeNet models and trained SVM classifier on them. They obtained highest AUC in Shenzhen dataset using ensemble of individual SVM classifiers. Lakhani and Sundaram <cit.> employed AlexNet and GoogLeNet on a combined dataset of four different databases and performed ensemble on the trained models. They do not report test results on Shenzhen dataset and thus it was not shown in the table.The DCN based methods shown in table <ref> have comparable or higher accuracy and lower AUC than the results already present in the literature. The VGG-16 model obtains highest sensitivity and AlexNet model obtains highest specificity. The sensitivity and specificity measures for Jaeger's, Hwang's and Lopes and Valiati's paper are not shown in the table as they were not reported in the respective papers. In terms of accuracy and AUC, our ensemble method obtains highest values of 90% and 0.94, respectively. This accuracy is obtained when classifier threshold is set to 0.74. When classifier threshold is set to 0.50, the accuracy obtained is 88%. Thus, we report a 5 percentage point higher accuracy and 1 percentage point higher AUC compared to nearest Lopes and Valiati's paper.§ CONCLUSION In summary, we have explored DCNN based abnormality detection in frontal chest XRays. We have found the existing literature to be insufficient for making comparison of various detectiontechniques either due to studies reported on private datasets or not reporting the test scores in proper detail <cit.>. In order to overcome these difficulties, we have used the publicly available Indiana chest X-Ray dataset and studied the performance of various DCN architectures on different abnormalities. We have found that the same DCNN architecture doesn't perform well across all abnormalities. When the number of training examples is low, a consistent detection result can be achieved by doing multiple train-test with random data split and the average values are used as the accuracy measure. Shallow features or earlier layersconsistently provide higher detection accuracy compared to deep features. We have also found ensemble models to improve classification significantly compared to single model when only DCNN models are used. Combining DCNN models with rule based models degraded the accuracy. Combining these insights, we have reported the highest accuracy on a number of chest X-Ray abnormality detection where comparison could be made. For the cardiomegaly classification task, the deep learning method improves the accuracy by a staggering 17 percentage point. Using the same method developed in the paper, we achieve the highest accuracy on the Shenzen dataset for Tuberculosis detection. We have also performed localization of features responsible for classification decision. We found that for spatially spread out abnormalities like cardiomegaly and pulmonary edema, the network can localize the abnormalities successfully most of the time. However, the localization fails for pointed features like lung nodule or bone fracture. One remarkable result of the cardiomegaly localization is that the heart and its surrounding region is most responsible for cardiomegaly detection. This is counterintuitive considering the usual method of using the ratio of heart and lung area as a measure for cardiomegaly. However, expert radiologists often conclude upon cardiomegaly by looking at the heart's shape rather than using a quantitative method. We believe that through deep learning based classification and localization, we will discover many more interesting features that are not considered traditionally. While finishing this paper, we became aware of a new dataset announcement and paper focused on similar problem <cit.>. It would be interesting to apply the techniques discussed our paper on the new dataset becomes available.§ ACKNOWLEDGEMENT This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Thanks Leonid Oliker at NERSC for sharing his allocation on the OLCF Titan supercomputer with us on project CSC103. Thanks to Hoo-Chang Shin for correspondence regarding the Indiana chest X-Ray dataset.IEEEbib§ SUPPLEMENTARY MATERIALS In this section, first a detailed description about the segmentation of the CXRs is given. Then the rule based machine learning model is described. After that we show classification results of 20 chest X-Ray abnormalities. And finally we show more localization results with additional insights.§.§ Rule based Approach in Detecting Cardiomegaly This method uses existing CXRs and their radiologist marked lung/heart boundaries as models, and estimates the lung/heart boundary of a patient X-ray by registering the model X-rays to the patient X-ray. <cit.>§.§.§ Using Radon Transform and Bhattacharyya Distance to find visually similar images We use the publicly available chest x-ray dataset (JSRT) <cit.> with reference boundaries given in the SCR dataset <cit.>. For a given test image, the radon transform of that image is calculated at radial coordinates, ranging from 0 to 90 degree. The radon function computes projections of an image along specified directions. Bhattacharyya distance <cit.> is calculated between radon transform of test CXR and the sample CXRs to find 5 visually similar samples from the JSRT dataset. We use the most similar images from the dataset to register to the test image. As mentioned by Candemir et al. <cit.>, the main objective of similarity measurement is to increase the correspondence performance and reduce the computational cost during registration.§.§.§ Calculating correspondence between test CXR and model CXRs using SIFTFlow We compute the correspondence map between the test CXR and visually similar CXR models by calculating local image features and matching the most similar locations. We employ the SIFT-flow algorithm which matches densely sampled SIFT features between two images. The computed correspondence map is a transformation from model X-ray to the patient X-ray. Finally, the computed transformation matrix is applied on the model CXR's lung-heart boundary to generate an approximate lung-heart segmentation of the test image.§.§.§ Rule based feature extractionRule based features are extracted using <ref>, <ref> and <ref> and SVM was used to classify between cardiomegaly and normal CXR images. §.§ Classification Results on the 20 chest X-Ray abnormalities In this section, we report the classification accuracy, sensitivity and specificity using the ResNet-152 model. We hope that these numbers will set a benchmark to compare against other machine learning methods on this dataset.§ ADDITIONAL EXAMPLES OF LOCALIZATION In this section we show more examples of localization. Few localization samples are shown in Fig. <ref>. It can be observed that, in the CXRs with Cardiomegaly (Fig. <ref>(a) and (b)) a fine localization around the heart is observed. In the normal CXRs (Fig. <ref>(c) and (d)) such localization is not observed. Rather the lowest 20% probabilities are spread out in the CXR image. It is interesting to note that, the localization algorithm gets low probability where the heart is enlarged during cardiomegaly, but the proportion is small compared to the localization in other areas of normal CXRs. In order to observe the performance of the heat map we computed histograms of heat maps of each of the 100 CXRs in the test set for Cardiomegaly detection and average histograms are shown in Fig. <ref>(e) and (f) for CXRs with Cardiomegaly and normal CXRs, respectively. It is to be noted that, the histograms include both success and failure cases. It can be observed that, for CXRs with Cardiomegaly the classifier is highly sensitive toward Cardiomegaly detection even under occlusion. This indicates that, the classifier primarily looks for local features in a CXR instead of some feature that is spread out in the entire CXR. However, the classifier is not sensitive toward normal CXRs under occlusion. Rather, the probabilities are spread out in the probability spectrum. After that, we analyzed the failure cases where the classifier is unable to classify the image correctly. Two such examples of failure cases are shown in Fig. <ref>. The localized CXR shown in Fig. <ref>(a) contains Cardiomegaly whereas the classifier detects it as normal. However, the localization shows that it localizes around heart quite well despite the in accurate classification. On the other hand, Fig. <ref>(b) shows an example of normal image which has been classified as Cardiomegaly by the classifier. There is stronger localization around the hear that that is observed for normal images as in Fig. <ref>(c) and (d), however, like those images the localization is spread out.In a similar fashion, additional localization results for Pulmonary Edema is shown in Fig. <ref>. In Fig. <ref>(a) and (b) localization of two examples of CXRs with Pulmonary Edema is shown. As stated earlier the classifier localizes in the lung region. This is not the case when normal images are used to localize Pulmonary Edema as seen in Fig. <ref>(c) and (d). The localizations are obtained in random dense locations such as the sternum or heart. Like the cardiomegaly case, the histogram averages for CXRs with pulmonary Edema (Fig. <ref>(e)) shows a sensitivity toward pulmonary edema detection while the normal CXRs shows a spread out detection. It is interesting to note that, in the histogram of normal images high probability (>0.85) is non-existent, thus ensuring low false positive rate. In the test set none of the normal images have been diagnosed as Pulmonary Edema. The failure cases are shown in Fig. <ref>. These CXRs are with Pulmonary Edema. However, the localization algorithm shows that one of them localizes in lungs whereas the other one shows a localization pattern similar to that obtained in normal CXRs. | http://arxiv.org/abs/1705.09850v3 | {
"authors": [
"Mohammad Tariqul Islam",
"Md Abdul Aowal",
"Ahmed Tahseen Minhaz",
"Khalid Ashraf"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170527175957",
"title": "Abnormality Detection and Localization in Chest X-Rays using Deep Convolutional Neural Networks"
} |
Universal Protocols for Information DisseminationUsing Emergent Signals [=========================================================================== We present a Gaussian kernel loss function and training algorithm for convolutional neural networks that can be directly applied to both distance metric learning and image classification problems. Our method treats all training features from a deep neural network as Gaussian kernel centres and computes loss by summing the influence of a feature's nearby centres in the feature embedding space. Our approach is made scalable by treating it as an approximate nearest neighbour search problem. We show how to make end-to-end learning feasible, resulting in a well formed embedding space, in which semantically related instances are likely to be located near one another, regardless of whether or not the network was trained on those classes. Our approach outperforms state-of-the-art deep metric learning approaches on embedding learning challenges, as well as conventional softmax classification on several datasets. Metric Learning, Deep Learning, Transfer Learning, Image Classification, Gaussian Kernel § INTRODUCTIONMetric learning aims to learn a transformation from the image space to a feature embedding space, in which distance is a measure of semantic similarity. Feature embeddings from semantically similar images will be located nearby, while those of semantically dissimilar images will be located far apart. Applications for such effective feature embeddings include transfer learning, retrieval, zero- and few-shot learning, clustering and weakly or self supervised learning.Image classification is the task of categorising an image into one of a set of classes. Applications include object and scene recognition.Classification and metric learning are generally treated as separate problems. As such, metric learning approaches have struggled to reach the classification performance of state-of-the-art classifiers. Likewise, classification approaches fail to learn feature spaces that represent inter- and intra-class similarities to the standard of metric learning approaches. Outside of zero- and few-shot learning, some metric learning algorithms have been applied to classification <cit.>, although approaches that perform well in both domains remain uncommon. We propose a novel loss function and training algorithm for convolutional neural networks that can be applied to both metric learning and image classification problems, outperforming conventional approaches in both domains. Our approach defines training set feature embeddings as Gaussian kernel centres, which are used to push or pull features in a local neighbourhood, depending on the labels of the associated training examples. Fast approximate nearest neighbour search is used to provide an efficient and scalable solution. Our approach differs from kernel or radial basis function neurons <cit.>, as our kernel centres are not learned network parameters, but are defined to be the locations of the training set features in the embedding space. Additionally, we use kernels only in the loss function and classifier, not as activation functions throughout the network. Beyond activation functions, kernels have also been used in neural network classifiers as support vector machines <cit.>. Our approach is related to NCA <cit.>, but introduces per exemplar weights, makes training feasible through the introduction of periodic asynchronous updates of the kernel centres, and is made scalable for a large number of training examples and a high embedding dimension. Additionally, we explore the importance of the embedding space dimensionality.The best success on embedding learning tasks has been achieved by deep metric learning methods <cit.>, which make use of deep neural networks. The majority of these approaches use or generalise a triplet architecture with hinge loss <cit.>, although including global loss terms can also be beneficial <cit.>.Triplet networks take a trio of inputs; an anchor image, an image of the same class as the anchor and an image of a different class. Triplet approaches aim to map the anchor nearer the positive example than the negative example, in the feature space.Such approaches may indiscriminately pull examples of the same class together, regardless of the local structure of the space. In other words, these methods aim to form a single cluster per class, limiting the intra- and inter-class similarities that can be represented. In contrast, our approach considers only the local neighbourhood of a feature, allowing multiple clusters to form for a single class, if that is appropriate.Our approach outperforms state-of-the-art deep metric learning approaches on embedding learning challenges.The most common approach to image classification is a convolutional neural network trained with softmax loss, which transforms activations into a distribution across class labels <cit.>. However, softmax is inflexible as classes must be axis-aligned and the number of classes is baked into the network. Our approach is free to position clusters such that the intrinsic structure of the data can be better represented. Our metric learning approach to classification outperforms softmax on several datasets, while simultaneously representing the intra- and inter-class similarities sought by metric learning approaches. Metric learning also allows for new classes to be added on-the-fly, with no updates to the network weights required to obtain reasonable results. The advantages of our approach are as follows: * Training is made feasible by introducing periodic asynchronous updates of the kernel centres (Section <ref>). * End-to-end learning can be made scalable by leveraging fast approximate nearest neighbour search (Section <ref>). * Our approach can be applied to two separate problems; image classification and metric learning. * Our approach outperforms state-of-the-art deep metric learning algorithms on the Stanford Cars196 and CUB Birds200 2011 datasets (Section <ref>). * Finally, our approach outperforms a conventional softmax classifier on the fine-grained classification datasets CUB Birds200 2011, Stanford Cars196, Oxford 102 Flowers and Leafsnap (Section <ref>). § APPROACHA Gaussian kernel returns a value that depends only on the distance between a point 𝐱 and the Gaussian centre 𝐜. The Gaussian kernel f is calculated as:f(𝐱,𝐜) = exp ( -𝐱 - 𝐜 ^ 2/2 σ ^2 ),where σ sets the kernel width. We define the feature embeddings of each training set example as Gaussian kernel centres. Specifically, in a deep neural network we take the layer immediately before the loss function or classifier as the embedding layer. For example, in a VGG architecture, this may be FC7 (fully connected layer 7), forming a 4096 dimension embedding. In general, however, the embedding may be of any size. An overview of this approach is seen in Figure <ref>. §.§ Classifier and Loss Function A classifier is formed by the weighted sum of the kernel distance calculations between a feature embedding and the centres. Classification of an example is achieved by passing the input through the network, resulting in a feature embedding in the same space as the centres. A probability distribution over class labels is found by summing the influence of each centre and normalising. A centre contributes only to the class of the training example coupled to that centre. For example, the probability that feature embedding 𝐱 has class label Q is:Pr(𝐱∈Q) = ∑_i ∈ Q w_i f(𝐱,𝐜_𝐢)/∑_j=1^m w_j f(𝐱,𝐜_𝐣) ,where f is the kernel, i ∈ Q are the centres with label Q, m is the number of training examples and w_i is a weight for centre i, which is learned end-to-end with the network weights. Note that a global σ value is shared by all kernels. If an example is in the training set, the distance calculation to itself is omitted during the computation of the classification distribution, the loss function and the derivatives.The loss function used is the summed negative logarithm of the probabilities of the true class labels. For example, the loss for example 𝐱 with ground truth label R is -ln(Pr(𝐱∈R) ). The same loss function is used for both classification and metric learning problems.§.§ Nearest Neighbour Gaussian KernelsEquation <ref> is calculated by summing over all kernels. However, since the centres are attached to training examples, of which there can be any large number, computing that sum is both intractable and unnecessary. Most kernel values for a given example will be effectively zero, as the feature will lie only within a subset of the Gaussian windows. As such, we consider only the local neighbourhood of a feature embedding. Considering the nearest Gaussian centres to a feature ensures that most of the distance computations are pertinent to the loss calculation. The classifier equation becomes:Pr(𝐱∈Q) = ∑_i ∈ Q ∩𝒩 w_i f(𝐱,𝐜_𝐢)/∑_j ∈𝒩 w_j f(𝐱,𝐜_𝐣) ,where 𝒩 is the set of approximate nearest neighbours for example 𝐱 and i ∈ Q ∩𝒩 is the set of approximate nearest neighbours that have label Q. Again, training set examples exclude their own centre from their nearest neighbour list.In the interest of providing a scalable solution, we use approximate nearest neighbour search to obtain candidate nearest neighbour lists. This allows for a trade off between precision and computational efficiency. Specifically, we use a Fast Approximate Nearest Neighbour Graph (FANNG) <cit.>, as it provides the most efficiency when needing a high probability of finding the true nearest neighbours of a query point. Importantly, FANNG provides scalability in terms of the number of dimensions and the number of training examples. §.§ Training the Network The Gaussian centre locations change as the network weights are updated each training iteration. Although required to compute the derivatives, it is intractable to find the true locations of each example's neighbouring centres online during training. However, we find that it is not necessary for the centres to be up to date at all times in order for the model to converge. We store a bank of the Gaussian centres and perform periodic asynchronous updates of all centres at a fixed interval.As the centres change, the nearest neighbours also change. Again, it is intractable to compute the correct nearest neighbours each time the network weights are updated. This is remedied by considering a larger number of nearest neighbours than would be required if all centres and neighbour lists were up-to-date at all times. The embedding space changes slowly enough that it is highly likely many of the previously neighbouring centres will remain relevant. Since the Gaussian kernel decays to zero as the distance between the points becomes large, it does not matter if a centre that is no longer near the example remains a candidate nearest neighbour. We call the interval at which the Gaussian centres are updated and the nearest neighbours computed during training the update interval.This interval is training set dependant and we find intervals between 1 and 10 epochs work well in our experiments. Note that the stored Gaussian centres do not have dropout <cit.> applied, but the current training embeddings may.§ EXPERIMENTS §.§ Distance Metric LearningWe evaluate our approach on Stanford Cars196 (16,185 images of 196 car models) <cit.> and CUB Birds200 2011 (11,788 images of 200 bird species) <cit.>. In this problem, the network is trained and evaluated on different sets of classes. Following the set-up in <cit.>, we train on the first half of classes and evaluate on the remaining classes. Stochastic gradient descent optimisation is used. Images are resized to 256x256 and data is augmented by random cropping and horizontal mirroring. The object bounding boxes are not used. GoogLeNet <cit.> with ImageNet <cit.> pre-trained weights is used as the model. We use 100 nearest neighbours, an update interval of 10 epochs, batch size of 20, base learning of0.00001 and weight decay of 0.0002. The Gaussian σ used depends on the number of dimensions of the feature embedding; values between 10 and 30 work well for this task. We evaluate on two metrics. The first, Normalised Mutual Information (NMI) <cit.>, is a clustering metric that finds the ratio of mutual information and average entropy of a set of clusters and labels. The second, Recall@K (R@K), defines a true positive as an example feature embedding that has at least one out of its true K nearest neighbours with the same class as itself. We first investigate the importance of the feature embedding dimension, which is set by the output dimensionality of a final fully connected layer.A similar study in <cit.> suggests that the number of dimensions is not important for triplet networks, in fact, increasing the number of dimensions can be detrimental to performance. We compare our method with increasing dimension size against triplet loss <cit.> and lifted structured embedding <cit.>, both taken from the study in <cit.>. Figures <ref> and <ref> show the effect of the embedding size on NMI score. While increasing the number of dimensions does not necessarily improve performance for triplet-based networks, higher dimensionality can be utilised by our approach, as the NMI score improves as the dimensionality increases. Similar behaviour is seen in Figures <ref> and <ref>, which show the Recall@K metric for our approach. Again, this shows that our approach can take advantage of a higher dimensionality. Our approach is compared to the state-of-the-art in Tables <ref> and <ref>, with the compared results taken from <cit.> and <cit.>. Since, as discussed above, the dimensionality does not have much impact on the other approaches, all results in <cit.> and <cit.> are reported using 64 dimensions. For fair comparison, we report our results at 64 dimensions, but also at the better performing higher dimensions. Our approach outperforms the other methods in both the NMI and Recall@K measures, at all embedding sizes presented. Our approach is able to produce better compact embeddings than existing methods, but can also take advantage of a larger embedding space.§.§ Image ClassificationWe compare classification performance with conventional softmax loss. Images are resized to 256x256 and random cropping and horizontal mirroring is used for data augmentation. Unlike in Section <ref>, we crop Birds200 and Cars196 images using the provided bounding boxes before resizing. The same classes are used for training and testing. All datasets are split into training, validation and test sets.For all approaches, we select hyperparameters that minimise the validation loss. For our approach with a VGG <cit.> or AlexNet <cit.> architecture, the FC7 layer (4096 dimensions), with dropout and without a ReLU, is used as the embedding layer. For a ResNet architecture <cit.>, we use the final pooling layer (2048 dimensions). We find that following the ResNet embedding layer with a dropout layer results in a small performance gain for both our approach and softmax. A batch size of 20, update interval of 10 epochs and base learning rate of 0.00001 are used for our approach. We use stochastic gradient descent optimisation. A Gaussian σ of around 100 is found to be suitable for the 4096 dimension VGG16 embeddings on Birds200. Networks are initialised with ImageNet <cit.> pre-trained weights. We first evaluate on the Birds200 dataset. Since there is no standard validation set for this dataset, we take 20% of the training data as validation data. In Table <ref>, we evaluate with three network architectures; AlexNet <cit.>, VGG16 <cit.> and ResNet50 <cit.>. Additionally, the effect of the number of training examples per class is shown in Figure <ref>. Our approach outperforms softmax loss at all numbers of training images, with a particularly large gain when training data is scarce. Further, we investigate the importance of the per kernel weights, e.g. w_i from Equation <ref>, and find that learning the weights end-to-end with the network results in a 0.69% increase in accuracy, compared with fixing the weights at a value of one. We further evaluate our approach on three other fine-grained classification datasets; Oxford 102 Flowers <cit.>, Stanford Cars196 <cit.> and Leafsnap <cit.>. We use the standard training, validation and test splits for Oxford 102 Flowers. For Stanford Cars196, we take 30% of the training set as validation data. We use the challenging field images from Leafsnap, which are taken in uncontrolled conditions. The dataset contains 185 classes of leaf species and we split the data into 50%, 20% and 30% for training, validation and testing, respectively. 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Nov, pp. 2579–2605, 2008.§ SUPPLEMENTARY MATERIALIn this section we provide supplementary qualitative and quantitative results to further evaluate our proposed approach. §.§ Ablation StudyTable <ref> shows an ablation study for our proposed approach. For each of the five arrangements of settings, the value of the Gaussian kernel σ is first tuned to minimise validation loss. The impact of learning the Gaussian kernel weights and fine-tuning network weights is shown. §.§ Neighbourhood SizeFigure <ref> shows the impact of the number of nearest neighbours used for each example during training. There is a clear lower bound required for good performance. This is because, as discussed in Section <ref>, the network weights are constantly being updated, but the stored kernel centres are not. As such, we need to consider a larger number of neighbours than if the centres were always up-to-date. Figure <ref> shows the average distance from each training example to its 200 nearest Gaussian kernel centres, at different points during training. Similarly, Figure <ref> shows the average Gaussian kernel value (from Equation <ref>) between training examples and their 200 nearest centres. These experiments use a VGG16 architecture. §.§ Embedding Space VisualisationA t-SNE <cit.> visualisation of the learned embedding space for the Birds200 dataset is shown in Figure <ref>. Similarly, Figure <ref> shows a visualisation for the Cars196 dataset. The visualised embeddings are the test set examples from the transfer learning task in Section <ref>. The classes shown in the visualisations are withheld and unseen by the network during training. Despite belonging to novel classes, examples are still well clustered based on class and attributes. | http://arxiv.org/abs/1705.09780v3 | {
"authors": [
"Benjamin J. Meyer",
"Ben Harwood",
"Tom Drummond"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170527073448",
"title": "Deep Metric Learning and Image Classification with Nearest Neighbour Gaussian Kernels"
} |
End-to-end Global to Local CNN Learning for Hand Pose Recovery in Depth Data[This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.] 16cmMeysam Madadi^1,2, Sergio Escalera^1,3, Xavier Baró^1,4, Jordi Gonzàlez^1,2^1 Computer Vision Center, Edifici O, Campus UAB, 08193 Bellaterra (Barcelona), Catalonia Spain^2 Dept. of Computer Science, Univ. Autònoma de Barcelona (UAB), 08193 Bellaterra, Catalonia Spain^3 Dept. Mathematics and Informatics, Universitat de Barcelona, Catalonia, Spain^4 Universitat Oberta de Catalunya, Catalonia, Spain ================================================================================================================================================================================================================================================================================================================================================================================================================================================= In the present paper, we treat random matrix products on the general linear group (V), where V is a vector space defined on any local field, when the top Lyapunov exponent is simple, without irreducibility assumption. In particular, we show the existence and uniqueness of the stationary measure ν onthat is relative to the top Lyapunov exponent and we describe the projective subspace generated by its support. We observe that the dynamics takes place in a open set ofwhich has the structure of askew product space. Then,we relate this support to the limit set of the semigroup T_μ of(V) generated by the random walk. Moreover, we show that ν has Hölder regularity and give some limit theorems concerning the behavior of the random walk and the probability of hitting a hyperplane. These results generalize known oneswhenT_μ acts strongly irreducibly and proximally (i-p to abbreviate) on V.In particular, when applied to the affine groupin the so-called contracting case or more generally when the Zariski closure of T_μ is not necessarily reductive, the Hölder regularity of the stationary measure together with the description of the limit set are new. We mention that we don't use results from the i-p setting; rather we see it as a particular case. MSC2010: 37H15, 60B15, 20P05End-to-end Global to Local CNN Learning for Hand Pose Recovery in Depth Data[This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.] 16cmMeysam Madadi^1,2, Sergio Escalera^1,3, Xavier Baró^1,4, Jordi Gonzàlez^1,2^1 Computer Vision Center, Edifici O, Campus UAB, 08193 Bellaterra (Barcelona), Catalonia Spain^2 Dept. of Computer Science, Univ. Autònoma de Barcelona (UAB), 08193 Bellaterra, Catalonia Spain^3 Dept. Mathematics and Informatics, Universitat de Barcelona, Catalonia, Spain^4 Universitat Oberta de Catalunya, Catalonia, Spain ================================================================================================================================================================================================================================================================================================================================================================================================================================================= § INTRODUCTION Let V be a finite dimensional vector space over a local field k and μ a probability measure on the general linear group (V). Random Matrix Products Theory studies the behavior of a random walk on (V) whose increments are taken independently with respect to μ. This theory is well-developed when the sub-semigroup T_μ generated by the support of μ is strongly irreducible (algebraic assumption) and contains a proximal element (dynamical assumption) <cit.>, <cit.>, <cit.>, <cit.>.The latter framework, which will be abbreviated by i-p,had shown to beapowerful tool for understanding the actions of reductive algebraic groups <cit.>, <cit.>, <cit.>, <cit.>...One reason is that a great informationon the structure of a reductive algebraic group is encoded in itsirreducible and proximal representations. This setting had also proved its efficiency in the solution to some fundamental problems involving stochastic recursions <cit.>, <cit.>.In this article, we extend this theory from the i-p setting to a more general and natural framework. More precisely, we consider a probability measure μ on (V) and assume only that its first Lyapunov exponent is simple; in some sense we keep the dynamical condition and assume no algebraic conditionon the support of μ.Recall that by a fundamental theorem of Guivarc'h-Raugi <cit.>, our setting includes the i-p setting.But it also includes new settings asrandom walkson the affine group in the called contracting case or more generallyany probability measure on a subgroup G of (V) that may fixsome proper subspace L of V provided the action on L isless expanding than that on the quotient V/L. Our goal is then to obtainlimit theorems concerning the random walk and the existence, uniqueness and regularity of stationary probability measure on the projective space of V.Our results give also new informationabout the limit sets of some non irreducible linear groups. In our proofs we don't use results from the i-p setting but rather see it as a particular case where our assumption concerning the Lyapunov exponent is satisfied. When applied to a probability measure on the affine group in the contracting case, the regularity of the stationary probability measure as well as the description of its support using the limit set of T_μ are new. More generally, we show that the dynamics takes place on an open subset ofwhich has essentially the structure of a skew product space with basis a projective space andfiberanaffine space. We believe that this generalizationcan be useful to treat random walks on non necessarily reductive algebraic groups just as the i-p setting has proved its efficiency.Here is the structure of the article. * In Section 2 we state formally our results. We note that Section <ref> shows the geometry behind our results and gives main examples that can be guiding ones through our paper. * Section 3 consists of some preliminary results concerning orthogonality in non-Archimedean local fields and some results on Lyapunov exponents.* In Section 4, we show the existence and uniqueness of the stationary measure on the projective space whose cocycle average is the top Lyapunov exponent (Theorem <ref> stated in Section 2). In addition, we describe the projective subspace generated by its support and show that it is not degenerate on it.The existence appeals to Oseledets theorem. The uniqueness is explicit:we show in Proposition <ref>that when λ_1>λ_2, every limit point of theright random walk (R_n)_n∈^* suitable normalizedis almost surely of rankone, and the projection of its image inis a random variable oflaw ν.* In Section 5, we make more precise the results of Section 4 byrelating the support of our unique stationary measure to the limit set of T_μ (Theorem <ref> stated in Section 2). * In Section 6, we show the Hölder regularity of the stationary measure (stated in Theorem <ref>). Moreover, we describe an important related large deviation estimate for the hitting probability of a hyperplane (Proposition <ref>). § ACKNOWLEDGEMENTSBoth authors have the pleasure to thank Emmanuel Breuillard for fruitful discussions.It is also a pleasure to thank Çağri Sert for enlightening discussionson the jointspectralradius/spectrum.Part of this project was financed by theEuropean Research Council, grant no 617129.RA thanksalso UFR Mathématiques, Université de Rennes 1 for the facilities given in January 2017.§ STATEMENT OF THE RESULTS§.§ Uniqueness of the Stationary MeasureFrom now on,k is a local field of any characteristic, V a finite dimensional vector space defined over k. Denote bythe projective space of V. We consider a probability measure μ on the general linear group (V) and denote by T_μ (resp. G_μ) the semigroup (resp. subgroup) of (V) generated by the support of μ. We define on the same probabilistic space (Ω, 𝒜,) a sequence (X_i)_i∈^* of independent identically distributed random variables of law μ. The right (resp. left) random walka time n is by definition the random variable R_n=X_1⋯ X_n (resp. L_n=X_n⋯ X_1).Endow V with any norm ||·|| and keep for simplicity the same symbol for the operator norm on End(V). We will always assume that μ has a moment of order one, i.e. (log^+||X_1^± 1||)<+∞and denote by λ_1(μ)≥⋯≥λ_d(μ) the Lyapunov exponents of μ defined recursively by:λ_1(μ)+⋯ + λ_i(μ) = lim_n→ +∞1/n (log||⋀^i L_n||)= lim_n→ +∞1/nlog||⋀^i L_n||,the last equality is an almost sure equality and is guaranteed by the subadditive ergodic theorem of Kingman <cit.>.In most of the paper, there will no be confusion about theprobability measure and therefore we will omit specifying μ when writing the Lyapunov exponents. For every finite dimensional representation (ρ,W) of G_μ, we denote byλ(ρ,W) the top Lyapunov exponent relative to the pushforward probability measure ρ(μ) of μ by the map ρ, when the latter has a moment of order one. When there is no confusion on the action of G_μ on W, we will simply denote this exponent byλ(W). To simplify, we will refer to it as the Lyapunov exponent of W.By convention, if (ρ,W) is the null representation, then λ(ρ,W)=-∞. Finally recall that if T is a topological semigroup acting continuously on a topological space X and μ is a Borel probability measure on T, then a Borel probability measure ν on X is said to be μ-stationary, or μ-invariant,if for every continuous realfunction f defined on X, the following equality holds:∬_G× Xf(g· x) dμ(g)dν(x) = ∫_Xf( x) dν(x). Let 𝒲 be the set of all G_μ-stable vector subspaces of V ordered by inclusion. Letℒ_μ:=∑_λ(W)<λ_1W∈𝒲W. Then ℒ_μ is a proper G_μ-stable subspace of V whose Lyapunov exponent is less that λ_1, and is thegreatest element of 𝒲 with these properties.We will check thisProposition/Definition in Section <ref> (Lemma <ref>) and give additional information of the subspace ℒ_μ. The motivation of this definition comes from the followingresult of Furstenberg-Kifer. <cit.> Let μ be a probability on (V) thathas a moment of order one.Then there exists r∈{1, ⋯, d},a sequenceof T_μ-invariant subspaces(ℒ_i=ℒ_i(μ))_i=0^r {0}=ℒ_r⊂ℒ_r-1⊂⋯⊂ℒ_1 ⊂ℒ_0=Vand a sequence of real values λ_1(μ)=β^1(μ) > β^2(μ) > ⋯ > β^r(μ) such that if x∈ℒ_i-1∖ℒ_i, then almost surely,lim_n→ +∞1/nlog||L_n x|| =β^i(μ).* It is immediate that the subspace ℒ_μ defined in Proposition/Definition <ref> coincides with the subspaceℒ_1(μ) defined in the theorem above.Hence we will be using in the rest of article, the following useful equivalence:x∉ℒ_μ⟺a.s. lim_n→ +∞1/nlog||L_n x|| = λ_1.* Furstenberg and Kifer gave actually an expression of λ_1 in terms of the “cocycle average” of stationary measures.More precisely, let N be the set of all μ-stationary measures on . For every ν∈ N, let α(ν)=∬_(V)×log||g v||/||v||dμ(g) dν([v]). Then, they showed that* λ_1=sup{α(ν); ν∈ N}. * ℒ_μ={0}, if and only if,α(ν) is the same for all ν∈ N (and hence equal to λ_1). * Note that the filtration given by Furstenberg and Kifer is deterministic, unlike the one given by Oseledets theorem.The set {β_1(μ), ⋯, β_r(μ) } is included in the Lyapunov spectrum {λ_1(μ), ⋯, λ_d(μ)} but the inclusion may be strict. For x∈ V∖{0} fixed, the growth of ||L_n x|| is almost surely as exp(n β_i(μ)) for some i=1, ⋯, r.. Hence, the Lyapunov exponents that are distinct from the β_i(μ)'s do not characterize the growth of the norm of ||L_n x|| if we fix first x and then perform a random walk. However they do characterize norm growth if we perform a random walk and choose x in a random subspace of the filtration given by Oseledets theorem.For every non zero vector x (resp. non zero subspace W) of V, we denote by [x] (resp. [W]) its projection on .Ourfirst result describes the stationary measures on . Let μ be a probability measure on (V) such that λ_1>λ_2. Then,a)There exists aunique μ-stationary probability measure ν onwhich satisfies ν([ℒ_μ])=0.b)The projective subspace ofgenerated by the support of ν is [𝒰_μ], where𝒰_μ:=⋂_λ(W)=λ_1W∈𝒲W.Moreover, ν is non degenerate on [𝒰_μ] (i.e. ν gives zero mass to every proper projective subspace of[𝒰_μ]).c) (, ν) is aμ-boundary in the sense of Furstenberg (<cit.>) , i.e. there exists a random variable ω↦ [Z(ω)]∈ such that,for :=μ ^⊗-almost every ω:=(g_n)_n∈∈(V)^, g_1⋯ g_n ν converges weakly to the Dirac probability measure δ_[Z(ω)]. An immediate corollary is the followingKeep the notation of Theorem <ref>. Suppose that for every i=1, ⋯, r the exponent β_i(μ) is simple when seen as a top Lyapunov exponent for the restriction of the random walk to ℒ_i-1(μ). Then there are exactly r distinct ergodic μ-stationary measures on . Theassumption of Corollary <ref> is equivalent to saying that, for every i=1, ⋯, r, β_i(μ) is simple as a top Lyapunov exponent of _i-1(μ)/_i(μ). Hence, by Guivarc'h-Raugi's theorem <cit.>, a sufficient condition for the finiteness of ergodic μ-stationary measures on is that each quotient _i-1(μ)/_i(μ) isstrongly irreducible and proximal. Definitely, another sufficient condition is the simplicity of theLyapunov spectrum, i.e. λ_1(μ)>⋯ > λ_d(μ).After finishing this paper, it came to our knowledge that Benoist and Bruère have studied recently and independently the existence and uniqueness of stationary measures on projective spaces overin a non irreducible context, in order to study recurrence on affine grassmannians. We will state one of themain results of the authors, namely <cit.>, then discuss the similarities and differences with Theorem <ref> stated above. In<cit.>, the authors consider a real vector space V, G Zariski connected algebraic group subgroup G of (V), W a G-invariant subspace of V such that W has no complementary G-stable subspace,the actionof G on W and the quotient V/W is i-p and such that the representations of G in W and V/W are not equivalent.Then for every probability measure μ such that λ(V/W)>λ(W) and whose support iscompact and generates a Zariski dense subgroup of G,the authors show that there exists a uniqueμ-stationary probability measure ν on the open set ∖ [W] and that the Cesaro mean 1/n∑_j=1^nμ^*j⋆δ_x converges weakly to ν.Theorem <ref> recovers the aforementioned result.Indeed, μ hasa moment of order one since its support is assumed to be compact. The conditions on the Lyapunov exponents imply that ℒ_μ=W andλ_1>λ_2. Moreover, since W has no complementary G-stable subspace, then𝒰_μ=V. Theorem <ref> permits actuallyto relax the i-p assumption on the action on W in the previous statement; only the condition i-p on the quotient and λ(V/W)>λ(W) is enough.Moreover, there is no need for the compactness of the support of μ; a moment of order one is enough. Furthermore, μ^*j⋆δ_x converges weakly to ν (see Remark<ref>), not only in average.In addition, the vector space V can be defined on any local field k. We note that,in the rest of the present paper, we will be interested in understanding further properties of this stationary measure. Namely inTheorem <ref>(Section <ref>) below,wedescribe more precisely the support of νin terms of thethe limit set of T_μ and weprove its Hölder regularity in Theorem <ref>(Section <ref>).It is worth-mentioning that in <cit.>, the authorsshow that whenλ(W)≥λ(V/W),there is no μ-stationary probability measure on ∖ [W] and that the above Cesaro mean converges weakly to zero. This says somehow that G-stable subspaces with top Lyapunov exponent guide the dynamics. This information is not disjoint from the one given by Part b) of Theorem <ref> saying that the projective subspace generated by the support of ν is [𝒰_μ]. The techniques used in the two papers are highly different.In the present paper we obtain the existence of such astationary measure via Oseledets theorem while Benoist and Bruère use Banach-Alaoglu theorem and a method developed in <cit.> for the situation of locally symmetric spaces. Concerning the uniqueness of the stationary measure, Benoist and Bruère's proof is by contradiction via a beautiful argument of joining measure and previous results on stationary measures on the projective space by Benoist-Quint <cit.>.Here we use methods of <cit.> and <cit.> based on the μ-boundary property. Our method is more explicit as it was described in the introduction (see Propositions <ref> and Proposition <ref>).§.§ The geometry behind Theorem <ref> and guiding Examples§.§.§The geometry behind Theorem <ref> By Theorem <ref>, our dynamics takes place in the open dense subset ∖ [_μ] of . Here we understand further this dynamics by considering as a compactification of ∖ [_μ]and identifyingtopologically ∖ [_μ] with a compactquotient of the product space _μ× S(V/_μ). We willcheck that this productspace is dynamicallya skew product space with base the unit sphere S(V/_μ) and fibers _μseenas an affine space. Hence, for each random walk, we are choosing a “model”or a “realization” ofthat depends on the G_μ-stable space _μ. This point of view will be used in Section <ref>.Formally,let 𝒮_ be the unit sphere of the local field (, |·|), L a proper subspace of V and G a subgroup of (V) that stabilizes L.Let ||·|| be a norm on the quotient V/L such that (V/L, ||·||) is an inner product space (resp. orthogonalizable) whenis Archimedean (resp. whenis non-Archimedean, see Section <ref>).Denoteby S(V/L) the unit sphere of V/L. Fix a supplementary L̃ of L in V. Identifying V/Land L̃ in the usual way, the map (t,ξ)∈ L × S(V/L)⟼[t+ξ]∈∖ [L] yields a homeomorphism between ∖ [L] and the orbit space X/𝒮_,where X isthe product spaceX:= L × S(V/L)and𝒮_acts on X in the natural way.Using this bijection,the space X/𝒮_ is endowed with a natural structure of G-space such that the natural mapX/𝒮_≃∖ [L] ψ⟶P(V/L) is G-equivariant. The action of G on X/𝒮_ can be lifted to an action of G on X which commutes with the naturalaction of 𝒮_ on X, as we explain hereafter. Every element g∈ G can be written in a basis compatible with the decomposition V= L ⊕L̃ in the form[ A B; 0 C ] with A (resp. C) is a square matrix representing the action of g on L (resp. on the quotient vector space V/L) andB is a rectangular matrix.To write down equations properly, one has to make a choice in normalizing non zero vectors inV/L. We write a polar decomposition of (V/L)∖{0}:(V/L)∖{0}=^*_+× S(V/L) when = or = and(V/L)∖{0}= ϖ^× S(V/L) whenis non-Archimedean(for a fixed uniformizer ϖ and a fixed discrete valuation on ).Let N: (V/L)∖{0}⟶∖{0} such that N(x) is the unique ^*_+ or ϖ^-partofthe non zero vector x of V/L in its polar decomposition. In the Archimedean case, one has simply that N(x)=||x||.One can then check that the following formula defines anaction of G on X=L× S(V/L) that lifts the action of G on X/𝒮_ andcommutes with the action of 𝒮_ on X: [ A B; 0 C ]· (t,ξ) =(At+ Bξ/N(Cξ), Cξ/N(Cξ)).We observe that the G-space X has a skew product structure given by the above formula with base the unit sphere of the vector space V/L. Considering L as an affine space, the fiberwise action is given by affine maps, as forg=[ A B; 0 C ]∈ G and ξ∈ S(V/L) fixed,the mapσ(g,ξ) : t ⟶At+Bξ/N(Cξ) is an affine transformation of the affine space L. Moreover,themap σ: G × S(V/L) ⟶Aff(L) is a cocycle, i.e. σ(g_1g_2, ξ)=σ(g_1,g_2 ·ξ) ∘σ(g_2, ξ)whereg ·ξ = C ξ/N(C ξ) .Let now μ be a probability measure on (V) whose top Lyapunov exponent is simple and such thatG=G_μ and L=ℒ_μ.The μ-random walk on X is then given by the following recursive stochastic equation:t_n= A_n t_n-1 + B_n ξ_n-1/N(C_n ξ_n-1) , ξ_n=C_n ξ_n-1/N(C_n ξ_n-1)where{( [ A_n B_n; 0 C_n ]); n∈} is a sequence of independent random variables on (V) of same law μ. The result of Theorem <ref> translates in saying that there exists a unique μ-stationary probability measure ν on X/𝒮_. This measure can be lifted to a probability measure ν̃ on X which is μ-stationary, 𝒮_-invariant and unique for these properties.Note that the pushforward measure ψ⋆ν of ν (resp. ψ̃⋆ν̃)by the natural map ∖ [L] ψ⟶P(V/L)(resp. Xψ̃⟶S(V/L)) isalso a μ-stationary probability measure on P(V/L) (resp. S(V/L)).Since the top Lyapunov exponent ofπ(μ), projection of μ on (V/L), is also simple and satisfies ℒ_π(μ)={0} (see item 3. of Remark <ref>), Theorem <ref> applies again on V/L and implies thatψ⋆ν(resp. ψ̃⋆ν̃) is the unique μ-stationary probability measure on P(V/L) (resp. on S(V/L) which is 𝒮_-invariant).We note that when 𝒰_μ=V,the condition λ_1>λ_2forces the action on V/L to bestrongly irreducible and to containa proximal element (see Lemma <ref>). Hence, the uniqueness of the probability measure ψ⋆ν on P(V/L)can be seen in this caseas a corollary of Guivarc'h-Raugi's work<cit.>based on techniques developed by Furstenberg <cit.>. Finally, note thatstochastic recursions similar to (<ref>)appeared recently in <cit.>, with (ℒ_μ)=1,as a crucial tool to prove the homogeneity at infinity of the measure ν, in the affine situation.§.§.§ Guiding Examples The guiding examples through this article are the following. The first two (i-p setting and the affine one) are standard and we just checkthat our general framework include them. The third example is an interesting new one that mixes somehow the first two. Together with the simulations of Section <ref>, they illustrateour newgeometric setting and the dynamic on it. * The irreducible linear groups. If T is a sub-semigroup of (V) that acts irreducibly on V, then for every probability measure μ such that T_μ=T, we have by irreducibility ℒ_μ={0} and 𝒰_μ=V. Bya theorem of Guivarc'h-Raugi<cit.>,the condition λ_1>λ_2 is equivalent to saying that T is i-p(strongly irreducible and contains a proximal element).The results given byTheorem <ref>are known in this case and are due also to Guivarc'h and Raugi in the same paper. With the notation of Section <ref>, X is just the unit sphere of V (for a fixed norm). * The affine group. Let L be a hyperplane of V and T a sub-semigroup of (V)that stabilizes L. Assume for the simplicity that the action on V/L is trivial. Hence, in a suitable basis of V, all the elements ofT have a matrix of the form([ A b; 0 1; ]) with A representing the action on the vector space L.The projective spaceis seen as a compactification of the affine space L with L an affine chart (the action on the base of the product space X is trivial). It will be clear in the following discussion whether L is seen as a subspace of V or as an affine space. Let μ be a probability measure on (V) such that T_μ=T. We denote bya_1 (resp. a_2) thetop (resp. second) Lyapunov exponent of the probability measure A(μ), relative to the linear part of μ. Then by Lemma <ref> and Corollary <ref> below, the following equalities holdλ_1(μ)=max{a_1,0}, λ_2(μ)= min{a_1,max{a_2,0}}.The subspacesℒ_μand 𝒰_μ of V dependon the measure μ, unlike the previous example. More precisely,* Contracting case (a_1<0). In this case, 0=λ_1>λ_2=a_1 andℒ_μ=L.If we assume moreover that T does not fix any proper affine subspaces of L, then this translates to the linear action by saying that every T-stable vector space of V is included in L.In particular we have 𝒰_μ=V.We can then apply Theorem <ref>. Its contenttranslates back to the affine action by saying that there is a unique μ-stationaryprobability measure on L and that this measure gives zero mass toany affine subspace.This result is well known (see for instance <cit.>, <cit.>).* Expansive case (a_1>0): In this case,a_1=λ_1 and λ_2=max{a_2, 0}. Assume for simplicity that the sub-semigroup A_T generated by A(μ)acts irreducibly on the vector space L.Hence the condition λ_1>λ_2 is equivalent to A_T being i-p, which we assume to hold in the sequel.Assume moreover that T does not fix a point in L. With these assumptions,𝒰_μ=L and ℒ_μ={0}.In this case,Theorem <ref>says thatthere exists a unique μ-stationary probability measure on the (compactified) affine space L and that it is concentrated on the hyperplane at infinity. This probability measure corresponds to the unique A(μ)-stationary probability measure on the projective space P(L) of L (we are back to Example 1).We note that our results do not apply to the interesting case a_1=0, called the critical case. * The Automorphism group of the Heisenberg group.Let L be a one-dimensional subspace of ^3 and G the group of automorphisms of V that stabilizes L. In a suitable basis of ^3, we can identify G with the following matrix group: G={g=(a_g b_g0C_g ); a_g∈∖{0}; b_g∈^2; C_g∈GL_2() }⊂GL_3(). The group G can be thought of a dual of the affine group on^2. In this context, random walks on G appeared naturally in<cit.> as we have mentioned in the previous section. Also, if one imposes the condition |a|=(g)in the definition of G, then by letting the continuous Heisenberg group ℋ_3 act on its Lie algebra,it can be proved (see <cit.>) that G is isomorphic to the automorphism group ofℋ_3; the one dimensional fixed subspace of ^3 being the center ofℋ_3. With the notation of Section <ref>, X= × S^1 and the projective plane P^2() isseen as a one-point compactification of X/{± 1}. Recall that by formula (<ref>), Xhas a structure of skew-product space whose base is a circle and fibers the affine line L. Now let μ be a probability measure on G. Assume that:* the action of T_μ on ^3/L is irreducible* ∫_Glog|a_g| dμ(g) < λ_1(^3/L).In this case, λ_1>λ_2, if and only if, the action of T_μ on ^3/L is strongly irreducible and proximal (i-p) (see Lemma <ref>). By the irreducibility of the action onthe quotient,ℒ_μ=L. Moreover,𝒰_μ=^3, if and only if, there does not exist a G_μ-invariant decomposition ^3=L⊕ W. With these conditions, the content of Theorem <ref> is new.The stationary measure given by the aforementioned theorem projects onto the projective line to theμ-stationary probability measure relative to the i-p semigroup of _2(), projection ofT_μ on ^3/L.We referto the simulations of Section <ref>.Note that when ∫_Glog|a_g| dμ(g) > λ_1(^3/L) and L has no G_μ-invariant supplementary in ^3, we have also λ_1>λ_2 but _μ={0}.Theorem <ref> applies and implies that the unique μ-stationary probability measure on P^2() is [L], i.e. the point at infinity in X/{± 1}. This case is similar to the expansive one in the affine situation.§.§ The support of the stationary measure and Limit Sets Our next goal will be torelate the support of the stationary measure ν obtained above with the limit set of T_μ.We refer to<cit.>and <cit.> when such a study is conducted in the strong irreducible and proximal case.We begin by some notations for a general semigroup T⊂(V) and two T-invariant subspaces L and U of V such that U⊄L.Denote by[g]∈PGL(V) the projective map associated to a linear automorphism g∈(V) and by PT:={[g]; g∈ T}⊂PGL(V) the projection of T onto PGL(V). We will need the notion and some properties of quasi-projective transformationintroduced by <cit.> anddeveloped in <cit.>. Recall that a quasi-projective transformation is amap fromto itself obtained by a pointwise limit of a sequence ofprojective transformations.Denote by 𝒬 the set of quasi-projective maps.* We denote by T⊂𝒬 the set of quasi-projective transformations𝔮: ⟶, pointwise limitsof projective maps [g_n]∈PT with the following property: there exists a proper projective subspace [W] ofsuch that [U]⊄[W] and for every y ∉[W], 𝔮(y) is point p(𝔮)∈ [U]. Let Λ(T)={p(𝔮); 𝔮∈T}⊂ [U].We will check in Lemma <ref> that this is a closed T-invariant subset of . We will call it the limit set of T (note that it depends onthe subspace U).* We consider the T-space O=∖ [L] and we endow it with the topology induced from that of . If X⊂ O, we denote by X its closure inand by X^O its closure in O. LetΛ^a(T)=Λ(T)∩ O so that Λ^a(T) is a closed T-invariant subset of O.* Let T_0 (resp. T^a_0) the subset of T which consists of elements g with a simple and unique dominant eigenvalue corresponding to a direction p^+(g)∈ [U] (resp. p^+(g)∈ [U∖ L]). The choice of the superscript “a” in the definition above refers to “affine” in line with the description givenin Section <ref>.Indeed, suppose that U=V, fix a norm on the quotient V/Land letg∈ T_0^a. In a suitable basis of V, g can be represented as a matrixg =(AB0C )∈ Twith A the restriction of g to L,C aproximal elementand λ_top(g)=λ_top(C), where λ_top(·) denotes the top eigenvalue.Pick a normalized eigenvector ξ_0 of C.Then the point p^+(g)∈ can be identified with 𝒮_(t_0, ξ_0) with t_0∈ L being the unique fixed point of the affinemap t↦At +Bξ_0/λ_top(C) of L (seen as an affine space).Note that this affine map is equal tothe map σ(g,ξ_0) introduced in Section <ref>, up to an element in 𝒮_.When L is a hyperplane of V and C is trivial, then p^+(g) represents exactly thefixed point of the affine map t↦ At +B of L. Let T be a semigroup of (V), L and U be T-invariant subspaces such that U⊄L. Letμ be a probability measure on (V) such that λ_1>λ_2, T_μ=T, ℒ_μ=L and 𝒰_μ=U. Let ν be the unique stationary measure on ∖ [L]. Then, * T_0^a≠∅ and Supp(ν)=p^+ (T_0) = p^+(T_0^a).* Supp(ν)=Λ(T).* Forany [x] ∈∖ [L], we have Λ(T) ⊂T· [x]. In particular, Λ^a(T) is the unique T-minimal subsetof ∖ [L].We easily deduce the following characterization of the compactness of Supp(ν) when seen in the open subset O=∖ [L] of . The following are equivalent:* Supp(ν)∩ O is compact* Supp(ν) is a T-minimal subset of . * There exists [x]∈ O such that T ·[x]^O is compact * (assume in this part that U=V and use the notation of Remark <ref>) There exists c>0 such that for everyg =(AB0C )∈ T_0^a, one has ||(A-λ_top(C) I)^-1 (B ξ_C)||<c,where||·|| is a fixed norm on L, I is the identity matrix,λ_top(C)∈ is the top eigenvalue of C andξ_C is any eigenvector of C corresponding to λ_top(C) of norm one. * When _μ={0} (as in the i-p case or in the expansive cases of Examples 2 and 3in Section <ref>),it follows from Corollary <ref> that Supp(ν) is the unique T-minimal subset of .When _μ≠{0} (as the contracting cases of Examples 2 and 3) andSupp(ν)∩ O is compact, then the latter is a T-minimal subset ofbut never the unique such one as [L] is a compact T-invariant subset of that does not intersectSupp(ν). In particular, Supp(ν) is the unique T-minimal subset of , if and only if,_μ={0}. * It follows from Theorem <ref> that the support of νdepends only on T and not on μ (providedT_μ=T, λ_1(μ)>λ_2(μ) and 𝒰_μ=U, in which case L=ℒ_μ is uniquely determined as U⊄L).* We assume U=V and adopt the notation of Remark <ref>.It follows from item 3 of Theorem <ref> that a sufficient condition for the non compactness of Supp(ν) in O is the existence of at least one proximal element g∈ T with an attracting direction p^+(g)∈ [L]. We will check in Lemma <ref> that proximality is not needed, i.e. if there existsg= ( AB 0C) ∈ T with ρ_spec(A)>ρ_spec(C), then Supp(ν)∩ O is not compact. For the situation where T isa non degeneratesemigroup of the affine transformations of the real line in the contracting case, this boils down to the well-known factthat the support of the unique stationary measure ν on the affine line is non compact when there exists at least one transformation x↦ ax +b with |a|>1. * We continue the previous remark. It is easy to see that if Supp(μ) is a bounded subset of affinities of the real line such that |a|<1 for every x↦ ax+b in Supp(μ), then the support of the unique stationary measure on the real line is compact (the well-known example of Bernoulli convolutions fits in this category, see Remark <ref> ).In our situation,having ρ_spec(A)<ρ_spec(C) for everyg= ( AB 0C) in T is not sufficient to insure the compactness of the support of ν in O. We refer to Example 3. of Section <ref>.* We give in Section <ref> a sufficient condition for the compactness of the support of ν in O in the case dim(_μ)=1 (see Example 3 of Section <ref>) using the notion of joint spectral radius and the geometric setting of Section <ref>.Note that the joint spectral radius is known to play a role in the existence of anattractor to affine iterated functions systems (IFS) and that projective IFS are gaininga lot of importance recently (see for instance <cit.>). * It is definitely interesting to conduct a study concerning the tail of ν when the latteris not compact in O.We refer to <cit.> for the case of the affine line. §.§ Regularity of the stationary measureThe following result shows that the unique stationary measureν given by Theorem <ref> has Hölder regularity when μ has an exponential moment, i.e. when∫_(V)||g^± 1||^τ dμ(g)<+∞ for someτ>0.We denote by δ the Fubini-Study metric on the projective space(see Definition <ref>).We recall thatthe projective subspace ofgenerated by ν is [𝒰_μ] and that ν is non degenerate on it. Hence, the following result gives a precision of that fact.Let μ be a probability measure on (V) such that λ_1>λ_2. If μ has an exponential moment, then there exists α>0 such thatsup_H hyperplane of 𝒰_μ∫δ^-α([x],[H]) dν([x])<+∞.* We note that we will give a slightly more general statement in Theorem <ref>involving the distance to any projective hyperplane of .* Assume that𝒰_μ is not a one dimensional subspace of V (otherwise ν is a Dirac probability measure).Theorem <ref> implies then, through Markov's inequality,that ν isα-Hölder, i.e. there exists D>0 such that for every ϵ>0, and for ν-almost every [x]∈ [𝒰_μ], ν(B([x],ϵ))<D ϵ^α, where B(·, ·) denotes the open ball in the metric space ([𝒰_μ], δ).In particular, the Hausdorff dimension of ν is greater or equal to α.In the i-p case, Theorem <ref> is known and is due toGuivarc'h <cit.>. When applied to the affine group it is new. More precisely,Let μ be a probability measure on the group of affinities of an affine space L whose support does not fix any proper affine subspace. Assume that the Lyapunov exponent of the linear part of μ is negative (contracting case). Then the unique μ-stationary probability measure ν on L has a positive Hausdorff dimension. We note that the problem ofestimation of the Hausdorff dimension of ν was initially considered by Erdös (see for instance <cit.>) if T⊂Aff() preserves an interval of the line. Itled recently to deep results in similar situations (see <cit.>, <cit.> for example).In the more general situation of this paper, we get only qualitative results on the dimension of ν.One of the important estimates in random matrix products theory is the probability of return of the random walk to hyperplanes. It is well studied in the i-p case and leads to fundamental spectral gap results <cit.>, <cit.>, <cit.>, <cit.>.... The general setting studied in this paper leads to new estimates in this direction. Let V a finite dimensional vector space andμ be a probability measure on (V) with an exponential moment such that λ_1>λ_2. Then, for everyϵ>0, there exist β=β(ϵ)>0, n_0=n_0(ϵ)∈ such that for every n≥ n_0,x∈ V∖ℒ_μ and every f∈ V^*∖_μ̌,[ δ(L_n[x], [Ker(f)])≤exp(-ϵ n)]≤exp(-n β)/δ([x], [_μ]) δ([f], [_μ̌]). In this statement V^* denotes the dual space of V andμ̌ isthe pushforward probability measure of μ by the map g ∈(V)⟼ g^t∈(V^*).§ PRELIMINARIES §.§ Linear algebra preliminariesOur proofs rely onsuitable choice of norms on our vector spaces and on theexpression of the distance between a point and a projective subspace of(Lemma <ref> below).For the convenience of the reader, we recall in Section <ref> basic facts about orthogonality in non-Archimedean vector spaces (c.f. <cit.> for instance).The reader interested only in vector spaces over Archimedean fields can check directly Section <ref>. §.§.§ Non-Archimedean orthogonality Let (k,|·|) be a non-Archimedean local field.We denote by 𝒪_k={x∈k; |x|≤ 1} its ring of integers and 𝒪_^×={x∈; |x|=1} the group of units of 𝒪_. LetV bea vector space overof dimension d∈^* and B_0=(e_1, ⋯, e_d) a fixed basis of V. We consider the following norm on V:||x||:=max{|x_i|; i=1, ⋯, d} wherethe x_i's are the coordinates of the vector x in the basis B_0.Every suchfinite dimensional normed vector space over a non-Archimedean local field will said to be orthogonalizable. We say that two subspacesE and F of V areorthogonal when ||v+w||=max{||v||, ||w||} for everyv∈ E and w∈ F.A family of vectors (v_1,⋯, v_r) in V is said to be orthogonal if for every α_1,⋯, α_r∈,||α_1 v_1+⋯ +α_r v_r||=max{|α_1 |||v_1||, ⋯, |α_r| ||v_r||}.We recall that _d(𝒪_k) is the subgroup of the general linear group _d(k) formed by thematricesg such that g and g^-1have coefficients in 𝒪_k; which is equivalent to impose that g has coefficients in 𝒪_k and that(g)∈𝒪_k^×. One can show that_d(𝒪_k) is a maximal compact subgroup of _d(k). The following lemma gives crucial results of orthogonality in non-Archimedean vector spacessimilar to the classicalonesin the Archimedean setting. * For every basis B=(v_1,⋯, v_d) of V, the following statements are equivalent:i. B is orthonormal ii.Thetransition matrix from B_0 to B belongs to _d(𝒪_k) iii. B is a basis of the 𝒪_k-module𝒪_ke_1⊕⋯⊕𝒪_k e_d≃𝒪_k^d.* Every subspace E of V has an orthonormal basis and admits anorthogonal complement E^⊥. Without loss of generality,V=^d and B_0 the canonical basis. One can easily show that _d(𝒪_k)isthe isometry group of(V,||·||).* The equivalence between items i., ii. and iii. is an easy consequence of the fact that _d(𝒪_k) acts by isometries on V.2.,3.Let r be the dimension of E as a k-vector space,M=𝒪_k^d and E'=E∩ M. Then M is a free𝒪_k-module of rank d and E' is a submodule.Since k is a local field, then 𝒪_k is a Principal Ideal Domain (PID). Then the structure theorem of modules over PID's gives a basis B=(v_1,⋯, v_n) of M, r∈^* and scalars d_1,⋯, d_k∈𝒪_k such that (d_1v_1,⋯, d_r v_r) is a basis of E' as a 𝒪_k-module.The set B is clearly also a basis of the k-vector space V, r the dimension of E as k-vector spaceand (v_1,⋯, v_r) a basis of the subspace E of V.By the equivalence between 1.i. and 1.iii.,B is orthonormal. Hence items 2 and 3 follow immediately. Unlike the Archimedean case, a subspace mayhave more than one orthogonal complement in V. Indeed, consider =_2, V=^2 and the one dimensional subspaces E, E_1 and E_2 of V generated respectively by (1,0), (0,1)and (1,1). ThenE_1 and E_2 are two distinct orthogonal complements of E because the identity matrix and the matrix([ 1 1; 0 1; ]) belong to SL_2(_2).Let E be a subspace of V and ||·|| the quotient norm in V/E, i.e. for everyx∈ V/E,||x||:=inf{||x+y||; y∈ E}. One can easily showthat for any orthogonal supplementary E^⊥ of E in V, and for every x∈ V/E, the following holds:||x||=||π_E^⊥(x)||.Here π_E^⊥ denotes the projection onto E^⊥ with kernel E. §.§.§ The Fubini-Study metric Now (k,|·|) is a local field and V a vector space over k ofdimension d≥ 2 and B_0 a fixed basis of V.Whenis Archimedean, we endow V with the canonical norm ||·|| for which (V,||·||) is an inner product space andB_0 is anorthonormal basis. Whenis non-Archimedean,we endow V with the norm described in the previous section. We consider the norm on ⋀^2 V, which will be denotedalso by||·||,such that (e_i∧ e_j)_1≤ i<j ≤ d is an orthonormal basis of ⋀^2 V.(Fubini-Study metric) Let (V,||·||) as above andthe projective space of V.* For every [x],[y]∈, we set:δ([x],[y]):=||x ∧ y||/||x|| ||y||.Then δ defines a metric on , called the Fubini-Study metric (see for instance <cit.>). * For every subset Y ofand [x]∈, letδ([x],Y)=inf_[y]∈ Yδ([x],[y]). The following are easy facts.* Whenis non-Archimedean, δ is actually ultrametric. * The metric δ is bounded by one. * If x and y are orthogonal, then δ([x],[y])=1.The following lemma will be fundamental for us. Let k be a localfield and (V,||·||) as above.Let E be a subspace of Vand E^⊥ an orthogonal complement (see Lemma <ref> when k is non-Archimedean).We denote by π_E^⊥ theprojection onto E^⊥ with kernel E. By abuse of notation, we denote also by ||·|| the quotient norm on V/E (see Remark <ref>).Then, for every non zero vector x of V, δ([x],[E])=||π_E^⊥(x)||/||x||=||x||/||x||. By Remark <ref>, it is remaining to prove the left equality only.Let [x]∈. WLOG x∉E.We write x=x_1+x_2, with 0≠ x_1∈ E and π_E^⊥(x)= x_2∈E^⊥. On the one hand, δ([x],[E])≤δ([x],[x_1])= ||x ∧ x_1||/||x|| ||x_1|| =||x_2 ∧ x_1||/||x|| ||x_1||≤||x_2||/||x||.This proves that δ([x],[E]) ≤||π_E^⊥(x)||/||x||.On the other hand, let B be an orthonormal basis of V obtained by concatenating a orthonormal basis,say (v_1,⋯, v_r) of Eandan orthonormal basis, say (v_r+1,⋯, v_d), of E^⊥(see Lemma <ref> when k is non-Archimedean). Let y∈ E∖{0}.By writing x_1,x_2 and y in the basis B, we seethat x_1∧ y belongs to subspace of ⋀^2 V generated by(v_i∧ v_j)_1≤ i< j ≤ r and x_2∧ y to the one generated by (v_i∧ v_j)_(i,j)∈{1,⋯, r}×{r+1,⋯, d}.The basis (v_i∧ v_j)_1≤ i<j ≤ d is also orthogonal in ⋀^2 V. Hence||x ∧ y||= ||x_1∧ y + x_2∧ y||≥||x_2∧ y||. Since x_2 and y are orthogonal in V, we have that ||x_2∧ y||=||x_2|| ||y||. Hence, for every y∈ E∖{0}, ||x ∧ y||≥ ||x_2|| ||y||. Hence,δ([x],[E])=inf_[y]∈ [E]||x ∧ y||/||x|| ||y||≥||x_2||/||x||=||π_E^⊥(x)||/||x||.The left equality of (<ref>) is proved.§.§ Preliminaries on Lyapunov exponents In Lemma <ref>, we recall a crucial result due to Furstenberg-Kiferthatreduces the computation of the top Lyapunov exponentof a random walk on a group of upper triangular block matrices to the top Lyapunov exponents of the random walks induced on the diagonal parts. For the reader's convenience, we include a proof. The, wededuce Corollary <ref> which showsthat all the other Lyapunov exponents of μ can be also read on the diagonal part with the right multiplicity. <cit.>, <cit.> Let k be a local field, V a finite dimensional vector space defined over k,μ be a probability on (V) having a moment of order one. Consider a G_μ-invariant subspace W of V. Then the first Lyapunov exponent λ_1 of μ is given by: λ_1= max{λ_1(W), λ_1(V/W)}.Since we deal here onlywith only top Lyapunov exponents, we will omit the subscript 1 in the notation.Without loss of generality, all the elements of G_μ are represented byd× d invertiblematrices of the form ([ A B; 0 C ]) where A represents the action of G_μ on W andCthe action on the quotient V/W. We use the canonical norm on V and the associated operator norm on End(V). Only the inequalityλ≤λ̃:= max{λ(W), λ (V/W)} requires a proof.For every n∈^*, write L_n=([ A_n B_n; 0 C_n; ]) for the left random walk at time n. Denote by (L'_n)_n∈^* the sequence of random variables defined by L'_n:=X_2n⋯ X_n+1 so that L_2n=L'_n L_n.Writing L'_n=([ A'_n B'_n;0 C'_n;]), we have that: B_2n=A'_nB_n+B'_nC_n.Fix for now ϵ>0 and η∈ (0,1).We know that the sequences of random variables(1/nlog ||L_n||)_n∈^* (1/nlog ||A_n||)_n∈^* and (1/nlog ||C_n||)_n∈^* converge in probability respectively to λ, λ(W) and λ(V/W). Moreover, A'_n (resp. B'_n) has the same law as A_n (resp. B_n) for every n∈^*. We deduce that there exists n_0=n_0(ϵ, η), such that for every n≥ n_0, all the following four real number are greater than 1-η: ( ||B_n||≤ e^n λ+nϵ),( ||C_n||≤ e^n λ(V/W)+nϵ),( ||A'_n||≤ e^n λ(W)+nϵ) and( ||B'_n||≤ e^n λ+nϵ). Using now identity (<ref>) and the inequality λ̃≤λ, we deduce that for n≥ n_0, ( ||B_2n||≤ 2 e^n (λ + λ̃)+ 2 n ϵ)≥ 1-4η.But since 2λ(W)≤λ+λ and 2λ(V/W)≤λ +λ, we obtain two other estimates similar to(<ref>) by replacing B_2n with A_2n and C_2n respectively (andtaking again n_0 bigger if necessary). Hence,for every n≥ n_0, ( ||L_2n||≤4 e^n (λ +λ̃ )+ 2n ϵ)≥ 1-6η. Butby the convergence of(1/nlog ||L_n||)_n∈^* in probability to λ, we can impose that for n≥ n_0, ( ||L_2n||≥e^2 nλ - nϵ)≥ 1-η, so that for n≥ n_0,(e^2nλ - nϵ≤ ||L_2n|| ≤ 4 e^n (λ + λ) + 2n ϵ)≥ 1- 7 η.Choosing any η∈ (0,1/7), and letting n→ +∞ and then ϵ→ 0, we get that λ≤λ̃. Consider the same situation as in the previous lemma.Denote by S_1 (resp. S_2) the set of Lyapunov exponentsassociated to the probability measure induced on W (resp. V/W). Then the set of Lyapunov exponents associated to μis S_1∪ S_2. Also the multiplicity of an exponent for the random walk in (V) is the sumof its multiplicity as an exponentfor the restricted random walk in(W) (if any) and as an exponent for the random walk in(V/W)(if any).First note that if E and F are two G_μ-invariant finite dimensional vector spaces, then λ_1(⋀^2 E)=λ_1(E)+λ_2(E) and λ_1(E⊗ F)=λ_1(E)+λ_1(F). Let now W be a supplementary of W in V. Let k∈{2,⋯, d}.The following decomposition holds ⋀^k V = i+j=k0≤ i,j≤ k⊕(⋀^i W ⊗⋀^jW).For every p∈{0,⋯ k}, let F_p:=i+j=k0≤ j≤ p⊕(⋀^i W ⊗⋀^jW). This isa G_μ-invariant subspace of ⋀^k Vand the quotientF_p/F_p-1 is isomorphic as G_μ-representation to⋀^k-pW ⊗ ⋀^p(V/W) (with the convention F_-1={0}). Since{0}=F_-1⊆ F_0⊆⋯⊆ F_k-1⊆ F_k=⋀^k V is a filtration of ⋀^kV,we applyLemma <ref>at most k+1 times and use the observations at the beginning of the proof in order to get the following identity: λ_1+⋯ + λ_k = max{λ_1(W)+⋯ + λ_k-p(W) + λ_1(V/W)+⋯ + λ_p(V/W);p=0, ⋯, k}. In the previous equation, we usedthe convention λ_i(W)=-∞ (resp. λ_i(V/W)=-∞) if i exceeds the dimension of W (resp. V/W). Note that for k=1,(<ref>) boilsdown to Theorem <ref>. Let m_1 bethe multiplicity of the top Lyapunov exponent λ_1 (as an exponent in (V)).Applying(<ref>) for k=1, ⋯, 1+m_1 gives two informations: first that λ_2 is the second largest number in the set S_1∪ S_2 and second that the multiplicity of λ_1 in (V) is the sum of its multiplicity as an exponent in (W) and in (V/W). Recursively, one shows the desired property for the all the other Lyapunov exponents.§.§ On the subspaces ℒ_μ and 𝒰_μLetμ be a probability measure on (V). In Definition <ref>, we introduced the following subspace of V: ℒ_μ:=∑_λ(W)<λ_1W∈𝒲W.In the statement of Theorem<ref>, we introduced the following subspace of V: 𝒰_μ:=⋂_λ(W)=λ_1W∈𝒲W.In this section, we state some useful properties of these subspaces that follow immediately from their definition. ℒ_μ is a proper G_μ-stable subspace of V whose Lyapunov exponent is less that λ_1, and is thegreatest element of 𝒲 with these properties.When λ_1>λ_2, 𝒰_μ⊄ℒ_μ. Inparticular, 𝒰_μ is non zero in this case and isthesmallest G_μ-subspace whose Lyapunov exponent isλ_1. The subspace ℒ_μ has the claimed property because on the one hand the sum that defines it can be made a finite one and on the other hand if W_1 and W_2 are two G_μ-stable subspaces of V, then one can easily prove that λ(W_1+W_2)=max{λ(W_1),λ(W_2)}.Assume now that λ_1>λ_2 and consider two G_μ-stable subspaces W_1 and W_2of V such that λ(W_1)=λ(W_2)=λ_1. We will prove that λ(W_1∩ W_2)=λ_1; and the claim concerning 𝒰_μ will immediately follow. Indeed, assume that λ(W_1∩ W_2)< λ_1. Then byLemma <ref> and Corollary <ref>, we deducethat the top Lyapunov exponent of E:=V/W_1∩ W_2 is simple and is equal to λ_1.The same holds for the subspaces W_1/W_1∩ W_2 and W_2/W_1∩ W_2 of E. By simplicity of λ_1 in E, we deduce that(W_1/W_1∩ W_2) ∩(W_2/W_1∩ W_2)≠{0}, contradiction.The followingeasy lemma will be crucial for us. For every g∈(V), we denote by g^t∈(V^*) the transpose linear map on the dual V^* of V, i.e. (g^t f) (x)=f(g x) for every g∈(V), f∈ V^* and x∈ V. For every subspace W of V, we denote by W^0⊆ V^* its annihilator, i.e. W^0={f∈ V^*; f_|_W=0}.(Duality between ℒ_μ and 𝒰_μ) Let μ be a probabilitymeasure on (V) such thatλ_1>λ_2. Denote byμ̌ the probability measure on (V^*) defined as the law of X_1^t, where X_1 has law μ. Then, ℒ_μ^0 = 𝒰_μ̌,𝒰_μ^0 =ℒ_μ̌. If W⊆ V is a G_μ-stable subspace of V, then W^0 is a G_μ̌-stable subspace of V^* which isisomorphic as G_μ̌-space to (V/W)^*.Hence λ_1(W^0, μ̌) = λ_1((V/W)^*, μ̌)=λ_1(V/W, μ).Hence, by Lemma <ref>, λ_1=max{λ_1(W, μ), λ_1(W^0, μ̌)}. Using Corollary <ref>, we deduce that when λ_1>λ_2, one and only one ofthe numbers λ_1(W, μ) and λ_1(W^0, μ̌) is equal to λ_1. Also, we deduce that W^0 contains a G_μ̌-stable subspace of V^* of μ̌-Lyapunov exponent is equal to λ_1, if and only, W is included in a G_μ-stable subspace whose μ-Lyapunov exponent is less than λ_1. Applying the previous remarks for W:=ℒ_μ, we get thatℒ_μ^0 is a G_μ̌-stable subspace of V^* whose Lyapunov exponent for μ̌ is equal to λ_1 and is the smallest such subspace. Since μ and μ̌ have the same Lyapunov exponents,Lemma <ref> yieldsthe identityℒ_μ^0 = 𝒰_μ̌. The equality 𝒰_μ^0 =ℒ_μ̌ follows also. During the proofs, we will frequently go back to the case where 𝒰_μ is the whole space V. We referto three guiding examples of Section <ref> where this condition was always satisfied, thanks to a “natural” geometric conditionimposed at each time.The following lemma reformulates this condition in different ways.Assume that λ_1>λ_2. The following propertiesare equivalent:* 𝒰_μ=V.* For every G_μ-stableproper subspace W of V,λ(W)<λ_1 * ℒ_μ is thegreatest element of 𝒲∖{V}, i.e. every G_μ-stable subspace of V iseither V or is included in ℒ_μ.* ℒ_μ̌={0}.Moreover, when one of these conditions is fulfilled, the action of T_μ on the quotient V/ℒ_μ is strongly irreducible and proximal.The equivalence between (1), (2), (3) and (4) is easy to prove by definition of ℒ_μ and 𝒰_μ, and by Lemmas <ref> and <ref>. We prove now the last statement. Assumethat (3) holds. It follows that the action of T_μ on the quotient V/ℒ_μ is irreducible. But byLemma <ref> and Corollary <ref>,the top Lyapunov exponent of V/ℒ_μ is simple. It is enough now to recall the following known result from <cit.> (see also <cit.>):ifE is a vector space defined over a local field andη is a probability measure on (E)such that T_η is irreducible, then T_η is i-p if and only if the top Lyapunov exponent relative toη is simple. This ends the proof. If ρ: G_μ⟶(𝒰_μ) is the restriction map to 𝒰_μ, then it is easy to see that 𝒰_ρ(μ)=𝒰_μ and that ℒ_ρ(μ)=ℒ_μ∩𝒰_μ. Observe also that it follows from Lemma <ref> that the action of T_μ on 𝒰_μ/ℒ_μ∩𝒰_μ is strongly irreducible and proximal. We will frequently usethe representationρ to go back to the case 𝒰_μ=V.* Another case for which estimates are easier to handle is the caseℒ_μ={0} (i.e. 𝒰_μ̌=V^*). This conditionappeared in <cit.> (see also <cit.>) as a sufficient conditionto ensure the continuity of the function μ↦λ(μ).Moreover, it corresponds to a unique cocycle average (see Remark <ref>). Recall that by Section <ref> this condition is satisfied for random walks in irreducible groups and in the affine group in the expansive case.However weinsiston the fact that one of the novelty of the present paper is to give limit theorems,when λ_1>λ_2, inthe case ℒ_μ≠{0} (as for instance random walks on the affine group in the contracting case, see Section <ref>).We refer also to <cit.> where limit theorems for cocycles are given depending on their cocycle average(s).* Note that if λ_1>λ_2, then it follows from Lemmas <ref> and <ref> that the following statements are equivalent:* ℒ_μ={0}.* For every G_μ-stableproper subspace W of V,λ(W)=λ_1 * Every G_μ-stable proper subspaceof V contains 𝒰_μ. * 𝒰_μ̌=V^*. * If π: G_μ⟶(V/ℒ_μ) is the morphism of the projection ontoV/ℒ_μ, then ℒ_π(μ)={0} and 𝒰_π(μ)=π(𝒰_μ). Observe also that if λ_1>λ_2, then byCorollary <ref> the top Lyapunov exponent of V/ℒ_μ is equal to λ_1 and is also simple. § STATIONARY PROBABILITY MEASURES ON THE PROJECTIVE SPACEIn this section, we prove Theorem <ref>.This will be donethrough different steps. In Section <ref> below, we show that if a stationary measure ν onsuch that ν([ℒ_μ])=0 exists,then this determines the projective subspace generated by its support. In Section <ref>, we show the existence of such a measure via Oseledets theorem. InSection <ref> we prove that it is unique in a constructive way. More precisely, we show in Proposition <ref>that ν is the law of a random variable [Z(ω)]∈ characterized in the following way: every limit point of the rightrandom walk(R_n)_n∈^* suitably normalized is almost surely of rank one with image that projects to [Z(ω)] in .We recall that k is a local field, V is a vector space over k of dimension d≥ 2 anddenotes the projective subspace of V.We endow V with the norm ||·|| described in Section <ref>.If μ is a probability measure on (V), then T_μ (resp. G_μ) denotes the sub-semigroup (resp. subgroup) of (V) generated by the support of μ. We denote by𝒲 the set of all G_μ-stable subspaces of V and for every W∈𝒲, λ(W) denotes the Lyapunov exponent relative to W.For every g∈(V), we denote by g^t∈(V^*) its transpose map. We denote by W^0⊆ V^* the annihilator of a subspace W of V. §.§ On the support of stationary probability measures Let μ be a probability measure on(V) such that λ_1>λ_2 and ν a stationary probability measure of the projective spacesuch thatν([ℒ_μ])=0. Let 𝒰_μ:=⋂_λ(W)=λ_1W∈𝒲W.Then,* The projective subspace generated by the support of ν is [𝒰_μ]. * Theprobability measure ν is non degenerate in[𝒰_μ]i.e. its gives zero mass to every proper projective subspace of[𝒰_μ]. The proof of this proposition will be done through different intermediate steps. First, we give belowa criterion insuring that a stationary measure on the projective space is non degenerate. When G_μ is strongly irreducible, Furstenberg has shown that every μ-stationary probability measure on the projective space is non degenerate. The proof of Furstenberg yields in fact the following general result. It will be used inLemma <ref> in order to identify non degenerate stationary measures outside the strongly irreducible case. Let E be a finite dimension vector space, μ a probability measure on (E) and ν a μ-stationary probability measure on the projective space P(E) of E.Then there exists a projective subspace of P(E) whose ν-measure is non zero, of minimal dimension and whose G_μ-orbit is finite. Equivalently, there exists a finite index subgroup G_0 of G_μ such that at least one of the projective subspaces of P(E) charged by ν is stable under G_0. Let Λ be the set of projective subspaces of P(E) charged by ν and of minimal dimension, say l. Letr=sup{ν([W]); [W]∈Λ}.By minimality of l, two distinct subspaces[W_1] and [W_2] of Λ satisfy ν([W_1∩ W_2])=0. Since ν is of total mass 1, we deduce that there are only finitely many subspaces [W]∈Λ such that ν([W])≥r/2. In particular, r=max{ν([W]); [W]∈Λ}. Consider then the following non-empty finite set:Γ:={[W]∈Λ; ν([W])=r}.We claim that Γ isstable under G_μ, which is sufficient to show the desired lemma.Indeed, since ν is a μ-stationary probability measure, then for every[W]∈Γand n∈:r=ν([W])= ∬1_[W] (g· [x]) dμ(g) dν([x]) = ∫ν (g^-1· [W])dμ^n(g).Let b be any probability measure onwith full support.By replacing if necessary μ by∑_i=1^+∞b(i)μ^iin the equality above, we can assumewithout loss of generalitythat the support of μ isthe semigroupT_μ:=∪_n∈Supp(μ^n).By combining this remark, together with equality(<ref>)and the maximality of r, we obtain that∀ g∈ T_μ, ν(g^-1· [W])=ri.e. g^-1·[W]∈Γ.Hence for every g∈ T_μ, g^-1Γ⊂Γ.Since Γ is finite, we deduce that for every g∈ T_μ, g Γ =Γ.It follows thatΓ is T_μ-stable (or equivalently G_μ-stable). We know that whenλ_1>λ_2, G_μ is irreducible if and only if G_μ is strongly irreducible (see<cit.>). Here's below a generalization. Let E be a finite dimension vector space and μ a probability measure on (E) such thatλ_1>λ_2. * If ℒ_μ={0}, then G_μ cannot fix any finite union ofnon zero subspaces of E unless they all contain 𝒰_μ.* Dually, if 𝒰_μ=E, then G_μ cannot fix a finite union of proper subspaces ofE unless they are all contained in ℒ_μ. In particular, if 𝒰_μ=E,then aμ-stationary probability measure ν onP(E) is non degenerate if and only if ν([ℒ_μ])=0.It is enough to showstatement 1. Indeed, the first part of statement 2 is actually equivalent to the first oneby passing to the dual E^* of E, thanks to Lemma <ref>, the fact that μ and μ̌ have the same Lyapunov exponents and finally to the fact that G_μ stabilizes a finite union {V_1, ⋯, V_r} of subspaces of E if and only if G_μ̌ stabilizes{V_1^0, ⋯, V_r^0} in E^*. The last part of the second statement is a consequence of the first part of the same statement and ofLemma<ref>. Now we prove the first statement. Arguing by contradiction, we let r to be the integerin {1,⋯, d-1} defined as the minimal dimension of a non zero subspace V of E such that 𝒰_μ⊄V and such that V belongs to some finite G_μ-invariant set of subspaces of E. Let V be such a subspace of E with dimension r and L:={g V; g∈ G_μ} be the orbit of V under G_μ. This is a finite G_μ-invariant set of subspaces of E all having the same dimension r and all not containing 𝒰_μ (as the latter is a G_μ-invariant subspace of E). Moreover, the cardinality s of L is greater or equal to 2 because the assumption ℒ_μ={0} implies that any G_μ-invariant subspace of V contains 𝒰_μ (see item 2. of Remark <ref>, dual of Lemma <ref>). Let then L:={V_1, ⋯, V_s} with the V_i's pairwise distinct and consider the following non empty set below:Γ:={V_i∩ V_j; 1≤ i <j ≤ s}.It is immediate that Γ is a finite G_μ-invariant set of subspaces ofE, all of them not containing 𝒰_μ and of dimension <r. By minimality ofr, we deduce that ∀ i≠ j,V_i ∩ V_j ={0} In particular, the projective subspaces [V_i]'s of P(E) are disjoint, so that we can define the following positive real number: α:=inf{δ([V_i], [V_j]); 1≤ i<j≤ s}>0. Letx∈ V_1∖{0} and y∈ V_2∖{0}. For every g∈ G_μ, there exist i=i(g) ,j= j(g) ∈{1,⋯, s} such thatg x∈ V_i and g y∈ V_j. We claim that i≠ j for every g∈ G_μ. Indeed, if i=j, then by denoting byk the unique integer such that g^-1V_i=V_k, we would havex∈ V_1∩ V_k and y∈ V_2∩ V_k. This contradicts (<ref>).We deduce that ∀ g∈ G_μ, δ(g [x], g[y])≥α.But since ℒ_μ={0}and since λ_1>λ_2, we have by<cit.> (see Theorem <ref>)that:δ(L_n [x], L_n [y])≤||⋀ ^2L_n||||x ∧ y||/||L_n x ||||L_n y||n→ + ∞a.s.⟶0,which contradicts(<ref>). First, we prove that [𝒰_μ] contains the projective subspace S ofgenerated by the support of ν. LetE be a G_μ-stable subspace of V such that λ(E)=λ_1. We want to show that ν([E])=1. First, we check that for every [x]∈∖ [_μ],the following almost sure convergence holds: δ(L_n[x], [E])n→ +∞a.s.⟶ 0.Indeed, consider the quotient norm on V/E.By Lemma <ref>,the following holds for every [x]∈:δ([x], [E])= ||x||/||x||.But since λ(E)=λ_1>λ_2, Lemma <ref> implies that λ(V/E)<λ_1.Hence, ∀ x ∈ V, a.s., lim sup1/nlog || L_n x|| < λ_1.Combining(<ref>),(<ref>) and Theorem <ref> gives, for any [x]∈∖ [_μ], the almost sure convergence(<ref>).Let now ϵ>0.Since ν is μ-stationary, we have for every n∈^*, ν{[x]∈; δ([x],[E])>ϵ} = ∫_(δ(L_n[x], [E])>ϵ) dν([x]).Since ν([_μ])=0,(<ref>) holds for ν-almost every [x]∈. In particular, for ν-almost every [x]∈, the following holds (δ(L_n[x], [E])>ϵ)n→ +∞⟶ 0. By Fubini's theorem and(<ref>), we deduce that ν([x]∈; δ([x], [E])>ϵ )=0. This being true for every ϵ>0, we deduce that ν([E])=1.This being true for every such stable subspace E, and since the intersection defining 𝒰_μ can be made a finite one (the dimension of V is finite), we deduce that ν([𝒰_μ])=1. Since [𝒰_μ] is closed in , we deduce that S⊂ [𝒰_μ].In order to prove the other inclusion,write S=[E] for some subspace E of V.Recall that Supp(ν) is T_μ-invariant, i.e. ∀ g∈ T_μ, g ·Supp(ν) ⊂Supp(ν).It follows from(<ref>) that E is a G_μ-invariant subspace of V. Moreover,since ν([_μ])=0, Theorem <ref> implies that the Lyapunov exponent relative to E is λ_1.By definition of 𝒰_μ, we deduce that 𝒰_μ⊂ E and then that [𝒰_μ]⊂ S.Item (1) of the proposition is then proved.In order to prove point (2) of the proposition, we set for simplicity of notation E=𝒰_μ and denote by ρ the restricted representation G_μ⟶(E). It follows from abovethat ν is a ρ(μ)-stationary probability measure on P(E). By definition of E, we have the following equalities: λ(E)=λ_1 , ℒ_ρ(μ)=ℒ_μ∩ E , 𝒰_ρ(μ) = E.By Lemma<ref>, the first and the third equalities above show that the probability measureρ(μ) on (E) satisfies the assumptions of Lemma <ref>. Since ν([ℒ_μ])=0, the second equality above gives ν([ℒ_ρ(μ)])=0. By Lemma <ref> again, ν is non degenerate on P(E).§.§ Oseledets theorem and stationary measures In this section, we prove that given a probability measure μon (V) such that λ_1>λ_2, there existsa μ-stationary probability measure ν on the projective spacethat satisfies the equality ν([ℒ_μ])=0 and the conclusions of Proposition <ref>. Our proof is constructive: we use Oseledets theorem to derivea random variable [Z]∈ of law ν from the random walk associated to μ.Since λ_1>λ_2, such a stationary measure will immediately be a μ-boundary. We note that the existence of such a probability measure holds even if λ_1=λ_2. This can be proved using themethods developed in<cit.>. Since the framework of the latter article is very general, the method is not constructive.Letμ be a probability measure on (V) such that λ_1>λ_2. Then, there exists aμ-stationary probability measure ν onsuch that ν([ℒ_μ])=0. By Proposition<ref>, ν is non degenerate on[𝒰_μ]. Moreover, ( ∖ [ℒ_μ], ν) is a μ-boundary. Such a measure will be obtained thanks to Oseledets theorem, and more precisely the equivariance equality we recall below.<cit.> Let (Ω,θ,) be an ergodic dynamical system. LetA: Ω⟶(V) be a measurable application such that log||A|| and log||A^-1|| are integrable. Then there exist l∈^*, m_1, ⋯, m_l∈^* and real numbers λ_1=⋯ = λ_m_1 > ⋯ > λ_m_l-1+1= ⋯ =λ_m_lsuch that for -almostevery ω∈Ω, there existsubspaces E=E^1_ω⊃⋯⊃ E^l_ω⊃ E^l+1_ω={0} such that: * Equivariance equality: forevery1≤ i ≤ l, A(ω)· E^i_ω =E^i_θ(ω)* for everyi=1,⋯, l and every non zero vector v of E, v∈ E^i_ω∖ E^i+1_ω if and only iflim1/nlog||A(θ^n-1 (ω))⋯ A(θ (ω))A(ω) v|| = λ_m_i.* m_i=dim(E^i_ω) - dim(E^i+1_ω), for everyi=1, ⋯, l.If, moreover,θ is invertiblethen there exists a splitting V=F_ω^1⊕⋯⊕F_ω^l such that4. Equivariance equality: forevery1≤ i ≤ l, A(ω)· F^i_ω =F^i_θ(ω). 5.foreveryi=1,⋯, l and every non zero vectorv∈F^i_ω, lim_n → +∞1/nlog||A(θ^n-1 (ω))⋯ A(θ(ω))A(ω) v|| = λ_m_i andlim_n→ +∞1/nlog||A^-1(θ^-n (ω))⋯ A^-1(θ^-1(ω))v|| = -λ_m_i.6. E_ω^i=⊕_j=i^lF_ω^j, for everyi=1, ⋯, l.Moreover, the subspaces E_ω^i and F_ω^i are unique -almost everywhere, andthey depend measurably on ω. Let d=dim(V),Ω=GL(V)^^*, =μ^⊗^*, θ the shift operator and A: Ω⟶ G, ω=(g_i)_i∈^*⟼ g_1.The distinct Lyapunov exponents relative to the measure μ will be denoted by λ_1=⋯ = λ_m_1>⋯λ_m_l-1+1=⋯ = λ_m_l=λ_d. The ones relative to the reflected measure μ̌, law of g_1^-1, are -λ_m_l>⋯ >-λ_m_1. We will construct ν as the law of the least expanding vector R_n^-1 given by Oseledets theorem. More precisely, applying Oseledets theorem for the dynamical system (Ω, , θ) and the transformation A^-1 (and not A),we obtain for the sameintegers l, m_1, ⋯, m_l above and for the same exponents λ_m_i's,a random filtration E^0_ω={0}⊂ E^1_ω⊂⋯⊂ E^l_ω=V such that for -almost every ω=(g_i)_i∈^*∈Ω:*E^i_ω= g_1 · E^i _θ(ω),for every1≤ i ≤ l.* For every non zero vectorv of V and every i=1, ⋯, l: v∈ E^i_ω∖ E^i-1_ω⟺lim_n→ +∞1/nlog||R_n^-1 v || = -λ_m_i,whereR_n(ω)=g_1⋯ g_n is the right random walk.* For every i=1, ⋯, l,m_i=dim(E^i_ω) - dim(E^i-1_ω).Under the assumption λ_1>λ_2, we havem_1=1 so that by(<ref>) kZ(ω):=E^1_ω is a line for -almost every ω∈Ω. Letν be the law of the random variableZ: Ω⟶, ω⟼ [Z(ω)] on the projective space. The probability ν is μ-stationary. Indeed, for every real valuedmeasurable functionf on,∫_f([x]) dν([x]) = ∫_Ωf(E^1_ω) d(ω)= ∫_Ωf ( g_1· E^1_θ(ω) ) d(ω)= ∫_G[ ∫_Ωf( γ·E^1_θ(ω) ) d(ω)] dμ(γ)= ∫_G×f(γ· x)dμ(γ)dν([x]) .Equality(<ref>)is straightforward consequence of the equivariance equality (<ref>);(<ref>) is due to the independence of g_1 and θ(ω)=(g_2, g_2, ⋯, ) while(<ref>) is true because θ preserves the measure .Finally, we show that ν([ℒ_μ])=0. Let E be a proper G_μ-stablesubspace such thatλ(E)<λ_1. Fixω∈Ω.Then, since the least Lyapunov exponent of m̌ǔ restricted to E is equal to -λ(E), ∀ v ∈ E, lim1/nlog||R_n^-1(ω) v ||>- λ_1.Taking if necessaryω in a measurable subset of Ω of -probability 1, assertion(<ref>)gives∀ v∈ E,v∉kZ(ω) ,i.e. [Z(ω)]≠ [v].Hence ν([E])=0.The fact that ν is a μ-boundary is also a consequence of the equivariance equality(see for example <cit.>, <cit.>, <cit.>). §.§ Uniqueness of the stationary measureIn this section, we prove that the stationary measure given by Proposition<ref> is the unique μ-stationary probability measure on ∖ [_μ]. We fix an orthonormal basis (e_1, ⋯, e_d) of V (see Section <ref> for the non-Archimedean case).The dual vector space V^* of V will be equipped with the dual norm and with the dualbasis (e_1^*. ⋯, e_d^*). We keep the same notation as Lemma <ref> concerning other duality notation. Recall that if K denotes the isometry group of (V,||·||) and A the subgroup of(V) consisting of diagonal matrices in the chosen basis, then the following decomposition holds G=KAK. For g∈_d(k), we write g=k(g) a(g) u(g) a KAK decomposition ofg. We note a(g):=(a_1(g), ⋯, a_d(g)).Note that g^t=u(g)^t a(g)^t k(g)^t is a KAK decomposition of g^t in (V^*).When = or =,one can impose that a_1(g)≥⋯≥a_d(g)>0. When is non-Archimedean, one can choosea_1(g), ⋯, a_d(g)∈ϖ ^ (with ϖ a fixed uniformizer of ) and sort them in ascending order of their valuation.With this choice,a(g) is unique and we can definethe mapN: (V) ⟶∖{0}, g ↦ a_1(g). Note that a similar map N: V∖{0}⟶∖{0} was definedin Section<ref>.It will be clear from the context whether N is applied to a non zero element of V orto an automorphism of V. Recall that in the Archimedean case, one has simply N(x)=||x||, N(g)=||g|| for x∈ V∖{0} and g∈(V).Let μ be a probability measure on such that λ_1>λ_2 and ν a μ-stationary probability onsuch that ν([ℒ_μ])=0. Then there exists a random variable ω↦ [Z(ω)]∈ of law ν such that:* almost surely, every limit point ofR_n/N(R_n)in End(V) is a matrix of rank one 1 whose image inis equal to [Z]. * k(R_n)[e_1] converges almost surely to [Z].In particular, ν is the unique such probability measure. In item i. below we provethe proposition in the particular case 𝒰_μ=V. In item ii. we check that this is enough to deduce theuniqueness of the stationary measure on ∖ [ℒ_μ]. Finally, in item iii. we prove the limit theorems claimed in the proposition in the general case.i.Assume first that𝒰_μ=V. By Proposition <ref>, ν is non degenerate on . Let ω∈Ω and A(ω) a limit point of R_n(ω)/N(R_n(ω)). We writeR_n_k(ω)/ N (R_n_k(ω))k→ +∞⟶ A(ω). Since ν isnon degenerate, the pushforward measure A(ω)ν onis well defined and we have the following vague convergence:R_n_k(ω) νk →∞vague⟶ A(ω)ν.Since λ_1>λ_2, the KAK decomposition of R_n(ω) shows that, taking if necessary ω in ameasurable subset of Ω of -probability 1, the matrix A(ω) has rank 1. Hence, if we denote by kZ(ω) its image, then A(ω)ν=δ_[Z(ω)], so that R_n_k(ω) νn →∞vague⟶δ_[Z(ω)]. But using Doob's theorem on convergence of bounded martingales, Furstenbergshowed in <cit.>that there exists for -almost every ω, a probability measure ν(ω) on such that R_n(ω) νn →∞⟶ν_ω and∀ f∈𝒞(),(∫f dν_ω) = ∫f dν.By(<ref>) and (<ref>), we obtain the following relation:ν_ω=δ_[Z(ω)]. In particular [Z(ω)]does not depend on the subsequence (n_k)_k∈^*. By (<ref>), ν is the law of the random variable ω⟶ [Z(ω)] on . This proves the uniqueness of ν, together withitem 1 in the case 𝒰_μ=V.Item2 is an immediate consequence of the KAK decomposition.ii.Now if 𝒰_μ≠ V, we apply the previous part for the restriction ρ: T_μ⟶(𝒰_μ) on 𝒰_μ. Since 𝒰_ρ(μ)=𝒰_μ, ℒ_ρ(μ)=𝒰_μ∩ℒ_μ (see Remark <ref>) and since the top Lyapunov exponent of ρ(μ) is simple, we obtain using item i. a unique μ-stationary probability measure on [𝒰_μ] ∖ [𝒰_μ∩ℒ_μ]. But by Proposition <ref>, any μ-stationary probability measure on ∖ [ℒ_μ] gives total mass to [𝒰_μ], then such a probability measure is unique.iii. It is left to prove the limit theoremsin the first and second claims of Proposition <ref> even if 𝒰_μ≠ V.For every n∈, let k_n (resp. u_n) be the left (resp. right) K part of R_n in the KAK decomposition. The following holds almost surely: ∀ x∈ V, R_n x/N(ρ(R_n)) =e_1^* (u_n x)N(R_n)/N (ρ(R_n)) k_n e_1 + O(a_2(n)/||ρ(R_n)||). Butthe Lyapunov exponent of ρ is λ_1 and loga_2(n)/n converges almost surely to λ_2<λ_1. Hence a_2(n)/||ρ(R_n)|| converges (exponentially fast)to zero,so that almost surely,∀ x∈ V, R_n x/N(ρ(R_n)) =e_1^* (u_n x)N(R_n)/N (ρ(R_n)) k_n e_1 + o(1).Let now ω∈Ω andk_∞ be a limit point of (k_n)_n∈^*. We write k_∞ = l→ +∞limk_n_l. Passing to a subsequence if necessary, we may assume that ρ(R_n_l)/N (ρ(R_n_l)) converges to the non zero endomorphismA(ω) of 𝒰_μ.Choose any x∈𝒰_μ∖Ker(A(ω)).In particular,R_n_l x/N (ρ(R_n_l))=ρ(R_n_l x)/N (ρ(R_n_l))l → +∞⟶ A(ω) x ∈ V∖{ 0}. Since K acts by isometry on V, (<ref>) givesthen that|e_1^*(u_n_l x)N(R_n_l)/N (ρ(R_n_l))|l→ +∞⟶ ||A(ω) x||>0.In particular, passing if necessary to a subsequence, we may assume that the sequence( e_1^* (u_n_l x)N(R_n_l)/N (ρ(R_n_l)))_l∈^* converges into some α(ω)∈∖{0}. By (<ref>) again, we deduce thatR_n_l x/N(ρ(R_n_l))l→ +∞⟶α(ω) k_∞ e_1 ∈ V∖{0} .In particular, R_n_l[x] ⟶ k_∞ [e_1] in . But since x∈𝒰_μ∖Ker(A(ω)), item i. shows that R_n_l[x]⟶ [Z(ω)]. Hence k_∞ [e_1]=[Z(ω)].This being true for all limit points of (k_n[e_1])_n∈^*, we deduce thatk_n [e_1] converges almost surely to [Z]. This proves part 2 of the proposition in the general case. Since λ_1>λ_2, part 1 is an easy consequence of the KAK decomposition. Let μ be a probability measure on (V) such that λ_1>λ_2. Then,* For every sequence ([x_n])_n inthat converges to some [x]∈∖[ℒ_μ], we have almost surely,inf_n∈^*||L_n x_n||/||L_n||||x_n|| >0. * 1/n(log||L_n x||/||x||) converges to λ_1 uniformly on compact subsets of ∖[ℒ_μ].* There exists a random variable [Z]∈ of law ν such thatfor every [x]∈∖ [ℒ_μ], the sequence of random variables (R_n [x])_n∈^* converges in probability to [Z].* Let([x_n])_n∈^* bea sequence inthat converges to [x]∈∖ [ℒ_μ]. Write L_n=K_nA_nU_n the KAK decomposition of L_n. Since λ_1>λ_2,||L_n x_n||/||L_n||||x_n||=||A_n U_n x_n||/|a_1(n)|||x_n||=|U_n^t e_1^* ( x_n/N(x_n))| +o(1).Since U_n^t=k(L_n^t)=k(X_1^t ⋯ X_n^t), and sincethe Lyapunov exponents of μ̌ coincide with those of μ,item 2 of Proposition <ref> applied to μ̌ shows then that||L_n x_n||/||L_n||||x_n||n→ +∞⟶| Ž (x)|, with ||Ž||=1and [Ž] being a random variable onwith law the unique μ̌-stationary probability measure ν̌onP( V^*)∖ [ℒ_μ̌].LetH= ( x)^0⊂ be the hyperplane orthogonal to x. Since x∉ℒ_μ, ℒ_μ^0⊄H, i.e. by Lemma <ref> 𝒰_μ̌⊄H.Since, by proposition <ref> ν̌([𝒰_μ̌])=1 andν̌is non degenerate on [𝒰_μ̌],we deduce thatν̌([H])=ν̌([H ∩𝒰_μ̌])=0.Hence, almost surely, | Ž (x) | ≠ 0. Item 1.is then proved. * To prove item 2, take a compact subset K of ∖ [ℒ_μ]. By compactness of K, it is enough to show that for any sequence ([x_n])_n in K that converges to some [x]∈ K, one has that1/n(log||L_n x_n||/||x_n||) ⟶λ_1. By the previous item 1.,we deduce that 1/nlog||L_n x_n||/||x_n|| converges to λ_1. But by the law of large numbers, it is easy to see that the sequence {1/nlog||L_n x_n||/||x_n||, n≥ 1} is uniformly integrable. This is enough to conclude.* Now we prove item 3. We claim that for every compact subset K of ∖ [ℒ_μ], sup_[x],[y]∈ K (δ(L_n[x],L_n[y])) n→ +∞⟶ 0.Admit for a while(<ref>) and let us indicate how to conclude. By the proof of item 1 of Proposition <ref>, there exists a random variable [Z]∈ of law ν such that, almost surely, R_n νvaguen→ +∞⟶δ_[Z]. Let ϵ>0. Since ν is a probabilitymeasure on the Polish space ∖ [_μ], one can find a compact subset K_0=K_0(ϵ) of∖ [ℒ_μ] such that ν(K_0)>1-ϵ.Now we write for every n∈^*: ( δ(R_n[x], [Z])) =∫_( δ(R_n[x], [Z])) dν([y])≤ ∫_( δ(R_n[x], R_n[y])) dν([y]) +∫_( δ(R_n[y],[Z])) dν([y])≤ ϵ+ ∫_K_0( δ(R_n[x], R_n[y])) dν([y]) +∫_( δ(R_n[y],[Z])) dν([y])≤ ϵ + sup_[x],[y]∈ K_0∪{x}(δ(R_n[x], R_n[y]))+( ∫_δ(R_n[y], [Z]) dν([y])) In the third line we used δ≤ 1 and,in the last line, we used Fubini's theorem.The second term of the right hand side converges to zero as n tends to infinity by(<ref>)and the fact that R_n and L_n have the same law for every n. The last term converges to zeroby the dominated convergence theorem and the fact that, almost surely, R_n νvague⟶δ_[Z]. Since ϵ>0 was arbitrary, we deduce that( δ(R_n[x], [Z]))n→ +∞⟶ 0 and a fortiori thatR_n[x] converges in probability to [Z] as desired. Finally, we prove(<ref>).Fix a compact subset K of ∖ [ℒ_μ]. For every sequences([x_n])_n∈^*,([y_n])_n∈^* of elements in K that converge in K, item 1. shows thatalmost surely: δ(L_n[x_n], L_n[y_n])≤||⋀^2 L_n||/||L_n^2|| ||L_n||||x_n||/||L_n x_n||||L_n||||y_n||/||L_n y_n||n→ +∞⟶ 0.By the dominated convergence theorem and the compactness of K, we deduce that for every sequence of elements [x_n],[y_n] in K, (δ(L_n[x_n], L_n[y_n])) n→ +∞⟶0. This implies(<ref>). We deduce from the previous corollary that: if λ_1>λ_2, then* [x]∈sup1/n(log||L_n x||/||x||) n→ +∞⟶λ_1.* if ℒ_μ={0}, then [x]∈inf1/n(log||L_n x||/||x||)n→ +∞⟶λ_1. This is coherent with the result of Furstenberg-Kifer saying that ℒ_μ={0} if and only if there exists a unique cocycle average (see Remark <ref>).We end this section by noting that Corollary <ref> is obtainedby applyingTheorem <ref> on each subspace ℒ_i given by Theorem <ref>. Indeed,ℒ_i = ℒ_ρ_i(μ) where ρ_i is the restriction to ℒ_i. § THE LIMIT SET AND THE SUPPORT OF THE STATIONARY MEASURE In this section, we understand further the support of the unique μ-stationary measure given by Theorem <ref>, by relating it to the limit set of T=T_μ. We will adapt the proof of <cit.> to our setting. Finally we give two concrete examples by simulating the limit set of two non irreducible subgroups of _3().§.§ Proof of Theorem <ref> We keep the same notation as in Section <ref> concerning the set 𝒬 of quasi-projective maps of , the limit set Λ(T) ⊂[U] of T relative to the subspaces L and U, and the subsets p^+(T_0) (resp. p^+(T^a_0) ) ofof attractive points of proximal elements of T in [U] (resp. in [U∖ L]). First we check that following property that we claimed to hold: Λ(T) is a closed T-invariant subset of [U].Only the closed part needs a proof. Let y_i∈Λ(T) be a sequence in Λ(T) that converges into some y. Clearly y∈ [U]. For each y_i=p(𝔮_i), find a projective subspace [W_i] of , a sequence of projective maps {[g_i,n]}_n∈ such that[g_i,n] converges pointwise, when n tends to infinity, to 𝔮_i with 𝔮_i that maps ∖ [W_i] to y_i. Since by <cit.>,𝒬 is sequentially compact for the topology of pointwise convergence, there exists asubsequence of the 𝔮_i's that converges to some quasi-projective map 𝔮. To simplify notations, we willwrite 𝔮=i → +∞lim𝔮_i.LetW:=lim inf_i→ +∞W_i={x∈ V; ∃ i(x); ∀ i≥ i(x), x∈ W_i}=⋃_n⋂_k≥ nW_k.It is clear that W is asubspace of V and hencethat the union above is a finite one.Taking the latter fact into account and the fact that U ⊄W_i for every i, we deduce thatU⊄W. Let now [x]∉[W]. By definition of W, one can finda subsequence (W_i_j)_j∈^* such that x∉W_i_j for every j. Hence 𝔮 [x]=i → +∞lim𝔮_i [x]= j → +∞lim𝔮_i_j[x]=j → +∞limy_j=y.It is left to show that 𝔮 is a pointwise limit of projective transformations that belong to PT⊂PGL(V). Since each 𝔮_i is such a map and since 𝔮 is the limit of the 𝔮_i's, this follows from <cit.>.We are nowable to prove Theorem <ref>. In item 1. of the following proof,we use the same notation as the invertible version of Oseledets theorem(Theorem <ref>).Also, any linear transformation of V=^d will be identified with its matrix in the canonical basis B_0=(e_1, ⋯, e_d). The set of linear maps between two vector spaces V_1 and V_2will be denoted by ℒ(V_1, V_2). * Consider thedynamical system ( Ω =GL(V)^, =μ^⊗, θ) with θ the shift θ((g_i)_i∈):=(g_i+1)_i∈. ApplyingOseledets theorem in the invertible case, and using equivariance property (<ref>), we get thatthe cocycle ×Ω⟶GL(V), L_n(ω)=A(θ^n-1(ω)) ⋯ A(ω)=g_n-1⋯ g_0 iscohomologous to a block diagonal one. More precisely, denote byω↦ϕ_ω∈End(V)the random transition matrix from B_0 to a measurable adapted basis of the splitting V=⊕_i=1^lF_ω^i. Then there exists a random block diagonal matrix Δ_n such that the following identity holds for -almost every ω∈Ω: L_n =(ϕ∘θ^n)Δ_n ϕ^-1 For every ω∈Ω, let v_n(ω):=ϕ(θ^n ω) ϕ^-1(ω). As in<cit.>, using Poincaré recurrence theorem,we can findalmost surelyarandom subsequencen_k(ω)=n_k such that k→ +∞limv_n_k(ω)=I. Butsince λ_2<λ_1, (<ref>) gives thatΔ_n e_1= λ_n^1 e_1 with λ_n^1 being a random non zero scalar such thatlim1/nlog|λ_n^1|=λ_1 almost surely.Also, since 1/nlog ||Δ_n x|| ≤λ_2<λ_1 for every x∈ V∖ e_1, we deduce that (ϕΔ_n ϕ^-1)/λ_n^1converges almost surely tothe random projection endomorphism Π on the line [Z]:=[ϕ (e_1)] parallel to Ker(Π)=⊕_i=2^lF_ω^i. Combining the previous fact with(<ref>), we get that almost surelyλ_n_k^-1L_n_kk→ +∞⟶Π.Let ω∈Ω where the previous convergence holds. Since Π(ω) is arank one projection, it is proximal. By a perturbation argument, L_n_k(ω) is also proximal for all large k with a dominant eigenvector p^+(L_n_k) close to [Z(ω)].By (<ref>), [Z(ω)] ∉[L]. Hence for all large k, p^+(L_n_k)∉[L]. Let us checkthat p^+(L_n_k)∈ [U]. Indeed, the largest eigenvalue of L_n_k is either an eigenvalue of its restriction to U withits corresponding eigenvector being that of the restriction operator,or is an eigenvalue of its projectionon V/U. But the latter eigenvalue grows at most asexp(n_k λ_2), while it follows from(<ref>) that the spectral radius of L_n_k growth as the norm of ||L_n_k||, i.e. as exp(n_k λ_1). Since λ_2<λ_1, we deduce that p^+(L_n_k)∈ [U], for all large k. Hence L_n_k∈ T_0^a so that[Z (ω)]∈p^+(T^a_0). In particular, p^+(T^a_0)≠∅.Now the law of the random variable [Z] on P(V) is the stationary measure ν.Indeed, by (<ref>) and the uniqueness part of Oseledets theorem, we deduce thatthe filtration {0}⊂F_ω^1⊂F_ω^1⊕F_ω^2⊂⋯⊕_i=1^l-1F_ω^i⊂ V depends only on the past of ω=(g_i)_i∈ i.e. on (⋯, g_-2, g_-1).Hence [Z] is an independent copy ofthe least expanding vector of R_n^-1(ω)=g_n^-1⋯ g_1^-1 given by Oseledets theorem. The proof of Proposition <ref> shows then that [Z] has law ν.Hence ν(p^+(T^a_0))=( [Z] ∈p^+(T^a_0))=1,so thatSupp(ν) ⊂p^+(T^a_0)⊂p^+(T_0). Conversely, let h∈ T_0. Then h^n/||h^n|| converges to the projection η on the line generated by p^+(h)∈ [U] and parallel to some h-invariant subspace of V. In particular, U⊄Ker(η). Since by Theorem <ref> ν is nondegenerate in [U], we havethat ν([U] ∩ Ker(η))=0 so thath^nνn→ +∞weakly⟶δ_p^+(h). SinceSupp(ν) is T-invariant,we get p^+(h)∈Supp(ν).Hencep^+(T_0)⊂Supp(ν) andSupp(ν)⊃p^+(T_0)⊃p^+(T^a_0).Inclusions(<ref>)and(<ref>) show item 1.* Let ω∈Ω and W:= Ker(Π(ω)). By the previous item,we deduce that the sequenceof projective maps ([L_n_k])_k∈ converges pointwise on ∖ [W] to the constant map [x]↦ [Z(ω)].Since Π(ω) isa rankoneproximal endomorphism of V, Im(Π(ω))⊄Ker(Π(ω)). But Im(Π(ω))=[Z(ω)]∈ [U], sothat [U]⊄[W].Considering the sequence r_n_k:=sup{N(L_n_k x);x∈ W, ||x||=1} in , we see thatthe sequence(r_n_kL_n_k_|_W) of linear mapsfrom W to Vadmits a subsequence that converges in ℒ(W,V) to a linear map Π'(ω) such that W':=Ker(Π'(ω))⫋ W. In particular, [L_n_k]admits a subsequencethat converges pointwise on[W]∖ [W'] and hence on ∖ [W']⫌∖ [W].Repeating this procedure at most dim(W) times,we obtaina subsequenceof[L_n_k] thatconverges pointwise ontoa quasi projective map 𝔮such that 𝔮 maps ∖ [W] to p(𝔮):=[Z(ω)]∈ [U].Hence 𝔮∈T and [Z(ω)]∈Λ(T). Since by Lemma <ref> Λ(T) is closed inand since [Z] has law ν, we deduce thatSupp(ν)⊂Λ(T). Conversely,let y=p(𝔮)∈Λ(T) and ([g_n])_n∈ a sequence of projective maps convergingpointwise to 𝔮, together with a projective subspace [W] ofthat does not contain [U] and such that with𝔮 maps ∖ [W] to the point y of . Since ν is non degenerate on [U] and since [W] does not contain [U],we deduce that ν([U] ∩ [W])=0so that 𝔮ν is the Dirac measure on y. We concludethatg_n νn→ +∞weakly⟶δ_y. Since Supp(ν) is T-invariant, we deduce that y∈Supp(ν). Consequently,Supp(ν)⊃Λ(T).Item 2 is then proved.* Let [x]∈∖ [L]. By Theorem <ref>,there exists Ω_x⊂Ω such that for every ω∈Ω_x, lim_n→ +∞1/nlog||L_n(ω) x ||=λ_1. Reducing if necessary Ω_x to a subset of -probability one, we deduce from (<ref>) that x∉⊕_i=2^lF_ω^i=Ker(Π(ω)) for every ω∈Ω_x. By (<ref>), we deduce that for every ω∈Ω_x there exists a random subsequence (n_k)_k such that L_n_k[x] k→ +∞⟶ [Z(ω)]; so that [Z(ω)]∈T ·[x]. Since (Ω_x)=1 and [Z] has law ν, we deduce that Supp(ν)⊂T·[x].One canalso useCorollary <ref> to prove item 3. above with the right random walk. Indeed it follows from item 3. of Corollary <ref> that there existsa non random subsequence (n_k) such thatR_n_k[x] converges almost surely to a random variable of law ν.We deduce easily the proof of Corollary <ref> stated in Section <ref>. The implication 1⟹2 follows immediately from the last part of item 3. of Theorem <ref>. The implication 2 ⟹1is an easy consequence of the fact that [L] is T-invariant.The equivalence between 1 and 3 follows directly from the first part ofitem 3. of Theorem <ref>. Finally, we check the equivalence 1 ⟺ 4. By item 1. of Theorem <ref>, Supp(ν) is compact if and only p^+(T_0^a) is precompact in O. As in Section <ref>, identify O with the quotient ofX=L× S(V/L) by the action of the unit sphere 𝒮_ of . A straightforward computation for any eigenvector of an upper triangular bloc matrix shows that an element ofp^+(T_0^a) is identified with 𝒮_(t_0, ξ_0), where[ A B; 0 C ]∈ T, C is proximal, ρ_spec(C)>ρ_spec(A),ξ_0a chosen normalized eigenvector of C, andt_0 a fixed point of the affine map t↦At +Bξ_0/λ_top(C) of L.Since ρ_spec(A)<ρ_spec(C), this affine map has a unique fixed point in L, namely -(A-λ_top(C) I)^-1 (B ξ_0). The desired result follows then from the compactnessof the unit sphere of V/L. The last part in the proof of item 1. of Theorem <ref>gives actually a stronger result and a sufficient non-compactness criterion for the support of ν, seen in O=∖ [L].Elements of T are represented by matrices in a suitable basis of V where the first diagonal bloc is the restriction to L. Assume U=V. If there exists g=(A B0 C )∈ T such that ρ_spec(A)>ρ_spec(C),then Supp(ν)⊆ O is not compact.Here ρ_spec(·) denotes the spectral radius evaluated in some finite extension of the local field k.Indeed, let g∈ T be such an automorphism.It is enough to checkthat there exists a projective subspace [E] ofsuch that for every [x]∈∖ [E], every limit point of g^n[x] belongs to [L]. Indeed,since U=V, the probability measure ν onis non-degenerate (Theorem <ref>) so that ν([E])=0. In particular, every limit point of the sequence of probability measures (g^n ν)_n∈^* ongives total mass to [L]. Since Supp(ν) is T-invariant, we deduce that Supp(ν)∩ [L]≠∅ and then that Supp(ν)∩ O is not compact. Now we check our claim.Assume first that the characteristic polynomial of g splits over .Sinceρ_spec(A)>ρ_spec(C), thegeneralized highest eigenspace W for gcorresponding to the top eigenvaluecoincides with the one forA corresponding to the same eigenvalue.In particular, W⊆ L. Writingnowg in its Jordan canonical form in (V), we deduce from the inequality ρ_spec(A)>ρ_spec(C), the existence ofa g-invariant supplementary E of W in V, such for every [x]∈P(V)∖ [E], every limit point of g^n[x]in P(V) lies in [W]⊆ [L]. This is what we wanted to prove. It is left to check that the same holds whendoes not contain all the eigenvalues of . This can be done by applying the previous reasoning to a finite extension ' of , and then use the natural embedding P(V)P(V⊗_ '), where P(V⊗') is the '-projective space of the '-vector space V⊗_ '.§.§ Compactness criterion, examples andsimulations In this section, we illustrate our results for semigroups of linear transformations of ^3 for which _μ is a line, i.e. thoserelative toExample 3. of Section <ref>. Note that when λ_1>λ_2, this is essentially theonly case to illustratesince if _μ is a plane, we are essentially in the contracting case of the affine situation and, if _μ={0}, we are either in the i-p case (if the action is irreducible) or in the expansive case of the affine case or in the degenerate case explained at the end of Example 3. of Section <ref>.In section <ref> below, we give a sufficient compactness criterion for the support of the stationary measure in the open dense subsetO=P^2()∖ [_μ] of the projective plane using the notion of joint spectral radius and Section <ref>. In Section <ref>, we give three examples, simulate the limit sets of two sub-semigroups of _3() and justify our observations using the techniques we have developed. §.§.§ Compactness criterion We recall the classical notion of joint spectral radius of a bounded set of square matrices introduced by Rota and Strang in<cit.>.Let d∈^*, ℳ_d() the set d× d matricesand denote by ρ_spec(A) the spectral radius of A∈ℳ_d(). A subset of ℳ_d()is said to be bounded if it is bounded when ℳ_d()is endowed with one, or equivalently any, norm on ℳ_d().Let d≥ 2 and Σ⊂ℳ_d() be a bounded set.Let ||·|| be any norm on ^dand denote by the same symbol the operator norm it induces on ℳ_d(). The joint spectral radius ℜ(Σ) of Σ is the following non negative real number, independent of the chosen norm:ℜ(Σ):=inf_n∈^*max{ ||A_1⋯ A_n ||^1/n; A_i∈Σ}= sup_n∈^*max{ρ_spec(A_1⋯ A_n)^1/n; A_i∈Σ}.The last equality was proved byBerger-Wang <cit.> after a question of Daubechies-Lagarias <cit.>.Now we state our compactness criterion when _μ is a line. Let μ be a probability measure on _3() with compact support. Assume that λ_1>λ_2 and that _μ is a line L. In a suitable basis of ^3=L⊕L̃, μ is seen as a probability measure on thegroupG:={ g=(a_gb_g 0 C _g ); a_g∈^*, b_g∈^2, C_g∈_2() }⊂_3().Letr:=ℜ({|a_g| C_g^-1; g∈Supp(μ)}).If r<1, then the support of the unique μ-stationary probability measure ν on O=P^2()∖ [_μ] is compact. In the following proof, Σ denotes the support of μ,T=T_μ the semigroup generated by the support of μ and, for every n∈^*, Σ^n denotes the subset of T that consists of all theelements that can be written g_1⋯ g_n with g_i∈Σ^n (so Σ^n=Supp(μ^n)). The set Aff() of all affine maps of the real line is identified topologically with the product space ^*×. We adopt also the notation and setting of Section <ref> namely: X:=L× S(^3/L)≃× S^1 (with ^3/L ≃L̃ and ^3/L endowed with its Euclidean structure), T is considered acting on X by the formula (<ref>)and the open subset O of P^2() is identified with X/{± 1}. Suppose r<1. Let ||·|| be the canonical norm on ^2. By the definition of the joint spectral radius, there exists n_0∈ such that for everyg∈Σ^n_0, |a_g| ||C_g^-1|| < 1.This implies that for every ξ∈ S^1 and every g∈Σ^n_0,the affine mapσ(g,ξ): t ↦a_g t+b_g ξ/||C_g ξ|| of the real line has linear part strictly less than 1 in absolute value.We will check hereafter that this implies that there exists a compact subset K_1⊆ stabilized by the family {σ(g,ξ) ;g∈Σ^n_0; ξ∈ S^1}. Let us indicate first how to conclude.Let K_2:=K_1 × S^1. This is a compact subset of X=× S^1 stabilized by all elements of Σ^n_0. Hence, by settingK_3:=K_1 ∪ ⋃_g∈n≤n_0∪Σ^ng· K_1, we obtain anothercompact subset of X which is stabilized by T.In particular, K:=K_3/{± 1} is a compact subset of X/{± 1}≃ O stabilized by T. This implies that there exists aμ-stationary probability measure on K. By the uniqueness part of Theorem <ref>, the aforementioned probability measure on K coincides with ν. HenceSupp(ν)∩ O is compact. Now we check the missing part of our proof.We arein the following general situation: Y is a compact topological space(here Y= Σ^n_0× S^1), σ is a continuous map σ : y∈ Y ↦σ_y( t)=a_y t+ b_y from Y to Aff() such that |a_y|<1 for every y∈ Y. It is standard that σ(Y) stabilizes a compact subset of .Let us check it for the completeness of the proof.For every point p∈ and every R>0, letB(p,R):=[p-R, p+R] be the closed interval of center p and radius R. Fix y_0∈ Y. For every y∈ Y,let f_y∈ be the unique fixed point of the affine map σ_y and R_y:=1+|a_y|/1-|a_y| |f_y-f_y_0|. Forevery y∈ Y and every R>R_y,σ_y (B(f_y_0,R)) ⊆ σ_y ( B(f_y, R+|f_y-f_y_0|) )⊆B(f_y, |a_y|(R+|f_y-f_y_0)) ⊆B(f_y, R - |f_y-f_y_0|) ⊆B(f_y_0,R). Estimate(<ref>)follows from our choice of R_y. Hencefor every R>R_y, B(f_y_0,R) is stable under f_y. Since y↦ R_y is a continuous map on the compact space Y, then R:=sup{R_y; y∈ Y} is finite so that σ(Y) stabilizes the compact subset B(f_y_0,R) of .This is what we wanted to prove.The condition r<1 is equivalent tothe existence of some norm ||·||_1 on ^2 such that |a_g| ||C_g^-1||_1<1 for all g in the support of μ. This is a consequence of a theorem of Rota-Strang <cit.> which asserts that, for a bounded subset Σ⊂ℳ_d(),ℛ(Σ)=inf_||·||∈𝒩max{||A||; A∈Σ},where 𝒩 is the set of norms on ^d.Hence, although the condition r<1 involves the eigenvalues/norms of the elements in the semigroup generated by the support of μ, it can be read also on the support of μ solely.(Generalizations) * Let K⊆ S^1 be any lift of the limit set in P^1()of the strongly irreducible and proximal semigroup T_π(μ), whereπ: G⟶(^3/L)≃_2(). The proof of Proposition <ref> yields the following stronger statement.Let ||·|| be the canonical norm on ^2. If|a_g| < ||C_g t|| for every every t∈ K and every g∈Σ^n_0 (for some n_0 large enough) then the support of ν is compact. In view of Rota-Strang's theorem, we can formulate the criterion above in the following way:if one can find a norm ||·||_1 on ^2 such that |a_g| <||C_g t||_1 for every g∈Σand every t∈ K, then the support of ν is compact in O. * When dim(_μ)>1, theproof of Proposition <ref> generalizes easilyin view of Rota-Strang's theorem and reads as follows:if one can find norms ||·||_1 and ||·||_2 such thatfor every g∈Σ,||A_g||_1||C_g^-1||_2< 1, then Supp(ν)∩ O is compact. However, formulating such a criterion in a more intrinsic way (i.e. using eigenvalues of elements of T), requires taking into accountthe data given by all the eigenvalues of elements in the semigroup living in the product group (_μ)×(^d/_μ), and not consideringthe joint spectral radii of the diagonal parts A and C separately(otherwise a criterion is valid but istoo restrictive).This can be done using the very recent notion of joint spectrum of a bounded set of matrices introduced recently by Breuillard and Sert <cit.>,<cit.>. We refrain from doing it here as it goes beyond the scope of the paper. §.§.§ Examples and Simulations All our vectors will be written using thecartesian coordinates in ^3. Also, automorphisms of ^3 will be identified with their matrices in the canonical basis of ^3. We endow ^3 with the canonical norm.Example 1: non compact support in OConsider the following matrices of _3(): g_1=( 1 2 30 11 0 0 1), g_2=( -1 1 20 0-1 0 1 0). Let L=(1,0,0) and μ be the uniform probability measure on the set {g_1, g_2}. The projection of T_μ on the quotient ^3/L is a non-compactstrongly irreducible sub-semigroup of _2(). Hence by Furstenberg theorem <cit.>,λ_1(^3/L)>0 (this can be also deduced from Guivarc'h-Raugi theorem <cit.>as the action on the quotient is i-p). In particular, ∫_Glog |a_g| dμ(g)<λ_1(^3/L)and the top Lyapunov exponent on the quotient is simple. By Lemma <ref>, λ_1(μ)>λ_2(μ).By the irreducibility of the action on the quotient, _μ=L. Let us check that𝒰_μ=^3. All we need to prove is that there does not exist a G_μ-invariant plane W in ^3 such that ^3=L⊕ W.Arguing by contradiction, suppose that such a plane W exists. Writing g_2 in its (real) Jordan canonical form, we deduce the existence of a g_2-invariant plane W' such that ^3=L⊕ W'. The non zero g_2-subspace W∩ W' cannot be a line because [L] is the only real eigenspace of g_2.Hence W=W' and in particular g_1 stabilizes W'. However, adirect computation shows that W'=Span( (3,0,2), (-1,2,0)) and that g_1(3,0,2)^t = (9,2,2)^t ∉W'. Contradiction.Let X:=× S^1 embedded in ^3 using the cartesian coordinates (t,ξ) with t∈ and ξ∈ S^1. Abounded portion of X is represented in the graycylinder ofFigure 1. Its axis is L=_μ=(1,0,0) and the symmetry with respect to the origin of ^3 (the centroid of the cylinder in Figure 1) acts naturally on X. The space X is a two-fold cover of the open dense subset O=P^2()∖ [L] of the projective plane. The latter is then thought as a one-point compactification of X/{± 1}, the direction of the line L beingthe point at infinity. Identifying /L with the yz-plane plane L̃, we let G act on Xby the formula (<ref>) of Section <ref>. By theorem <ref> and Section <ref>, there exists aunique μ-stationary stationary measure ν̃ onX which is {± 1} invariant.By Theorem <ref>, Supp(ν̃) is a two-fold cover of the limit set Λ(T) of T_μ (Theorem <ref>), and the unique such one which is ± 1 invariant. The blue points in Figure 1 below live on the cylinder X and represent the points ± Z(ω) where Z(ω)∈ X and[Z(ω)] is given by Theorem <ref>.Hence, the picture is a simulation of ν̃. The points that are in the transparent face of the cylinder are exactly identical to the visible ones by symmetry. In Figure 2, we plot the projection of the points of Figure 1 on S^1.By the discussion of Section <ref>, Picture 2 is then a simulation of a two-fold cover of the limit set of the i-p semigroup of SL_2() generated by π(g_1) and π(g_2), projection of g_1 and g_2 on (^3/L).Weobserve in Figure 1the fibered structured described in Section <ref>:each fiberis contained in an affine line with (horizontal) direction L and projects to a point of Figure 2. < g r a p h i c s >figureSimulation of ν̃< g r a p h i c s >figureProjection on S^1The support of ν is not compact in O as suggested by Figure 1.Indeed, the unipotent structure of g_1 gives that g_1^n =( 1ψ_1(n)ψ_2(n)0 1ψ_3(n) 0 0 1), with ψ_1(n) = O(n), ψ_2(n)=O(n^2) and ψ_3(n)=O(n). In particular, for every [x]∈ O=P^2()∖ [L], every limit point of (g_1^n[x])_n∈ belongs to the pointat infinity [L]. Hence, the closure of the orbit of any point of O under T_μ meets [L].We conclude by item 3. of Corollary <ref>.Example 2: compact support in OWe replace g_1and g_2 of Example 1 withg_1=( 0.5 2 30 11 0 0 1), g_2=(0.5 1 20 0-1 0 1 0) instead of g_1 and g_2.Again λ_1>λ_2, ℒ_μ=L and 𝒰_μ=^3 (same argument as above withW'=Span((0,0,1), (2,-1,0)) instead and g_1(0,0,1)^t = (3,1,1)∉W'). We obtain the following simulation of the unique ± 1-stationary measure ν̃ on the cylinder X.< g r a p h i c s > figureSimulation of ν̃ Note that the projection of ν̃ on S^1 is identical to the one given by Figure 2 above as the two semigroups of Example 1 and 2 have the same projection on (^3/L). The support of ν is compact in O as suggested by the picture.Indeed, by Proposition <ref>, it is enough to check thatfor C_1:= ( 11 0 1) and C_2:= (0-11 0), one has that ℜ({C_1^-1, C_2^-1})<2. Consider now the the operator norm on ℳ_2() induced by the L^2 norm on ^2. We have:||C_1^-1||=||C_1||=√(ρ_spec(C_1 C_1^t))= √(3+√(5)/2)<2. Clearly, ||C_2||=1. Hence max{||C_1^-1||, ||C_2^-1||}<2.But by definition of the joint spectral radius, we have ℜ(Σ)≤max{||g||; g∈Σ} for every bounded subset Σ of ℳ_d() and for every matrix norm. Hence (<ref>) is fulfilled and the support of ν is compact in O.Observe that the freedom in the choice of the norm was crucial in the previous proof (as for instance neither the L^1 nor the L^∞ norm would help fulfilling the criterion).Example 3Here we present another example for a behavior similar to the one of Example 1 in order to justifyitem 4. of Remark <ref>.Let μ be the uniform probability measure on the set {g_1, g_2, g_3} with g_1= [0.523;041;00 0.25 ], g_2= [ 0.5 1 2; 0 0-1; 0 1 0 ] andg_3= [0.511;0 0.25 -1;004 ]. We have also λ_1>λ_2, _μ=L and 𝒰_μ=^3.We will show that the support of the unique stationary measure on Ois not compact, although ρ_spec(A)≤1/2 < 1 ≤ρ_spec(C) for every[ A B; 0 C ]∈ T.Denote by C_i∈ GL_2() the projection of g_i on ^3/L and observe that C_3=C_1^-1.Since g_3 isa proximal element of _3() that belongs to T, we haveby Theorem <ref> that p^+(g_3)∈Supp(ν). We will check thatthe orbit of p^+(g_3) under the cyclic group generated by g_1 is not compact in O. This is enough to conclude as Supp(ν) is T-invariant.To do so, we use the identificationO ≃ X/{± 1} of Section <ref> and write p^+(g_3)=± (t_0, ξ_0), with t_0 = 22/ 7√(241)∈ L≃ and ξ_0=1/√(241) (-4,15)∈ S^1 a normalized eigenvector for the top eigenvalue 4 ofC_3=C_1^-1.But ξ_0 is also a fixed point of C_1 for the natural action on S^1 as ξ_0 is an eigenvector of C_1for its least eigenvalue 0.25>0.Hence, by the cocycle property shared by σ (notation of Section <ref>): ∀ n∈, g_1^n(t_0, ξ_0) = (σ(g_1, ξ_0)^n(t_0) , ξ_0).Now σ(g_1, ξ_0) is the affine map of the real line t ↦ 8t+4 ⟨ (2,3), ξ_0⟩,with ⟨·, ·⟩ the Euclidean inner product. Since its linear parthas absolute value >1 and since t_0 is not equal to its unique fixed point (by a direct computation), then σ(g_1, ξ_0)^n(t_0) n→ +∞⟶ +∞ in . In particular, g_1^n p^+(g_3) n→ +∞⟶ [L] in P^2(). This ends the proof.§ REGULARITY OF THE STATIONARY MEASURE§.§ IntroductionLet V be a vector space over the local fieldof dimension d≥ 2. We use the same notation as Section <ref> concerning the choice of the norm on Vand the Fubini-Study metric on .In this section, we prove that under an exponential moment of μ, the stationary measure given by Theorem <ref>has Hölder regularity, and more precisely the followingLet μ be a probability measure on (V) with an exponential moment such that λ_1>λ_2. Let ν be the unique μ-stationary probability measure onsuch thatν([ℒ_μ])=0. Then there exists α>0 such that, sup_[f] ∈P(V^*)∖ [_μ̌]δ([f], ∖ [ℒ_μ̌])∫_δ^-α([x],[Ker(f)]) dν([x])<+∞ A crucial step is to show thatthe random walk converges exponentially fast towards its stationary measure uniformly on compact subsets of ∖ [_μ], namely: Let μ be a probability measure on (V) with an exponential moment such that λ_1>λ_2. Let ν be the unique μ-stationary measure onsuch that ν([ℒ_μ])=0. Then, there exists a random variable [Z] on with law ν, there exist β>0 andn_0∈^* such that for everyn≥ n_0 and every [x]∈∖ [_μ], ( δ(R_n[x], [Z] ))≤exp(-n β)/δ([x], [ℒ_μ]).The above statement is stronger than just saying that ν is a μ-boundary. Indeed, it implies that, for any probability measure η onthat gives zero mass to [ℒ_μ], μ^n ⋆ηn→ +∞weakly⟶ν.When T_μ is irreducible (or equivalently i-p), Theorem <ref> was shownin <cit.> using the spectral gap property <cit.>. Other alternative proofs were then proposed <cit.>,<cit.>. When T_μ is a non degenerate sub-semigroup of the affine group of ^d, our result on Hausdorff dimension is new. Here arethe main ingredients of the proof. A first step is Theorem <ref> above. It consists of showing that R_n[x] converges exponentially fast towards the stationary measure, with exponential speed and uniformly on compact subsets of ∖[ℒ_μ]. In the i-p case, this is known (see <cit.> for the convergence and <cit.> for the speed). For affine groups in the contracting setting, this is straightforward by direct computation. When λ_1>λ_2 andG_μ is any group of upper triangular matrix blocs, such as a subgroup of the automorphism group of the Heisenberg group (see Section <ref>), this result is new. The second step is the deterministic Lemma <ref>. This lemma will implythat estimating the distance from R_n[x] to a fixed hyperplane H consists, with probability exponentially close to one, of establishing large deviation estimates of the ratioof norms ||R_n^t f||/||f|| ||R_n|| uniformly on f∈ V^*. In both steps, we need large deviation inequalities for norms ratios. This is done using a classical cocycle lemma (see Lemma <ref> below). Since we do not need the more delicate large deviation estimates for the norms themselves, we do not aim to give the optimalformulation(see Corollary <ref>). We refer to <cit.> for related estimates for cocycles.In terms of techniques,we note that even though our result applies to the interesting case ℒ_μ≠{0} (as the contracting case in the context of affine groups),our proofuses heavily different passages through the easier case ℒ_μ={0}(as the expansive case for affine groups or the irreducible groups)viagroup representations.We refer toRemark <ref> for more on this condition.§.§ CocyclesWe begin by recalling a cocycle lemma:Lemma <ref> below. The case a)allows us to obtain large deviations estimates of cocycles whose average is negative. It is due to Le Page <cit.> and was crucial in order to establish fine limit theorems for the norm of matrices.Case b) treats the case where the average of the cocycle is zero and appears in <cit.>, <cit.>. (Cocycle lemma)<cit.> Let G be a semigroup acting on a space X, s an additive cocycle onG × X, μ a probability measure on G such thatthere existsτ>0 satisfying:(exp(τ sup_x∈ X|s(X_1,x)|) )< ∞.Set l=n→∞lim 1/nsup_x∈ X (s(R_n,x)).a) If l<0, then there existλ>0, r_0>0, n_0∈^* such that for every 0<r<r_0 and n> n_0:sup_x∈ X (exp (rs(R_n,x) ) )≤exp(-n r λ).b)If l=0, then for every γ>0, there exist r(γ)>0, n (γ)∈^* such that for every 0<r<r(γ) andn>n(γ): sup_x∈ X (exp (rs(R_n,x) ) )≤exp(n r γ). (Controlling ratio norms) Let μ be a probability measure on (V) such that μ has an exponential moment andλ_1>λ_2. Then, for every ϵ>0, there exist β=β(ϵ)>0, n_0=n(ϵ)∈^* such that for every n≥ n_0 and every [x]∈∖ [ℒ_μ], ( ||L_n x||/||L_n|| ||x||≤exp(-nϵ) )≤exp(-n β)/δ([x], [ℒ_μ]).Endow V/_μ with the quotient norm. Let (e_1, ⋯, e_d) be an orthonormalbasis of V. For every x∈ V, denote by x its projection on the quotient vector space V/ℒ_μ. Let π be the morphism action of G_μ on V/ℒ_μ. Recall that δ([x],[ℒ_μ])=||x||/||x||(Lemma <ref>) and ||gx||≥ ||π(g)x|| for every g∈(V) and x∈ V∖{0}.Fix now [x]∈∖ [_μ] andϵ>0. Then for every r>0 and every n∈^*, ( ||L_n x||/||L_n|| ||x||≤exp(-nϵ) )=[ (||L_n|| ||x||/||L_nx ||)^r≥exp(nϵ r) ]≤ exp(-nϵ r)[ (||L_n|| ||x||/||L_n x||)^r]≤ exp(-nϵ r)/δ^r([x],[ℒ_μ]) [ (||L_n|| ||x||/||π(L_n) x||)^r]. Since ||g||≤ d max{||g e_i||; i=1, ⋯, d} for every g∈(V) and since the expectation of the maximum of d random real variables is less than d times the maximum of the expectations, we get that: ( ||L_n x||/||L_n|| ||x||≤exp(-nϵ) ) ≤ d^r+1exp(-n ϵ r)/δ^r([x],[ℒ_μ])max_i=1, ⋯ , d[ (||L_n e_i|| ||x||/||π(L_n) x||)^r]≤ d^r+1exp(-n ϵ r)/δ^r([x],[ℒ_μ])sup_z ∈P(V/ℒ_μ) ×[exp(r s (R_n, z))], where s is the function defined on G_μ×(P(V/ℒ_μ)×)bys( g,([x],[y])):=log||gy|| |||x||/||π(g)x|| ||y||. Now let G_μ act naturally on the product space Z:=P(V/ℒ_μ)×. It isimmediate to see thats: G_μ× Z ⟶ is a cocycle. Since μ has an exponential moment, thencondition(<ref>) of Lemma <ref> is satisfied. With the notations of the aforementioned lemma, let us show that l=0.Since ℒ_π(μ)={0} (see Remark <ref>) and λ_1(π(μ))=λ_1(μ)=λ_1, then Corollary<ref> (and Remark <ref> part 2.) shows that:inf_[x]∈P(V/ℒ_μ)1/n(log||π(L_n)x||/||x||) n→ +∞⟶λ_1.Moreover, by Remark <ref> part 1., [y]∈sup1/n(log||L_ny||/||y||) n→ +∞⟶λ_1. Hence,l:=n→∞lim 1/nsup_z∈ Z (s(R_n,z)) = 0.Applying the cocycle lemma for γ:=ϵ/2 gives some r=r(ϵ)>0, n=n(ϵ)∈^* such that for every 0<r<r(ϵ) and everyn>n(ϵ), sup_z ∈P(V/ℒ_μ) ×[exp(r s (R_n, z))]≤exp(n r ϵ/2).Without loss of generality, one can assume 0<r<1. Since the Fubini-Study metric is bounded by one, we obtain the desired estimate by combining (<ref>) and (<ref>).§.§ Exponential convergence in direction In this section, we prove Theorem <ref> stated above. Step 1: First, we checkthat it is enough to show the following statement: there exists β>0, n_0∈^* such that for every [x]∈∖ [ℒ_μ], and every n≥ n_0,( δ(R_n[x], R_n+1[x] ))≤exp(-n β)/δ([x], [ℒ_μ]).Indeed(<ref>) would imply that for every x∉ℒ_μ,(R_n[x])_n∈^* is almost surely a Cauchy sequence in the complete space . Hence, it converges to a random variable [Z_x]∈. By item 3. of Corollary <ref>, [Z_x]=[Z] is almost surely independent of x and has law ν. Now(<ref>) would follow immediately from(<ref>) by applying Fatou's lemma and the triangular inequality.Step 2:Next, we give an upper bound of the leftside of estimate(<ref>). We denote by π: G_μ⟶(V/ℒ_μ) the morphism of the projection of G_μ onto V/ℒ_μ.Let [x]∈∖ [ℒ_μ].For every n∈^*, the following almost sure estimates hold: δ(R_n[x], R_n+1[x] ) = δ (R_n[x], R_n X_n+1[x])=||⋀^2 R_n (x ∧ X_n+1 x)||/||R_n x||||R_n X_n+1 x||≤ ||⋀^2 R_n (x ∧ X_n+1 x)||/||π(R_n) x|| ||π(R_n)π( X_n+1)x||We let G_μ act naturally on Z:=P(⋀^2 V) ×P(V/ℒ_μ)^2 and set, for every z=([a ∧ b], [c], [d])∈ Z and every g∈ G_μ, s(g,z):= log||⋀^2 g (a ∧ b)||||c|| ||d||/||a∧ b||||π(g)c||||π(g)d||. Hence if Y_n denotes the following random variable in Z, Y_n:= ([x ∧ X_n+1 x], [x], [π(X_n+1)x] ), (<ref>) becomes,δ(R_n[x], R_n+1[x] ) ≤exp(s(R_n, Y_n)) × ||x ∧ X_n+1 x||/||x|| ||π(X_n+1) x||.By combining(<ref>), the equality δ([x], ℒ_μ) =||x||/||x|| and the inequalities ||x ∧ y||≤ ||x|| ||y||, ||gx||≥||x||/||g^-1||, ||π(g)||≤ ||g|| true for every x,y∈ V∖{0} and g∈ G_μ, we obtain the following almost sure inequality ([x] is always fixed): δ(R_n[x], R_n+1[x] ) ≤1/δ([x], [ℒ_μ])^2 ||X_n+1|| ||X_n+1^-1|| exp( s (R_n, Y_n)).Using Cauchy-Schwarz inequality and the fact that R_n=X_1 ⋯ X_n and Y_n are independent random variables, we deduce that for every α>0 (to be chosen in Step 3 below),(δ^α/2(R_n[x], R_n+1[x] ) )≤1/δ([x],[ℒ_μ])^α √((||X_n+1||^α ||X_n+1^-1||^α)) √(z∈ Zsup (exp(α s (R_n, z)))).Step 3: Finally, we check that we are in the case b) ofthe cocycle lemma (Lemma <ref>). The map s: G_μ× Z ⟶ is clearly a cocycle on G_μ× Z. Since μ has an exponential moment, condition (<ref>) is fulfilled. Moreover the representation π satisfies ℒ_π(μ)=0. Hence, by Corollary <ref>,inf_[c]∈P(V/ℒ_μ) ||π(R_n) c||/||c||n→ + ∞⟶λ_1. Consequently, l ≤λ_2-λ_1<0. The cocycle lemma gives then α_1>0 and β>0 such that for every α∈ [0,α_1) and every large n,sup_z∈ Z(exp( α s (R_n, z)) )≤exp(-β n).Since μ has an exponential moment, there exists α_2>0 such that for every α∈ [0,α_2), ( ||X_n+1||^α ||X_n+1^-1||^α)<+∞. Apply now(<ref>) forα=min{α_1,α_2,1}. SincetheFubini-Study metric δ is boundedby one,we obtain the desired estimate(<ref>). Theorem <ref> is then proved. §.§ Proof of the regularity of the stationary measureWe begin with the following deterministic lemma. Letbe a local field, V a vector space overof dimension d≥ 2 endowed with the norm described inSection <ref>, L a subspace of V and F an orthonormal basis of an orthogonal supplement to L in V(see Section <ref> whenis non-Archimedean). Let C(,d)=1/√(d+1) ifis Archimedean and C(,d)=1 otherwise. Then for any g∈(V) such that g(L)=L and for any f∈ V^*∖{0}, there exists x∈ F such that: δ(g[x],[Ker(f)]) ≥ C||g^t f||/||g^t|| ||f|| 1_C||g^t f||/||g^t|| ||f|| > ||g_| L||/||g||.Let d'=(L),L^⊥ an orthogonal of L in V, B={e_1, ⋯, e_d} an orthonormal basis of V such that B':=(e_1, ⋯, e_d') is a basis of L and F=B∖ B' a basis forL^⊥. Assume first thatis non-Archimedean. Then the following relation is true for every g∈(V), ||g^t f ||/||g^t|| ||f|| = max_1≤ i≤ d|f(ge_i)|/||g^t||||f||=max{max_1≤ i ≤ d'|f(ge_i)|/||g||||f|| , max_d'+1≤ i ≤d|f(ge_i)|/||g||||f||}≤ max{||g_| L||/||g|| , max_d'+1≤ i ≤dδ(g[e_i],[Ker(f)])} Equality (<ref>)holds because||g^t||=||g|| and inequality(<ref>) is true becausefor any x∈ V, |f(x)|/||f||≤ ||x|| andδ(g [x],[Ker(f)])= |f(gx)|/||gx|| ||f||≥|f(gx)|/||g|| ||f|| ||x||.Estimate(<ref>)shows that the lemma is true for C=1. When isArchimedean, estimate(<ref>) is replaced by: ( ||g^t f ||/||g^t|| ||f||)^2 ≤(||g_| L||/||g||)^2+ ∑_i=d'+1^dδ(g[e_i],[Ker(f)])^2.Hence, for any C<1,C ||g^t f||/||g^t|| ||f||≥||g_|L||/||g|| ⟹ d'≤ i≤ dmaxδ(g[e_i], [Ker(f)])≥√(1-C ^2/ d)||g^t f||/||g^t|| ||f||. The constant C(,d):=1/√(d+1 ) solves the equationC=√(1-C^2/d).Let [Z]∈ be the random variable given by Theorem <ref>. Let f ∈ V^* ∖ℒ_μ̌ and H=Ker(f).Since the Lyapunov exponent of the restriction to ℒ_μ is less than λ_1, one can show using the same techniques as the proof of Corollary <ref>that there exists β_1>0 such that||R_n_| L||/||R_n||≤exp(-β_1 n), with probability tending to one exponentially fast. Corollary <ref> applied to the measure μ̌, together with δ≤ 1, show then thatfor any C>0 there exists β_2>0 and n_0∈ (both independent of f) such that for all n≥ n_0, (C ||R_n^t f||/||R_n^t|| ||f||≥||R_n_| L||/||R_n||)≥ 1-exp(-β_2 n)/δ([f], [ℒ_μ̌]).Take now C to be the constant C(,d) given by Lemma<ref>. The aforementioned lemmatogether with estimate(<ref>) imply that for every n≥ n_0:(∃ x∈ F; δ(R_n[x],[H]) ≥ C ||R_n^t f||/||R_n^t|| ||f||)≥ 1 - exp(-β_2 n)/δ([f], [ℒ_μ̌]).Hence, for every ϵ>0, (δ([Z], [H])≤exp(-ϵ n))≤exp(-β_2 n)/δ([f], [ℒ_μ̌]) + ∑_x∈ F( C ||R_n^t f||/||R_n^t|| ||f||≤exp(-ϵ n)+ δ(R_n[x],[Z])). But by Theorem <ref> and Markov's inequality,one deduces thatthere existβ_3=β_3(ϵ),β_4=β_4(ϵ)>0 such that for all n large enough,(δ([Z],[H])≤exp(-ϵ n) )≤exp(-β_3 n) /δ([f], [ℒ_μ̌])+ ∑_x∈ F( ||R_n^t f||/||R_n^t|| ||f||≤exp(-β_4 n)).Using Corollary <ref> and the fact that [Z] has law ν, we deduce finally that for every ϵ>0 there exists β=β(ϵ)>0 and n_0=n_0(ϵ)∈(both independent of f∈ V^*∖ [_μ̌]) such that for every n≥ n_0 ,ν{[x]∈; δ([x],[H]) ≤exp(-ϵ n) }≤exp(-β n)/δ([f], [ℒ_μ̌]).Let now A_n:={[x]∈; δ([x],[H])∈ ( e^-(n+1), e^-n ] }, n∈. On the one hand, (A_n)_n∈ cover ∖ [H].On the other hand, ν( [H])=ν([H] ∩ [𝒰_μ])=0 because ν is not degenerate on [𝒰_μ] (Theorem <ref>) andH⊄𝒰_μ (as f∉_μ̌, see Lemma <ref>). Estimate(<ref>) applied for ϵ =1 gives thensome β>0 and some n_0∈ (both independent on f)such that for any α>0, ∫_δ^-α([x],[H]) dν([x])= ∑_n=0^n_0-1∫_A_nδ^-α([x],[H]) dν([x]) +∑_n=n_0^+∞∫_A_nδ^-α([x],[H]) dν([x]) ≤n_0 exp(α n_0) +1/δ([f], [ℒ_μ̌])∑_n=n_0^+∞exp( α (n+1)) exp(-β n )≤ 1/δ([f], [ℒ_μ̌])( n_0 exp(α n_0 ) +exp(α ) ∑_n=n_0^+∞exp(- (β-α) n))Hence δ([f], [ℒ_μ̌])∫_δ^-α([x],[H]) dν([x]) is finite (and independent of [f]∈∖ [ℒ_μ̌]) as soonas 0<α< β. Finally, we show howto conclude easily from Theorem <ref>the proof of some results stated in Section <ref>. 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"authors": [
"Richard Aoun",
"Yves Guivarc'h"
],
"categories": [
"math.DS",
"math.GR",
"math.PR",
"60BXX, 37-XX, 37AXX, 60-XX"
],
"primary_category": "math.DS",
"published": "20170526142513",
"title": "Random matrix products when the top Lyapunov exponent is simple"
} |
UWThPh-2017-11 Renormalization and radiative corrections to masses in a general Yukawa modelM. FoxE-mail: [email protected] , footnote1 W. GrimusE-mail: [email protected] M. LöschnerE-mail: [email protected] University of Vienna, Faculty of Physics Boltzmanngasse 5, A–1090 Vienna, AustriaOctober 2, 2017 ====================================================================================================================================================================================================================================================== We propose an object detection method that improves the accuracy of the conventional SSD (Single Shot Multibox Detector), which is one of the top object detection algorithms in both aspects of accuracy and speed. The performance of a deep network is known to be improved as the number of feature maps increases.However, it is difficult to improve the performance by simply raising the number of feature maps. In this paper, we propose and analyze how to use feature maps effectively to improve the performance of the conventional SSD.The enhanced performance was obtained by changing the structure close to the classifier network, rather than growing layers close to the input data, e.g., by replacing VGGNet with ResNet.The proposed network is suitable for sharing the weights in the classifier networks, by which property, the training can be faster with better generalization power. For the Pascal VOC 2007 test set trained with VOC 2007 and VOC 2012 training sets, the proposed network with the input size of 300 × 300 achieved 78.5% mAP (mean average precision) at the speed of 35.0 FPS (frame per second), while the network with a 512 × 512 sized input achieved 80.8% mAP at 16.6 FPS using Nvidia Titan X GPU. The proposed network shows state-of-the-art mAP, which is better than those of the conventional SSD, YOLO, Faster-RCNN and RFCN. Also, it is faster than Faster-RCNN and RFCN.§ INTRODUCTIONObject detection is one of the main areas of researches in computer vision. In recent years, convolutional neural networks (CNNs) have been applied to object detection algorithms in various ways, improving the accuracy and speed of object detection <cit.>.Among various object detection methods, SSD <cit.> is relatively fast and robust to scale variations because it makes use of multiple convolution layers for object detection. Although the conventional SSD performs good in both the speed and detection accuracy, it has a couple of points to be supplemented. First, as shown in Fig. <ref>, which shows the overall structure of conventional SSD, each layer in the feature pyramid[The term feature pyramid is used to denote the set of layers that are directly used as an input to the classifier network as shown in Fig. <ref>.] is used independently as an input to the classifier network. Thus, the same object can be detected in multiple scales.Consider a certain position of a feature map in a lower layer (say, Conv4-3) is activated. This information can affect entire scales up to the the last layer (Conv11-2), which means that the relevant positions in the higher layers have a good chance to be also activated.However, SSD does not consider the relationships between the different scales because it looks at only one layer for each scale. For example, in Fig.<ref>(a), SSD finds various scale boxes for one object.Second, SSD has the limitation that small objects are not detected well. This is not the problem only for SSD but the problem for most object detection algorithms. To solve this problem, there have been various attemptssuch as replacing the base network with more powerful one, e.g., replacing VGGNet with ResNet<cit.> or increasing the number of channels in a layer <cit.>. Fig.<ref>(b) shows that SSD has a limitation in detecting small objects. Especially, in the two figures, persons on the boat and small cows are not detected, respectively.In this paper, we tackle these problems as follows. First, the classifier network is implemented considering the relationship between layers in the feature pyramid. Second, the number of channels (or feature maps) in a layer is increased efficiently. More specifically, only the layers in the feature pyramid are allowed to have increased number of feature maps instead of increasing the number of layers in the base network.The proposed network is suitable for sharing weights in the classifier networks for different scales, resulting in a single classifier network. This enables faster training speed with advanced generalization performance. Furthermore, this property of single classifier network is very useful in a small database. In the conventional SSD, if there is no object at a certain size, the classifier network of that size cannot learn anything. However, if a single classifier network is used, it can get information about the object from the training examples in different scales.Using the proposed architecture, our version of SSD can prevent detecting multiple boxes for one object as shown in Fig.<ref>(c). In addition, the number of channels can be efficiently increased to detect small objects as shown in Fig.<ref>(d).The proposed method shows state-of-the-art mAP (mean average precision) with a slightly degraded speed compared to the conventional SSD. The paper is organized as follows. In Section <ref>, we briefly review the related works in the area of object detection, and then propose a different version of SSD, which is called as the rainbow SSD, in Section <ref>. The experiments and the evaluation of the proposed algorithm are presented in Section <ref>.In Section <ref>, a discussion with some exemplary results of the proposed method is made. Finally, the paper is concluded in Section <ref>.§ RELATED WORKSA wide variety of methods using deep learning have been applied to the problem of object detection and it continues to show performance improvements. In the earlier works pioneered by R-CNN (region-based CNN) <cit.>, the candidate region was proposed through a separate algorithms such as selective search <cit.> or Edge boxes <cit.> and the classification was performed with deep learning. Although R-CNN improved the accuracy using deep learning, speed was still a problem, and end-to-end learning was impossible. The region proposal network (RPN) was first proposed in faster R-CNN, which improved the speed of the object detector significantly and was able to learn end-to-end <cit.>. YOLO (you only look once) greatly improved the speed by dividing a single image into multiple grids and simultaneously performing localization and classification in each grid <cit.>. While YOLO performed object detection by concentrating only on speed, an enhanced version of YOLO, which is denoted as YOLO2, removed the fully connected layers and used anchor boxes to improve both the speed and the accuracy <cit.>. On the other hand, SSD creates bounding box candidates at a given position and scale and obtains their actual bounding box and score for each class <cit.>. Recently, to improve the accuracy of SSD, especially for small object, DSSD (deconvolutional SSD) that uses a large scale context for the feature pyramid was proposed <cit.>. DSSD applied a deconvolution module to the feature pyramid and used ResNet instead of VGGNet. DSSD succeeded in raising accuracy at the expense of speed.Besides the fields of object detection, the fields of segmentation have also been much developed by the application of deep neural networks. Many of them use pixel-based object classification in combination with upsampling or deconvolution to obtain the segmentation results in the same size as the input image <cit.>. In addition, features are engineered in various ways by concatenation, element-wise addition or element-wise product with bypass, to obtain improved performance. This paper is inspired by the fact that we get improved abstract representation of the original image from features in different scales <cit.>.§ METHOD As mentioned above, our strategy of improving the accuracy of SSD is to let the classifier network fully utilize the relationship between the layers in the feature pyramidwithout changing the base network that is closely located to the input data. In addition, it also increases the number of channels in the feature pyramid efficiently. Figure <ref> shows several ways of increasing the number of feature maps in different layers for the classifier networks to utilize the relationship between layers in the feature pyramid. To enable this, in Fig. <ref>(a), feature maps in the lower layers are concatenated to those of the upper layers through pooling. In this way, the classifier networks with large receptive fields can have enriched representation power for object detection.On the other hand, Fig. <ref>(b) shows the method of concatenating the feature maps of the upper layers to the lower layer features through deconvolution or upsampling. Fig. <ref>(c) shows the feature map concatenation method that utilize both the lower layer pooling and the upper layer deconvolution.One thing to note is that before concatenating feature maps, a normalization step is inevitable. This is because the feature values in different layers are quite different in scale. Here, batch normalization <cit.> is applied for each filter before concatenation. All of the above methods have the advantage that end-to-end learning is possible. More details about each are described below. §.§ Concatenation through pooling or deconvolutionIn the structure of SSD, generally, the numbers of channels in the lower layers are larger than those in the upper layer. To make explicit relationship between feature pyramid and to increase the number of channels effectively, we concatenate feature maps of the upper layers through pooling or concatenate feature maps of the lower layers through deconvolution. Unlike DSSD <cit.>, which uses deconvolution module consisting of 3 convolution layer, 1 deconvolution layer, 3 batch normalization layer, 2 Relu and elementwise product, our model of concatenation through deconvolution performs only deconvolution with batch normalization and does not needelementwise product. The advantage of these structure is that object detection can be performed with information from the other layers. On the other hand, the disadvantage is that information flows unidirectional and the classifier network cannot utilize other directional information. §.§ Rainbow concatenationAs shown in Fig. <ref>(c), in the rainbow concatenation, pooling and deconvolution are performed simultaneously to create feature maps with an explicit relationship between different layers. After pooling or deconvolving features in every layers to the same size, we concatenate them. Using this concatenated features, detection is performed considering all the cases where the size of the object is smaller or larger than the specific scale. That is, it can have additional information about the object larger than or smaller than the object.Therefore, it is expected that the object in a specific size is likely to be detected only in an appropriate layer in the feature pyramid as shown in Fig. <ref>(c). In addition, the low-layer features that has been with limited representation power are enriched by the concatenation of higher-layer features, resulting in good representation power for small object detection as in DSSD <cit.> without much computational overhead.By rainbow concatenation, each layer of feature pyramid contains 2,816 feature maps (concatenation of 512, 1024, 512, 256, 256, and 256 channels) in total. Because each layer in the feature pyramid now has the same number of feature maps, weights can be shared for different classifier networks in different layers. Each classifier network of the conventional SSD checks 4 or 6 default boxes and the proposed rainbow SSD can unify default boxes in different layers with weight sharing. As shown in Table <ref>, conventional SSD makes a 8,732 total boxes. On the other hand, in the shared classifier with 4 default boxes for each layer, the number becomes 7,760, likewise, for the shared classifier with 6 default boxes, it becomes 11,640 in total.§.§ Increasing number of channelsIt is known that the larger the number of channels, the better the performance becomes. To demonstrate how efficient our method is, we present a comparison model, for which we simply change the number of channel in the original base network as shown in Table <ref>. As in SSD, reduced VGG-16 pre-trained model is used as the base network. The number of channels in each convolution layer are set to be 2 to 8 times larger than the original network. From now on, this comparison network will be denoted as I-SSD, which stands for increased-channel SSD.§ EXPERIMENTS In order to evaluate the performance of the proposed algorithm, we tested various versions of SSD for PASCAL VOC2007 dataset<cit.>.The VOC dataset consists of 20 object classes with the annotated ground truth location and the corresponding class information for each image.We trained our model with VOC2007 and VOC2012 `trainval' datasets. The SSDs with 300× 300 input were trained with a batch size of 8, and the learning rate was reduced from 10^-3 to 10^-6 by 10^-1. With each learning rate, we trained 80K, 20K, 20K, and 20K iterations respectively. Thus the total number of iteration for each model was 140K. The 512 × 512 input models were performed with a batch size of 4 and the learning rate was equal to that for 300 × 300 models.In the case of speed, it is measured by using the forward path of the network with a batch size of 1. The experiments were done with cuDNN v5.1 using CAFFE time function. Therefore, if the detection time is measured from the pre-processing (resizing image and so on), it may take longer. The experimental results are shown in Table <ref>. In the table, the performances of YOLO <cit.>, YOLOv2 <cit.>, Faster R-CNN <cit.>, R-FCN <cit.>, and DSSD <cit.> were obtained from their homepage [YOLO and YOLOv2 : http://pjreddie.com/darknet] or the respective paper. To see the performance of various feature augmentation methods, we performed experiments using features concatenated through pooling (SSD pooling) and deconvolution (SSD deconvolution) as described in Section <ref>. We also tested ISSD described in Section <ref> for comparison. Three types of R-SSD was tested. The first one utilizes separate classifier networks for different scales and the rest two use a common classifier with 4 or 6 default boxes for a scale as described in Section <ref>. The conventional SSD was also trained and tested by ourselves.ISSD:For the 300 input model, we experimented ISSD by increasing the number of the channels from 2 to 8 times for different layers as Table <ref>. As a result, there is a 0.4% mAP improvement in accuracy with 78.1% mAP compared to conventional SSD. However, due to the increased number of channels, the speed drops to 26.9 FPS Concatenation through pooling or deconvolution:For the 300 input model, the concatenation through pooling and concatenation through deconvolution result in mAP of 77.1%and 77.3% respectively which is a degraded performance than that of conventional SSD by 0.6% and 0.4%. In addition, due to the increased complexity, the speed was also degraded to 48.3 FPS and 39.9 FPS, respectively. R-SSD:For the 300 input model, there is a 0.8% improvement in accuracy with 78.5% mAP compared to conventional SSD. However, due to the increased computational complexity, the speed drops to 35.0 FPS. For the 512 input model, it results in mAP of 80.8% which is 1% better than conventional SSD. However its speed drops to 16.6 FPS. In particular, comparing the two SSD 512 models, precision increases 2.9% at recall value 0.8 and 8.2% at recall of 0.9. In the case of single classifier model with 300 input model, it has a 76.2% and 77.0% mAP when they use four and six default boxes, respectively.§ DISCUSSION§.§ New evaluation method The most commonly used evaluation technique for object detection is mAP. AP (Average Precision) is a concept of integrating precision as recall is varied from 0 to 1 and mAP is defined as the average of AP for all the object classes. In Figure <ref>, we show the recall vs. average precision graph for PASCAL VOC 2007 test data. Note that average precision (vertical axis) here is averaged value for all the object classes. In the figure, we can see that different versions of SSD (SSD 300, SSD 512, R-SSD 300, R-SSD 512) have almost the same average precision for small (<0.5) recall values. Due to this, even if the precisions for large recall values show significant difference between algorithms (around 14% difference for SSD 300 and R-SSD 512), the difference in mAP is relatively small (around 3% for SSD 300 and R-SSD 512).Considering the use case of most object detection algorithms such as for autonomous vehicles, the precision value measured at high (> 0.7) recall is more important than that measured at small recall value. Therefore, in Table <ref>, we show the mAP computed by averaging only the APs at recall of 0.7 or higher. In the table, we can see more clearly the effectiveness of R-SSD over SSD. At recall of 0.9, R-SSD (512) outperformed SSD (512) by more than 8%. Note that the APs at recall of 1 is 0 for all the cases. This is due to the fact that we cannot recall all the objects in the test images regardless how we lower the score threshold.§.§ Concatenation by pooling or deconvolution These two models are both inferior in accuracy and speed compared to conventional SSD, although they made explicit relationship between multiple layers and increased the number of channels. These two models need to perform more operations, therefore the speed can drop. As for accuracy, the reason can be conjectured that the layers sharing the same feature maps with other layers can be affected by the loss of other scales and do not fully focus on the scale. That is, they cannot learn properly on their scale. §.§ Single classifier vs. Multiple classifiers Unlike the conventional SSD, because R-SSD have the similar feature maps for different layers only different in size, the classifier network can be shared. Here, the experiments with a single classifier network by unifying the number of channels in each scale of feature pyramid. As shown in the Table <ref>, there is a difference in the number of boxes, but there is little difference in speed. In comparison, performance was 1.2 % and 0.7 % lower than that of conventional SSD. However, the advantage of a single classifier is that learning can be effective especially when there are significant imbalance between the numbers of training samples for different sizes. In this case, conventional SSD cannot train the classifier for a scale with small number of samples. However, in R-SSD, this problem is avoided because the classifier network is shared. Furthermore, single classifier is faster at the early stage of training. Therefore, even for a large dataset, R-SSD can be trained fast by training a single classifier in the early stage and at some point, the classifiers can be trained separately for different scales. §.§ Accuracy vs. Speed The conventional SSD is one of the top object detection algorithms in both aspects of accuracy and speed. For SSD300 or SSD512, it has 77.7 % mAP and 79.8 % mAP respectively and has 61.1 FPS and 25.2 FPS. Looking at the results of the ISSD for comparison with our algorithm, ISSD had a 0.4 % accuracy gain, but the speed dropped to 26.9 FPS. In our experiments, R-SSD shows improved accuracy with a bit slow speed. Compared to the ISSD, R-SSD shows higher accuracy and faster speed. Moreover, it shows about 1% mAP improvement over the conventional SSD. At a speed of 15 fps or higher, our R-SSD shows 80.8 % mAP. Compared to R-FCN with similar accuracy, R-SSD is about three times faster.§.§ Performances for different scales Table <ref> shows the recall of each object size <cit.>.Normally, the AP or AR (Average Recall) should be obtained, but, the VOC2007 test set has a total of 12,032 objects, of which 567 are small objects. In evaluating the performance for small objects, there are several classes with no object at all. Therefore, we integrate all the objects in measuring the recall.When the object size is small, R-SSD300 and R-SSD512 detect more number of objects than SSD300 and SSD512, respectively. It can be shown that R-SSD misses a few small objects. At medium size, R-SSD300 has even more recall than SSD512.Furthermore, when the object size is large, recall of all models show high value over 0.93. The difference of R-SSD300, SSD512, and R-SSD512 is less than 10 out of 7,641. § CONCLUSIONIn this paper, we presented a rainbow concatenation scheme that can efficiently solve the problems of the conventional SSD. The contribution of the paper is as follows. First, it creates a relationship between each scale of feature pyramid to prevent unnecessary detection such as multiple boxes in different scales for one object. Second, by efficiently increasing the number of feature maps of each layer in the feature pyramid, the accuracy is improved without much time overhead. Finally, the number of feature maps for different layers are matched so that a single classifier can be used for different scales. By using a single classifier, improvement on the generalization performance can be expected, and it can be effectively used for datasets with size imbalance or for small datasets. The proposed R-SSD was created considering both the accuracy and the speed simultaneously, and shows state-of-the-art mAP among the ones that have speed of more than 15 FPS.§ ADDITIONAL EXPERIMENTSTo show that the proposed method performs well especially for a small training dataset, we trained the networks using the train dataset in VOC2007 and compared the proposed methods with others.Also, we trained the networks using PASCAL VOC2012<cit.> datasets and show the detection results here. §.§ Small Train Dataset in VOC2007 Table <ref> is experimental results of different networks that were trained using the train dataset in VOC2007 which consists of a relatively small number of images (2,501 images in total). Each network was trained with these 2,501 images and the mAP was measured with VOC 2007 test dataset. The conventional SSD <cit.> shows a mAP of66.1%, while R-SSD achieves 66.9% which is 0.8% better than that of SSD. Furthermore, R-SSD with one classifier achieves an even better mAP of 67.2%. It shows that training a single classifier is better not only in generalization power resulting in a faster training speed but also in detection performance when the training dataset is small.§.§ VOC2012 Table <ref> shows the results of VOC2012. All experiments were trained with VOC07++12 train dataset.YOLO <cit.> and YOLOv2 <cit.> have mAPs of 57.9% and 73.4%, respectively. SSD300 <cit.> and DSSD321 <cit.> show higher mAPs of 75.6% and 76.5%, respectively. Our R-SSD shows 76.6% mAP, which is higher than those of SSD300 and DSSD321. R-SSD is also faster than DSSD321. §.§ MS COCO example images by R-SSD We test MS-COCO dataset and Fig.<ref> shows several resultant images among the dataset. R-SSD can prevent detecting multiple boxes for one object as shown in Fig.<ref>(b). Furthermore, it can efficiently increase the detection rate forsmall objects as shown in Fig.<ref>(d). | http://arxiv.org/abs/1705.09587v1 | {
"authors": [
"Jisoo Jeong",
"Hyojin Park",
"Nojun Kwak"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170526140741",
"title": "Enhancement of SSD by concatenating feature maps for object detection"
} |
[ Convergent Tree Backup and Retrace with Function Approximation equal*Ahmed Touatiudem,fb Pierre-Luc Baconmcgill Doina Precupmcgill,cifar Pascal Vincentudem,fb,cifar fbFacebook AI Research cifarCanadian Institute for Advanced Research (CIFAR) udemMILA, Université de MontréalmcgillMILA, McGill UniversityAhmed [email protected] Machine Learning, ICML0.3in ] Off-policy learning is key to scaling up reinforcement learning as it allows to learn about a target policy from the experience generated by a different behavior policy. Unfortunately, it has been challenging to combine off-policy learning with function approximation and multi-step bootstrapping in a way that leads to both stable and efficient algorithms. In this work, we show that the Tree Backup and Retrace algorithms are unstable with linear function approximation, both in theory and in practice with specific examples. Based on our analysis, we then derive stable and efficient gradient-based algorithms using a quadratic convex-concave saddle-point formulation. By exploiting the problem structure proper to these algorithms, we are able to provide convergence guarantees and finite-sample bounds. The applicability of our new analysis also goes beyond Tree Backup and Retrace and allows us to provide new convergence rates for the GTD and GTD2 algorithms without having recourse to projections or Polyak averaging.§ INTRODUCTIONRather than being confined to their own stream ofexperience, off-policy learning algorithms are capable ofleveraging data from a different behavior than the one being followed, which can provide many benefits: efficient parallel exploration as in <cit.> and <cit.>, reuse of past experience with experience replay <cit.> and, in many practical contexts, learning form data produced by policies that are currently deployed, but which we want to improve (as in many scenarios of working with an industrial or health care partner). Moreover, a single stream of experience can be used to learn about a variety of different targets which may take the form of value functions corresponding to different policies and time scales <cit.> or to predictingdifferent reward functions as in <cit.> and <cit.>. Therefore, the design and analysis of off-policy algorithms using all the features of reinforcement learning, e.g. bootstrapping, multi-step updates (eligibility traces), and function approximation has been explored extensively over three decades. While off-policy learning and function approximation have been understood in isolation, their combination with multi-steps bootstrapping produces a so-called deadly triad <cit.>, i.e., many algorithms in this category are unstable.A convergent approach to this triad is provided by importance sampling, which bends the behavior policy distribution onto the target one <cit.>. However, as the length of the trajectories increases, the variance of importance sampling corrections tends to become very large. The Tree Backup algorithm <cit.> is an alternative approach which remarkably does not rely on importance sampling ratios directly. More recently, <cit.> introduced the Retrace algorithm which also builds on Tree Backup to perform off-policy learning without importance sampling. Until now, Tree Backup and Retrace(λ) had only been shown to converge in the tabular case, and their behavior with linear function approximation was not known. In this paper, we show that this combination with linear function approximation is in fact divergent. We obtain this result by analyzing the mean behavior of Tree Backup and Retrace using the ordinary differential equation (ODE) <cit.> associated with them. We also demonstrate this instability with a concrete counterexample.Insights gained from this analysis allow us to derive a new gradient-based algorithm with provable convergence guarantees. Instead of adapting the derivation of Gradient Temporal Difference (GTD) learning from <cit.>, we use a primal-dual saddle point formulation <cit.> which facilitates the derivation of sample complexity bounds. The underlying saddle-point problem combines the primal variables, function approximation parameters, and dual variables through a bilinear term. In general, stochastic primal-dual gradient algorithms like the ones derived in this paper can be shown to achieve O(1/k) convergence rate (where k is the number of iterations). For example, this has been established for the class of forward-backward algorithms with added noise <cit.>. Furthermore, this work assumes that the objective function is composed of a convex-concave term and a strongly convex-concave regularization term that admits a tractable proximal mapping. In this paper, we are able to achieve the same O(1/k) convergence rate without having to assume strong convexity with respect to the primal variables and in the absence of proximal mappings. As corollary, our convergence rate result extends to the well-known gradient-based temporal difference algorithms GTD <cit.> and GTD2 <cit.> and hence improves the previously published results.The algorithms resulting from our analysis are simple to implement, and perform well in practice compared to other existing multi-steps off-policy learning algorithms such as GQ(λ) <cit.> and AB-Trace(λ) <cit.>. § BACKGROUND AND NOTATION In reinforcement learning, an agent interacts with its environment which we model as discounted Markov Decision Process (, , γ, P, r) with state space , action space , discount factor γ∈ [0, 1), transition probabilities P : ×→ (→ [0,1]) mapping state-action pairs to distributions over next states, and reward function r : (×) →. For simplicity, we assume the state and action space are finite, but our analysis can be extended to the countable or continuous case. We denote by π(as) the probability of choosing action a in state s under the policy π : → (→ [0, 1]). The action-value function for policy π, denoted Q^π:×→, represents the expected sum of discounted rewards along the trajectories induced by the policy in the MDP: Q^π(s, a) = ∑_t=0^∞γ^t r_t(s_0, a_0)=(s, a), π.Q^π can be obtained as the fixed point of the Bellman operator over the action-value function ^π Q = r + γ P^π Q where r is the expected immediate reward and P^π is defined as:(P^π Q)(s,a) ≜∑_s' ∈∑_a' ∈ P(s's, a) π(a's') Q(s', a'). In this paper, we are concerned with the policy evaluation problem <cit.> under model-free off-policy learning. That is, we will evaluate a target policy π using trajectories (i.e. sequences of states, actions and rewards) obtained from a different behavior policy μ. In order to obtain generalization between different state-action pairs, Q^π should be represented in a functional form. In this paper, we focus on linear function approximation of the form:Q(s, a) ≜θ^⊤ϕ(s, a),where θ∈Θ⊂^d is a weight vector and ϕ: ×→^d is a feature map from a state-action pairs to a given d-dimensional feature space.Off-policy learning <cit.> provided a unified perspective on several off-policy learning algorithms, namely: those using explicit importance sampling corrections <cit.> as well asTree Backup (TB(λ)) <cit.> and Q(λ)^π <cit.> which do not involve importance ratios. As a matter of fact, all these methods share a general form based on the λ-return <cit.> but involve different coefficients κ_i in :G_k^λ ≜ Q(s_k, a_k) + ∑_t=k^∞ (λγ)^t-k(∏_i=k+1^t κ_i) × (r_t +γ_π Q(s_t+1, ·) - Q(s_t, a_t))=Q(s_k, a_k) + ∑_t=k^∞ (λγ)^t-k(∏_i=k+1^t κ_i) δ_t ,where _π Q(s_t+1, .) ≜∑_a ∈π(as_t+1)Q(s_t+1, a) and δ_t ≜ r_t +γ_π Q(s_t+1, .) - Q(s_t, a_t) is the temporal-difference (TD) error. The coefficients κ_i determine how the TD errors would be scaled in order to correct for the discrepancy between target and behavior policies. From this unified representation, <cit.> derived the Retrace(λ) algorithm. Both TB(λ) and Retrace(λ) consider this form of return, but set κ_i differently. The TB(λ) updates correspond to the choice κ_i = π(a_is_i) while Retrace(λ) sets κ_i = min(1, π(a_is_i)/μ(a_is_i)), which is intended to allow learning from full returns when the target and behavior policies are very close.The importance sampling approach <cit.> converges in the tabular case by correcting the behavior data distribution to the distribution that would be induced by the target policy π. However, these correction terms lead to high variance in practice. Since Q(λ) does not involve importance ratios, this variance problem is avoided but at the cost of restricted convergence guarantees satisfiedonly when the behavior and target policies are sufficiently close. The analysis provided in this paper concerns TB(λ) and Retrace(λ), which are convergent in the tabular case, but have not been analyzed in the function approximation case. We start by noting that the Bellman operator [We overload our notation over linear operators and their corresponding matrix representation.]ℛ underlying these these algorithms can be written in the following form: ( Q)(s,a)≜ Q(s, a) + _μ[ ∑_t=0^∞ (λγ)^t(∏_i=1^t κ_i)× (r_t+ γ_π Q(s_t+1, ·) - Q(s_t, a_t)) ]= Q(s,a) + (I- λγ P^κμ)^-1 (^π Q - Q)(s, a) ,where _μ is the expectation over the behavior policy and MDP transition probabilities and P^κμ is the operator defined by:(P^κμ Q)(s, a) ≜∑_s' ∈a' ∈ P(s's, a) μ(a's') κ(s', a') Q(s', a'). In the tabular case, these operators were shown to be contraction mappings with respect to the max norm <cit.>. In this paper, we focus on what happens to these operators when combined with linear function approximation.§ OFF-POLICY INSTABILITY WITH FUNCTION APPROXIMATION When combined with function approximation, the temporal difference updates corresponding to the λ-return G_k^λ are given by θ_k+1= θ_k + α_k (G_k^λ - Q(s_k, a_k) )∇_θ Q(s_k, a_k)= θ_k + α_k (∑_t=k^∞ (λγ)^t-k(∏_i=k+1^t κ_i) δ_t^k )ϕ(s_k, a_k)where δ_t^k = r_t + γθ_k^⊤_πϕ(s_t+1, ·) - θ_k^⊤ϕ(s_t, a_t) and α_k are positive non-increasing step sizes. The updates (<ref>) implies off-line updating as G_k^λ is a quantity which depends on future rewards. This will be addressed later using eligibility traces: a mechanism to transform the off-line updates into efficient on-line ones. Since (<ref>) describes stochastic updates, the following standard assumption is necessary: The Markov chain induced by the behavior policy μ is ergodic and admits a unique stationary distribution, denoted by ξ, over state-action pairs. We write Ξ for the diagonal matrix whose diagonal entries are (ξ(s,a))_s ∈, a ∈.Our first proposition establishes the expected behavior of the parameters in the limit.If the behavior policy satisfies Assumption <ref> and (θ_k)_k≤ 0 is the Markov processdefined by (<ref>) then:[θ_k+1θ_0] = (I + α_k A )[θ_k θ_0]+ α_k b,where matrix A and vector b are defined as follows:A≜Φ^⊤Ξ(I - λγ𝑃^κμ)^-1(γ𝑃^π - I) Φ, b≜Φ^⊤Ξ(I - λγ P^κμ)^-1r. θ_k+1= θ_k +α_k (∑_t=k^∞ (λγ)^t-k(∏_i=k+1^t κ_i) ϕ(s_k, a_k)×( [γ𝔼_πϕ(x_t+1, ·) - ϕ(x_t, a_t)]^⊤θ_k+ r_t ) )= θ_k + α_k (A_k θ_k+ b_k ).So, [θ_k+1θ_k] = (I + α_k A) θ_k + α_k b where A =[A_k] and b = [b_k] The ODE (Ordinary Differential Equations) approach <cit.> is the main tool to establish convergence in the function approximation case <cit.>. In particular, we useProposition 4.8 in <cit.>, which states that under some conditions, θ_k converges to the unique solution θ^* of the system A θ^* + b = 0. This crucially relies on the matrix A being negative definite i.e y^⊤ A y < 0, ∀ y ≠ 0. In the on-policy case, when μ = π, we rely on the fact that the stationary distribution is invariant under the transition matrix P^π i.e d^⊤ P^π = d^⊤ <cit.>. However, this is no longer true for off-policy learning with arbitrary target/behavior policies and the matrix A may not be negative definite: the series θ_k may then diverge.We will now see that the same phenomenon may occur with TB(λ) and Retrace(λ). Counterexample: We extend the two-states MDP of <cit.>, originally proposed to show the divergence of off-policy TD(0), to the case of function approximation over state-action pairs. This environment has only two states, as shown in Figure <ref>, and two actions: left or right. In this particular case, both TB(λ) and Retrace(λ) share the same matrix P^κμ and P^κμ = 0.5 P^π:P^π = (0 1 0 00 1 0 0 1 0 0 0 1 0 0 0),(P^π)^n =(0 1 0 00 1 0 0 0 1 0 0 0 1 0 0) ∀ n ≥ 2If we set β := 0.5 γλ, we then have:(I - λγ P^κμ )^-1= (1β/1-β0 001/1-β0 0β β^2/1-β1 0β β^2 /1-β0 1), A = ( 6 γ - β -5/1- β03 (γβ - β^2 - β - γ)/1-β-5) .Therefore, ∀ γ∈ (5/6, 1) and ∀ λ∈ [0, min(1, 12 γ -10/γ)), the first eigenvalue e_1 = 6 γ - β -5/1- β of A is positive. The basis vectors (1,0)^⊤ and (0,1)^⊤ are eigenvectors of A associated with e_1 and -5, then if θ_0 = (η_1, η_2)^⊤, we obtain [θ_kθ_0] = (η_1∏_i=0^k-1(1 + α_ie_1), η_2 ∏_i=0^k-1(1 -5 α_i))^⊤ implying that || [θ_kθ_0] || ≥ |η_1| ∏_i=0^k-1(1 + α_i e_1). Hence, as ∑_kα_k →∞, || [θ_kθ_0] || →∞ if η_1 ≠ 0. § CONVERGENT GRADIENT OFF-POLICY ALGORITHMSIf A were to be negative definite, Retrace(λ) or TB(λ) with function approximation would converge to θ^*= -A^-1b. It is known <cit.> that Φθ^* is the fixed point of the projected Bellman operator :Φθ^* = Π^μℛ(Φθ^*),where Π^μ = Φ (Φ^⊤ΞΦ)^-1Φ^⊤Ξ is the orthogonal projection onto the space S = {Φθ | θ∈ℝ^d } with respect to the weighted Euclidean norm ||.||_Ξ. Rather than computing the sequence of iterates given by the projected Bellman operator, another approach for findingθ^* is to directly minimize <cit.> the Mean Squared Projected Bellman Error (MSPBE):𝐌𝐒𝐏𝐁𝐄(θ) = 1/2||Π^μℛ(Φθ)- Φθ ||^2_Ξ . This is the route that we take in this paper to derive convergent forms of TB(λ) and Retrace(λ). To do so, we first define our objective function in terms of A and b which we introduced in Proposition <ref>.Let M ≜Φ^⊤ΞΦ = [ΦΦ^⊤] be the covariance matrix of features. We have:𝐌𝐒𝐏𝐁𝐄(θ) = 1/2 || A θ + b||^2_M^-1(The proof is provided in the appendix.) In order to derive parameter updates, we could compute gradients of the above expression explicitly as in <cit.>, but we would then obtain a gradient that is a product of expectations. The implied double sampling makes it difficult to obtain an unbiased estimator of the gradient. <cit.> addressed this problem with a two-timescale stochastic approximations. However, the algorithm obtained in this way is no longer a true stochastic gradient method with respect to the original objective. <cit.> suggested an alternative which converts the original minimization problem into a primal-dual saddle-point problem. This is the approach that we chose in this paper.The convex conjugate of a real-valued function f is defined as:f^*(y)= sup_x ∈𝑋 ( ⟨ y, x ⟩- f(x)) ,and f is convex, we have f^** = f. Also, if f(x) = 1/2 ||x||_M^-1, then f^*(x) = 1/2||x||_M. Note that by going to the convex conjugate, we do not need to invert matrix M.We now go back to the original minimization problem:min_θ𝐌𝐒𝐏𝐁𝐄(θ)⇔min_θ1/2 || A θ + b||_M^-1^2 ⇔min_θmax_ω( ⟨ Aθ+b, ω⟩ - 1/2 || ω||_M^2 )The gradient updates resulting from the saddle-point problem (ascent in ω and descent in θ) are then:ω_k+1= ω_k + η_k (Aθ_k + b - Mω_k) ,θ_k+1= θ_k - α_k A^⊤ω_k .where {η_k} and {α_k} are non-negative step-size sequences. As the A, b and M are all expectations, we can derive stochastic updates by drawing samples, which would yield unbiased estimates of the gradient. On-line updates: We now derive on-line updatesby exploiting equivalences in expectation between forward views and backward views outlined in <cit.>.Let e_k be the eligibility traces vector, defined as e_-1 = 0 and :e_k= λγκ(s_k, a_k) e_k-1 + ϕ(s_k, a_k) ∀ k ≥ 0 .Furthermore, letÂ_k = e_k (γ𝔼_π[ϕ(s_k+1, .)] - ϕ(s_k, a_k)])^⊤,b̂_k=r(s_k, a_k) e_k,M̂_k= ϕ(s_k, a_k) ϕ(s_k, a_k)^⊤. Then, we have [Â_k] = A, [b̂_k] = b and [M̂_k] = M.(The proof is provided in the appendix.) This proposition allows us to replace the expectations in Eq. (<ref>) by corresponding unbiased estimates. The resulting detailed procedure is provided in Algorithm <ref>.§ CONVERGENCE RATE ANALYSISIn order to characterize the convergence rate of the algorithm <ref>, we need to introduce some new notations and state new assumptions.We denote by A≜sup_x=1 Ax the spectral norm of the matrix A and by c(A) = A A^-1 its condition number. If the eigenvalues of a matrix A are real, we use λ_max(A) and λ_min(A) to denote respectively the largest and the smallest eigenvalue.If we set η_k = βα_k for a positive constant β, it is possible to combine the two iterations present in our algorithm as a single iteration using a parameter vector z_k ≜( θ_k1/√(β)ω_k ) where : z_k+1 = z_k - α_k (Ĝ_k z_k - ĝ_k)where:Ĝ_k ≜(0√(β)Â_k^⊤- √(β)Â_kβM̂_k )ĝ_k ≜(0√(β)b̂_k )Let G ≜[ Ĝ_k] and g = [ ĝ_k]. It follows from the proposition <ref> that G and g are well defined and more specifically:G = (0√(β) A^⊤- √(β) Aβ M )g = (0√(β) b )Furthermore, let ℱ_k = σ(z_0, Ĝ_0, ĝ_0 …, z_k, Ĝ_k, ĝ_k, z_k+1) be the sigma-algebra generated by the variables up to time k. With these definitions, we can now state our assumptions.The matrices A and M are nonsingular. This implies that the saddle-point problem admits a unique solution (θ^⋆, ω^⋆) = (-A^-1b, 0) and we define z^⋆≜ (θ^⋆, 1/√(β)ω^⋆). The features and reward functions are uniformly bounded. This implies that the features and rewards have uniformly bounded second moments. It follows that there exists a constant σ such that:[Ĝ_k z_k - ĝ_k^2 | ℱ_k-1] ≤σ^2 (1 +z_k^2) Before stating our main result, the following key quantities needs to be defined:ρ≜λ_max(A^⊤M^-1A), δ≜λ_min(A^⊤ M^-1A ), L_G ≜[ Ĝ_k^⊤Ĝ_kℱ_k-1]The following proposition characterize the convergence in expectation of z_k - z^⋆^2 = θ_k - θ^⋆^2 + 1/βw_k^2 Suppose assumptions <ref> and <ref> holds and if we choose β = 8 ρ/λ_min(M) and α_k = 9^2 × 2 δ/8 δ^2 (k+2) + 9^2 ζ where ζ =2 × 9^2 c(M)^2 ρ^2 + 32 c(M) L_G. Then the mean square error [z_k - z^⋆^2] is upper bounded by:9^2 × 8 c(M) {(8δ + 9ζ)^2 [z_0 - z^⋆^2] /(8^2δ^2 k + 9^2ζ)^2+ 8 σ^2 (1 +z^⋆^2)/(8^2δ^2 k + 9^2ζ) } The beginning of our proof relies on <cit.> which shows the linear convergence rate of deterministic primal-dual gradient method for policy evaluation. More precisely, we make use of the spectral properties of matrix G shown in the appendix of this paper. The rest of the proof follows a different route exploiting the structure of our problem.The above proposition <ref> shows that the mean square error [z_k - z^⋆^2] at iteration k is upper bounded bytow terms. The first bias term tells that the initial error [z_0 - z^⋆^2] is forgotten at a rate O(1/k^2) and the constant depends on the condition number of the covariance matrix c(M). The second variance term shows that noise is rejected at a rate O(1/k) and the constant depends on the variance of estimates σ^2 and c(M). The overall convergence rate is O(1/k). Existing stochastic saddle-point problem results:<cit.> provides a comprehensive review of stochastic saddle-point problem. When the objective function is convex-concave, the overall convergence rate is O(1/√(k)). Although several accelerated techniques could improve the dependencies on the smoothness constants of the problem in their convergence rate, the dominant term that depends on the gradient variance still decays only as O(1/√(k)).When the objective function is strongly convex-concave, <cit.> and <cit.> showed that stochastic forward-backward algorithms can achieve O(1/k) convergence rate. Algorithms in this class are feasible in practice only if their proximal mappings can be computed efficiently. In our case, our objective function is strongly concave because of the positive-definiteness of M but is otherwise not strongly convex. Because our algorithms are vanilla stochastic gradient methods, they do not rely on proximal mappings. Singularity: If assumption <ref> does not hold, the matrix G is singular and either G z + g = 0 has infinitely many solutions or it has no solution. In the case of many solutions, we could still get asymptotic convergence. In <cit.>, it was shown that under some assumptions on the null space of matrix G and using a simple stabilization scheme, the iterates converge to the Drazin <cit.> inverse solution of Gz+g = 0. However, it is not clear how extend our finite-sample analysis because the spectral analysis of the matrix G <cit.> in our proof assumes that the matrices A and M are nonsingular.§ RELATED WORK AND DISCUSSIONConvergent Retrace: <cit.> have recently introduced the ABQ(ζ) algorithm which uses an action-dependent bootstrapping parameter that leads to off-policy multi-step learning without importance sampling ratios. They also derived a gradient-based algorithm called AB-Trace(λ) which is related to Retrace(λ). However, the resulting updates are different from ours, as they use the two-timescale approach of <cit.> as basis fortheir derivation. In contrast, our approach uses the saddle-point formulation, avoiding the need for double sampling. Another benefit of this formulation is that it allows us to provide a bound of the convergence rate (proposition <ref>) whereas <cit.> is restricted to a more general two-timescale asymptotic result from <cit.>. The saddle-point formulation also provides a rich literature on acceleration methods which could be incorporated in our algorithms. Particularly in the batch setting, <cit.> recently introduced Stochastic Variance Reduction methods for state-value estimation combining GTD with SVRG <cit.> or SAGA <cit.>. This work could be extended easily to ours algorithms in the batch setting. Existing Convergence Rates: Our convergence rate result <ref> can apply to GTD/GTD2 algorithms. Recall that GTD/GTD2 are off-policy algorithms designed to estimate the state-value function using temporal difference TD(0) return while our algorithms compute the action-value function using Retrace and Tree Backup returns. In both GTD and GTD2, the quantities Â_k and b̂_kinvolved in their updates are the same and equal to Â_k = ϕ(s_k) (γϕ(s_k+1) - ϕ(s_k))^⊤,b̂_k = r(s_k, a_k) ϕ(s_k) while the matrix M̂_k is equal to ϕ(s_k) ϕ(s_k)^⊤ for GTD2 and to identity matrix for GTD.The table <ref> show in chronological order the convergence rates established in the literature of Reinforcement learning. GTD was first introduced in <cit.> and its variant GTD2 was introduced later in <cit.>. Both papers established the asymptotic convergence withRobbins-Monro step-sizes. Later, <cit.> provided the first sample complexityby reformulating GTD/GTD2 as an instance of mirror stochastic approximation <cit.>. <cit.> showed that with high probability, 𝐌𝐒𝐏𝐁𝐄(θ̅_k) ∈ O(1/√(k)) where θ̅_k ≜∑_k α_k θ_k/∑_k α_k. However, they studied an alternated version of GTD/GTD2 as they added a projection step into bounded convex set and Polyak-averaging of iterates. <cit.> studied also the same version as <cit.> but for the case of Markov noise case instead of the i.i.d assumptions. They prove that with high probability 𝐌𝐒𝐏𝐁𝐄(θ̅_k) ∈ O(∑_k α_k^2/∑_k α_k) when the step-size sequence satisfies ∑_k α_k = ∞, ∑_k α_k^2/∑_k α_k < ∞. The optimal rate achieved in this setup is then O(1/√(k)). Recently, <cit.> improved on the existing results by showing for the first time that [θ̅_k - θ^⋆^2] ∈ O(1/k) without projection step. However, the result still consider the Polyak-average of iterates. Moreover, the constants in their bound depend on the data distribution that are difficult to relate to the problem-specific constants, such as those present in our bound <ref>. Finally, <cit.> studied sparsily projected version of GTD/GTD2 and they showed that for step-sizes α_k = 1/k^1-c, η_k = 1/k^(2/3)(1-c)where c ∈ (0, 1), θ_k - θ^⋆∈ O(k^-1/3 + c/3) with high probability. The projection is called sparse as they project only on iterations which are powers of 2. Our work is the first to provide a finite-sample complexity analysis of GTD/GTD2 in its original setting, i.e without assumption a projection step or Polyak-averaging and with diminishing step-sizes. § EXPERIMENTAL RESULTSEvidence of instability in practice: To validate our theoretical results about instability, we implemented TB(λ), Retrace(λ) and compared them against their gradient-based counterparts GTB(λ) and GRetrace(λ) derived in this paper. The first one is the 2-states counterexample that we detailed in the third section and the second is the 7-states versions of Baird's counterexample <cit.>. Figures <ref> and <ref> show the MSBPE (averaged over 20 runs) as a function of the number of iterations. We can see that our gradient algorithms converge in these two counterexamples whereas TB(λ) and Retrace(λ) diverge. Comparison with existing methods: We also compared GTB(λ) and GRetrace(λ) with two recent state-of-the-art convergent off-policy algorithms for action-value estimation and function approximation: GQ(λ) <cit.> and AB-Trace(λ) <cit.>. As in <cit.>, we also consider a policy evaluation task in the Mountain Car domain. In order to better understand the variance inherent to each method, we designed the target policy and behavior policy in such a way that the importance sampling ratios can be as large as 30. We chose to describe state-action pairs by a 96-dimensional vector of features derived by tile coding <cit.>. We ran each algorithm over all possible combinations of step-size values (α_k, η_k) ∈ [0.001, 0.005, 0.01, 0.05, 0.1]^2 for 2000 episodes and reported their normalized mean squared errors (NMSE):𝐍𝐌𝐒𝐄(θ) = Φθ - Q^π^2_Ξ/ Q^π^2_Ξwhere Q^π is estimated by simulating the target policy and averaging the discounted cumulative rewards overs trajectories. As AB-Trace(λ) and GRetrace(λ) share both the same operator, we can evaluate them using the empirical 𝐌𝐒𝐏𝐁𝐄= 1/2 ||Âθ + b̂||^2_M̂^-1 where Â, b̂ and M̂ are Monte-Carlo estimates obtained by averaging Â_k, b̂_k and M̂_k defined in proposition <ref> over 10000 episodes. Figure <ref> shows that the best empirical 𝐌𝐒𝐏𝐁𝐄 achieved by AB-Trace(λ) and GRetrace(λ) are almost identical across value of λ. This result is consistent with thefact that they both minimize the 𝐌𝐒𝐏𝐁𝐄 objective function. However, significant differences can be observed when computing the 5th percentiles of NMSE (over all possible combination of step-size values) for different values of λ in Figure <ref>. When λ increases, the NMSE of GQ(λ) increases sharply due to increased influence of importance sampling ratios. This clearly demonstrate the variance issues of GQ(λ) in contrast with the other methods based on the Tree Backup and Retrace returns (that are not using importance ratios). For intermediate values of λ, AB-Trace(λ) performs better but its performance is matched by GRetrace(λ) and TB(λ) for small and very large values of λ. In fact, AB-Trace(λ) updates the function parameters θ as follows:θ_k+1 = θ_k - α_k ( δ_k e_k - Δ_k )where Δ_k ≜γ w_k^⊤ e_k (𝔼_πϕ(s_k+1, .) - λ∑_a κ(s_k, a) μ(as_k) ϕ(s_k, a)) is a gradient correction term. When the instability is not an issue, the correction term could be very small and the update of θ would be essentially θ_k+1∼θ_k - α_k δ_k e_k so that θ_k+1 follows the semi-gradient of the mean squared error Φθ - G^λ_k^2_Ξ.To better understand the errors of each algorithm and their robustness to step-size values, we propose the box plots shown in Figure <ref>. Each box plot shows the distribution of NMSE obtained by each algorithm for different values of λ. NMSE distributions are computed over all possible combinations of step-size values. GTB(λ) has the smallest variance as it scaled its return by the target probabilities which makes it conservative in its update even with large step-size values. GRetrace(λ) tends to more more efficient than GTB(λ) since it could benefit from full returns. The latter observation agrees with the tabular case ofTree Backup and Retrace <cit.>. Finally, we observe that AB-Trace(λ) has lower error, but at the cost of increased variance with respect to step-size values.§ CONCLUSIONOur analysis highlighted for the first time the difficulties of combining the Tree Backup and Retrace algorithms with function approximation. We addressed these issues by formulating gradient-based algorithm versions of these algorithms which minimize the mean-square projected Bellman error. Using a saddle-point formulation, we were also able to provide convergence guarantees andcharacterize the convergence rate of our algorithms GTB and GRetrace. We also developed a novel analysis method which allowed us to establish a O(1/k) convergence rate without having to use Polyak averaging or projections (which might also make implementation more difficult). Furthermore, our proof technique is general enough that we were able to apply it to the existing GTD and GTD2 algorithms. Our experiments finally suggest that the proposed GTB(λ) and GRetrace (λ) are robust to step-size selection and have less variance than both GQ(λ) <cit.> and AB-Trace(λ) <cit.>.icml2018§ PROOF OF PROPOSITION <REF>We compute[A_k] and [b_k] where expectation are over trajectories drawn by executing the behavior policy: s_k, a_k, r_k, s_k+1, … s_t, a_t, r_t, s_t+1… where s_k, a_k ∼ d,r_t =r(s_t, a_t),s_t+1∼ p(· s_t, a_t). We note that under stationarity of d, [A_k] = E[A_0] and [b_k] = [b_0]. Let θ, θ' ∈^d and let Q = Φθ and Q' = Φθ' their respective Q-functions. θ'^⊤[A_k] θ= [∑_t=0^∞ (λγ)^t(∏_i=1^t κ_i) Q'(s_0, a_0) [γ_π Q(s_t+1, .) - Q(s_t, a_t) ]^⊤]= ∑_t=0^∞ (λγ)^t _s_0:t+1a_0:t[ Q'(s_0, a_0)(∏_i=1^t κ_i)[γ_π Q(s_t+1, .) - Q(s_t, a_t) ]^⊤]= ∑_t=0^∞ (λγ)^t _s_0:ta_0:t[ Q'(s_0, a_0) (∏_i=1^t κ_i) ( γ_s_t+1[_π Q(s_t+1, .)| s_t, a_t] - Q(s_t, a_t)) ]= ∑_t=0^∞ (λγ)^t _s_0:ta_0:t[ Q'(s_0, a_0)(∏_i=1^t κ_i) ( γ∑_s' ∈∑_a' ∈ p(s'|s_t, a_t) π(a'|s') Q(s', a') - Q(s_t, a_t))]=∑_t=0^∞ (λγ)^t_s_0:ta_0:t[ Q'(s_0, a_0)(∏_i=1^t κ_i) (γ P^πQ(s_t, a_t) - Q(s_t, a_t))]=∑_t=0^∞ (λγ)^t _s_0:t-1a_0:t-1[ Q'(s_0, a_0) (∏_i=1^t-1κ_i)∑_s' ∈∑_a' ∈ p(s'|s_t-1, a_t-1) κ(a', s') μ(a'|s') (γ P^πQ(s', a') - Q(s', a'))]=∑_t=0^∞ (λγ)^t_s_0:t-1a_0:t-1[ Q'(s_0, a_0) (∏_i=1^t-1κ_i) P^κμ(γ P^π - I)Q(s_t-1, a_t-1) ]= _s_0, a_0[ Q'(s_0, a_0)∑_t=0^∞ (λγ)^t( P^κμ)^t(γ P^π - I)Q(x_0, a_0)]=_s_0, a_0[ Q'(s_0, a_0)(I - λγ P^κμ)^-1( γ P^π - I)Q(s_0, a_0) ]= ∑_s ∈∑_a ∈ξ(s, a) Q'(s, a)(I - λγ P^κμ)^-1( γ P^π - I)Q(s, a)= Q'^⊤Ξ (I - λγ P^κμ)^-1( γ P^π - I)QSo, θ'^⊤[A_k] θ = θ'^⊤Φ^⊤Ξ (I - λγ P^κμ)^-1(γ P^π - I) Φθ∀θ, θ' ∈^d, which implies that:[A_k] = Φ^⊤Ξ (I - λγ P^κμ)^-1(P^π - I) Φ θ^⊤[b_k] = [∑_t=0^∞ (λγ)^t(∏_i=1^t κ_i) r_t Q(s_0, a_0)]= ∑_t=0^∞ (λγ)^t_s_0:ta_0:t[Q(s_0, a_0)(∏_i=1^t κ_i) r(s_t, a_t)]= ∑_t=0^∞ (λγ)^t_s_0:t-1a_0:t-1[Q(s_0, a_0)(∏_i=1^t-1κ_i)∑_s' ∈∑_a' ∈ p(s'|s_t-1, a_t-1) κ(a', s') μ(a'|s') r(s', a')]= ∑_t=0^∞ (λγ)^t_s_0:t-1a_0:t-1[Q(s_0, a_0)(∏_i=1^t-1κ_i)P^κμr(s', a')] = _s_0, a_0[ Q(s_0, a_0) (I - λγ P^κμ)^-1 r(s_0, s_0)]= ∑_s ∈∑_a ∈ξ(s, a) Q(s, a)(I - λγ P^κμ)^-1 r(s,a) = Q^⊤Ξ (I - λγ P^κμ)^-1 rSo, θ^⊤[b_k]= θ^⊤Φ^⊤Ξ (I - λγ P^κμ)^-1 r ∀θ∈^d, which implies that:[b_k] =Φ^⊤Ξ (I - λγ P^κμ)^-1 r § PROOF OF PROPOSITION <REF> 𝐌𝐒𝐏𝐁𝐄(θ) = 1/2||Π^μℛ(Φθ)- Φθ ||^2_Ξ = 1/2||Π^μ(ℛ(Φθ)- Φθ)||^2_Ξ = 1/2(Π^μ(ℛ(Φθ)- Φθ))^⊤Ξ(Π^μ(ℛ(Φθ)- Φθ) )= 1/2(Φ^⊤Ξ(ℛ(Φθ)- Φθ))^⊤ (Φ^⊤ΞΦ)^-1Φ^⊤Ξ(Φ (Φ^⊤ΞΦ)^-1Φ^⊤Ξ(ℛ(Φθ)- Φθ) )= 1/2 ||Φ^⊤Ξ((Φθ)- Φθ) ||^2_M^-1 = 1/2|| Φ^⊤Ξ( (I - λγ P^μπ)^-1 (^π - λγ P^μπ) Φθ - Φθ)||^2_M^-1 = 1/2|| Φ^⊤Ξ(I - λγ P^μπ)^-1(γ𝑃^π - I) Φθ + Φ^⊤Ξ(I - λγ P^μπ)^-1 r||^2_M^-1 =1/2 || A θ + b||^2_M^-1 § PROOF OF PROPOSITION <REF>Let's show that 𝔼[Â_k] = A. Let's Δ_t denotes [γ𝔼_πϕ(s_t+1, .)^⊤- ϕ(s_t, a_t)^⊤]A= [ ∑_t=k^∞ (λγ)^t-k(∏_i=k+1^t κ_i) ϕ(s_k, a_k) Δ_t ]=[ ϕ(s_k, a_k) Δ_k +∑_t=k+1^∞ (λγ)^t-k(∏_i=k+1^t κ_i) ϕ(s_k, a_k) Δ_t ]= [ ϕ(s_k, a_k) Δ_k + ∑_t=k^∞ (λγ)^t-k+1(∏_i=k+1^t+1κ_i) ϕ(s_k, a_k) Δ_t+1] = [ ϕ(s_k, a_k) Δ_k + λγκ(s_k+1, a_k+1)ϕ(s_k, a_k) Δ_k+1 +∑_t=k+1^∞ (λγ)^t-k+1(∏_i=k+1^t+1κ_i) ϕ(s_k, a_k) Δ_t+1] (⋆)=[ ϕ(s_k, a_k) Δ_k + λγκ(s_k, a_k)ϕ(s_k-1, a_k-1) Δ_k +∑_t=k+1^∞ (λγ)^t-k+1(∏_i=k+1^t+1κ_i) ϕ(s_k, a_k) Δ_t+1]= [ Δ_k (ϕ(s_k, a_k) + λγκ(s_k, a_k)ϕ(s_k-1, a_k-1)+ (λγ)^2 κ(s_k, a_k) κ(s_k-1, a_k-1)ϕ(s_k-2, a_k-2) + ...) ] =[Δ_k (∑_i=0^k(λγ)^k-i(∏_j=i+1^kκ_j) ϕ(x_i,a_i)) ]=[Δ_k e_k]= 𝔼[Â_k]we have used in the line (⋆) the fact that[κ(s_k+1, a_k+1)ϕ(s_k a_k) Δ_k+1] = [κ(s_k, a_k)ϕ(s_k-1 a_k-1) Δ_k] thanks to the stationarity of the distribution d.we have also denote by e_k the following vector:e_k = ∑_i=0^k(λγ)^k-i(∏_j=i+1^kκ_j) ϕ(s_i,a_i) = λγκ_k ( ∑_i=0^k-1(λγ)^k-1-i(∏_j=i+1^k-1κ_j) ϕ(s_i,a_i)) + ϕ(s_k, a_k)=λγκ_k e_k-1 + ϕ(s_k, a_k)Vector e_k corresponds to the eligibility traces defined in the proposition. Similarly, we could show that 𝔼_μ[b̂_k] = b.§ TRUE ON-LINE EQUIVALENCEIn <cit.>, the authors derived a true on-line update for GTD(λ) that empirically performed better than GTD(λ) with eligibility traces. Based on this work, we derive true on-line updates for our algorithm. The gradient off-policy algorithm was derived by turning the expected forward view into an expected backward view which can be sampled. In order to derive a true on-line update, we sample instead the forward view and then we turn the sampled forward view to an exact backward view using Theorem 1 in <cit.>. If k denotes the time horizon, we consider the sampled truncated interim forward return: ∀ t < k,Y_t^k = ∑_i=t^k-1 (λγ)^i-t(∏_j=t+1^i κ_j) δ_iwhere δ_i = r_i + θ_t^⊤_πϕ(s_t+1, ·) - θ_t^⊤ϕ(s_t, a_t), which gives us the sampled forward update of ω:∀ k < t, ω_t+1^k = ω_t^k + α_t(Y_t^k - ϕ(x_t, a_t)^⊤ω_t^k)ϕ(x_t, a_t)For any k, the parameter ω_k^k defined by the forward view (<ref>) is equal to ω_k defined by the following backward view:e^ω_-1 = 0, ∀ k ≥ 0 e^ω_k = λγκ_k e^ω_k-1 + α_k (1- λγκ_k ϕ(s_k, a_k)^⊤ e^ω_k-1 ) ϕ(s_k, a_k)ω_k+1= ω_k + δ_k e^ω_k - α_t ϕ(s_k, a_k)^⊤ω_k ϕ(s_k, a_k)The return's temporal differenceY_t^k+1 - Y_t^k are related through:∀ t<k,Y_t^k+1 -Y_t^k=∑_i=t^k (λγ)^i-t (∏_j=t+1^i κ_j) δ_i - ∑_i=t^k-1 (λγ)^i-t (∏_j=t+1^i κ_j) δ_i= (λγ)^k-t(∏_j=t+1^k κ_j) δ_k= λγκ_k+1((λγ)^k-(t+1)(∏_j=t+2^k κ_j) δ_k )=λγκ_k+1( Y_t+1^k+1 -Y_t+1^k )We could then apply Theorem 1 of <cit.> that give us the following backward view:e_0= α_0 ϕ(x_0, a_0) e_t = λγκ_t e_t-1 + α_t (1- λγκ_k ϕ(s_t, a_t)^⊤ e_t-1 ) ϕ(s_t, a_t) ∀ t > 0ω_t+1= ω_t + ( Y_t^t+1 -Y_t^t) e_t + α_t (Y_t^t - ϕ(s_t, a_t)^⊤ω_t)ϕ(s_t, a_t)(⋆)=ω_t + δ_t e_t - α_t ϕ(s_t, a_t)^⊤ω_t ϕ(s_t, a_t)We used in the line (⋆) that Y_t^t+1 = δ_t and Y_t^t = 0 The resulting detailed procedure is provided in Algorithm <ref>.Note that when λ is equal to zero, the Algorithm 1 and 2 both reduce to the same update:ω_k+1= ω_k + α_k ( δ_k - ϕ(s_k, a_k)^⊤ω_k) ϕ(s_k, a_k)θ_k+1= θ_k - α_k ϕ(s_k, a_k)^⊤ w_k(γ𝔼_π[ϕ(s_k+1, .)] - ϕ(s_k, a_k)]) § CONVERGENCE RATE ANALYSISLet's recall the key quantities defined in the main article:ρ≜λ_max(A^⊤M^-1A), δ≜λ_min(A^⊤ M^-1A ),L_G ≜[ Ĝ_k^⊤Ĝ_kℱ_k ]We will make use of spectral properties of the matrix G provided in the appendix A of<cit.>. it was shown that if we set β = 8 ρ/λ_min(M), the matrix G is diagonalizable with all its eigenvalues real and positive. It is a straightforward application of result from <cit.> Moreover, it was proved that G can be written as:G= Q Λ Q^-1 where Λ is a diagonal matrix whose diagonal entries are the eigenvalues of G and Q consists of it eigenvectors as columns such that the condition number of Q is upper bounded by the one of M as follows:c(Q)^2 ≤ 8 c(M)Finally, the paper showed upper and lower bounds for the eigenvalues of G:λ_max(G)≤ 9 c(M) ρ λ_min(G)≥8/9δLet's recall our updates:z_k+1 = z_k - α_k (Ĝ_k z_k - ĝ_k)By subtracting z^⋆ from both sides on the later equation and using the optimality condition G z^⋆+ g = 0:Δ_k+1 = Δ_k - α_k G Δ_k + α_k [ G z_k - g - (Ĝ_k z_k - ĝ_k)] whereΔ_k ≜ z_k - z^⋆By multiplying both sides by Q^-1 and using the fact that Q^-1G = Λ Q^-1:Q^-1Δ_k+1= Q^-1Δ_k - α_k Q^-1 G Δ_k + α_k Q^-1[ G z_k - g - (Ĝ_k z_k - ĝ_k)]= (I - α_kΛ) Q^-1Δ_k + α_k Q^-1[ G z_k - g - (Ĝ_k z_k - ĝ_k) ] [ Q^-1Δ_k+1^2 ℱ_k-1]=[ (I - α_k) Q^-1Δ_k + α_k Q^-1[ G z_k-g - (Ĝ_k z_k - ĝ_k) ] ^2 ℱ_k-1]= [(I - α_kΛ) Q^-1Δ_k ^2 ℱ_k-1] =1+ 2[⟨(I - α_k) Q^-1Δ_k,α_k Q^-1[ G z_k - g - (Ĝ_k z_k - ĝ_k) ] ⟩ℱ_k-1] =1 +α_k^2[Q^-1[ G z_k - g - (Ĝ_k z_k - ĝ_k) ] ^2 ℱ_k-1]= (I - α_kΛ) Q^-1Δ_k ^2+ α_k^2[Q^-1[ G z_k - g - (Ĝ_k z_k - ĝ_k) ]^2 ℱ_k-1]≤I - α_kΛ^2Q^-1Δ_k ^2 + α_k^2 [Q^-1(Ĝ_k z_k - ĝ_k) ^2 ℱ_k-1]=I - α_kΛ^2Q^-1Δ_k ^2 + α_k^2 [Q^-1( Ĝ_kΔ_k + Ĝ_k z^⋆ - ĝ_k) ^2 ℱ_k-1] ≤I - α_kΛ^2Q^-1Δ_k ^2 + 2 α_k^2 [Q^-1Ĝ_kΔ_k ^2 ℱ_k-1]+ 2 α_k^2 [ Q^-1(Ĝ_k z^⋆ + ĝ_k) ^2 ℱ_k-1]we use in the third line the fact that[ Ĝ_k ℱ_k-1] = G and [ ĝ_k-1ℱ_k-1] = g.I - α_kΛ^2 = max{ |1 - α_k λ_min(G)|^2, | 1 - α_k λ_max(G) |^2 }≤ 1 - 2 α_k λ_min + α_k^2 λ_max^2 ≤ 1 - 2 α_k 8/9δ + α_k^2 9^2 c(M)^2 ρ^2 ≤ 1 -2 α_k δ' + α_k^2 9^2 c(M)^2 ρ^2whereδ' ≜8/9δ [Q^-1Ĝ_kΔ_k ^2 ℱ_k-1] ≤ Q^-1^2[ Ĝ_kΔ_k ^2 ℱ_k-1]=Q^-1^2 [ Δ_k^⊤Ĝ_k^⊤Ĝ_kΔ_kℱ_k-1] = Q^-1^2 Δ_k^⊤[ Ĝ_k^⊤Ĝ_kℱ_k-1] Δ_k≤ Q^-1^2 [ Ĝ_k^⊤Ĝ_kℱ_k ] ^2 Δ_k^⊤Δ_k ≤ Q^-1^2L_G Δ_k^2=Q^-1^2L_G Q Q^-1Δ_k^2 ≤ Q^-1^2 Q^2L_G Q^-1Δ_k^2 ≤ c(Q)^2 L_G Q^-1Δ_k^2So, we have: [Q^-1Δ_k+1^2]≤ (1 -2 α_k δ' + α_k^2 9^2 c(M)^2 ρ^2 + 16 α_k^2 c(M) L_G)[ Q^-1Δ_k^2 ] + 2 α_k^2 Q^-1^2 [ Ĝ_k z^⋆ - ĝ_k) ^2 ] By selecting α_k = 2 δ'/δ'^2 (k+2) + 2× 9^2 c(M)^2 ρ^2 + 32 c(M) L_G= 2 δ'/δ'^2 (k+2) + ζ with ζ =2 × 9^2 c(M)^2 ρ^2 + 32 c(M) L_G, we get:[Q^-1Δ_k+1^2]≤ (1 - δ' α_k) [ Q^-1Δ_k^2 ] + 2 α_k^2 Q^-1^2 [ Ĝ_k z^⋆ - ĝ_k ^2 ]= δ'^2 k + ζ/δ'^2(k+2) + ζ[Q^-1Δ_k^2] + 8 δ'^2/(δ'^2(k+2) + ζ)^2 Q^-1^2 [ Ĝ_k z^⋆ + ĝ_k) ^2 ] ≤( ∏_i = 0^k δ'^2 i + ζ/δ'^2(i+2) + ζ) [Q^-1Δ_0^2] =1 + 8 δ'^2 ∑_i = 0^k ( ∏_j = i^k δ'^2 j + ζ/δ'^2(j+2) + ζ) 1/(δ'^2(i+2) + ζ)^2 Q^-1^2 [ Ĝ_i z^⋆ + ĝ_i) ^2 ]= ζ(δ'^2 + ζ) /(δ'^2(k+1) + ζ) ( (δ'^2(k+2) + ζ)[Q^-1Δ_0^2]=1 + 8 δ'^2 ∑_i = 0^k (δ'^2(i+1) + ζ)(δ'^2i + ζ) /(δ'^2(k+1) + ζ) ( (δ'^2(k+2) + ζ)1/(δ'^2(i+2) + ζ)^2 Q^-1^2 [ Ĝ_i z^⋆ - ĝ_i^2 ]≤ζ(δ'^2 + ζ) /(δ'^2(k+1) + ζ) ( δ'^2(k+2) + ζ)[Q^-1Δ_0^2]=1 + 8 δ'^2 ∑_i = 0^k 1/(δ'^2(k+1) + ζ) ( δ'^2(k+2) + ζ) Q^-1^2 [ Ĝ_i z^⋆ - ĝ_i ^2 ]≤(δ' + ζ)^2/(δ'^2 (k+1) + ζ)^2 [Q^-1Δ_0^2] =1 + 8 δ'^2 (k+1) /(δ'^2(k+1) + ζ) ( δ'^2(k+2) + ζ) Q^-1^2 sup_i = 0 … k[ Ĝ_i z^⋆ + ĝ_i) ^2 ] ≤(δ' + ζ)^2/(δ'^2 (k+1) + ζ)^2 [Q^-1Δ_0^2]+ 8/(δ'^2(k+1) + ζ)Q^-1^2 sup_i = 0 … k[ Ĝ_i z^⋆ - ĝ_i ^2 ] ≤(δ' + ζ)^2Q^-1^2/(δ'^2 (k+1) + ζ)^2 [ Δ_0^2]+ 8 σ^2Q^-1^2/(δ'^2(k+1) + ζ)(1 +z^⋆^2) Moreover, we have [ Δ_k+1^2] = [Q Q^-1Δ_k+1^2] ≤ Q^2 [ Q^-1Δ_k+1^2]. Then, we get:[ Δ_k+1^2] ≤(δ' + ζ)^2 c(Q)^2/(δ'^2 (k+1) + ζ)^2 [ Δ_0^2]+ 8 σ^2 c(Q)^2/(δ'^2(k+1) + ζ)(1 +z^⋆^2) ≤8 (δ' + ζ)^2 c(M)/(δ'^2 (k+1) + ζ)^2 [ Δ_0^2]+ 8^2 σ^2 c(M)/(δ'^2(k+1) + ζ)(1 +z^⋆^2)= 8 9^2 (8δ + 9ζ)^2 c(M)/(8^2δ^2 (k+1) + 9^2ζ)^2 [ Δ_0^2]+ 9^2 × 8^2 σ^2 c(M)/(8^2δ^2(k+1) + 9^2ζ)(1 +z^⋆^2)= 9^2 × 8 c(M) {(8δ + 9ζ)^2 [ Δ_0^2] /(8^2δ^2 (k+1) + 9^2ζ)^2 + 8 σ^2 (1 +z^⋆^2)/(8^2δ^2(k+1) + 9^2ζ) }The overall convergence rate is then equal to O(1/k). | http://arxiv.org/abs/1705.09322v4 | {
"authors": [
"Ahmed Touati",
"Pierre-Luc Bacon",
"Doina Precup",
"Pascal Vincent"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170525183755",
"title": "Convergent Tree Backup and Retrace with Function Approximation"
} |
APS/[email protected] San Diego, CA 92121, USANon-relativistic quantum mechanics is reformulated here based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. This idea, combining with the emphasis that measurement of a quantum system is a bidirectional interaction process, leads to a new framework to calculate the probability of an outcome when measuring a quantum system. In this framework, the most basic variable is the relational probability amplitude. Probability is calculated as summation of weights from the potential alternative measurement configurations. The properties of quantum systems, such as superposition and entanglement, are manifested through the rules of counting the alternatives. Wave function and reduced density matrix are derived from the relational probability amplitude matrix. They are found to be secondary mathematical tools that equivalently describe a quantum system without explicitly calling out the measuring system. Schrödinger Equation is obtained when there is no entanglement in the relational probability amplitude matrix. Feynman Path Integral is used to calculate the relational probability amplitude, and is further generalized to formulate the reduced density matrix. In essence, quantum mechanics is reformulated as a theory that describes physical systems in terms of relational properties.Keywords Relational Quantum Mechanics, Measurement Probability, Summation of Alternatives,Entanglement, Quantum Reference FrameQuantum Mechanics from Relational PropertiesPart I: Basic Formulation Jianhao M. Yang December 30, 2023 ======================================================================= § INTRODUCTION Although quantum mechanics is one of the most successful physical theories and has been experimentally confirmed extensively, there are many fundamental questions still left unanswered. For instance, the origin of probability in quantum mechanics is not clearly understood. It is still a curiosity why the probability is calculated as the absolute square of a complex number. The meaning of wave function, especially the interpretation of wave function collapse in a measurement, has been always a debated topic. These questions were not fully addressed by the traditional Copenhagen Interpretation <cit.>. Over the years in the modern history of quantum physics, many more theories and interpretations have been developed. These include the many-worlds interpretation <cit.>, consistent histories <cit.>, decoherent theory <cit.>, relational interpretations <cit.>, quantum Bayesian theory <cit.>, and many others. Along the development of these interpretations, one noticeable idea is the realization that a quantum state is relative in nature. That is, an observer independent quantum state is not necessarily the basic description of a quantum system. In the early days of quantum mechanics, Bohr had already emphasized that the description of a quantum system depends on the measuring apparatus <cit.>. Ref. <cit.> recognized that a quantum state of a subsystem is only meaningful relative to a given state of the rest of the system. Similarly, in developing the theory of decoherence induced by environment, Ref. <cit.> concluded that correlation information between two quantum systems is more basic than the properties of the quantum systems themselves. Relational Quantum Mechanics (RQM) has pursued this idea to the furthest extend. RQM is inspired by the basic principle from Einstein's Special Relativity. In the context of RQM, a quantum system should be described relative to another system, there is no absolute state for a quantum system. Specifically, the main idea of RQM is stated as following,Quantum mechanics is a theory about the physical description of physical system relative to other systems, and this is a complete description of the world <cit.>.This statement appears radical but reflects the fact that quantum mechanics was originally developed as a theory to describe the experimental observations of a quantum system in a measurement. When we state that the observing system records the measurement results of a variable of the observed system, it means that a correlation between the observed system and the observing system is established through physical interaction. By reading the pointer variable in the observing system, one can infer the value of variable in the observed system. In this sense, quantum theory does not describe the independent properties of a quantum system. Instead, it describes the relation among quantum systems, and how correlation is established through physical interaction during measurement. The reality of a quantum system is only meaningful in the context of measurement by another system.The idea of RQM is thought provoking. It essentially implies two aspects of relativity. The first aspect of RQM is to insist that a quantum system must be described relative to a reference system. The reference system is arbitrarily selected. It can be an apparatus in a measurement setup, or another system in the environment. A quantum system can be described differently relative to different reference systems. The reference system itself is also a quantum system, which is called a quantum reference frame (QRF). There are extensive research activities on QRF, particularly how to ensure consistent descriptions when switching QRFs <cit.>. Noticeably, Ref. <cit.> completely abandon any external reference system and the concept of absolute state. Physical description is constructed using relational variables from the very beginning within the framework of traditional quantum mechanics. In addition, all reference systems are treated as quantum systems instead of some kinds of abstract entities. Treating a reference frame as a classical system, such as how the relativity theory does, should be considered as an approximation of a more fundamental theory that is based on QRF.The second aspect of RQM is more fundamental. Since the relational properties between two quantum systems are considered more basic than the independent properties of one system, the relational properties, instead of the independent properties, of quantum systems should be considered as a starting point for constructing the formulation of quantum mechanics itself. Questions associated with this aspect of RQM include how to quantify the relational properties between two quantum systems, and how to reconstruct a quantum mechanics theory from relational properties. Note that the relational properties themselves are relative to a QRF. Different observers can ascribe a quantum system with different sets of relational properties relative to their choices of QRFs. It is this second aspect of RQM that inspires our works here. Traditional quantum mechanics always starts with an observer-independent quantum state. It is of interest to see if a quantum theory constructed based on relational properties can address some of the unanswered fundamental questions mentioned earlier. Such reconstruction program was initiated in Ref. <cit.> and had some successes, for example, in deriving the Schrödinger Equation. This reconstruction is based on quantum logic approach. Alternative reconstruction that follows the RQM principle but based on information theory is also developed <cit.>. These reconstructions appear rather abstract, not closely connect to the physical process of a quantum measurement. We believe that the relational properties should be identified in a measurement event given the idea that the reality of a quantum system is only meaningful in the context of measurement by another system.The goal of this paper is to continue the program of reconstructing the formulation of quantum mechanics with the starting point that the relational properties are the most basic elements. What is novel in our approach is a new framework for calculating the probability of an outcome when measuring a quantum system. Such a framework is fundamental in deriving basic laws of quantum mechanics, so we briefly describe it here. In searching for the appropriate relational properties as the starting elements for the reconstruction, we recognize that a physical measurement is a probe-response interaction process between the measured system and the measuring apparatus. This important aspect of measurement process seems being overlooked in other reconstruction efforts. Our framework for calculating the probability, on the other hand, explicitly models this bidirectional process. As such, the probability can be derived from product of two quantities and each quantity is associated with a unidirectional process. We call such quantity relational probability amplitude. When two quantum systems interact, there are many alternative configurations for such two-way process. Each alternative is assigned with a weight that is a product of two relational probability amplitudes associated with the configuration. The probability of a measurement outcome is then postulated to be proportional to the summation of such weights from all the applicable configurations. Thus, the task of calculating the probability is reduced to counting the applicable alternatives. The properties of the measured system are manifested through the rules to count the alternatives. Another aspect of novelty of this framework is the introduction of the concept of entanglement to the relational properties. When the quantum system is entangled with the apparatus, the rule of counting the alternatives is adjusted accordingly due to the availability of inference information. Furthermore, the entanglement measure quantifies the difference between time evolution and quantum measurement. Lastly, we show that such framework to calculate probability amplitude can be explicitly implemented using Feynman Path Integral <cit.>.The impacts of this framework are fundamental as it is the basis for deriving the formulations that are mathematically equivalent to the laws in traditional quantum mechanics. The formulation for calculating the probability of finding the system in an eigenstate is equivalent to Born's rule, but with a new insight: the fact it is an absolute square of a complex number is a consequence that a quantum measurement is a bidirectional process. Wave function is found to be a mathematical tool representing the summation of relational probability amplitude. Thus, the notion of wave function collapse during measurement is just a consequence of changes of relational properties. Schrödinger equation can be derived when the entanglement measure between the observed quantum system and the observing system is zero and unchanged. On the other hand, when there is change in the entanglement measure, quantum measurement theory is obtained. Although the formulation presented here is mathematically equivalent to the traditional quantum mechanics, the theory presented here provides new understanding on the origin of quantum probability. It shows that an essence of quantum mechanics is a new set of rules to calculate the measurement probability from an interaction process. The most important outcome of this paper is that quantum mechanics can be constructed with the relational properties among quantum systems as the most fundamental building blocks.The paper is organized as following. We first clarify the definitions of key terminologies in Section <ref>. Section <ref> gives the main results of the paper. The postulates and frameworks to calculate quantum probability is provided in Section <ref> and <ref>. In Section <ref> to <ref> formulation for time evolution of a quantum system is developed, and the conditions when Schrödinger equation can be recovered are analyzed. In Section <ref>, we provide a comparison between this works and the original RQM theory, discuss the limitations, and summarize the conclusions. An explicit calculation of the relational probability amplitude using Feynman Path Integral formulation is presented in Section <ref>.§ DEFINITIONS OF KEY TERMINOLOGIES §.§ Quantum System, Apparatus, and Observer To avoid potential confusion, it is useful to define several key terms before proceeding. A Quantum System, denoted by symbol S, is an object under study and follows the postulates that will be introduced in next section. An Apparatus, denoted as A, can refer to the measuring devices, the environment that S is interacting with, or the system from which S is created. It is another quantum system that can interact with S, can acquire or encode information from S. We will strictly follow the assumptions that all systems are quantum systems, including any apparatus. Depending on the selection of observer, the boundary between a system and an apparatus can change. For example, in a measurement setup, the measuring system is an apparatus A, the measured system is S. However, the composite system S+A as a whole can be considered a single system, relative to another apparatus A'. In an ideal measurement to measure an observable of S, the apparatus is designed in such a way that at the end of the measurement, the pointer state of A has a distinguishable, one to one correlation with the eigenvalue of the observable of S.The definition of Observer is associated with an apparatus. An observer, denoted as O, is an intelligent entity who can operate and read the pointer variable of the apparatus. This can be a human being, or an artificial intelligent computer. The distinction between an observer and an apparatus is that an apparatus directly interacts with S, while an observer does not. Whether or not this observer is a quantum system is irrelevant in our formulation. However, there is a restriction that is imposed by the principle of locality. An observer is defined to be physically local to the apparatus he associates with. This prevents the situation that O can instantaneously read the pointer variable of the apparatus that is space-like separated from O. Receiving the information from A at a speed faster than the speed of light is prohibited. This locality requirement is crucial in resolving the EPR argument <cit.>. An observer cannot be associated with two or more apparatuses in the same time if these apparatuses are space-like separated. A Quantum Reference Frame (QRF) is a quantum system where all the descriptions of the relational properties between S and A is referred to. There can be multiple QRFs. How the descriptions are transformed when switching QRFs is not in the scope of this study. But we expect the theories developed in Ref. <cit.> can be applicable here. In this paper, we only consider the description relative to one QRF, denoted as F. It is also possible to choose A as the reference frame. In that case, F and A are the same quantum system in a measurement <cit.>. Fig. 1 shows a schematic illustration of the entities in the relational formulation. §.§ Quantum Measurement Given the hypothesis that a quantum system should be described relative to another system, the first question to ask is which another system the description is relative to? A quantum system, at any given time, is either being measured by an apparatus, or interacting with its environment, or is in an isolated environment. It is intuitive to select a reference system that has been previously interacting with the quantum system. A brief review of the traditional quantum measurement theory is helpful since it brings important insights on the meaning of a quantum state. In the traditional quantum measurement theory proposed by von Neumann<cit.>, both the quantum system and the measuring apparatus follow the same quantum mechanics laws. Von Neumann further distinguished two separated measurement stages, Process 1 and Process 2. Mathematically, an ideal measurement process is expressed as|Ψ⟩_SA= |ψ_S⟩|a_0⟩ = ∑_i c_i|s_i⟩|a_0⟩⟶∑_i c_i|s_i⟩|a_i⟩⟶ |s_n⟩|a_n⟩Initially, both S and A are in a product state described by |Ψ⟩_SA. In Process 2, referring to the first arrow in Eq.(<ref>), the quantum system S and the apparatus A interact. However, as a combined system they are isolated from interaction with any other system. Therefore, the dynamics of the total system is determined by the Schrödinger Equation. Process 2 establishes a one to one correlation between the eigenstate of observable of S and the pointer state of A. After Process 2, there are many possible outcomes to choose from. In the next step which is called Process 1, referring to the second arrow in Eq.(<ref>), one of these possible outcomes (labeled with eigenvalue n) emerges out from the rest[Traditional quantum mechanics does not provide a theoretical description of Process 1. In the Copenhagen Interpretation, this is considered as the “collapse" of the wave function into an eigenstate of the measured observable. The nature of this wave function collapse has been debated over many decades.]. An observer knows the outcome of the measurement via the pointer variable of the apparatus. Both systems encode information each other, allowing an observer to infer measurement results of S by reading pointer variable of A. This observation is also applicable in the case that a quantum system is prepared in a particular state. The term preparation refers to the situation that S is measured by an apparatus, or is prepared with a particular lab setup (for instance, a spin half particle passes through a Stern-Gerlach Apparatus), such that its state is explicitly known to an observer. The measuring system, and the environment that S interacts with, are collectively termed as the apparatus A. Because of the correlation established between S and A during the state preparation process, it is natural to describe the state of S in reference to A. After the state preparation, suppose the interaction Hamiltonian between S and A vanishes, S starts its unitary time evolution. During time evolution, S can still be described in reference to the original apparatus A. The dynamics is deterministically governed by the Schrödinger Equation, but there is no change of correlation between them because there is no interaction. When the next measurement occurs, or when the unitary time evolution stops because S starts to interact with another apparatus A', the relational properties are updated. As a result, the quantum state of S is updated in reference to A'. After the interaction finishes, S enters unitary time evolution again. This process can be repeated continuously. The key insight of quantum measurement is that it is a question-and-answer bidirectional process. The measuring system interacts (or, disturbs) the measured system. The interaction in turn alters the state of the measuring system. As a result, a correlation is established, allowing the measurement result for S to be inferred from the pointer variable of A.§.§ Quantum StateThe notion of information in Ref. <cit.>, is closely related to concept of correlation. Information exchange between the observed system and the observing apparatus implies change of correlation between the two systems. Correlation is relational and observer-dependent. There are many ways to mathematically define correlation, one of them is introduced in Section <ref>. However, in this paper, we use the notion of information in a more general sense. It can be understood as data that represents values attributed to parameters or properties, or knowledge that describes understanding of physical systems or abstract concepts. Correlation is one type of information.A Quantum State of S describes the complete information an observer O can know about S. From the examination on the measurement process and the interaction history of a quantum system, we consider a quantum state encodes the information relative to the measuring system or the environment that the system previously interacted with. In this sense, the quantum state of S is described relative to A.The idea that a quantum state encodes information from previous interactions is also proposed in Ref <cit.>. The information encoded in the quantum state is the complete knowledge an observer can say about S, as it determines the possible outcomes of the next measurement. When the next measurement with another apparatus A' is completed, the description of quantum state is updated to be relative to A'. In traditional quantum mechanics, the quantum state is described through an observer-independent variable, the wave function |ψ⟩. Its meaning is assigned through the Born's rule <cit.>, which states that the probability to find S in an eigenvector |s_i⟩ is given by p_i= |⟨ s_i|ψ⟩|^2, and ∑ p_i=1. However, in this paper we consider observer-dependent relational properties more basic. By developing a framework to calculate the quantum probability, the meaning of |ψ⟩ is naturally emerged as a secondary concept, as shown in Section <ref>. With the clarifications of the key terminologies, we can proceed to introduce the postulates and start the reformulation of quantum mechanics.§ RESULTS§.§ Probability in a Quantum Measurement Suppose there is no quantum mechanics formulation yet and the goal is to construct a quantum theory that describes a quantum system S in the context of measurement by an apparatus A. We start the reconstruction process by asking a basic question - what are the possible outcomes if one performs a measurement on S using apparatus A? More specifically, if one measures a variable q of S by referring a pointer variable q' of A, what are the expected outcomes?To begin with, we assume a fixed QRF, F, is chosen to describe the quantum measurement event. From experimental observations, the measurement yields multiple possible outcomes randomly. Each potential outcome is obtained with a certain probability. We call each measurement with a distinct outcome a quantum event. Denote these alternatives events with a set of kets {|s_i⟩} for S, where (i=0,… ,N-1), and a set of kets {|a_j⟩} for A, where (j=0,… ,M-1). A potential measurement outcome is represented by a pair of kets (|s_i⟩, |a_j⟩). The ket |s_i⟩ is introduced not to represent a quantum state of S, instead as an abstract notation for a quantum event. They reflect the experimental observations that there can be many distinct measurement outcomes when a variable of S, q, is measured. |s_i⟩ is associated with one of the outcomes with a certain probability, with q_i as the measured value for variable q. Similarly, a ket |a_j⟩ represents a measurement outcome when the pointer variable q' equals q'_j. Here finite number of measurement outcomes is assumed. It is always possible to extend the notation to infinite number of events.With such a representation, the next step is to develop a mathematically framework to calculate the probabilities of possible events. This prediction is carried out before a measurement is actually performed. For instance, what is the probability of a joint event |s_i⟩ and |a_j⟩, denoted as p_ij? It is subtle to assign a probability of an outcome from a quantum measurement process. As mentioned earlier, a measurement is an inferring process that depends on the physical interaction between S and A. The interaction process consists A probing (or, disturbing) S, and S in the same time altering A. In other words, it is a bidirectional process. We denote this as A⇌ S. Accordingly, p_ij is called an interactional probability. This process is true for measurement in either classical or quantum mechanics. The difference is that in classical mechanics, the measurement can be setup such that there is only one measurement outcome deterministically. This means there is a one-to-one correlation between the macroscopic state of measured object and the pointer variable in the measuring device. The probability of this correlation always equals to one. On the other hand, in quantum mechanics, measurement of a variable q of the quantum system S yields multiple possible results. To calculate the probability of the joint event |s_i⟩, and |a_j⟩, the process A⇌ S at the macroscopic level should be replaced by |a_j⟩⇌ |s_i⟩ at the quantum level. This is a two-way process, or, a questioning and answering pair in terms of quantum logic approach <cit.>.Although the bidirectional interaction process is well known, the following realization is not fully appreciated in the research literature.Because a quantum measurement is a bidirectional process, the calculation of the probability of a measurement outcome must faithfully model such bidirectional process.The bidirectional process does not necessarily imply two sequential steps. Instead, the probing and responding processes are understood as two aspects of a complete process in a measurement event. We can use a classical probability problem to analogize this. Suppose tossing a special coin gets a face up with probability of p. Let us consider a measurement process that requires tossing two such coins in the same time, and the measurement is successful if one coin facing up and one coin facing down. We ask what is the probability of a successful measurement event. The answer is to multiple two probability quantities together, p(1-p). In the similar manner, given the bidirectional process in a quantum measurement event, the observable measurement probability should be calculated as a product of two quantities of weights. One weight quantity is associated with the probing process from A→ S, denoted as Q^A→ S (|a_j⟩∩ |s_i⟩), and the other is associated with the responding process from S→ A, denoted as R^S→ A (|s_i⟩∩ |a_j⟩), so thatp_ij∝ Q^A→ S (|a_j⟩∩ |s_i⟩)R^S→ A (|s_i⟩∩ |a_j⟩)Here we assume process of each direction is independent from each other. The requirements for the interactional probability p_ij can be summarized as following: * p_ij should be a product of two numbers that are associated with a bidirectional process.* p_ij should be symmetric with respect to either S or A. What this means is that the probability is the same for both processes |a_j⟩→ |s_i⟩→ |a_j⟩ that is viewed from A and |s_i⟩→ |a_j⟩→ |s_i⟩ that is viewed from S.* p_ij should be a non-negative real number.To satisfy requirement 2, we rewrite these two quantities as matrix elements, i.e., Q^A→ S (|a_j⟩∩ |s_i⟩) = Q^AS_ji, and R^S→ A (|s_i⟩∩ |a_j⟩) = R^SA_ij. Eq.(<ref>) becomes p_ij∝ Q_ji^ASR_ij^SA.The probability for the process |s_i⟩→ |a_j⟩→ |s_i⟩ is p_ij∝ R_ij^SAQ_ji^AS, the same as Eq.(<ref>). Thus, requirement 2 is satisfied.Now let's consider the third requirement for p_ij that it should be a non-negative real number. We should assume the weakest possible restrictions to the variables Q_ji^AS and R_ij^SA. The three requirements for p_ij are not necessarily applicable to Q_ji^AS and R_ij^SA. First, a unidirectional process |a_j⟩→ |s_i⟩ does not constitute a complete physical measurement process. We should not consider these variables themselves as probability quantities in the classical sense. This is, Q_ji^AS and R_ij^SA are not necessarily non-negative real number. They can be complex numbers. In other words, a probabilistic quantity is a non-negative real number only when it is associated with an actual physical measurement. Such a requirement does not need to be true for probabilistic quantity associated with an incomplete, one-way process. A similar argument can be found in Ref. <cit.>. We summarize this crucial but subtle point as following:Measurement probability of an observable event must be a non-negative real number. However, the requirement of being a non-negative real number is not applicable to non-measurable quantities for sub-processes that constitute a quantum measurement.There is also a temptation to express the relational variable as Q^A→ S (|a_j⟩| |s_i⟩), making it looks like a conditional probability quantity. However, we choose the expression Q^A→ S (|a_j⟩∩ |s_i⟩) because it better represents a relational quantity for a joint event. Second, there is no reason to assume R^SA_ij = Q^AS_ji. The direction from S to A is significant here and explicitly called out in the superscript[In this notation, index i is reserved for S while index j is reserved for A.]. To see this, let us analyze the factors that the weight Q^A→ S (|a_j⟩∩ |s_i⟩) depends on. Intuitively, this quantity depends on three factors: * Likelihood of finding system S in state |s_i⟩ without interaction;* Likelihood of finding system A in state |a_j⟩ without interaction;* A factor that alters the above two likelihoods due to the passing of physical elements such as energy and momentum from A→ S in the probing process.Similarly, the other weight quantity R^S→ A (|s_i⟩∩ |a_j⟩) depends on the similar first two factors, and the third factor that is due to the passing of physical elements from S→ A in the responding process.The three dependent factors for Q_ji^AS and R_ij^SA are related to each other respectively. The likelihood of finding system S in state |s_i⟩ and system A in state |a_j⟩ without interaction are the same. The third factor is triggered by passing physical elements during interaction. There are conservation laws such as energy conservation and momentum conservation during interaction. Conceivably, the third factors for Q_ji^AS and R_ij^SA must be equal in absolute value, but may be different in phase. With all these considerations, it is reasonable to assume |Q_ji^AS| = |R_ij^SA|. We chooseQ_ji^AS = (R_ij^SA)^*so that p_ij=Q_ji^ASR_ij^SA is a non-negative real number. Written in a different format, Q_ji^AS = (R^SA)^†_ji. This means Q^AS = (R^SA)^†. Eq.(<ref>) then becomes p_ij = |R^SA_ij|^2/Ωwhere Ω is a real number normalization factor. Eq.(<ref>) can be intuitively understood as this: viewed from A or viewed from S, the probabilistic quantities have the same magnitude, but different in phase. Physically it ensures there is no preferred choice of S and A in defining the relational variables[When the correlation between S and A are established, both systems are effectively measuring each other (see similar remark in Ref <cit.>). Change occurs either in S or in A will be reflected by the relational matrix element. But there should not have a preference of considering S or A as a measuring system.]. Eq.(<ref>) is a weaker version of requirement for R^AS_ij compared to the second requirement for p_ij.Q_ji^AS and R_ij^SA are called relational probability amplitudes. In Section <ref>, we will give an explicit calculation of R^SA_ij, using the Path Integral formulation. Given the relation in Eq.(<ref>), we will not distinguish the notation R versus Q, and only use R. The relational matrix R^SA gives the complete description of S. It provides a framework to derive the probability of future measurement outcome. We summarize the ideas presented in this section with the following two postulates.Postulate 1 A quantum system S is completely described by a matrix R^SA relative to an apparatus A, where the matrix element R^SA_ij is the relational probability amplitude for the joint events |s_i⟩ and |a_j⟩. Postulate 2 Probability of a measurement outcome is calculated by modeling the potential interaction process, i.e., by multiplying two relational probability amplitudes representing the bidirectional process.There are two important notes. First, R^SA_ij is probabilistic quantity, not a quantity associated with certain physical variable. R^SA_ij should not be considered as certain coupling strength between S and A. In Section <ref>, in the context of path integral,R^SA_ij is defined as the sum of quantity e^iS_p/ħ, where S_p is the action of the composite system S+A along a path. Physical interaction between S and A may cause change of S_p, which is the phase of the probability amplitude. But e^iS_p/ħ itself is a probabilistic quantity. Second, although R^SA_ij is a probability amplitude, not a probability real number, we assume it follows certain rules in the classical probability theory, such as multiplication rule, and sum of alternatives in the intermediate steps.§.§ Wave Function and Born's Rule So far, we have not yet introduced the notion of quantum state for S. We only describe S and A with sets of events and the relational matrix R^SA. The next step is to derive the properties of S based on R^SA. This can be achieved by examining how the probability of measuring S with a particular outcome of variable q is calculated. We will take a move on mathematical notation before proceeding further. It is more convenient to introduce a Hilbert space for the quantum system S. The set of kets {|s_i⟩}, previously considered as representing distinct measurement events for S, can be considered as eigenbasis of Hilbert space H_S with dimension N, and |s_i⟩ is an eigenvector. Since each measurement outcome is distinguishable, ⟨ s_i|s_j⟩ = δ_ij. Similarly, the set of kets {|a_j⟩} is eigenbasis of Hilbert space H_A with dimension N for the apparatus system A. The bidirectional process |a_j⟩⇌ |s_i⟩ is called a potential measurement configuration. A potential measurement configuration comprises possible eigen-vectors of S and A that involve in the measurement event, and the relational weight quantities. It can be represented by Γ_ij: {|s_i⟩, |a_j⟩, R^SA_ij, Q^AS_ji}.In the previous section, we argue that the probability of the joint events |s_i⟩ and |a_j⟩ is given by p_ij = Q_ji^ASR_ij^SA=|R_ij^SA|^2, because the corresponding measurement configuration is |a_j⟩→ |s_i⟩→ |a_j⟩. Here we clearly specify that the probability is for the joint event |s_i⟩ and|a_j⟩. But there is a limitation for such specification if we wish to calculate the probability of measuring S with a particular outcome of variable q. In such case, the measurement configuration used earlier |a_j⟩→ |s_i⟩→ |a_j⟩ over-describe the configuration because no measurement is actually performed. We do not know that which event will occur to the quantum system A since it is completely probabilistic. The only way an observer can determine which event occurs is to perform actual measurement, or to infer from another system. Therefore, it is legitimate to generalize the potential measurement configuration as |a_j⟩→ |s_i⟩→ |a_k⟩. In other words, the measurement configuration in the joint Hilbert space starts from |a_j⟩, but can end at |a_j⟩, or any other event, |a_k⟩. Correspondingly, we generalize Eq.(<ref>) by introducing a quantity for such configurationW_jik^ASA = Q^AS_jiR^SA_ik = (R^SA_ij)^*R^SA_ik.The second step utilizes Eq.(<ref>). We interpret this quantity as a weight associated with the potential measurement configuration |a_j⟩→ |s_i⟩→ |a_k⟩. The probability for a measurement outcome can be calculated by identifying the appropriate alternatives and summing up their weights. The classical macroscopic configuration A→ S→ A can be considered as a special case when the dimension of the Hilbert space is one for either S or A. Indeed, the most general form of measurement configuration in a bipartite system can be |a_j⟩→ |s_m⟩→ |s_n⟩→ |a_k⟩, and its weight is given by W_jmnk^ASSA=Q^AS_jmR^SA_nk.The indeterminacy on which event will occur to a quantum system influences the way possible measurement configurations can be arranged. Consequently, it influences how the applicable configurations are counted and then how the probability is calculated[The situation when inference information is available is discussed in Section <ref>. In probability theory, it is crucial not to under-count or over-count applicable alternatives when calculating probability. When a quantum system is in a superposition state, although each eigenvector is labeled with a different ket, each ket should be considered indistinguishable for counting purpose because there is no information to determine exactly which ket the system is in. It is an under-count if only considering |a_j⟩→ |s_i⟩→ |a_j⟩. There is similar example in statistical physics. When counting the number of microscopic states of an ensemble consisting of identical particles, one strategy is to first over-count by assuming the particles are distinguishable, then divide the counting result by a factor to offset the over-counting.]. Suppose we do not perform actual measurement and inference is not available, the probability of finding S in a future measurement outcome can be calculated by summing W_jmnk^ASSA from all applicable alternatives of measurement configurations. Such generalized framework of calculating probability is stated by extending Postulate 2.Postulate 2e Probability of a measurement outcome is calculated by modeling the potential interaction process. The probability is proportional to the sum of weights from all applicable measurement configurations, where the weight is defined as the product of two relational probability amplitudes corresponding to the configuration.With this framework, the remaining task to calculate the probability is to correctly count the applicable alternatives of measurement configuration. This task depends on the expected measurement outcome. Some typical cases are analyzed next.Case 1. Suppose the expected outcome of an ideal measurement is event |s_i⟩, i.e., measuring variable q gives eigenvalue q_i. The probability of event |s_i⟩ occurs, p_i, is proportional to the summation of W_jmnk^ASSA from all the possible configurations related to |s_i⟩. Mathematically, we select all W_jmnk^ASSA with m=n=i, sum over index j and k, and obtain the probability p_i.p_i ∝∑_j,k=0^M (R^SA_ij)^*R^SA_ik.To see why this quantity can be considered a probability number, we note that Eq.(<ref>) is symmetric with respect to the swap of index j ↔ k. It can be rewritten asp_i ∝∑_j (R^SA_ij)^* ∑_k R^SA_ik= |∑_j R^SA_ij|^2.It is a positive real number. Normalization condition is given by∑_i |∑_j R_ij|^2 =∑_jk∑_i R_ijR_ik^* =∑_jk(R^† R)_jk = 1.A notation move is made in the above equation by omitting the superscript in R^SA, with the convention that R refers to the relational matrix from S to A. The definition of the wave function naturally emerges out from Eq. (<ref>). Define a variable φ_i = ∑_j R_ij, then p_i=|φ_i|^2. The quantum state can be described either by the relational matrix R, or by a set of variables {φ_i}. We call φ_i the wave function for eigenvector |s_i⟩. The quantum state of S is a vector representation of the variable set {φ_i}, i.e., the vector state of S relative to A, is |ψ⟩_S^A = (φ_0, φ_1,…, φ_N)^T where superscript T is the transposition symbol. In summary,|ψ⟩_S^A = (φ_0, φ_1,…, φ_N)^Twhere φ_i = ∑_j R_ij.Note that we have not yet written |ψ⟩_S^A as linear combination of φ_i. Case 2. Suppose the expected ideal measurement outcome is that S in a superposition of eigenvectors |s_0⟩ and |s_1⟩. This means one cannot determine event |s_0⟩ or |s_1⟩ occurs. The compute the probability, the applicable weights should include not only ∑_jkR_0j^*R_0k=|∑_jR_0j|^2=|φ_0|^2 and ∑_jkR_1j^*R_1k=|∑_jR_1j|^2=|φ_1|^2, but also the terms that index 0 and 1 are inter-exchanged due to the indeterminacy, i.e., ∑_jkR_0j^*R_1k and ∑_jkR_1j^*R_0k. Adding these terms together, the probability isp_{0,1}= |∑_jR_0j + ∑_jR_1j|^2 = |φ_0 + φ_1|^2Eq.(<ref>) captures the characteristics of superposition. The wave function for a superposition of eigenvectors |s_0⟩ and |s_1⟩ is the linear combination of φ_0 and φ_1. Based on this observation, it is mathematically convenient to write the state vector of S as linear combination of φ_i |s_i⟩|ψ⟩_S^A = ∑_iφ_i |s_i⟩where φ_i = ∑_j R_ij.The justification for the above definition is that the probability can be calculated from it by defining a projection operator P̂_i=|s_i⟩⟨ s_i|. Noted that {|s_i⟩} are orthogonal eigenbasis, the probability is rewritten as:p_i=⟨ψ|P̂_i|ψ⟩ = |φ_i|^2Similarly, introducing a projection operator P̂_{0,1}=(|s_0⟩+|s_1⟩)(⟨ s_0|+⟨ s_1|), we can rewrite the probability asp_{0,1}=⟨ψ|P̂_{0,1}|ψ⟩ = |φ_0 + φ_1|^2.Eq.(<ref>) and (<ref>) give the equivalent results as what Born's Rule states, but with more physical insights on how the quantum measurement probability is calculated based on detailed analysis on the interaction process during measurement. Case 3. Given a relational matrix R and that the correspondent state vector of S is |ψ⟩, suppose the expected measurement outcome is described by another relational matrix Q and the correspondent state vector of S is |χ⟩, the probability is p(Q|R) =∑_i,j(Q^† R)_i,j.The poof is given in Section <ref>. Using the state vector notation of S, the probability can be equivalently expressed as p(χ | ψ)= ⟨ψ|P̂_χ|ψ⟩ = ⟨χ | ψ⟩, where P̂_χ = |χ⟩⟨χ |. This is a generalization of Eq.(<ref>).Although the introduction of wave function φ_i brings much mathematical convenience, the relational matrix R is a more fundamental quantity. φ_i is introduced as a byproduct of the derivation instead of as a fundamental variable.Eqs.(<ref>) and (<ref>) are introduced on the condition that there is no correlation between quantum system S and A. If there is correlation between them, the summation in Eq.(<ref>) over-counts the applicable alternatives of measurement configurations and should be modified accordingly. But first, from the relational matrix R, how to determine whether there is a correlation between S and A? §.§ EntanglementCorrelation between two quantum system means one can infer the information on one system from information on the other system. The relational variable R_ij itself does not quantify an inference relation between S and A. Quantity |R_ij|^2 is the measurement probability for the joint events |s_i⟩ for S and |a_j⟩ for A. However, given R_ij, one cannot infer that event |s_i⟩ occurs to S from knowing event |a_j⟩ occurs to A.We need to define a different parameter that can quantify the quantum correlation between S and A. The capability of inferring information of a quantum state of one system from information of the other system is defined as entanglement. Since S and A both are quantum systems, they form a bipartite quantum system. Entanglement between two composite system is quantified by an entanglement measure E. There are many forms of entanglement measure <cit.>, the simplest one is the von Neumann entropy. Given the relational matrix R, the von Neumann entropy is defined as following. For reason that will become obvious in Section <ref>, we first define a product matrix ρ = RR^†, and denote the eigenvalues of ρ as {λ_i}, then the von Neumann entropy for the relational matrix R isH(R)= -∑_iλ_i lnλ_i.A change in H(R) implies there is change of entanglement between S and A. Unless explicitly pointed out, we only consider the situation that S is described by a single relational matrix R. In this case, the entanglement measure E=H(R).The definition of H(R) enables us to distinguish different quantum dynamics. Given a quantum system S and its referencing apparatus A, there are two types of the dynamics between them. In the first type of dynamics, there is no physical interaction and no change in the entanglement measure between S and A. S is not necessarily isolated in the sense that it can still be entangled with A, but the entanglement measure remains unchanged. This type of dynamics is defined as time evolution. In the second type of dynamics, there is a physical interaction and correlation information exchange between S and A, i.e., the von Neumann entropy H(R) changes. This type of dynamics is defined as quantum operation. Quantum measurement is a special type of quantum operation with a particular outcome. Whether the entanglement measure changes distinguishes a dynamic as either a time evolution or a quantum operation. This is summarized in the following postulate.Postulate 3 In a time evolution process, the entanglement measure of relational matrix is unchanged, while in a quantum operation process, there is change in the entanglement measure of relational matrix.The following theorem allows us to detect whether relational matrix R is entangled. The theorem will be used extensively later.H(R)=0 if and only if the matrix element R_ij can be decomposed as R_ij=c_id_j, where c_i and d_j are complex numbers.The proof is left to the Section <ref>. The wave function in this case is simplified to φ_i=∑_j c_id_j=c_i∑_jd_j=c_id where d is a constant. If we choose ∑_i|c_i|^2=1, then d=e^iϕ. For simplicity, let d=1 so that φ_i=c_i. When there is entanglement between S and A, A and S can infer information from each other. The way probability is calculated in Eqs.(<ref>) and (<ref>) must be modified because the summation in Eq.(<ref>) over counts the alternatives that are due to indeterminacy. Some of the potential measurement configurations should be excluded in order to calculate the probability correctly. To see this more clearly, we decompose the relational matrix R to R=UDV by virtue of the singular value decomposition, where U and V are two unitary matrices, and D is a diagonal matrix. Applying the two unitary matrices is equivalent to changing eigenbasis of S and A to |s̃_i⟩ and |ã_i⟩ such that R is diagonal. D is an irreducible diagonal matrix. H(R)>0 implies that D has more than one diagonal matrix elements. Each element corresponds to a one to one correlation between |s̃_i⟩ and |ã_i⟩. One can infer S is in |s̃_i⟩ from knowing A is in |ã_i⟩, and vice versa. Effectively, neither S nor A is in a superposition state anymore. The contributions in calculating probability due to indeterminacy of eigenvectors must be excluded. This results in a different rule to count the applicable alternatives.Case 1e. When there is an entanglement between S and A, to calculate the probability of finding S in eigenvector |s_i⟩, one should only select measurement configuration |a_j⟩→ |s_i⟩→ |a_j⟩. The corresponding weight is R_ij^*R_ij=|R_ij|^2. Summing all possible index j give the probabilityp_i=∑_j|R_ij|^2 Case 2e. Suppose we want to calculate the probability of finding S in eigenvectors |s_0⟩ or |s_1⟩ when there is an entanglement between S and A. Since there is inference information on whether S is in eigenvector |s_0⟩ or |s_1⟩, to calculate the probability, we can only count the weights ∑_jR^*_0jR_0j and ∑_jR^*_1jR_1j, and not to include the interference terms such as ∑_jR^*_0jR_1j.p_{0,1}=∑_j|R_0j|^2 + ∑_j|R_1j|^2 = p_0+p_1. It is worth to mention that the rule to calculate the probability when there is an entanglement between S and A echoes the idea from Feynman's path integral formulation. In path integral formulation <cit.>, the probability amplitude is calculated by summing possible alternatives. When the alternatives can be indistinguishable, they are called “interfering alternatives". When the alternatives are distinguishable such as in the case of entanglement between S and A, they are called “exclusive alternatives". Supposed each alternative is assigned a weight, the rules to calculate the probability can be summarized asProbability for Alternatives To calculate the probability for interfering alternatives, one first takes the summation of weight for each alternative, then takes the modulus square of the summation. To calculate the probability for exclusive alternatives, one first takes the modulus square of weight for each alternative, then takes the summation.In the other words, the order of taking modulus square and taking summation is swapped for both cases, as clearly shown in (<ref>) and (<ref>). This rule is introduced as a postulate in path integral <cit.> and the justification for the step of taking modulus square is not provided. Here we complete the justification by explaining that the modulus square is due to the bidirectional measurement process, as shown in the derivation of (<ref>).As a consequence of entanglement, we cannot define a wave function as in Eq.(<ref>) to describe the state of S. To describe S without explicitly referencing A when S and A are entangled, alternative formulation is needed. This is the reduced density matrix approach. §.§ Reduced Density Matrix To describe S without explicitly referencing A when S and A are entangled, we first describe the composite system S+A as an isolated system such that Eq.(<ref>) is applicable. We need to describe S+A relative to another measurement apparatus A' that is unentangled with S+A. Suppose an observer O_E is local to apparatus A', and has the same information of the relational matrix R and using the same reference frame F. O_E wishes to describe the composite system S+A using Postulate 2e. In order to describe a quantum state of a composite system, another postulate is needed, which is commonly found in standard textbooks, for example,Postulate 4 Let S_12 be the composite system of quantum system S_1 and S_2 with Hilbert spaces H_1 and H_2. Then the associated Hilbert space of S_12 is a tensor product Hilbert space H_1⊗ H_2. A physical variable of S_1 represented by Hermitian operator A_1 on H_1 is identified with the physical variable of S_12 represented by A_1⊗ I_2 on H_1⊗ H_2, where I_2 is the identity operator on H_2 <cit.>.An eigenvector denotes a distinct quantum event that a measurement of a variable yield a distinct eigenvalue. If there are N orthogonal eigenvectors for S, {|s_i⟩}, and M orthogonal eigenvectors for A, {|a_i⟩}, according to Postulate 4, the orthogonal basis set for the composite S+A system should have N× M eigenvectors, {|s_i⟩⊗ |a_j⟩}. O_E would describe S+A with a higher order relational matrix, denoted as R', with matrix element R'_mn. Index m is defined in Hilbert space H_S⊗ H_A, (m=0, … ,NM-1), while index n is defined in Hilbert space H_A', (n=0, … ,M'-1) and M'= H_A'. Since there is no entanglement between A' and S+A, R' can be used to define the wave function of the composite system as φ_m^A' = ∑_n R^'_mn according to Eq.(<ref>). However, there is restrictions on R^'_mn because the relational matrix between S and A has been established. The relational matrix R' must satisfy the following condition[From Postulate 2e, the probability of finding the composite system S+A in an eigenvector |m⟩ is p_m=|∑_n R^'_mn|^2. From Postulate 4, |m⟩ can be rewritten to be |s_i⟩|a_j⟩ by renumbering index m to i,j since m is defined in the Hilbert space H_S⊗ H_A. Therefore p_m is the probability for the combined events |s_i⟩ for S and |a_j⟩ for A, i.e., p_m=p_ij. But p_ij=|R_ij|^2 so that |∑_n R^'_mn|^2=|R_ij|^2. This gives ∑_n R^'_mn=e^iϕR_ij where e^iϕ is an unimportant phase factor.] φ_m=∑_n R^'_mn =R_ij.Therefore, relative to O_E, the state vector of the composite system S+A is|Ψ⟩= ∑_m φ_m|m⟩ = ∑_ijφ_ij |s_i a_j⟩ = ∑_ijR_ij|s_i⟩|a_j⟩.Next, we ask how to describe S itself. To answer this, we examine how the probability of the system S in an eigenvector |s_i⟩ can be calculated. We know that the probability of S in eigenvector |s_i⟩ and A in eigenvector |a_j⟩ is p_ij=|R_ij|^2. If event 1.)S in eigenvector |s_i⟩ and A in eigenvector |a_j⟩, and event 2.)S in eigenvector |s_i⟩ and A in eigenvector |a_k⟩, are mutually exclusive, the probability of S in eigenvector |s_i⟩ is then just the sum of p_ij over index j, i.e., p_i=∑_ip_ij=∑_j|R_ij|^2. It gives the desired result as Eq.(<ref>) when S and A are entangled. This is not a surprise since the assumption that event-1 and event-2 are mutually exclusive implies there is no event such that S is in eigenvector |s_i⟩ while A is in eigenvector |a_j⟩ and |a_k⟩. In other words, the mutual exclusivity of event-1 and event-2 eliminates the potential measurement configuration |a_j⟩→ |s_i⟩→ |a_k⟩, thus satisfies the requirement for calculating probability when there is entanglement between S and A. The mathematical tool to implement this requirement is the reduced density operator of S, defined asρ̂_S =Tr_A|Ψ⟩⟨Ψ| = ∑_ii'(∑_kR_ikR^*_i'k)|s_i⟩⟨ s_i'|=∑_ii'(RR^†)_ii'|s_i⟩⟨ s_i'|.The partial trace over A, Tr_A(.) = ∑_k⟨ a_k|.|a_k⟩, ensures the mutual exclusivity of event-1 and event-2 since only the diagonal elements are selected in the sum. To obtain the desired probability p_i=∑_j|R_ij|^2, we define a projection operator P̂_i=|s_i⟩⟨ s_i|, so thatp_i = Tr_S(P̂_iρ̂_S) = ∑_j|R_ij|^2.Since the information of A is traced out in ρ̂_S, we find a mathematical tool to describe the state of S without explicitly referring to A. Eq.(<ref>) gives a clear meaning of the matrix product RR^† as the reduced density matrix of S, i.e., ρ_S=RR^†. Thus, the entanglement measure defined in (<ref>) is the von Neumann entropy for the reduced density matrix of S. Similarly, the probability of event |a_j⟩ for A is p^A_j=∑_ip_ij=∑_i|R_ij|^2. This can be more elegantly written by introducing a partial projection operator I^S⊗P̂_j^A where P̂_j^A=|a_j⟩⟨ a_j|. It is easy to verify thatp_j^A =⟨Ψ|I^S⊗P̂_j^A|Ψ⟩ =⟨Ψ|a_j⟩⟨ a_j|Ψ⟩ =∑_i|R_ij|^2 To calculate the probability of finding S in |s_0⟩ or |s_1⟩, we use the projection operator defined as P̂_{01}=|s_0⟩⟨ s_0|+|s_1⟩⟨ s_1|, then p_{0,1} = Tr_S(P̂_{01}ρ̂_S)= ∑_j|R_0j|^2 + ∑_j|R_1j|^2 = p_0+p_1.which is the same as Eq. (<ref>) in Case 2e. The trace operation over S in Eqs. (<ref>) and (<ref>) takes only diagonal matrix elements, effectively eliminates the indeterminacy with respect to eigenvector |s_i⟩ in the information acquisition flows. Together with the partial trace operation in the definition of ρ̂_S, they exclude the interference terms in the summation of weights for calculation of probability, thus effectively factor in the inference information between S and A, and yield the same results as deduced from Postulate 2e.Normalization of |Ψ⟩ requiresTr(ρ_S) = Tr(RR^†)= ∑_i(∑_jR_ijR^†_ji)= ∑_ij|R_ij|^2 =1Note Eqs.(<ref>) and (<ref>) cannot be true in the same time. Eq.(<ref>) is true only when the relational matrix R is unentangled. When S+A is entangled, Eq.(<ref>) cannot be used to describe S. This is evident if we rewrite Eq.(<ref>) in the density matrix operator format, ρ̂_S^'= |ψ⟩_S⟨ψ|= ∑_ii'(∑_jj'R_ijR^*_i'j')|s_i⟩⟨ s_i'| = ρ̂_S + ∑_ii'(∑_j j'R_ijR^*_i'j')|s_i⟩⟨ s_i'|.Clearly, ρ̂_S^'ρ̂_S in general. The second term in Eq.(<ref>) comes from the indeterminacy of eigenvector for A. This term should be discarded when H(R)>0. This confirms that S should be described by Eq.(<ref>) instead of Eq.(<ref>) when H(R)>0. The second term in Eq.(<ref>) is related to the coherence of the quantum state of S. When H(R)=0, it turns out both density matrices are mathematically equivalent, as shown in the following theorem.If the entanglement measure H(R)=0, ρ̂_S^' = ρ̂_S.Proof is left to Section <ref>. Essentially, when H(R)=0, the coherence term in Eq.(<ref>) is equal to ρ̂_S multiplied by a constant, effectively making ρ̂_S^' and ρ̂_S differ only by a constant. We see that there are three mathematical tools to describe a quantum system, namely, the relational matrix R, the reduced density matrix ρ_S, and the wave function |ψ⟩_S. They are equivalent in terms of calculating the probability of future measurement outcome. The wave function can only be used when H(R)=0. The reduced density matrix, on the other hand, can describe the quantum state of S regardless H(R)=0 or H(R)>0. It is more generic in quantum mechanics formulation. However, in the case of H(R)=0, the wave function defined in Eq.(<ref>) reflects better the physical meaning of a superposition quantum state. Both ρ_S and |ψ⟩_S are derived from R. This confirms that R is a more fundamental variable in quantum mechanics formulation.In deriving Eq.(<ref>), we assume observer O_E who is local to apparatus A' has the latest information of the relational matrix R and using the same reference frame F. O_E then comes to an equivalent description of S using the reduced density matrix, as shown in Eq.(<ref>). This is significant since it gives the meaning of objectivity of a quantum state. Objectivity can be defined as the ability of different observers coming to a consensus independently <cit.>. On the other hand, if O_E is out of synchronization on the latest information of R, for instance, there is update on R due to measurement and not known to O_E, O_E can have different descriptions of S. This synchronization of latest information is operational, but it is necessary. One can argue that the quantum state is absolute to any observer, but the statement is non-operational if two observers are space-like separated, and causes the EPR paradox <cit.>. §.§ Operator Although R^SA_ij itself is not a probability quantity, we assume it follows some of the rules for probability calculation. For example, the multiplication rule, as seen in Eq.(<ref>). Another important rule is the summation of alternatives in the intermediate steps. Let's denote the initial relational matrix for S is R^SA_init. Suppose there is a dynamic (either a local operation or a time evolution) that changes S to a new state. The effect of the dynamics connects the initial state and new state through a matrix R^SS_p. The new relational matrix element between the A and S is(R^SA_new)_ij = ∑_k(R^SS_p)_ik(R^SA_init)_kjFigure <ref> in page fig:1 shows the meaning of Eq. (<ref>). The new matrix element (R^SA_new)_ij is obtained by multiplying the initial relational matrix element (R^SA_init)_kj and the local dynamics matrix element (R^SS_p)_ik, then summing over all possible intermediate steps. With the notation of wave function φ_i and reduced density matrix ρ_S, it is mathematically convenient to rewrite Eq.(<ref>) without referring to A. Defined an operator M̂ in Hilbert space H_S as ⟨ s_i|M|s_k⟩ = (R^SS_p)_ik, Eq.(<ref>) becomes (R^SA_new)_ij = ∑_kM_ik(R^SA_init)_kj, or R_new = MR_init.If R is not an entangled matrix, we can sum over index j in both sides of the above equation.Referring to the definition of φ_i we obtain (φ_i)_new = ∑_k M_ik(φ_k)_init. Substitute this into Eq.(<ref>), |ψ⟩_new=M̂|ψ⟩_initwhich is a familiar formulation. If R is an entangled matrix, we use the reduced density formulation,ρ_new = R_new(R_new)^† = Mρ_initM^†.Once again, we see that change of a quantum system can be described by either the relational matrix R, or the reduced density matrix that traces out the information of the reference system. Both descriptions are equivalent. §.§ General Formulation of Time EvolutionWithout loss of generality, we will only consider discrete time evolution here that describes state change from initial time t_0 to some finite time later at t. By definition, there is no physical interaction between S and A, S and A are evolving independently. According to Eq. (<ref>), the new state that S is changed to is related to the original state through a local evolution matrix R_p^SS. R_p^SS is independent of A since there is no interaction between S and A. Similarly, the new state that A is changed to is related to the original state through a local evolution matrix R_p^AA. R_p^AA is independent of S. To simplify the notation, we rewrite Q(t-t_0)=R_p^SS and O(t-t_0)=R_p^AA, and denote the initial relational matrix between S and A is R(t_0). The time evolution of the relational matrix R(t) is depicted in Figure <ref> of page fig:2. Matrix element at time t, R_ij(t), shown in the dot line in Figure <ref>, is calculated by summation of all the possible intermediate steps between eigenvector |s_i(t)⟩ and eigenvector |a_j(t)⟩:R_ij^S_tA_t(t) = ∑_m,n Q_im^S_tS_0(t-t_0) R_mn^S_0A_0(t_0) O_nj^A_0A_t(t_0-t)= (Q(t-t_0)R(t_0)O(t_0-t))_ijThe superscripts ensure the consistency of notation for the process (S_t→ S_0 → A_0 → A_t), and in the last step, they are omitted. Thus, the general formulation of the time evolution for the relational matrix is given byR(t) = Q(t-t_0)R(t_0)O^† (t-t_0),where we assume the property[This property is clearer when O is the representation of a unitary operator. In that case, operator Ô(t-t_0)=e^-iĤ(t-t_0)/ħ where Ĥ is a Hermitian operator. Reverting the parameter of time gives Ô(t_0-t)=e^-iĤ(t_0-t)/ħ=e^iĤ(t-t_0)/ħ= O^† (t-t_0).] O(t_0-t)=O^† (t-t_0). For simplicity, let t_0=0, the reduced density matrix at time t is ρ (t)=R(t)R^† (t)=Q(t)R(0)O^†(t)O(t) R^† (0) Q^† (t). According to Postulate 3, in time evolution the entanglement measure is unchanged. This means the von Neumann entropy is an invariance during time evolution, H(R(t))=H(R(0)). One sufficient condition to meet such requirement is that Q(t) and O(t) are unitary matrices, consequently ρ(t) and ρ(0) are unitary similar matrices and have the same von Neumann entropy. However, the converse statement is not necessarily true. The condition H(R(t))=H(R(0)) is too weak to lead to the conclusion that Q(t) and O(t) are unitary matrices. We wish to find additional conditions such that Q(t) and O(t) are unitary[Mathematically, the Specht's Theorem and its improved version Pearcy's Theorem give the necessary and sufficient conditions for two matrices to be similar <cit.>. This allows one to determine if ρ(t) and ρ(0) are unitary similar matrices. However, how such condition is related to whether Q(t) and O(t) are unitary matrices is not obvious. It requires further investigation.].§.§ Schrödinger Equation In the case that the initial state for S and A are unentangled, the eigenvalue of ρ(0) has only one value λ=1 and H(R(0))=0. From Theorem 1, R_mn(0)=c_md_n, Eq. (<ref>) becomesR_ij(t)= ∑_m,n Q_im(t) c_md_n O_nj^†(t)= (∑_m Q_im(t) c_m)( ∑_n d_nO_nj^†(t)).The last expression of R_ij(t) shows it can be still decomposed into the product of two separated terms, therefore H(R(t))=0 as expected. By definition, the initial wave function is φ_m(0)=∑_nc_md_n=c_md_0. At time t it becomesφ_i(t) = ∑_jR_ij(t) = ∑_m Q_im(t) c_m ∑_j,n d_nO_nj^†(t) = d(t)∑_m Q_im(t) φ_m(0)where d(t)=∑_jn (d_n/d_0)O_nj^†(t) is a constant independent of S. Defined linear operator Q̂(t) in Hilbert space H_S as ⟨ s_i|Q̂(t)|s_k⟩=Q_ik(t) and substituted Eq.(<ref>) to Eq.(<ref>), the state vector|ψ(t)⟩=d(t)Q̂(t)|ψ(0)⟩.Since the total probability should be preserved, ⟨ψ(t)|ψ(t)⟩ =|d|^2⟨ψ(0) |Q̂^†Q̂ |ψ(0)⟩ = 1. This is true for any initial sate |ψ(0)⟩, thus, Q̂^†Q̂ = I/|d|^2. There is an undetermined constant |d|.In general, one cannot conclude that Q(t) is a unitary matrix unless choosing |d|=1. If |d|=1, d=e^iϕ(t) is an arbitrary phase. Rewritten Q̂ as Û, Eq.(<ref>) becomes|ψ(t)⟩=e^iϕ(t)Û(t)|ψ(0)⟩ = e^iϕ(t)e^-iĤt/ħ|ψ(0)⟩where Ĥ is a Hermitian operator for S. Omitting the arbitrary phase, Eq.(<ref>) is the Schrödinger Equation. The derivation here does not give the actual expression of the Hamiltonian operator, but it manifests the fact that there is no change of entanglement measure between the observed system and the observing apparatus.The above derivation depends on several conditions. First, there is no physical interaction between S and A; Second, S and A are not entangled; Third, the total probability is preserved; Lastly, we choose |d(t)|=1. The first two conditions are usually what one refers as S is in an isolated state. In summary, given H(R(t))=H(R(0)), if two more conditions are added, H(R(0))=0 and |d(t)|=1, matrix Q(t) is unitary, which leads to the Schrödinger Equation.A special case for Eq. (<ref>) to be reduced to the Schrödinger Equation is when O(t)=I. With O(t)=I, R(t)=U_S(t)R(0). Since H(R)=0, we can use Eq.(<ref>) to calculate the wave functionφ_i(t)= ∑_jR_ij(t) = ∑_m Q_im(t) ∑_jR_mj(0)= ∑_m Q_im(t) φ_m(0)which is the same as Eq.(<ref>) with d(t)=1. The same reasoning from Eq.(<ref>) to Eq.(<ref>) is applied here. O(t)=I is a very strong condition, it may not be physically sensible because any quantum system evolves as time elapses. However, this may be considered an approximation that, for a macroscopic classical apparatus, the change as a ratio to its overall state is so infinitesimal in magnitude compared to the change for a microscopic quantum state, that it can be ignored. §.§ Generalized Differential Equation Next, we consider the more general case that S and A are not interacting but initially entangled, i.e., H(R(0)) > 0. Since entanglement measure is unchanged, H(R(t))=H(R(0)) > 0. S and A stay entangled at time t. S is not in an isolated state. We need to describe the composite system S+A as a whole relative to another unentangled apparatus A'. To proceed further the following theorem is introduced.Applying operator Q̂⊗Ô over the composite system S+A is equivalent to change the relational matrix R to R'=QRO^T, where the superscript T represents a transposition.The proof is left to Section <ref>. Since the composite system S+A is in isolated state, based on the result in Section <ref>, the overall time evolution operator Û_SA is unitary. The state vector of the composite system at time t should be |Ψ(t)⟩=Û_SA|Ψ(0)⟩=Û_SA∑_ijR^SA_ij|s_i⟩|a_j⟩. Let Û_SA=exp(-iĤ_SAt/ħ) where Ĥ_SA is the Hamiltonian of the composite system. Since there is no interaction between S and A, Ĥ_SA=Ĥ_S+Ĥ_A where Ĥ_S and Ĥ_A are the Hamiltonian operators in their respective Hilbert spaces. As shown in Section <ref>, Û_SA can be decomposed into Û_SA=Û_S⊗Û_A. According to Theorem 3, this effectively change the relational matrix to R(t)=U_S(t)R(0)U_A^T(t). Note that U_A^T(t) is also a unitary matrix. This is equivalent to the general time evolution equation (<ref>) with the condition that both Q̂(t) and Ô(t) are unitary.Let's rewrite the general time evolution dynamics, Eq. (<ref>), in operator notation by introducing a linear operator R̂=∑_ijR_ij|s_i⟩⟨ a_j|. Since Q̂(t)=Û_S(t)=exp{-iĤ_St/ħ} and Ô^†(t)=Û_A^T(t)=exp{-i(Ĥ_A^T)t/ħ}, Eq. (<ref>) becomesR̂(t) = e^-iĤ_St/ħR̂(0)e^-i(Ĥ_A^T)t/ħ.Because H(R)>0, the wave function φ(t) of S cannot be defined. Consequently we cannot obtain a dynamics equation of wave function. Instead, a dynamics equation for R̂ can be derived. Taking the derivative over t of both sides of Eq.(<ref>) and noting [exp{i(Ĥ_A^T)t/ħ}, Ĥ_A^T]=0, one getsiħdR̂(t)/dt= Ĥ_SR̂(t) + R̂(t)Ĥ_A^T.Note that [R̂,Ĥ_A^T]0, i.e., R̂ andĤ_A^T are non-commutative[It is easier to realize the non-commutation if using the matrix representation of Eq. (<ref>): iħ (dR(t)/dt)= H_SR(t) + R(t)H_A^T. Since R is a N× M matrix while H_A^T is a M× M matrix, matrix multiplication H_A^T× R is even not possible when N M.]. Eq. (<ref>) is a more general form of differential equation that describes the time evolution of R. Once solving the above equation and obtaining R(t), one can calculate the probability according to Postulate 2e.To derive a differential equation without explicitly referring to the apparatus A, we should use the reduced density matrix approach since S and A can be entangled. Given the dynamics of the relational matrix is R(t)=U_S(t)R(0)U_A^T(t), the reduced density matrix of S is ρ(t) = R(t)R^†(t) =U_S(t)ρ(0)U_S^†(t). Defining density operator ρ̂(t) for S such that ⟨ s_i|ρ̂(t)|s_j⟩ = [R(t)R^† (t)]_ij, we can rewrite the equation toρ̂(t)= e^-iĤ_St/ħρ̂(0)e^iĤ_St/ħTaking the derivative over t of both sides, we obtain the Liouville-von Neumann equationiħdρ̂(t)/dt= Ĥ_Sρ̂(t) - ρ̂(t)Ĥ_S = [Ĥ_S, ρ̂(t)].Eqs.(<ref>) and (<ref>) give equivalent descriptions of the dynamics of quantum state of S. Eq.(<ref>) has the advantage of describing the time evolution of S without referencing to A and therefore mathematically more elegant. However, it leads to the impression that the quantum system can be described independent of the reference system. Eqs.(<ref>) and (<ref>) also confirm two equivalent methodologies to describe the change of quantum state of S: 1.) Calculate the change of relational matrix R and compute the quantum probability based on Postulate 2e; 2.) Derive the wave function of the composite state for S+A, then trace out A over the composite state to obtain ρ_S. § DISCUSSION AND CONCLUSION§.§ Hypotheses in the ReconstructionThe reconstruction of quantum theory presented in this paper is based on two hypotheses. First, the relational properties between two quantum systems are more basic than the properties of one system. We take this hypothesis as a starting point to reformulate quantum mechanics. This reference system is not arbitrary. It is the apparatus, or environment, A, that the system S has previously interacted with. Although the reference system A is unique and objectively selected, it is possible that another observer does not have complete information of the interaction (or, measurement) results between A and S. In such case she can describe S differently using a different set of relational properties between S and A. It is in this sense that we say the relational properties themselves are observer-dependent. This is indeed the main thesis of Ref. <cit.>. In the example of ideal measurement described by Eq.(<ref>), supposed the measurement outcome is correspondent to eigenvector |s_n⟩. For an observer that operates and reads the pointer variable of A, she knows the measurement outcome. At the end of the measurement, her relational description is given by |s_n⟩|a_n⟩. On the other hand, for another observer who only knows there is interaction between S and A, but does not know the measurement outcome, the relational description is given by ∑_i c_i|s_i⟩|a_i⟩. Both descriptions are based on relational properties, and they are observer-dependent. Thus, there are two layers of relativity here. In this paper, we assume observers share the same information of the relational matrix, and focus on formulating quantum mechanics based on the relational probability amplitude. The observer-dependent aspect of the formulation is more relevant to measurement theory, which will be analyzed in an upcoming paper.The second hypothesis is due to the realization that a real physical measurement is a bidirectional process. It is a question-and-answer, or a probe-and-response, interaction process. This bidirectional aspect of a physical measurement seems overlooked in other quantum mechanics formulations. Here we mandate that a framework of calculating probability for a potential outcome from a physical interaction must explicitly model the bidirectional process. A variable that only models unidirectional of the process cannot be considered as a real probability number because a one-way process does not model an actual measurement. Instead, such unidirectional quantity is called probability amplitude and is not necessarily a real non-negative number. The distinction of unidirectional versus bidirectional process allows us to relax the mathematical requirement on the probability amplitude and to consider it as a complex number. However, we assume it still follows some of rules in probability calculation, such as multiplication, and sum over alternatives of intermediate steps.With these two hypotheses, a framework is developed such that the task of calculating probability in a specific measurement setup is reduced to counting the applicable measurement configurations in the joint Hilbert space for the measured system and the apparatus. It is interesting to notice that the two hypotheses philosophically echo the ideas expressed in Ref. <cit.> that the physical world is made of processes instead of objects, and the properties are described in terms of relationships between events. §.§ The Apparatus System Although the relational properties between the quantum system S and the measurement apparatus A, such as the probability amplitude matrix R, are considered as the most basic variables, there are mathematical tools that allow a quantum system S to be described without explicitly calling out the apparatus system A. When S and A are unentangled, S is described by a wave function that sums out the information of the reference system. When S and A are entangled, S is described by a reduced density matrix that traces out the information of A. These mathematical tools enable us to develop the formulations for time evolution and measurement theory that are equivalent to those in the traditional quantum mechanics.Except some special scenario such as that is described in the EPR argument, there is no need to explicitly call out the apparatus quantum system A. Mathematically it is more convenient and elegant to trace out the information of the apparatus system. However, explicitly including the apparatus system allows us to develop a framework to explain the origin of the quantum probability and to quantify the difference between time evolution and quantum measurement. It is interesting to notice that Ref. <cit.> also proposes to use relational logic and category theory to deduce the basic laws of quantum mechanics. However, the formulation in Ref. <cit.> is rather abstract. Quantum probability is introduced purely from mathematical perspective, instead of associating it with the process of actual physical measurement. How entanglement influences the probability calculation is not discussed and formulated in Ref. <cit.>. §.§ Comparison with the Original RQM TheoryThe works presented here is inspired by the main idea of the original RQM theory <cit.>. However, there are several significant improvements that should be pointed out. The works of Refs. <cit.> establish the idea that relational properties are more basic, and a quantum system must be described relative to another quantum system. However, they do not provide a clear formulation on how a quantum system should be described relative to another system and what the basic relational properties are. On the other hand, our works gives a clear quantification of the relational property, which is the relational probability amplitude. The introduction of the relational probability amplitude is based on a detailed analysis of measurement process. It enables us to develop a framework to calculate probability during quantum measurement. We further show that the relational probability amplitude can be calculated using Feynman path integral in Section <ref>. The second improvement in this works comes from the introduction of the concept of entanglement to the RQM theory. We recognize not only that a quantum system must be described relative to another quantum system, but also that the entanglement between these two systems plays a crucial role in how the formulation the observed system is described. If there is no entanglement, the observed system can be described by a pure wave function. If there is entanglement, a reduced density matrix is more appropriate mathematical tool. In addition, entanglement measure plays a pivot role in determining a system is undergoing a time evolution or measurement process. This allows us to reconstruct both the Schrödinger equation and the measuring theory[The reconstruction of quantum measurement theory is submitted in an upcoming paper.]. When one states that a quantum system must be described relative to another quantum system, one can further quantify this relativity via the entanglement measure between these two systems. However, the concept of entanglement is not presented in Ref <cit.>. The reconstruction attempts in Ref <cit.> to derive the laws of quantum mechanics based on quantum logic is rather limited since only the Schrödinger equation is reconstructed. Thirdly, although a quantum system must be described relative to another quantum system, our works show that there are mathematical tools that can describe the observed system without explicitly calling out the observing system, such as the wave function and the reduced density matrix as shown in Section <ref>. Therefore, RQM and traditional QM are compatible mathematically. This is important because it confirms that although the main idea of RQM seems radical, it does not change the practical application of quantum mechanics. Again, this point was not clear in Ref <cit.>. §.§ Compared to the Transactional Interpretation The bidirectional measurement process which is important in the derivation of the measurement probability appears sharing some common ideas with the transaction model <cit.>. In particular, the transaction model requires a handshake between a retarded “offered wave" from an emitter and an advanced “confirmation wave" from an absorber to complete a transaction in a quantum event. This is a bidirectional process. While it is encouraging to note that the bidirectional nature of a quantum event has been recognized in the transaction model, there are several fundamental differences between the transaction model and the bidirectional measurement framework presented here. First, the transaction model considers the offered wave and the confirmation wave as real physical waves. In our framework, we do not assume such waves existing. Instead, the probing and responding are just two aspects in a measurement event, and we require the probability calculation to faithfully model such bidirectional process. Second, in the transaction model, the randomness of measurement outcomes is due to the existence of different potential future absorbers. Thus, the randomness in quantum mechanics depends on the existence of absorbers. There is no such assumption in our framework. Third, the transaction model derives that the amplitude of the confirmation wave at the emitter is proportional to the modulus square of the amplitude of the offer wave, which is related to the probability of completing a transaction with the absorber. This appears to be ambiguous since it suggests the confirmation wave is also a probability wave. In our formulation, we only focus on how the measurement probability is calculated, and clearly point out that the wave function is just a mathematical tool. §.§ Limitations and Outlook The framework to calculate the measurement probability in Section <ref> is the key to our reformulation. However, it is essentially based on an operational model from a detailed analysis of bidirectional measurement process, instead of being derived from more fundamental physical principles. In particular, there may be better justification for Eq. (<ref>). The current model is only served as a step to deepen our understanding of relational quantum mechanics. It is desirable to continue searching for more fundamental physical principles to justify the calculation of measurement probability. The fact the R^SA_ij is a complex number means that this variable actually comprises two independent variables, the modulus and the phase. This implies that more degrees of freedom are needed to have a complete description of a quantum event. Stochastic mechanics, for instance, introduces forward and backward velocities instead of just one classical velocity to describe the stochastic dynamics of a point particle. With the additional degree of freedom, two stochastic differential equations for the two velocities are derived. Then, through a mathematical transformation of two velocity variables in ℝ into one complex variable in ℂ, the two differential equations turn into the Schrödinger equation <cit.>. It will be interesting to investigate if R^SA_ij can be implemented in the context of stochastic mechanics, where we expect R^SA_ij will be decomposed to two independent variables in ℝ and each of them is a function of velocity variables.As discussed in the introduction section, the RQM principle consists two aspects. First, we need to reformulate quantum mechanics relative to a QRF which can be in a superposition quantum state, and show how quantum theory is transformed when switching QRFs. In this aspect, we accept the basic quantum theory as it is, including Schrödinger equation, Born's rule, von Neunman measurement theory, but add the QRF into the formulations and derive the transformation theory when switching QRFs <cit.>. Second, we go deeper to reformulate the basic theory itself from relational properties, but relative to a fixed QRF. Here the fixed QRF is assume to be in a simple eigen state. This is what we do in the present work. A complete RQM theory should combine these two aspects together. This means one will need to reformulate the basic quantum theory from relational properties and relative to a quantum reference frame that exhibits superposition behavior. Therefore, a future step is to investigate how the relational probability amplitude matrix should be formulated when the QRF possesses superposition properties, and how the relational probability amplitude matrix is transformed when switching QRFs.There are other limitations that are worth to mention here. The formulation assumes a finite Hilbert space for either the observed quantum system or the observing apparatus. It is desirable to extend the formulation for Hilbert space with infinite dimension. It is mathematically more cumbersome to calculate the wave function from a relational matrix than to just assume an observer independent wave function. Mathematical rigorousness is needed for some of the derivations. For instance, given the general time evolution dynamics in Eq. (<ref>), it is left unanswered on what the sufficient and necessary condition should be in order to keep the entanglement measure as an invariance. Section <ref> only gives several sufficient conditions that lead to the Schrödinger Equation. Furthermore, implementing the relational properties between S and A with one definite matrix implies that the composite system S+A is in a pure state. S+A can be in a mixed state and described by an ensemble of relational matrices. A rigorous mathematical treatment for mixed state is desirable, especially when ρ_SA is not a separable mixed state. It should be also noted that only non-relativistic quantum mechanics is considered here. §.§ ConclusionsWe have shown that quantum mechanics can be constructed by shifting the starting point from the independent properties of a quantum system to the relational properties among quantum systems. This idea, combined with the emphasis that a physical measurement is a bidirectional interaction process, enables us to propose a framework to calculate the probability of outcome when measuring a quantum system. Quantum probability is proportional to the summation of weights representing the bidirectional measurement process from all applicable configuration in the joint Hilbert space of the measured and measuring composite system. This postulate explains why the quantum probability is the absolute square of a complex number when there is no entanglement. The wave function of the observed system is simply the summation of relational probability amplitudes. If there is entanglement between the measured and measuring composite system, the way probability is calculated is adjusted due to the presence of correlation. In essence, quantum mechanics demands a new set of rules to calculate probability of a potential outcome from a physical interaction in the quantum world. Quantum theory does not describe directly measurable physical properties such as force, length, etc. Instead it deals with quantity such as probability amplitude, and provides a set of rules to connect to those measurable physical properties. In this sense, quantum mechanics is a probability theory for describing the process of measuring a quantum system through interaction. Based on the postulates, formulations for time evolution and quantum measurement can be reconstructed. Schrödinger Equation is derived when the observing system is in an isolated state. Although the theory developed in this work is mathematically equivalent to the traditional quantum mechanics, there are several significant implications of this formulation. First, the reformulation shows that relational property can be the most fundamental element to construct quantum mechanics. Second, it brings new insight on the origin of the quantum probability. Third, path integral formulation is generalized to formulate the reduced density matrix of a quantum system. This may pave the way to extend the reformulation to quantum field theory and deserves further research. Finally, as with other efforts of reformulating quantum mechanics, it is always interesting to recognize a new perception on a traditional theory. The hope is that the reformulation presented here can be one step towards a better understanding of quantum mechanics.§ METHODS§.§ Proof of Eq.(<ref>) To prove Eq.(<ref>), we perform a transformation of eigenbasis. The initial eigenbasis for S is { |s_i⟩} and the relational matrix is R. If we introduce another set of eigenbasis { |s^'_i⟩} such that the first eigenvector is |s^'_0⟩ is |χ⟩. Denote the unitary matrix that relates the two sets of eigenbasis as U. We have U|χ⟩ = |s^'_0⟩, or, |χ⟩ = U^†|s^'_0⟩From the definition of wave function, we have |χ⟩ = {ϕ_0, ϕ_1, …, ϕ_N }^T, where ϕ_i = ∑_jQ_ij. Substitute this into the above equation, we get U^†_i0 = ϕ_i, i.e., U_0i=∑_jQ^*_ijIn the new eigenbasis, the original relational matrix R is transformed to R^' = UR. The probability of finding S described by state vector χ is correspondent to the probability to find S in engeinvector |s^'_0⟩, which according to Eq.(<ref>) isp(χ|ψ) = |∑_jR^'_0j|^2 = |∑_j(UR)_0j|^2 = |∑_j∑_iU_0iR_ij|^2= |∑_ij∑_kQ^*_ikR_ij|^2= |∑_jk(∑_iQ^†_kiR_ij)|^2= |∑_jk(Q^† R)_kj|^2= |∑_i (∑_kQ^*_ik)(∑_jR_ij)|^2= |∑_i ϕ^*_iψ_i|^2 = ⟨χ | ψ⟩.In the first step of the second line, we use the relation Eq.(<ref>). §.§ Proof of Theorem 1 According to the singular value decomposition, the relational matrix R can be decomposed to R = UDV, where D is rectangular diagonal and both U and V are N× N and M× M unitary matrix, respectively. This gives ρ = RR^† = U(DD^†) U^†. If H(R)=0, matrix ρ is a rank one matrix, therefore DD^† is diag{1,0,0...}. This means D is a rectangular diagonal matrix with with only one eigenvalue e^iϕ. Expanding the matrix product R=UDV gives R_ij=∑_nmU_inD_nmV_mj=U_i1e^iϕV_1j.We just choose c_i=U_i1 and d_j=e^iϕV_1j to get R_ij=c_i d_j. Conversely, if R_ij=c_i d_j, R can be written as outer product of two vectors,R = [ c_1 c_2 … c_n ]^T ×[ d_1 d_2 … d_m ].Considering vector C_1={c_1, c_2, …,c_n} as an eigenvector in Hilbert space H_S, one can use the Gram-Schmidt procedure <cit.> to find orthogonal basis set C_2, …, C_n. Similarly, considering vector D_1={d_1, d_2, …,d_m} as an eigenvector in Hilbert space H_A, one can find orthogonal basis set D_2, …, D_m. Under the new orthogonal eigenbasis, R becomes a rectangular diagonal matrix D=diag{1,0,0...}. Therefore R=UDV where U and V are two unitary matrices associated with the eigenbasis transformations. Then ρ = RR^†=U(DD^†) U^†, and DD^†=diag{1,0,0...} is a square diagonal matrix. Since the eigenvalues of similar matrices are the same, the eigenvalues of ρ are (1, 0, ...), thus H(R)=0.§.§ Proof of Theorem 2 Assuming S+A is in a pure state, we use the Von Neumann entropy H(R) as entanglement measure. If H(R)=0, by virtue of Theorem 1, R_ij=c_id_j. Assuming both O_I and O_E share the same knowledge of R_ij. The reduced density matrices relative to each observer are calculated asρ̂_S^'= ∑_ii'(∑_jj'R_ijR^*_i'j')|i⟩⟨ i'|= ∑_ii'c_i(c_i')^*(∑_jj'd_jd_j')|i⟩⟨ i'| =d_A∑_ii'c_i(c_i')^*|i⟩⟨ i'|ρ̂_S = ∑_ii'(∑_kR_ikR^*_i'k)|i⟩⟨ i'|=∑_ii'c_i(c_i')^*(∑_k|d_k|^2)|i⟩⟨ i'| = d_A'∑_ii'c_i(c_i')^*|i⟩⟨ i'|where d_A and d_A' are two constant. ρ̂_S and ρ̂_S^' only differ by a constant when H(R)=0. Since Tr(ρ̂_S^')=Tr(ρ̂_S)=1, we can simply choose d_A=d_A' so that ρ̂_S=ρ̂_S^'. §.§ Proof of Theorem 3 Denote the initial state vector of the composite system as |Ψ_0⟩=∑_ijR_ij|s_i⟩|a_j⟩. Apply the composite operator Q̂(t)⊗Ô(t) to the initial state,|Ψ_1⟩= (Q̂⊗Ô) ∑_ij R_ij |s_i⟩⊗ |a_j⟩ = ∑_ijR_ijQ̂|s_i⟩⊗Ô|a_j⟩ = ∑_ij∑_mnR_ijQ_miO_nj|s_m⟩⊗ |a_n⟩ = ∑_mn(∑_ijQ_miR_ijO^T_jn)|s_m⟩⊗ |a_n⟩ =∑_mn(QRO^T)_mn|s_m⟩|a_n⟩where T represents the transposition of matrix. Compared the above equation to Eq.(<ref>) for the definition of |Ψ_1⟩, it is clear that the relational matrix is changed to R'=QRO^T. §.§ Decomposition of the Unitary Operator of a Bipartite System Here we show that if there is no interaction between S and A, a global unitary operator for the composite system S+A is decomposed into the tensor product of two local unitary operators. Let {|s_i⟩} be the orthogonal eigenbasis of Ĥ_S, Ĥ_S|s_i⟩=E_i^S|s_i⟩. Recall that the definition of a function of operator Ĥ isf(Ĥ) = ∑_i f(E_i)|s_i⟩⟨ s_i|Based on this definition, Û_S = exp{-(i/ħ)Ĥ_St} = exp{-(i/ħ)E_i^St}|s_i⟩⟨ s_i|. Similarly, let {|a_j⟩} be the orthogonal eigenbasis of Ĥ_A, Ĥ_A|a_j⟩ =E_j^A|a_j⟩ and Û_A = exp{-(i/ħ)E_j^At}|a_j⟩⟨ a_j|. When there is no interaction between S and A, Ĥ_SA=Ĥ_S+Ĥ_A where Ĥ_S and Ĥ_A are the Hamiltonian operators in their respective Hilbert spaces, thus Û_SA= exp{-(i/ħ)(Ĥ_S+Ĥ_A)t)}. According to Postulate 4, the set {|s_i⟩|a_j⟩} forms the orthogonal eigenbasis for Ĥ_SA, so that Ĥ_SA|s_i⟩|a_j⟩=(E_i^S+E_j^A)|s_i⟩|a_j⟩ and exp{-(i/ħ)(Ĥ_S+Ĥ_A)t)|s_i⟩|a_j⟩ = exp{-(i/ħ)(E_i^S+E_j^A)t)|s_i⟩|a_j⟩. From the definition of operator function,Û_SA= ∑_ijf(E_ij)|s_i⟩|a_j⟩⟨ s_i|⟨ a_j| =∑_ijexp{-(i/ħ)(E_i^S+E_j^A)t)|s_i⟩|a_j⟩⟨ s_i|⟨ a_j| = ∑_i exp{-(i/ħ)E_i^St}|s_i⟩⟨ s_i| ⊗∑_j exp{-(i/ħ)E_j^At}|a_j⟩⟨ a_j|= Û_S ⊗Û_A. §.§ Path Integral Implementation This section briefly describes how the relational probability amplitude can be calculated using the Path Integral formulation. Without loss of generality, the following discussion just focuses on one dimensional space-time quantum system. In the Path Integral formulation, the probability to find a quantum system moving from a point x_a at time t_a to a point x_b at time t_b is the absolute square of a probability amplitude, i.e., P(b, a)=|K(b, a)|^2. The probability amplitude is postulated as the sum of the contribution of phase from each path <cit.>:K(b, a)=1/N∑_pathe^(i/ħ)S_p(x(t))where N is a normalization constant, and S_p (x(t)) is the action along a particular path from point x_a to point x_b. The action is defined as S_p (x(t))=∫_t_a^t_bL(ẋ, x, t)dt where L is the Lagrangian of the system. Since there is infinite number of possible paths from point x_a to point x_b, more precisely the summation in Eq.(<ref>) should be replaced by an integralK(b, a)=∫_a^be^(i/ħ)S_p(x(t))Dx(t)where Dx(t) denotes integral over all possible paths from point x_a to point x_b. It is the wave function for S moving from x_a to x_b <cit.>. The wave function of the particle at position x_b isφ(x_b, t_b)=∫_-∞^∞K(x_b, t_b; x_a, t_a)φ(x_a, t_a)dx_awhere φ(x_a, t_a) is the wave function of the particle at position x_a. Eq.(<ref>) is the integral form of the Schrödinger Equation (<ref>).Now let's consider how the relational matrix element can be formulated. At a particular time t_a, we denote the matrix element as R(x_a; y_a). Here the coordinates x_a and y_a act as indices to the system S and apparatus A, respectively. From time t_a to t_b, suppose S moves from x_a to x_b, and A moves from y_a to y_b, the relational matrix element is written as R(x_b, x_a; y_b, y_a). Borrowing the ideas described in Eq.(<ref>), we propose thatR(x_b, x_a; y_b, y_a)= ∫_a^b ∫_a^be^(i/ħ)S^SA_p(x(t), y(t))×Dx(t)Dy(t)where the action S_p^SA(x(t), y(t)) consists three termsS^SA_p(x(t), y(t))= S^S_p(x(t)) + S^A_p(y(t))+ S^SA_int(x(t), y(t)).The last term is the action due to the interaction between S and A when each system moves along its particular path. Eq.(<ref>) is considered an extension of Postulate 1. We can validate Eq.(<ref>) by deriving formulation that is consistent with traditional path integral. Suppose there is no interaction between S and A. The third term in Eq.(<ref>) vanishes. Eq.(<ref>) is decomposed to product of two independent terms,R(x_b, x_a; y_b, y_a) = ∫_a^b e^(i/ħ)S^S_p(x(t)) Dx(t)×∫_a^b e^(i/ħ)S^A_p(y(t)) Dy(t)Noticed that the coordinates y_a and y_b are equivalent of the index j in Eq.(<ref>), the wave function of S can be obtained by integrating y_a and y_b over Eq.(<ref>)φ(x_b, x_a) = ∫∫^∞_-∞R(x_b, x_a; y_b, y_a)dy_ady_b ={∫_a^b e^(i/ħ)S^S_p(x(t)) Dx(t)}×{∫∫^∞_-∞∫_a^b e^(i/ħ)S^A_p(y(t)) Dy(t)dy_ady_b}=c∫_a^b e^(i/ħ)S^S_p(x(t)) Dx(t)where constant c is the integration result of the second term in step two. The result is the same as Eq.(<ref>) except an unimportant constant.Next, we consider the situation that there is entanglement between S and A as a result of interaction. The third term in Eq.(<ref>) does not vanish. We can no longer define a wave function for S. Instead, a reduced density matrix should be used to describe the state of the particle, ρ = RR^†. From Eq.(<ref>), the element of the reduced density matrix is ρ(x_b, x'_b; x_a, x'_a)= ∑_ y_a,y_b∫_x_a^x_b∫_x'_a^x'_b∫_y_a^y_b∫_y_a^y_be^(i/ħ)Δ S×Dx(t)Dx'(t)Dy(t)Dy'(t) whereΔ S= S^S_p(x(t)) - S^S_p(x'(t))+ S^A_p(y(t))- S^A_p(y'(t))+ S^SA_int(x(t), y(t))- S^SA_int(x'(t), y'(t)).The path integral over Dy'(t) takes the same end points y_a and y_b as the path integral over Dy(t). After the path integral, a summation over y_a and y_b is performed. Eq.(<ref>) is equivalent to the J function introduced in Ref <cit.>. We can rewrite the expression of ρ using the influence functional, F(x(t), x'(t)),ϱ(x_b, x'_b; x_a, x'_a)= 1/Z∫_x_a^x_b∫_x'_a^x'_be^(i/ħ)[S^S_p(x(t)) - S^S_p(x'(t))]× F(x(t), x'(t))Dx(t)Dx'(t) F(x(t), x'(t)) = ∑_ y_a,y_b∫_y_a^y_b∫_y_a^y_be^(i/ħ)Δ S'×Dy(t)Dy'(t) whereΔ S'= S^A_p(y(t))- S^A_p(y'(t))+ S^SA_int(x(t), y(t))- S^SA_int(x'(t), y'(t)).Where Z=Tr(ρ) is a normalization factor to ensure Tr(ϱ) = 1. The reduced density matrix allows us to calculate the probability of the system changing from one state to another, for instance, the probability of the system initially in a state χ(x_a) transitioning to another state ψ(x_b). This is similar to calculate the probability of an ideal measurement that specifies the initial state is χ(x_a) and the final state is ψ(x_b). Defining a project operator P=|χ(x_a)ψ(x_b)⟩⟨χ(x_a)ψ(x_b)|, the probability is calculated, similar to Eq.(<ref>), as p(χ,ψ)=Tr(ϱP) =∫∫∫∫ψ^*(x'_b)ψ(x_b)ϱ(x_b, x'_b; x_a, x'_a) ×χ(x_a)χ^*(x'_a)dx_adx_bdx'_adx'_bThis is equivalent to the result in Ref <cit.>. To find the particle moving from a particular position x̅_a at time t_a to another particular position x̅_b at time t_b, we substitute χ(x_a)=δ(x_a-x̅_a) and χ(x_b)=δ(x_b-x̅_b) into Eq.(<ref>),p(x̅_b, x̅_a)= ∫∫∫∫ϱ(x_b, x'_b; x_a, x'_a)δ (x_b - x̅_b) ×δ (x'_b - x̅_b) δ (x_a - x̅_a) δ (x'_a - x̅_a)× dx_b dx'_b dx_a dx'_a= ϱ(x̅_b, x̅_b; x̅_a, x̅_a). 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D, 1774-1786 (1983) Yang2021J. M. Yang, Stochastic Quantization Based on Information Measures, arXiv:2102.00392 SmolinL. Smolin, Three Roads to Quantum Gravity. Basic Books, New York (2017) Nicolaidis09A. Nicolaidis, Categorical Foundation of Quantum Mechanics and String Theory, Int. J. Mod. Phys. A24, 1175-1183 (2009)§ AUTHOR INFORMATIONAffiliations Qualcomm, 5775 Morehouse Drive, San Diego, CA 92121, USAJ. M. Yang ContributionsJ.M.Y. designed the study, conceived the ideas, performed the mathematical calculation, and wrote the manuscript.Competing interestsThe author declares no competing interests as defined by Nature Research, or other interests that might be perceived to influence the results and/or discussion reported in this paper. | http://arxiv.org/abs/1706.01317v8 | {
"authors": [
"Jianhao M. Yang"
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"categories": [
"quant-ph"
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"primary_category": "quant-ph",
"published": "20170527065058",
"title": "Quantum Mechanics from Relational Properties, Part I: Basic Formulation"
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APS/123-QED [email protected]^1Department of Physics, Tohoku University, Sendai 980-8578, Japan. ^2Department of Materials Science and NanoEngineering, Rice University, Houston, TX 77005-1892, USA. The optical properties of a multilayer system of dielectric media with arbitrary N layers is investigated. Each layer is one of two dielectric media, with thickness one-quarter the wavelength of light in that medium, corresponding to a central frequency. Using the transfer matrix method, the transmittance T is calculated for all possible 2^N sequences for small N. Unexpectedly, it is found that instead of 2^N different values of T at the central frequency (T_0), there are either (N/2+1) or (N+1) discrete values of T_0 for even or odd N, respectively. We explain the high degeneracy in the T_0 values by defining new symmetry operations that do not change T_0. Analytical formulae were derived for the T_0 values and their degeneracy as functions of N and an integer parameter for each sequence we call “charge”. Additionally, the bandwidth of the transmission spectra at f_0 is investigated, revealing some asymptotic behavior at large N. PACS numbers .42.25.Bs,78.20.Bh,78.67.Pt Hidden symmetries in N-layer dielectric stacks Riichiro Saito^1 December 30, 2023 ==============================================§ INTRODUCTION The advances in electronics have enabled us to control electron transport through materials, allowing us to develop electronic devices, such as transistors and diodes. However, due to the temporal and spatial limitations of electrons, transporting information using electrons for long distance is not efficient. Light can be used for such purposes as the carrier of information instead of electrons <cit.>. Nowadays, controlling the propagation of light has been the main subject in optical engineering, which is known as photonics <cit.>. Similar to electronics, in photonics, one tries to modify how light propagates through materials, including how one can allow or prevent the propagation of light, or localize the light <cit.>, which can be useful to amplify the electric field. To achieve this, one can use a multilayer system consisting of N layers of dielectric media varying in one-dimension. We will refer to this system as a multilayer stack. By manipulating the sequence of dielectric media in one dimension, one can control how light propagates through it <cit.>.Previous studies have shown that sequences of dielectric media with a periodic structure, known as photonic crystals (PCs), or generated based on fractal patterns can control the propagation properties of light through the multilayer stack, such as transmission (T) spectra, group velocity and dispersion <cit.>. One can expect some desirable optical properties, such as a very sharp and localized peak in the T spectrum or high electric field enhancement at a specific point, which allow us to develop optical filters, widely known as Fabry-Perot resonators <cit.>, and optical switches <cit.>. One can also realize perfect dielectric mirrors based on a multilayer stack, which are known as Bragg reflectors <cit.>, as well as structures based on them called Bragg-grating filters <cit.>. Furthermore, if one adds a conducting layer inside the multilayer stack, such as metal or graphene, one can also control the absorption of the light intensity as a function of Fermi energy of the metal layer <cit.> and we can expect enhancement of the absorption due to the high electric field enhancement inside the multilayer system. However, these previous studies focus only on some specific sequences of dielectric media, such as alternating and periodic sequences or Fibonacci and Cantor sequences <cit.>. The general optical properties for any arbitrary sequence in a multilayer stack have not yet been discussed as far as we know, simply because we did not have a systematic analysis to understand the phenomena for any arbitrary sequence.In this study, we investigate the optical properties of a multilayer system consisting of arbitrary sequences of N layers, in particular the transmittance of light T through the system. In this work, the N layers are made of two kinds of dielectric media. In contrast to the previous studies, which discussed only very specific sequences, we calculate T for all possible 2^N sequences. Hence, our system includes all previously mentioned sequences. One might think that there is no pattern in T for an arbitrary sequence of the N-layer stack. In this work we found that instead of 2^N different values, at a particular central (or resonant) frequency f_0, there are either (N/2+1) or (N+1) discrete values for even or odd number N, respectively, provided we select the thickness of each layer to be one-quarter the wavelength of light in that layer corresponding to f_0. This high degeneracy generally implies the existence of hidden symmetry operations for exchanging the dielectric layers in the stack. In particular, we will define a new integer parameter called “charge” which is invariant for the operations. We will show that all T values at f_0 are given by the “charge” and understanding the origin of these hidden symmetries and patterns can be useful for finding and designing optimal sequences, especially for systems with large N. Our paper is organized as follows. In Sec. II we will describe our method to calculate the T of a multilayer system with arbitrary N layers of dielectric media. In Sec. III we will show our results and explain the symmetry operations of the multilayer system. We will also provide the analytical formula of T at f_0 as a function of “charge” in Sec. III, as well as briefly discuss the bandwidth of the T spectra (i.e. how sharp the peak at f_0 is). We will give our conclusion in Sec. IV. All the mathematical proofs are given in the Appendix.§ METHOD In Fig. <ref>, we show a schematic picture of our multilayer system consisting of N layers of dielectric media where the i-th layer, L_i, is one of two dielectric media that are labeled by A and B, with refraction indices n_A and n_B, respectively. The thickness of L_i is selected as ℓ_i=λ_0/4n_i, where λ_0=c/f_0 is the wavelength of light in vacuum with frequency f_0, which is chosen as the central frequency, and n_i is either n_A and n_B. Hence, we have 2^N possible sequences of L_1L_2...L_N, for example with N=6, we have 64 different possible sequences.We assume that the incident light I is normal to the surface of the layer. The reflectance and transmittance of light, R and T respectively, can be calculated by the transfer matrix method <cit.>. By using the transfer matrix method, we can relate the electromagnetic (EM) fields of light between any two different positions without knowing the multiple reflection processes between them in detail. This method has been used in previous studies of propagation of a wave inside varying media <cit.>. We will briefly show the transfer matrix method as below.In Fig. <ref> we define the electric field of left- (-) and right- (+) going waves from z=0 to z=∑_i=1^Nℓ_i. The light propagates in the z-direction and the electric field is chosen to be in the x-direction. In this case, the magnetic field lies in the positive (negative) y-direction which we show as red dots (crosses) in Fig. <ref>. E^(i)_+ and E^(i)_- are electric field amplitude at the leftmost edge of L_i for right- and left-going waves respectively. In this paper, L_0 and L_N+1 are taken as vacuum. Therefore, E^(0)_+ and E^(0)_- denote the incident and reflected electric fields, respectively, while E^(N+1)_+ is the transmitted field.The electric field in L_i as a function of z_i (local z-coordinate, z_i=0 at the leftmost edge of L_i) is given byE^(i)(z_i)= E_+^(i)e^ik_iz_i+E_-^(i)e^-ik_iz_i,where k_i=2π n_i/λ is the wavevector of light with wavelength λ in L_i. Eq. (<ref>) means that the electric field can be written as a superposition of right- and left-going electromagnetic waves. The magnetic field is related to the electric field by the following equation: H^(i)(z_i)=iωε_0 n_i^2 ∫E^(i) dz_i. Thus the magnetic field in L_i as a function of z_i is given byH^(i)(z_i) =ωε_0 n_i^2/k_i(E_+^(i)e^ik_iz_i-E_-^(i)e^-ik_iz_i). The total electric and magnetic fields are continuous at the interface between L_i and L_i+1, so in terms of amplitude, E^(i) = E^(i+1) and H^(i) = H^(i+1). Let us take for example the interface between layer 0 and layer 1 shown in Fig. <ref>. Using Eqs. (<ref>)-(<ref>) and the above, we getE_+^(0)+E_-^(0) =E_+^(1)+E_-^(1), E_+^(0)-E_-^(0) =k_0/k_1(n_1/n_2)^2(E_+^(1)-E_-^(1)).From Eq. (<ref>) and (<ref>), we can form a matrix that relates the electric fields across the interface,[ [ E_+^(0); E_-^(0) ]]= 1/2[ [ 1+β_0 1-β_0; 1-β_0 1+β_0 ]] [ [ E_+^(1); E_-^(1) ]], where β_i denotesβ_i=k_i/k_i+1(n_i+1/n_i)^2=n_i+1/n_i. After entering L_1, the right-going light propagates through L_1 until it hits another interface with L_2. During the propagation inside L_1 (from z=0 to z=ℓ_1), the electric field changes only by its phase. From Eq. (<ref>), we can form another matrix that relate the electric field at z=0 and z=ℓ_1 inside L_1,[ [ E_+^(1); E_-^(1) ]]= [ [ e^-ik_1ℓ_10;0e^ik_1ℓ_1 ]] [ [ E_+^(1'); E_-^(1') ]] ,where E_+^(1') and E_-^(1') are the electric fields in L_1 at z=ℓ_1. We can combine the matrices of Eq. (<ref>) and Eq. (<ref>) to get[ [ E_+^(0); E_-^(0) ]]= M_0 P_1 [ [ E_+^(1'); E_-^(1') ]],whereM_i = 1/2[ [ 1+β_i 1-β_i; 1-β_i 1+β_i ]], P_i = [ [ e^-ik_iℓ_i0;0e^ik_iℓ_i ]].M_i and P_i are called matching and propagation matrices, respectively. The product of M_0P_1 in Eq. (<ref>) is known as the transfer matrix <cit.>.This transfer matrix describes the propagation of incident light from vacuum L_0 through L_1. If we have multiple layers, we can continue the multiplication of M_i-1 and P_i for i=2,…,N. We can write the transfer matrix for an N-layer system as follows,[ [ E_+^(0); E_-^(0) ]]= M_0P_1M_1⋯ P_NM_N [ [ E_+^(N+1); 0 ]],where we do not expect any left-going light coming to the system at L_N+1. The product of M_i-1 and P_i in Eq. (<ref>) can be expressed by a 2× 2 matrix as follows,[ [ E_+^(0); E_-^(0) ]] = [ [ a b; c d ]] [ [ E_+^(N+1); 0 ]]which gives us the transmittance of light T,T =|E_+^(N+1)/E_+^(0)|^2=|1/a|^2.Using the transfer matrix method, we can calculate the T for any arbitrary sequence. In the next section we show numerically calculated T's for all 2^N different sequences. § RESULTS AND DISCUSSION§.§ Transmittance of lightThe transmittance T as a function of incident frequency f was calculated by MATLAB. We choose dielectric constants (or relative permittivities) ε_A and ε_B of the two dielectric media to be 4 and 2.25 respectively. Index of refraction is n = √(με), with relative permeability μ≈ 1, so n_A and n_B are taken to be √(ε_A)=2 and √(ε_B)=1.5, respectively, and used throughout this paper for simplicity. Examples of common real materials with refractive indices very close to these include silicon nitride (Si_3N_4) for n_A <cit.> and silica or acrylic glass for n_B. As mentioned before, the thickness of each layer is one-quarter the central wavelength λ_0 in that layer, i.e. ℓ_i=λ_0/4n_i. If we choose f_0=2 THz, then ℓ_A=18.75 μm and ℓ_B=25 μm. Note as a matter of convention that in this paper, the sequence L_i's are represented by a string of A's and B's, such as ABAABBA.In Fig. <ref>, T is plotted as a function of frequency normalized to central frequency f_0 for all 16 possible 4-layer sequences. We also show the same plot for 6-layer sequences in Fig. <ref>. It is noted that the shape of the spectra does not change for different f_0's, sinceℓ_i's also change accordingly, hence why we plot T as a function of f/f_0. The first thing that is noticed in Fig. <ref> is that there are not actually 16 unique spectra, but only 10, by counting the number of curves on the graph. Upon investigation, it is realized that sequences that are mirrored versions of each other, e.g. AABA and ABAA, would produce identical T spectra. This is not too surprising, since light propagating through the sequence one way is essentially equivalent to light propagating through the mirrored sequence the other way (or the time-reversal symmetry of T <cit.>). Detailed proofs of this mirror symmetry can be found in Appendix B1.It is also noticed that the curves seem to converge at three points at f_0. To investigate this further, the T spectra for all 64 possible 6-layer sequences are calculated, and can be seen in Fig. <ref>. We found that due to mirror symmetry, there are only 36 unique spectra. By considering the number of symmetric or “palindromic" sequences, which are invariant under mirror symmetry, we determine the number of unique spectra for an N-layer system to be:Even N2^N/2+ 1/2(2^N-2^N/2)Odd N2^N+1/2+ 1/2(2^N-2^N+1/2). With the greater number of spectra, it is clear that they are converging to 4 T's at f_0, in the case of N = 6. This cannot be accounted for only by “mirror symmetry”, because different spectra give the same T at f_0, hereafter denoted T_0. This is a surprising result, because by changing one layer from A to B, for example, one would expect the complex interactions of internal multiple reflections to completely change, and thus have a completely different T. Indeed, this is the behavior at frequencies other than f_0, where we see many non-degenerate spectra. The high degeneracy at f_0 implies there are hidden symmetries besides simple mirror symmetry to be foundin the sequences, giving rise to the degeneracy.In order to begin finding patterns and understanding this phenomenon, the number of unique T_0 values as a function of N is calculated and listed for N=1 through 12, and shown in Table <ref>.There is a clear pattern in the number of T_0 values, but different patterns for even and odd N. It was conjectured that the number of T_0 values for even N is N/2+1, and for odd N, N+1, which are also shown in Table <ref>. These numbers are proved in subsection C Formula for T_0.The sequences which all give the same T_0 are manually tabulated in Table <ref> for N = 6, all 64 sequences. The number of sequences at each T_0, which we may call the degeneracy, is listed as well. As an example, there are 20 6-layer sequences that gave a T_0 of 1.0 (perfect transmittance). These are listed in the first part of Table <ref>.It is not at all obvious what these sequences with the same T_0 have in common, i.e. how they are related by symmetry operations like mirroring, hence hidden symmetries is an apt name. As we begin to find these symmetries, it is clear that even and odd N do not have the same symmetries. Since the symmetries for even N seemed less elusive, we focus our efforts on finding all the hidden symmetries for even N in the next subsection that can explain how sequences give the same T_0. These 20 sequences also serve as a prototypical example demonstrating why all the symmetry operations are needed. §.§ Symmetry for even NIn the previous subsection, we discuss how the mirror of a sequence produces the same T spectrum as the original sequence for all frequency, and so in particular, it also produces the same T_0 value. Mirror symmetry is schematically represented in Fig. <ref>(a), and can explain how 12 of the 20 sequences (in 6 pairs) are related. This symmetry exists in both even and odd N. The rest of the symmetries are valid only at f_0 and for even N. The first of the hidden symmetries is conjectured by looking at groups of sequences like AAAABB, AAABBA, and AABBAA. These are cyclic permutations of one another, so we call this “cyclic symmetry”. A schematic representation is seen in Fig. <ref>(b). Straight away, this is a more complex form of symmetry than mirror or inversion, because it's not a single symmetry operation. Rather, it's a set of symmetries dependent on the number of layers you cycle, from 1 to N-1 (cycling N layers would be the identity operation). The invariance of T_0 under cyclic symmetry can be proven directly and rather elegantly from properties of the transfer matrix, as shown in Appendix B2. The next symmetry is swapping all A's with B's and vice versa, which we call “inversion symmetry”, and schematically represented in Fig. <ref>(c). Inversion symmetry can explain how sequences like AABAAB and BBABBA, which seem very different at first glance, are related to each other. It turns out that this symmetry is very difficult to prove directly from operations on the transfer matrices, which is how mirror and other symmetries are proven, and we are not able to do it. Instead, the proof of inversion symmetry becomes trivial with the formula for T_0 given in subsection C.Having cyclic symmetry, along with mirror and inversion, the 20 sequences that gives T_0=1.0 can be split into three “cycles”, represented by AAAAAA, AAAABB, and AABAAB, such that any two sequences in the same cycle can be related by at most two of these symmetry operations. An example would be AAABBA and BBAABB, both in the cycle AAAABB, and related by an inversion and cyclic permutation. However, we still have no symmetry operation to related sequences from different cycles.The last two symmetries essentially explain how to connect between these “cycles”. They are rather unusual, in that there aren't analogous operations in discussing, for example, point symmetry group of molecules, which does not change numbers of A and B. The first is arbitrary permutations of double layers, that is, break up the sequence into N/2 two-layer segments (remember this symmetry is only for even N), such as (AA)(BA)(AB), and permute those segments arbitrarily, shown in Fig. <ref>(d). We may call this “double permutations”. This along with cyclic permutation allows us to get between AAAABB cycles and AABAAB as follows: we permute (AA)(BA)(AB) to (AA)(AB)(BA), then cyclical permute one place to the right to AAAABB. The proof that double permutation does not change T_0 is given in Appendix B3.The last symmetry needed is inversion of a pair of layers that are the same either (AA) or (BB), which we call “pair inversion”, shown in Fig. <ref>(e). This is easy to understand, and allows us to get between AAAABB and AAAAAA, where the BB inverted to AA. AABAAB can also be turned in AAAABB by inverting the second AA in the first sequence so that it becomes AABBBB, inverting the whole sequence to BBAAAA, the cyclically permuting two places to the left. Like for inversion symmetry, the proof that pair inversion does not change T_0 is trivial once we define the formula for T_0 seen in subsection C.These symmetry operations and their products can now relate any two sequences with even N that have the same T_0 value. However, we have not yet investigated what those actual T_0 values are. In the next section, an analytical formula for T_0 shall be derived, for both even and odd N, as functions of a parameter associated with each sequence we shall call “charge”. §.§ Formula for T_0Looking at all the sequences for even N, with different T_0 values, two obvious patterns immediately jump out. The two unvarying sequences (AAAA... and BBBB...) always have the highest T_0=1.0. On the other hand, the two alternating sequences (ABAB... and BABA...), and only those, always have the lowest T_0, decreasing as N increases. This structure is known as a dielectric mirror or Bragg reflector, since having the lowest T_0 means it has the highest R=1-T <cit.>. This is the starting point and clue that lead to the theory of “charge” for sequences in general. The definition of “charge” can be drawn by considering that the unvarying sequence can be thought of as being composed of blocks of AA or BB repeated. Similarly, the alternating sequences are blocks of AB or BA repeated. These are the two extremes, and every sequence can be thought of as being composed of some combination of these four blocks, observing that their T_0 values fall somewhere in between as well.The basic idea is that we assign a “charge” to each of these blocks: AB is +1, BA is -1, and AA and BB are both 0 as shown in Fig. <ref>(a). Note that this is not inherently related to electrical charge (though we are investigating a potential link), but the trichotomy of values and, as will be seen, the behavior of charges “cancelling” is entirely analogous, so “charge” is an apt name. Given these assignments, the total “charge” of an even N sequence is straightforwardly defined by adding together the charge of each 2-layer block, as illustrated in Fig. <ref>(a). We shall denote the total “charge” of a sequence as q.Then it can be seen that all sequences with the same T_0 also have the same |q| as shown in Table <ref>. For example, the 20 prototypical 6-layer sequences with T_0=1.0 have |q|=0. Moreover, each |q| value corresponds of only one T_0 value. We now see that all of the symmetries operations discussed in the previous section simply preserve |q|. In fact, any operation defined in subsection B on a sequence that doesn't change |q| would be a symmetry operation that doesn't change T_0.We know that the lowest |q| = 0 gives the highest T_0=1.0. We also know that the highest |q|=N/2, corresponding to the alternating sequences, gives the lowest T_0. Through some intuition and careful algebraic manipulation, the following formula for T_0 as a function of |q| (and fixed ε_A, ε_B) was derived.T_0 = 4(ε_Aε_B)^|q|/(ε_A^|q|+ε_B^|q|)^2 A full proof is given in Appendix C. This equation can be considered a generalization of the equation for reflectance given by Orfanidis in Chap. 6 of <cit.>, wherein only the max |q| is considered. With |q|=0, T_0 = 4(1)/(1+1)^2 = 1.0, as expected. As |q| increases, the square in the denominator makes it increase faster than the numerator, meaning T_0 decreases.Proof of inversion symmetry follows as a direct consequence, since Eq. (<ref>) is symmetric with respect to ε_A and ε_B. Pair inversion is simply replacing AA with BB and vice versa, which has no effect on q and thus T_0, since both have 0 “charge”. It is also clear now why there are (N/2 + 1) T_0 values for even N, as conjectured. Given even N layers, each sequence is composed of N/2 blocks, so the maximum |q| is N/2. Every integer from 0 to N/2 is a possible |q| value, so there are N/2+1 values, each corresponding to a different T_0 value.We finally tackle the question of odd N sequences. We found that the patterns are too difficult and non-obvious to study in terms of symmetry. However, the theory of “charge” offers a simpler yet more powerful tool to understand the patterns. First, we extend the definition of total “charge” q to odd N. The first N-1 even number of layers can have “charge” assigned exactly as for even N sequences. All that remains is one extra layer, which is either A or B. We assign a “charge” of 0 to A and -1 to B, which is added on to the “charge” of the first N-1 layers, to get the total “charge” q as shown in Fig. <ref>(b). (Note that the assignment of 0 and -1 was somewhat arbitrary, it could also work with 1 and 0, but the formula below would be slightly different.)The formula for T_0 for odd N is somewhat more tricky, but we get the following expression, where proof is given in Appendix C. T_0 = 4ε_A^|q+1|ε_B^|q|/(ε_A^|q+1|+ε_B^|q|)^2 We first note that it is not symmetric with respect to ε_A and ε_B, explaining the lack of inversion symmetry that was noticed when initially looking for patterns. Second, there is no value of q for which the expression reduces to 1 for any ε_A and ε_B except for ε_A=ε_B=1 (vacuum), like |q|=0 with even N, meaning perfect transmittance is not guaranteed. The fact that we get T_0=0.99988 very close to 1, was purely a coincidence in our choice of ε_A and ε_B. Relatedly, T_0 eventually decreases as |q| increases, though not monotonically as with even N.Third, the function is not even in q, i.e. +|q| and -|q| give different T_0 values. Finally, we can explain why there are N+1 T_0 values for odd N, as conjectured. The first N-1 layers have can have “charge” of 0, ±1, …, ±N-1/2, a total of 2N-1/2+1=N values. The final layer either doesn't change charge (if A) or decreases it by 1 (if B). For almost all of them, decreasing by 1 simply gives the charge below, no extra q values, except for the lowest charge, -N-1/2, where decreasing by 1 produces q = -N+1/2. Hence there are (N+1) q values, each corresponding to a different T_0 value. §.§ DegeneracyThe last unsolved question is, for a given N, how many sequences there are for each T_0 value, which we call the degeneracy at that T_0 (d_N for even N and d'_N for odd N). The first step to understanding the pattern is noticing a connection to Pascal's triangle and the binomial coefficients. In particular, the number of sequences at T_0=1.0 for every even N seemed to be a central binomial coefficient: 1, 2, 6, 20, 70, …. With this as the starting point, the following combinatorial formulae were inductively derived, by calculating degeneracies at each T_0 for increasing N. d_N(|q|) = NN/2 for|q| = 0 2NN/2+|q| for|q| = 1, 2, …, N/2d'_N(q) =NN+1/2+qforq = 0, ±1, …, ±N-1/2, -N+1/2 Armed with our understanding of “charge”, the proof of this becomes a problem of combinatorics. Essentially, we can count the number of ways to get q in N layers given the number of ways to get q-1, q, and q+1 in N-2 layers, form a recurrence relation, then relate this to the binomial coefficients. A full proof is given in Appendix D. §.§ Bandwidth In this section, we investigate the how the sequence affects the bandwidth of T spectra at f_0. Finding sequences with the sharpest peak (i.e. narrowest bandwidth) at f_0 is useful for application such as optical filter. However, we will not discuss about how the optical filter is realized, rather we will discuss about the behavior of the bandwidth and the pattern that gives the sharpest T spectrum.Firstly, we are only looking at sequences with the highest T_0, that is, with |q| = 0 for even N-layer sequences, where T_0=1.0. For odd N, there is in general no value of q that gives T_0= 1.0. The “charge” q that gives the highest T_0 varies as a function of ε_A and ε_B. To find this q, we differentiate Eq. (<ref>) (T_0 for odd N) to find the maximum, and getq_max=round(log(ε_A)/log(ε_B/ε_A)),rounding because q can only take integer values. Then after setting ε_A and ε_B, we consider only sequences with this q_max for odd N. We define fractional bandwidth of a spectrum normalized to f_0 as Δ F=Δ f/f_0, where Δ f is the full width at half maximum (FWHM). In Fig. <ref> we plot the minimum Δ F for each N as a function of N on a log-log plot. The first thing to notice is that the minimum Δ F (the narrowest spectrum) decreases with increasing N. So on a very general level, to get a sharper peak in T spectrum at f_0, we need to have more layers, as can be expected.At first, the Δ F values appear to follow roughly a straight line on the log-log plot, indicating a power law relationship. However, even in Fig. <ref>, the points clearly start to curve. So Δ F is calculated for larger N and shown in Fig. <ref> as a log-linear plot. The values of ε_A and ε_B were also varied to see how Δ F changes. As seen in Fig. <ref>, the linear relationship between Δ F and N in log-linear plot immediately jumped out, indicating exponential relationship in the linear plot. We plot only even N for clarity, as odd N has the same long-term linear behavior, parallel to even N but shifted upwards slightly.Although the points clearly don't follow a straight line for small N, they do show very regular behavior as N gets larger. The exponential fit (with ε_A=n_A^2 and ε_B=n_B^2 fixed) for the asymptotic behavior was found to be the very simple equation:Δ F = e^-ρ N-2,whereρ = n_A-n_B/n_A+n_Bis called the (elementary) reflection coefficient <cit.>.Several things may be taken from this. First, it is useful in the design of optical filters. For example, if we need a filter with a specified Q factor of 10^5 (i.e. Δ F = 10^-5), and knowing the refractive indices n_A and n_B of the materials we have available and thus ρ, we can easily solve Eq. (<ref>) for an estimate of the minimum number of layers N required. This is graphically represented by the dotted lines in Fig. <ref>. Second, it demonstrates and moreover explains why having materials with a greater difference in refractive index is better for narrower filters, since that maximizes ρ, thereby minimizing Δ F for a given N.It is worth pointing out that an expression for bandwidth for this type of filter is given by Macleod in <cit.>. However, our expression is considerably simpler, making it much easier and quicker to solve for N, the only tradeoff being a worse fit for small N. We would like to emphasize that Eq. (<ref>) is an “empirical" fit, however, its simplicity and accuracy suggests it should be possible to derive analytically with some suitable approximations to account for its asymptotic nature. Though we offer no such derivation in this paper, we would conjecture it can be derived from the expressions given in <cit.>.We also found the pattern for which sequence gives the narrowest Δ f for any given even N. It is explained in Table <ref>. This pattern holds for any ε_A and ε_B with ε_A > ε_B (otherwise simply swap A's and B's).It may be of interest to note that the second narrowest sequence for any given even N follows a very simple pattern too. Simply replace the middle two layers with AA if it's BB, and vice versa. For example, for N = 8, the narrowest spectrum is given by the sequence ABABBABA, the second narrowest is given by ABAAAABA. These in fact exactly correspond to the high-index and low-index cavity all-dielectric filters described by Macleod in <cit.>, and we have now conclusively shown, by calculating all 2^N sequences, that they are the “best" possible filters (in terms of bandwidth) for a given N.A similar albeit more complicated pattern was found for odd N. However, because the q that gives the highest T_0 varies as a function of ε_A and ε_B, so too does this pattern. Thus, we feel it is not worth describing here the rule for odd N, since it only works for some particular values of ε_A and ε_B, along with the fact that the narrowest bandwidth for any odd N is larger than that for the even N-1.§ CONCLUSION In conclusion, we have found that, somewhat unexpectedly, the transmittance T of N-layer dielectric stacks are highly degenerate and discrete at the central frequency f_0. We have found all hidden symmetry operations to sufficiently explain how all even N sequences with the same T_0 are related. Furthermore, T_0 depends only on the “charge” q of a sequence, with formulae for T_0, for both even and odd N, derived as functions of q. This is a simpler, more elegant way to explain why different sequences have the same T_0 value. The degeneracy at each T_0 is explained by combinatorics, again with formulae derived as functions of q.There is a lot of potential for future work, in various directions stemming from this initial discovery and investigation. A well-established mathematical tool to analyze and understand symmetries is group theory. In fact, we have already started in this endeavor, trying to form a group of symmetry operations both for N=2 and for N=4, then analyzing the structure using representation theory to extract the degeneracies. However, we run into issues such as not being able to include some of the more exotic operations in the group, and the predicted degeneracies of irreducible representation do not match the degeneracies that we calculated. These problem might be related to the fact that we discuss transmittance but not transmission coefficient or any eigenvalue of linear operators that commute with symmetry operations. Alternatively, the symmetry operations could potentially have the structure of a groupoid, a generalization of a group.Recalling that PCs may be used for optical filters, we want sequences with both high T and a sharp peak at f_0. Now that we understand how to find T_0 just by looking at the sequence, we only have to consider a much smaller subset of sequences, those with low q and high T_0. The next big step is to continue our preliminary investigation into how bandwidth depends on the sequence, e.g. we would want a sharp peak for a filter. If we can fully understand how that changes under the symmetry operations as well, we could imagine creating an algorithm to find the optimal sequence for any kind of T spectrum desired for a given N, or designing sequences satisfying some given requirements (e.g. Q factor > some value).An interesting and potentially fruitful area to investigate is whether there is any physical meaning to this artificial value associated with a sequence we call “charge”. Again bringing it back to physical applications, PCs can also have E field enhancement, which is useful in the enhanced Raman spectroscopy to get a stronger signal. Preliminary investigations suggest that there may be a relationship between “charge” or “cumulative charge” in the sequence, and the E field within the PC. For example, a sequence like ABABAB...BABABA has overall |q|=0 so T_0=1.0. But right at the middle of the sequence, it has very high “cumulative charge”, and correspondingly, a very high E field at the middle point. Further investigation and understanding could allow us to design PC sequences with E field enhancement at any position we desire.Before finishing the story, we would like to point out similarities of the present story to a general physics in which odd and even number of particles give a different symmetry (or statistics). Although it is beyond our ability, it is our pleasure if the reader has an interest in such hidden symmetries for applying to general physics. H. L. thanks J. Kono, C. J. Stanton, S. Phillips, K. Packard, K. Ogawa, and U. Endo formaking the Nakatani RIES program possible. M.S.U. is supported by the MEXT scholarship. R.S. acknowledges JSPS KAKENHI Grant Numbers JP 25107005 and JP 25286005. § Appendix omitted in this version (waiting until publication). | http://arxiv.org/abs/1705.09535v1 | {
"authors": [
"Haihao Liu",
"M. Shoufie Ukhtary",
"Riichiro Saito"
],
"categories": [
"physics.optics",
"math-ph",
"math.MP",
"physics.app-ph"
],
"primary_category": "physics.optics",
"published": "20170526112728",
"title": "Hidden symmetries in $N$-layer dielectric stacks"
} |
Discrete Boltzmann method with Maxwell-type boundary condition for slip flow Zhihua Chen^2 December 30, 2023 ============================================================================ § INTRODUCTIONGravitational plane waves arise generically as the Penrose limits<cit.> of an arbitrary curved spacetimeand an associated null geodesic γ. They are a truncation of the full background spacetime which captures the essential geometry of geodesic deviation around γ.Expressed in Brinkmann coordinates, the metric is ds^2 = 2 du dv + h_ij(u) x^i x^j du^2 + (dx^i)^2.These Brinkmann coordinates are identified as Fermi normal coordinates associated with thenull geodesic γ in the original spacetime <cit.>, while the profile function h_ij(u) is given by the curvature component R_uiujwhich appears in the Jacobi equation describing geodesic deviation.This is precisely the feature of the geometry required to describe vacuum polarisation, or loop, effects on the propagation of quantum fields in curved spacetime <cit.>. This is evident in a worldline formalism, or equivalently using the Schwinger-de Witt representation, where the propagator is expressed in terms of classical trajectories. Loop corrections are determined by the geometry of the set of geodesics neighbouring the classical path – that is, by the geodesic congruence in which the classical null geodesic is embedded.In previous work, quantum loop effects have been shown to lead to a variety of novel effects related to the induced violation of the strong equivalence principle, including ultra-high energy scattering <cit.> and the generation of matter-antimatter asymmetry<cit.>, challenging conventional understanding of how fundamental concepts such as causality, analyticity and unitarity are realised in quantum field theory in curved spacetime <cit.>.The propagation of quantum fields in a general curved spacetime backgroundis therefore determined by the geometry of (null) geodesic congruences in thecorresponding gravitational plane wave realised in the Penrose limit. This is the central motivation for the study of plane wave geometry and null congruences presented here, although the novel geometry we will describe is in principle of much wider relevance. As discussed briefly in section <ref>, this could also include potential applications to the detection of astronomicalgravitational waves. Conventionally, studies of the propagation of, for example, electromagnetic fields in curved spacetime have centred on electromagnetic plane waves, in a geometric optics description where the tangent vector fields characterising the null rays are given as the derivative of the plane-wave phase, k_ = ∂_Θ, the rays being normal to surfaces of constant phase. Such `hypersurface-forming' rays are therefore by definition a gradient, rotationless, flow. However, in general, geodesic congruences are described by the Raychoudhuri equations in terms of three optical scalars θ,and ω, representing expansion, shear, and rotation or `twist'.Clearly the twist vanishes for a gradient flow.In this paper, we establish the mathematical framework to allow a more general analysis of quantum field propagation in curved spacetime by developing the geometry of twisted null congruences in gravitational plane wave backgrounds. An important element is the relation of Brinkmann and Rosen coordinates.Whereas the originalBrinkmann coordinates for the plane wave metric (<ref>) are more fundamental, the Rosen description is tailored to a choice of geodesic congruence. This explains why many calculations of loop effects in QFT are most simply and elegantly performed in Rosen coordinates. Here, we introduce the Rosen description appropriate to a twisted null congruence, in which case the metric takes the novel form,ds^2 = 2 du dV - 2 X^a _ab dX^b du + dX^a _ab(u) dX^b.Here _ab = (E^T E)_ab, where E^i_a(u) is the zweibein relating thetransverse Brinkmann and Rosen coordinates, x^i = E^i_a(u) X^a, and thetwist enters through = E^t ω E. The new feature is of course the off-diagonal term involving the twist.The origin of twist and the subtleties of the Brinkmann-Rosen relation are first explained, then we study the Rosen metric (<ref>) in detail in its own right, finding the geodesics, wave equation solutions, the nature of the isometries and the explicit form of the van Vleck-Morette matrix.The van Vleck-Morette (VVM) determinant, or more generally the VVM matrix, is perhapsthe most important geometric quantity influencing quantum field propagation in curvedspacetime. It enters the Schwinger-de Witt representation of the propagator and all loop effects, including modified dispersion relations, are formally expressedin terms of it <cit.>. Since it is essentially linked to the geodesic congruence, its evaluationis fundamentally altered in the presence of twist. Here, we derive the general form of theVVM matrix related to twisted null congruences, showing the equivalence of the Brinkmannand Rosen descriptions. Another important theme of the paper is the analysis of isometries. We first give an account of theisometry algebra, and its extension for the homogeneous plane waves discussed below, using the Brinkmann metric (<ref>), and show how the gravitational plane wave spacetime may be represented as a coset space <cit.>.Then, we reconsider the isometriesfrom the point of view of the Rosen metric (<ref>) and show how the form of the correspondingKilling vectors and commutation relations depend on the twist, generalising the analyses of <cit.> for the twist-free case.A particularly important class of gravitational plane waves are the homogeneous plane waves (HPWs).These are of special interest from several points of view, including their rôle as exactly solvable,time-dependent string backgrounds (see e.g. <cit.>). Our discussion of the geometry of twist in these backgrounds may therefore be of relevance also in string theory. They occur importantly as Penrose limits – in particular, HPWs of type II (see below) arise in the near-singularity region of a broad class of black hole and cosmologicalspacetimes <cit.>, <cit.>.Homogeneous plane waves (HPWs) have been classified by Blau and O'Loughlin <cit.> in a paper which is key to the present work. They fall into two types, generalising the symmetric plane waves(Cahen-Wallach spacetime <cit.>) and singular homogeneous plane waves, respectively. The corresponding metrics are, in Brinkmann coordinates,ds^2 = 2 du dv +(e^ u h_0e^- u)_ij x^i x^jdu^2 + (dx^i)^2, ( type I)andds^2 = 2 du dv + ( e^log u h_0/u^2e^-log u)_ij x^i x^jdu^2 + (dx^i)^2, ( type II)for a constant 2× 2 matrix (h_0)_ij, where ϵ_ij is the usual antisymmetric symbol. Compared to the general plane wave metric (<ref>), whose isometries satisfy a Heisenberg algebra (see section <ref>), these HPWs are characterised by an additional symmetryrelated to translations in the lightcone coordinate u. This extra degree of symmetry underliestheir importance in many applications, especially in string theory.In this paper, we focus on the geometry of twisted null congruences and the Rosen description for metrics of type I, though our methods and conclusions will carry over more or less directly to type II metrics as well. A special place in classical general relativity belongs to the Ricci-flat HPW metric of type I. This is known as Ozsváth-Schücking spacetime <cit.>, originally presented in the literature in the form (our notation, see appendix <ref>)ds^2 = 2 dU dW -2√(2) z^1 dz^2dU + (z^1)^2dU^2 + (dz^i)^2 .Clearly, while making obvious three of the isometries (translations in U, W and z^2) associated with the extended Heisenberg algebra, this choice of coordinates disguises the fact that this is a gravitational radiation metric. This becomes manifest only after evaluating the Newman-Penrosecurvature scalars and noting that it is indeed a Petrov type N spacetime. Its interest in general relativity is due to the fact that it is a vacuum (Ricci-flat) solution of Einstein's field equations which is geodesically complete and singularity free, yet is non-trivial in the sense that it is not simply flat, Minkowski spacetime. In this sense, it violates Mach's principle and we follow <cit.> in referring to it as an “anti-Mach” spacetime.As we shall see, this type of spacetime very naturally contains geodesics which fall into twisted null congruences, and we study their geometry in considerable detail in appendix <ref>.The property of twist automatically leads to null geodesics which are periodic in u, and this underlies the suggestionthat the metric (<ref>) allows the existence of closed null geodesics <cit.> (see also <cit.>).Our analysis does not confirm this, however, and we find,as expected on general grounds <cit.>, that these type N homogeneous plane wave spacetimes are indeed causal. The paper is organised as follows. In section <ref>, we introduce the essential geometry andgeodesics for a general gravitational plane wave metric in Brinkmann form, explain the origin of twist, and derive the new Rosen metric (<ref>). Homogeneous plane waves are introduced in section <ref> and their isometries and coset structure are explored.We return to the twisted Rosen metric in section <ref> and, taking it on its own merits, discuss the geodesic equations and solutions of the wave equation in this background. The Rosen isometries are described at length from different points of view in section <ref>.Section <ref> contains the evaluation of theVVM matrix in both Brinkmann and twisted Rosen coordinates and the demonstration of their equivalence. Some concluding remarks are given in section <ref>. Finally, in an extensive appendix, we give a detailed account of twisted null congruences in the the generalised Ozsváth-Schücking model,introducing co-rotating coordinates and a Newman-Penrose basis adapted to the null geodesics. § PLANE WAVES, TWISTED NULL CONGRUENCES AND OPTICAL SCALARS We begin by studying the geometry of a general gravitational plane wave and its null geodesics in theBrinkmann description, extending the conventional description to allow the possibility of twisted congruences.[ There is an extensive literature in general relativity on gravitational plane waves and theirsymmetries and geodesic structure. Some classic references on exact solutions of Einstein's equations including gravitational plane waves are <cit.>, while a comprehensive set of references may also be found in <cit.>.See also <cit.> for an early discussion of quantum field theory effects and <cit.> for recent studies of null geodesics and gravitational lensing in gravitational wave backgrounds. ]The optical scalars and their Raychoudhuri equations are introduced, and the metric is rewitten in a novel Rosen form appropriate to twisted null congruences.§.§ Brinkmann coordinates The starting point is the gravitational plane wave metric in four dimensions in Brinkmann coordinates x^ = (u,v,x^i),ds^2 = 2 du dv + h_ij(u) x^i x^j du^2 + (dx^i)^2,where the Riemann curvature tensor R_uiuj = -h_ij is determined directly by the profile function. The corresponding null geodesic equations are,ẍ^i - h^i_j x^j= 0,v̈ + 12ḣ_ij x^i x^j + 2 h_ijẋ^i x^j= 0.Here, we have immediately taken u as the affine parameter, the overdot denotes differentiationwith respect to u, and the transverse indices are i = 1,2.The solutions to (<ref>) can be written in terms of a zweibein E^i_a(u), with a=1,2, as follows:x^i= E^i_a(u) X^a, v= V - 12Ω_ab(u) X^a X^b,where we define the key quantity Ω_ij(u) byΩ_ij = (Ė E^-1)_ij ,and Ω_ab = (E^T Ω E)_ab. The zweibein E^i_a and Ω_ij are related to the profile function h_ij in the metric byh_ij = Ω̇_ij + (Ω^2)_ij .Notice that nowhere in this construction have we assumed that Ω_ij is symmetric.Now consider the null congruence of geodesics labelled by X^a, V, centred on a reference geodesic γ, with tangent vectors k^ = dx^/du.Evaluating the first integrals of thegeodesic equations, we findk^ = ẋ^ = [1; -12 x (Ω̇ + Ω^2 + Ω^T Ω) x;(Ω x)^i;]Now let z^ be the connecting vector between γ and other elements of the congruence. By definition, the Lie derivative of z^ along γ vanishes, i.e.ℒ_k z^ = k^ D_ z^ - (D_ k^) z^ ,where D_ is the ordinary covariant derivative, and sok^ D_ z^= Ω^_ z^ ,where we define[Although at this point we only require the transverse projection of Ω_, we will need the full component form later. This is readily evaluated using the covariant derivatives, with Christoffel symbolsΓ^v_uu = 12ḣ_ij x^i x^j, Γ^v_ui = h_ij x^j, Γ^i_uu = - h^i_j x^j,and we findΩ_ = [ẋΩẋ0 - (ẋΩ)_j;000; - (Ωẋ)_i0 Ω_ij;]where ẋ = Ω x.Note that this is not symmetric unless Ω = Ω^T. ]Ω_ = D_ k_ .It is then clear that with this definition, Ω_ij≡ D_j k_i coincides with theform (<ref>) above. This shows very directly how Ω_ij determines how the transverse connecting vector is parallel transported along the reference geodesic and therefore characterises the null congruence. Note that the order of indices is important in (<ref>) since Ω_ij is not necessarily symmetric.The optical scalars for this congruence are defined from Ω_ij asΩ_ij = 12θ_̣ij + _ij + _ij ,where θ is the expansion, the symmetric trace-free tensor _ij is the shear and the antisymmetric tensor _ij is the rotation or twist.Note immediately that in the case where k^ represents a gradient flow, i.e.k_ = ∂_Θ, then Ω_ij is symmetric and the congruence has vanishing twist.Such a congruence is then said to be “hypersurface-forming”. It is realised in the familiar case where we consider k_ to be the wave vector of, say, an electromagnetic wave propagating in thespacetime (<ref>), in which case Θ represents the phase; k_ is then normal to the surface of constant phase.Here, we relax this condition and allow k_ to be a general vector field. In this more general situation, Ω_ij need not be symmetric and the null congruencemay have a non-vanishing twist, _ij≠ 0.The variation of the optical scalars along the congruence is described by the Raychaudhuri equations. In fact, these can be obtained simply as a rewriting of (<ref>) as Ω̇_ij = h_ij - (Ω^2)_ij .Substituting (<ref>), we find after a short calculation the individual variations for theoptical scalars themselves, viz.θ̇ = - 12θ^2 -^2 -^2 - R_uu ,_ij = - θ_ij - C_uiuj ,_ij = - θ_ij ,where C_uiuj is the Weyl tensor.Note also the useful formula = - Ω - Ω^T,which follows directly from (<ref>) with the condition that h_ij is symmetric.§.§ Rosen coordinates An important alternative description of the gravitational plane wave is in terms of Rosen coordinates. Although the Rosen form of the metric in general has unphysical coordinate singularities, it isadapted to the nature of the geodesic congruence we wish to consider and so plays a key role in our discussion. Clearly, since it is tied to a particular choice of congruence, the Rosen metric is not unique.To make this coordinate transformation, we take the X^a, V from the geodesic solutions (<ref>) and define Rosen coordinates (u, V, X^a) asX^a= (E^-1)^a_i x^i, V= v + 12 x^i Ω_ij x^j.Expressing the Brinkmann metric (<ref>) in terms of these coordinates, we find after a shortcalculation,ds^2= 2 du dV - X(Ω̇ - E^T h E - Ė^T Ė) Xdu^2 - [dX (Ω - E^T Ė) X + X(Ω - Ė^T E) dX ] du+ dX E^T E dX.Then, since from its definition as Ω_ab = (E^T Ω E)_ab,[As far aspossible, we use boldface notation for quantities such as , ,etc. inRosen coordinates related in this way to the fundamental definitions in Brinkmann coordinates.] we haveΩ̇ = Ė^T Ė + E^T Ω^2 E + E^T Ω̇E,and using (<ref>) for h_ij, we see that the coefficient of du^2 vanishes as in the conventional construction. As usual, we also identify _ab = (E^T E)_ab as the transverse Rosen metric. The novelty comes in the remaining term, where we must distinguish Ω = E^T Ė from its transpose Ω^T = Ė^T E when the zweibein is describing a congruence with twist. Defining ω_ab = (E^T ω E)_ab, we therefore find the Rosen metricds^2 = 2 du dV - 2 X^a ω_ab dX^b du + dX^a _ab(u) dX^b. This differs from the usual form of the Rosen metric for a plane wave spacetime due to the rotation term involving the twist ω. Notice immediately that its equivalence to theBrinkmann metric (<ref>) requiresto be constant. This follows from the symmetry of the profile function, since ω̇_ab = E^T (ω̇ + Ωω + ωΩ^T)E = 0,using (<ref>). We emphasise however that with this proviso, (<ref>) still describes the same spacetime as the Brinkmann metric (<ref>) and the equivalent standard Rosen metric. It is simply expressed in terms of a different choice of the non-unique Rosen coordinates adapted to the description of twisted null congruences.The Rosen metric (<ref>) will be the basis of many further developments later in the paper, especially in the context of the special class of homogeneous plane waves. First, however, we look in more detail at the construction of the zweibeins and the origin of twist in the corresponding null congruences.§.§ Origin of twist In order for (<ref>) to be a solution of the geodesic equations, the zweibein E^i_a(u) must satisfythe oscillator equationË^i_a - h^i_j E^j_a = 0.Viewed as a second-order differential equation for vectors labelled by the index i, there are four linearly independent solutions (therefore a total of 8 solutions for the components) which we split into two sets f^i_(r)(u) and g^i_(r)(u), r = 1,2, satisfying the canonical boundary conditions[ It is interesting to note the correspondence with the quantities A(x,x') and B(x,x')introduced, for example, in (3.18) of ref.<cit.> to characterise geodesic deviation. Here,A and B satisfy `parallel' and `spray' boundary conditions in correspondencewith f and g. Note also that in this paper we have chosen the opposite sign convention h_ij = - R_uiuj from <cit.> for the Brinkmann profile function.Also note that our f^i_(r)(u), g^i_(r)(u) are the functions denoted b_i^(k)(x_0^+), b_i^*(k)(x_0^+) in <cit.>.]f^i_(r)(0)= ^̣i_r, ḟ^i_(r)(0) = 0, g^i_(r)(0)= 0, ġ^i_(r)(0) = ^̣1_r.An important role in what follows is played by the Wronskian. For example, choosing two solutions f^i_(r)(u) and g^i_(s)(u), their Wronskian is[ The Wronskian W_rs is independent of u by virtue of the fact that f^i_(r) and g^i_(s) satisfythe oscillator equation and that h_ij is symmetric, sinceẆ = f^i_(r) h_ij g^j_(s) - h_ij f^j_(r) g^i_(s) = 0.W_rs may therefore be evaluated at any value of u, and the result (<ref>) followsimmediately by using the boundary conditions (<ref>) at u=0.]W_rs = ∑_i (f^i_(r)ġ^i_(s)- ḟ^i_(r) g^i_(s)) = _̣rs ,whereas the Wronskian of two f^i_(r) or two g^i_(r) vanishes.The zweibein E^i_a which determines the geodesics is an arbitrary linear combination of the f^i_(r) and g^i_(r), i.e. the zweibeins use half of the complete set of solutions to the oscillator equation. It follows that the nature of the null congruence depends on the particular linear combination chosen.This brings us to the key point. The Wronskian associated with a particular choice E^i_a(u) for the zweibein isW_ab = (E^T Ė)_ab - (Ė^T E)_ab= (E^T (Ω - Ω^T) E)_ab= (Ω - Ω^T)_ab= 2 ω_ab .in other words, the twist ω_ab = (E^T ω E)_abin the Rosen metric.This identification of the twist with the Wronskian of the zweibeins explains how the formalism developed here generalises the corresponding discussion in <cit.>, especially in the appendix where the transformation between Brinkmann and Rosen coordinates and the link with Killing vectors was explored in detail. In particular, the `somewhat mysterious' symmetry condition ĖE^T = E Ė^T, already associated in <cit.>with the vanishing of the Wronskian(corresponding to choosing the zweibeins using oscillator solutions generating a maximal set of commuting Killing vectors) is now seen to be the restriction to congruences with vanishing twist. Indeed, the symmetry condition must hold for consistency in passing from the usual form of the Rosen metricto Brinkmann since the twist term involving ω_ab in the generalised Rosen metric is omitted, thereby assuming vanishing twist a priori.As anticipated above, the oscillator equation solutions f^i_(r) and g^i_(r) also play a key role in constructing the Killing vectors characterising the extended Heisenberg isometry algebra of the special class of homogeneous plane waves, which we discuss in section <ref>.§.§ Newman-Penrose tetrad and the Penrose limit The standard Newman-Penrose basis for a plane wave spacetime is the null tetradℓ^, n^, m^, m̅^ satisfying ℓ . n = -1, m.m̅=1,built around ℓ_ = ∂_ u, i.e. the normal to the null hypersurfaces u =const. In terms of this basis, the metric may be writteng_ = -ℓ_ n_ - ℓ_ n_ + m_m̅_ + m_m̅_ .A straightforward construction then gives,ℓ^ = [ 0; 1; 0 ] , 1.5cm n^ = [-1; 12 x^i h_ij x^j; 0 ] , 1.5cm m ^ = 1/√(2)[0;0; ^̣i1 + i^̣i2 ] . The only non-vanishing NP curvature scalars (see, e.g. <cit.>) areΦ_22 = -12 R_ n^ n^ = - 12 R_uu = 12 h_ij ,and Ψ_4 = -C_n^m̅^ n^̅̊m^ = - C_uiujm̅^i m̅^j= 12(h_11 - h_22) - i h_12 ,where notably both Φ_00 and Ψ_0 vanish.This characterises a Petrov type N spacetime.In previous work <cit.>, it proved useful to construct a Newman-Penrose basis adapted to thegeodesic γ. Here, taking k^ from (<ref>), we may define the basis vectors as[ For reference,L_ = k_ = [ 12x(h - Ω^T Ω)x, 1,(Ω x)_i;] , N_ = [ -1,0, 0; ] , M_ = 1√(2)[ - (Ω x)_1 - i (Ω x)_2, 0,_̣i1 + i _̣i2;] . ]L^ = k^ = [ 1; -12 x (h + Ω^T Ω) x; (Ω x)^i; ] , 0.4cm N^ = [0; -1;0 ] , 0.4cm M^ = 1/√(2)[ 0; -(Ω x)^1 - i (Ω x )^2; ^̣i1 + i ^̣i2;]and check explicitly that they satisfy the metric conditions L.N = -1, M.M̅ = 1. Importantly, we also impose that they are parallel-transported along the geodesic, i.e.L^ D_ L^ = 0 , L^ D_ N^ = 0, L^ D_ M^ = 0.The first merits further comment. From its definition in (<ref>), it follows thatL^ D_ L_ =(Ω - Ω^T)_ L^ + 12 D_ L^2,where the final term vanishes since L^ is null. Normally this would immediately imply the vanishing of L^ D_ L_, which (in affine parametrisation) is the geodesic equation for $̧.However, if we allow congruences with twist,Ω_is not symmetric and thisis no longer obvious. Nevertheless, using the results quoted in footnote <ref> we can check explicitly that(Ω - Ω^T)_L^ = [ 0 0 -(ẋω)_j; 0 0 0; -(ωẋ)_i 0ω_ij ][ 1;v̇; ẋ_j ] = 0,confirming the direct calculation ofL^D_L_using the covariant derivative. Although much of this workcentred on exploiting plane wave geometries as the Penrose limitsassociated with wave propagation along null geodesics in a more general spacetime, it is clearly of interest to consider the plane wave in its own right and ask what is its Penrose limit given a particular geodesic$̧. In particular, we will allow for $̧ to be a null geodesic in atwisted congruence.In <cit.> we introduced an elegant method of determining Penrose limits based on the Newman-Penrose tetrad formalism. The method relies on the correspondence of the NP tetrad associated with the chosen geodesic$̧, with the basis vectors parallel-transported along $̧, and Fermi null coordinates (FNCs). The construction in terms of FNCs <cit.> gives perhaps the best insight into the nature and properties of the Penrose limit,making clear how it captures the geometry of geodesic deviation.In particular, as shown in <cit.>, the Brinkmann coordinates describing the Penrose limitplane wave are identified as FNCs. The upshot is that the profile functionĥ_ijof the Penrose limit metric is given in terms of the components of the Weyl and Ricci tensors in the NP basis associated with$̧ as:ĥ_ij = - [12(C_LMLM + C_LM̅ L M̅) + 12 R_LL-i2( C_LMLM - C_LM̅ L M̅);;-i2( C_LMLM - C_LM̅ L M̅) -12(C_LMLM + C_LM̅ L M̅) + 12 R_LL ] . The next step is to write the NP basis L^, N^, M^, M̅^ in terms of the standard basisintroduced above. We find,L^ = - 12 x Ω^T Ω x ℓ^ - n^ + [ 1√(2)((Ω x)_1 - i (Ω x)_2 ) m^ +h.c.], N^ = - ℓ^ , M^ = - 1√(2)((Ω x)_1 + i (Ω x)_2 ) ℓ^ + m^ .Then, recalling that only the curvatures Φ_22 = -12 R_nn and Ψ_4 = - C_nm̅ n m̅ are non-vanishing in this type N spacetime, we can evaluate (<ref>). We find the elegant general resultĥ_ij = [ Re Ψ_4 + Φ_22 -Im Ψ_4;;- Im Ψ_4 - Re Ψ_4 + Φ_22 ] ,that is,ĥ_ij =h_ij .This shows the satisfying result that the Penrose limit metric is the same as for the original plane wave.An important point is that this construction of the Penrose limit does not appear to involve the twist directly. This is to be expected. The Penrose limit encodes the geometry of geodesic deviation around a chosen geodesic $̧ and so depends on the background spacetime and the geodesic$̧. On the other hand, twist is a property of the congruence, not an individual geodesic. So while the Penroselimit may depend on $̧, twist itself should play no rôle. § HOMOGENEOUS PLANE WAVES AND ISOMETRIES I – BRINKMANN As described in section <ref>, homogeneous plane waves fall into two classes <cit.> specified by particular forms for the profile functionh_ij(u). Here, we focus on the first, for which the metric isds^2 = 2 du dv + (e^ uh_0e^- u)_ij x^i x^j du^2 + (dx^i)^2,where_ijis the usual antisymmetric symbol,_ij = [01; -10 ], andh_0is a constant symmetric2 ×2matrix, which we may take to be diagonal. Notice immediately that with this form,ḣ(u) = [,h(u)].The special case wheretr h_0 = 0is Ricci-flat and is known in the literature as Ozsváth-Schücking spacetime. It is “anti-Machian”in the sense that itis a geodesically complete, singularity-free, vacuum solution of Einstein's equations which nevertheless is not Minkowski spacetime <cit.>.Compared with the symmetric space plane wave, or Cahen-Wallach space, for whichh_ij = const., the metric (<ref>) has an extra isometry. We therefore begin with a discussion of the Killing vectors and isometry algebra for this metric.§.§ Killing vectors and the isometry algebra The isometries of a Riemannian manifold are generated by Killing vector fieldsK, which are defined such that the Lie derivative of the metricg_with respecttoKvanishes,i.e. ℒ_K g_ ≡ K^g̊_, + K^_, g_ + K^_, g_= D_ K_ + D_ K_= 0.It follows that the quantityK^dx_/dłis conserved along a geodesicx^(ł), wherełis the affine parameter,i.e. d/dł(K^dx_/dł) = 0. The commutators of the Killing vectors define the isometry algebra of the spacetime. In the case of a general plane wave, with arbitrary profile functionh(u), this is theHeisenberg algebra for generatorsQ_r, P_randZ:[Q_r,Q_s]= 0, [P_r,P_s] = 0, [Q_r, P_s] = _̣rs Z,[Z,Q_r]= 0, [Z,P_r] = 0,while for the homogeneous plane wave (<ref>) this is extended with a further generatorX,related tou-translations, satisfying[X, Q_r]=_rs Q_s+h_rs(0) P_s,[X, P_r]=Q_r+_rs P_s,[X, Z]= 0.Clearly, omitting the_rsterms recovers the isometry algebra for the Cahen-Wallach spacetime withh_rs(0)identified as the constant profile functionh_ij. To see how this arises, consider the coordinate transformations which leave the Brinkmann plane wave metric invariant. First, the metric is evidently invariant under translations inv, with correspondingKilling vectorK_Z = ∂_v,i.e. vv +, K_Z = ∂_v, K^_Z = [ 0; 1; 0 ] .We can also check that for arbitraryh_ij(u), there is an invariance underuu, vv - Ḟ^i x_i, x^ix^i +F^i,providedF^iis a solution of the oscillator equation,F̈^i = h^i_j F^j. A convenient choice is to use the canonical solutionsf^i_(r),g^i_(r)given above to define the generatorsQ_r,P_rrespectively, with corresponding Killing vectors:K_Q= - ḟ^i_(r) x_i ∂_v + f^i_(r)∂_i, K^_Q = [0; -ḟ^i_(r) x_i;f^i_(r) ] , K_P= - ġ^i_(r) x_i ∂_v + g^i_(r)∂_i, K^_Q = [0; -ġ^i_(r) x_i;g^i_(r) ] .The corresponding conserved quantities are easily identified. For example, takinguas the affine paramter, we haveK^_Q ẋ_≡ g_ K^_Q ẋ^ = -ḟ^i_(r) x_i + f^i_(r)ẋ_i,and clearly, d/d u(K^_Q ẋ_) = - f̈^i_(r) x_i + f^i_(r)ẍ_i = 0,using the oscillator equation forf^i_(r)and the geodesic equation (<ref>) forx^i(u).Finally, for the homogeneous plane wave (<ref>), there is a further invariance involving translations inu,viz. uu +, vv, x^ix^i +^i_j x^j,which is easily checked using the relation (<ref>) forḣ_ij(u). The corresponding Killing vector is K_X = ∂_u + ( x)^i ∂_i, K^_X = [1;0; ( x)^i ] .The conserved quantity here is K^_X ẋ_ = g_uu + v̇ + ẋx= 12 x (h - Ω^T Ω - 2 Ω) x,and it can easily be checked using the formulae in section 2 that this is conserved along the geodesic. The invariance of the metric under the associated Lie derivative is also readily confirmed, for example:ℒ_K_X g_uu = K^u_X g_uu,u + K^i_X g_uu,i = x ḣ x + ( x)^i ( h_ij x^j + x^j h_ji) = x ( ḣ + [h,]) x = 0. The commutation relations follow directly from the form of the Killing vectors. For example, [K_Q_r,K_P_s]= f^i_(r)∂_i ( - ġ^j_(s) x_j ) ∂_v - g^j_(s)∂_j (-ḟ^i_(r) x_i) ∂_v = - _̣ij( f^i_(r)ġ^j_(s) - ḟ^i_(r) g^j_(s)) ∂_v= - W_rs(f,g) K_Z.Since the Wronskian isu-independent, it may be evaluated atu=0where we may use thecanonical boundary conditions (<ref>) for the solutionsf^i_(r),g^i_(s)specifying the Killing vectorsK_Q_randK_P_srespectively. This givesW_rs(f,g) = _̣rs, as given in(<ref>).Next, we readily find[ K_X, K_Q_r] = - ^i_(r) x_i ∂_v + ^i_(r)∂_i,where ^i_(r) = ḟ^i_(r) - ^i_j f^j_(r) .The r.h.s. is clearly of the same form as the Killing vectorsK_QandK_Pand can be written as alinear combination of them <cit.>. To determine this, note that since^I_(r)satisfies the oscillatorequation,[From the definition (<ref>), ^i_(r) = h^i_j f^j_(r) -^i_j ḟ^J_(r) ,and so, using(<ref>),^i_(r) = [,h]_ij f^j_(r) + h^i_j ḟ^j_(r) - ^i_j h^j_k f^k_(r)=h^i_j ( ḟ^j_(r) - ^j_k f^k_(r)) = h^i_j ^j_(r) . ] it can be written as a linear combination of the basis setf^i_(r),g^i_(r)as^i_(r) = a_rs f^i_(s) + b_rs g^i_(s) .The constant coefficientsa_rsandb_rsare determined by evaluating^i_(r)and^i_(r)atu=0and using the boundary conditions forf^i_(r)andg^i_(r).This gives^i_(r) = _rs f^i_(s) + h_rs(0) g^i_(s) ,and so we find[K_X, K_Q_r] = _rs K_Q_s + h_rs(0) K_P_s ,as shown in (<ref>). The corresponding result for[K_X, K_P_r]follows similarly.Notice for future use that the form of (<ref>) involving the Wronskian of the two solutions characterising theQorPtype Killing vectors, and the form of the solutionin theircommutator withK_Xin (<ref>), did not depend at that stage on the specific choice of the functionsf^i_(r)andg^i_(r), but would hold for Killing vectors defined with any solutions of the oscillator equation. Later, we will consider the commutation relations of Killing vectors with other choices of these solutions, particularly with the relatedconstruction based on the Rosen metric (<ref>). §.§ Homogeneous plane wave as a coset space As the name indicates, homogeneous plane waves are example of homogeneous spaces and as such can be described as a coset spaceG/H, whereGis the isometry group andHis the isotropy subgroup. In the model considered here, the isometry groupGis generated by the set{X, Z, Q_r, P_r}describing theextended Heisenberg algebra (<ref>), (<ref>). The isotropy group may be taken asH = {P_r}.The elements of the coset spaceG/Hare then in one-to-onecorrespondence with the four-dimensional manifold described by the metric (<ref>). The metric for the homogeneous plane wave of Ozsváth-Schücking type (<ref>) may be constructed from a knowledge of the isometry algebra𝔤of(<ref>), (<ref>) in astandard way. (See <cit.>for an analysis of the singular homogeneous plane wave (<ref>) and <cit.> for discussions of the general formalism.) The starting point is to regard the isometry groupGas a principalH-fibre bundle over the coset manifoldG/Hand define a sectionℓ(x) ∈Gin terms of the `broken' generators ( i.e. those generators inGbut not inH) as follows:ℓ(x) = e^u X e^v Z e^y.Q ,wherex^= (u,v,y^i)are coordinates onG/Hand we abbreviatey.Q = y^i Q_r ^̣r_i.Notice that the choice of section is not unique – different choices, for example in the ordering of the factors in (<ref>), correspond to different coordinate choices. We then construct the Maurer-Cartan 1-formℓ^-1 dℓ∈𝔤and expand in terms of the generators ofGasℓ^-1 dℓ = e^X X +e^Z Z +e^r Q_r +ω^r P_r,where𝐞^A = e^A_dx^are the frame 1-forms onG/H(e^A_are the corresponding vielbeins)andω^iis a localH-connection.The metric for the coset spaceG/His then ds^2 = g_AB e^A_ e^B_ dx^ dx^ ,whereg_ABis aG-invariant metric, which may be chosen to reflect the Minkowskilight-cone coordinates,g_XZ = g_ZX = 1,g_rs = _̣rs.Following through this construction for the algebra (<ref>), (<ref>), we first writeℓ^-1 dℓ = e^-y.Q X e^y.Q du + Z dv + Q.dy,where we have used the commutators[Z,X] = 0and[Z,Q_r]=0. Now we need the general result e^-y.Q X e^y.Q = X + [X,y.Q] + 12[[X,y.Q],y.Q] + …and find, using the commutators[X,y.Q]= - y..Q + y.h(0).P [[X,y.Q],y.Q]= y.h(0).y Z,that the series then terminates since[Z,Q_r]=0. This leavesℓ^-1 dℓ = (X - y..Q + y.h(0).P - 12 y.h(0).y Z) du + Zdv + Q.dy.We therefore identify the frame and connection 1-forms ase^X= due^Z= 12y^i h_ij(0) y^j du+ dv e^r= (^i_j y^j du+ dy^i) ^̣r_iω^r= (h^i_j(0) y^j du )^̣r_i. The metric is then given by (<ref>) asds^2 = 2 du dv + y^i h_ij(0) y^j du^2 +(dy^i + ^i_j y^j du ) (dy_i + _iky^k du).The final step to recover the Brinkmann metric in standard form is to make the change of variabley = e^-u xsuch thatdy = - y du + e^-u dx. The metric (<ref>) then becomes simplyds^2 = 2 du dv + x e^ u h_ij(0) e^-e u x du^2 + ( dx)^2,recovering the Oszváth-Schücking metric in Brinkmann form (<ref>). This confirms that this spacetime is indeed a homogeneous spaceG/Hdefined by the extended Heisenberg algebra.§ TWISTED ROSEN METRIC FOR PLANE WAVES We now return to the discussion of twisted null congruences in general plane wave spacetimes in section <ref> and focus here on their description in terms of Rosen coordinates. Our starting point is therefore the plane wave Rosen metric (<ref>) in the form adapted to twisted congruences:ds^2 = 2 du dV - 2 X^a ω_ab dX^b du + dX^a _ab(u) dX^b.First, taking the metric (<ref>) on its own merits, we derive the corresponding null geodesics and optical scalars. This discussion will apply to any plane wave, not necessarily homogeneous. As in section <ref>, we then specialise to homogeneous plane waves and describe the Rosen form of the Killing vectors and isometry algebra.§.§ Rosen metric and geodesics The null geodesic equations following from the Rosen metric (<ref>) are[ The non-vanishing Christoffel symbols for the metric (<ref>) areΓ^v_uu =X ^-1 X, Γ^v_ua = X^-1+ 12 X ^-1Ċ , Γ^v_ab = - 12 ,Γ^a_uu = ^-1 X, Γ^a_ub = ^-1 + 12^-1 .where for generality we have quoted the results including a u dependence for . ] Ẍ + ^-1( + 2 ) Ẋ + ^-1 X = 0 ,V̈ - 12ẊẊ + X ^-1( + 2 ) Ẋ + X ^-1 X= 0.We immediately restrict to thecase=0so that the Rosen and Brinkmann metrics are equivalent. Then, noting that + 2= 2 ,these may be written in compact form asẌ + 2 Ẋ = 0, V̈ - ẊẊ - X Ẍ = 0. The geodesic equation for the transverse coordinates can be written in the convenient form,d/du(Ẋ + 2X) = 0.The first integrals of the geodesic equations (<ref>), (<ref>) are therefore,Ẋ + 2^-1 X= ^-1ξ V̇ = η - 1/2ξ Xwith integration constantsηandξ_a, the latter equation following following fromV̈ = ( -1/2Ẋ + X ) Ẍ= -1/2ξẌ .Substituting back into the metric, we see that for anull geodesic we requireη= 0.To solve (<ref>), we introduce the path-ordered exponentialP(u,a)defined by P(u,a) =T_- exp∫_a^u dt 2 ^-1(t),withaarbitrary, whereT_-denotes anti-uordering.[ Note that the corresponding u ordered exponential P_+ defined with T_+ would satisfy Ṗ_+(u,a) =2 ^-1(u)P_+(u,a).We will also use the relation,P(u,a) ≡ T_- exp∫_a^u dt 2 ^-1(t) = T_+ exp[-∫_u^a dt 2 ^-1(t) ]. ]Its derivative has the key property,Ṗ(u,a) =P(u,a) 2 ^-1(u).This is the required integrating factor for (<ref>). WritingPẊ + Ṗ X =P ^-1ξ ,and integrating, we therefore findP(u,a)X(u) -P(u',a)X(u') = ∫_u'^u du_1 P(u_1,a) ^-1(u_1)ξ . This simplifies if we seta = u, sinceP(u,u) = 1and we findX(u) = ∫_u'^u du_1 P(u_1,u) ^-1(u_1) ξ + P(u',u) X(u').Now, recalling its definition in (<ref>), the constantξmay be evaluated for anyvalue ofu, so in particular we may writeξ = (u') Ẋ(u') + 2X(u').Substituting into (<ref>) we then haveX(u) = ∫_u'^u du_1P(u_1,u) ^-1(u_1) (u') Ẋ(u') + [ P(u',u) + ∫_u'^u du_1P(u_1,u) 2 ^-1(u_1)] X(u').Finally, using the relation (<ref>) to simplify the integral in square brackets, we find the elegant result,X(u) = ∫_u'^u du_1 P(u_1,u)^-1(u_1)(u') Ẋ(u') + X(u').We can of course now check by explicit differentiation that (<ref>) does indeedsatisfy the geodesic equation. This form of the solution is particularly useful since it isolates the dependence on the twist entirely in the path-ordered exponential. It is also convenient at this point to introduce some simplified notation, which will also make the Rosen description of the Killing vectors and isometriesin the following section more transparent.[The notation here is chosen to be as close as possible to refs. <cit.>to allow an easy comparison with the corresponding results for zero twist. The zero-twist analogue of the function p H(u)p was denoted by ψ(u) in <cit.>.] We therefore define the key functionH^ab(u)asH(u) = ∫_u'^u du_1P(u_1,u) ^-1(u_1),or equivalently,H(u) = ∫_u'^u du_1T_+exp[-∫_u_1^u dt 2 ^-1(t) ].By construction, this satisfiesḦ + 2 Ḣ = 0,together with the important identity,Ḣ + 2H =1 . We also denote the integration constants representing the Rosen position and velocity at the reference pointu'byp_a = _ab(u') Ẋ^b(u'), 2cma^a = X^a(u').Note though that for non-vanishing twist, the conserved integral of motion is actuallyξ = Ẋ + 2X = p + 2a.The geodesic solution (<ref>) is then written in compact form asX(u) =H(u) p + a. Now return to the geodesic equations (<ref>) and (<ref>) forV.Integrating (<ref>) and fixing the integration constant atu=u', immediately givesV(u) = -1/2ξ (X(u) - X(u') ) + η(u-u') + V(u').Rewriting in the simplified notation above, and withd = V(u'), we may then showV(u) = - 1/2 pH(u) p - pH ^T(u)a + η (u-u') + d.Compared with the twist-free case <cit.>,the twisttherefore enters the geodesic solution forVexplicitly as well asimplicitly through the form ofH(u). §.§ Twisted null congruences and optical scalars The simplest null congruence to consider is the original one described in Brinkmann coordinates in section <ref>, whereV,X^aare constant. The corresponding tangent vector^and covariant vector_= g_ ^are then^ = [ 1; 0; 0 ] , 2cm _ = [ 0; 1; X ] .Defining_ ≡D__, we find all the components vanish except for_ab = ∂_b _a - Γ^v_ba_v= _ab + 1/2_ab = _ab .For this congruence, therefore, we recover the natural relation for the twist, = . It is interesting to compare this with the null congruence defined with the more general solutions to the Rosen geodesic equations described above. These solutions (forη= 0) define a null congruence with tangent vector^ = [ u̇; V̇;Ẋ ] = [1; -1/2ξ^-1(ξ - 2X);^-1(ξ - 2X) ] .and_ = [ V̇ - X Ẋ;1;Ẋ + X ] = [ -1/2(ξ + 2 X ) ^-1(ξ - 2X);1; ξ -X ] ,and we readily check^2 = 0.The optical scalars for this congruence, defined entirely within theRosen metric framework, are constructed as above. Specifically, we have[The complete resultfor _≡ D__ can be written analogously to footnote <ref> in section <ref> as _ ≡ D__ = [Ẋ^T Ẋ0- Ẋ^T;000; - ^T Ẋ0 ^T ] . ] _ab ≡ D_b_a = ∂_b _a - Γ^v_ba_v= -_ab + 1/2_ab = (^T)_ab .With the usual decomposition into optical scalars, = 12θ̂ 1 + σ̂ + , we find that the twist for this congruence is given by = - . In section <ref>, we see how these two different congruences are related to theQandPgenerators of the isometry algebra, and how this different result for the Rosen twistsandis reflected in the commutation relations. §.§ Wave equation and twist We now consider solutions of the wave equation in the gravitational plane wave background described by the twisted Rosen metric (<ref>).The d'Alembertian in this metric is= 1/√(-g)∂_(√(-g) g^∂_),whereg_ = [01 -X;100;X0] ,2cm g^ = [ 0 1 0; 1 -X^-1 XX^-1; 0 - ^-1 X ^-1 ] ,and√(-g) = .It follows that ∂_u log√(-g) = 1/2 ^-1= 1/2(Ω + Ω^T) = θ ,whereθis the expansion scalar (<ref>). Collecting terms, we find the followingelegant form,= 2 ∂_u∂_V + (∂_X + X∂_V ) ^-1(∂_X -X ∂_V)+ θ ∂_V. Generalising from the twist-free case, we look for solutionsϕ(x)of the formϕ(x) =A(u)F(u,X)e^iV ,whereVis the usual phase and the the amplitude factorA(u)satisfies∂_u logA = - 1/2θ .This shows how the amplitude reflects the overall expansion of the null congruence, in the way familiar from conventional geometric optics. We then haveϕ = A(u)e^iV [2 i ∂_u + (∂_X + i X ) ^-1(∂_X - iX ) ]F(u,X).Clearly,ϕ(x)is a solution of the wave equation ifF(u,X)solves[2 i ∂_u + (∂_X + i X ) ^-1(∂_X - iX) ] F(u,X) = 0,which we can write as [ 2i∂_u + D_a (^-1)^ab D_b ] F(u,X) = 0,withD_a = ∂_a - i _abX^b. TheseD_aare in essence covariantderivatives with abelian connection_abX^b.Provided that = 0, this corresponds to a constant field strengthF_ab =-2 _ab. We see immediately that this takes the form of a heat equation (foruimaginary)and so may readily be solved using techniques from the heat kernel or Schwinger proper-time formalisms. We can therefore write a standard form of solution asF(u,X) = Φ(X,Y)exp[i/2 (X-Y)^a A_ab(u) (X-Y)^b + B(u)],whereΦ(X,Y)is the path-dependent phase factor,Φ(X,Y) = exp[ i ∫_Y^X dZ^a _abZ^b ].The rôle of the phase factor in this type of solution (see,e.g. ref. <cit.>) is to convert the covariant derivatives to ordinary derivatives and field strengths as they actthroughΦto the exponential term in (<ref>). This follows from the key property(for the special case of constant field strength),D_a Φ(X,Y) = - i Φ(X,Y)_ab (X-Y)^b,where we have taken the path fromYtoXat constantu.In the present case with = 0, however, this geometrically natural constructionis not needed in its full generality, since choosingY=0we can show that the phase factor is trivial,Φ(X,0) = 1. Our ansatz for the solution to (<ref>) is then simply,[Note that while a linear term in X may in principle be included in this ansatz, the wave equation requires it to vanish provided B is real.] F(u,X) = exp[i/2X^a A_ab(u) X^b + B(u) ],where, after equating real and imaginary terms, theu-dependent functionsA_ab(u)andB(u)satisfyd/duA +( A +) ^-1(A - ) = 0d/duB + 1/2 ^-1 A = 0.The terms involvingAandBtherefore modify the phase and amplitude ofϕ(x)respectively.Equation (<ref>) is a non-linear, first order ODE for the symmetric matrixA_ab(u), which we recognise as a Riccati equation. This motivates converting it into a linear, second order ODE with an appropriate change of variables. We therefore introduce the matrixM^ab(u)such that Ṁ = ^-1(A - )M.Differentiating, and using = 2(- )and = 0,we findM̈ = [-2 ^-1( - ) - ^-1(A + ) + ^-1(A - ) ] Ṁ= - 2 ^-1 Ṁ ,with the term ofO(A^2)vanishing by construction. That is,Mis the solution ofM̈ + 2 ^-1 Ṁ = 0. We recognise this as the defining equation (<ref>) for the key functionH(u)given in (<ref>). Inverting (<ref>), we therefore solve forA_abin the form,A = ḢH^-1 + = H^-1 - ,using (<ref>). The final solution for the wave equation is then,ϕ(x) = Ã(u) exp[i (V + 12 XH^-1(u) X ) ],with amplitudeÃ(u) = exp[-1/2∫ du (θ + ( H)^-1)]. It is interesting to consider this solution in the special case of zero twist, where a geodesic with tangent vectork^can be given in terms of the derivative of the phase of a solution of the wave equation,i.e.k_= ∂_Θ. Of course this is not possible for a congruence with non-vanishing twist, which cannot be described as a gradient flow. Here, takingΘas the phase in (<ref>), this would givek_ = [ 12 X Ḣ^-1 X; 1;H^-1 X ] = [ -12 p ^-1 p; 1; p ] ,since with= 0, we haveḢ = ^-1and we have setX = H(u) p. This reproduces the zero-twist limit of the expression found in (<ref>) directly from the geodesic equations.§ HOMOGENEOUS PLANE WAVES AND ISOMETRIES II – ROSEN In this section, we reconsider the symmetries of gravitational plane waves, and the extended symmetries of homogeneous plane waves, this time from the point of view of the twisted Rosen metric (<ref>). We derive the explicit Rosen forms of the isometries and Killing vectors and show how the extended Heisenberg algebra of section <ref> arises entirely within the framework of Rosen coordinates.§.§ Killing vectors The first isometry of the Rosen metric (<ref>), which we display here again for ease of reference,ds^2 = 2 du dV - 2 X^a ω_ab dX^b du + dX^a _ab(u) dX^b,is of course invariance under translations ofV, with generatorZand Killing vectorK_Z,i.e. uu, VV + α , XX; K_Z = ∂_V. The basic Heisenberg algebra is completed by the generatorsQ_r,P_rwhich involve a choice of independent solutionsF^i_(r),G^i_(r)respectively of the oscillator equation. Unlike section <ref>, however, we will not immediately commit to the choicef,gsatisfying the canonical boundary conditions (<ref>). Focusing now onQ_r, and converting from the Brinkmann transformation (<ref>), we expect the Rosen metric to be invariant under the transformationsuu, VV - [Ḟ^T E- F^T E^T-1( - ) ]X , XX + F^T E^T-1 .Here,[ The conversion from Brinkmann to Rosen coordinates,u = u, v = V - 1/2 XX, x = E X,suppressing indices, implies∂_u = ∂_u, ∂_v = ∂_V, ∂_x = E^T-1(∂_X + ( - )X ∂_V) . ]we use an abbreviated matrix notationF = F^i_(r)and parameters= ^r,so that written in full theXtransformation, for example, isX^a ^r F_(r)^i (E^-1)_i^a. The corresponding Killing vector is thereforeK_Q_r =F^TE^T-1 ∂_X -[Ḟ^T E- F^T E^T-1( - ) ]X ∂_V,with the analogous form forK_P_r,viz. K_P_r =G^TE^T-1 ∂_X -[Ġ^T E- G^T E^T-1( - ) ]X ∂_V, The direct proof that (<ref>) leaves the Rosen metric invariant requires some calculation. Useful intermediate results includedXdX + ( Ḟ^T E - F^T E^T-1^T ) ^-1 du,and dVdV- [Ḟ^T E- F^T E^T-1( - ) ]dX- [(Ḟ^T E- F^T E^T-1^T)^-1. . + F̈^T E - F^T E^T-1 + F^T E^T-1^T ^-1] X du,provided = 0, as required for the equivalence of the Brinkmann and Rosen metrics. Then using the fact thatFis an oscillator solution to writeF̈^T E = F^T E^T-1 ,together with the analogue of (<ref>),=- ^T ^-1 ,we can verify that the variation of the Rosen metric indeed vanishes under (<ref>). This general form of theQ_randP_risometries is not at all evident from inspection of the Rosen metric. However, we can make particular choices of the oscillator solutionsF^i_(r)andG^i_(r)which correspond more directly to the natural invariances of themetric.First, we define the generatorQ_aby takingF^i_(r) = E^i_a, since by construction the zweibeinE^i_ais itself a solution of the oscillator equation. The corresponding isometry, with parameterc^a, is then simplyuu, VV + cX, X → X + c,since in this caseḞ^T E = ^T, with Killing vectorK_Q = ∂_X +X ∂_V.The transformation (<ref>) is manifestly a symmetry of the twisted Rosen metric.Notice the key point that a simpleXtranslation is not an isometry, but must beaccompanied by a twist-dependent transformation ofV.Next, note that a further oscillator solution is given byG^i_(r) = E^i_a H^abprovidedH^ab(u)satisfies the equation,Ḧ + 2 ^-1Ḣ = 0.This follows immediately fromG̈ = Ë H + 2 ĖḢ + E Ḧ= h G + E (Ḧ + 2 ^-1 H ).We then look for a corresonding isometry of the form,u → u, V → V - X N b - 1/2 b M b, X → X + Hb,with parameterb_a, for functionsH^ab(u),N_a^b(u)andM^ab(u)to be determined. Here, we have included a term ofO(b^2)to identify a finite transformation, although this will not be present in the infinitesimal transformations defining the Killing vectors. To check that this is indeed an invariance of the metric, we substitute (<ref>) together with dXdX + Ḣbdu,anddVdV - dX N b - X Ṅ bdu - 1/2 b Ṁ bdu,into the Rosen metric. A short calculation shows that the metric is invariant providedH(u)satisfies (<ref>) andN= Ḣ +H,Ṁ = Ḣ^T ( Ḣ + 2H ). Notice now that the equation (<ref>) forH^ab(u)is precisely the same asthat satisfied by the functionH^ab(u)involving the path-ordered integrating factor introduced in (<ref>) in the discussion of Rosen geodesics.We therefore chooseH ≡Hand, using the further identity (<ref>), define theP^aisometry to be,uu, VV - X (1 -H) b - 1/2 b Ḣ b, XX +Hb,with corresponding Killing vector,K_P =H^T ∂_X - ( 1 +H^T ) X ∂_V.It is now easy to check that this precisely reproduces the general form (<ref>) for the particular choiceG = E H.The remaining generatorXof the extended Heisenberg algebra only corresponds to an isometry whenthe Rosen metric describes a homogeneous plane wave. This is encoded through the functionsandin the metric, and therefore implicitly through the zweibeinEwhich isdetermined by the Brinkmann profile functionh(u). Proving that this is indeed an isometry therefore relies on the use of the homogeneous plane wave condition (<ref>),ḣ(u) = [,h(u)].Again starting from the equivalent Brinkmann transformation (<ref>), we expect the Rosen metric tobe invariant under uu +, VV +1/2 XX, XX + X,with corresponding Killing vector,K_X = ∂_u + X ^T ∂_X + 1/2 XX ∂_V,where= -^-1( - ), =- ^2 + [,].Here,= ^a_b, = _aband for clarity we have omitted the^-1factors incontracting Rosen indices in the expression for.Notice that while theVtransformation only involves its symmetric part,itself is not symmetric when the twist is non-vanishing. Instead, we have the useful identity,1/2( - ^T) = ^T+. To check the invariance of the twisted Rosen metric under (<ref>), we need theu-derivatives,= - ^-1 , = -2 ^-1 .ThendX dX + dX -^-1 Xdu, dV dV + 1/2 X( + ^T) dX -X ^-1 A Xdu,and invariance of the metric follows directly.Again, we see that in contrast to the Brinkmann description, the enhanced isometry for the homogeneous plane wave involves a compensating transformation ofVas well asXto balance theu-translation.§.§ Isometry algebra Having identified the isometries of the twisted Rosen metric, the next step is to find the corresponding algebra by evaluating the commutators of the Killing vectorsK_Z, K_QK_PandK_X. We perform these calculations initially with arbitrary oscillator solutions definingQ_rP_r, and work entirely in Rosen coordinates.First, the Killing vectorK_Z = ∂_Vnaturally commutes with all the others, so we have simply,[Z,Q_r] = 0, [Z,P_r] = 0, [Z,X] = 0. Next, we show that[Q_r, Q_s] = -W_rs(F,F)Z, [P_r,P_s] = - W_rs(G,G)Z ,and[Q_r, P_s] = - W_rs(F,G)Z,in terms of the appropriate Wronskian,e.g. W_rs(F,G)=F^T Ġ - Ḟ^T G.This follows readily from the expressions (<ref>), (<ref>) for the Killing vectors, which imply[K_Q_r, K_P_s]=- [(Ġ^T E - G^T E^T-1( - ) ) E^-1 F ]_sr ∂_V- [ F ↔ G ]_rs ∂_V = - (F^T Ġ - Ḟ^T G)_rs ∂_V = -W_rs(F,G) K_Z,with the terms not involving derivatives ofFandGcancelling.The evaluation of the[X,Q_r]commutator in the Rosen case involves a lengthier calculation. First, using the definition (<ref>) for, we eventually find[K_X, K_Q_r] = (Ḟ^T + F^T ϵ) E^T-1 ∂_X + F^T E^T-1[ ^T ^-1 -( - ) ^-1( - ) - 1/2( + ^T) ] X ∂_V.Then, from the definition (<ref>) forand the relation (<ref>), we can show[K_X,K_Q_r] = ^T E^T-1 ∂_X- [ ^T E - ^T E^T-1( - ) ] X ∂_V,where= Ḟ - ϵ F,is itself a solution of the oscillator equation. We recognise the r.h.s. as being of the same form asK_Q_rbut with the oscillator solutionF^i_(r)replaced by^i_(r).This relation thereforereproduces the Brinkmann expression (<ref>) for[K_X, K_Q]for a general solutionF. The same analysis also applies of course to the[K_X, K_P]commutator with the solutionG.At this point, it is clear that by choosing the oscillator solutionsF,GdefiningQ_r,P_rto be the canonical basisf,gas in section <ref>, we recover the isometry algebra(<ref>), (<ref>) precisely. Only the[Q_r,P_s]commutator has a non-vanishingWronskian, while decomposing the solutionas in (<ref>) we recover the[X,Q]commutator from (<ref>), with similar results for[X,P].However, it is more constructive to consider the particular choice of generatorsQ_a,P^amade above. We can evaluate the commutators either directly from the expressions for the Killing vectors in(<ref>) and (<ref>), or from the Wronskian forms (<ref>), (<ref>) with oscillator solutionsF = EandG = E H. It is straightforward to show:[K_Q_a, K_Q_b]= - 2 _ab ∂_V = - W(E,E) ∂_V , [K_Q_a, K_P^b]= - _̣a^b∂_V = - W(E,E H) ∂_V ,[K_P^a, K_P^b]=(H -H^T + 2H^TH)^ab = - W(E H, E H)∂_V.Using the various identities above for the derivatives ofE^i_a(u)andH^ab(u), we can verify explicitly that the term in brackets in (<ref>) is independent ofu, as it must be from its representation as a Wronskian. We are therefore free to evaluate it for any value ofuand choosingu=u', whereH(u') = 0, we see that it must vanish. We therefore find thefollowing extremely simple form for the modified Heisenberg algebra:[Q_a, Q_b] = -2 _abZ, [P^a, P^b] = 0,and[Q_a, P^b] = - _̣a^bZ.TheQ_agenerators therefore develop a non-vanishing commutation relation in the presence of twist. This is in line with our discussion in section <ref> on the origin of twist. It is interesting here to contrast the commutators for[Q_a,Q_b]and[P^a,P^b], the Wronskian for the former being2and0for the latter. This seems to reflect the difference in the twist for the two congruences described insection <ref>. The oscillator solutionF = Especifying theQ_agenerators correspondsto (<ref>) (since thereX =constant) and defines a congruence with twist, whereas the solutionG = EH(corresponding toX = H p) is characteristic of thesecond type of congruence (<ref>) with twist-.Finally, we need to consider the commutators[X, Q_a]and[X,P^a]in the special case of homogeneous plane waves. Again, we can evaluate these directly using the explicit expressions (<ref>), (<ref>) and (<ref>) for the Killing vectorsK_Q_a,K_P^aandK_X, or alternatively using the general result (<ref>) with the oscillator solutionsFandGdefiningQ_aandP^a. Starting with the generatorQ_a, we easily show that withF=E,= Ḟ - ϵ F = - E, = E^T-1(A -B),then from (<ref>) we have[K_X, K_Q_a] = - ^T ∂_X - (^T + ^T ) X ∂_V,as can also be derived directly from the definitions ofK_XandK_Q_a.The next step is to write this in the form[X, Q_a] = a_a^b Q_b + b_ab P^b,for constanta,b.Comparing the r.h.s. of (<ref>) with the definitions (<ref>), (<ref>) forK_QandK_P, we requirea + b H^T = - ^T, b(1 + 2H^T ) = ^T.Differentiating, and using the identities (<ref>) and (<ref>) forḢand,, we readily findȧ + ḃ H^T = 0, ḃ( 1 + 2H^T ),so conclude thataandbare indeed independent ofu. Once again evaluating atu=u'where the terms involvingH(u')vanish, we therefore determine the commutator[X, Q_a] = - ^T(u') Q + ^T (u') P. This just leaves the[X,P]commutator. A similar calculation shows= Ġ - ϵ G = E (Ḣ -H), = - E^T-1[ ( + ) Ḣ -(- )H],and from either the analogue of (<ref>) with, or directly from the definitions ofK_XandK_P^a, we obtain[X,P^a] = (Ḣ^T -H^T ^T) ∂_X - (Ḣ^T+ ( 1 +H^T )+ 12 H^T ( + ^T) )X∂_V.This can then be written in the form[K_X, K_P^a]= c^ab Q_b + d^a_b P^b,withc + dH^T = Ḣ^T -H^T ^T, d( 1 + 2H^T ) = Ḣ^T ( + 2 )+H^T ^T.Then, differentiating (<ref>), we can show after some calculation thatċ + ḋ H^T = 0, ḋ - 2 ċ = 0,and so verify thatcanddare constants. Evaluating atu=u', we therefore find,[X, P^a] = ^-1(u') Q + ( (u') + 2^-1(u')) P. At this point, we can make the natural consistency check that these commutation relations satisfy the Jacobi identity. From (<ref>) and (<ref>), together with (<ref>) and(<ref>), we have (with all functions evaluated atu'),[[X,Q_a],P^b] - [[X, P^b],Q_a] + [[Q_a,P^b],X]= (^T)_a^b Z +(2 (^-1)^b_a - ( + 2 ^-1)^b_a )Z + 0= 0.Notice especially the necessity of the twist appearing in the commutator for[Q_a,Q_b]of the Heisenberg algebra to ensure the self-consistency of the extended isometry algebrafor homogeneous plane waves.The commutators (<ref>) and (<ref>) may be written out explicitly in terms of(u'),(u'),(u')and(u')by simply substituting the definitions ofand. The resulting expressions display the dependence of the commutators[X,Q_a]and[X,P^a]on the congruence defining the Rosen metric through the zweibeinE^i_a(u')(determining,and) and its derivativeĖ^i_a(u')(determining) at the reference pointu'. In this explicit form, however, they are quite lengthy and we will not write them here.In order to complete our self-consistency checks and make contact with thecanonical set of commutation relationsdescribed in section <ref>, we may without loss of generality chooseu'=0and consider the congruence withE^i_a(0) = ^̣i_aandĖ^i_a(0) = 0. Notice though that this is a twist-free congruence, since these conditions imply(0) = 0and sinceis independent ofuit is therefore zero. (Note that this does not apply to the expansion or shear.) In this twist-free case, the commutators (<ref>) and (<ref>) then simplify to[X,Q_a]= ϵ_a^bQ_b + h_ab(0) P^b,[X, P^a]= ^̣abQ_b + ϵ^a_b P^b,reproducing (<ref>).§.§ Geodesics and isometries Finally, it is interesting to see how theseQ_aandP^asymmetries for a general gravitational plane wave act on the geodesic solutions themselves.This discussion follows that given in <cit.>in the twist-free casewith the conventional Rosen metric and geodesics, and we have made this sectionas self-contained as possible to facilitate comparison. For simplicity, we have also suppressed the index notation below.Recall the general geodesic solutions from section <ref>,X(u) = H(u) p + a, 4cmV(u) = - 1/2 pH(u)p - pH^T(u)a + η(u-u') + d,wherep^a,a^aanddare constants,η= 0for null geodesics, and the key functionH(u), which incorporates the twist-dependent path-ordered exponentialP(u,a)of (<ref>), satisfiesH(u') = 0at the reference pointu'.First, under aZtransformation,uu, VV + f, XX,the geodesics obviously retain the same form with the simple parameter shift,pp, aa, dd + f. Next, under theQ_aisometry,uu, VV + cX, XX + c,we see that the geodesic solutions transform asX(u) H(u) p + a + c, V(u) - 1/2 pH(u) p - pH^T(u) (a + c) + η(u-u') - ac,so again the form of the geodesics is preserved, with the parameter shifts,pp, aa+c, dd - ac. Finally, consider theP^aisometry,uu, VV - X( 1 -H(u) ) b- 1/2 bH(u) b.A short calculation shows that here,X(u) H(u) (p + b) + a, V(u) -1/2 (p+b)H(u) (p+b) - (p+b)H^T(u)a + η (u-u')+ d - ab + 1/2 p [H -H^T + 2H^TH] b.We now recognise the term in square brackets as that occurring in (<ref>).As explained there, we can verify that its derivative w.r.t.uvanishes, so that it is independent ofu, then evaluating atu=u'withH(u') = 0we see that it vanishes. The form of the geodesics is then once again preserved by the isometry, with theparameter transformations,pp + b, aa, dd-ab. Collecting all this, we therefore find that under a general isometry of the twisted Rosen metric, the constant parameters specifying the geodesics transform as,(p, a, d, η) (p+b, a+c, d - a c - a b + f),generalising the result of <cit.> (see eqs. (III.8), (IV.17) respectively)where the corresponding symmetry was identified with a restricted Carroll group. § VAN VLECK - MORETTE MATRIX FOR TWISTED NULL CONGRUENCES One of the most important geometrical quantities characterising geodesic congruences is the van Vleck-Morette (VVM) determinant or, more generally, the VVM matrix<cit.>.It encodes information on the nature of the geodesic flow and plays a key rôle in the constructionof Green functions and heat kernels for QFTs in curved spacetime (for reviews,seee.g. <cit.>. In particular, zeroes of the VVM determinant correspond to conjugate points on the congruence where the geodesics focus; in turn, this influences the analytic structure of the corresponding Green functions which is implicitly related to the realisation of causality in the QFT<cit.>.Here, we generalise the construction of the VVM matrix for plane wave spacetimes previously given in refs. <cit.>to the case of null geodesic congruences with twist. First we give a derivation in terms of the original Brinkmann coordinates then show how theresult may be obtained directly using the Rosen form (<ref>) of the plane wave metric adapted to twisted null congruences.§.§ Brinkmann construction In Brinkmann coordinates, we may define the transverse components of the VVM matrix by _ij(x,x') = ∂^2(x,x')/∂ x^i∂ x^' j ,where(x,x')is the geodetic interval,(x,x') = 1/2∫_0^1 dg_d x^/d d x^/dalong the geodesicx^(). For a general plane wave (not necessarily a homogeneous plane wave as we have been discussing elsewhere), this is(x,x')= 1/2∫_u'^u du_1 (2 v̇ + x^i h_ijx^j + (ẋ^i)^2) = (u-u') (v-v') + 1/2(u-u') [x_iẋ^i]_u'^u,where we have used the transverse geodesic equation (<ref>) to writed/du(x_iẋ^i) =x^i h_ijx^j + (ẋ^i)^2. As in our earlier work <cit.>,we expand the transverse geodesic solutionsx^i(u)(in other words, the Jacobi fields) asx^i(u) = B^i_j(u,u')x^j(u') + A^i_j(u,u') ẋ^j(u'),whereAandBare solutions of the oscillator/geodesic equation ( i.e.Ä - hA = 0,B̈ - hB = 0) with “spray” and “parallel” boundary conditions:A^i_j(u',u')= 0, 2cmȦ^i_j(u',u') = ^̣i_j, B^i_j(u',u')= ^̣i_j, 1.8cmḂ^i_j(u',u') = 0,respectively.These functions are of course closely related to the optical scalars characterising the null congruence. To see this, recallẋ^i = Ω^i_j x^jandx^i = E^i_a X^aso that, suppressing indices, we havex(u)= B(u,u')x(u') + A(u,u') Ω(u')x(u') = E(u) E^-1(u')x(u').Then, from∫_u'^u du_1 Ω(u_1) = ∫_u'^u du_1 Ė(u_1) E^-1(u_1) = [ log E ]_u'^u,we find the required relation <cit.>[As a special case, if we restrict to a geodesic spray congruence and the corresponding optical scalars, eq.(<ref>) reduces to ∂_u log A(u,u') = Ω(u) 1cm ⇒1cm ∂_u log A =Ω = θ .For a twist-free congruence, using the relation (<ref>) between A_ij and the VVM matrix _ij, this directly implies the well-known relation <cit.>between the VVM determinant and the expansion scalar:(u,u') = (u-u')^2 e^-∫_u'^u du^ θ(u^). ] B(u,u') + A(u,u') Ω(u')= exp∫_u'^u du_1 Ω(u_1) = exp∫_u'^u du_1 [ 1/2θ 1 ++ ω]. Now, we can express the geodetic interval in terms ofA(u,u')andB(u,u')as follows:(x,x') = (u-u')(v-v') - 1/2(u-u')[x(u)(A^-1,T(u,u') - A^-1(u',u))x(u') .+ x(u) (A^-1(u',u) B(u',u) ) x(u) .- x(u') (A^-1(u,u') B(u,u')) x(u')] ,and so the VVM matrix is _ij(u,u') = - 1/2 (u-u')[ A^-1,T(u,u') - A^-1(u',u) ].Notice immediately though that we can not assume thatA^T(u,u') = - A(u',u)in the case of acongruence with twist, unlike the conventional twist-free casedescribed in <cit.>.Determining the VVM matrix therefore reduces to finding the solutionA(u,u')of theoscillator/geodesic equation with spray boundary conditions. We already have one solutionin the form of the zweibeinE^i_a(u), so we can find a second solution using the Wronskian,i.e. E^T(u) Ȧ(u,u') - Ė^T(u) A(u,u') = W(u'),whereW_aj(u')is independent ofusince the oscillator equation is second order with no terms linear in derivatives.Now rearrange to find a differential equation forE^-1A. First, from (<ref>) we haveE^-1Ȧ = E^-1Ω^T A + ^-1 Wrecalling= E^T E. ExpressingΩ^T = Ω- 2ω, some further manipulation then givesd/du(E^-1 A) +2 ^-1(E^-1 A) = ^-1 W,in terms of the Rosen twist_ab.To solve this, we need the path-ordered exponentialP(u,a)introduced in (<ref>). Since this is an integrating factor for the differential equation (<ref>), the general solution is P(u,a) E^-1(u) A(u,u') -P(u^,a) E^-1(u^) A(u^,u') =∫_u^^u du_1 P(u_1,a)^-1(u_1) W.Choosingu^=u'and using the boundary conditions (<ref>) to setA(u',u') = 0, then takinga=u, we find the solution in the convenient form,E^-1(u) A(u,u') = ∫_u'^u du_1P(u_1,u) ^-1(u_1)W.Now, sinceWis independent ofu', we may evaluate its definition (<ref>) atu=u'to findW(u') = E^T(u') Ȧ(u',u') - Ė^T(u') A(u',u') = E^T(u'),again imposing the boundary conditions (<ref>). Finally, therefore, we findA(u,u') = E(u)∫_u'^u du_1 P(u_1,u)^-1(u_1) E^T(u'),or more explicitly,A(u,u') = E(u)( ∫_u'^u du_1 T_+exp[-∫_u_1^u dt 2^-1(t) ] ^-1(u_1))E^T(u').The difference from our previous result <cit.>for twist-free congruences is clearly the inclusion of the twist-dependent integrating factor. The VVM matrix then follows from (<ref>). §.§ Rosen construction In Rosen coordinates, we define the transverse VVM matrix_ab(u,u')in terms of the geodetic interval as_ab(u,u') = E_ a^T i(u) _ij(u,u') E^j_b(u') = ∂^2(x,x')/∂ X^a∂ X^'b .From the Rosen metric (<ref>) or (<ref>), we have(x,x')= 1/2(u-u')∫_u'^u du_1 (2V̇ - 2 X^a _abẊ^b+ Ẋ^a _abẊ^b) = (u-u') (V-V') + 1/2(u-u') [X (Ẋ + 2X) ]_u'^u,where, assumingindependent ofuas in (<ref>) to maintain the equivalence with theBrinkmann congruence, we have used the transverse geodesic equation (<ref>),d/du (Ẋ + 2X) = 0.which implies (<ref>),Ẋ + 2 ^-1 X = ^-1ξ ,for constantξ^a. As discussed in section <ref>, the solution is P(u,a)X(u) -P(u',a)X(u') = ∫_u'^u du_1 P(u_1,a) ^-1(u_1)ξ .Inverting now givesξ = ( ∫_u'^u du_1 P(u_1,a) ^-1(u_1) )^-1 [P(u,a)X(u) -P(u',a)X(u') ]. It then follows from the expression (<ref>) for the geodetic interval that[Note that the presence of the twist-dependent integrating factor in (<ref>) means that the geodetic interval is not a simple quadratic form in (X-X'), in contrast to the twist-free case <cit.> where(x,x') = (u-u')(V-V') + 1/2 (X-X')^a _ab(u,u') (X-X')^b. ] (x,x')= (u-u') (V-V') + 1/2 (u-u') (X-X') ( ∫_u'^u du_1 P(u_1,a) ^-1(u_1) )^-1 6cm ×[P(u,a)X(u) -P(u',a)X(u') ].From its definition (<ref>), the VVM matrix is then_ab(u,u') = - 1/2 (u-u') [ ^-1,T(u,u') - ^-1(u,u') ],where^-1(u',u) = - ( ∫_u'^u du_1 P(u_1,a) ^-1(u_1) )^-1P(u',a).That is, combining the integrals, or equivalently choosinga = u', (u,u')= ∫_u'^u du_1P(u_1,u') ^-1(u_1) ≡∫_u'^u du_1T_+ exp[-∫_u_1^u dt2^-1(t) ]^-1(u_1) .Finally, lowering indices with the Rosen metric and with(u,u') = E^T(u) A(u,u') E(u'),we recover the Brinkmann form of the VVM matrix derived above. § DISCUSSION In this paper, we have established the mathematical framework to describe the geometry of twisted null congruences in gravitational plane wave spacetimes, with a special focus on homogeneousplane waves. Since these metrics arise as Penrose limits, our results are of sufficient generalityto encompass any application where the essential physics is governed by the geometry ofgeodesic deviation. Notably, this includes loop effects in quantum field theory in general curved spacetimes, with applications ranging from ultra-high energy particle scattering to the origin of matter-antimatter asymmetry.Existing studies of quantum field propagation in gravitational backgrounds have almostentirely been restricted to the conventional geometric optics description associated withplane waves, where the classical null rays form a gradient flow.That is, the corresponding null geodesic congruence exhibits only expansion and shear. While little is known at present about the nature of quantum field theoretic effects associatedwith twisted null congruences (for some related classical studies,seee.g. <cit.>), our aim here has been to develop a comprehensive geometric toolkit to enable future work in this area.Moreover, the importance of gravitational plane waves as string backgrounds in itself motivatesthe most intensive exploration of the geometry of these spacetimes. It may be hoped thatour novel description of the rôle of twist in this geometry may find useful applications in string theory itself.A key focus of our work was the relation of Rosen coordinates, which reflect the nature of a chosen null congruence, to the more fundamental Brinkmann coordinates, and we derived the generalised Rosen metric (<ref>) adapted to a twisted congruence. The modifications due to twist of many geometrical constructions relevant toloop calculations in QFT in curved spacetime were discussed in detail, notably the generalised form of the van Vleck-Morette matrix and its Brinkmann-Rosen correspondence, and a thorough description of isometries of (homogeneous) plane waves in both theBrinkmann and twisted Rosen descriptions was presented.Geodesic deviation is also central to the detection of gravitational waves in an astronomicalcontext. At the most basic level, the passage of a gravitational wave may be detectedby its effect on a ring of freely-falling test particles. This “Tissot circle”<cit.> is simply a cross-section of a (timelike) geodesic congruence, and the squeezing and squashing measured by the detector, for example the arms of an interferometer such as LIGO <cit.> or eLISA <cit.>,is the expansion and shear of the congruence in the gravitational wave metric. The geometric results presented here would therefore be relevant if a detector would in addition be sensitive to a rotation of the particles on the Tissot circle,i.e. the twist of the corresponding geodesic congruence. In principle, recent proposals<cit.> to detect gravitational waves through measurements of the induced effective Doppler shifts on ultra-precise optical lattice atomic clocks in space may ultimately allow this possibility to be realised. These astrophysical gravitational waves would be in the form of short duration bursts or pulses, in which the plane wave profile functionh_ij(u)vanishes outside a given range. As discussed recently in <cit.>(see also references therein)the shape of this profile function encodes information on the nature of the source, with its iterated integrals, in both Brinkmann and Rosen coordinates, distinguishing different phenomena such as the memory effect.In conclusion, this geometry of twisted null congruences in (homogeneous) gravitationalplane waves is mathematically elegant and a natural extension of existing results in thisimportant area of general relativity. Our hope is that it will provide an impetus to further explorations of the physics of twist in quantum field theory and string theory in curved spacetime as well as for gravitational plane waves in astronomy.0.7cmAcknowledgments I am grateful to Tim Hollowood for many discussions and collaboration on plane wave geometry, and Gary Gibbons for bringing refs.<cit.> to my attention. This work was supported in part by STFC grant ST/L000369/1. 1cm § NULL CONGRUENCES IN AN ANTI-MACH SPACETIME In this appendix, we study in detail the null geodesics and congruences in the homogeneous plane wave metric <ref>. We describe explicit solutions to the geodesic equations for this generalisedOzsváth-Schücking (OS) metric in different coordinate systems, including a Newman-Penrose basis, and discuss the Raychoudhuri equations and optical scalarsfor a twisted null congruence. §.§ Generalised Ozsváth-Schücking metric and co-rotating coordinates We can write the generalised OS metric in the alternative forms,ds^2= 2 du dv + (e^ u h_0 e^- u)_ij x^i x^j du^2 + (dx^i)^2 = 2 du dv + xO(u) h_0 O^T(u)xdu^2 + (dx^i)^2,where O(u) = e^ u = [cos usin u; -sin ucos u ] ,is an orthogonal matrix. We will use these two representations interchangeably in what follows, depending on which is most transparent at a given step.We consider first an arbitrary profile functionh_0 = [ a 0; 0 b ]witha,bconstant. The non-vanishing Riemann, Ricci and Weyl curvature components areR_uiuj = - h_ij , 2cmR_uu = - h_0 = -(a+b), C_uiuj= -12 (a-b) [cos 2u -sin 2u; -sin 2u -cos 2u ] ,whereR_uiuj = C_uiuj + 12R_uu _̣ij. From (<ref>), (<ref>) we then have the NP curvature scalars in the standard basis,Φ_22 = 12 (a+b), 2cm Ψ_4 = 12(a-b) e^2iu .Evidently the metric witha=bis conformally flat, whilea=-bgives a Ricci flat spacetime. The original OS, anti-Mach metric is the Ricci-flat solutionwitha=1,b=-1.It is clear from <ref> that a natural choice of transverse coordinates would be to take out the rotation in the profile function and definez^i = (e^- u x)^i = O^T(u)x .The metric in these co-rotating, or stationary, coordinates becomes ds^2= 2 du dv + ( h_0 + 1)_ij z^i z^j du^2- 2 _ij z^i dz^j du+ (dz^i)^2= 2 du dv + ( (a+1) (z^1)^2 + (b+1)(z^2)^2 ) du^2 + 2(z^2 dz^1 - z^1 dz^2) du+ (dz^i)^2 . The curvature tensors are especially simple in these coordinates. The Riemann and Weyl tensors are R_uiuj = - (h_0)_ij = - [ a 0; 0 b ] ,2cm C_uiuj = -12 (a-b) [10;0 -1 ] ,while of courseR_uu = -(a+b)as before.TheXisometry also simplifies.The manifest invariance of the metric (<ref>) under translationsu →u+implies the Killing vector is now justK_X = ∂_u, the accompanyingrotation on thex^icoordinate in (<ref>) being removed by the transformation to co-rotating coordinates.[For the Ricci-flat, Ozsváth-Schücking metric a=1, b=-1, a further coordinate transformation U= √(2)u,W = 1√(2) (v + z^1 z^2) brings the metric to ds^2 = 2 dU dW -2√(2) z^1 dz^2dU + (z^1)^2dU^2 + (dz^i)^2 ,which, up to normalisation factors, is the form given in the original OS paper <cit.>. Evidently, this has 3 commuting isometries corresponding to translations in U, W and z^2. In terms of the generators in (<ref>) and (<ref>), these are X, Z and the linear combinationQ_2 + P_1, which for b=-1 commutes with both Z and X<cit.>. ]The null geodesic equations, written explicitly in terms of the co-rotating coordinates, are[ In the co-rotating transverse coordinates z^i, the Christoffel symbols areΓ^v_uu = (a-b) z^1 z^2, 1cm Γ^v_ui = (h_0)_ij z^j, 1cm Γ^i_uu = -(h_0 +1)_ij z^j, 1cm Γ^i_uj = ^i_j. ] v̈ + 2a z^1 ż^1 + 2b z^2 ż^2 + (a-b)z^1 z^2 = 0,z̈^1 + 2 ż^2 - (a+1) z^1 = 0,z̈^2 - 2 ż^1 - (b+1) z^2 = 0. It is also useful to note the explicit form of the standard Newman-Penrose tetrad in these coordinates. A straightforward construction starting fromℓ_= ∂_ugivesℓ^ = [ 0; 1; 0 ] , 1.5cm n^ = [ -1; 12z(h_0 +1)z;0 ] , 1.5cm m^ = 1/√(2)[ 0;-z^2 + i z^1; ^̣i1 + i ^̣i2 ]The corresponding NP curvature scalars are then simply Φ_22 = - 12R_n^ n^ = 12 tr h_0 = 12(a+b), 1.5cm Ψ_4 = - C_nm̅nm̅ = 12(a-b).§.§ Oscillator solutions In the original coordinates, the null geodesic equations are given in (<ref>). Importantly, the equationsfor the transverse coordinatesx^iare solutions of the oscillator equation and we focus on these.That is, we look for explicit solutions of the oscillator equation,F̈^i = h^i_j(u) F^j,withh = e^ u[ a 0; 0 b ]e^- u =O(u) [ a 0; 0 b ] O^T(u).We can immediately write these solutions in the form (suppressing indices)F = e^ uf = O(u)f ,where f̈ + 2ḟ - (h_0 +1) f = 0.This corresponds to the geodesic equations (<ref>) for the co-rotating transverse coordinates.To solve these equations, we first make the ansatzf = P e^ł u X = P O(ł u) X ,whereXis a constant vector,P = [ 0; 0 ]and,$̱, ł are to be determined. Without loss of generality, we can immediately rescale so that = 1. Substituting in (<ref>),we requiref̈ + 2ḟ - (h_0 +1) f = [ -ł^2 + 2ł PP^-1 - (h_0 +1)] f = 0,so we find a solution if $̱,łsatisfy[It will be useful in later calculations to eliminate $̱ to obtainłdirectly as a solution ofł^4 + (a+b-2)ł^2 + (a+1)(b+1) = 0.ł^2 + 2ł̱+ a + 1= 0,ł^2 + 2/ł + b + 1= 0.In the OS (a=1,b=-1) metric, we have=̱±√(2)andł = ∓√(2).The remaining two solutions are not so simple in general, but if we specialise to theOS metric we can show thatf = [ 1 0; u 1 ] X,withXa constant vector, also solves (<ref>).To summarise, in the most interesting case of the Ricci-flat, OS spacetime, we have a complete set of solutions to the oscillator equation:f^ i_(1) = O(u) [ 1 0; 0 -√(2) ] O(√(2) u)[ 1; 0 ] ,f^ i_(2) = O(u) [ 1 0; 0 -√(2) ] O(√(2) u)[ 0; 1 ] ,f^ i_(3) = O(u) [ 1 0; u 1 ][ 1; 0 ] = O(u) [ 1; u ] ,f^ i_(4) = O(u) [ 1 0; u 1 ][ 0; 1 ] = O(u) [ 0; 1 ] .The solutionsf^i_(r),g^i_(r),r = 1,2introduced in section <ref> are linear combinations of these solutions chosen to satisfy the canonical boundary conditions (<ref>). They may be compared directly with the solutions (3.40) of ref. <cit.>.§.§ Twisted null congruence and optical scalars As discussed in section <ref>, to select a null congruence with the geodesic equations satisfiedby transverse coordinatesx^i = E^i_a(u) X^a, we choosehalf of the oscillator solutionsto form the zweibeinE^i_a. Here, instead of the canonical choice, we study the natural congruence picked out by the first two solutions above and defineE^i_a = F^i_(a),a=1,2. This impliesx^i = E^i_a X^a = O(u) [ 1 0; 0 ] O(ł u)[ X^1; X^2 ] ,for the general metric, with the integration constantsX^ainterpreted as Rosen coordinates.The corresponding Wronskian is therefore (recalling that we may evaluateatu=0becauseE^i_asatisfy the oscillator equation)W_ab = (E^T Ė)_ab - (Ė^T E)_ab= (2+̱(^̱2 + 1)ł) _ab .This null congruence therefore has a non-vanishing twist, _ab = (+̱12(^̱2 + 1)ł) _ab .in Rosen coordinates. The corresponding Brinkmann twist isω_ij = ((E^T)^-1 E^-1)_ij= (+̱12(^̱2 + 1)ł) O(u) [10;0 1/ ][10;0 1/ ] O^T(u) = Ł _ij ,whereŁ = (1+ 12(+̱ 1/)ł). For the OS metric,Ł = -1/2.We can construct the full set of optical scalars directly from the general formulae in section <ref>. From the definition (<ref>), we findΩ_ij = (Ė E^-1)_ij= O(u) [0 1 + ł/ -(1+ł)̱0 ] O^T(u),then read off the optical scalars from the decomposition (<ref>). EvidentlyΩ_ijis traceless so the expansion scalarθvanishes. Its symmetric part gives the shear, and we check that the antisymmetric part reproduces the twist (<ref>). Simplifying the resulting expressions using the defining equations (<ref>) for$̱ and ł,we eventually find,θ = 0, 1.5cm_ij = 14 (a-b) [sin 2ucos 2u;cos 2u -sin 2u ] , 1.5cmω_ij = Ł _ij ,with Ł = 12(1 - ł^2 - 12(a+b)).The Raychaudhuri equations (<ref>) simplify as a result of the vanishing of θ. We immediately have θ̇ = 0,_ij = 12(a-b) [cos 2u -sin 2u; -sin 2u -cos 2u ] = - C_uiuj ,ω̇_ij = 0,where for the first we need to verify^2 +^2 = - R_uu = (a+b), which followsusing the identity in footnote <ref>.§.§ Null geodesics Through the Raychaudhuri equations, the solutions of the geodesic equations for the transverse coordinatescontrol the essential features of the null congruence. Now, we want to focus on the properties of anindividual null geodesic, so we also require the solution for the coordinate v. This discussion is best made in terms of the co-rotating coordinates, so we start from the set of geodesic equations (<ref>).First note that a first integral of the geodesic equation for v follows immediately from imposing the null condition on the metric (<ref>), giving2 v̇ + z^i (h_0 +1)_ij z^j - 2 _ij z^i ż^j+ (ż^i )^2= 0.The transverse geodesic equations can be written in compact form asz̈^i + 2 ^i_j ż^j - (h_0 +1)_ij z^j = 0.The geodesics (<ref>) forming the twisted null congruence are given byz = [ 1 0; 0 ] O(ł u) X,which impliesż = ł[ 1 1/- 0 ] z, 2cm z̈ = -ł^2 z.A short calculation gives the consistency checkz̈ + 2 ż - (h_0 +1) z =- [ ł^2 + 2ł̱+ a+10;0ł^2 + 2ł +b+1 ] z = 0,by virtue of the equations (<ref>) defining $̱ andł.Next, substituting the explicit solution forżinto (<ref>) givesv̇ = - 12(^̱2 - 1) ł^2 z [ 1 0; 0 - 1/^̱2 ] z =- 12(^̱2 - 1) ł^2 X O^T(ł u) [10;0 -1 ]O(ł u) X.Collecting earlier results, we can verify that this is consistent with the original form (<ref>) forv, which impliesv̇ = -12 X^a Ω̇_ab X^b =-12 X E^T (h + Ω^T Ω) E X. Integrating to findvitself, and writing out the solutions explicitly, we finally findv= V -14(^̱2 -1) ł X [sin 2ł ucos 2ł u;cos 2ł u -sin 2ł u ] X, z = [cosł usinł u; -s̱i̱ṉł u c̱o̱s̱ł u ] X. To visualise these geodesics, it is sufficient to select an individual element of the congruence by choosing values for the Rosen coordinatesV, X^1, X^2. The corresponding curves(withV=0,X^1=1andX^2=0in the Ricci-flat OS metrica=1,b=-1)are plotted in Figure <ref>.In the transverse space, as the geodesic progresses alongu, thecoordinatesz^1, z^2describe an ellipse with periodu=2π/ł. Meanwhile, the null coordinatevis oscillating sinusoidally with half the period. The full geodesic is therefore periodic inuwith period2π/ł, as can be seen in the right-hand figure. < g r a p h i c s > 0.3cm< g r a p h i c s > The left-hand plot highlights (in red) an element of the twisted null congruence propagating in the null u-direction. The right-hand plot illustrates the periodicity in u of the geodesic, whichtherefore appears closed when plotted in z^1, z^2, v coordinates. Evidently, this geodesic is part of a twisted null congruence, withω_ij = Ł_ijandŁ = -1/2. There is no focusing, and there are no conjugate points, consistent with the vanishing of the expansionoptical scalarθfor the congruence. The shear_ijis apparent in the differentamplitudes for the oscillations inz^1andz^2for≠̱1. (Recall from (<ref>) that the shear is non-vanishing for a profile functionh_ijwitha≠ b, which from (<ref>) implies≠̱1.)The absence of conjugate points is worth noting. Generically, null congruences in a plane wave spacetime will focus to conjugate points provided the Ricci tensor satisfies the null energy condition (R_uu≥ 0in our conventions) by virtue of the negativity of the r.h.s. of the Raychaudhuri equation (<ref>) for the expansion scalarθ. These conjugate pointsplayed a key rôle in our work on quantum loop effects in wave propagation in curved spacetime, where they are associated with singularities in the relevant Green functionsand determine key features of the refractive index in the quantum field theory<cit.>. However, this only holds in the absence of twist. Sinceω_ij = Ł_ij, the term ω^2in (<ref>) isnegative.[Note that we use the notation (ω^2)_ij = ω_ikω^k_j, which introduces a minus sign in ω^2 relative to acommon convention for the Raychaudhuri equations (seee.g. <cit.>.] So even for non-Ricci flat spacetimes, expansion-free null congruences can be supported by balancing the twist contribution to (<ref>) against the shear and Ricci terms.§.§ Newman-Penrose tetrad in co-rotating coordinates As we have seen, the Newman-Penrose tetrad associated with a null geodesic is a powerful tool for analysing the geometry of gravitational plane waves. Here, we find the Newman-Penrose basis for thenull geodesic congruence as described above in the co-rotating coordinates and use this to find the corresponding Penrose limit. Naturally this should reproduce the result of section <ref>, though the construction involves some interesting subtleties which were observed in the original description <cit.> of the Ozsváth-Schücking spacetime.A short calculation shows that the tetrad[The corresponding covectors areL_ = (v̇ + z(h_0 +1)z + z^2 ż^1 - z^1 ż^2 , 1, z^2 + ż^1, -z^1 + ż^2 ), N_ = (-1, 0, 0, 0), 1cm a_ = (-ż^1, 0, 1, 0), 1cm b_ = (-ż^2, 0, 0, 1). ] L^ = k^ = [ 1;v̇; ż^1; ż^2 ] , 0.8cm N^ = [0; -1;0;0 ] , 0.8cm a^ = [0; -ż^1 - z^2;1;0 ] , 0.8cm b^ = [0; -ż^2 + z^1;0;1 ]0.8cmwithm^ = 1√(2)(a^ + i b^), satisfies the required Newman-PenroseconditionsL^2 = N^2 = m^2 = 0,L.N=-1,m.m̅ = 1,etc. The first derivativesv̇, ż^1, ż^2can be eliminated in favour ofz^1, z^2immediately using (<ref>) and (<ref>).However, we still need to check that this tetrad is parallel-transported along the geodesic$̧.We do indeed find L^ D_ L^ = 0 and L^ D_ N^, the former especiallyrequiring care.[For example, using thenon-vanishing Christoffel symbols in footnote <ref>, we find for the L^ component,L^ D_ L^ = ż^i ∂_i L^ + Γ^v_uu + 2 Γ^v_uiż^i = z^1 z^2 (a - b + 2a ł/ - 2b ł̱+ 2 ł^3/ - 2 ł̱^3 ) = 0,after using (<ref>) to simplify the terms involving ł^3.] This is not true, however,for the transverse vectors, where we findL^ D_ a^ = [0; -(1+ł)̱z^1;0; -1 ] , 2cm L^ D_ b^ = [ 0; -(1+ł/)̱z^1; 1; 0 ] .The resolution is to define new transverse vectors[ A^; B^ ] = O(u) [ a^; b^ ] ,for which we findL^ D_ A^ = cos u (L^ D_ a^ + b^) + sin u (L^ D_ b^ - a^)= 0, L^ D_ B^ = cos u (L^ D_ b^ - a^) - sin u (L^ D_ a^ + b^)= 0,since the bracketed contributions vanish by comparing (<ref>) and (<ref>) using (<ref>). The full set of basis vectors L^, N^ and M^ = 1√(2)(A^ + B^)now satisfies all the required properties for a Newman-Penrose tetrad parallel-transported along $̧.What this shows is that even working in the stationary coordinate system(u,v,z^1,z^2)for the generalised OS metric, requiring that the transverse Newman-Penrose vectors are parallel-transportedalong the null geodesicreintroduces the rotationO(u)which is manifest in the Brinkmanncoordinate description of the metric.This also resolves what at first sight seems mysterious in deriving the Penrose limit associated witha null geodesic in the generalised OS metric described in the stationary coordinates,viz. how does the Penrose limit reproduce the homogeneous plane wave metric including the rotation factorO(u)as shown in section <ref>. In the Newman-Penros formalism, the resolution is especially elegant. Writing (<ref>) in terms ofA^, B^rather thanM^, M̅^, wehave the Penrose limit profile functionĥ_ijin the form,ĥ_ij = - [C_LALA + 12R_LL C_LALB;; C_LBLA C_LBLB + 12 R_LL ]= - O(u) [C_LaLa + 12R_LL C_LaLb;; C_LbLa C_LbLb + 12 R_LL ]O^T(u).Evaluating the Ricci and Weyl tensor components using (<ref>), we then quickly findĥ_ij = O(u)[ a 0; 0 b ]O^T(u) = h_ij .That is, the correct choice of parallel-transported Newman-Penrose tetrad automatically reinstates theimplicit rotation in the Brinkmann coordinate description of the homogeneous plane wave.This confirms in this explicit example of geodesics belonging to a twisted null congruence in the generalised OS spacetime that the associated Penrose limit simply reproduces the original homogeneous plane wave metric.§.§ Isometries In section <ref>, we described the extended isometry algebra for a homogeneous plane wave, with the generatorsQ_randP_rdefined in (<ref>) with oscillator solutionsf^i_(r)andg^i_(r)satisfying canonical boundary conditions (<ref>). We are free, however, to choose any independent linear combination of these to defineQ_r,P_rand such a redefinition will of course change the standard form of the algebra (<ref>), (<ref>).As we saw when describing the isometries in terms of Rosen coordinates, a particularly natural choice is to define the generatorQ_rwith an oscillator solutionF^i_(r) = E^i_a ^̣a_r, since this reflects the nature of the twisted congruence. ForP_r, we require a second, independent solutionG^i_(r).As we now show, an interesting choice in this model is to takeG = Ė - ϵ E.By definition then, from (<ref>), (<ref>) we immediately have[X, Q_r] = P_r,for the commutator with the extra generatorXrelated tou-translations.To find the commutator ofXwithP_r, we need to iterate this construction and determine the combinationĠ - ϵ G. Considering thegeneralised OS model and using the explicit expressions forE,Ωandωfrom section <ref>, we can easily show,Ġ - ϵ G= (h - 1 - 2 ϵΩ) E = O(u) [ a+1 + 2λ 0;0 (b + 1 + 2 λ/) ] O(λ u)= - λ^2 E,using the usual equations (<ref>) forλ, $̱. It follows directly that[X, P_r] = - λ^2 Q_r. For the remaining commutators involving Q_r and P_r, we need to evaluate therelevant Wronskians. A short calculation using the zweibein of section <ref> gives first G = λ E ϵ and then,W(F,F) = 2, W(F,G) = 2 λϵ , W(G,G) = 2 λ^2.The corresponding commutators are (compare (<ref>)),[Q_r,Q_s] =- W_rs(F,F)Z, [P_r,P_s] = - W_rs(G,G) Z,[Q_r,P_s] = - W_rs(F,G)Z.In particular, this shows how the twist enters into the non-vanishing commutators of Q_r (and P_r) with itself.An alternative presentation using the model-dependent constants λ, $̱ andΛdefined in section <ref>(where=̱±√(2),λ = ∓√(2), Λ = -1/2for the Ricci-flatOS metric) is then[Q_r,Q_s] =2Λ̱ ϵ_rs Z, [P_r,P_s] = 2 λ^2 Λ̱ ϵ_rs Z,[Q_r,P_s] = - 2λΛ̱ _̣rs Z. 99Penrose:1965rx R. Penrose, “A remarkable property of plane waves in general relativity,” Rev. Mod. Phys.37 (1965) 215. Penrose R. Penrose, “Any Space-Time has a Plane Wave as a Limit,” in `Differential Geometry and Relativity: A Volume in Honour of André Lichnerowicz on his 60th Birthday', 271, Springer Netherlands, Dordrecht (1976). Blau:2006ar M. Blau, D. Frank and S. 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[LIGO Scientific and Virgo Collaborations], “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett.116 (2016) no.6,061102,[arXiv:1602.03837 [gr-qc]].Abbott:2016nmj B. P. Abbottet al. [LIGO Scientific and Virgo Collaborations], “GW151226: Observation of Gravitational Waves from a 22-Solar-MassBinary Black Hole Coalescence,” Phys. Rev. Lett.116 (2016) no.24,241103,[arXiv:1606.04855 [gr-qc]].AmaroSeoane:2012je P. Amaro-Seoaneet al., “Low-frequency gravitational-wave science with eLISA/NGO,” Class. Quant. Grav.29 (2012) 124016,[arXiv:1202.0839 [gr-qc]].Loeb:2015ffa A. Loeb and D. Maoz, “Using Atomic Clocks to Detect Gravitational Waves,” arXiv:1501.00996 [astro-ph.IM].Kolkowitz:2016wyg S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth and J. Ye, “Gravitational wave detection with optical lattice atomic clocks,” Phys. Rev. D94 (2016) no.12,124043,[arXiv:1606.01859 [physics.atom-ph]].] | http://arxiv.org/abs/1705.09533v2 | {
"authors": [
"Graham M. Shore"
],
"categories": [
"gr-qc",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170526111548",
"title": "A New Twist on the Geometry of Gravitational Plane Waves"
} |
1]Hazim Shakhatreh 1]Abdallah Khreishah 2]Jacob Chakareski 3]Haythem Bany Salameh 4]Issa Khalil [1]Department of Electrical and Computer Engineering, New Jersey Institute of Technology[2]Department of Electrical and Computer Engineering, University of Alabama [3]Department of Telecommunications Engineering, Yarmouk University [4]Qatar Computing Research Institute On The Continuous Coverage Problem for a Swarm of UAVs [ Version December 30, 2023 ====================================================== Unmanned aerial vehicles (UAVs) can be used to provide wireless network and remote surveillance coverage for disaster-affected areas. During such a situation, the UAVs need to return periodically to a charging station for recharging, due to their limited battery capacity. We study the problem of minimizing the number of UAVs required for a continuous coverage of a given area, given the recharging requirement. We prove that this problem is NP-complete. Due to its intractability, we study partitioning the coverage graph into cycles that start at the charging station. We first characterize the minimum number of UAVs to cover such a cycle based on the charging time, the traveling time, and the number of subareas to be covered by the cycle. Based on this analysis, we then develop an efficient algorithm, the cycles with limited energy algorithm. The straightforward method to continuously cover a given area is to split it into N subareas and cover it by N cycles using N additional UAVs.Our simulation results examine the importance of critical system parameters: the energy capacity of the UAVs, the number of subareas in the covered area, and the UAV charging and traveling times. We demonstrate that the cycles with limited energy algorithm requires 69%-94% fewer additional UAVs relative to the straightforward method, as the energy capacity of the UAVs is increased, and 67%-71% fewer additional UAVs, as the number of subareas is increased.Unmanned aerial vehicles, charging, continuous coverage. § INTRODUCTION In 2005, Hurricane Katrina in the United States caused over 1,900 deaths, 3 million land-line phone interruptions, and more than 2,000 base stations going out of service <cit.>. Another example of a large-scale interruption of telecommunications service is the World Trade Center attack in 2001, when it took just minutes for the nearby base stations to be overloaded. The attacks caused the disturbance of a phone switch with over 200,000 lines, 20 cell sites, and 9 TV broadcast stations <cit.>. These incidents demonstrate the need for a quick/efficient deployment network for emergency cases. The authors in <cit.> proposed a UAV-based replacement network during disasters, where the UAVs serve as aerial wireless base stations. However, this study did not consider how the UAVs will guarantee a continuous coverage when they need to return to the charging station for recharging. Though a UAV has limited energy capacity and needs to recharge its battery before running out of power during the coverage process, only few studies have considered this constraint in the UAV coverage problem. Concretely, the author in <cit.> determined the minimum number of UAVs that can provide continuous coverage for a single area using UAVs with uniform and non-uniform energy capacity. However, no consideration has been made for the case when there are multiple subareas that need to be covered, which is the typical scenario during disasters. The authors in <cit.> formulated the Mobile Charging Problem, in which multiple mobile chargers collaborate to charge static sensors with minimum number of mobile chargers subject to speed and energy limits of the mobile chargers, such that the energy consumption of the mobile chargers is minimized. Note that the mobile charging problem is different from the problem we study. In the mobile charging problem, the chargers will not cover the sensors continuously. The mobile charger will visit the sensor and stay for a specific time to charge the sensor and after finishing the charging process, it will visit other sensors. In <cit.>, the authors studied the continuous coverage problem for mobile targets, where during the coverage process a UAV that runs out of energy is replaced by a new one. Many studies <cit.> focused on minimizing the total transmission power of the UAVs during the coverage of a geographical area, however, no limits on the UAV energy capacity and the need for recharging have been considered. The work in <cit.> reported that the energy consumption during data transmission and reception is much smaller than the energy consumption during the UAV hovering, i.e., it only constitutes 10%-20% of the UAV energy capacity. Thus, it is important to conduct studies that take into account the energy consumption during the UAV hovering rather than focusing on minimizing the energy consumption during data transmission and reception.Contrary to the related work above, we integrate the recharging requirements into the coverage problem and examine the minimum number of required UAVs for enabling continuous coverage under that setting (see Figure <ref>). To the best of our knowledge, this is the first study that jointly considers the coverage and recharging problems where multiple subareas are to be covered. Our main contributions in this context are: 1) We formulate the problem of minimizing the number of UAVs required to provide continuous coverage of a given area, given the recharging requirement. 2) We prove that this problem is NP-complete. 3) Due to the intractability of the problem, we study partitioning the coverage graph into cycles that start at the charging station. 4) Based on this analysis, we develop an efficient algorithm.The rest of this paper is organized as follows. Section <ref> presents our system model including the problem formulation and the proof of the NP completeness. In Section <ref>, we show how to find the minimum number of additional UAVs that are required to guarantee the continuous coverage. Then, we present our proposed algorithm. In Section <ref>, we present our experimental results. Finally, Section <ref> concludes the paper.§ SYSTEM MODEL§.§ Problem StatementConsider a geographical area G={g_1,...,g_N}, where g_i represents a subarea i, the subarea g_1 ∈ G includes the charging station and all subareas except subarea g_1 need to be covered G_0=G \ g_1. We want to find the minimum number of UAVs that can provide a continuous coverage over G_0 by placing the UAVs at locations where each UAV will provide full coverage for one subarea. In the continuous coverage problem, we assume: (1) Time slotted system in which the slot duration is 1 time unit and the total coverage duration is T. (2) All UAVs start the coverage process from the charging station and they need to return to the charging station after they complete the coverage process. (3) Each UAV has limited energy capacity E and it needs to return to the charging station to recharge the battery before running out during the coverage process. (4) Each UAV can move (from the charging station to location i), (from location i to location j) or (from location j to the charging station) and this process will take one time slot. (5) Each UAV covers a given subarea for one or multiple time slots. (6) Each subarea will be covered by only one UAV. (7) The UAV cannot travel to the charging station or to any other location until the handoff process is completed in which another UAV arrives to cover the subarea such that the continuous coverage is guaranteed. (8) The recharging process takes T_charge at the charging station. §.§ Problem Formulation Now, we formulate the continuous coverage problem. In order to present the problem formulation, we introduce thebinary variable x_m that takes the value of 1 if the UAV m visits any subarea from charging station during the coverage duration T and equals 0 otherwise; the binary variable y_ij,m^t that takes the value of 1 if the if the UAV m moves through edge ij during the time slot t and equals 0 otherwise; the binary variable z_j,m^t that takes the value 1 if the UAV m covers the subarea j at time slot t and equals 0 otherwise. Table I provides a list of the major notations used in this paper.[ min∑_m∈ M^ x_m; subject to;y_ij,m^t≤ x_m ∀ i,j ∈ G,∀ t ∈ [0,T], ∀ m∈ M(1); z_j,m^t ≤x_m ∀ j ∈ G_0,∀ t ∈ (0,T),∀ m∈ M(2); ∑_m∈ M^ y_1j,m^0 = 1 ∀ j ∈ G_0(3);∑_m∈ M^ z_j,m^t =1 ∀ j ∈ G_0,∀ t ∈ (0,T)(4);∑_i∈ G,i≠ j∑_m∈ M^ y_ij,m^t≤1 ∀ j ∈ G_0,∀ t ∈ [0,T](5); ∑_i_1∈ G^ y_i_1j,m_1^t=∑_i_2∈ G^ y_ji_2,m_2^t+1 ∀ j ∈ G_0,; ∀ t ∈ (0,T),m_1≠ m_2(6); ∑_m∈ M∑_t∈ [0,T)^∑_i∈ G^ y_ij,m^t ≤∑_m∈ M∑_t∈ (0,T)^ z_j,m^t;∀ j ∈ G_0(7);∑_j∈ G_0^∑_τ∈ T_charge^ [y_j1,m^t+y_1j,m^t+τ]≤1; ∀ m∈ M,∀ t ∈ (0,T)(8); ∑_m∈ M∑_t∈ [0,T]^∑_j∈ G_0^ y_1j,m^t= ∑_m∈ M∑_t∈ [0,T]^∑_i∈ G_0^ y_i1,m^t(9); ∑_t∈ [t_1,t_2]^∑_i,j∈ G^ E_ij^Travel y_ij,m^t+∑_t∈ [t_1,t_2]^∑_j∈ G^ E_j^Cover z_j,m^t;≤ E ∀ m∈ M,∀ [t_1,t_2]∈ [0,T],t_1=arg y_1j,m^t;t_2=arg y_i1,m^t,t_2 > t_1, ∀ t_3∈(t_1,t_2); t_3≠ arg y_1j,m^t ≠ arg y_i1,m^t (10) ] The objective is to minimize the number of UAVs that are needed to provide a continuous coverage during the coverage duration T subject to various design constraints. Constraints (1) and (2) ensure that the UAV can travel and cover the subareas only if we select it to participate in the coverage process. Constraint (3) ensures that all subareas will be covered at the first time slot. Constraint (4) guarantees the continuous coverage for each subarea. Constraint (5) allows the UAV to visit a new subarea (when y_ij,m^t=1) or to continue covering the current subarea (when y_ij,m^t=0). Constraint (6) characterizes the handoff process between the UAVs, when the UAV m_1 wants to visit the subarea j from subarea i_1 at time t (y_i_1j,m_1^t=1), the UAV m_2 that covers the subarea j will travel to subarea i_2 at time t+1 (y_ji_2,m_2^t+1=1). Constraint (7) describes the relation between the traveling process and the covering process, where the number of times that the subarea j is covered will be greater than or equal the number of times that it is visited. Constraint (8) shows that the recharging process will take T_charge at the charging station. Constraint (9) ensures that the number of UAVs outgoing from the charging station and the number of UAVs incoming to charging station are the same after we complete the coverage process. Constraint (10) shows that the energy capacity of the UAV can cover the wasted energy during the traveling and the covering processes in each cycle where t_1 represents the time that the UAV travels from the charging station with full energy capacity and t_2 represents the time that the UAV arrives to the charging station to charge the battery. Now we will prove that the continuous coverage problem is an NP-complete. §.§ NP completeness The Continuous Coverage Problem is NP-complete. The number of constraints is polynomial in terms of the number of subareas, the number of UAVs and the number of time slots. Given any solution for our problem, we can check thesolution’s feasibility in polynomial time, then the problem is NP.To prove that the problem is NP-hard, we reduce the Bin Packing Problem which is NP-hard <cit.> to a special case of our problem. The special case will be the discrete coverage problem in which each subarea will be visited one time by one UAV during the coverage process. In the Bin Packing Problem, we have p items where each item has volume z_p. All items must be packed into a finite number of bins (b_1, b,...,b_B), each of volume V in a way that minimizes the number of bins used. The reduction steps are: 1) The b-th bin in the Bin Packing Problem is mapped to the m-th UAV in our problem (where the volume V for each bin is mapped to the energy capacity of the UAV E). 2) The p-th item is mapped to the n-th subarea, (where the volume for each item p is mapped to the energy consumed when the UAV (visits and covers) subarea n. 3) All UAVs have the same energy capacity E. 4) The energy consumed (during the traveling and the covering processes) when the UAV visits subarea j from any subarea (i ∈ G \ {j}) will be constant. 5) The energy required for the UAV to return to the charging station from any subarea i will be zero (E_i1^Travel=0). 6) The time that the UAV needs to recharge the battery at the charging station will be infinity. 7) Each subarea will be visited one time by one UAV during the coverage process (discrete coverage). If there exists a solution to the bin packing problem with cost C, then the selected bins will represent the UAVs that are selected and the items in each bin will represent the subareas that the UAV must visit and the total cost of our problem is C. If there exists a solution to our problem with cost C, then the selected UAVs will represent the bins that are selected and the subareas that the UAV must visit will represent the items in each bin and the total cost of the bin packing problem is C. We prove that there exists a solution to the bin packing problem with cost C iff there exists a solution to our problem with cost C. § HEURISTIC ALGORITHMDue to the intractability of the problem, we study partitioning the coverage graph into cycles that start at the charging station. We first characterize the minimum number of UAVs to cover each cycle based on the charging time, the traveling time, and the number of subareas to be covered by the cycle. Our analysis based on the uniform coverage in which the UAV covers each subarea in a given cycle for a constant time. Based on this analysis, we then develop an efficient algorithm, the cycles with limited energy algorithm, that minimizes the required number of UAVs that guarantees a continues coverage.§.§ Analysis It is obvious that we need N UAVs to cover N subareas atany given time, but the question here is how many additional UAVs are needed to guarantee a continuous coverage. In this subsection, we assume that the UAV visits the subareas based on a cycle that starts from the charging station and ends at the charging station for charging process. We also assume that a given UAV covers the subareas in the cycle uniformly, in which the UAV covers each subarea in a given cycle for a constant time.In Theorem <ref>, we find the minimum number of additional UAVs that are needed to guarantee a continuous coverage for a cycle, which will help us while developing Algorithm 1. The minimum number of additional UAVs k that are required to provide continuous and uniform coverage for a cycle that contains n subareas must satisfy this inequality: kT_Coverage/n≥ (n+1)T + T_Chargewhere T_Coverage is the time that the UAV allocates to cover all subareas in the cycle, T is the time that the UAV needs to travel from subarea i to subarea j and T_charge is the time that the UAV needs to recharge the battery at the charging station. Consider that all n subareas in the cycle are covered by n UAVs and the UAV that covers the last subarea want to return to the charging station to recharge its battery. The handoff process needs to begin between one of the additional UAVs from the charging station and the UAV that covers the first subarea in the cycle.The UAV that covers the last subarea needs to wait (n-1) T to do the handoff process, during this time the additional UAVs are covering the first subarea. After the handoff process is completed, the UAV needs T time units to return to the charging station, T_charge to recharge the battery and T to visit the first subarea in the cycle again. Then, we have: kT_Coverage/n≥ (n-1)T +T+ T_Charge+T§.§ The cycles with limited energy algorithmThe straightforward method (SM) to continuously cover N subareas is to allocate two UAVs for each subarea. At the first time slot, N UAVs cover the N subareas. Then, any UAV wants to return to the charging station to recharge the battery will do the handoff process with one of the additional UAVs that are available at the charging station. By applying SM, we need N additional UAVs and we have N cycles to cover all the subareas.Our proposed algorithm, the cycles with limited energy algorithm (CLE), is inspired by the nearest neighbor algorithm, the nearest neighbor algorithm is used to solve the Traveling Salesman Problem <cit.>, in which the salesman keeps visiting the nearest unvisited vertex until all the vertices are visited. In our algorithm, the UAV (salesman) has limited energy capacity and before visiting any new subarea, we must check if the remaining energy is enough to return to the charging station from the new location or not. In the previous subsection, we show how to find the minimum number of additional UAVs that are required to guarantee the continuous coverage for a given cycle, we use the Theorem <ref> to find the minimum number of additional UAVs that are required to provide the continuous coverage for a given area, by finding the cycles that need only one additional UAV. The pseudo code of this algorithm is shown in Algorithm 1. § PERFORMANCE EVALUATION §.§ Power Consumption Models In this section, we quantify the power consumption by each UAV when it is hovering, traveling and transmitting data. §.§.§ Power consumption during hovering The power consumption in watt by the UAV during hovering can be given by <cit.>: P=4T_h^3/2/√(2QS) + p where T_h is the fourth of the quadcopter total weight in N, Q is the density of the air in kg/m^3, S is the rotor swept area in m^2 and p is the power consumption of electronics in watt. §.§.§ Power consumption during traveling The power consumption in kW by the UAV during traveling can be given by <cit.>: P=(m_p+m_v)v/370η r + p where m_p is the payload mass in kg, m_v is the vehicle mass in kg, r is the lift-to-drag ratio (equals 3 for vehicle that is capable of vertical takeoff and landing), η is the power transfer efficiency for motor and propeller, p is the power consumption of electronics in kW and v is the velocity in km/h. §.§.§ Power consumption during data transmission The power consumption in dB by the UAV during data transmission can be given by <cit.>: P_t(dB)=P_r(dB)+ L̅(R,h) L̅(R,h)=P(LOS)× L_LOS+P(NLOS)× L_NLOS P(LOS)=11+α.exp(-β[180/πθ-α]) L_LOS(dB)=20log(4π f_cdc)+ξ_LOS L_NLOS(dB)=20log(4π f_cdc)+ξ_NLOS In equation (5), P_t is the transmit power, P_r is the required received power to achieve a SNR greater than threshold γ_th, L̅(R,h) is the average path loss as a function of the altitude h and coverage radius R. In equation (6), P(LOS) is the probability of having line of sight (LOS) connection at an evaluation angle of θ, P(NLOS) is the probability of having non LOS connection and equal (1-P(LOS)), L_LOS and L_NLOS are the average path loss for LOS and NLOS paths. In equations (7), (8) and (9), α and β are constant values which depend on the environment, f_c is the carrier frequency, d is the distance between the UAV and the user, c is the speed of the light , ξ_LOS andξ_NLOS are the average additional loss which depend on the environment. Actually, the power consumed by the UAV during data transmission and reception is much smaller than the power consumed during the UAV hovering or traveling <cit.>. In this paper, we assume that the power wasted during data transmission is constant. §.§ Simulation Setup Given a geographical area G , the number of the subareas that we need to cover and the density of the users, the question here is how to find the optimal boundaries of the subareas that to be covered by the UAVs. To answer this question, the authors of <cit.> used transport theory to find the optimal boundaries of the subareas. Unfortunately, this approach needs to solve N2 non-linear equations at each iteration, where N is the number of subareas. In this paper, we divide the geographical area uniformly and apply the SM and CLE algorithm to find the minimum number of additional UAVs that provides the continuous coverage. We study the effect of the UAV energy capacity, the grid size of the geographical area, the charging time and the traveling time on the number of the additional UAVs. Table II lists the parameters used in the numerical analysis <cit.>. In Figure <ref>, we uniformly divide the geographical area into 16 subareas and apply the CLE algorithm to find the cycles with minimum number ofadditional UAVs. From the figure, we notice that 5 cycles are needed to cover all subareas with 5 additional UAVs. Also, we note that the paths of the cycles are intersected in many locations. To avoid the collisions between the UAVs, we operate the paths (cycles) at different altitudes with small altitude differences. In Figure <ref>, we study the effect of the UAV energy capacity on the number of theadditional UAVs needed to cover the subareas. Actually, when increasing the energy capacity of the UAV and apply SM, the number of additional UAVs needed will not change because each subarea is covered by one cycle and two UAVs. When increasing the energy capacity of the UAVs, only the coverage time of each UAVincreases. on the other hand, increasing the energy capacity of each UAV results in minimizing the number of additional UAVs that needed using CLE. This is because increasing the energy capacity of each UAV gives the UAV a chance to visit and to cover more subareas, which minimizes the number of the cycles that are needed to cover the subareas. In Figure <ref>, the slope of the line produced by SM is greater than the curve of CLE. When applying SM, the number of additional UAVs increases linearly with the grid size. This is because the number of additional UAVs equals the grid size. Also, when applying the CLE, the number of additional UAVs increases with the grid size. This is because more cycles are needed to cover more subareas and each cycle will need one additional UAV. In Figure <ref>, we study the effect of the charging time on the number of additional UAVs needed. Changing the charging time will not affect the number of additional UAVs needed when applying SM. This is because the coverage time of each UAV will cover the time that the UAV needs to return to the charging station to recharge the battery and to visit the subarea again. On the other hand, when applying CLE, it will be a critical issue (see Theorem <ref>). Actually, charging the battery of the UAV takes long time. For this reason, each UAV has a replacement battery <cit.>. In this paper, we assumethe time needed to replace the battery for each UAV is 5 minutes. In Figure <ref>, we study the effect of the traveling time on the number of additional UAVs. Changing the traveling time will not affect the number of additional UAVs when applying SM. On the other hand, it will be a critical issue to choose the appropriate traveling time when applying CLE. When increasing the traveling time, the wasted energy during traveling will increase and the coverage time will decrease. Hence, the chance to visit other subareas will decrease. § CONCLUSION In this paper, we study the continuous coverage problem with minimum number of replacement UAVs and prove that it is NP complete. We design an efficient algorithm to solve it, the cycles with limited energy algorithm. The proposed algorithm covers the N subareas by cycles, in which each cycle needs one additional UAV to ensure continuous coverage. We showed that the energy capacity of the UAVs, the number of subareas in the affected area, and the UAV charging and traveling times will all impact the required number of UAVs. Our simulation results showed that applying the cycles with limited energy algorithm, can efficiently reduce the number of additional UAVs needed relative to the straightforward method. As future work, we will study the continuous coverage problem using UAVs with non-uniform energy capacities and the use of green energy. § ACKNOWLEDGMENTThe work of Jacob Chakareski has been partially supported by the NSF under award CCF-1528030.IEEEtran | http://arxiv.org/abs/1705.09766v1 | {
"authors": [
"Hazim Shakhatreh",
"Abdallah Khreishah",
"Jacob Chakareski",
"Haythem Bany Salameh",
"Issa Khalil"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170527053432",
"title": "On The Continuous Coverage Problem for a Swarm of UAVs"
} |
[ 1slp_12slp_2p_Mp_M' [email protected]@yonsei.ac.kr^1 Department of Physics, Universidad Técnica Federico Santa María, Valparaíso, Chile ^3Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea We consider the lepton number violating decays B →μ^±μ^±π^∓ and B →D^(*)μ^±μ^±π^∓ which may be detected at LHCb and Belle-II experiments; andB →μ^±μ^± e^∓ν and B → D^(*)μ^±μ^± e^∓ν decays which may be detected at Belle-II experiment. The projected total number of produced B mesons is 4.8 × 10^12 at LHCb upgrade and 5 × 10^10 at Belle-II. For the case that the above decays are not detected, we deduce the new upper bounds (sensitivity limits) for the mixing parameter |U_μ N|^2 of heavy sterile neutrino with sub-eV light neutrino, as a function of the sterile neutrino mass in the interval 1.75 GeV < M_N < 5.0 GeV. We take into account the probability of decay of the sterile neutrino N within the detector, taking as the effective detector lengthL=2.3m at LCHb upgrade and L=1m at Belle-II. In the interval 1.75 GeV < M_N < 3 GeV, the most stringent bounds can be obtained with the decays B →μ^±μ^±π^∓ at LHCb upgrade. The sensitivity limits are expected to be in general more stringent at LHCb upgrade than at Belle-II, principally because the number of produced B mesons in LHCb upgrade is expected to be by about two orders of magnitude larger than at Belle-II. We conclude that the LHCb upgrade and Belle-II experiments have the potential to either find a new heavy Majorana neutrino N, or to improve significantly the sensitivity limits (upper bounds) on the heavy-light mixing parameter |U_μ N|^2, particularly in the mass range 1.75 GeV < M_N < 3 GeV. This work is a continuation and refinement of our previous work [Phys. Rev. D94, 053001 (2016);ibid 95, 039901(E) (2017)] on the subject.14.60St, 13.20HeSensitivity limits on heavy-light mixing |U_μ N|^2from lepton number violating B meson decays[v4: typos in Eqs. (10), (11) and (17b) corrected; the correct expressions had been used in calculations, results unchanged.] C. S. Kim^2 December 30, 2023 ============================================================================================================================================================================================================================§ INTRODUCTION The existence of sterile neutrinos has not been proven yet. However, their existence is suggested by various scenarios which can explain the detected differences of masses of the three known light neutrinos. Furthermore, most of such scenarios suggest that the neutrinos are Majorana fermions. Since Majorana fermions, unlike the Dirac fermions, are their own antiparticles, they can participate not just in thelepton number conserving (LNC) processes, but also in the lepton number violating (LNV) processes. LNV processes are appreciable if the Majorana neutrinos are sufficiently massive. Various scenarios suggest that mixing of sterile neutrinos with the known Standard Model (SM) flavor neutrinos leads to neutrinos which are significantly heavier than the known light neutrinos. The main questions facing the neutrino physics beyond the SM are: (1) Are the neutrinos Majorana or Dirac? (2) How heavy are the new mass eigenstatesN? (3) What are the values of theheavy-light mixing parameters U_ℓ N, i.e., the mixing parameters of a massive N neutrino with the SM flavor neutrinos ν_ℓ (ℓ=e, μ, τ)?Whether the neutrinos are Majorana particles can be determined in neutrino experiments with various LNV processes. Among the most known such experiments are those with the neutrinoless double beta decay (0νββ) <cit.>, rare LNV decays of mesons <cit.> and of τ lepton <cit.>, and specific scattering processes <cit.>.Observation of neutrino oscillations <cit.> can determine (small) mass differences between neutrinos, and thus prove that the neutrinos have mass. The neutrino oscillations of the SM flavor neutrinos have been observed <cit.>. If sterile neutrinos exist and if their mixing with the SM flavor neutrinos leads to almost degenerate heavy neutrinos, also such neutrinos can oscillate among themselves <cit.>.The neutrino sector can also have CP violation <cit.>, which plays an important role in the leptogenesis <cit.>. Resonant CP violation of neutrinos appears when we have two heavy almost degenerate neutrinos. It can appear in scattering processes <cit.>, in semileptonic rare meson decays <cit.>, and in purely leptonic rare meson decays <cit.>. Among the models with almost degenerate heavy neutrinos arethe neutrino minimal standard model (νMSM) <cit.> and low-scale seesaw models <cit.>.As mentioned, extended sectors of Majorana neutrinos appear in models which explain the very small masses of the three light neutrinos. Such models are the original seesaw models <cit.> (the heavy neutrinos there have masses M_N ≫ 1 TeV), and seesaw models with heavy neutrinos with lower masses M_N ∼ 1 TeV <cit.>, and M_N ∼ 1 GeV <cit.>. In such models, the heavy-light mixing parameters are in general less suppressed than in the original seesaw models. In this work, we will work in a generic framework where we have one massive neutrino N which mixes with theSM flavor neutrinos ν_ℓ (ℓ=e, μ, τ). We will evaluate the rates of some rare decays of B mesons at the future LHCb upgrade and Belle-II experiments, namely, the LNV decays with one on-shell Majorana massive neutrino N:B → (D^(*)) μ^± N →(D^(*)) μ^±μ^± X^∓, where X^∓ is either a pion π^∓, or a lepton-neutrino pair ℓν_ℓ(this latter option only at Belle-II). This work is based on our previous work <cit.>, but now the obtained results are more specific and directly applicable to the calculation of the sensitivity limits on the |U_μ N|^2 mixing parameter, as a function of mass M_N, achievable at LHCb upgrade and at Belle-II, where the projected total number of produced B mesons is 4.8 × 10^12<cit.> and 5 × 10^10<cit.>, respectively. Unlike in Ref. <cit.>, here we do not make any assumptions on the size of the probability P_N of the produced neutrino N to decay within the detector (in <cit.> we assumed that either P_N ≈ 1 or P_N ≪ 1). Detailed explanation on this issue is given in Sec. <ref> and in Appendix <ref>.Similar analyses for the upper bounds on |U_μ N|^2 from the absence of the rare B-meson decays were made for the Belle-I mesurements in Ref. <cit.>, and for LHCb (run I) measurements in Refs. <cit.> and reconsideration thereof in Ref. <cit.>.In Sec. <ref> we summarize the framework in which we work, and the decay widths which are relevant for the decay rates that we want to obtain. The summarized formulasfor these decay widths are presented in subsections of Sec. II and Appendix <ref>. In Sec. <ref> we present the probability P_N of the produced on-shell neutrino N to decay within the detector, and the integration formulas which account for the effect of this probability on the effective rate for the mentioned LNV decays. In Appendix <ref> we present detailed formulas for the Lorentz factors and the probabilities P_N for the various considered decays. In Sec. <ref> we present the results of the numerical evaluations, in the form of the obtained sensitivity limits on |U_μ N|^2, as a function of M_N, that can be achieved by LHCb upgrade and Belle-II experiments. In Sec. <ref> we discuss the obtained results and make conclusions.§ DECAY WIDTHS FORB → (D^(*)) ℓ_1 N →(D^(*)) ℓ_1 ℓ_2 X Here we briefly summarize the results of Ref. <cit.> for the decay widths of the rare decays of B mesons via on-shell sterile neutrino N. The on-shellness of N implies the factorization Γ( B → (D^(*)) ℓ_1 N → (D^(*)) ℓ_1 ℓ_2 X ) =Γ( B → (D^(*)) ℓ_1 N ) Γ(N →ℓ_2 X)/Γ_N .Here, ℓ_j (j=1,2) are generical names for charged leptons; later we will use ℓ_1 = ℓ_2 = μ^±. The second factor on the right-hand side of Eq. (<ref>) represents the effect of the subsequent decay of the produced heavy on-shell neutrino N into ℓ_2 + X, where X will be either a charged pion π, or a leptonic pair ℓ_3 ν_3.The first factor in Eq. (<ref>), Γ( B → (D^(*)) ℓ_1 N ), is well known when no D^(*) meson is produced; when D^(*) is produced, this factor was obtained and evaluated in Ref. <cit.>. The formulas for this factor are summarized in subsections A-C, as well as some (here relevant) differential decay widths for these decays B → (D^(*)) ℓ_1 N. The second factor in Eq. (<ref>) includes the exclusive decay width Γ(N →ℓ_2 X) which is well known, either for X=π or X=ℓ_3 ν_3. For both cases, the expressions for these decay widths are summarized in subsections D-E. The denominator of the second factor in Eq. (<ref>), namely the total decay width Γ_N of neutrino N, was evaluated numerically in <cit.> for the case of Majorana N (cf. also <cit.> for the case of N Majorana or Dirac); the expression for Γ_N and its evaluation is presented in Appendix <ref>.All the mentioned decay widths involve the (suppressed) heavy-light mixing parameters U_ℓ N (ℓ=e, μ, τ) appearing in the coupling of the heavy N neutrino with the W boson and ℓ lepton. These parameters are part of the (extended) Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, i.e., the light flavor neutrino states ν_ℓ (with flavor ℓ = e, μ, τ) are the following combination of the three light mass eigenstates ν_k and of the heavy mass eigenstate N: ν_ℓ = ∑_k=1^3 U_ℓν_kν_k + U_ℓ N N. §.§ Decay width Γ(B →ℓ_1 N) The decay width for the processB →ℓ_1 N, where ℓ_1 is a charged lepton (ℓ_1=e, μ, τ) and N is a (massive) neutrino, is Γ(B^±→ℓ_1^± N) =|U_ℓ_1 N|^2(B^±→ℓ_1^± N) ,where the canonical decay width , i.e., the part without the heavy-light mixing factor, is(B^±→ℓ_1^± N) = G_F^2 f_B^2/8 π |V_u b|^2 M_B^3 λ^1/2(1,y_N,y_1) [ (1 - y_N) y_N + y_1 (1 + 2 y_N - y_1) ].Here, G_F is the Fermi coupling constant (G_F = 1.1664 × 10^-5GeV^-2), f_B is the decay constant of the B-meson, V_u b its CKM matrix element, and in the mass dependent parts the following notations are used:y_N= M_N^2/M_B^2 ,y_1 = M_1^2/M_B^2 ,λ^1/2(x,y,z)= [ x^2 + y^2 + z^2 - 2 x y - 2 y z - 2 z x ]^1/2.We denote the mass of ℓ_1 as M_1 throughout this paper. We use the values |V_ub|=0.00409 and f_B=0.1871 GeV <cit.> (cf. also <cit.>). §.§ Decay width Γ(B → D ℓ_1 N) We now consider the decay B → D ℓ_1 N, cf. Fig. <ref>.For the general case of a massive neutrino N (and a massive charged lepton ℓ_1), the general expression for the decay width of the process B → D ℓ_1 N was obtained in Ref. <cit.>. There, the differential decay width d Γ(B^- → D^0 ℓ_1^- N̅)/d q^2 was presented.Here we present the “more differential” cross section d Γ(B^- → D^0 ℓ_1^- N̅)/(d q^2 d Ω_q̂' d Ω_p̂_1), which is needed for calculation of the effective (true) branching ratio Br_ eff(B → D ℓ_1 N → D ℓ_1 ℓ_2 X) of Eq. (<ref>). The differential of the decay width isd Γ(B^- → D^0 ℓ^-_1 N) = 1/2 M_B1/(2 π)^5 d_3 | T|^2,where d_3 is the differential for the three-particle final phase spaced_3 =d^3 p⃗_D/2 E_D(p⃗_D)d^3 p⃗_1/2 E_ℓ_1(p⃗_1)d^3 p⃗_N/2 E_N(p⃗_N)δ^(4)( p_B - p_D - p_1 - p_N ) = d_2 ( B^- → D^0(p_D) W^*(q) ) d q^2 d_2 ( W^*(q) →ℓ_1(p_1) N(p_N) ),and the two-particle final phase space differentials ared_2(B^- → D^0(p_D) W^*(q)) =1/8λ^1/2( 1, M_D^2/M_B^2, q^2/M_B^2) d Ω_q̂', d_2(W^*(q) →ℓ^-_1(p_1) N(p_N)) =1/8λ^1/2( 1, M_1^2/q^2, M_N^2/q^2) d Ω_p̂_1.The decay amplitude T appearing in Eq. (<ref>) isT=U_ℓ_1 N V_c bG_F/√(2)[u_(ℓ_1)(p_1) γ_μ (1 - γ_5) v_(N)(p_N) ] {[ (2 p_D + q)^μ - (M_B^2-M_D^2)/q^2 q^μ] F_1(q^2) + (M_B^2-M_D^2)/q^2 q^μ F_0(q^2) },where F_1(q^2) and F_0(q^2) are the form factors of the B-D transition, and we consider them to be real.In terms of the reduced canonical decay amplitude T defined via the relation|T |^2 = 4 |U_ℓ_1 N|^2 |V_c b|^2 G_F^2 | T|^2,we can then express the differential decay width (<ref>) in a somewhat more explicit form,[In v3, the right-hand side of Eq. (<ref>) has a typo (a superfluous factor 1/4), but in the calculations the correct expression was used.]d Γ(B^- → D^0 ℓ^-_1 N)/d q^2 d Ω_q̂' d Ω_p̂_1=|U_ℓ_1 N|^2 |V_c b|^2 G_F^2/M_B (4 π)^5 | T|^2λ^1/2( 1, M_D^2/M_B^2, q^2/M_B^2)λ^1/2( 1, M_1^2/q^2, M_N^2/q^2),where p̂_1 is the direction ofℓ^-_1 in the W^*-rest frame (Σ), and q̂' is the direction of W^*- (ℓ^-_1 N pair) in the B-rest frame (Σ'). We use the expression (<ref>) for the decay amplitude, and calculate the square of its absolute magnitude, | T|^2, summing over the helicities of the final particles. We then obtain for the square of the reduced canonical amplitude, | T|^2, introduced via Eq. (<ref>), the following expression:| T|^2= 1/q^2 F_1(q^2) (F_0(q^2)-F_1(q^2)) (M_B^2-M_D^2) [M_1^2 (-4 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0p_1^0)+2 M_B^2-2 M_D^2+2 M_N^2-q^2)+M_N^2 (4 (cosθ_1 |p⃗_D||p⃗_N|+p_D^0p_1^0)-M_N^2+q^2)-M_1^4]-1/2 F_1(q^2)^2[M_1^2( 8 (cosθ_1 |p⃗_D||p⃗_N|+p_D^0 p_1^0)-4 M_B^2-2 M_N^2+3q^2)-8 M_B^2 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0 p_1^0)+M_D^2 (8 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0 p_1^0)-4 M_N^2+4 q^2) -8 M_N^2 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0 p_1^0) +8q^2 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0 p_1^0)+16 (cosθ_1 |p⃗_D| |p⃗_N|+p_D^0p_1^0)^2+M_1^4+M_N^4-M_N^2q^2]+1/2 (q^2)^2 (F_0(q^2)-F_1(q^2))^2 (M_B^2-M_D^2)^2 [-M_1^4+M_1^2 (2M_N^2+q^2)-M_N^4+M_N^2 q^2].Here, we denoted as p_1 the 4-momentum of ℓ_1 (in W^*-rest frame Σ), and θ_1 is the angle between p⃗_1 and ẑ = q̂'. We also used in Eq. (<ref>)the following quantities:|p⃗_N| = |p⃗_1| =1/2√(q^2) λ^1/2( 1, M_1^2/q^2, M_N^2/q^2), |p⃗_D| =M_B^2/2 √(q^2) λ^1/2( 1, M_D^2/M_B^2, q^2/M_B^2) = M_B |q⃗'⃗|/√(q^2), p_1^0 =1/2 √(q^2) (q^2 - M_N^2 + M_1^2), p_D^0 =1/2 √(q^2) (M_B^2 - M_D^2 - q^2).They are all in the W^*-rest frame (Σ). We can see from these expressions that the absolute square of the reduced canonical amplitude, | T|^2, and thus the differential decay width (<ref>), depend only on the variables q^2 (square of the invariant mass of W^*) and on cosθ_1 [note: d Ω_p̂_1 = d ϕ_1 d (cosθ_1)]. They are thus independent of the direction q̂', i.e., of the direction of W^* in the B-rest frame.The expressions (<ref>) and (<ref>) contain two form factors, F_1 and F_0. The form factor F_1(q^2) is well known <cit.> and can be expressed in terms of a variable w(q^2) w =(M_B^2 + M_D^2 - q^2)/2 M_B M_D , z(w) =√(w+1) - √(2)/√(w+1) + √(2) .According to Ref. <cit.>, F_1(q^2) has the following power expansion in z(w(q^2)):F_1(q^2) = F_1(w=1) ( 1 - 8 ρ^2 z(w) + (51 ρ^2 - 10) z(w)^2 - (252 ρ^2 - 84) z(w)^3 ).The free parameters ρ^2 and F_1(w=1) in this expansion have been determined by the Belle Collaboration, Ref. <cit.>ρ^2= 1.09 ± 0.05, |V_cb| F_1(w=1)=(48.14 ± 1.56) × 10^-3 .In our numerical evaluations we use the above central values, and |V_cb|=40.12 × 10^-3<cit.>.The form factor F_0(q^2) is not well known at present, principally because it contributes only when the masses of N and ℓ_1 are not very small as can be deduced from Eq. (<ref>).[It can be checked that the difference [ | T|^2 - | T|^2(F_0 ↦ 0)] is zero when M_1=M_N=0.] In our case F_0(q^2) is important, and it was presented in Ref. <cit.> by usingthe truncated expansion for F_0 in powers of w(q^2) - 1 of Ref. <cit.> F_0(q^2) =(M_B+M_D)/2 √(M_B M_D)[ 1 - q^2/(M_B+M_D)^2] f_0(w(q^2)), f_0(w)≈f_0(w=1) [ 1 - ρ_0^2 (w - 1) + (0.72 ρ_0^2 - 0.09) (w - 1)^2 ].Here,[In v3, as well as in our previous work <cit.>, the expression (<ref>) was written with a typo [+ρ_0^2 (w - 1) instead of - ρ_0^2 (w - 1)], but the correct expression was used in the calculations here and in <cit.>.]we use the value f_0(w=1) ≈ 1.02<cit.> which is obtained from the heavy quark limit. The other free parameter ρ_0 in Eq. (<ref>) is then fixed by requiring the absence of spurious poles at q^2=0: F_0(0)=F_1(0) (≈ 0.690). This yields the value ρ_0^2 ≈ 1.102 and (0.72 ρ_0^2 - 0.09) ≈ 0.704.For the curves of these form factors F_1(q^2) and F_0(Q^2), as a function of positive q^2, we refer to Ref. <cit.> (Fig. 2 there). §.§ Decay width Γ(B → D^*ℓ_1 N) We now consider the decay B → D^*ℓ_1 N, i.e., the same type of decay as in the previous Sec. <ref>, but now instead of the (pseudoscalar) D meson we have vector meson D^*. The expressions for the (differential) decay widths are now more complicated, because D^* is a vector particle. For the case of massive neutrino N (and massive lepton ℓ_1), these expressions were obtained in Ref. <cit.>, using the approach of Ref. <cit.>. The needed differential decay width, after summation over helicities and over the three polarizations of D^*, turns out to be <cit.>d Γ/dq^2 d Ω_q̂' d Ω_p̂_1=1/8^4 π^5|U_ℓ_1 N|^2 |V_cb|^2 G_F^2/M_B^2^1/2 2 |q⃗'⃗|q^2 {[2 (1 - (M_N^2+M_1^2)/q^2) - sin^2 θ_1] ( (H̅_+1)^2 + (H̅_-1)^2 ) - η2 ^1/2cosθ_1 ( (H̅_+1)^2 - (H̅_-1)^2 ) + 2 [ (1 - (M_N^2+M_1^2)/q^2) - cos^2 θ_1 ] (H̅^3)^2 + 4 ( M_N^2-M_1^2/q^2) ^1/2cosθ_1 H̅^0H̅^3 + 2 [ - (M_N^2-M_1^2/q^2)^2 + (M_N^2+M_1^2)/q^2] (H̅^0)^2 } .Here, the factor η=± 1 appears at one term proportional to cosθ_1; η=+1 if ℓ^-_1 is produced, and η=-1 if ℓ^+_1 is produced.[The quantity (<ref>) is written in Ref. <cit.> in Eq. (C19) for the case η=-1; the quantity d Γ/d q^2 used there is independent of η.] Further, the following notations are used:|q⃗'⃗|= 1/2 M_B λ^1/2( 1,M_^2/M_B^2, q^2/M_B^2), ≡ λ( 1, M_1^2/q^2, M_N^2/q^2),and H̅_± 1, H̅^0 and H̅^3 are expressions containing the form factors V and A_j (j=0,1,2,3) appearing in the B-D^* matrix elements H̅_± 1 =(M_B+M_) A_1(q^2) ∓ V(q^2) |q⃗'⃗| 2 M_B/(M_B+M_) , H̅^3=M_B^2/2 M_√(q^2)[(M_B+M_) A_1(q^2) (1 - (q^2+M_^2)/M_B^2)- 4 A_2(q^2) |q⃗'⃗|^2/(M_B+M_)], H̅^0=M_B |q⃗'⃗|/M_√(q^2)[ (M_B+M_) A_1(q^2)- (M_B- M_) A_2(q^2) + 2 M_( A_0(q^2) - A_3(q^2) ) ].A_3 form factor is not independent, it is a linear combination of A_1 and A_2 A_3(q^2) = (M_B+M_)/2 M_ A_1(q^2) - (M_B-M_)/2 M_ A_2(q^2).Among the other four form factors, three (V, A_1 and A_2) are well known, they were recently determined to a high precision<cit.> in terms of the parametrization of Ref. <cit.> A_1(q^2) =1/2 R_* (w+1) F_*(1) [ 1 - 8 ρ_*^2 z(w) + (53 ρ_*^2 - 15) z(w)^2 - (231 ρ_*^2 - 91) z(w)^3 ], V(q^2) = A_1(q^2)2/R_*^2 (w+1)[ R_1(1) - 0.12 (w-1) + 0.05 (w-1)^2 ], A_2(q^2) = A_1(q^2) 2/R_*^2 (w+1)[ R_2(1) + 0.11 (w-1) - 0.06 (w-1)^2 ].The notationR_* = 2 √( M_B M_)/(M_B+M_) is used here, and w=w(q^2) and z=z(w(q^2)) are given in Eqs. (<ref>) (with M_D ↦ M_D^*). The values of the three parameters in Eqs. (<ref>) were determined in Ref. <cit.>ρ_*^2 = 1.214(± 0.035),10^3 F_*(1) |V_cb| = 34.6(± 1.0), R_1(1) =1.401(± 0.038),R_2(1) = 0.864(± 0.025).We use the central values in the present work.The form factor A_0, on the other hand, is not well known. It is relevant only if the masses of N or ℓ_1 are nonnegligible, which is the case here. Employing the heavy quark limit relations between A_1 and A_2, the relation (<ref>) gives a relation between A_2 and A_3. Using this relation in the heavy quark limit relation A_0 ≈ A_2, we then obtain the following approximation for the form factor A_0 in terms of A_3:A_0(q^2) ≈ A_3(q^2)/[1 - q^2/2 M_ (M_B+M_)] = (M_B+M_)^2/( 2 M_ (M_B+M_) - q^2 )( 1 - (M_B-M_)/(M_B+M_)A_2(q^2)/A_1(q^2)) A_1(q^2),This relation satisfies the relation A_0(0)=A_3(0) which is obligatory since it reflects the absence of the pole at q^2=0 in the B-D^* matrix elements. We refer for any further details on these points to Ref. <cit.>. §.§ Decay width for N →ℓ^±π^∓ The decay width Γ(N →ℓ^±π^∓) is proportional to the heavy-light mixing factor |U_ℓ N|^2Γ(N →ℓ^±π^∓) = |U_ℓ N|^2 (N →ℓ^±π^∓).Here, the canonical decay widthis (e.g., cf. Refs. <cit.>) (N →ℓ^±π^∓) = 1/16 π |V_u d|^2 G_F^2 f_π^2 M_N^3 λ^1/2(1, x_π, x_ℓ) [ 1 - x_π - 2 x_ℓ - x_ℓ(x_π-x_ℓ) ],where f_π (≈ 0.1304 GeV) is the decay constant of pion, and we use the notationsx_π = M_π^2/M_N^2 ,x_ℓ=M_ℓ^2/M_N^2 .§.§ Decay width for N →ℓ_2 ℓ_3 ν If the heavy neutrino N is produced by the decay B → (D^(*)) ℓ_1^± N, the neutrino can decay into various leptonic channels ℓ_2 ℓ_3 ν. We can have the leptonic decays of N of the lepton number conserving (LNC) typeN →ℓ_2^∓ℓ_3^±ν_ℓ_3, and of the lepton number violating (LNV) type N →ℓ_3^±ℓ_2^∓ν_ℓ_2Γ^ (LNC)(N →ℓ_2^∓ℓ_3^±ν_ℓ_3) = |U_ℓ_2 N|^2(N →ℓ_2 ℓ_3 ν),Γ^ (LNV)(N →ℓ_3^±ℓ_2^∓ν_ℓ_2) = |U_ℓ_3 N|^2(N →ℓ_2 ℓ_3 ν).Here, the charged leptons can be μ, e or τ. The canonical decay widths (N →ℓ_2 ℓ_3 ν) have in the general case (with masses of leptons) the following form <cit.>: Γ(N →ℓ_2 ℓ_3 ν) =G_F^2 M_N^2/192 π^3 F(x_2,x_3),where we denoted x_j = M_j^2/M_N^2 (M_j is the mass of ℓ_j), and the function F is <cit.> F(x_2,x_3) = {λ^1/2 (1, x_2, x_3)[ (1 + x_2) (1 -8 x_2 + x_2^2)- x_3 (7 - 12 x_2 + 7 x_2^2) - 7 x_3^2 (1 + x_2)+ x_3^3] - 24 (1 - x_3^2) x_2^2 ln 2 +12[ - x_2^2 (1 - x_3^2) ln x_2+ (2 x_2^2 -x_3^2 (1 + x_2^2)) ln (1 + x_2 + λ^1/2 (1, x_2, x_3)- x_3) + x_3^2 (1 - x_2^2) ln( (1 - x_2)^2 + (1-x_2) λ^1/2 (1, x_2, x_3) - x_3 (1+x_2)/x_3)]} .The function F is symmetric under the exchange of the two arguments. When one lepton is massless (or almost massless, i.e., lepton e), this expression reduces to the well-known resultF(x,0) = F(0,x)= f(x) = 1 - 8 x + 8 x^3 - x^4 - 12 x^2 ln x. § DECAY PROBABILITY OF HEAVYNEUTRINO IN THE DETECTOR; EFFECTIVE BRANCHING RATIO If all the neutrinos N decay within the detector with probability one, then the decay width Eq. (<ref>) is also the effective (true) decay width, and the effective branching ratio is obtained by dividing it by the decay width of the B meson Γ_B. However, since the neutrino N is weakly coupled to SM particles, it often does not decay within the detector and, consequently, the mentioned decays B → (D^(*)) ℓ_1 ℓ_2 X are not observed although N may be produced in the B-decays. The effect of the decay of N can be accounted for by multiplying the above decay width Eq. (<ref>) by the decay (nonsurvival) probability P_N of N within the detectorP_N = 1 - exp[ - L/τ_N γ_N β_N] = 1 - exp[ - L Γ_N/γ_N β_N]where L is the maximum possible flight length of N within the detector, β_N is the velocity of N in the lab frame, τ_N = 1/Γ_N is the lifetime of N in its rest frame, and γ_N =(1 - β_N^2)^-1/2 is the Lorentz time dilation factor <cit.>.For Belle-II, the B meson pairs will be produced in SuperKEKB in central collisions of e^-(p_1) and e^+(p_2), which will produce a moving Υ(4S), the latter decaying into a B meson pair (either B^+ B^- or B^0 B̅^0). In the lab frame, the e^± have the momentap_j = ( E_j,0,0,(-1)^j+1 E_j )(j=1,2),with the values E_1=7.007 GeV and E_2 = 3.993 GeV. This then produces the invariant mass (p_1+p_2)^2 = M^2_Υ(4 S), where M_Υ(4 S)=10.579 GeV <cit.>. The kinetic energy of the produced Υ(4 S) is K_Υ=E_1 + E_2 -M_Υ(4 S) = 0.421 GeV, which is semirelativistic, leading to the Lorentz factor in the lab frame γ_Υ=(E_1+E_2)/M_Υ(4 S) = 1.0398⇒ β_Υ=(1 - 1/γ_Υ^2)^1/2 = 0.274.When Υ(4 S) produces B meson pair, the kinetic energy of the produced B mesons is about 0.010 GeV in the Υ(4 S)-rest frame, which is negligible. Therefore, we consider the velocity of the produced B mesons in the lab frame to be the same as the velocity ofΥ(4 S)β_B = β_Υ = 0.274, γ_B = γ_Υ = 1.0398,(p_B)_ lab=M_B β_B γ_B = 1.504 GeV.In the decays B → D^(*)ℓ_1 N, we will denote the rest frame of the off-shell W^* (i.e., of ℓ_1 N pair) as Σ; the B-rest frame as Σ'; the laboratory frame as Σ”. With these notations, the effective (true) branching ratio is calculatedBr_ eff(B → D^(*)ℓ_1 N → D^(*)ℓ_1 ℓ_2 X) =∫ d q^2 ∫ d Ω_q̂'∫ d Ω_p̂_1d Γ(B → D^(*)ℓ_1 N)/ d q^2 d Ω_q̂'d Ω_p̂_1Γ(N →ℓ_2 X) /Γ_N Γ_B ×{ 1 - exp[- L Γ_N/√((E”_N(q^2;q̂',p̂_1)/M_N )^2 - 1 )] },where in the denominator inside the exponent we have the Lorentz factor β_N^”γ_N^” = √((E”_N(q^2;q̂',p̂_1)/M_N )^2 - 1 ) ,in the laboratory frame, which is a function of W^* (=ℓ_1 N) momentum q' (in the B-rest frame)[Note that q'^2=q^2 is frame independent.] and of the direction p̂_1 of the momentum p_1 of the produced charged lepton ℓ_1 (in the W^*-rest frame). The expression (<ref>) as an explicit function of q^2, q̂^' and p̂_1 is derived in Appendix <ref>. It depends on the angle θ_q between the direction of β̂_B (in the lab frame Σ”) and q̂' of W^* (in the B-rest frame Σ'), as well as on the spherical angles θ_1 and ϕ_1 of the vector p⃗_1 of ℓ_1 in the W^*-rest (Σ) frame, in a specific 3-dimensional system of coordinates in the frame Σ (cf. Fig. <ref> in Appendix <ref>). On the other hand, the differential decay width d Γ(B → D^(*)ℓ_1 N)/( d q^2 d Ω_q̂' d Ω_p̂_1) depends only on q^2 and θ_1, as shown in subsections <ref>-<ref>. Due to the mentioned dependence in the decay (nonsurvival) factor P_N, integration over these momenta is needed, as indicated in Eq. (<ref>). The differential decay widths d Γ(B → D^(*)ℓ_1 N)/(d q^2 d Ω_q̂' d Ω_p̂_1) are given in subsections <ref>-<ref>. All this implies that the integration Eq. (<ref>) has the following form: ∫_(M_N+M_1)^2^(M_B - M_D^(*))^2 d q^2 2 π∫_-1^+1d (cosθ_q) ∫_-1^+1 d (cosθ_1) ∫_0^2 π d ϕ_1 f(q^2, θ_q, θ_1, ϕ_1) . If no mesons D^(*) are produced in the decays, then the differential decay width is even simpler, as it depends only on the direction p̂'_N of the on-shell N in the B-rest frame, and the expression (<ref>) simplifiesBr_ eff(B →ℓ_1 N →ℓ_1 ℓ_2 X) =∫ d Ω_p̂'_Nd Γ(B →ℓ_1 N)/ d Ω_p̂'_NΓ(N →ℓ_2 X) /Γ_N Γ_B ×{ 1 - exp[- L Γ_N/√((E”_N(p̂'_N)/M_N )^2 - 1 )] }.The differential decay width is d Γ(B →ℓ_1 N)/d Ω_p̂'_N = Γ(B →ℓ_1 N)/(4 π) since B is a pseudoscalar, and the expression of Γ(B →ℓ_1 N) is given in subsection II A. The nonsurvival probability P_N is in the case of Eq. (<ref>) also simpler, because it (and the energy of N in the lab frame, E”_N) depends only on the direction p̂'_N of N in the B-rest frame. The expression E”_N(p̂'_N) is given in Appendix <ref>.On the other hand, in the LHCb experiment, the entire procedure described in this Section, designed for a given momentum (p_B)_ lab≡ p_B^” of B in the laboratory frame [cf. Eq. (<ref>) for Belle-II where p_B=1.504 GeV], has to be repeated for various values of momenta p_B^”. The obtained effective branching ratios then have to be averaged over these momenta p_B^”. We took into account that the lab momentum p_B^” of the produced B mesons in LHCb is distributed over a large interval, cf. the shaded curve in Fig. <ref>(a).[We thank Sheldon L. Stone (LHCb Collaboration) for providing us with the distribution, from Ref. <cit.>, appearing here as Fig. <ref>(a).] [b].49 < g r a p h i c s > [b].49 < g r a p h i c s >(a) (left-hand figure) The lab momentum(p_B^”) distribution of the produced B^0 mesons in LHCb <cit.>. We take the shaded figure as the representative case; (b) (right-hand figure) the distribution of the left-hand shaded curve in ten bins of equal weight (equal number of events). We separated this distribution in ten bins of equal weight (equal number of events), cf. Fig. <ref>(b), and calculated the results of Figs. <ref>(a)-(d) by averaging over these ten bins. For each bin, we took in our evaluations the value of the B meson momentum to be such that, within the bin interval, the number of events to the left and to the right of it [according to the shaded curve of Fig. <ref>(a)] are equal; e.g., in the last bin, 223 GeV < p_N < 403 GeV, the average momentum value taken is p = 273 GeV.§ NUMERICAL RESULTS FOR SENSITIVITY LIMITS ON |U_Μ N|^2AT LHCB UPGRADE AND BELLE-II We assume that in the considered decays, the produced on-shell neutrino N has the available length of L=1m for flight within the detector, at Belle-II and L=2.3m at LHCb upgrade.[This length L is considered here to be independent of the position of the vertex where N is produced and independent of the direction in which the produced N travels. It can be called here the effective detector length for the neutrino N. In the case of LHCb, the length of the Vertex Locator (VELO) is about 1m<cit.>; the effective detector length could be extended beyond that locator, to L=2.3m<cit.>.] We consider that at Belle-II, the total number of 5 × 10^10B-mesons will be produced <cit.>, and at LHCb upgrade this number will be about 4.8 × 10^12<cit.>. We assume that there are no background events for the considered lepton number violating (LNV) decays B → D^(*)μ^± N → D^(*)μ^±μ^± X^∓; and B^±→μ^± N →μ^±μ^± X^∓. Here, X^± stands either for π^± (LHCb and Belle-II), or the lepton pair e^±ν_e(Belle-II), and B stands for B^0, B̅^0 or B^±. In these events, we have no QED background because no μ^+ μ^- pairs appear in the final states. The effective branching ratios of the mentioned decay modes depend crucially on the heavy-light mixing parameter |U_μ N|^2. The sensitivity limit on |U_μ N|^2 at 95 % confidence limit is obtained for N_ events=3.09<cit.>. Therefore, the sensitivity limits on |U_μ N|^2 are obtained by requiring ⟨ Br_ eff⟩ = 3.09/(4.8 × 10^12) at LHCb upgrade, and⟨ Br_ eff⟩ = 3.09/(5 × 10^10) at Belle-II, where we recall that the projected total number of produced B mesons at LHCb upgrade and at Belle-II is 4.8 × 10^12 and 5 × 10^10, respectively.The values of ⟨ Br_ eff(B → D^⋆μμ X) ⟩ (X=π or e ν_e) are obtained by taking the arithmetic average of the values of Br_ eff for the four LNV decay modes: B^- → D^⋆ 0μ^- μ^- X^+, B̅^0 → D^⋆ +μ^- μ^- X^+ and their charge conjugates. Analogously, ⟨ Br_ eff(B → D μμ X ) ⟩ is the arithmetic average over the four analogous LNV decays as mentioned before, having now D instead of D^⋆. We note that the total decay widths of B^0 and B^± differ somewhat, Γ_B^0/Γ_B^+ =1.078<cit.>, and we took this into account. In our calculations we neglected, however, the small difference between the masses of D^+ and D^0 (about 5 MeV), and between the masses of D^⋆ + and D^⋆ 0 (about 3 MeV); we used m_D ≈ 1.865 GeV and m_D^⋆≈ 2.010 GeV.Further, for the LNV decays of B without D^(*) mesons, B →μμ X, we do not have four, but only two modes, due to the electric charge restriction: B^±→μ^±μ^± X^∓. For such decays, the average ⟨ Br_ eff(B →μμ X) ⟩ is taken only over these two LNV modes. In these latter cases, we have to take into account that the total number of produced charged B mesons is only half of the total number of produced B mesons. Hence,the sensitivity limits on |U_μ N|^2 are obtained in these cases by requiring ⟨ Br_ eff (B →μμ X) ⟩ = 3.09/(2.4 × 10^12) at LHCb upgrade, and⟨ Br_ eff⟩ = 3.09/(2.5 × 10^10) at Belle-II.We note that the charge-conjugated versions of the decays, i.e., the decays of B^0 vs B̅^0, and of B^+ vs B^-, give in general the same results. The only exception are the decays in which D^* vector meson is produced. This is so because of the factor η=± 1 in the expression (<ref>), in one term there proportional to cosθ_1, which changes sign. The effect of this sign change does not entirely cancels out in the integration (<ref>) for the effective branching ratio, because the expression E”_N(q^2;q̂',p̂_1) in the neutrino N decay probability also has dependence on cosθ_1.We assume in our formulas that only the mixings |U_μ N|^2 are nonzero; if other mixings (|U_e N|^2, |U_τ N|^2) are nonzero, the obtained upper bounds for|U_μ N|^2 are in general less restrictive (higher).[ If N̅ (and N) were Dirac, it would produce, e.g., a pair μ^+ μ^- or a pair e^+ e^-, which have a strong QED background, and would thus not be useful. Or it could produce a pair μ^± e^∓; this could give important contribution, but only in the scenario where both U_μ N and U_e N are nonnegligible, i.e., the scenario not considered here.]The results for the decays with π^± in the final state, for LHCb upgrade, are given in Figs. <ref>(a)-(d). In Figs. <ref>(a)-(c), the present direct experimental bounds are included for comparison, along with our results - the obtained prospective sensitivity limits for LHCb upgrade. Fig. <ref>(d) shows the LHCb sensitivity limits for the three considered decays, for mutual comparisons. Further, we note that the decay modes B → (D^(*)) μ^±μ^± e^∓ν_e cannot be detected at LHCb. The results for the considered decays at Belle-II,either with π^± or with e^±ν_e in the final state, are given in Figs. <ref>(a)-(d). In Figs. <ref>(a)-(c), the present experimental bounds are included for comparison. In Fig. <ref>(d), the prospective Belle-II sensitivity limits for all the six considered decays are presented, for mutual comparisons.§ DISCUSSIONS AND CONCLUSIONS From Figures <ref> and <ref>, we can see that the decays where D^* and D are produced give quite strong new sensitivity limits on |U_μ N|^2 in the mass interval 1.75 GeV < M_N < 3 GeV. This is a reflection of the fact that the presence of D^(*) mesons leads to a significantly weaker CKM suppression in the decay rates, because |V_cb|^2 ≈ 10^2 |V_ub|^2. However, when M_N > 3 GeV, such decays are kinematically suppressed, and then only the (CKM-suppressed) decays B →μμ X give useful sensitivity limits, as seen in Figs. <ref>(c), (d) and Figs. <ref>(c), (d). Further, we see in Figs. <ref> that in general the sensitivity limits are more restrictive (lower) when X=e ν than when X=π.Comparing Figs. <ref> with Figs. <ref>, we can see that the decays B → (D^(*)) μ^±μ^±π^∓, which can be measured at both LHCb and Belle-II experiments, give more stringent (lower) sensitivity limits on |U_μ N|^2 at LHCb upgrade experiment. This is so primarily because the expected number of produced B mesons at LHCb upgrade (4.8 × 10^12) is by two orders of magnitude larger than the number at Belle-II (5 × 10^10). Yet another factor contributing to the more stringent bounds is the effective detector length, which is assumed to be larger at LHCb upgrade (L=2.3mvs L=1m at Belle-II). The difference between the two sets of the sensitivity limits is somewhat reduced by the fact that the lab energy of the produced B mesons in LHCb is significantly higher than in Belle-II; as a consequence, the produced on-shell N neutrinos move in the LHCb case faster and are thus less likely to decay within the detector. If, on the other hand, the acceptance factors decrease the effective number N of produced B mesons, or if the effective detector length L turns out to be smaller, the sensitivity limits for |U_μ N|^2 go up, in general as approximately proportional to 1/√(N L) for not very heavy neutrinos (M_N < 3 GeV).This approximate proportionality comes from the following behavior. For the values of |U_μ N|^2 which are of the order of magnitude of the presented upper bounds, we have at M_N ≲ 2.5 GeV small N-decay probabilities, P_N ≪ 1, and therefore our expressions imply in such a case the approximate proportionality Br_ eff∝ |U_μ N|^4 L. However, for M_N ≳ 4.5 GeV we have P_N ≈ 1 and thus the approximate proportionality Br_ eff∝|U_μ N|^2 (and L-independent). We verified these approximate proportionalities also numerically with our expressions. Approximate L-independence of Br_ eff occurs already at M_N ≳ 3 GeV.In Ref. <cit.>, a similar analysis was made for the decay B^+ →μ^+ N →μ^+ μ^- π^- at Belle-II, where the same total number of B meson pairs was assumed as here, 5 × 10^10. They obtained lower, i.e., more restrictive sensitivity limits on |U_μ N|^2 than we do for this decay for Belle-II. The reason for the difference cannot be the fact that they did not take into account the movement of B-mesons in the lab frame (this effect changes the sensitivity limits only weakly). The reason for the difference lies possibly in the evaluated values of the total decay width Γ_N as a function of M_N. We evaluated this decay width according to the formulas and Figures in Appendix <ref>, based on Refs. <cit.>, and we applied those evaluations in Refs. <cit.>. The experimental bounds on |U_μ N|^2 presented in Figs. <ref>(a)-(c)and Figs. <ref>(a)-(c) are from various experiments: DELPHI <cit.>,BEBC <cit.>, NuTeV <cit.>, NA3 <cit.>, CHARM II <cit.>, and Belle <cit.>. On the basis of the obtained results, Figs. <ref> and <ref>, we conclude that the LHCb upgrade and Belle-II experiments have the potential to either find a new heavy Majorana neutrino N, or to improve significantly the sensitivity limits (upper bounds) on the heavy-light mixing parameter |U_μ N|^2, particularly in the mass range 1.75 GeV < M_N < 3 GeV where the LNV decays of B mesons involving D or D^* mesons and an on-shell neutrino N are possible.If N is not Majorana but Dirac particle, then clear sensitivity limits cannot be obtained for |U_μ N|^2, but rather for the product |U_e N U_μ N|; this is a less attractive possibility, principally because the present upper bounds for |U_e N|^2 in the mentioned mass range, coming from the neutrinoless double beta decay experimental data <cit.>, are more restrictive (lower) than those for |U_μ N|^2. § ACKNOWLEDGMENTS The work of C.S.K. was supported by the NRF grant funded by the Korean government of the MEST (No. 2016R1D1A1A02936965). We thank Y.J. Kwon and Sheldon L. Stone for providing us with valuable information on Belle-II and LHCb upgrade experiments, respectively.§ TOTAL DECAY WIDTH OF NEUTRINO N We summarize here the formulas needed for evaluation of the total decay width of a massive sterile neutrino N, Γ(N → all).The formulas for the widths for leptonic decays and semileptonic decay are given in Ref. <cit.> (Appendix C there), for M_N ≲ 1 GeV. For higher masses M_N, the calculation of the semileptonic decay widths cannot be performed in this way because not all the resonances are known. For such higher masses, the decay widths for semileptonic decays were calculated in Refs. <cit.> by the inclusive approach based on duality. In this approach, the various (pseudoscalar and vector) meson channels were calculated by quark-antiquark channels. This was applied for M_N ≥ M_η^'≈ 0.958 GeV. Below we write the expressions given in Ref. <cit.> for the decay width channels. In some of these formulas, twice the decay width appears [2 Γ(N →…)], where the factor two is applied if N is a Majorana neutrino, and factor one if it is Dirac neutrino. This is so because when Majorana neutrino decays to charged particles, the decay in charge conjugate channel is equally possible; this is not possible if N is Dirac particle.The leptonic decays are2 Γ(N →ℓ^- ℓ^'+ν_ℓ^') = |U_ℓ N|^2 G_F^2/96 π^3 M_N^5 I_1(y_ℓ,0, y_ℓ^') (1 - δ_ℓℓ^' ),Γ(N →ν_ℓℓ^'-ℓ^'+) = |U_ℓ N|^2 G_F^2/96 π^3 M_N^5[ (g_L^( lept) g_R^( lept) + δ_ℓℓ^' g_R^( lept)) I_2(0,y_ℓ^',y_ℓ^') + ( (g_L^( lept))^2 + (g_R^( lept))^2 +δ_ℓℓ^' (1 + 2 g_L^( lept)) ) I_1(0,y_ℓ^',y_ℓ^')] ∑_ν_ℓ∑_ν^'Γ(N →ν_ℓν^'ν̅^') =∑_ℓ |U_ℓ N|^2 G_F^2/96 π^3 M_N^5The factor 2is included in Eq. (<ref>) when N is Majorana, because in such a case both decays, N →ℓ^- ℓ^'+ν_ℓ^' and N →ℓ^+ ℓ^'-ν_ℓ^' are contributing (ℓ≠ℓ^').Further, the following semileptonic decays contribute when M_N < M_η^'≈ 0.968 GeV, involving pseudoscalar (P) and vector (V) mesons:2 Γ(N →ℓ^- P^+) = |U_ℓ N|^2 G_F^2/8 π M_N^3 f_P^2 |V_P|^2 F_P(y_ℓ, y_P)Γ(N →ν_ℓ P^0) = |U_ℓ N|^2 G_F^2/64 π M_N^3 f_P^2 (1 - y_P^2)^2 2 Γ(N →ℓ^- V^+) = |U_ℓ N|^2 G_F^2/8 π M_N^3 f_V^2 |V_V|^2 F_V(y_ℓ, y_V)Γ(N →ν_ℓ V^0) = |U_ℓ N|^2 G_F^2/2 π M_N^3 f_V^2 κ_V^2 (1 - y_V^2)^2 (1 + 2 y_V^2) .Again, the factor 2 appears in the charged meson channels if N is Majorana. The factors V_P and V_V appearing in the above expressions stand for the CKM matrix elements of the valence quarks of the mesons. Ths constants f_P and f_V are the corresponding decay constants of these mesons. Their values are given in Table 1 of Ref. <cit.>.The contributing pseudoscalar mesons here are: P^± = π^±, K^±; P^0 = π^0, K^0, K̅^0, η. The contributing vector mesons here are: V^± = ρ^±, K^* ±; V^0= ρ^0, ω, K^*0, K̅^*0. On the other hand, for higher mass M_N ≥ M_η^' (=0.9578 GeV), the quark-hadron duality is used and the sum of the widths of the semileptonic decay modes are represented by the following widths into quark-antiquark decay modes <cit.>:2 Γ(N →ℓ^- U D̅) = |U_ℓ N|^2 G_F^2/32 π^3 M_N^5 |V_UD|^2 I_1(y_ℓ,y_U,y_D)Γ(N →ν_ℓ q q̅) = |U_ℓ N|^2 G_F^2/32 π^3 M_N^5 [ g_L^(q) g_R^(q) I_2(0,y_q,y_q)+( (g_L^(q))^2+(g_R^(q))^2 ) I_1(0,y_q,y_q) ]In all the formulas (<ref>)–(<ref>), the notationsy_Y≡ M_Y/M_N(Y = ℓ, ν_ℓ,P, V, q)are used. We denoted in Eq. (<ref>): U=u,c; D=d,s,b; q=u,d,c,s,b. The used values of the quark masses in our evaluations are: M_u=M_d = 3.5 MeV; M_s=105 MeV; M_c=1.27 GeV; M_b=4.2 GeV.As mentioned earlier, in the evaluation of the total decay width Γ_N, if N is Majorana we add the expressions (<ref>) and (<ref>); if N is Dirac, the expressions should be added, but with the expressions (<ref>) multiplied by 1/2. The same is valid in the case when we sum the expressions (<ref>) and (<ref>).In Eqs. (<ref>) and (<ref>), the following SM neutral current couplings appear:g_L^( lept) =- 1/2 + sin^2 θ_W,g_R^( lept) =sin^2 θ_W g_L^(U)=1/2 - 2/3sin^2 θ_W,g_R^(U) = - 2/3sin^2 θ_W g_L^(D)= - 1/2 + 1/3sin^2 θ_W,g_R^(U) =1/3sin^2 θ_W In Eq. (<ref>), the neutral current couplings κ_V (for the neutral vector mesons) are κ_V =1/3sin^2 θ_W (V=ρ^0, ω)κ_V = - 1/4 +1/3sin^2 θ_W(V=K^*0, K̅^*0) Further, in the above expressions,the following expressions I_1, I_2, F_P and F_V were used:I_1(x,y,z) = 12 ∫_(x+y)^2^(1-z)^2 ds/s (s - x^2 - y^2) (1 + z^2 -s) λ^1/2(s,x^2,y^2) λ^1/2(1,s,z^2) I_2(x,y,z) = 24 y z ∫_(y+z)^2^(1-x)^2 ds/s (1 + x^2 - s) λ^1/2(s,y^2,z^2) λ^1/2(1,s,x^2) F_P(x,y) =λ^1/2(1,x^2,y^2) [(1 + x^2)(1 + x^2-y^2) - 4 x^2 ] F_V(x,y) =λ^1/2(1,x^2,y^2) [(1 - x^2)^2 + (1 + x^2) y^2 - 2 y^4 ] .Here, the λ^1/2 function is given in Eq. (<ref>). All these formulas then give the total decay width Γ(N → all) as a function of M_N. This total decay width can be written in the following form:Γ_N =Γ_N(M_N).The corresponding canonical (i.e., without the heavy-light mixing factors) decay width expression isΓ_N(M_N) ≡G_F^2 M_N^5/96 π^3 .The factorin Eq. (GNwidth) contains the dependence on the heavy-light mixing factors, and it has the form(M_N) ≡ =N_e N|U_e N|^2 +N_μ N|U_μ N|^2 +N_τ N|U_τ N|^2The dimensionless coefficients N_ℓ N(M_N) here are numbers ∼ 1-10 which are functions of the mass M_N, and they are determined by the above formulas given in this Appendix. We present in Figs. <ref> the resulting coefficients N_ℓ N(M_N) as a function of neutrino mass M_N, for the case of Dirac and Majorana neutrino N. The figures are from Ref. <cit.> for Majorana N, and <cit.> for Dirac N.It is interesting to notice a small kink in the curves of Figs. <ref> at M_N=M_η^' (=0.9578 GeV). The kink is there because atM_N ≥ M_η^' the use of quark-hadron duality is made, i.e., we replace the semileptonic decay channel contributions by those of the quark-antiquark channel. As a consequence, we can conclude that the quark-hadron duality works well at M_N ≥ M_η^'. A partial exception isthe case ℓ = τ because τ lepton has a large mass. § LORENTZ FACTORS OF ON-SHELL N IN LABORATORY FRAME In this Appendix we calculate the energy E”_N of the produced heavy neutrino N in the laboratory frame Σ” [the rest frame of Υ(4S)] in the reaction B → D^(*)ℓ_1 N, cf. Sec. <ref>. We recall that our notations are: Σ is the rest frame of the virtual W^* (i.e., of the ℓ_1-N pair); Σ' is the rest frame of the B meson; and Σ” is the laboratory frame.As explained in Sec. <ref>, the velocity of the produced mesons B in the laboratory (Σ”) frame, β⃗_B, is (practically) the same as the velocity of Υ(4S) there, Eqs. (<ref>)-(<ref>). The momentum p_N transforms between the Σ” (lab) frame and the Σ' (B-rest) frame in the following way:E”_N =γ_B ( E'_N + β_B (p⃗'⃗_⃗N⃗·β̂_B) ), (p⃗”⃗_⃗N⃗·β̂_B) =γ_B ( (p⃗'⃗_⃗N⃗·β̂_B) + β_B E'_N ), ( p⃗”⃗_⃗N⃗ )_⊥ = ( p⃗'⃗_⃗N⃗ )_⊥,where in the last line (…)_⊥ denotes the component of the vector perpendicular to β̂_B≡ẑ^”, i.e., perpendicular to the direction of movement of B [↔ of Υ(4S)] in the lab frame Σ”.[Strictly speaking, we should use the notation β⃗”⃗_⃗B⃗ for the velocity of B meson in the lab, but we prefer the simplified notationβ⃗_B for this vector.]The momentum p_N transforms between the B rest frame Σ' and the W^* rest frame Σ in the following way:E'_N =γ_W(q^2) ( E_N(q^2) - β_W(q^2) |p⃗_N(q^2)| cosθ_1 ), (p⃗'⃗_⃗N⃗·q̂') =γ_W(q^2) ( -|p⃗_N(q^2)| cosθ_1 + β_W(q^2) E_N(q^2) ).Here, θ_1 is the angle bewteen q̂' ≡ẑ and p⃗_1 of ℓ_1 in the Σ frame of ℓ_1-N. The corresponding quantities in the Σ frame, as a function of the squared invariant mass of W^*, Q^2, areE_N =1/2 √(q^2) (q^2 + M_N^2 - M_1^2), |p⃗_N| = |p⃗_1|= 1/2√(q^2)λ^1/2( 1, M_1^2/q^2, M_N^2/q^2),the Lorentz factors for the transition between Σ' and Σ are γ_W(q^2) = ( 1 + |q⃗'⃗|^2/q^2)^1/2, β_W(q^2) = ( q^2/|q⃗'⃗|^2 + 1 )^-1/2,where the magnitude |q⃗'⃗| of the 3-momentum of W^* in Σ' (B-rest frame) is|q⃗'⃗| = 1/2 M_B λ^1/2( 1,M_^2/M_B^2, q^2/M_B^2).In order to combine all these relations Eqs. (<ref>)-(<ref>) to obtain E”_N as a function of q^2, q̂' and p̂_1, we must express the B-meson velocity direction β̂_B in a 3-dimensional coordinate system in Σ. We introduce such a system in the following way: ẑ is defined as ẑ=ẑ'̂ = q̂', i.e., the direction of W^* in the B-rest frame (Σ'). Then the vectors q̂' and β̂_B define a plane, the angle between q̂'(=ẑ) and β̂_B is θ_q(0 ≤θ_q ≤π), and the axis x̂ in this plane is such that (β̂_B)_x = sinθ_q (>0). We recall that β̂_B is the direction vector of B in Σ” (lab) frame. The axis ŷ is then obtained in the usual way, ŷ = ẑ×x̂, cf. Fig. <ref>.As a result, we have β̂_B =sinθ_q x̂ + cosθ_q q̂'⇒(p⃗'⃗_⃗N⃗·β̂_B) = ( p⃗'⃗_⃗N⃗·q̂' ) cosθ_q + (p⃗'⃗_⃗N⃗·x̂) sinθ_q .We can now take into account that p⃗'⃗_⃗N⃗·x̂ = p⃗_⃗N⃗·x̂, because these are components perpendicular to the boost direction q̂'=ẑ between Σ' and Σ. Since in Σ we have p⃗_1 = - p⃗_N, we thus have p⃗'⃗_⃗N⃗·x̂ = p⃗_⃗N⃗·x̂ = - p⃗_1·x̂ = - |p⃗_1| sinθ_1 cosϕ_1 = - |p⃗_N| sinθ_1 cosϕ_1,where θ_1 and ϕ_1 are the spherical coordinates of p⃗_1 in Σ (0 ≤θ_1 ≤π; 0 ≤ϕ_1 < 2 π), cf. Fig. <ref>. Substitution of Eq. (<ref>) into Eq. (<ref>), and taking into account the relation (<ref>) then gives(p⃗'⃗_⃗N⃗·β̂_B) = [ γ_W(q^2) ( - |p⃗_N(q^2)| cosθ_1 + β_W(q^2) E_N(q^2) ) cosθ_q - |p⃗_N(q^2)| sinθ_1 cosϕ_1 sinθ_q ].Using this expression, and the expression for E'_N of Eq. (<ref>), in the Lorentz transformation (<ref>), we finally obtain the energy E”_N of the N neutrino in the lab frame in terms of q^2, q̂' (i.e., θ_q) and p̂_1 (i.e., θ_1 and ϕ_1)E”_N(q^2; θ_q; θ_1, ϕ_1)= γ_B {γ_W(q^2) ( E_N(q^2) - β_W(q^2) |p⃗_N(q^2)| cosθ_1 ) + β_B [ γ_W(q^2) ( - |p⃗_N(q^2)| cosθ_1 + β_W(q^2) E_N(q^2) ) cosθ_q - |p⃗_N(q^2)|sinθ_1 cosϕ_1 sinθ_q ] }.Here, the expressions γ_W(q^2) and β_W(q^2) are given in Eq. (<ref>), and the expressions for E_N(q^2), |p⃗_N(q^2)|and |q⃗'⃗| are given in Eqs. 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"authors": [
"Gorazd Cvetic",
"C. S. Kim"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526004108",
"title": "Sensitivity limits on heavy-light mixing $|U_{μN}|^2$ from lepton number violating $B$ meson decays"
} |
by 2theoremTheorem theorembox 500 lemmaLemma lemmabox 500 corollaryCorollary corollarybox 500 proofProof proofbox 500definitionDefinition definitionbox 500 propositionProposition propositionbox 500@nat@width>@nat@width A New Look at Physical Layer Security, Caching, and WirelessEnergy Harvesting for Heterogeneous Ultra-dense Networks Lifeng Wang, Member, IEEE,Kai-Kit Wong, Fellow, IEEE,Shi Jin, Member, IEEE, Gan Zheng, Senior Member, IEEE,and Robert W. Heath, Jr., Fellow, IEEE L. Wang, and K.-K. Wong are with the Department of Electronic and Electrical Engineering, University College London, WC1E 7JE, London, UK (E-mail: {lifeng.wang, kai-kit.wong}@ucl.ac.uk). S. Jin is with National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (Email: [email protected]). G. Zheng is with the Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Leicestershire, LE11 3TU, UK (Email: [email protected]). Robert W. Heath, Jr. is with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Texas, USA (E-mail: [email protected]).December 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Heterogeneous ultra-dense networks enable ultra-high data rates and ultra-low latency through the use ofdense sub-6 GHz and millimeter wave (mmWave) small cells with different antenna configurations.Existing work has widely studied spectral and energy efficiency in such networks and shown that high spectral and energy efficiency can be achieved. This article investigates thebenefits of heterogeneous ultra-dense network architecture from the perspectives of three promising technologies, i.e., physical layer security, caching, and wireless energy harvesting, and provides enthusiastic outlook towards application of these technologies in heterogeneous ultra-dense networks. Based on the rationale of each technology,opportunities and challenges areidentified to advance the research in this emerging network. § INTRODUCTIONFuture wireless networks such as 5G are designed to support increasing mobile traffic while reducing energy consumption. To meet these targets, new radio-access technologies such asmassive multiple-input multiple-output (MIMO) and millimeter wave (mmWave) will be key components of future wireless access <cit.>. Meanwhile, the new use cases such as drone delivery, autonomous driving, and smart homes require ultra-high reliability and ultra-low latency, and such requirementsentail deployment of ultra-dense low-power small cells, since user equipment (UEs) need to be much closer to the network.Moreover,since blockage can severely deteriorate mmWave transmissions, mmWave small cells need to be densely deployed in an attempt to provide seemless coverage <cit.>.Therefore, next-generation wireless networks will accommodate network densification and a variety of radio-access technologies with different antenna configurations in sub-6 GHz and mmWave bands <cit.>. Heterogeneous ultra-dense networkconsisting of dense sub-6 GHz and mmWave small cells isa key driver to fulfil critical requirements in terms of reliability and latency, and support enormous connectivity triggered by massive Internet-of-things (IoT) and cellular vehicle-to-everything (C-V2X),which is also spotlighted by the industry <cit.>. Existing contributions have widely investigated the spectral and energy efficiency of heterogeneous ultra-dense networks <cit.>. This article aims to provide a comprehensive overview of physical layer security, caching, and wireless energy harvesting in such networks, and deliver insightful guidelines for creating new efficient solutions for security, content delivery and energy.Fig. 1 illustrates a heterogeneous ultra-dense network architecture that has the capabilities of these new technologies, where physical layer security can overcome malicious eavesdropping, caching can offload the core network traffic, and wireless energy harvesting can prolong the lifetime of UE's battery. The remainder of this article is organized as follows. Section II identifies the key features of heterogeneous ultra-dense networks. Sections III, IV and IV illustrate physical layer security, caching and wireless energy harvesting in such networks, respectively, and provide new opportunities and challenges. Finally, conclusions are drawn in Section VI. § KEY FEATURES IN HETEROGENEOUS ULTRA-DENSE NETWORKSIn traditional heterogeneous networks (HetNets), the main difference between tiers is the level of transmit power.The configuration of BSs in heterogeneous ultra-dense networks will be much diverse, with different tiers providing different levels of transmit power, antenna array gain, bandwidth and backhaul cost. The key features of future heterogeneous ultra-dense networks can be identified as follows: * Channel Model: Sub-6 GHz and mmWave links will coexist in heterogeneous ultra-dense networks. Compared to the high diffraction and penetration characteristics of sub-6 GHz links, the detrimental effect of blockage on mmWave links is significant, and mmWave links undergo more dramatic swings between line-of-sight (LoS) and non-line-of-sight (NLoS) due to high level of shadowing.The coherence time in mmWave frequencies is about an order of magnitude shorter than that in sub-6 GHz frequenciessince the Doppler shift linearly scales with frequency. Moreover, unlike rich scattering sub-6 GHz scenarios, there only exists limited spatial selectivity in mmWave environment, which makes traditional small-scale fading models invalid.* Frequency Band: In heterogeneous ultra-dense networks, multiple frequency bands will be leveraged, and UEsmayexperience different pathloss conditions,since mmWave channels are more sensitive to blockages.* BS Densification: A very large number of small cells will be deployed in heterogeneous ultra-dense networks, to support huge amount of connectivity. In such networks, communication distances become much shorter than ever before, which reducespathloss and thus UEs' transmit power during information transmission. Meanwhile, there are some unique features in such networks. For example, densifying the sub-6 GHz tier (usually interference-limited) leads to constant positivedownlink signal-to-interference-plus-noise ratio (SINR) [The near-field pathloss exponent is larger than 2.] when all BSs with the same transmit power are active and none of interference mitigation approaches are utilized <cit.>. However, densifyingthe mmWave tier can significantly enhance the SINR, thanks to directional pencil-beams and its sensitivity of blockage <cit.>. * Beamforming/Precoding: Each BS will be equipped with dozens or hundreds of antennas in an attempt to achieve large array gain or multiplexing gain. More importantly, mmWave BSs can pack more antennas than sub-6 GHz BSs because of shorter mmWave wavelengths. Due to hardware constraints, mmWavebeamforming/precoding may be hybrid analog/digitial, in contrast to the digital sub-6 GHz beamforming/precoding designs. Hence the array gains or multiplexing gains achieved by sub-6 GHz and mmWave BSs may be distinct.* Self Backhaul:It is challenging to let each backhaul link be fiber-optic in such networks, and a large number of small cells are anticipated to access to the core networks via wireless backhaul. By leveraging massive MIMO and mmWave,high-speed wireless backhaul links can be realized. In addition, multi-hop wireless self-backhauling is also required to enable flexible extension of the coverage range, which has been established in technical specification of 5G system by 3GPP <cit.>. § PHYSICAL LAYER SECURITY-ENABLED HETEROGENEOUS ULTRA-DENSE NETWORKSTraditional cryptographic techniques may result in high-latency and communication failure in fast changing networks, due to higher-layer key distribution and management. Physical layer security is a low-complexity approach to keep messages confidential in the presence of malicious eavesdroppers <cit.>. The principle of physical layer security is to exploit the randomness of wireless channels to achieve secrecy on physical layer. In heterogeneous ultra-dense networks,wireless channels are more diverse and random than ever before,as indicated in Section II. Therefore, the application of physical layer security in such networks is promising.We will explain how heterogeneous ultra-dense networks can be powerfully combat malicious eavesdropping with the assistance ofBS densificaition and large antenna arrays.BS Densificaition: One of the main benefits in heterogeneous ultra-dense networks is that by deploying a very large number of BSs, significant BS densification gains can be achieved to boost the spectral efficiency <cit.>, due to lower pathloss. In particular, only the active UEs can reap such benefits, and eavesdroppers that intend to overhear certain UEs' transmissions are not benefited but receive more inter-cell interference from interfering BSs. This can be understood by the fact that in the practical cellular networks, UEs are connected to the BSs based on specific user association rule such as maximum receive signal strength (max-RSS) and make handovers between adjacent BSs <cit.>, however, eavesdroppers have no choice but to undergo arbitrarily varying channel conditions when intercepting a specific UE. The densities of BSs and active UEs determine the amount of BS densification gains and inter-cell interference. The density of eavesdroppers determines the level of eavesdropping ability in the networks, i.e., there may exist stronger eavesdropping channels when more eavesdroppers are incertain UEs' proximity. Large Antenna Arrays: Since eavesdroppers cannot obtain any transmit antenna array gains from multi-antenna BSs,BSs equipped with large antenna arrays can provide legitimate UEs with large array gains and thus significantly enhance secrecy performance. The use of large antenna arrays also allows BSs or UEs to further cut transmit power, which in return reduces the received signal power of eavesdroppers. In heterogeneous ultra-dense networks, sub-6 GHz BSs and mmWave BSs may provide different levels of array gains based on different beamforming/precoding designs. Because BS can fit more mmWave antennas than sub-6 GHz antennas for a fixed antenna aperture, the mmWave beams may be much narrower than the sub-6 GHz counterpart, which degrades mmWave eavesdropping channels. Interference: In heterogeneous ultra-dense networks, enormous connectivity results in severe interference. For UEs, the obtained BS densification gains and array gains can well combat the interference. However, eavesdroppers can only alleviate the harm of interference by increasing their receive antenna array gains or colluding with each other. As mentioned before, an interesting feature of heterogeneous ultra-dense networks is that when densifying the sub-6 GHz tier in which all the BSs are assumed to use the same transmit power and near-field pathloss is larger than 2, the SINRs of UEs nearly remain unchanged[In reality, small cells may be more lightly loaded, and densifying the sub-6 GHz tier will improve the SINRs of UEs <cit.>.] but the SINRs of eavesdroppers will significantly decrease due to very strong interference. In this case, the SINRs of eavesdroppers tend to be zero as the density of sub-6 GHz BSs goes to infinity, and the SINRs of UEs can still be improved by obtaining more array gains from their associated BSs, which means that perfect secrecy is very likely to be achieved. Unlike sub-6 GHz tier, densifying the mmWave tier improves the SINRs of UEs,i.e., the inter-cell interference power grows at a lower speed compared to the UEs' signal power, due to the fact that mmWave signals are transmitted via narrow beams and are susceptible to blockage. Densifying the mmWave tier will inflict more interference on eavesdroppers, which degrades the eavesdropping channels. Fig. 2 shows the effect of different densities of nodeson the average transmission rate and average secrecy rate in a heterogeneous ultra-dense network where UEs can be connected to sub-6 GHz BSs or mmWave BSs under max-RSS based user association. It is seen that the average secrecy rate significantly increases with the density of mmWave BSs, and the average secrecy rate converges to the average transmission rate when the mmWave tier becomes ultra-dense, which indicates that little information will be overheard by eavesdroppers. The reason is that mmWave interference power can still be large when the mmWave tier is extremely dense, which deteriorates the eavesdropping channels. Moreover, the average transmission rate and average secrecy rate slowly increase with density of mmWave BSs after a critical point, due to the fact that ultra-dense mmWave tier can still be interference-limited <cit.>. In addition, Fig. 2 shows that when the sub-6 tier is more dense, the average secrecy rate decreases because UEs will be more likely to select sub-6 GHz links with stronger signal strength and smaller bandwidth. The aforementioned benefits have shown that heterogeneous ultra-dense networks with physical layer security can well preserve security and privacy of connectivity services. Moreover, advanced interference mitigation designs in heterogeneous ultra-dense networks may still need to be developed for security enhancement, particularly in sub-6 GHz tier which is usually interference-limited. It should be noted that whether the mmWave tier is noise-limited or interference-limited depends on the specific mmWave carrier frequency, mmWave beam pattern, blockage probability of the setting, and density of mmWave BSs. Thus, interference mitigation in mmWave tier may still be required when it is interference-limited <cit.>. In addition, to create more flexible and scalable physical layer security designs in heterogeneous ultra-dense networks, researchers are encouraged to study new risks and security threats involving new applications and use cases such as critical infrastructure and industry processes.§ CACHE-ENABLED HETEROGENEOUS ULTRA-DENSE NETWORKSVideo-based services and personal data storage applications have been instrumental for massive mobile traffic growth, and efficient content delivery is required infuture wireless networks. Recently, 3GPP has released the technical specification for services requirements in 5G system, in which 5G shall support content caching applications and operators are required to place the content caches close to UEs <cit.>. We will show that the content services will be delivered with very high reliability and very low latency in cache-enabled heterogeneous ultra-dense networks, where a very large number of BSs with caches have the ability of storing contents requested by UEs.Cache-enabled BSs can deliver the cached contents to UEs at a short distance in heterogeneous ultra-dense networks, which significantly reduces the latency. The connectivity in such networks is highly reliable at a high-speed data rate, owing to the BS densification gains,array gains, and large mmWave bandwidths.Therefore, in heterogeneous ultra-dense networks, whether UEs' requested contents are cached by the servingBSs or not dominates the performance of the network. As indicated in Fig. <ref>,the average delay for cached content delivery is lower than self-backhauled content delivery. The average delay for cached content delivery increases with increasing the cache size.In contrast, the average delay for self-backhauled content delivery decreases with increasing the cache size. The reason is that larger cache size results in higher hit probability, and more SBSs can provide cached content delivery, which results in more inter-SBS interference over thefrequency band allocated to the cached content delivery, and less inter-SBS interference over the frequency band allocated to the self-backhauled content delivery. Hit probability is an important metric to characterize the probability that a requested content file is stored by an arbitrary BS. A lower hit probability means that more content services are provided via wired/wireless backhaul. Different content placement strategies result in different hit probabilities, since the content placement mechanismdetermines whether a content should be cached by a BS or not. Content placement mechanism is mainly designed based on content popularity, and caching the most popular contents (MPC) at each BS is an intuitive andlow-complexity approach, which achieves the highest hit probability for fixed cache size.However, MPC caching may not be optimal in dense HetNets, and existing contributions have attempted to propose some optimal content placement solutions <cit.>, to maximize the probability that the requested content files are not only cached by the serving BSs but also successfully delivered to the associated UEs. In light of the aforementioned heterogeneous ultra-dense system features, caching design in such networks may need to take into account thefollowing important aspects: * User Association: In contrast to the conventional user association schemes that mainly aim to improve spectral and energy efficiency, UEs may be given priority to be associated with the nearby BSs that have cached the requested contents, to reduce backhaul cost and latency. The reason is that under the conventional user association schemes such as max-RSS,UEs may be connected to the BSs without caching the requested contents, which will result in more backhaul and inevitably increase the latency and the backhaul cost. Therefore, content-aware user association needs to be developed. In addition, the throughput and energy efficiency differencesbetween sub-6 GHz link and mmWave link will play a key role in determining which type of link to deliver the same contents, since spectral and energy efficiency are two of the key performance indicators in 5G. Fig. 4 gives an example to illustrate the effect of user association on the network throughput under MPC caching. It is observed that different user association schemes achieve different levels of network throughput, and deploying more SBSs can improve the network throughput, due to BS densification gains. Expanding the cache size improves network throughput since the contents requested by UEs are more likely cached at nearby BSs, which means that the core network traffic can be greatly offloaded. The proportionally fair based user association outperformsmax-RSS, which can be explained by the fact that in conventionalmax-RSS based user association scheme, UEs only select the strongest channels and the effect of cache hit is ignored. * Cache Size: BSs in different tiers may have different cache sizes, and such heterogeneity also has a big impact on efficient content delivery. In cache-enabled heterogeneous ultra-dense networks, the appropriate cache size of a tier heavily depends on the density of BSs, latency and backhaul cost, i.e., tiers with low latency and low backhaul cost may only need to fit small caches, since the uncachedcontents can be obtained via wired/wireless backhaul, however, tiers with high latency or high backhaul cost shall have more BSs with large caches. * Dual Connectivity:UEs may have the ability of dual radio capabilities to be connected to multiple BSs in different types of tiers at the same time, e.g., in 5G systems, a UE can operate at sub-6 GHz band and mmWave band simultaneously <cit.>. In such cases, content placement optimization designs may be more complicated, since the associated BSs can cooperatively serve a UE. As shown in Fig. 5, there are two types of cooperative caching, i.e., Type I: the associated BSs transmit the same content file to a UE, to achieve transmit diversity; and Type II: each associated BS caches partition of a content file and UE receives all the partitions from its associated BSs, to achieve content diversity. It should be noted that for Type II cooperative caching, the partition size of a content may be different between the associated sub-6 GHz BS and mmWave BS, since mmWave BS with gigahertz bandwidth supports much higher data rate.* Integrated Access and Backhaul:Content placement has a significant effect on the backhaul load, due to the fact that uncached contents have to be obtained via backhaul. The self-backhaul feature of heterogeneous ultra-dense networks creates a need for integrating access and backhaul links to improve the efficiency of content delivery. Therefore, load balancing problem in cache-enabled heterogeneous ultra-dense networks with integrated access and backhaul links needs to be addressed. Since the access and backhaul links may be operated on the same or different frequencies, spectrum allocation solutions for integrated access and backhaul links are also required to improve spectral efficiency in content delivery.Cache-aware resource allocation optimization methods canbe developed to enhance the efficiency of content delivery. However, it is challenging to provide the holistic design in cache-enabled heterogeneous ultra-dense networks with very large numbers of BSs in practice, because it is hard to obtain all the BSs' channel station information (CSI). The application of mean-field theory could be a tractable approach, which allows BSs to make their own decision based on local CSI while abstracting other BSs' strategies using a mean-field <cit.>. To date, there are few research results available yet for presenting resource allocation designs in cache-enabled heterogeneous ultra-dense networks, and researchers are encouraged to pay attention to this area. Security and privacy are of great importance in cache-enabled heterogeneous ultra-dense networks, since the content delivery needs to be protected from being intercepted or attacked by illegitimate UEs. Meanwhile, the growing demand for content services such as video puts pressure on network management when implementing traditional content encryption schemes, and the multi-hop characteristic of backhaul links in heterogeneous ultra-dense networks gives rise to more complexity in higher-layer key distribution and management. More importantly, the public cares deeply about their privacy such as locations and browsing history when requesting content services. Hence new security designs need to be developed for facilitating network management under various security concerns. As suggested in Section III, physical layer security may be a cost-effective security solution for dealing with security issues in cache-enabled heterogeneous ultra-dense networks, particularly securing access and backhaul links. Currently, the research on securing cache-enabled heterogeneous ultra-dense networks is in its infancy, and more research efforts need to be made. § WIRELESS ENERGY HARVESTING-ENABLED HETEROGENEOUS ULTRA-DENSE NETWORKSWireless energy harvesting is a much more controllable approach to prolong the lifetime of UEs or devices in IoT, compared to the traditional renewable energy harvesting that heavily depends on the conditions of the environments. The rationale behind it is that radio-frequency (RF) energy can be harvested via microwave radiationin sub-6 GHz or mmWave frequency. RF-to-DC conversionand pathloss are two key factors that influence the efficiency of wireless energy harvesting. Sophisticated rectifier circuit hardware needs to be designed to achieve high RF-to-DC conversion efficiency such that more RF energy can be harvestedduring RF-to-DC conversion. Pathlossimposes a limit on the received signal power, and thus the distance between UEs and BSs (i.e., RF energy sources) cannot be too long since the amount of received RF energy is required to be large enough to activate the harvesting circuit. We believe that heterogeneous ultra-dense networks provide a wealth of opportunities for wireless energy harvesting based on the following advantages: * Large Antenna Arrays: The use of large antenna array forms very sharp signal beams, to achieve significant array gains and redeem pathloss. The obtained array gains from sub-6 GHz BSs and mmWave BSs may be different, and mmWave BSs may provide very large array gains given the fact that very shorter mmWave wavelengths enable BSs to fit a very large number of mmWave antennas.* Network Densification: Compared to the today's networks, UEs will experience less pathloss in heterogeneous ultra-dense networks, which is of great importance for wireless energy harvesting. Moreover, interference power from enormous access and self-backhaul links could also be harvested by UEs. * Dual Connectivity:Since a UE can be associated with multiple BSs (multiple RF energy sources) at the same time, it can be cooperatively powered by its serving BSs, to obtain more array gains and BS densification gains. Therefore, dual connectivity can be an appealing approach to quickly recharge high-power UEs via BS cooperation. It should be noted that dual connectivity for wireless power transfer will inevitably result in more energy loss since different serving BSs undergo different pathloss conditions and far-away serving BSs have higher pathloss. Therefore, power cost issue needs to be addressed in realistic networks.There are two types of RF energy sources in wireless energy harvesting-enabled heterogeneous ultra-dense networks, namely the associated BSs (as dedicated RF energy sources) that provide the directed power transfer, and the remaining BSs that act as ambient RF energy sources. Since the associated BSsdeliver much higher amount of RF energy to the UEs via directed beams than other BSs <cit.>, high-power UEs have to be powered by their associated BSs. Therefore, to maximize the directed transferred energy, high-power UEs are associated with BSs under max-RSS based user association. Considering the fact that enormous connectivity results in high interference power in heterogeneous ultra-dense networks,a lower-power UEor device in IoT can harvest such energy and thus it is unnecessary to be associated with a specific BS. In addition, a high-power UE with dual-mode may ask for directed power transfer from a sub-6 GHz BS or mmWave BS. Fig. <ref> shows the effect of BS density on power transfer user association. It is observed that UEs are more likely to select mmWave links to deliver RF energy when mmWave tier is ultra-dense, and densifying the sub-6 GHz tier means that the probability for UEs being powered by sub-6 GHz BSs increases. In addition, adding moreantennas can improve sub-6 GHz power transfer association probability because of more array gains.In wireless powered heterogeneous ultra-dense networks, UEs first harvest the RF energy from their associated BSs, and then utilize the harvested energy to transmit messages.In such networks, user association may need to be designed by considering the below aspects: * Coupled: In this case,UEs are associated with the same BSs in both downlink and uplink phases, and it could be downlink-based or uplink-based. The downlink-based user association can maximize the harvested energy, and the uplink-based user association can strengthen the received information signal power for energy saving. Therefore, it is important to justify the coupled association policy based on downlink or uplink.* Decoupled: In this case, UEs can be associated with different BSs in downlink and uplink phases. Although such user association mechanism can enable UEs to select the best links for energy harvesting and information transmissions, channel reciprocity in massive MIMO or mmWave links will be lost. Therefore, link budget issues need to be carefully addressed.In addition, the ratio of BS density to active UE density is one of the key system design parameters, which determines the amount of obtained BS densification gains in order to alleviate the harm of uplink interference, since dense active UEs can still result in high uplink interference in wireless powered dense networks. Moreover, to avoid using complicated and time-consuming interference mitigation schemes, one appealing alternative could be that UEsharvest RF energy from sub-6 GHz BSs or mmWave BSs, but only choose mmWave links to deliver information messages, considering the fact that the uplink of mmWave tier tends to be noise-limited when active UEs are not extremely dense <cit.>.Downlink simultaneous wireless information and power transfer (SWIPT) also needs to be investigated in heterogeneous ultra-dense networks. There are two commonly-considered SWIPT protocols, namely power splitting and time switching <cit.>, and it would be interesting to study which SWIPT protocol is more suitable in such networks. Moreover, rectifier circuit at different frequencies may achieve different levels ofRF-to-DC conversion efficiency, and it is essential to select appropriate rectifier circuit in multi-band heterogeneous ultra-dense networks. § CONCLUSIONSHeterogeneous ultra-dense networks fuse various technologies including ultra-dense small cells, massive MIMO and mmWave. The unique features of such network architecture provide physical layer security, caching, and wireless energy harvesting with new opportunities.We have illustrated the benefits of using physical layer security, caching, and wireless energy harvesting in heterogeneous ultra-dense networks, and identified technical challenges, respectively. Since security, content services, and energy are of paramount importance (e.g., 5G service requirements <cit.>), the new solutions introduced by this article can help engineersform the basis of efficient future networks. IEEEtran10url@samestyle F_Boccardi_5G F. Boccardi, R. W. Heath Jr., A. 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Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013. Lifeng Wang [M] is the postdoctoral research fellow in the Department of Electronic and Electrical Engineering, University College London (UCL). He received the Ph.D. degree in Electronic Engineering at Queen Mary University of Londonin2015. His research interests include massive MIMO, millimeter wave, dense HetNets, edge caching,physical-layer security, and wireless energy harvesting. He received the Exemplary Reviewer Certificate and the Exemplary Editor Certificate of the IEEE Communications Letters in 2013 and 2016, respectively.Kai-Kit Wong [F] is Professor of Wireless Communications at the Department of Electronic and Electrical Engineering, University College London, United Kingdom. He received the BEng, the MPhil, and the PhD degrees, all in Electrical and Electronic Engineering, from the Hong Kong University of Science and Technology, Hong Kong, in 1996, 1998, and 2001, respectively. He is Fellow of IEEE and IET. He is Senior Editor of IEEE Communications Letters and also IEEE Wireless Communications Letters.Shi Jin [SM] is Professor at the faculty of the National Mobile Communications Research Laboratory, Southeast University. His research interests include 5G and beyond, random matrix theory, and information theory. He serves as an Associate Editor for the IEEE Transactions on Wireless Communications. He has been awarded the 2011 IEEE Communications Society Stephen O. Rice Prize Paper Award in the field of communication theory, and the 2016 GLOBECOM Best Paper Award. Gan Zheng [SM] is a Senior Lecturer in Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, UK. He received the BEng and MEng from Tianjin University, China, and the PhD degree from The University of Hong Kong in 2008. His research interests include edge caching, full-duplex radio, wireless power transfer, and physical-layer security. He received the 2013 IEEE Signal Processing Letters Best Paper Award, and the 2015 GLOBECOM Best Paper Award.Robert W. Heath Jr. [F] is a Cullen Trust for Higher Education Endowed Professor in the Department of ECE at The University of Texas at Austin and Director of UT SAVES. He has received several awards including the 2017 Marconi Prize Paper Award and the 2017 EURASIP Technical Achievement Award. He is a licensed Amateur Radio Operator, a registered Professional Engineer in Texas, and a Private Pilot. | http://arxiv.org/abs/1705.09647v3 | {
"authors": [
"Lifeng Wang",
"Kai-Kit Wong",
"Shi Jin",
"Gan Zheng",
"Robert W. Heath Jr"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170526164113",
"title": "A New Look at Physical Layer Security, Caching, and Wireless Energy Harvesting for Heterogeneous Ultra-dense Networks"
} |
=-0.6in =-0.80in =-0.3in =0.00in=220mm =165mm =0.1in ReIm ł=12pt plain =16pt Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4× 4 Lax pair on a finite intervalZhenya Yan[ Email address: [email protected]]Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China =13pt=15ptIn this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii systemwith a 4× 4 Lax pair on the finite interval x∈ [0, L] by extending the Fokas unified transform approach.The solution of this system can be expressed in terms of the solution of a 4× 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k),S(k),S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations. Keywords: Integrable spin-1 Gross-Pitaevskii system; Initial-boundary value problem; Riemann-Hilbert problem; Global relation;Finite interval; GLM representation; Maps between Dirichlet and Neumann problemsMathematics Subject Classification numbers: 37K15; 35Q15; 34A55=15pt § INTRODUCTIONAt the end of the 1960s, the well-known inverse scattering transform (IST) <cit.> (also called nonlinear Fourier transform) was first presented to solve theinitial value problem for the KdV equation starting from its two linear eigenvalue equations (also called the Lax pair <cit.>). After that, the IST was used to solve many integrablenonlinear evolution equations (NLEEs) with Lax pairs (see, e.g., Ref. <cit.> and references therein), in which the initial conditions were usually chosen as the constants or plane waves to generate bright or dark solitons and breathers. In the early 1990s, Deift and Zhou <cit.> developed the IST method to present the powerful nonlinear steepest descent method to analytically study the long-time asymptotics of the Cauchy problems of the (1+1)-dimensional integrable NLEEs such as the mKdV equation and nonlinear Schrödinger equation. At the end of the 1990s, Fokas <cit.> further extended the idea of the IST to present a unified method studyinginitial-boundary value problems for both linear and nonlinear integrable equations with Lax pairs (see Refs. <cit.> for the details). Particularly, the Fokas method is used to study integrable nonlinear PDEs by means of the simultaneous spectral analysis of both parts of the Lax pairs and the global relation among the spectral functions. This differs from the classical IST, in which the spectral analysis of only one part of the Lax pairs was considered. The Fokas method well unites the key ideas of the IST with the Riemann-Hilbert problems <cit.>.The Fokas method, as a novel and effective method, has been used to study the initial-boundary value (IBV) problems for the linear PDEs and some integrable NLEEs with 2× 2 Lax pairs on the half-line and the finite interval such as, the nonlinear Schrödinger (NLS) equation <cit.>, the sine-Gordon equation <cit.>, the KdV equation <cit.>, the modified KdV equation <cit.>, the derivative NLS equation <cit.>, and etc. (see, e.g., Refs. <cit.> and references therein). Recently, Lenells further developed the Fokas method to analyze the IBV problems for integrable NLEEs with 3× 3 Lax pairs on the half-line <cit.>. After that, the modified approach was applied in the IBV problems of other integrable NLEEs with 3× 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation <cit.>, the Sasa-Satsuma equation <cit.>, the coupled NLS equations <cit.>, and the Ostrovsky-Vakhnenko equation <cit.>.More recently, we <cit.> successfully extended the ideas of the Fokas method <cit.> for the 2× 2 Lax pairs and its extension <cit.> for the 3× 3 Lax pairs to study the IBV problems for the integrable spin-1 GP system <cit.> with a 4× 4 Lax pair on the half-line 0<x<∞{[ ị∂ q_1/∂ t+ ∂^2 q_1/∂ x^2 -2α(|q_1|^2+2|q_0|^2)q_1-2αβ q_0^2q̅_-1=0,; ị∂ q_0/∂ t+ ∂^2 q_0/∂ x^2-2α(|q_1|^2+|q_0|^2+|q_-1|^2)q_0-2αβ q_1q_-1q̅_0=0,; ị∂ q_-1/∂ t+ ∂^2 q_-1/∂ x^2-2α(2|q_0|^2+|q_-1|^2)q_-1-2αβ q_0^2q̅_1=0, ]. α^2=β^2=1, with the IBV conditions {[Initial conditions: q_j(x, t=0)=q_0j(x), j=1,0,-1, 0<x<∞,; Dirichletboundaryconditions: q_j(x=0, t)=u_0j(t), j=1,0,-1, 0<t<∞,; Neumannboundaryconditions:q_jx(x=0, t)=u_1j(t), j=1,0,-1, 0<t<∞, ]. where the overbar stands for the complex conjugate. The spin-1 GP system (<ref>) can describe soliton dynamics of an F=1 spinor Bose-Einstein condensates <cit.>. The four types of parameters: (α, β)={(1,1), (1, -1), (-1, 1), (-1, -1)} in the spin-1 GP system (<ref>)correspond to the four roles of the self-cross-phase modulation (nonlinearity) and spin-exchange modulation,respectively, that is, (attractive, attractive), (attractive, repulsive), (repulsive, attractive), and (repulsive, repulsive). In this paper, we extend the idea in Ref. <cit.> from the half-line to the finite interval 0<x<L<∞. The aim of this paper is to develop a methodology for analyzing the integrable spin-1 GP system (<ref>) withthe following IBV problem {[ Initial data:q_j(x, t=0)=q_0j(x),0<x<L,;Dirichletboundarydata:q_j(x=0, t)=u_0j(t),q_j(x=L, t)=v_0j(t),0<t<T,;Neumannboundarydata: q_jx(x=0, t)=u_1j(t), q_jx(x=L, t)=v_1j(t),0<t<T, ].j=1,0,-1, on the finite interval Ω={(x,t) |x∈ [0, L],t∈ [0, T]} with L>0 and T>0 being the fixed finite length and time, respectively, the initial data q_0j(x), j=1,0,-1and boundary data {u_0j(t), u_1j(t),j=1,0, -1} are sufficiently smooth and compatible at points (x,t)=(0, 0). where the initial data q_0j(x), j=1,0,-1and boundary data {u_sj(t), v_sj(t),s=0,1; j=1,0,-1} are sufficiently smooth and compatible at points (x,t)=(0, 0),(L, 0), respectively.The main steps for analyzing the IBV problem for the integrable spin-1 GP system (<ref>) with Eq. (<ref>) are listed as follows: Step 1. Suppose that a sufficiently smooth solution {q_j(x,t),j=1,0,-1},0<x<L,0<t<T of the integrable spin-1 GP system (<ref>) exists, we implement the direct spectral analysis of the corresponding Lax pair in order to explore these points: *Introduce appropriate solutions of the exact one-form of the modified Lax pair (<ref>), which are bounded and analytic for the isospectral parameter k in domains to form a partition of the Riemann sphere. *Introduce the following matrix-valuedspectral functions (a)s(k) is determined using the initial data q_j(x, 0)=q_0j(x),j=1,0,-1,0<x<L; (b)S(k) is generated using the boundary data at x=0,q_j(x=0, t)=u_0j(t),q_jx(x=0, t)=u_1j(t), j=1,0,-1,0<t<T; (c)S_L(k) is determined using the boundary dataat x=L, q_j(x=L, t)=v_0j(t),q_jx(x=L, t)=v_1j(t), j=1,0,-1,0<t<T; *Show that these above-mentioned spectral functions satisfy a global relation, which implies thatthe initial-boundary value conditions can not be chosen arbitrary. Step 2. Use the spectral functions {s(k),S(k),S_L(k)} to determine a regular Reimann-Hilbert problem, whose solution can generate a solution of the spin-1 GP system (<ref>). Step 3. The Gel'fand-Levitan-Marchenko (GLM) representations can also given for the IBV problem of system (<ref>). The rest of this paper is organized as follows. In Sec. 2, we introduce the 4× 4 Lax pair of Eq. (<ref>) and explore its spectral analysis such as the eigenfunctions, the jump matrices, and the global relation. Sec. 3 exhibits the corresponding 4× 4 matrix RH problem in terms of the jump matrices found in Sec. 2. The global relation is found to generate the map between the Dirichlet and Neumann boundary values in Sec. 4. Particularly, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. In Sec. 5, we give the GLM representations of the eigenfunctions in terms of the global relation. Moreover, we also show that the GLM representations are equivalent toones in Sec. 4 and present the linearizable boundary conditions for the GLM representations. Finally, we give the conclusions and discussions. § A 4 × 4 LAX PAIR AND ITS SPECTRAL ANALYSIS§.§2.1.The closed one-formThe integrable spin-1 GP system(<ref>) can be regarded as a compatibility condition of the following 4 × 4 Lax pair formulation <cit.>{[ψ_x+ikσ_4ψ=U(x,t)ψ,; ψ_t+2ik^2σ_4ψ=V(x,t,k)ψ, ].where ψ≡ψ(x,t,k) is a 4× 1 column vector-valued or 4×4 matrix-valued eigenfunction, k∈ℂ is an iso-spectral parameter, σ_4= diag(1,1,-1,-1), and the 4 × 4 matrix-valued functions U(x,t) and V(x,t,k) are given asU(x,t)=([ 0 0q_1(x,t)q_0(x,t); 0 0β q_0(x,t) q_-1(x,t);αq̅_1(x,t) αβq̅_0(x,t) 0 0;αq̅_0(x,t) αq̅_-1(x,t) 0 0 ]), U̅^T(x,t)=α U(x,t),and [ V(x,t,k)=2kU+iσ_4(U_x-U^2); =([-iα(|q_1|^2+|q_0|^2) -iα(β q_1q̅_0+q_0q̅_-1) iq_1x+2kq_1 iq_0x+2kq_0; -iα(β q_0q̅_1+q_-1q̅_0) -iα(|q_-1|^2+|q_0|^2) β (iq_0x+2kq_0) iq_-1x+2kq_-1; α(-iq̅_1x+2kq̅_1)αβ(-iq̅_0x+2kq̅_0) iα(|q_1|^2+|q_0|^2)iα(β q_-1q̅_0+q_0q̅_1);α (-iq̅_0x+2kq̅_0) α(-iq̅_-1x+2kq̅_-1)iα(β q_0q̅_-1+q_1q̅_0)iα(|q_-1|^2+|q_0|^2) ]). ] where the potential function {q_j(x,t),j=1,0,-1} satisfies the spin-1 GP equations (<ref>). Introduce a new eigenfunction μ(x,t,k) defined byμ(x,t,k)=ψ(x,t,k)e^i(kx+2k^2t)σ_4, such that the Lax pair (<ref>) becomes an equivalent form {[ μ_x+ik[σ_4,μ]= U(x,t)μ,; μ_t+2ik^2[σ_4,μ]=V(x,t,k)μ, ].where [σ_4, μ]≡σ_4μ-μσ_4.Let σ̂_4 denote the commutator with respect to σ_4 and the operator acting on a 4× 4 matrix A by σ̂_4A=[σ_4, A]=σ_4A-Aσ_4 such that e^σ̂_4A=e^σ_4Ae^-σ_4, then the Lax pair (<ref>) can be written as a full derivative formd[e^i(kx+2k^2t)σ̂_4μ(x,t,k)]=W(x,t,k),where W(x,t,k) is the exact one-form defined by W(x,t,k)=e^i(kx+2k^2t)σ̂_4ł[U(x,t)μ(x,t,k)dx+V(x,t,k)μ(x,t,k)dt]̊.Notice that Eq. (<ref>) can be used to obtain an expression for μ(x,t,k) via the fundamental theorem of calculus.§.§2.2. The eigenfunctions {μ_j(x,t,k)}_1^4For any point (x,t)∈Ω={(x,t)| x∈ [0, L],t∈ [0, T]},we assume that {γ_j}_1^4 denote the four contours connecting fours vertexes(x_1, t_1)=(0, T),(x_2, t_2)=(0, 0),(x_3, t_3)=(L, 0),(x_4, t_4)=(L, T), of the rectangle Ω to the point (x,t) (see Fig. <ref>). Thus for any point (ξ, τ) ∈γ_j,j=1,2,3,4, we have the relations on the contours: [γ_1: x-ξ≥ 0, t-τ≤ 0,;γ_2: x-ξ≥ 0, t-τ≥ 0,; γ_3=-γ_1: x-ξ≤ 0, t-τ≥ 0,; γ_4=-γ_2: x-ξ≤ 0, t-τ≤ 0, ] where the negative sign in γ_3=-γ_1 and γ_4=-γ_2 denotes the opposite directions.We assume that system (<ref>) possesses a smooth complex-valued solution {q_j(x,t),j=1,0,-1} in the domain Ω (if T=∞ then we assume that the solution {q_j(x,t),j=1,0,-1} is a sufficient decay as t→∞). It follows from the Lax pair (<ref>) that we can define its four eigenfunctions (sectionally holomorphic functions) {μ_j(x,t,k)}_1^4 on the four contours {γ_j}_1^4[μ_j(x,t,k)=I+∫_γ_je^-i(kx+2k^2t)σ̂_4W_j(ξ,τ,k);= I+∫_(x_j, t_j)^(x,t)e^-i(kx+2k^2t)σ̂_4W_j(ξ,τ,k), j=1,2,3,4. ] in terms of the Volterra integral equations, where 𝕀= diag(1,1,1,1), the integral is over a piecewise smooth curve from (x_j, t_j) to (x,t),W_j(x,t,k)'s are defined by Eq. (<ref>) with μ(x,t,k) replaced by μ_j(x,t,k)'s. Since the one-forms W_j(x,t,k)'s are closed, thus μ_j(x,t,k)'s are independent of the path of integration. The integral Eq. (<ref>) reduces to [μ_j(x,t,k)=I+∫_x_j^x e^-ik(x-ξ)σ̂_4(Uμ_j)(ξ,t,k)dξ;+̣ e^-ik(x-x_j)σ̂_4∫_t_j^te^-2ik^2(t-τ)σ̂_4 (Vμ_j)(x_j,τ,k)dτ,j=1,2,3,4. ] if the paths of integration are chosen to be parallel to the x and t axes.Eq. (<ref>) implies that the four columns of the matrix μ_j(x,t,k) contain, respectively, these exponentials [μ_j(x,t,k)]_s : e^2i[k(x-ξ)+2k^2(t-τ)], e^2i[k(x-ξ)+2k^2(t-τ)],s=1,2; j=1,2,3,4, [μ_j(x,t,k)]_s :e^-2i[k(x-ξ)+2k^2(t-τ)], e^-2i[k(x-ξ)+2k^2(t-τ)],s=3,4; j=1,2,3,4, To analyze the bounded regions of the eigenfunctions {μ_j(x,t,k)}_1^4, we need to use the curve K={k∈ℂ |( f(k))( g(k))=0,f(k)=ik,g(k)=ik^2}, to separate the complex k-plane into four domains (see Fig. <ref>): {[ D_1={k∈ℂ |f(k)<0and g(k)<0},; D_2={k∈ℂ |f(k)<0and g(k)>0},; D_3={k∈ℂ |f(k)>0and g(k)<0},;D_4={k∈ℂ |f(k)>0 and g(k)>0}, ].which imply that D_1 and D_3 (D_2 and D_4) are symmetric about the origin of the complex k-plane. Therefore, it follows from Eqs. (<ref>), (<ref>) and (<ref>) that the regions, where the distinct columns of eigenfunctions {μ_j(x,t,k)}_1^4 are bounded and analytic in the complex k-plane, are given below: {[ μ_1(x,t,k): (f_- ∩ g_+,f_- ∩ g_+,f_+ ∩ g_-,f_+ ∩ g_-)=: (D_2, D_2, D_3, D_3),; μ_2(x,t,k): (f_- ∩ g_-,f_- ∩ g_-,f_+ ∩ g_+,f_+ ∩ g_+)=: (D_1, D_1, D_4, D_4),; μ_3(x,t,k): (f_+ ∩ g_-,f_+ ∩ g_-,f_- ∩ g_+,f_- ∩ g_+)=: (D_3, D_3, D_2, D_2),; μ_4(x,t,k): (f_+ ∩ g_+,f_+ ∩ g_+,f_- ∩ g_-,f_- ∩ g_-)=: (D_4, D_4, D_1, D_1), ].where f_+=:f(k)>0,f_-=: f(k)<0,g_+=:g(k)>0, and g_-=: g(k)<0. §.§2.3. The new matrix-valued functions M_n(x,t,k)'sWe introduce the matrix-valued solutions M_n(x,t,k), n=1,2,3,4 of Eq. (<ref>) in the form(M_n)_lj(x,t,k)=δ_lj+∫_(γ^n)_lj[e^-i(kx+2k^2t)σ̂_4W_n(ξ,τ,k)]_lj, l,j=1,2,3,4,k∈ D_n, via the Volterra integral equations, where δ_ij=1 for l=j and δ_ij=0 for l≠j,W_n(x,t,k) is defined byW_n(x,t,k)=e^i(kx+2k^2t)σ̂_4ł[U(x,t)M_n(x,t,k)dx+V(x,t,k)M_n(x,t,k)dt]̊. and the contours (γ^n)_lj's are defined as(γ^n)_lj={[ γ_1,if f_l(k)<f_j(k)and g_l(k)≥ g_j(k),; γ_2,if f_l(k)<f_j(k)and g_l(k) <g_j(k),; γ_ 3,if f_l(k) ≥ f_j(k)and g_l(k) ≤ g_j(k),;γ_4, if f_l(k)≥ f_j(k) and g_l(k)≥ g_j(k), ].k∈ D_n, l,j=1,2,3,4,where f_1,2(k)=-f_3,4(k)=-ik and g_1,2(k)=-g_3,4(k)=-2ik^2. Remark. To distinguish (γ^n)_lj's to be the contour γ_3 or γ_4 for the special cases, f_l(k)= f_j(k) and g_l(k)= g_j(k), we choose them as γ_3 (or γ_4) in these cases if we can determine that γ_3 (or γ_4) must appear in other positions of the matrices γ^n.The definition (<ref>) of (γ^n)_ljcan generate theexplicit expressions of γ^n(n=1,2,3,4)as [ γ^1=( [ γ_4 γ_4 γ_4 γ_4; γ_4 γ_4 γ_4 γ_4; γ_2 γ_2 γ_4 γ_4; γ_2 γ_2 γ_4 γ_4 ]), γ^2=( [ γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3; γ_1 γ_1 γ_3 γ_3; γ_1 γ_1 γ_3 γ_3 ]),; γ^3=(γ^2)^T=( [ γ_3 γ_3 γ_1 γ_1; γ_3 γ_3 γ_1 γ_1; γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3 ]), γ^4=(γ^1)^T=( [ γ_4 γ_4 γ_2 γ_2; γ_4 γ_4 γ_2 γ_2; γ_4 γ_4 γ_4 γ_4; γ_4 γ_4 γ_4 γ_4 ]), ]Proposition 2.1. For the matrix-valued functions M_n(x,t,k),n=1,2,3,4 defined by Eq. (<ref>) for k∈D̅_n and (x,t)∈Ω and any fixed point (x_0,t_0), M_n(x_0,t_0,k)'s are the bounded and analytic function of k∈ D_n away from a possible discrete set of singularity {k_j} at which the Fredholm determinants vanish. Furthermore, M_n(x,t,k)'s also have the bounded and continuous extensions to D̅_n and M_n(x,t,k)=𝕀+O(1/k),k∈ D_n,n=1,2,3,4,k→∞,Proof. Following the proof for the 3× 3 Lax pair in Ref. <cit.>, we can also show the bounedness and analyticity of the 4× 4 matrix M_n(x,t,k). The substitution of the eigenfunction μ(x,t,k)=M_n(x,t,k)=M_n^(0)(x,t)+∑_j=1^∞M_n^(j)(x,t)/k^j, k →∞,n=1,2,3,4,into the x-part of the Lax pair (<ref>) can obtain Eq. (<ref>). □ Notice that the above-defined functions M_n(x,t,k)'s can be used to formulate a 4× 4 matrix Riemann-Hilbert problem. §.§2.4.The minors of eigenfunctions and Lax pairThe cofactor matrix X^A (or the transpose of the adjugate) of a 4× 4 matrix X is given byadj(X)^T=X^A=([m_11(X) -m_12(X)m_13(X) -m_14(X); -m_21(X)m_22(X) -m_23(X)m_24(X);m_31(X) -m_32(X)m_33(X) -m_34(X); -m_41(X)m_42(X) -m_43(X)m_44(X) ]),where m_ij(X) denotes the (ij)th minor of X,and (X^A)^TX = adj(X) X= X.It follows from the Lax pair (<ref>) that the cofactor matrices {μ_j^A(x,t,k)}_1^4 of the matrices {μ_j(x,t,k)}_1^4 satisfy the modified Lax equation {[ μ_j,x^A(x,t,k)-ikσ̂_4μ_j^A(x,t,k)= -U^T(x,t)μ_j^A(x,t,k),; μ_j,t^A(x,t,k)-2ik^2σ̂_4μ_j^A(x,t,k)=-V^T(x,t,k)μ_j^A(x,t,k), ]. whose solutions can also be expressed as [ μ_j^A(x,t,k)= 𝕀-∫̣_γ_je^i[k(x-ξ)+2k^2(t-τ)]σ̂_4[U^T(ξ, τ)dξ+V^T(ξ, τ, k)dτ]μ_j^A(ξ, τ, k); =I-∫_x_j^x e^ik(x-ξ)σ̂_4(U^Tμ_j^A)(ξ,t,k)dξ; -̣e^ik(x-x_j)σ̂_4∫_t_j^te^2ik^2(t-τ)σ̂_4 (V^Tμ_j^A)(x_j,τ,k)dτ, j=1,2,3,4 ] using the Volterra integral equations, where U^T(x,t,k) and V^T(x,t,k) denote thetransposes of U(x,t,k) and V(x,t,k) given by Eq. (<ref>), respectively.Therefore, the regions of boundedness of μ_j^A(x,t,k),j=1,2,3,4 are given by {[ μ_1^A(x,t,k)isbounded for k∈ (D_3, D_3, D_2, D_2),; μ_2^A(x,t,k)isbounded for k∈ (D_4, D_4, D_1, D_1),; μ_3^A(x,t,k)isbounded for k∈ (D_2, D_2, D_3, D_3),; μ_4^A(x,t,k)isbounded for k∈ (D_1, D_1, D_4, D_4), ].which are symmetric ones of μ_j about the k-axis (cf. Eq. (<ref>)). §.§2.5.Symmetries of eigenfunctionsLet Ǔ(x,t, k)=-ikσ_4+U(x,t,k),V̌(x,t, k)=-2ik^2σ_4+V(x,t,k).SinceP_±Ǔ(x,t, k̅)P_±=-Ǔ(x,t,k)^T,P_±V̌(x,t, k̅)P_±=-V̌(x,t,k)^T,whereP_±=([±α 0 0 0; 0±α 0 0; 0 0 ∓ 1 0; 0 0 0 ∓ 1 ]),P_±^2=𝕀,α^2=1 According to Eq. (<ref>), we have the following proposition:Proposition 2.2. The eigenfunction ψ(x,t,k) of the Lax pair (<ref>) and μ_j(x,t,k) of the Lax pair (<ref>) both possess the same symmetric relation [ψ^-1(x,t,k)=P_±ψ(x,t,k̅)^TP_±,; μ_j^-1(x,t,k)=P_±μ_j(x,t,k̅)^TP_±, j=1,2,3,4, ]Moreover, in the domains where μ_j is bounded, we have μ_j(x,t,k)=𝕀+O(1/k), k→∞,j=1,2,3, 4anddet [μ_j(x,t,k)]=1,j=1,2,3, 4since the traces of the matricesU(x,t) and V(x,t,k) are zero. §.§2.6. The spectral functionsSince μ_j(x,t,k) are linearly dependent, thus we can have six relations {S(k), s(k), 𝕊(k), S_L(k), s_T(k), 𝔖(k)} between any two eigenfunctions μ_j (see Fig. <ref>), however, we know that these six relations are not dependent such that we only introduce three of those, that is, the three 4× 4 matrix-valued functions S(k),s(k) and 𝕊(k) between μ_j(x,t,k), j=1,3,4 and μ_2(x,t,k) as [ μ_1(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S(k),; μ_3(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4s(k),; μ_4(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4𝕊(k), ]Evaluatingsystem (<ref>) at (x,t)=(0,0) and the three equations in system (<ref>) at (x, t)=(0, T), (L, 0), (L, T), respectively, we find [ S(k)=μ_1(0,0,k)=e^2ik^2Tσ̂_4μ_2^-1(0,T,k),;s(k)=μ_3(0,0,k)=e^ikLσ̂_4μ_2^-1(L,0,k),; 𝕊(k)=μ_4(L,0,k)=e^2ik^2Tσ̂_4μ_3^-1(L,T,k), ]It follows from Eqs. (<ref>) and (<ref>) that we can find the relations among {S(k), s(k), 𝕊(k), S_L(k), s_T(k), 𝔖(k)}:(I) the relation between μ_3(x,t,k) and μ_1(x,t,k)[ μ_3(x,t,k)= μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4𝔖(k), ] with 𝔖(k)=S^-1(k)s(k),which generates the relation of the three edges of the triangle consisting of (x_j, t_j),j=1,2,3.(II) the relation between μ_4(x,t,k) and μ_3(x,t,k)[ μ_4(x,t,k)= μ_3(x,t,k)e^-i[k(x-L)+2k^2(t-T)]σ̂_4μ_3^-1(L, T, k); = μ_3(x,t,k)e^-i(kx+2k^2t)σ̂_4[s^-1(k)𝕊(k)]; = μ_3(x,t,k)e^-i[k(x-L)+2k^2t]σ̂_4S_L(k), ] withS_L(k)=μ_4(L, 0,k)=e^2ik^2Tσ̂_4μ_3^-1(L, T, k)=e^-ikLσ̂_4[s^-1(k)𝕊(k)],that is, 𝕊(k)=s(k)e^ikLσ̂_4S_L(k),which generates the relation of the three edges of the triangle consisting of (x_j, t_j),j=2,3,4. (III) the relation between μ_1(x,t,k) and μ_4(x,t,k)[ μ_4(x,t,k)= μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4s_T(k), ] withs_T(k)=S^-1(k)𝕊(k)=𝔖(k)e^ikLσ̂_4S_L(k)=S^-1(k)s(k)e^ikLσ̂_4S_L(k),which generates the relations of the three edges of the triangle consisting of (x_j, t_j),j=1,2,4, of the three edges of the triangle consisting of (x_j, t_j),j=1,3,4, and of the four edges of the rectangle consisting of (x_j, t_j),j=1,2,3,4, respectively.According to the definition (<ref>) of μ_j and Eqs. (<ref>) and (<ref>), we have [S(k)= I-∫_0^T e^2ik^2τσ̂_4(Vμ_1)(0, τ,k)dξ=[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ]^-1,;s(k)=I-∫_0^L e^ikξσ̂_4(Uμ_3)(ξ,0,k)dξ = [𝕀+∫_0^L e^ikξσ̂_4(Uμ_2)(ξ, 0,k)dξ]^-1 ,;S_L(k)= I-∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ =[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_3)(L,τ,k)dτ]^-1,;][ 𝕊(k)=I-∫_0^L e^ikξσ̂_4(Uμ_4)(ξ,0,k)dξ -e^ikLσ̂_4∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ; = [𝕀+e^2ik^2Tσ̂_4∫_0^L e^ikξσ̂_4(Uμ_2)(ξ,T,k)dξ+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ]^-1; =ł̣[𝕀-∫_0^L e^ikξσ̂_4(Uμ_3)(ξ,0,k)dξ]̊ e^ikLσ̂_4ł[𝕀-∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ]̊; = [𝕀+∫_0^L e^ikξσ̂_4(Uμ_2)(ξ, 0,k)dξ]^-1 e^ikLσ̂_4[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_3)(L,τ,k)dτ]^-1,; ][ 𝔖(k)= ł̣[𝕀-∫_0^T e^2ik^2τσ̂_4(Vμ_1)(0, τ,k)dξ]̊^-1ł[ 𝕀-∫_0^L e^ikξσ̂_4(Uμ_3)(ξ,0,k)dξ]̊; =[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ][𝕀+∫_0^L e^ikξσ̂_4(Uμ_2)(ξ, 0,k)dξ]^-1,; ][ s_T(k)= ł̣[𝕀-∫_0^T e^2ik^2τσ̂_4(Vμ_1)(0, τ,k)dξ]̊^-1ł[ 𝕀-∫_0^L e^ikξσ̂_4(Uμ_3)(ξ,0,k)dξ]̊; ×̣e^ikLσ̂_4ł[𝕀-∫_0^T e^2ik^2τσ̂_4(Vμ_4)(L,τ,k)dτ]̊, ] where μ_j_2(0,t,k), j_2=1,2, μ_j_3(L, t, k),j_3=3,4, μ_j_1(x,0,k),j_1=2,3,4,0<x<L,0<t<T satisfy the integral equations [μ_j(0,t,k)=I+∫_T^te^-2ik^2(t-τ)σ̂_4 (Vμ_j)(0,τ,k)dτ, j=1,2,; μ_j(L,t,k)=I+∫_0^te^-2ik^2(t-τ)σ̂_4 (Vμ_j)(L,τ,k)dτ,j=3,4,;μ_j(x,0,k)=I+∫_0^x e^ikξσ̂_4(Uμ_j)(ξ,0,k)dξ, j=2,3,; μ_4(x,0,k)=I+∫_L^x e^ikξσ̂_4(Uμ_4)(ξ,0,k)dξ-e^-ik(x-L)σ̂_4∫_0^Te^2ik^2τσ̂_4 (Vμ_4)(L,τ,k)dτ. ]It follows from Eqs. (<ref>) and (<ref>) that s(k),S(k) and S_L(k) are determined by U(x,0,k), V(0,t,k), andV(L,t,k), that is, s(k) is decided by the initial data q_j(x, t=0), and S(k) and S_L(k) arefound by the Dirichlet-Neumann boundary data q_j(x, t) and q_jx(x, t),j=1,0,-1 at x=0 and x=L, respectively. In fact, μ_3(x,0,k) and {μ_1(0,t,k), μ_4(L,t,k)} satisfy the x-part and t-part of the Lax pair (<ref>) at t=0 and x=0,L, respectively, that is,x- part: {[ μ_x(x,0,k)+ikσ̂_4μ(x,0,k)=U(x, t=0)μ(x,0,k), 0<x<L,;μ̣(L,0,k)=𝕀, ].t- part: {[ μ_t(x_j,t,k)+2ik^2σ̂μ(x_j,t,k)=V(x=0,t,k)μ(x-j,t,k),0<t<T,j=1,4,; μ(x_j,0,k)=𝕀, μ(x_j,T,k)=𝕀, x_1=0,x_4=L, ].It follows from the properties of μ_j and μ_j^A that the functions{S(k),s(k), 𝕊(k),S_L(k), 𝔖(k),s_T(k)} and {S^A(k),s^A(k), 𝕊^A(k),S_L^A(k), 𝔖^A(k),s_T^A(k)} have the following boundedness:{[ S(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_1∪ D_3, D_1∪ D_3),; s(k) isbounded for k∈(D_3∪ D_4, D_3∪ D_4, D_1∪ D_2, D_1∪ D_2),; 𝕊(k) isbounded for k∈(D_4, D_4, D_1, D_1),; S_L(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_1∪ D_3, D_1∪ D_3),; 𝔖(k) isbounded for k∈(D_4, D_4, D_1, D_1),; s_T(k) isbounded for k∈(D_4, D_4, D_1, D_1),;].and {[S^A(k) isbounded for k∈(D_1∪ D_3, D_1∪ D_3, D_2∪ D_4, D_21∪ D_4),; s^A(k) isbounded for k∈(D_1∪ D_2, D_1∪ D_2, D_3∪ D_4, D_3∪ D_4),; 𝕊^A(k) isbounded for k∈(D_2, D_2, D_3, D_3),; S_L^A(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_1∪ D_3, D_1∪ D_3),; 𝔖^A(k) isbounded for k∈(D_2, D_2, D_3, D_3),; s_T^A(k) isbounded for k∈(D_2, D_2, D_3, D_3), ]. Proposition 2.3. The new matrix-valued functions S_n(k)=(S_n^ij(k))_4× 4,n=1,2,3,4 introduced by M_n(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k),k∈ D_n, can be found uing the entries of the matrix-valued spectral functions s(k)=(s_ij(k))_4× 4,S(k)=(S_ij(k))_4× 4, and 𝕊(k)=(𝕊_ij(k))_4× 4=s(k)e^ikLσ̂_4S_L(k)given by Eq. (<ref>) as follows: [ S_1(k)=([ m_22(𝕊(k))n_33,44(𝕊(k)) m_21(𝕊(k))n_33,44(𝕊(k)) 𝕊_13(k) 𝕊_14(k); m_12(𝕊(k))n_33,44(𝕊(k)) m_11(𝕊(k))n_33,44(𝕊(k)) 𝕊_23(k) 𝕊_24(k); 0 0 𝕊_33(k) 𝕊_34(k); 0 0 𝕊_43(k) 𝕊_44(k) ]),; S_2(k)=([ S_2^11(k) S_2^12(k) s_13(k) s_14(k); S_2^21(k) S_2^22(k) s_23(k) s_24(k); S_2^31(k) S_2^32(k) s_33(k) s_34(k); S_2^41(k) S_2^42(k) s_43(k) s_44(k) ]),; S_3(k)=([ s_11(k) s_12(k) S_3^13(k) S_3^14(k); s_21(k) s_22(k) S_3^23(k) S_3^24(k); s_31(k) s_32(k) S_3^33(k) S_3^34(k); s_41(k) s_42(k) S_3^43(k) S_3^44(k) ]),; S_4(k)=([ 𝕊_11(k) 𝕊_12(k) 0 0; 𝕊_21(k) 𝕊_22(k) 0 0; 𝕊_31(k) 𝕊_32(k) m_44(𝕊(k))n_11,22(𝕊(k)) m_43(𝕊(k))n_11,22(𝕊(k)); 𝕊_41(k) 𝕊_42(k) m_34(𝕊(k))n_11,22(𝕊(k)) m_33(𝕊(k))n_11,22(𝕊(k)) ]), ] where n_i_1j_1,i_2j_2(X) denotes the determinant of the sub-matrix generated by choosing the cross elements of i_1,2th rows and j_1,2th columns of a 4× 4 matrix X, that is, n_i_1j_1,i_2j_2(X)=|[ X_i_1j_1 X_i_1j_2; X_i_2j_1 X_i_2j_2 ]|.and {[ S_2^1j(k)=n_1j,2(3-j)(S)m_2(3-j)(s)+n_1j,3(3-j)(S)m_3(3-j)(s)+n_1j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^2j(k)=n_2j,1(3-j)(S)m_1(3-j)(s)+n_2j,3(3-j)(S)m_3(3-j)(s)+n_2j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^3j(k)=n_3j,1(3-j)(S)m_1(3-j)(s)+n_3j,2(3-j)(S)m_2(3-j)(s)+n_3j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^4j(k)=n_4j,1(3-j)(S)m_1(3-j)(s)+n_4j,2(3-j)(S)m_2(3-j)(s)+n_4j,3(3-j)(S)m_3(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4), ]. j=1,2, {[ S_3^1j(k)=n_1j,2(7-j)(S)m_2(7-j)(s)+n_1j,3(7-j)(S)m_3(7-j)(s)+n_1j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^2j(k)=n_2j,1(7-j)(S)m_1(7-j)(s)+n_2j,3(7-j)(S)m_3(7-j)(s)+n_2j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^3j(k)=n_3j,1(7-j)(S)m_1(7-j)(s)+n_3j,2(7-j)(S)m_2(7-j)(s)+n_3j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^4j(k)=n_4j,1(7-j)(S)m_1(7-j)(s)+n_4j,2(7-j)(S)m_2(7-j)(s)+n_4j,3(7-j)(S)m_3(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4), ]. j=3,4,where 𝒩([S]_1[S]_2[s]_3[s]_4)= det([S]_1, [S]_2, [s]_3, [s]_4) denotes the determinant of the matrix generated by choosing the first and second columns of S(k) and the third and fourth columns of s(k), and 𝒩([s]_1[s]_2[S]_3[S]_4)= det([s]_1, [s]_2, [S]_3, [S]_4), that is, 𝒩([S]_1[S]_2[s]_3[s]_4)= det([S]_1, [S]_2, [s]_3, [s]_4)=|[ S_11(k) S_12(k) s_13(k) s_14(k); S_21(k) S_22(k) s_23(k) s_24(k); S_31(k) S_32(k) s_33(k) s_34(k); S_41(k) S_42(k) s_43(k) s_44(k) ]|. 𝒩([s]_1[s]_2[S]_3[S]_4)= det([s]_1, [s]_2, [S]_3, [S]_4)=|[ s_11(k) s_12(k) S_13(k) S_14(k); s_21(k) s_22(k) S_23(k) S_24(k); s_31(k) s_32(k) S_33(k) S_34(k); s_41(k) s_42(k) S_43(k) S_44(k) ]|.Proof.Introduce the 4× 4 matrix-valued functions R_n(k), S_n(k), T_n(k), and P_n(k) by the eigenfunctions M_n(x,t,k) and μ_j(x,t,k), j=1,2,3,4{[ M_n(x,t,k)=μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4R_n(k),; M_n(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k),; M_n(x,t,k)=μ_3(x,t,k)e^-i(kx+2k^2t)σ̂_4T_n(k),; M_n(x,t,k)=μ_4(x,t,k)e^-i(kx+2k^2t)σ̂_4P_n(k), ]. It follows from Eq. (<ref>) that we have the relations {[R_n(k)=e^2ik^2Tσ̂_4M_n(0,T,k),;S_n(k)=M_n(0,0,k),; T_n(k)=e^ikLσ̂_4M_n(L,0,k),; P_n(k)=e^i(kL+2k^2T)σ̂_4M_n(L,T,k), ].and {[ S(k)=μ_1(0,0,k)=S_n(k)R_n^-1(k),; s(k)=μ_3(0,0,k)=S_n(k)T_n^-1(k),; 𝕊(k)=μ_4(0,0,k)=S_n(k)P_n^-1(k), ].which can in general deduce the functions {R_n, S_n, T_n, P_n} for the given functions {s(k), S(k), 𝕊(k)}.Moreover, we can also determine some entries of {R_n, S_n, T_n, P_n} by using Eqs. (<ref>) and (<ref>) {[(R_n(k))_ij=0, if(γ^n)_ij=γ_1,;(S_n(k))_ij=0,if (γ^n)_ij=γ_2,;(T_n(k))_ij=δ_ij, if (γ^n)_ij=γ_3,; (P_n(k))_ij=δ_ij,if (γ^n)_ij=γ_4, ]. Thus it follows from systems (<ref>) and (<ref>) that we can deduce Eq. (<ref>) by the direct calculation. □ §.§2.7.The residue conditionsSince μ_2(x,t,k) is an entire function, it follows from Eq. (<ref>) that M_n(x,t,k)'s only have singularities at the points where the S_n's have singularities. We know from the expressions of S_n given byEq. (<ref>) that the possible singularities of M_n in the complex k-plane are as follows: * [M_1(x,t,k)]_j,j=1,2 could possess poles in D_1 at the zeros of n_33,44(𝕊)(k);* [M_2(x,t,k)]_j, j=1,2 could admit poles in D_2 at the zeros of 𝒩([S]_1[S]_2[s]_3[s]_4)(k);* [M_3(x,t,k)]_j,j=3,4 could have poles in D_3 at the zeros of 𝒩([s]_1s]_2[S]_3[S]_4)(k);* [M_4(x,t,k)]_j,j=3,4 could be of poles in D_4 at the zeros of n_11,22(𝕊)(k). We introduce the above-mentioned possible zeros by {k_j}_1^N and suppose that they satisfy the following assumption.Assumption 2.4. Supposethat* n_33,44(𝕊)(k) admits n_1 possible simple zeros {k_j}_1^n_1 in D_1; * 𝒩([S]_1[S]_2[s]_3[s]_4)(k) has n_2-n_1 possible simple zeros {k_j}_n_1+1^n_2 in D_2; * 𝒩([s]_1[s]_2[S]_3[S]_4)(k) is of n_3-n_2 possible simple zeros {k_j}_n_2+1^n_3 in D_3; * n_11,22(𝕊)(k) has N-n_3 possible simple zeros {k_j}_n_3+1^N in D_4,and that none of these zeros coincide. Moreover, none of these functions are assumed to have zeros on the boundaries on the D_n's (n=1,2,3,4).Lemma 2.5. For a 4× 4 matrix X=(X_ij)_4× 4, e^θσ̂_4X is given bye^θσ̂_4X=e^θσ_4Xe^-θσ_4=([X_11X_12X_13e^2θX_14e^2θ;X_21X_22X_23e^2θX_24e^2θ; X_31e^-2θ X_32e^-2θX_33X_34; X_41e^-2θ X_42e^-2θX_43X_44 ]),We can deduce the residue conditions at these zeros in the following expressions:Proposition 2.6.Let {M_n(x,t,k)}_1^4 be the eigenfunctions given by Eq. (<ref>) and suppose that the set {k_j}_1^N of singularities are as the above-mentioned Assumption 2.4. Then we have the following residue conditions: [ Ṛẹṣ_k=k_j[M_1(x,t,k)]_l= m_2(3-l)(𝕊)(k_j)𝕊_24(k_j)-m_1(3-l)(𝕊)(k_j)𝕊_14(k_j)ṅ_33,44(𝕊)(k_j)n_13,24(𝕊)(k_j)e^2θ(k_j)[M_1(x,t,k_j)]_3;+m_1(3-l)(𝕊)(k_j)𝕊_13(k_j)-m_2(3-l)(𝕊)(k_j)𝕊_23(k_j)ṅ_33,44(𝕊)(k_j)n_13,24(𝕊)(k_j)e^2θ(k_j)[M_1(x,t,k_j)]_4,; for 1≤ j≤ n_1, l=1,2, k∈ D_1, ][ Res_k=k_j[M_2(x,t,k)]_l= S_2^1l(k_j)s_24(k_j)-S_2^2l(k_j)s_14(k_j)𝒩̇([S]_1[S]_2[s]_3[s]_4)(k_j)n_13,24(s)(k_j)e^2θ(k_j)[M_2(x,t,k_j)]_3;+S_2^2l(k_j)s_13(k_j)-S_2^1l(k_j)s_23(k_j)𝒩̇([S]_1[S]_2[s]_3[s]_4)(k_j)n_13,24(s)(k_j)e^2θ(k_j)[M_2(x,t,k_j)]_4,;for n_1+1≤ j≤ n_2, l=1,2,k∈ D_2,;][ Res_k=k_j[M_3(x,t,k)]_l= S_3^1l(k_j)s_22(k_j)-S_3^2l(k_j)s_12(k_j)𝒩̇([s]_1[s]_2[S]_3[S]_4)(k_j)n_11,22(s)(k_j)e^2θ(k_j)[M_3(x,t,k_j)]_1;+S_3^2l(k_j)s_11(k_j)-S_3^1l(k_j)s_21(k_j)𝒩̇([s]_1[s]_2[S]_3[S]_4)(k_j)n_11,22(s)(k_j)e^2θ(k_j)[M_3(x,t,k_j)]_2,; for n_2+1≤ j≤ n_3, l=3,4, k∈ D_3,;][ Res_k=k_j[M_4(x,t,k)]_l= m_4(7-l)(𝕊)(k_j)𝕊_42(k_j)-m_3(7-l)(𝕊)(k_j)𝕊_32(k_j)ṅ_11,22(𝕊)(k_j)n_31,42(𝕊)(k_j)e^2θ(k_j)[M_4(x,t,k_j)]_1;+m_3(7-l)(𝕊)(k_j)𝕊_31(k_j)-m_4(7-l)(𝕊)(k_j)𝕊_41(k_j)ṅ_11,22(𝕊)(k_j)n_31,42(𝕊)(k_j)e^2θ(k_j)[M_4(x,t,k_j)]_2,; for n_3+1≤ j≤ N, l=3,4, k∈ D_4, ] where the overdot denotes the derivative with resect to the parameter k and θ(k)=-i(kx+2k^2t).Proof.According to Lemma 2.5, it follows from Eqs. (<ref>) and (<ref>) that the four columns of M_1(x,t,k) are obtained by the matrices μ_2 and S_1(k) [M_1]_1=[μ_2]_1 m_22(𝕊)n_33,44(𝕊)+[μ_2]_2m_12(𝕊)n_33,44(𝕊),[M_1]_2=[μ_2]_1 m_21(𝕊)n_33,44(𝕊)+[μ_2]_2m_11(𝕊)n_33,44(𝕊),[M_1]_3=[μ_2]_1 𝕊_13e^2θ +[μ_2]_2𝕊_23e^2θ+[μ_2]_3𝕊_33+[μ_2]_4𝕊_43,[M_1]_4=[μ_2]_1 𝕊_14e^2θ +[μ_2]_2𝕊_24e^2θ+[μ_2]_3𝕊_34+[μ_2]_4𝕊_44,the four columns of M_2(x,t,k) are found bythe matrices μ_2 and S_2(k) [M_2]_1=[μ_2]_1 S_2^11 +[μ_2]_2S_2^21+[μ_2]_3e^-2θS_2^31+[μ_2]_4e^-2θS_2^41,[M_2]_2=[μ_2]_1 S_2^12 +[μ_2]_2S_2^22+[μ_2]_3e^-2θS_2^32+[μ_2]_4e^-2θS_2^42,[M_2]_3=[μ_2]_1e^2θ s_13 +[μ_2]_2e^2θs_23+[μ_2]_3s_33+[μ_2]_4s_43,[M_2]_4=[μ_2]_1e^2θ s_14 +[μ_2]_2e^2θs_24+[μ_2]_3s_34+[μ_2]_4s_44,the four columns of M_3(x,t,k) are given bythe matrices μ_2 and S_3(k) [M_3]_1=[μ_2]_1 s_11 +[μ_2]_2s_21+[μ_2]_3e^-2θs_31+[μ_2]_4e^-2θs_41,[M_3]_2=[μ_2]_1 s_12 +[μ_2]_2s_22+[μ_2]_3e^-2θs_32+[μ_2]_4e^-2θs_42, [M_3]_3=[μ_2]_1e^2θ S_3^13 +[μ_2]_2e^2θS_3^23+[μ_2]_3S_3^33+[μ_2]_4S_3^43,[M_3]_4=[μ_2]_1e^2θ S_3^14 +[μ_2]_2e^2θS_3^24+[μ_2]_3S_3^34+[μ_2]_4S_3^44,and the four columns of M_4(x,t,k) are given bythe matrices μ_2 and S_4(k) [M_4]_1=[μ_2]_1 𝕊_11 +[μ_2]_2𝕊_21+[μ_2]_3𝕊_31e^-2θ+[μ_2]_4𝕊_41e^-2θ,[M_4]_2=[μ_2]_1 𝕊_12 +[μ_2]_2𝕊_22+[μ_2]_3𝕊_32e^-2θ+[μ_2]_4𝕊_42e^-2θ,[M_4]_3=[μ_2]_3 m_44(𝕊)n_11,22(𝕊)+[μ_2]_4m_34(𝕊)n_11,22(𝕊),[M_4]_4=[μ_2]_3 m_43(𝕊)n_11,22(𝕊)+[μ_2]_4m_33(𝕊)n_11,22(𝕊), For the case that k_j∈ D_1 is a simple zero of n_33,44(𝕊)(k), it follows from Eqs. (<ref>) and (<ref>) that we have [μ_2]_1 and [μ_2]_2[ [̣μ_2]_1=[M_1]_3𝕊_24-[M_1]_4𝕊_23 +[μ_2]_3(𝕊_23𝕊_34-𝕊_33𝕊_24) +[μ_2]_4(𝕊_23𝕊_44-𝕊_43𝕊_24)/𝕊_13𝕊_24-𝕊_14𝕊_23e^-2θ,; [̣μ_2]_2=[M_1]_3𝕊_14-[M_1]_4𝕊_13 +[μ_2]_3(𝕊_14𝕊_33-𝕊_13𝕊_34) +[μ_2]_4(𝕊_14𝕊_43-𝕊_13𝕊_44)/𝕊_13𝕊_24-𝕊_14𝕊_23e^-2θ, ]and then substitute them into Eqs. (<ref>) and (<ref>) yields [M_1]_1=m_22(𝕊)𝕊_24-m_12(𝕊)𝕊_14n_33,44(𝕊)n_13,24(𝕊)e^-2θ[M_1]_3 +m_12(𝕊)𝕊_13-m_22(𝕊)𝕊_23n_33,44(𝕊)n_13,24(𝕊)e^-2θ[M_1]_4+m_42(𝕊)n_13,24(𝕊)e^-2θ[μ_2]_3 +m_32(𝕊)n_13,24(𝕊)e^-2θ[μ_2]_4,[M_1]_2=m_21(𝕊)𝕊_24-m_11(𝕊)𝕊_14n_33,44(𝕊)n_13,24(𝕊)e^-2θ[M_1]_3 +m_11(𝕊)𝕊_13-m_21(𝕊)𝕊_23n_33,44(𝕊)n_13,24(𝕊)e^-2θ[M_1]_4+m_41(𝕊)n_13,24(𝕊)e^-2θ[μ_2]_3 +m_31(𝕊)n_13,24(𝕊)e^-2θ[μ_2]_4,whose residues at k_j yield Eq. (<ref>) for k_j∈ D_1, respectively.Similarly, we solve Eqs. (<ref>) and (<ref>) for [μ_2]_1 and [μ_2]_2, and then substitute them into Eqs. (<ref>) and (<ref>) to yield [M_2]_1=S_2^11s_24-S_2^21s_14𝒩([S]_1[S]_2[s]_3[s]_4)n_13,24(s)e^-2θ[M_2]_3 +S_2^21s_13-S_2^11s_23𝒩([S]_1[S]_2[s]_3[s]_4)n_13,24(s)e^-2θ[M_2]_4+m_42(s)n_13,24(s)e^-2θ[μ_2]_3 +m_32(s)n_13,24(s)e^-2θ[μ_2]_4, [M_2]_2=S_2^12s_24-S_2^22s_14𝒩([S]_1[S]_2[s]_3[s]_4)n_13,24(s)e^-2θ[M_2]_3 +S_2^22s_13-S_2^12s_23𝒩([S]_1[S]_2[s]_3[s]_4)n_13,24(s)e^-2θ[M_2]_4 +m_41(s)n_13,24(s)e^-2θ[μ_2]_3 +m_31(s)n_13,24(s)e^-2θ[μ_2]_4,whose residues at k_j yield Eq. (<ref>) for k_j∈ D_2, respectively.Similarly, we can verify Eq. (<ref>) for k_j∈ D_3 and Eq. (<ref>) for k_j∈ D_4 by studying Eq. (<ref>)-(<ref>).□ §.§2.8.The global relationThe definitions of the above-mentioned spectral functions S(k), s(k), S_L(k), and 𝕊(k) imply that they are dependent. It follows from Eqs. (<ref>) and (<ref>) that [ μ_4(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4𝕊(k); = μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4[s(k)e^ikLσ̂_4S_L(k)]; = μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4[S^-1(k)s(k)e^ikLσ̂_4S_L(k)], ] which leads to theglobal relationc(T,k)=μ_4(0, T, k)=e^-2ik^2Tσ̂_4[S^-1(k)s(k)e^ikLσ̂_4S_L(k)],by evaluating Eq. (<ref>) at the point (x,t)=(0, T) and using μ_1(0, T, k)=𝕀.§.§2.9.The jump matrices with explicit (x, t)-dependence It follows from Eq. (<ref>) that the spectral functions S_n(k),n=1,2,3, 4 is given by S_n(k)=M_n(x=0,t=0,k),k∈ D_n, n=1,2,3,4.Let M(x,t,k) stand for the sectionally analytic function on the Riemann k-spere, which is equivalent to M_n(x,t,k) for k∈ D_n. Then M(x,t,k) solves the jump equations M_n(x,t,k)=M_m(x,t,k)J_mn(x,t,k), k∈D̅_n∩D̅_m,n,m=1,2,3,4, n≠ m,with the jump matrices J_mn(x,t,k) defined byJ_mn(x,t,k)=e^-i(kx+2k^2t)σ̂_4[(S_m^-1(k)S_n(k)]. § THE 4× 4 MATRIX RIEMANN-HILBERT PROBLEMBy using the district contours γ_j(j=1,2,3,4), the integral solutions of the revised Lax pair (<ref>), and S_n(k),n=1,2,3,4 due to {S(k), s(k), 𝕊(k), S_L(k)}, we have defined the sectionally analytic function M_n(x,t,k)(n=1,2,3,4), which solves a 4× 4 matrix Riemann-Hilbert (RH) problem. This matrix RH problem can be formulated on basis of the initial and boundary data of the functions q_j(x,t), j=1,0,-1. Thus the solution of Eq. (<ref>) for all values of x,t can be refound by solving the RH problem.Theorem 3.1.Suppose that {q_j(x,t), j=1,0,-1} is a solution of system (<ref>) in the interval regionΩ={(x,t)| x∈ [0, L],t∈ [0, T]}. Then it can be refound from the initial data defined by q_j(x, t=0)=q_0j(x), j=1,0,-1,and Dirichlet and Neumann boundary values defined by[Dirichletboundarydata:q_j(x=0, t)=u_0j(t),q_j(x=L, t)=v_0j(t), j=1,0,-1,;Neumannboundarydata: q_jx(x=0, t)=u_1j(t), q_jx(x=L, t)=v_1j(t), j=1,0,-1, ] We use the initial and boundary data to construct the jump matrices J_mn(x, t, k),n, m = 1,..., 4, by Eq. (<ref>) as well as the spectral functions S(k), s(k) and(𝕊_ij(k))_4× 4=s(k)e^ikLσ̂_4S_L(k)by Eq. (<ref>). Assume that the possible zeros {k_j}^N_1 of the functions n_33,44(𝕊)(k), 𝒩(S_1S_2s_3s_4)(k), 𝒩(s_1s_2S_3S_4)(k) and n_11,22(𝕊)(k) are as in Assumption 2.4. Then the solution (q_1(x,t),q_0(x,t), q_-1(x,t)) of system (<ref>) is found using M(x,t,k) in the form {[q_1(x,t)=2̣ilim_k→∞(kM(x,t,k))_13,; q_0(x,t)=2̣ilim_k→∞(kM(x,t,k))_14=2iβlim_k→∞(kM(x,t,k))_23,; q_-1(x,t)=2̣ilim_k→∞(kM(x,t,k))_24, ].where M(x,t,k) satisfies the following 4× 4 matrix Riemann-Hilbert problem: * M(x,t,k) is sectionally meromorphic on the Riemann k-sphere with jumpsacross the contours D̅_n∪D̅_m, (n, m = 1,..., 4) (see Fig. <ref>). *Across the contours D̅_n∪D̅_m(n, m = 1,..., 4), M(x, t, k) satisfies thejump condition (<ref>). *The residue conditions of M(x,t,k) are satisfied in Proposition 2.6. * M(x, t, k) = I+O(k^-1) as k→∞. Proof.System (<ref>) can be deduced from the large k asymptotics of the eigenfunctions. We can follow the similar one in Refs. <cit.>to complete the rest proof of the Theorem. □§ NONLINEARIZABLE BOUNDARY CONDITIONSThe key difficulty of initial-boundary value problems is to find the boundary values for a well-posed problem.All boundary conditions are required for the definition of S(k) and S_L(k), and hence for the formulating theRiemann-Hilbert problem. Our main conclusion exhibits the unknown boundary condition on basis of the prescribed boundary condition and the initial conditionin terms of the solution of a system of nonlinear integral equations.§.§4.1.The revisited global relationBy evaluating Eqs. (<ref>) and (<ref>) at the point (x,t)=(0, t), we have c(t,k)=μ_2(0,t,k)e^-2ik^2tσ̂_4[s(k)e^ikLσ̂_4S_L(k)], which and Eq. (<ref>) lead to [c(t,k)= μ_2(0,t,k)e^-2ik^2tσ̂_4[s(k)e^ikLσ̂_4e^2ik^2tσ̂_4μ_3^-1(L,t,k)],;= μ_2(0,t,k)[e^-2ik^2tσ̂_4s(k)][e^ikLσ̂_4μ_3^-1(L,t,k)], ] Let š(t,k)=(š_ij(t,k))_4×4=e^-2ik^2tσ̂_4s(k) and μ_3(L,t,k)=(ϕ_ij(t,k))_4× 4, then Eq. (<ref>) can be expanded as[[c(t,k)]_l= ϕ̅_l1(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j1(t,k) +ϕ̅_l2(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j2(t,k); -α e^-2ikLϕ̅_l3(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j3(t,k); -α e^-2ikLϕ̅_l4(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j4(t,k), l=1,2,;][ [c(t,k)]_l= ϕ̅_l3(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j3(t,k)+ϕ̅_l4(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j4(t,k); -α e^2ikLϕ̅_l1(t,k̅)∑_j=1^4μ_2(0,t,k)]_jš_j1(t,k); -α e^2ikLϕ̅_l2(t,k̅)∑_j=1^4[μ_2(0,t,k)]_jš_j2(t,k),l=3,4, ] Thus, the column vectors [c(t,k)]_l,l=1,2 are analytic and bounded in D_4 away from the possible zeros of n_11,22(𝕊)(k) and of order O(1+e^-2ikL/k) as k→∞, and the column vectors [c(t,k)]_l,l=3,4 are analytic and bounded in D_1 away from the possible zeros of n_33,44(𝕊)(k) and of order O(1+e^2ikL/k) as k→∞, §.§4.2. The asymptotic behaviors of eigenfunctions and global relationIt follows from the Lax pair (<ref>) that the eigenfunctions {μ_j(x,t,k)}_1^4 possess the following asymptotics as k→∞ (see Appendix) [ μ_j=I+∑_l=1^2 1/k^l([ μ_j,11^(l) μ_j,12^(l) μ_j,13^(l) μ_j,14^(l); μ_j,21^(l) μ_j,22^(l) μ_j,23^(l) μ_j,24^(l); μ_j,31^(l) μ_j,32^(l) μ_j,33^(l) μ_j,34^(l); μ_j,41^(l) μ_j,42^(l) μ_j,43^(l) μ_j,44^(l) ]) +O(1/k^3);= I+1/k([ ∫̣_(x_j, t_j)^(x,t)Δ_11 ∫̣_(x_j, t_j)^(x,t)Δ_12-̣i/2q_1 -i/2q_0; ∫̣_(x_j, t_j)^(x,t)Δ_21 ∫̣_(x_j, t_j)^(x,t)Δ_22 -̣iβ/2q_0 -̣i/2q_-1; %̣ṣ/̣%̣ṣiα2q̅_1%̣ṣ/̣%̣ṣiαβ2q̅_0 ∫̣_(x_j, t_j)^(x,t)Δ_33 ∫̣_(x_j, t_j)^(x,t)Δ_34; %̣ṣ/̣%̣ṣiα2q̅_0%̣ṣ/̣%̣ṣiα2q̅_-1 ∫̣_(x_j, t_j)^(x,t)Δ_43 ∫̣_(x_j, t_j)^(x,t)Δ_44 ]) +1/k^2([ μ_j,11^(2) μ_j,12^(2) μ_j,13^(2) μ_j,14^(2); μ_j,21^(2) μ_j,22^(2) μ_j,23^(2) μ_j,24^(2); μ_j,31^(2) μ_j,32^(2) μ_j,33^(2) μ_j,34^(2); μ_j,41^(2) μ_j,42^(2) μ_j,43^(2) μ_j,44^(2) ]); +̣O(1/k^3), ] where we have introduced the following functions {[Δ_11= -Δ_33 =%̣ṣ/̣%̣ṣiα2(|q_1|^2+|q_0|^2)dx+α/2∑_j=0,1(q_jq̅_jx-q_jxq̅_j)dt,;Δ_22=-Δ_44^(1)=%̣ṣ/̣%̣ṣiα2(|q_-1|^2+|q_0|^2)dx+α/2∑_j=-1,0(q_jq̅_jx-q_jxq̅_j)dt,;Δ_12= -Δ̅_21=-Δ_34=Δ̅_43;= iα2(β q_1q̅_0+q_0q̅_-1)dx+α2(β q_1q̅_0x-β q_1xq̅_0+q_0q̅_-1x-q_0xq̅_-1)dt, ].and {[ μ_j,13^(2)=%̣ṣ/̣%̣ṣ14q_1x+1/2i(q_1μ_j,33^(1)+q_0μ_j,43^(1)); = %̣ṣ/̣%̣ṣ14q_1x+1/2i[q_1∫_(x_j,t_j)^(x,t)Δ_33+q_0∫_(x_j,t_j)^(x,t)Δ_43],; μ_j,14^(2)=%̣ṣ/̣%̣ṣ14q_0x+1/2i(q_1μ_j,34^(1)+q_0μ_j,44^(1)); = %̣ṣ/̣%̣ṣ14q_0x+1/2i[q_1∫_(x_j,t_j)^(x,t)Δ_34+q_0∫_(x_j,t_j)^(x,t)Δ_44],; μ_j,23^(2)= %̣ṣ/̣%̣ṣβ4q_0x+1/2i(β q_0μ_j,33^(1)+q_-1μ_j,43^(1)); =%̣ṣ/̣%̣ṣβ4q_0x+1/2i[β q_0∫_(x_j,t_j)^(x,t)Δ_33+q_-1∫_(x_j,t_j)^(x,t)Δ_43],; μ_j,24^(2)=%̣ṣ/̣%̣ṣ14q_-1x+1/2i(β q_0μ_j,34^(1)+q_-1μ_j,44^(1)); = %̣ṣ/̣%̣ṣ14q_-1x+1/2i[β q_0∫_(x_j,t_j)^(x,t)Δ_34+q_-1∫_(x_j,t_j)^(x,t)Δ_44], ]. {[ μ_j,31^(2)=%̣ṣ/̣%̣ṣα4q̅_1x+iα/2(q̅_1μ_j,11^(1)+βq̅_0μ_j,21^(1)); =%̣ṣ/̣%̣ṣα4q̅_1x+iα/2[q̅_1∫_(x_j,t_j)^(x,t)Δ_11+βq̅_0 ∫_(x_j,t_j)^(x,t)Δ_21],; μ_j,32^(2)= %̣ṣ/̣%̣ṣαβ4q̅_0x+iα/2(q̅_1μ_j,12^(1)+βq̅_0μ_j,22^(1)); = %̣ṣ/̣%̣ṣαβ4q̅_0x+iα/2[q̅_1∫_(x_j,t_j)^(x,t)Δ_12+βq̅_0 ∫_(x_j,t_j)^(x,t)Δ_22],; μ_j,41^(2)=%̣ṣ/̣%̣ṣα4q̅_0x+iα/2(q̅_0μ_j,11^(1)+q̅_-1μ_j,21^(1)); =%̣ṣ/̣%̣ṣα4q̅_0x+iα/2[q̅_0∫_(x_j,t_j)^(x,t)Δ_11+βq̅_-1∫_(x_j,t_j)^(x,t)Δ_21],; μ_j,42^(2)= %̣ṣ/̣%̣ṣα4q̅_-1x+iα/2(q̅_0μ_j,12^(1)+q̅_-1μ_j,22^(1)); = %̣ṣ/̣%̣ṣα4q̅_-1x+iα/2[q̅_0∫_(x_j,t_j)^(x,t)Δ_12+βq̅_-1∫_(x_j,t_j)^(x,t)Δ_22], ].The functions {μ^(i)_jl=μ^(i)_jl(x,t)}_1^4,i=1, 2 are independent of k.We introduce the matrix-valued function Ψ(t,k)=(Ψ_ij(t,k))_4× 4 as [ μ_2(0, t,k)=Ψ(t,k)=([ Ψ_11(t, k) Ψ_12(t, k) Ψ_13(t, k) Ψ_14(t, k); Ψ_21(t, k) Ψ_22(t, k) Ψ_23(t, k) Ψ_24(t, k); Ψ_31(t, k) Ψ_32(t, k) Ψ_33(t, k) Ψ_34(t, k); Ψ_41(t, k) Ψ_42(t, k) Ψ_43(t, k) Ψ_44(t, k) ]); = I+∑_s=1^21/k^s([ Ψ_11^(s)(t) Ψ_12^(s)(t) Ψ_13^(s)(t) Ψ_14^(s)(t); Ψ_21^(s)(t) Ψ_22^(s)(t) Ψ_23^(s)(t) Ψ_24^(s)(t); Ψ_31^(s)(t) Ψ_32^(s)(t) Ψ_33^(s)(t) Ψ_34^(s)(t); Ψ_41^(s)(t) Ψ_42^(s)(t) Ψ_43^(s)(t) Ψ_44^(s)(t) ]) +O(1/k^3), ]Based on the asymptotic of Eq. (<ref>) and the boundary data (<ref>) at x=0, we find {[ Ψ_13^(1)(t)=-%̣ṣ/̣%̣ṣi2u_01(t), Ψ_14^(1)(t)=βΨ_23^(1)(t)=-i/2u_00(t),Ψ_24^(1)(t)=-%̣ṣ/̣%̣ṣi2u_0-1(t),; Ψ_13^(2)(t)=%̣ṣ/̣%̣ṣ14u_11(t)-i/2[u_01(t)Ψ_33^(1)+u_00(t)Ψ_43^(1)],; Ψ_14^(2)(t)=%̣ṣ/̣%̣ṣ14u_10(t)-i/2[u_01(t)Ψ_34^(1)+u_00(t)Ψ_44^(1)],;Ψ_23^(2)(t)=%̣ṣ/̣%̣ṣβ4u_10(t)-i/2[β u_00(t)Ψ_33^(1)+u_0-1(t)Ψ_43^(1)],; Ψ_24^(2)(t)=%̣ṣ/̣%̣ṣ14u_1-1(t)-i/2[β u_00(t)Ψ_34^(1)+u_0-1(t)Ψ_44^(1)],;Ψ_33^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=0,1[u̅_0j(t)u_1j(t)-u_0j(t)u̅_1j(t)]dt,;Ψ_44^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=-1, 0[u̅_0j(t)u_1j(t)-u_0j(t)u̅_1j(t)]dt,;Ψ_34^(1)(t)=α/2∫^t_0[β u_11(t)u̅_00(t)-β u_01(t)u̅_10(t)+u_10(t)u̅_0-1(t)-u_00(t)u̅_1-1(t)]dt,;Ψ_43^(1)(t)=α/2∫^t_0[β u_10(t)u̅_01(t)-β u_00(t)u̅_11(t)+u_1-1(t)u̅_00(t)-u_0-1(t)u̅_10(t)]dt,; ]. Thus we obtain the Dirichlet-Neumann boundary conditions at x=0 by using the spectral function: {[u_01(t)= 2iΨ_13^(1)(t),u_00(t)=2iΨ_14^(1)(t)=2iβΨ_23^(1)(t),u_0-1(t)= 2iΨ_24^(1)(t),;u_11(t)=4Ψ_13^(2)(t)+2ił[u_01(t)Ψ_33^(1)(t)+u_00(t)Ψ_43^(1)(t)]̊;u_10(t)=4Ψ_14^(2)(t)+2ił[u_01(t)Ψ_34^(1)(t)+u_00(t)Ψ_44^(1)(t)]̊; =4βΨ_23^(2)(t)+2iβł[β u_00(t)Ψ_33^(1)(t)+u_0-1(t)Ψ_43^(1)(t)]̊,; u_1-1(t)=4Ψ_24^(2)(t)+2ił[β u_00(t)Ψ_34^(1)(t)+u_0-1(t)Ψ_44^(1)(t)]̊, ]. Similarly, we assume that the asymptotic formula of μ_3(L, t, k)=ϕ(t,k)={ϕ_ij(t,k)}_i,j=1^4 is of the from [ μ_3(L, t,k)=ϕ(t,k)=([ ϕ_11(t, k) ϕ_12(t, k) ϕ_13(t, k) ϕ_14(t, k); ϕ_21(t, k) ϕ_22(t, k) ϕ_23(t, k) ϕ_24(t, k); ϕ_31(t, k) ϕ_32(t, k) ϕ_33(t, k) ϕ_34(t, k); ϕ_41(t, k) ϕ_42(t, k) ϕ_43(t, k) ϕ_44(t, k) ]); = I+∑_s=1^21/k^s([ ϕ_11^(s)(t) ϕ_12^(s)(t) ϕ_13^(s)(t) ϕ_14^(s)(t); ϕ_21^(s)(t) ϕ_22^(s)(t) ϕ_23^(s)(t) ϕ_24^(s)(t); ϕ_31^(s)(t) ϕ_32^(s)(t) ϕ_33^(s)(t) ϕ_34^(s)(t); ϕ_41^(s)(t) ϕ_42^(s)(t) ϕ_43^(s)(t) ϕ_44^(s)(t) ]) +O(1/k^3), ]By using the asymptotic of Eq. (<ref>) and the boundary data (<ref>) at x=L, we find {[ ϕ_13^(1)(t)=-%̣ṣ/̣%̣ṣi2v_01(t), ϕ_14^(1)(t)=βϕ_23^(1)(t)=-i/2v_00(t),ϕ_24^(1)(t)=-%̣ṣ/̣%̣ṣi2v_0-1(t),; ϕ_13^(2)(t)=%̣ṣ/̣%̣ṣ14v_11(t)-i/2[v_01(t)ϕ_33^(1)+v_00(t)ϕ_43^(1)],; ϕ_14^(2)(t)=%̣ṣ/̣%̣ṣ14v_10(t)-i/2[v_01(t)ϕ_34^(1)+v_00(t)ϕ_44^(1)],;ϕ_23^(2)(t)=%̣ṣ/̣%̣ṣβ4v_10(t)-i/2[β v_00(t)ϕ_33^(1)+v_0-1(t)ϕ_43^(1)],; ϕ_24^(2)(t)=%̣ṣ/̣%̣ṣ14v_1-1(t)-i/2[β v_00(t)ϕ_34^(1)+v_0-1(t)ϕ_44^(1)],;ϕ_33^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=0,1[v̅_0j(t)v_1j(t)-v_0j(t)v̅_1j(t)]dt,;ϕ_44^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=-1, 0[v̅_0j(t)v_1j(t)-v_0j(t)v̅_1j(t)]dt,;ϕ_34^(1)(t)=α/2∫^t_0[β v_11(t)v̅_00(t)-β v_01(t)u̅_10(t)+v_10(t)v̅_0-1(t)-v_00(t)v̅_1-1(t)]dt,;ϕ_43^(1)(t)=α/2∫^t_0[β v_10(t)v̅_01(t)-β v_00(t)v̅_11(t)+v_1-1(t)v̅_00(t)-v_0-1(t)v̅_10(t)]dt,; ].which generates the Dirichlet-Neumann boundary data at x=L using the spectral function {[v_01(t)= 2iϕ_13^(1)(t),v_00(t)=2iϕ_14^(1)(t)=2iβϕ_23^(1)(t),v_0-1(t)= 2iϕ_24^(1)(t),;v_11(t)=4ϕ_13^(2)(t)+2ił[v_01(t)ϕ_33^(1)(t)+v_00(t)ϕ_43^(1)(t)]̊;v_10(t)=4ϕ_14^(2)(t)+2ił[v_01(t)ϕ_34^(1)(t)+v_00(t)ϕ_44^(1)(t)]̊; =4βϕ_23^(2)(t)+2iβł[β v_00(t)ϕ_33^(1)(t)+v_0-1(t)ϕ_43^(1)(t)]̊,; v_1-1(t)=4ϕ_24^(2)(t)+2ił[β v_00(t)ϕ_34^(1)(t)+v_0-1(t)ϕ_44^(1)(t)]̊, ]. For the vanishing initial values, it follows from Eqs. (<ref>) and (<ref>) that we have the following asymptotic of c_1j(t,k), c_j1(t,k), j=3,4, c_24(k) and c_42(k).Proposition 4.1. Let the initial and Dirichlet boundary data be compatible at points x=0, L (i.e., q_0j(0)=u_0j(0) at x=0 andq_0j(L)=v_0j(0) at x=L, j=1,0,-1), respectively. Then, the global relation (<ref>) in the vanishing initial data case implies that the large k behaviors ofc_1j(t,k),c_j1(t,k),j=3,4, c_24(t,k), and c_42(t,k) can be given as follows:[ c_13(t,k)=%̣ṣ/̣%̣ṣΨ_13^(1)k+Ψ_13^(2)+Ψ_13^(1)ϕ̅_33^(1) +Ψ_14^(1)ϕ̅_34^(1)/k^2 +O(1/k^3);-α[ϕ̅_31^(1)/k+ϕ̅_31^(2)+Ψ_11^(1)ϕ̅_31^(1) +Ψ_12^(1)ϕ̅_32^(1)/k^2+O(1/k^3)]e^2ikL, as k→∞, ] [c_14(t,k)=Ψ_14^(1)/k+Ψ_14^(2)+Ψ_14^(1)ϕ̅_44^(1) +Ψ_13^(1)ϕ̅_43^(1)/k^2 +O(1/k^3); -α[ϕ̅_41^(1)/k+ϕ̅_41^(2)+Ψ_11^(1)ϕ̅_41^(1) +Ψ_12^(1)ϕ̅_42^(1)/k^2+O(1/k^3)]e^2ikL, k→∞, ] [c_24(t,k)= Ψ_24^(1)/k+Ψ_24^(2)+Ψ_24^(1)ϕ̅_44^(1)+Ψ_23^(1)ϕ̅_43^(1)/k^2 +O(1/k^3); -α[ϕ̅_42^(1)/k+ϕ̅_42^(2) +Ψ_21^(1)ϕ̅_41^(1) +Ψ_22^(1)ϕ̅_42^(1)/k^2+O(1/k^3)]e^2ikL, as k→∞, ] [ c_31(t,k)= Ψ_31^(1)/k+Ψ_31^(2)+Ψ_31^(1)ϕ̅_11^(1) +Ψ_32^(1)ϕ̅_12^(1)/k^2 +O(1/k^3);-α[ϕ̅_13^(1)/k+ϕ̅_13^(2)+Ψ_33^(1)ϕ̅_13^(1) +Ψ_34^(1)ϕ̅_14^(1)/k^2+O(1/k^3)]e^-2ikL, ask→∞, ] [ c_41(t,k)= Ψ_41^(1)/k+Ψ_41^(2)+Ψ_41^(1)ϕ̅_11^(1) +Ψ_42^(1)ϕ̅_12^(1)/k^2 +O(1/k^3);-α[ϕ̅_14^(1)/k+ϕ̅_14^(2)+Ψ_43^(1)ϕ̅_13^(1) +Ψ_44^(1)ϕ̅_14^(1)/k^2+O(1/k^3)]e^-2ikL, ask→∞, ] [ c_42(t,k)= Ψ_42^(1)/k+Ψ_42^(2)+Ψ_41^(1)ϕ̅_21^(1) +Ψ_42^(1)ϕ̅_22^(1)/k^2 +O(1/k^3);-α[ϕ̅_24^(1)/k+ϕ̅_24^(2)+Ψ_43^(1)ϕ̅_23^(1) +Ψ_44^(1)ϕ̅_24^(1)/k^2+O(1/k^3)]e^-2ikL, ask→∞, ]Proof. It follows from the global relation (<ref>) under the vanishing initial data that we get c_13(t,k)=-α e^2ikL[Ψ_11(t,k)ϕ̅_31(t,k̅)+Ψ_12(t,k)ϕ̅_32(t,k̅)] +Ψ_13(t,k)ϕ̅_33(t,k̅)+Ψ_14(t,k)ϕ̅_34(t,k̅), c_14(t,k)=-α e^2ikL[Ψ_11(t,k)ϕ̅_41(t,k̅)+Ψ_12(t,k)ϕ̅_42(t,k̅)] +Ψ_13(t,k)ϕ̅_43(t,k̅)+Ψ_14(t,k)ϕ̅_44(t,k̅),c_24(t,k)=-α e^2ikL[Ψ_21(t,k)ϕ̅_41(t,k̅)+Ψ_22(t,k)ϕ̅_42(t,k̅)] +Ψ_23(t,k)ϕ̅_43(t,k̅)+Ψ_24(t,k)ϕ̅_44(t,k̅), Recalling the t-part of the Lax pair (<ref>) μ_t+2ik^2[σ_4, μ]=V(x,t,k)μ, It follows from Eq. (<ref>) that the first column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) is {[ Ψ_11,t(t,k)= 2k(u_01Ψ_31+u_00Ψ_41)+i(u_11Ψ_31+u_10Ψ_41);-iα[(|u_01|^2+|u_00|^2)Ψ_11+(β u_01u̅_00+u_00u̅_0-1)Ψ_21],; Ψ_21,t(t,k)= 2k(β u_00Ψ_31+u_0-1Ψ_41)+i(β u_10Ψ_31+u_1-1Ψ_41); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_11+(|u_0-1|^2+|u_00|^2)Ψ_21],; Ψ_31,t(t,k)= 4ik^2Ψ_31+2α k(u̅_01Ψ_11+βu̅_00Ψ_21) -iα(u̅_11Ψ_11+βu̅_10Ψ_21); +iα[(|u_01|^2+|u_00|^2)Ψ_31+(β u_0-1u̅_00+u_00u̅_01)Ψ_41]; Ψ_41,t(t,k)=4ik^2Ψ_41+2α k(u̅_00Ψ_11+u̅_0-1Ψ_21)-iα(u̅_10Ψ_11+u̅_1-1Ψ_21); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_31+(|u_0-1|^2+|u_00|^2)Ψ_41], ].the second column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) yields {[ Ψ_12,t(t,k)= 2k(u_01Ψ_32+u_00Ψ_42)+i(u_11Ψ_32+u_10Ψ_42);-iα[(|u_01|^2+|u_00|^2)Ψ_12+(β u_01u̅_00+u_00u̅_0-1)Ψ_22],; Ψ_22,t(t,k)= 2k(β u_00Ψ_32+u_0-1Ψ_42)+i(β u_10Ψ_32+u_1-1Ψ_42); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_12+(|u_0-1|^2+|u_00|^2)Ψ_22],; Ψ_32,t(t,k)= 4ik^2Ψ_32+2α k(u̅_01Ψ_12+βu̅_00Ψ_22) -iα(u̅_11Ψ_12+βu̅_10Ψ_22); +iα[(|u_01|^2+|u_00|^2)Ψ_32+(β u_0-1u̅_00+u_00u̅_01)Ψ_42]; Ψ_42,t(t,k)=4ik^2Ψ_42+2α k(u̅_00Ψ_12+u̅_0-1Ψ_22)-iα(u̅_10Ψ_12+u̅_1-1Ψ_22); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_32+(|u_0-1|^2+|u_00|^2)Ψ_42], ].the third column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) is of {[Ψ_13,t(t,k)= -4ik^2Ψ_13+2k(u_01Ψ_33+u_00Ψ_43)+i(u_11Ψ_33+u_10Ψ_43);-iα[(|u_01|^2+|u_00|^2)Ψ_13+(β u_01u̅_00+u_00u̅_0-1)Ψ_23],;Ψ_23,t(t,k)= -4ik^2Ψ_23+2k(β u_00Ψ_33+u_0-1Ψ_43)+i(β u_10Ψ_33+u_1-1Ψ_43); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_13+(|u_0-1|^2+|u_00|^2)Ψ_23],;Ψ_33,t(t,k)=2α k(u̅_01Ψ_13+βu̅_00Ψ_23) -iα(u̅_11Ψ_13+βu̅_10Ψ_23); +iα[(|u_01|^2+|u_00|^2)Ψ_33+(β u_0-1u̅_00+u_00u̅_01)Ψ_43];Ψ_43,t(t,k)= 2α k(u̅_00Ψ_13+u̅_0-1Ψ_23)-iα(u̅_10Ψ_13+u̅_1-1Ψ_23); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_33+(|u_0-1|^2+|u_00|^2)Ψ_43], ].and the fourth column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) is {[Ψ_14,t(t,k)= -4ik^2Ψ_14+2k(u_01Ψ_34+u_00Ψ_44)+i(u_11Ψ_34+u_10Ψ_44);-iα[(|u_01|^2+|u_00|^2)Ψ_14+(β u_01u̅_00+u_00u̅_0-1)Ψ_24],;Ψ_24,t(t,k)= -4ik^2Ψ_24+2k(β u_00Ψ_34+u_0-1Ψ_44)+i(β u_10Ψ_34+u_1-1Ψ_44); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_14+(|u_0-1|^2+|u_00|^2)Ψ_24],;Ψ_34,t(t,k)=2α k(u̅_01Ψ_14+βu̅_00Ψ_24) -iα(u̅_11Ψ_14+βu̅_10Ψ_24); +iα[(|u_01|^2+|u_00|^2)Ψ_34+(β u_0-1u̅_00+u_00u̅_01)Ψ_44];Ψ_44,t(t,k)= 2α k(u̅_00Ψ_14+u̅_0-1Ψ_24)-iα(u̅_10Ψ_14+u̅_1-1Ψ_24); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_34+(|u_0-1|^2+|u_00|^2)Ψ_44], ]. Suppose that Ψ_j1(t,k)'s, j=1,2,3,4 are of the form ([ Ψ_11(t,k); Ψ_21(t,k); Ψ_31(t,k); Ψ_41(t,k) ]) =(a_10(t)+a_11(t)/k+a_12(t)/k^2+⋯)+(b_10(t)+b_11(t)/k +b_12(t)/k^2+⋯)e^4ik^2t,where the 4× 1 column vector functions a_1j(t),b_1j(t)(j=0,1,...,) are independent of k.By substituting Eq. (<ref>) into Eq.(<ref>) and using the initial conditions a_10(0)+b_10(0)=(1, 0,0,0)^T, a_11(0)+b_11(0)=(0, 0, 0, 0)^T, we have [ ([ Ψ_11; Ψ_21; Ψ_31; Ψ_41 ]) = ([ 1; 0; 0; 0 ]) +1/k([ Ψ_11^(1); Ψ_21^(1); Ψ_31^(1); Ψ_41^(1) ]) +1/k^2([ Ψ_11^(2); Ψ_21^(2); Ψ_31^(2); Ψ_41^(2) ]) +O(1/k^3);+[1/k([ 0; 0; -iα/2u̅_01(0); -iα/2u̅_00(0) ])+O(1/k^2)]e^4ik^2t, ] Similarly, it follows from Eqs. (<ref>)-(<ref>) that we have the asymptotic formulae for Ψ_ij,i=1,2,3,4; j=2,3,4 in the forms [ ([ Ψ_12; Ψ_22; Ψ_32; Ψ_42 ]) = ([ 0; 1; 0; 0 ]) +1/k([ Ψ_12^(1); Ψ_22^(1); Ψ_32^(1); Ψ_42^(1) ]) +1/k^2([ Ψ_12^(2); Ψ_22^(2); Ψ_32^(2); Ψ_42^(2) ]) +O(1/k^3);+[1/k([0;0; -iαβ/2u̅_00(0); -iα/2u̅_0-1(0) ])+O(1/k^2)]e^4ik^2t, ][ ([ Ψ_13; Ψ_23; Ψ_33; Ψ_43 ]) = ([ 0; 0; 1; 0 ]) +1/k([ Ψ_13^(1); Ψ_23^(1); Ψ_33^(1); Ψ_43^(1) ]) +1/k^2([ Ψ_13^(2); Ψ_23^(2); Ψ_33^(2); Ψ_43^(2) ]) +O(1/k^3); +[1/k([i/2u_01(0); iβ/2u_00(0); 0; 0 ])+O(1/k^2)]e^-4ik^2t, ] and [ ([ Ψ_14; Ψ_24; Ψ_34; Ψ_44 ]) = ([ 0; 0; 0; 1 ]) +1/k([ Ψ_14^(1); Ψ_24^(1); Ψ_34^(1); Ψ_44^(1) ]) +1/k^2([ Ψ_14^(2); Ψ_24^(2); Ψ_34^(2); Ψ_44^(2) ]) +O(1/k^3);+[1/k([i/2u_00(0); i/2u_0-1(0); 0; 0 ])+O(1/k^2) ]e^-4ik^2t, ]The substitution of Eqs. (<ref>)-(<ref>) into Eq. (<ref>) yields Eq. (<ref>). Similarly, we can also get Eqs. (<ref>) and (<ref>).Similar to Eqs. (<ref>)-(<ref>) for μ_2(0,t,k), we also know that the function μ(x,t,k)=μ_3(L, t,k) at x=L satisfy the t-part of Lax pair (<ref>) such that we have the first column of Eq. (<ref>) with μ=μ_3(L,t,k)=ϕ(t,k){[ ϕ_11,t(t,k)= 2k(v_01ϕ_31+v_00ϕ_41)+i(v_11ϕ_31+v_10ϕ_41);-iα[(|v_01|^2+|v_00|^2)ϕ_11+(β v_01v̅_00+v_00v̅_0-1)ϕ_21],; ϕ_21,t(t,k)= 2k(β v_00ϕ_31+v_0-1ϕ_41)+i(β v_10ϕ_31+v_1-1ϕ_41); -iα[(β v_00v̅_01+v_0-1u̅_00)ϕ_11+(|v_0-1|^2+|v_00|^2)ϕ_21],; ϕ_31,t(t,k)= 4ik^2ϕ_31+2α k(v̅_01ϕ_11+βv̅_00ϕ_21) -iα(v̅_11ϕ_11+βv̅_10ϕ_21); +iα[(|v_01|^2+|v_00|^2)ϕ_31+(β v_0-1v̅_00+v_00v̅_01)ϕ_41]; ϕ_41,t(t,k)=4ik^2ϕ_41+2α k(v̅_00ϕ_11+v̅_0-1ϕ_21)-iα(v̅_10ϕ_11+v̅_1-1ϕ_21); +iα[(β v_00v̅_0-1+v_01v̅_00)ϕ_31+(|v_0-1|^2+|v_00|^2)ϕ_41], ].the second column of Eq. (<ref>) with μ=μ_3(L,t,k)=ϕ(t,k){[ ϕ_12,t(t,k)= 2k(v_01ϕ_32+v_00ϕ_42)+i(v_11ϕ_32+v_10ϕ_42);-iα[(|v_01|^2+|v_00|^2)ϕ_12+(β v_01v̅_00+v_00v̅_0-1)ϕ_22],; ϕ_22,t(t,k)= 2k(β v_00ϕ_32+v_0-1ϕ_42)+i(β v_10ϕ_32+v_1-1ϕ_42); -iα[(β v_00v̅_01+v_0-1v̅_00)ϕ_12+(|v_0-1|^2+|v_00|^2)ϕ_22],; ϕ_32,t(t,k)= 4ik^2ϕ_32+2α k(v̅_01ϕ_12+βv̅_00ϕ_22) -iα(v̅_11ϕ_12+βv̅_10ϕ_22); +iα[(|v_01|^2+|u_00|^2)ϕ_32+(β v_0-1v̅_00+v_00v̅_01)ϕ_42]; ϕ_42,t(t,k)=4ik^2ϕ_42+2α k(v̅_00ϕ_12+v̅_0-1ϕ_22)-iα(v̅_10ϕ_12+v̅_1-1ϕ_22); +iα[(β v_00v̅_0-1+v_01v̅_00)ϕ_32+(|v_0-1|^2+|v_00|^2)ϕ_42], ].the third column of Eq. (<ref>) with μ=μ_3(L,t,k)=ϕ(t,k){[ϕ_13,t(t,k)= -4ik^2ϕ_13+2k(v_01ϕ_33+v_00ϕ_43)+i(v_11ϕ_33+v_10ϕ_43);-iα[(|v_01|^2+|v_00|^2)ϕ_13+(β v_01v̅_00+v_00v̅_0-1)ϕ_23],;ϕ_23,t(t,k)= -4ik^2ϕ_23+2k(β v_00ϕ_33+v_0-1ϕ_43)+i(β v_10ϕ_33+v_1-1ϕ_43); -iα[(β v_00v̅_01+v_0-1v̅_00)ϕ_13+(|v_0-1|^2+|v_00|^2)ϕ_23],;ϕ_33,t(t,k)=2α k(v̅_01ϕ_13+βv̅_00ϕ_23) -iα(v̅_11ϕ_13+βv̅_10ϕ_23); +iα[(|v_01|^2+|v_00|^2)ϕ_33+(β v_0-1v̅_00+v_00v̅_01)ϕ_43];ϕ_43,t(t,k)= 2α k(v̅_00ϕ_13+v̅_0-1ϕ_23)-iα(v̅_10ϕ_13+v̅_1-1ϕ_23); +iα[(β v_00v̅_0-1+v_01v̅_00)ϕ_33+(|v_0-1|^2+|v_00|^2)ϕ_43], ].and the fourth column of Eq. (<ref>) with μ=μ_3(L,t,k)=ϕ(t,k){[ϕ_14,t(t,k)= -4ik^2ϕ_14+2k(v_01ϕ_34+v_00ϕ_44)+i(v_11ϕ_34+v_10ϕ_44);-iα[(|v_01|^2+|v_00|^2)ϕ_14+(β v_01v̅_00+v_00v̅_0-1)ϕ_24],;ϕ_24,t(t,k)= -4ik^2ϕ_24+2k(β v_00ϕ_34+v_0-1ϕ_44)+i(β v_10ϕ_34+v_1-1ϕ_44); -iα[(β v_00v̅_01+v_0-1v̅_00)ϕ_14+(|v_0-1|^2+|v_00|^2)ϕ_24],;ϕ_34,t(t,k)=2α k(v̅_01ϕ_14+βv̅_00ϕ_24) -iα(v̅_11ϕ_14+βv̅_10ϕ_24); +iα[(|v_01|^2+|v_00|^2)ϕ_34+(β v_0-1v̅_00+v_00v̅_01)ϕ_44];ϕ_44,t(t,k)= 2α k(v̅_00ϕ_14+v̅_0-1ϕ_24)-iα(v̅_10ϕ_14+v̅_1-1ϕ_24); +iα[(β v_00v̅_0-1+u_01v̅_00)ϕ_34+(|v_0-1|^2+|v_00|^2)ϕ_44], ].Similarly, we can also obtain the asymptotic formulae for ϕ_ij,i,j=1,2,3,4. The substitution of these formulae into Eq. (<ref>) and using the assumption the initial and boundary data are compatible at x=0 and x=L, we find the asymptotic result (<ref>) of c_13(t,k) for k→∞. Similarly we can also show Eqs. (<ref>) and (<ref>) for c_j4(t,k),j=1,2 as k→∞. Similarly, it follows from the global relation (<ref>) under the vanishing initial data that we have c_31(t,k)=Ψ_31(t,k)ϕ̅_11(t,k̅)+Ψ_32(t,k)ϕ̅_12(t,k̅)-α e^-2ikL[Ψ_33(t,k)ϕ̅_13(t,k̅)+Ψ_34(t,k)ϕ̅_14(t,k̅)], c_41(t,k)=Ψ_41(t,k)ϕ̅_11(t,k̅)+Ψ_42(t,k)ϕ̅_12(t,k̅)-α e^-2ikL[Ψ_43(t,k)ϕ̅_13(t,k̅)+Ψ_44(t,k)ϕ̅_14(t,k̅), c_42(t,k)=Ψ_41(t,k)ϕ̅_21(t,k̅)+Ψ_42(t,k)ϕ̅_22(t,k̅)-α e^-2ikL[Ψ_43(t,k)ϕ̅_23(t,k̅)+Ψ_44(t,k)ϕ̅_24(t,k̅)],such that we can show Eqs. (<ref>)-(<ref>) by means of c_j1(t,k),j=3,4 and c_42(t,k) as k→∞.□ §.§4.3.The map between Dirichlet and Neumann problemsIn what follows we mainly show that the spectral functions S(k) and S_L(k) can be expressed in terms of the prescribed Dirichlet and Neumann boundary data and the initial data using the solution of a system of integral equations.For simplicity, we define the notations as F_± (t,k)=F(t,k)± F(t, -k), Σ_±=Σ_±(k)=e^2ikL± e^-2ikL.The sign ∂ D_j, j=1,...,4 stands for the boundary of the jth quadrant D_j, oriented so that D_j lies to the left of ∂ D_j. ∂ D_3^0 denotes the boundary contour which has not contain the zeros of Σ_-(k) and ∂ D_3^0=-∂ D_1^0. Theorem 4.2. Let q_0j(x)=q_j(x,t=0)∈ C^∞[0, L],j=1,0,-1 be an initial data of Eq. (<ref>) on the finite interval x∈ [0, L] andT<∞.For the Dirichlet problem, the smooth boundary data u_0j(t) and v_0j(t)(j=1,0,-1) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),(j=1,0,-1) at points (x_2, t_2)=(0, 0) and (x_3, t_3)=(L, 0), respectively, i.e., u_0j(0)=q_0j(0),v_0j(0)=q_0j(L), j=1,0,-1.For the Neumann problem, the smooth boundary data u_1j(t) and v_1j(t)(j=1,0,-1) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),(j=1,0,-1) at the origin (x_2, t_2)=(0, 0) and (x_3, t_3)=(L, 0), respectively, i.e., u_1j(0)=∂_x q_0j(0),v_1j(0)=∂_x q_0j(L), j=1,0,-1. For simplicity, let n_33,44(𝕊)(k) have no zeros in the domain D_1. Then the spectral functions S(k) and S_L(k) are defined byS(k)=([Ψ_11(T,k̅)Ψ_21(T,k̅)-αΨ_31(T,k̅)e^4ik^2T-αΨ_41(T,k̅)e^4ik^2T;Ψ_12(T,k̅)Ψ_22(T,k̅)-αΨ_32(T,k̅)e^4ik^2T-αΨ_42(T,k̅)e^4ik^2T; -αΨ_13(T,k̅)e^-4ik^2T -αΨ_23(T,k̅)e^-4ik^2TΨ_33(T,k̅)Ψ_43(T,k̅); -αΨ_14(T,k̅)e^-4ik^2T -αΨ_24(T,k̅)e^-4ik^2TΨ_34(T,k̅)Ψ_44(T,k̅) ]), S_L(k)=([ϕ_11(T,k̅)ϕ_21(T,k̅)-αϕ_31(T,k̅)e^4ik^2T-αϕ_41(T,k̅)e^4ik^2T;ϕ_12(T,k̅)ϕ_22(T,k̅)-αϕ_32(T,k̅)e^4ik^2T-αϕ_42(T,k̅)e^4ik^2T; -αϕ_13(T,k̅)e^-4ik^2T -αϕ_23(T,k̅)e^-4ik^2Tϕ_33(T,k̅)ϕ_43(T,k̅); -αϕ_14(T,k̅)e^-4ik^2T -αϕ_24(T,k̅)e^-4ik^2Tϕ_34(T,k̅)ϕ_44(T,k̅) ]), and the complex-valued functions {Ψ_ij(t,k)}_i,j=1^4 can be expressed using the following system of integral equations {[ Ψ_11(t,k)=1+∫_0^t {-iα[(|u_01|^2+|u_00|^2)Ψ_11+(β u_01u̅_00+u_00u̅_0-1)Ψ_21].; +. (2ku_01+iu_11)Ψ_31+(2ku_00+iu_10)Ψ_41}(t',k)dt',; Ψ_21(t,k)=∫_0^t{-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_11+(|u_0-1|^2+|u_00|^2)Ψ_21].; +. (2k(β u_00+iβ u_10)Ψ_31+ (2ku_0-1+iu_1-1)Ψ_41}(t',k)dt',; Ψ_31(t,k)= ∫_0^te^4ik^2(t-t'){2α k(u̅_01Ψ_11+βu̅_00Ψ_21) -iα(u̅_11Ψ_11+βu̅_10Ψ_21).;.+iα[(|u_01|^2+|u_00|^2)Ψ_31+(β u_0-1u̅_00+u_00u̅_01)Ψ_41]} (t',k)dt',; Ψ_41(t,k)= ∫_0^te^4ik^2(t-t')[ 2α k(u̅_00Ψ_11+u̅_0-1Ψ_21)-iα(u̅_10Ψ_11+u̅_1-1Ψ_21).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_31+(|u_0-1|^2+|u_00|^2)Ψ_41]](t',k)dt', ]. {[Ψ_12(t,k)=∫_0^t{-iα[(|u_01|^2+|u_00|^2)Ψ_12+(β u_01u̅_00+u_00u̅_0-1)Ψ_22].; .+2k(u_01Ψ_32+u_00Ψ_42)+i(u_11Ψ_32+u_10Ψ_42)}(t',k)dt',;Ψ_22(t,k)= 1+∫_0^t{-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_12+(|u_0-1|^2+|u_00|^2)Ψ_22].;. +2k(β u_00Ψ_32+u_0-1Ψ_42)+i(β u_10Ψ_32+u_1-1Ψ_42)}(t',k)dt',;Ψ_32(t,k)= ∫_0^t e^4ik^2(t-t'){2α k(u̅_01Ψ_12+βu̅_00Ψ_22) -iα(u̅_11Ψ_12+βu̅_10Ψ_22).;. +iα[(|u_01|^2+|u_00|^2)Ψ_32+(β u_0-1u̅_00+u_00u̅_01)Ψ_42]}(t',k)dt',;Ψ_42(t,k)=∫_0^t e^4ik^2(t-t'){2α k(u̅_00Ψ_12+u̅_0-1Ψ_22)-iα(u̅_10Ψ_12+u̅_1-1Ψ_22).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_32+(|u_0-1|^2+|u_00|^2)Ψ_42]}(t',k)dt', ]. {[Ψ_13(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(|u_01|^2+|u_00|^2)Ψ_13+(β u_01u̅_00+u_00u̅_0-1)Ψ_23].,; .+2k(u_01Ψ_33+u_00Ψ_43)+i(u_11Ψ_33+u_10Ψ_43)}(t',k)dt',;Ψ_23(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_13+(|u_0-1|^2+|u_00|^2)Ψ_23].;. +2k(β u_00Ψ_33+u_0-1Ψ_43)+i(β u_10Ψ_33+u_1-1Ψ_43)}(t',k)dt',;Ψ_33(t,k)= 1+∫_0^t{2α k(u̅_01Ψ_13+βu̅_00Ψ_23) -iα(u̅_11Ψ_13+βu̅_10Ψ_23).; .+iα[(|u_01|^2+|u_00|^2)Ψ_33+(β u_0-1u̅_00+u_00u̅_01)Ψ_43]}(t',k)dt',;Ψ_43(t,k)=∫_0^t{2α k(u̅_00Ψ_13+u̅_0-1Ψ_23)-iα(u̅_10Ψ_13+u̅_1-1Ψ_23).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_33+(|u_0-1|^2+|u_00|^2)Ψ_43]}(t',k)dt', ].and {[ Ψ_14(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(|u_01|^2+|u_00|^2)Ψ_14+(β u_01u̅_00+u_00u̅_0-1)Ψ_24].; .+2k(u_01Ψ_34+u_00Ψ_44)+i(u_11Ψ_34+u_10Ψ_44)}(t',k)dt',; Ψ_24(t,k)=∫_0^te^-4ik^2(t-t'){-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_14+(|u_0-1|^2+|u_00|^2)Ψ_24].;.+ 2k(β u_00Ψ_34+u_0-1Ψ_44)+i(β u_10Ψ_34+u_1-1Ψ_44)}(t',k)dt',; Ψ_34(t,k)=∫_0^t[2α k(u̅_01Ψ_14+βu̅_00Ψ_24) -iα(u̅_11Ψ_14+βu̅_10Ψ_24).; .+iα[(|u_01|^2+|u_00|^2)Ψ_34+(β u_0-1u̅_00+u_00u̅_01)Ψ_44]}(t',k)dt',; Ψ_44(t,k)= 1+∫_0^t{2α k(u̅_00Ψ_14+u̅_0-1Ψ_24)-iα(u̅_10Ψ_14+u̅_1-1Ψ_24).; . +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_34+(|u_0-1|^2+|u_00|^2)Ψ_44]}(t',k)dt', ].The functions {ϕ_ij(t,k)}_i,j=1^4 are of the same integral equations (<ref>)-(<ref>) by replacing the functions {u_0j(t),u_1j(t)} with {v_0j(t),v_1j(t)},(j=1,0,-1), respectively, that is, ϕ_ij(t,k)=Φ_ij(t,k)|_{u_0l(t)→ v_0l(t),u_1l(t)→ v_1l(t), Ψ_ij→ϕ_ij},(i,j=1,2,3,4;l=1,0,-1). (i) For the given Dirichlet boundary data, the unknown Neumann boundary data {u_1j(t),j=1,0,-1} at x=0 and {v_1j(t),j=1,0,-1} at x=L, 0<t<T can be found as[ u_11(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2Σ_+(k)/iΣ_-(k)(kΨ_13-+iu_01) -(u_01ϕ̅_33-+u_00ϕ̅_34-) +4i/Σ_-(k)(α kϕ̅_31-+i v_01)]dk; +∫̣_∂ D_3^04k/iπΣ_-(k){[Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34]e^-2ikL-αł[(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]̊}_-dk;+%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(u_01Ψ_33-+u_00Ψ_43-)dk, ][ u_10(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2Σ_+(k)/iΣ_-(k)(kΨ_14-+iu_00) -(u_01ϕ̅_43-+u_00ϕ̅_44-) +4i/Σ_-(k)(α kϕ̅_41-+i v_00)]dk;+∫̣_∂ D_3^04k/iπΣ_-(k){[Ψ_13ϕ̅_43+Ψ_14(ϕ̅_44-1)]e^-2ikL-α [(Ψ_11-1)ϕ̅_41+Ψ_12ϕ̅_42]}_-dk;+%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(u_01Ψ_34-+u_00Ψ_44-)dk, ][ u_1-1(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2Σ_+(k)/iΣ_-(k)(kΨ_24-+iu_0-1) -(u_00ϕ̅_43-+u_0-1ϕ̅_44-) +4i/Σ_-(k)(α kϕ̅_42-+i v_0-1)]dk;+∫̣_∂ D_3^04k/iπΣ_-(k){[Ψ_23ϕ̅_43+Ψ_24(ϕ̅_44-1)]e^-2ikL-α [Ψ_21ϕ̅_41+(Ψ_22-1)ϕ̅_42]}_-dk; +%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(β u_00Ψ_34-+u_0-1Ψ_44-)dk, ] and[v_11(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2iΣ_+(k)/Σ_-(k)(kϕ_13-+iv_01)-(v_01Ψ̅_33-+v_00Ψ̅_34-) -4i/Σ_-(k)(α kΨ̅_31-+i u_01)]dk;+∫̣_∂ D_3^04k/iπΣ_-(k){α [(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32] -[ϕ_13(Ψ̅_33-1)+ϕ_14Ψ̅_34]e^2ikL}_-dk;+%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(v_01ϕ_33-+v_00ϕ_43-)dk, ] [v_10(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2iΣ_+(k)/Σ_-(k)(kϕ_14-+iv_00)-(v_01Ψ̅_43-+v_00Ψ̅_44-) -4i/Σ_-(k)(α kΨ̅_41-+i u_00)]dk;+∫̣_∂ D_3^04k/iπΣ_-(k){α [(ϕ_11-1)Ψ̅_41+ϕ_12Ψ̅_42] -[ϕ_13Ψ̅_43+ϕ_14(Ψ̅_44-1)]e^2ikL}_-dk;+%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(v_01ϕ_34-+v_00ϕ_44-)dk, ] [v_1-1(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0[2iΣ_+(k)/Σ_-(k)(kϕ_24-+iv_0-1)-(v_00Ψ̅_43-+v_0-1Ψ̅_44-) -4i/Σ_-(k)(α kΨ̅_42-+i u_0-1)]dk;+∫̣_∂ D_3^04k/iπΣ_-(k){α [ϕ_21Ψ̅_41+(ϕ_22-1)Ψ̅_42] -[ϕ_23Ψ̅_43+ϕ_24(Ψ̅_44-1)]e^2ikL}_-dk; +%̣ṣ/̣%̣ṣ2π∫_∂ D_3^0(β v_00ϕ_34-+v_0-1ϕ_44-)dk, ] (ii) For the given Neumannboundary data, the unknown Dirichlet boundary data {u_0j(t),j=1,0,-1} at x=0 and {v_0j(t),j=1,0,-1} at x=L, 0<t<T can be obtained as[u_01(t)=∫̣_∂ D_3^0Σ_+(k)Ψ_13+-2αϕ̅_31+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){[Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34]e^-2ikL-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]}_+dk, ] [u_00(t)=∫̣_∂ D_3^0Σ_+(k)Ψ_14+-2αϕ̅_41+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){[Ψ_13ϕ̅_43+Ψ_14(ϕ̅_44-1)]e^-2ikL-α [(Ψ_11-1)ϕ̅_41+Ψ_12ϕ̅_42]}_+dk, ] [ u_0-1(t)=∫̣_∂ D_3^0Σ_+(k)Ψ_24+-2αϕ̅_42+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){[Ψ_23ϕ̅_43+Ψ_24(ϕ̅_44-1)]e^-2ikL-α [Ψ_21ϕ̅_41+(Ψ_22-1)ϕ̅_42]}_+dk, ] and[v_01(t)= ∫̣_∂ D_3^0-Σ_+(k)ϕ_13++2αΨ̅_31+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){α [(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32]-[ϕ_13(Ψ̅_33-1) +ϕ_14Ψ̅_34]e^2ikL}_+dk, ] [v_00(t)= ∫̣_∂ D_3^0-Σ_+(k)ϕ_14++2αΨ̅_41+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){α [(ϕ_11-1)Ψ̅_41+ϕ_12Ψ̅_42]-[ϕ_13Ψ̅_43 +ϕ_14(Ψ̅_44-1)]e^2ikL}_+dk, ] [ v_0-1(t)= ∫̣_∂ D_3^0-Σ_+(k)ϕ_24++2αΨ̅_42+/πΣ_-(k)dk; +∫̣_∂ D_3^02/πΣ_-(k){α [ϕ_21Ψ̅_41+(ϕ_22-1)Ψ̅_42]-[ϕ_23Ψ̅_43 +ϕ_24(Ψ̅_44-1)]e^2ikL}_+dk, ] where Ψ_13=Ψ_13(t,k), ϕ̅_33=ϕ_33(t, k̅)=ϕ̅_33(t, k̅), and other functions have the similar expressions.Proof. We canproveEqs. (<ref>) and (<ref>) by means of Eqs. (<ref>) and (<ref>) with replacing T by t, that is,S(k)=e^-2ik^2tσ̂_4μ_2^-1(0,t,k) and S_L(k)=e^-2ik^2tσ̂_4μ_3^-1(L,t,k) and the symmetry relation (<ref>). Moreover, Eqs. (<ref>)-(<ref>) for Ψ_ij(t,k),i,j=1,2,3,4 can be given in terms of the Volteral integral equations of μ_2(0,t,k). Similarly, the expressions of ϕ_ij(t,k),i,j=1,2,3,4 can be obtained via the Volteral integral equations of μ_3(L,t,k).In the following we will certify Eqs. (<ref>)-(<ref>).(i) Applying the Cauchy's theorem to Eq. (<ref>), we have [ -iπ2Ψ_33^(1)(t)=∫_∂ D_2[Ψ_33(t,k)-1]dk=∫_∂ D_4[Ψ_33(t,k)-1]dk,; -iπ2Ψ_43^(1)(t)=∫_∂ D_2Ψ_43(t,k)dk=∫_∂ D_4Ψ_43(t,k)dk,; -iπ2Ψ_13^(2)(t)=∫_∂ D_2[kΨ_13(t,k)+i/2u_01(t)]dk=∫_∂ D_4[kΨ_13(t,k)+i/2u_01(t)]dk, ]We further find[iπΨ_33^(1)(t)= -̣(∫_∂ D_2+∫_∂ D_4)[Ψ_33(t,k)-1]dk = (∫_∂ D_1+∫_∂ D_3)[Ψ_33(t,k)-1]dk; = ∫̣_∂ D_3[Ψ_33(t,k)-1]dk-∫_∂ D_3[Ψ_33(t,-k)-1]dk =∫_∂ D_3Ψ_33-(t,k)dk, ][iπΨ_43^(1)(t)= -̣(∫_∂ D_2+∫_∂ D_4)Ψ_43(t,k)dk =∫̣_∂ D_3Ψ_43-(t,k)dk, ] and [iπΨ_13^(2)(t)=(∫_∂ D_1+∫_∂ D_3)[kΨ_13(t,k)+i/2u_01(t)]dk; = ∫̣_∂ D_3[kΨ_13(t,k)+i/2u_01(t)]_-dk; = ∫̣_∂ D_3^0{kΨ_13(t,k)+i/2u_01(t)+2e^-2ikL/Σ_-(k)[kΨ_13(t,k)+i/2u_01(t)]}_-dk+C_1(t); = ∫̣_∂ D_3^0Σ_+(k)/Σ_-(k)(kΨ_13-+iu_01)dk+C_1(t), ] where we have introduced the function C_1(t) asC_1(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_13(t,k)+i/2u_01(t)]}_-dk, We use the global relation (<ref>) to further reduce C_1(t) in the form [ C_1(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_13(t,k)+i/2u_01(t)]}_-dk; = ∫̣_∂ D_3^0{2e^-2ikL/Σ_-[Ψ_13^(1)-kc_13(t,k)+ Ψ_13^(1)ϕ̅_33^(1)+Ψ_14^(1)ϕ̅_34^(1)/k-αϕ̅_31^(1)e^2ikL]}_-dk; -∫̣_∂ D_3^0{2e^-2ikL/Σ_-[ Ψ_13^(1)ϕ̅_33^(1)+Ψ_14^(1)ϕ̅_34^(1)/k+α(kϕ̅_31-ϕ̅_31^(1))e^2ikL]}_-dk; +∫̣_∂ D_3^0{2ke^-2ikL/Σ_-[ Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34-α ((Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32)e^2ikL]}_-dk, ]By applying the Cauchy's theorem and asymptotic (<ref>) to Eq. (<ref>), we find that the terms on the right-hand side of Eq. (<ref>) are of the form [C_1(t)= -iπΨ_13^(2)-∫̣_∂ D_3^0[i/2(u_01ϕ̅_33-+u_00ϕ̅_34-) +2α/Σ_-(k)(kϕ̅_31-+iα v_01)]dk;+∫̣_∂ D_3^02k/Σ_-{[Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34]e^-2ikL-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]}_-dk, ]It follows from Eqs. (<ref>) and (<ref>) that we have [2iπΨ_13^(2)(t)= ∫̣_∂ D_3^0[Σ_+/Σ_-(kΨ_13-+iu_01) -i/2(u_01ϕ̅_33-+u_00ϕ̅_34-) -2/Σ_-(α kϕ̅_31-+i v_01)]dk;+∫̣_∂ D_3^02k/Σ_-{[Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34]e^-2ikL-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]}_-dk, ] Thus substituting Eqs. (<ref>), (<ref>) and (<ref>) into the third one of system (<ref>), we can get Eq. (<ref>).Applying the Cauchy's theorem to Eq. (<ref>), we have [ iπΨ_14^(2)(t)= (∫_∂ D_1+∫_∂ D_3)[kΨ_14(t,k)+i/2u_00(t)]dk;=∫̣_∂ D_3[kΨ_14(t,k)+i/2u_00(t)]_-dk;= ∫̣_∂ D_3^0{kΨ_14(t,k)+i/2u_00(t) +2e^-2ikL/Σ_-(k)[kΨ_14(t,k)+i/2u_00(t)]}_-dk+C_2(t);=∫̣_∂ D_3^0Σ_+(k)/Σ_-(k)(kΨ_14-+iu_00)dk+C_2(t), ] where we have introduced the function C_2(t) asC_2(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_14(t,k)+i/2u_00(t)]}_-dk, We apply the Cauchy's theorem, asymptotic (<ref>), and the global relation (<ref>) to C_2(t) to have [ C_2(t)=-∫̣_∂ D_3^0{2e^-2ikL/Σ_-[kΨ_14(t,k)+i/2u_00(t)]}_-dk; = ∫̣_∂ D_3^0{2e^-2ikL/Σ_-[Ψ_14^(1)-kc_14(t,k)+ Ψ_13^(1)ϕ̅_43^(1)+Ψ_14^(1)ϕ̅_44^(1)/k-αϕ̅_41^(1)e^2ikL]}_-dk; -∫̣_∂ D_3^0{2e^-2ikL/Σ_-[ Ψ_13^(1)ϕ̅_43^(1)+Ψ_14^(1)ϕ̅_44^(1)/k+α(kϕ̅_41-ϕ̅_41^(1))e^2ikL]}_-dk;+∫̣_∂ D_3^0{2ke^-2ikL/Σ_-[ Ψ_13ϕ̅_43+Ψ_14(ϕ̅_44-1)-α ((Ψ_11-1)ϕ̅_41+Ψ_12ϕ̅_42)e^2ikL]}_-dk; =-iπΨ_14^(2)-∫̣_∂ D_3^0[i/2(u_01ϕ̅_43-+u_00ϕ̅_44-) +2/Σ_-(k)(α kϕ̅_41-+i v_00)]dk;+∫̣_∂ D_3^02k/Σ_-{[Ψ_13ϕ̅_43+Ψ_14(ϕ̅_44-1)]e^-2ikL-α [(Ψ_11-1)ϕ̅_41+Ψ_12ϕ̅_42]}_-dk, ]It follows from Eqs. (<ref>) and (<ref>) that we have [2iπΨ_14^(2)(t)= ∫̣_∂ D_3^0[Σ_+/Σ_-(kΨ_14-+iu_00) -i/2(u_01ϕ̅_43-+u_00ϕ̅_44-) -2/Σ_-(α kϕ̅_41-+i v_00)]dk;+∫̣_∂ D_3^02k/Σ_-{[Ψ_13ϕ̅_43+Ψ_14(ϕ̅_44-1)]e^-2ikL-α [(Ψ_11-1)ϕ̅_41+Ψ_12ϕ̅_42]}_-dk, ] We further find iπΨ_j4^(1)(t)=∫̣_∂ D_3Ψ_j4-(t,k)dk, j=3,4,from Eq. (<ref>). Thus substituting Eqs. (<ref>) and (<ref>) into the fourth one of system (<ref>), we can deduce Eq. (<ref>). Similarly, we can also show Eq. (<ref>).To use Eq. (<ref>) to show Eq. (<ref>) for v_11(t) we need to find these functions ϕ_33^(1)(t,k), ϕ_43^(1)(t,k), and ϕ_13^(2)(t,k).Applying the Cauchy's theorem to Eq. (<ref>), we have [iπϕ_13^(2)(t)= ∫̣_∂ D_3[kϕ_13(t,k)-ϕ_13^(1)]_-dk; = ∫̣_∂ D_3^0[kϕ_13(t,k)-ϕ_13^(1)-2e^2ikL/Σ_-(k)ł(kϕ_13-ϕ_13^(1))̊]_-dk+C_3(t); = ∫̣_∂ D_3^0-Σ_+(k)/Σ_-(k)[kϕ_13--ϕ_13^(1)]dk+C_3(t), ] where the function C_3(t) is defined asC_3(t)=∫̣_∂ D_3^0{2e^2ikL/Σ_-[kϕ_13(t,k)-ϕ_13^(1)]}_-dk, We need to further reduce C_3(t) by using the asymptotic (<ref>), the global relation (<ref>) and the Cauchy's theorem such that we find that C_3(t) can be reduced to [ C_3(t)= ∫̣_∂ D_3^0{-2/Σ_-[α kc̅_31(t,k̅)+ϕ_13^(1)e^2ikL+ ϕ_13^(1)Ψ̅_33^(1)+ϕ_14^(1)Ψ̅_34^(1)/ke^2ikL-αΨ̅_31^(1)]}_-dk;+∫̣_∂ D_3^0{2/Σ_-[ϕ_13^(1)Ψ̅_33^(1)+ϕ_14^(1)Ψ̅_34^(1)/ke^2ikL +α(kΨ̅_31-Ψ̅_31^(1))]}_-dk; +∫̣_∂ D_3^0{2k/Σ_-[α ((ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32) -[ϕ_13(Ψ̅_33-1)+ϕ_14Ψ̅_34]e^2ikL]}_-dk; =-iπϕ_13^(1)+∫̣_∂ D_3^0[1/2i(v_01Ψ̅_33-+v_00Ψ̅_34-) +2/Σ_-(α kΨ̅_31-+i u_01)]dk;+∫̣_∂ D_3^02k/Σ_-{α [(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32] -[ϕ_13(Ψ̅_33-1)+ϕ_14Ψ̅_34]e^2ikL}_-dk,; ]Eqs. (<ref>) and (<ref>) imply that [ 2iπϕ_13^(2)(t)= ∫̣_∂ D_3^0[Σ_+/Σ_-(ϕ_13^(1)-kϕ_13-) +1/2i(v_01Ψ̅_33-+v_00Ψ̅_34-)+2/Σ_-(α kΨ̅_31-+i u_01)]dk;+∫̣_∂ D_3^02k/Σ_-{α [(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32] -[ϕ_13(Ψ̅_33-1)+ϕ_14Ψ̅_34]e^2ikL}_-dk,; ]We also have iπϕ_j3^(1)(t)=∫̣_∂ D_3ϕ_j3-(t,k)dk, j=3,4,from Eq. (<ref>).Thus substituting Eqs. (<ref>) and (<ref>) into the third one of system (<ref>) yields Eq. (<ref>). Similarly, we can also showEqs. (<ref>) and (<ref>). (ii) We now deduce the Dirichlet boundary value problems given by Eqs. (<ref>)-(<ref>) at x=0 from the known Neumann boundary value problems. It follows from the first one of Eq. (<ref>) that u_01(t) can be expressed by means of Ψ_13^(1). Applying the Cauchy's theorem to Eq. (<ref>) yields [iπΨ_13^(1)(t)= (∫_∂ D_1+∫_∂ D_3)Ψ_13(t,k)dk =∫_∂ D_3Ψ_13-(t,k)dk; = ∫̣_∂ D_3^0[Ψ_13-(t,k)+2/Σ_-(k)(e^-2ikLΨ_13)_+]dk+C_4(t); =∫̣_∂ D_3^0Σ_+(k)/Σ_-(k)Ψ_13+dk+C_4(t), ] where we have introduced the function C_4(t) asC_4(t)=-∫̣_∂ D_3^02/Σ_-(k)(e^-2ikLΨ_13)_+dk,By applying the global relation (<ref>), the Cauchy's theorem and asymptotics (<ref>) to Eq. (<ref>), we find [ C_4(t)= -∫̣_∂ D_3^02/Σ_-(e^-2ikLΨ_13)_+dk; = ∫̣_∂ D_3^02/Σ_-[-c_13(t,k)e^-2ikL-αϕ̅_31]_+dk;+∫̣_∂ D_3^02/Σ_-{[Ψ_13(ϕ̅_33-1)+Ψ_14ϕ̅_34]e^-2ikL-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]}_+dk; =-iπΨ_13^(1)-∫̣_∂ D_3^02α/Σ_-ϕ̅_31+dk; +∫̣_∂ D_3^02/Σ_-{-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]+[Ψ_13(ϕ̅_33-1) +Ψ_14ϕ̅_34]e^-2ikL}_+dk, ]Eqs. (<ref>) and (<ref>) imply that [ 2iπΨ_13^(1)(t)= ∫̣_∂ D_3^0[Σ_+(k)/Σ_-(k)Ψ_13+-2α/Σ_-ϕ̅_31+]dk; +∫̣_∂ D_3^02/Σ_-{-α [(Ψ_11-1)ϕ̅_31+Ψ_12ϕ̅_32]+[Ψ_13(ϕ̅_33-1) +Ψ_14ϕ̅_34]e^-2ikL}_+dk, ] Thus, substituting Eq. (<ref>) into the first one of Eq. (<ref>) yields Eq. (<ref>). Similarly, by applying the expressions of Ψ_j4^(1)(t),j=1,2 to the second and third ones of Eq. (<ref>), we can obtain Eqs. (<ref>) and (<ref>).We now derive the Dirichlet boundary value problems (<ref>)-(<ref>) at x=L from the known Neumann boundary value problems. It follows from the first one of Eq. (<ref>) that v_01(t) can be expressed by means of ϕ_13^(1). Applying the Cauchy's theorem to Eq. (<ref>) yields [ iπϕ_13^(1)(t)=(∫_∂ D_1+∫_∂ D_3)ϕ_13(t,k)dk =∫_∂ D_3ϕ_13-(t,k)dk;= ∫̣_∂ D_3^0[ϕ_13-(t,k)-2/Σ_-(k)(e^2ikLϕ_13)_+]dk+C_5(t);=∫̣_∂ D_3^0-Σ_+(k)/Σ_-(k)ϕ_13+dk+C_5(t), ] where we have introduced the function C_5(t) asC_5(t)=∫̣_∂ D_3^02/Σ_-(k)(e^2ikLϕ_13)_+dk, By applying the global relation (<ref>), the Cauchy's theorem and asymptotics (<ref>) to Eq. (<ref>), we find [C_5(t)=∫̣_∂ D_3^02/Σ_-(e^2ikLϕ_13)_+dk;= ∫̣_∂ D_3^02α/Σ_-[-c̅_31(t,k̅)+Ψ̅_31]_+dk; +∫̣_∂ D_3^02/Σ_-{α[(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32]-[ϕ_13(Ψ̅_33-1) +ϕ_14Ψ̅_34]e^2ikL}_+dk;= -iπϕ_13^(1)+∫̣_∂ D_3^02α/Σ_-Ψ̅_31+dk;+∫̣_∂ D_3^02/Σ_-{α[(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32]-[ϕ_13(Ψ̅_33-1) +ϕ_14Ψ̅_34]e^2ikL}_+dk, ]Eqs. (<ref>) and (<ref>) imply that [ 2iπϕ_13^(1)(t)=∫̣_∂ D_3^0[-Σ_+(k)/Σ_-(k)ϕ_13++2α/Σ_-Ψ̅_31+]dk; +∫̣_∂ D_3^02/Σ_-{α[(ϕ_11-1)Ψ̅_31+ϕ_12Ψ̅_32]-[ϕ_13(Ψ̅_33-1) +ϕ_14Ψ̅_34]e^2ikL}_+dk , ] Thus, substituting of Eq. (<ref>) into the first one of Eq. (<ref>) yields Eq. (<ref>). Similarly, by applying the expression of ϕ_j4^(1)(t),j=1,2 to the second and third ones of Eq. (<ref>), we can obtain Eqs. (<ref>) and (<ref>). □ §.§4.3.The effective characterizations For the given Dirichlet boundary data {u_0j(t),v_0j(t),j=1,0,-1}, substituting Eqs. (<ref>)-(<ref>) into Eqs. (<ref>)-(<ref>) and the similar expressions for {ϕ_ij(t,k)}_i,j=1^4 yields a system of quadratic nonlinear integral equations for {Ψ_ij(t,k), ϕ_ij(t,k)}_i,j=1^4. The nonlinear integral system gives an effective characterization of the spectral functions {Ψ(t,k), ϕ(t,k)} for the given Dirichlet problem. In what follows we use the perturbation expression approach to exhibit them in detail.Substituting these perturbated expressions {[Ψ_ij(t,k)= Ψ_ij^[0](t,k)+ϵΨ_ij^[1](t,k)+ϵ^2Ψ_ij^[2](t,k)+⋯, i,j=1,2,3,4,;ϕ_ij(t,k)= ϕ_ij^[0](t,k)+ϵϕ_ij^[1](t,k)+ϵ^2ϕ_ij^[2](t,k)+⋯, i,j=1,2,3,4,;u_0j(t)=ϵ u_0j^[1](t)+ϵ^2 u_0j^[2](t)+ϵ^3 u_0j^[3](t)+⋯, j=1,0,-1,;v_0j(t)=ϵ v_0j^[1](t)+ϵ^2 v_0j^[2](t)+ϵ^3 v_0j^[3](t)+⋯, j=1,0,-1,;u_1j(t)=ϵ u_1j^[1](t)+ϵ^2 u_1j^[2](t)+ϵ^3 u_1j^[3](t)+⋯, j=1,0,-1,;v_1j(t)=ϵ v_1j^[1](t)+ϵ^2 v_1j^[2](t)+ϵ^3 v_1j^[3](t)+⋯, j=1,0,-1, ].into Eqs. (<ref>)-(<ref>), where ϵ>0 is a small parameter, we find these terms of O(1) and O(ϵ) of Ψ_ij(t,k) asO(1): {[ Ψ_jj^[0]=1, j=1,2,3,4,; Ψ_ij^[0]=0, i,j=1,2,3,4,i≠j, ].O(ϵ): {[ Ψ_11^[1]=Ψ_12^[1]=Ψ_21^[1]=Ψ_22^[1]=Ψ_33^[1]=Ψ_34^[1]=Ψ_43^[1]=Ψ_44^[1]=0,; Ψ_13^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_01^[1]+iu_11^[1])(t')dt',; Ψ_14^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_00^[1]+iu_10^[1])(t')dt',;Ψ_23^[1](t,k)=β̣∫_0^te^-4ik^2(t-t')(2ku_00^[1]+iu_10^[1])(t')dt',; Ψ_24^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_0-1^[1]+iu_1-1^[1])(t')dt',; Ψ_31^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_01^[1]-iu̅_11^[1])(t')dt',;Ψ_32^[1](t,k)=α̣β∫_0^te^4ik^2(t-t')(2ku̅_00^[1]-iu̅_10^[1])(t')dt',; Ψ_41^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_00^[1]-iu̅_10^[1])(t')dt',; Ψ_42^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_0-1^[1]-iu̅_1-1^[1])(t')dt', ].and other terms O(ϵ^s),s>1 of Ψ_ij(t,k), which are omitted here.Similarly, we can also obtain the analogous expressions for {ϕ_ij^[l]}_i,j=1^4, l=0,1 by means of the boundary values at x=L, that is, {v_ij^[l]}, i=0,1; j=1,2; l=0,1.If we assume that n_33,44(𝕊) has no zero points, then we expand Eqs. (<ref>)-(<ref>) to have {[u_11^[1](t)=∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_13-^[1]+iu_01^[1]) +4i/πΣ_-(α kϕ̅_31-^[1]+i v_01^[1])]dk,;u_10^[1](t)=∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_14-^[1]+iu_00^[1]) +4i/πΣ_-(α kϕ̅_41-^[1]+i v_00^[1])]dk,; u_1-1^[1](t)=∫̣_∂ D_3^0[2Σ_+/iπΣ_-(kΨ_24-^[1]+iu_0-1^[1]) +4i/πΣ_-(α kϕ̅_42-^[1]+i v_0-1^[1])]dk,;v_11^[1](t)=∫̣_∂ D_3^0[2iΣ_+/πΣ_-(kϕ_13-^[1]+iv_01^[1]) -4i/πΣ_-(α kΨ̅_31-^[1]+i u_01^[1])]dk,;v_10^[1](t)=∫̣_∂ D_3^0[2iΣ_+/πΣ_-(kϕ_14-^[1]+iv_00^[1]) -4i/πΣ_-(α kΨ̅_41-^[1]+i u_00^[1])]dk,; v_1-1^[1](t)=∫̣_∂ D_3^0[2iΣ_+/πΣ_-(kϕ_24-^[1]+iv_0-1^[1]) -4i/πΣ_-(α kΨ̅_42-^[1]+i u_0-1^[1])]dk, ]. It further follows from Eq. (<ref>) that we have {[Ψ_13-^[1](t,k)=4̣k∫_0^te^-4ik^2(t-t')u_01^[1](t')dt',;Ψ_14-^[1](t,k)=4̣k∫_0^te^-4ik^2(t-t')u_00^[1](t')dt',; Ψ_24-^[1](t,k)=4̣k∫_0^te^-4ik^2(t-t')u_0-1^[1](t')dt',;Ψ_31-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')u̅_01^[1](t')dt',;Ψ_41-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')u̅_00^[1](t')dt',; Ψ_42-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')u̅_0-1^[1](t')dt', ].Similarly, we have {[ ϕ_13-^[1](t,k)=4̣k ∫_0^te^-4ik^2(t-t')v_01^[1](t')dt',; ϕ_14-^[1](t,k)=4̣k ∫_0^te^-4ik^2(t-t')v_00^[1](t')dt',;ϕ_24-^[1](t,k)=4̣k ∫_0^te^-4ik^2(t-t')v_0-1^[1](t')dt',;ϕ_31-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')v̅_01^[1](t')dt',;ϕ_41-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')v̅_00^[1](t')dt',; ϕ_42-^[1](t,k)=4̣α k∫_0^te^4ik^2(t-t')v̅_0-1^[1](t')dt', ]. Therefore, the unknown Nuemannboundary problem can now be solved perturbatively as follows: for the given n_33,44(𝕊) without zero points and Dirichletboundary data u_0j^[1] and v_0j^[1],j=1,0,-1 at x=0, we can find these functions {Ψ_13-^[1], Ψ_14-^[1], Ψ_24-^[1], Ψ_31-^[1], Ψ_41-^[1], Ψ_42-^[1], ϕ_13-^[1], ϕ_14-^[1], ϕ_24-^[1], ϕ_31-^[1], ϕ_41-^[1], ϕ_42-^[1]} in terms of Eqs. (<ref>) and (<ref>). And then we can obtain u_1j^[1](t) and v_1j^[1](t),j=1,0,-1 from Eq. (<ref>). Finally, we have Ψ_ij^[1] from Eq. (<ref>). Similarly, we can also findϕ_ij^[1].Similarly, it follows from Eqs. (<ref>)-(<ref>) that we have {[u_01^[1](t)=∫̣_∂ D_3^0[Σ_+(k)/πΣ_-(k)Ψ_13+^[1] -2α/πΣ_-ϕ̅_31+^[1]]dk,; u_00^[1](t)=∫̣_∂ D_3^0[Σ_+(k)/πΣ_-(k)Ψ_14+^[1]-2α/πΣ_-ϕ̅_41+^[1]]dk,;u_0-1^[1](t)=∫̣_∂ D_3^0[Σ_+(k)/πΣ_-(k)Ψ_24+^[1]-2α/πΣ_-ϕ̅_42+^[1]]dk,;v_01^[1](t)=∫̣_∂ D_3^0[-Σ_+(k)/πΣ_-(k)ϕ_13+^[1]+2α/πΣ_-Ψ̅_31+^[1]]dk,;v_00^[1](t)=∫̣_∂ D_3^0[-Σ_+(k)/πΣ_-(k)ϕ_14+^[1]+2α/πΣ_-Ψ̅_41+^[1]]dk,; v_0-1^[1](t)=∫̣_∂ D_3^0[-Σ_+(k)/πΣ_-(k)ϕ_24+^[1]+2α/πΣ_-Ψ̅_42+^[1]]dk, ]. It further follows from Eq. (<ref>) that we have {[Ψ_13+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')u_11^[1](t')dt',;Ψ_14+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')u_10^[1](t')dt',; Ψ_24+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')u_1-1^[1](t')dt',;Ψ_31+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')u̅_11^[1](t')dt',;Ψ_41+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')u̅_10^[1](t')dt',; Ψ_42+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')u̅_1-1^[1](t')dt', ]. Similarly we get{[ϕ_13+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')v_11^[1](t')dt',;ϕ_14+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')v_10^[1](t')dt',; ϕ_24+^[1](t,k)=2̣i∫_0^te^-4ik^2(t-t')v_1-1^[1](t')dt',;ϕ_31+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')v̅_11^[1](t')dt',;ϕ_41+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')v̅_10^[1](t')dt',; ϕ_42+^[1](t,k)=-̣2iα∫_0^te^4ik^2(t-t')v̅_1-1^[1](t')dt', ].Therefore, the unknown Dirichletboundary problem can now be solved perturbatively as follows: for the given n_33,44(𝕊) without zero points and Neumannboundary data at x=Lu_1j^[1] and v_1j^[1]j=1,0,-1, we can determine these functions {Ψ_13+^[1], Ψ_14+^[1], Ψ_24+^[1], Ψ_31+^[1], Ψ_41+^[1], Ψ_42+^[1], ϕ_13+^[1], ϕ_14+^[1], ϕ_24+^[1], ϕ_31+^[1], ϕ_41+^[1], ϕ_42+^[1]}from Eqs. (<ref>) and (<ref>). Moreover, we can further find u_0j^[1] and v_0j^[1]j=1,0,-1 from Eq. (<ref>). Finally, we can have Ψ_ij^[1] from Eq. (<ref>). Similarly, we can also findϕ_ij^[1].In fact, the above-obtained recursive formulae can be continued indefinitely. We assume that they hold for all 0≤ j≤ n-1, then for n>0, the substitution of Eq. (<ref>) into Eqs. (<ref>)-(<ref>) yields the terms of O(ϵ^n) asu_11^[n](t)=∫̣_∂ D_3^02/iπΣ_-[Σ_+(kΨ_13-^[n]+iu_01^[n])-2(α kϕ̅_31-^[n]+i v_01^[n])]dk+lowerorderterms, u_10^[n](t)=∫̣_∂ D_3^02/iπΣ_-[Σ_+(kΨ_14-^[n]+iu_00^[n])-2(α kϕ̅_41-^[n]+i v_00^[n])]dk+lowerorderterms,u_1-1^[n](t)=∫̣_∂ D_3^02/iπΣ_-[Σ_+(kΨ_24-^[n]+iu_0-1^[n])-2(α kϕ̅_42-^[n]+i v_0-1^[n])]dk+lowerorderterms,where `lower order terms' stands for the result involving known terms of lower order.The terms of O(ϵ^n) forΨ_ij in Eqs. (<ref>)-(<ref>) and the similar equations for ϕ_ij yield {[ Ψ_13^[n](t,k)=∫_0^t e^-4ik^2(t-t')(2ku_01^[n]+iu_11^[n])(t')dt'+lowerorderterms,; ϕ̅_31^[n](t,k̅)=α̣∫_0^t e^-4ik^2(t-t')(2kv_01^[n]+iv_11^[n])(t')dt'+lowerorderterms,; Ψ_14^[n](t,k)=∫̣_0^t e^-4ik^2(t-t')(2ku_00^[n]+iu_10^[n])(t')dt' +lowerorderterms,; ϕ̅_41^[n](t,k̅)=α̣∫_0^te^-4ik^2(t-t')(2kv_00^[n]+iv_10^[n])(t')dt' +lowerorderterms,; Ψ_24^[n](t,k)=∫̣_0^t e^-4ik^2(t-t')(2ku_0-1^[n]+iu_1-1^[n])(t')dt' +lowerorderterms,; ϕ̅_42^[n](t,k̅)=α̣∫_0^te^-4ik^2(t-t')(2kv_0-1^[n]+iv_1-1^[n])(t')dt' +lowerorderterms, ].which leads to {[Ψ_13-^[n](t,k)=4̣k∫_0^t e^-4ik^2(t-t')u_01^[n](t')dt'+lowerorderterms,;ϕ̅_31-^[n](t,k̅)=4̣α k∫_0^t e^-4ik^2(t-t')v_01^[n](t')dt'+lowerorderterms,;Ψ_14-^[n](t,k)=4̣k∫_0^t e^-4ik^2(t-t')u_00^[n](t')dt'+lowerorderterms,;ϕ̅_41-^[n](t,k̅)=4̣α k∫_0^t e^-4ik^2(t-t')v_00^[n](t')dt'+lowerorderterms,; Ψ_24-^[n](t,k)=4̣k∫_0^t e^-4ik^2(t-t')u_0-1^[n](t')dt'+lowerorderterms,; ϕ̅_42-^[n](t,k̅)=4̣α k∫_0^t e^-4ik^2(t-t')v_0-1^[n](t')dt'+lowerorderterms, ].It follows from system (<ref>) that Ψ_1j-^[n], Ψ_24-^[n], ϕ̅_j1-^[n], ϕ̅_42-^[n],j=3,4 can be obtained at each step from the known Dirichlet boundary data u_0j^[n] and v_0j^[n],j=1,0,-1 such that we know that the Neumann boundary data at x=0u_1j^[n],j=1,0,-1can then be found by Eqs. (<ref>)-(<ref>). Similarly, we also show that the Neumann boundary data at x=Lv_1j^[n],j=1,0,-1 can then be determined by the known Dirichlet boundary data u_0j^[n] and v_0j^[n],j=1,0,-1 .Similarly, the substitution of Eq. (<ref>) into Eqs. (<ref>)-(<ref>) yields the terms of O(ϵ^n) as u_01^[n](t)=∫̣_∂ D_3^01/πΣ_-(k)[Σ_+(k)Ψ_13+^[n]-2αϕ̅_31+^[n]]dk+lowerorderterms, u_00^[n](t)=∫̣_∂ D_3^01/πΣ_-(k)[Σ_+(k)Ψ_14+^[n]-2αϕ̅_41+^[n]]dk+lowerorderterms, u_0-1^[n](t)=∫̣_∂ D_3^01/πΣ_-(k)[Σ_+(k)Ψ_24+^[n]-2αϕ̅_42+^[n]]dk+lowerorderterms, Eq. (<ref>) implies that {[Ψ_13+^[n](t,k)=2̣i∫_0^t e^-4ik^2(t-t')u_11^[n](t')dt' +lowerorderterms,;ϕ̅_31+^[n](t,k̅)=2̣iα∫_0^t e^-4ik^2(t-t')v_11^[n](t')dt'+lowerorderterms,;Ψ_14+^[n](t,k)=2̣i∫_0^t e^-4ik^2(t-t')u_10^[n](t')dt' +lowerorderterms,;ϕ̅_41+^[n](t,k̅)=2̣iα∫_0^te^-4ik^2(t-t')v_10^[n](t')dt' +lowerorderterms,; Ψ_24+^[n](t,k)=2̣i∫_0^t e^-4ik^2(t-t')u_1-1^[n](t')dt' +lowerorderterms,; ϕ̅_42+^[n](t,k̅)=2̣iα∫_0^te^-4ik^2(t-t')v_1-1^[n](t')dt' +lowerorderterms, ].It follows from system (<ref>) that Ψ_1j+^[n], Ψ_24+^[n], ϕ̅_j1+^[n], ϕ̅_42+^[n],j=3,4 can be determined at each step from the known Neumann boundary data at x=0u_1j^[n] and v_1j^[n],j=1,0,-1 such that we know that the Dirichlet boundary data u_0j^[n],1,0,-1 can then be given by Eqs. (<ref>)-(<ref>). Similarly, we also show that the Dirichlet boundary data at x=Lv_0j^[n], j=1,0,-1 can then be determined by the known Neumann boundary data at x=Lu_1j^[n] and v_1j^[n],j=1,0,-1. §.§4.4. The large L limitThe formulae for u_0j(t) and u_1j(t),j=1,0,-1 of Theorem 4.2 in the limit L→∞ can reduce to the corresponding ones on the half-line. Since when L→∞, [ v_0j→ 0, v_1j→ 0, j=1,0,-1, ϕ_ij→δ_ij, %̣ṣ/̣%̣ṣΣ_+(k)Σ_-(k)→ 1as k→∞ in D_3, ]Thus, according to Eq. (<ref>), the L→∞ limits of Eqs. (<ref>)-(<ref>), (<ref>)-(<ref>) yield the unknown Neumann boundary data{[ u_11(t)=-̣1/π∫_∂ D_3^0[2i(kΨ_13-+iu_01)+u_01ϕ̅_33--2(u_01Ψ_33-+u_00Ψ_43-)]dk,; u_10(t)=-̣1/π∫_∂ D_3^0[2i(kΨ_14-+iu_00)+u_00ϕ̅_44--2(u_01Ψ_34-+u_00Ψ_44-)]dk,;u_1-1(t)= -̣1/π∫_∂ D_3^0[2i(kΨ_24-+iu_0-1)+u_0-1ϕ̅_44--2(β u_00Ψ_34-+u_0-1Ψ_44-)]dk, ].for the given Dirichlet boundary problem, and the unknown Dirichlet boundary data u_01(t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_13+dk,u_00(t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_14+dk, u_0-1(t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_24+dk,for the given Neumann boundary problem.§ THE GLM REPRESENTATIONS AND EQUIVALENCENowadays we rededuce the eigenfunctions Ψ(t,k) and ϕ(t,k) in Sec. 4 by means of the Gel'fand-Levitan-Marchenko (GLM) approach <cit.>. Moreover, the global relation can be used to find the unknown Neumann (Dirichlet) boundary values from the given Dirichlet (Neumann) boundary values in terms of the GLM representations. Moreover, the GLM representations are shown to be equivalent to the ones obtained in Sec. 4. Finally, the linearizable boundary conditions are presented fortheGLM representations.§.§5.1. The GLM representations for Ψ(t,k) and ϕ(t,k) Proposition 5.1. The eigenfunctions Ψ(t,k) and ϕ(t,k) defined in Sec. 4 have the GLM representationsΨ(t,k)=𝕀_4× 4 +∫̣_-t^t[L(t,s)+(k+i/2U^(0)σ_4)G(t,s)]e^-2ik^2(s-t)σ_4ds, ϕ(t,k)=𝕀_4× 4+∫̣_-t^t[ℒ(t,s)+(k+i/2𝒰^(L)σ_4)𝒢(t,s)]e^-2ik^2(s-t)σ_4ds,where the 4× 4 matrix-valued functions L(t, s)=(L_ij)_4× 4 and G(t, s)=(G_ij)_4× 4,-t≤ s≤ t satisfy a Goursat system {[L_t(t, s)+σ_4L_s(t,s)σ_4= iσ_4U_x^(0)L(t,s) -12{(U^(0))^3+iU̇^(0)σ_4+ł[U_x^(0),U^(0)]̊}G(t,s),;G_t(t, s)+σ_4G_s(t,s)σ_4= 2U^(0)L(t,s)+iσ_4U_x^(0) G(t,s), ].with the initial conditions {[ L_11(t,-t)= L_12(t,-t)=L_21(t,-t)=L_22(t,-t)=L_33(t,-t); = L_34(t,-t)=L_43(t,-t)=L_44(t,-t)=0,; G_11(t,-t)= G_12(t,-t)=G_21(t,-t)=G_22(t,-t)=G_33(t,-t); = G_34(t,-t)=G_43(t,-t)=G_44(t,-t)=0,;G_13(t,t)=u_01(t), G_24(t,t)=u_0-1(t), G_14(t,t)=u_00(t), G_23(t,t)=β u_00(t),;G_31(t,t)=αu̅_01(t), G_42(t,t)=αu̅_0-1(t),G_32(t,t)=αβu̅_00(t), G_41(t,t)=αu̅_00(t),;L_13(t,t)= i/2u_11(t), L_24(t,t)=i/2u_1-1(t),L_14(t,t)=i/2u_10(t), L_23(t,t)=i/2β u_10(t),;L_31(t,t)= -i/2αu̅_11(t), L_42(t,t)=-i/2αu̅_1-1(t),L_32(t,t)=-i/2αβu̅_10(t),L_41(t,t)=-i/2αu̅_10(t), ].and [ U^(0)=([00u_01(t)u_00(t);00β u_00(t) u_0-1(t);αu̅_01(t) αβu̅_00(t)00;αu̅_00(t) αu̅_0-1(t)00 ]), U̇^(0)=ddtU^(0),;U_x^(0)=([00u_11(t)u_10(t);00β u_10(t)u_11(t);αu̅_11(t) αβu̅_10(t)00;αu̅_10(t) αu̅_1-1(t)00 ]), ] Similarly, ℒ(t,s), 𝒢(t,s) satisfy the similar Eqs. (<ref>) and (<ref>) with L(t,s)→ℒ(t,s),G(t,s)→𝒢(t,s),u_0j(t)→ v_0j(t),u_1j(t)→ v_1j(t), U^(0)→𝒰^(L)=U^(0)|_u_0j(t)→ v_0j(t),U_x^(0)→𝒰_x^(L)=U_x^(0)|_u_1j(t)→ v_1j(t),j=1,0,-1.Proof. We assume thatthe function ψ(t,k)=e^-2ik^2tσ_4+∫̣_-t^t[L_0(t,s)+kG(t,s)]e^-2ik^2sσ_4ds,satisfies the time-part of Lax pair (<ref>) with the boundary data ψ(0,k)=𝕀 at x=0, where L_0(t,s) and G(t,s) are the unknown 4× 4 matrix-valued functions. We substitute Eq. (<ref>) into the time-part of Lax pair (<ref>) with the boundary data (<ref>) and use the identity ∫̣_-t^tF(t,s)e^-2ik^2sσ_4ds=i/2k^2[F(t,t)e^-2ik^2tσ_4-F(t,-t)e^2ik^2tσ_4- ∫̣_-t^tF_s(t,s)e^-2ik^2sσ_4ds]σ_4,where the function F(t,s) is a 4× 4 matrix-valued function. As a result, we have {[L_0(t, -t)+σ_4L_0(t,-t)σ_4=-iU^(0)G(t,-t)σ_4,;G(t, -t)+σ_4G(t,-t)σ_4=0,;L_0(t, t)-σ_4L_0(t,t)σ_4=iU^(0)G(t,t)σ_4+V_0^(0),; G(t, t)-σ_4G(t,t)σ_4=2U^(0),; L_0t(t, s)+σ_4L_0s(t,s)σ_4=-iU^(0)G_s(t,s)σ_4+V_0^(0)L_0(t,s),; G_t(t, s)+σ_4G_s(t,s)σ_4=2U^(0)L_0(t,s)+V_0^(0)G(t,s), ].where U^(0) is given by Eq. (<ref>) and [ V_0^(0)=iσ_4[U_x^(0)-(U^(0))^2];= i([-α(|u_01|^2+|u_00|^2) -α(β u_01u̅_00+u_00u̅_0-1) u_11 u_10; -α(β u_00u̅_01+u_0-1u̅_00) -α(|u_00|^2+|u_0-1|^2) β u_10u_1-1;-αu̅_11 -αβu̅_10 α(|u_01|^2+|u_00|^2) α(β u_0-1u̅_00+ u_00u̅_01);-αu̅_10 -αu̅_1-1α(β u_00u̅_0-1+u_01u̅_00)α(|u_0-1|^2+|u_00|^2) ]). ]We further introduce the new matrix L(t,s) byL(t,s)=L_0(t,s)-i/2U^(0)σ_4G(t,s),such that we can simplify the first four equations of system (<ref>) as {[ L(t, -t)+σ_4L(t,-t)σ_4=0,; G(t, -t)+σ_4G(t,-t)σ_4=0,; L(t, t)-σ_4L(t,t)σ_4=V_0^(0),;G(t, t)-σ_4G(t,t)σ_4=2U^(0), ].which leads to Eq. (<ref>), and from the last two equations of system (<ref>) we have Eq. (<ref>). By usingthe transformation (<ref>), that is, μ_2(0, t, k)=Ψ(t,k)=ψ(t,k)e^2ik^2tσ_4, we know that Ψ(t,k) is given by Eq. (<ref>). Similarly, we can also show that Eq. (<ref>) holds. □ For convenience, we rewrite a 4× 4 matrix C=(C_ij)_4× 4 asC=(C_ij)_4× 4=([ C̃_11 C̃_12 C̃_21 C̃_22 ]),with C̃_11=([ C_11 C_12C_21 C_22 ]), C̃_12=([ C_13 C_14C_23 C_24 ]),C̃_21=([ C_31 C_32C_41 C_42 ]), C̃_22=([ C_33 C_34C_43 C_44 ]). The Dirichlet and Neumann boundary values at x=0, L can simply be rewritten as u_j(t)=([u_j1(t) β u_j0(t)u_j0(t) u_j-1(t) ]),v_j(t)=([v_j1(t) β v_j0(t)v_j0(t) v_j-1(t) ]),j=1,2, For a matrix-valued function F(t,s), we introduce the F̂(t,k) by F̂(t,k)=∫̣_-t^tF(t,s)e^2ik^2(s-t)ds,Thus, the GLM expressions (<ref>) and (<ref>) for {Ψ_ij(t,k), ϕ_ij(t,k)} can be rewritten as{[ ̣̃Ψ_11(t,k)=𝕀_2× 2+L̂̃̂_11-i/2u_0^T(t)Ĝ̃̂_21+kĜ̃̂_11,;̣̃Ψ_12(t,k)=L̂̃̂_12-i/2u_0^T(t)Ĝ̃̂_22+kĜ̃̂_12,;̣̃Ψ_21(t,k)=L̂̃̂_21+iα/2u̅_0(t)Ĝ̃̂_11+kĜ̃̂_21,; ̣̃Ψ_22(t,k)=𝕀_2× 2+L̂̃̂_22+iα/2u̅_0(t)Ĝ̃̂_12+kĜ̃̂_22, ].{[ ̣̃ϕ_11(t,k)=𝕀_2× 2+ℒ̂̃̂_11-i/2v_0^T(t)𝒢̂̃̂_21+k𝒢̂̃̂_11,;̣̃ϕ_12(t,k)=ℒ̂̃̂_12-i/2v_0^T(t)𝒢̂̃̂_22+k𝒢̂̃̂_12,;̣̃ϕ_21(t,k)=ℒ̂̃̂_21+iα/2v̅_0(t)𝒢̂̃̂_11+k𝒢̂̃̂_21,; ̣̃ϕ_22(t,k)=𝕀_2× 2+ℒ̂̃̂_22+iα/2v̅_0(t)𝒢̂̃̂_12+k𝒢̂̃̂_22, ]. Proposition 5.2. ḷịṃ_t'→ t∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')(F̃_12(t,k)e^-2ikL)_-dk=∫̣_∂ D_1^0[i k/2u_0^TĜ̃̂_22+ k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^T+k/Σ_-(F̃_12(t,k)e^-2ikL)_-]dk,ḷịṃ_t'→ t∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')F̃_21-(t,k)dk= ∫̣_∂ D_1^0[iα k/2v_0^T𝒢̂̃̂_22+α k/2iv_0^TĜ̅̃̅̂̅_22^T+k/Σ_-F̃_21-(t,k)]dk,ḷịṃ_t'→ t∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')(F̃_12(t,k)e^-2ikL)_+dk= ∫̣_∂ D_1^01/Σ_-(F̃_12(t,k)e^-2ikL)_+dk,ḷịṃ_t'→ t∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')F̃_21+(t,k)dk= ∫̣_∂ D_1^01/Σ_-F̃_21+(t,k)dk,where the 2× 2 matrix-valued functions F̃_12(t,k) and F̃_21(t,k) are given by[F̃_12(t,k)=-̣i/2u_0^T(t)Ĝ̃̂_22+i/2𝒢̅̂̅̃̅̂̅_11^Tv_0^T(t)e^2ikL -αł(Ψ̃_11-𝕀)̊ϕ̅̃̅_21^Te^2ikL +Ψ̃_12ł(ϕ̅̃̅_22(t,k̅)^T-𝕀)̊;= -̣i/2u_0^T(t)Ĝ̃̂_22+i/2𝒢̅̂̅̃̅̂̅_11^Tv_0^T(t)e^2ikL; +̣(L̂̃̂_12-i/2u_0^T(t)Ĝ̃̂_22+kĜ̃̂_12)(ℒ̅̂̅̃̅̂̅_22^T-iα/2𝒢̅̂̅̃̅̂̅_12^Tv_0^T(t)+k𝒢̅̂̅̃̅̂̅_22^T);-̣α e^2ikL(L̂̃̂_11-i/2u_0^T(t)Ĝ̃̂_21+kĜ̃̂_11) (ℒ̅̂̅̃̅̂̅_21^T-iα/2𝒢̅̂̅̃̅̂̅_11^Tv_0^T(t) +k𝒢̅̂̅̃̅̂̅_21^T), ][ F̃_21(t,k)= -̣iα/2Ĝ̅̃̅̂̅^T_11u_0^T(t)+iα/2v_0^T(t)𝒢̂̃̂_22e^2ikL +ł(ϕ̃_11-𝕀)̊Ψ̅̃̅^T_21(t,k̅)-αϕ̃_12ł(Ψ̅̃̅_22^T(t,k̅)-𝕀)̊e^2ikL; = -̣iα/2Ĝ̅̃̅̂̅^T_11u_0^T(t)+iα/2v_0^T(t)𝒢̂̃̂_22e^2ikL;+̣(ℒ̂̃̂_11-i/2v_0^T(t)𝒢̂̃̂_21+k𝒢̂̃̂_11) (L̅̂̅̃̅̂̅^T_21-iα/2Ĝ̅̃̅̂̅^T_11u_0^T(t)+kĜ̅̃̅̂̅^T_21);-̣α e^2ikL(ℒ̂̃̂_12-i/2v_0^T(t)𝒢̂̃̂_22+k𝒢̂̃̂_12)(L̅̂̅̃̅̂̅^T_22-iα/2Ĝ̅̃̅̂̅^T_12u_0^T(t)+kĜ̅̃̅̂̅^T_22), ]Proof. Similar to the proof of Lemma 4.3 in Ref. <cit.>, we here show Eq. (<ref>). We multiply Eq. (<ref>) by k/Σ_-e^4ik^2(t-t') with 0<t'<t and integrate along along ∂ D_1^0 with respect to dk to have [ ∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')(F̃_12e^-2ikL)_-dk= ∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_22dk -∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_12𝒢̅̂̅̃̅̂̅_22^Tdk;-∫̣_∂ D_1^0 k e^4ik^2(t-t')(L̂̃̂_12-i/2u_0^TĜ̃̂_22) (ℒ̅̂̅̃̅̂̅_22^T-iα/2𝒢̅̂̅̃̅̂̅_12^Tv_0^T)dk;+̣∫̣_∂ D_1^0k^2Σ_+/Σ_-e^4ik^2(t-t')[(L̂̃̂_12-i/2u_0^TĜ̃̂_22)𝒢̅̂̅̃̅̂̅_22^T +(ℒ̅̂̅̃̅̂̅_22^T-iα/2𝒢̅̂̅̃̅̂̅_12^Tv_0^T)Ĝ̃̂_12]dk;-∫̣_∂ D_1^02α k^2/Σ_-e^4ik^2(t-t')[(L̂̃̂_11-i/2u_0^TĜ̃̂_21)𝒢̅̂̅̃̅̂̅_21^T+ (ℒ̅̂̅̃̅̂̅_21^T-iα/2𝒢̅̂̅̃̅̂̅_11^Tv_0^T)Ĝ̃̂_11]dk, ]To further analyse the above equation, the following identities are introduced ∫̣_∂ D_1ke^4ik^2(t-t')F̂(t,k)dk={[ %̣ṣ/̣%̣ṣπ2 F(t, 2t'-t), 0<t'<t,; %̣ṣ/̣%̣ṣπ4F(t, t),0<t'=t, ].and ∫̣_∂ D_1^0k^2/Σ_-e^4ik^2(t-t')F̂(t,k)dk= 2∫̣_∂ D_1^0k^2/Σ_-[∫_0^t'e^4ik^2(t-t')F̂(t,2τ-t)dτ-F(t, 2t'-t)/4ik^2]dk,which also holds for the case that k^2Σ_- is taken place by k^2Σ_+Σ_- or k^2.It follows from the first integral on the right hand side (RHS) of Eq. (<ref>) and Eq. (<ref>) that we findḷịṃ_t'→ t∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_22dk =lim_t'→ tiπ/2 u_0^TG̃_22(t, 2t'-t) =iπ/4u_0^TG̃_22(t, t),ḷịṃ_t'→ t∫̣_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_22dk =∫̣_∂ D_1^0i k/2u_0^TĜ̃̂_22dk =iπ/8u_0^TG̃_22(t,t), Therefore, we know that the first integral on the RHS of Eq. (<ref>) yields the following two terms ḷịṃ_t'→ t∫_∂ D_1^0i k/2e^4ik^2(t-t')u_0^TĜ̃̂_22dk =∫_∂ D_1^0i k/2u_0^TĜ̃̂_22dk|_(<ref>) +∫_∂ D_1^0i k/2u_0^TĜ̃̂_22dk|_(<ref>), Nowadays we study the second integral on the RHS of Eq. (<ref>). It follows from the second integral on the RHS of Eq. (<ref>) and Eq. (<ref>) that we have [ -∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_12𝒢̅̂̅̃̅̂̅_22^Tdk =-2∫̣_∂ D_1^0 k^3∫_0^te^4ik^2(τ-t')G̃_12(t, 2τ-t)𝒢̅̂̅̃̅̂̅_22^Tdτ dk; = -2∫̣_∂ D_1^0 k^3[∫_0^t'e^4ik^2(τ-t')G̃_12(t, 2τ-t)dτ-G̃_12(t, 2t'-t)/4ik^2]𝒢̅̂̅̃̅̂̅_22^Tdk, ] Therefore we take the limit t'→ t of Eq. (<ref>) to get-lim_t'→ t∫̣_∂ D_1^0 k^3e^4ik^2(t-t')Ĝ̃̂_12𝒢̅̂̅̃̅̂̅_22^Tdk =-∫̣_∂ D_1^0 k^3Ĝ̃̂_12𝒢̅̂̅̃̅̂̅_22^Tdk +∫̣_∂ D_1^0k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^Tdk Finally, following the proof in Ref. <cit.> we can show the limits t'→ t of the rest three integrals (i.e., the third, fourth and fifth integrals) of Eq. (<ref>) can be deduced by simply making the limit t'→ t inside the every integral, that is, no additional terms arise in these integrals. For example, lim_t'→ t∫̣_∂ D_1^0 k e^4ik^2(t-t')(L̂̃̂_12-i/2u_0^TĜ̃̂_22) (ℒ̅̂̅̃̅̂̅_22^T-iα/2𝒢̅̂̅̃̅̂̅_12^Tv_0^T)dk =∫̣_∂ D_1^0 k (L̂̃̂_12-i/2u_0^TĜ̃̂_22) (ℒ̅̂̅̃̅̂̅_22^T-iα/2𝒢̅̂̅̃̅̂̅_12^Tv_0^T)dk.Thus we complete the proof of Eq. (<ref>). Similarly, we can show that Eqs. (<ref>), (<ref>) and (<ref>) also hold. □Theorem5.3. Let q_0j(x)=q_j(x,t=0)=0,j=1,0,-1 be the initial data of Eq. (<ref>) on the interval x∈ [0, L] andT<∞. For the Dirichlet problem, the boundary data u_0j(t) and v_0j(t)(j=1,0,-1) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_j0(x)(j=1,0,-1) at the points (x_2, t_2)=(0, 0) and(x_3, t_3)=(L, 0), respectively. For the Neumann problem, the boundary data u_1j(t) and v_1j(t)(j=1,0,-1) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x)(j=1,0,-1) at the points (x_2, t_2)=(0, 0) and(x_3, t_3)=(L, 0), respectively. For simplicity, let n_33,44(𝕊)(k) have no zeros in the domain D_1. Then the spectral functions S(k) and S_L(k) are defined by Eqs. (<ref>) and (<ref>) with Ψ(t,k) and ϕ(t,k) given by Eq. (<ref>) and (<ref>).(i) For the given Dirichlet boundary values u_0(t) and v_0(t), the unknown Neumann boundary values u_1(t) and v_1(t) are given by[u_1^T(t)= ([u_11(t) u_10(t)β u_10(t) u_1-1(t) ]);= %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_12(t,t)+ i/2u_0^T(t)] -%̣ṣ/̣%̣ṣ2αΣ_-[k^2𝒢̅̂̅̃̅̂̅_21^T(t, t)+iα/2v_0^T(t)].; .+%̣ṣ/̣%̣ṣi k2u_0^TĜ̃̂_22+k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^T+k/Σ_-[F̃_12(t,k)e^-2ikL]_-}dk, ] [ v_1^T(t)=([v_11(t) v_10(t)β v_10(t) v_1-1(t) ]); = %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{-Σ_+/Σ_-[k^2𝒢̂̃̂_12(t,t)+i/2v_0^T(t)] +%̣ṣ/̣%̣ṣ2αΣ_-[k^2Ĝ̅̃̅̂̅_21^T(t, t)+iα/2u_0^T(t)].; .+%̣ṣ/̣%̣ṣi k2v_0^T𝒢̂̃̂_22+k/2iv_0^TĜ̅̃̅̂̅_22^T +α k/Σ_-F̃_21-(t,k)}dk, ](ii) For the given Neumann boundary values u_1(t) and v_1(t), the unknown Dirichlet boundary values u_0(t) and v_0(t) are given by[ u_0^T(t)=([u_01(t) u_00(t)β u_00(t) u_0-1(t) ])= %̣ṣ/̣%̣ṣ2π∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_12-2α/Σ_-ℒ̅̂̅̃̅̂̅_21^T +1/Σ_-(F̃_12(t, k)e^-2ikL)_+]dk, ] [ v_0^T(t)=([v_01(t) v_00(t)β v_00(t) v_0-1(t) ])= %̣ṣ/̣%̣ṣ2π∫_∂ D_1^0[ 2α/Σ_-L̅̂̅̃̅̂̅^T_12-1/Σ_-ℒ̂̃̂_21 +α/Σ_-F̃_21+(t, k)]dk, ] where F̃_12(t,k) and F̃_21(t,k) are defined by Eqs. (<ref>) and (<ref>).Proof. According to the global relation (<ref>) and Proposition 5.1, we can show that the spectral functions S(k) and S_L(k) are defined by Eqs. (<ref>) and (<ref>) with Ψ(t,k) and ϕ(t,k) given by Eq. (<ref>) and (<ref>). (i) we study the Dirichlet problem. It follows from the global relation (<ref>) with the vanishing initial data c(t,k)=([ c̃_11(t,k) c̃_12(t,k); c̃_21(t,k) c̃_22(t,k) ]) =μ_2(0, t,k)e^ikLσ̂_4μ_3^-1(L, t,k),that we havec̃_12(t,k)=([ c_13(t,k) c_14(t,k); c_23(t,k) c_24(t,k) ]) =-αΨ̃_11ϕ̅̃̅_21^T(t,k̅)e^2ikL+Ψ̃_12ϕ̅̃̅_22^T(t,k̅),c̃_21(t,k)=([ c_31(t,k) c_32(t,k); c_41(t,k) c_42(t,k) ])=Ψ̃_21ϕ̅̃̅_11^T(t,k̅)-αΨ̃_22ϕ̅̃̅_12^T(t,k̅)e^-2ikL,We substitute Eqs. (<ref>) and (<ref>) into Eq. (<ref>) to have-L̂̃̂_12+αℒ̅̂̅̃̅̂̅_21^Te^2ikL=kĜ̃̂_12 -α k𝒢̅̂̅̃̅̂̅_21^Te^2ikL+F̃_12(t,k)-c̃_12(t,k),where F̃_12(t,k) is given by Eq. (<ref>).Eq. (<ref>) with k→ -k yields-L̂̃̂_12+αℒ̅̂̅̃̅̂̅_21^Te^-2ikL=-kĜ̃̂_12 +α k𝒢̅̂̅̃̅̂̅_21^Te^-2ikL+F̃_12(t, -k)-c̃_12(t, -k), It follows from Eqs. (<ref>) and (<ref>) that we find ̣̂L̃_12=kΣ_+(k)/Σ_-(k)Ĝ̃̂_12 -2α k/Σ_-(k)𝒢̅̂̅̃̅̂̅_21^T+1/Σ_-(k){[F̃_12(t, k)-c̃_12(t, k)]e^-2ikL}_-. We multiply Eq. (<ref>)by k e^4ik^2(t-t') with 0<t'<t and integrate them along ∂ D_1^0 with respect to dk, respectively to yield [∫̣_∂ D_1^0k e^4ik^2(t-t')L̂̃̂_12dk= ∫̣_∂ D_1^0 e^4ik^2(t-t')k^2Σ_+/Σ_-Ĝ̃̂_12dk -∫̣_∂ D_1^0e^4ik^2(t-t')2α k^2/Σ_-𝒢̅̂̅̃̅̂̅_21^Tdk;+∫̣_∂ D_1^0 e^4ik^2(t-t')k/Σ_-ł[F̃_12(t,k)e^-2ikL]̊_-dk, ] where we have used ∫̣_∂ D_1^0 k e^4ik^2(t-t')c̃_12-(t, k)dk =∫̣_∂ D_1^0k e^4ik^2(t-t')ł[c̃_12(t, k)e^-2ikL]̊_-dk=0since these two matrix-valued functionsk e^4ik^2(t-t')c̃_12-(t, k), k e^4ik^2(t-t')(c̃_12(t, k)e^-2ikL)_-are bounded and analytic in D_1^0.By using these conditions given by Eqs. (<ref>) and (<ref>), Eq. (<ref>) can be reduced to [%̣ṣ/̣%̣ṣπ2L̃_12(t, 2t'-t)= 2∫̣_∂ D_1^0k^2Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')G̃_12(t,2τ-t)dτ-G̃_12(t, 2t'-t)/4ik^2]dk; -4∫̣_∂ D_1^0α k^2/Σ_-[∫_0^t'e^4ik^2(t-t')𝒢̅̃̅_21^T(t,2τ-t)dτ-𝒢̅̃̅_21^T(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')ł[F̃_12(t, k)e^-2ikL]̊_-dk, ]We consider the limits t'→ t of Eq. (<ref>) with the initial data (<ref>) and Proposition 5.2 to find [ %̣ṣ/̣%̣ṣπ2L̃_12(t, t)=2ḷịṃ_t'→ t∫_∂ D_1^0k^2Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')G̃_12(t,2τ-t)dτ-G̃_12(t, 2t'-t)/4ik^2]dk; -4ḷịṃ_t'→ t∫_∂ D_1^0α k^2/Σ_-[∫_0^t'e^4ik^2(t-t')𝒢̅̃̅_21^T(t,2τ-t)dτ-𝒢̅̃̅_21^T(t, 2t'-t)/4ik^2]dk; +ḷịṃ_t'→ t∫_∂ D_1^0k/Σ_-e^4ik^2(t-t')ł[F̃_12(t, k)e^-2ikL]̊_-dk;= ∫̣_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_12(t,t)+i/2G̃_12(t, t)] -%̣ṣ/̣%̣ṣ2αΣ_-[k^2𝒢̅̂̅̃̅̂̅_21^T(t, t)+i/2𝒢̅̃̅_21^T(t, t)].;.+%̣ṣ/̣%̣ṣi k2u_0^TĜ̃̂_22+ k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^T+k/Σ_-ł[F̃_12(t,k)e^-2ikL]̊_-}dk, ] Since the initial data (<ref>) are of the form L̃_12(t, t)=i/2u_1^T(t)=i/2([ u_11(t) u_10(t); β u_10(t)u_1-1(t) ]),then we have Eq. (<ref>) by using Eqs. (<ref>) and (<ref>).To show Eq. (<ref>) we rewrite Eq. (<ref>) in the form c̅̃̅^T_21(t,k̅)=([ c̅_31(t,k̅) c̅_41(t,k̅); c̅_32(t,k̅) c̅_42(t,k̅) ])=ϕ̃_11Ψ̅̃̅^T_21(t,k̅)-αϕ̃_12Ψ̅̃̅_22^T(t,k̅)e^2ikL, We substitute Eqs. (<ref>) and (<ref>) into Eq. (<ref>) to have-L̅̂̅̃̅̂̅^T_21+αℒ̂̃̂_12e^2ikL=kĜ̅̃̅̂̅^T_21 -α k𝒢̂̃̂_12e^2ikL+F̃_21(t,k)-c̅̃̅^T_21(t,k̅),where F̃_21(t,k) is givenby Eq. (<ref>).Eq. (<ref>) with k→ -k yields-L̅̂̅̃̅̂̅^T_21+αℒ̂̃̂_12e^-2ikL=-kĜ̅̃̅̂̅^T_21 +α k𝒢̂̃̂_12e^-2ikL+F̃_21(t,-k)-c̅̃̅^T_21(t,-k̅), It follows from Eqs. (<ref>) and (<ref>) that we have α̣ℒ̂̃̂_12=2k/Σ_-Ĝ̅̃̅̂̅^T_21 -α kΣ_+/Σ_-𝒢̂̃̂_12+1/Σ_-[F̃_21(t, k)-c̅̃̅_21^T(t, k̅)]_- We multiply Eq. (<ref>)by k e^4ik^2(t-t') with 0<t'<t, integrate them along ∂ D_1^0 with respect to dk, and use these conditions given by Eqs. (<ref>) and (<ref>) to yield [%̣ṣ/̣%̣ṣαπ2ℒ̃_12(t, 2t'-t)=-2α∫̣_∂ D_1^0k^2Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')𝒢̃_12(t,2τ-t)dτ-𝒢̃_12(t, 2t'-t)/4ik^2]dk;+4∫̣_∂ D_1^0 k^2/Σ_-[∫_0^t'e^4ik^2(t-t')G̅̃̅_21^T(t,2τ-t)dτ-G̅̃̅_21^T(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')F̃_21-(t, k)dk, ] where we have used ∫̣_∂ D_1^0k/Σ_-e^4ik^2(t-t')c̅̃̅^T_21-(t, k̅)dk=0since the matrix-valued function k/Σ_-e^4ik^2(t-t')c̅̃̅^T_21-(t, k̅)is bounded and analytic in D_1^0. We consider the limit t'→ t of Eq. (<ref>) with the initial data (<ref>) and Proposition 5.2 to obtain [%̣ṣ/̣%̣ṣαπ2ℒ̃_12(t, t)= -̣α∫_∂ D_1^0{Σ_+/Σ_-[k^2𝒢̂̃̂_12(t,t)+i/2𝒢̃_12(t, t)] +%̣ṣ/̣%̣ṣ2Σ_-[k^2Ĝ̅̃̅̂̅_21^T(t, t)+i/2G̅̃̅_21^T(t, t)].; .+%̣ṣ/̣%̣ṣiα k2v_0^T𝒢̂̃̂_22+α k/2iv_0^TĜ̅̃̅̂̅_22^T+k/Σ_-F̃_21-(t,k)}dk, ] Since the initial data are of the form 𝒢̃_12(t, t)=i/2v_1^T(t)=i/2([ v_11(t) v_10(t); β v_10(t)v_1-1(t) ]),then we have Eq. (<ref>) by using Eqs. (<ref>) and (<ref>).(ii) We now consider the Neumann problem. It follows from Eqs (<ref>), (<ref>), (<ref>) and (<ref>) that we havệG̃_12=1/kΣ_-(k){Σ_+(k)L̂̃̂_12-2αℒ̅̂̅̃̅̂̅_21^T+[(F̃_12(t, k)-c̃_12(t, k))e^-2ikL]_+}, ̣̂𝒢̃_12=α/kΣ_-(k){2L̅̂̅̃̅̂̅^T_12-αΣ_+(k)ℒ̂̃̂_12+[F̃_21(t, k)-c̅̃̅^T_21(t, k̅)]_+}.We multiply Eqs. (<ref>) and (<ref>) by k e^4ik^2(t-t') with 0<t'<t, integrate them along ∂ D_1^0 with respect to dk, and use these conditions given by Eqs. (<ref>) and (<ref>) to yield[%̣ṣ/̣%̣ṣπ2G̃_12(t, 2t'-t)=∫̣_∂ D_1^02Σ_+/Σ_-[∫_0^t'e^4ik^2(t-t')L̃_12(t,2τ-t)dτ-L̃_12(t, 2t'-t)/4ik^2]dk; -∫̣_∂ D_1^04α/Σ_-[∫_0^t'e^4ik^2(t-t')ℒ̅̃̅_21^T(t,2τ-t)dτ-ℒ̅̃̅_21^T(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')ł[F̃_12(t, k)e^-2ikL]̊_+dk, ][ %̣ṣ/̣%̣ṣπ2𝒢̃_12(t, 2t'-t)= ∫̣_∂ D_1^04α/Σ_-[∫_0^t'e^4ik^2(t-t')L̅̃̅^T_12(t,2τ-t)dτ-L̅̃̅^T_12(t, 2t'-t)/4ik^2]dk; -∫̣_∂ D_1^02 /Σ_-[∫_0^t'e^4ik^2(t-t')ℒ̃_21(t,2τ-t)dτ-ℒ̃_21(t, 2t'-t)/4ik^2]dk;+∫̣_∂ D_1^0α/Σ_-e^4ik^2(t-t')F̃_21+(t, k)dk, ] where we have used ∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')(c̃_12(t, k)e^-2ikL)_+dk=∫̣_∂ D_1^01/Σ_-e^4ik^2(t-t')c̅̃̅^T_21+(t, k̅)dk=0since the matrix-valued functions 1/Σ_-e^4ik^2(t-t')(c̃_12(t, k)e^-2ikL)_+, 1/Σ_-e^4ik^2(t-t')c̅̃̅^T_21+(t, k̅)are bounded and analytic in D_1^0.We study the limits t'→ t of Eqs. (<ref>) and (<ref>) with the initial data (<ref>) and Proposition 5.2 to find[ %̣ṣ/̣%̣ṣπ2G̃_12(t, t)= ∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_12-2α/Σ_-ℒ̅̂̅̃̅̂̅_21^T +1/Σ_-(F̃_12(t, k)e^-2ikL)_+]dk, ][ %̣ṣ/̣%̣ṣπ2𝒢̃_12(t, t)= ∫̣_∂ D_1^0[2α/Σ_-L̅̂̅̃̅̂̅^T_12 -1/Σ_-ℒ̂̃̂_21+α/Σ_-F̃_21+(t, k)]dk, ] Since the initial data are of the form [ G̃_12(t, t)=u_0^T(t)=([ u_01(t) u_00(t); β u_00(t)u_0-1(t) ]),; 𝒢̃_12(t, t)=v_0^T(t)=([ v_01(t) v_00(t); β v_00(t)v_0-1(t) ]), ] then we find Eqs. (<ref>) and (<ref>) by Eqs. (<ref>) and (<ref>). This completes the proof of the Theorem. □§.§5.2. The equivalence of the two distinct representations We here show that the above-mentionedGLMrepresentations for the Dirichlet and Neumann boundary data in Theorem 5.3 are equivalent to ones in Theorem 4.2.Case i. From the Dirichlet boundary data to the Neumann boundary data Eqs. (<ref>) and (<ref>) imply that Ĝ̃̂_12=1/2kΨ̃_12-,𝒢̂̃̂_12=1/2kϕ̃_12-,Ĝ̃̂_22=1/2kΨ̃_22-,𝒢̂̃̂_22=1/2kϕ̃_22-, The substitution of Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields[ u_1^T(t)= %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_12(t,t)+i/2u_0^T(t)] -%̣ṣ/̣%̣ṣ2αΣ_-[k^2𝒢̅̂̅̃̅̂̅_21^T(t, t)+iα/2v_0^T(t)].; .+%̣ṣ/̣%̣ṣi k2u_0^TĜ̃̂_22+ k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^T+k/Σ_-ł[F̃_12(t,k)e^-2ikL]̊_-}dk; = %̣ṣ/̣%̣ṣ4iπ∫_∂ D_1^0{Σ_+/Σ_-[k^2Ĝ̃̂_12(t,t)+i/2u_0^T(t)] -%̣ṣ/̣%̣ṣ2αΣ_-[k^2𝒢̅̂̅̃̅̂̅_21^T(t, t)+iα/2v_0^T(t)].; .+ịk u_0^TĜ̃̂_22+ k/2iu_0^T𝒢̅̂̅̃̅̂̅_22^T+k/Σ_-[Ψ̃_12(ϕ̅̃̅_22^T-𝕀)e^-2ikL -α(Ψ̃_11-𝕀)ϕ̅̃̅_21^T]_-}dk; = ∫̣_∂ D_1^0{2Σ_+/iπΣ_-[kΨ̃_12-+iu_0^T(t)] +%̣ṣ/̣%̣ṣ4iπΣ_-[α kϕ̅̃̅^T_21-+i v_0^T(t)]+2/πu_0^TΨ̃_22-1/πu_0^Tϕ̅̃̅_22^T.;.+̣4k/iπΣ_-[Ψ̃_12(ϕ̅̃̅_22^T-𝕀)e^-2ikL -α(Ψ̃_11-𝕀)ϕ̅̃̅_21^T]_- }dk, ] Since [ Ψ̃_11=([ Ψ_11 Ψ_12; Ψ_21 Ψ_22 ]),Ψ̃_12=([ Ψ_13 Ψ_14; Ψ_23 Ψ_24 ]), Ψ̃_22=([ Ψ_33 Ψ_34; Ψ_43 Ψ_44 ]),; ϕ̅̃̅^T_21=([ ϕ̅_31 ϕ̅_41; ϕ̅_32 ϕ̅_42 ]),ϕ̅̃̅^T_22=([ ϕ̅_33 ϕ̅_43; ϕ̅_34 ϕ̅_44 ]), ] and the integrand in Eq. (<ref>) is an odd function about k, which makes sure that the contour ∂ D_1^0 can be replaced by ∂ D_3^0, thus we can find the same Neumann boundary data u_1j(t),j=1,0,-1 at x=0 given by Eqs. (<ref>)-(<ref>) from Eqs. (<ref>) and (<ref>). Similarly, we can also find the Neumann boundary data v_1j(t), j=1,0,-1 at x=L given by Eqs. (<ref>)-(<ref>) from Eq. (<ref>).Case ii. From theNeumannboundary data to the Dirichlet boundary data It follows from Eqs. (<ref>) and (<ref>) that we have L̂̃̂_12=1/2Ψ̃_12+(t,k)+i/2u_0^TĜ̃̂_22,ℒ̅̂̅̃̅̂̅^T_21=1/2ϕ̅̃̅^T_21+(t,k)+i/2𝒢̅̂̅̃̅̂̅^T_11v_0^T,The substitution of Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields [ u_0^T(t)= %̣ṣ/̣%̣ṣ2π∫̣_∂ D_1^0[Σ_+/Σ_-L̂̃̂_12-2α/Σ_-ℒ̅̂̅̃̅̂̅_21^T +1/Σ_-(F̃_12(t, k)e^-2ikL)_+]dk; =̣̣∫_∂ D_1^0{Σ_+/πΣ_-Ψ̃_12+-2α/πΣ_-ϕ̅̃̅_21+^T .; .+2/πΣ_-[ Ψ̃_12ł(ϕ̅̃̅_22(t,k̅)^T-𝕀)̊e^-2ikL-α(Ψ̃_11-𝕀)ϕ̅̃̅_21^T ]_+}dk, ]Since the integrand in Eq. (<ref>) is an odd function about k, which makes sure that the contour ∂ D_1^0 can be replaced by ∂ D_3^0, thus the substitution of Eq. (<ref>) into Eq. (<ref>) yields the Dirichlet boundary values u_0j(t),j=1,2 again. Similarly, we can also deduce the Dirichlet boundary values v_0j(t),j=1,2 from Eq. (<ref>).§.§5.3. The linearizable boundary conditionsIn the following we investigate the linearizable boundary conditions for the above-mentioned representations.Theorem 5.4. Let q_j(x, t=0)=q_0j(x),j=1,0,-1 be the initial data of the spin-1 GP system (<ref>) on the interval x∈ [0, L], and one of the following boundary data, either(i) the Dirichlet boundary data q_j(x=0,t)=u_0j(t)=0 and q_j(x=L,t)=v_0j(t)=0,j=1,0,-1,or(ii) the Robin boundary data q_jx(x=0,t)-χ q_j(x=0,t)=u_1j(t)-χ u_0j(t)=0,j=1,0,-1 and q_jx(x=L,t)-χ q_j(x=L,t)=v_1j(t)-χ v_0j(t)=0,j=1,0,-1, where χ is a real parameter.Then the eigenfunctions Ψ(t,k) and ϕ(t,k) can be given by(i)Ψ(t,k)=𝕀+([ L̂̃̂_11 L̂̃̂_12; L̂̃̂_21 L̂̃̂_22 ]), ϕ(t,k)=𝕀+([ ℒ̂̃̂_11 ℒ̂̃̂_12; ℒ̂̃̂_21 ℒ̂̃̂_22 ]),where the 4× 4 matrix-valued function L(t, s)=(L_ij)_4× 4satisfies a reduced Goursat system {[L̃_11t+L̃_11s=iu_1^TL̃_21,;L̃_12t-L̃_12s=iu_1^TL̃_22,; L̃_21t-L̃_21s=-iαu̅_1L̃_11,; L̃_22t+L̃_22s=-iαu̅_1L̃_12, ].with the initial data (cf. Eq. (<ref>)) L̃_11(t, -t)=L̃_22(t, -t)=0, L̃_12(t, t)=i/2u_1^T(t),L̃_21(t, t)=-i/2αu̅_1(t),Similarly, the 4× 4 matrix-valued function ℒ(t, s)=(ℒ_ij)_4× 4satisfies the analogous system (<ref>) with the initial data (<ref>) by replacing u_1(t) with v_1(t). (ii)Ψ(t,k)=I+([ L̂̃̂_11 L̂̃̂_12; L̂̃̂_21 L̂̃̂_22 ])+([ -%̣ṣ/̣%̣ṣi2u_0^TĜ̃̂_21kĜ̃̂_12;kĜ̃̂_21%̣ṣ/̣%̣ṣiα2u̅_0Ĝ̃̂_12 ]),ϕ(t,k)=I+([ ℒ̂̃̂_11 ℒ̂̃̂_12; ℒ̂̃̂_21 ℒ̂̃̂_22 ])+([ -%̣ṣ/̣%̣ṣi2v_0^T𝒢̂̃̂_21k𝒢̂̃̂_12;k𝒢̂̃̂_21%̣ṣ/̣%̣ṣiα2v̅_0𝒢̂̃̂_12 ]),where the 4× 4 matrix-valued functions L(t, s)=(L_ij)_4× 4 and G(t, s)=(G_ij)_4× 4satisfy the reduced nonlinear Goursat system {[ L̃_11t+L̃_11s=iχ u_0^TL̃_21+%̣ṣ/̣%̣ṣ12(iu̇_0^T-α u_0^Tu̅_0u_0^T)G̃_21,; L̃_12t-L̃_12s=iχ u_0^TL̃_22,; L̃_21t-L̃_21s=-iχαu̅_0L̃_11,; L̃_22t+L̃_22s=-iχαu̅_0L̃_12-%̣ṣ/̣%̣ṣ12(iαu̇̅̇_0+u̅_0u^T_0u̅_0)G̃_12,; G̃_12t-G̃_12s=2u_0^TL̃_22,; G̃_21t-G̃_21s=2αu̅_0L̃_11, ].with the initial data (cf. Eq. (<ref>)) {[L̃_11(t, -t)=L̃_22(t, -t)=0,; L̃_12(t, t)=%̣ṣ/̣%̣ṣi2χ u_0^T(t),; L̃_21(t, t)=-%̣ṣ/̣%̣ṣi2αχu̅_0(t),; G̃_12(t, t)=u_0^T(t),; G̃_21(t, t)=αu̅_0(t), ].Similarly, the 4× 4 matrix-valued functions ℒ(t, s)=(ℒ_ij)_4× 4 and 𝒢(t, s)=(𝒢_ij)_4× 4satisfy the similar system (<ref>) with the initial data (<ref>) by replacing u_0(t) with v_0(t).Proof. Let us proof that the linearizable boundary data can be regarded as the special cases of Proposition 5.1. Case i.Dirichlet zero boundary data: q_j(x=0,t)=u_0j(t)=0,j=1,0,-1.It follows from the second one of system (<ref>) that G̃_ij(t,s),i,j=1,2 satisfy {[G̃_11t+G̃_11s=iu_1^TG̃_21,;G̃_12t-G̃_12s=iu_1^TG̃_22,; G̃_21t-G̃_21s=-iαu̅_1G̃_11,; G̃_22t+G̃_22s=-iαu̅_1G̃_12, ].with the initial data (cf. Eq. (<ref>)) G̃_11(t, -t)=G̃_22(t, -t)=0, G̃_12(t, t)=G̃_21(t, t)=0,Therefore, the unique solution of Eq. (<ref>) is trivial, that is, G̃_ij(t,s)=0,i,j=1,2 such that Eq. (<ref>) reduces to Eq. (<ref>) and the condition (<ref>) with Eq. (<ref>) becomesEq. (<ref>) with Eq. (<ref>). Similarly, for the Dirichlet zero boundary data q_j(x=L,t)=v_0j(t)=0,j=1,0,-1, we can also have Eq. (<ref>). Case ii. the Robin boundary data q_jx(x=0,t)-χ q_j(x=0,t)=u_1j(t)-χ u_0j(t)=0, j=1,0,-1, imply that the Dirichlet and Neumann boundary data have the linear relation u_1(t)=χ u_0(t).Introduce the new 4× 4matrix Q(t, s)=(Q_ij)_4× 4 by {[ ̣̃Q_11(t,s)=L̃_11(t,s)-iχ/2G̃_11(t,s),; ̣̃Q_12(t,s)=L̃_12(t,s)-iχ/2G̃_12(t,s),; ̣̃Q_21(t,s)=L̃_21(t,s)+iχ/2G̃_21(t,s),; ̣̃Q_22(t,s)=L̃_22(t,s)+iχ/2G̃_22(t,s), ]. It follows from Eqs. (<ref>) and (<ref>) with Eq. (<ref>) that Q̃_ij(t,s), G̃_ij(t,s),i,j=1,2 satisfy {[ Q̃_11t-Q̃_11s=(-α/2u_0^Tu̅_0u_0^T+i/2u̇_0^T+χ^2/2u_0^T)G̃_21,; Q̃_12t-Q̃_12s=(-α/2u_0^Tu̅_0u_0^T+i/2u̇_0^T+χ^2/2u_0^T)G̃_22,; Q̃_21t-Q̃_21s=(-1/2u̅_0u_0^Tu̅_0-iα/2u̇̅̇_0+αχ^2/2u̅_0)G̃_11,; Q̃_22t-Q̃_22s=(-1/2u̅_0u_0^Tu̅_0-iα/2u̇̅̇_0+αχ^2/2u̅_0)G̃_12,; G̃_11t+G̃_11s=2u_0^T Q̃_21,; G̃_12t-G̃_12s=2u_0^T Q̃_22,; G̃_21t-G̃_21s=2αu̅_0 Q̃_11,; G̃_22t+G̃_22s=2αu̅_0 Q̃_12, ].with the initial conditions (cf. Eq. (<ref>)) {[ G̃_11(t, -t)=G̃_22(t, -t)=0,;G̃_12(t, t)=u_0^T(t),;G̃_21(t, t)=αu̅_0(t),; Q̃_12(t, t)=Q̃_21(t, t)=0,; Q̃_11(t, -t)=Q̃_22(t, -t)=0, ]. Therefore, the unique solution of Eq. (<ref>) is also trivial, that is, Q̃_12(t,s)=Q̃_21(t,s) =G̃_11(t,s)=G̃_22(t,s)=0. As a result, Eq. (<ref>) reduces to Eq. (<ref>) and the condition (<ref>) with Eq. (<ref>) becomes Eq. (<ref>) with Eq. (<ref>).Similarly, for the Robin boundary dataq_jx(x=L,t)-χ q_j(x=L,t)=v_1j(t)-χ v_0j(t)=0,j=1,0,-1, that is, v_1(t)=χ v_0(t), we can also findEq. (<ref>). □According to Theorem 5.3 and Theorem 5.4, we have the following Proposition.Proposition 5.5. For the linearizable Dirichlet boundary data u_0(t)=v_0(t)=0, we have the Neumann boundary data u_1(t) and v_1(t): u_1^T(t)=%̣ṣ/̣%̣ṣ4iπ∫̣_∂ D_1^0kΨ̃_12(ϕ̅̃̅^T_22-𝕀)dk,v_1^T(t)=%̣ṣ/̣%̣ṣ4iπ∫̣_∂ D_1^0kϕ̃_12(Ψ̅̃̅^T_22-𝕀)dk,where {[ Ψ̃_12t+4ik^2Ψ̃_12=iu_1^T(Ψ̃_22+𝕀),; Ψ̃_22t=-iαu̅_1Ψ̃_12,; ϕ̃_12t+4ik^2ϕ̃_12=iv_1^T(ϕ̃_22+𝕀),; ϕ̃_22t=-iαv̅_1ϕ̃_12. ].Remark 5.6. The analogous analysis of the Fokas unified method will be extended to analyze the IBV problems for other integrable nonlinear evolution PDEs with 4× 4 Lax pairs both on the the half-line and the finite interval, such as the three-component NLS equations <cit.>, the three-component coupled derivative NLS equations, etc.. § CONCLUSIONS AND DISCUSSIONSIn conclusion, we have extended the Fokas method to explore the initial-boundary value problem for the integrable spin-1 GP equations (<ref>) with a 4× 4 Lax pair on the finite interval x∈ [0, L]. We find that the solution of the system can be generated by means of the solution of the 4× 4 matrix RH problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found using the spectral functions {s(k),S(k),S_L(k)} related to the initial data and the Dirichlet-Neumann boundary data at x=0 and x=L, respectively. We present the global relation to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. In particular, the obtained results for the boundary value problems on the finite interval can reduce to ones on the half-line as the length L approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representations. It is still open problem that how to further study the obtained matrix RH problem. The nonlinear steepest descent method <cit.> and the numerical method <cit.> may be chosen to explore it, which will be considered in the future. The idea used in this paper can also be extended to other integrable NLEES with 4× 4 Lax pairs on the finite interval.AcknowledgmentsThis work was partially supported by the NSFC under Grant No.11571346 and the Youth Innovation Promotion Association, CAS.Appendix. The asymptotic behavior of the eigenfunction μ(x, t, k) in the Lax pair (<ref>).We rewrite the matrices in the Lax pair (<ref>) as [ σ_4=([𝕀_2× 2 0; 0 -𝕀_2× 2 ]),U(x,t)=([ 0 ℚ^T; αℚ̅ 0 ]),ℚ=([ q_1 β q_0; q_0q_-1 ]),;V(x,t,k)=2kU+V_0, V_0=([ -iαℚ^Tℚ̅ iℚ^T_x;-iαℚ̅_xiαℚ̅ℚ^T ]), ] where[ ℚ^Tℚ̅=([|q_1|^2+|q_0|^2 β q_1q̅_0+q_0q̅_-1; β q_0q̅_1+q_-1q̅_0 |q_-1|^2+|q_0|^2 ]),; ℚ̅ℚ^T=([|q_1|^2+|q_0|^2 β q_-1q̅_0+q_0q̅_1; β q_0q̅_-1+q_1q̅_0 |q_-1|^2+|q_0|^2 ]), ]Let the eigenfunction μ(x,t,k) of the Lax pair (<ref>) be of the form μ(x,t,k)=D^(0)(x,t)+D^(1)(x,t)/k+D^(2)(x,t)/k^2+D^(3)(x,t)/k^3+⋯,where the 4× 4 matrices D_j(x,t)'s are the functions of (x,t) to be determined, then the substitution of Eq. (<ref>) into the Lax pair (<ref>) yields the recurrence relations [ x -part:{[ O(k):[σ_4, D^(0)]=0,; O(k^-j):D_x^(j)+i[σ_4, D^(j+1)]=U^(j), j=0,1,2,...;].; t -part:{[ O(k^2): [σ_4, D^(0)]=0,; O(k): i[σ_4, D^(1)]=UD^(0),; O(k^-j): D_t^(j)+2i[σ_4, D^(j+2)]=2UD^(j+1)+V_0D^(j), j=0,1,2,...; ]. ]Forconvenience, we write a 4× 4 matrix D^(j)=(D^(j)_ls)_4× 4 as [ D^(j)=([ D̃^(j)_11 D̃^(j)_12; D̃^(j)_21 D̃^(j)_22 ]),D̃^(j)_11=([ D^(j)_11 D^(j)_12; D^(j)_21 D^(j)_22 ]),D̃^(j)_12=([ D^(j)_13 D^(j)_14; D^(j)_23 D^(j)_24 ]),;D̃^(j)_21=([ D^(j)_31 D^(j)_32; D^(j)_41 D^(j)_42 ]),D̃^(j)_22=([ D^(j)_33 D^(j)_34; D^(j)_43 D^(j)_44 ]), ] It follows from O(k^2) andO(k) in the t-part of Eq. (<ref>) that we have {[ D̃_12^(0)=D̃_21^(0)=0,; ̣̃D_12^(1)=-i/2ℚ^TD̃_22^(0),;̣̃D_21^(1)=iα/2ℚ̅D̃_11^(0), ].From O(k^-j),j=0,1,2,... in the t-part of Eq. (<ref>), we have (cf. Eq. (<ref>)) [ ([ D̃_11t^(j) D̃_12t^(j); D̃_21t^(j) D̃_22t^(j) ]) +4i([0D̃_12^(j+2); -D̃_21^(j+2)0 ]) =2([ ℚ^TD̃_21^(j+1) ℚ^TD̃_22^(j+1); αℚ̅D̃_11^(j+1) αℚ̅D̃_12^(j+1) ]); +i([ ℚ_x^TD̃_21^(j)-αℚ^Tℚ̅D̃_11^(j) ℚ_x^TD̃_22^(j)-αℚ^Tℚ̅D̃_12^(j); αℚ̅ℚ^TD̃_21^(j)-αℚ̅_xD̃_11^(j) αℚ̅ℚ^TD̃_22^(j)-αℚ̅_xD̃_12^(j) ]), ] which leads to {[D̃_11t^(j)=2ℚ^TD̃_21^(j+1) +i[ℚ_x^TD̃_21^(j)-αℚ^Tℚ̅D̃_11^(j)],;D̃_22t^(j)=2αℚ̅D̃_12^(j+1) +i[αℚ̅ℚ^TD̃_22^(j)-αℚ̅_xD̃_12^(j)],; ̣̃D_12^(j+2) =i/4D̃_12t^(j)-i/2ℚ^TD̃_22^(j+1) +1/4[ℚ_x^TD̃_22^(j)-αℚ^Tℚ̅D̃_12^(j)],; ̣̃D_21^(j+2)=-i/4D̃_21t^(j)+i/2αℚ̅D̃_11^(j+1) +1/4[αℚ̅_xD̃_11^(j)-αℚ̅ℚ^TD̃_21^(j)], ]. Eq. (<ref>) with j=0and Eq. (<ref>) yields {[ D̃_11t^(0)=2ℚ^TD̃_21^(1)-iαℚ^Tℚ̅D̃_11^(0)=0,; D̃_22t^(0)=2αℚ̅D̃_12^(1)+iαℚ̅ℚ^TD̃_22^(0)=0,; ̣̃D_12^(2)=-i/2ℚ^TD̃_22^(1)+1/4ℚ_x^TD̃_22^(0),; ̣̃D_21^(2)=iα/2ℚ̅D̃_11^(1)+α/4ℚ̅_xD̃_11^(0), ].Eq. (<ref>) with j=1 yields {[D̃_11t^(1)=%̣ṣ/̣%̣ṣα2(ℚ^Tℚ̅_x-ℚ^T_xℚ̅)D̃_11^(0),;D̃_22t^(1)=%̣ṣ/̣%̣ṣα2(ℚ̅ℚ^T_x-ℚ̅_xℚ^T)D̃_22^(0),;̣̃D_12^(3) =1/8[ℚ^T_t+iαℚ^Tℚ̅ℚ^T]D̃_22^(0) -i/2ℚ^TD̃_22^(2)+1/4ℚ^T_xD̃_22^(1),; ̣̃D_21^(3)=1/8[αℚ̅_t-iℚ̅ℚ^Tℚ̅]D̃_11^(0) +iα/2ℚ̅D̃_11^(2)+1/4αℚ̅_xD̃_11^(1), ].Eq. (<ref>) with j=2 yields {[D̃_11t^(2)= 2ℚ^TD̃_21^(3) +i[ℚ_x^TD̃_21^(2)-αℚ^Tℚ̅D̃_11^(2)];= %̣ṣ/̣%̣ṣ14[αℚ^Tℚ̅_t-i(ℚ^Tℚ̅)^2+iαℚ^T_xℚ̅_x]D̃_11^(0) +α/2(ℚ^Tℚ̅_x-ℚ^T_xℚ̅)D̃_11^(1),;D̃_22t^(2)=2αℚ̅D̃_12^(3)+i[αℚ̅ℚ^TD̃_22^(2)-αℚ̅_xD̃_12^(2)];=%̣ṣ/̣%̣ṣ14[αℚ̅ℚ^T_t+i(ℚ̅ℚ^T)^2-iαℚ̅_xℚ^T_x]D̃_22^(0)+α/2(ℚ̅ℚ^T_x-ℚ̅_xℚ^T)D̃_22^(1), ]. Similarly, it follows from O(k^-j),j=0,1,2,... in the x-part of Eq. (<ref>) that we have ([ D̃_11x^(j) D̃_12x^(j); D̃_21x^(j) D̃_22x^(j) ]) +2i([0D̃_12^(j+1); -D̃_21^(j+1)0 ])=([ ℚ^TD̃_21^(j) ℚ^TD̃_22^(j); αℚ̅D̃_11^(j) αℚ̅D̃_12^(j) ]),which generates {[ D̃_11x^(j)=ℚ^TD̃_21^(j),; D̃_22x^(j)=αℚ̅D̃_12^(j),; ̣̃D_12^(j+1) =i/2D̃_12x^(j)-i/2ℚ^TD̃_22^(j),; ̣̃D_21^(j+1)=-i/2D̃_21x^(j)+i/2αℚ̅D̃_11^(j), ]. Thus, Eq. (<ref>) with j=0 yields {[ D̃_11x^(0)=D̃_22x^(0)=0,; ̣̃D_12^(1)=-i/2ℚ^TD̃_22^(0),;̣̃D_21^(1)=iα/2ℚ̅D̃_11^(0), ].Eq. (<ref>) with j=1 yields {[ ̣̃D_11x^(1)=ℚ^TD̃_21^(1)=iα/2ℚ^Tℚ̅D̃_11^(0),;̣̃D_22x^(1)=αℚ̅D̃_12^(1)=-iα/2ℚ̅ℚ^TD̃_22^(0),; ̣̃D_12^(2) =i/2D̃_12x^(1)-i/2ℚ^TD̃_22^(1)= -i/2ℚ^TD̃_22^(1)+1/4ℚ_x^TD̃_22^(0),; ̣̃D_21^(2)=-i/2D̃_21x^(1)+i/2αℚ̅D̃_11^(1)= iα/2ℚ̅D̃_11^(1)+α/4ℚ̅_xD̃_11^(0), ].and Eq. (<ref>) with j=2 yields {[ ̣̃D_11x^(2)=ℚ^TD̃_21^(2)=iα/2ℚ^Tℚ̅D̃_11^(1)+α/4ℚ^Tℚ̅_xD̃_11^(0),; ̣̃D_22x^(2)=αℚ̅D̃_12^(2)= -iα/2ℚ̅ℚ^TD̃_22^(1)+α/4ℚ̅ℚ_x^TD̃_22^(0), ]. It follows from Eqs. 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Department of Physics, George Mason University , Fairfax, VA 22030, USA Spatial profile of the Majorana fermion wave functionin a one-dimensional p-wave superconductors (PWS) with quasi periodic disorder is shown to exhibitspatial oscillations. These oscillations damp outin the interior of the chain and are characterized by a period that has topological originand is equal to the Chern number determining the Hall conductivity near half-filling ofa two-dimensional electron gas in a crystal.This mapping unfolds in view of acorrespondence between the critical point for the topological transition in PWS and the strong coupling fixed point of the Harper's equation.Oscillatory character of these modes persist in a generalized modelrelated to an extended Harpersystem where the electrons also tunnel to the diagonalsof asquare lattice.However, beyond a bicritical point, the Majorana oscillationsoccur with a random period,characterized byan invariant fractal set. 03.75.Ss,03.75.Mn,42.50.Lc,73.43.Nq Topology InducedOscillations in Majorana Fermions in a Quasiperiodic Superconducting Chain Indubala I SatijaDecember 30, 2023 ============================================================================================§ INTRODUCTION Majorana Fermions – predicted in1937 byEttore Majorana<cit.>, arecharge-neutral fermions that are their own antiparticles. These mysterious particles remain elusive, although there are many speculativetheories about their incarnationin a varietyof problems that includeneutrino oscillations, supersymmetry, dark matter and also in topological excitations in superconductors<cit.>. Following a pioneering work by Kitaev <cit.>, the one-dimensional p-wave superconducting quantum wire has emerged as one of the key system for studyingMajorana Fermions. Kitaev's proposal has broadened the scope of exploration of these particles, beyond the perimeter ofparticle physics to the condensed matter, energizingboththe study of these exotic particles and also thefield of superconductivity.The iconic system underlying these studiesis given by the following one dimensional Hamiltonian, H_sc=∑_ n = -∞^∞J (c_n+1^†c_n^ + Δ c_n+1^† c^†_n) + c.c. + μ (c_n^† c_n^ -1/2). Here c_n^† is the fermion creation operator at the site n of the superconducting chain. The parameter J is the nearest-neighbor hopping amplitude for the fermions, Δ the superconducting gap function (assumed real), and μ the on-site chemical potential.Modeling a PWS, the system exhibits a quantum phase transition as one tunes μ from μ > J (topologically trivial) to μ < J (topological nontrivial) with μ=J being the point oftopological transition <cit.>.The above Hamiltonian also describes a spin chain, the transverse field Ising-like model<cit.>, H_spin =∑_ n = -∞^∞ [ J_xσ^x_n σ^x_n+1 + J_y σ^y_nσ^y_n+1 +μσ^z_n ] Here J_x and J_yare the exchange interactions along the x and the y directions in spin spacewith spin space anisotropy J_y-J_x and μ is the transverse magnetic field. The two models given by Eqs (<ref>) and (<ref>) are equivalent in view ofthe Jordan-Wigner transformation<cit.> that maps the spin operators to fermion operators, with J_x = J(1-Δ) and J_y= J(1+Δ).Thus the superconducting gap parameter Δ in PWS plays the role of the spin space anisotropy in the Ising chain.There has been numerous studies of the spin chain<cit.> with random or quasi periodic disorder modeled by either the exchange interaction J or the magnetic field μbeing site dependent. These two ways of introducing inhomogenity in the system are somewhat equivalent reflecting the well known duality of the Ising model model<cit.>.The eigenvalue equation for the system where μ is site dependent and denotedbelow as μ_n is given bythe followingset of coupledequations, (1-Δ) ψ^a_n+1+(1+Δ)ψ^a_n-1 +2μ_n ψ^a_n= E ψ^b_n (1+Δ) ψ^b_n+1+(1-Δ) ψ^b_n-1 + 2μ_n ψ^b_n=E ψ^a_n Here we have set J equal to unity, that isμ and E are scaled by J. In this paper, we revisit the study of quasi periodicsuperconducting chain with sinusoidal modulation in μ_n described by,μ_n = 2 λcos ( 2πϕ n + δ).Here ϕ is an irrational numberand δ is a phase factor.Analogous to a periodic chain, such a quasi periodic system has been shown to exhibit the Ising transition to magnetic long range order<cit.>,. For the corresponding superconducting chain, this corresponds to the topological transition where thecritical value λ_Tis given by, λ_T = 1+Δ In other words, the superconducting chain with quasiperiodic sinusoidal modulationsupports Majorana modes for λ < λ_T = 1+Δ. Motivation to study such a quasi periodic system was partially due to the fact that for Δ=0, theEq. (<ref>) reduces to a well known tight binding equationfor ψ=ψ^a=ψ_b,known as the Harper equation<cit.> ψ_n+1+ψ_n-1 + 2 λcos ( 2 πϕ n+ δ) ψ_n = E ψ_n, Harper equation models the iconic problem of a two-dimensional electron gas (2DEG) in a crystal in a magnetic field.The parameter ϕ is the magnetic flux ( measured in units of flux quantum ħ/e ).The system exhibits all possible integer quantum Hall states<cit.> of noninteracting fermions summarized in angraph known as the Hofstadter butterfly<cit.>. In its pure form, the Hamiltonian of such a system can be written as<cit.>,H = cos p + λcos x,where [x, p] = i ϕ. In other words,the parameter ϕ,the magnetic flux , measured in units of the flux quantum ħ/e,threading every unit cell of a square lattice plays the role of the Planck constant. This one-dimensional mathematical representation of the two-dimensional problem is a consequence of thegauge choiceoriginating from the fact thatthere is an inherent freedom in choosing the vector potential that determines the magnetic field. Withinthe Landau gauge, the two dimensional wave function decouples as Ψ_n,m= e^ i k_y mψ_n(k_y) where the Bloch vector k_y appears as a phase factor δ in the above equations.For rational flux value ϕ=p/q,eigenstates ofthe Harper's equation are Bloch states and the energy spectrumconsists of q bands . However,for irrational values of ϕ,the system exhibits localization transition<cit.>: for λ < 1 states are Bloch states while for λ > 1 states are exponentially localized The critical point λ=1 has been extensively studied by various renormalization group (RG) analysis<cit.>. These studies for quadratic irrational values of ϕ such as the golden mean describe self-similar spectral properties of the system.Somewhat less known is the strong coupling(λ→∞) fixed point of the Harper's equation, where the fractal fluctuationsabout exponentially decaying wave functions exhibit self-similar characteristics with universal power law scaling<cit.>. Interestingly,thestrong coupling limit of the Harper equation has a direct correspondence with the quasi periodic PWS chain. In 1995 paper by Ketoja and Satija<cit.>,it was pointed out thatλ> 1,universal properties of the Harper equation describes the onset totopological phase transition in the superconducting chain.This relationship (briefly reviewed in section II below) is the key in identifyingChern numbers of quantum Hall problem described by the Harper's equation as a topological length scale associated with the Majorana wave functions in a quasi periodic PWS.In this paper, we show that quasi periodicity induces oscillations in the Majorana modesthat are dampedas one moves in the interior of the chain. The system was very briefly touched in our recent paper<cit.> where the effect of quasi periodicity on the Majorana wave function was described as the spitting of the Majorana peaks. Here, we explore the subject in greater detail and bring out additional richness and clarity to elucidatethe effectsof quasi periodic disorder on the topological Majorana mode as well as the topological critical point of the superconducting chain. Our study is extended to a generalized chain which is doubly quasi periodic and exhibits a new type of interplay between topology and quasi periodicity.Section III describes the Majoranamodes in the quasi periodic superconducting chain. By studyingthe edge modes that exist in a hidden dimension related to translational invariance of the quasi periodic disorder,we show that the topological length scale is the Chern number of the 2DEG problem described by the Harper's equation. The correspondence between the strong coupling limitof Harper ( that does not alter the topological landscapeof the system) and the critical point of the topological phase transition in PWS is the key in identifying what we refer as the Chern dressing of the Majorana wave function.In section IV, we study an extended Harper<cit.> and the corresponding superconducting chain and show that the relationship betweenthe Majorana spreading and theChern number persists below a bicritical point.Topological and fractal characteristics of the system above this bicritical pointare described by anew universality class as was found to be the case also in an extended Harperby Thouless<cit.>.Section V examines the scaling propertiesof the zero energy mode at the onset to topological phase transition in PWS beyond the bicritical pointand show that they are described by an invariant fractal set. In appendix, we show the general mapping between the extended Harper and doubly quasi periodic PWS where diagonal hopping along two diagonals of a square lattice may not be equal. This mapping that includes the special case of triangular latticereveals a kind of anomalous term in the spin Hamiltonian and may berelevant to a earlier studies of mathematical properties of extended Harper system<cit.>. §STRONG COUPLING FIXED POINT OF HARPER EQUATION AND TOPOLOGICAL PHASE TRANSITION IN P-WAVE SUPERCONDUCTING CHAIN We briefly review the relationship between the strong coupling fixed point ofthe Harper's equation and the critical point of the PWS chain describing topological phase transition.For λ > 1, the Harper's equation for incommensurate flux exhibits localized state.As describedin Ref.(<cit.>) fractal character ofthese exponentially localized statesemerge oncewe factorout the exponentially decaying part of the wave function in Eq. (<ref>) as, ψ_n≡e^-nξη_n where the localization length ξ is given by,ξ =1/ ln λHence η_n describes fluctuations in the exponentially decaying localized wave function of the Harper equation. From Eq. (<ref>), it follows that the fluctuations η_nsatisfy the following equation,e^-ξη_n+1 + e^ ξη_n-1 + 2 λcos( 2π (ϕ n + k_y)) η_n = E η_n Since e^-ξ = λ, the limit λ→∞,leads to the following strong coupling form of the Harper equation, η_n-1 + 2cos 2π (ϕ n + k_y) η_n = E/λη_n We note that for E=0,Eq. (<ref>) reduces to the Eq. (<ref>) for E=0 at its critical pointfor Δ =1asλ_T = 1+ Δ = 2. It turns out that even for Δ 1, theuniversal aspects of the E=0 wave function at criticality is described by the Eq. (<ref>)<cit.>. In other words, the strong coupling fixed point of the Harper's equation describes the critical point of the topological phase transition in PWS chains, for all valuesof the gap parameter Δ.Figure (<ref>)shows theself similar fractal fluctuations of the E=0 mode of the Harper's equation for inverse golden mean flux. We note that every sub peak is highly asymmetrical,reminiscent of the exponential decay of Majorana modes in the topological phase as described in next section.This is in sharp contrast to the correspondingself-similarcritical wave function of the Harper equation as shown in Fig. (<ref>). §MAJORANA MODES INA QUASIPERIODICPWS We now discuss topological phase of the quasi periodic PWS characterized by Majorana modes.Analogous to a periodic chain, the sinusoidally modulated quasi periodic PWSwith μ_n given by Eq. (<ref>) supportsMajorana excitationsfor λ < λ_T = 1+Δ.Fig. (<ref>) showsthe Majorana wave function,for ϕ equals to the inverse golden mean that we denote as ϕ_g.In sharp contrast to a periodic chain,the edge states penetrate the bulk exhibiting damped oscillations characterized by a spatial periodicity.The inserts in these graphs are the corresponding energy spectrum showing edge modes near E=0. It should be noted that these spectral insets show energy as a function of the phase parameter δ and in that sense the edge states exist in a fictitious dimension. The key point to be noted here is that the number of the edge modes near E=0 coincide with the spatial periodicity of the oscillating Majorana wave function. In other words, the spatial length scale characterizing recurrence of damped oscillations in the Majorana mode hasa topological origin.The damped Majorana peaks are spaced with a spatial pattern of 4,4,4, 5 that continues throughout the chain as highlighted in the log scale graph. The emergence of spacing5 after three successive spacings of 4 is an exampleof competing length scales as the topological length 4 tries to “adapt" to quasi periodic pattern which for the golden mean ( ϕ = ϕ_g)case leads to peaks separated by rational approximates of ϕ_g^3<cit.>.This type of competition between topology and quasi periodic wasdiscussed inin our earlier paper<cit.>. We also note that the characteristic pattern of the Majorana wave function is identical to the pattern seen at the topological transition pointas shown in Fig. (<ref>). Fig (<ref>)shows theanalogous spatial profiles of the Majorana for two other values of the commensurability parameter ϕ, confirming the relationship between the spatial period associated with the Majorana and thenumber of edge modes in the spectral graph.We next argue that this topological length scale encoded in the oscillatory profile of the Majorana mode is related to the Chern numbers of thetwo dimensional electron gas (2DEG) problem described the Harper's equation<cit.>.The Harper spectrum evolves smoothly into the quasiparticle excitation spectrum of the superconducting chain, particularly near E=0 as there are no band crossings as shown in the Fig. <ref>). This allows us to assign integers to the gaps of the superconducting butterfly spectrum. Figure<ref> also showsthe correspondence between edge modes<cit.> in Harper and PWSfor ϕ close to the inverse golden-mean where the Chern 4 state is the dominant state near E=0 in the Harper equation and we see four edge modes in Harper as well as in the superconducting chain as illustrated in the figure.Unlike Harper equation where the number of edge modes represent the topological quantum number of Hall conductivity,the physical significance of the topological integer in the Ising chain or PWS remains elusive, specially in view of the fact that edge statesexists in a fictitious dimension.Remarkable fact that the Majorana modes are shadowed by the edge modes existingin a nearby gap in a hidden dimension encoded in the phase factor δ is consistent with the reasoning that emerged from recent studies<cit.>arguing the fact thatquasicrystalsare topological states of matter. Thistopological character was unveiledby edge modes existing in a fictitious dimension – the phase factor,related to the translational invariance that shifts the origin of quasiperiodicity and manifests as an additional degree of freedom . This hidden degree of freedom was used to relate the quasi periodic systemsto higher dimensional periodic systems<cit.> and associate topological feature to quasi periodic systems, assigningtopological insulator<cit.> status to quasi periodic systems. An explicit demonstration of edge transport in the quasicrystalsmediated by the edge modes, was demonstrated by pumping light across photonic quasicrystals<cit.>.§ EXTENDED HARPER AND DOUBLY QUASIPERIODIC SUPERCONDUCTING CHAIN We next consider a generalized Harper<cit.>, commonly referred as the extended Harper described by the Hamiltonian,H^ext = cos x +λcos p + αcos ( p+x) +βcos(p-x),where [x, p] = i ϕ.The parameters α and β describe the strengths of the hoping along the two diagonals of the rectangular lattice, in units of nearest-neighbor hopping. For β=0, it describes the system in a triangular lattice. The eigenvalue equation for the extended Harper can be written as,[1+ 2 αcos (2 πϕ (n-1/2)-k_y)] ψ_n-1+2 λcos ( 2 πϕ n + k_y) ψ_n +[1+ 2 αcos (2 πϕ (n+1/2)- k_y)] ψ_n+1= E ψ_n Fig. (<ref>) shows the schematic phase diagramof the extended model<cit.> for irrational fluxvalueswith α=β. For λ=1 and α=β, system has square symmetrybut there are two quite different regimes with this symmetry, according to whetherλ or α dominates exhibiting an interesting bicritical point at λ=α=1 that separates the two different regimes. Below we discuss only this symmetric case with α = β. For α< 1, the system belongs to the university class of the Harper equation. However, above the bicritical point, we see critical behavior characterized by a Cantor set spectrum existing in a parameter space of finite measure. Labeled as the regime II in the Fig. (<ref>), this regime of “spectral collapse"has been the subject of detailed mathematical analysis.<cit.>In close analogy with our analysis of a relationship between the Harper equation and the superconducting chain as described earlier,we now seek a generalized superconductingor spin chain that may relate to the extended Harper model. Firstly, we note that using Jordan Wigner transformation, theextended Harper model equation can be shown to map to a one-dimensional spin chain, given by the followingHamiltonian, H^ext_0 =∑_ n = -∞ J_n ( σ^x_n σ^x_n+1 + σ^y_nσ^y_n+1) + 2 λcos (2 πϕ n+ k_y)σ^z_n where J(n)= (1+ 2αcos (2 πϕ n+δ)). This shows thatthe extended Harper systemdescribes in fermionic representation,a spin chain where both the exchangespin interaction as well as the magnetic field are quasi periodic. Following the analogy between the PWS problem and the Harper equation as discussed earlier, we now introducethe parameter Δ in the extended spin modelthat describes spin space anisotropy. H^ext_spin =∑_ n = -∞[ (J_n - Δ) σ^x_n σ^x_n+1+(J_n+Δ) σ^y_nσ^y_n+1)+2 λcos (2 πϕ n+k_y)σ^z_n ]Theresulting tight binding model, corresponding to this spin chain is,a coupled set of equations involving two component wave functions (ψ^a, ψ^b).(J_n-1 +Δ) ψ^a_n-1 + (J_n+Δ)ψ^a_n-1 + 2μ_n ψ^a_n= E ψ^b_n(J_n-1 - Δ) ψ^b_n-1 +(J_n +Δ) ψ^b_n-1 + 2μ_n ψ^b_n= E ψ^a_n These coupled set of equations describe a superconducting chain where both the nearest-neighbor fermion hoping as well as the chemical potential are quasi periodic.Our detailed study of this model shows a superconducting chain exhibitingtopological phase supporting Majorana modes that delocalize at a critical point λ_T^ext of the transition that depends on both Δ and α. In the topological phase, these damped modes exhibit a characteristic topological length scale only below the bicritical point, as shown in the figure.Above this bicritical point,the Majorana oscillations exhibit a random pattern, that is,varioussub peaksare spacedsomewhat randomly. This feature persists also characterizesthe critical point of the topological phase transition above the bicritical point.§UNIVERSAL “BOW-TIE" INVARIANT SET We now show a kind of hidden order in the random oscillations in E=0 wave function in the extended PWS chainabove the bicritical point at the onset to topological phase transition by studying the strong coupling limit of the extended Harper.Following Ref.<cit.>, we describe the universal scaling properties of the zero energy wave function for the generalized superconducting chain at the onset to topological transitionby monitoring the amplitude of the wave function atthe Fibonacci sites where ϕ is chosen to be the inverse golden mean. System exhibits a periodic cycle of period six<cit.>as long as the parameter space is below or at the bicritical point as shown with two distinct cycles (in red and green) in Fig. (<ref>) . However, above the bicritical point, the amplitudesof the wave function at Fibonacci sites vary randomly.Interestingly, the random set of amplitudes converge on an invariant set as we vary α > 1. Reminiscent of the “orchid flower" ( that has also been analyzedmathematically<cit.>),for the band end energy<cit.>,the invariant set for E=0 state has a shape of a bow-tie. This shows an intriguing example of order and complexity at the topological critical point when subjected to quasi periodic disorder in both the hopping as well as the chemical potential.§SUMMARY Damped oscillations of the Majorana modes in a quasi periodic PWS chains where the period of oscillations has topological root describes a novel interplay between two distinct types of topologies.Interestingly, this Chern dressing of the wave functionpersists even at the onset to the topological phase transition.Such“trail marks"left behind by the Majorana as it disappears at the critical point may facilitate in the detection of Majorana modes in laboratories<cit.>.Our study of a generalized model where scaling associated with the multi fractal wave functions at the topological phase transition are described by a strange attractor raises interesting mathematical question aboutthe role of topological states in determining such stable strange sets. §SPIN MAPPING FOR EXTENDED HARPER WITH ΑΒ We now describe spin-Fermion mapping for the extended Harper for αβ. The generalized tight binding model can be written as,[1+α e^- 2 πi ϕ (n-1/2)- i k_y+β e^- 2 πiϕ (n-1/2)+i k_y] ψ_n-1+2 t_b cos ( 2 πiϕ n + k_y) ψ_n+[1+α e^- 2 πiϕ (n+1/2)- i k_y+β e^- 2 πiϕ (n+1/2)+i k_y] ψ_n+1= E ψ_nJust like Harper model, we now explore the corresponding spin and superconducting chainshidden in this fermionic Hamiltonian.Using Jordan Wigner transformation, the equation can be shown to map to theone-dimensional spin chain, given by the followingHamiltonian,H_s=∑_ n = -∞1/2 [ 1 + ( α+β) cos (2 πϕ n+k_y) ] ( σ^x_n σ^x_n+1+ σ^y_nσ^y_n+1)+ 1/2 ( α-β)sin (2 πϕ n+k_y)( σ^x_n σ^y_n+1 - σ^y_nσ^x_n+1) + 2 λcos (2 πϕ n+k_y)σ^z_nNote that for αβ,the spectral collapse disappears<cit.>. This mapping may be relevant in recentmathematical studies (<ref>) exploring the the absence of Cantor sets of zero measure existing in extended Harper model.99Maj Majorana, E. Nuovo Cimento 5, 171184 (1937).SA Clara Moskowitz, Scientific American,October 2 (2014) 201.Will F.Wilczek, Nature Physics5 ( 2009) 614.Kitaev A. Kitaev, Ann. Phys. (NY) 321 , 2 (2006)LM E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).DS M. Doria and I. Satija, Phys. Rev. Lett. 60, 444 (1988).CH. A. Ceccatto, Phys Rev Lett 62 ( 1989) 203. Fradkin Eduardo Fradkin and Leonard Susskind,Phys. Rev. D17 , 2637 (1978).IISDI. Satija and M. Doria, PRB(R) 39, (1989) 9757;J. Ketoja and I. Satija, Physica A 219 (1995) 212.SNI. Satija and G. Naumis, Phys Rev B 88 054204 (2013). Harper P. G. Harper, Proc. Phys. Soc. A 68 874. QHE von Klitzing K, Dorda G and Pepper M , Phys. Rev. Lett 45 494 (1980).TKNN Thouless, D., Kohmoto, P., Nightingale, and M. den Nijs, Physical Review Letters 49 (1982), p. 405.DRH D. Hofstadter, Phys Rev B, 14 (1976) 2239.book I. I. Satija, Butterfly in the Quantum World (IOP Concise, Morgan and Claypool, San Raffael, CA, 2016), Chap. 10.Wil M. Wilkinson,J. Phys. A: Math. Gen. 20 4337 ( 1987).AA S. Aubry and G. Andre, Ann. Isr. Phys. Soc. 3, 133 (1980). RG1 S. Ostlund and R. Pandit, Phys Rev B, 29 1394 (1984).RG2 Ketoja and Satija, Phys Lett A 194 ( 1994) 64. KSJ. A. Ketoja and I. I. Satija, Physical Review Letters 75, 2762 (1995).FC Francisco Claro, phys. stat. sol. (b) 104KB1 (1981)Thou Han J H, Thouless D J, Hiramoto H and Kohmoto M Phys. Rev. B 50(1994) 11365.GHarperA. Avila, S.Jitomikskaya and C. A. Marxarxiv. 1602.05111 (2016).KS2 Ketoja J A and Satija I I J. Phys. Condens. Matter 9 ( 1997) 1123.Hat Hatsugai and Kohmoto, Phys Rev B, 42, (1990), 8282QCti Yaacov E. Kraus,1 Yoav Lahini,2 Zohar Ringel,1 Mor Verbin,2 and Oded Zilberberg, PRL, 109 106402 (2012).TI Kane C L and Mele E JScience 314 1692, 2006.QCT2 M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013); Y. E. Kraus and O. Zilberberg, Phys. Rev. Lett. 109, 116404 (2012).Orchid Ben Mestel and Andrew H. Osbaldestin, Journal of Mathematical Physics 45(2004) 12.MFexpt A. Das et al , arXiv:1205.7073 (2012);L. Rokhinson et al, Nature Physics (2012). ;Phys.org. October 2, 2014. Retrieved 3 October 2014; Nadj-Perge, Stevan; Drozdov, Ilya K.; Li, Jian; Chen, Hua; Jeon, Sangjun; Seo, Jungpil; MacDonald, Allan H.; Bernevig, B. Andrei; Yazdani, Ali (2 October 2014) Science. 346: 602607 ( 2014) . | http://arxiv.org/abs/1705.09130v1 | {
"authors": [
"Indubala I Satija"
],
"categories": [
"cond-mat.dis-nn",
"nlin.CD"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170525113430",
"title": "Topology Induced Oscillations in Majorana Fermions in a Quasiperiodic Superconducting Chain"
} |
XXVIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions(Quark Matter 2017) Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000 The chiral magnetic effect (CME) is a macroscopic transport effect resulting from the chiral anomaly.We review the recent progress in theoretical understanding theproperties of chiral plasmas, in which the CME and other anomaly-induced transports take place. In particular, the nontrivial interplay of anomalous currents anddynamical electromagnetic fields is discussed.We also review the theoretical status of the modeling of anomalous transport effects in heavy-ion collisions.Chiral Magnetic EffectChiral Vortical EffectMagnetohydrodynamicsHeavy Ion Collisions § INTRODUCTIONMacroscopic transport effects arising from the chiral anomaly have been attracting much attention in recent years. For example,magnetic fields generate dissipationless electric currents when they are applied to chirally imbalanced media, in which thenumbers of left-handed and right-handed fermions are different.This is the chiral magnetic effect (CME) <cit.>. Theoretically, the existence of CME can be derived by a number of ways such as the perturbationtheory, lattice QCD & QED simulations <cit.>, and holography <cit.>. Moreover,the CME and other chiral transport effects constitute an integral part of relativistic hydrodynamics. Those effects are not only allowed but required from the consistency with the second law of thermodynamics <cit.>.Recently, the first experimental observation of CME usinga Dirac semimetalis reported <cit.>.Heavy-ion collisions offer the opportunity to observe anomalous transport effects as well.In this talk, we would like to discuss the interplay of anomalous chiral effects and dynamical electromagnetic fields.The CME currents are generated by applied magnetic fields.The currents in turn produceelectromagnetic fields, that affect the configuration of the electromagnetic fields. We are interested in the fate of such coupled systems of chiral media and electromagnetic fields.For the complete understanding of various phenomena in such systems,one needs a consistent framework to describe both the chiral plasma and the electromagnetic fields. Chiral magnetohydrodynamics (MHD) is such a theory.An ordinary MHD is a low-energy theory for electrically conducting fluids.It can describe the time evolution of the coupled system of the conducting fluids and electromagnetic fields in a consistent way.In chiral MHD, the fluid is a chiral one, which includes the anomalous chiral effects like CME as a medium response.It has been pointed out that the chiral plasmadevelops an instability <cit.>.Chiral MHD can answer the eventual fate of the instability.Description in terms of chiral MHD is appropriate and important not only for the heavy-ion collisions, but also forearly Universe before electroweak phase transition. For example, the interplay of chiral fermions with dynamical gaugefields leads to a formation of large-scale magnetic fields like we see in the current Universe. In this contribution, we reviewthe recent theoretical progress understanding the nature of chiral plasmas.We also review recent hydrodynamic attempts at describing CME in heavy-ion collisions.§ ANOMALOUS CHIRAL EFFECTS AND DYNAMICAL ELECTROMAGNETIC FIELDS§.§ Chiral anomaly and the topology of magnetic fieldsLet us start by explaining a relation of the topology of magnetic fields and fermions.The conservation of the axial current j^μ_A is broken by a quantum effect, the extent which is quantified by the anomaly equation,_μ j^μ_A = C_AE · B,where E and B are electric and magnetic fields. After a spatial integration, the anomaly equation (<ref>) takes the following form, d/dt[ ℋ + ℋ_F ] = 0 ,where we have defined ℋ≡∫ d^3xA · B, ℋ_F ≡2/C_A∫ d^3 xn_A .Here A is the vector potential, and n_A is the axial charge density. We have introduced two helicities:ℋ is so-called magnetic helicity, and ℋ_F is the total fermionic helicity. Equation (<ref>) tells us thatthe total “chirality” is a constant, althoughit can be stored either in magnetic fields or in fermions.When the chirality is stored in the fields, the field takes a topologically nontrivial form. Indeed, it is well known that the magnetic helicity is a measure of the topology of magnetic fields. When the system is made of magnetic flux tubes, the magnetic helicity can be written in terms of topological invariants as ℋ= ∑_i𝒮_i φ^2_i+ 2 ∑_i,jℒ_ijφ_i φ_j ,where φ_i is the magnetic flux of the i-th flux tube,𝒮_i is the Călugăreanu-White self-linking number, and ℒ_ij is the Gauss linking number<cit.>. The integrated anomaly equation (<ref>) tells us that fermions can change the magnetic helicity, hence the topology of B fields. §.§ Inverse cascade Let us find how the fermions affect the B field topology.Coupled system of chiral matter and electromagnetism have been studied using the Maxwell-Chern-Simons theory <cit.>. The total helicity is conserved and this constrains the dynamics of the system.Remarkably, such systems exhibit the so-called “inverse cascade,” in which the energy is transferred from smaller to larger scales.This leads to large structures of magnetic fields as the system evolves.It also turns out that the evolution is self-similar <cit.>, although the existence of such solution might depend on the choice of equation of state <cit.>.This discussion can be extended to include the degrees of freedom of the fluid. Fluid can also share the helicity in the form of fluid helicity, ∫ d^3 x v ·ω, where ω≡∇× v is the vorticity of the fluid.The turbulent spectrum in chiral MHD is discussed <cit.> and self-similar inverse cascade remains. §.§ Quantized CME An explicit formula that connects the change in the topology of B field and CME current has been derived recently <cit.>, ∑_i∮_C_iΔ J · dx = - e^3/2 π^2Δℋ ,where Δ J is the generated CME current,C_i are the trajectories of the magnetic flux tubes,Δℋ is the change in magnetic helicity,and e is the electric charge.Equation (<ref>) indicates that a change in the topology of the magnetic fields (Δℋ) necessarily results in the generation of a CME current (see Fig. 1 as a sketch). This relation can be extended to include vortices <cit.>.Namely, the reconnections of magnetic fields and vortices can also lead to the generation of CME currents in a chiral fluid. §.§ Equations of motion of (chiral) MHD Let us turn to the dynamical description of a chiral plasma. A low-energy theory of a conducting fluid and electromagnetic fields is MHD, equations of motions of which are given by [ The formulation of MHD has recently been revisited in Refs. <cit.> ]_μ T^μν_ tot= 0 , _μF̃^μν =0.where T^μν_ tot is the energy-momentum tensor of whole system, and the latter equation is the Bianchi identity. Ideal MHD is characterized by the constitutive relation for the electric field, E^μ_(0) = 0.This corresponds to the limit of large conductivity. E^μ is the electric field in the frame of the fluid element, so the observer on the fluid element does not feel any electric field. In ideal MHD, the magnetic helicity is conserved and the topology of B fields is unchanged, which can be seen as follows.The magnetic helicity is the volume integral of the zero-th component of the Chern-Simons current h^μ_B = F̃^μν A_ν.The divergence of h^μ_B reads _μh^μ_B =8 E_μ B^μ ,and since E^μ = 0 in ideal MHD,h^μ_B is conserved in this limit.Although anomalous chiral effects are dissipationless, they do not appear in ideal MHD.Anomalous effects enter if one considers the contribution from finite conductivity σ.The correction from resistivity to the electric field reads E^μ_(1) = λϵ^μναβ_α[ u_ν B_β] - ϵ_B B^μ- ϵ_ωω^μ,where λ≡ 1/σ is the resistivity, ϵ_B = σ_B / σ, andϵ_ω = σ_ω / σ. Coefficients σ_B and σ_ω are chiral magnetic/vortical conductivities.The latter two terms in Eq. (<ref>) are anomalous effects. §.§ Linear excitations in chiral MHD In idea MHD, the magnetic field is “frozen in” to the fluid. The fluid is pierced by magnetic fields and they move together.Because of the tension of the magnetic field and the moment of inertia of the fluid,oscillatory motion of the magnetic field line happens, and itpropagates along the magnetic field. This is the Alfven wave. The nature of the Alfven wave is affected by anomalous chiral effects.If we take the wave vector parallel to B, the dispersion relation readsω = ± v_A k_||- i/2[ (η̅+ λ)k_||^2 - s ϵ_B k_||],where the Alfven velocity v_A is defined by v_A^2 ≡ B^2 / (e+p+B^2), η̅≡η / (e+p+ B^2) is a normalized shear viscosity,and s is the helicity of the wave (there are left-handed and right-handed Alfven waves). Because of the contribution proportional to σ_B, helicity-dependent instability appears. For example, if σ_B>0, the positive helicity modes with k<k_c is unstable, wherek_ c = σ_B/1+ η̅σ.This instability generates helical flows in the presence of chirality imbalance, hence is a mechanism to transfer helicity from fermions to fluid flow <cit.>.§ ANOMALOUS HYDRODYNAMIC MODELING OF HEAVY-ION COLLISIONS§.§ Chirality production and CME in the glasmaLet us turn to the anomalous chiral effects in heavy-ion collisions.Shortly after the collisions, the matter is in a nonequilibrium statecalled glasma, which consists of highly occupied gluons. Then, fermions are created from those fields and the system will reachthe local equilibrium state described by hydrodynamics.In the context of the CME search,the glasma dynamics is important, because chromo E^a · B^a creates chirality imbalance, that is necessary for CME current to be generated <cit.>.The amount of axial charge in the initial stage of hydrodynamicsstrongly affect the value of the final observable, and its quantitative estimation is important for the experimental detection of CME. The chirality generation from nonequilibrium color fields has been studied via real-time classical lattice simulations.In Ref. <cit.>, the sphaleron rate is measured in a nonequilibrium non-Abelian plasma, and it isfound to be enhanced compared to the equilibrium values. This means that a glasma can more efficiently produce chirality imbalance than an equilibrium plasma. Furthermore, CME itself can also be happening in non-equilibrium states of glasma and fermions.Indeed, CME is simulated in real-time lattice simulations, in which U(1) magnetic fields are applied in addition to color fields <cit.>. Figure 2shows the time evolution of axial and vector charge densities.A sphaleron transition creates axial charges, and later CME current develops in the direction of the applied magnetic field.Since magnetic fields are stronger at earlier times, the contribution of the CME current in glasmas can be important forexperimental search of CME. Those pre-hydro CME currents should enter in the initial conditions ofthe subsequent anomalous hydrodynamic stage. In those works, the backreaction from the generated fermions to the fields is not included, but it can also be incorporated. In Ref. <cit.>, such a study is performed in the case of an Abelian plasma.§.§ Anomalous hydrodynamic calculations To reach a decisive conclusion about the existence of anomalous chiral effects in heavy-ion collisions <cit.>, we need a tool to describe this phenomena quantitatively.For this purpose, hydrodynamic models with anomaly-induced transports have been developed <cit.>.In Ref. <cit.>, charge transport from anomalous currents are studied on a background solution of second order viscous fluid in 2+1 dimensions (VISHNew). The authors incorporated the effects of resonance decays, which contribute as a background effect, and simulations are performed on an event-by-event basis. Figure 3shows <cit.> the centrality dependence of the so-called H correlation <cit.>, which shows a very similar trend with the STAR data. Another approach for describing heavy-ion collisions is chiral kinetic theory <cit.>.Examples for such calculations have been reported<cit.> in this Quark Matter. §.§ MHD and magnetic fields In the search of CME in heavy-ion collisions,one of the biggest uncertainties is the strength and life time of magnetic fields. The MHD-type description is also useful in investigatingelectromagnetic properties of the plasma <cit.>.Recently, MHD simulations for heavy-ion collisions are performed <cit.> and the effects on observables like v_2 is discussed.In Fig. 4, the values of magnetic fields are plotted as a function of Bjorken time <cit.>. Compared to vacuum evolution (dotted line), the result from MHD simulation (solid line) shows slower decay as an effect of the medium. The initial B field for the MHD calculation (which starts from τ = 0.4fm) is given by the solution of Maxwell equations at finite conductivity <cit.>. In most of the calculations so far, the sources of electromagnetic fields are treated as classical ones. The importance of quantum effects in the estimation of magnetic fields is pointed out in Ref. <cit.> The authors treated the sources as wave packets satisfying the Dirac equation, and the obtained field configurations turned out to show different behavior from classical treatment. §.§ Vorticity Formation of vortices in heavy-ion collisions is attracting renewed attentions since the report of finite Λ polarization from STAR <cit.>.The fluids formed after collisions naturally have global vortical structure pointing in the direction of the angular momentum.In addition, in event-by-event initial conditions of fluid calculated from transport models (HIJING/UrQMD), more complex vortex structures are found <cit.>.Since the life-time and magnitude of vorticities are less uncertain compared to magnetic fields, detection of CVE in heavy-ion collisions can be more feasible than CME. Since vorticities are larger at lower collisions energies, the search for CVE can benefit from the Beam Energy Scan II at RHIC.Event-by-event anomalous hydrodynamic analysis of γ correlation has been reported <cit.> in this conference. §.§ Isobaric collisions γ correlation can be contaminated with background effects,such as transverse momentum conservation <cit.>, charge conservation <cit.> and cluster particle correlations <cit.>.Those effects are “flow driven” because their contributions are proportional to v_2.In order to identify the contributions from anomalous transport,RHIC is planning to perform the collisions of isobars using^96_44 Ru and ^96_40 Zr in 2018. Since those isobars have the same mass number, the geometry of the collisions of Ru+ Ru and Zr+ Zr are the same.But the numbers of protons are different and the strength of the magnetic fields can be varied without changing the flow. In Ref. <cit.>, it is shown that the two types of collisions can indeed give rise to sizable (∼ 20 %) difference in the observables. Figure 5shows ⟨ B^2 cos[2 (Ψ_ B - Ψ_ RP)] ⟩ as a function of centrality for the two types of collisions. The quantity⟨ B^2 cos[2 (Ψ_ B - Ψ_ RP)] ⟩ is a good measure for the anomalous contribution for the following reasons:γ correlation from anomalous transport should scale as | B|^2.Since γquantifies charge separation in the out-of-plane direction, if the direction of B field (Ψ_ B) is decorrelated with Ψ_ RP, the signal should vanish. The quantity ⟨ B^2 cos[2 (Ψ_ B - Ψ_ RP)] ⟩ captures this. § SUMMARY In summary, the interplay of chiral fluids and dynamical electromagnetic fields leads to a rich variety of phenomena.The fermions affect the topological configuration of magnetic fields and fluid velocities.There are ongoing efforts for more sophisticated description of anomalous chiral effects aiming at the detection of those effects in heavy-ion collisions.§ ACKNOWLEDGEMENTS This worksupported by theU.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract No. DE-SC0012704 and within the framework of the Beam Energy Scan Theory (BEST) Topical Collaboration.elsarticle-num | http://arxiv.org/abs/1705.09409v1 | {
"authors": [
"Yuji Hirono"
],
"categories": [
"hep-ph",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170526014213",
"title": "Chiral magnetohydrodynamics for heavy-ion collisions"
} |
On Mntz-type formulas related to the Riemann zeta function Hlder Lima==========================================================Abstract. The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical Mntz formula <cit.>. These formulas are derived in the critical strip and in the half-plane (s)<0. As particular cases, integral representations for products of the gamma and zeta functions are exhibited.Keywords: Arithmetic functions, Dirichlet series, Mellin transform, Riemann zeta function, Euler gamma function, Mntz formula, Mntz-type formulas.AMS Subject Classifications: 11M06, 11M26, 33B15, 42A38, 42B10, 44A05.§ INTRODUCTIONThe Mellin transform <cit.> of a function f is defined byf^*(s)=∫_0^∞f(x)x^s-1dxand its inverse transform is given by f(x)=1/2π i∫_σ-i∞^σ+i∞f^*(s)x^-sds. The following proposition establishes reciprocity between (<ref>) and (<ref>) under favourable conditions (see section 1.29 of <cit.>).Suppose that a,b∈, a<b, f^*(s) is an analytic function in the strip a<(s)<b such that, for each a<σ<b, f^*(s)∈ L_1(σ± i∞)≡ L_1(σ- i∞,σ+ i∞) and f(x) is defined by (<ref>). Then f(x)x^σ-1∈ L_1(0,∞), for all a<σ<b, and the Mellin transform of f is equal to f^*(s) in the strip a<(s)<b. As it is known, the Riemann zeta function (ζ(s)) <cit.> is analytic in the entire complex plane except the point s=1, where it has a simple pole such that _s=1ζ(s)=lim_s→ 1(s-1)ζ(s)=1. Moreover, we have the following representations of expressions involving the Riemann zeta function in form of Dirichlet series absolutely convergent in the half-plane (s)>1:ζ(s)=∑_n=1^∞1/n^s;1/ζ(s)=∑_n=1^∞μ(n)/n^s;ζ^k(s)=∑_n=1^∞d_k(n)/n^s, k∈;ζ(s)/ζ(2s) =∑_n=1^∞|μ(n)|/n^s;ζ^2(s)/ζ(2s) =∑_n=1^∞2^ω(n)/n^s;ζ^3(s)/ζ(2s) =∑_n=1^∞d(n^2)/n^s;ζ^4(s)/ζ(2s) =∑_n=1^∞(d(n))^2/n^s. These Dirichlet series involve several arithmetic functions. The Mbius function (μ(n)) is defined by μ(1)=1; μ(n)=(-1)^k, if n is the product of k distinct primes; and μ(n)=0, if there exists any prime p such that p^2| n. The function ω(n) represents the number of distinct prime factors of n. The Dirichlet divisor function (d(n)) expresses the number of divisors of n. For any fixed k∈, d_k(n) denotes the number of different ways of writing n as the product of k natural factors where expressions with the same factors in different orders are counted as distinct.Observe that d_2(n)=d(n).Finally (see <cit.>), fixed t_0>1, there exists M∈^+ such that, for all t≥ t_0, |ζ(σ± it)|≤ M,if σ≥ 2;Mln(t),if 1≤σ≤ 2;M t^1-σ/2ln(t),if 0≤σ≤ 1;M t^1/2-σln(t),if σ≤ 0. § A NEW FAMILY OF CLASSES OF FUNCTIONSThe Mntz-type class of functions _α, where α>1, is introduced in <cit.>. Here we generalise this class, defining the following family of classes of functions. A function f(x), defined for x∈_0^+, belongs to the generalized Mntz-type class of functions _α,k, where α>1 and k∈_0, if f∈𝒞^(k)(_0^+) and f^(j)(x)=(x^-α-j), x→∞, for all j=0,1,⋯,k. This definition is a generalisation of the class _α because _α,2=_α, for any α>1.Note that, if k≥ l and β≥α, then _β,k⊆_α,l.The next theorem shows that the Mellin transform of a function in a _α,k class is analytic in the strip -k< s<α except some finite (at most k) singularity points. Let f∈_α,k. Then, for any n=0,1,⋯,k, f^(n)(x)x^σ+n-1∈ L_1(0,∞), for all -n<σ<α, and the Mellin transform of f, f^*(s), is an analytic function in the strip 0<(s)<α, which can be analytically continued to the strip -n<(s)<α byf^*(s)=(-1)^n/(s)_n∫_0^∞ f^(n)(x)x^s+n-1dx,where (s)_n is the Pochhammer symbol, defined by (s)_0=1 and (s)_n=s(s+1)⋯(s+(n-1)).Moreover, f^*(s) is analytic in the strip -k<(s)<α, except at the points s=-n, with n=0,1,⋯,k-1, where f^*(s) either has a simple pole with residue f^(n)(0)/n!, if f^(n)(0)≠ 0, or has a removable singularity, if f^(n)(0)=0. Proof. Fix n=0,1,⋯,k. Then, if -n<σ<α, f^(n)(x)x^σ+n-1= (x^σ+n-1), x→ 0 and f^(n)(x)x^σ+n-1= (x^σ-α-1), x→∞, so f^(n)(x)x^σ+n-1∈ L_1(0,∞).As a consequence, the integral ∫_0^∞f^(n)(x)x^s+n-1dx defines an analytic function in the strip -n<(s)<α. In particular, if n=0, it can be deduced that f^*(s) is analytic in the strip 0<(s)<α and its derivatives are obtained differentiating inside the integral (<ref>).Now we derive (<ref>) in the strip 0<(s)<α, for all n=0,1,⋯,k.If n=0, (<ref>) coincides with the definition of f^*(s).Otherwise, if n=1,2,⋯,k, (<ref>) can be deduced from the case n-1, using integration by parts and eliminating the integrated terms due to the asymptotic behaviour of f^(n-1)(x) at the infinity.Moreover, (<ref>) gives the analytic continuation of f^*(s) to the strip -k<(s)<α, except at the zeros of (s)_k=s(s+1)⋯(s+k-1): the points s=-n, n=0,1,⋯,k-1. Furthermore, lim_s→ -n(s+n)f^*(s) =lim_s→ -n(-1)^n+1(s+n)/(s)_n+1∫_0^∞f^(n+1)(x)x^s+ndx. Besides that, (s+n)/(s)_n+1=1/(s)_n and (-n)_n=(-1)^nn! so lim_s→-n(-1)^n+1(s+n)/(s)_n+1 =(-1)^n+1/(-n)_n=-1/n! and the integral ∫_0^∞f^(n+1)(x)x^s+ndx defines an analytic function in the strip -(n+1)<(s)<α so lim_s→-n∫_0^∞f^(n+1)(x)x^s+ndx =∫_0^∞f^(n+1)(x)dx=-f^(n)(0). Therefore lim_s→-n(s+n)f^*(s)=f^(n)(0)/n!and, as a result, either s=-n is a simple pole of f^*(s) with residue f^(n)(0)/n!, if f^(n)(0)≠ 0, or it is a removable singularity, if f^(n)(0)=0.The following proposition establishes an upper bound for the Mellin transform of a function in a _α,k class and, as a result, it gives us sufficient conditions for the absolute convergence of its integral over vertical lines of the complex plane. Let f∈_α,k. Then, for any -k<σ<α, there exists C(σ)∈ such that |f^*(σ+it)|≤C(σ)/|t|^k, for all t∈\{0}.Moreover, if k≥ 2, f^*(s)∈ L_1(σ± i∞), for any -k<σ<α such that σ≠ 0,-1,⋯,-(k-1), and f(x) can be represented by (<ref>), for any 0<σ<α.Proof.For any s∈, |(s)_k|≥|(s)|^k so, if -k<(s)<α and (s)≠ 0, we replace n=k in (<ref>) to deduce that|f^*(s)|≤1/|(s)_k|∫_0^∞|f^(k)(x)x^s+k-1|dx ≤1/|(s)|^k∫_0^∞|f^(k)(x)|x^(s)+k-1dx. Fix -k<σ<α. Then, by theorem <ref>, f^(k)(x)x^σ+k-1∈ L_1(0,∞), so we can define C(σ)=∫_0^∞|f^(k)(x)|x^σ+k-1dx and we derive that |f^*(σ+it)|≤C(σ)/|t|^k, for all t∈\{0}.Furthermore, if σ≠ 0,-1,⋯,-(k-1), f^*(s) is continuous on the line (s)=σ.As a result, if k≥ 2, f^*(s)∈ L_1(σ± i∞) and, because f^*(s) is analytic in the strip 0<(s)<α, f(x) is the Mellin inverse transform of f^*(s) in that strip.§ MNTZ-TYPE FORMULAS IN THE CRITICAL STRIPFor functions f with suitable properties (see section 2.11 of <cit.>), the Mntz formulaζ(s)∫_0^∞f(y)y^s-1dy =∫_0^∞(∑_n=1^∞f(nx) -1/x∫_0^∞f(t)dt)x^s-1dxis valid in the critical strip 0<(s)<1 . Here we derive several identities similar to (<ref>) in the _α,k classes.The following theorem generates, for each Dirichlet series exhibited in the end of our introduction, an equality between an integral over a vertical line in the half-plane (s)>1 and a series where appears an arithmetic function (see <cit.>). Suppose that f∈_α,k, k≥ 2, ϕ(n) is an arithmetic function and Φ(s) is defined in the half-plane (s)>1 by the absolutely convergent Dirichlet series ∑_n=1^∞ϕ(n)/n^s.Then, for any 1<σ<α, Φ(s)f^*(s)∈ L_1(σ± i∞),1/2π i∫_σ-i∞^σ+i∞Φ(s)f^*(s)x^-sds=∑_n=1^∞ϕ(n)f(nx)and the Mellin transform of (∑_n=1^∞ϕ(n)f(nx)) in the strip 1<(s)<α is Φ(s)f^*(s).Proof. Fix 1<σ<α. By definition of Φ(s),1/2π i∫_σ-i∞^σ+i∞Φ(s)f^*(s)x^-sds =1/2π i∫_σ-i∞^σ+i∞∑_n=1^∞ϕ(n)/n^sf^*(s)x^-sds. Moreover, the Dirichlet series that defines Φ(s) is absolutely convergent in the half-plane (s)>1 and, by proposition <ref>, f^*(s)∈ L_1(σ± i∞), so Φ(s)f^*(s)∈ L_1(σ± i∞), because∫_σ-i∞^σ+i∞|Φ(s)f^*(s)ds| ≤∫_σ-i∞^σ+i∞∑_n=1^∞|ϕ(n)/n^sf^*(s)ds| ≤∑_n=1^∞|ϕ(n)|/n^σ∫_σ-i∞^σ+i∞|f^*(s)ds|<∞. Next we change the order of summation and integration to obtain1/2π i∫_σ-i∞^σ+i∞Φ(s)f^*(s)x^-sds =∑_n=1^∞ϕ(n)/2π i∫_σ-i∞^σ+i∞f^*(s)(xn)^-sds. Furthermore, again by proposition <ref>,f(x) is equal to the inverse Mellin transform of f^*(s) over the vertical line (s)=σ, so we can deduce (<ref>) from the previous formula. Finally, using proposition <ref>, we deduce that the Mellin transform of (∑_n=1^∞ϕ(n)f(nx)) exists and is equal to Φ(s)f^*(s) in the strip 1<(s)<α.To derive our Mntz-type formulas it will be necessary to move integrals of the type on (<ref>), first to the critical strip and later to half-plane (s)<0. To this purpose, we show the following result similar to the residue theorem for integrals over vertical lines of the complex plane. Suppose that a,b,c,d∈, with c<a<b<d, F(s) is an analytic function in the strip c<(s)<d except at a point s=x_0, with a<x_0<b, and there exists t_0∈^+ and a continuous function g(t) integrable at the infinity such that |F(s)|≤ g((s)), for all s∈ such that a≤(s)≤ b and |(s)|≥ t_0.Then F(s)∈ L_1(a± i∞), F(s)∈ L_1(b± i∞) and, for any x∈^+,_s=x_0(F(s)x^-s) =1/2π i∫_b-i∞^b+i∞F(s)x^-sds -1/2π i∫_a-i∞^a+i∞F(s)x^-sds.Proof. The functions F(a+it) and F(b+it) are continuous and upper bounded at infinity by the integrable function g(t), thus F(s)∈ L_1(a± i∞) and F(s)∈ L_1(b± i∞).Fix x∈^+. For any T∈^+, we define Ω_T as the rectangle (positively oriented) whose sides are segments of the vertical lines (s)=a and (s)=b and the horizontal lines (s)=T and (s)=-T. Then, using the Cauchy residue theorem, we obtain_s=x_0(F(s)x^-s) =1/2π i∫_Ω_TF(s)x^-sds =1/2π i(∫_b-iT^b+iT-∫_a+iT^b+iT -∫_a-iT^a+iT+∫_a-iT^b-iT)F(s)x^-sds. Besides that, as g(t) is continuous and integrable at the infinity, lim_t→∞g(t)=0 so|∫_a± iT^b± iTF(s)x^-sds| ≤∫_a^b|F(u± iT)|x^-udu≤ (b-a)x^-a g(T),ifx≥ 1(b-a)x^-b g(T),ifx≤ 1 0. Finally we pass to the limit T→∞ on the formula for _s=x_0(F(s)x^-s) to deduce (<ref>). §.§ Mntz-type formulas in the critical strip involving ζ^k(s)Here we derive a family of Mntz-type formulas where appears ζ^k(s), k∈, in the critical strip 0<(s)<1.We observe that the case k=1 of these identities is the classical Mntz formula and we determine explicitly the case k=2, deducing a Mntz-type formula involving ζ^2(s).Fix k∈ and suppose that f∈_α,m, α>1 and m∈_0.Then ζ^k(s)f^*(s) is analytic in the strip 0<(s)<α, except at the point s=1 where it has a pole of order at most k (or a removable singularity).Moreover, if m≥ 2, using (<ref>) and theorem <ref>, we claim that, for all 1<σ<α, ζ^k(s)f^*(s)∈ L_1(σ± i∞) and, if x∈^+,1/2π i∫_σ-i∞^σ+i∞ζ^k(s)f^*(s)x^-sds =∑_n=1^∞d_k(n)f(nx). The following proposition establishes sufficient conditions to move the integral in (<ref>) to the left, using theorem <ref>. Let k∈ and f∈_α,m, m≥ 1+k/2.Then, if we fix t_0,u_1,u_2∈ such that t_0>1 and 1/2-m-1/k<u_1<u_2<α, there exists a continuous function g_k(t) integrable at the infinity such that |ζ^k(s)f^*(s)|≤ g_k((s)), if u_1≤(s)≤ u_2 and |(s)|≥ t_0. Proof. Fix t_0,u_1,u_2∈ such that t_0>1 and1/2-m-1/k<u_1<u_2<α. For any u_1≤ u≤ u_2,∫_0^∞|f^(m)(x)|x^u+m-1dx ≤∫_0^1|f^(m)(x)|x^u_1+m-1dx +∫_1^∞|f^(m)(x)|x^u_2+m-1dx=:C,so, remembering (<ref>), we obtain, for all t>0,|f^*(u± it)| ≤1/t^m∫_0^∞|f^(n)(x)|x^u+m-1dx ≤C/t^m. By (<ref>), there exists M∈^+ such that, for all t≥ t_0: |ζ(u± it)|≤ Mln(t),if u≥ 1; |ζ(u± it)|≤ M t^1-u/2ln(t), if 0≤ u≤ 1;and |ζ(u± it)|≤ Mt^1/2-uln(t), if u≤ 0.Therefore, for any t≥ t_0 and u_1≤ u≤ u_2, |ζ^k(u± it)f^*(u± it)|≤ g_k(t)=M^kC(ln(t))^k t^p(u_1), where p(u_1)=-m, if 1≤ u_1<α; p(u_1)=-m+k/2(1-u_1),if 0≤ u_1≤ 1;and p(u_1)=-m+k(1/2-u_1),if 1/2-m-1/k<u_1≤ 0. Finally, p(u_1)<-1, for any of the possible values for u_1, so g_k(t) is a (continuous) function integrable at the infinity.Suppose that f∈_α,m, m≥ 1+k/2. Fix 0<c_0<1 and 1<σ<α. Then theorem <ref> and proposition <ref> can be used to deduce that ζ^k(s)f^*(s)∈ L_1(c_0± i∞) and1/2π i∫_c_0-i∞^c_0+i∞ζ^k(s)f^*(s)x^-sds =1/2π i∫_σ-i∞^σ+i∞ζ^k(s)f^*(s)x^-sds -_s=1(ζ^k(s)f^*(s)x^-s). Besides that, _s=1(ζ^k(s)f^*(s)x^-s) =∫_0^∞f(xy)P_k-1(ln(y))dy,where P_k-1(x) is a certain monic polynomial of degree k-1 (see <cit.>).Therefore, remembering (<ref>), we obtain1/2π i∫_c_0-i∞^c_0+i∞ζ^k(s)f^*(s)x^-sds =∑_n=1^∞d_k(n)f(nx) -∫_0^∞f(xy)P_k-1(ln(y))dy. Finally, applying proposition <ref> to (<ref>), we derive the following theorem.Let k∈ and f∈_α,m, m≥ 1+k/2. Then the Mntz-type formulaζ^k(s)f^*(s) =∫_0^∞(∑_n=1^∞d_k(n)f(nx) -∫_0^∞f(xy)P_k-1(ln(y))dy)x^s-1dxis valid in the critical strip 0<(s)<1.If k=1, P_0(x)=1 so_s=1(ζ(s)f^*(s)x^-s) =∫_0^∞f(xy)dy =1/x∫_0^∞f(t)dt =f^*(1)/x. Therefore, because d_1(n)=1, for all n∈, we deduce from the previous theorem that the Mntz formula (<ref>) is valid in the critical strip, for any function f∈_α,m, m≥ 2.If k=2, P_1(x)=x+2γ (where γ is the Euler-Mascheroni constant <cit.>) so_s=1(ζ^2(s)f^*(s)x^-s) =∫_0^∞f(xy)P_1(ln(y))dy =2γ∫_0^∞f(xy)dy+∫_0^∞f(xy)ln(y)dy. Besides that, making a change of variable t=xy,∫_0^∞f(xy)ln(y)dy =1/x∫_0^∞f(t)ln(t)dt -ln(x)/x∫_0^∞f(t)dt =(f^*)'(1)/x-f^*(1)ln(x)/x. As a result,_s=1(ζ^2(s)f^*(s)x^-s) =∫_0^∞f(xy)P_1(ln(y))dy =1/x (((f^*)'(1)+2γ f^*(1))-f^*(1)ln(x))and, as d_2(n)=d(n), for all n∈, we derive the Mntz-type formulaζ^2(s)f^*(s) =∫_0^∞(∑_n=1^∞d(n)f(nx)+1/x (f^*(1)ln(x)-((f^*)'(1)+2γ f^*(1))))x^s-1dxvalid in the critical strip 0<(s)<1, for any function f∈_α,m, m≥ 2.§.§ Mntz-type formulas in the critical strip involving ζ^k(s)/ζ(2s)In this section we derive Mntz-type formulas where appears ζ^k(s)/ζ(2s), k=1,2,3,4.Let k∈ and f∈_α,m. Then, because 1/ζ(2s) is analytic in the half-plane (s)>1/2, ζ^k(s)/ζ(2s)f^*(s)is an analytic function in the strip 1/2<(s)<α except at the point s=1, where it has a pole of order at most k (or a removable singularity).Moreover, if m≥ 2, remembering formulas (<ref>)-(<ref>) and theorem <ref>,we deduce that, for all 1<σ<α, ζ^k(s)/ζ(2s)f^*(s)∈ L_1(σ± i∞) and, for any x∈^+,1/2π i∫_σ-i∞^σ+i∞ζ(s)/ζ(2s)f^*(s)x^-sds =∑_n=1^∞|μ(n)|f(nx);1/2π i∫_σ-i∞^σ+i∞ζ^2(s)/ζ(2s)f^*(s)x^-sds =∑_n=1^∞2^ω(n)f(nx);1/2π i∫_σ-i∞^σ+i∞ζ^3(s)/ζ(2s)f^*(s)x^-sds =∑_n=1^∞d(n^2)f(nx);1/2π i∫_σ-i∞^σ+i∞ζ^4(s)/ζ(2s)f^*(s)x^-sds =∑_n=1^∞(d(n))^2f(nx). Let u_1∈ and s∈ such that u:=(s)≥ u_1>1/2.Then, using (<ref>) and (<ref>), |1/ζ(2s)| ≤∑_n=1^∞|μ(n)/n^2s| ≤∑_n=1^∞1/n^2u =ζ(2u)≤ζ(2u_1). Therefore |ζ^k(s)/ζ(2s)f^*(s)| ≤ζ(2u_1)|ζ^k(s)f^*(s)| and we may derive the following result from proposition <ref>, defining g̃_k(t):=ζ(2u_1)g_k(t), where g_k(t) is the function obtained in proposition <ref>. Let k∈ and f∈_α,m, m≥ 1+k/2. Then, if we fix t_0,u_1,u_2∈ such that t_0>1 and 1/2<u_1<u_2<α, there exists a continuous function g̃_k(t) integrable at the infinity such that |ζ^k(s)/ζ(2s)f^*(s)|≤g̃_k((s)), if u_1≤(s)≤ u_2 and |(s)|≥ t_0.Suppose that f∈_α,m, m≥ 1+k/2. Fix 1/2<c_0<1 and 1<σ<α. Then theorem <ref> and proposition <ref> can be used to deduce that ζ^k(s)/ζ(2s)f^*(s)∈ L_1(c_0± i∞) and1/2π i∫_c_0-i∞^c_0+i∞ζ^k(s)/ζ(2s)f^*(s)x^-sds =1/2π i∫_σ-i∞^σ+i∞ζ^k(s)/ζ(2s)f^*(s)x^-sds -_s=1(ζ^k(s)/ζ(2s)f^*(s)x^-s). Next we calculate this residue for k=1 and k=2.If k=1, ζ(s)/ζ(2s)f^*(s)x^-s has a simple pole (or a removable singularity) at the point s=1 and then, remembering (<ref>) and ζ(2)=π^2/6,_s=1(ζ(s)/ζ(2s)f^*(s)x^-s) =1/ζ(2)_s=1(ζ(s)f^*(s)x^-s) =6f^*(1)/π^2 x. If k=2, ζ^2(s)/ζ(2s)f^*(s)x^-s has (at most) a double pole at the point s=1, so_s=1(ζ^2(s)/ζ(2s)f^*(s)x^-s) =lim_s→ 1d/ds((s-1)^2ζ^2(s)/ζ(2s)f^*(s)x^-s)=1/ζ(2)_s=1(ζ^2(s)f^*(s)x^-s) +1/x f^*(1)(lim_s→ 1((s-1)ζ(s)))^2 .d/ds(1/ζ(2s))|_s=1. Besides that ζ'(2)=π^2/6(γ+ln(2π/A^12)), where A is the Glaisher-Kinkelin constant, so.d/ds(1/ζ(2s))|_s=1 =-2ζ'(2)/ζ^2(2) =12/π^2(ln(A^12/2π)-γ). Therefore, remembering (<ref>), we obtain_s=1(ζ^2(s)/ζ(2s)f^*(s)x^-s) =6/π^2x((f^*)'(1)+f^*(1)ln(A^24/4π^2 x)). Finally, the following theorem can be derived from (<ref>), (<ref>)-(<ref>), (<ref>)-(<ref>) and proposition <ref>. Let f∈_α,m, m≥ 2.Then the Mntz-type formulasζ(s)/ζ(2s)f^*(s) =∫_0^∞(∑_n=1^∞|μ(n)|f(nx) -6f^*(1)/π^2 x)x^s-1dx,and ζ^2(s)/ζ(2s)f^*(s) =∫_0^∞(∑_n=1^∞2^ω(n)f(nx) +6/π^2x(f^*(1)ln(4π^2x/A^24) -(f^*)'(1)))x^s-1dxare valid in the strip 1/2<(s)<1. Analogously, using (<ref>), (<ref>) and (<ref>), one may deduce the following theorem (but we will not calculate the residues appearing here). Let f∈_α,m, m≥ 3.Then the Mntz-type formulasζ^3(s)/ζ(2s)f^*(s) =∫_0^∞(∑_n=1^∞d(n^2)f(nx) -_s=1(ζ^3(s)/ζ(2s)f^*(s)x^-s) )x^s-1dxandζ^4(s)/ζ(2s)f^*(s) =∫_0^∞(∑_n=1^∞d^2(n)f(nx) -_s=1(ζ^4(s)/ζ(2s)f^*(s)x^-s) )x^s-1dxare valid in the strip 1/2<(s)<1. This section ends with a remark about how the Riemann hypothesis may affect the strip of validity of the formulas exhibited above.If the Riemann hypothesis holds true, the formulas given by theorems <ref> and <ref> are not only valid in the strip 1/2<(s)<1 but in the entire strip 1/4<(s)<1.§ MNTZ-TYPE FORMULAS IN THE HALF-PLANE (S)<0In the previous section we moved some integrals of the type on (<ref>) to the critical strip 0<(s)<1.Now we move the integrals of ζ^k(s)f^*(s)x^-s (k∈) to the half-plane (s)<0 and we derive some Mntz-type formulas in that half-plane.Suppose that f∈_α,m, α>1 and m∈. By theorem <ref>, f^*(s) is analytic in the strip -m<(s)<α except at the points s=-j, j=0,1,⋯,m-1, where f^*(s) either has a simple pole, if f^(j)(0)≠ 0, or a removable singularity, if f^(j)(0)=0. Besides that, ζ(s) is analytic in the entire complex plane except at the point s=1 and ζ(-2n)=0, for all j∈, so these zeros cancel the eventual simple poles of f^*(s) at the points s=-2n, 1<2n<m. Therefore, for any fixed k∈, ζ^k(s)f^*(s)x^-s is an analytic function in the strip -m<(s)<1, except at the points s=-j, with j=0 or j=2n-1<m (n∈), where ζ^k(s)f^*(s)x^-s either has a simple pole, if f^(j)(0)≠ 0, or a removable singularity, if f^(j)(0)=0. Moreover, we can use theorem <ref> to calculate the residues of ζ^k(s)f^*(s)x^-s at these possible singularities, because_s=-j(ζ^k(s)f^*(s)x^-s) =ζ^k(-j) x^j_s=-jf^*(s) =ζ^k(-j)/j! f^(j)(0) x^j,for all j∈_0 such that j<m. In particular, if j=0, then, because ζ(0)=-1/2,_s=0(ζ^k(s)f^*(s)x^-s) =(-1)^k/2^k f(0). Now we move the integral of ζ^k(s)f^*(s)x^-s to the strip -1<(s)<0 to obtain a Mntz-type formula in that strip. Let k∈ and f∈_α,m, m≥ 1+3k/2. Fix -1<σ_0<0 and 0<c_0<1. Using proposition <ref> andtheorem <ref>, we claim that ζ^k(s)f^*(s)∈ L_1(σ_0± i∞) and1/2π i∫_σ_0-i∞^σ_0+i∞ζ^k(s)f^*(s)x^-sds =1/2π i∫_c_0-i∞^c_0+i∞ζ^k(s)f^*(s)x^-sds -_s=0(ζ^k(s)f^*(s)x^-s). Then, remembering (<ref>) and (<ref>), we deduce that1/2π i∫_σ_0-i∞^σ_0+i∞ζ^k(s)f^*(s)x^-sds =∑_n=1^∞d_k(n)f(nx) -∫_0^∞f(xy)P_k-1(ln(y))dy +(-1)^k+1/2^k f(0). Therefore, the following theorem can be obtained applying proposition <ref> to (<ref>). Let k∈ and f∈_α,m, m≥ 1+3k/2. Then the Mntz-type formulaζ^k(s)f^*(s) =∫_0^∞(∑_n=1^∞d_k(n)f(nx) -∫_0^∞f(xy)P_k-1(ln(y))dy +(-1)^k+1/2^k f(0))x^s-1dxis valid in the strip -1<(s)<0. Replacing k=1 and k=2 on the previous theorem and remembering (<ref>) and (<ref>), we derive the Mntz-type formulasζ(s)f^*(s)=∫_0^∞(∑_n=1^∞f(nx)-f^*(1)/x +f(0)/2)x^s-1dxand ζ^2(s)f^*(s)=∫_0^∞(∑_n=1^∞d(n)f(nx) +1/x (f^*(1)ln(x)-((f^*)'(1)+2γ f^*(1))) -f(0)/4)x^s-1dxvalid in the strip -1<(s)<0, for any function f∈_α,m, where m≥ 3 and m≥ 4, respectively.Next we move the integral of ζ^k(s)f^*(s)x^-s to the half-plane (s)<-1 to deduce Mntz-type formulas in strips of that half-plane.Let k,m∈ and f∈_α,l, l≥ 1+k(2m+3/2). Fix σ_n, n=0,1,⋯,m, such that -1<σ_0<0 and -2n-1<σ_n<-2n+1, for all n=1,2,⋯,m. Using proposition <ref> and theorem <ref>, we claim that, for any n=1,2,⋯,m, ζ^k(s)f^*(s)∈ L_1(σ_n± i∞) and1/2π i∫_σ_n-i∞^σ_n+i∞ζ^k(s)f^*(s)x^-sds =1/2π i∫_σ_n-1-i∞^σ_n-1+i∞ζ^k(s)f^*(s)x^-sds -_s=1-2n(ζ^k(s)f^*(s)x^-s). Then, remembering formulas (<ref>) and (<ref>), we may deduce that1/2π i∫_σ_m-i∞^σ_m+i∞ζ^k(s)f^*(s)x^-sds=(P_k,mf)(x),where (P_k,mf)(x) (x∈^+) is defined as∑_n=1^∞d_k(n)f(nx) -∫_0^∞f(xy)P_k-1(ln(y))dy +(-1)^k+1f(0)/2^k -∑_n=1^mζ^k(1-2n)/(2n-1)! f^(2n-1)(0) x^2n-1.Finally, the following theorem can be obtained applying proposition <ref> to (<ref>). Let k,m∈ and f∈_α,l, l≥ 1+k(2m+3/2). Then, if we define (P_k,mf)(x) as above, the Mntz-type formulaζ^k(s)f^*(s)=∫_0^∞(P_k,mf)(x)x^s-1dxis valid in the strip -2m-1<(s)<-2m+1. Fixing m∈, replacing k=1 and k=2 on the previous theorem and remembering (<ref>) and (<ref>), we derive the Mntz-type formulasζ(s)f^*(s)=∫_0^∞(∑_n=1^∞f(nx)-f^*(1)/x+f(0)/2 -∑_n=1^mζ(1-2n)/(2n-1)! f^(2n-1)(0) x^2n-1)x^s-1dxandζ^2(s)f^*(s)=∫_0^∞(P_2,mf)(x)x^s-1dx,where (P_2,mf)(x) (x∈^+) is defined as equal to∑_n=1^∞d(n)f(nx) +1/x (f^*(1)ln(x)-((f^*)'(1)+2γ f^*(1))) -f(0)/4 -∑_n=1^mζ^2(1-2n)/(2n-1)! f^(2n-1)(0) x^2n-1,valid in the strip -2m-1<(s)<-2m+1, for any function f∈_α,l, where l≥ 2m+3 and l≥ 4m+4, respectively.This section ends with a remark about the relation between the Mntz-type formulas in the strip -1<(s)<0 and some classical summation formulas.The classical Poisson and Voronoi summation formulas (see <cit.> and <cit.>, respectively) can be derived in the _α,m classes, replacing k=1 and k=2, respectively, on (<ref>) and using the functional equation of the Riemann zeta function <cit.> to calculate 1/2π i∫_σ_0-i∞^σ_0+i∞ζ^k(s)f^*(s)x^-sds, -1<σ_0<0. § IDENTITIES INVOLVING THE GAMMA AND ZETA FUNCTIONSPerhaps the most important example of a function belonging to the generalized Mntz-type classes of functions is f(x)=e^-x∈_α,k, for all α>1 and k∈_0. Its Mellin transform is the gamma function Γ(s). In this section we replace f(x)=e^-x and f^*(s)=Γ(s) in the previously derived formulas to obtain integral representations in vertical strips of the complex plane for products of the gamma and zeta functions.Observe that, if f(x)=e^-x and f^*(s)=Γ(s), then f^*(1)=Γ(1)=1 and∑_n=1^∞f(nx) =∑_n=1^∞e^-nx=1/e^x-1. Therefore we can derive from the Mntz formula the following integral representation for ζ(s)Γ(s) in the critical strip 0<(s)<1, which may be found in section 2.7 of <cit.>,ζ(s)Γ(s)=∫_0^∞(1/e^x-1-1/x)x^s-1dx. Similarly, we may deduce more formulas relating the gamma and zeta functions. For instance, making f(x)=e^-x and f^*(s)=Γ(s), we have f^*(1)=1 and (f^*)'(1)=Γ'(1)=-γ, so f^*(1) ln(x)/x-(2γ f^*(1)+(f^*)'(1))1/x=ln(x)/x-γ/x.As a result, we obtain from (<ref>) the integral representationζ^2(s)Γ(s) =∫_0^∞x^s-1(∑_n=1^∞d(n)e^-nx +ln(x)/x-γ/x)dxvalid in the crtical strip 0<(s)<1.Analogously, we derive from theorem <ref> the integral representationsζ(s)/ζ(2s)Γ(s) =∫_0^∞x^s-1(∑_n=1^∞|μ(n)|e^-nx -6/π^2 x)dxandζ^2(s)/ζ(2s)Γ(s) =∫_0^∞x^s-1(∑_n=1^∞2^ω(n)e^-nx +6/π^2x(ln(4π^2x/A^24) +γ))dxvalid in the strip 1/2<(s)<1 (or in the strip 1/4<(s)<1, if the Riemann hypothesis holds true).Moreover, if f(x)=e^-x then f(0)=1, so we deduce from (<ref>) and (<ref>) the integral representationsζ(s)Γ(s)=∫_0^∞(1/e^x-1-1/x+1/2)x^s-1dxandζ^2(s)Γ(s) =∫_0^∞(∑_n=1^∞d(n)e^-nx +ln(x)/x-γ/x-1/4)x^s-1dxvalid in the strip -1<(s)<0.Finally, if f(x)=e^-x then, for any j∈_0, f^(j)(x)=(-1)^je^-x so f^(j)(0)=(-1)^j. In particular, f^(2n-1)(0)=-1, for all n∈. Therefore, for any fixed m∈, we obtain from (<ref>) and (<ref>) the integral representationsζ(s)Γ(s)=∫_0^∞(1/e^x-1-1/x+1/2+∑_n=1^mζ(1-2n)/(2n-1)! x^2n-1)x^s-1dxandζ^2(s)Γ(s) =∫_0^∞(∑_n=1^∞d(n)e^-nx +ln(x)/x-γ/x-1/4+∑_n=1^mζ^2(1-2n)/(2n-1)! x^2n-1)x^s-1dxvalid in the strip -2m-1<(s)<-2m+1. §.§ AcknowledgementsThe author is deeply grateful to Semyon Yakubovich for fruitful discussions of these topics and his useful suggestions, which rather improved the presentation of this paper.plain | http://arxiv.org/abs/1705.09386v1 | {
"authors": [
"Hélder Lima"
],
"categories": [
"math.CA",
"11M06, 11M26, 33B15, 42A38, 42B10, 44A05, 44A20"
],
"primary_category": "math.CA",
"published": "20170525225910",
"title": "On Müntz-type formulas related to the Riemann zeta function"
} |
PVEs: Position-Velocity Encoders for Unsupervised Learning of Structured State Representations Rico Jonschkowski1,2, Roland Hafner1, Jonathan Scholz1, and Martin Riedmiller11DeepMind, 2Robotics and Biology Laboratory at Technische Universität Berlin December 30, 2023 ==============================================================================================================================================================We propose position-velocity encoders (PVEs) which learn—without supervision—to encode images to positions and velocities of task-relevant objects. PVEs encode a single image into a low-dimensional position state and compute the velocity state from finite differences in position. In contrast to autoencoders, position-velocity encoders are not trained by image reconstruction, but by making the position-velocity representation consistent with priors about interacting with the physical world. We applied PVEs to several simulated control tasks from pixels and achieved promising preliminary results.§ INTRODUCTIONWhile position and velocity are fundamental components of state representations in robotics, robots cannot directly sense these properties. Instead, they need to extract task-relevant position and velocity information from sensory input. For robots to be versatile, they must be able to learn such representations from experience. Deep learning allows to learn position-velocity representations in principle—but most existing approaches depend crucially on labelled data. In this paper, we investigate how robots could learn position-velocity representations without supervision. We approach this problem by using robotics-specific prior knowledge about interacting with the physical world, also known as robotic priors <cit.>, in order to learn an encoding from high-dimensional sensory observations to a low dimensional state representation. Our contribution is to split the state representation into a velocity state and a position state and to incorporate robotic priors about position and velocity in the form of model constraints and learning objectives. Our method, the position-velocity encoder (PVE), implements a hard model constraint by estimating velocity states from finite differences in position states. This constraint fixes the relation between these two parts of the state representation. Additionally, PVEs include soft objectives that measure consistency with robotic priors. These objectives are optimized during learning and shape which information is encoded and how multiple state samples relate to each other. Both ingredients work together to learn an encoding into a structured state representation that includes position states, which describe information from a single observation, and velocity states, which describe how this information changes over time.Figure <ref> shows the position encoder that maps observations (blue rectangles) to position states (blue dots). The velocity state—the time derivative of the position state—is approximated from finite differences in position. This structured state space allows us to formulate new robotic priors, specifically for positions and velocities, in the form of learning objectives.PVEs learn to encode observations into states without state labels and without learning a decoder. Instead, they learn the encoding by making position states and their derivatives consistent with different robotic priors. Inconsistency with each prior is measured in a loss function. PVEs learn by minimizing a weighted sum of these losses using gradient descent. The gradients can be imagined as forces in the state space that pull state samples together (when they should be similar) or push them apart (when they should be different). Backpropagation transforms these forces into parameter changes in the encoder (see pink and purple arrows in Fig. <ref>).In our preliminary experiments, we apply position-velocity encoders to simulated control tasks from pixels. We show that PVEs are able to discover the topology and the dimensionality of the task, that they can learn equivalent representations from different camera perspectives, that they capture information about the true positions and velocities of physical objects in the scene, and that reinforcement learning based on the learned position-velocity state can produce precise control. § RELATED WORK This paper extends work on learning state representations with robotic priors by Jonschkowski and Brock <cit.>, which introduced the idea of robotic priors and their implementation in the form of objectives for representation learning. Our extension to position-velocity states is inspired by work on physics-based priors in model-based reinforcement learning by Scholz et al. <cit.>, which proposed to learn a physically plausible dynamics model given a position-velocity representation. Here, we turn their approach around and ask: How could we learn a position-velocity representation from sensory input without specifying which positions and velocities are relevant for the task?The answer we are proposing, the position-velocity encoder, works by incorporating prior knowledge in two different ways, which fit Mitchell's categorization of inductive biases into restriction biases and preference biases <cit.>. Restriction biases restrict the hypothesis space that is considered during learning, such as our constraint in the PVE model to estimate velocities from finite differences in position (rather than learning to extract velocity information from a sequence of observations). Preference biases express preferences for certain hypothesis, such as our loss functions for training PVEs, which measure inconsistency with robotic priors.Other examples of restriction biases in the visual representation learning literature include convolutional networks <cit.>, spatial transformer networks <cit.>, and spatial softmax <cit.>, which incorporate priors about visual input as architectural constraints of a neural network, as well as backprop Kalman Filters <cit.> and end-to-end learnable histogram filters <cit.>, which incorporate the structure of the Bayes' filter algorithm for recursive state estimation. SE3-nets <cit.> implement assumptions about rigid body transformations. While these approaches can regularize learning, unsupervised learning (also) requires suitable preference biases.Preference biases for internal representations are commonly expressed indirectly via other learnable functions based on the representation. For example, others train representations by using them to learn image reconstruction <cit.>, prediction of future states <cit.>, or other auxiliary tasks <cit.>. A powerful but underexplored alternative is to express preference biases directly on the learned representation, an approach which can enable to learn symbol grounding <cit.> or perform label-free supervised learning <cit.>. Direct preference biases are also the focus of metric learning, e.g. learning representations of faces using the fairly general triplet loss <cit.>, but there is large potential for formulating more informative robotic priors in the form of direct preference biases, as we do in this work.§ POSITION-VELOCITY ENCODERS Position-Velocity Encoders (PVEs) learn to map raw observations into a structured state space that consists of a position part and a velocity part. PVEs are trained without state labels and they do not need to learn auxiliary functions such as reconstructing observations or predicting state transitions. PVEs achieve this by combining two key ideas: * PVEs encode the current observation into a position state and estimate a velocity state from finite differences in position (more details in Sec. <ref>).* PVEs are trained by optimizing consistency with robotic priors about positions, velocities, and accelerations (more details in Secs. <ref> & <ref>). §.§ ModelThe PVE model consists of a convolutional network and a numerical velocity estimation. The convolutional network ϕ encodes a visual observation o_t into a low-dimensional position-state s^(p)_t, where superscript (p) stands for position.s^(p)_t= ϕ( o_t).From the difference of the last two position states s^(p)_t and s^(p)_t-1, the model estimates the velocity state s^(v)_t:s^(v)_t= α( s^(p)_t -s^(p)_t-1),where α is ahyperparameter that subsumes 1/timestep and scales velocity states. It is important that velocity states have the right scale relative to position states in order to create a sensible metric in the combined state s_t, which we construct by stacking the position state and the velocity state.s_t= [ s_t^(p); s_t^(v) ].We can also use finite differences to estimate acceleration (or jerk, jounce, etc.). We do not include these derivatives in the state because we assume that the robot controls accelerations by its actions. But we do use the acceleration state in some loss functions. We compute the acceleration state s^(a)_t in the same way as the velocity state but we omit the scaling since we will not use accelerations in the combined state space:s^(a)_t=s^(v)_t -s^(v)_t-1.§.§ Robotic Priors and Learning ObjectivesThe encoder ϕ is trained by making the combined state space consistent with a set of robotic priors, which we will describe in this section. These priors use the structured state space and are specific to positions, velocities, and accelerations. Consistency with these priors is defined in the form of loss functions that are minimized during learning. The following list of robotic priors should be understood as an exploration into this matter, not as a final answer.§.§.§ VariationPositions of relevant things vary. As the robot explores its task and manipulates its environment, the positions of task-relevant objects (including itself) will vary—otherwise there is not much that the robot could learn. If we assume that positions of relevant objects vary in the robot's experience, the internal representation of such positions must also vary; random pairs of position states should not be similar. Therefore, we optimize consistency with the variation prior by minimizing the expected similarity between random pairs of position states,L_variation = [e^-‖ s_a^(p) -s_b^(p)‖],where we use e^-distance as a similarity measure that is 1 if the distance is 0 and that goes to 0 with increasing distance between the position states, which is exactly what we want.§.§.§ SlownessPositions change slowly <cit.>. Physical objects do not teleport; they do not change their position arbitrarily from one second to the next. To make the internal position state consistent with the slowness prior, we minimize the expected squared distance between consecutive position states,L_slowness = [‖ s_t^(p) -s_t-1^(p)‖^2].Since this change in position is directly connected to the rate of position change (or velocity), we can also write down the same loss using the velocity state.L_slowness = [‖ s_t^(v)/α‖^2],where α is the scaling hyperparameter defined earlier. This reformulation hints at a different interpretation of slowness, which is simply: velocities are low[Note that defining the slowness prior to mean velocities are low translates to the loss function L_slowness = [( s_t^(v))^2] = [(α( s^(p)_t -s^(p)_t-1))^2], which depends on the scaling parameter α. We use the other formulation to make this loss independent of α because we want to change α during training without affecting this loss (see Sec <ref> for more details).].§.§.§ InertiaVelocities change slowly. Since physical objects have inertia, they resist changes to their velocity (both in direction or magnitude). If we assume limited forces to overcome this resistance, velocities should only change by small amounts. Note how the inertia prior corresponds to the slowness prior applied to velocities. L_inertia = [‖ s_t^(v) -s_t-1^(v)‖^2] = [‖ s_t^(a)‖^2].This formulation of the inertia prior focuses on large velocity changes due to the square in the loss function. Alternatively, we can define the loss function based on absolute changes.L_inertia (abs) = [‖ s_t^(a)‖].Small changes in velocity have a higher weight in the second loss compared to the first loss. We found that combining both losses leads to better results than using either one of them.§.§.§ ConservationVelocity magnitudes change slowly. This prior derives from the law of conservation of energy, which states that the total energy in a closed system remains constant. As the robot applies forces to the environment, we do not have a closed system. Additionally, we cannot estimate, e.g. kinetic energy without knowing the masses of objects, let alone potential energy stored in springs etc. Still, we want to enforce the same idea of keeping the absolute amount of energy, or in our case "movement" similar in consecutive time steps.L_conservation = [(‖ s_t^(v)‖ - ‖ s_t-1^(v)‖)^2]. §.§.§ ControlabilityControllable things are relevant. The objects that can be controlled by the robot are likely relevant for its task. If the robot acts by applying forces, controllable things could be those whose accelerations correlate with the actions of the robot. Accordingly, we can define a loss function per action dimension i to optimize covariance between action dimension i and accelerations in a state dimension i.L_controlability (i) = e^-Cov( a_t,i,s_t+1,i^(a))= e^-[(a_t,i - [a_t,i])(s_t+1,i^(a) - [s_t+1,i^(a)])].Note that we used this loss in only one of the tasks—ball in cup—because the above priors were insufficient. The results for this task are still preliminary. A complete solution of this task and a deeper investigation into other formulations of controlability are part of future work.§.§ Training Procedure We train PVEs by minimizing a weighted sum of the loss functions described above using gradient descent. This section explains the training procedure in detail.§.§.§ Data Gathering First, the robot gathers data by exploring its environment. Since we are using a reinforcement learning setting, the data consist of sequences of observations, actions, and rewards. Most of the presented loss functions only use observations, the controlability loss also uses actions, but none of our current losses uses the reward signal.§.§.§ Loss ComputationWe iterate through the collected data in mini batches, which consist of a small set of short sequences. For each mini-batch, we compute the loss functions by replacing expectations with statistical averages.[For the variation loss, we sample all pairs of experiences with the same time step in different sequences of the mini batch. For all other losses we consider all samples in the mini batch.]§.§.§ Loss CombinationWe combine these losses in a weighted sum. Finding the right weights is important because they balance how much each prior is enforced during learning. We determined these weights empirically by adjusting them until the gradients in the encoder parameters had similar magnitudes for all priors. Future work should try to automate this process of weight tuning, potentially by applying the same heuristic in an automated fashion.§.§.§ Parameter UpdatesFor each mini-batch, we compute the gradient[Some of the gradients can only be computed after adding small Gaussian noise to the encoded states.] of the combined loss with respect to the encoder parameters using symbolic auto-differentiation <cit.> and perform an update using the Adam optimizer <cit.>. We iterate this process until convergence.§.§.§ Velocity Scaling CurriculumWhile training PVEs, we follow a curriculum that in the beginning focuses on positions and only later also takes velocities into account. This curriculum is implemented by changing the velocity scaling parameter α. In the first phase, we train with α=0 until convergence. In the second phase, we increase α linearly from 0 to its final value and train until convergence again. In phase one, only the first two priors, variation and slowness, are active. Surprisingly, these two are powerful antagonists that can unfold the topology of the position-state space. The second phase mainly smooths the state space such that velocities can be accurately estimated from finite differences.§.§.§ HyperparametersWe used the following hyperparemeters in our experiments. The convolutional network had three convolutional layers with 16, 32, and 64 channels, kernel size 5x5, and stride 2, followed by three fully connected layers of sizes 128, 128, and 5 (for a 5-dimensional position state). Every layer except the last one was followed by a ReLu nonlinearity <cit.>. The mini-batch size was 32 sequences of 10 steps each. The maximum velocity scaling α was 10. The weights for the different losses are shown in Table <ref>.§ EXPERIMENTS AND RESULTS We applied PVEs to a series of simulated control tasks from raw pixel input (see Fig. <ref>). All tasks use the MuJoCo simulator <cit.>. For each task, we collected a batch of training data that consists of 1000 short trajectories of 20 steps by randomly sampling start configurations with different positions and velocities and applying a random policy. §.§ Tasks §.§.§ Inverted PendulumThe inverted pendulum is a stick that is fixed to a rotary joint at one end. The goal is to swing it up and balance it upright by applying forces at the joint. However, the motor is not strong enough to pull the pendulum up all at once. Instead the pendulum must be swung back and forth to generate enough (but not too much) momentum.§.§.§ Cart-PoleThe cart-pole task is an extension of the inverted pendulum task. Since the pole is attached to a cart with a passive joint, it can only be swung up by accelerating the car correctly, which requires precise control.§.§.§ Ball in CupThis task includes a cup and a ball attached to the bottom of the cup with a string. The goal is to move the cup in such a way that the ball lands in the cup. In our version of the task, cup and ball can only move in the plane. §.§ Learned Position-Velocity RepresentationsFor each task, we will now look at the learned state representations. We visualize 5-dimensional position-states by projecting to their principal components.§.§.§ Inverted PendulumThe state representation learned by the PVE is shown in Figure <ref>, where we can see the encoding of test observations into the position-state space. Each dot is the position encoding of a single image. The color denotes the amount of reward that was achieved in that instance.The plot shows a number of interesting results. First, observations that correspond to similar rewards are encoded close together in position space. Second, the position states form a circle, which makes sense because the inverted pendulum moves in a circle. Third, all principal components after the first two are close to zero. This means that the circular encoding lies on a plane in the five-dimensional space—the PVE discovered that the task is two dimensional[Even though the task only has one positional degree of freedom (the angle of the pendulum), we need at least two dimensions if we want a Euclidean metric to make sense in this space, such that there are no jumps as from 360 to 0 degrees in an angular representation.].Next, we will look at the estimated velocities in the learned space. In Figure <ref>, we overlayed encoded training data colored by reward with the encoding of a single sequence of observations shown in Figure <ref>. The position states are marked with black dots and the velocity state vectors are drawn as lines. In the observation sequence, the pendulum swings from the left side to the top and then to the right side. Similarly, the encoded positions move from a medium reward region via the high-reward region (red color) to the medium reward region on the other side. During this motion, the velocity estimations are tangential to the circle in the position space with minimal noise, which should be useful for controlling the pendulum (see video links in Fig. <ref>). §.§.§ Cart-Pole Here, we compare PVEs on two different observations: 1) using a moving camera that follows the cart by rotating sideways, 2) using a static camera that covers the entire region in which the cart moves. Figure <ref> shows the learned position representations for both perspectives.This experiment demonstrates how PVEs can learn equivalent internal representations (compare Figs. <ref> and <ref>) from observations that look very different (Figs. <ref>, <ref>). For both kinds of observations, the state samples form a tube, the length of which corresponds to the position of the cart, while the circular part represents the position of the pole. Here, the PVE uses three of the five dimensions and thereby discovers the three-dimensional nature of the given task.The observation sequence from the moving camera (Fig. <ref>) shows the cart moving to the left while the pole falls down on the right side. The PVE represents this trajectory (Fig. <ref>) by moving from the high-reward red region to the blue region, which reflects the movement of the pole, and to the right side, which corresponds to sideways movement of the cart. The observation sequence from the static camera (Fig. <ref>) shows the pole swinging up while the cart moves to the right. Accordingly, the encoded trajectory (Fig. <ref>) goes to the red region and to the right side (right and left a swapped between these two representations).§.§.§ Ball in Cup The results for this task are preliminary. The task is challenging due to the movement of the cup, which is inconsistent with some of our robotic priors. The cup is confined to a small region and controlled by the robot allowing rapid movements and changes of direction. The cup can be moved from one end of its position range to the other end in a few time steps. Therefore, the slowness prior does not hold here (unless we sampled observations at a higher frequency). Additionally, the robot can apply large forces on the cup, leading to large accelerations and jerky movements, which are again inconsistent with many of our priors on changes in velocity. As a result, PVEs struggle with encoding the cup, which we will quantify in the following section.To approach this problem, we added the controllability prior, which enforces that things controlled by the robot are encoded into the state. This improved the resulting state representation (see Fig. <ref>). While the semantics of the state representation are not as clear as for the previous tasks, the representation uses four dimensions, which makes sense for two objects in a plane. Additionally, the goal states (ball in cup) are clearly separated from the other states. As we will see in the following section, the information about the cup is still very noisy, which is probably why reinforcement learning based on PVEs does not reach the same performance as in the other tasks. This result makes the ball in cup task a good candidate for a next step on extending PVEs by revising and adding robotic priors.§.§ Regression to True Positions and Velocities To measure the quality of the learned position-velocity representation, we performed regression from the learned postion-velocity state to true positions and velocities of relevant objects. Here, we trained a fully connected neural network with 3 hidden ReLu layers of 256 units each for 200 steps with Adam. We normalized the true positions and velocities and performed supervised learning from the learned position-velocity state to the true features minimizing mean squared error. After training on observations from 1000 times 20 steps, we tested with 100 times 20 test samples. The resulting test errors are shown in Table <ref>.When we compare these errors, we find that the errors are lowest for the pendulum task, which makes sense because the range of possible observations is so small in this task, that it is well covered by the training data. For the cart-pole the errors are still very low for position, but higher for the estimated velocities because noise in the position states is increased when computing velocities from finite differences. Also, the errors double when we go from the moving camera setting to the static camera setting. From this difference, we can predict that control should be easier in the first setting. Finally, for ball in cup, the errors are again much larger for the reasons discussed earlier. The estimation of the cup velocity is particularly challenging. Note that we performed this regression test to measure how well these properties are encoded in the state. We do not use the state labels for training the representation and we do not use them for learning control. In the following section, we will measure the utility of the learned representation by reinforcement learning performance based on these representations. §.§ Enabling Reinforcement Learning In this experiment, we learn control for these tasks with neural fitted Q-iteration (NFQ, <cit.>) based on the encoding learned by PVEs. As a baseline, we use untrained PVEs with randomly initialized encodings in this preliminary work (we will thoroughly compare to other methods in future work). For the policy, we used a fully connected neural network with two hidden layers of 250 sigmoid units per layer. We trained it two times for 30 episodes after each training epoch. We rescaled rewards to be non-positive and ommitted discounting. We repeated actions for multiple time steps (4 for the pendulum and cart-pole tasks, 6 for the ball in cup task).The resulting learning curves are shown in Figure <ref>. The blue curves show the baselines with random encodings, which do not allow learning any of the three tasks. The green and red curves show reinforcement learning based on PVEs that were trained on a batch of 1000 trajectories of 20 steps. For the inverted pendulum and for the cart-pole task, the green curves reach optimal performance after only 50 and 300 epochs. The red curve, which shows the performance based on the static camera perspective does not reach optimal performance, probably due to the more noisy state estimation discussed in the previous section. At this point, it is not clear whether this issue comes from the low resolution in the input or from the fact that the position of the pole and the cart are more strongly coupled in these observations which makes learning the state encoding more difficult. Lastly, for the ball in cup task, the learned control beats the baseline consistently and (as the light green maximum shading shows) more successful control using the learned representation is possible. But due to the noisy state estimation, this is not sufficient for solving the task consistently. Future work could start from here and investigate which priors are missing to solve this and more realistic robotic tasks. § CONCLUSION AND FUTURE WORK We have presented position-velocity encoders (PVEs), which are able to learn state representations structured into a position and a velocity part without supervision and without requiring image reconstruction. The keys to PVEs are to constrain the model to estimate velocities in the correct way from positions and to train the position encoder by optimizing consistency with robotic priors, which are specific to positions, velocities, and accelerations. We have shown how structuring the state space into positions and velocities opens up new opportunities for formulating useful constraints and learning objectives. In future research, we will work towards adding further structure into the state space, revising and extending the list of robotic priors, and combining these approaches with end-to-end reinforcement learning. plain | http://arxiv.org/abs/1705.09805v3 | {
"authors": [
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"Roland Hafner",
"Jonathan Scholz",
"Martin Riedmiller"
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"published": "20170527111749",
"title": "PVEs: Position-Velocity Encoders for Unsupervised Learning of Structured State Representations"
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=1Ubboldmn= = 0.2in = 0.0in | http://arxiv.org/abs/1705.09611v1 | {
"authors": [
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"Xueda Wen",
"Onkar Parrikar",
"Chang-Tse Hsieh",
"Bo Han",
"Taylor L. Hughes",
"Robert G. Leigh"
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"categories": [
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"published": "20170526151607",
"title": "Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory"
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OT1pzcmit | http://arxiv.org/abs/1705.09179v3 | {
"authors": [
"Sachin Kumar",
"Zafar Ahmed"
],
"categories": [
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"primary_category": "quant-ph",
"published": "20170525135052",
"title": "Pseudo-symmetric random matrices: semi-Poisson and sub-Wigner statistics"
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^1 Department of Computer Science, School of Mathematics, Statistics, and Computer Science, University of Tehran, Tehran, Iran ^2 School of Biological Sciences, Institute for Research in Fundamental Sciences (IPM) , Tehran, Iran ^3 CerCo UMR 5549, CNRS — Université Toulouse 3, France First-spike based visual categorization using reward-modulated STDP Mohammad Ganjtabesh^1,2,[Corresponding author.Email addresses: [email protected] (MM),[email protected] (SRK),[email protected] (TM), [email protected] (AND),[email protected] (MG).]=================================================================================================================================================================================================================================== Reinforcement learning (RL) has recently regained popularity, with major achievements such as beating the European game of Go champion. Here, for the first time, we show that RL can be used efficiently to train a spiking neural network (SNN) to perform object recognition in natural images without using an external classifier. We used a feedforward convolutional SNN and a temporal coding scheme where the most strongly activated neurons fire first, while less activated ones fire later, or not at all. In the highest layers, each neuron was assigned to an object category, and it was assumed that the stimulus category was the category of the first neuron to fire. If this assumption was correct, the neuron was rewarded, i.e. spike-timing-dependent plasticity (STDP) was applied, which reinforced the neuron's selectivity. Otherwise, anti-STDP was applied, which encouraged the neuron to learn something else. As demonstrated on various image datasets (Caltech, ETH-80, and NORB), this reward modulated STDP (R-STDP) approach extracted particularly discriminative visual features, whereas classic unsupervised STDP extracts any feature that consistently repeats. As a result, R-STDP outperformed STDP on these datasets. Furthermore, R-STDP is suitable for online learning, and can adapt to drastic changes such as label permutations. Finally, it is worth mentioning that both feature extraction and classification were done with spikes, using at most one spike per neuron. Thus the network is hardware friendly and energy efficient. Keywords: Spiking Neural Networks, Reinforcement Learning, Reward-Modulated STDP, Visual Object Recognition, Temporal Coding, First-Spike Based Categorization.§ INTRODUCTIONNeurons in the brain are connected by synapses that can be strengthened or weakened over time. The neural mechanisms behind long-term synaptic plasticity, which is crucial for learning, have been under investigation for many years. Spike-timing-dependent plasticity (STDP) is an unsupervised form of synaptic plasticity, observed in different brain areas <cit.>, in particular in the visual cortex <cit.>. STDP works by considering the time difference between pre- and post-synaptic spikes. According to this rule, if the pre-synaptic neuron fires earlier (later) than the post-synaptic one, the synapse is strengthened (weakened). Studies have shown that STDP results in coincidence detectors, by which a neuron gets selective to a frequent input spike pattern leading to an action potential whenever the pattern is presented <cit.>. STDP works well in finding statistically frequent features, however, as any unsupervised learning algorithm, it faces with difficulties in detecting rare but diagnostic features for important functionalities such as decision-making.Several studies suggest that the brain's reward system plays a vital role in decision-making and forming behaviors. This is also known as reinforcement learning (RL), by which the learner is encouraged to repeat rewarding behaviors and avoid those leading to punishments <cit.>.It is found that dopamine, as a neuromodulator, is one of the important chemical substances involved in the reward system <cit.>, where its release is proportional to the expected future reward <cit.>. It is also shown that dopamine, as well as some other neuromodulators influences the synaptic plasticity, such as changing the polarity <cit.> or adjusting the time window of STDP <cit.>.One of the well-studied ideas to model the role of the reward system is to modulate or even reverse the weight change determined by STDP, which is called reward-modulated STDP (R-STDP) <cit.>. R-STDP stores the trace of synapses that are eligible for STDP and applies the modulated weight changes at the time of receiving a modulatory signal; a reward or punishment (negative reward).In 2007, Izhikevich <cit.> proposed a R-STDP rule to solve the distal reward problem, where the reward is not immediately received. He solved the problem using a decaying eligibility trace, by which the recent activities are considered to be more important. He showed that his model can solve both classical and instrumental conditionings <cit.>. In the same year, Farries and Fairhall <cit.> employed R-STDP to train neurons for generating particular spike patterns. They measured the difference between the output and target spike trains to compute the value of the reward. Also, Florian <cit.> showed that R-STDP is able to solve the XOR task by either rate or temporal input coding and learning a target firing rate. A year later, Legenstein et al. <cit.> investigated conditions, under which R-STDP achieves a desired learning effect. They demonstrated the advantages of R-STDP by theoretical analysis, as well as practical applications to biofeedbacks and a two-class isolated spoken digit recognition task. Vasilaki et al. <cit.> examined the idea of R-STDP on problems with continuous space. They showed that their model is able to solve the Morris water maze quite fast, while the standard policy gradient rule failed. Investigating capabilities of R-STDP continued by research from Frémaux et al. <cit.>, in which conditions for a successful learning is theoretically discussed. They showed that a prediction of the expected reward is necessary for R-STDP to learn multiple tasks simultaneously. Studying the RL mechanism in the brain has gathered attentions in recent years, and researchers try to solve more practical tasks by reward-modulated synaptic plasticity <cit.>.Visual object recognition is a sophisticated task, at which humans are expert. This task requires both feature extraction, that is done by the brain's visual cortex, and decision-making on the category of the object, for which higher brain areas are involved. Spiking neural networks (SNNs) have been widely used in computational object recognition models. In terms of network architecture, there are several models with shallow <cit.>, deep <cit.>, recurrent <cit.>, fully connected <cit.>, and convolutional structures <cit.>. Some use rate-based coding <cit.>, while others use the temporal coding <cit.>. Various kinds of learning techniques are also applied to SNNs, from backpropagation <cit.>, tempotron <cit.>, and other supervised techniques <cit.>, to unsupervised STDP and STDP-variants <cit.>. Although STDP-enabled networks provide a more biological plausible means of visual feature extraction, they need an external readout, e.g. support vector machines <cit.>, to classify input stimuli. Additionally, STDP tends to extract frequent features which are not necessarily suitable for the desired task. In this research, we present a hierarchical SNN equipped with R-STDP to solve the visual object recognition in natural images, without using any external classifier. Instead, we put class-specific neurons that are reinforced to fire as early as possible if their target stimulus is presented to the network. Thus, the input stimuli are classified solely based on the first-spike latencies in a fast and biologically plausible way. R-STDP enables our network to find task-specific diagnostic features, therefore, decreases the computational cost of the final recognition system.Our network is based on Masquelier and Thorpe’s model <cit.> with four layers. The first layer of the network converts the input image into spike latencies based on the saliency of its oriented edges. This spike train goes under a local pooling operation in the second layer. The third layer of the network includes several grids of integrate-and-fire neurons that combine the received information of oriented edges and extract complex features. This is the only trainable layer in our network which employs R-STDP for synaptic plasticity. The signal (reward/punishment) for modulation of synaptic plasticity is provided by the fourth layer, in which the decision of the network is made. Our network only uses the earliest spike emitted by the neurons in the third layer to make a decision, without using any external classifier. If its decision is correct (incorrect) a global reward (punishment) signal is generated. Besides, in order to increase the computational efficiency, each cell in the network is allowed to spike only once per image. The motivation for at most one spike per neuron is not only computational efficiency, it is also biological realism <cit.>. Decision-making without any classifiers with at most one spike per neuron, turns the proposed method into a well-suited candidate for the hardware implementation.We performed two toy experiments to illustrate the abilities of R-STDP. We showed that the network employing R-STDP finds informative features using fewer computational resources than STDP. We also showed that R-STDP can change the behavior of a neuron, if needed, by encouraging it to unlearn what it has learned before. Thus, reusing a computational resource that is no longer useful. Moreover, we evaluated the proposed network on object recognition in natural images, using three different benchmarks, that are Caltech face/motorbike (two classes), ETH-80 (eight classes), and NORB (five classes). The results of the experiments demonstrate the advantage of employing R-STDP over STDP in finding task-specific discriminative features. Our network reached the performances (recognition accuracies) of 98.9% on Caltech face/motorbike, 89.5% on ETH-80, and 88.4% on NORB datasets.The rest of this paper is organized as follows: A precise description of the proposed network is provided in Section 2. Then, in Section 3, the results of the experiments are presented. Finally, in Section 4, the proposed network is discussed from different points of view and the possible future works are highlighted. § MATERIALS AND METHODSIn this section, we first describe the structure of the proposed network and the functionality of each layer. We then explain R-STDP, by which the neurons achieve reinforced selectivity to a specific group of input stimuli. Finally, we give a detailed description on the classification strategy that is used to evaluate the network's performance. §.§ Overall StructureSimilar to Masquelier and Thorpe’s model <cit.>, our network consists of two simple and two complex layers, that are alternately arranged in a feed-forward manner (see Fig. <ref>).The first layer of the network (S1) is a simple layer whose cells detect oriented edges in the input image. These cells emit a spike with a latency that is inversely proportional to the saliency of the edge. After S1, there is a complex layer (C1), which introduces some degrees of position invariance by applying local pooling operation. A C1 neuron propagates the earliest spike in its input window.The second simple layer (S2) is made of integrate-and-fire (IF) neurons. A neuron in this layer, that detects a complex feature, receives its inputs from C1 neurons and generates a spike once its membrane potential reaches the threshold. For synaptic plasticity, we use a learning rule based on three factors: (1) pre-synaptic spike time, (2) post-synaptic spike time, and (3) a reward/punishment signal. This kind of synaptic plasticity provides the ability to control the behavior of the neurons in terms of their selectivity to input patterns.The second complex layer (C2) of our network is the decision-making layer. Each neuron in this layer is assigned to a category and performs a global pooling operation over S2 neurons in a particular grid. Using a rank-order decoding scheme, the neuron which fires first, indicates the network's decision about the input image. According to the decision made by the network, a reward/punishment signal is then generated, which drives in the synaptic plasticity of S2 neurons.Implementation of the network is mainly done with C# and the code is available on ModelDB[<https://senselab.med.yale.edu/ModelDB/>]. §.§ Layer S1The goal of this layer is to extract oriented edges from the gray scaled input image and turn them into spike latencies. To this end, the input image is convolved with Gabor filters of four different orientations. Thus, this layer includes four feature maps, each representing the saliency of edges in a particular preferred orientation.Let I be the grayscaled input image and G(θ) represent a Gabor filter (convolution kernel) with window size 5× 5, wavelength 2.5, effective width 2, and orientation θ. Then, the lth feature map of layer S1 is generated using the following equations:S_1^l= |I⊗ G(θ_l)|, θ_l = (l - 1)×π/4 +π/8,where ⊗ is the convolution operator and l ∈{ 1, 2, 3, 4 }. In order to introduce invariance to image negative operation, the absolute value of the convolution is used. Also, since vertical and horizontal edges are very common in natural images, a π/8 offset is applied to relax this bias <cit.>.For each of the feature maps (orientations), we put a 2D grid of the same size containing dummy neurons to propagate spikes. Using an intensity-to-latency encoding scheme, the obtained feature maps are converted to the spike latencies that are inversely proportional to the saliency of edges. In other words, the more salient the edge, the earlier the corresponding spike is propagated.We implemented the proposed network in an event-based manner, where the spikes are sorted by their latencies in ascending order and propagated sequentially (i.e. the first spike is propagated in time step t = 1, the second one in t = 2, and so on). §.§ Layer C1Our first complex layer is a local pooling layer over the spikes coming from layer S1. Here, there are four 2D neuronal grids corresponding to each of the orientations. Each C1 neuron performs a local pooling operation over a window of size ω_c1×ω_c1 and stride r_c1 (here we set r_c1 = ω_c1 - 1) on S1 neurons in a particular grid, after which, it emits a spike immediately after receiving its earliest input spike. This pooling operation decreases the redundancy of layer S1, and shrinks the number of required neurons, which consequently increases the computational efficiency. It also adds a local invariance to the position of oriented edges.Let 𝒫_c1(i) be the set of all pre-synaptic neurons of the ith neuron in layer C1. Then, the firing time of this neuron is computed as follows: t_c1^f(i) = min_j ∈𝒫_c1(i){t_s1^f(j)},where t_s1^f(j) denote the firing time of the jth neuron in 𝒫_c1(i).Additionally, two kinds of lateral inhibition mechanisms are employed, which help the network to propagate more salient information. If a neuron located at position (x,y) of the ith grid (orientation) fires, (1) the other neurons at the same position, but in other grids are prevented from firing, and (2) the latencies of the nearby neurons in the same grid are increased by a factor relative to their mutual Euclidean distance. In our experiments, inhibition is done for distances from 1 to 5 pixel(s) (floating-point distances are truncated to integer values) with inhibition factors 15%, 12%, 10%, 7%, and 5%, respectively.§.§ Layer S2This layer combines the incoming information about oriented edges and turns them into meaningful complex features. Here, there are n 2D grids of IF neurons with the threshold 𝒯. Each neuron receives its inputs from a ω_s2×ω_s2× 4 window of C1 neurons through plastic synapses. A weight sharing mechanism is also applied to the neurons belonging to the same grid. This mechanism provides the ability of detecting a particular feature over the entire spatial positions. To be precise, let 𝒫_s2(i) be the set of all pre-synaptic neurons corresponding to the ith neuron. Then, the membrane potential of this neuron at time step t is updated by the following equation:v_i(t) = v_i(t - 1) + ∑_j ∈𝒫_s2(i)^W_ij×δ(t-t^f_c1(j)),where W_ij denotes the synaptic weight, δ is the Kronecker delta function, and t^f_c1(j) is the firing time of the jth cell in layer C1. For each input image, a neuron in S2 fires if its membrane potential reaches the threshold 𝒯. Also, these neurons have no leakage and are allowed to fire at most once while an image is being presented.As the neurons fire, their synaptic weights - the feature they are detecting - are being updated based on the order of pre- and post-synaptic spikes, as well as a reward/punishment signal (see section reward-modulated STDP). This signal is derived from the activity of the next layer, that indicates the network's decision. Besides, initial weights of the synapses are randomly generated, with mean 0.8 and standard deviation 0.05. Note that choosing small or midrange values for mean results in inactive, thus untrained, neurons. Moreover, large values for variance increase the impact of network's initial state. Accordingly, high mean value with small variance is a suitable choice <cit.>. §.§ Layer C2This layer contains exactly n neurons, each is assigned to one of the S2 neuronal grids. A C2 neuron only propagates the first spike that is received from its corresponding neuronal grid. To put it differently, let 𝒫_c2(i) define the set of S2 neurons in the ith neuronal grid (for i ∈{1,2,...,n}). Then, the firing time of the ith C2 neuron is computed as follows: t_c2^f(i) = min_j ∈𝒫_c2(i){t_s2^f(j)},where t_s2^f(j) denote the firing time of the jth neuron in layer S2.As mentioned before, the activity of C2 neurons indicates the decision of the network. To this end, we divide C2 neurons into several groups and assign each group to a particular category of input stimuli. Then, the network's decision on the category of the input stimulus is assumed to be the one whose group propagates the earliest spike among other C2 groups.Assume that there are m distinct categories for the input stimuli, labeled from 1 to m, and n neuronal grids in layer S2. Accordingly, there are exactly n neurons in layer C2, that are divided into m groups. Let g: {1,2,...,n}↦{1,2,...,m} denote a function that returns the group's index of a C2 neuron, and let t^f_c2(i) denote the firing time of the ith neuron in layer C2. Then, the network's decision 𝒟 is made by ℱ = min_i{ t^f_c2(i) | 1≤ i ≤ n }, 𝒟 = g(ℱ),where ℱ is the index of a C2 neuron which fires first. The network receives reward (punishment) if its decision matches (does not match) the correct category of the input stimulus. If none of the C2 neurons fire, no reward/punishment signal is generated, thus, no weight-change is applied. Moreover, if more than one neuron fire early (with the minimum spike time), the one with the minimum index (i) is selected. §.§ Reward-Modulated STDP (R-STDP)We propose a reinforcement learning mechanism to update the pre-synaptic weights of S2 neurons. Here, the magnitude of weight change is modulated by a reward/punishment signal, which is received according to the correctness/incorrectness of the network's decision. We also applied a one-winner-takes-all learning competition among the S2 neurons, by which the one with the earliest spike is the winner and the only one which updates its synaptic weights. Note that this neuron is the one determining the network's decision.To formulate our R-STDP learning rule, if a reward signal is received, thenΔ W_ij= a_r^+×W_ij×(1-W_ij) t^f_c1(j) - t^f_s2(i) ≤ 0, a_r^-×W_ij×(1-W_ij) t^f_c1(j) - t^f_s2(i) > 0,and in case of receiving a punishment signal, we haveΔ W_ij= a_p^+×W_ij×(1-W_ij) t^f_c1(j)-t^f_s2(i) > 0, a_p^-×W_ij×(1-W_ij) t^f_c1(j) - t^f_s2(i) ≤ 0,where i and j refer to the post- and pre-synaptic cells, respectively, Δ W_ij is the amount of weight change for the synapse connecting the two neurons, and a_r^+, a_r^-, a_p^+, and a_p^- scale the magnitude of weight change. Furthermore, to specify the direction of weight change, we set a^+_r, a^+_p > 0 and a^-_r, a^-_p < 0. Here, our learning rule does not take into account the exact spike time difference and uses an infinite time window. According to this learning rule, the punishment signal reverses the polarity of STDP (a.k.a anti-STDP). In other words, it swaps long-term-depression (LTD) with long-term-potentiation (LTP), which is done to conduct the effect of aversion (avoid repeating a bad behavior), and a^+_p is there to encourage the neuron to learn something else. §.§ Overfitting AvoidanceIn reinforcement learning problems, there is a chance of being trapped intolocal optima or overfitting to acquiring the maximum possible reward over the training examples. In order to help the network, exploring other possible solutions that are more general to cover both seen and unseen examples, we apply two additional mechanisms during the training phase. These techniques are only used for object recognition tasks.§.§.§ Adaptive learning rateSince the initial weights of the neurons are randomly set, the number of misclassified samples is relatively high at the beginning of the training phase (i.e. the performance is at the chance level). As training trials go on, the ratio of correctly classified samples to the misclassified ones increases. In the case of high rate of misclassification, the network receives more punishment signals, which rapidly weakens synaptic weights and generates dead or highly selective neurons that cover a small number of inputs. Similarly, when the rate of correct classification gets higher, the rate of reward acquisition increases as well. In this case, the network prefers to exclude misclassified samples by getting more and more selective to correct ones and remain silent for the others. In either cases, the overfitting happens due to the unbalanced impact of reward and punishment.To tackle this problem, we multiply an adjustment factor to the amount of weight modification, by which the impact of correct and incorrect training samples is balanced over the trials. Assume that the network sees all of the training samples on each training iteration and let N_hit and N_miss denote the number of samples that are classified correctly and incorrectly in the last training iteration, respectively. If N is the number of all training samples, then, the weight changes for the current training trial are modified as follows:W_ij = W_ij +(N_miss/N)Δ W_ij ,(N_hit/N)Δ W_ij .Note that N_hit + N_miss≤ N, since there may be some samples for which none of the S2 neurons is active.§.§.§ DropoutIn a reinforcement learning scenario, the goal of the learner is to maximize the expected value of reward acquisition. In our case, since the network only sees the training samples, it may find a few number of features that are sufficient to correctly classify almost all of the training samples. This issue appears to cause severe overfitting in face of complex problems and the network prefers to leave some of the neurons untrained. These neurons decrease the hit rate of the network over the testing samples, as they blindly fire for almost all of the stimuli.Here, we employ the dropout technique <cit.>, which causes a C2 neuron to be temporary turned off with the probability of p_drop. This technique gives rise to the overall involvement rate of the neurons, which in turn, not only increases the chance of finding more discriminative features, but also decreases the rate of blind firings (see Supplementary Materials: Dropout). §.§ ClassificationAs mentioned before, the activity of the last layer, particularly the earliest spike in layer C2, is the only information that our network uses to make its final decision on the input stimuli. This way, we do not need external classifiers and increase the biological plausibility of the network at the same time.To setup the network for a classification task with m categories, we put n = k × m neuronal grids in layer S2, where k is the number of features associated to each category. Then, we assign each C2 neurons to a category by the association function g: {1,2, ..., n}↦{1, 2, ..., m} defined as follows:g(i) = ⌊ (i - 1)/k ⌋ + 1.Then, the network uses equation (<ref>) to classify the input stimuli. During the training phase, each network's decision is compared to the label of stimulus and a reward (punishment) signal is generated, if the decision matches (mismatches) the label. §.§ Comparison of R-STDP and STDPIn object recognition tasks, we make a comparison between our model, SNN with R-STDP, and the one that uses STDP. To this end, we first train the network using STDP and let the network extract features in an unsupervised manner. Next, we compute three kinds of feature vectors of length n from layer S2: * The first-spike vector. This is a binary vector, in which all the values are zeros, except the one corresponding to the neuronal grid with earliest spike. * The spike-count vector. This vector saves the total number of spikes emitted by neurons in each grid. * The potential vector. This vector contains the maximum membrane potential among the neurons in each grid, by ignoring the threshold. After extracting feature vectors for both training and testing sets, K-nearest neighbors (KNN) and support vector machine (SVM) classifiers are used to evaluate the performance of the network. Moreover, the learning strategy and the STDP formula is the same as <cit.>, and to make a fair comparison, we use the same values for parameters in both models. The only parameters that are explored for the STDP are the magnitudes of LTP and LTD. § RESULTSTo evaluate the proposed network and learning strategy, we performed two types of experiments. First, we used a series of hand-made problems to show the superiority of R-STDP over STDP. Second, we assessed the proposed network on several object recognition benchmarks. §.§ R-STDP Increases Computational Efficiency Using STDP, when a neuron is exposed to input spike patterns, it tends to find the earliest repetitive sub-pattern by which the neuron reaches its threshold and fires <cit.>. This tendency to favor early input spikes can be troublesome in case of distinguishing spike patterns that have temporal differences in their late sections.Assume that there are several categories of input stimuli that possess the same spatial configuration (Fig. <ref>). They also have identical early spikes. These patterns are repetitively presented to a group of IF neurons, for which the synaptic plasticity is governed by STDP and the one-winner-takes-all mechanism. If the neurons have low thresholds, one of them gets selective to the early common part of the input stimuli and inhibits the other neurons. Since the early parts are spatio-temporally the same among all of the input stimuli, there is no chance for the other neurons to fire and win the synaptic plasticity. Consequently, the overall activity of the neuronal group is the same for all of the input stimuli and classifies them into a single category. As we will see below (Fig. <ref>), there are also some STDP-based solutions for this problem, however they are inefficient in using computational resources. For example, if we increase the size of receptive fields along with the thresholds, neurons gain the opportunity to receive the last spikes as well as the early ones. Another possible solution is to use many neurons that locally inhibit each other and drop the one-winner-takes-all constraint. This way, regarding the initial random weights, there is a chance for the neurons to learn other parts of the input stimuli.Here, we show that the R-STDP learning rule solves this issue in a more efficient way than STDP. For this purpose, we designed an experiment containing two 3× 11 input stimuli. The inputs are spatially similar, which means that spikes are propagated from similar locations of both inputs. As illustrated in Fig. <ref>, each input is a 2D grid of white and gray squares. By white (gray) squares we denote locations, from which a spike is (is not) propagated. At the time of presenting any of these patterns to the network, spikes are propagated with a temporal order that is defined by the numbers written on the squares. According to this ordering, spikes with lower numbers are propagated earlier.Since the input stimuli are artificial spike patterns, there was no need to apply Gabor filters, thus, they were fed directly into the layer S2. There, we put two neuronal grids with parameters ω_s2 = 3 and 𝒯 = 3. Therefore, each grid contained 1× 9 neurons to cover the entire input stimuli. We also set a_r^+ = 0.05, a_r^- = -0.05, a_p^+ = 0.1, and a_p^- = -0.1. The goal of the task was that the first (second) C2 neuron fires earlier for the first (second) pattern. We examined both STDP and R-STDP learning rules to see if the network finds discriminative features or not.As shown in Fig. <ref>, using STDP, the network extracted a non-discriminative feature, the shared one between both input stimuli. On the other side, the proposed reinforcement learning mechanism guided the neurons to extract features whose temporal order of appearance is the only thing leading to a successful pattern discrimination. We repeated this experiment for 100 times using different random initial weights. Results showed that our network succeeded in 98% of the times, while there were no chance for STDP to find the discriminative features. When we increased the threshold to 4 (requiring at least two sub-patterns) and the size of the receptive fields to 11× 11 (covering the entire pattern), the network employing the STDP could also find discriminative features (see Fig. <ref>) in 80% of the times. §.§ Plastic NeuronsAs mentioned earlier, the brain reward system plays an important role in the emergence of a particular behavior. In this part of the paper, we demonstrate the R-STDP's capability of re-adjusting neurons' behavior in an online manner.We designed an experiment, in which the pre-defined desired behavior of the neurons is changed during the simulation. The experimental setup is very similar to the “Temporal Discrimination" task with similar input stimuli and parameter values, except that we swapped the target input stimuli during the training iterations (see Task 1 and Task 2 in Fig. <ref>). As shown in Fig. <ref>, at the beginning of the simulation, the desired behavior was that the neurons belonging to the first grid respond to the first stimulus earlier than those in the second grid, and vice versa. After 200 iterations, when the convergence is fulfilled, we swapped the target stimuli. At this stage, since the neurons were exclusively sensitive to the previous target stimuli, they began to generate false alarms. Consequently, the network was receiving high rates of punishments for around 80 iterations (see iterations 200 to 280 in Fig <ref>), which in turn swapped LTD and LTP (see Materials and Methods: Reward-modulated STDP). As the network received punishments, the previously weakened (strengthened) synapses got stronger (weaker). Therefore, the sensitivity diminished for a while, and the neurons regained the possibility of learning something new. After iteration 300, neurons found their new target stimulus and, once again, converged to the discriminative features (see the plots of synaptic weights in the top two rows in Fig. <ref>).In summary, R-STDP enables the neurons to unlearn what they have learned so far. This ability results in neurons with flexible behavior (plastic neurons), that are able to learn rewarding behavior in changing environments. This ability also helps the neurons to forget and escape from the local optima in order to learn something that earns more reward. Applying STDP in such a scenario does not work at all, since there is no difference between Task 1 and Task 2 from an unsupervised point of view. §.§ Object RecognitionIn this section, the performance of our network on categorization of natural images is evaluated. We begin with a description of the datasets that are used in our experiments. Then, we show how the network benefits from the reinforcement learning mechanism to extract features from natural images, followed by comparing R-STDP and STDP in object recognition tasks. Finally, we illustrate how the dropout and adaptive learning techniques reduce the chance of overfitting to the training samples.§.§.§ DatasetsWe used three well-known object recognition benchmarks to evaluate the performance of the proposed network. The first and easiest one is Caltech face/motorbike which is mainly used for demonstration purposes. The next two that are used to evaluate the proposed network are ETH-80 and small NORB. These datasets contain images of objects from different view points which make the task harder (see supplementary Fig. S1). §.§.§ Reinforced SelectivityThe previous experiments showed that R-STDP enables the network to find informative and discriminative features, both spatially and temporally. Here, we show that R-STDP encourages the neurons to become selective to a particular category of natural images. To this end, we trained and examined the network on images from two categories of face and motorbike from the Caltech dataset.In this experiment, we put 10 neuronal grids for each category that were reinforced to win the first-spike competition in response to the images from their target categories. Therefore, the desired behavior of the network was that the neurons of the first 10 grids get selective to the face category, while those in the other grids get selective to the motorbikes.Fig. <ref> illustrates the behavior of the network over the training iterations. Since the early iterations contained rapid changes, they are plotted wider. During early iterations, strong synaptic weights (see Materials and Methods: Layer S2) and 50% dropout probability resulted in an unstable networkwhose neurons responded to random input stimuli. This chaotic behavior can be easily spotted on early iterations in the middle plot (see Fig. <ref>). As the network continues training iterations, reward/punishment signals made neurons more and more selective to their target categories. As shown in Fig. <ref>, after 200 iterations, a quite robust selectivity appeared for the training samples, while on the testing samples, it is elongated for 300 more iterations. This quick convergence on training samples is due to the fact that the network is relatively fast in finding features that successfully discriminate seen samples (see Fig. <ref>). These primary features need to converge more to be applicable on testing samples, which requires even more iterations because of the adaptive learning rates. Moreover, we do not let the learning rate drops below 20% of the values of parameters a_r^+, a_r^-, a_p^+, and a_p^-. This allows the network to continue convergence with a constant rate even if all of the training samples are correctly categorized (see Fig. <ref>).We repeated the experiment 30 times with random initial weights and different training and testing samples and the performance achieved by the proposed network is 98.9± 0.4% (mean ± std). When we tried a same network structure with STDP, 97.2% was its best achievement (see Table <ref>).§.§.§ PerformanceWe have shown how the proposed network successfully classified faces from motorbikes with high accuracy. Here, we examined the performance of the proposed network on the ETH-80 and NORB datasets that are more challenging (see Supplementary Materials: Datasets). The performance of the network is tested over the entire testing set after each training iteration, in which the network receives all of the training samples in random order.For ETH-80 dataset, we configured the network to extract 10 features per category, which resulted in 8 × 10 = 80 features in total. The receptive field of each neuron in layer S2 was set in a way that it covered the whole input image. Here, nine instances of each category were presented to the network as the training samples, and the remaining were employed in the test phase. After performing 250 training and testing iterations, the best testing performance of the network was reported.Again, we repeated this experiment 30 times, each time using a different training and testing set. As before, the network successfully extracted discriminative features (see Supplementary Fig. 2) and reached the performance of 89.5± 1.9% (mean ± std). We also applied STDP to a network with the same structure. To examine the STDP performance, we used support vector machines with linear kernel and KNNs (K was changed from 1 to 10). According to the results, the accuracy achieved by this network is 84.5%, when the maximum potentials were used as the feature vectors and the classifier was KNN. Considering that the proposed network classifies input patterns solely based on the first-spike information, R-STDP definitely outperforms STDP. Table <ref> provides the details of the comparison made between R-STDP and STDP.By looking at confusion matrices (see Supplementary Fig. 3a), we found that both R-STDP and STDP agree on the most confusing categories, that are cow, dog, and horse. However, thanks to the reinforcement learning, R-STDP not only decreased the confusion error, but also provided a more balanced error distribution.The same experiment was also performed on the NORB dataset. Again, we put 10 neuronal grids for each of the five categories, whose neurons are able to see the entire incoming stimuli. The proposed network with R-STDP reached the performance of88.4± 0.5% (mean ± std) on testing samples, whereas STDP achieved 66% at most. By reviewing confusion matrices of both methods, we found that both networks encountered difficulties mostly in distinguishing four-leg animals from humans, as well as cars from trucks (see Supplementary Fig. 3b). As before, R-STDP resulted in a more balanced error distribution.Additionally, we compared the proposed network to convolutional neural networks (CNNs). Although the proposed network is not able to beat pre-trained deep CNNs such as VGG16 <cit.> (see Supplementary Materials: Comparison with Deep Convolutional Neural Networks), comparing it to a shallow CNN, with a similar network structure and same input would be a fair point. We repeated all of the object categorization experiments using a shallow CNN implemented with Keras neural networks API and Tensorflow as its backend. As shown in Table <ref>, the proposed network successfully outperformed the supervised CNN in both of the ETH-80 and NORB datasets.§.§.§ Overfitting ProblemOverfitting is one of the most common issues in supervised or reinforcement learning scenarios. This problem got even worse by the emergence of deep learning algorithms. There are many studies focused on developing techniques that increase the generalization power of the learning algorithms. One of the mechanism that has shown promising empirical results on deep neural networks is the dropout technique <cit.>. This technique temporary reduces the complexity of the network by suppressing the activity of a specific number of neurons. This reduction in neuronal resources forces the network to generalize more in order to reduce the prediction error. The proposed network is not an exception and has shown tendencies to overfit on the training samples through our examinations. Therefore, we adopted the dropout technique in our experiments. We also found that an steady learning rate does increase the chance of overfitting. Thus, we made use of dynamic learning rates with respect to the performance of the network (see Material and Methods: Overfitting Avoidance).To show the impact of the aforementioned mechanisms, we repeated the object recognition experiments with different dropout probabilities and steady learning rates. Fig. <ref> simultaneously shows the impact of both mentioned mechanisms on categorization of test samples. It is clear that when the adaptive learning rate mechanism is applied, the network achieved higher performances (solid lines). It is also shown that the dropout probability must be chosen according to the complexity of the dataset as well as the network. Since the NORB dataset contains more complex samples than the ETH-80, it tends more to overfitting on training samples. As a consequence, it needs more dropout rate to overcome this issue. The magnitude of this tendency is even clearer when the steady learning rates are used. To put it differently, faster convergence rate along with the complexity of the samples induce more overfitting, which in turn needs more dropout rate. § DISCUSSIONMammals are fast and accurate at visual object recognition. Their visual cortex processes the incoming data in a hierarchical manner, through which the complexity of neuronal preference is gradually increased. This hierarchical processing provides a robust and invariant object recognition <cit.>. Computational modeling of the mammalian visual cortex has been under investigation for many years. Developing a biologically plausible model not only enables scientists to examine their hypotheses with low cost, but also provides a human-like vision for artificially intelligent machines <cit.>.Deep convolutional neural networks (DCNNs) are the most successful works in this area <cit.>. The idea behind these networks is inspired by the hierarchical structure of the visual cortex. Despite the promising results obtain by DCNNs, they are not biologically plausible because of using supervised learning rules. In addition, they employ rate-based encoding scheme, which is both energy and resource consuming. There is another group of studies trying to use spiking neurons along with the unsupervised STDP learning rule <cit.>. These models are more biologically plausible, but they cannot beat DCNNs in terms of accuracy. In theory, spiking neural networks (SNNs) have more computational power than DCNNs, however, they are harder to control because of the complex dynamics and high dimensional space of effective parameter. Furthermore, since most of them are trained in an unsupervised manner, the classification step is done by an external classifier or statistical methods.Here, we solved the object recognition task using a hierarchical SNN equipped with a reinforcement learning rule called R-STDP <cit.>. There are several studies showing that the brain uses RL to solve the problem of decision-making <cit.>. Therefore, it is a suitable choice for training class-specific neurons that are able to decide on the class of the input image. Therefore, we put one step further developing a more biologically plausible model which is able to perform the visual categorization totally on its own. The proposed network functions in the temporal domain, where the information is encoded by spike times. The input image is first convolved with oriented Gabor filters and a spike train is generated based on a latency-to-intensity coding scheme. The resulting spikes are then propagated toward the feature extraction layer. Using R-STDP, the proposed network successfully found task-specific diagnostic features using neurons that were pre-assigned to the class labels. In other words, each neuron was assigned to a class a priori, where its desired behavior was to respond early for the instances belonging to the specified class. To decrease the computational cost even more, neurons were forced to fire at most once for an input image and the latency of their spike is considered as the measure of stimulus preference. Therefore, if a neuron fired earlier than the others, it would have received its preferred stimulus. This measure of preference served as an indicator for the network's decision. That is to say, when a neuron belonging to a particular class fired earlier, the network's decision was considered to be that class.Through our experiments, we compared R-STDP to STDP from different aspects. We showed that R-STDP can save computational resources. This was clarified by a hand-designed discrimination task, in which the order of spikes was the only discriminative feature. R-STDP solved the problem using minimal number of neurons, synapses, and threshold, whereas STDP needed more neurons, more synapses, and higher thresholds. This drawback for STDP is due to the fact that it tends to find statistically frequent features <cit.>, which are not necessarily the diagnostic ones. As a consequence, one needs to use either more neurons or more synapses to ensure that the diagnostic features will be eventually found. On the other hand, since R-STDP informs the neurons about their outcomes, they can function better using minimal resources.After having demonstrated the advantages of R-STDP in finding diagnostic features, we investigated how well it can be combined with a hierarchical SNN for solving both visual feature extraction and object categorization in a biologically plausible manner. We evaluated the proposed network and a similar network which uses STDP, as well as a CNN with the same structure, on three datasets of natural images Caltech Face/Motorbike, ETH-80 and NORB. The last two contain images of objects from different viewpoints, which made the task harder. When we compared the performances obtained by the networks, we found that R-STDP strongly outperforms STDP and the CNN with same structure. An even more interesting point is that the proposed network achieved this superiority decisions solely based on the first-spikes, while in the case of the others, even the powerful classifiers like SVMs and error back-propagation were not of any help.To compare R-STDP with STDP, both networks used the same values for parameters except the learning rate (see Materials and Methods: Comparison of the R-STDP and STDP). However, one can use STDP with higher number of neurons and tuned thresholds to compensate the blind unsupervised feature extraction and achieve better performances <cit.>. Again, we conclude that R-STDP helps the network to act more efficiently in consuming computational resources.Putting everything together, the proposed network has the following prominent features: * Robust object recognition in natural images. * Each neuron is allowed to spike only once per image. This results in a huge reduction of energy consumption. * Decision-making (classification) is performed using the first-spike latencies instead of powerful classifiers. Therefore, the biological plausibility of the model is increased. * Synaptic plasticity is governed by RL (the R-STDP rule), for which supporting biological evidence can be found <cit.>, and which allows to extract highly diagnostic features. Our network can be interesting for neuromorphic engineering <cit.>, since it is both biologically plausible and hardware-friendly. Although hardware implementation and efficiency is out of the scope of the current paper, we believe that the proposed network can be implemented in hardware in an energy-efficient manner for several reasons. Firstly, SNNs are more hardware friendly than classic artificial neural networks, because the energy-consuming “multiply-accumulator" units can be replaced by more energy-efficient “accumulator" units. For this reason, studies on training deep convolutional SNNs (DCSNNs) <cit.> and converting DCNNs into DCSNNs <cit.>, as well as restricted DCNNs <cit.> have gained interests in recent years. Secondly, most SNN hardwares use event-driven approaches by considering spikes as events. This way, energy consumption increases with the number of spikes. Thus, by allowing at most one spike per neuron, the proposed model is as efficient as possible.Finally, the proposed learning rule is more suitable for online, on-chip learning than error backpropagation in deep networks, where updating weights based on high-precision gradients brings difficulties for hardware implementation.To date, we could not find any other works possessing the aforementioned features. To mention one of the closest attempts, Gardner et al. <cit.> tried to classify Poisson-distributed spike trains by a readout neuron equipped with R-STDP. Although their method is working, it cannot be applied on natural images as it is, because of their time-based encoding and target labeling. There is another related work by Huerta and Nowotny <cit.>. In this work, the authors designed a model of the RL mechanism which occurs in the mushroom body. They applied their RL mechanism on a pool of randomly connected neurons with 10 readout neurons to classify handwritten digits. Our work is different from theirs in several aspects. First, we used a hierarchical structure based on the mammalian visual cortex, while they used randomly connected neurons. Second, we used the R-STDP learning rule, whereas they employed a probabilistic approach for the synaptic plasticity. Third, the input of our network were natural images using intensity-to-latency encoding, while they used binary encoding with a threshold on artificial images.Although the results of the proposed network were significantly better than the network employing STDP with external classifiers, they are still not competitive to the state-of-the-art deep learning approaches. One of the limitations to the current method is using only one trainable layer. Besides, the receptive field of the neurons in the last layer are set to be large enough to cover an informative portion of the image. As a result, the network cannot resist high rates of variations in the object, unless using more and more number of neurons. Extending the number of layers in the current network is one of the directions for future research. Going deeper seems to improve the performance by providing a gradual simple to complex feature extraction. However, deeper structure needs more parameter tuning, and a suitable multi-layer synaptic plasticity rule. Recent studies have also shown that combining deep networks and RL can lead to outstanding results <cit.>.Another direction for the future research is to use the RL for learning semantic associations. For example, STDP is able to extract features for different kinds of animals in different viewpoints, but it is not able of relating all of them into the category of “animal", because different animals have no reason to co-occur. Or, it can extract features for the frontal and profile face, but it cannot generate an association putting both in the general category of “face". 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"authors": [
"Milad Mozafari",
"Saeed Reza Kheradpisheh",
"Timothée Masquelier",
"Abbas Nowzari-Dalini",
"Mohammad Ganjtabesh"
],
"categories": [
"q-bio.NC",
"cs.CV"
],
"primary_category": "q-bio.NC",
"published": "20170525113816",
"title": "First-spike based visual categorization using reward-modulated STDP"
} |
1USRA, 7178 Columbia Gateway Drive, Columbia, MD 21046, [email protected] 2Arecibo Observatory, HC-3 Box 53995, Arecibo PR 00612 3Physics Department, University of Memphis, Memphis, TN 38152 We have expanded upon earlier work that investigates the relative importance of coronal loops with isothermal versus multithermal cross-field temperature distributions. These results are important for determining if loops have substructure in the form of unresolved magnetic strands. We have increased the number of loops targeted for temperature analysis from 19 to 207 with the addition of 188 new loops from multiple regions. We selected all loop segments visible in the 171-Å images of the Atmospheric Imaging Assembly (AIA) that had a clean background. 86 of the new loops were rejected because they could not be reliably separated from the background in other AIA filters. 61 loops required multithermal models to reproduce the observations. 28 loops were effectively isothermal, that is, the plasma emission to which AIA is sensitive could not be distinguished from isothermal emission, within uncertainties. 10 loops were isothermal. Also part of our inventory were one small flaring loop, one very cool loop whose temperature distribution could not be constrained by the AIA data, and one loop with inconclusive results. Our survey can confirm an unexpected result from the pilot study: we found no isothermal loop segments where we could properly use the 171-to-193 ratio method, which would be similar to the analysis done for many loops observed with TRACE and EIT. We recommend caution to observers who assume the loop plasma is isothermal, and hope that these results will influence the direction of coronal heating models and the efforts modelers spend on various heating scenarios. § INTRODUCTION Recent results from solar X-ray and EUV imagers and spectrometers inspired a series of workshops which were designed to resolve the coronal loop controversy, where new observations were in conflict with the predictions of classical heating models. The model of Rosner et al. (1978), for example, assumed that a loop was single magnetic flux tube with no internal structure. The tube was filled with high-temperature, low-density plasma in hydrostatic equilibrium where the temperature could vary along but not across the field. These predictions were challenged by the results of Schmelz et al. (2001), who used spectral line data from the Coronal Diagnostics Spectrometer (CDS) on the Solar and Heliospheric Observatory and broadband data from the Soft X-ray Telescope (SXT) on Yohkoh to do Differential Emission Measure (DEM) analysis at several positions along their target loop. The resulting temperature distributions were clearly inconsistent with isothermal plasma in the cross-field direction. Subsequent analysis focusing on background subtraction (Schmelz et al. 2005; Schmelz & Martens 2006), modeling (Martens et al. 2002; Weber et al. 2005), and similar results from other CDS loops (Cirtain et al. 2007; Schmelz et al 2007) supported the original claim that multithermal plasma was required to explain the CDS data. In parallel with the analysis described above, isothermal results continued to be published in the literature. Early papers used a simple ratio analysis, like the one employed by Kano & Tsuneta (1996) for loops observed by SXT. This ratio analysis, however, assumes an isothermal plasma, so obviously, the results presented in these papers could not be used as evidence of isothermal loop cross sections. Later analysis used DEM methods with strong high- and low-temperature constraints and data from the EUV Imaging Spectrometer (EIS) and the Atmospheric Imaging Assembly (AIA) to show that loops had both isothermal and multithermal cross sections (Warren et al. 2008; Schmelz et al. 2013a,b; Brooks et al. 2011, 2012). Some of these examples could be described as marginally multithermal, and others require a fairly broad DEM distribution (Schmelz et al. 2010b; 2011b). In fact, Schmelz et al. (2014a) found that the DEM width correlated with the DEM-weighted temperature where the cooler the loop, the narrower the DEM required to model the data.Throughout this paper, we apply the definitions that were agreed upon by the participants of the first coronal loops workshop, which took place in Paris in 2002. A loop is a distinct configuration in an observation and a strand is an elementary flux tube. The observational results described above indicate that the loops analyzed in these various studies could not be a simple flux tube and that they must have some internal structure. These observations are consistent with the predictions of nanoflare coronal heating models (Cargill 1994; Cargill & Klimchuk 1997), where unresolved strands reconnect and release small amounts of energy. Klimchuk (2009) describes the concept of a nanoflare storm. During the storm, bundles of unresolved strands are heated impulsively as reconnections release energy. But as the storm ends, reconnections cease, and since the cooling timescale tends to increase with decreasing temperature, hot strands cool relatively quickly and all strands spend more time cooling slowly through the lower coronal temperature range. Since there is now a preponderance of cool strands, the DEM is expected to narrow with time. This effect was observed with AIA where the temperature evolution of a target loop was followed as it cooled. The DEM distribution required to model the data was broad early in the loop lifetime, but isothermal before the loop faded from view (Schmelz et al. 2014b).Based on the cooling timescale argument outlined above, one might expect to find many more isothermal or marginally multithermal DEMs than broad DEMs. The ratio of these populations might reveal something about the heating timescale for nanoflare storms. In an effort to determine which type of DEM distribution - isothermal or multithermal - dominates the loop populations, Schmelz et al. (2015) examined AR 11294, which was observed by AIA on 2011 September 15. In this original loop inventory paper, they examined the 171-Å image of the target active region and selected all loop segments that were visible against a clean background. There were a variety of results: two segments were isothermal, six were effectively isothermal (a cooler AIA channel would be required for a definitive conclusion), one had both an isothermal transition region as well as a multithermal coronal solution, five required multithermal DEMs, and five could not be separated reliably from the background in other AIA channels. The rate of occurrence of a nanoflare on a strand must be faster than the cooling timescale to explain these results. To take an extreme example: if the timescale was very long then most of the plasma would be allowed to cool substantially and hence the DEMs would be more isothermal than multithermal. In this paper, we continue the loop inventory analysis for six new regions. We hope that these results will affect the course of coronal heating models and the endeavors of modelers on diverse heating schemes.§ OBSERVATIONS The Solar Dynamics Observatory was launched on 2010 February 11 with the purpose of increasing the understanding of the solar magnetic field. It contains three instruments: AIA, the Extreme Ultraviolet Variability Explorer, and the Helioseismic and Magnetic Imager. The excellent resolution and cadence of the AIA allows for spectacular images of the Sun including a few wavelengths that were previously rarely observed. Specifically, the AIA has a field of view with 1.28 solar radii in the EW and NS directions and 0.6 arcsecond pixels, and the cadence is around 10 seconds. Of the ten filters in the AIA, observations were taken from the six filters centered on ionized iron including: Fe VIII (131 Å), Fe IX (171 Å), Fe XII (193 Å), Fe XIV (211 Å), Fe XVI (335 Å), and Fe XVIII (94 Å). Some filters contain flaring lines while others, e.g., the 94-Å filter, contain lower-temperature lines. The former should not contribute to these observations, but the latter almost certainly do. Nevertheless, all of these lines are included in the appropriate response function (see below). These six filters used in conjunction allow for both isothermal and multithermal analysis, which is a major boon to solar physics and, in particular, the study of the corona.Level 1 AIA data were obtained from the Virtual Solar Observatory[http://sdac.virtualsolar.org/cgi/search] website.Using the AIA_prep program included in SolarSoft[http://www.lmsal.com/solarsoft/] on the data then ensured proper alignment, rotation, and image scaling.Occasionally when examining the images, it became apparent that a shift by a pixel in one filter would provide much better alignment with the other filters. In order to perform an effective temperature analysis, it is necessary to obtain a response function for each filter. To this end, the effective areas, the instrument platescale, and gain provided by the instrument teams were obtained from SolarSoft. A synthetic solar spectrum was constructed with atomic data and ionization equilibria from CHIANTI 7.1 (Dere et al. 1997; Landi et al. 2013) and the set of coronal element abundances from Schmelz et al. (2012). The resulting response functions have units of DN s^-1 pixel^-1per unit emission measure.§ ANALYSIS Normally, when we select AIA loops for DEM analysis (e.g., Schmelz et al. 2011a,c), we would cycle through the images from all the coronal filters searching for targets. We would want these loops to be visible in at least three filters, the minimum required for DEM, and there would need to be a nearby area of reasonably clean background. The Loop Inventory process is a bit different, however. In this analysis, we are following the procedure outlined in the paper by Schmelz et al. (2015). Only the AIA 171-Å images were used to select loop segments for detailed temperature analysis. There are several reasons for this. AIA results from the literature indicate that the loops seen in the 171-Å images: (1) appear sharper and more readily discernable; (2) are likely to be cooling; and (3) are more prone to have isothermal cross-sections. This third point is especially important because, as described in the Introduction, our multithermal loop cross-sections led in part to the coronal loop controversy. These temperature results were not only new and unexpected, but also in conflict with the predictions of classical heating models. It was our responsibility to show that the data actually required these more complex temperature solutions and to make every effort to understand why different types of observations were giving different results. In this work, we have done this due diligence by deliberately skewing our selection criteria toward the loop population that had traditionally shown isothermal cross-sections. Although we are not selecting out multithermal cases with this approach - our pilot study already shows that broad DEMs can have cool components - we are making every effort to incorporate isothermal cases in our sample. This criterion is consistent with the coronal cooling timescales, which as noted in the Introduction, increase with decreasing temperature, and with the results of Schmelz et al. (2014a,b), which show evidence for narrowing DEMs with decreasing temperature.AIA 171-Å images of the six regions studied here are shown in Figure 1. Each small box marks a different loop segment, which was selected using the criteria described above. Table 1 lists the active regions, observation dates, solar coordinates, and the number of loop segments identified. Table 2 lists the individual loops with the same numbering scheme as the Figure. After these targets were selected, the other AIA filters were examined. We describe the appearance of each loop in each filter in Table 2.Our data set reveals several examples of the simplest type of temperature analysis, where the loop is clearly visible in the AIA 171-Å filter but not visible in the other filters. Figure 2 shows a Rainbow Plot of Region A Loop 11 where the 131-Å AIA image is shown in purple, 171 in blue, 193 in green, 211 in yellow, 225 in orange, and 94 in red. The position of the target, Loop 11, is designated with arrows. It is isothermal, within the temperature resolution of AIA, with Log T ≃ 5.8, near the peak response of the 171-Å AIA filter. These results are listed in the last two columns of Table 2.With only a superficial examination of the Rainbow Plot in Figure 3, one might conclude that Region F Loop 7 is isothermal for reasons similar to those described in the last paragraph. There is an important difference, however. Although this loop is visible only in the 171-Å image, it is not necessarily because the loop is not present in the other images. Rather, it sits in a crowded arcade and the background has come up in some of the other filters, possibly masking the appearance of the target with the increased emission measure of many unresolved structures. As a result, this loop and the others like it are not candidates for temperature analysis because they cannot be separated from the surrounding background. In the last two columns of Table 2, they are not assigned a temperature and are categorized as Background.There are several examples of loops that we describe as effectively isothermal where the plasma to which AIA is sensitive could not be distinguished from isothermal emission, within measurement uncertainties (e.g., Schmelz et al. 1996; 2014b). We illustrate this category with Region E Loop 9. Figure 4 shows that this loop is visible in the 131- and 171-Å filters, but not visible in the hotter filters. This loop and other like it are categorized as effectively isothermal because although they could be multithermal at cool (T < 1 MK) temperatures, we cannot know this without additional data. These temperatures can be found with ratio analysis, which we describe below.The data in Table 2 indicate that there are many loops that are eligible for temperature analysis. With each candidate, 10 loop pixels and 10 background pixels are selected and averaged.Once we calculated the standard deviations, we then subtracted the background and propagated the errors. These values were then normalized by the appropriate exposure time, resulting in units of DN s^-1 pixel^-1 for each filter. If I is the average intensity, then I ∝ ∑ Resp(T) × DEM(T) Δ T , where Resp(T) is the instrument response function, DEM is the differential emission measure, and T is the temperature. If the isothermal approximation applies, then we can pull the response out of the summation, and the equation simplifies toI ∝ Resp(T) × ∑ DEM(T) Δ T ∝ Resp(T) × EM . EM is the plasma emission measure, and we can use the ratio method to find the temperature: I_131 I_171=Resp_131(T)Resp_171(T) The ratio method is illustrated in Figure 5. The curve with the peak at Log T = 5.55 is the 131-Å response divided by the 171-Å response. The flat lines show the observed intensity ratio (solid) and uncertainties (dashed). Each panel shows a different example from the data set. In general, the flat lines intersect the curve at two locations, both of which are candidates for the plasma temperature. These values and the associated uncertainties are listed in Table 2. In some cases, the solid (Region E Loop 07) and/or dashed (Region C Loop 13) line misses the curve, indicating that the data do not have a high enough signal-to-noise to pin down either the temperature value or the associated limit. These cases are indicated by a double dashed in Table 2.There are many examples of loops in our data sets that appear in multiple filters. The rainbow plot in Figure 6 is for Region B Loop 43, which is visible or barely visible in all the AIA coronal filters. For this and any loop segment that is visible in more than two filters, we cannot assume that the plasma is isothermal (although it may be if its temperature falls in a range where those filters have significant overlap), and we cannot use the ratio method to find the temperature. For the analysis of these loops, we require DEM techniques. Schmelz et al. (2010a, 2011a,b,c) describe a method called DEM_manual, which uses forward fitting. One option available in the DEM_manual program is to find the best isothermal fit to the data. We used this option for all the loops that were visible in three or more filters. We first input a spike-shaped DEM at Log T = 5.30, and the program determines the height of the spike that provides the best fit to the available data. We then move to the next temperature bin and repeat the process. In general, for quiescent loops, we continue to Log T = 6.60, but can go higher for flares. The best isothermal fit is the spike DEM with the lowest reduced χ^2. The results for several different loops are shown in Figure 7, where the blue spike is the best isothermal fit and the reduced χ^2, also in blue, is listed in the upper right corner. The predicted-to-observed intensity ratios for these DEM_manual results are plotted as open diamonds in Figure 8. We use the Rainbow Plot color coding which corresponds to the temperature sequence for these loops: 131 (purple), 171 (blue), 193 (green), 211 (yellow), 225 (orange), and 94 (red). In some cases, this method finds a good fit to the data with reduced χ^2 < 1. One example is for Region E Loop 25, which is plotted in the first panel of Figures 7 and 8. The loop is visible in the three coolest coronal filters, and the analysis done with DEM_manual shows that isothermal plasma can reproduce the observations, within the uncertainties. This loop is effectively isothermal (since it is visible in the 131-Å filter), and the resulting temperature is recorded in Table 2.Many cases, however, show that even the best isothermal result is not a good fit to the data. The reduced χ^2 values in Figure 7 are too high and the predicted-to-observed intensity ratios for some filters in Figure 8 are significantly different from one. For these cases, we use XRT_dem_iterative2 (Weber et al. 2004; Schmelz et al. 2009) to generate 100 multithermal Monte-Carlo realizations. These are shown in red in the panels of Figure 7. The black curve is the best fit and the corresponding reduced χ^2 is listed in the upper right corner, also in black. In these cases, the DEM-weighted temperature is listed in Table 2. The predicted-to-observed intensity ratios for the best-fit results from XRT_dem_iterative2 are shown as filled squares in Figure 8.§ DISCUSSION The results of our analysis discussed in the previous section and summarized in Table 2 support and expand the findings of the loop inventory pilot study of Schmelz et al. (2015). We have added 188 targets to the original 19 from the pilot. The most common entry in the comments column of Table 2 is background, which indicates that the target loop cannot be reliably separated from the background in one or more of the AIA coronal channels required for temperature analysis. This occurred in 86 of the loops in Table 2 and five from the pilot study. We can therefore confirm earlier results that separating the loop from the coronal background represents a serious and, in many cases, insurmountable challenge, although some improvement might be expected using observations made with a spectrometer. This finding comes with an associated caution to other researchers. Although there may be a distinct AIA 171-Å loop segment, it could be part of an arcade that gets more densely populated with increasing temperature. The target may indeed be present in the 193- and 211-Å images, but the background may be too dense to see it (see, e.g., Brickhouse & Schmelz 2006). An unintended result of an automatic background subtraction algorithm might be that a high background might be subtracted from the embedded loop leaving essentially zero flux. This is one way to misidentify loops as isothermal rather than multithermal.The next most common entry in Table 2 is multithermal. This is despite trying to bias our selection criteria toward loops with isothermal cross sections. Our 61 examples here can be added to the original five from the pilot study. These are the loop segments that appeared in at least three of the AIA coronal filters and where the background subtracted intensities could not be reproduced with an isothermal model, within uncertainties. We ran both DEM_manual and XRT_dem_iterative2 on these data and got good results, with small reduced χ^2 values. The DEM-weighted temperature for these examples is listed in the last column of Table 2.The third most common entry in Table 2 is effectively isothermal. This tally includes 28 loops from this study plus six from the pilot. This descriptive indicates that the emission to which AIA is sensitive could not be distinguished from isothermal emission, within measurement uncertainties. These loops may, however, be multithermal at transition region temperatures, but we would require data from a different instrument, perhaps the EUV Imaging Spectrometer or the Interface Region Imaging Spectrograph, to determine the true cross-field temperature distribution of these examples. We encourage observers with expertise in the analysis of data from these instruments to look into this question. Ten entries from this study join two from the pilot in the isothermal category from Table 2. These are cases where the loop is visible only at 171-Å and there is clean background in 131- and 193-Å images. Our survey also appears to have caught one rather small flare, Region C Loop 6, which was not visible in GOES data. DEM analysis was done on these data, and a good result was obtained. We also identified one loop with so much transition-region temperature plasma that the AIA data alone were not able to constrain the DEM model, Region B Loop 35. This loop is labeled unconstrained in the comments column of Table 2. We complete our survey with a single inconclusive entry, Region D Loop 12. Despite multiple attempts at pixel selection for both the loop and the background, we were not able to find a DEM model that successfully reproduced the fluxes. One possible explanation for this might be that there was something behind our loop that we were not resolving properly.Our survey can confirm an unexpected finding from the pilot study. We also found no loop segments where we could assume the plasma was isothermal and properly use the 171-to-193 ratio method, which would be similar to the analysis done for many loops observed with TRACE and EIT. This would require loops to be visible in the 171- and 193-Å images only with clean background in the 131- and 211-Å images. In our examples and in those of the pilot study, the 171-Å segment is either not visible in the 193-Å filter, masked by complex background in the 193-Å filter, or visible in the 193-Å filter but also visible in other filters. The size of the present study indicate that this is probably not just a coincidence, and is more likely to reflect how the loops are cooling. All of the isothermal DEMs identified here are below 1 MK. The plasma starts to cool more quickly below this temperature as the peak of the radiative loss curve is reached. Consequently, one may expect to see fewer purely isothermal DEMs/loop segments simply because the plasma cools relatively quickly below 1 MK and so there are fewer examples to be observed. It could also be that plasma tends to be re-heated before the DEMs approach isothermality. Evidence of a correlation between the DEM-weighted temperature and the cross-field DEM width for coronal loops was found by Schmelz et al. (2014a). Their AIA, XRT, and EIS data showed that warmer loops require broader DEMs. A target loop analyzed by Schmelz et al. (2014b), which was cooling through the AIA passbands, was also evolving from a broad DEM to a narrow DEM.These results are consistent with the radiative loss function in the coronal temperature range. If loops are indeed composed of bundles of unresolved magnetic strands, then our results could indicate that fewer strands are emitting in the later cooling phase, consistent with the nanoflare storm model (Klimchuk 2009), and potentially giving us more insight into the long standing isothermal versus multithermal component of the coronal loop controversy.§ CONCLUSIONS We have expanded the results of the pilot study done by Schmelz et al. (2015) to include multiple regions. The original Loop Inventory included only one active region, AR 11294, and 19 loops. We have added 188 loops to the analysis. In both the pilot as well as the current study, we examined the AIA 171-Å images and selected all loop segments with a clean background. Many of these targets, five in the pilot and 86 here, could not be separated from the coronal background in higher temperature images, and were therefore, not candidates for temperature analysis. Five loops from the pilot and 61 here were visible in three or more AIA filters, and DEM analysis revealed that they had multithermal cross sections. Six loops from the pilot and 28 here were effectively isothermal, and two plus 10 were isothermal. The pilot had one loop that had both isothermal and multithermal solutions. The current work included one small flare, one example where the data available could not constrain the DEM, and one example where the results were inconclusive. The results of our inventory indicate that even a loop seen in a 171-Å image is significantly more likely than not to have a multithermal cross section. This implies that both observers and modelers should exercise caution when making simplistic assumptions about the temperature structure of coronal loops. The data may require DEM methods, which are much more difficult to use than simplistic ratio techniques. These results should also influence the direction of coronal heating models and the efforts that models spend investigating the fundamental properties of loops. This work was inspired by discussions at the Coronal Loops Workshops. The Atmospheric Imaging Assembly on the Solar Dynamics Observatory is part of NASA's Living with a Star program. CHIANTI is a collaborative project involving the NRL (USA), the Universities of Florence (Italy) and Cambridge (UK), and George Mason University (USA). Solar physics research at the University of Memphis is supported by a Hinode subcontract from NASA/SAO. [Brickhouse & Schmelz(2006)]2006ApJ...636L..53B Brickhouse, N.S., Schmelz, J.T. 2006, , 636, L53 [Brooks et al.(2012)]2012ApJ...755L..33B Brooks, D. H., Warren, H. P., Ugarte-Urra, I. 2012, , 755, L33 [Brooks et al.(2011)]2011ApJ...730...85B Brooks, D. H., Warren, H. 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R., Dere, K.P., Del Zanna, G., Mason, H.E. 2013, , 763, 86 [Martens et al.(2002)]2002ApJ...577L.115M Martens, P.C.H., Cirtain, J.W. & Schmelz, J.T. 2002, , 577, 115 [Rosner et al.(1978)]1978ApJ...220..643R Rosner, R., Tucker, W.H., Vaiana, G.S. 1978, , 220, 643 [Schmelz et al.(2011)]2011ApJ...731...49S Schmelz, J.T., Jenkins, B.S., Worley, B.T., Anderson, D.J., Pathak, S., Kimble, J.A. 2011a, , 731, 49 [Schmelz et al.(2013)]2013ApJ...770...14S Schmelz, J.T., Jenkins, B.S. & Pathak, S. 2013a, , 770, 14 [Schmelz et al.(2010)]2010ApJ...725L..34S Schmelz, J.T.; Kimble, J.A.; Jenkins, B.S.; Worley, B.T.; Anderson, D.J.; Pathak, S.; Saar, S.H. 2010a, , 725, 34 [Schmelz & Martens(2006)]2006ApJ...636L..49S Schmelz, J.T., Martens, P.C.H. 2006, , 636, 49 [Schmelz et al.(2007)]2007ApJ...658L.119S Schmelz, J.T., Nasraoui, K., Cirtain, J.W., DeLuca, E.E., Del Zanna, G., Mason, H.E. 2007, , 658, 119 [Schmelz et al.(2005)]2005ApJ...627L..81S Schmelz, J.T., Nasraoui, K., Richardson, V.L., Hubbard, P.J., Nevels, C.R., Beene, J.E. 2005, , 627, 81 [Schmelz et al.(2014)]2014ApJ...795..171S Schmelz, J.T., Pathak, S., Brooks, D.H., Christian, G.M., Dhaliwal, R.S.S. 2014a, apj, 795, 171 [Schmelz et al.(2015)]2015ApJ...813...71S Schmelz, J.T., Pathak, S., Christian, G.M., Dhaliwal, R.S.S., Paul, K. 2015, , 813, 71 [Schmelz et al.(2014)]2014ApJ...795..139S Schmelz, J.T., Pathak, S., Dhaliwal, R.S.S., Christian, G.M., Fair, C.B. 2014b, , 795, 139 [Schmelz et al.(2013)]2013ApJ...764...53S Schmelz, J.T., Pathak, S. Jenkins, B.S., Worley, B.T. 2013b, , 764, 53 [Schmelz et al.(2012)]2012ApJ...755...33S Schmelz, J.T., Reames, D.V., von Steiger, R., Basu, S. 2012, , 755, 33[Schmelz et al.(2011)]2011ApJ...738..146S Schmelz, J.T., Rightmire, L.A., Saar, S.H., Kimble, J.A., Worley, B.T. 2011b, , 738, 146 [Schmelz et al.(2009)]2009ApJ...693L.131S Schmelz, J.T., Saar, S.H., DeLuca, E.E., Golub, L., Kashyap, V.,L., Weber, M.A., Klimchuk, J.A. 2009, , 693, L131 [Schmelz et al.(2010)]2010ApJ...723.1180S Schmelz, J.T., Saar, S.H., Nasraoui, K., Kashyap, V.L., Weber, M.A., DeLuca, E.E., Golub, L. 2010b, , 723, 1180 [Schmelz et al.(1996)]1996ApJ...473..519S Schmelz, J.T., Saba, J.L.R., Ghosh, D., Strong, K.T. 1996, , 473, 519 [Schmelz et al.(2001)]2001ApJ...556..896S Schmelz, J.T., Scopes, R.T., Cirtain, J.W., Winter, H.D. & Allen, J.D. 2001, 556, 896 [Schmelz et al.(2011)]2011ApJ...739...33S Schmelz, J.T., Worley, B.T., Anderson, D.J., Pathak, S., Kimble, J.A., Jenkins, B.S.,Saar, S.H 2011c, , 739, 33 [Warren et al.(2008)]2008ApJ...686L.131W Warren, H.P., Ugarte-Urra, I., Doschek, G.A., Brooks, D.H. & Williams, D.R. 2008, , 686, L131 [Weber et al.(2004)]2004IAUS..223..321W Weber, M. A., De Luca, E. E., Golub, L., & Sette, A. L. 2004, Multi-Wavelength Investigations of Solar Activity (Proc. IAU Symp. 223), ed. A. V. Stepanov, E. E. Benevolenskaya, & A. G. Kosovichev (Cambridge: Cambridge Univ. Press), 321 [Weber et al.(2005)]2005ApJ...635L.101W Weber, M.A., Schmelz, J.T., DeLuca, E.E., Roames, J.K. 2005, , 635, 101 lllll 0ptAIA Active RegionsRegion Date Coordinates # Loops A 11158 2011 February 15 S21 W28 37 B 11166 2011 March 06 N10 E27 51 C 11330 (east) 2011 October 27 N13 E13 13 D 11330 (west) 2011 October 27 N06 E01 30 E 11944 (east) 2014 January 06 S11 E22 26 F 11944 (west) 2014 January 06 S08 E08 31lllllllllllll0ptAIA Loop Segments and Properties#131 171 193 211 335 94 Comments TempRegion A 1 visible visible visible visible visible background multithermal 6.06 2 visible visible visible visible background background multithermal 5.96 3 background visible background background background background background 4 barely visible visible visible visible background background multithermal 6.25 5 barely visible visible visible visible background background multithermal 6.38 6 visible visible background background background background background 7 visible visible background background background background background 8 background visible background background background background background 9 not visible visible not visible not visible not visible not visible isothermal 5.80 10 visible visible visible visible not visible not visible multithermal 6.20 11 not visible visible not visible not visible not visible not visible isothermal 5.80 12 visible visible visible visible barely visible not visible multithermal 6.12 13 not visible visible visible background not visible not visible background 14 barely visible visible visible visible background not visible multithermal 6.30 15 not visible visible visible visible background not visible background 16 not visible visible background background background not visible background 17 not visible visible background background not visible not visible background 18 not visible visible not visible not visible background background isothermal 5.80 19 not visible visible not visible not visible not visible not visible isothermal 5.80 20 visible visible visible barely visible background background multithermal 6.31 21 visible visible not visible not visible not visible not visible eff isothermal 5.43^+.01_-.02, 5.78^+.04_-.03 22 background visible visible visible background background multithermal 6.98 23 background visible visible visible visible background multithermal 6.64 24 not visible visible not visible not visible not visible not visible isothermal 5.80 25 not visible visible background not visible not visible not visible background 26 barely visible visible background not visible not visible not visible background 27 barely visible visible background background not visible not visible background 28 visible visible background background not visible not visible background 29 background visible background background not visible not visible background 30 background visible background background not visible not visible background 31 visible visible background barely visible not visible not visible background 32 not visible visible background background not visible not visible background 33 not visible visible background not visible not visible not visible background 34 not visible visible background not visible not visible not visible background 35 barely visible visible barely visible background not visible not visible multithermal 6.32 36 not visible visible visible background not visible not visible background 37 background visible not visible not visible not visible not visible background Region B 1 visible visible background not visible not visible not visible background 2 visible visible not visible not visible not visible not visible eff isothermal 5.38^+.02_-.04, 5.88^+.05_-.04 3 not visible visible not visible not visible not visible not visible isothermal 5.80 4 not visible visible not visible not visible not visible not visible isothermal 5.80 5 visible visible visible visible not visible barely visible multithermal 6.05 6 visible visible visible background not visible background background 7 barely visible visible background not visible not visible not visible background 8 visible visible visible visible background background multithermal 6.01 9 barely visible visible visible visible not visible not visible multithermal 6.16 10 not visible visible not visible not visible not visible not visible isothermal 5.80 11 visible visible not visible not visible not visible not visible eff isothermal 5.37^+.01_-.02, 5.91^+.05_-.03 12 visible visible visible visible visible barely visible multithermal 5.96 13 visible visible barely visible not visible not visible background eff isothermal 6.40 14 barely visible visible not visible not visible not visible not visible eff isothermal 5.41^+.03_-.07, 5.81^+.16_-.06 15 barely visible visible not visible not visible not visible not visible eff isothermal 5.41^+.03_-.07, 5.81^+.16_-.06 16 background visible visible background background background background 17 background visible background not visible not visible not visible background 18 background visible background background background background background 19 visible visible visible visible visible visible multithermal 6.33 20 not visible visible background background background background background 21 visible visible not visible not visible not visible not visible background 22 background visible background background not visible background background 23 barely visible visible not visible not visible not visible background eff isothermal 5.35^+.01_-.04, 5.96^+.05_-.06 24 not visible visible background background not visible not visible background 25 barely visible visible not visible not visible background background eff isothermal 5.35^+.01_-.04, 5.96^+.05_-.06 26 barely visible visible background background background background background 27 background visible not visible not visible not visible not visible background 28 barely visible visible background background not visible background background 29 barely visible visible barely visible background not visible not visible multithermal 6.09 30 background visible not visible not visible not visible not visible background 31 visible visible not visible not visible not visible not visible eff isothermal 5.36^+.02_-.06, 5.93^+.17_-.06 32 visible visible visible visible background not visible multithermal 6.45 33 not visible visible not visible not visible not visible not visible isothermal 5.80 34 not visible visible visible background not visible not visible background 35 barely visible visible visible barely visible background background unconstrained 36 not visible visible visible barely visible not visible not visible multithermal 6.14 37 visible visible visible visible barely visible background multithermal 6.70 38 background visible not visible not visible not visible not visible background 39 barely visible visible visible visible background not visible multithermal 6.33 40 barely visible visible background background background not visible background 41 barely visible visible not visible not visible not visible not visible eff isothermal 5.37^+.03_-.05, 5.90^+.11_-.06 42 not visible visible visible barely visible not visible not visible multithermal 6.00 43 visible visible visible visible visible barely visible multithermal 5.91 44 visible visible barely visible barely visible background not visible eff isothermal 5.70 45 background visible visible barely visible barely visible not visible background 46 visible visible not visible not visible not visible not visible eff isothermal 5.38^+.03_-.11, 5.88^+–_-.06 47 barely visible visible visible barely visible background background multithermal 6.60 48 visible visible background not visible not visible not visible background 49 visible visible visible visible barely visible not visible multithermal 5.98 50 visible visible visible barely visible barely visible background multithermal 6.49 51 visible visible visible background not visible not visible background Region C 1 visible visible visible barely visible background not visible background 2 visible visible visible background background not visible multithermal 6.14 3 visible visible not visible not visible not visible not visible eff isothermal 5.35^+.01_-.04, 5.96^+.05_-.06 4 barely visible visible background background background background background 5 visible visible visible visible background not visible multithermal 6.28 6 visible visible visible visible visible background small flare 6.74 7 visible visible visible visible background background multithermal 6.60 8 visible visible visible barely visible not visible not visible multithermal 6.01 9 not visible visible background not visible not visible not visible background 10 background visible barely visible not visible not visible not visible background 11 background visible visible barely visible not visible not visible background 12 not visible visible background background not visible not visible background 13 barely visible visible not visible not visible not visible not visible eff isothermal 5.32^+.05_—, 6.00^+–_-.10 Region D 1 visible visible visible visible background background multithermal 6.22 2 visible visible visible background background barely visible multithermal 6.13 3 visible visible visible background not visible background background 4 barely visible visible background background background background background 5 visible visible visible background background barely visible multithermal 6.30 6 visible visible visible background background not visible multithermal 6.11 7 visible visible visible visible barely visible visible multithermal 6.20 8 visible visible visible visible barely visible visible multithermal 6.50 9 visible visible visible background not visible not visible multithermal 6.22 10 background visible visible background background background background 11 visible visible visible visible visible visible multithermal 6.30 12 barely visible visible visible visible barely visible barely visible inconclusive 13 barely visible visible visible background background background background 14 visible visible visible visible visible visible multithermal 6.05 15 background visible background background background background background 16 visible visible visible visible background visible multithermal 6.21 17 barely visible visible visible visible not visible background multithermal 6.26 18 barely visible visible visible visible background barely visible multithermal 6.18 19 visible visible visible visible background not visible multithermal 6.18 20 visible visible visible barely visible background barely visible background 21 visible visible visible visible background background multithermal 6.06 22 background visible background background background background background 23 visible visible visible background background background background 24 background visible not visible not visible not visible background background 25 visible visible background not visible not visible not visible background 26 visible visible visible visible visible barely visible multithermal 5.96 27 barely visible visible barely visible background not visible not visible background 28 background visible barely visible barely visible background not visible background 29 background visible background background background background background 30 background visible background not visible not visible background background Region E 1 background visible background background background not visible background 2 visible visible not visible not visible not visible barely visible eff isothermal 6.10 3 barely visible visible background not visible not visible not visible background 4 background visible barely visible not visible not visible not visible background 5 not visible visible not visible not visible not visible not visible isothermal 5.80 6 not visible visible background not visible not visible not visible background 7 barely visible visible not visible not visible not visible background eff isothermal 5.28^+.09_—, -.–^+–_— 8 visible visible visible background not visible background multithermal 6.11 9 visible visible not visible not visible not visible background eff isothermal 5.35^+.02_-.07, 5.94^+–_-.07 10 visible visible visible barely visible background background background 6.05 11 visible visible visible barely visible background background multithermal 6.06 12 barely visible visible not visible not visible not visible background eff isothermal 5.36^+.01_-.02, 5.93^+.05_-.05 13 barely visible visible visible barely visible not visible background multithermal 6.07 14 visible visible not visible not visible not visible background eff isothermal 5.35^+.03_-.03, 5.95^+.10_-.04 15 visible visible visible visible background background multithermal 6.06 16 background visible background background not visible background background 17 visible visible visible visible visible visible multithermal 6.05 18 barely visible visible visible background background barely visible multithermal 6.29 19 background visible background background background background background 20 barely visible visible not visible not visible not visible not visible eff isothermal 5.36^+.02_-.21, 5.93^+–_-.06 21 barely visible visible not visible not visible not visible not visible eff isothermal 5.37^+.03_-.02, 5.88^+.08_-.04 22 barely visible visible background not visible not visible not visible background 23 visible visible visible visible visible visible multithermal 6.15 24 barely visible visible not visible not visible not visible background eff isothermal 5.36^+.02_-.34, 5.94^+–_-.07 25 visible visible barely visible not visible not visible not visible eff isothermal 6.05 26 background visible background background background background background Region F 1 visible visible visible barely visible barely visible barely visible multithermal 5.93 2 barely visible visible background not visible not visible background background 3 visible visible visible visible background background multithermal 6.04 4 visible visible background background not visible not visible background 5 barely visible visible not visible not visible not visible not visible eff isothermal 5.35^+.03_-.30, 5.94^+–_-.07 6 background visible visible visible background background multithermal 6.14 7 background visible not visible not visible not visible not visible background 8 barely visible visible not visible not visible not visible not visible eff isothermal 5.37^+.03_-.04, 5.90^+.10_-.05 9 barely visible visible background background not visible not visible background 10 barely visible visible barely visible barely visible barely visible not visible multithermal 6.31 11 visible visible visible barely visible background background background 12 background visible background background not visible not visible background 13 visible visible visible barely visible background not visible background 14 barely visible visible background not visible not visible background background 15 visible visible visible visible background not visible multithermal 6.01 16 visible visible visible visible visible visible multithermal 6.50 17 barely visible visible background background not visible not visible background 18 background visible background background background background background 19 visible visible not visible not visible not visible not visible eff isothermal 5.35^+.02_-.15, 5.96^+–_-.13 20 barely visible visible visible visible barely visible visible multithermal 6.07 21 visible visible visible visible background background multithermal 6.38 22 barely visible visible background background background background background 23 background visible visible barely visible background background background 24 barely visible visible visible barely visible background background background 6.05 25 background visible not visible not visible not visible not visible background 26 barely visible visible background background not visible not visible background 27 visible visible background not visible not visible not visible background 28 visible visible not visible not visible not visible not visible eff isothermal 5.36^+.02_-.05, 5.92^+.15_-.05 29 background visible background background background background background 30 visible visible not visible not visible not visible not visible eff isothermal 5.37^+.02_-.02, 5.89^+.07_-.03 31 background visible visible visible not visible not visible multithermal 6.20 | http://arxiv.org/abs/1705.09360v1 | {
"authors": [
"J. T. Schmelz",
"G. M. Christian",
"R. A. Chastain"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170525210219",
"title": "The Coronal Loop Inventory Project: Expanded Analysis and Results"
} |
=8.5truein =11trueinarrowspackedItem packedEnum17cm 22.4cm theoremTheorem corollary[theorem]Corollary lemma[theorem]Lemma proposition[theorem]Proposition obs[theorem]Observation conjecture[theorem]Conjecture definition question[theorem]Question definition[theorem]Definition example[theorem]Example remark[theorem]Remark fact[theorem]Fact equationsection Reconfiguration graphs of shortest pathsThis project was initiated as part of the REUF program at AIM, NSF grant DMS 1620073. John AsplundDepartment of Technology and Mathematics, Dalton State College, Dalton, GA 30720, USA [email protected] EdohDepartment of Mathematics North Carolina Agricultural and Technical State University Greensboro, NC 27411, USA [email protected] Ruth Haas Partially supported by Simons Foundation Award Number 281291Department of Mathematics, University of Hawaii at Manoa,Honolulu, Hawaii 96822, [email protected] Smith College, Northampton MA 01063Yulia HristovaDepartment of Mathematics and Statistics,University of Michigan - Dearborn,Dearborn, MI 48128, [email protected] NovickDepartment of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA [email protected] WernerDepartment of Mathematics, Computer Science & Cooperative Engineering, University of St. Thomas, Houston, TX 77006, USA [email protected] ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= For a graph G and a,b∈ V(G),theshortest path reconfiguration graph of G with respect to a and b is denoted by S(G,a,b). The vertex set of S(G,a,b) is the set ofall shortest paths between a and b in G. Two vertices in V(S(G,a,b)) are adjacent,if their corresponding paths in G differ by exactly one vertex. This paper examines theproperties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater.We also show that the shortest path graph of a grid graph is an induced subgraph of a lattice. § INTRODUCTION The goal of reconfiguration problems is to determine whether it is possible to transform one feasible solution s into a target feasible solution t in a step-by-step manner (a reconfiguration) such that each intermediate solution is also feasible. Such transformations can be studied via the reconfiguration graph, in which the vertices represent the feasible solutions and there is an edge between two vertices when it is possible to get from one feasible solution to another in one application of the reconfiguration rule. Reconfiguration versions of vertex coloring <cit.>, independent sets <cit.>, matchings <cit.>, list-colorings <cit.>, matroid bases <cit.>, and subsets of a (multi)set of numbers <cit.>, have been studied.This paper concerns the reconfiguration of shortest paths in a graph. Let G be a graph with distinct vertices a and b. The shortest path graph of G with respect to a and b is the graph S(G,a,b) in which every vertex U corresponds to a shortest path in G between a and b, and two vertices U,W ∈ V(S(G,a,b)) are adjacent if and only if their corresponding paths in G differ in exactly one vertex.While there have been investigations into shortest path reconfiguration in<cit.>, these papers focused on the complexity of changing one shortest path into another[The shortest path graph is denotedby SP(G,a,b) in <cit.>.]. It was found in <cit.> that the complexity of this problem is PSPACE-complete.In contrast, the focus of our work is on the structure of shortest path graphs, rather than algorithms. Our main goal is to understand which graphs occur asshortest path graphs.A similar study on classifying color reconfiguration graphs can be found in <cit.>. The paper is organized as follows. Some definitions and notations are provided in Section <ref>. Section <ref>contains some useful properties and examples.In particular, we show that paths and complete graphs are shortest path graphs.In Section <ref>we show that the family of shortest path graphs is closed underdisjoint union and under Cartesian products. We establisha decomposition resultwhich suggests that, typically, 4-cycles are prevalent in shortest path graphs.Thus, we would expect the structure of shortest path graphs containing no 4-cycles to be rather simple. This is substantiated in Section <ref>,where we give a remarkably simple characterization of shortest path graphs with girth 5 or greater.In the process of establishing this characterization, we show that the claw and the odd cycle C_k,for k>3 are, in a sense, forcing structures. As a consequence, we determine precisely which cycles are shortest path graphs;that the claw, by itself, is not a shortest path graph;and that a tree cannot be a shortest path graph unless it is a path. In contrast, our main theorem in the final section of the paper involves a class of shortest path graphs which contain many 4-cycles. We establish that the shortest path graph of a grid graph is an induced subgraph of the lattice. One consequence of our construction is that the shortest path graph of the hypercube Q_n with respect to two diametric vertices is a Cayley graph on the symmetric group S_n. § PRELIMINARIESLet G be a graph with distinct vertices a and b.A shortest a,b-path in G is a path between a and b of length d_G(a,b).When it causes no confusion, we write d(a,b) to mean d_G(a,b).We often refer to a shortest path as a geodesic and to a shortest a,b-path as an a,b-geodesic. Note that any subpath of a geodesic is a geodesic.If the paths corresponding totwo adjacentvertices U, W in S(G,a,b) are a v_1⋯ v_i-1v_i v_i+1⋯ v_pb and av_1⋯ v_i-1v_i'v_i+1⋯ v_pb, we say that U and W differ in the i^ th index, or that i is the difference index of the edge UW. We call thegraph Gthe base graph of S(G,a,b), and we say that a graph H is a shortest path graph, if there exists a graph G with a,b∈ V(G) such that S(G,a,b)≅ H. Several examples are given in Figure <ref>. With a slight abuse of notation, a label for a vertex in the shortest path graph will often also represent the corresponding path in itsbase graph. To avoid confusion between vertices in G and vertices in S(G,a,b), throughout this paper,we will use lower case letters to denote vertices in the base graph,and upper case letters to denote vertices in S(G, a,b).It caneasily be seen that several base graphs can have the same shortest path graph. For example,ife∈ E(G) and e is an edge not in any a,b-geodesic, then S(G, a, b) ≅ S(G∖ e, a, b). To this end, wedefine the reduced graph, (G,a,b),to be the graph obtained from G by deletingany edge or vertex that does not occur in any a,b-geodesic,and contracting any edge that occursin all a,b-geodesics. If the reduced graph (G, a, b) is again G then G iscalled areduced graph with respect to a, b.We may omit the reference to a,b when it is clear from context. We conclude this section with a review ofsome basic definitions. If G_1 and G_2 are graphs then G_1∪ G_2 is defined to be the graph whose vertex set is V(G_1)∪ V(G_2) and whose edge set is E(G_1) ∪ E(G_2).When V(G_1)∩ V(G_2) =∅ we say that G_1 and G_2 are disjoint, andrefer to G_1∪ G_2 as the disjoint union of G_1 and G_2. For two graphs G_1 and G_2, the Cartesian product G_1 □G_2 is a graph with vertex set V(G_1)× V(G_2) and edge set {(v_1,v_2)(u_1,u_2) :v_i,u_i∈ V(G_i)fori∈{1,2} and either v_1=v_2 andu_1∼ u_2,orv_1∼ v_2andu_1=u_2}. If U_1 is a v_0,v_ℓ-path and U_2 is a v_ℓ,v_m-path, where U_1 and U_2 have only one vertex in common,namely v_ℓ, then the concatenation of U_1 and U_2 is the v_0,v_m-path U_1∘ U_2=v_0v_1 … v_ℓ v_ℓ+1… v_m. A hypercube of dimension n,denoted Q_n, is the graph formed by labeling a vertex with each of the 2^n binary sequences of length n, and joining two vertices with an edge if and only if their sequences differ in exactly one position.§ GENERAL PROPERTIES, EXAMPLES,AND CONSTRUCTIONS In this section we answer some natural questions as to which classes of graphs are shortest path graphs. We easily see that theempty graph is a shortest path graph. We show that paths and complete graphs are shortest path graphs as well. Let G be a graph formed by joiningtpaths of equal length greater than 2,each having the same end vertices,a and b,and with all other vertices between any two paths being distinct. Then S(G,a,b)=K_t.Before finding shortest path graphs with edges, we make the following simple observationwhich willbe used implicitly throughout. Let H be a shortest path graph. If U_1U_2U_3 is an induced pathin H then U_1U_2 and U_2U_3 have distinct difference indices. We use this observation to construct a family of graphs whose shortest path graphs are paths. For any k≥ 1, the path P_k is a shortest path graph.For k≥ 1,define the graphG_kby V(G_k)= {a,b,v_0, v_1, … , v_⌊ k/2 ⌋, v_0',v_1', … v'_⌈ k/2 ⌉} and E(G_k)= {av_i,v_iv_i' :0≤ i ≤⌊ k/2 ⌋}∪{v_i-1v_i':1≤ i ≤⌈ k/2 ⌉}∪{v_i'b :0≤ i ≤⌈ k/2 ⌉}. One checks thatP_k ≅ S(G_k,a,b). For any n≥ 1 the complete graph K_n is a shortest path graph. Let a and b be the vertices on one side of the bipartition of K_2,n. Then S(K_2,n, a,b) ≅ K_n. In fact, as we show in the proof ofTheorem <ref>, any graphfor which the shortest path graph is a complete graph must reduce toK_2,n.We will see later that in generalthere can be different reduced graphs that have the same shortest path graph. S(G,a,b)=K_n for some n∈,if and only if each pair of a,b-geodesics in G differs at the same index.If every pair of a,b-geodesics differs only at the i^ th index, for some i, then it is clear that S(G,a,b)=K_n,where n is the number of a,b-geodesics in G.Now suppose that U,V and W∈ V(S(G,a,b)) are such that UV and VW have distinct difference indices. Then the paths U and W differ at two vertices, so there could be no edge between them in S(G,a,b). Hencethe reduced graph of G is K_2,n. It is clear that if two graphs give the same reduced graph with respect to a, b then they have the same shortest path graph. It will be useful to be able to construct different graphs with the same reduced graph. The next resultinvolves, in a sense, an operation which is the reverseof forming areduced graph. If H=S(G,a,b) and d_G(a,b) = k, then for any k'≥ k there exists a graph G' with vertices a, b'∈ G' such that d_G'(a, b') = k' and H≅ S(G',a,b').Suppose H= S(G, a, b). Define G' as follows:V(G')= V(G) ∪{x_1, x_2, … x_k'-k-1, b'}, and E(H') = E(H)∪{ bx_1, x_1x_2, …x_k'-k-2 x_k'-k-1,x_k'-k-1b' }.It is clear that H≅ S(G', a, b'). § DECOMPOSITIONS AND SUMS In the previous sectionweconstructed a few special classes of shortest path graphs.In the present sectionwe establish two methods of obtaining newshortest path graphs from old.In particular,we show that the family of shortest path graphs is closed underdisjoint unions and isclosed underCartesian products. IfH_1 and H_2areshortest path graphs, then H_1∪ H_2 is a shortest path graph.By Proposition <ref> we can choosedisjoint base graphs G_i for H_i, i∈{1,2}, such that {a_i,b_i}∈ G_i, with d_G_1(a_1,b_1) = d_G_2(a_2,b_2) and with H_i ≅ S(G_i,a_i,b_i). Construct a graph G as follows. Let V(G) = V(G_1) ∪ V(G_2) ∪{a, b} andE(G) = E(G_1) ∪ E(G_2) ∪{aa_1,aa_2, b_1b, b_2b}. It is clear by the construction of G that every a,b-geodesic corresponds to an a_1,b_1-geodesic through G_1 or an a_2,b_2-geodesic through G_2. In addition, if two shortest paths are adjacent in S(G_i,a_i,b_i), i ∈{1,2}, they are still adjacent in S(G,a,b). If U_1 and U_2 are a,b-geodesics in G between a and b where V(U_2)∩ V(G_1)≠∅ and V(U_1)∩ V(G_2)≠∅,then since a_1, b_1∈ V(U_1)and a_2, b_2∈ V(U_2) we have U_1≁U_2.Thus the result holds. Proposition <ref> concernsthe structure of thesubgraph of a shortest path graph H≅ S(G,a,b) induced by all a,b-geodesics containing a givenvertex v. Let G be a connected graph with a, b∈ V(G) and d=d(a,b) ≥ 2. LetH=S(G,a,b)and let v be a vertex of G which is on at least one a,b-geodesic.Let H' be the subgraph of Hinduced by allvertices corresponding to a,b-geodesics which contain v.ThenH' ≅ S(G, a, v)□ S(G, v,b).Furthermore, ifG_1is any subgraph of Gcontainingall a,v-geodesics, and G_2isany subgraph of Gcontainingall v,b-geodesics, then H' is isomorphic to S(G_1,a,v) □ S(G_2,v,b).Let H' be the subgraph of H induced by all elements of V(H) corresponding to a,b-geodesics in G which contain the vertex v. These a,b-geodesics are precisely the concatenations T_av∘ T_vb where T_av is an a,v-geodesic and T_vb is a v,b-geodesic.Hence we speak interchangeably about the elements in the vertex set of H' and geodesic paths in G of the formT_av∘ T_vb.By definition, the vertex set of S(G,a,v)□S(G,v,b) is the collection of ordered pairs (T_av,T_vb) where T_av is a vertex of S(G,a,v), and T_vb is a vertex of S(G,v,b). It is clear that the mapping f: V(H') → V(S(G,a,v)□S(G,v,b)) given by f(T_av∘ T_vb) = (T_av,T_vb), is a bijection.We claim that this bijection is edge-preserving.Indeed, let (U_1,R_1)∼ (U_2,R_2)in S(G,a,v)□S(G,v,b). Then by definition of Cartesian product,either (i) U_1 ∼ U_2 in S(G,a,v) and R_1 = R_2 or (ii) U_1 = U_2 and R_1 ∼ R_2 in S(G,v,b). In the former caseU_1 and U_2 differ in exactly one index while V(R_1)=V(R_2), so U_1∘ R_1 ∼ U_2∘ R_2 in H'.An analogous argument holds in case (ii). Now assume that (U_1,R_1) is neither equal to nor adjacent to (U_2,R_2)in S(G,a,v)□S(G,v,b). Then one of the following occurs: U_1=U_2, in which caseR_1 and R_2 differ in at least two indices; R_1=R_2 in which case U_1 and U_2 differ in at least two indices;or U_1≠ U_2 and R_1≠ R_2 in which case U_1 and R_1 differ with U_2 and R_2 in at least one index respectively.In each of these cases R_1∘ U_1 and R_2∘ U_2 differ in at least two indices and hence are not adjacent, as required.To complete the proof, we simply note that S(G,a,v)≅ S(G_1,a,v) and that S(G,v,b)≅ S(G_2,v,b).For two graphs G_1 and G_2 with vertex sets such that V(G_1)∩ V(G_2)={c},the one-sum of G_1 and G_2 isdefined to be the graph G with vertex set V(G_1)∪ V(G_2) and edge set E(G_1)∪ E(G_2).Theorem <ref> characterizes the shortest path graph of the one-sum of two graphs. Let G_1 and G_2 be graphs with vertex sets such that V(G_1)∩ V(G_2)={c}. Let G be the one-sum of G_1 and G_2. Then for any a∈ V(G_1)∖{c} and any b∈ V(G_2)∖{c},S(G,a,b)≅ S(G_1,a,c) □S(G_2,c,b). Because c is a cut-vertex, every a,b-geodesic in G must contain c.The result now follows immediately from Proposition <ref>.LetH_1 and H_2 be shortest path graphs. Then H_1 □ H_2 is also a shortest path graph.Let H_1 = S(G_1,a_1,b_1) and H_2 = S(G_2,a_2,b_2),where G_1 and G_2 are reduced graphs. Identify b_1 with a_2 to obtain a graph (G,a_1,b_2) for which H_1 □ H_2 is the shortest path graph. The construction of Theorem <ref> leads toa family of graphs whose shortest path graphs are hypercubes. Let J_k be the graph formed by taking one-sums of k copies of C_4 as follows. For i=1,…, k let a_i and b_i be antipodal vertices in the i^ th copy ofC_4.Form J_kby identifying b_i and a_i+1 for i=1, … , k-1. See Figure <ref>. For J_k as defined above, S(J_k,a_1,b_k)≅ Q_k where Q_k is a hypercube of dimension k.For k=1, S(G,a_1,b_1)≅ P_1≅ Q_1. The proof follows by induction on k and from the statement and proof ofCorollary <ref>. The next result gives a decomposition of a shortest path graph into a disjoint set of one sums with additional edges. Note thatthe following theorem, holds for all1≤ i< d(a,b). Hence, there are actually d(a,b)- 1different decompositions of this sort. Let H be a shortest path graph with reduced base graph (G,a,b),where d(a,b)≥ 2. Fix an index i,with 1≤ i < d(a,b).Let {v_i_1,v_i_2, … , v_i_k} be the set of k vertices in V(G) of distance i from a, and let E_i be the set of all edges UW ∈ E(H) having difference index i. Then (a) the result of deleting the edges E_i from H yields a graph having k disjoint components, each of which is a Cartesian product: H∖ E_i = ⋃_j=1^kD_i_j,where D_i_j=S(G,a,v_i_j)□ S(G,v_i_j, b) and (b) For any two subgraphs D_i_j and D_i_ℓ, the edges in E_i between V(D_ij) and V(D_i ℓ) form a partial matching.For each j∈{1, 2, …, k},it follows fromProposition <ref>that the set of vertices in Hcorresponding to a,b-geodesics containing v_i_j induce a subgraph isomorphic to S(G,a,v_i_j)□ S(G,v_i_j,b). Since each a,b-geodesic in G contains precisely one vertex v_i_j∈{v_i_1, … ,v_i_k},these k induced subgraphsof H are vertex disjoint.Furthermore,any pair U, W of adjacent vertices in H whose corresponding a,b-geodesics contain distinct vertices in {v_i_1, … ,v_i_k},must differ in index i.We conclude that UW has difference index i and is in E_i. This establishes part (a).Each vertex U in D_i_jcorresponds to a path with vertex v_i_j at the i^ th index, and each vertex W in D_i_ℓ corresponds to a path with vertex v_i_ℓ at the i^ th index. Thus for each vertex U in D_i_j there is at most one vertex in D_i_ℓ adjacent to U.Note that 4-cycles occur very often in Cartesian products: Take any edge UWin H_1and any edge XYin H_2. Then the set of vertices { (U,X), (U,Y), (W,X), (W,Y)}induces a 4-cycle in H_1 □ H_2. From Theorem <ref>, part (a), and the fact that4-cycles are ubiquitous in Cartesian products ofgraphs, we conclude that 4-cycles are prevalent in shortest path graphs.In view of the fact that Theorem <ref> holds for everyindex i, Observation <ref> is especially strong:we expect shortest path graphs having no 4-cycles to have a relativelysimple structure, and we predictthe study of shortest path graphs with no such restriction to be more challenging.We conclude this section with another way to combine base graphs. Let G_1 and G_2 be graphs with edge sets such that E(G_1)∩ E(G_2) = {e},where e=xy,and V(G_1)∩ V(G_2) = {x,y}. The two-sum of G_1 and G_2 is defined to bethe graph G with vertex set V(G_1)∪ V(G_2) and edge set E(G_1)∪ E(G_2).Theorem <ref> characterizes the shortest path graph of the two-sum of two graphs.Let G_1 and G_2 be graphs with edge sets such that E(G_1)∩ E(G_2) = {e},where e=xy,and V(G_1)∩ V(G_2) = {x,y}. Let G be thetwo-sum of G_1 and G_2. Let a∈ V(G_1) and b∈ V(G_2) where {a,b}∩{x,y} = ∅.Then S(G, a,b) is isomorphic to one of the following: (i) the disjoint union S(G_1,a,x) □S(G_2,x,b) ⋃S(G_1,a,y) □S(G_2,y,b) plus additional edges which comprise a matching between the two, in the case that d(a,x)=d(a,y) and d(x,b)=d(y,b);(ii) S(G_1,a,x) □S(G_2,x,b), in the case that d(a,x)≤ d(a,y) and d(x,b) < d(y,b), or d(a,x)<d(a,y) and d(x,b) ≤ d(y,b);(iii) S(G_1,a,y) □S(G_2,y,b), in the case thatd(a,y)≤ d(a,x) and d(y,b) < d(x,b),or d(a,y)<d(a,x) and d(y,b) ≤ d(x,b);(iv) and otherwise S(G_1,a,x) □S(G_2,x,b) ⋃ S(G_1,a,y) □S(G_2,y,b),where vertices common toS(G_1,a,x) □S(G_2,x,b) and S(G_1,a,y) □S(G_2,y,b)correspond precisely to a,b-geodesics containing the edge e. Note that every a,b-geodesic has non-empty intersection with{x,y}.Case (i)Suppose,i=d(a,x)=d(a,y) and d(x,b)=d(y,b).In this case, the vertices x and yare the only vertices at distance i from a in G to be used in any a,b-geodesic. Let E_i be the set of all edges in S(G,a,b) having difference index i. If we notethat S(G,a,x),S(G,a,y), S(G,x,b) and S(G,y,b) are, respectively,isomorphic to S(G_1,a,x),S(G_1,a,y), S(G_2,x,b) and S(G_2,y,b),then it follows immediately from Theorem <ref>, part (a),thatS(G,a,b)/E_i ≅S(G_1,a,x) □S(G_2,x,b) ⋃S(G_1,a,y) □S(G_2,y,b).From part (b) of that same theorem,it follows directly that the edges connectingthe vertex disjoint components S(G_1,a,x) □S(G_2,x,b) and S(G,a,y) □S(G,y,b)form a matching. This completes the proof for Case 1.Case (ii) Either d(a,x)≤ d(a,y) and d(x,b) < d(y,b),or d(a,x)<d(a,y) and d(x,b) ≤ d(y,b). Every a,b-geodesic in G contains the vertex x, and the result follows directly from Theorem <ref>.Case (iii) Either d(a,y)≤ d(a,x) and d(y,b) < d(x,b), or d(a,y)<d(a,x) and d(y,b) ≤ d(x,b). Every a,b-geodesic in G contains the vertex y, and the result follows directly from Theorem <ref>. Case (iv) Consider firstwhend(a,x) > d(a,y)and d(x,b) < d(y,b).Since d(x,y)=1,we have that d(a,x)=d(a,y)+1 and d(x,b)+1=d(y,b). By Proposition <ref>,the vertices of S(G,a,b) which correspond to paths containing x induce a subgraph isomorphic to S(G_1,a,x)□ S(G_2, x,b), and those which correspond to paths containing y induce a subgraph isomorphic to S(G_1,a,y)□ S(G_2, y,b). Note that some a,b-geodesics contain the edge e=xy,and hence the two induced subgraphs described abovehave non-empty intersection.Now letU be a vertex in V(S(G,a,b))which corresponds to an a,b-geodesic containing x and not y,and let W be a vertex in V(S(G,a,b))which corresponds to an a,b-geodesic containing y but not x. Then U and W differ in both indexd(a,y) and indexd(a,y)+1 and hence are non-adjacent. The case d(a,x)< d(a,y) and d(x,b) > d(y,b) is handled analogously.This completes the proof. Note that in the proof of Theorem <ref>,the edge e is used only in Case (iv).Hence, if Case (i),(ii), or (iii) holds in the statement of that theorem,then S(G∖ e,a,b) ≅ S(G,a,b). Also note that results similar to Theorem <ref> can be obtained by considering joining two graphs at two verticeswith no edges between the pair; or indeed joining graphs on more than two vertices.§ SHORTEST PATH GRAPHS OF GIRTH AT LEAST 5 In this section, we completely classify all shortest path graphs with girth 5 or greater.In the process,we characterize precisely which cycles are shortest path graphs and we show that the claw is not a shortest path graph. The following simple observation will be crucial. Let H be a shortest path graph. Let U_1,U_2,U_3 be distinct vertices in H such that U_1U_2U_3 is an induced path. If the difference indices of U_1U_2 and U_2U_3 are i and j, respectively, where j∉{i-1,i,i+1}, then H has an induced C_4 containing U_1U_2U_3.Let U_1 = av_1 … v_pb, U_2 = av_1… v_i'… v_pb, and U_3 = av_1 … v_i-1v_i'v_i+1… v_j'… v_p b.Then there is a shortest path U_4=av_1 … v_j'… v_p b in G, creating the 4-cycle (U_1,U_2,U_3,U_4). The next result says that any shortest path graph containing an induced odd cycle larger than a 3-cycle must necessarily contain an induced C_4. Theorem <ref>establishes the same result for induced claws. Let H be a shortest path graph that contains an induced C_k for odd k > 3. Then H contains an induced C_4.Let (U_1, …, U_k) be an induced C_k with odd k>3 in H, and suppose that H does not contain an induced C_4. Let i be the difference index of U_1U_2. By Proposition <ref>, the difference index of U_2U_3 is either i-1 or i+1. In particular, if i is odd then U_2 and U_3 differ at an even index, and if i is even then U_2 and U_3 differ at an odd index. The same is true at every step, that is,the parityof the difference indexalternates around the cycle. This is impossible if k is odd. In contrast, C_3 and every even cycle are shortest path graphs. C_k is a shortest path graph, if and only if k is even or k=3.We have already seen that C_3 and C_4 are shortest path graphs, see Figure <ref> and Corollary <ref>. From Lemma <ref>, it follows that C_k is not a shortest path graph for odd k > 3.We now construct a graph whose shortest path graph is C_2n. Define Gwith2n+2 vertices namely V(G)= {a,b,v_0,v_1,…,v_n-1,v_0',v_1',…,v_n-1'} and edge setE(G)={av_i : i∈ℤ_n}∪{bv_i' : i∈ℤ_n}∪{v_iv_i': i ∈ℤ_n}∪{v_iv_i+1' :i∈ℤ_n},where indices are calculated modulo n. There are exactly 2n a,b-geodesics of G, namely the set {av_iv_i'b,av_iv'_i+1b:i=0,…,n-1}. It is easy to check that the shortest path graph S(G,a,b)=C_2n.If a shortest path graph H has an induced claw, K_1,3, then H must have a 4-cycle containing two edges of the induced claw. In particular, K_1,3 is not a shortest path graph.Let H be a shortest path graph that contains an induced claw with vertices U_0,U_1,U_2,U_3, such that U_0 is adjacent to U_1, U_2, and U_3.Leti_j be the difference index of U_0U_j for j∈{1, 2, 3}.Since the claw is induced, these difference indices must be distinct. Suppose that no three vertices of the claw are part of an induced 4-cycle. By Proposition <ref>, since U_0, U_1, U_2 is not a part of an induced 4-cycle, it follows that i_2= i_1 ± 1. Without a loss of generality let i_2=i_1+1. Similarly, because U_0, U_1, U_3 is not a part of an induced 4-cycle, we havei_3= i_1 ± 1. Since the indices i_j are distinct, it must be that i_3=i_1-1. By Proposition<ref> it follows thatU_0, U_2, U_3 is in an induced 4-cycle in S(G,a,b).An immediate consequence of Theorem <ref> is the following observation.If H is a tree and a shortest path graph, then H is a path. Next we establish a characterization of when C_k can be an induced subgraph of some shortest path graph. C_k is an induced subgraph of some shortest path graph if and only if k ≠ 5.Assume to the contrary, that S(G,a,b) contains an induced C_5, sayU=(U_1,U_2,U_3,U_4,U_5). Consider the difference indices along the edges of the cycle. Every difference index that occurs must occur at least twice in order to return to the original shortest path.Thus a 5-cycle can use at most 2 distinct difference indices.Furthermore, if a difference index occurs twice in a row, say forU_1U_2 and for U_2U_3, then the edge U_1U_3 is also in S(G, a,b). Therefore, C_5 is not an induced subgraph of a shortest path graph.To finish the proof, we show that for any k ≠ 5, C_k is an induced subgraph of some shortest path graph. In Theorem <ref> we saw that C_k is itself a shortest path graph when k=3 or when k is even. Thus, we only need to consider odd k>6. Suppose that k=2p+1 and let G_2p+1 be a graph with vertex set V(G_2p+1)={a,b,v_1,v_2,…,v_p,v_1',v_2',…,v_p',v_1”} and edge setE(G_2p+1)= {av_1, av_1', bv_p, bv_p',av_1”,v_1”v_2',v_1”v_2}∪{v_iv_i+1,v_i'v_i+1',v_iv_i+1',v_i'v_i+1 : i∈{1,2,…,n-1}}Then the following pathsof G_2p+1 induce a C_2p+1 inS(G_2p+1, a, b):[ av_1v_2v_3⋯ v_pb,;av_1'v_2v_3⋯ v_pb,; av_1'v_2'v_3⋯ v_pb,;…,; av_1'v_2'v_3'⋯ v_p'b,; av_1”v_2'v_3'⋯ v_p'b,;av_1”v_2v_3'⋯ v_p'b,; av_1”v_2v_3v_4'⋯ v_p'b,;…,; av_1”v_2v_3⋯ v_p-1 v_p'b,;av_1v_2v_3⋯ v_p-1v_p'b; ] Let H be a graph with (H) ≥ 5. Then H is a shortest path graph,if and only if each nontrivial component of H is a path or a cycle of even length greater than 5.If (H) ≥ 5, by Theorem <ref> there is no vertex U ∈ V(H) with degree _H(U) ≥ 3.Thus each vertex in H must have degree 0, 1, or 2. From this, it follows that every nontrivial component of H is a path or cycle. By Lemma <ref>, any induced odd cycle forces an induced C_4. Therefore all of these cycles must have even length.By Lemma <ref>, a path of any length is attained. In Theorem <ref>, it was shown how to construct a shortest path graph that is a cycle of even length. Finally, by Theorem <ref>, the disjoint union of any set of shortest path graphs is again a shortest path graph. Now that shortest path graphs ofgirth 5 or morehave been characterized, a natural next step would be to work towards a characterization of girth 4 shortest path graphs.The prevalence of Cartesian products in shortest path graphs tends to indicate that 4-cycles will play a large and challenging role in the study of these graphs. We leave this challenge to a future paper and instead characterizethe shortest path graphsof grid graphs, which haveparticularly nice structure. We study these in the next section.§ SHORTEST PATHS IN GRID GRAPHS An m-dimensional grid graph is the Cartesian product of m paths, P_n_1□⋯□ P_n_m. We denote the vertices of P_n_1□⋯□ P_n_m with the usual Cartesian coordinates on the m-dimensional lattice, so the vertex set is given by V(P_n_1□⋯□ P_n_m) = {(x_1,x_2,…, x_m) : x_i ∈ℤ, 0 ≤ x_1 ≤ n_1, …, 0 ≤ x_m ≤ n_m}.In what follows, we consider the geodesics between two diametric vertices of a grid graph, i.e., the shortest paths between the origin and the vertex (n_1,…,n_m). Because these two vertices will always be the vertices of consideration for grid graphs, we will denote the shortest path graph of P_n_1□⋯□ P_n_m with respect to them simply by S(P_n_1□⋯□ P_n_m). A two-dimensional grid graph and the diametric vertices under consideration are shown in Figure <ref>.For convenience of notation, we will consider the shortest paths in P_n_1□⋯□ P_n_m as a sequence of moves through the grid in the following way. For 1 ≤ i ≤ m, let 𝐞_𝐢 be the i^ th standard basis vector in ℝ^m. A move from a vertex (x_1,…,x_i,…,x_m) ∈ V(P_n_1□⋯□ P_n_m) in the 𝐞_𝐢 direction means that the next vertex along the path is (x_1,…,x_i+1,…,x_m).Note that a shortest path in P_n_1□⋯□ P_n_m from (0,…,0) to (n_1,…,n_m) consists of exactly N=∑_i=1^m n_i moves, n_i of which are in the 𝐞_𝐢 direction. Furthermore, observe that any m-ary sequenceof length N in which the symbol i occurs exactlyn_i times corresponds to a geodesic in P_n_1□⋯□ P_n_m andthat there are (∑_i=1^m n_i)!/n_1!⋯ n_m! such shortest paths. Explicitly, a geodesic Uwill be denoted by them-ary sequence U= s_1 …s_N∈ℤ_m^N, where s_j = i ∈{1,…,m} if the j^ th move in U is in the 𝐞_𝐢 direction. In this way, the symbol 1 corresponds to a move in the 𝐞_1 direction, 2 corresponds to a move in the 𝐞_2 direction, etc. We will refer to U as the sequence representation of U and use B_n_1,…,n_m⊂ℤ_m^N to denote the set of allsequences with each i occurring exactly n_i times.Two shortest paths in S(P_n_1□⋯□ P_n_m) are adjacent if and only if they differ by a single vertex, i.e., if one path can be obtained from the other by swapping a single pair of two consecutive moves in different directions (see Figure <ref>). Thus, if U and W are two shortest paths, then U ∼ W if and only if their sequence representations inB_n_1,…, n_m, U and W, respectively, have the forms U = s_1 … s_is_i+1… s_N and W = s_1 … s_i+1s_i… s_N where s_i ≠ s_i+1.It follows that UW∈ E( S(P_n_1□⋯□ P_n_m)) if and only if thetwo sequences U, W∈ B_n_1,…,n_m can be obtained from eachother by switching two different consecutive symbols.The main result of this section is that S(P_n_1□⋯□ P_n_m) isisomorphic to an induced subgraph of the integer lattice graphℤ^M, where M = ∑_i=2^m(i-1)n_i. To prove this result we define a mapping from B_n_1,…,n_m⊂ℤ_m^N to coordinates in ℤ^M. The shortest path graph of P_n_1□⋯□ P_n_m is isomorphic to an induced subgraph of the integerlattice graphℤ^M, where M = ∑_i=2^m(i-1)n_i.Consider the m-dimensional grid graphP_n_1□⋯□ P_n_m, withB_n_1,…,n_m⊂ℤ_m^N the set of allm-ary sequences corresponding to itsgeodesics. Define a map ϕ: B_n_1,…,n_m→ℤ^M. For a sequence U∈ B_n_1,…,n_m, let [ ϕ(U):= (a_121,…, a_12n_2,; a_131,…, a_13n_3,a_231,…, a_23n_3,; …; a_1m1,…, a_1mn_m,a_2m1,…, a_2mn_m,… , a_(m-1)m1,…, a_(m-1)mn_m), ]where a_ijk is the number of i's following the k^ th j in U. For example, if U=32121231 ∈ B_3,3,2, then ϕ(U) =(3,2,1,3,1,3,0). Also, ϕ maps the sequence 1⋯ 1 2⋯2 ⋯ m⋯ m to the origin. Thus, ϕ maps B_n_1,…,n_m into a set of vectors (a_ijk) ∈ℤ^M such that 2≤ j ≤ m,1 ≤ i < j, and 1 ≤ k ≤ n_j, in the order indicated. Note that for all i,j,k, a_ijk≤ n_i since there are at most n_i i's following a j, and a_ijk≥ a_ij(k+1) since at least as many i's appear after thek^ th j than after the (k+1)^ st j.To see that ϕ is injective, consider two distinct sequences U = s_1… s_N and W = s_1'… s_N' in B_n_1,…,n_m, and denote their images under ϕ by A and A', respectively. Let r be the first index where the entries of U and W differ. Without loss of generality, assume that s_r = j > s_r' = i and that s_r is the k^ th j occurring in U. Then, the k^ th j inW will appear after s_r' = i, so the number of i's in W following the k^ th j will be at least one less as compared to the sequence U. Therefore, if a_ijk and a_ijk' are the ijk-components of A and A', respectively, then a_ijk > a_ijk', showing ϕ(U) ≠ϕ(W).To finish the proof, we need to show that ϕ preserves adjacency. Suppose U and W are adjacent sequences in B_n_1,…,n_m. Then U and W have the forms U=s_1… s_rs_r+1… s_N and W=s_1… s_r+1s_r … s_N, where s_r ≠ s_r+1. Let s_r+1=j> s_r = i. Now, suppose that s_r+1 is the k^ th j appearing in U. Then the only difference in the vectors ϕ(U) and ϕ(W) is that the ijk-component of ϕ(W) is increased by one unit, so ϕ(U) and ϕ(W) are adjacent vertices in ℤ^M. To see that ϕ^-1 also preserves adjacency, consider two adjacent vertices, A and A', in the image of ϕ. Let a_ijk be the ijk-component of A, and without loss of generality, assume that A' is obtained from A by increasing a_ijk to a_ijk+1. Because A' is in the image of ϕ, it follows that the symbol directly preceding the k^ th j in ϕ^-1(A) is i. To see this, first note that there must be at least one i preceding the k^ th j. Otherwise, a_ijk = n_i and cannot be increased. If there is any other symbol between the k^ th j and the i preceding it, other components of A must be changed in order to increase a_ijk. However, A and A' have only one different component. Thus, ϕ^-1(A') can be obtained from ϕ^-1(A) by switching the k^th j and the i directly preceding it. Therefore, ϕ^-1(A) and ϕ^-1(A') are adjacent vertices in B_n_1,…,n_m. The shortest path graph of a two-dimensional grid graph P_n_1□ P_n_2 is particularly easy to characterize, as demonstrated in the following corollary. First, we need an additional definition. The staircase graph S_n_1,n_2 is an induced subgraph of the grid graph on the integer lattice ℤ^n_2.S_n_1,n_2 has vertex set V(S_n_1,n_2) = {(a_1,…,a_n_2) : a_k ∈ℤ, n_1≥ a_1 ≥ a_2 ≥⋯≥ a_n_2≥ 0} (see Figure <ref>). The shortest path graph of P_n_1□ P_n_2 is isomorphic to the staircase graph S_n_1,n_2. We have seen that the shortest path graph S(P_n_1□ P_n_2) can be described as a graph on theset of binary strings B_n_1,n_2. Furthermore, from the proof of Theorem <ref>, the mapping ϕ:B_n_1,n_2→ℤ^n_2 defined by ϕ(U) := (a_1, a_2…,a_n_2), where a_k is the number of 1's following the k^ th occurrence of 2 in U, is injective and adjacency preserving. Therefore, this corollary follows if we can show ϕ(B_n_1,n_2) = V(S_n_1,n_2). Let U∈ B_n_1,n_2 and let ϕ(U) = (a_1, a_2…,a_n_2). From the definition of ϕ, it follows that n_1≥ a_k ≥ a_k+1≥ 0 for all k ∈{ 1,…,n_2-1}, since the number of 1's following the k^ th 2 is greater than or equal to the number of 1's following the (k+1)^ st 2. Thus, ϕ(U) ∈ V(S_n_1,n_2). Conversely, for anyA=(a_1,…,a_n_2) ∈ V(S_n_1,n_2), let U be the sequence in B_n_1,n_2 that has exactly a_k 1's following the k^ th 2 for k∈{1,…,n_2}. Then, ϕ(U) = A showing V(S_n_1,n_2) ⊆ϕ(B_n_1,n_2).Theorem <ref> implies that the dimension of the lattice graph ℤ^M of which S(P_n_1□⋯□ P_n_m) is an induced subgraph depends on the ordering of n_1,…, n_m. SinceM=∑_i=2^m(i-1)n_i, the least value forM will occur when n_1,⋯, n_m arelisted in decreasing order. It is a direct consequence of our discussion on grid graphs that the path of length k is the shortest path graph of P_k□ P_1.For the grid G=P_k□ P_1, S(G)≅ P_k.Earlier we made the comment that two base graphs may produce the same shortest path graph. In fact, even two reduced graphs can have the same shortest path graph, e.g., thegraphs P_k□ P_1 and G_k given inLemma <ref> have the same shortest path graph yet they are reduced and non-isomorphic.Another special grid graph is the m-dimensionalhypercube, Q_m=P_1 □⋯□ P_1. We shall observein Proposition <ref>, that S(Q_m) is isomorphic to a Cayley graph of the symmetric group S_m. We firstrecall some material from elementary group theory and algebraic graph theory. See<cit.> for more detail. Let (Γ, ·) be a group. Let S be a generating set of Γ that does not contain the identity element and such thatfor each g∈ S, g^-1 is also in S. The Cayley graph of Γ with generating set S, denoted by(Γ; S),is the graph whose vertices are the elements of Γ, and which has an edge between two vertices x and y if and only if x· s = y for some s∈ S.The symmetric group S_m is the group whose elements are the permutations on the set {1, 2, …, m}. An element ofS_m is a bijection fromthe set {1, 2, …, m} to itself.Denote by s_1s_2 … s_m the permutation σ given by σ(i)=s_i, 1≤ i ≤ m.The group operation in S_m is the composition of permutations defined by (στ )(j)=σ(τ(j)), for σ, τ∈ S_m,1≤ j≤ m. An adjacent transposition is a permutation τ_i such thatτ_i(i) = i+1, τ_i(i+1) = i , and τ_i(k) = k fork∉{i,i+1}. It is well known that every permutation can be represented as the composition of finitely many adjacent transpositions. Therefore, the set of adjacent transpositions, T, generates S_m. We also note that each adjacent transposition is its own inverse. Hence, we can define the Cayley graph of S_m with generating set T, (S_m; T ).[This Cayley graph was studied by Bacher <cit.> and is also known, in other contexts, as a Bubble Sort Graph.]To gain some insight into the structure of(S_m; T),consider the effect of the composition of an element σ∈ S_m with an adjacent transposition. Let σ = s_1s_2 … s_m and 1≤ i≤ m-1. Thenστ_i(i) = σ(i+1), στ_i(i+1) = σ(i),and στ_i(k) =σ( k),ifk ∉{i,i+1},or simplyστ_i = s_1s_2 … s_i+1j_i … s_m.Thus, the effect of the composition στ_i is the switching of the two consecutive elements s_i and s_i+1in σ. We can conclude that the neighborhood of a vertex σ in (S_m; T ) is the collection of all (m-1) permutations on {1,…,m} obtained from σ by interchanging two consecutive elements. Nowconsider the m-dimensional hypercube, Q_m=P_1 □⋯□ P_1. The vertices ofP_1 □⋯□ P_1 correspond to all binary strings of length m.A shortest path in Q_m is a sequence of exactly one move in each direction.In thediscussion preceding Theorem <ref> we introduced acorrespondence between the geodesics of P_n_1□⋯□ P_n_m and the set of sequences of moves B_n_1,…,n_m. For Q_m, this is a bijection between the geodesics in P_1 □⋯□ P_1 and the sequences in B_1,…,1. Sincein each geodesic a vertex coordinate changes exactly once, B_1,…,1 coincides with the set of permutations on {1, 2, …, m}.The sequence representation of U is denoted by U = s_1s_2 … s_m,where each element of {1, …, m} appears precisely once.Recall that we defined two sequences in B_1,…,1 as adjacent, if and only if one can be obtained from the other by switching two (different) consecutive symbols. This is equivalent to two permutations being adjacent if and only if one can be obtained from the other using an adjacent transposition. Thus, the set B_1,…,1 together with the adjacency relation is isomorphic to (S_m; T ).Hence we observe: Let S_m be the symmetric group,let T be the set of adjacent transpositions, and let a and b be diametric vertices on Q_m.Then S(Q_m,a,b) ≅(S_m; T ). Although the function ϕ introduced in the proof of Theorem <ref> is not needed in Proposition <ref>,that function has an interesting interpretation in the case of Q_m.Here the domain of ϕ is B_1,…,1,which as we have discussed,is isomorphic to S_m,the set of permutations of {1,2, …, m}.Referring to the definition of ϕ in the proof of Theorem <ref>,and using the notation introduced there, the image ofϕ is a sequence whose elements are a_ijk,where 1≤ i <j ≤ m and 1≤ k ≤ n_j. In the case of Q_m, n_j =1 for each j. Hence, for a sequence U∈ B_1, …, 1, corresponding to a permutation s_1s_2 ⋯ s_m,it makes sense to simplify our notation to ϕ(U):=(a_12, a_13, a_23,…, a_1m,a_2m, …, a_(m-1)m),where a_ij is equal to the number of i's following the 1^ st (and only) j in U. FromTheorem <ref>, the length of the sequence ϕ(U) isM = ∑_i=2^m(i-1)n_i= ∑_i=1^m(i-1)= m2. Since every element in {1,2, … ,m} occurs precisely once in the permutation s_1s_2 … s_m,we have that for every pair i,j with 1≤ i<j≤ m, a_ij =0 ifi occurs before j and1 if i occurs after j.In the case of Q_m, one may interpret ϕ(U)as the edge set of a complete directed graph on m vertices as follows.For each pair of vertices i,j with1≤ i<j ≤ m,theedgeij is oriented from i to jif a_ij=0, andfrom j to i if a_ij=1.A complete directed graph having this transitive property is called a transitive tournament.We conclude that for the hypercube Q_m,the image ofϕ corresponds precisely to the set of m! transitive tournaments.amsplain | http://arxiv.org/abs/1705.09385v1 | {
"authors": [
"John Asplund",
"Kossi Edoh",
"Ruth Haas",
"Yulia Hristova",
"Beth Novick",
"Brett Werner"
],
"categories": [
"math.CO",
"05C38, 05C75"
],
"primary_category": "math.CO",
"published": "20170525225415",
"title": "Reconfiguration graphs of shortest paths"
} |
Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, JapanWe study the effective spin-orbital model for honeycomb-layered transition metal compounds, applying the second-order perturbation theory to the three-orbital Hubbard model with the anisotropic hoppings. This model is reduced to the Kitaev model in the strong spin-orbit coupling limit. Combining the cluster mean-field approximations with the exact diagonalization, we treat the Kugel-Khomskii type superexchange interaction and spin-orbit coupling on an equal footing to discuss ground-state properties. We find that a zigzag ordered state is realized in the model within nearest-neighbor interactions. We clarify how the ordered state competes with the nonmagnetic state, which is adiabatically connected to the quantum spin liquid state realized in a strong spin-orbit coupling limit. Thermodynamic properties are also addressed. The present work should provide another route to account for the Kitaev-based magnetic properties in candidate materials.Role of the spin-orbit coupling in the Kugel-Khomskii model on the honeycomb latticeJoji Nasu December 30, 2023 ======================================================================================Orbital degrees of freedom have been studied as a central topic of strongly correlated electron systems as they possess own quantum dynamics and are strongly entangled with other degrees of freedom such as charge and spin <cit.>. Recently, multiorbital systems with strong spin-orbit (SO) couplings have attracted considerable attention <cit.>. One of the intriguing examples is the series of the Mott insulators with honeycomb-based structures such as A_2 IrO_3 (A= Na, Li) <cit.>, and β-Li_2IrO_3 <cit.>. In these compounds, a strong SO coupling for 5d electrons lifts the triply degenerate t_2g levels and the low-energy Kramers doublet, which is referred to as an isospin, plays an important role at low temperatures. Furthermore, anisotropic electronic clouds intrinsic in the t_2g orbitals result in peculiar exchange couplings and the system is well described by the Kitaev model for the isospins <cit.>. The ground state of this model is a quantum spin liquid (QSL), and hence a lot of experimental and theoretical works have been devoted to the iridium oxides in this context <cit.>. Very recently, the ruthenium compound α-RuCl_3 with 4d electrons has been studied actively as another Kitaev candidate material <cit.>. In general, the SO coupling in 4d orbitals is weaker than that in 5d orbitals and is comparable with the exchange energy. Therefore, it is highly desired to deal with SO and exchange couplings on an equal footing although the magnetic properties for honeycomb-layered compounds have been mainly discussedwithin the isospin model with the Kitaev and other exchange couplings including longer-range interactions <cit.>.In this Letter, we study the role of the SO coupling in the Mott insulator with orbital degrees of freedom. We examine the localized spin-orbital model with the Kugel-Khomskii type superexchange interactions between nearest-neighbor sites and onsite SO couplings on the two-dimensional honeycomb lattice. In the strong SO coupling limit, this model is reduced to the Kitaev model and the QSL state is realized. On the other hand, a conventional spin-orbital ordered state may be stabilized in the small SO coupling case. To examine the competition between the magnetically disordered and ordered states in the intermediate SO coupling region, we first use the cluster mean-field (CMF) theory <cit.> with the exact diagonalization (ED). We determine the ground state phase diagram in the model and clarify that a zigzag magnetically ordered state is realized due to the competition between distinct exchanges. Calculating the specific heat and entropy in terms of the thermal pure quantum (TPQ) state <cit.>, we discuss how thermodynamic properties characteristic of the Kitaev model appear in the intermediate SO coupling region.We start with the three-orbital Hubbard model on the honeycomb lattice. This should be appropriate to describe the electronic state of the t_2g orbitals in the compounds A_2 IrO_3 and α-RuCl_3 since there exists a large crystalline electric field for the d orbitals. The transfer integral t between the t_2g orbitals via ligand p orbitals are evaluated from the Slater-Koster parameters, where the neighboring octahedra consisting of six ligands surrounding transition metal ions share their edges. Note that the transfer integrals involving one of the three t_2g orbitals vanish due to the anisotropic electronic clouds <cit.>. We refer to this as an inactive orbital and the other orbitals as active ones. These depend on three inequivalent bonds, which are schematically shown as the distinct colored lines in Fig. <ref>. Moreover, we consider the onsite intra- and inter-orbital Coulomb interactions, U and U', Hund coupling K, and pair hopping K' in the conventional manner. In the following, we restrict our discussions to the conditions U=U'+2K and K'=K, which are lead by the symmetry argument of the degenerate orbitals.We use the second-order perturbation theory in the strong coupling limit since the Mott insulating state is realized in the honeycomb-layered compounds. We then obtain the Kugel-Khomskii-type exchange model, assuming that five electrons occupy the t_2g orbitals in each site. By taking the SO coupling into account, the effective Hamiltonian is explicitly given asH=∑_⟨ ij⟩_γ H_ij^ ex(γ)-λ∑_iL_i· S_i,where λ is the SO coupling, and S_i and L_i are spin and orbital angular-momentum operators at the ith site, respectively. The exchange Hamiltonian H_ij^ ex(γ), which depends on the bond γ(=x,y,z) of the honeycomb lattice (see Fig. <ref>), is given asH^ ex(γ)_ij= H_1;ij^(γ)+ H_2;ij^(γ)+ H_2;ij^(γ),with H_1;ij^(γ) =2J_1( S_i· S_j+3/4) [ τ_ix^(γ)τ_jx^(γ)-τ_iy^(γ)τ_jy^(γ)-τ_iz^(γ)τ_jz^(γ) +1/4τ_i0^(γ)τ_j0^(γ)-1/4(τ_i0^(γ)+τ_j0^(γ)) ],H_2;ij^(γ) = 2J_2( S_i· S_j-1/4) [ τ_ix^(γ)τ_jx^(γ)-τ_iy^(γ)τ_jy^(γ)-τ_iz^(γ)τ_jz^(γ) +1/4τ_i0^(γ)τ_j0^(γ)+1/4(τ_i0^(γ)+τ_j0^(γ)) ],H_3;ij^(γ) = -4/3(J_2-J_3)( S_i· S_j-1/4) [ τ_ix^(γ)τ_jx^(γ)+τ_iy^(γ)τ_jy^(γ)-τ_iz^(γ)τ_jz^(γ) +1/4τ_i0^(γ)τ_j0^(γ)], where we follow the notation of Ref. <cit.>, and J_1=2t^2/U [1-3K/U]^-1, J_2=2t^2/U [1-K/U]^-1, J_3=2t^2/U [1+2K/U]^-1 are the exchange couplings between nearest neighbor spins.Here, we have newly introduced the orbital pseudospin operators τ_l^(γ) with l=x,y,z,0. Note that its definition depends on the direction of the bond (γ-bond) between the nearest neighbor pair ⟨ ij⟩. τ_l^(γ) is represented by the 3× 3 matrix based on the three orbitals: the 2× 2 submatrix on the two active orbitals is given by σ_l/2 for l=x,y,z and the identity matrix for l=0, and the other components for one inactive orbital are zero, where σ_l is the Pauli matrix. We here note that Hamiltonian H_1 enhances ferromagnetic correlations, while H_2 and H_3 lead to antiferromagnetic correlations. Therefore, spin frustration should play an important role for the ground state in the small K/U region, where J_1∼ J_2 ∼ J_3. What is the most distinct from ordinary spin-orbital models is that the present system describes not only spin-orbital orders but also the QSL state realized in the Kitaev model. When the SO coupling is absent, the system is reduced to the standard Kugel-Khomskii type Hamiltonian. In the large Hund coupling case, the Hamiltonian H_1;ij^(γ) is dominant. Then, the ferromagnetically ordered ground state should be realized despite the presence of orbital frustration. In the smaller case of the Hund coupling, the ground state is not trivial due to the existence of spin frustration, discussed above. On the other hand, in the case λ→∞, the SO coupling lifts the degeneracy at each site and the lowest Kramers doublet, |σ̃⟩=( |xy,σ⟩∓|yz,σ̅⟩ +i|zx,σ̅⟩)/√(3), plays a crucial role for low temperature properties. Then, the model Hamiltonian Eq. (<ref>) is reduced to the exactly solvable Kitaev model with the spin-1/2 isospin operator S̃, as H_ eff=-J̃∑_⟨ ij⟩_γS̃_iγS̃_jγ(γ=x,y,z), where J̃[=2(J_1-J_2)/3] is the effective exchange coupling <cit.>. It is known that, in this effective spin model, the QSL ground state is realized with the spin gap. At finite temperatures, a fermionic fractionalization appears together with double peaks in the specific heat <cit.>. In the following, we set the exchange coupling J_1 as a unit of energy. We then study ground-state and finite-temperature properties in the spin-orbital system with parameters K/U and λ/J_1. First, we discuss ground state properties in the spin-orbital model by means of the CMF method <cit.>. In the method, the original lattice model is mapped to an effective cluster model, where spin and orbital correlations in the cluster can be taken into account properly. Intercluster correlations are treated through several mean-fields at ith site, ⟨ S_ik⟩, ⟨τ_il^(γ)⟩and ⟨ S_ikτ_il^(γ)⟩, where k=x,y,z and l=x,y,z,0. These mean-fields are determined via the self-consistent conditions imposed on the effective cluster problem. The method is comparable with the numerically exact methods if the cluster size is large, and has successfully been applied to quantum spin <cit.> and hard-core bosonic systems <cit.>. To describe some possible ordered states such as the zigzag and stripy states <cit.>,we introduce two kinds of clusters in the honeycomb lattice, which are shown as distinct colors in Fig. <ref>(a). Using the ED method, we self-consistently solve two effective cluster problems. To discuss magnetic properties at zero temperature, we calculate spin and orbital moments, m^α_S=|∑_i(-1)^δ^α_i⟨ S_i⟩|/N and m^α_L=|∑_i(-1)^δ^α_i⟨ L_i⟩|/N, where N is the number of sites and δ^α_i is the phase factor for an ordered state α.When λ=0, the spin and orbital degrees of freedom are decoupled. Here, we show in Fig. <ref> the spin moments m^f_S and m^z_S for the ferromagnetically and zigzag ordered states, respectively, which are obtained by means of the ten-site CMF method (CMF-10). Namely, we have confirmed that other ordered states such as antiferromagnetic and stripy states are never stabilized in the present calculations, and thereby we do not show them in Fig. <ref>. Meanwhile, the local orbital moment disappears in the case λ=0. In the system with the large Hund coupling, the exchange coupling J_1 is dominant, and the ferromagnetically ordered ground state is realized with the fully-polarized moment m^f_S=0.5, as shown in Fig. <ref>.On the other hand, in the smaller K region, the exchange couplings J_2 and J_3 are comparable with J_1. Since H_2 and H_3 should enhance antiferromagnetic correlations, the ferromagnetically ordered state becomes unstable. We find that a zigzag magnetically ordered state is realized with finite m^z_S around K/U∼ 0.12. To study the competition between these ordered states, we show the ground state energies in the inset of Fig. <ref>. We clearly find the hysteresis in the curves, which indicates the existence of the first-order phase transition. By examining the crossing point, we clarify that the quantum phase transition between ferromagnetically and zigzag ordered states occurs at K/U∼ 0.15. In the case with K/U<0.1, due to strong frustration, it is hard to obtain the converged solutions. This will be interesting to clarify this point in a future investigation.The introduction of λ couples the spin and orbital degrees of freedom. The spin moments slightly decrease in both states, as shown in Fig. <ref>. The zigzag and ferromagnetically ordered states are stable against the small SO coupling and the first-order transition point has little effect on the SO coupling. To discuss the stability of these states against the strong SO coupling, we calculate the spin and orbital moments in the system with K/U=0.12 and 0.3, as shown in Fig. <ref>.The introduction of the SO coupling slightly decreases the spin moment, as discussed above. By contrast, the orbital moment is induced parallel to the spin moment. Therefore, the total magnetic moment m_μ^α=|∑_i(-1)^δ_i^α⟨ 2 S_i +L_i ⟩|/N increases. When K/U=0.12, the zigzag ordered state becomes unstable and the first-order phase transition occurs to the ferromagnetically ordered state at λ/J_1∼ 0.4. Further increase of the SO coupling decreases the total moment m_μ^f. Finally, a jump singularity appears around λ/J_1∼ 0.8 (1.8) in the system with K/U=0.12 (0.3). It is also found that the magnetic moment is almost zero and each orbital is equally occupied as in the isospin states |σ̃⟩ in the larger SO coupling region. Therefore, we believe that this state isessentially the same as the QSL state realized in the Kitaev model.By performing similar calculations, we obtain the ground state phase diagram, as shown in Fig. <ref>.The disordered (QSL) state is realized in the region with large λ/J_1. The ferromagnetically ordered state is realized in the region with small λ/J_1 and large K/U. The decrease of the Hund coupling induces spin frustration, which destabilizes the ferromagnetically ordered state. We wish to note that the zigzag ordered state is stable in the small SO coupling region, which is not directly taken into account in the Kitaev model. Next, we discuss thermodynamic properties in the system. It is known that, in the Kitaev limit (λ→∞), the excitations are characterized by two energy scales, which correspond to localized and itinerant Majorana fermions. This clearly appears in the specific heat as two peaks at T/J̃=0.012 and 0.38 <cit.>. To clarify how the double peak structure appears in the intermediate SO coupling region, we make use of the TPQ state for the twelve-site cluster with the periodic boundary condition [see Fig. <ref>(b)]. According to the previous study <cit.>, the double peak structure appears in the spin-1/2 Kitaev model even with the twelve-site cluster. Therefore, we believe that thermodynamic properties in the system can be discussed, at least, qualitatively in our calculations.Here, we fix the Hund coupling as K/U=0.3 to discuss finite temperature properties in the system with the intermediate SO coupling. Figure <ref> shows the specific heat and entropy in the system with λ/J_1=0, 1, 2, 4 and 10.In this calculation, the quantities are deduced by the statistical average of the results obtained from, at least, twenty independent TPQ states. When λ=0, we find a broad peak around T/J_1=0.4 in the curve of the specific heat. In addition, most of the entropy is released at T/J_1∼ 0.1, as shown in Fig. <ref>(b). This can be explained by the fact that ferromagnetic correlations are enhanced and spin degrees of freedom are almost frozen. The appearance of the large residual entropy should be an artifact in the small cluster with the orbital frustration. The introduction of the SO coupling leads to interesting behavior. It is clearly found that the broad peak shifts to higher temperatures. This indicates the formation of the Kramers doublet and a part of the entropy S=log(6)-log(2) is almost released, as shown in Fig. <ref>(b). In addition, we find in the case λ/J_1≥ 2, two peaks in the specific heat at lower temperatures. The corresponding temperatures are little changed by the magnitude of the SO coupling and the curves are quantitatively consistent with the results for the isospin Kitaev model on the twelve sites, which are shown as dashed lines. Therefore, we believe that the Kitaev physics appears in the region. On the other hand, when λ/J_1=1, a single peak structure appears in the specific heat, indicating that the Kitaev physics is hidden by the formation of the Kramers doublet due to the competition between the exchange interaction and SO coupling. We have used the TPQ states to clarify how the double peak structure inherent in the Kitaev physics appears, in addition to the broad peak for the formation of the Kramers doublet at higher temperatures.To conclude, we have studied the effective spin-orbital model obtained by the second-order perturbation theory. Combining the CMF theory with the ED method, we have treated the Kugel-Khomskii type superexchange interaction and SO coupling on an equal footing to determine the ground-state phase diagram. We have clarified how the magnetically ordered state competes with the nonmagnetic state, which is adiabatically connected to the QSL state realized in a strong SO coupling limit. Particularly, we have revealed that a zigzag ordered state is realized in this effective spin-orbital model with finite SO couplings. 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"authors": [
"Akihisa Koga",
"Shiryu Nakauchi",
"Joji Nasu"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170526180000",
"title": "Role of the spin-orbit coupling in the Kugel-Khomskii model on the honeycomb lattice"
} |
[ \begin@twocolumnfalseDeep Neural Network Probabilistic Decoder for Stabilizer Codes Stefan Krastanov^1, Liang Jiang^2 December 30, 2023 ==============================================================^1Department of Physics, Yale University, Yale Quantum Institute^2Department of Applied Physics, Yale University, Yale Quantum InstituteCorrespondence to [email protected] networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks to naturally finding the most probable error and accounting for correlations between errors. \end@twocolumnfalse ] Constructing a physical computing machine, whether a classical or a quantum one, requires, inescapably, the implementation of an error correcting mechanism that guards against the noise picked up from the environment and the imperfections in the operations being performed <cit.>. Early in the development of both classical and quantum computers, “threshold” theorems were proven to show the existence of encoding schemes which reliably store and process information in a “logical” set of bits (or qubits), by encoding it redundantly on top of a bigger set of less reliable “physical” bits (or qubits), as long as the error rate on the physical layer is smaller than a fixed threshold <cit.>. The vast majority of quantum error correcting codes fall in the class of stabilizer codes (a generalization of the classical linear codes) <cit.>. They are characterized by the group of stabilizer operators that preserve the logical states (similarly to the list of constraints represented by the parity check matrix H for classical linear codes). The list of nontrivial stabilizer operator measurements (or violated parity constraints for a classical code) is called the syndrome of the error. While providing for efficient encoding, linear and stabilizer codes do not necessarily have known efficient decoding algorithms that can deduce from a given syndrome what errors have occurred.In the general case decoding a stabilizer code is an NP-hard problem. An active area of research is the design of codes with some additional algebraic structure that permits efficient decoders, but still retains high rates (ratio of logical to physical qubits) with acceptable distances (maximal number of correctable errors on the physical qubits). Schemes like the CSS approach <cit.> permit the creation of quantum codes from classical codes, but they do not guarantee that the decoder that worked for the classical code still works for the quantum one. A particularly interesting example is the class of LDPC codes <cit.> which are high-performing classical codes with efficient decoders, however those decoders do not work for the quantum LDPC codes <cit.>.Here we present a decoding algorithm that can be applied to any stabilizer code — the decoder employs machine learning techniques to “learn” any structures that would make the approximate decoding problem easier than the general NP-hard decoding problem: it “learns” the probability distributions of errors conditioned on a given syndrome and efficiently uses samples from that distribution in order to predict probable errors. The conditional probability distribution is encoded in a deep neural network. The “learning” involves training the neural network on pairs of errors and corresponding syndromes (generated from an error model for the physical qubits and a parity check matrix for the code in use). We test the algorithm on the toric code (ToricCodeDecodera) definied on a two-dimensional lattice on a torus <cit.>. Since the toric code has low-weight local stabilizers, it is also a quantum LDPC code with structure that impedes typical belief propagation algorithms. Our decoder significantly outperforms the standard “minimal-weight perfect matching” (MWPM) decoder <cit.>. Moreover, it has comparable threshold with the best renormalization group decoders <cit.>. For code-sizes up to 200 physical qubits the decoder is practical and we discuss how to extend our neural network architecture to negate the inefficiencies that kick in at that stage.Machine learning techniques, specifically neural networks, have been gaining popularity over the last year, in particular with the recent developments in using restricted Boltzmann machines for describing the ground state of many-body systems <cit.> or convolutional networks for identifying phases of matter <cit.>. A preprint on the use of restricted Boltzmann machines to decoding the toric code has been available for a few months as well <cit.>, however that architecture does not yet outperform known decoders like MWPM and has been tested only on the Z syndrome on lattices no bigger than 5-by-5. At the time of submission of this manuscript two other related preprints were made available: a fast neural network decoder for small surface codes, that however also does not significantly outperform MWPM <cit.>, and a recurrent neural network decoder outperforming MWPM as evaluated on a 17 qubit surface code <cit.>. It is worth noting as well that over the last few months work has started on deep learning methods for decoding classical algebraic codes <cit.>.§ RESULTS For testing purposes we trained our neural decoder (depicted in ToricCodeDecoderb) on the toric code, which already has a number of known decoding algorithms specifically tuned for its lattice structure. The evaluation was done under the depolarization error model. Our algorithm significantly outperforms the standard MWPM decoder. The comparison of the two decoders in threshold shows a threshold single-qubit error which is nearly 2 percentage points higher for the new algorithm (around 16.4% for the depolarization error model), and the fraction of correctly decoded errors is consistently around 10 percentage points higher than the fraction of errors corrected by MWPM. Furthermore, the neural decoder threshold compares favorably to renormalization group decoders <cit.> (threshold of 15.2%), and decoders explicitly tuned to correlations between Z and X errors <cit.> (threshold of 13.3% for a triangular lattice, as it is tuned for asymmetric codes). To our knowledge only a renormalization group decoder <cit.> enhanced by a sparse code decoder <cit.> reaches a similar threshold (16.4%). It is worth noting that the sampling procedure in our decoder makes it impractically slow for codes of more than 200 physical qubits, while other decoders remain practical. On the other hand, the neural architecture is versatile enough to be applied to any stabilizer code, unlike the other decoders discussed here, which are limited to only topological codes. The best of both worlds — record threshold and fast decoding — should be achievable if we couple the renormalization decoder of <cit.> with our neural decoder (instead of the currently suggested sparse code decoder <cit.>), however this will be applicable only to topological codes. We discuss other ways to avoid the inefficiencies in our decoder without compromising its ability to “learn” to decode any stabilizer code. After being properly trained for a given error rate of a particular error model, the neural network at the heart of our decoder becomes a compact approximate representation of the probability distribution of errors that can occur. The decoding algorithm consist of inputing the measured syndrome in the neural network, interpreting the output as a probability distribution of the errors conditioned on the given syndrome, and repeatedly sampling from that distribution. The performance of the decoder scales monotonically with the size of the network, up to a point of diminishing returns where using more than about 15 hidden layers (for a distance 5 code) stops providing improvements.The significant gain in the threshold value relative to some known decoders can be traced to two characteristics of the neural network (discussed in more details in the Methods section). Firstly, the neural network is trained on (stabilizer, error) pairs generated from the error model, therefore it is optimized directly for producing “most probable error”, not for finding an imperfect proxy like “error with lowest energy” as is the case for MWPM. Secondly (depicted in correlations), it learns the Z and X stabilizers together, hence it can encode correlations between them in its structure. Namely, in a typical depolarization error models, one third of the errors are Y errors (equivalent to both X and Z error happening), therefore the knowledge of this correlation can be a useful resource for decoding. Other decoders need significant modifications to even partially employ those correlations in decoding <cit.>.§ METHODS Neural networks are particularly efficient tools for function approximation <cit.>, where a function f: x→ f(x) is to be learned from large amount of training data given in the form of pairs (x, f(x)). The input x is set as the value of the input layer of neurons. Each of those neurons is connected through axons with each neuron of the next layer (the first “hidden” layer). Multiple hidden layers of neurons can be connected together in this fashion in order to construct a deeper neural network. The last layer of the network is the output layer - its value represents f_learned(x). The value of a neuron (i.e. its activation value) is calculated as a weighted sum of the activation values of the neurons connected to it from the previous layer. That sum is then passed through a non-linear function (called the activation function). This activation value is then further passed on to the neurons of the next layer, where the process is repeated until it reaches the output layer. The weights in the sums (i.e. the strength of connections between the neurons) are parameters which are optimized through stochastic gradient descent in order to minimize the distance between f_learned and f calculated on the training data. The choice of activation function, the size of the hidden layers, and the step size for gradient descent (also called the hyperparameters) are decided in advance, before training. Current best practices include performing a random search to find the best hyperparameters.In the particular case of decoding a stabilizer quantum error correcting code we want to map syndromes to corresponding physical errors, hence, we take the input layer to be the syndrome (obtained from measuring the stabilizers). For instance, for a toric code of lattice size 9-by-9 we have to measure 81 plaquette operators and 81 star operators for a total of 162 input neurons (having value 0 if the syndrome is trivial and 1 if not). Similarly, we set the output layer to be the prediction for what physical errors occurred (typically represented in the Heisenberg picture, thanks to the Gottesman–Knill theorem). Using the same example, we have 162 physical qubits and we need to track their eigenvalues under both Z and X operators, requiring a total of 324 output neurons (having value 0 if no error has occurred and value 1 otherwise).To completely define the neural network architecture we set the activation functions of the hidden layers to tanh and the activation of the output layer to the sigmoid function σ(x)=(1-e^-x)^-1∈[0,1]. The size of the hidden layer was set to four times the size of the input layer. These decisions were reached after an exhaustive search over possible hyperparameters tested on toric codes of distance 3 to 6, and proved to work well for bigger codes as well. The number of hidden layers was varied - deeper networks produce better approximations up to a point of diminishing returns around 15 layers. The step size for the gradient descent (a.k.a. the learning rate) was annealed - gradually lowered, in order to permit rapidly reaching the minimum. The distance measure between training and evaluation data that is being minimized by the gradient descent is their binary crossentropy (a measure of difference between two probability distributions discussed below).The training was done over one billion (syndrome, error) pairs in batches of 512, taking about a day of GPU wall time for a 5-by-5 toric code. The pairs were generating on the fly, by first generating a sample error from the given error model (this training set can also be efficiently generated directly on the experimental hardware), and then obtaining the corresponding syndrome by a dot product with the parity check matrix. The error model used for each physical qubit was qubit depolarization, parametrized by qubit fidelity p - the probability of no error happening on a given qubit, or equivalently depolarization rate 1-p. Under this model, Z, X, and Y (consecutive Z and X) errors had equal probabilities of 1/3(1-p). For each value of p we trained a new network, however the results showed some robustness to testing a neural network at an error rate different from the one at which it was trained.The performance of the network was improved if we normalize the input values to have an average of 0 and a standard deviation of 1. For a depolarization error rate 1-p, the rate at which a Z eigenvalue flips is P_e=2/3(1-p) and independently the rate for X flips is the same. In the example of the toric code the rate of non-trivial stabilizer measurements will be the same for Z and for X, namely P_s=4q^3(1-q)+4q(1-q)^3 and the variance will be V_s=P_s-P_s^2.At this point we have not discussed yet how to use the fully trained neural network in decoding. A trained network can efficiently evaluate the approximation of the decoding function (from here on referred to as DECODE:syndrome→error), so all Alice needs to do in order to perform error correction on her quantum memory is to measure the syndrome and run the neural network forward to evaluate DECODE(syndrome). However, the neural network is a continuous function and an imperfect approximation, therefore the values in DECODE(syndrome) will not be discrete zeros and ones, rather they will be real numbers between zero and one. A common way to use and interpret those values is to view them as a probability distribution over possible errors, i.e. the i-th value in the array DECODE(syndrome) is a real number between zero and one equal to the probability of the i-th qubit experiencing a flip (half of the array corresponds to Z errors and half of the array corresponds to X errors). This interpretation is reinforced by our use of binary crossentropy as an optimization target during training. In order to deduce what error has occurred we sample this probability distribution. We verify the correctness of the sample by computing the syndrome that the predicted error would cause - if it differs from the given syndrome we resample. This sampling procedure is present in <cit.> as well, however we further employ a simple “hard decision belief propagation / message passing” sampling, which can speed up the sampling process by an order of magnitude: we resample only the qubits taking part in the stabilizer measurement corresponding to the incorrect elements of the syndrome (see Sampling-the-neural-network). *Data AvailabilityThe code for building, training, and evaluating the neural network decoder is publicly available on the authors' web page, and shell scripts with the parameters for the presented figures are available upon request. Pretrained neural networks can be provided as well.§ DISCUSSION On first sight our decoder implementation can look like a look-up table implementation, however we would like to stress the immense compression of data that the neural network achieves. Firstly, one can consider the size of the neural network itself. For a code on N physical qubits the number of parameters needed to describe a neural decoder of L layers will be 𝒪(N^2L) or on the order of thousands for the codes we tested. Moreover, the size of the training dataset for the codes we tested did not exceed 10 billion, and it can be made orders of magnitude smaller if we reuse samples in the stochastic gradient descent (a common approach taken in training). On the other hand, the size of a complete lookup table would be on the order of 𝒪(4^N). Even if we take only the most probable errors (and discard the errors that have less than 5% chance of occurring), at depolarization rate of 0.1 we need a lookup table bigger than 10^12 for a distance 5 toric code (50 qubits), bigger than 10^23 for distance 7 toric code (98 qubits), and bigger than 10^37 for distance 9 toric code (162 qubits).Thanks to this compression, to the direct optimization for most probable error, and to the ease of including knowledge of error correlations in the decoding procedure, the algorithm presented here is one of the best choices for decoding stabilizer codes of less than 200 qubits. While we used the toric code for our testing, there is nothing in our design that has knowledge of the specific structure of that code - the neural decoder can be applied to the decoding of any stabilizer code.Due to the probabilistic nature of the sampling, the decoder becomes impractically inefficient for codes bigger than roughly 200 qubits as one can see in Sampling-giveup. This can be attributed to two characteristics of our algorithm: we use a simple hard-decision message passing algorithm in our sampling instead of a more advanced belief propagation algorithm seeded by output of the neural network; additionally, our neural network learns only the marginal probabilities for errors on each qubit, without providing the correlations between those errors. A more advanced neural network could address this problem by providing correlation information in its output layer. Our focus forward goes beyond that: we can consider recurrent generative networks <cit.> that have the belief propagation as part of their recurrent structure.While this decoder is general and it can be applied to any stabilizer code, one can also design neural network architectures that specifically exploit the lattice structure and translational symmetry of the toric code. For instance, convolutional neural networks are well adapted for processing 2D data. Moreover thanks to the translational symmetry one can envision a decoder that is trained on a fixed patch of the code and it can be used for toric codes of any size. As already mentioned, our decoder can readily replace the sparse code decoder <cit.> used as part of the renormalization group decoder of <cit.>, hence providing great decoding speed and high threshold values.§ ACKNOWLEDGEMENTS We acknowledge the stimulating discussions with Kasper Duivenvoorden, Steven Flammia, Steven Girvin, Alexandar Makelov, and Mehmet Tuna Uysal. We thank the Yale HPC staff for the provided computing resources. We acknowledge support from the ARL-CDQI (W911NF-15-2-0067), ARO (W911NF-14-1-0011, W911NF-14-1-0563), ARO MURI (W911NF-16-1-0349 ), AFOSR MURI (FA9550-14-1-0052, FA9550-15-1-0015), NSF (EFMA-1640959), Alfred P. Sloan Foundation (BR2013-049), and Packard Foundation (2013-39273).§.§.§ Additional Information SK contributed the code for the project. Design, analysis, and manuscript preparation was contributed jointly by SK and LJ.Competing Financial Interests statement: There is no competing interest.unsrt § SUPPLEMENTARY MATERIALS §.§.§ Performance of sampling through “hard-decision message passing” In the main text we mention that sampling from the neural network can be done in two different manners: either through naive resampling in which the entire “candidate error vector” sample is scraped if it does not reproduce the measured syndrome or through a more advanced “hard-decision message passing” which provides for a significant speedup. The message passing works by checking which qubits belong to the violated syndrome components (i.e. which variable nodes are connected to the active check nodes on the Tanner graph) - in the resampling step, only those qubits are resampled, hence preserving the already properly decoded components of the error vector. The following figure, similar to the figures in the main text, demonstrates the speedup for a resampling performed on a deep network decoding the 5x5 code. In order to also compare the hard-decision message sampling on its own, without the use of a neural network (i.e. the way message passing is currently used in classical LDPC decoders), we also show a curve where the message passing sampler is run on an untrained network (as expected, it performs poorly - it is known that message passing on its own does not work well for quantum LDPC codes due to the presence of 4-cycles in the Tanner graph).< g r a p h i c s > §.§.§ Performance vs depth of the neural network The following figure, similar in style to figures in the main text, shows the growth in performance as we increase the depth of the neural network (the example is for a 5x5 toric code). As discussed, a point of diminishing returns is reached, where the expense of adding more layers outweighs the minor gains in performance.< g r a p h i c s > §.§.§ Software package The software described in the main text is available from the authors. It is based on the Keras and Tensorflow NN libraries and can run on GPU accelerators. The software provides independent command line utilities that can be used to design and evaluate deep neural network decoders with arbitrary hyperparameters. Upon request we would be happy to provide pretrained networks for any reasonably sized code of interest. | http://arxiv.org/abs/1705.09334v3 | {
"authors": [
"Stefan Krastanov",
"Liang Jiang"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170525191335",
"title": "Deep Neural Network Probabilistic Decoder for Stabilizer Codes"
} |
Extension of the input-output relation for a Michelson interferometer to arbitrary coherent-state light sources: — Gravitational-wave detector and weak-value amplification —Kouji Nakamura[E-mail address: [email protected]], and Masa-Katsu Fujimoto[E-mail address: [email protected]]Received: date / Accepted: date ============================================================================================================================================================================================§ INTRODUCTION After 50 years of their discovery, pulsars are still fascinating objects. They possess the strongest magnetic fieldsin the universe. Magnetars is a special kind of pulsars. They may have the strongest magnetic field in all pulsars, with magnetic field as high as 10^15G <cit.>. Therefore, magnetars are new specimen to study the physics of normal pulsars and accretion-powered X-ray pulsars <cit.>. For isolated magnetars, their distribution on the period period-derivative diagram of pulsars is shown in figure <ref>. As can be seen this figure, both pulsars and magnetars are spinning down. One fundamental question of pulsar study is: what's the braking mechanism of pulsars and magnetars? In both pulsar and magnetar studies, the magnetic dipole braking mechanism is usually assumed. However, this is only a crude approximation since it assumes a rotating dipole in vacuum.A physical braking mechanism must consider the presence of the pulsar magnetosphere. There are many models in this direction. Among them are the wind braking of pulsars <cit.> and wind braking of magnetars <cit.>.Observationally, magnetars have varying spin-down torque <cit.>, or decreasing torque during outburst <cit.>, or increasing torque <cit.>. Their spectra also contain a significant fraction of nonthermal emissions <cit.>. Therefore, a physical picture is that: the magnetic energy of magnetars is first converted to a system of particles. These particles are responsible for both the X-ray emissions and the spin-down torque. Then, it is natural that correlations between radiation and timing are seen ubiquitously in magnetars. This means that the spin-down of magnetars may bedominated by the outflowing particles. This is dubbed as the “wind braking of magnetars” <cit.>. Recently, there are many interesting observations of magnetars. It is found that many of them can be understood safely in the wind braking model. Furthermore, the discovery of possible accreting magnetars enable people to study the magnetosphere of accreting magnetars. In the following, the recent observations of isolated magnetars and accreting magnetars are discussed.Understandings of these observations from the magnetospheric point of view are presented.§ ISOLATED MAGNETARS: WIND BRAKING AND WIND NEBULA§.§ Decreasing torque during outbursts In the wind braking scenario, magnetars are neutron stars with strong multipole field. The strong multipole field is responsible for the X-ray emission, bursts, and spin-down of the central magnetar. The rotation energy of the magnetar is carried away mainly the outflowing particles. Their surface dipole field can be very high or not very high. The discovery of low magnetic field magnetars is consistent with the picture of wind braking of magnetars. For the second low magnetic field magnetar Swift J1822.3-1606, different authors gave different timing solutions. Ref. <cit.> applied the wind braking model to Swift J1822.3-1606. If the particle outflow decreases as that of the X-ray luminosity, then a decreasing period derivative is expected. This may solve the observational ambiguities. Ref. <cit.> predicted a long term period derivative of 1.9× 10^-14 (last paragraph in section 2 there).Subsequent timing found a long term period derivate of (2.1± 0.2) × 10^-14 <cit.>. This is consistent with the theoretical prediction. Alternative explanation is discussed in Ref. <cit.>. Recent observations of the first radio-loud magnetar XTE J1810-197 found a decreasing spin-down torque, accompanied by a decreasing X-ray luminosity and radio luminosity. The radio luminosity finally disappears <cit.>. The spin-down torque of XTE J1810-197 decreased by a factor of three during the process. While its radio luminosity decrease by factor of ten <cit.>. Similar radiation and timing behavior is also seen in the third radio-loud magnetar PSR J1622-4950 <cit.>.For PSR J1622-4950, its spin-down decrease by about a factor of 10. During this process, the radio flux decrease by a factor about 100 <cit.>. In the wind braking scenario, the spin-down rate is proportional to the square-root of the particle luminosity: Ṗ∝ L_ p^1/2 <cit.>. These particles may be also responsible for the radio and X-ray emission. One possibility is that a constant fraction of the particle luminosity is converted to X-ray and radio emission. Then the decrease in radio luminosity and decrease in spin-down torque for these two radio emitting magnetars are quantitatively consistent with each other. X-ray observations of XTE J1810-197 found a decrease in spin-down torque by a factor of eight <cit.> during the whole outburst epoch. While the X-ray flux decreased by a factor of 50 <cit.>. This is also consistent with the anticipation in the wind braking model. Furthermore, in the wind braking scenario, the magnetars is expect to have smaller dipole magnetic field than their apparent characteristic magnetic field. Then during quiescent state, the magnetar may be below the radio death line of pulsars, i.e. they are radio quiet. During outburst, when the magnetosphere of magnetars is perturbed significantly, the magnetar may be radio-loud temporally. After the outburst, when the magnetar return back to the quiescent state, it will become radio-quiet again (Ref. <cit.> obtained a similar conclusion using a different model for magnetars). The final radio disappearance of these two radio-loud magnetars <cit.> are consistent the general expectations in the wind braking model.§.§ Anti-glitch, anti-correlation between radiation and timing Glitches in normal pulsars and magnetars are sudden spin-up events <cit.>. However, a spin-down glitch (i.e. anti-glitch) is observed in one magnetar 1E 2259+586 <cit.>. Meanwhile, the X-ray luminosity also increases by about a factor of two during this epoch. The X-ray luminosity contains a significant non-thermal component. The increase in X-ray luminosity should be accompanied by an increase in the particle luminosity. This will result in an enhanced rotational energy loss rate. It is found that the rotational energy carried away by the outflowing particles may explain theso-called “anti-glitch” <cit.>. In this case, there is no “anti-glitch”. It is is just a period of enhanced spin-down caused by an enhanced particle wind. In the original work for anti-glitch <cit.>, the net spin-down glitch of PSR J1846-0258 is also discussed. Later observations of spin-down glitches in magnetar 4U 0142+61 etc are consistent with the wind braking model <cit.>. Timing of the Galactic center magnetar SGR J1745-2900 found a decreasing X-ray luminosity accompanied by an increasing spin-down torque <cit.>. For a constant particle luminosity, a change in the polar cap geometry may explain the anti-correlation between X-ray luminosity and spin-down torque <cit.>. Subsequent timing of SGR 1806-20 also found an increasing spin-down torque <cit.>. It is qualitatively consistent with the change in emission geometry. Detailed modeling is needed for this source. In summary, there are two typical changes in the polar caps of magnetar magnetospheres: (1) The total particle luminosity changes while the geometry is unchanged. The luminosity can increase or decrease, while the pulse profile is almost constant. This will result in a positive correlation between radiation and spin-down torque. This may correspond to the decreasing spin-down torque during the outburst of magnetars. And an enhanced spin-down period may result in a net spin-down of the magnetar.(2) The geometry of the polar cap changes while the particle luminosity is almost constant. This will result in a anti-correlation between the radiation and timing of the magnetar. Changes in the pulse profile is also possible. In reality, both the particle luminosity and geometry may change, this may corresponding to the diverse behaviors of magnetars radiation and timing.§.§ Two predictions: wind nebula and braking index There are two predictions of the wind braking model: a magnetism-powered wind nebula and a braking index smaller than three <cit.>. The particle luminosity of magnetars may be comparable with their X-ray luminosities, about 10^35erg s^-1. It is usually much higher than the magnetar's rotational energy loss rate. However, it is lower thanthe rotational energy loss rate of young pulsars, which is from about 10^36ergs^-1 to 10^38erg s^-1. Pulsar wind nebulae are always seen around young pulsars. Therefore, the particle outflow of magnetars may also be seen as a wind nebula around the magnetar. Since the particle outflow originates from the magnetic energy release, the corresponding wind nebula is powered by the magnetic energy rather than the magnetar's rotational energy. Because the particle luminosity is lower than that of the young pulsars', the magnetar wind nebula may also be relatively weak. This may explain why magnetar wind nebula is not detected until 2016 <cit.>. The possible wind nebula around magnetar Swift J1834.9-0846 <cit.> may be a magnetism-powered wind nebula <cit.>. The structure and evolution of magnetar wind nebula need further studies. For both normal pulsars and magnetars, a particle wind will result in a braking index about one. Observationally magnetars are more noisy <cit.>. Therefore, the braking index of magnetars may be difficult to measure. However, there are some indirect evidences. The rotation-powered pulsar PSR J1846-0258 showed some kind of magnetar-like activities. Later timing found a smaller braking index <cit.>. The radio pulsar PSR J1119-6127 have braking index measurement. Recently, it also showed some kind of magnetar-like activities <cit.>. One anticipation of the wind braking model is that: PSR J1119-6127 will have a smaller braking index after the outburst. This may be tested by future observations.§ ACCRETING MAGNETARS Magnetar is a special kind of pulsars. Since there are both isolated pulsars and accretion-powered X-ray pulsars in binary systems, then both isolated magnetars and accreting magnetars should exist. For accreting magnetars, the question is: which accreting neutron star is an accreting magnetar? In order to answer this question and finding accreting magnetars, we must find the difference between accreting normal neutron stars and accreting magnetars. By analyzing previous studies of pulsars and magnetars, we proposed that accreting magnetars can have two signatures <cit.>: (1) a magnetar-like outburst, (2) a hard X-ray tail above 100keV. Ultraluminous X-ray pulsars are accretion-powered X-ray pulsars with luminosity as high as or higher than 10^40ergs^-1 <cit.>. With the discovery of ultraluminous X-ray pulsars, it is proposed that they may also be accreting magnetars <cit.>. Considering that the magnetar in the binary system may be an old magnetar. Old magnetar are more likely to be low magnetic field magnetars. In the frame of accreting low magnetic field magnetar, both the radiation and timing behaviors of ultraluminous X-ray pulsar can be explained. Up to now, three ultraluminous X-ray pulsars are discovered <cit.>. The radiation and evolution of accreting magnetars need more studies. Neutron stars are born in supernova explosions. During this process, some of the ejecta may fallback and form a disk, i.e. a fallback disk. Neutron stars accreting from a fallback disk is proposed as an alternative model to the magnetar model. Some evidence for a fallback disk is also found. Since magnetars are also born in supernova explosions, then they can also have fallback disks.The high magnetic field of magnetars, combined with a relatively low accretion rate from the fallback disk, will result in a long equilibrium period. The central compact object inside supernova remnant RCW 103 is identified as a magnetar with a possible rotational period of 2× 10^4 s <cit.>. This super-slow magnetars may have a fallback disk in the past <cit.>. This can account for its super-slow rotation. Now the disk has faded away, and the central neutron star can have magnetar activities. Pulsars and magnetars with fallback disks may explain a wide range of peculiar observations.§ SUMMARY By studying recent observations of magnetars, we may get the following conclusions: * Anomalous X-ray pulsars and soft gamma-ray repeaters may be magnetars. The spin-down of magnetars may be dominated by the particle wind. In the wind braking model of magnetars, it is natural that there are correlations between the radiation and timing of magnetars.* Magnetars may also generate a wind nebula. This wind nebula can be powered by the magnetic energy of magnetars.* Central compact objects may be magnetars in waiting or magnetars with fallback disks.* Ultraluminous X-ray pulsars may be another manifestation of accreting magnetars. In summary, magnetar is just a special kind of pulsars. It is special because it can have a higher magnetic field and can be powered by the magnetic energy. At the same time, magnetar can also have wind nebula, fallback disk, and accretion from a binary companion, similar to that of normal pulsars.§ ACKNOWLEDGEMENT H. Tong would to thank his collaborators very much, including R. X. Xu, W. Wang, F. F. Kou etc, and C. H. Lee etc for discussions. 99Mereghetti2008S. Mereghetti, A&ARv 15, 225 (2008)Mereghetti2015 S. Mereghetti, J. A. 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"authors": [
"H. Tong"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170527054323",
"title": "Understanding recent observations of isolated magnetars and accreting magnetars from the magnetospheric point of view"
} |
[E-mail: ][email protected] University of Oslo, Centre for Materials Science and Nanotechnology, N-0318 Oslo, Norway Center for Physical Sciences and Technology, Vilnius LT-10257, Lithuania This work explores the Zn vacancy in ZnO using hybrid density functional theory calculations. The Zn vacancy is predicted to be an exceedingly deep polaronic acceptor that can bind a localized hole on each of the four nearest-neighbor O ions. The hole localization is accompanied by a distinct outward relaxation of the O ions, which leads to lower symmetry and reduced formation energy. Notably, we find that initial symmetry-breaking is required to capture this effect, which might explain the absence of polaronic hole localization in some previous hybrid density functional studies. We present a simple model to rationalize our findings with regard to the approximately equidistant thermodynamic charge-state transition levels. Furthermore, by employing a one-dimensional configuration coordinate model with parameters obtained from the hybrid density functional theory calculations, luminescence lineshapes were calculated. The results show that the isolated Zn vacancy is unlikely to be the origin of the commonly observed luminescence in the visible part of the emission spectrum from n-type material, but rather the luminescence in the infrared region. The Zn vacancy as a polaronic hole trap in ZnO A. Alkauskas December 30, 2023 ==============================================§ INTRODUCTION Although there is a plethora of density functional theory (DFT) studies addressing various aspects of intrinsic defects in ZnO, precise determination of properties like the formation energy and thermodynamic charge-state transition levels has been difficult, and the results scatter widely in the literature. This is especially evident for the Zn vacancy (V_Zn), where the reported thermodynamic charge-state transition levels spread over more than 2 eV <cit.>. The main reason for this large variation can be divided into two categories <cit.>: (i) The choice of exchange-correlation functional, and (ii) The choice (or omission) of finite-size corrections. Despite this spread in results, DFT calculations have provided valuable insights into the properties of many defects in ZnO. The majority of studies conclude that V_Zn is a deep acceptor—the dominant “native” acceptor-type defect—acting as as a compensating center in n-type material <cit.>. However, theoretical studies that relied on semi-local and local density functionals, while providing valuable information, could not properly describe the localization of holes at V_Zn as observed experimentally in, e.g., electron paramagnetic resonance (EPR) studies <cit.>. By employing the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional <cit.>, which intermixes a portion of screened Hartree-Fock (HF) exchange with the standard GGA-PBE exchange-correlation functional, we are able to capture the hole localization at V_Zn, which drastically modifies its properties. Furthermore, by using a one-dimensional configuration coordinate model <cit.>, defect luminescence lineshapes and positions for all optical transitions involving V_Zn and the band edges are calculated. The results show that the isolated V_Zn is unlikely to be the origin of the luminescence in the visible part of the emission spectrum from n-type material. This paper is organized as follows. In Section <ref>, we present computational details, outline how the various quantities are calculated and elaborate on the accuracy of the calculations. In Section <ref>, the results are presented and discussed, including thermodynamics, electronic structure, a simple model for the energetics of V_Zn and optical properties. Section <ref> concludes the paper. § THEORETICAL FRAMEWORK §.§ Computational details All calculations were performed using the projector-augmented wave (PAW) method <cit.>, as implemented in the Vienna ab-initio Simulation Package (VASP) <cit.>, using a plane-wave energy cutoff of 500 eV. The Zn 3d, 4s, 4p, and O 2s, 2p electrons were considered as valence electrons.The α-tuned HSE hybrid functional was used with a screening parameter <cit.> of 0.2 Å^-1, and the amount of exact exchange was set to α = 37.5 %. The resulting lattice parameters for wurtzite ZnO (a=3.244 Å and c=5.194 Å) and band gap (3.42 eV) are in excellent agreement with experimental data. Defect calculations were performed with a 96-atom-sized supercell by relaxing all ionic positions, but keeping its shape and volume fixed to that of the pristine supercell. Ionic optimization was performed until all forces were smaller than 5 meV/Å, and the break condition for the electronic self consistent loop was set to 10^-6 eV. Due to the periodic boundary conditions, defect wave functions may overlap causing an artificial dispersion of defect states. This may lead to an error in the defect formation energy for small supercells, particularly if a Γ-only k-point sampling is used <cit.>. In this work, a special off-Γ k-point at k=(1/4,1/4,1/4) was employed in order to minimize this error within the bounds of computational cost. This setup was checked for several defects in ZnO against a 72 atom supercell with a 2x2x2 Γ-centered k-mesh, yielding an average energy difference of merely 0.02 eV. Spin-polarized calculations were performed for all charge-states. §.§ Defect thermodynamics The formation energy of a defect in charge-state q is given by <cit.> E^f(q) = E^tot_defect(q)-E^tot_bulk-∑_iΔ n_iμ_i+q(ε_VBM+ε_F)+E^FNV, where E^tot are the electronic total energies, Δ n_i is the change in the number of atoms i (Zn,O) with chemical potential μ_i, ε_F is the Fermi level relative to the bulk valence band (VB) maximum ε_VBM, and E^FNV is an electrostatics-based finite-size correction term used to obtain the formation energy of the isolated defect from the finite-sized supercell calculation. We have employed the extended FNV correction scheme <cit.> E^FNV=E^PC+qΔ V^PC_q/bulk|far, where E^PC is the anisotropic Madelung energy for a periodic array of point charges immersed in a neutralizing background charge. The potential alignment term Δ V^PC_q/bulk|far is the difference between the defect-induced potential and point charge potential in a region far away from the defect Δ V^PC_q/bulk|far=(V_defect,q-V_bulk)-V^PC_q. Since the atomic structure is allowed to relax when defects are introduced, the atomic site electrostatic potential is used as a potential marker, as discussed in detail by Kumagai et al. <cit.>. A small uncertainty in the alignment-like term arises due to the limited supercell size (less than 0.1 eV). The Madelung energy is estimated by an Ewald summation, and the macroscopic dielectric constant, valid for cubic systems, is replaced by a dielectric tensor. Both ion-clamped and ionic contributions to the dielectric tensor of the bulk system were calculated from self-consistent response of the system to a finite electric field <cit.>, resulting in ϵ_⊥=7.19 and ϵ_∥=8.23. These values are lower than the experimental ones <cit.>, but closer than those obtained from density functional perturbation theory with the GGA-PBE functional <cit.>. By varying the chemical potential, different experimental conditions can be explored. Upper and lower bounds are given by the stability of the phases that constitute the reservoir, which is expressed by the thermodynamic stability condition Δ H^f(ZnO)=μ_Zn+μ_O. The upper bound of μ_O (and thus the lower bound of μ_Zn) is given by half the total energy of an O_2 molecule, and corresponds to O-rich conditions. Likewise, the lower limit of μ_O (and the upper limit of μ_Zn) is given by the reduction of ZnO to metallic Zn, corresponding to Zn-rich conditions <cit.>. The hybrid functional yields a ZnO heat of formation of Δ H^f(ZnO) = -3.49 eV, which is close to the experimental value of -3.61 eV <cit.>. From the calculated defect formation energies, thermodynamic charge-state transition levels are given by the Fermi level position for which the formation energy of the defect in two charge-states q_1 and q_2 is equal, i.e., <cit.> ε(q_1/q_2)=E^f(q_1;ε_F=0)-E^f(q_2;ε_F=0)/q_2-q_1.§.§ Defect luminescence Defect luminescence lineshapes were calculated by using the methodology described in Ref. Alkauskas2012, wherein the multidimensional vibrational problem is mapped onto an effective one-dimensional configuration coordinate diagram <cit.> (Fig. <ref>). The parameters that enter the model are the effective phonon frequencies Ω_g/e, the zero phonon line E_ZPL and the configuration coordinate ΔQ which is defined as (ΔQ)^2=MΔ R^2=∑_iαm_α(R_e,iα-R_g,iα)^2. Here, M is the effective modal mass in atomic units and Δ R is the magnitude of the displacement in Å for all atoms α in the supercell (i={x,y,z}). Thus, the configuration coordinate represents the collective motion of all atoms in the supercell between the different charge-states, meaning that the various individual vibrational modes are replaced by a single effective mode. All parameters are obtained from the hybrid DFT calculations by using finite differences. Huang-Rhys (HR) factors <cit.> describe the average number of phonons that are involved in a transition, and can be expressed as S_g = Δ E_g/ħΩ_g for emission. Δ E_g is the relaxation energy, often referred to as the Franck-Condon shift. The effective one-dimensional configuration coordinate model is a good approximation for broad luminescence bands with strong electron-phonon coupling (S≫ 1), as demonstrated in Refs. Alkauskas2012,Alkauskas2016. §.§ The self-interaction error While the value of the fraction of HF exchange used (37.5%) reproduces the lattice parameters and bulk band gap of ZnO, this does not necessarily mean that the defect states of V_Zn are described correctly <cit.>. In order to elaborate on possible over-localization, all charge-states were also calculated using the original HSE06 functional (25% HF exchange). However, the results remained qualitatively unaffected regarding the localization of holes. The energy positions of the polaronic Kohn-Sham (KS) states, relative to the average electrostatic potential, were almost unchanged. Moreover, we found the so-called non-Koopmans' energy for V_Zn^0, defined in Refs. Ivady2013,Lany2009 as E_NK=ε(N)-E_A=ε(N)-(E(N+1)-E(N)), to be small (0.12eV). Here, ε(N) is the KS quasiparticle energy of the polaronic state and E_A is the electron addition energy of the system, i.e., the difference in total energy between the (N+1)- and N-electron system, keeping the ions fixed to their N-electron ground-state positions. The electrostatic finite-size correction was applied only to E(N+1), using the ion-clamped dielectric tensor, since the two remaining terms are for the neutral defect. E_NK may still contain a small finite-size error. We conclude, however, that the self-interaction error is small in the calculations, and, importantly, the qualitative results do not hinge on the specific value used for the exchange parameter.§ RESULTS AND DISCUSSION In wurtzite ZnO, four O ions form a tetrahedron around every Zn ion and vice versa. In these tetrahedra, we shall refer to the ions in the three corners of the basal plane as azimuthal ions, while the ion in the fourth corner will be referred to as the axial ion. When a Zn vacancy is formed, four Zn–O bonds are broken. The dangling O 2p bonds that remain are partially filled by six electrons, and can accommodate two more. This simple chemical picture dictates that V_Zn acts as a double acceptor. However, it can also trap holes in polaronic states, as will be shown in Sections <ref> and <ref>. §.§ Thermodynamics of the Zn vacancy Fig. <ref> shows the formation energy of V_Zn as a function of the Fermi level position under O-rich conditions. The formation energy approaches ∼0.2 eV near the CB minimum, which indicates that V_Zn should be the dominating intrinsic acceptor in n-type ZnO. This is in agreement with positron annihilation spectroscopy (PAS) measurements <cit.> and previous DFT studies <cit.>. However, the majority of previous DFT studies have only included acceptor charge-states of V_Zn (two examples are shown in Fig. <ref>). As shown previously <cit.>, V_Zn can display positive charge-states as well. Indeed, we find both the + and 2+ state, with thermodynamic transitions located at 0.25 (2+/+), 0.89 (+/0), 1.40 (0/-) and 1.96 eV (-/2-) above the VB maximum (note that the transitions are approximately equidistant). The emergence of both positive and negative charge-states means that V_Zn is an amphoteric defect. Although the formation energy is rather high in p-type material, V_Zn is predicted to act as a compensating donor in a frozen-in, out-of-equilibrium scenario. One might question why the previous hybrid calculations by Oba et al. <cit.> deviate so much from our results. The main reason for this is the fact that spin-polarization was taken into account for V_Zn^- only in Ref. Oba2008 (due to computational constraints), but also because a plane-wave energy cutoff of 300 eV and a Γ-only k-point sampling were used. We elaborate on this in Section <ref>. The calculated position of the (0/-) transition agrees well with photo-EPR data; Evans et al.<cit.> inferred that the threshold energy to excite an electron from V_Zn^- to the CB (to observe the EPR signal of V_Zn^0) is ∼2.5 eV. We obtain an absorption energy of 2.64 eV for V_Zn^-, as shown in Fig. <ref> (b). This is somewhat higher than the photo-EPR data, but the onset shifts down due to vibrational broadening. This can be shown by simulating the absorption profile, as demonstrated in Ref. Alkauskas2016. The O vacancy (V_O) is also included in Fig. <ref> for comparison, since the (2+/0) transition level of this defect has become a benchmark case for defects in ZnO <cit.>. In fact, the defect state of V_O is fairly well described by a wide range of different functionals. While there is a large spread in the reported thermodynamic charge-state transition levels relative to the VB maximum, the agreement becomes decent when they are aligned to a common reference level, such as the average electrostatic potential <cit.>. We obtain 2.1 eV above the VB maximum for the (2+/0) transition, which is in good agreement with previous calculations based on hybrid functionals <cit.>. The defect wave function of V_Zn, on the other hand, is poorly described at the (semi)local level, implying that the energy positions of thethermodynamic charge-state transition levels depend strongly on the choice of functional. §.§ Polaronic hole localization The emergence of the positive charge-states of V_Zn can be understood by taking a closer look at its electronic and atomic structure. First V_Zn^2- is considered, where the O 2p dangling bond states are completely filled with electrons. By examining the spd- and site-projected wave function character of each KS state, one can deduce that the dangling bonds introduce three states in the band gap close to the VB maximum, and one resonant with the VB. When one electron is removed, the spin-degeneracy of the resulting half-filled defect state is broken, and the empty state moves deep into the band gap. As more electrons are removed, additional empty states exhibiting polaronic nature appear deep in the band gap, until all four dangling bond states are half-filled (V_Zn^2+). The probability density of the empty defect states, shown in Fig. <ref>, illustrates that each hole localizes onto a single nearest-neighbor O^2- ion in the form of a small hole polaron. This spontaneous localization of holes is accompanied by a distinct outward relaxation of the O^- ion, which further moves the polaronic state into the band gap, lowering the total defect energy. The azimuthal O^- ions with trapped holes move away from the vacancy by approximately 14% of the bulk Zn–O bond length, which is about twice as far as the azimuthal O^2- ions without trapped holes. This behavior (hole localization with local lattice distortion) is common for many oxide semiconductors <cit.>. As pointed out by Janotti et al. <cit.>, the (semi)local functionals are unable to describe the Zn vacancy (and thus fail to stabilize V_Zn^+ and V_Zn^2+). This is because of the self-interaction error; a lower total energy occurs by dividing the hole between multiple O ions. Lany and Zunger <cit.> removed this delocalization bias by using a hole-state potential to enforce fulfillment of the generalized Koopmans' condition, ensuring a linear behavior of the total energy and a constant behavior of the highest-occupied single-particle level with respect to fractional occupation <cit.>. This correction stabilized V_Zn^+ and V_Zn^2+, but it does not remedy the severe band gap underestimation of GGA (E_g=0.73 eV), leading to an ambiguity in the energy position of the thermodynamic charge-state transition levels with respect to the band edges. It must be noted, however, that the overall result of Lany and Zunger is in good agreement with our result. Incorporating a fraction of exact exchange, hybrid functionals cancel (at least in part) the self-interaction error, and provide accurate band gaps. However, previous hybrid DFT studies employing the HSE, PBE0 and sX functionals did still not reveal the positive charge-states of V_Zn <cit.>. Here, we demonstrate that initial symmetry-breaking operations, like moving the O ions slightly or specifying their initial magnetic moment, are a prerequisite to obtain localization of the holes onto single O ions. It is also crucial that the ions are relaxed with the hybrid functional, and that spin-polarized calculations are performed for all charge-states. Otherwise, the ground-state will not be obtained; the holes may instead delocalize over more than one O ion, with no polaronic effects. By breaking the symmetry, all metastable localized hole configurations were investigated. The azimuthal configuration of holes, shown in Fig. <ref>, was found to be the most stable one—in agreement with EPR data <cit.>. In addition, a seperation of 3.69 Å between the two O^- ions with trapped holes in V_Zn^0 was obtained, which is close to the 3.75 Å inferred from EPR measurements <cit.>. Finally, we find that the high-spin configuration of V_Zn is energetically preferred, which means that S=1/2 for V_Zn^-, S=1 for V_Zn^0, S=3/2 for V_Zn^+ and S=2 for V_Zn^2+. Polaronic hole localization is not unique for V_Zn in ZnO. While lattice deformation alone is not sufficient to induce hole localization <cit.>, polarons can form when an acceptor-like defect exists at a neighboring Zn site <cit.>. Indeed, substitutional Group-I impurities (Li_Zn, Na_Zn) exhibit the same tendency to stabilize anion trapped hole polarons <cit.>. Moreover, anion site substitutional impurities can lead to deep atomic-like localized states <cit.>. These effects, in combination with the very low position of the ZnO VB on an absolute energy scale and the heavy hole effective masses, render p-type doping of ZnO challenging at equilibrium conditions. §.§ A simple model for the Zn vacancy energetics Here, a simple model to explain the energetics of V_Zn, with regard to the approximately equidistant charge-state transition levels, is presented. First, assume the Fermi level position at the VB maximum (ε_F=0 eV in Fig. <ref>), and as a starting point consider V_Zn^2-. Adding a hole to the defect lowers the formation energy by ε_0, which includes the polaron formation energy. Subsequently E^f(V_Zn^-)=E^f(V_Zn^2-) - ε_0. According to Eq. (<ref>), this translates into the ε(-/2-) charge-state transition level being located at ε(-/2-)=ε_0 above the VB maximum. Adding another hole lowers the formation energy of V_Zn^- by ε_0-U, where U is the hole-hole repulsion energy, that is ε(0/-)=ε_0-U. Adding a third hole lowers the energy even less, as now the hole is repelled by two other holes: ε(+/0)=ε_0-2U. Similarly for the fourth hole ε(2+/+)=ε_0-3U. This simple model explains why the V_Zn charge-state transition levels are approximately equidistant, since ε(-/2-)-ε(0/-) ≅Uε(0/-)-ε(+/0) ≅Uε(+/0)-ε(2+/+) ≅U. Of course, this is only approximately true. The hole-hole repulsion U corresponds to the so-called Hubbard correlation energy <cit.>. By taking the average of Eq. (12–14), the value can be derived as U≃0.57 eV. Additionally, the fact that the in-plane configuration of holes is energetically preferred explains why the separation between the (2+/+) and (+/0) level is somewhat larger than the other two; the fourth hole must localize on the remaining axial O ion (lower hole addition energy). Taking this into account, i.e., considering only Eq. (12) and (13), the hole-hole repulsion energy becomes U≃0.53 eV. These considerations help rationalize our findings with reference to the energetics of V_Zn. In fact, by inspecting the results of Ref. Lyons2015, we suggest that a similar model applies to the Ga vacancy (V_Ga) in GaN, which can also trap up to four holes at the nearest neighbor N ions. §.§ Optical properties of the Zn vacancy Optical transitions involving V_Zn and the band edges have been investigated. The configuration coordinate diagrams and calculated lineshapes and positions are shown in Fig. <ref>, and all the effective parameters for the various transitions are given in Table <ref>. We find effective normal mode frequencies ħΩ_g/e between 24–34 meV, total mass-weighted distortions between 2.6–3.0 amu^1/2Å and large HR factors between 23–28, resulting in broad luminescence lines. This is expected for optical transitions involving polaronic acceptors like V_Zn, because of the sizeable changes in the atomic geometry between different charge-states <cit.>. Indeed, the main contribution to ΔQ comes from the four nearest Zn ions to the O^- ions with trapped holes. Considering ZnO as primarily an n-type material, optical transitions involving V_Zn^+ and V_Zn^2+ require V_Zn^2- to rapidly trap three and four holes, respectively. This is perhaps an unlikely scenario (unless the concentration of photogenerated holes is extremely high). Accordingly, we restrict primarily our following discussion to transitions involving V_Zn^2-, V_Zn^- and V_Zn^0. Capture of an electron located at the CB minimum by V_Zn^- results in a broad luminescence lineshape peaking at an energy of 0.71 eV. Note, however, that the excited- and ground-state normal modes overlap close to the minimum of the excited state (illustrated in Fig. <ref> (a)). The energy barrier from the minimum of the excited state up to the point of intersection is only 90 meV, implying that the transition is likely to be nonradiative. Hole capture by V_Zn^0 and V_Zn^+ is expected to be nonradiative for the same reason, and have been omitted from Fig. <ref>. In fact, in the latter case, the effective normal modes intersect at Q<ΔQ, i.e., before the minimum of the excited state is reached. In contrast, electron capture by V_Zn^0 will have both a radiative and nonradiative component, since the energy barrier is 0.48 eV. The resulting luminescence lineshape peaks at 1.24 eV. Capture of a hole located at the VB maximum by V_Zn^2- results in a somewhat narrower luminescence band peaking at 1.40 eV (Fig. <ref> (e)). Strictly speaking, since the VB in ZnO and the defect states of V_Zn both have O 2p character, this transition should be forbidden. However, just like for hole capture by V_Ga^3- in GaN <cit.>, the transition may be allowed because of the strong polaronic relaxation. Nevertheless, this transition has to compete with shallower negatively charged acceptors like Li_Zn^-, which captures holes nonradiatively in an efficient manner <cit.>. Following this logic, the luminescence might be weak in reality, depending on the purity of the sample and the concentration of V_Zn. Finally, it should be pointed out that the simulation results in Fig. <ref>, revealing prevalent V_Zn-related luminescence at low energies close to the infrared region in n-type material, are consistent with recent experimental data by Dong et al. <cit.> and Knutsen et al. <cit.>. Through a combination of cathodoluminescence and PAS measurements, it was found in Ref. Dong2010 that the emission from small V_Zn clusters peaks at ∼1.9 eV and shifts to lower energies with decreasing cluster size. Similarly, using samples irradiated with electrons having energies below and above the threshold for displacement of Zn atoms, as well as samples annealed in Zn-rich and O-rich ambients, Knutsen et al. <cit.> demonstrated that the luminescence in the near infrared region arises from V_Zn, or defects containing V_Zn. Especially in the case where V_Zn forms a complex with a donor, e.g., V_O <cit.>, the most negative thermodynamic charge-state transition levels would be passivated by the donor electrons. Hence, one could speculate that a transition similar to c) in Fig. <ref>, peaking in the 1.6–1.9 eV range, would prevail for such a complex, consistent with the experimental data <cit.>.§ CONCLUSION Based on the present calculations, we conclude that V_Zn in ZnO is a deep polaronic acceptor that can bind a localized hole on each of its four nearest-neighbor O ions. The distinct outward relaxation of these O ions is a key feature of the polaronic nature of V_Zn—in agreement with experimental EPR data <cit.>. By employing a one-dimensional configuration coordinate model <cit.>, luminescence positions and lineshapes from V_Zn were simulated. In contrast to what has been previously suggested <cit.>, the present results show that the isolated V_Zn is unlikely to be a major source of luminescence in the visible range for n-type material. All transitions involving V_Zn^2-, V_Zn^- and V_Zn^0 are nonradiative and/or lead to luminescence lineshapes that are very low in energy (near-infrared region). Electron capture by V_Zn^+ and V_Zn^2+ leads to red and green luminescence, respectively, but these transitions are unlikely to occur in n-type material unless the concentration of photogenerated holes is extremely high. These results are consistent with recent experimental data <cit.>. The luminescence lines arising from V_Zn are broad. This is because of the large relaxation associated with hole capture by one of the nearest-neighbor O ions, i.e., the strong electron-phonon coupling. This highlights the important role of hybrid functionals, which, unlike (semi)local functionals, are able to predict charge localization associated with local lattice distortions around defects <cit.>. We wish to thank Prof. F. Oba for helpful discussions. Financial support was kindly provided by the Research Council of Norway and University of Oslo through the frontier research project FUNDAMeNT (no. 251131, FriPro ToppForsk-program). K. M. Johansen would like to thank the Research Council of Norway for support to the DYNAZOx project (no. 221992). A.A. was supported by Marie Skłodowska-Curie Action of the European Union (project Nitride-SRH, Grant No. 657054). UNINETT Sigma2 and the Department for Research Computing at the University of Oslo are acknowledged for providing computational resources and support under Projects No. NN4604K, NN9180K and NN9136K. | http://arxiv.org/abs/1705.09215v1 | {
"authors": [
"Y. K. Frodason",
"K. M. Johansen",
"T. S. Bjørheim",
"B. G. Svensson",
"A. Alkauskas"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170525151057",
"title": "Zn vacancy as a polaronic hole trap in ZnO"
} |
21cm 16cm-1.4cm0.4cm 0.4cmThTheorem PropProposition CoCorollary LmLemma LmaLemma[section] DfiDefinition RmRemark ConOpen Problem | http://arxiv.org/abs/1705.09848v2 | {
"authors": [
"Tristan Rivière"
],
"categories": [
"math.DG",
"math.AP",
"49Q05, 53A10, 58E12, 49Q10"
],
"primary_category": "math.DG",
"published": "20170527173525",
"title": "Minmax Hierarchies and Minimal Surfaces in Manifolds"
} |
Department of Mathematics, UC Davis, Davis, CA 95616 [email protected] Department of Mathematics, UCLA, Los Angeles, CA 90095 and Department of Mathematics, Koç University, 34450,Istanbul, Turkey [email protected][2000]The second author was partially supported by a research grant of the Scientific and Technological Research Council of Turkey. We develop a technique forgluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtaintrisection diagrams forclosed 4-manifolds.As an application,we describe atrisection of any closed 4-manifold which admits a Lefschetz fibration over S^2 equipped with a section of square -1, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtainexplicit trisection diagrams fora pair ofclosed 4-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented S^2-bundle over any closed surface and in particularwe draw thecorresponding diagrams for T^2 × S^2 and T^2 ×̃ S^2 using our gluing technique. Furthermore, we provide analternate proof of a recent result of Gay and Kirby which says that every closed 4-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry. Trisections of 4-manifolds via Lefschetz fibrations Nickolas A. Castro and Burak Ozbagci December 30, 2023 ===================================================§ INTRODUCTION Recently, Gay and Kirby <cit.> proved that every smooth, closed, oriented, connected 4-manifold X admits a trisection, meaning thatfor every X, there exist non-negative integers g ≥ k such that X is diffeomorphic to the union X_1 ∪ X_2 ∪ X_3 of three copies of the 4-dimensional 1-handlebody X_i ≅♮^k (S^1 ×B^3), intersecting pairwise in 3-dimensional handlebodies, with triple intersection a closed, oriented, connected 2-dimensional surface _g of genus g.Such a decomposition of X is called a (g,k)-trisection or simply a genus g trisection, since k is determined by g using the fact that χ (X)=2+g-3k, where χ (X) denotes the Euler characteristic of X. Moreover, they showed that the trisection data can be encodedas a 4-tuple (_g, , ,̱), which is called a (g,k)-trisection diagram,such that each triple (_g, , )̱, (_g, ,̱), and(_g, , ) is a genus g Heegaard diagram for#^k (S^1 × S^2). Furthermore, they proved that trisection of X (and its diagram)is unique up to a natural stabilization operation. On the other hand, various flavors of Lefschetz fibrations have been studied extensively in the last two decades to understand the topology of smooth 4-manifolds. Suppose that a closed 4-manifold X admits a Lefschetz fibration over S^2, whose regular fiber is a smooth, closed, oriented, connectedsurface _p of genus p. The fibration induces a handle decomposition of X, where the essential data can be encoded by a finite set of ordered simple closed curves (the vanishing cycles) on a surface diffeomorphic to _p. The only condition imposed on the set curves is that the product of right-handed Dehn twists along these curves is isotopic to the identity diffeomorphism of _p. In addition, every 4-manifold W with nonempty boundary has arelative trisection and under favorable circumstances W also admits an achiral Lefschetz fibration over B^2 with bounded fibers. The common feature shared by these structures is that each induces a natural open book on W. To exploit this feature in the present paper, we develop a technique to obtaintrisection diagrams for closed 4-manifolds by gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary. The precise result is stated in Proposition <ref>,which is too technical to include in the introduction. Nevertheless, our gluing technique has several applications — one of which is the following result. Theorem 3.7. Suppose that X is asmooth, closed, oriented, connected 4-manifoldwhich admits a genus p Lefschetz fibration over S^2 with n singular fibers, equipped with a section of square -1. Then, an explicit(2p+n+2, 2p)-trisection of X can be describedby acorresponding trisection diagram,which isdetermined by the vanishing cycles of the Lefschetz fibration. Moreover, if X denotes the 4-manifold obtained from X by blowing down the section of square -1, then we also obtain a (2p+n+1, 2p)-trisection of X along with a corresponding diagram.In particular, Theorem <ref> provides a description of a (46, 4)-trisection diagram of the Horikawa surface H'(1) (see <cit.> for its definition and properties),a simply connectedcomplex surface of general type which admits a genus 2 Lefschetz fibration over S^2 with 40 singular fibers, equipped with a section of square -1.This section is the unique sphere in H'(1) with self-intersection -1 so thatby blowing it down,we obtain a trisection diagram for the simply connectedminimal complex surface H'(1)of general type.To the best of our knowledge, none of the existing methods in the literature can be effectively utilized to obtain explicit trisection diagrams for complex surfaces of general type. For example,Gay and Kirby describetrisections of S^4, , , closed 4-manifolds admittinglocally trivial fibrations over S^1 or S^2 (including of course S^1 × S^3, S^2 × S^2 and S^2 ×̃ S^2) andarbitraryconnected sums of these in<cit.>. Note that, by Freedman's celebrated theorem, the Horikawa surface H'(1) is homeomorphic to 5 # 29 (and also to the elliptic surface E(3)), since it is simply connected, nonspin and its Euler characteristic is 36, while its signature is -24.On the other hand, since H'(1) is a simply connected complex surface (hence Kähler) with b_2^+ (H'(1)) >1, it has non-vanishing Seiberg-Witten invariants,while 5 # 29 has vanishing Seiberg-Witten invariants which follows from the fact that 5 # 29= # (4 # 29 ). Hence, we conclude thatH'(1) is certainly not diffeomorphic to 5 # 29. As a consequence, we obtain explicit (46,4)-trisection diagrams for a pair ofclosed 4-manifolds, the Horikawa surface H'(1) and5 # 29, which are homeomorphic but not diffeomorphic. Note that 5 # 29 has a natural (34,0)-trisection diagram (obtained by the connected sum of the standard (1,0)-trisection diagrams ofand ), which can be stabilized four times to yield a(46,4)-trisection diagram.More generally,Theorem <ref> can be applied to a large class of 4-manifolds. A fundamental result of Donaldson <cit.> says that every closed symplectic4-manifold admits a Lefschetz pencil over ℙ^1 ≅ S^2. By blowing up its base locus the Lefschetz pencil can be turned into a Lefschetz fibration over S^2, so that each exceptional sphere becomes a (symplectic)section of square -1. Conversely,any 4-manifold X satisfying the hypothesis of Theorem <ref> must carry a symplectic structure where the section of square -1 can be assumed to be symplectically embedded.Therefore, X is necessarilya nonminimal symplectic 4-manifold. In <cit.>, Gay describes a trisection for any closed 4-manifold X admitting a Lefschetz pencil, although he does not formulate the trisection of X in terms of the vanishing cycles of the pencil (see <cit.>). He also points out that his technique does not extend to cover the caseof Lefschetz fibrations on closed 4-manifolds <cit.>.We would like to point out that Theorem <ref> holds true for any achiral Lefschetz fibrationπ_X:X → S^2equipped with a section of square -1. In this case, X is not necessarily symplectic. We opted to state our result only for Lefschetz fibrations to emphasize their connection to symplectic geometry.Nextwe turn our attention to another natural application of ourgluing technique where we find trisections ofdoubles of 4-manifolds with nonempty connected boundary.It is well-known (see, for example, <cit.>) that there aretwo oriented S^2-bundles overa closed, oriented,connected surface _h of genus h: the trivialbundle _h × S^2 and the twisted bundle_h ×̃ S^2. The former is the double of any B^2-bundle over _h with even Euler number, while the latter isthe double of any B^2-bundle over _h with odd Euler number. We obtaintrisections ofthese S^2-bundles by doubling the relative trisections of the appropriateB^2-bundles. In particular, we draw thecorresponding (7,3)-trisection diagram for T^2 × S^2 and the (4,2)-trisection diagram for T^2 ×̃ S^2 using our gluing technique. For anyh ≥ 1, the twisted bundle _h ×̃ S^2 is not covered by the examples in <cit.>, while our trisection for _h × S^2has smaller genus compared to that of given in <cit.>.We discuss the case oforiented S^2-bundles over nonorientable surfaces in Section <ref>. Finally, we providea simple alternate proof of the following result due to Gay and Kirby. Theorem 5.1.Every smooth, closed, oriented, connected 4-manifold admits a trisection.Our proof is genuinely different from the two original proofs due to Gay and Kirby <cit.>, one with Morse 2-functions and one with ordinary Morse functions,since not only contact geometry plays a crucial role in our proof, but we also employ a technique for gluing relative trisections.After the completion of our work, we learned that Baykur and Saeki <cit.>gave yet another proof of Theorem <ref>, setting up a correspondence between broken Lefschetz fibrations and trisections on 4-manifolds, using a method which is very different from ours. In particular, they prove theexistence of a (2p+k+2, 2k)-trisectionon a 4-manifold X which admits a genus p Lefschetz fibration over S^2 with k Lefschetz singularities — generalizing the first assertion in our Theorem <ref>, but without providing the corresponding explicit diagram for the trisection.They also give examples of trisections (without diagrams) on a pair of closed 4-manifolds (different from ours)which are homeomorphic but not diffeomorphic.In addition, for any h ≥ 0, they give small genus trisections (again without the diagrams) for _h × S^2.Conventions: All 4-manifolds are assumed to be smooth, compact, oriented and connected throughout the paper. The corners which appear in gluing manifolds are smoothed in a canonical way. § GLUING RELATIVE TRISECTIONSWe first review somebasic results about trisections and their diagrams (cf. <cit.>).Let Y^+_g,k∪ Y^-_g,k denote the standard genus g Heegaard splitting of #^kS^1× S^2 obtained by stabilizing the standard genus k Heegaard splitting g-k times. A (g,k)-trisection of a closed 4-manifold X is a decomposition X = X_1 ∪ X_2 ∪ X_3 such that for each i=1,2,3, * there is a diffeomorphism φ_i: X_i →♮^k S^1 × B^3, and * taking indices mod 3, φ_i(X_i ∩ X_i+1) = Y^+_g,k and φ_i(X_i ∩ X_i-1) = Y^-_g,k. It follows thatX_1 ∩ X_2 ∩ X_3 is a closed surface of genus g.Also note that g and k determine each other, since the Euler characteristic χ(X) isequal to 2+g-3k, which can be easily derived by gluing X_1 and X_2 first and then gluing X_3. Suppose that each of η and ζ is a collection of m disjoint simple closed curves on some compact surface Σ. We say that twosuch triples (, η, ζ) and (', η', ζ') are diffeomorphism and handleslide equivalent if there exists a diffeomorphism h: →' such that h(η) is related to η' by a sequence of handleslides and h(ζ) is related to ζ' by a sequence of handleslides.A (g,k)-trisection diagram is an ordered 4-tuple (Σ, α, β, γ) such that * Σ is a closed genus g surface, * each of α, β, andγ is anon-separating collection of g disjoint, simple closed curves on _g, * each triple (Σ, α, β), (Σ, β, γ), and (Σ, α, γ) is diffeomorphism and handleslide equivalentto the standard genus g Heegaard diagram of #^k S^1× S^2 depicted in Figure <ref>. According to Gay and Kirby <cit.>, every closed 4-manifold admits a trisection, which in turn, can be encoded by a diagram. Conversely,every trisection diagram determines a trisected closed 4-manifold,uniquely up to diffeomorphism.Next we recall the analogous definitions of relativetrisections and their diagramsfor 4-manifolds with nonempty connected boundary (cf. <cit.>). Suppose that W is a 4-manifold with nonempty connected boundary W. We would like to find a decompositionW=W_1 ∪ W_2 ∪ W_3, such that each W_i is diffeomorphic to ♮^kS^1 × B^3 for some fixed k.Since ∂ W ≠∅, it would be natural to require that part of each W_i contribute to W.Hence, we need a particular decomposition of (♮^kS^1 × B^3 )=#^k S^1× S^2 to specify a submanifoldof W_i to be embedded in ∂ W. With this goal in mind, we proceed as follows to develop the language we will use throughout the paper.Suppose thatg,k,p,b are non-negative integerssatisfying b >0 andg+p+b-1 ≥ k ≥ 2p+b-1. Let Z_k = ♮^k S^1× B^3 and Y_k=∂ Z_k = #^k S^1× S^2 for k ≥ 1,andZ_0 = B^4, Y_0=S^3.We denote by P a fixed genus p surface with b boundary components. LetD={re^iθ∈ℂ| r ∈ [0,1]-π/3≤θ≤π/3}be a third of the unit disk in the complex plane whose boundary is decomposed as ∂ D = ∂^-D ∪∂^0 D∪∂^+D, where ∂^-D= {re^iθ∈∂ D |θ=-π/3} ∂^0D= {e^iθ∈∂ D} ∂^+D={re^iθ∈∂ D |θ=π/3}The somewhat unusual choice of the disk D will be justified by the construction below.We setU = P × D, which is indeed diffeomorphic to ♮^2p+b-1S^1× B^3. Then, ∂ U inherits a decomposition ∂ U = ∂^-U ∪∂^0U ∪∂ ^+U, where∂^0U=(P ×∂^0D) ∪ (∂ P × D) and ∂^±U = P ×∂^±D.Let ∂(S^1× B^3) = H_1 ∪ H_2 be the standard genus one Heegaard splitting of S^1× S^2. For any n>0, let V_n = ♮^n(S^1× B^3), where the boundary connected sum is taken in neighborhoods of the Heegaard surfaces, inducing the standard genus n Heegaard splitting of ∂ V_n =#^n S^1× S^2 = ∂^-V_n ∪∂^+V_n. Stabilizing this Heegaard splitting s times we obtain a genus n+s Heegaard splitting of ∂ V_n = ∂^-_s V_n ∪∂^+_s V_n. We set s=g-k+p+b-1 andn=k-2p-b+1.Note that we can identify Z_k = U♮ V_n,where the boundary connected sum again takes place along the neighborhoods of points in the Heegaard surfaces. We now have a decomposition of ∂ Z_k as follows: ∂ Z_k = Y_k = Y^+_g,k;p,b∪ Y^0_g,k;p,b∪ Y^-_g,k;p,b,where Y^±_g,k;p,b = ∂^± U ♮∂^±_s V_n and Y^0_g,k;p,b= ∂^0 U. A (g,k;p,b)-relative trisection of a 4-manifold W with non-empty connected boundary is a decomposition W= W_1 ∪ W_2 ∪ W_3 such that for each i=1,2,3, * there is a diffeomorphism φ_i: W_i → Z_k = ♮^k S^1 × B^3, and * taking indices mod 3, φ_i(W_i ∩ W_i+1) = Y^+_g,k;p,b and φ_i(W_i ∩ W_i-1) = Y^-_g,k;p,b. As a consequence, W_1 ∩ W_2 ∩ W_3 is diffeomorphic toY^-_g,k;p,b∩ Y^+_g,k;p,b, which is a genus g surface with b boundary components.Note that the Euler characteristic χ(W) is equal tog-3k+3p+2b-1, which can be calculated directly from the definition of a relative trisection. We also give alternate method to calculate χ(W)in Corollary <ref>.According to <cit.>, every4-manifold W with nonempty connected boundary admits a trisection. Moreover, there is a natural open book induced on ∂ W, whose page is diffeomorphic to P, which is anessential ingredient in our definition ofY^±_g,k;p,b. Informally, the contribution of each W_i to W isone third of an open book. This is because the part of each W_i that contributes to W is diffeomorphic toY^0_g,k;p,b = ∂^0U=(P ×∂^0D) ∪ (∂ P × D),whereP ×∂^0D is one third of the truncated pages, while∂ P × D is one third of the neighborhood of the binding. In other words,not onlywe trisect the 4-manifold W, but we also trisect its boundary W. Conversely,if an open book is fixed on ∂ W, then W admits a trisection whose induced open book coincides with the given one. A (g,k;p,b)-relative trisection diagram is anordered 4-tuple (Σ, α, β, γ) such that * Σ is a genus g surface with b boundary components, * each of α, β and γ is a collection of g-p disjoint, essential, simple closed curves, * each triple (Σ, α, β), (Σ, β, γ), and (Σ, α, γ) is diffeomorphism and handleslide equivalent to the diagram depicted inFigure <ref>.It was shown in <cit.> that every relative trisection diagram determines uniquely,up to diffeomorphism, (i)a relatively trisected 4-manifold W with nonempty connected boundary and (ii) the open book on W induced by the trisection. Moreover, the page and the monodromy of the open book on∂ W is determined completely by the relative trisection diagram by an explicit algorithm, which we spell out below. Suppose that (, , ,̱)is a(g,k;p,b)-relative trisection diagram, which represents a relative trisection of a4-manifold W with nonempty connected boundary.The page of the induced open bookon W is given by _, which is the genus p surface with b boundary components obtained fromby performing surgery along thecurves. This means that to obtain _,we cut openalong eachcurve and glue in disks to cap off the resulting boundaries. Now that we have a fixed identification of the page ofas _, we use Alexander's trickto describe the monodromy μ : _→_ of . Namely,we cut _ into a single disk via two distinct ordered collections of 2p+b-1 arcs, so that for each arc in one collection there is an arc in the other collection with the same endpoints. As a result,we get a self-diffeomorphism of S^1 that takes one collection of arcs to the other respecting the ordering of the arcs, and equals to the identity otherwise. This diffeomorphism uniquely extends to aself-diffeomorphism ofthe disk, up to isotopy.Therefore,we get a self-diffeomorphism μof _ fixing _ pointwise, which is uniquely determined up to isotopy. Next, we provide some more details (see <cit.>) about how to obtain the aforementioned collection of arcs. Let _α be any ordered collection of disjoint, properly embedded 2p+b-1 arcs in Σ disjoint from , such that the image of _ in Σ_ cuts Σ_αinto a disk. We choose a collection of arcs _$̱,and a collection of simple closed curves'̱disjoint from_$̱ insuch that (, _)̱ is handleslide equivalent to (, _), and '̱ is handleslide equivalent to $̱. This means that_$̱ arcs are obtained by sliding _ arcs overcurves, and '̱ is obtained by sliding$̱curves over$̱ curves. Next we choose a collection of arcs _, and a collection of simple closed curves ' disjoint from_ insuch that ('̱, _) is handleslide equivalent to ('̱, _)̱, and ' is handleslide equivalent to . This means that _ arcs are obtained by sliding _$̱ arcs over'̱curves,and'is obtained by slidingcurves overcurves. Finally,we choose a collection of arcs, and a collection of simple closed curves'disjoint frominsuch that(', )is handleslide equivalent to(', _), and'is handleslide equivalent to. This means thatarcs are obtained by sliding_arcs over'curves,and'is obtained by slidingcurves overcurves. It follows that(', )is handleslide equivalent to(, _*)for some collection of arcs_*disjoint fromin. We call the triple (_, _,̱_) a cut system of arcs associated to the diagram(, , ,̱). Now we have two ordered collections of2p+b-1arcs_and_*in∖, such that each of their images in_cuts_into a disk. Then, as we explained above, there is a unique diffeomorphismμ:Σ_α→Σ_α, up to isotopy,which fixes∂_pointwise such thatμ(_α)= _*. It is shown in <cit.> that, up to isotopy,the monodromy of the resulting open book is independent of the choices in the above algorithm.Next,we give a very simple version of the general gluing theorem <cit.> for relatively trisected4-manifolds. Here we present a different proof — where we use the definition of a relative trisection as given in <cit.> instead of <cit.>— for the case of a single boundary component.Suppose thatW and W' are4-manifolds such that ∂ W and ∂ W' are both nonempty and connected.Let W=W_1 ∪ W_2 ∪ W_3 and W'=W_1' ∪ W_2'∪ W_3'be (g, k; p, b)- and (g', k';p', b')-relative trisections withinduced open booksand ' on ∂ W and ∂ W', respectively. If f: ∂ W →∂ W' is an orientation-reversing diffeomorphism which takes to ' (and hence p'=p and b'=b), then the relative trisections on W and W' can be glued together to yield a (G,K)-trisection of the closed 4-manifold X= W ∪_f W', where G=g+ g' + b-1 andK = k + k' - (2p + b-1). Since there is an orientation-reversing diffeomorphismf : ∂ W →∂ W' which takesthe open bookto the open book ', we have p'=p and b'=b. Let W=W_1 ∪ W_2 ∪ W_3 and W'=W_1' ∪ W_2'∪ W_3' be (g, k;p,b)- and (g', k';p,b)-relative trisections, respectively. Then, byDefinition <ref>,there arediffeomorphisms φ_i : W_i →Z_k = ♮^k S^1 × B^3 and φ'_i : W_i'→ Z_k' = ♮^k' S^1 × B^3 for i =1,2,3. Let B⊂ W and B' ⊂ W' be the bindings ofand ', where π :W ∖ B → S^1 and π' :W' ∖ B' → S^1 are the projection maps of these open books, respectively.Since the gluing diffeomorphism f:W → W' takesto ' by our assumption, we have π' (f(π^-1(t)))=t, for all t ∈ S^1.Moreover,we may assume thatf_i = φ_i' ∘ f ∘φ_i^-1 :Y^0_g, k;p,b→ Y^0_g', k';p,b is a diffeomorphism for each i=1,2,3.Informally, we identify each third ofon Wwith the appropriate third of ' on W' via the gluing map f.This allows us to define X_i = W_i f∪ W_i' = W_i ∪ W_i' / ∼ where x ∼ y ifx ∈φ^-1_i(Y^0_g, k;p,b) ⊂ W_i and y =f(x)∈ W_i', where φ_i' (y) ∈Y^0_g', k';p,b. We claim that X= W ∪_f W' = X_1 ∪ X_2 ∪ X_3 is a (G,K)-trisection, whereG=g+ g' + b-1 K = k + k' - (2p + b-1).In order to prove our claim,we first need to describe, for each i=1,2,3,a diffeomorphism Φ_i: X_i→ Z_K = ♮^K S^1 × B^3.The diffeomorphism Φ_i is essentially obtained by gluing the diffeomorphisms φ_i : W_i →Z_k and φ_i' : W_i' →Z_k' using the diffeomorphism f_i:W_i → W_i', as we describe below.Toconstruct the desired diffeomorphism Φ_i: X_i→ Z_K = ♮^K S^1 × B^3, it suffices to describe how to glue Z_k with Z_k' to obtain Z_K by identifyingY^0_g, k;p,b⊂ Z_kwithY^0_g', k';p,b⊂ Z_k' using the gluing map f_i.By definition,Z_k = U ♮ V_n, and similarly Z_k'= U' ♮ V_n', where U = P × D and U' = P' × D.Note that P is diffeomorphic to P' via f_i. To glue Z_kto Z_k' we identify Y^0_g, k;p,b⊂ U withY^0_g', k';p,b⊂ U' via the diffeomorphism f_i : Y^0_g, k;p,b→ Y^0_g', k';p,b.Next, we observe thatby gluing U and U' along the aforementioned parts of their boundaries using f_i,we get ♮^l S^1 × B^3, where l =2p+b-1.To see this,we view U = P × D as P × I_1 × I_2, and similarly U' = P' × D as P' × I_1 × I_2, where I_1=I_2=[0,1]. We glueP × I_1 with P' × I_1 and then take its product with I_2. To glue P × I_1 with P' × I_1 we identifyP ×{1} with P' ×{1} using f_i. The result ofthis identification is diffeomorphic to P × [0,2] ≅♮^l S^1 × B^2. However, to complete the identification dictated by f_i, we have to take the quotient ofP × [0,2] by the relation (x,t) ∼ (x, 2-t) for all x ∈ P.Note that we suppressed f_i here since we have already identified P with P' via f_i. The result is still diffeomorphic to the handlebody ♮^l S^1 × B^2. Therefore, the gluing of U and U' is diffeomorphic to♮^l S^1 × B^3, since it is a thickening of ♮^l S^1 × B^2 by taking its product with I_2.As a consequence,the result of gluing Z_k to Z_k' is diffeomorphic to (♮^l S^1 × B^3) ♮ V_n ♮ V_n'≅♮^l+n+n' S^1 × B^3 ≅♮^K S^1 × B^3 ≅ Z_K, since l+n+n' = 2p+b-1 + k-(2p+b-1)+k'-(2p+b-1)= k+k'-(2p+b-1)=K.To finish the proof of the lemma,we need to show that taking indices mod3,Φ_i (X_i ∩ X_i+1) = Y^+_G,KΦ_i (X_i ∩ X_i-1) = Y^-_G,K,whereY^+_G,K∪ Y^-_G,K isthe standard genus G Heegaard splitting of #^KS^1× S^2 = Y_K=Z_K.We observe that Φ_i (X_i ∩ X_i+1) = φ_i (W_i ∩ W_i+1)f_i∪φ_i' (W_i' ∩ W_i+1') =((Y^+_g,k;p,b∪ Y^+_g', k';p,b) / ∼) ⊂ Y_K where x∼ y if x ∈ (∂ P ×∂^+D) ∪ (P ×{e^iπ/3}) ⊂ Y^+_g,k;p,band y =f_i(x) ∈ Y^+_g', k';p,b. Note that by definition, Y^+_g, k;p,b = ∂^+U ♮∂^+_sV_n and similarlyY^+_g', k';p,b= ∂^+U' ♮∂^+_s'V_n'. Since the boundary connected sums are taken along the interior of Heegaard surfaces, the identification ∼ does not interact with ∂^+_sV_n and ∂^+_s'V_n' and hence (Y^+_g,k;p,b∪ Y^+_g', k';p,b) /∼ ≅((∂^+U ∪∂^+U' ) /∼) ♮ (∂^+_sV_n♮∂^+_s'V_n').But we seethat (∂^+U ∪∂^+U') /∼is diffeomorphic to ♮^l S^1 × B^2, by exactly the same argument used above when we discussed the gluing of U with U'.Therefore, we have (Y^+_g,k;p,b∪ Y^+_g', k';p,b) /∼ ≅((∂^+U ∪∂^+U') / ∼) ♮ (∂^+_sV_n♮∂^+_s'V_n')≅ (♮^lS^1× B^2) ♮ (∂^+_sV_n♮∂^+_s'V_n')≅ (♮^lS^1× B^2)♮^s+n+s'+n'S^1× B^2≅♮^G S^1× B^2since l+s+n+s'+n'= g+g'+b-1=G.Similarly, we haveΦ_i (X_i ∩ X_i-1) =φ_i (W_i ∩ W_i-1)f_i∪φ_i' (W_i' ∩ W_i-1')= (Y^-_g,k;p,b∪ Y^-_g', k';p,b) / ∼) ⊂ Y_K where x∼ y if x ∈ (∂ P ×∂^-D) ∪ (P ×{e^-iπ/3}) ⊂ Y^-_g,k;p,band y =f_i(x) ∈ Y^-_g', k';p,b. Thus we obtain (Y^-_g,k;p,b∪ Y^-_g', k';p,b) /∼ ≅♮^GS^1× B^2. Moreover, ((Y^+_g,k;p,b∪ Y^+_g', k';p,b) / ∼) ∪_((Y^-_g,k;p,b∪ Y^-_g', k';p,b) / ∼) = Y_K.Therefore,Y^+_G,K = (Y^+_g,k;p,b∪ Y^+_g', k';p,b) / ∼ and Y^-_G,K = (Y^+_g,k;p,b∪ Y^+_g', k';p,b) / ∼ givesthe standard genus G Heegaard splitting of #^KS^1× S^2 = Y_K, as desired.To summarize, we showed that there is a diffeomorphism Φ_i: X_i→ Z_K = ♮^K S^1 × B^3, for each i=1,2,3, andmoreover, taking indices mod 3, Φ_i (X_i ∩ X_i+1) = Y^+_G,K and Φ_i (X_i ∩ X_i-1) = Y^-_G,K. Therefore, we conclude that X= W ∪_f W' = X_1 ∪ X_2 ∪ X_3 is a (G,K)-trisection.Here is an immediate corollary of Lemma <ref>. Suppose that W=W_1 ∪ W_2 ∪ W_3 is a (g, k;p,b)-relative trisection of a 4-manifold W with nonempty connected boundary. Let DW denote the double of W, obtained by gluing W and W (meaning W with the opposite orientation) by the identity map ofthe boundary W. Then DW admits a (2g+b-1, 2k-2p-b+1)-trisection. If W=W_1 ∪ W_2 ∪ W_3 is a (g, k;p,b)-relative trisection of W with the induced open book on W, thenW=W_1 ∪W_2 ∪W_3 is a (g, k;p,b)-relative trisection of W with the induced open book on W, whereis obtained frombyreversing the orientation of the pages. Sincethe identity map from W to Wis an orientation-reversing diffeomorphism which takesto , weobtain the desired result about DW byLemma <ref>.The point of Corollary <ref> is that one does not need to know the monodromy of the open book onW,to describe a trisection onDW.Let E_n,h denote the B^2-bundle over _h with Euler number n ∈ℤ. In <cit.>, there is a description of a (|n| + h, |n| + 2h-1; h, |n|)-relative trisection of E_n,hfor n ≠ 0, and a(h+2, 2h+1; h, 2)-relative trisection of E_0,h=_h × B^2.Since the double of E_n,h is _h × S^2 or _h ×̃ S^2 depending on n modulo 2, we get a (2h+3|n|-1, 2h+|n|-1)-trisection of _h × S^2 (resp. _h ×̃ S^2) for any even (resp. odd) nonzero integer n, by Corollary <ref>.In particular,by doubling the(h+2, 2h+1; h, 2)-relative trisection of _h × B^2, we obtain a (2h+5, 2h+1)-trisection of _h × S^2 which is smaller compared to the (8h+5, 4h+1)-trisection presented in <cit.>, provided that h ≥ 1.Similarly, by setting n=±1, we obtain a(2h+2, 2h)-trisection for _h ×̃ S^2, which is not covered by the examples in <cit.>, except for h=0.Note that there is also a (2,1;0,2)-relative trisection of E_n,0 given in <cit.> for each n ∈ℤ. Since the double of E_n,0 is S^2 × S^2 or S^2 ×̃ S^2 depending on n modulo 2, we getinfinitely many(5,1)-trisections of S^2 × S^2 and S^2 ×̃ S^2.If W=W_1 ∪ W_2 ∪ W_3 is a (g, k;p,b)-relative trisection of a 4-manifold W with nonempty connected boundary, then the Euler characteristic χ(W) is equal to g-3k+3p+2b-1. Using Corollary <ref>, we compute χ(W)=1/2χ(DW)= 1/2 (2+ 2g+b-1 -3(2k-2p-b+1)))= g-3k+3p+2b-1. One can of course derive the same formula directly from the definition of arelative trisection.Since every relatively trisected4-manifold with connected boundary is determined by some relative trisection diagram, it would be desirable to have a version of Lemma <ref>, where one “glues" the relative trisection diagrams corresponding toW=W_1 ∪W_2 ∪W_3andW'=W_1' ∪W_2' ∪W_3'to get a diagram corresponding to the trisectionX= W ∪_f W' =X_1 ∪X_2 ∪X_3.This is the content of Proposition <ref>, but first we develop some language to be used in its statement.Let(, , ,̱ )and(', ', '̱, ')be(g, k;p,b)- and(g', k';p,b)-relative trisection diagrams corresponding tothe relative trisectionsW= W_1 ∪W_2 ∪W_3andW'=W_1' ∪W_2' ∪W_3', with induced open booksand'onWandW', respectively.Suppose that there is an orientation-reversing diffeomorphismf:∂W →∂W'whichtakesto'.We observe thatsince the orientation-reversing diffeomorphismf : W →W'takesto', it descends to an orientation-reversingdiffeomorphism, denoted again byffor simplicity,fromthe page_ofonto the page'_'of'. Therefore,frestricted to_is an orientation-reversing diffeomorphism from the binding_ofto the binding'_'of'.Let(_α, _β, _γ)denote a cut system of arcsas in Definition <ref> associated to the diagram (Σ, α, β, γ).We denote the arcs in_as{ a_1, …, a_l}, the arcs in_$̱ as { b_1, …, b_l}, and the arcs in _ as { c_1, …, c_l}, where l= 2p+b-1. We choose a cut system (_', _'̱, _') of arcsassociated to the diagram (Σ', α', β', γ') as follows.Since the collection _ of arcs cuts _ into a disk, the collection _' = { a'_1, …, a'_l} of arcs, where a'_i = f(a_i) for i=1, …, l,cuts '_' into a disk as well.Then we obtain _'̱= { b'_1, … ,b'_l} and _' ={ c'_1, …,c'_l} from _' as in Definition <ref>.In particular, we see that a'_i = f(a_i), b'_i = f(b_i)and c'_i = f(c_i) foreach i = 1, …, l. LetΣ^* denote the closed, orientedsurface obtained by gluingΣ and ' along their boundaries using the orientation-reversing diffeomorphism f := _→'_' =' defined above.It follows that _i = a_i ∪_∂ a'_i, _i =b_i ∪_∂ b'_i and _i = c_i ∪_∂ c'_i are simple closed curves in ^*, for i = 1, …, l. Then the collection of G= g+g'+b-1 disjoint, simple closed curves ^* = {_1, …, _G}⊂^* is defined as followsα^*_i:= {[ _i1≤ i≤ g-p _i-g+ p g-p+1≤ i≤ G-g'+pα_i'G-g'+p+1 ≤ i≤ G;].We write^* = ∪∪'. The collection of curves ^̱* = ∪̱∪'̱ and ^* = ∪∪' are defined similarly.We say that (Σ^*, α^*, β^*, γ^*)is obtained by gluing (, , ,̱) and (', ', '̱, ') by the map f. Let W and W' be 4-manifolds such that ∂ W and ∂ W' are both nonempty and connected.Suppose that (Σ, α, β, γ) and (Σ', α', β', γ') are (g, k;p,b)- and (g', k';p,b)-relative trisection diagrams corresponding tothe relative trisectionsW= W_1 ∪ W_2 ∪ W_3 and W'=W_1' ∪ W_2' ∪ W_3', with induced open booksand ' on ∂ W and∂ W',respectively. Suppose further that there is an orientation-reversing diffeomorphism f:∂ W →∂ W' whichtakesto '.Then (Σ^*, α^*, β^*, γ^*), which is obtained by gluing (Σ, α, β, γ) and (Σ', α', β', γ') by the map f as in Definition <ref>,is a (G,K)-trisection diagram corresponding to the trisection X= W ∪_f W' =X_1 ∪ X_2 ∪ X_3 described in Lemma <ref>, where G=g+g'+b-1 and K= k+k' -(2p+b-1).We claim that (Σ^*, α^*, β^*, γ^*) is a (G,K)-trisection diagram representing the trisection X= W ∪_f W' = X_1 ∪X_2 ∪ X_3 described in Lemma <ref>. By construction, ^* is a closed, orientedsurface ofgenus(g-p) + (2p+b-1) + (g'-p) = g+g'+b-1 =Gand each of ^*, ^̱* and ^* is a non-separating collection of G disjoint simple closed curves on ^*. To finish the proof, we need to show that ^* bounds disks in X_1 ∩ X_2,^̱* bounds disks in X_2 ∩ X_3,^* bounds disks in X_1 ∩ X_3.We know thatcurves bound disks in W_1 ∩ W_2,$̱ curves bound disks inW_2 ∩W_3,andcurves bound disks inW_1 ∩W_3. Similarly,'curves bound disks inW'_1 ∩W'_2,'̱curves bound disks inW'_2 ∩W'_3,and'curves bound disks inW'_1 ∩W'_3.Therefore∪'curves bound disks inX_1 ∩X_2,∪̱'̱curves bound disks inX_2 ∩X_3,and∪'curves bound disks inX_1 ∩X_3.Hence, all we need to show is thatcurvesbound disks inX_1 ∩X_2,curves bound disks inX_2 ∩X_3,andcurves bound disks inX_1 ∩X_3. But this follows by the fact thatα curves bound disks in the handlebody∂^+U ∪∂^+U'/∼,whereasβ curves bound disks in the handlebody∂^-U ∪∂^-U'/∼.Similar statement holds for the pairs(, )and(, ).Suppose that W=W_1 ∪ W_2 ∪ W_3 is a (g, k;p,b)-relative trisection of a 4-manifold Wwith nonempty connected boundary and let (, , ,̱) be a corresponding relative trisection diagram.Then the (2g+b-1, 2k -2p-b+1)-trisection of the double DW of W described in Corollary <ref> has a corresponding diagram (^*, ^*, β^*, ^*) which is obtained by gluing (, , ,̱) and (, , ,̱) by the identity mapfrom W to W.If W=W_1 ∪ W_2 ∪ W_3 is a (g, k;p,b)-relative trisection of W with the induced open book on W, whose corresponding relative trisection diagram is (, , ,̱), thenW=W_1 ∪W_2 ∪W_3 is a (g, k;p,b)-relative trisection of W with the induced open book on W, whose corresponding relative trisection diagram can be canonically given by(, , ,̱).The proof is complete by observing thatthe identity map from W to Wis an orientation-reversing diffeomorphismtakingto .In Sections <ref>and <ref>, using Corollary <ref>, wedraw explicit diagramsfor the (7, 3)-trisection of T^2 × S^2and the (4, 2)-trisection ofT^2 ×̃ S^2 given in Example <ref>, respectively. § TRISECTING LEFSCHETZ FIBRATIONSIn this section, weprove Theorem <ref>.We refer to <cit.> for the definitions and properties of Lefschetz fibrations andopen books. We need some preliminary results. The following lemma is well-known (cf. <cit.>).Suppose that a closed 4-manifold X admits a genus p Lefschetz fibration π_X :X → S^2 with a section of square -1 so that thevanishing cycles of π_X are given by the ordered set of simple closed curves {ł_1, ł_2, …, ł_n}. Let V denote a regular neighborhood of the section union a nonsingular fiber and let Wdenote the 4-manifold with boundary obtained from X by removing the interior of V. Then π_X: X → S^2 descends to a Lefschetz fibration π_W: W → B^2 whose regular fiber is a genus p surface _p,1with connected boundary. Moreover, the monodromy of π_W (and hence the open book naturally induced on W) is given byD()̣ = D(ł_n) D(ł_n-1) ⋯ D(ł_1) where $̣ is a boundary parallel curve in_p,1. Let V denote a regular neighborhood of the section of square -1 union a nonsingular fiber as in Lemma <ref>. Then there is an achiral Lefschetz fibration π_V:V → B^2 whose regular fiber is a surface _p,1of genus p with connected boundary such that π_V has only one singular fiber carrying two singularities. Moreover,the monodromy of the open book on V induced by π_V is a single left-handed Dehn twist along a boundary parallel curve in _p,1. Let _p denote the regular fiber of the Lefschetz fibration π_X : X → S^2 constructed in Lemma <ref>. Then the4-manifold V can be described as theplumbing of the disk bundle over S^2 with Euler number -1 with the trivial disk bundle B^2 ×_p. Applying the algorithm in <cit.>, we obtain an achiralLefschetz fibration π_V : V → B^2 whose regular fiber is a surface _p,1 of genus p with connected boundary. Moreover, π_V has only one singular fiber carrying two vanishing cycles: a homotopically trivial curve(so π_V is not relatively minimal) with framing -1, and a boundary parallel curve $̣ in_p,1with framing+1. It follows that the monodromy ofthe open book on Vis a single left-handed Dehn twistD^-1()̣. Let V denote the 4-manifold obtained from V by blowing down the (-1)-sphere. Then there is an achiral Lefschetz fibration π_V : V→ B^2 whose regular fiber is _p,1, and which contains only one singular fiber whose vanishing cycle is δ. This can be most easily seen by drawing a Kirby diagram of V and simply blowing down the sphere with framing -1.<cit.> Let π_W : W→ B^2be an achiral Lefschetz fibration with regular fiber a surface _p,bof genus p with b boundary components and with n vanishing cycles. Then there is a (p+n, 2p+b-1; p, b)-relative trisection of W realizing the natural open book on W induced from the Lefschetz fibration π_W. Moreover, the corresponding trisection diagram can be described explicitly, based on the vanishing cycles of π_W.Here we briefly sketch a proof of Lemma <ref>. We start with describing a relative trisection of the neighborhood_p,b× B^2of a nonsingular fiber_p,binπ_W.Letπ: _p,b× B^2 →B^2denote the projection onto the second factor. Trisectingthe baseB^2into three wedges and taking the union of the inverse images of these pieces underπgives a(p, 2p+b-1; p,b)-relative trisection of_p,b× B^2, such that the open book on ( _p,b× B^2 )is the trivialone, whose page is_p,band monodromy is the identity map. Note that the open book on ( _p,b× B^2 )induced by the trivial fibration_p,b× B^2 → B^2is the same as the one induced by the relative trisection.Moreover, the corresponding relative trisection diagram is empty, i.e.,it is a genuspsurface withbboundary components with no,$̱ orcurves on it.It is well-known that the total space W of the achiral Lefschetz fibration π_W : W→ B^2 is obtained by attaching a 2-handle to the product _p,b× B^2 for each vanishing cycle.The 2-handle is attached along the vanishing cycle with framing ± 1 with respect to the surface framing. Therefore,to find adescription ofa relative trisection of W,it sufficesto extendthe relative trisection on_p,b× B^2 over any 2-handle attachment as described above. It turns out that such an extension is possiblein a more general setting (cf. <cit.>), and we describe its diagrammatic version below. Suppose that(, , ,̱) is a relative trisection diagram of a 4-manifold N which admits an achiral Lefschetz fibration π_N : N → B^2 such that the open books induced by the relative trisection and the Lefschetz fibration agree on N. Let ł be a simple closed curve onwhich is disjoint fromand transverse to$̱ and.Hence, we can viewłas a curve on_, which we identified as the page of the aforementioned open book on N.If we attach a2-handle toNalongłwith framing± 1with respect to the page_, then a relative trisection diagram(, , , ^±)of the resulting4-manifold is described as follows. The surfaceis obtained frombyremovingan annular neighborhood ofłfromand insertingina two-holed torus. The two-holed torus carries three curves (colored red, blue and green) with two options for the green curve corresponding to the attaching framing of the2-handle,as shown in Figure <ref>.We defineto beunionthe new red curve,to be$̱ unionthe new blue curve, and ^± to be union the new green curve on the bottom left (resp. right) in Figure <ref>.In order to draw the relative trisection diagram ofan achiralLefschetz fibration over B^2 described by theordered set of vanishing cycles ł_1, …, ł_n ⊂_p,b,we proceed as follows. First we draw ł_1 on the surface _p,b and replace an annularneighborhood of it by one of the two-holed tori shown at the bottom of Figure <ref> depending on the sign of the surgery coefficient, to obtain a surface _p+1,b decorated with one red, one blue and one green curve. Then we isotope ł_2so that it does not intersect the red or the blue curve but only intersects the green curve transversely, say m ≥ 0 times,in the diagram. Next,we remove an annular neighborhood of ł_2, and thereforewe also remove mdisjoint arcs of the green curve. Now we plug inthe appropriatetwo-holed torus shown in Figure <ref>, so that each arc cut from the first green curve is replaced by a green arc in the glued in two-holed torus with the same end points,disjoint from both the new blue and the green curves and transversely intersecting the new red curve once.As a result, we obtain a surface _p+2,b decorated with two red curves, two blue curves and two green curves,so that two curves with the same color are disjoint. Then we isotope ł_3suchthat it does not intersect the red or the blue curves but only intersects the green curves transverselyandapply the same procedure as we implementedfor ł_2. It is now clear how to iterate this procedure for the rest of the vanishing cycles. It is important to note that for a different ordering of the same set of vanishing cycles we get a different relative trisectiondiagram, in general. This is of course consistent with the fact that the total space of the Lefschetz fibration (and therefore the open book on the boundary) is determined by theorderedset of vanishing cycles.The Euler characteristic χ(W) can be calculated simply as χ(B^2)χ (_p,b) +n=2-2p-b +n, using the achiral Lefschetz fibration π_W : W→ B^2 described in Lemma <ref>.On the other hand, using the (p+n, 2p+b-1; p, b)-relative trisection of W, we have χ(W) = p+n - 3(2p+b-1) + 3p + 2b -1= 2-2p-b+n, as expected, by the formula in Corollary <ref>.Suppose that X is asmooth, closed, oriented, connected 4-manifoldwhich admits a genus p Lefschetz fibration over S^2 with n singular fibers, equipped with a section of square -1. Then, an explicit(2p+n+2, 2p)-trisection of X can be describedby acorresponding trisection diagram,which is determined by the vanishing cycles of the Lefschetz fibration.Moreover, if X denotes the 4-manifold obtained from X by blowing down the section of square -1, then we also obtain a (2p+n+1, 2p)-trisection of X along with a corresponding diagram.Suppose that π_X: X → S^2is a genus p Lefschetz fibrationwith n singular fibers, equipped with a section of square -1. Let V denote a regular neighborhood of the section union a nonsingular fiber as in Lemma <ref>.Then the 4-manifold W obtained by removing the interior of V from X admits aLefschetz fibration π_W: W → B^2 whose regular fiber is a genus p surface _p,1 with connected boundary. Note that the monodromy of the open book on the boundary W (oriented as the boundary of W) is given byD()̣∈_p,1. ApplyingLemma <ref> we get a (p+n, 2p; p, 1)-relative trisection on W realizing the open book on W.On the other hand, there is an achiral Lefschetz fibration π _V : V → B^2 as described in Lemma <ref>. It follows, by Lemma <ref>, that V admits a (p+2, 2p; p, 1)-relative trisectionrealizing the open book on V (oriented as the boundary of V) whose monodromy is given by D^-1()̣∈_p,1.To get a (2p+n+2, 2p)-trisection on X we just glue the (p+n, 2p; p, 1)-relative trisection on W with the(p+2, 2p; p, 1)-relative trisection on V along the identical open book on their common boundary W= - V, using Lemma <ref>.In addition, using Proposition <ref>, the corresponding relative trisection diagrams can be glued together diagrammatically to obtain a (2p+n+2, 2p)-trisection diagram of X.To prove the last statement in Theorem <ref>, we first observe by Remark <ref>thatX ≅ W ∪_ V ≅ W∪_ (V#) ≅ (W ∪_V) #≅X# .In particular, we have X=W ∪_V, and henceusing Lemma <ref>we obtain a(2p+n+1, 2p)-trisection of X by gluing the(p+n, 2p; p, 1)-relative trisection on W and the (p+1, 2p; p, 1)-relative trisection on Vcorresponding to the Lefschetz fibration π_V :V→ B^2 with only one vanishing cycle, along the identical open book on their boundaries. The corresponding trisection diagram for X can be obtained using Proposition <ref>. § TRISECTION DIAGRAMS In our figures, if there is no indicated boundary, then by adding the point at infinity we obtain aclosed oriented surface.Two small black disks labeled with the same white number or letter represents surgery on the sphere S^0 consisting of the centers of these disks. This means that we remove these two disks from the underlying oriented surface and glue in a cylinder I × S^1 to get an oriented surface of one higher genus.Any arc connecting twoblack disks with the same label represents a simple closed curve obtainedby joining the two ends of this arc by I × p ⊂ I × S^1 for some p ∈ S^1.Therefore, we refer to these kind of arcs as curves in our figures and reserve the term arc for the properly embedded arcs appearing in the cut systems.In the trisection diagrams, the , $̱ andcurves are drawn (and also referred to) as red, blue and green curves, respectively. We follow the same coloring convention for the cut system(_, _,̱_)of arcs for the relative trisection diagrams.Namely, the arcs in_,_$̱ and _ aredrawn (and also referred to) as red, blue and green arcs, respectively. Notation:We use _p,b to denote the mapping class group of the genus p surface _p,b with b boundary components, and D(ł) to denote the right-handed Dehn twist along a simple closed curve ł⊂_p,b. We use functional notation for the products of Dehn twists in _p,b.§.§ The elliptic surface E(1) Our goal in this subsection is toillustrate the method of proof of Theorem <ref>by constructing an explicit (16,2)-trisection diagram of the elliptic surface E(1), based on the standard elliptic fibration E(1)→ S^2 with 12 singular fibers. This is not the minimal genus trisection of E(1), however, since E(1) is diffeomorphic to # 9, whichadmits a genus 10 trisection obtained from the connected sum of genus one trisections ofandgiven in <cit.>.It is well-known that the relation D()̣ = (D(b)D(a))^6 ∈_1,1, where a and b denotethe standard generators of the first homology of _1,1 and $̣ is a curveparallel to_1,1, describes the elliptic Lefschetz fibrationE(1) → S^2with12singular fibers, equipped with a section of square-1.According to our notation in Section <ref>,E(1) = W ∪_ V, whereVdenotes a regular neighborhood of the section union a nonsingular fiber.By Lemma <ref>, there isan achiral Lefschetz fibrationπ_V : V → B^2, with two vanishing cycles on_1,1, a homotopicallytrivial curvewith framing-1and a boundary parallel curve$̣ with framing +1.Note that the Dehn twist D() is isotopic to the identity sinceis homotopically trivial, and hence it does not contribute to the monodromy.However, we still have to take the vanishing cycleinto account while drawing the corresponding (3,2;1,1)-relative trisection diagram of V shown in Figure <ref>, which we obtained by applyingRemark <ref>. In Figure <ref>,the surgeries labeled by 1 and 2 correspond to the vanishing cyclesand $̣, respectively.Next, wedecorate the relative trisection diagram forVwitha cut system(^V_, ^V_,̱^V_)of arcs as follows.First note that surgeries along the two red curves cancel out the surgeries labeled by1and2in Figure <ref> and henceΣ_is the genus one surface with one boundary component,represented by the surgery labelled byh. By definition, the set^V_consists of two red arcs thatcutΣ_into a disk. An obvious choice of^V_is depicted in Figure <ref>.Then we obtain^V_$̱ consisting of the two blue arcs in Figure <ref>,simply by taking parallel copies of the red arcs.We do not need any handleslides since the resulting blue arcs are clearly disjoint from the blue curves. Finally, the two green arcs belonging to ^V_ in Figure <ref> areobtained by applying some handleslides to the parallel copies of the blue arcs over the blue curve associated to the surgerylabeled by 2. Now we turn our attention to W, which is obtained by removing the interior of V from the elliptic surface E(1).According to Lemma <ref>, the standard elliptic fibration on E(1) with 12 singular fibers inducesa Lefschetz fibrationπ_W : W→ B^2 whose vanishing cycles are depicted in Figure <ref>. The corresponding (13,2;1,1)-relative trisection diagram for W (see Remark <ref>)is depicted in Figure <ref>.The labeling of the surgeries from 1 to 12 inFigure <ref> corresponds to the ordering ofthe 12 vanishing cycles a_1,b_1,a_2, …,b_6 of the Lefschetz fibrationπ_W : W→ B^2 shown in Figure <ref>.We choosea cut system(^W_, ^W_,̱^W_) of arcs (see Figure <ref>) for the relative trisection diagram for W as follows.We first choose^W_ (the two red arcs) cutting _ into a disk so that the end points of these two red arcs match with the end points of the two red arcs in ^V_ in Figure <ref>. Next,we obtain the blue arcs in ^W_$̱ by taking parallel copies ofthe red arcs without any need for handleslides. Finally,we obtain the green arcs in^W_by applying some handleslides to the parallel copies of the blue arcs. According to Proposition <ref>, in orderto obtain a(16,2)-trisection diagram forE(1) = W ∪_ V, wejust have to glue the(13,2;1,1)-relative trisection diagram forWdecorated withthe cut system(^W_, ^W_,̱^W_)of arcs depicted in Figure <ref> with the(3,2;1,1)-relative trisection diagram ofVdecorated withthe cut system(^V_, ^V_,̱^V_)of arcsdepicted in Figure <ref> so that the end points of the red, blue and green arcs in Figure <ref>are identified with the end points of the the red, blue and green arcs in Figure <ref>, respectively,on the common boundary circle. We depicted a planar version of theresulting(16,2)-trisection diagram forE(1) = W ∪_ Vin Figure <ref>.Note that there is also agenus 10 trisection diagram of E(1) viewed as theconnected sum #9, which wedepicted in Figure <ref>. The trisection diagrams of E(1) shown in Figure <ref> and Figure <ref> are related by stabilization, handleslides and diffeomorphism by <cit.>.The elliptic surfaceE(n)forn ≥ 2would be the next natural candidate to trisect. Unfortunately, our method of proof of Theorem <ref>does notimmediately extend to coverany of these4-manifolds. The elliptic fibration onE(n)has a section of square-n. The regular neighborhood of the union of a non-singular fiber and the section of square-nis a plumbing which admits an achiral Lefschetz fibrationas in Lemma <ref>, but whenn ≥ 2,the open book on the boundary does not match the open book coming from the Lefschetz fibration in the complement. On the other hand,E(2) #2 admits a genus2Lefschetz fibration overS^2with30singular fibers, equipped with a section of square-1(cf. <cit.>). An implementation of Theorem <ref> gives a(36, 4)-trisection ofE(2) #2 . In the next subsection, we turn our attention to another well-known genus2Lefschetz fibration on the Horikawa surfaceH'(1). §.§ The Horikawa surface H'(1)Our goal in this subsection is to present an explicit(46,4)-trisection diagram of the Horikawa surfaceH'(1), which is defined as the desingularization of the double branched cover of the Hirzebruch surface𝔽_2(cf. <cit.>). As we pointed out in theintroduction,H'(1)is a simply-connected complex surface of general type, which is homeomorphic but not diffeomorphic to5 # 29 . Therelation1 = (D(c_1)D(c_2)D(c_3)D(c_4))^10∈_2where the curvesc_1,c_2,c_3,c_4are shown in Figure <ref>,defines a genus2Lefschetz fibration overS^2. In <cit.>, Fuller showed that the total space of this genus2Lefschetz fibration is diffeomorphic to the Horikawa surfaceH'(1). Moreover, it is easy to see that this fibration admits a section of square-1, since the relation above lifts to the chain relationD()̣= (D(c_1)D(c_2)D(c_3)D(c_4))^10∈_2,1.According to <cit.>,H'(1)admits a unique sphere of square-1, and therefore by blowing it down,we also obtain a trisection diagram for the minimal simply connected complex surfaceH'(1)of general type,by Theorem <ref>. IfVdenotes a regular neighborhood of the section union a nonsingular fiber, then we havea decomposition ofH'(1)as W ∪_ V. By Lemma <ref>, there isan achiral Lefschetz fibrationπ_V : V → B^2with regular fiber_2,1, which has two vanishing cycles:a homotopicallytrivial curvewith framing-1and a boundary parallel curve$̣ with framing +1.By Remark <ref>, we obtain the corresponding (4,4;2,1)-relative trisection diagram of V shown in Figure <ref>, which is decoratedwiththe cut system(^V_, ^V_,̱^V_) of arcsin Figure <ref>.On the other hand, we obtain the corresponding (42,4;2,1)-relative trisection diagram of W shown in Figure <ref> using the Lefschetz fibration π_W: W → B^2 with monodromy (D(c_1)D(c_2)D(c_3)D(c_4))^10.Note that we implemented Remark <ref> for the first 8 vanishing cycles c_4, c_3, c_2, c_1, c_4, c_3, c_2, c_1 labeled by the surgeries 1 through 8 and the last 4 vanishing cycles c_4, c_3, c_2, c_1 labeled by the surgeries 37 through 40.We hope that the pattern is clear for the reader. In Figure <ref>, we depictedthe (42,4;2,1)-relative trisection diagram of W decorated withthe cut system(^W_, ^W_,̱^W_) of arcs.Finally, in orderto obtain a (46,4)-trisection diagram for H'(1) = W ∪_ V using Proposition <ref>, wejust have to glue the (42,4;2,1)-relative trisection diagram for W decorated withthe cut system(^W_, ^W_,̱^W_) of arcs depicted in Figure <ref> with the(4,4;2,1)-relative trisection diagram of V decorated withthe cut system(^V_, ^V_,̱^V_) of arcsdepicted in Figure <ref> so that the end points of (^W_, ^W_,̱^W_) are identified with the end points of (^V_, ^V_,̱^V_) on the common boundary circle. The end result is depicted in Figure <ref>. Similar to the diagram depicted in Figure <ref>, there is indeed a(34,0)-trisection diagram of5 # 29which can be stabilized fourtimes to yield a(46,4)-trisection diagram.As a consequence, we obtain explicit (46,4)-trisection diagrams for a pair of closed 4-manifolds, the Horikawa surface H'(1) and5 # 29, which are homeomorphic but not diffeomorphic. §.§ Trivial S^2 bundle over _hIn <cit.>, there is a description of a (8h+5, 4h+1)-trisection for any oriented _h-bundle over S^2, including of course _h × S^2, without an explicit diagram. In Example <ref>, however, we obtained a (2h+5, 2h+1)-trisection of _h × S^2. Here we illustrate our doubling technique by drawing a (7,3)-trisection diagram for T^2 × S^2. We first observe that T^2 × B^2 admits an achiral Lefschetz fibration over B^2 with fiber_1,2, which carriestwo vanishing cycles _̣1 and _̣2, each of which is parallel to one boundary component of _1,2. The monodromy of the this fibration is given by D(_̣1)D^-1(_̣2). ImplementingRemark <ref>, we obtain the (3,3;1,2)-relative trisection diagram for T^2 × B^2 shown in Figure <ref> (see also<cit.>). A cut system of arcs for this trisection diagram of T^2 × B^2 is depicted in Figure <ref>. To double the (3,3;1,2)-relative trisection diagram decorated with the cut system of arcs inFigure <ref>, we simply draw its mirror image (for the orientation reversal)next to it and identifythe inner and outer boundary components, respectively,to each other. It is clear how to identify the outer boundary components and glue the arcs with the same color, as we illustrated many times so far in this paper. To identify the inner boundary components and still draw a planar diagram, we replace the inner boundaries by the surgery labeled by 5 in Figure <ref>. As a result we obtain the (7,3)-trisection diagram for T^2 × S^2 as shown in Figure <ref>, where we isotoped some of the curves after implementing Corollary <ref>.One can similarly draw a(2h+5, 2h+1)-trisection diagram of_h × S^2 for any h ≥ 2.§.§ Twisted S^2-bundle over _h In Example <ref>, we obtained a (2h+2, 2h)-trisection of the non-trivial S^2-bundle _h ×̃ S^2. Except for S^2 ×̃ S^2 ≅#, which has a (2,0)-trisection,these bundles are not covered by the examples in <cit.>.In the following, we illustrate our doubling technique described in Corollary <ref>by drawing a (4,2)-trisection diagram for T^2 ×̃ S^2.Note that E_1,1, the B^2-bundle over T^2 with Euler number +1, admits an achiral Lefschetz fibration over B^2 whose regular fiber is _1,1 and which has only one singular fiber whose vanishing cycle is theboundary parallel curve ⊂̣_1,1.Therefore, by Lemma <ref>, E_1,1 has a (2,2;1,1)-relative trisection, whose diagram, decorated with the cut system of arcs, is depictedin Figure <ref>. Since the double ofE_1,1 isT^2 ×̃ S^2, we obtain the(4,2)-trisection diagram forT^2 ×̃ S^2 in Figure <ref>,by doubling the diagram in Figure <ref>.This is done by drawingFigure <ref> and its mirror image next to it and gluing the arcs of the same color. One can similarly draw atrisection diagram of _h ×̃ S^2for any h ≥ 2. Note thatfor h=0 our doubling technique gives the standard(2,0)-trisection diagram for #, which is indeed diffeomorphic to S^2 ×̃ S^2.§.§ Oriented S^2-bundles over non-orientable surfacesThere are two oriented S^2-bundles over any closed connectednon-orientable surface N_h = #^h ℝℙ^2 of genus h, both of which are obtained as doubles of B^2-bundles over N_h. The disk cotangent bundle DT^*N_h of N_h, which is diffeomorphic to the B^2-bundle over N_h with Euler number h-2, admits a Lefschetz fibration over B^2 whose regular fiber is _0, 2h+2and which has h+2 singular fibers (cf. <cit.>). According to Lemma <ref>,DT^*N_h has a (h+2, 2h+1, 0, 2h+2)-relative trisection, and one can draw the corresponding diagram by Remark <ref>. It follows that for any h ≥ 1, one of the oriented S^2-bundles over N_h has a (4h+5, 2h+1)-trisection by Corollary <ref>, since it is the double of DT^*N_h.In particular,a (3,3,0,4)-relative trisection diagram of the disk cotangent bundle of ℝℙ^2,a rational homology ball with boundary the lens space L(4,1), is depicted in <cit.>.Thedouble of this rational homology ball isan oriented S^2-bundle over ℝℙ^2, which has a (9,3)-trisection whose corresponding diagram can be obtained by doubling the (3,3,0,4)-relative trisection diagramas illustrated in the previous sections. § ALTERNATE PROOF OF THE GAY-KIRBY THEOREMIn this section, we provide an alternate proof of Theorem <ref>. We refer to <cit.> for the definitions and properties of Lefschetz fibrations, open books, contact structures, etc. Every smooth, closed, oriented, connected 4-manifold admits a trisection. Suppose that X is a closed 4-manifold. According to Etnyre and Fuller <cit.>, there is an embedded 3-manifold M ⊂ X satisfying the following properties:*There exists a decomposition X= W∪_M W', where W = M = - W'. *Each ofW and W'admits an achiral Lefschetz fibration over B^2 with bounded fibers. *The two open books induced by the respective achiral Lefschetz fibrations on W and W' coincide on M. As we described in Lemma <ref>,there is a straightforward method to turn the total space of an achiral Lefschetz fibration over B^2 into a relative trisection so that the respective open books induced by the relative trisection and the Lefschetz fibration agree on the boundary. Consequently, we have a decomposition of X into two pieces W and W',each of which has an explicit relative trisection so that the induced open books on the boundary coincide with the appropriate orientations.We rely on Lemma <ref>to finish the proof of Theorem <ref>. Here we briefly outline the proof of the aforementioned result of Etnyre and Fuller to indicate how contact geometry enters the scene.Suppose that the 4-manifold X is given by a handle decomposition. We first realize the union of 0-, 1-, and 2-handlesas an achiral Lefschetz fibration over B^2, whose vanishing cycles can be explicitlydescribed using the technique in <cit.>. Similarly, the union of 3- and 4-handles also admits a Lefschetz fibration over B^2. But there is indeed no reason for the open books on the boundary to coincide at this point. However, by stabilizing both achiral Lefschetz fibrations several times, the contact structures supported by the resulting open books on the boundary become homotopic as oriented plane fields. Moreover, both contact structures may be assumed to be overtwisted by negatively stabilizing, if necessary. Therefore, we conclude that the contact structures are isotopic by Eliashberg's classification <cit.>.We achieve the desired decomposition of X,using a fundamental result of Giroux <cit.>, which says that if two open books support isotopic contact structures on a closed 3-manifold, then they have a common (positive) stabilization. Suppose that we have a splitting of a closed 4-manifold X into two achiral Lefschetz fibrations over B^2— not necessarily by the method of Etnyre and Fuller as outlined above — inducing the same open book on their commonboundary.Then the next two steps in the proof of Theorem <ref> provide a trisection of X. On the other hand, afurther positive or negative stabilization ofthe common open book on the boundary yieldsa stabilization of each of the Lefschetz fibrationsin the splitting of X, which in turn,induces a new (higher genus) trisection of X.This new trisection of X can be obtained bya stabilization ofthe initial trisection. The decomposition theorem of Etnyre and Fuller that we quoted at the beginning of our proof of Theorem <ref>was improved by Baykur <cit.> who showed that the Lefschetz fibration on W (resp. W') can be assumed to have only positively (resp. negatively) oriented Lefschetz singularities.Acknowledgement: This work was initiated at the “Trisections and low-dimensional topology" workshop at the American Institute of Mathematics (AIM). We would like express our gratitude to AIM for its hospitality.We would also like thank D. T. Gay and R. İ. Baykur for their helpful comments on a draft of this paper. 10ao S. Akbulut and B.Ozbagci, On the topology of compact Stein surfaces. Int. Math. Res. Not. 2002, no. 15, 769–782.bR. İ. Baykur, Kähler decomposition of 4-manifolds.Algebr. Geom. Topol. 6 (2006), 1239–1265. bsR. İ. Baykur and O. Saeki,Simplifying indefinite fibrations on 4-manifolds.arXiv:1705.11169c N. A. Castro,Trisecting smooth 4-dimensional cobordisms. arXiv:1703.05846cgp N. A. Castro, D. T. Gay,and J. Pinzón-Caicedo,Diagrams for Relative Trisections. arXiv:1610.06373don S. K. Donaldson,Lefschetz pencils on symplectic manifolds. J. Differential Geom. 53 (1999), no. 2, 205–236. e Y. Eliashberg,Classification of overtwisted contact structures on 3 -manifolds. Invent. Math.98(1989),no. 3, 623–637.et J. B. Etnyre, Lectures on open book decompositions and contact structures. Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006. ef J. B. Etnyre and T. Fuller,Realizing 4-manifolds as achiral Lefschetz fibrations. Int. Math. Res. Not.2006, Art. ID 70272, 21 pp.f T. Fuller,Diffeomorphism types of genus 2 Lefschetz fibrations.Math. Ann. 311 (1998), no. 1, 163–176.g D. T. Gay,Trisections of Lefschetz pencils.Algebr. Geom. Topol. 16 (2016), 3523–3531.gk D. T. Gay and R. Kirby,Trisecting 4-manifolds. Geom. Topol.20(2016),no. 6, 3097–3132.gi E. Giroux,Géométrie de contact: de la dimension trois vers les dimensions supérieures. (French) [Contact geometry: from dimension three to higher dimensions]Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002),405–414, Higher Ed. Press, Beijing, 2002. gs R. E. Gompf and A. I. Stipsicz,4–manifolds and Kirby calculus.Graduate Studies inMathematics,20.American Mathematical Society, Providence, RI,1999.hJ. L. Harer, Pencils of curves on 4-manifolds, Ph.D. thesis, University of California, Berkeley, 1979.m Y. Matsumoto,Lefschetz fibrations of genus two–a topological approach. Topology and Teichmüller spaces (Katinkulta, 1995), 123–148, World Sci. Publ., River Edge, NJ, 1996. oo T. Oba and B. Ozbagci, Canonical contact unit cotangent bundle. arXiv:1601.05574v2, to appear in Advances in Geometry. os B. Ozbagci andA. I. Stipsicz,Surgery on contact 3-manifolds and Stein surfaces.Bolyai Society Mathematical Studies, 13. Springer-Verlag, Berlin; János Bolyai Mathematical Society, Budapest, 2004.s Y. Sato, 2-spheres of square -1and the geography of genus-2 Lefschetz fibrations.J. Math. Sci. Univ. Tokyo15(2008),no. 4, 461–491. | http://arxiv.org/abs/1705.09854v2 | {
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Victor LakhnoPeculiarities in the concentration dependence of the superconducting transition temperature ...Institute of Mathematical Problems of Biology RAS the Branch of Keldysh Institute of Applied Mathematics of Russian Academy of Sciences142290, Vitkevicha str. 1,Pushchino, Moscow Region, [email protected] IN THE CONCENTRATION DEPENDENCE OF THE SUPERCONDUCTING TRANSITION TEMPERATURE IN THE BIPOLARON THEORY OF COOPER PAIRS VICTOR LAKHNO October 2, 2017 ===================================================================================================================================It is shown that the bipolaron theory of Cooper pairs suggests that there is a possibility for a superconducting phase to exist at low and high levels of doping and be absent at intermediate level of doping. The results obtained imply possibly universal character of 1/8 anomaly.The results of paper,<cit.> where a Cooper pair was demonstrated to be nothing but a bipolaron, actualize the problem of a bipolaron in a polaron gas.There consideration was given to a problem of electron-phonon interaction (EPI) between two electrons in Coopers formulation<cit.> when the Fermi energy exceeds the EPI one: E_F>|E_pol|, whereE_polis a polaron energy. In high-temperature superconductors (HTSC), however, of importance is the case of E_F<|E_pol|(HTSC with low level of doping). We will show that these two cases lead to two qualitatively different pictures.a) The case ofE_F>|E_pol|Let us consider the limit case when E_F>>|E_pol|and above the Fermi surface there is one electron taking part inelectron-phonon interaction. In view of Pauli principle the interaction of this electron with the electron occuring below the Fermi surface can be neglected. Hence, in this case we have the polaron problem for an electron occuring near the Fermi surface. Because of EPI, the energy of this electron should be below the Fermi surface at the depth of E_pol. But the same will be valid for all the electrons occuring at the Fermi level: owing to EPI their energy will be decreased by E_pol. Hence, if we denote the Fermi energy in the absence of EPI byE_F^0, then in the presence of EPI the renormalized value of the Fermi energy will be:E_F=E^0_F+E_pol. The masses of electrons whose energies occur near E_Fwill also undergo a relevant polaron renormalization. Therefore in the energy layer(E_F+E_pol, E_F) we will have a polaron gas.Let us consider the case of two electrons above the Fermi surface. From the aforesaid it follows that now the Fermi surface is determined by E_F rather than byE^0_F. Now the presence of EPI cannot decrease the energy of either of the two electrons by the value ofE_pol since in view of EPI, the energy of the electrons occuring on the Fermi surface is already decreased by this value. If the two electrons form a paired state, the energy of the state will be below the Fermi surface E_Fat the depth ofE_bp, whereE_bpis the bipolaron energy for any value of the EPI constant α. This result is in complete agreement with Coopers conclusion<cit.> about instability of theFermi surface with respect to formation of pairs for arbitrarily small values ofα, which makes the polaron theory in metals qualitatively different from that in polar dielectrics. Accordingly, the value of the gap near the Fermi surface will be not |E_bp-2E_pol|as it is resume there,<cit.> but |E_bp|. Hence, only the electrons whose energies occur in the (E_F+E_pol, E_F) layer take part in the formation of bipolaron paired states, which differentiates the bipolaron theory of superconductivity<cit.> from the BCS theory<cit.> which implies that forT=0 all the electrons are in the paired state.b) The case of E_F<|E_pol|In this case we cannot neglect the interaction between the excess electrons and electrons below the Fermi surface. If , as in the previous case, we believe that the electrons below the Fermi surface are polarons then the Cooper problem will correspond to the problem of a bipolaron in a polaron gas. As a bipolaron is placed in a polaron gas, an additional energy difference arises in view of the fact that a bipolaron is a Bose particle while a polaron is a Fermi particle.Hence if we take Δ E=E_bp-2E_pol (corresponding to the energy gain for Δ E<0 or energy failure forΔ E>0in the absence of a polaron gas) as a reference point of energy in the absence of polaron gas, then in a polaron gasE_polshould include the additional termsE_F=p^2_F/2m_pol ( p_F=(3π^2)^1/3ħ n^1/3 is the Fermi momentum, n is the concentration of current carriers) andE_exch, whereE_exch=-e^2p_F/πħϵis the exchange energy in the Hartree-Fock approximation,<cit.>m_pol is the polaron mass,ϵ is the dielectric permittivity. As a result, the bipolaron stability criterion takes on the form:<cit.> Δ E<2E_F+2E_exch.This implies that on condition that: (m_pole^2/πħϵ)^2>-m_polΔ E>0 the bipolarons are stable in two regions:(0,p_F1)and (p_F2,∞), p_F1,2=m_pole^2/πħϵ±√((m_pole^2/πħϵ)^2+m_polΔ E), where: p_F1corresponds to the sign (-), andp_F2 - to the sign (+). The region (0, p_F1) corresponds to small concentration of current carriers (low doping) while the region (p_F2, ∞ ) corresponds to large concentration (high doping). If we believe that the presence of bipolarons at T=0immediately leads to superconductivity, then the existence of two different regions of bipolaron stability will correspond to the existence of two different regions of superconductivity occurrence.If the converse condition is fulfilled: (m_pole^2/πħϵ)^2<-m_polΔ E<0 the bipolarons are stable at any level of doping. In this case the concentration dependence of the critical temperature of the superconducting transition T_Cwill have a local minimum.This dependence is actually realized in superconductors La_2-xM_xCuO_4, M=(Sr,Ba) , where the high-temperature superconductivity was observed for the first time. For example, in La_2-xSr_xCuO_4 the optimal level of doping is equal to x≈0,16. As x decreases, T_Cis lowered too. This behavior remains unchanged up to x≈1/8 , whenT_Creaches its minimum. As x further decreases,T_C grows achieving some maximum and then decrease vanishing at small x. In La_2-xBa_xCuO_4this behavior is still more pronounced: there exists a sharp dip in the T_c-x phase diagram, indicating that bulk superconductivity is greatly suppressed in narrow range around x=1/8.The emergence of minimum in the concentration dependence is known as 1/8 anomaly which probably has universal character being observed in other HTSC materials.<cit.>§ ACKNOWLEDGEMENTSThe work was supported by RFBR, N 16-07-00305 and RSF, N 16-11-10163.0 1 V. D. Lakhno, Mod. Phys. Lett. B 30 (2016) 1650365. 2 L. N. Cooper, Phys. Rev. 104 (1956) 1189. 3 V. D. Lakhno, SpringerPlus 5 (2016) 1277; V. D. Lakhno, arXiv:1510.04527 [cond-mat.supr-con]. 4 J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys. Rev. 108 (1957) 1175. 5 N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). 6 A. A. Shanenko et al., Sol. St. Comm. 98 (1996) 1091. 7 M. A. Smondyrev et al., Phys. Rev. B 63 (2000) 024302. 8 A. R. Moodenbaugh et al., Phys. Rev. B 38 (1988) 4596. 9 S. A. Kivelson et al., Rev. Mod. Phys. 75 (2003) 1201. 10 M. Vojta, Adv. Phys. 58 (2009) 699. 11 M. Hücker et al., Phys. Rev. B 83 (2011) 104506. | http://arxiv.org/abs/1705.09534v1 | {
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"title": "Peculiarities in the concentration dependence of the superconducting transition temperature in the bipolaron theory of Cooper pairs"
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On-chip generation of photon-triplet states Stephan Krapick,^1,2 Benjamin Brecht,^1 Harald Herrmann,^1 Viktor Quiring,^1 and Christine Silberhorn^1,3 December 30, 2023 =============================================================================================================empty§Investigating relationships between words offers insights into both the structure of language and the influence of cognition on linguistic tasks <cit.>. As a result, cognitive network science is rapidly emerging at the interface between network theory, statistical mechanics, and cognitive science <cit.>. The field is influenced by the seminal work of Collins and Quillian<cit.>, who assumed that concepts in the human mind are cognitive units, each representable as a node linked to associated elements. These connections represent a complex cognitive system known as the mental lexicon <cit.>. Extensive empirical research has shown that relationships in the lexicon can be modelled as a network of mental pathways influencing both how linguistic information is acquired <cit.>, stored <cit.>, and retrieved <cit.>. The cognitive role of quantifying lexical navigability as distances in a network finds empirical support in several experiments related to word identification and retrieval tasks <cit.>. For instance, Collins and Loftus<cit.> showed a correlation between network topology of semantic networks and word processing times: words farther apart in the network require longer identification times, thus indicating higher cognitive effort. More recently, the structural organisation of mental pathways among words was analysed in several large-scale investigations, considering similarity of words in terms of their semantic meaning <cit.>, their phonology <cit.>, or their taxonomy <cit.>. Remarkably, all these networks, based on different definitions of relationships between words, were found to be highly navigable: words were found to be clustered with each other and separated by small network distances (sometimes called small-world networks <cit.>). This may suggest a universal structure of language organisation related to minimising cognitive load while maximising navigability of words <cit.>.The above studies, however, have not yet attempted to use multi-relational information for characterising and quantifying the mental lexicon, instead focusing on only one relationship at a time <cit.>. Some researchers have considered the aggregation of several of these relationships into single-layer networks <cit.> and others have considered multi-relational models but only to capture the syntactic structure of language <cit.>. The above approaches offer only limited insight into the cognitive complexity that allow individuals to use language <cit.> with diversity and ease.More information about the lexical structure can indeed be obtained by accounting, simultaneously, for multiple types of word-word interactions. A natural and suitable framework for this purpose are multilayer networks <cit.>. Multilayer networks simultaneously encode multiple types of interaction among units of a complex networked system. Therefore, they can be used to extract information about linguistic structures beyond information available from single-layer network analysis <cit.>. The usefulness of multiplex representations has recently been shown for diverse applications including the human brain<cit.>, social network analysis<cit.>, transportation<cit.> and ecology <cit.>.Here, on an unprecedented scale and from a multi-relational perspective, we investigate the semantics, phonology, and taxonomy of the English lexicon as a model of distinct layers of a multiplex network (see Fig. 1). We study the evolution of multiplex connectivity over the developmental period from early childhood (2 years of age) to adulthood (21 years of age) also through the use of word attributes (e.g. word frequency, length, etc.) influencing lexical acquisition <cit.>.The proposed multiplex representation provides a powerful framework for the analysis of the mental lexicon, allowing for the capture of sudden structural changes that can not be identified by traditional methods. More specifically, when modelling lexical growth, we observe an explosive emergence of a cluster of words in the lexicon around the age of 7 years, which is not observed in single-layer network analyses. We show that this cluster is beneficial from a cognitive perspective, as its sudden appearance facilitates word processing across connected network pathways across all lexicon layers. This boost to cognitive processing also enhances the resilience of the lexicon network when individual words become progressively inaccessible, such as what may happen in cognitive disorders like anomia<cit.>. These findings represent the first quantitative confirmation and interpretation of previous conjectures about the presence and cognitive impact of a core in the human mental lexicon <cit.>. § RESULTS §.§ Structure of the Multiplex Lexical Representation Our multilayer lexical representation (MLR) of words in the mind is a multiplexnetwork<cit.> made of N=8531 words and four layers. Each layer encodes a distinct type of word-word interaction (cf. Fig. <ref> (a)): (i) empirical free associations <cit.>, (ii) synonyms <cit.>, (iii) taxonomic relations <cit.>, and (iv) phonological similarities <cit.>. As shown in Fig. <ref> (b), different relationships can connect words that would otherwise be disconnected in some single-layer representations. We considered these relationships with the aim of building a representation accounting for different types of semantic association, either from dictionaries (i.e. synonyms and taxonomic relations) or from empirical experiments (i.e. free associations). We also include sound similarities (i.e. phonological similarities) as they are involved in lexical retrieval <cit.>. This set of relationships represents a first approximation to the multi-relational structure of the mental lexicon. Compared to previous work on multiplex modelling of language development <cit.>, our multiplex representation is enriched with node-level attributes related to cognition and language: (i) age of acquisition ratings <cit.>, (ii) concreteness ratings <cit.>, (iii) identification times in lexical decision tasks <cit.>, (iv) frequency of word occurrence in Open Subtitles <cit.>, (v) polysemy scores, i.e. the number of definitions of a word in WordNet, used to approximate polysemy in computational linguistics <cit.> (cf. Methods and SI Sect. 12) and (vi) word length <cit.>. The analysis of structural reducibility of our multiplex model (cf. SI Sect. 2) quantifies the redundancy of the network representation <cit.>. Results suggest that no layers should be aggregated, as each network layer contributes uniquely to the structure of the multiplex representation, confirming the suitability of the multiplex framework for further investigation.As already discussed, investigating navigation on linguistic networks has proved insightful <cit.>. Hence we focus on analysing the navigability of our multiplex network <cit.>, identifying word clusters that are fully navigable on every layer, i.e. where any word can be reached from any other word on every layer when considered in isolation. An example is reported in Fig. <ref> for a representative multiplex network with two layers. In network theory, these connected subgraphs are also called viable clusters <cit.> (cf. Methods). Notice that the largest viable cluster of a single-layer network coincides with its largest connected component <cit.>, i.e. the largest set of nodes that can all be reached from each other within one layer. In multiplex networks the two concepts are distinct, as viable clusters are required to be connected on every layer when considered individually. Removing this constraint of connectedness on every layer leads to the more general definition of multi-layer connected components<cit.>, i.e. the largest set of nodes all connected to each other when jumps across layers are allowed. Fig. <ref> (c-e) conveys the idea that the emergence of viable clusters can be due to the addition of particular links in the network. Our multiplex model contains a single non-trivial (i.e. with more than two nodes) viable cluster composed of 1173 words, about 13.8% of the network size. In the following we refer to this cluster as the largest viable cluster (LVC). For easier reference, we indicate words in the empirical LVC as “LVC-in words” and words outside of the empirical LVC as “LVC-out words”. Reshuffling network links while preserving word degrees leads to configuration model-layers <cit.> that still display non-trivial LVCs (cf. LVC Rew. in Tab. 1). Further, on average 98.1 ± 0.1 % of LVC-in words persist in the viable cluster after rewiring 5% of all the intra-layer links at random. We conclude that the LVC does not break but rather persists also in the case of potentially missing or erroneous links in the network dataset (e.g. spurious free associations or mistakes in phonological transcriptions).In order to further test correlations between network structure and word labels, we also consider a full reshuffling null model (see SI Sect. 4), in which word labels are reshuffled independently on every layer and thus word identification across layers is not preserved. Hence, full reshuffling destroys inter-layer correlations but preserves network topology. Fully reshuffled multiplex networks did not display any non-trivial viable clusters, emphasizing the important role of inter-layer relationships for the presence of the LVC in the empirical data.In the next section we analyse the evolution of the LVC during language learning over a time period of more than 15 years. We demonstrate the existence of an explosive phase transition <cit.> in the emergence of the LVC and explore the significance of this transition from the perspective of cognitive development.§.§ Emergence of the Largest Viable Cluster To study the emergence of the LVC during cognitive development, we simulate probabilistic normative word orderings by smearing the age of acquisition dataset<cit.>. We refer to these orderings as normative acquisition. Smearing allows us to account for the variance in age of acquisition across individuals by introducing a probabilistic interpretation of these orderings (see Methods). We compare the trajectories of normative acquisition against five null models: (i) random word learning (i.e. words are acquired at random), (ii) frequency word learning (i.e. higher frequency words are acquired earlier), (iii) polysemy-scores word learning (i.e. words with a higher count of context-dependent meanings are learned earlier) and (iv) multidegree word learning (i.e. words with more connections –across all layers– are learned earlier) and (v) word length learning (i.e. shorter words are learned earlier). We investigate if modelling the development of the mental lexicon as growth of the empirical multiplex representation according to a given learning scheme matches the explosive transition observed in normative learning. Results are reported in Fig. 2 (a).Normative acquisition indicates a sudden emergence of the LVC around age 7.7 ± 0.6 years, almost four years earlier than expected if learning words at random. Further analysis reveals two distinct patterns. Firstly, this sudden appearance is robust to fluctuations in word rankings in the age of acquisition ratings (AoA): in all simulations based on AoA reports, after roughly 2500 words have been acquired, an LVC with at least 260 words suddenly appears with the addition of just a single word to the lexicon. Secondly, the average magnitude of this explosive change is Δ L_AoA=(420± 50) words. These patterns suggest an explosive phase transition<cit.> in the structural development of the mental lexicon. To the best of our knowledge, this work is the first detection of an explosive change in lexicon structure in cognitive network science during vocabulary growth. Explosive behaviour in the emergence of the LVC is not observed in the random acquisition null model (see Methods and SI Sect. 7-11), with only a few cases (χ_Ran=32%) displaying a discontinuity of more than ten words. Further, the average magnitude of the LVC size change is only Δ L_Ran = (30± 10) words, a full order of magnitude smaller than in the normative cases. Therefore explosiveness characterises normative acquisition as a genuine pattern of language learning. Is the explosive appearance of the LVC due to the acquisition of specific links or rather to specific words? In order to test this, we focus on the set of “critical” words, i.e. the single words whose addition allows for the sudden emergence of the LVC. We then compare features of these critical words with features of words already within the LVC at the time of its emergence. We test features like node-attributes (e.g. frequency, polysemy scores, etc.) and node degree. At a 95% confidence level, no difference was found for any feature (sign test, p-value = 0.007). This lack of difference suggests that the emergence of the LVC is indeed due to higher-order link correlations rather than local topological features (such as degree) or psycholinguistic attributes. Hence, it is the global layout of links that ultimately drive the explosive appearance of the LVC. As shown also in Fig. <ref> (c-e), links crucial to the formation of the viable cluster might be acquired earlier (Fig. <ref> (c)) but the LVC might appear only later (Fig. <ref> (e)), after some key pathways completing the viable cluster are added to the network (Fig. <ref> (d)). The explosive emergence of the LVC has an interesting cognitive interpretation. Work in psycholinguistics suggests that frequency is the single most influential word feature affecting age of acquisition <cit.> (mean Kendall τ ≈ -0.47 between frequency and AoA). We thus test whether the LVC growth can be reproduced through early acquisition of highly frequent words, with frequency counts gathered from Open Subtitles<cit.>. All simulations on the frequency-based ordering display an explosive emergence of an LVC (χ_fre=100%), however, the magnitude of the explosive transition is Δ L_fre = 280± 30 words, which is only 2/3 of the normative one. At a confidence level of 95%, the distribution of frequency-based LVC magnitude changes differs from the normative one (sign test, p-value = 0.01). The distribution of ages at which the LVC emerges in the frequency null model overlaps in 21% of cases with the analogous normative one. However, we observe that the frequency null model differs from the normative one not only quantitatively (i.e. magnitude and appearance of explosiveness) but also qualitatively: the frequency null model displays a second explosive phase transition in LVC-size later in development, at around 10± 0.2 years of age. This second transition might be due to the merging of different viable clusters, since we focused only on the largest viable cluster, rather than on viable clusters of non-trivial size. Further analysis reveals that the multiplex network has only one viable cluster, which suddenly expands through a second explosive transition in the frequency-based vocabulary growth model (but not in the normative AoA model). The above differences provide strong evidence that explosiveness in the mental lexicon is not an artefact of correlation of word frequency with language learning patterns.We next test preferentially learning words with high degree in the multiplex network to see if the LVC emerges earlier than in normative acquisition. Learning higher degree words first makes more links available in the multiplex network. As we said above, it is links that drive the LVC emergence, hence we expect an earlier LVC appearance. The multidegree null model confirms this expectation and it displays a distribution of explosive transitions with average magnitude of 430± 30 but happening almost two years earlier than in normative acquisition, around age 5.8 ± 0.1, cf. Fig. <ref>. The distribution of critical ages overlaps with the normative one only for 2% of the time. We conclude that the degree acquisition is significantly different from the empirical case (mean Kendall τ≈ -0.31 between multidegree and AoA).Also word length influences lexical processing <cit.> and acquisition <cit.>. Acquiring shorter words first leads to the sudden emergence of the LVC around age 6.6 ± 0.6, similarly to what happens for the polysemy curve. The LVC appears explosively with an initial size of 330±50 words, a value lower than the normative one (mean Kendall τ≈ 0.24 between word length and AoA). Differently from what happens with the polysemy curve, the growth of the LVC for shorter words is considerably faster compared to the normative case.Another feature that can influence language acquisition is polysemy <cit.>, i.e. how many different definitions a word can have. We estimate word polysemy through polysemy scores <cit.>, including homonymy and also different meanings: the number of word definitions listed in the Wolfram dataset WordData <cit.>, which mostly coincides with WordNet. For a discussion about the caveats of using polysemy scores as we have defined above for quantifying polysemy we refer to SI Sect. 12. When words with higher polysemy scores are acquired earlier, we find the appearance of the LVC at around age 6.6 ± 0.6 years, with an average magnitude of 470±60 words, close to the normative one. The distribution of critical ages at which the LVC emerges in the polysemy null model displays the highest overlap (35%) with the analogous distribution from the normative case across all the null models we tested. Despite polysemy scores displaying a smaller correlation with the age of acquisition (mean Kendall τ≈ -0.26) when compared to frequency or multidegree, it actually provides the highest overlap in terms of age at which the LVC emerges. This indicates that polysemy might play a role in driving the LVC emergence. Another attribute that could impact language development is concreteness, i.e. how tangible a given concept according to human judgements <cit.>. Experimental research has shown that children tend to learn words earlier if a word is rated higher on concreteness <cit.>. In order to test how concreteness can influence the LVC evolution, we develop a partial reshuffling null model (cf. Methods) where thetopology of words is fixed but node attributes are reshuffled at random. Partial reshuffling destroys the correlations between word features and the network topology, such that we can quantify the role of the relational structure in the absence of correlation with word features. Partial reshuffling gives rise to LVCs of the same size but containing words that are less concrete and less polysemous than in normative acquisition, cf. Fig. 2 (b). Partial reshuffling of word frequency leads to a gap in frequency of similar size as we see for concreteness (cf. SI Sect. 9). The gap in polysemy scores between the empirical and the reshuffled LVCs is five times larger than the analogous concreteness gap, suggesting that polysemy has a greater influence than concreteness over the emergence of the LVC. We also notice a peak in polysemy scores: the “backbone” of the LVC (i.e. the LVC emerging around 8 yr) is composed of significantly more polysemous words compared to the LVC at age 20 (cf. Fig. 2 (b), sign test, p-value = 0.001 < 0.05). This early peak is absent in the partial reshuffling null model for polysemy scores. Furthermore, frequency (cf. SI Sect. 9) and concreteness do not display peaks early on after the LVC emergence. Such an early richness in high-polysemy words further indicates the idea that polysemy strongly influences the emergence of the LVC.Even though potentially causing ambiguity in communication, polysemy is a universal property of all languages <cit.>. Conventionally when constructing semantic networks<cit.> word senses and meanings can be represented by links and polysemic words can have links related to different semantic areas (e.g. “character” is linked to “nature” in the context of complexion but also to “font” in the context of typography). Randomly Reshuffling word labels for all the neighbourhoods in the network evidently disrupts semantic relationships, thus destroying polysemy. We call this reshuffling “full” as it preserves the structure of local connections in the layers while fully destroying both intra-layer correlations at the endpoints of links and inter-layer correlations of words. We use full reshuffling as a null model (see Methods and SI) for testing how important polysemy is in determining the presence of the LVC. We fully reshuffle 2025 high-polysemy words (i.e. the words making up the heavy tail of the polysemy distribution) and compute the LVC size in the resulting reshuffled multiplex networks. Results are compared against a reference case in which the same number of low-polysemy words are fully reshuffled. No viable cluster emerges on the multiplex networks with fully reshuffled high-polysemy words, while the LVC only shrinks by roughly 13% in case of fully reshuffling low-polysemy words. We conclude that correlations between network structure and polysemy scores are indeed necessary in determining the presence of the LVC. The above results indicate that polysemy does increase lexicon navigability by ultimately giving rise to the LVC, i.e. a relatively small cluster of words that is fully navigable under both semantic, taxonomic, and phonological relationships in the mental lexicon. Such view is in agreement with previous works<cit.>, which point out how polysemy provides long-range connections in the lexicon which can increase navigability through different word clusters on semantic single-layer networks <cit.>.§.§ Psycholinguistic characterisation of the Largest Viable Cluster (LVC) Next, we explore the impact of the presence of the LVC on cognitive aspects of language such as word processing. Our aim is to explore if words belonging to the empirical LVC (LVC-in) are processed differently than those words not in the LVC (LVC-out), more from a language use perspective rather than a developmental one (which was analysed with the previous null models). Hence, we turn to large-scale datasets of node attributes (see Tab. 1 and Methods). We find (cf. Tab. 1) that words in the largest viable cluster (i) are more frequent in the Open Subtitles dataset<cit.>, (ii) acquired earlier according to AoA reports <cit.>, (iii) quicker to identify as words in lexical decision tasks <cit.>, (iv) rated as more concrete concepts <cit.> andthus more easily memorised <cit.> and (v) represent more meanings in different semantic areas<cit.> when compared to LVC-out words.In Fig. 3 (a-e), we report the cumulative probabilities of finding a word with a given feature less than a certain value for a set of particular node-level attribute within and outside of the LVC. The difference between LVC-in and LVC-out further indicates how different the words in the LVC are compared to LVC-out words. For instance, let us consider reaction times, which indicate how quickly people classify stimuli as words or nonwords in lexical decision tasks <cit.>. The probability of finding at random an LVC-in word correctly identified in less than 500 ms is 0.48 while the same probability is less than half, 0.2, for LVC-out words. Hence the LVC is rich in words identified more quickly. Analogous results hold for all the tested attributes.Since LVC-in words have a higher degree compared to LVC-out words (see SI Sect. 3) and degree correlates with many of the psycholinguistic attributes used in our study, it is interesting to quantify to what extent the difference between LVC-in and LVC-out is due to correlations with degree. Results shown below the thick line, in the lower part of Tab. 1, suggest that the degree effect does not fully explain the observed psycholinguistic features of the LVC: a sign test indicates that all the median node-attributes of LVC-in words are higher than those of LVC-out words, at 95% confidence level. Notice that thecomparison that does not account for degree is still important since one could easily argue that degree itself can be interpreted as a cognitive component that affects word processing <cit.>. Tab. 1 also compares the statistics of the LVC against its single-layer counterparts, i.e. the largest connected components <cit.> (LCC-In). We also consider multiplex alternatives to the LVC such as: the intersection across all layers of words in the LCC of each layer (LCC Int, cf. SI Sect. 8) and the LVC-in configuration models (LVC Rew.), which consist on average of 40% more words. The empirical LVC consists of words with the most distinct linguistic features compared to the other tested sets of words, in terms of all tested node attributes. Even rewiring all links does not completely disrupt such distinctness (cf. LVC Rew.). These differences in linguistic attributes suggest that the LVC is a better measure of “coreness” for words in the mental lexicon than either the LCCs or their intersection, an idea we test further in the next section. §.§ Robustness of the multiplex lexicon and LVC to cognitive impairmentsThe LVC has been characterised as a set of higher degree words that differ in psycholinguistic features when compared to words located outside the LVC in our multiplex. This suggests that the higher degree, and cognitive correlations, of the LVC may be because the LVC is acting as a core for the mental lexicon. Let us denote the total number of links on a given layer as L and the link density as p. As shown in Fig. <ref> (a), there are more links within the LVC (Lp_In/In) across all layers than outside of it (Lp_Out/Out) or at the interface of the LVC (Lp_In/Out). Further, across all individual layers the inequality p_In/In>p_In/Out>p_Out/Out holds, denoting the presence of a core-periphery structure for the node partition {In,Out} <cit.>. In order to better interpret both the coreness and cognitive impact of the LVC, we perform a resilience analysis of the MLR by means of numerical experiments. Random word failure provides a plausible toy model for progressive anomia<cit.> driven by cognitive decline, where words become progressively non-accessible on all the lexicon levels without a clear trend <cit.>. To simulate progressive anomia, we randomly remove LVC-in and LVC-out words in separate experiments. The maximum number of removed words is 1173, corresponding to the size of the LVC. As a proxy for robustness, we consider the average multiplex closeness centrality, which correlates with the average cognitive effort for identifying and retrieving words within the lexicon <cit.> and plays a prominent role in early word acquisition as well <cit.>. The results of this analysis are shown in Fig. <ref> (b).We find that the multiplex representation is robust to random LVC-out word removal: removing almost 1170 LVC-out words only reduces average closeness, a measure previously linked to cognitive navigation <cit.>, to a level that is still within a 95% confidence level of the original multiplex. Therefore failure of LVC-out words does not impact the cognitive effort in identifying and retrieving words within the lexicon. Instead, the multiplex lexicon is fragile to random LVC-in word removal: removing 50% of words from the LVC leads to a decrease in closeness 20 times larger than the drop observed for LVC-out words. While considering random removal in both cases, it is true that in general LVC-in words have higher degree than LVC-out words, which might influence the robustness results from a technical perspective. The discrepancy in closeness degradation is only partly due to the higher degree of LVC-in words. Performing degree-corrected LVC-out word deletions still leads to less of a decrease in navigability as compared to LVC-in word deletion, as evident from Fig. <ref> (b). In summary, the multiplex lexicon is fragile to word failures of LVC-in words and robust to random failures of LVC-out words. This difference is a strong indicator that the LVC provides the necessary short-cuts for efficient navigation – with high closeness and thus low cognitive effort – of the mental lexical representation. It is worth remarking that the network's navigability is expected to increase in the presence of cores<cit.>, further supporting the interpretation that the LVC acts as a core of the multiplex structure. It has been conjectured that the mental lexicon has a core set of concepts <cit.>; we show here how various cognitive metrics can be correlated with the LVC, suggesting that future work may benefit from considering the LVC as a quantification of lexical core structure.§ DISCUSSION Previous literature from psycholinguistics has conjectured the existence of a core set of words in the lexicon <cit.>. Here, for the first time, we give large-scale quantitative evidence to support these conjectures. In fact, we identify the largest viable cluster (LVC) of words which: (i) favours the emergence of connectivity allowing for navigation across all layers at once and (ii) acts as a core for the multiplex lexical representation. Words within the LVC display distinct cognitive features, being (i) more frequent in usage <cit.>, (ii)learned earlier <cit.>, (iii) more concrete <cit.> and thus easily memorised <cit.> and activating perceptual regions of the brain <cit.>, (iv) more context-dependent meanings <cit.> and (iv) more easily identified in lexical decision tasks <cit.> and (v) of shorter length <cit.> than words outside the LVC.Remarkably, the explosive emergence of the LVC happens around 7 years of age, which is also a crucial stage for cognitive development in children. According to Piaget's theory of cognitive development<cit.>, age 7 is the onset of the concrete operational stage, in which children develop more semantic and taxonomic relationships among concepts (e.g. recognising that their cat is a Siamese, that a Siamese is a type of cat and that a cat is an animal, thus drawing the conclusion that their cat is an animal among several). Experimental evidence <cit.> has also shown that, in this developmental stage, children display an increased ability of mental planning and usage of context-dependent words in a connected discourse such as narratives <cit.>. Interestingly, age 7-8 is also the onset of the so-called orthographic stage for the cognitive model of reading acquisition by Frith<cit.>. Around age 7-8 years, children start recognising a large number of words automatically and instantly access their meaning, matching words to an internal lexicon that they have built up in the previous years. As a result, reading becomes much faster, as documented in experimental setups<cit.>. Age 7-8 is found to be crucial for cognitive development also by the empirical work of Gentner and Toupin<cit.>, who showed how at that age the analogical reasoning improved dramatically in children. The emergence of the lexical core represented by the LVC around age 7 might support analogical reasoning through the acquisition of more metaphorical relationships. Once in place, the lexical core may improve the ability to acquire and connect new abstract words based on analogy at later stages. All these findings can be interpreted in terms of an increased ability to navigate context-dependent meanings in the mental lexicon, which we quantitatively link to the explosive emergence of LVC core structure above. This indicates that the multiplex lexical network is a powerful representation of the mental lexicon: the network structure can indeed capture and translate well-documented mental processes driving cognitive development into quantifiable information. Notice that the current study does not test whether the LVC causes such changes butquantifies for the first time a change in the multiplex network structure that agrees with well documented developmental shifts in language learning and processing. Ad hoc longitudinal studies in children around age 7 are needed in order to better relate the LVC emergence with specific psycholinguistic tasks related to proficiency in memory and language use. From a psycholinguistic perspective, in our robustness experiments one could point out that removal of LVC-in words might increase the overall degree similarity of the remaining words, thus impairing retrieval of similar forms due to retrieval and recall issues, such as lemma selection <cit.>. While this effect agrees with the impairment expressed by the decrease in closeness, this drop cannot be attributed exclusively to increases in the similarity of degrees among words, due to removal of high degree LVC-in words. In fact, when we remove words with the same degrees both in the LVC and outside of it, closeness drops significantly more when removing LVC-in words. This strongly suggests that lemma selection issues due to degree similarities alone cannot explain the drop in closeness and the related “coreness” of concepts in the LVC.One limitation of our current approach is that we do not consider lexical restructuring over time, i.e. the adults' representation of word relationships could be different compared to children's or adolescents'. Previous work on the phonological level <cit.> showed partial differences in phonological neighbourhoods between pre-schoolers and pre-adolescents. However, we show that the LVC persists even when all connections are randomly rewired and the LVC still identifies relevant words, e.g. more frequent, more concrete, etc. suggesting that the role of the LVC may still hold even with restructuring. Link rewiring also allows consideration of the variance in word learning due to individual differences. Individual difference modelling may be especially important for quantification, diagnosing, explaining, and correcting various language learning and usage issues <cit.>.Another limitation is that the network representation might not be exact, e.g. there might be spurious links in the empirical free association layer or mistaken phonetic transcriptions in the phonological layer. In order to address this issue, we randomly reshuffle 10% of word labels, 2.5% on each layer separately, and find that the largest viable clusters are 10% smaller than the empirical LVC (t-test, p-value=0.009). However, the LVC after reshuffling exhibits analogous performance in the features discussed in Tab. 1 (sign test, p-value=0.96). Together with the random rewiring experiments, this is an indication that the LVC structure is robust to small perturbations due to errors in the annotation of links or word labels.Core/periphery network organisation is commonly found in many real-world systems <cit.>, even though the definition of cores in multiplex networks remains an open challenge. We interpret the robustness experiments as quantitative indication that the LVC is acting as a core for the whole multiplex lexical network, increasing navigability in two ways. Within the LVC, words must be connected to each other, implying navigability from every word within the LVC across all individual layers. Outside of the LVC, connections to the viable cluster facilitate network navigation by making words closer to each other. Since closeness correlates with the cognitive effort in word processing <cit.>, the LVC can be considered as facilitating mental navigation through pathways of the mental lexicon. This quantitative result is in agreement with previous conjectures about multiple meanings facilitating mental navigation of words <cit.>. Additionally, our results also indicate that the LVC acts as a multiplex core. The core is robust to node failure due to densely entwined links and connections which allow for navigation even in cases where words become inaccessible, as in cognitive disorders like progressive anomia <cit.>. It is worth remarking that we identify such a core with the largest LVC as no other non-trivial viable cluster exists in the multilayer lexical representation.Indeed, identifying a core in the mental lexicon provides quantitative evidence supporting previous claims<cit.> about the existence of a core of highly frequent and concrete words in the lexicon that facilitates mental navigation and thus word retrieval in speech production experiments <cit.>. Alongside the cognitive perspective, interpreting the LVC as a lexicon core provides support for further previous findings about the presence of a “kernel lexicon” in language <cit.>, a set of a few thousand words which constitute almost 80% of all written text <cit.> and can define every other word in language <cit.>. Previous works on semantic <cit.>, taxonomic <cit.> and phonological <cit.> single-layer networks identified a kernel lexicon for the English language with roughly 5000 words which has not changed in size during the evolution of languages. This kernel lexicon was identified with the largest connected component of the English phonological network <cit.>. The LVC we present here is: (i) a subset of the phonological largest connected component and (ii) it also persists across semantic and taxonomic aspects of language. Hence, the LVC represents a further refinement of the kernel lexicon that (i) is rich in polysemous words, (ii) facilitates mental navigation and (iii) is robust to rewiring or cognitive degradation. These three features suggest an interpretation of the LVC as a linguistic core of tightly interconnected concepts facilitating mental navigation through key words.While the framework presented here has been applied only for the English language, comparison with other languages and linguistic representations to assess how universal the LVC core is remains an exciting challenge for future experimental and theoretical work.§ METHODS§.§ Dataset and cognitive interpretation The datasets used in this work come from different sources and thus the resulting multiplex network representation is based on independent studies. For the MLR we construct four layers that model semantic, taxonomic, and phonological relationships. We further distinguish semantic relationships in free associations and synonyms. For free associations, e.g. “A reminds one of B”, we used the Edinburgh Associative Thesaurus <cit.>. For both, taxonomic relations (e.g. “A is a type of B”) and synonyms (e.g. “A also means B”) we used WordData <cit.> from Wolfram Research, which mostly coincides with WordNet 3.0 <cit.>. For phonological similarities we used the same dataset analysed in <cit.> based on WordNet 3.0 <cit.>. We treat every layer as undirected and unweighted. Words in the multiplex representation are required to be connected on at least one layer.Free associations indicate similarities within semantic memory, i.e. when given a cue word “house”, human participants respond with words that remind them of “house”, for example “bed” or “home”. Networks of free associations play a prominent role in capturing word acquisition in toddlers <cit.> and also word identification <cit.>. Networks of synonyms are also found to play a role in lexical processing <cit.>. The hierarchy provided by taxonomic relationships deeply affects both word learning and word processing <cit.>. Phonological networks provide insights about the competition of similar sounding words for confusability in word identification tasks <cit.>. For the linguistic attributes we combine several different sources. We source word frequency from OpenSubtitles <cit.>, a dataset of movie subtitles whose word frequencies were found to be superior to frequencies from classical sources in explaining variance in the analysis of reaction times from lexical decision experiments <cit.>. Concretess scores <cit.> and age of acquisitions ratings <cit.> were gathered from Amazon Turk experiments, allowing for large-scale data collection and confirmation of previous findings based on small-scale experiments <cit.>. Concreteness ratings indicate how individual concepts are rated as abstract (on a scale of 1 - “abstract” to 5 - “concrete”)<cit.>. Polysemy scores were quantified as the number of different definitions for a given word in WordData from Wolfram Research which coincides with WordNet <cit.>. Reaction times were obtained from the British Lexicon Project <cit.> and indicate the response time in milliseconds for the identification of individual words were compared against non-words. §.§ Smearing normative acquisition Smearing is a technique used in statistics for generalisation of data samples <cit.>. We smear the age of acquisition data from Kuperman et al.<cit.>, where the average age of acquisition a_i and standard deviation σ_a(i) around each word are provided, e.g. a_aim=6.72 yrs,σ_a(aim)=2.11 yrs. In our case, smearing consists of sampling possible age of acquisitions for word i from a Gaussian distribution 𝒩[a_i,σ_a(i)] rather than considering only the average value. Sampling independently an age of acquisition for each word in the dataset, we can build multiple artificial acquisition rankings from empirical data. Hence, smearing enables our analysis to account for not only the average ages of acquisition of words but also for their variability across individuals, thus adding robustness against individual variability to our results.§.§ Lexicon growth experiments We simulate lexicon growth over time t(n) by considering subgraphs of the multiplex lexicon where the first n ≤ 8531 words in a given ranking r are considered. 8531 is the total number of words in our network. Rankings indicate the way words are acquired in the lexicon over time and can be based on word features or age of acquisition reports. The rankings we use are based on: (i) smeared age of acquisition <cit.>, (ii) frequency<cit.> (higher frequency words are learned earlier), (iii) multidegree<cit.> (words with more links across all layers are learned earlier), and (iv) polysemy (words with more definitions are learned earlier). As a randomised null model, we consider random word rankings. When the first n words in a ranking are considered, a subgraph of the multiplex lexicon with these words is built and its LVC is detected. By using the non-smeared age of acquisitions, we relate the number of learned words to the developmental stage in years t(n), e.g. n=1000 corresponds to t=5.5 years. The size of the LVC L(t) is then obtained as a function of developmental stage t(n) for every specific type of ranking. Results for the smeared age of acquisitions and the random null model are averaged over an ensemble of 200 iterations. Results for the frequency, degree, and polysemy orderings are averaged over 200 iterations where words appearing in ties are reshuffled. Results are reported in Fig. 2.Each iteration represents the evolution of the LVC size through the acquisition of an individual word. This acquisition trajectory may be related to different developmental stages. For every iteration, we detect the magnitude of the transition on the LVC size due to its appearance when adding words one by one to the network. We then compute the fraction χ of iterations presenting a discontinuity of more than 10 words entering into the LVC. We also compute the average magnitude of the explosive transition Δ L. Comparisons of the empirical distributions of ages at which the LVC emerges considers the overlapping coefficient <cit.>, i.e. the overlap of two distributions normalised by the maximum overlap obtained when shifting the central moment of one of the distributions. An overlap of 100% means that one distribution is fully contained in the other one. An overlap of 0% means that the distributions have no overlap.§.§ Robustness experiments We carried out robustness testing via word/node removal: individual words are removed at random across all layers. Closeness centrality is then measured by considering shortest paths across the whole multiplex network structure, i.e. also including jumps between layers. We consider closeness centrality as a measure for the spreading of information and the mental navigability of the lexicon <cit.>. In our case closeness is well defined, since even the deletion of the whole LVC leaves the multiplex network connected<cit.>. We consider a multiplex network as connected if it is possible to reach any pair of nodes by allowing for traversal along links on any layers. With reference to Fig. 3, we perform random attacks of words within the LVC (LVC-in) and outside of it (LVC-out). Since LVC-in words are more connected compared to words outside, we also perform degree corrected attacks: random words within the LVC and words of equivalent degree outside the LVC are removed. This degree correction (LVC-out - Deg. Corr.) allows for the attack of LVC-out words but reduces the number of links by the same amount as LVC-in attacks. §.§ Data availability and Additional Information No new datasets were generated during the current study. The list of LVC-in and LVC-out words is available online at https://goo.gl/Dd9eC6. The authors declare no competing financial interests. Material requests should be addressed to the corresponding author.§ ACKNOWLEDGEMENTS M.S. was supported by an EPSRC Doctoral Training Centre grant (EP/G03690X/1). M.D.D. acknowledges financial support from the MINECO (Spain) program “Juan de la Cierva” (IJCI-2014-20225).§ AUTHOR CONTRIBUTIONS STATEMENT M.S., N.B., M.B. and M.D.D. conceived the experiments, M.S. overlapped and cleaned the data, M.S. performed the experiments, M.S., N.B., M.B. and M.D.D. analysed the results. All authors reviewed the manuscript. | http://arxiv.org/abs/1705.09731v3 | {
"authors": [
"Massimo Stella",
"Nicole M. Beckage",
"Markus Brede",
"Manlio De Domenico"
],
"categories": [
"physics.soc-ph",
"cs.CL",
"cs.SI",
"nlin.AO"
],
"primary_category": "physics.soc-ph",
"published": "20170526221839",
"title": "Multiplex model of mental lexicon reveals explosive learning in humans"
} |
argmin sgn ⟨ ⟩ x y z | http://arxiv.org/abs/1705.09412v2 | {
"authors": [
"Haoran Sun",
"Xiangyi Chen",
"Qingjiang Shi",
"Mingyi Hong",
"Xiao Fu",
"Nicholas D. Sidiropoulos"
],
"categories": [
"cs.IT",
"eess.SP",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170526022152",
"title": "Learning to Optimize: Training Deep Neural Networks for Wireless Resource Management"
} |
Neural Decomposition of Time-Series Data for Effective GeneralizationLuke B. Godfrey and Michael S. Gashler Department of Computer Science and Computer EngineeringUniversity of ArkansasFayetteville, AR 72701Email: {lbg002, mgashler}@uark.edu ======================================================================================================================================================================================== We present a neural network technique for the analysis and extrapolation of time-series data called Neural Decomposition (ND). Units with a sinusoidal activation function are used to perform a Fourier-like decomposition of training samples into a sum of sinusoids, augmented by units with nonperiodic activation functions to capture linear trends and other nonperiodic components. We show how careful weight initialization can be combined with regularization to form a simple model that generalizes well. Our method generalizes effectively on the Mackey-Glass series, a dataset of unemployment rates as reported by the U.S. Department of Labor Statistics, a time-series of monthly international airline passengers, the monthly ozone concentration in downtown Los Angeles, and an unevenly sampled time-series of oxygen isotope measurements from a cave in north India. We find that ND outperforms popular time-series forecasting techniques including LSTM, echo state networks, ARIMA, SARIMA, SVR with a radial basis function, and Gashler and Ashmore's model. § INTRODUCTION The analysis and forecasting of time-series is a challenging problem that continues to be an active area of research. Predictive techniques have been presented for an array of problems, including weather <cit.>, traffic flow <cit.>, seizures <cit.>, sales <cit.>, and others <cit.>. Because research in this area can be so widely applied, there is great interest in discovering more accurate methods for time-series forecasting.One approach for analyzing time-series data is to interpret it as a signal and apply the Fourier transform to decompose the data into a sum of sinusoids <cit.>. Unfortunately, despite the well-established utility of the Fourier transform, it cannot be applied directly to time-series forecasting. The Fourier transform uses a predetermined set of sinusoid frequencies rather than learning the frequencies that are actually expressed in the training data. Although the signal produced by the Fourier transform perfectly reproduces the training samples, it also predicts that the same pattern of samples will repeat indefinitely. As a result, the Fourier transform is effective at interpolation but is unable to extrapolate future values. Another limitation of the Fourier transform is that it only uses periodic components, and thus cannot accurately model the nonperiodic aspects of a signal, such as a linear trend or nonlinear abnormality.Another approach is regression and extrapolation using a model such as a neural network. Regular feedforward neural networks with standard sigmoidal activation functions do not tend to perform well at this task because they cannot account for periodic components in the training data. Fourier neural networks have been proposed, in which feedforward neural networks are given sinusoidal activation functions and are initialized to compute the Fourier transform. Unfortunately, these models have proven to be difficult to train <cit.>.Recurrent neural networks, as opposed to feedforward neural networks, have been successfully applied to time-series prediction <cit.>. However, these kinds of networks make up a different class of forecasting techniques. Recurrent neural networks also have difficulty handling unevenly sampled time-series. Further discussion about recurrent neural networks and other classes of forecasting techniques is provided in Section <ref>.We claim that effective generalization can be achieved by regression and extrapolation using a model with two essential properties: (1) it must combine both periodic and nonperiodic components, and (2) it must be able to tune its components as well as the weights used to combine them. We present a neural network technique called Neural Decomposition (ND) that demonstrates this claim. Like the Fourier transform, it decomposes a signal into a sum of constituent parts. Unlike the Fourier transform, however, ND is able to reconstruct a signal that is useful for extrapolating beyond the training samples. ND trains the components into which it decomposes the signal represented by training samples. This enables it to find a simpler set of constituent signals. In contrast to the fast Fourier transform, ND does not require the number of samples to be a power of two, nor does it require that samples be measured at regular intervals. Additionally, ND facilitates the inclusion of nonperiodic components, such as linear or sigmoidal components, to account for trends and nonlinear irregularities in a signal.In Section <ref>, we demonstrate that the simple innovations of ND work together to produce significantly improved generalizing accuracy with several problems. We tested with the chaotic Mackey-Glass series, a dataset of unemployment rates as reported by the U.S. Department of Labor Statistics, a time-series of monthly international airline passengers, the monthly ozone concentration in downtown Los Angeles, and an unevenly sampled time-series of oxygen isotope measurements from a cave in north India. We compared against long short-term memory networks (LSTM), echo state networks, autoregressive integrated moving average (ARIMA) models, seasonal ARIMA (SARIMA) models, support vector regression with a radial basis function (SVR), and a model recently proposed by Gashler and Ashmore <cit.>. In all but one case, ND made better predictions than each of the other prediction techniques evaluated; in the excepted case, LSTM and echo state networks performed slightly better than ND.This paper is outlined as follows. Section <ref> provides a background and reviews related works. Section <ref> gives an intuitive-level overview of ND. Section <ref> provides finer implementation-level details. Section <ref> shows results that validate our work. Finally, Section <ref> discusses the contributions of this paper and future work.§ RELATED WORK §.§ Models for Time-Series Prediction Many works have diligently surveyed the existing literature regarding techniques for forecasting time-series data <cit.>. Some popular statistical models include Gaussian process <cit.> and hidden Markov models <cit.>.Autoregressive integrated moving average (ARIMA) models <cit.> are among the most popular approaches. The notation for this model is ARIMA(p, d, q), where p is the number of terms in the autoregressive model, d is the number of differences required to take to make the time-series stationary, and q is the number of terms in the moving average model. In other words, ARIMA models compute the dth difference of x(t) as a function of x_t-1, x_t-2 , ..., x_t-p and the previous q error terms.Out of all the ARIMA variations that have been proposed, seasonal ARIMA (SARIMA) <cit.> is considered to be the state of the art “classical” time-series approach <cit.>. Notation for SARIMA is ARIMA(p, d, q)(P, D, Q)[S], where p, d, q are identical to the normal ARIMA model, P, D, Q are analogous seasonal values, and S is the seasonal parameter. For example, an ARIMA(1,0,1)(0,1,1)[12] uses an autoregressive model with one term, a moving average model with one term, one seasonal difference (that is, x'_t = x_t - x_t - 12), and a seasonal moving average with one term. This seasonal variation of ARIMA exploits seasonality in data by correlating x_t not only with recent observations like x_t-1, but also with seasonally recent observations like x_t - S. For example, when the data is a monthly time-series, S=12 correlates observations made in the same month of different years, and when the data is a daily time-series, S=7 correlates observations made on the same day of different weeks.In the field of machine learning, three high-level classes of techniques (illustrated in Figure <ref>) are commonly used to forecast time-series data <cit.>. Perhaps the most common approach, (A), is to train a model to directly forecast future samples based on a sliding window of recently collected samples <cit.>. This approach is popular because it is simple to implement and can work with arbitrary supervised learning techniques.A more sophisticated approach, (B), is to train a recurrent neural network <cit.>. Several recurrent models, such as LSTM networks <cit.>, have reported very good results for forecasting time-series. In an LSTM network, each neuron in the hidden layer has a memory cell protected by a set of gates that control the flow of formation through time<cit.>. Echo state networks (ESNs) have also performed particularly well at this task <cit.>. An ESN is a randomly connected, recurrent reservoir network with three primary meta-parameters: input scaling, spectral radius, and leaking rate <cit.>. Although they are powerful, these recurrent models are only able to handle time-series that are sampled at a fixed interval, and thus cannot be directly applied to unevenly sampled time-series.Our model falls into the third category of machine learning techniques, (C): regression-based extrapolation. Models of this type fit a curve to the training data, then use the trained curve to anticipate future samples. One advantage of this approach over recurrent neural networks is that it can make continuous predictions, instead of predicting only at regular intervals, and can therefore be directly applied to irregularly spaced time-series. A popular method in this category is support vector regression (SVR) <cit.>. Many models in this category decompose a signal into constituent parts, providing a useful mechanism for analyzing the signal. Our model is more closely related to a subclass of methods in this category, called Fourier neural networks (see Section <ref>), due to its use of sinusoidal activation functions. Models in the first two categories, (A) and (B), have already been well-studied, whereas extrapolation with sinusoidal neural networks remains a relatively unexplored area. §.§ Harmonic Analysis The harmonic analysis of a signal transforms a set of samples from the time domain to the frequency domain. This is useful in time-series prediction because the resulting frequencies can be used to reconstruct the original signal (interpolation) and to forecast values beyond the sampled time window (extrapolation). Harmonic analysis, also known as spectral analysis or spectral density estimation, has been well-studied for decades <cit.>.Perhaps the most popular method of harmonic analysis is the distrete Fourier transform (DFT). The DFT maps a series of N complex numbers in the time domain to the frequency domain. The inverse DFT (iDFT) can be applied these new values to map them back to the time domain. More interestingly, the iDFT can be used as a continuous representation of the originally discrete input. The transforms are generally written as a sum of N complex exponentials, which can be rewritten in terms of sines and cosines by Euler's formula.The DFT and the iDFT are effectively the same transform with two key differences. First, in terms of sinusoids, the DFT uses negative multiples of 2 π / N as frequencies and the iDFT uses positive multiples of 2 π / N as frequencies. Second, the iDFT contains the normalization term 1/N applied to each sum.In general, the iDFT requires all N complex values from the frequency domain to reconstruct the input series. For real-valued input, however, only the first N/2 + 1 complex values are necessary (N/2 frequencies and one bias). The remaining complex numbers are the conjugates of the first half of the values, so they only contain redundant information. Furthermore, in the real-valued case, the imaginary component of the iDFT output can be discarded to simplify the equation, as we do in Equation <ref>. This particular form of the iDFT (reconstructing a series of real samples) can therefore be written as a real sum of sines and cosines.The iDFT is as follows. Let R_k and I_k represent the real and imaginary components respectively of the kth complex number returned by the DFT. Let 2 π k / N be the frequency of the kth term. The first frequency yields the bias, because cos(0) = 1 and sin(0) = 0. The second frequency is a single wave, the third frequency is two waves, the fourth frequency is three waves, and so on. The cosine with the kth frequency is scaled by R_k, and the sine with the kth frequency is scaled by I_k. Thus, the iDFT is sufficiently described as a sum of N/2 + 1 terms, with a sin(t) and a cos(t) in each term and a complex number from the DFT corresponding to each term: x(t) = ∑_k = 0^N/2 R_k · cos( 2 π k/N t ) - I_k · sin( 2 π k/N t ) Equation <ref> is useful as a continuous representation of the real-valued discrete input. Because it perfectly passes through the input samples, one might naively expect this function to be a good basis for generalization. In order to choose appropriate frequencies, however, the iDFT assumes that the underlying function always has a period equal to the size of the samples that represent it, that is, x(t + N) = x(t) for all t. Typically, in cases where generalization is desirable, the period of the underlying function is not known. The iDFT cannot effectively model the nonperiodic components of a signal, nor can it form a simple model for series that are not periodic at N, even if the series is perfectly periodic.Figure <ref> illustrates the problems encountered when using the iDFT for time-series forecasting. Although the model generated by the iDFT perfectly fits the training samples, it only has periodic components and so is only able to predict that these samples will repeat to infinity, without taking nonperiodicity into account. Our approach mimics the iDFT for modeling periodic data, but is also able to account for nonperiodic components in a signal (Figure <ref>).Because of these limitations of the DFT, other approaches to the harmonic analysis of time-series have been proposed. Some of these other approaches perform sinusoidal regression to determine frequencies that better represent the periodicity of the sampled signal <cit.>. Our approach similarly uses regression to find better frequencies. §.§ Fourier Neural NetworksUse of the Fourier transform in neural networks has already been explored in various contexts <cit.>. The term Fourier neural network has been used to refer to neural networks that use a Fourier-like neuron <cit.>, that use the Fourier transform of some data as input <cit.>, or that use the Fourier transform of some data as weights <cit.>. Our work is not technically a Fourier neural network, but of these three types, our approach most closely resembles the third.Silvescu provided a model for a Fourier-like activation function for neurons in neural networks <cit.>. His model utilizes every unit to form DFT-like output for its inputs. He notes that by using gradient descent to train sinusoid frequencies, the network is able to learn “exact frequency information” as opposed to the “statistical information” provided by the DFT. Our approach also trains the frequencies of neurons with a sinusoidal activation function.Gashler and Ashmore presented a technique that used the fast Fourier transform (FFT) to approximate the DFT, then used the obtained values to initialize the sinusoid weights of a neural network that mixed sinusoidal, linear, and softplus activation functions <cit.>. Because this initialization used sinusoid units to model nonperiodic components of the data, their model was designed to heavily regularize sinusoid weights so that as the network was trained, it gave preference to weights associated with nonperiodic units and shifted the weights from the sinusoid units to the linear and softplus units. Use of the FFT required their input size to be a power of two, and their trained models were slightly out of phase with their validation data. However, they were able to generalize well for certain problems. Our approach is similar, except that we do not use the Fourier transform to initialize any weights (further discussion on why we do not use the Fourier transform can be found in Section <ref>).§ HIGH LEVEL APPROACHIn this section, we describe Neural Decomposition (ND), a neural network technique for the analysis and extrapolation of time-series data. This section focuses on an intuitive-level overview of our method; implementation details can be found in Section <ref>. §.§ Algorithm Description We use an iDFT-like model with two simple but important innovations. First, we allow sinusoid frequencies to be trained. Second, we augment the sinusoids with a nonperiodic function to model nonperiodic components. The iDFT-like use of sinusoids allows our model to fit to periodic data, the ability to train the frequencies allows our model to learn the true period of a signal, and the augmentation function enables our model to forecast time-series that are made up of both periodic and nonperiodic components.Our model is defined as follows. Let each a_k represent an amplitude, each w_k represent a frequency, and each ϕ_k represent a phase shift. Let g(t) be an augmentation function that represents the nonperiodic components of the signal. x(t) = ∑_k = 1^N( a_k · sin( w_k t + ϕ_k ) )+ g(t) Note that in our model, compared to the iDFT, two indexing changes have been made: 1) the lower index of the sum has changed from k = 0 to k = 1, and 2) the upper index of the sum has changed from N/2 to N. The lower index has changed because ND can account for bias in the augmentation function g(t), so the 0 frequency is not necessary. The upper index has changed to simplify the equation as a sum of N sines rather than a sum of N/2 sines and cosines.If the phase shifts are set so that sin(t + ϕ) is transformed into cos(t) and -sin(t), the frequencies are set to the appropriate multiples of 2 π, the amplitudes are set to the output values of the DFT, and g(t) is set to a constant (the bias), then ND is identical to the iDFT. However, by choosing a g(t) better suited to generalization and by learning the amplitudes and tuning the frequencies using backpropagation, our method is more effective at generalization than the iDFT. g(t) may be as simple as a linear equation or as complex as a combination of linear and nonlinear equations. A discussion on the selection of g(t) can be found in Section <ref>.We use a feedforward artificial neural network with a single hidden layer to compute our function (see Figure <ref>). The hidden layer is composed of N units with a sinusoid activation function and an arbitrary number of units with other activation functions to calculate g(t). The output layer is a single linear unit, so that the neural network outputs a linear combination of the units in the hidden layer.We initialize the frequencies and phase shifts in the same way as the inverse DFT as described above. Rather than use the actual values provided by the DFT as sinusoid amplitudes, however, we initialize them to small random values (see Section <ref> for a discussion on why). Weights in the hidden layer associated with g(t) are initialized to approximate identity, and weights in the output layer associated with g(t) are randomly perturbed from zero.We train our model using stochastic gradient descent with backpropagation. This training process allows our model to learn better frequencies and phase shifts so that the sinusoid units more accurately represent the periodic components of the time-series. Because frequencies and phase shifts are allowed to change, our model can learn the true period of the underlying function rather than assuming the period is N. Training also tunes the weights of the augmentation function.ND uses regularization throughout the training process to distribute weights in a manner consistent with our goal of generalization. In particular, we use L^1 regularization on the output layer of the network to promote sparsity by driving nonessential weights to zero. Thus, ND produces a simpler model by using the fewest number of units that still fit the training data well.By pre-initializing the frequencies and phase shifts to mimic the inverse DFT and setting all other parameters to small values, we reduce time-series prediction to a simple regression problem. Artificial neural networks are particularly well-suited to this kind of problem, and using stochastic gradient descent with backpropagation to train it should yield a precise and accurate model.The neural network model and training approach we use is similar to those used by Gashler and Ashmore in a previous work on time-series analysis <cit.>. Our work builds on theirs and contributes a number of improvements, both theoretically and practically. First, we do not initialize the weights of the network using the Fourier transform. This proved to be problematic in their work as it used periodic components to model linear and other nonperiodic parts of the training data. By starting with weights near zero and learning weights for both periodic and nonperiodic units simultaneously, our model does not have to unlearn extraneous weights. Second, their model required heavy regularization that favored using linear units rather than the initialized sinusoid units. Our training process makes no assumptions about which units are more important and instead allows gradient descent to determine which components are necessary to model the data. Third, their training process required a small learning rate (on the order of 10^-7) and their network was one layer deeper than ours. As a result, their frequencies were never tuned, their results were generally out of phase with the testing data, and their training times were very long. Because our method facilitates the training of each frequency and allows a larger learning rate (10^-3 in our experiments), our method yields a function that is more precisely in phase with the testing data in a much shorter amount of time. Thus, our method has simplified the complexity of the model's training algorithm, minimized its training time, and improved its overall effectiveness at time-series prediction. The superiority of our method is demonstrated in Section <ref> and visualized in Figure <ref>. §.§ Comparison to iDFTNeural Decomposition has a number of benefits over the iDFT for time-series prediction. One is that, unlike the FFT-approximated iDFT, ND does not require the number of samples to be a power of two. In order to use the FFT on any input size that is not a power of two, the input must be padded with zeros (or some other arbitrary placeholder) to make it a power of two in size. Although this is acceptable in some applications, it sabotages generalization by training a model to reconstruct these arbitrary values. The removal of a power of two restriction maximizes the amount of information that ND is able to effectively utilize.Additionally, our approach does not make the generally false assumption that the input is periodic at N. The iDFT predicts that the input series will repeat itself indefinitely and cannot handle fractional periods. ND uses flexible frequencies that enable it to learn the actual period of the underlying function, even if the input series contains a fractional part of a signal's periodic components. Our method effectively harnesses the information provided by the entire input series, including the fractional part. The iDFT, however, is not able to use this extra information. In fact, the fractional part introduces unnecessary complexity into the model generated by the iDFT.A third advantage ND has over the iDFT is that ND does not require samples taken at regular intervals. Although many real-world datasets are sampled regularly, there are a number of applications that are not. Any time-series data obtained from a mobile device, for example, may contain irregular samples due to power consumption or loss of signal <cit.>.The iDFT can be used to model time-series as a sum of sinusoids, which is ideal for periodic data. To model any nonperiodic components of a time-series, however, the iDFT has to use several sinusoid units, resulting in an unnecessarily complicated function representing the closed form of the series. ND is able to account for nonperiodic components of a signal using nonperiodic functions, resulting in a simpler model.The most important benefit of our approach is that it is able to generalize. The iDFT yields a model that perfectly fits the input samples, but it generalizes poorly for nonperiodic data, or for data that is periodic at a point other than at N. Because it has flexible frequencies and can model nonperiodic components, ND can generalize for both periodic and nonperiodic time-series, regardless of where the periodicity is.§.§ Toy Problem for Justification Figure <ref> demonstrates that flexible frequencies and an appropriate choice for g(t) are essential for effective generalization. We compare three ND models using the equation x(t) = sin(4.25 π t) + sin(8.5 π t) + 5t to generate time-series data. This is a sufficiently interesting toy problem because it is composed of periodic and nonperiodic functions and its period is not exactly N (otherwise, the frequencies would have been multiples of 2 π). We generate 128 values for 0 ≤ t < 1.0 as input and 256 values for 1.0 ≤ t < 3.0 as a validation set. Powers of two are not required, but we used powers of two in order to compare our approach with using the inverse DFT (approximated by the inverse FFT).One of the compared ND models freezes the frequencies so that the model is unable to adjust them. Although it is able to find the linear trend in the signal, it is unable to learn the true period of the data and, as a result, makes predictions that are out of phase with the actual signal. This demonstrates that the ability to adjust the constituent parts of the output signal is necessary for effective generalization.Another of the compared ND models has flexible frequencies, but uses no augmentation function (that is, g(t) = 0). This model canlearn the periodic components of the signal, but not its nonperiodic trend. It tunes the frequencies of the sinusoid units to more accurately reflect the input samples, so that it is more in phase than the second model. However, because it cannot explain the nonperiodic trend of the signal, it also uses more sinusoid units than the true underlying function requires, resulting in predictions that are not perfectly in phase. This model shows the necessity of an appropriate augmentation function for handling nonperiodicity.The final ND model compared in Figure <ref> is ND with flexible frequencies and augmentation function g(t) = wt + b. As expected, it learns both the true period and the nonperiodic trend of the signal. We therefore conclude that an appropriate augmentation function and the ability to tune components are essential in order for ND to generalize well. §.§ Toy Problem AnalysisIn Figure <ref>, we plot the weights over time of our g(t) = wt + b model being trained on the toy problem. Weights in Figure <ref>(a) are the frequencies of a few of the sinusoids in the model, initialized based on the iDFT, but tuned over time to learn more appropriate frequencies for the input samples, andweights in Figure <ref>(b) are their corresponding amplitudes. The training process tunes frequencies w_A and w_B to more accurately reflect the period of the underlying function and adjusts the corresponding amplitudes ϕ_A and ϕ_B so that only the sinusoids associated with these amplitudes are used in the trained model and all other amplitudes are driven to zero. This demonstrates that ND tunes frequencies it needs and learns amplitudes as we hypothesized. It is also worth noting that after the first 2500 training epochs, no further adjustments are made to the weights. This suggests that ND is robust against overfitting, at least in some cases, as the “extra” training epochs did not result in a worse prediction.Gashler and Ashmore utilized the FFT to initialize the sinusoid amplitudes so that the neural network immediately resembled the iDFT <cit.>. Using the DFT in this way yields an unnecessarily complex model in which nearly every sinusoid unit has a nonzero amplitude, either because it uses periodic functions to model the nonperiodic signal or because it has fixed frequencies and so uses a range of frequencies to model the actual frequencies in the signal <cit.>. Consequently, the training process required heavy regularization of the sinusoid amplitudes in order to shift the weight to the simpler units (see Section <ref>). Training from this initial point often fell into local optima, as such a model was not always able to unlearn superfluous sinusoid amplitudes.Figure <ref> demonstrates why using amplitudes provided by the Fourier transform is a poor initialization point. The actual underlying function only requires two sinusoid units (found by ND), but the Fourier transform uses every sinusoid unit available to model the linear trend in the toy problem. Instead of tuning two amplitudes, a model initialized with the Fourier transform has to tune every amplitude and is therefore far more likely to fall into local optima.ND, by contrast, does not use the FFT. Sinusoid amplitudes (the weights feeding into the output layer) and all output-layer weights associated with g(t) are initialized to small random values. This allows the neural network to learn the periodic and nonperiodic components of the signal simultaneously. Not only does this avoid unnecessary “unlearning” of the extra weights used by the DFT, but also avoids getting stuck in the local optima represented by the DFT weights. Without the hindrance of having to unlearn part of the DFT, the training process is able to find more optimal values for these weights. Figure <ref> shows a comparison of our trained model with the frequencies used by the iDFT, omitting the linear component learned by ND. §.§ Chaotic SeriesIn addition to the toy problem, we applied ND to the Mackey-Glass series as a proof-of-concept. This series is known to be chaotic rather than periodic, so it is an interesting test for our approach that decomposes the signal as a combination of sinusoids. Results with this data are shown in Figure <ref>. The blue points on the left represent the training sequence, and the red points on the right half represent the testing sequence. All testing samples were withheld from the model, and are only shown here to illustrate the effectiveness of the model in anticipating future samples. The green curve represents the predictions of the trained model. The series predicted by Neural Decomposition exhibits shapes similar to those in the test data, and has an RMSE of 0.086. Interestingly, neither the shapes in the test data nor those exhibited within the model are strictly repeating. This occurs because the frequencies of the sinusoidal basis functions that ND uses to represent its model may be tuned to have frequencies with no small common multiple, thus creating a signal that does not repeat for a very long time. Our model does not capture all the high-frequency fluctuations, but it is able to approximate the general shape and some of the dynamics of the chaotic series.To determine whether Neural Decomposition merely predicts a periodic function, we tried our experiment again but set g(t) = 0 rather than using nonlinear, nonperiodic components for g(t). We found that with these changes, our model was unable to capture the subtle dynamics of the Mackey-Glass series. As in the toy problem, omitting g(t) resulted in poorer predictions, and the resulting predictions had an RMSE of 0.14 (a 63% increase in error). This indicates that ND does more than predict a strictly periodic function, and is able to capture at least some of the nonlinear dynamics in some chaotic systems.Although preliminary tests on the toy problem and the Mackey-Glass series were favorable to Neural Decomposition, not all of our tests were as successful. In particular, we applied ND to another chaotic series: samples from the Lorenz-63 model. We found that ND was unable to effectively model the dynamics of this chaotic system. This seems to indicate that although ND does well with some problems, it should not be expected to anticipate all the subtle variations that occur in chaotic systems.§ IMPLEMENTATION DETAILSIn this section, we provide a more detailed explanation of our approach. A high level description of Neural Decomposition can be found in Section <ref>. For convenience, an implementation of Neural Decomposition is included in the Waffles machine learning toolkit <cit.>. §.§ Topology We use a feedforward artificial neural network as the basis of our model. For an input of size N, the neural network is initialized with two layers: 1 → m and m → 1, where m = N + |g(t)| and |g(t)| denotes the number of nodes required by g(t). The first N nodes in the hidden layer have the sinusoid activation function, sin(t), and the rest of the nodes in the hidden layer have other activation functions to compute g(t).The augmentation function g(t) can be made up of any number of nodes with one or more activation functions. For example, it could be made up of linear units for learning trends and sigmoidal units to fit nonperiodic, nonlinear irregularities. Gashler and Ashmore have suggested that softplus units may yield better generalizing predictions compared to standard sigmoidal units <cit.>. In our experiments, we used a combination of linear, softplus, and sigmoidal nodes for g(t). The network tended to only use a single linear node, which may suggest that the primary benefit of the augmentation function is that it can model linear trends in the data. Softplus and sigmoidal units tended to be used very little or not at all by the network in the problems we tested, but intuitively it seems that nonlinear activation functions could be useful in some cases. §.§ Weight Initialization The weights of the neural network are initialized as follows. Let each of the N sinusoid nodes in the hidden layer, indexed as k for 0 ≤ k < N, have a weight w_k and bias ϕ_k. Let each w_k represent a frequency and be initialized to 2 π⌊ k/2 ⌋. Let each ϕ_k represent a phase shift. For each even value of k, let ϕ_k be set to π/2 to transform sin(t + ϕ_k) to cos(t). For each odd value of k, let ϕ_k be set to π to transform sin(t + ϕ_k) to -sin(t). A careful comparison of these initialized weights with Equation <ref> shows that these are identical to the frequencies and phase shifts used by the iDFT, except for a missing 1/N term in each frequency, which is absorbed in the input preprocessing step (see Subsection <ref>).All weights feeding into the output unit are set to small random values. At the beginning of training, therefore, the model will predict something like a flat line centered at zero. As training progresses, the neural network will learn how to combine the hidden layer units to fit the training data.Weights in the hidden layer associated with the augmentation function are initialized to approximate the identity function. For example, in g(t) = wt + b, w is randomly perturbed from 1 and b is randomly perturbed near 0. Because the output layer will learn how to use each unit in the hidden layer, it is important that each unit be initialized in this way. §.§ Input PreprocessingBefore training begins, we preprocess the input data to facilitate learning and prevent the model from falling into a local optimum. First, we normalize the time associated with each sample so that the training data lies between 0 (inclusive) and 1 (exclusive) on the time axis. If there is no explicit time, equally spaced values between 0 and 1 are assigned to each sample in order. Predicted data points will have a time value greater than or equal to 1 by this new scale. Second, we normalize the values of each input sample so that all training data is between 0 and 10 on the y axis.This preprocessing step serves two purposes. First, it absorbs the 1/N term in the frequencies by transforming t into t/N, which is why we were able to omit the 1/N term from our frequencies in the weight initialization step. Second, and more importantly, it ensures that the data is appropriately scaled so that the neural network can learn efficiently. If the data is scaled too large on either axis, training will be slow and susceptible to local optima. If the data is scaled too small, on the other hand, the learning rate of the machine will cause training to diverge and only use linear units and low frequency sinusoids.In some cases, it is appropriate to pass the input data through a filter. For example, financial time-series data is commonly passed through a logarithmic filter before being presented for training, and outputs from the model can then be exponentiated to obtain predictions. We use this input preprocessing method in two of our experiments where we observe an underlying exponential growth in the training data. §.§ Regularization Regularization is essential to the training process. Prior to each sample presentation, we apply regularization on the output layer of the neural network. Even though we do not initialize sinusoid amplitudes using the DFT, the network is quickly able to learn how to use the initialized frequencies to perfectly fit the input samples. Without regularizing the output layer, training halts as soon as the model fits the input samples, because the measurable error is near zero. By relaxing the learned weights, regularization allows our model to redistribute its weight over time. We find that regularization amount is especially important; too much prevented our model from learning, but too little caused our model to fall into local optima. In our experiments, setting the regularization term to 10^-2 avoided both of these potential pitfalls.Another important function of regularization in ND is to promote sparsity in the network, so that the redistribution of weight produces as simple a model as the input samples allow. We use L^1 regularization for this reason. Usually, the trained model does not require all N sinusoid nodes in order to generalize well, and this type of regularization enables the network to automatically discard unnecessary nodes by driving their amplitudes to zero. L^2 regularization is not an acceptable substitute in this case, as it would distribute the weights evenly throughout the network and could, like the DFT, try to use several sinusoid nodes to model what would more appropriately be modeled by a single node with a nonperiodic activation function.It is worth noting that we only apply regularization to the output layer of the neural network. Any regularization that might occur in the hidden layer would adjust sinusoid frequencies before the output layer could learn sinusoid amplitudes. By allowing weights in the hidden layer to change without regularization, the network has the capacity to adjust frequencies but is not required to do so.Backpropagation with stochastic gradient descent tunes the weights of the network and accomplishes the redistribution of weights that regularization makes possible. In our experiments, we use a learning rate of 10^-3.§ VALIDATIONIn this section, we report results that validate the effectiveness of Neural Decomposition. In each of these experiments, we used an ND model with an augmentation function made up of ten linear units, ten softplus units, and ten sigmoidal units. It is worth noting that g(t) is under no constraint to consist only of these units; it could include other activation functions or only contain a single linear node to capture trend information. We use a regularization term of 10^-2 and a learning rate of 10^-3 in every experiment to demonstrate the robustness of our approach; we did not tune these meta-parameters for each experiment.In our experiments, we compare ND with LSTM, ESN, ARIMA, SARIMA, SVR, Gashler and Ashmore's model <cit.>. We used PyBrain's implementation of LSTM networks <cit.> with one input neuron, one output neuron, and one hidden layer. We implemented a grid-search to find the best hidden layer size for the LSTM network for each problem and used PyBrain's RPROP- algorithm to train the network. We used Lukoševičius' implementation of ESN <cit.> and implemented a grid-search to find the best parameters for each problem. We used the R language implementation for ARIMA, SARIMA, and SVR <cit.>. For the ARIMA models, we used a variation of themethod that performs a grid-search to find the best parameters for each problem. For SVR, we used themethod, which also performs a grid-search for each problem. Although these methods select the best models based on the amount of error calculated using the training samples, the grid-search is a very slow process. Gashler and Ashmore's model did not require a grid-search for parameters because it has a default set of parameters that are automatically tuned during the training process. With ND, no problem-specific parameter tuning was performed.In each figure, the blue points in the shaded region represent training samples and the red points represent withheld testing samples. The curves on the graph represent the predictions made by the four models that made the most accurate predictions (only two models are shown in the fourth experiment because only two models could be applied to an irregularly sampled time-series). The actual error for each model's prediction is reported for all experiments and all models in Table <ref> and Table <ref>.The LSTM network tended to fall into local optima, and was thus extremely sensitive to the random seed. Running the same experiment with LSTM using a different random seed yielded very different results. In each experiment, therefore, we tried the LSTM model 100 times for each different topology tested in our grid-search and selected the result with the highest accuracy to present for comparison with ND. Conversely, ND consistently made approximately identical predictions when run multiple times, regardless of the random seed.In our first experiment, we demonstrated the effectiveness of ND on real-world data compared to widely used techniques in time-series analysis and forecasting. We trained our model on the unemployment rate from 1948 to 1969 as reported by the U.S. Bureau of Labor Statistics, and predicted the unemployment rate from 1969 to 1977. These results are shown in Figure <ref>. Blue points on the left represent the 258 training samples from January 1948 to June 1969, and red points on the right represent the 96 testing samples from July 1969 to December 1977. The four curves represent predictions made by ND (green), LSTM (orange), ESN (cyan), and SARIMA (magenta); ARIMA, SVR, and Gashler and Ashmore's model yielded poorer predictions and are therefore omitted from the figure. Grid-search found ARIMA(3,1,2) and ARIMA(1,1,2)(1,0,1)[12] for the ARIMA and SARIMA models, respectively. ARIMA, not shown, did not predict the significant rise in unemployment. SARIMA, shown in magenta, did correctly predict a rise in unemployment, but underestimated its magnitude, and did not predict the shape of the data well. SVR, not shown, correctly predicted that unemployment would rise, then fall again. However, it also underestimated the magnitude. ESN, shown in cyan, predicted a reasonable mean value for the general increase in unemployment, but failed to capture the dynamics of the actual data. The best LSTM network topology, found by grid-search, had a hidden layer with 16 neurons. LSTM, shown in orange, predicted the first peak in the data, but leveled off to predict only the mean. Gashler and Ashmore's model, not shown, predicted the rise and fall in unemployment, but underestimated its magnitude and the model's predictions significantly diverge from the subsequent testing samples. It is also worth noting that Gashler and Ashmore's model took about 200 seconds to train compared to ND, which took about 30 seconds to train.Results with Neural Decomposition (ND) are shown in green. ND successfully predicted both the depth and approximate shape of the surge in unemployment. Furthermore, it correctly anticipated another surge in unemployment that followed. ND did a visibly better job of predicting the nonlinear trend much farther into the future.Our second experiment demonstrates the versatility of Neural Decomposition by applying to another real-world dataset: monthly totals of international airline passengers as reported by Chatfield <cit.>. We use the first six years of data (72 samples) from January 1949 to December 1954 as training data, and the remaining six years of data (72 samples) from January 1955 to December 1960 as testing data. The training data is preprocessed through a log(x) filter and the outputs are exponentiated to obtain the final predictions. As in the first experiment, we compare our model with LSTM, ESN, ARIMA, SARIMA, SVR, and the model proposed by Gashler and Ashmore. The predictions of the four most accurate models (ND, LSTM, ESN, and SARIMA) are shown in Figure <ref>; ARIMA, SVR, and Gashler and Ashmore's model yielded poorer predictions and are therefore omitted from the figure. SVR, not shown, predicts a flat line after the first few time steps and generalizes the worst out of the four predictive models. The ARIMA model found by grid-search was ARIMA(2,1,3). ARIMA, not shown, was able to learn the trend, but failed to capture any of the dynamics of the signal. Grid-search found ARIMA(1,0,0)(1,1,0)[12] for the SARIMA model. Both SARIMA (shown in magenta) and ND (shown in green) are able to accurately predict the shape of the future signal, but ND performs better. Unlike SARIMA, ND learns that the periodic component gets bigger over time. Gashler and Ashmore's model makes meaningful predictions for a few time steps, but appears to diverge after the first predicted season. ESN, shown in cyan, performs similarly to the ARIMA model, only predicting the trend and failing to capture seasonal variations. The LSTM network, with a hidden layer of size 64 found by grid-search, failed to capture any meaningful seasonality in the training data. Instead, LSTM immediately predicted a valley and a peak that did not actually occur, followed by a poor estimation of the mean.The third experiment uses the monthly ozone concentration in downtown Los Angeles as reported by Hipel <cit.>. Nine years of monthly ozone concentrations (152 samples) from January 1955 to December 1963 are used as training samples, and the remaining three years and eight months (44 samples) from January 1964 to August 1967 are used as testing samples. The training data, as in the second experiment, is preprocessed through a log(x) filter and output is exponentiated to obtain the final predictions. Figure <ref> compares the SARIMA, ESN, LSTM, and ND models on this problem; ARIMA, SVR, and Gashler and Ashmore's model yielded poorer predictions and are therefore omitted from the figure. The ARIMA and SARIMA models found by grid-search were ARIMA(2,1,2) and ARIMA(1,1,1)(1,0,1)[12], respectively. ARIMA and SVR resulted in flat-line predictions with a high amount of error, and Gashler and Ashmore's model diverged in training and yielded unstable predictions. SARIMA (shown in magenta), ESN (shown in cyan), LSTM (shown in orange), and ND (shown in green), on the other hand, all forecast future samples well. LSTM and ESN yielded the most accurate predictions, but SARIMA and ND yielded good results as well.Our fourth experiment demonstrates that ND can be used on irregularly sampled time-series. We use a series of oxygen isotope readings in speleothems in a cave in India from 1489 AD to 1839 AD as reported by Sinha et. al <cit.>. Because the time intervals between adjacent samples is not constant (the interval is about 1.5 years on average, but fluctuates between 0.5 and 2.0 years), only ND and SVR models can be applied. ARIMA, SARIMA, Gashler and Ashmore's model, ESN, and LSTM cannot be applied to irregular time-series because they assume a constant time interval between adjacent samples; these five models are therefore not included in this experiment. Figure <ref> shows the predictions of ND and SVR. Blue points on the left represent the 250 training samples from July 1489 to April 1744, and red points on the right represent the 132 testing samples from August 1744 to December 1839. SVR, shown in orange, predicts a steep drop in value that does not exist in the testing data. ND, shown in green, accurately predicts the general shape of the testing data.Table <ref> presents an empirical evaluation of each model for the four real-world experiments. We use the mean absolute percent error (MAPE) as our error metric for comparisons <cit.>. MAPE for a set of predictions is defined by the following function, where x_t is the actual signal value (i.e. it is an elementof the set of testing samples) and x(t) is the predicted value: MAPE = 1/n∑_t=1^n| x_t - x(t)/x_t| Using MAPE, we compare Neural Decomposition to ARIMA, SARIMA, SVR with a radial basis function, Gashler and Ashmore's model, ESN, and LSTM. We found that on the unemployment rate problem (Figure <ref>), ND yielded the best model, followed by LSTM and ESN. On the airline problem (Figure <ref>), ND performed significantly better than all of the other approaches. On the ozone problem (Figure <ref>), LSTM and ESN were the best models, but ND and SARIMA also performed well. On the oxygen isotope problem (Figure <ref>), ND outperformed SVR, which was the only other model that could be applied to the irregular time-series. Table <ref> presents the results of our experiments, and Table <ref> presents the same data using the root mean square error (RMSE) metric instead of MAPE. In each problem, the accuracy of the best algorithm is shown in bold.§ CONCLUSIONIn this paper, we presented Neural Decomposition, a neural network technique for time-series forecasting. Our method decomposes a set of training samples into a sum of sinusoids, inspired by the Fourier transform, augmented with additional components to enable our model to generalize and extrapolate beyond the input set. Each component of the resulting signal is trained, so that it can find a simpler set of constituent signals. ND uses careful initialization, input preprocessing, and regularization to facilitate the training process. A toy problem was presented to demonstrate the necessity of each component of ND. We applied ND to the Mackey-Glass series and was found to generalize well. Finally, we showed results that demonstrate that our approach is superior to popular techniques LSTM, ESN, ARIMA, SARIMA, SVR, and Gashler and Ashmore's model in some cases, including the US unemployment rate, monthly airline passengers, and an unevenly sampled time-series of oxygen isotope measurements from a cave in north India. We also showed that in some cases, our approach is at least comparible to these other techniques, as in the monthly time-series of ozone concentration in Los Angeles. We predict that ND will similarly perform well on a number of other problems.This work makes the following contributions to the current knowledge: * It empirically shows why the Fourier transform provides a poor initialization point for generalization and how neural network weights must be tuned to properly decompose a signal into its constituent parts. * It demonstrates the necessity of an augmentation function in Fourier and Fourier-like neural networks and shows that components must be adjustable during the training process, observing the relationships between weight initialization, input preprocessing, and regularization in this context. * It unifies these insights to describe a method for time-series forecasting and demonstrates that this method is effective at generalizing for some real-world datasets. The primary area of future work is to apply ND to new problems. The preliminary findings on the datasets in this paper show that ND can generalize well for some problems, but the breadth of applications for ND not yet known. Some interesting areas to explore are traffic flow <cit.>, sales <cit.>, financial <cit.>, and economic <cit.>.IEEEtran | http://arxiv.org/abs/1705.09137v2 | {
"authors": [
"Luke B. Godfrey",
"Michael S. Gashler"
],
"categories": [
"cs.NE"
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"primary_category": "cs.NE",
"published": "20170525115537",
"title": "Neural Decomposition of Time-Series Data for Effective Generalization"
} |
leer.epsgsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore | http://arxiv.org/abs/1705.09408v1 | {
"authors": [
"C. E. Fontoura",
"J. Haidenbauer",
"G. Krein"
],
"categories": [
"nucl-th",
"hep-ph"
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"primary_category": "nucl-th",
"published": "20170526014026",
"title": "SU(4) flavor symmetry breaking in D-meson couplings to light hadrons"
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The entanglement properties of the time periodic Kitaev chain with nearest neighbor and next nearest neighbor hopping, is studied. The cases of the exact eigenstate of the time periodic Hamiltonian, referred to as the Floquet ground state (FGS), as well as a physical state obtained from time-evolving an initial state unitarily under the influence of the time periodic drive are explored. Topological phases are characterized by different numbers of Majorana zero (ℤ_0) and π (ℤ_π) modes, where the zero modes are present even in the absence of the drive, while the π modes arise due to resonant driving. The entanglement spectrum (ES) of the FGS as well as the physical state show topological Majorana modes whose number is different from that of the quasi-energy spectrum. The number of Majorana edge modes in the ES of the FGS vary in time from |ℤ_0-ℤ_π| to ℤ_0+ℤ_π within one drive cycle, with the maximal ℤ_0+ℤ_π modes appearing at a special time-reversal symmetric point of the cycle. For the physical state on the other hand, only the modes inherited from the initial wavefunction, namely the ℤ_0 modes, appear in the ES. The ℤ_π modes are absent in the physical state as they merge with the bulk excitations that are simultaneously created due to resonant driving. The topological properties of the Majorana zero and π modes in the ES are also explained by mapping the parent wavefunction to a Bloch sphere.Department of Physics, New York University, 726 Broadway, New York, NY, 10003, USA Entanglement properties of the time periodic Kitaev Chain Aditi Mitra December 30, 2023 =========================================================§ INTRODUCTIONTopological insulators (TIs) and metals are now a major part of condensed matter research <cit.>. While traditional topological systems such as integer <cit.> and fractional <cit.> quantum Hall states have a clear transport signature, the consequences of topology for almost all other topological systems is far more subtle. The bulk-boundary correspondence implies protected edge states, however these states do not have as dramatic a signature on transport as quantum Hall systems. This has lead to more creative ways to identify topological systems in the laboratory, such as via direct probe of edge states by ARPES <cit.>, or more sophisticated proposals for exhibiting braiding statistics from exchanging edge modes <cit.>.A new class of TIs are those that arise due to periodic driving <cit.>. Almost all topological insulators and metals are argued to have a time-periodic version <cit.>. The manifestation of topology in these systems is even more complex <cit.> because for an out of equilibrium system, the state may be far from an exact eigenstate of the Hamiltonian, and coupling to a low temperature reservoir does not always ensure the appearance of a Gibbs' state <cit.>.Due to all this, an appealing way of characterizing the topology, that does not rely on specific transport signatures, nor assumptions requiring the system to be in thermal equilibrium, is via a study of the wavefunction or the system density matrix itself. This can be done by employing various information theoretic measures. For wavefunctions that are ground states of static Hamiltonians, the entanglement entropy (EE) shows area law due to the gapped spectrum, much like ground state wavefunctions of generic gapped Hamiltonians. This is not so useful from the point of view of topology, barring a few exceptions where the EE shows subleading corrections due to topology <cit.>. In contrast, the bulk-boundary correspondence for the spectrum of the Hamiltonian remarkably also translates to a bulk boundary correspondence in the spectrum of the reduced density matrix, which is referred to as the entanglement spectrum (ES) <cit.>. A natural question to ask is, how does the bulk boundary correspondence manifest in the entanglement properties of time-periodic systems? This issue has been addressed for Floquet Chern insulators <cit.>. In this paper, we address this for the time-periodic Kitaev chain <cit.>.We study a Kitaev chain representing a mean-field p-wave superconductor, and allow for nearest-neighbor (NN) and next nearest-neighbor (NNN) hopping. The chemical potential is made to vary periodically in time. In the absence of a periodic drive, the static Kitaev chain preserves time-reversal and particle hole symmetry, and falls into the BDI class of the Altland Zirnbauer (AZ)classification <cit.>. The topological invariant is an integer ℤ corresponding to ℤ Majorana zero modes. Periodic drive makes the “energy” or quasi-energy spectrum periodic, and elevates the topological invariant to <cit.> ℤ×ℤ, where the first ℤ represent the number of zero quasi-energy Majorana modes (MZM), while the second integer represents ℤ Majorana modes at the Floquet zone boundaries, the so called Majorana π modes (MPM).In this paper we explore the fate of the MZM and MPM on the ES. We study two states, one is the exact eigenstate of the time-periodic Hamiltonian, also known as the Floquet mode. We refer to this state as the Floquet Ground State (FGS). The second state which we study is one that is the ground state of the static Kitaev chain, but unitarily time-evolved under the influence of a time-periodic drive. We refer to this as the physical or quenched state as the dynamics is under the influence of a rapid switch on protocol of the periodic drive.The new topological features of the drive arise entirely due to resonant band-crossings, resulting in MPMs. We define a resonant process as one where the frequency of the drive is such that an on-shell processes can connect the ground state to the excited state of the static Hamiltonian. Since there is no adiabatic limit for a resonant drive <cit.>,the main physics we uncover, namely which topological modes are present in the ES of the physical state and which are absent, does not depend on how fast the drive has been switched on. The paper is organized as follows. Section <ref> introduces the model and outlines construction of the Floquet Ground State (FGS), and the physical state arising from a quench.This section also presents the quasi-energy spectrum and quasi-modes for a finite wire with open boundary conditions, in order to highlight the appearance of MZM and MPM modes in the physical boundary. This will form a helpful point of comparison with the ES and the Schmidt states that reside at the entanglement cut.Section <ref> discusses the construction of the entanglement Hamiltonian. Section <ref> presents results for the entanglement properties for the ground state wavefunction of the static Hamiltonian, including an analytic solution for the MZM on the entanglement cut. Section <ref> presents the entanglement properties for the FGS. The main features of the ES of the FGS are elucidated using a spinor representation in Section <ref>. Finally section <ref> presents the entanglement properties for the physical state obtained from unitary time evolution under the effect of the periodic drive. We present our conclusions in Section <ref>.§ MODEL Our Hamiltonian is the Kitaev model, with the addition of NNN hopping,H= ∑_i -t_h ( c_i^† c_i+1 + c_i+1^† c_i ) -Δ( c_i^† c_i+1^† +c_i+1 c_i) -μ(t) ( c_i^† c_i - 1/2)-t_h'(c_i^† c_i+2 + c_i+2^† c_i) -Δ'( c_i^† c_i+2^† + c_i+2 c_i).Unless otherwise stated, results will be presented with NNN hopping turned off (Δ' = t_h' = 0) except in Section <ref>. We drive the system with a periodic chemical potential,μ(t) = μ_0 + ξsin (Ω t).The system is symmetric under μ→ - μ; we enforce μ_0 ≥ 0. Working with the static case for now, H(t) → H(ξ=0), we can diagonalize the system with a Fourier transform,H(ξ=0) =∑_k[ c_k^†c_-k ] H_ BdG(k) [c_k; c_-k^† ] ,where H_ BdG(k) is the Bogoliubov-de-Gennes (BdG) Hamiltonian,H_ BdG(k)= -(Δsin (k)+ Δ' sin(2k)) σ_y- ( t_h cos (k) + t_h' cos(2k) + μ/2) σ_z= h⃗_k ·σ⃗,where the momenta k are in units of the lattice spacing. §.§ Ground State Wave function of static Hamiltonian The static BdG Hamiltonian can be fully diagonalized via a Bogoliubov transformation,[d_k; d_-k^† ] = [uv; -v^*u^* ][c_k; c_-k^† ],where we let u = cosθ_k, v = -i sinθ_k, andθ_k = 1/2arctan(Δsin(k)/t_hcos (k) + μ/2).The result of the transformation is,H(ξ=0)= ∑_ksgn[h_k,z]ϵ_k (d_k^† d_k - d_-kd_-k^†)= ∑_k sgn[h_k,z] 2 ϵ_k d_k^† d_k,and ϵ_k isϵ_k = √(( t_h cos k + μ_0/2)^2 + Δ^2 sin^2 k).We are free to redefine d_k^†↔ d_k for each |k| sector toensure that sgn[ h_k,z] ϵ_k>0, resulting in,H(ξ=0) = ∑_k 2ϵ_k d_k^† d_k.The ground state is the dressed vacuum,|GS⟩ = ∏_k d_k |0⟩= ∏_k>0[d_-kd_k] |0⟩.We can also express this in terms of the original operators. When h_k,z>0, the ground state is:|GS⟩_|k| = [i sinθ_k c_k^† c_-k^† + cosθ_k ],and when h_k,z<0:|GS⟩_|k| = [cosθ_k c_k^† c_-k^†+ i sinθ_k ].Considering these states as vectors in the (c_k^† c_-k^†, ) basis, the ground state at each point |k| is simply the numerical ground state of the BdG Hamiltonian.§.§.§ Topology of the static system We discuss the anti-unitary symmetries present in our model. The particle-hole symmetry (PHS) is manifest in (<ref>), σ_x H_ BdG^*(-k) σ_x = -H_ BdG(k). Physically, this corresponds to the excitation energy of a quasi-particle being equivalent to the addition of a quasi-hole.Our model also has time reversal symmetry (TRS). This corresponds to complex conjugation 𝒦 for spinless fermions, and is clear in the position space definition of the model (<ref>). While TRS may be a confusing issue for spin-less fermions, note that the model can also be derived from a spin-chain where the spinless fermions of our model are simply the Jordan-Wigner fermions <cit.>. With both PHS and TRS, our model falls into the BDI Altland Zirnbauer (AZ) class <cit.>, which for one dimension, has a topological index of ℤ. This index counts the number of Majorana edge modes.Writing H_ BdG=h⃗_k·σ⃗, PHS imposes h_x(k)=-h_x(-k), h_y(k)=-h_y(-k), h_z(k)=h_z(-k). In addition, TRS corresponds to invariance under complex conjugation and k→ -k, implying h_x=0. This is equivalent to the statement that the vector h⃗_k has no σ_x component and thus lies in the σ_y,z plane. It therefore admits a well defined winding number definition via the pseudo-vector h⃗_k encircling the origin. The winding number canbe conveniently extracted by writing the eigenstates of our two level system at each k as a spinor (cosα/2, e^i βsinα/2), which rests on the unit sphere parameterized by α and β. The topological state consists of the spinors in the Brillouin zone connecting the north and south poles. We will use this spinor analogy later when we discuss the topological features in the ES of the FGS.When one considers only a single wire with no NNN terms (t_h'=0,Δ'=0), the original construction of the Kitaev model<cit.> only considered a ℤ_2 invariant consistent with class D <cit.> (PHS but no TRS). The fact that it is now considered to be in class BDIappears contradictory, but turns out to not matter as the topological indices are ± 1,0 when only NN hopping is present. So for the purposes of counting edge states, ℤ and ℤ_2 will predict the same number and there is no contradiction.In our paper we specifically would like to lift the ambiguity between BDI and D. Therefore we also present results with NNN hopping in section <ref>. This longer range hopping generates more Majorana edge states clearly placing our model in the BDI category. Consequently we will discuss the stability of the observed edge modes in the ESto TRS and PHS preserving perturbations.§.§ Floquet Ground State We now consider the time-dependent problem, and hence restore the time-dependence of the chemical potential in Eq. (<ref>). For the time-periodic system we solve the Schrödinger equation with the Floquet ansatz |ψ(t)⟩ = e^-i ϵ_a t|a(t)⟩, where |a(t+T)⟩ = |a(t)⟩ is time periodic, and ϵ_a are the quasi-energies. We call the states |ψ(t)⟩ the Floquet modes, they are eigenstates of the propagator when time evolved an integer number of periods. We find the quasi-energies and quasi-modes by solving the Floquet-Bloch equation[ H_k(t) - i∂_t ] |a_k(t)⟩ = ϵ_a |a_k(t)⟩.The operator H_F=H_k(t) - i∂_t is termed the Floquet Hamiltonian.In order to calculate the Floquet modes, we expand the Floquet Hamiltonian into the expanded Hilbert space<cit.> ℋ⊗ℒ_𝒯, whereℒ_T is the space of periodic functions. The result of the expansion is,H_F(k) ≡∑_n,m[ 1/T∫_0^T dt' H_k(t') e^-i (n-m)Ω t' + δ_n,m mΩ].This leads to a time independent matrix indexed by the photon numbers n,m. The elements of the matrix are various frequency expansions of the original time-dependent Hamiltonian. Denoting the m-th Fourier expansion of H_k(t) as H_k^(m)=1/T∫_0^T dt' H_k(t') e^-i m Ω t', for our model, the matrix is triple-banded, with the center-most bands for a given row n, from left to right being, H_k^(-1), H_k^(0) + n Ω, and H_k^(1), where H_k^(± 1)= ± i ξσ_z/4 and H_k^(0) is the static Hamiltonian.In the limit of a weak drive (ξ/t_h≪ 1), and a highly off-resonant frequencymuch larger than the bandwidth of H_0 (Ω/t_h≫ 1), the spectrum of the expanded Floquet Hamiltonian will largely be copies of the static Hamiltonian repeated at integer multiples of Ω. The expanded Floquet Hamiltonian contains a large amount of redundancies, as the majority of the eigenstates from its spectrum will produce the same Floquet modes in the traditional Hilbert space. To avoid over-counting, we restrict ourselves to the Floquet Brillouin Zone (FBZ) of the eigenstates contained within the quasi-energy range ϵ_a<|Ω/2|.The diagonalization in the extended space effectively creates a time-periodic unitary (Bogoliubov) transformation on our initially static Nambu spinor.This transformation will diagonalize our Floquet Hamiltonian and result in the following, where the time dependence is absorbed into the new operators,2ϵ_k (d̃_k^†(t)d̃_k(t) + d̃_-k^†(t) d̃_-k(t) -1) |a_k(t)⟩ = ϵ_k |a_k(t)⟩. There are four possible eigenstates |FGS⟩ = d̃_-kd̃_k, d̃_k^†|FGS⟩, d̃_-k^†|FGS⟩, and d̃_-k^†d̃_k^†|FGS⟩. We can disregard the odd-parity states here as our ground state is even-parity i.e., we allow for only doubly occupied or empty sites. These states are equivalent to finding the two component (numerical) eigenvectors in the basis, ( c_k^† c_-k^† |0⟩, |0⟩), we denote the even parity excited state as |FES⟩ = d̃_k^†d̃_-k^†|FGS⟩.§.§.§ Finite wire To better understand the bulk-boundary correspondence in the ES, we include here the quasi-energy spectrum and edge states for physical edges on a finite wire. The spectrum and eigenstates of a driven finite Majorana chain are shown in figures <ref>, <ref>, <ref>, <ref>. The calculation of these figures consisted of the tight-binding finite system analogs of the bulk Floquet quantities outlined above <cit.>.Figures <ref> and <ref> clearly show the appearance of additional π edge states that can only occur in a spectrum that is periodic. Comparing the spectrum with the eigenstates plotted in figures <ref> and <ref> shows that these states are indeed edge states. Furthermore, the MPM states in figure <ref> occur in the topologically trivial region of the static wire, showing that periodic driving can induce topological phase transitions. These topological phase transitions arise due to resonant gap-closing and re-opening at ϵ = ±Ω/2, with such resonances introducing MPMs into the system. Figure <ref> shows that with a drive, one can now have three different scenarios. One is a phase where only MZM exist and this corresponds to a high frequency off-resonant drive. Second, a phase where only MPM exist, this corresponds to a resonant drive. And finally a phase where MZM and MPM coexist, also arising due to a resonant drive.§.§ Physical StateKnowing the Floquet modes, we can construct the propagator:U_k(t) = ∏_|k|∑_a e^-i ϵ_a,k t|a_k(t)⟩⟨a_k(0)|.The physical time-evolved state we will consider is one which is the ground state of the static Hamiltonian H(ξ=0), but unitarily time-evolved under the influence of a sudden switch on of the periodic drive at t=0. Thus the physical state is,|Ψ(t)⟩ = ∏_|k| U_k(t) |GS⟩_k= ∏_|k|[ e^-2iϵ_|k|tρ_|k|,↑d̃_-k^†(t)d̃_k^†(t)|FGS(t)⟩_|k|..+ e^2iϵ_|k|tρ_|k|,↓|FGS(t)⟩_|k|].Where ρ_↕ are the time-independent overlaps of the Floquet states with the ground state at the instant the periodic drive was switched on:ρ_↓,|k| = ⟨FGS_|k|(0)|GS_|k|⟩, ρ_↑,|k| = ⟨FES_|k|(0)|GS_|k|⟩. §.§.§ Topology of the time-periodic system There are two states of interest in the driven setting, the FGS and the quenched state. As the FGS is the ground state of the effectively time-independent Floquet Hamiltonian, the topology of this state can be understood via methods analogous to analyzing topologies in conventional static Hamiltonians. The topology of the quenched state is not as clear to discern and will be discussed later.For now, using approaches valid for the static case, let us understand the topologies of the Floquet Hamiltonian. For this, we must identify the anti-unitary symmetries. In the driven setting, it is clear that PHS holds at all times during the drive. TRS is a more delicate question to answer. One can define TRS for a Floquet system as, 𝒯H(τ - t)𝒯^-1 = H(t), for some τ and for all t <cit.>. Such a condition does hold for our drive with τ=π/Ω=T/2. In addition there are two special times during the drive when the two TRS points coincide i.e., t=τ-t, t=τ+T-t. For our drive this happens at t^*=T/4,3T/4. We will show later in the paper that the number of Majorana edge modes in the ES show special behavior at t^*.While TRS and PHS hold for the Floquet Hamiltonian, the new feature of Floquet systems is that we have two gaps, one at ϵ = 0 and the other at the zone boundaries ϵ = ±Ω/2. We thus have the possibility of edge states spanning either or both gaps when the system is placed in a finite geometry. We expect the BDI classification to persist, so the topological classification in the driven setting becomes <cit.> ℤ×ℤ. The first integer counts the number of Majorana edge modes at ϵ = 0 and the second integer counts the number of Majorana edge modes at ϵ = Ω/2. Examples showcasing the ℤ×ℤ index for the eigenstates are shown in figures <ref> and <ref>. For the physical or quenched state, the topological features and bulk-boundary correspondence is not clear. Introducing a physical boundary can non-trivially modify the time-evolution, and create system dependent excitations. To avoid doing this, we will explore the bulk-boundary correspondence in the physical state by introducing an entanglement cut in the spatially periodic physical state.§ ENTANGLEMENTWe consider the entanglement cut shown in figure <ref>. Our full state is a pure state which could be one of the following three with periodic boundary conditions. One is the ground state of the static Hamiltonian, the second is the FGS, and the third is the time evolved quenched state. The entanglement cut is in real space, and the reduced density matrix is obtained from tracing out the complementary degrees of freedom.Carrying out such a partial trace is not a simple task. However for quadratic Hamiltonians, there exists a relation between the reduced density matrix and a matrix of single particle correlations on the section of the lattice of interest <cit.>. By Wick's theorem, all free-fermion cumulants will factor, including those that occur within the entanglement cut. Since our reduced density matrix must recreate this decomposition, the reduced density matrix must have the following form,ρ∝ e^-ℋ,where ℋ is quadratic in the fermionic operators. To derive the ES which is the eigenvalues of ℋ, we will resort to the Majorana basis description<cit.>.Our “complex" fermions in the region of interest (say A) are broken into the Majorana basis much like real and imaginary parts of a complex number,c_i= 1/2( a_2i-1 + i a_2i), c_i^† = 1/2(a_2i-1 - i a_2i).We have {a_2i,a_2j} = {a_2i-1, a_2j-1} =2 δ_i,j and {a_2i,a_2j-1} = 0.In order to study the Majorana correlation matrix,C̃_n,m = [ ρ a_n a_m ] = ⟨Ψ(t)|a_n a_m |Ψ(t)⟩,it is convenient to define C_i,j=[ ρ c_i^† c_j ]=⟨Ψ(t)| c_i^† c_j |Ψ(t)⟩ and F_i,j=[ ρ c_i^† c_j^†]=⟨Ψ(t)|c_i^† c_j^†|Ψ(t)⟩, so that,C̃_2i-1,2j-1 = δ_i,j + 2 i [C_i,j +F_i,j], C̃_2i-1,2j = i δ_i,j - 2 i [C_i,j - F_i,j], C̃_2i,2j-1 = -i δ_i,j + 2 i [C_i,j + F_i,j], C̃_2i,2j = δ_i,j + 2i [C_i,j - F_i,j].We group together the neighboring matrix elements into a single matrix,ℂ_i,j = (C̃ - 1)_i,j= i [2[ C_i,j +F_i,j]δ_i,j - 2 [C_i,j- F_i,j]; -δ_i,j +2 [C_i,j + F_i,j] 2 [C_i,j - F_i,j] ],where i,j index the physical sites within the entanglement cut. ℂ is hermitian and purely imaginary.When the system is the static ground state, we can assume that the C,F are purely real. In finding the spectrumof the static system we can rearrange the rows and columns inside the determinant to bring it to the form,[ℂ- λ] =[- λi -2 i [ C -F ]; -i +2 i [ C +F ] -λ ].Using the Schur complement[ A B; C D ] =D ( A - B D^-1 C ),we arrive at the following characteristic equation,( λ^2 - (1 - 2 ( C + F) ) (1 - 2 (C - F) )) = 0,which is the same as the eigenvalue equation found in <cit.>, and was shown to give the static entanglement spectra. We now proceed to show that the more general Majorana correlation matrix in equation (<ref>) will give the ES in the time dependent case.A general quadratic Hamiltonian, such as our entanglement Hamiltonian, can be written in terms of Majorana fermions as,ℋ = i ∑_m,n w_m,n a_m a_n.Where w is real and anti-symmetric; this means that there exists an orthogonal transformation that will bring ℋ into a block diagonal form,a_n = ∑_m O_n,mγ_m.The transformation being orthogonal is important as our new operators are still Majorana fermions γ^†= γ. This transformation will bring the entanglement Hamiltonian to the following form,ℋ = i a⃗· w·a⃗ = i γ⃗· O^T· w· O·γ⃗,= i/4γ⃗·( ∑_i ε_i i σ_y ) ·γ⃗,= i/2∑_i ε_i γ_2i-1γ_2i.The Pauli matrix acts on the 2i-1, 2i sub-basis for each i. As always we are free to add a constant energy to our Hamiltonian,ℋ = 1/2∑_i ε_i (1+ i γ_2i-1γ_2i).This is the same form if we had used a Bogoliubov transformation on the complex fermions and then performed the transformation to the Majorana basis afterwards.In terms of γ, the reduced density matrix has the diagonal form,ρ = ∏_i [e^-ε_i/2( 1 + i γ_2i-1γ_2i) /1 + e^-ε_i].We now insert this form into our correlation matrix definition,C̃_n,m = ∑_p,q[ ∏_i [e^-ε_i/2( 1 + i γ_2i-1γ_2i) /1 + e^-ε_i]O_n,p O_m,qγ_p γ_q ].The only terms that survive the trace are when p = q, when p = 2j, q = 2j -1 and when p = 2j -1, q = 2j. The operator i γ_2j-1γ_2j will measure the fermion parity at the site j, thus we perform the trace and arranging the sum into the even-odd matrix notation as before,C̃_n,m =∑_jO_n,j[ 1 -i tanh( ε_j/2); itanh( ε_j/2) 1 ] O^T_j,m.The orthogonal transformation that was performed on the entanglement Hamiltonian also block-diagonalizes C̃ -1.O^TC̃O = C̃' = ∑_i[1 + σ_y tanh(ε_i/2) ],so that spectrum of C̃ - 1 will yield the ES.Defining E = tanhε_k/2 our spectrum will lie between -1, 1 and the entanglement entropy (EE) will take the form,S = -∑_k [ 1+E/2log(1+E/2)+1-E/2log(1-E/2)].A value of E = 0 corresponds to a maximally entangled Schmidt state, whereas E = ± 1 corresponds to a state that is minimally entangled.In what follows we will discuss only the ES and not the EE. This is because <cit.>, the EE of the FGS behaves generically as that of a ground state wavefunction of a gapped Hamiltonian by showing area law scaling. The quenched or physical state on the other hand, at long times after the quench, shows a saturation to a volume law scaling for the EE. The volume law reflects the finite density of excitations that are always present when the periodic drive is resonant, and is again a behavior generic to excited states. Since the EE does not show any features of topology, in the remaining paper we will study the ES alone. § ENTANGLEMENT SPECTRA OF THE STATIC GROUND STATE For the static case, the C,F matrices are purely real, and take the form,C_i,j = δ_i,j/2 + ∫_0^πdk/2 πcos (k(i-j))(μ_0/2 + t_h cos k)/√(( μ_0/2 +t_h cos k)^2 + Δ^2 sin^2 k),F_i,j = -∫_0^πdk/2 πΔsin (k(i-j)) sin k/√((μ_0/2 +t_h cos k )^2 +Δ^2 sin^2(k)).Inserting these relations into our Majorana correlation matrix (<ref>), yields the ES for the static ground state upon diagonalization.Figure <ref> shows the ES of the ground state.Comparing to the off-resonant section of figure <ref>, the ES correctly recreates the energy level spectrum of the Kitaev chain with physical boundaries, but with the bands flattened<cit.>. We also note that figure <ref> has a large “entanglement gap" which will remain open in the absence of bulk excitations<cit.>. Figure <ref> shows that the zero energy states are edge states.All levels away from the singular points of the topological phase transition show a double degeneracy. This corresponds to the inversion operation with respect to the center of the entanglement cut. The regions close to the transition correspond to diverging length scales and hence the anti-symmetric and symmetric states separate.Figure <ref> shows the Schmidt states within a window |.4| of zero entanglement energy tanh(ϵ/2). This window reliably captures the topological Majorana zero modes, and is employed for all plots showing Schmidt states. Note that even the trivial phase could host edge-modes, but these edge modes are non-topological in that they are composed of complex or Dirac fermions rather than Majorana fermions, and are located at an entanglement energy closer to the bulk entanglement energy ± 1 (and therefore do not appear in Figure <ref>). Weak perturbations such as disorder, not considered in this paper, will not protect such complex fermions from merging with the bulk.To understand what states will be protected in a more general system, we must investigate the effect of allowed, non-interacting couplings. Since the entanglement Hamiltonian is closely related to a band-flattened version of the parent Floquet Hamiltonian if placed in finite geometry, we consider the robustness of edge states in the ES to perturbations which preserve the anti-unitary symmetries of PHS and TRS.PHS dictates that the positive entanglement energy states are related to the negative entanglement energy states through particle-hole conjugation. Since under particle-hole conjugation, a Majorana state transforms to itself, ϵ = 0, ±Ω/2 are the only viable energy levels for Majorana states in the quasi-energy spectrum. In the entanglement Hamiltonian, only ϵ = 0 is possible. Thus any coupling between Majorana states and a bulk state away from ϵ = 0 is not allowed. If such a coupling did exist and successfully gapped out the Majorana state, then the newly gapped state would also have to be described by a complex fermion, but there are no extra Majorana operators for the coupled Majorana to join together with and form a complete fermion. Thus PHS symmetry preserving couplings cannot move the zero energy Majorana level.Now we discuss stability with respect to TRS preserving couplings. Breaking a fermion into its Majorana operators c_i ∝ a_i + i b_i (a/b ↔ odd/even sites), TRS is the pair of statements, 𝒯a_i 𝒯^-1 = a_i and 𝒯 b_i 𝒯^-1 = - b_i. Thus, if we are to have a TRS entanglement Hamiltonian, couplings of the form i a_i a_j and i b_i b_j are prohibited. Since a Majorana mode corresponds to an a-type fermion on one edge, and a b-type fermion on the other, local couplings such as i a_i b_j, while TR preserving, cannot affect the Majorana mode.Thus in summary, for the static ground state ES, the topological Majorana edge states are protected by a large gap, PHS and TRS. In the trivial phase, the edge states are complex fermions and thus are sensitive to simple perturbations that couple to their occupation. §.§ Analytic solution for edge states in the ESWe would like to understand how the edge states are created in the ES analytically. This can be done at some special points and mirror the reasoning provided in Kitaev's original paper <cit.>, but for the entanglement cut.We set Δ/t_h = 1 and probe the trivial region by letting μ_0→∞, and the topological region by setting μ_0=0. In the trivial region (μ_0→∞) the two correlators become,C_i,j ≈δ_i,j,F_i,j →0.Plugging this into our Majorana correlator, (<ref>),ℂ_i,j = [0 -i δ_i,j;i δ_i,j0 ].Our Majorana correlator becomes block-diagonal with degenerate bands at ± 1, thus reproducing the ES of the trivial phase.Now we consider the topological phase (μ_0 = 0),C_i,j = δ_i,j/2 + ∫_0^πdk/2 πcos (k(i-j))cos k,F_i,j = -∫_0^πdk/2 πsin (k(i-j)) sin k.The above implies,[C + F]_i,j = δ_i,j/2 + δ_i-j,-1/2, [C - F]_i,j = δ_i,j/2 + δ_i-j,1/2,which when inserted into (<ref>), gives,ℂ_i,j = [0 -i δ_i-j,1;iδ_i-j,-10 ].This corresponds to the same block-diagonal matrix as for the trivial case with the modification of an empty top and bottom row. It is the same matrix in the bulk with degenerate bands at ± 1, butwe now have null vectors that occupy the first and last sites. These null vectors denote the Majorana edge modes in the ES. Thus we have re-derived the Kitaev picture, but now for our entanglement cut. § ENTANGLEMENT SPECTRA OF FGSWe now wish to calculate the correlators as a function of time. We can construct the C,F matrices from our knowledge of the full time-evolved wave function. Denoting the Floquet ground and excited states as ↓ and ↑ respectively, such that |FGS(t)⟩_k = α_↓(k,t) c_k^† c_-k^† + β_↓(k,t), we find,C_i,j = 1/π∫_0^π dk cos(k(i-j)) [ |ρ_↓α_↓|^2 +.. |ρ_↑α_↑|^2 + e^i ϵ_k tρ_↓ρ_↑^*α_↓α_↑^* + e^-i ϵ_k tρ_↑ρ_↓^*α_↑α_↓^* ], F_i,j = i/π∫_0^π dk sin(k(i-j)) [ |ρ_↓|^2 β_↓α_↓^* + ..|ρ_↑|^2 β_↑α_↑^* + e^i ϵ_k tρ_↓ρ_↑^*β_↓α_↑^* + e^-i ϵ_k tρ_↑ρ_↓^*β_↑α_↓^* ]. We are interested in two driven states, the FGS, and a physical state obtained from a quench. The physical state will correspond to utilizing the full expressions for C and F with ρ_↑,ρ_↓ given in Eq. (<ref>). The FGS ES will be determined from the above after setting ρ_↓ = 1 and ρ_↑ = 0 for all k. The quench state discussion will follow the FGS in section <ref>.Unlike the quasi-energy spectrum, the ES is not periodic, and has only one gap. Any topological edge modes have to lie within this gap. In this sense, the ES of the FGS creates an edge spectrum with only “half" the information compared to the quasi-energy spectrum at physical boundaries. The loss of “half" the information will result in the MZM and MPM states both having the same entanglement energy, ϵ = 0. *Discussion of the FGS plots. We will now make a series of observations from the plots for the ES of the FGS, but will explain these observations in section <ref>. In particular, since the FGS ES is constructed from the pure FGS alone, we will find it convenient to explain some of the unusual features in the ES through a spinor description of the FGS in section <ref>.We first focus on the FGS ES in figures <ref> and <ref> forΩ/t_h = 5, which contain the three phases MZM, trivial, and MPM. A comparison with figures <ref>and <ref> shows that the ES correctly reproduces the topological edge states of the physical edges. The ES is a spectrally flattened version of the quasi-energy spectrum for the physical edges, with the important modification that the π modes now reside at zero entanglement energy. Further, the region of the ES that corresponds to off-resonant drives closely resembles the ES of the static ground state and the physics of this portion of the ES is similar to the static case. In addition, the MPM and MZM phases behave in a largely similar manner. Figure <ref> shows the exponentiallocalization of both the MPM and MZM modes, within a few lattice sites of the edges.The decay length is governed by the entanglement gap in the ES, which for figure <ref> is nearly constant throughout the period.We now shift our focus to figures <ref> and <ref>, where Ω/t_h = 3 and corresponds to a resonance occurring while still in the MZM phase. Here we have a MZM, MZM&MPM, and MPM phase for the range of μ_0 shown. We will now discuss the new MZM&MPM phase, as the remaining phases are largely the same as that of the example discussed above for Ω/t_h = 5.The MZM&MPM phase is interesting because the MZM and MPM states gap each other out in the ES. This is in contrast to figures <ref> and <ref> where the MZM&MPM phase is simply the combination of the individual phases, without any coupling between them, as predicted by the ℤ×ℤ index. We find that ℤ×ℤ, no longer holds in the ES as is evident by the gap opening. By studying the Schmidt states we note that the gap opening does not involve the MZM and MPM modes merging with the bulk. Rather, we still have edge modes, but the nature of the edge modes are different when MZM and MPM modes couple to each other.Figure <ref> shows the time dependence of the gap in the MZM&MPM phase. At most points during the drive, the phase is gapped, but at a special point, the gap closes and both the MZM and the MPM reside at zero entanglement energy. The time-dependence of the gap can be best understood through the spinor parameterization of the FGS which we discuss in the next section. Figure <ref> also highlights the lack of time dependence in the ES for the topologically protected states in the MZM and MPM phase, which as mentioned before behaves like the ES of the static Hamiltonian. The “bulk" states also show only a small amount of time dependence.As far as the nature of the Schmidt states are concerned in the MZM&MPM phase, when the gap closes (t=3T/4 in figure <ref>), we have two Majorana modes at each end of the cut. When the gap opens symmetrically around zero, this corresponds to a pair of edge modes that are related by charge conjugation. Thus the positive entanglement energy edge mode is particle like, and the negative entanglement energy edge mode is hole like.This observed coupling between 0 and π modes during a cycle leads to the conclusion that the ℤ×ℤ in the quasi-energy spectrum becomes |ℤ_0 - ℤ_π| × |ℤ_0 + ℤ_π| in the ES, where the two integers now denote the number of Majorana zero modes in the ES at the two TRS points t^*=T/4,3T/4. We strengthen this observation further in the next subsection <ref> where we generate more Majorana modes by introducing longer ranged hopping.§.§ Next-Nearest-Neighbor hopping With NN hopping, ℤ_0,ℤ_π take values 0,1 and cannot therefore differentiate between classes BDI and D. In order to lift this ambiguity,we would like to generate more MZMs and MPMs. The easiest way of creating more topologically protected edge modes is to introduce NNN hopping. Here we use the model studied in <cit.> and turn on the NNN parameters (t_h', Δ'). The main conclusion from this section of the paper is that the NNN hopping produces larger number of edge states in our system and the ES correctly detects these edge states. In particular, for the example in figure <ref>, we have three different phases. For the phase corresponding to μ_0/t_h>4.5, we have four π modes at zero entanglement energy at all times, of which two sit on one end of the entanglement cut and the other two at the other end. This phase has ℤ_0 =0, ℤ_π=2. The gaplessness at zero energy holds true at all times and reveals that the ES preserves |ℤ_0 - ℤ_π|=2.This point is further highlighted by studying the time-dependence of the gap. While the phases which contain only MZMs (1<μ_0/t_h<1.5) and only MPMs (4.5<μ_0/t_h) show no significant time-dependence, in the central region (2<μ_0/t_h<4) corresponding to a phase containing both MZMs and MPMs, the time-dependence is dramatic. Comparing figure <ref> with figure <ref> shows that for this 2 MPMs&1 MZM phase, a pair of levels remain at entanglement energy of zero throughout the drive (|ℤ_0 - ℤ_π|=1). The gapped out states oscillate during the drive and at a special point during the drive (t^*=3 T/4) also reside at zero entanglement energy, similar to the MZM&MPM phase in the NN hopping diagrams for Ω/t_h = 3 (figure <ref>).Employing a spinor description in section <ref> we will explain the above observations, and also argue that the reason why the the number of |ℤ_0 - ℤ_π| Majorana modes persist at other times besides the special TRS points is due to our particular drive, namely one that couples to the chemical potential. More generic TRS and PHS preserving periodic drives will give rise to additional couplings between the |ℤ_0 - ℤ_π| Majorana modes away from t^*, reducing the invariant in the ES during the period to ℤ_2. § SPINOR DESCRIPTION AND TOPOLOGY We show that the FGS ES can be better understood through the spinor representation of the FGS. We consider a spinor parameterized by the two angles α and β, ( cosα/2, e^i βsinα/2) and consider the Bloch sphere parameterized by these angles. Consider the static Hamiltonian for now. We know from PHS that the k= 0 and k = π points must lie at either the north or south poles, as the symmetries force the σ_x,y term in H_ BdG to be odd in k. TRS on the other hand forces the third Pauli matrix (in our case σ_x) to be absent so that the spinor lies in y-z plane of the Bloch sphere. This allows us to define a winding number for the number of times the spinor winds around in a given plane of the Bloch sphere.NNN terms via Δ', t_h'≠ 0, does not change this, but includes the possibility of introducing another point k^* different from 0 and π, where the spinor points either on the north or south pole. Thus with NNN terms, and TRS, there is a possibility of introducing additional windings, and hence additional MZMs. If the drive is resonant, then additional special points k_π appear where the spinors are constrained to be at the poles, but this constrain is true only at special TRS points of the drive. For our drive, these special TRS points during the cycle are t^*=T/4,3T/4. Thus at times t^* we find well defined windings of |ℤ_0-ℤ_π| and ℤ_0 + ℤ_π on the Bloch sphere. Since our drive couples to the chemical potential, it has the additional property that the spinors at k=0,π, k^*stay pinned to the poles at all times. This results in the |ℤ_0-ℤ_π| winding to be preserved at all other times besides the special t^* points. A slightly different drive which coupled to Δ' →Δ' + δsin(Ω t) would have the property that only the k=0,π are constrained at the poles at all times. This would imply that the ES will show ℤ_2 invariance, with |ℤ_0-ℤ_π| and ℤ_0 + ℤ_π appearing only intermittently in the ES at the two special t^* times.We now discuss this basic picture with a specific example, with only NN hopping. Figure <ref> shows how the FGS for all k wraps around the sphere for several times during the drive, highlighting the generic behavior for the following phases: trivial, MZM, MPM, and MZM&MPM. The trivial phase fails to connect the north and south poles as expected. Since the drive is highly off-resonant in the trivial phase, its only effect is to cause small deformations of the loop during the drive cycle.In contrast to the trivial phase, the MZM, MPM phases always connect the two poles at all times. The fact that these modes survive during the periodic drive is clear from the fact that the drive still constrains the spinors to stay pinned at the poles at k=0,π. The above picture explains why the ES shows static in time zero modes in the MZM and MPM phases.The MPM phase, in addition to being pinned at the poles, has an associated wrapping in the β angle as well. The rotation in β can be understood in the context of a rotating wave approximation. The rotating frame effectively maps the resonant time-periodic Hamiltonian (recall MPM phase is always associated with a resonance) into a Hamiltonian that appears like the static ground state in the topological phase. Thus we regain the topological winding in the α parameter, but when we rotate back into the lab frame, we acquire a relative phase between the two components of the spinor that is periodic with the drive.Now we turn to the case of the MZM&MPM phase. If we start in the MZM phase and increase μ_0, the eventual phase transition and introduction of the MPM, takes the end of the “string" at the north pole and relocates it to the south pole, thus ruining the non-trivial topology and gapping out the ES. However, focusing on figure <ref>, one finds that at the special TRS point t^*=3 T/4, this string lassos around and straddles the two poles.The spinors show nicely how the gap in the FGS ES changes with time. The gap in the ES in the MZM&MPM phase is sensitiveto the degree in which the string connects the two poles during this winding. For example, at the special time t^*=3T/4, the wrapping arranges the FGS to pass over the north pole at some point k_π, and the gap in the ES is closed.In conclusion, we see that since the FGS ES is constructed from the time-dependent modes, the spectrum is sensitive to the micro-motion of the states, and thus, unlike the quasi-energy spectrum, cannot rely on TRS arguments that depend on integration over the full period. At most points during the drive TRS appears instantaneously broken, except at T/4 and 3T/4. The π modes in general will “flip" the spinor configuration, at k = 0, π, k^*, which can be understood by the typical behavior of skyrmions under a band inversion. For times away from the special TRS points, the FGS string on the Bloch-sphere is free to adjust to this flip by making a trivial loop on the sphere leaving |ℤ_0 - ℤ_π| windings. However at the two special TRS points (T/4,3T/4), the FGS string is forced to lie on the y-z great circle. Thus we have |ℤ_0 - ℤ_π| for one of the TRS points and |ℤ_0 + ℤ_π| for the other TRS point. Due to the nature of the drive chosen by us, the |ℤ_0 - ℤ_π| winding is preserved throughout the period. However other drives such as one that drives the pairing term, could break the invariant down to ℤ_2 away from the TRS points.§ ENTANGLEMENT OF PHYSICAL OR QUENCHED STATE We have mainly discussed the entanglement properties of the FGS.We will now discuss the entanglement of the physical state obtained from time-evolving under a quench switch on protocol of the periodic drive. The entanglement properties of the physical state are interesting to study as the ES will reveal topological features without the need of introducing physical edges. Moreover unitary time evolution from some initial state is much simpler than dissipative dynamics in the presence of a reservoir as the latter can create further complications in a driven system.We will show that the quench ES will largely diverge from the FGS due to the presence of resonances. Since the resonances involve a gap-closing process, the exact nature of the switch-on will not alter the main conclusions of this section, as gap-closings do not have an adiabatic limit.For the quench, the ES is determined from equations (<ref>) and (<ref>) and inserted into the Majorana correlation matrix, (<ref>). The results are shown in figures <ref>, <ref> for the ES and in figures <ref> and <ref> for the Schmidt states. Since we are only interested in the ES long times after the quench occurred, terms with overlaps between the FGS and FES will yield zero after taking the momentum integral. This is an example of dephasing that drives spatially extended quenched systems into a diagonal ensemble. This also signifies that the ES at long times has no knowledge of the quasi-energy spectrum and the time dependencies and topologies enter the ES through the time-periodic Floquet modes.First focusing on figures <ref> and <ref>, we find that the off-resonant drive is largely equivalent to what is in the FGS ES. There is an increase in the bulk excitations, but the entanglement gap is still open. The resonant portion of the drive is however qualitatively new in the following manner. The physical state fails to “acquire" the MPM state and in its place we have highly excited bulk states, as expected <cit.>. We expect for larger system sizes, the bulk states will completely fill the resonant portion of the ES, andthe entanglement gap will close. The edge states in <ref> shows the states in the resonant portion of the ES are completely delocalized. In contrast, the off-resonant portion of the ES are only slightly modified from the FGS ES as we have minimal bulk excitations in this portion of phase space.Shifting attention to figures <ref> and <ref>, the off-resonant region is largely the same as the FGS. The region where we have a MPM phase in the FGS ES, corresponds to highly excited bulk states and no MPM, just as in the Ω/t_h = 5 case discussed above. The new region is the MZM&MPM phase in the FGS. In the physical state the ES shows a central edge state surviving the quench, and the appearance of highly excited bulk states. Essentially, the physical state fails to acquire the π modes, so there is no issue of the MZM&MPM gapping itself out and ES maintains the original (pre-quench) MZM topology of the state. Looking at figure <ref>, the Schmidt states for those levels are indeed edge states and remain sharply defined throughout the drive. Even though the central MZM persists in the physical state, it is not topologically robust because the gap between the topological and bulk states in the MZM&MPM phase can become smaller in the limit of larger entanglement cuts.To summarize, we find that the topology of the physical state is that of the initial state before unitary time-evolution. However, these inherited edge modes have weaker topological protection due to nearby bulk excitations created when the drive is resonant. § DISCUSSIONWe investigated the entanglement properties of a periodically driven Kitaev chain through the use of the entanglement spectrum (ES). The goal was to better understand the topological features of the eigenstates of a driven system in a general way without resorting to making assumptions of thermal equilibrium occupation of the states. To this end we studied the entanglement properties of the exact eigenstate of the Floquet Hamiltonian, which we call the Floquet Ground State (FGS), and also that of a physical state arising from unitary time-evolution following a quench of the periodic drive.We made use of a Majorana correlation matrix in order to construct the reduced density matrix, and the corresponding entanglement Hamiltonian. We carried this out for three parent states, one was the ground state of the static Hamiltonian, the second was the FGS, and the third was the physical state. Both numerical and analytical arguments were presented, demonstrating the bulk-boundary correspondence that exists for the entanglement Hamiltonian, where the boundary is now that of a fictitious entanglement cut.While the ES provides topological information about the state in question, we found that the ℤ_0 ×ℤ_π topological index for the Floquet Hamiltonian does not carry over to the entanglement Hamiltonian. In particular the number of topological Majorana modes in the ES vary during the period of the drive in a way such thatat the two time reversal symmetric points of the drive (t^*=T/4, 3T/4 in our example), the number of Majorana modes in the ES are |ℤ_0 - ℤ_π| and |ℤ_0 + ℤ_π| respectively.The essential reasoning behind this breakdown is that the ES is not a periodic quantity and thus the topological states all reside at zero entanglement energy, while the ℤ×ℤ classification required an energy separation for the 0 and the π modes. Thus now one may couple some of the MZM and MPM modes in the ES, reducing the number of Majorana modes.We also considered NNN hopping in order to generate more edges states to further demonstrate this result. The topology of the Floquet ground state and the corresponding ES was made explicit through usage of Bloch-sphere diagrams showing that the FGS spinor has well defined winding only at t^* where the |ℤ_0 ±ℤ_π| Majorana modes appear. The persistence of |ℤ_0 - ℤ_π| away from t^* despite the lack of a well defined winding of the spinor, was because the periodic drive was applied to the chemical potential. Periodic driving to the pairing on the other hand will reduce the Majorana modes in the ES at times other than t^* to ℤ_2.When the MZM&MPM modes couple, one obtains complex or Dirac fermions with entanglement energies lifted away from zero,but with Schmidt states still localized at the entanglement cut. However unlike accidental edge Dirac fermions that can always occur even in the trivial phase, the ones that arise in the topologcal phase are protected becausethe Dirac fermions again have to uncouple into two Majorana fermions at the special TRS points t^* of the drive.The study of the ES of the physical state, and the corresponding Schmidt states show that the π modes do not appear in the ES. This is because they co-occur with large number of bulk excitations and thus hybridize with them. In contrast the MZMs are still visible, where they are inherited from the wavefunction before the quench <cit.>. However we expect that these modes lose their topological protection due to nearby bulk excitations created by the resonant laser <cit.>.In a study of the entanglement properties of the Floquet Chern Insulator, it was found that <cit.> for a physical state obtained from taking half-filled graphene, and time-evolving it with a circularly polarized periodic drive, chiral modes appeared in the ES of the physical state even though they are absent in the initial state. This is in contrast to what we find here, where only the topological modes of the initial state persist during the time-evolution. This difference is due to the fact that graphene, being a semi-metal, is in a topologically critical state, and has its own edge-states. 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"authors": [
"Daniel J. Yates",
"Aditi Mitra"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170527110647",
"title": "Entanglement properties of the time periodic Kitaev Chain"
} |
1]John F. R. DuncanEmail:2]Andrew O'DeskyEmail:[1]Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, U.S.A. [2]Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.MSC2010: 11F37, 11F50, 17B69, 17B81, 20C35.Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine [==================================================================== The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen,we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups. § INTRODUCTIONEguchi–Ooguri–Tachikawa initiated a new phase in moonshine with theirobservation <cit.> that representations of the largest sporadic Mathieu group M_24 are visible in themultiplicities of irreducible superconformal algebra modules in the K3 elliptic genus.The generating function of these multiplicities is a mock modular form H^(2) of weight 1/2 (cf. <cit.>).Once twined counterparts H^(2)_g for g∈ M_24 had been identified <cit.> and characterized <cit.>, Gannon was able to confirm <cit.> that there is a corresponding graded M_24-module, for which the q-series of H^(2)=H^(2)_e is the graded dimension.But so far there has been no explicit construction of this Mathieu moonshine module, such as might be compared to the vertex operator algebra of monstrous moonshine <cit.> that was discovered by Frenkel–Lepowsky–Meurman <cit.>, and used to prove the monstrous moonshine conjectures by Borcherds <cit.>. The purpose of this paper is to solve a closely related construction problem, for some closely related instances of moonshine.To motivate our approach we recall thecuriouscircumstance that the McKay–Thompson series H^(2)_gof Mathieu moonshine may be repackaged into modular forms of different kinds. Indeed,if χ^(2)_g is the number of fixed points of g∈ M_24 in the defining permutation representation on 24 points, then Z^(2)_g(τ,z):= χ^(2)_gμ_2,0(τ,z)/μ_1,0(τ,z) +H^(2)_g(τ)θ_1(τ,z)^2/η(τ)^3is a weak Jacobi form of weight 0, index 1, and some level depending on g (where μ_m,0 is defined in (<ref>), and θ_1 and η are recalled in <ref>).The g=e case of(<ref>) expresses the K3 elliptic genus in terms of H^(2), and is the starting point of <cit.>. This suggests that the Mathieu moonshine module might be realized in terms of a suitably chosen K3 sigma model, but it was found in <cit.> that the symmetries of these objects are precisely the subgroups of the automorphism group of the Leech lattice—i.e., the Conway group, _0≃ 2._1—that fix a 4-space. In particular, M_24 does not appear. Interestingly,it has been found <cit.> that suitable trace functions attached to the moonshine module for Conway's group (see <cit.>) attach weak Jacobi forms of weight 0 and index 1 (with level) to 4-space-fixing automorphisms of the Leech lattice, and this construction recovers the K3 elliptic genus when applied to the trivial symmetry.More generally, many—but not all—of the Z^(2)_g appear in this way. So the Conway moonshine module serves as a kind of “fake” Mathieu moonshine module, with a closer connection to K3 sigma models (cf. <cit.>) than to Mathieu moonshine itself.As an alternative to (<ref>) we may consider the functions ψ^(2)_g:=-μ_1,0Z^(2)_g, which are meromorphic Jacobi forms of weight 1 and index 2 (cf. <ref>). Although this is a simple manipulationit seems to be essential for umbral moonshine<cit.>, since in this more general setting weak Jacobi form formulations of the McKay–Thompson series are only known in some cases, whereas meromorphic Jacobi forms ψ^(ℓ)_g may be constructed in a uniform way (cf. 4 of <cit.>, or <ref> of this work). In umbral moonshinevector-valued mock modular forms H^(ℓ)_g=(H^(ℓ)_g,r) are associated to (outer) automorphisms of Niemeier lattices (i.e., self-dual even positive definite lattices of rank 24 with roots), and Mathieu moonshine is recovered by specializing to the Niemeier lattice whose root system is A_1^⊕ 24. It has been proven <cit.> that theH^(ℓ)_gdefine modules for the groups to which they are attached, but except for the case of the Niemeier lattice E_8^⊕ 3 (see <cit.>), no explicit constructions of the H^(ℓ)_g as traces on algebraic structures are known. An extension of the method of <cit.> apparently requires a finer knowledge of the relationship between mock modular forms and indefinite lattices than is currently available. So here we promote the alternative approach of focusing on the meromorphic Jacobi forms ψ^(ℓ)_g rather than the vector-valued mock modular forms H^(ℓ)_g. We call this the meromorphic module problem for umbral moonshine. In this work we solve the meromorphic module problem for the cases of umbral moonshine corresponding to the Niemeier lattices A_6^⊕ 4 and A_12^⊕ 2 (corresponding to ℓ=7 and ℓ=13, respectively), and provide partial solutions for A_3^⊕ 8 and A_4^⊕ 6 (corresponding to ℓ=4 and ℓ=5, respectively). We achieve this by considering suitable tensor products of simple free field super vertex operator algebras, equipping them with suitable bigradings, and identifying suitable trace functions on their canonically twisted modules. This approach is motivated by the fact that many of the corresponding meromorphic Jacobi forms admit product formulas (cf. <ref>). For A_6^⊕ 4 and A_12^⊕ 2 the corresponding umbral groups act naturally, and all the corresponding ψ^(ℓ)_g are realized explicitly (see Theorems <ref> and <ref>). For A_3^⊕ 8 and A_4^⊕ 6 we find actions of certain maximal subgroups of the corresponding umbral groups, and realize most, but not all, of the corresponding ψ^(ℓ)_g (see Propositions <ref> and <ref>). It will be interesting to see if a modification of the methods presented here can solve the meromorphic module problem completely for A_3^⊕ 8 and A_4^⊕ 6. We expect that that would yield some useful insight into the broader question of constructing meromorphic umbral moonshine modules in general.The structure of the article is as follows. In <ref> we briefly recall the Clifford module and Weyl module constructions of super vertex operator algebras and their canonically twisted modules. In <ref> we recall the relationship between the mock modular forms H^(ℓ)_g and the meromorphic Jacobi forms ψ^(ℓ)_g, for the Niemeier lattices with root system of the form A_ℓ-1^⊕ d (i.e., the cases that ℓ-1 is a divisor of 24). Our new results are Theorems <ref> and <ref>, and Propositions <ref> and <ref>. They appear in <ref>. In <ref> we present the character tables of the umbral groups G^(ℓ) for ℓ∈{4,5,7,13}, and for the relevant maximal subgroups in case ℓ∈{4,5}. In <ref> we recall the explicit expressions for the ψ^(ℓ)_g that were used for the purpose of proving the (abstract) module conjectures for umbral moonshine in <cit.>. These expressions play a role in the proofs ofour results in <ref>. In <ref> we recall the definitions of the characters χ^(ℓ)_g and χ̅^(ℓ)_g which appear in the formula that relates H^(ℓ)_g to ψ^(ℓ)_g (cf. <ref>).§ SUPER VERTEX ALGEBRAS We briefly review the Clifford module and Weyl module constructions of super vertex operator algebras, and their canonically twisted modules in this section.The umbral moonshine modules we present in <ref> will be realized as tensor products of these simple free field super vertex operator algebras.We refer the reader to <cit.> for background on vertex algebra theory.§.§ Clifford Modules Let a be a complex vector space and let · ,· be a non-degenerate symmetric bilinear form on a. The Clifford algebra associated to this data is a:=T(a)/I where T(a):= 1⊕a⊕a^⊗ 2⊕⋯ is the tensor algebra of a, and I is the ideal of T(a) generated by the expressions a⊗ a'+a'⊗ a - a,a' 1 for a,a'∈a. The composition of natural maps a→ T(a)→a is an embedding, so we may regard a as a subspace of a. Let 1 also denote the unit in a. A polarization of a is a vector space splitting a=a^+⊕a^- for which the summands a^± are isotropic for the given bilinear form. Given such a splitting (this requires a to be even if it is finite) the induced module a⊗_a^+ is irreducible for a, when a^+ is the sub algebra of a generated by 1 and a^+, andis the unique unital a^+-module such that a=0 for every a∈a^+.Henceforth assume that a is finite and even. For r∈1/2 let a(r) be a vector space isomorphic to a. Choose an isomorphism a→a(r) for each r, and denote it a↦ a(r). Define â:=⊕_n∈a(n+1/2) and â_:=⊕_n∈a(n), and extend · ,· to â and â_ by requiring that a(r),a'(r')= a,a'δ_r+r',0 for a,a'∈a and r,r'∈1/2. Choose a polarization a=a^+⊕a^- of a, and define polarizations of â and â_ by setting â^+:=⊕_n≥ 0 a(n+12), â^-:=⊕_n<0 a(n+12), â_^+:=a^+(0)⊕⊕_n>0 a(n), â_^-:=a^-(0)⊕⊕_n<0 a(n).The Clifford module super vertex algebra associated to a and · ,· is the unique super vertex algebra structure on A(a):=â⊗_â^+ such thatis the vacuum, and Y(a(-1/2),z)=∑_n∈ a(n+1/2)z^-n-1 for a∈a. Note that A(a) is simple.Define A(a)_:=â_⊗_â_^+_ (where _ is the unique unital â_^+-module such that u_=0 for u∈â_^+). Then there is a unique structure of canonically twisted A(a)-module on A(a)_ such that Y_(a(-1/2),z)=∑_n∈a(n)z^-n-1/2 for a∈a. If {a_i^±} is a basis for a^± such that a_i^∓,a_j^±=δ_i,j then ω:=1/2∑_i ( a_i^+(-32)a_i^-(-12) - a_i^+(-12)a_i^-(-32) )is a Virasoro element for A(a) with central charge c=1/2a that makes A(a) a super vertex operator algebra. Set :=∑_i a_i^+(-12)a_i^-(-12). Write J(n) for the coefficient of z^-n-1 in Y(,z) or Y_(,z), and write L(n) for the coefficient of z^-n-2 in Y(ω,z) or Y_(ω,z). ThenJ(0) and L(0) commute, and act semisimply on A(a) and A(a)_, with finite-dimensional (simultaneous) eigenspaces. For the corresponding bigraded dimensions we have (y^J(0)q^L(0)-c/24|A(a))=q^-d/48∏_n>0(1+y^-1q^n-1/2)^d/2(1+yq^n-1/2)^d/2, (y^J(0)q^L(0)-c/24|A(a)_)=y^d/4q^d/24∏_n>0(1+y^-1q^n-1)^d/2(1+yq^n)^d/2,when d=a. Note that ω does not depend upon the choice of polarization a=a^+⊕a^-, butdoes.The group (a^+) acts naturally on A(a) and A(a)_, preserving ω and . For g∈(a^+) there is a unique g'∈(a^-) such that ga,g'a'= a,a' for all a∈a^+ and a'∈a^-. Abusing notation slightly, we write g also for the linear automorphism (g,g') on a=a^+⊕a^-. Then the action of (a^+) on A(a) is given by g· a_1(r_1)… a_n(r_n):=(ga_1)(r_1)… (ga_n)(r_n) for a_i∈a and r_i∈+1/2, and similarly for A(a)_. We then have Y(gu,z)gv=gY(u,z)v and Y_(gu,z)gw=gY_(u,z)w for u,v∈ A(a) and w∈ A(a)_, and also gω=ω and g=, so the bigradings of A(a) and A(a)_ are preserved.§.§ Weyl Modules The Weyl module construction runs in parallel with that of the previous section, but with an anti-symmetric bilinear form in place of a symmetric one. So let b be a complex vector space and let · ,· be a non-degenerate anti-symmetric bilinear form on b. The Weyl algebra associated to this data is b:=T(b)/I whereI is the ideal of T(b) generated by b⊗ b'-b'⊗ b - b,b' 1 for b,b'∈b. Just as for Clifford algebras we may naturally identify b as a subspace of b, and we write 1 also for the unit in b.Given a polarization b=b^+⊕b^- (so that b^± is isotropic for · ,·), the induced module b⊗_b^+ is irreducible for b. Assume now that b is finite. This forces b to be even.Define b̂:=⊕_n∈b(n+1/2) and b̂_:=⊕_n∈b(n), just as in the previous section, and extend · ,· to b̂ and b̂_ by requiring that b(r),b'(r')= b,b'δ_r+r',0 for b,b'∈b and r,r'∈1/2. Choose a polarization b=b^+⊕b^- of b, and define polarizations of b̂ and b̂_ by setting b̂^+:=⊕_n≥ 0 b(n+12), b̂^-:=⊕_n<0b(n+12), b̂_^+:=b^+(0)⊕⊕_n> 0b(n), b̂_^-:=b^-(0)⊕⊕_n< 0b(n).The Weyl module super vertex algebra associated to b and · ,· is the unique super vertex algebra structure on (b):=b̂⊗_b̂^+ such thatis the vacuum, and Y(b(-1/2),z)=∑_n∈ b(n+1/2)z^-n-1 for b∈b. Define (b)_:=b̂_⊗_b̂_^+_.Then there is a unique structure of canonically twisted (b)-module on (b)_ such that Y_(b(-1/2),z)=∑_n∈b(n)z^-n-1/2 for b∈b. If {b_i^±} is a basis for b^± such that b_i^∓,b_j^±=±δ_i,j then ω:=1/2∑_i ( b_i^+(-32)b_i^-(-12) - b_i^+(-12)b_i^-(-32) )is a Virasoro element for (b), with central charge c=-1/2b, that makes (b) a super vertex operator algebra. Note that although (b) is simple and C_2-cofinite, it is not rational (cf. <cit.>). Set :=∑_i b_i^+(-12)b_i^-(-12). Write J(n) for the coefficient of z^-n-1 in Y(,z) or Y_(,z), and write L(n) for the coefficient of z^-n-2 in Y(ω,z) or Y_(ω,z). Then, just as in the Clifford case,J(0) and L(0) commute, and act semisimply on (b) and (b)_, with finite-dimensional (simultaneous) eigenspaces. For the corresponding bigraded dimensions we have (y^J(0)q^L(0)-c/24|(b)) =q^d/48∏_n>0(1-y^-1q^n-1/2)^-d/2(1-yq^n-1/2)^-d/2, (y^J(0)q^L(0)-c/24|(b)_) =y^-d/4q^-d/24∏_n>0(1-y^-1q^n-1)^-d/2(1-yq^n)^-d/2,when d=b. Note that (1-X)^-1 should be interpreted as ∑_n≥ 0X^n in (<ref>) and (<ref>).Similar again to the Clifford case, the group (b^+) acts naturally on (b) and (b)_, preserving their bigradings. Explicitly, for g∈(b^+) write g also for the linear automorphism (g,g') on b=b^+⊕b^-, where g'∈(b^-) is determined by requiring gb,g'b'= b,b' for all b∈b^+ and b'∈b^-.The action of (b^+) on (b) is given by g· b_1(r_1)… b_n(r_n):=(gb_1)(r_1)… (gb_n)(r_n) for b_i∈b and r_i∈+1/2, and similarly for (b)_. Vertex operators are preserved by this action, as are ω and , just as in <ref>. § MEROMORPHIC JACOBI FORMSWe briefly review the relationship between meromorphic Jacobi forms and the mock modular forms of umbral moonshine in this section.The original reference for this is 4 of <cit.>. We refer the reader to <cit.> for more detailed and more general discussions of mock modular forms, mock Jacobi forms andmeromorphic Jacobi forms. Let X be a Niemeier root system. For simplicity we restrict to the pure type A case that X=A_m-1^⊕ d for some integer m>1 such that m-1 is a divisor of 24, and d:=24/m-1. Let N^(m) be the corresponding Niemeier lattice, and set G^(m):=(N^(m))/(N^(m)) where (N^(m)) is the subgroup of (N^(m)) generated by reflections in root vectors. Then umbral moonshine <cit.> attaches a 2m-vector-valued mock modular form H^(m)_g(τ)=(H^(m)_g,r(τ))_r 2m to each g∈ G^(m). One way to explain what this means is as follows. Let :={τ∈|(τ)>0} denote the upper half-plane, and set S:={(τ,aτ+b)∈×| a,b∈}. Define functions μ_m,0^k on ×∖ S for k 2 by setting μ_m,0^k(τ,z):=1/2(μ_m,0(τ,z)+(-1)^kμ_m,0(τ,z+1/2)), where μ_m,0(τ,z):=∑_ℓ∈y^2mℓq^mℓ^2yq^ℓ+1/yq^ℓ-1for y=e^2π i z and q=e^2π iτ. Also define θ_m,r(τ,z):=∑_ℓ=r 2my^ℓ q^ℓ^2/4m for r 2m. Then for χ̅_g^(m) and χ^(m)_g the characters of G^(m) defined in B.2 of <cit.> or <cit.> (or C of this work, for m∈{4,5,7,13}), the function ψ^(m)_g(τ,z):= -χ^(m)_gμ_m,0^0(τ,z) -χ̅^(m)_gμ_m,0^1(τ,z) +∑_r 2mH^(m)_g,r(τ)θ_m,r(τ,z)is a meromorphic Jacobi form with simple poles in τ+1/2. That is to say, we have ψ^(m)_g=ϕ_1/ϕ_2 for some (holomorphic) Jacobi forms ϕ_1 and ϕ_2, and for any fixed τ∈, the function z↦ψ^(m)_g(τ,z) is meromorphic on . Moreover, its poles are simple, and lie within the lattice τ+1/2. In the next section we will recover series expansions of the functions ψ^(m)_g as traces on twisted modules for explicitly constructed super vertex algebras, for all g∈ G^(m) for m=7 (see <ref>) and m=13 (see <ref>), and for all g in a maximal subgroup of G^(m) for m=4 (see <ref>) and m=5 (see <ref>). This will solve the meromorphic module problem for umbral moonshine for the root systems A_6^⊕ 4 and A_12^⊕ 2, and partially solve it for A_3^⊕ 8 and A_4^⊕ 6.§ MOONSHINE MODULESWe now present our main constructions.§.§ Lambency SevenLet e and a be 2-dimensional complex vector spaces equipped with non-degenerate symmetric bilinear forms, and letb be a 4-dimensional complex vector space equipped with a non-degenerate anti-symmetric bilinear form.Fix polarizations e=e^+⊕e^-, a=a^+⊕a^- and b=b^+⊕b^-, and let {e^±}, {a^±} and {b_i^±} be bases for e^±, a^± and b^±, respectively, such that e^-,e^+= a^-,a^+=1 and b_i^-,b_j^+=δ_i,j.Applying the constructions of <ref> we obtain a super vertex operator algebra W^(7), and a canonically twisted module for it W^(7)_ by settingW^(7) :=A(e)⊗ A(a) ⊗(b), W^(7)_ :=A(e)_⊗ A(a)_ ⊗(b)_,and equipping W^(7) with the usual tensor product Virasoro element ω^(7):=ω⊗⊗+⊗ω⊗+⊗⊗ω. To define bigradings on both spaces we set ^(7):=4⊗⊗+⊗⊗whereis defined for A(a) and (b) as in <ref>. . We also define _e:=⊗⊗. Then (e^+)⊗(a^+)⊗(b^+) acts naturally on W^(7) and W^(7)_, respecting the super vertex operator algebra module structures and preserving the bigradings.The character table of the umbral group G^(7) is Table <ref> in <ref>. Choose homomorphisms ϱ:G^(7)→(a^+) and :G^(7)→(b^+) such that the corresponding characters are χ_2 and χ_6 in Table <ref>, respectively.Since χ_6 is faithful the assignment g↦ I⊗ϱ(g) ⊗(g) defines faithful actions of G^(7) on W^(7) and W^(7)_. Set (-1)^F:=(-I)⊗(-I)⊗ I, and let J_e(0) denote the coefficient of z^-1 in Y_(_e,z). Let J(0) be the coefficient of z^-1 in Y_(^(7),z), and let L(0) be the coefficient of z^-2 in Y_(ω^(7),z). For g∈ G^(7) define a formal series ψ^(7)_g∈[y][[y^-1]][[q]] by settingψ^(7)_g:=-((g+g^-1)J_e(0)(-1)^Fy^J(0)q^L(0)|W^(7)_).For g∈ G^(7) the series ψ^(7)_g is the expansion of ψ^(7)_g in the domain 0<-(z)<(τ).Let g∈ G^(7). The action of g on a^+ is multiplication by a scalar, λ say, and there are a pair of eigenvalues {_1,_2} for its action on b^+. With this notation we haveψ^(7)_g=-y∏_n>0(1-q^n)^2(1-λ̅y^-4q^n-1)(1-λ y^4q^n)/∏_j=1^2(1-_jy^-1 q^n-1)(1- _j y q^n)-y ∏_n>0(1-q^n)^2(1- λ y^-4q^n-1)(1- λ̅y^4q^n)/∏_j=1^2(1-_jy^-1 q^n-1)(1- _j y q^n)where (1-X)^-1 is shorthand for ∑_k≥ 0X^k. This series converges in the given domain once we substitute q=e^2π i τ and y=e^2π i z, so we require to check that the right-hand side of (<ref>) agrees with the meromorphic Jacobi form ψ^(7)_g when viewed as a function of τ and z. This follows from a case by case check using the values of λ and _j in Table <ref> and the explicit descriptions of the ψ^(7)_g in (<ref>). For example, for g∈ 4A the right-hand side of (<ref>) becomes-2y∏_n>0(1-q^n)^2(1-y^-4q^n-1)(1- y^4q^n)/(1+ y^-2 q^2n-2)(1+y^2 q^2n) =-2iη(2τ)η(τ)θ_1(τ,4z)/θ_2(2τ,2z)whichis exactly the expression for ψ^(7)_4A that appears in (<ref>). The other cases are similar.§.§ Lambency Thirteen Let e and a be 2-dimensional complex vector spaces equipped with non-degenerate symmetric bilinear forms, and letb and b' be 2-dimensional complex vector spaces equipped with non-degenerate anti-symmetric bilinear forms.Fix polarizations e=e^+⊕e^-, a=a^+⊕a^-, b=b^+⊕b^- and b'=b'^+⊕b'^-, and let {e^±}, {a^±}, {b^±} and {b'^±} be bases for e^±, a^±, b^± and b'^±, respectively, such that e^-,e^+= a^-,a^+= b^-,b^+=b'^-,b'^+=1.Define a super vertex operator algebra W^(13), and a canonically twisted W^(13)-module W^(13)_ by settingW^(13) :=A(e)⊗ A(a) ⊗(b)⊗(b'), W^(13)_ :=A(e)_⊗ A(a)_ ⊗(b)_⊗(b')_.Equip W^(13) with the usual tensor product Virasoro element, denote it ω^(13),set _e:=⊗⊗⊗, and set ^(13):=6⊗⊗⊗+⊗⊗⊗+3⊗⊗⊗.Then (e^+)⊗(a^+)⊗(b^+)⊗(b'^+) acts naturally on W^(13) and W^(13)_ respecting the super vertex operator algebra module structures and preserving the bigradings. Theumbral group G^(13) is cyclic of order 4. (Cf. Table <ref>.) Define compatible actions of G^(13) on W^(13) and W^(13)_ by choosing a generator and mapping it to I⊗ (-I)⊗ (iI)⊗ (-iI) in (e^+)⊗(a^+)⊗(b^+)⊗(b'^+). Similar to <ref> we set (-1)^F:=(-I)⊗(-I)⊗ I,let J_e(0) denote the coefficient of z^-1 in Y_(_e,z), let J(0) be the coefficient of z^-1 in Y_(^(13),z), and let L(0) be the coefficient of z^-2 in Y_(ω^(13),z). Then for g∈ G^(13) we define a formal seriesin [y][[y^-1]][[q]] by settingψ^(13)_g:=-((g+g^-1)J_e(0)(-1)^Fy^J(0)q^L(0)|W^(13)_).For g∈ G^(13) the series ψ^(13)_g is the expansion of ψ^(13)_g in the domain 0<-(z)<(τ).Let g∈ G^(13). Then g acts by scalar multiplication on a^+, b^+ and b'^+. Let λ,andbe the respective scalars. Then we haveψ^(13)_g= -y∏_n>0 (1-q^n)^2(1-λ̅y^-6 q^n-1)(1-λ y^6 q^n)/ (1- y^-1q^n-1)(1-yq^n) (1-y^-3q^n-1)(1- y^3q^n) -y ∏_n>0 (1-q^n)^2(1-λ y^-6 q^n-1)(1-λ̅y^6 q^n)/ (1- y^-1q^n-1)(1- yq^n) (1- y^-3q^n-1)(1- y^3q^n)where, as before, (1-X)^-1 is shorthand for ∑_k≥ 0X^k. This convergence of this series, upon substitutingq=e^2π i τ and y=e^2π i z, is the same as in Theorem <ref>. So we just need to check that the right-hand side of (<ref>) agrees with the meromorphic Jacobi form ψ^(13)_g when viewed as a function of τ and z. This follows from a case by case comparison with (<ref>). For example, for g the involution in G^(13) the right-hand side of (<ref>) becomes-2y∏_n>0(1-q^n)^2(1-y^-6q^n-1)(1- y^6q^n)/(1+ y^-1 q^n-1)(1+y q^n)(1+ y^-3 q^n-1)(1+y^3 q^n)= -2iη(τ)^3θ_1(τ,6z)/θ_2(τ,z)θ_2(τ,3z)whichis precisely ψ^(13)_2A as it appears in (<ref>). We leave the remaining cases to the reader.§.§ Lambency Four This section and the next are similar to the previous two, except that we realize umbral moonshine only for maximal subgroups G^(4)_336 and G^(5)_24 of the umbral groups G^(4) and G^(5). For ℓ=4 let e be just as in <ref>,<ref>, let a be a 6-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form, and let b be an 8-dimensional complex vector space equipped with a non-degenerate anti-symmetric bilinear form. Choose polarizations e=e^+⊕e^-, a=a^+⊕a^- and b=b^+⊕b^-, and let {e^±}, {a^±_i} and {b_i^±} be bases for e^±, a^± and b^±, respectively, such that e^-,e^+=1 and a_i^-,a_j^+= b_i^-,b_j^+=δ_i,j. Define a super vertex operator algebra and a canonically twisted module for it by settingW^(4) :=A(e)⊗ A(a) ⊗(b), W^(4)_ :=A(e)_⊗ A(a)_ ⊗(b)_,and let ω^(4) denote the (tensor product) Virasoro element for W^(4).Set _e:=⊗⊗ and ^(4):=2⊗⊗+⊗⊗. As in <ref>, the group (e^+)⊗(a^+)⊗(b^+) acts naturally on W^(4) and W^(4)_, respecting the super vertex operator algebra module structures and preserving the bigradings defined by the zero modes of ω^(4) and ^(4). We write G^(4)_336 for a subgroup of G^(4) isomorphic to _2(7). Such subgroups are maximal and unique up to conjugacy, but note that there are two other conjugacy classes of maximal subgroups of order 336. The character tables of G^(4) and G^(4)_336 are Tables <ref> and <ref>, respectively. Table <ref> also gives the fusion of conjugacy classes with respect to an embedding ι: G^(4)_336→ G^(4). Choose homomorphisms ϱ:G^(4)_336→(a^+) and :G^(4)_336→(b^+) such that the corresponding characters are χ_2 and χ_8 in Table <ref>, respectively.Then the assignment g↦ I⊗ϱ(g) ⊗(g) defines faithful and compatible actions of G^(4)_336 on W^(4) and W^(4)_. Define (-1)^F, J_e(0), J(0) and L(0) just as in <ref>,<ref>. For g∈ G^(4)_336 we consider the formal series ψ^(4)_g defined in direct analogy with (<ref>) and (<ref>),ψ^(4)_g:=-((g+g^-1)J_e(0)(-1)^Fy^J(0)q^L(0)|W^(4)_).For g∈ G^(4)_336 the series ψ^(4)_g is the expansion of ψ^(4)_g in the domain 0<-(z)<(τ).The proof is very similar to that of Theorem <ref>. Let g∈ G^(4)_336. Let {λ_i} and {_j} be the eigenvalues for the actions of g on a^+ and b^+, respectively. Then we haveψ^(4)_g=-y∏_n>0(1-q^n)^2∏_i=1^3(1-λ̅_i y^-2q^n-1)(1- λ_i y^2q^n)/∏_j=1^4(1-_j y^-1 q^n-1)(1-_j y q^n)-y ∏_n>0(1-q^n)^2∏_i=1^3(1-λ_i y^-2q^n-1)(1- λ̅_i y^2q^n)/∏_j=1^4(1-_j y^-1 q^n-1)(1-_j y q^n).As in the proof of Theorem <ref> we just require to check that the right-hand side of (<ref>) agrees with the meromorphic Jacobi form ψ^(4)_g when viewed as a function of τ and z, and we achieve this by comparing in each case the values of λ_i and _j in Table <ref> with the explicit descriptions of the ψ^(4)_g in (<ref>). §.§ Lambency Five Let e, a and a' be 2-dimensional complex vector spaces equipped with non-degenerate symmetric bilinear forms, and letb be a 6-dimensional complex vector space equipped with a non-degenerate anti-symmetric bilinear form.Fix polarizations e=e^+⊕e^-, a=a^+⊕a^-, a'=a'^+⊕a'^- and b=b^+⊕b^-, and let {e^±}, {a^±}, {a'^±} and {b^±_i} be bases for e^±, a^±, a'^± and b^±, respectively, such that e^-,e^+= a^-,a^+=a'^-,a'^+=1 and b^-_i,b^+_j=δ_i,j. Similar to (<ref>), (<ref>) and (<ref>) we define a super vertex operator algebraand a canonically twistedmodule for it by settingW^(5) :=A(e)⊗ A(a) ⊗ A(a') ⊗(b), W^(5)_ :=A(e)_⊗ A(a)_ ⊗ A(a')_ ⊗(b')_.Equip W^(5) with the usual tensor product Virasoro element,set _e:=⊗⊗⊗ and^(5):=2⊗⊗⊗+3⊗⊗⊗+⊗⊗⊗. Then (e^+)⊗(a^+)⊗(a'^+)⊗(b^+) acts naturally on W^(5) and W^(5)_, respecting the super vertex operator algebra module structures and preserving the bigradings defined by the Virasoro element and ^(5).Thereis a unique conjugacy class of maximal subgroups of G^(5) with order 24. We choose a subgroup in this class, denote it G^(5)_24, and let g↦ι g denote the inclusion G^(5)_24→ G^(5). The character tables of G^(5) and G^(5)_24 are Tables <ref> and <ref>, respectively, and Table <ref> gives the fusion of conjugacy classes under g↦ι g. Choose homomorphisms ϱ:G^(5)_24→(a^+), ϱ':G^(5)→(a'^+) and :G^(5)→(b^+) such that the corresponding characters are χ_2, χ_3 and χ_7+χ_12 in Table <ref>, respectively. Define(-1)^F, J_e(0), J(0) and L(0) as in <ref>-<ref>, and to g∈ G^(5)_24 attach the formal series ψ^(5)_g:=-((g+g^-1)J_e(0)(-1)^Fy^J(0)q^L(0)|W^(5)_). For g∈ G^(5) the series ψ^(5)_g is the expansion of ψ^(5)_g in the domain 0<-(z)<(τ). The proof is directly similar to the proofs of Theorems <ref> and <ref>, and Proposition <ref>, and depends upon a verification that the natural product representations of the ψ^(5)_g for g∈ G^(5)_24 coincide with the meromorphic Jacobi forms ψ^(5)_g given in (<ref>). For the convenience of the reader we present the eigenvalues arising from the representations ϱ, ϱ' andin Table <ref>. § ACKNOWLEDGEMENTSThe authors thank Matthias Gaberdieland Miranda Cheng for comments, and discussions on closely related topics.J.D. gratefully acknowledges support from the Simons Foundation (#316779), and the U.S. National Science Foundation (DMS 1203162, DMS 1601306). § CHARACTER TABLES Here we give character tablesfor the groups thatappear in <ref>. We use the abbreviations a_n:=√(-n), b_n:=(-1+√(-n))/2 and r_n:=√(n). § UMBRAL JACOBI FORMS Here we recall from B of <cit.> the meromorphic Jacobi forms associated to the groups that we construct in <ref>. Some of these expressions were obtained earlier in <cit.>. To present the formulas we use the Dedekind eta and Jacobi theta functions,η(τ):=q^1/24∏_n>0(1-q^n), θ_1(τ,z):=-iq^1/8y^1/2∏_n>0(1-y^-1q^n-1)(1-yq^n)(1-q^n), θ_2(τ,z):=q^1/8y^1/2∏_n>0(1+y^-1q^n-1)(1+yq^n)(1-q^n),where q=e^2π i τ and y=e^2π i z. In what follows, the subscript in ψ^(ℓ)_nZ names a conjugacy class in the umbral group G^(ℓ), where the labelling of the conjugacy classes is as defined by the character tables in <ref>. For the cases that ℓ=4 and ℓ=5 we only recall formulas for the conjugacy classes that are represented by elements of the groups G^(4)_336 and G^(5)_24 (cf. <ref> and <ref>). ψ^(7)_1A(τ,z) :=2iη(τ)^3θ_1(τ,4z)/θ_1(τ,z)^2 ψ^(7)_2A(τ,z) :=-2iη(τ)^3θ_1(τ,4z)/θ_2(τ,z)^2 ψ^(7)_4A(τ,z) :=-2iη(τ)η(2τ)θ_1(τ,4z)/θ_2(2τ,2z) ψ^(7)_3A(τ,z) := -iη(3τ)/θ_1(3τ,3z)×(θ_1(τ,4z+13)θ_1(τ,z-13) +θ_1(τ,4z-13)θ_1(τ,z+13)) ψ^(7)_6A(τ,z) := -iη(3τ)/θ_2(3τ,3z)×(θ_1(τ,4z+13)θ_1(τ,z-16) +θ_1(τ,4z-13)θ_1(τ,z+16)) ψ^(13)_1A(τ,z) :=2iη(τ)^3θ_1(τ,6z)/θ_1(τ,z)θ_1(τ,3z) ψ^(13)_2A(τ,z) :=-2iη(τ)^3θ_1(τ,6z)/θ_2(τ,z)θ_2(τ,3z) ψ^(13)_4AB(τ,z) := -iη(2τ)^2θ_2(τ,6z)/η(τ)θ_2(2τ,2z)θ_2(2τ,6z)×(θ_1(τ,z+14)θ_1(τ,3z+14) -θ_1(τ,z-14)θ_1(τ,3z-14))ψ^(4)_1A :=2iη(τ)^3θ_1(τ,2z)^3/θ_1(τ,z)^4 ψ^(4)_2A :=2iη(τ)^3θ_1(τ,2z)^3/θ_2(τ,z)^4 ψ^(4)_4A :=-2iη(2τ)^2θ_1(τ,2z)θ_2(τ,2z)^2/η(τ)θ_2(2τ,2z)^2 ψ^(4)_3A :=2iη(τ)^3θ_1(3τ,6z)/θ_1(τ,z)θ_1(3τ,3z) ψ^(4)_6A :=-2iη(τ)^3θ_1(3τ,6z)/θ_2(τ,z)θ_2(3τ,3z) ψ^(4)_8A :=-2iη(τ)η(4τ)θ_1(τ,2z)θ_2(2τ,4z)/η(2τ)θ_2(4τ,4z) ψ^(4)_7AB :=-iη(7τ)/η(τ)^4θ_1(7τ,7z)× ( θ_1(τ,2z+17) θ_1(τ,2z+27)θ_1(τ,2z+47) θ_1(τ,z-17)θ_1(τ,z-27)θ_1(τ,z-47) +θ_1(τ,2z- 17)θ_1(τ,2z-27)θ_1(τ,2z-47) θ_1(τ,z+17)θ_1(τ,z+27)θ_1(τ,z+47) ) ψ^(4)_14AB :=iη(7τ)/η(τ)^4θ_2(7τ,7z)× ( θ_1(τ,2z+17) θ_1(τ,2z+27)θ_1(τ,2z+47)θ_2(τ,z-17)θ_2(τ,z-27)θ_2(τ,z-47) +θ_1(τ,2z- 17)θ_1(τ,2z-27)θ_1(τ,2z-47)θ_2(τ,z+17)θ_2(τ,z+27)θ_2(τ,z+47))ψ^(5)_1A(τ,z) :=2iη(τ)^3θ_1(τ,2z)θ_1(τ,3z)/θ_1(τ,z)^3 ψ^(5)_2A(τ,z) :=-2iη(τ)^3θ_1(τ,2z)θ_2(τ,3z)/θ_2(τ,z)^3 ψ^(5)_2B(τ,z) :=-2iη(τ)^3θ_1(τ,2z)θ_1(τ,3z)/θ_1(τ,z)θ_2(τ,z)^2 ψ^(5)_2C(τ,z) :=2iη(τ)^3θ_1(τ,2z)θ_2(τ,3z)/θ_1(τ,z)^2θ_2(τ,z) ψ^(5)_3A(τ,z) :=-2iη(3τ)θ_1(τ,2z)θ_1(τ,3z)/θ_1(3τ,3z) ψ^(5)_6A(τ,z) :=-2iη(3τ)θ_1(τ,2z)θ_2(τ,3z)/θ_2(3τ,3z) ψ^(5)_4AB(τ,z) := -iη(2τ)^2θ_2(τ,2z)/η(τ)θ_2(2τ,2z)^2×(θ_1(τ,z+14)θ_1(τ,3z+14)-θ_1(τ,z-14)θ_1(τ,3z-14)) ψ^(5)_12AB(τ,z) := iη(6τ)θ_2(τ,2z)/η(τ)^3θ_2(6τ,6z)×(θ_1(τ,z+112)θ_1(τ,z+14)θ_1(τ,z+512)θ_1(τ,3z-14) - θ_1(τ,z-112)θ_1(τ,z-14)θ_1(τ,z-512)θ_1(τ,3z+14)) § EULER CHARACTERS Here we tabulate the character values χ̅^(ℓ)_g and χ^(ℓ)_gfor each g∈ G^(ℓ), for each ℓ∈{4,5,7,13}. tocsectionReferences GHV10b[Abe07]MR2274534 Toshiyuki Abe. A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra. Math. Z., 255(4):755–792, 2007.[Bor92]MR1172696 Richard E. Borcherds. Monstrous moonshine and monstrous Lie superalgebras. Invent. Math., 109(2):405–444, 1992.[CD12]Cheng2011 Miranda C. N. Cheng and John F. R. Duncan. On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine. Commun. Number Theory Phys., 6(3):697–758, 2012.[CDH14a]UM Miranda C. N. Cheng, John F. R. Duncan, and Jeffrey A. Harvey. Umbral Moonshine. Commun. Number Theory Phys., 8(2):101–242, 2014.[CDH14b]MUM Miranda C. N. Cheng, John F. R. Duncan, and Jeffrey A. Harvey. Umbral Moonshine and the Niemeier Lattices. Research in the Mathematical Sciences, 1(3):1–81, 2014.[CDH17]mumcor M. C. N. Cheng, J. F. R. Duncan, and J. A. Harvey. Weight One Jacobi Forms and Umbral Moonshine. J. Phys. A: Math. Theor. 51 (2018) 104002.[CDR17]2017arXiv170403678C T. Creutzig, J. F. R. Duncan, and W. Riedler. 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Soc., 11(3):352–353, 1979.[TW17]2017arXiv170403813T A. Taormina and K. Wendland. The Conway Moonshine Module is a Reflected K3 Theory. ArXiv e-prints, April 2017. | http://arxiv.org/abs/1705.09333v2 | {
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[email protected] Universidade Federal do Pampa Rua Luiz Joaquim de Sá Brito, s/n, Promorar,Itaqui - RS, 97650-000, [email protected] de Física Teórica e ComputacionalUniversidade Cruzeiro do Sul – Rua Galvão Bueno 868 São Paulo, SP, Brazil, [email protected] Institutode Física Teórica–Universidade Estadual Paulista R. Dr. Bento Teobaldo Ferraz 271, Barra FundaSão Paulo - SP, 01140-070, [email protected] Física Teórica–Universidade Estadual Paulista R. Dr. Bento Teobaldo Ferraz 271, Barra FundaSão Paulo - SP, 01140-070, BrazilWe consider an extension of the standard model with three Higgs doublet model and S_3×ℤ_2 discrete symmetries. Two of the scalar doublets are inert due to the ℤ_2 symmetry. We have calculated all the mass spectra in the scalar and lepton sectors and accommodated the leptonic mixing matrix as well. We also show that the model has scalar and pseudoscalar candidates to dark matter. Constraints on the parameters of the model coming from the decay μ→ eγ were considered and we found signals between the current and the upcoming experimental limits, and from that decay we can predictthe one-loop μ→ eee̅ channel. 14.60.Pq,14.60.St,13.35.BvLepton masses and mixing in a scotogenic model V. Pleitez 05/26/17 ==============================================§ INTRODUCTIONAlthough since 2012 we know that there exist a neutral spin-0 resonance with properties (mass and couplings) that are compatible, within the experimental error, with the Higgs boson of the standard model (SM) <cit.>, the data do not exclude the existence of more scalar fields and almost all extensions of the SM include extra Higgs doublets. This is the reason for considering multi-Higgs models. Moreover, although many scalar doublets may exist in nature, it is possible that only one of them is the responsible for the electroweak spontaneous symmetry breaking and the generation of the charged fermion masses. In this case, the other scalar multiplets may be inert ones: they do not couple to fermions, do not contribute to the vector bosons masses, and interact only with vector bosons and with other scalars. This possibility was put forward many years ago in Ref. <cit.> in which a ℤ_2 symmetry was imposed to keep inert one of the doublets.On the other hand, an indication that there must be physics beyond the SM is the origin of the neutrino masses. In fact, the generation of masses smaller than 0.1 eV demand the introduction of new degrees of freedom even in the context of the gauge symmetries of the SM. An interesting possibility is that the extra scalar fields that may exist as an extension of the SM also induce the appropriate neutrino mass. In particular, neutrino mass generation in a model with two doublets, being one of them inert, was considered by Ma <cit.>. This is the so called scotogenic mechanism for generating neutrino masses through one-loop corrections involving the inert neutral components.One interesting feature of the mechanism is that it includes by construction one or more dark matter (DM) candidates, or the implementation of the baryon asymmetry in the Universe, relating in this way three of the more important problems in elementary particle physics: the generation of the neutrino masses, the nature of the DM, and the observed asymmetry between matter and anti-matter, see Ref. <cit.> and references therein. Moreover, the existence of many components of DM is interesting by their own. In this case DM may decay from heavier to lighter components, and also co-annihilate in two dark particles <cit.>. In fact, this might be the only possibility to accommodate several astrophysical observations. For instance, the electron/positron excesses observed by many experiments and recently confirmed by AMS-02 <cit.> and the excess of gamma rays peaking at energies of several GeVfrom the region surrounding the Galactic Center <cit.> must need at least two component DM. Moreover, the latter case can avoid some constraints on the one component DM from the AMS-02 data <cit.>. Notwithstanding there are alternative interpretations for these gamma rays excess, see <cit.>. It is interesting that the one-doublet inert model has at least two DM components. However see <cit.>.Here we will work out a similar mechanism but in the context of the model with two inert scalar doublets proposed in Ref. <cit.>. Moreover, the inert character is due to the ℤ_2 symmetry, and the S_3 symmetry makes the scalar potential more predictive and easier to be analyzed. Although three right-handed neutrinos are introduced, the active neutrino masses do not arise through the type-I seesaw mechanism but via the scotogenic mechanism, at the 1-loop level. We need to add two real singlet scalar fields in order to accommodate the charged lepton masses and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix. In the context of inert doublet models, the latter issue is to the best of our knowledge done for the first time.The outline of this paper is as follows. In the next section we discuss the model while in Sec. <ref> analyse the scalar sector. Lepton mass matrices and the leptonic mixing matrix is shown in Sec. <ref>. In Sec. <ref> we show that the model provides a multi-component DM spectra, although we do not consider the most general case. In Sec. <ref> we consider the decays μ→ eγ, Subsec. <ref>, and μ→ ee̅e in Subsec. <ref>.Our conclusions appear in Sec. <ref>. § THE MODELIn Ref. <cit.> it was proposed an extension of the electroweak SM with three Higgs scalars H_1,2,3 transforming as doublets under SU(2) and having Y=+1.One H_1≡ S transforms as singlet of S_3, and the others asdoublets, D=(H_2,H_3) ≡ (D_1,D_2). Here we will extend the model of Ref. <cit.> by adding three right-handed sterile neutrinos, N_1R transforming as singlet,and N_d=(N_2R,N_3R) transforming as doublets of S_3, and two real scalar singlets of SU(2) (Y=0) but doublets of S_3, ζ_d=(ζ_1,ζ_2). See the other quantum numbers in Table <ref>. The vacuum alignment is given by ⟨ S ⟩=v_SM/√(2), and⟨ D_1,D_2⟩=0, ⟨ζ_1,2⟩=v_ζ. In the charged lepton and quark sectors, all usual fields of SM transform as singlet under S_3.With these fields the Yukawa interactions in the lepton sector, invariant under the gauge, S_3 and ℤ_2 symmetries (see Table <ref>) are given by-ℒ^leptons_Yukawa =G^l_ijL̅_il_jRS+G^ν_idL̅_i [ N_d D̃]_1+1/Λ G^ν_isL̅_i [N_s[D̃ζ_d]_1^']_1+ 1/2 M_s N^c_s N_s + 1/2 M_d [N^c_d N_d]_1 +H.c.,where i,j=e,μ,τ (we omit summation symbols), L_i(l_iR) and N_s,d denote the usual left-handed lepton doublets (right-handed singlets) and the right-handed neutrinos, respectively;[D̃ ζ_d]_1^'=D̃_1ζ_2-D̃_2ζ_1. [N_d D]_1=N_2RD_1+N_3RD_2, according to the S_3 multiplication rules, and D̃_1,2=iτ^2D^*_1,2. Notice that the doublets D and ζ_d couple only with neutrinos. We assume that ⟨ζ_1,2⟩≲Λ, where Λ is an energy scale much larger than the electroweak one. It is also interesting to note that the right-handed neutrinos in the S_3 doublet are mass degenerated, with mass M_d, which is different from the mass of the right-handed neutrino in singlet of S_3, which has a mass denoted by M_s. Notice that at three level active neutrinos are still massless.§ THE SCALAR SECTORThe scalar sector of the model is presented as follows:S= ([ S^+; 1/√(2)(v_SM+ReS^0+iImS^0) ]), D_1,2=( [ D^+_1,2; 1/√(2)(ReD^0_1,2+iImD^0_1,2). ]),plus the singlets ζ_i=v_i+Reζ_i+iImζ_i, i=1,2.The scalar potential invariant under the gauge and S_3⊗ℤ_2 symmetries isV_S_3 = μ^2_sS^† S+μ^2_d [D^†⊗D]_1+ μ^2_ζ_D [ζ_D⊗ζ_D]_1 +λ_1 ([D^†⊗D]_1)^2+μ^2_12ζ_1ζ_2 +a_2 [[D^†⊗ D]_1^'[D^†⊗ D]_1^'] +a_3[(D^†⊗ D)_2^'(D^†⊗ D)_2^']_1 +a_4(S^† S)^2+ a_5[D^†⊗ D]_1 S^†S +a_6 [[S ^† D]_2^' [S^† D]_2^']_1+ H.c.]+ a_7 S^† [ D ⊗ D^†]_1 S+ b_1 S^† S [ ζ_D ⊗ζ_D]_1 + b_2[D^†⊗D]_1 [ ζ_D ⊗ζ_D]_1+ b_3 [[D^†⊗D]_2^' [ ζ_D ⊗ζ_D]_2^']_1 +c_1([ζ_D⊗ζ_D]_1)^2+ c_2 [[ζ_D⊗ζ_D]_2^'[ζ_D⊗ζ_D]_2^']_1,with μ^2_d>0 that is guaranteed by the ℤ_2 symmetry.We can write Eq. (<ref>) explicitly asV(S,D,ζ_d)=V^(2)+V^(4a)+V^(4b)+V^(4c),whereV^(2) = μ^2_SMS^† S+μ^2_d (D^†_1D_1+D^†_2 D_2)+μ^2_ζ(ζ^2_1+ζ^2_2 )+μ^2_12ζ_1ζ_2, V^(4a) = a_1 (D^†_1D_1+D^†_2 D_2)^2 +a_2 (D^†_1D_2-D^†_2D_1)^2+ a_3[(D^†_1D_2+D^†_2D_1)^2+(D^†_1D_1-D^†_2D_2)^2] +a_4(S^† S)^2+ a_5(D^†_1D_1+D^†_2 D_2)S^†S+ a_6[(S^† D_1S^† D_1+S^† D_2S^† D_2)+H.c.] + a_7 S^† (D_1D^† _1+D_2 D^†_2) S, V^(4b) =b_1S^† S(ζ^2_1+ζ^2_2)+b_2(D^†_1D_1+D^†_2D_2)(ζ^2_1+ζ^2_2)+ b_3[(D^†_1D_2+D^†_2D_1)(ζ_1ζ_2+ζ_1ζ_2)×(D^†_1D_1-D^†_2D_2)(ζ^2_1-ζ^2_2)+H.c.], V^(4c) = c_1(ζ^2_1+ζ^2_2)^2 + c_2[(ζ_1ζ_2+ζ_2ζ_1)^2+(ζ^2_1- ζ^2_2)^2],where we have used [ζ_d ζ_d]_2^'=(ζ_1ζ_2+ζ_2ζ_1,ζ_1ζ_1-ζ_2ζ_2). Notice that the term μ^2_12 breaks softly the S_3 symmetry but not the ℤ_2. Notice also that the ℤ_2 symmetry forbids trilinear terms in the scalar potential [D^†⊗D]_1ζ_i and ζ^3_i.From derivation of Eq. (<ref>), we obtain the constraint equations:v_SM[2μ^2_SM+2a_4v^2_SM+b_1(v^2_1+v^2_2) ] = 0,v_1[4μ^2_ζ +2b_1v^2_SM+4c(v^2_1+v^2_2) +4μ^2_12v_2/v_1] = 0,v_2[4μ^2_ζ+2b_1v^2_SM+4c(v^2_1+v^2_2) +4μ^2_12v_1/v_2] = 0,where we have defined c=c_1+c_2. Notice from (<ref>) that neither v_1=0 nor v_2=0 are allowed if μ^2_12≠0, hence we have that v_1≠0,v_2≠0, or v_1=v_2≡ v_ζ. We have chosen the latter case, so the constraint equations becomev_SM[μ^2_SM+a_4v^2_SM+b_1v^2_ζ]=0 ,v_ζ[2μ^2_ζ+b_1v^2_SM+4cv_ζ+2μ^2_12]=0. The scalar potential has to be bounded from below to ensure its stability. In the SM it is easy to ensure the stability of this potential,we just have to ensure that λ > 0. In theories in which the number of scalars is increased,it is more difficult to ensure that the potential is bounded from below, in all directions. A scalar potential has a quadratic form in the quadratic couplings, i.e. A_abϕ^2_a ϕ^2_b, where ϕ^2_a and ϕ^2_b represents the scalar fields, S, D and ζ_d. If the matrix A_ab is copositive it is possible to ensure that the potential has a global minimum.Assuming a quadratic form (e.g. considering only the quartic terms of the potential) is valid, even if there exist trilinear terms, because in the case where the fields assume large values, the terms of order 2 and 3 are negligible compared to the terms of order 4. For more detail see Refs. <cit.>. We consider all quartic couplings positive, i.e. all A_ij are positive. Below we will denote B=a_1-2a_2-a_3 and C=a_5-2a_6+a_7. And finally the following limits guarantee that the scalar potential is bounded the from below:a_4≥ 0,a_5/2≥ 0,C ≥ 0,b_1≥ 0,a_1+a_3≥ 0, b_1+b_2≥ 0,B ≥ 0,c_1+c_2≥ 0. Next, we consider the scalar mass spectra. In the CP-even sector the mass matrix becomes in block diagonal form with one 3×3, ℳ_1R sub-matrix and one 2×2 matrix, ℳ_2R. The first one in the basis (ReS^0, ζ_1,ζ_2) is given byℳ^2_1R=( [ 2a_4v^2_sb_1v_SMv_ζb_1v_SMv_ζ; -μ^2_12+2cv^2_ζμ^2_12+2cv^2_ζ; -μ^2_12+2cv^2_ζ ]).The respective eigenvalues arem^2_1= -2μ^2_12, m^2_2= 1/2[a_4v_SM^2+2cv^2_ζ -√(a_4^2v^4_SM + 2 (b_1^2 - 2 a_4 c)v^2_ζ v^2_SM + 4 c^2 v_ζ^2) ] ,m^2_3=1/2[[4a_4v_SM^2+8cv^2_ζ]^2 +√(a_4^2v^4_SM + 2 (b_1^2 - 2 a_4 c)v^2_ζ v^2_SM + 4 c^2 v_ζ^2) ].From m^2_1 we see that μ^2_12<0, hence m^2_1 may have large mass. The SM-like Higgs boson may be identified with the scalar with mass m_3 in Eq. (<ref>). In order to see this just make v^2_ζ = 0.The second mass matrix M_2R in the basis(ReD^0_1,ReD^0_2) readsℳ^2_2R=( [ μ^2_d+a^'/2 v_SM^2+b_2v^2_ζb_3v^2_ζ; μ^2_d+a^'/2 v_SM^2+b_2v^2_ζ; ]) ,with the eigenvaluesm^2_R1=2μ^2_d+a^' v_SM^2+2(b_2-b_3)v^2_ζ,m^2_R2=2μ^2_d+a^' v_SM^2+2(b_2+b_3)v^2_ζ. The mass matrix in the CP odd sector has the form in the basis (ImD^0_1,ImD^0_2) (the would-be Goldstone bosons has been already decoupled)M_I=( [ μ^2_d+a^''/2 v_SM^2+b_2v^2_ζ b_3v^2_ζ;μ^2_d+a^''/2 v_SM^2+b_2v^2_ζ ]),with eigenvaluesm^2_I1=2μ^2_d+a^'' v_SM^2+2(b_2-b_3)v^2_ζ, m^2_I2=2μ^2_d+a^'' v_SM^2+2(b_2+b_3)v^2_ζ.Above we have defined a^'=a_5+a_7+2a_6 and a^''=a_5+a_7-2a_6.The model allows four neutral scalars with different masses that could contribute to the DM relic density in different proportion: two CP even and two CP odd.Notice also that a_6 is the term in the scalar potential that transfer the L violation to the active neutrino sector.In the charged scalars sector, besides the charged would-be Goldstone boson, we have two charged scalar fields (we have already omitted the charged would-be Goldstone boson)M^2_C=([ μ^2_d+a_5/2 v_SM^2+ b_2v^2_ζ b_3v^2_ζ;μ^2_d+a_5/2v_SM^2+b_2v^2_ζ ]),with the non-zero eigenvalues given bym^2_+1=μ^2_d+a_5/2v_SM^2+(b_2-b_3)v^2_ζ , m^2_+2=μ^2_d+a_5/2v_SM^2+(b_2+b_3)v^2_ζ, Notice thatm^2_R1-m^2_+1=μ^2_d+(a_5/2 + a_7+2a_6 )v^2_SM +(b_2-b_3)v^2_ζ , m^2_R2-m^2_+2=μ^2_d+(a_5/2 + a_7+2a_6 )v^2_SM +(b_2+b_3)v^2_ζ. Notice that μ^2_d>0 does not disappear in the mass difference above because, in oder to reproduce the Klein-Gordon equation for each component, a real scalar has a 1/2 factor in the mass term related to the mass of a complex scalar.§ LEPTON MASSES AND THE PMNS MATRIXIn Sec. <ref> we have seen that at tree level the neutrinos are massless, thea_6 term in Eq. (<ref>) induce diagrams like those in Fig. <ref> and it is possible to implement the mechanism of Ref. <cit.> for radiative generation of neutrinos mass. In fact, the diagram in Fig. <ref> are exactly calculable from the exchange of ReD^0_1,2 and ImD^0_1,2(M_ν)_ij =∑_a,kY_ikY_jk M_k/32π^2[ m^2_Ra/m^2_Ra-M^2_klnm^2_Ra/M_k^2- m^2_Ia/m^2_Ia-M^2_klnm_Ia^2/M_k^2],where a=1,2; k=s,d; m_Ra and m_Ia are the masses of ReD^0_1,2 and ImD^0_1,2, respectively. In Eq. (<ref>) Y_ikY_jk corresponds to G^ν_isG^ν_js when the coupling is with the N_s; andY_ikY_jk corresponds to G^ν_idG^ν_jd when the coupling is with the N_d, and finally M_s is the mass of the right-handed neutrino N_s, and M_d is the common mass of theright-handed neutrino in the doublet of S_3, N_d i.e, M_2=M_3≡ M_d. We can define Δ^2_a= m^2_Ra- m^2_Ia= 4a_6v^2_SM, and m^2_0a=(m^2_Ra+m^2_aI)/2, a=1,2.If Δ^2≪ m^2_a0 we obtain(M_ν)_ij = a_6v^2_SM/16 π^2[ G^ν_idG^ν_jd M_d/m^2_01 - M_d^2(1 - M_d^2/m^2_01 - M_d^2lnm^2_01/M_d^2)+ G^ν_is G^ν_js M_s/m ^2_01-M^2_s(1-M^2_s/m ^2_01-M^2_slnm^2_01/M^2_s). . + G^ν_idG^ν_jd M_d/m^2_02 - M_d^2(1 - M_d^2/m^2_02 - M_d^2lnm^2_02/M_d^2)+ G^ν_is G^ν_js M_s/m ^2_02-M^2_s(1-M^2_s/m ^2_02-M^2_slnm^2_02/M^2_s) ],where M_s is the mass of the right-handed neutrino N_s and M_d is the common mass of the neutrinos N_d. Under the condition in which the scalars are mass degenerated i.e.,b_3=0 in (<ref>) we obtain just a factor 2 in Eq.(<ref>). Below, for simplicity, we will consider the case b_3=0.In order to obtain the active neutrinos masses we assume a normal hierarchy and, without loss of generality, that M_s ∼ M_d and will be represented from now on by M_R. M^ν is diagonalized with a unitary matrix V_L^ν i.e.,M̂^ν = V^ν T_LM^ν V^ν_L, where M̂^ν = diag (m_1, m_2, m_3)≈ (0,√(δ m^2_12), √(δ m^2_23)). Taken the central values in PDG we have M̂^ν≈ (0,x,y).In the charged lepton sector we assume their masses at the central values in PDGM̂^̂l̂= (0.510, 105.658, 1776.86 ) GeV. It is important to note from these considerations, that there exist a multitude of other possibilities which satisfy also the masses squared differences and the astrophysical limits in the active neutrino sector. Each one corresponds to different parameterization of the unitary matrices V^l_L,R,V^ν_L.We will obtain the neutrinos masses from Eq. (<ref>). We have as free parameters a_6, M_R and the Yukawas [m_+1=m_+2≡ m_+ are also still free but they will enter only in the leptonic decays considered in Sec. <ref>]. In the Fig. <ref> we show the dependence of a_6 with respect to the main Yukawas G_τ d^ν in (a) and G_es^ν in (b) for fixed M_R values. Notice that G_τ d^ν are essentially of the same order of magnitude, while the rest G_ed,μ d,μ s,τ s^νare suppressed by four orders of magnitude when comparing with any specific value of those G_τ d,es^ν.The mass matrices in the charged lepton sector M^l are diagonalized by a bi-unitary transformation M̂^l=V^l†_L M^l V^l_R and M̂^l = diag (m_e, m_μ, m_τ). The relation between symmetry eigenstates (primed) and mass (unprimed) fields are l^'_L,R=V^l_L,Rl_L,R and ν^'_L=V^ν_L ν_L, where l^'_L,R=(e^',μ^',τ^')^T_L,R, l_L,R=(e,μ,τ)^T_L,R,ν ^'_L=(ν_e,ν_μ,ν_τ)^T_L and ν_L=(ν_1,ν_2,ν_3)_L. Defining the lepton mixing matrix as V_PMNS=V^l†_LV^ν_L, it means that this matrix appears in the charged currents coupled to W_μ^+. We have tested the robustness of our fitting of the lepton masses and the leptonic mixing matrix by using several parametrization corresponding to the values of the Yukawa couplings given in Table <ref>. We omit the respective matrices V^l_L and V^ν_L but in all cases we have obtained:| V_PMNS|≈([ 0.815 0.565 0.132; 0.479 0.527 0.702; 0.327 0.635 0.700; ]),which is in agreement within the experimental error data at 3σ given by <cit.>| V_PMNS|≈([ 0.795-0.846 0.513-0.585 0.126-0.178;0.4205-0.543 0.416-0.730 0.579 - 0.808; 0.215 - 0.548 0.409 - 0.7250.567 -0.800; ]),and we see that it is possible to accommodate all lepton masses and the PMNS matrix. Here we do not consider CP violation.§ DARK MATTERAs we said before, the present model may have a multi-component DM spectrum, which means that many particles may contribute to the relic density of DM, but we will consider the simplest example where one of the CP even scalar, say R_1, and one of the CP odd scalar, say I_1, as the dark matter candidates, each case is considered separately for simplicity. A two inert doublet model without right-handed neutrinos and scalar singlet was considered in Ref. <cit.>.As usual, in order to determine the relic density, we solve the Boltzmann equation. Firstly, considering R as the candidate, we havedn_R/dt+3Hn_R=-⟨σ |v| ⟩ [(n_R)^2-(n_R^eq)^2],where ⟨σ |v| ⟩ is the annihilation cross section already thermally averaged and H is the Hubble constant. In the thermal equilibrium, the number density of DM <cit.> isn^eq_R=g(m_R T/2π)^3/2exp(-m_R/T),where g=1 for a scalar DM. When solving the Boltzmann equation we obtain the equation for the relic density:Ω_R h^2≈1.04× 10^9x_F/M_Pl√(g_*)(a+3b/x_F),where M_Pl=1.22× 10^19 GeV is the Planck mass, x_F= m_R/T_F, where m_R is the mass of the neutral scalar and T_F is the temperature at freeze-out, the terms a and b result from the partial wave expansion of σ |v|= a + bv^2. The number of relativistic degrees of freedom g_*=118.375is a result of the SM particles plus three right-handed neutrinos, five neutral scalars, two pseudo-scalars and two charged scalars. The evaluation of x_F leads tox_F=ln [c(c+2)√(45/8)g m_RM_Pl(a+6b/x_F)/2π^3√(g_*(x_F)) ],where theunitary parameter c≈ 5/4.Here we will consider the solution for the relic density which, at the same time, solves the charged lepton masses and neutrino masses given in Eq. (<ref>) for the sets of parameters showed in Tables <ref> and <ref>, in order to obtain the PMNS matrix.We call them scenario 1 and 2, when R_1 and I_1 is the DM candidate, respectively. In both scenarios the Yukawas values adjust the squared masses differences for the neutrinos and the PMNS.With the numbers in Tables <ref> and <ref> we obtain the mixing matrix, that is:V^ν_L ≈([0.00010637-10.0000444397;-1 -0.0001063748570.00007634160462; 0.000076336876851 0.000044447836932 1 ]),and the charged leptons has the following Yukawas: G^l_11=0.0004219, G^l_12 = 0.000515, G^l_22 = 0.00374772,G^l_13 = -0.000801115, G^l_23 = -0.0035675, G^l_33 = 0.00367627, and we obtain m_e = 0.510 MeV, m_μ = 105.658 MeV and m_τ = 1776.86 MeV, and the mixing matrix isV^l_L =([0.564902 0.527040.634914;0.814515 -0.479341 -0.326799; -0.132104 -0.7017560.700062 ]).As previously defined the mixing matrix for the leptonic sector V_PMNS=V^l†_LV^ν_L, From Eqs. (<ref>) and (<ref>) we obtain again Eq.(<ref>).To perform DM calculation we have usedpackage <cit.>. For instance, let us consider scenario 1, where R_1is the DM candidate. In the range of parameters used by us, DM annihilates mainly in W^+W^-.Once again we emphasize that other solutions in other annihilation channels do exist. We have chosen the following parameters for the couplings, and vacuum expected value: G_τ d = 2.81×10^-7, G_τ s = 1 × 10^-11, v_ζ≲Λ = 1000 GeV, λ^'=2.7× 10^-2, λ^''=0.34, μ^2_d=2.809(TeV)^2, for values of other parameters see Table <ref>. With this parameters choice m_R1 = 85.15 GeV. Theparameter dependence of m_R1 is presented in the last sections. So, the dominant contributions for Ω are 99% in R_1R_1→ W^+W^-. In this case, x_F∼ 23.8. The annihilation cross section is ⟨σ v⟩= 1.0× 10^-26 cm^3/s and the DM-nucleus cross section for spin-independent elastic scattering is numerically given by σ_SI^p=5.84× 10^-46 cm^2 and σ_SI^n=6.70× 10^-46 cm^2.In scenario 2 we consider I_1 as the DM candidate. In this case, as a result of the parameter choice (G_τ d =3.75×10^-7, G_τ s = 1 × 10^-11, v_ζ≲Λ = 1000 GeV, λ^'=1, λ^''=0.12, μ^2_d=2830.24 TeV^2),I_1 annihilates 97% in I_1 I_1→ hh and 2% in I_1 I_1→ bb. The value of other parameters can be seen in Table <ref>. The annihilation cross section is ⟨σ v⟩ = 4.55× 10^-28 cm^3/s and the DM-nucleus cross section for spin-independent elastic scattering is numerically given by σ_SI^p=6.40× 10^-45 cm^2 and σ_SI^n=7.35× 10^-45 cm^2.In these two scenarios, we had set R_1 andI_1 as DM candidates, making them lighter than the others possible neutral scalars. We emphasize that other choices for DM are possible so that other annihilation channels may also give interesting signatures. There is also the possibility that two, three or even four of the neutral scalars contribute partially to the DM density, but this case is beyond the scope of this paper.In Fig. <ref> we present the fluxes of photons, positrons and antiprotons in the scenario 1 with m_DM=85.15 GeV and the scenario 2 with m_DM=113.70 GeV, where the upper limits of the energy spectrum are determined by the DM masses since annihilation occurs near at rest. The model can accommodate DM candidtes with smaller masses than the values above.§ THE LEPTONIC DECAYS L_I→ L_JΓ AND L_I→ L_JL_KL̅_K.In this section we study the impact of the new particles, the charged scalars D_1,2^+ and the right-handed neutrinos N_s,2,3 in the lepton flavor violating processes l_i→ l_jγ. Here we will consider these rare decays in two cases: one in which we do not care with DM solutions and one in which we use the parameters for having a DM candidates that also give the correct lepton masses and the PMNS.In terms of the leptons mass eigenstates, the interactions with charged scalars from Eq. (<ref>) are written asℒ^l-N_Yukawa = -l̅_kLV^l†_ki[ G^ν_id(N_2RD^-_1+N_3RD^-_2)+v_ζ/Λ G^ν_is N_s(D^-_1-D^-_2)],i,k=e,μ,τ, the values of the entries of the matrices G_i,d^ν and G_i,s^ν are given in Table <ref>.In the model, the allowed lepton flavor violation (LFV) decays l_i→ l_jγ and l_i→ l_jl_kl̅_karise only at the1-loop level. These diagramsare generated by the known SM contribution W & ν_l and by the new content D_1^+ & N_2, D_2^+ & N_3, D_1^+& N_s and D_2^+& N_s.For l_i→ l_jγ, it is known that the SM contribution is extraordinarily suppressed with respect to the experimental capabilities of detection, see Table <ref>. As we will show below, the new particle content in the model predicts signals close to the experimental upper limits for the space of our considered allowed parameters. Regarding the three body decay l_i→ l_jl_kl̅_k, it arises when in l_i→ l_jγ we attach to the photon the γ ll̅ coupling. In the following we are interested in presenting the μ→ eee̅ channel, because it provides interesting results near the experimental upper limit, while all the other channels are out of the experimental interest region because the devices are unable of reaching such suppressed signals.In our study we have solved the amplitudes and the loop integrals with the help of , <cit.>, and <cit.>. §.§ Predictions of μ→ eγ and μ→ eee̅in the scotogenic model without dark matter.We start our numerical analysis of the decays in the scotogenic model without DM content. Accordingly to the Yukawa values derived in the Sec. <ref> (see Fig. <ref>), the obtained values of the Yukawas are G_ad,as^ν∈ [10^-11,10^-1], they satisfy the neutrino masses. Notice that G_τ d^ν≃ G_es^ν≫ G_ed,μ d,μ s,τ s^ν. We recall that all the previous analyse were done in the case of b_3=0 in which the charged scalars are mass degenerated. As starting point we consider the mass of the charged scalar in the range m_+∈ [80,750] GeV, N_2,3,s degenerated as well with values m_N∈ [250, 4000] GeV. We have tested the five different parameterizations of the V_L^l matrices derived in the Sec. <ref> (see Table <ref>). They can be separated into two sets which will have two different behaviours in the processes, the set A is conformed by the parameterizations P1 and P5, and the B by P2, P3 and P4.For the channel μ→ eγ such situation occurs when G_τ d^ν= G_es^ν∼10^-1 and G_ed,μ d,μ s,τ s^ν∼10^-5. In the Fig. <ref> (a)-(b) it is shown the Br(μ→ eγ) as function of the sterile neutrino mass m_N∈[250,4000] GeV with given values for the charged scalar m_D^+=80, 250, 500, 750 GeV. The current experimental upper limit Br(μ→ eγ)^Exp<4.2×10^-13, indicated with the red line in the plots, will constrain the right-handed neutrino mass m_N for given values of m_+ in order to respect such limit, those constraints are listed in the Table <ref>, where it is evident that the V_L^l parameterization A in Fig. <ref> (a) allows a lighter mass for the right-handed neutrino than the parameterization B in Fig. <ref> (b). Regarding to the subcase μ→ eee̅, we are able to predict the branching ratio from the neutrino right-handed mass constraints obtained for the μ→ eγ channel. These predictions are organized in the Table <ref>, and such values are indicated with the green line in the Fig. <ref> (c) and (d), being of the same order of magnitude ∼10^-15 for both scenarios.About the analogous tau decays, for the same space of parameter values than in the μ→ eγ case, and respecting the obtained mass bounds, we have found that Br(τ→ eγ)≤10^-12 and Br(τ→ eγ)≤10^-14, which are beyond the current and upcoming experimental capabilities of detection, see Table <ref> for comparison. §.§ Predictions of l_i→ l_jγ in the scotogenic model with dark matter As commented in the Sec. <ref>, the model can be extended to include DM. In order to estimate the consequences on the transition μ→ eγ, we consider the Yukawa values given in Table <ref>. The resulting prediction with G_τ d^ν=G_es^ν=10^-7 isBr(μ→ eγ)=10^-34,which is beyond the scope of detection.§ CONCLUSIONSHere we have considered an extension of the SM with three scalar doublets of SU(2) with S_3 and ℤ_2 symmetries. We had analysed all the mass spectra in the scalar sectors and used the scotogenic mechanism for generating neutrino masses. Moreover, we had obtained the PMNS matrix once the unitary matrices which diagonalize the lepton masses are obtained. Although the model can have many DM candidates, we have shown two cases in which the DM candidate is a CP even scalar (scenario-1) and other one in which the DM is composite of CP odd scalar (scenario-2).But we emphasize that other possible choicesfor DM candidates are possible, considering for example, smaller masses, since besides the SM-like scalar, we have eight additional neutral scalars in the model. The study of other candidates and other channels of annihilation will be done soon. We had exemplified in some range of parameters space, two DM candidates for the model. For the scenarios 1 and 2 presented, DM annihilates mainly in W^+W^- and hh respectively. We have also presented some fluxes for this model. Of course, there may be other possible scenarios which could explain the Galactic gamma ray excess, as well as the the PAMELA and AMS-02 results. These processes could tightly constrain the parameter space of this sort of scotogenic models.The considered scotogenic model without DM provides optimistic predictions for possible detection of the LFV decay μ→ eγ due to our solution space of the Yukawa values, which adjusts the squared masses differences for the neutrinos and the PMNS matrix. Our estimations predict a mass for the right-handed neutrinostarting from m_N>295 GeV, and from μ→ eγ we predict Br(μ→ eee̅)≲ 10^-15. On the other hand, considering DM content in the model we found that Br(μ→ eγ)∼ 10^-34, which is out of detection range.ACBM thanks CAPES for financial support, JM thanks to FAPESP for financial support under the processe number 2013/09173-5, andVP thanks to CNPqfor partial financial support.99 Aad:2012tfa G. Aad et al. 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Baldini:2013ke A. M. Baldini et al., “MEG Upgrade Proposal,” arXiv:1301.7225 [physics.ins-det]. Olive:2016xmw C. Patrignani et al. [Particle Data Group], “Review of Particle Physics,” Chin. Phys. C 40, no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001Aushev:2010bq T. Aushev et al., “Physics at Super B Factory,” arXiv:1002.5012 [hep-ex]. | http://arxiv.org/abs/1705.09414v1 | {
"authors": [
"E. C. F. S. Fortes",
"A. C. B. Machado",
"J. Montaño",
"V. Pleitez"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526023142",
"title": "Lepton masses and mixing in a scotogenic model"
} |
VI. First chromosphere model of a late-type [email protected] of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio al. 5, Vilnius LT-10221, Lithuania ZAH, Landessternwarte Königstuhl, D-69117 Heidelberg, GermanyAlthough observational data unequivocally point out to the presence of chromospheres in red giant stars, no attempts have been made so far to model them using 3D hydrodynamical model atmospheres. We therefore compute an exploratory 3D hydrodynamical model atmosphere for a cool red giant in order to study the dynamical and thermodynamic properties of its chromosphere, as well as the influence of the chromosphere on its observable properties. Three-dimensional radiation hydrodynamics simulations are carried out with themodel atmosphere codefor a star with the atmospheric parameters (≈4010 K, log g=1.5, =0.0), which are similar to those of the K-type giant star Aldebaran (α Tau).The computational domain extends from the upper convection zone into the chromosphere (7.4≥log≥ -12.8) and covers several granules in each horizontal direction.Using this model atmosphere, we compute the emergent continuum intensity maps at different wavelengths, spectral line profiles of Caii K, the Caii infrared triplet line at 854.2 nm, and Hα, as well as the spectral energy distribution (SED) of the emergent radiative flux. The initial model quickly develops a dynamical chromosphere that is characterised by propagating and interacting shock waves. The peak temperatures in the chromospheric shock fronts reach values on the order of up to 5 000 K although the shock fronts remain quite narrow. Like for the Sun, the gas temperature distribution in the upper layers is composed of a cool component due to adiabatic cooling in the expanding post-shock regions and a hot component due to shock waves. For this red giant model, the hot component is a rather flat high-temperature tail, which nevertheless affects the resulting average temperatures significantly.The simulations show that the atmospheres of red giant stars are dynamic and intermittent.Consequently, many observable properties cannot be reproduced with one-dimensional static models but demand for advanced 3D hydrodynamical modelling.Furthermore, including a chromosphere in the models might produce significant contributions to the emergent UV flux. Three-dimensional hydrodynamicalmodel atmospheres of red giant stars Sven Wedemeyer1 Arūnas Kučinskas2 Jonas Klevas2 Hans-Günter Ludwig36 January 2017; accepted 30 May 2017 ============================================================================ § INTRODUCTION The existence of chromospheres around red giant stars is suggested by avariety of observational facts. Mostly, the evidence comes from indirect indicators related to stellar activity, such as, for example, measurements of (variable) flux in the emission cores of CaII H & K and MgII h & k lines, asymmetries and shifts of the Hα line core <cit.>. Observations of red giants in the (sub)millimetre and centimetre ranges also point out to the presence of chromospheres <cit.>. From the theoretical side, however, the properties of red giant chromospheres are still relatively poorly understood. For example, 1D semi-empirical models have been constructed to account for various observable properties of these stars, such as the strengths of chromospheric spectral lines, fluxes in the milimeter wavelength range, and so on <cit.>. However, these models suffer from various shortcomings. For instance, earlier models were unable to account simultaneously for the required ultraviolet (UV) pumping of carbon monoxide (CO) resonance lines and cool excitation temperatures of these lines in the chromosphere of Aldebaran <cit.>.The inability of models to reproduce cool features (e.g., CO lines) and UV lines, led authors to suggest model atmospheres withmultiple components and the presence of shock waves in α Tau <cit.>. Detailed multi-component models with a realistic description of time-dependent shock waves require the utilisation of3D hydrodynamical model atmosphere codes. Unfortunately, to the best of our knowledge, red giant models with chromospheres computed with such codes have not been publishedso far. Three-dimensional hydrodynamical models are well-suited for studying non-stationary stellar chromospheres and thus may help to better understand the complex interplay between non-stationary physical phenomena (such as, e.g., shock wave activity) that shape their structures.In this work, we present the results of our first exploratory simulation of the red giant chromosphere performed with the 3D hydrodynamicalmodel atmosphere package. The atmospheric parameters of the model are similar to those of the K-type giant star Aldebaran, α Tau (see Sect. <ref> below). Our main goal is to study the dynamical and thermodynamic structure of the chromosphere, and to investigate its influence on the observable properties of this red giant.It must be emphasised that the simulations presented here are only a first step. The included physical effects restrict to what extent the model results can be interpreted. The exploratory model reproduces conditions as they should only be expected in rough approximation for very quiet regions in the low chromosphere. While this is already of interest,α Tau exhibits both CIV and OVI features <cit.>. These observations imply significantly higher temperatures than those achieved with shock heating in our 3D hydrodynamical model atmosphere and hints at yet-to-be-included physical (heating) mechanisms. The paper is structured as follows: after a brief description of the numerical codes and models in Sect. <ref>, we present the obtained results in Sect. <ref>, followed by a discussion and conclusions in Sects. <ref> and <ref>.§ METHODOLOGY§.§ Three-dimensional radiation hydrodynamic simulations The numerical simulations are carried out with the radiation hydrodynamics code<cit.>. The hydrodynamic equations and the frequency-dependent radiative transfer equations are solved time-dependently for a fully compressible plasma in a constant gravitational field. The lateral boundaries are periodic. The bottom boundary is open so that material can flow into and out of thecomputational box. The entropy of the inflowing material is prescribed and sets the effective temperature of the simulated star. The top boundary is transmitting, i.e., material can leave and enter themodel at the top. The equation of state is used in form of a look-up table, which is computedin advance under the assumption of thermodynamic equilibrium.For this table, partial ionisation of H and He, the formation and dissociation of H_2, and a representative metal are taken into account.The chemical element abundances are adopted from the CIFIST project<cit.>.Non-equilibrium effects like the time-dependence of the hydrogen ionization degree in chromospheres are not considered yet for this first exploratory model due to the high computational costs but should be included in more advanced future detailed models. Therefore, it has yet to be seen how much the atmospheric state for the modelled stellar type would be affected when these effects are taken into account. A long characteristics scheme with multi-group opacities is used for thesolution of the radiative transfer equation <cit.>. The opacity data is provided in form of a look-up table with five bins, which are constructed in advance using opacities from the MARCS model atmosphere package <cit.>.§.§ Numerical modelThe numerical model analysed here was produced with theradiativehydrodynamics code (Sect. <ref> above) in several steps. The initial model atmosphere is based on a snapshot of a well relaxed simulation that was computed using the following atmospheric parameters: , log g = 1.5, and =0.0. This snapshot was taken from the CIFIST 3D hydrodynamical model atmosphere grid <cit.>. The initial model includes the top of the convection zone and the photosphere with a total extent of4.6·10^9 m× 4.6·10^9 m× 2.8·10^9 m(x,y,z). The size of the initial modelwas increased to cover more granules by doubling the computational box in each horizontal direction, thus producing 2 × 2initially identical quadrants. A multi-scale random velocity field was added to the new model in order to accelerate the diversion of the four quadrants and thus remove the initial symmetry as fast as possible. The sequence was run for one million seconds of stellar simulation time until the initial symmetry was not longer visible in the granulation. Next, an initially homogeneous chromosphere with 130 grid layers and a vertical extent of 1.31·10^9 m was added on top. The chromosphere evolves much faster than the layers below and is therefore only added in the last step in order to save computational time. The final model consists of 280 × 280 × 280 grid cells and has a total spatial extent of 9.2·10^9 m × 9.2· 10^9 m × 4.1· 10^9 m. The grid spacing is constant in horizontal direction (Δ x = Δ y = 3.29·10^7 m) while the vertical grid cell size decreases from Δ z = 5.73·10^7 m) at the lower boundary in the convection zone toΔ z = 9.66·10^6 m in the atmosphere, i.e. for all heights above z = 0 m (except for a slight increase in the topmost layers). The simulation with a chromosphere is then advanced for 5.7·10^5 sec (157 hours) of stellar time. The model maintains an effective temperature of T_eff≈ (4010 ± 13) K.For comparison, we also use a model without a chromosphere, i.e., the relaxed model obtained before adding the extra 130 grid layers. Note that the entropy flux at the bottom boundary of the chromospheric model was adjusted to obtain the same total emergent radiative flux as that of the model without the chromosphere. In effect, this ensured that the effective temperatures of the two models are nearly identical. §.§ Intensity synthesis The radiative transfer code LINFOR3D[http://www.aip.de/Members/msteffen/linfor3d] is used to calculate the emergent continuum intensity at different wavelengths ranging from the ultraviolet to the millimeter range (see Sect. <ref> for the results).LINFOR3D is originally based on the Kiel code LINFOR/LINLTE and solves the radiative transfer equation in detail in 3D for an input atmosphere under the assumption of local thermodynamic equilibrium (LTE). The same model was used as input for the MULTI_3D code, i.e. the column-by-column version of the original MULTI_3D code<cit.>, which provides the detailed solution of the radiative transfer equation in non-LTE (non-local thermodynamic equilibrium).The output contains intensity cubes (I = f (x, y, λ)) for several spectral lines of hydrogen (e.g., Hα and Lyman α)and singly ionized calcium(e.g., Ca II H, K, and the infrared triplet) for each selected snapshot.§.§ Computation of the spectral energy distributionsThe spectral energy distribution (SED) of the emergent radiation field is calculated using the NLTE3D code <cit.>. The code is designed to compute departure coefficients for the different energy levels of a given model atom[The departure coefficient, b_ i, of atomic level i is defined as the ratio of population number densities, b_ i=n_ i^ NLTE/n_ i^ LTE, obtained under the assumptions of non-local thermodynamic equilibrium, NLTE, and local thermodynamic equilibrium, LTE.]. For this purpose, NLTE3D calculates mean intensities at each grid point,which we then use to compute the SED for the given model atmosphere, i.e., the radiative flux at its top, at different wavelengths. We compute SEDs for two model atmospheres, one with the chromosphere and one without it (see Sect. <ref>). Note that in both cases the two models shared identical atmospheric parameters, chemical composition, opacities, and equation of state. The SEDs were computed in the 150-10000 nm wavelength range. To account for the line opacities, we use LITTLE opacity distribution functions (ODFs) from the ATLAS9 model atmosphere package <cit.>.Continuum opacities are taken into account by using theIONOPA package <cit.>. The SEDs computed using the two model atmospheres are shown in Fig. <ref>. § RESULTSThe initially homogeneous atmosphere evolves quickly towards a new dynamic equilibrium state. The first shock waves have propagated through the whole chromosphere and reach the upper boundary of the model after 210 000 s. From then on, the whole chromosphereis characterised by a shock-induced dynamic pattern, which is visible in many different quantities such as, e.g., gas temperature and velocity (Fig. <ref>).The resulting chromosphere exhibits many features known from earlier (non-magnetic) solar models <cit.> although the details are quite different. Propagating shock waves act as a structuring agent and produce a hot mesh of filaments, which can beseen in horizontal and vertical cross-sections through the selected model snapshot presented in Fig. <ref> and Fig. <ref>, respectively.The resulting complex and dynamic topology exhibits variations on a large range of spatial scales, whichare in general larger than in solar models but also involve scales as small as the extent of the intergranular lanes in the photosphere below.The thermal structure of the atmosphere and the emergent continuum intensityare addressed in Sect. <ref> and Sect. <ref>,respectively.§.§ Atmospheric structure and dynamics The horizontal cross-sections for a selected model snapshot after 146.8 hours simulated time in Fig. <ref> demonstrate how the atmospheric structure changes with height in terms of gas temperature, logarithmic gas pressure, and velocity. The convection zone (see the panels a-c) is characterised by mostly hot upwelling gas with a mesh of cool sinking gas. The cool gas is mostly concentrated in plumes that have their origin in intergranular vertices in the photosphere above.These plumes can also be seen as areas of lower gas pressure and in the velocity, which sometimes reaches downward speeds in excess of –15 km s^-1. These flows can thus become supersonic and reach a Mach number of up to ∼ 2. The average gas temperature at the bottom of the models isand decreases monotonically to the effective temperature at the height where log = 0, i.e., at the transition to the photosphere.The low photosphere (see the panels d-f in Fig. <ref>) is characterised by a granulation pattern with hot granules, where gas is rising to the surface, and narrow intergranular lanes, where the cooled gas sinks down again into the convection zone. The typical size of a granule is around 1.3 - 1.8·10^9 mso that about 5 - 7 of such granules fit next to each other in the computational box. However, the distribution of granule sizes in this layer seems to be otherwise continuous with larger and smaller granules occurring, too. The intergranular lanes typically have widths of 0.2 - 0.3·10^9 m with larger downdraft areas located at granule vertices. There, the conservation of momentum in the downflowing plasma results in vortex flows, which seem to be an integral part of stellar surface convection (seeand , for vortex flows occurring in models of the Sun and M-dwarfs, respectively). Shock formation.The upwards propagating wave fronts steepen into shocks above 4·10^8 m and sometimes as low as 3·10^8 m and reach peak temperatures on the order of up to 5 000 K. The gas in the wake of shock waves is expanding adiabatically, which results in reduced gastemperatures as low as 2 000 K.The average gas temperature in the chromosphere is about 2 500 K. The velocities in shock fronts reach typical values on the order of 10 km s^-1, with more extreme values of up to more than 20 km s^-1. As can be seen in Fig. <ref>f, thecorresponding Mach numbers are typically on the order of 3 to 4 but can also reach values of 6 and - in very extreme, localised cases - up to 9. Upward propagating wave fronts can clearly be seen in vertical cross-section in Fig. <ref>e as regions of high upward directed velocity. It is also imminent that these wave fronts usuallyrun into material that is falling down from above at high speeds, which often exceed-20 km s^-1. The situation is also illustrated for a selected column (i.e., a singlehorizontal position in the computational model box) in Fig. <ref>. The column is taken from the vertical x-z cut shown in Fig. <ref> at a horizontal position of x =3.3·10^9 m. In that column, two major shock fronts can be seen at heights of approximately of z = 0.48·10^9 m andz = 1.30·10^9 m. The shock fronts can clearly be seen as temperature spikes and characteristic sawtooth profiles in the vertical velocity. The vertical velocity and the corresponding Mach number in Fig. <ref>c illustrate the existence of supersonic downflowing material. The fast downflows are caused by material that has been transported up by previous shock waves and then falls down again under the influence of gravity. These downflows themselves contribute to the continued formation of shocks throughout the whole model chromosphere.The same process has been found in earlier solar models <cit.>.Gas temperature distribution.An important consequence of the dynamic shock pattern in the chromosphere is that the thermal and kinetic state of the gas cannot be described well with average values only. In other words, the complicated and intermittent structure of the modelled atmosphere cannot be described correctly with a simple one-dimensional and static model atmosphere. The same is also true for the modelled convection zone. This result becomes obvious when looking at the histograms in Fig. <ref>. The displayed distributions for the same model snapshot as in Figs. <ref> and <ref> are very similar to the distributions for time steps spanning the second half of the chromospheric simulation. Hereafter, we refer to the latter. The gas temperature distribution for the chromosphere (Fig. <ref>g) has a peak at T_gas≈ 2 450 K whereas the average gas temperature is 2 670 K. The true peak of the distribution is still within one standard deviation (340 K) from the average value but there is a tail stretching to the maximum value of 4 600 K, which is not well presented by the average value at all. The high-temperature tail is due to the narrow chromospheric shock fronts, whereas the pronounced peak of the distribution is due to the cooler post-shock regions with temperatures down to ∼ 2 000 K. Due to the non-linear dependence of gas temperature and emergent intensity, the high-temperature tail significantly affects the average temperature derived from observed intensities as we will discuss in Sect. <ref>. Velocity distribution. The velocity distributions for selected heights in the model are shown in the rightmost column of Fig. <ref>. Again, the distributions for the selected model snapshot (red) are very close to the distributions for the whole second half of the simulation sequence, indicating that the selected snapshot is representative. For all selected heights, each distribution has a pronounced peak at downward directed velocity (v_z < 0) and a peak for upwards directed velocity (v_z > 0). In the convection zone (see Fig. <ref>c), roughly as many grid cells with downward velocities as grid cells with upward velocities are found, resulting in an average velocity close to zero. However, the peak for upward velocities is narrower with a maximum around <|v| × v_z/|v_z|> ≈ 2 km s^-1 whereas the downward velocities span a broader range of values with a maximum close to <|v|> ≈ -3 km s^-1 and a tail reaching beyond -10 km s^-1. The distribution is consistent with the rather smoothly upwelling hot gas and the cool gas shooting back into the convection zone in plumes. The same is in principle true for the “surface” layer at optical depth unity (logτ_Ross = 0.0, see Fig. <ref>f) although the distribution is more symmetric in the sense that the downward velocity peak is similar (but not identical) to the upward velocity part. The downward velocity distribution peaks around <|v|> ≈ -5 ... -6 km s^-1 and has a tail extending well beyond -10 km s^-1, even reaching -15 km s^-1 in extreme cases. The upward velocity distribution is slightly narrower, peaks around <|v|> ≈ +3 ... +4 km s^-1 and has a tail extending well beyond +10 km s^-1.The situation is reversed in the chromosphere (logτ_Ross = -6.0, see Fig. <ref>i) compared to the convection zone. In the chromosphere, slightly more grid cells exhibit downward velocities due to material falling down after having been lifted up by previous shock wave trains. The downward velocity part of the distribution peaks around<|v|> ≈ -10 ... -7 km s^-1 with extreme values reaching -20 km s^-1. Consequently, the arithmetic velocity average is negative, namely <|v|> ≈ -2 km s^-1. The upward velocity part of the distribution, which is due to upwards propagating shock waves, peaks around <|v|> ≈ +7 ... +10 km s^-1 with extreme values reaching +20 km s^-1. In summary, the model chromosphere is found to be highly dynamic with the matter often travelling atsupersonic speeds and the whole atmospheric structure thus varying significantly on rather short time scales. At a typical speed of 10 km s^-1, a shock wave (or equivalently downfalling material) propagates a distance of 10^9 km, which is on the order of the thickness of the modelled chromospheric layer, in on the order of 10^5 s.§.§ Properties of the radiation fieldSpectral energy distributions. A comparison of SEDs computed in Sect. <ref> and shown in Fig. <ref> reveals that the red giant model with the chromosphere emits significantly more flux in the UV (<270 nm) than does the model without it. This is caused by the increasing average emissivity temperature towards the lower optical depths in the chromosphere. Obviously, the chromosphere makes an important contribution towards the emergent flux in the UV so that the model without a chromosphere naturally produces a significantly lower UV flux than the model with a chromosphere. This expected finding illustrates the need for detailed models that incorporate all atmospheric layers. Continuum intensity maps. In Figure <ref> continuum intensity maps at different wavelengths for the same model snapshot as in Figs. <ref> and <ref> are presented. All maps represent a synthetic observation at disk center (μ =1.0), i.e., as if seen spatially resolved from top. The maps for the wavelengths 300 nm, 500 nm, 800 nm, 1.6 μm all clearly exhibit the granulation pattern in the model although mapping slightly different height ranges around log = 0. Consequently, the contrasts of the maps decrease from53.85 % at λ = 300 nm to only 10.93 % at λ = 1.6 μm. The average intensity is shown as function of wavelengths in Fig. <ref>f. It has a peak at 500 nm.The continuum intensity at millimeter wavelengths is another promising way to map the chromospheric layers. The large diagnostic potential of the Atacama Large Millimeter/submillimeter Array (ALMA) for this purpose has been demonstrated for the Sun <cit.> and other stars, e.g., α Cen <cit.> and Mira <cit.>.In Fig. <ref>e a corresponding intensity map for a wavelength of 1.0 mm is shown. In contrast to the other maps at shorter wavelengths, the continuum intensity at 1.0 mm emerges from the chromosphere. The map therefore exhibits a filamentary pattern with apparent cells with diameters on the order of 1-2·10^9 m and filament widths on the order of 0.1·10^9 m.The pattern in intensity corresponds to the chromospheric shock fronts and cooler post-shock regions.Spectral lines. In Figure <ref>, the consequences of the intermittent nature of the model chromosphere are illustrated for a few commonly used spectral lines, namely Ca II K, the Ca II infrared triplet line at λ = 854 nm, and Hα. The figure shows spatially averaged spectral line profiles and corresponding value ranges next to intensity maps for the line cores. Since the line cores are expected to form highest in the model chromosphere, the resulting line core maps should exhibit the intermittent structure already seen directly in the maps for gas temperature, pressure and velocity (see Fig. <ref>g-i).However, the different spectral lines are sensitive to different aspects of the model chromosphere due to non-linear dependencies in the line formation processes. This effect is particular obvious when comparing the Ca II line core maps in Fig. <ref>c and f with the continuum intensity map at λ = 1.0 mm (Fig. <ref>). The shock-induced pattern of bright filamentary threads and dark regions is as clearly seen in the mm map as in the gas temperature map. The Ca II K map emphasizes more the hottest filamentary parts, whereas the picture is less clear in theCa II 854 nm line core map. The contrast, i.e., the standard deviation of the quantity divided by its average, for the chromospheric gas temperature as shown in Fig. <ref>g is 14.9 %, whereas the contrasts for the Ca II K line and the Ca II 854 nm line core intensity maps are274.6 % and 64.5 %, respectively.The mm radiation provides a more direct measure of the actual gas temperatures in the model chromosphere, which can be calculated under the assumption of local thermodynamic equilibrium (LTE), whereas the Ca II line cores are subject to non-LTE effects and a more complicated relationship between the state of the atmospheric gas and the line core intensity. The spectral line profiles therefore vary significantly for the different locations. The intensity range covered by the individual Ca II K line profiles is illustrated in Fig. <ref>. For some horizontal positions, the line profiles exhibit Ca II K2 peaks in the core and equivalent for the Ca II 854 nm line core as can be seen from the upper 99 % percentile (upper dashed lines in panels b and e, respectively). However, such features are not as clearly visible for many other positions sothat, despite the large spatial variations, the spatially averaged line profile does not exhibit a strong central emission reversal peak. While this makes it difficult to deduce the existence of shock waves from averaged spectra, as they would be observed, it obviously does not mean that shock waves do not exist. Fitting an observed spectrum with a one-dimensional model atmosphere would in this case lead to wrong conclusions by underestimating the extent to which shock waves occur.The last row in Fig. <ref> shows the synthetic intensities for the Hα for the same model snapshot as discussed above. This spectral line is commonly used aschromospheric diagnostics and asactivity indicator. Observations in Hα on the Sun typically exhibit a complicated topology with fibrils that essentially outline the magnetic field<cit.>. The lack of such fibrils in the Hα line core map for the red giant model shown in Fig. <ref>i is not unexpected because no magnetic fields are included in the model. Instead, the resulting intensity maps are dominated by photospheric contributions and thus mostly show the granulation pattern, which can be seen in the continuum intensity images in Fig. <ref>, too. The information that can be retrieved from analysing the Hα line for this particular non-magnetic model is therefore quite different from the usual diagnostic use of the Hα line. § DISCUSSION§.§ Preliminary comparison to observationsIn Figure <ref>, the synthetic spectra for the presented red giant model are compared to observations ofAldebaran (α Tau), namely HARPS data[The HARPS data were obtained from .]<cit.>, UVES and VUES <cit.> data. A pipeline-reduced UVES spectrum (R∼80 000) was taken from the UVES POP spectral library <cit.>.It should be noted that are remaining uncertainties regarding the scale as compared to the synthetic spectrum due to the difficulties in precisely determining the continuum in the observed spectrum. The VUES spectrum was obtained at Moletai Astronomical Observatory (Lithuania) using the 1.65 m telescope and Vilnius University Echelle Spectrograph <cit.> in high-resolution mode (R∼67 000), taking 4 × 60 sec exposures. The spectrum was reducing using the standard VUES pipeline procedure, and was further continuum-normalized using a synthetic 1D LTE spectrum of Aldebaran. Apart from omission of the line blends in the synthetic spectra, there are substantial differences in terms of spectral line depths and widths for all three spectral lines, which are not unexpected for several reasons. First of all, the numerical model has an effective temperature of T_eff≈ (4010 ± 13) K, which is thus 83 K higher than the 3927 K for Aldebaran as stated by <cit.> <cit.>. More importantly, the exploratory model is still lacking physical ingredients that would influence the shapes of spectral lines formed in the chromosphere. In particular, the numerical simulation is purely hydrodynamic and thus represents an artificially quiet state while magnetic fields have recently been detected in α Tau and similar other stars<cit.>. An ultimate next, although computationally expensive step, will be the inclusion of a magnetic field, non-equilibrium ionization and excitation of hydrogen and other species, which should lead to more realistic models and thus may better fit the observational data (see also Sect. <ref>). Despite this, one should also keep in mind that modeling of hydrogen line formation in red giant atmospheres is anotoriously difficult task and that current models are consequently still not capable of reproducing observational data satisfactorily <cit.>. It may therefore still take some time until sufficiently realistic models will become available. Furthermore, the model presented here only includes a photosphere and a chromosphere whereas an observed spectrum naturally contains possible contributions from a corona,circumstellar gas, incl. stellar wind <cit.>, which in principle could (partially) account for differences between the modelled and observed spectra. It is nevertheless interesting to see that already the simplified but dynamic model chromosphere exhibits downward velocities that often exceed the upward velocities.Consequently, the line cores are accordingly shifted for lines of sight with a strong downward velocity component, which also leaves an imprint in the averaged spectra in Fig. <ref>. This effect is known from the solar spectrum <cit.>. A small net downward velocity on the order of 1-2 km/s has also been detected in the low chromosphere of α Tau <cit.>, which may hint at the existence of shock waves. More detailed, quantitative comparisons with observations are essential for the further improvement of numerical models and thus ultimately for a correct interpretation of the observations. For this purpose, complementary diagnostics that probe different aspects of stellar atmospheres should be combined. That includes different spectral lines and continua that are formed in different atmospheric layers and that are sensitive to different properties of the atmospheric gas (e.g., density, temperature, magnetic field).The centre-to-limb variation as far as it can be derived from stellar observations is another important test. For instance, <cit.>used lunar occultations of α Tauto investigatethe limb-darkening characteristics andfound substantial asymmetries in the photospheric brightness profile.Such measurements could be directly compared to corresponding synthetic observables (e.g., limb darkening profiles) as it will be presented in forthcoming publications.§.§ Average temperature stratification. As mentioned above, the shock waves are clearly visible as sawtooth-like profiles in the gas temperature and velocity (Figs. <ref>-<ref>). The shock wave signature, however, is intermittent and thus not visible in the horizontal arithmetic average of a single snapshot over even after averaging over a few time steps.The average gas temperature on the geometric height scale (z), as shown in Fig. <ref>a-b, appears to remain roughly constant throughout the model chromosphere. Like for the solar models, there is no distinct temperature minimum visible in the averaged temperature stratification. Also, averaging on a column mass scale (Fig. <ref>c-d) or optical depth scale (Fig. <ref>e-f) produces no substantial temperature rise. On the other hand, the shaded areas in all panels illustrate the large range of temperature values, which is most extreme when looking at the temperature as function of optical depth inFig. <ref>f. The extended value ranges, in line with the histograms in Fig. <ref>g, strongly suggest that the chromospheric gas temperature distribution cannot be described sufficiently by means of a simple arithmetic average alone. In the case of an atmosphere with significant spatial and/or temporal temperature variations, deriving an average temperature stratification by adjusting a model atmosphere to reproduce an intrinsically spatially average stellar spectrum only bears limited if not potentially misleading information. While this statement was proven to be true for the Sun (as described below), it should be noted that the model presented here is still of exploratory nature and more advanced simulations in the future are needed for a more realistic assessment of the thermal structure of red giant chromospheres.The non-linear dependence of the source function and thus the emergent intensity on the gas temperature, as it is for instance the case in the ultraviolet, shifts the derived average gas temperature towards high temperatures in shock fronts, which enter the average with a higher weight than the cold temperatures in the cooler post-shock regions. The average emissivity temperature <cit.>T_em (z) =⟨( ⟨κρ T^4 ⟩_x,y/⟨κρ⟩_x,y)^1/4⟩_ttakes this effect into account, although rather crudely. The average emissivity temperature is shown as red dot-dashed lines in all panels of Fig. <ref>. In case of the Sun, this effect explains apparently contradicting temperature stratifications derived from UV continua and lines on the one hand and infrared carbon monoxide lines on the other hand <cit.>.The high-temperature component of the chromospheric temperature distribution is less pronounced in the red giant model presented here as compared to (non-magnetic) solar models<cit.> so that the influence of hot chromospheric gas on the average temperature is not as strong.Nevertheless, the emissivity temperature average produces a slightly higher average temperature in the chromosphere on the geometrical height scale (Fig. <ref>b), a notable chromospheric temperature rise on the column mass scale (Fig. <ref>d), and temperatures that are higher by up to 2 100 K on the optical depth scale (Fig. <ref>f). The resulting difference between the averages on the optical depth scale is particularly remarkable because the average emissivity temperature exceeds the 95 % percentile in the uppermost layers, indicating that a minority of extraordinarily hot locations in the model chromosphere strongly influence the resulting average due to the non-linear relation between the observable intensity and the actual gas temperature. It is this effect that has to be considered in detail when trying to derive meaningful atmospheric temperature stratification from intrinsically averaged stellar observations. The model atmosphere derived by <cit.> for α Tau is shown in Fig. <ref> for comparison. The semi-empirical, plane-parallel, hydrostatic, one-component atmosphere model was constructed to reproduce observations of optical and ultraviolet photospheric and chromospheric spectral lines including, e.g., CaII H and K,CI, CII, SiII, MgII andCIV.There are small offsets of Δ z = 7 10^7 m and Δ lg m_c = -0.25, respectively, between the 3D numerical model and the McMurry model, which is not unexpected since the height of optical depth unity and the origin of the column mass scale vary for different horizontal positions in the 3D model so that the average scales depends on the exact way the average is calculated.On the column mass scale, the arithmetically averaged temperature from our simulation agrees well with the McMurry model for values of log m_c > -1. For lower values, i.e., above the low chromosphere the McMurry temperature stratification features a sharp chromospheric temperature rise and temperatures exceeding those in our model forlog m_c < -2 and z > 1.4 10^9 m, respectively. The McMurry model also includes a transition region at z = 6 10^9 m with a jump to 10^5 K which lies well above the upper boundary of the 3D simulation presented here. Although the 3D model does not exhibit a similarly strong temperature rise in the chromosphere, at least the average emissivity temperature begins to rise roughly in the same layer as the McMurry model although not as pronounced. As discussed above, plotting the temperatures on the optical depth scale[The optical depth scale for the McMurry model was calculated byconverting the column mass scale with the help of the relation between optical depth and column mass in the 3D model. The resulting scale should be reasonably accurate for preliminary comparisons as presented here.]results in a stronger apparent chromospheric increase of the average emissivity temperature. The corresponding 95 % percentile on the optical depth scale gives the impression of a sharp temperature rise similar to the McMurry model although this behaviour only applies the hottest chromospheric grid cells in the 3D model. Based on the exploratory 3D model, we conclude that (i) the chromospheric temperature rise has to be interpreted in view of potential averaging effects for a spatially intermittent atmosphere and that (ii) the discrepancy in temperature in the upper layers implies that the 3D simulations are still lacking physical mechanisms that are important for atmospheric heating. Overall the situation resembles the state of development of numerical model atmospheres of the Sun in the past for which the apparent controversy regarding the chromospheric temperature structure – in particular regarding its spatially and temporally intermittent nature – has been largely resolved by now<cit.>.§.§ Magnetic fields Magnetic fields have been omitted for the first simulation presented here but will be considered for future models. Based on experience withmodels with chromospheres for the Sunand other stellar types <cit.>, we expect that the inclusionof magnetic fields will change the chromospheric structure seen in the models significantly. While shock waves will also be excited in magnetohydrodynamic models, the details of their interaction with magnetic fields in the chromosphere will depend on the magnetic field strength and field topology in the model and change gradually from the non-magnetic shock-induced pattern like in the model presented here towards a magnetically dominated chromosphere with a mixture of open magnetic field lines and closed magnetic loops. Effects like, e.g, wave guiding, wave mode conversion, and initiation of torsional and rotational motions are expected and will not only affect the atmospheric structure but also the energy transport and thus the overall activity level of the model. In general, it is thought that stellar chromospheres are produced by a mixture of acoustic and magnetic heating mechanisms <cit.>. Although the results of the red giant simulation presented in this study reveal noticeable heating of its chromospheric layers due to shock waves, the importance of magnetic heating still needs to be assessed and will be considered in our future work. Interestingly, although we detect vigorous shock wave activity in the chromospheric layers of the red giant modeled here, red giants that are thoughtto be chromospherically inactive do not show spectroscopic signatures of shock waves <cit.>. This finding may suggest a magnetic origin of the heating of red giant chromospheres but, on the other hand, could simply mean that the shock signatures are by nature difficult to detect due to the intrinsic averaging over the whole stellar disk as illustrated for the Ca II lines in Fig. <ref>. On the other hand, there is observational evidence that suggests the presence of magnetic fields in the atmosphere of α Tau <cit.>, which thus motivates the development of magnetohydrodynamic 3D simulations of this star.§ CONCLUSIONSThe three-dimensional hydrodynamic model presented here is a first step towards more realistic model chromospheres for red giant stars that can support a meaningful in-depth analysis of stellar observations. Future models will include magnetic fields and eventually further physical ingredients like time-dependent hydrogen ionisation, which are important for the chromospheric gas properties and, e.g., the formation of a transition region (if existing for the modelled star) as natural upper boundary of a chromosphere.Despite the simplifications, the first model presented here already exhibits a very intermittent and dynamic chromosphere, which is in line with chromospheres modelled for other stellar types. 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"authors": [
"Sven Wedemeyer",
"Arunas Kucinskas",
"Jonas Klevas",
"Hans-Gunter Ludwig"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170526162743",
"title": "Three-dimensional hydrodynamical CO5BOLD model atmospheres of red giant stars VI. First chromosphere model of a late-type giant"
} |
Department of Physics and Astronomy, University of California – Los Angeles, Los Angeles, California, 90095, USAhas been identified as an attractive ion for quantum information processing due to the unique combination of its spin-1/2 nucleus and visible wavelength electronic transitions. Using a microgram source of radioactive material, we trap and laser-cool the synthetic A = 133 radioisotope of barium II in a radio-frequency ion trap. Using the same, single trapped atom, we measure the isotope shifts and hyperfine structure of the↔and↔electronic transitions that are needed for laser cooling, state preparation, and state detection of the clock-state hyperfine and optical qubits. We also report the↔electronic transition isotope shift for the rare A = 130 and 132 barium nuclides, completing the spectroscopic characterization necessary for laser cooling all long-lived barium II isotopes. Spectroscopy of a synthetic trapped ion qubit David Hucul, Justin E. Christensen,Eric R. Hudson,Wesley C. Campbell December 30, 2023 ============================================================================Since the demonstration of the first CNOT gate over 20 years ago <cit.>, trapped ion quantum information processing (QIP), includingquantum simulation,has developed considerably<cit.>, recently demonstrating fully-programmable quantum processors <cit.>. To date, qubits have been demonstrated in trapped ion hosts of all non-radioactive, alkaline-earth-like elements <cit.>. These ions possess a simple electronic structure that facilitates straightforward laser cooling as well as quantum state preparation, manipulation, and readout via electromagnetic fields.For the coherent manipulation of qubits, the phase of this applied electromagnetic field must remain stable with respect to the qubit phase evolution. Thus, atomic hyperfine structure is a natural choice for the definition of a qubit, as these extremely long-lived states can be manipulated with easily-generated, phase-coherent microwave radiation. In particular, qubits defined on the hyperfine structure of ions with half-integer nuclear spin possess a pair of states with no projection of the total angular momentum (F) along the magnetic field (m_F = 0). These so-called “clock-state" qubits are well-protected from magnetic field noise and can yield coherence times exceeding 10 minutes <cit.>. Further, for these species, F = 0 ground and excited states only occur when the nuclear spin I=1/2. This is desirable because the F^'=0 ↮F^''= 0 selection rule can be leveraged to produce fast, robust qubit state preparation and readout that relies solely on frequency selectivity <cit.>.Among the alkaline-earth-like elements, only three (Cd, Hg, Yb) have naturally occurring I = 1/2 isotopes. Mercury and cadmium ions require lasers in the deep ultraviolet portion of the electromagnetic spectrum, making it difficult to integrate them into a large-scale QIP architecture. Since ^171Yb^+ has the longest laser-cooling wavelength at 370 nm, it has been used in a wide variety of groundbreaking QIP experiments<cit.>. However, even at this ultraviolet wavelength, the use of photonics infrastructure developed for visible and infrared light is limited. For example, significant fiber attenuation limits the long-distance transmission of quantum information at 370 nm. Furthermore, in Yb^+, the short lifetime of the 5^2D_5/2 manifold (7 ms<cit.>), along with decays to a low-lying ^2F_7/2 manifold, complicate state-selective shelving of the hyperfine qubit with ultra-high fidelity readout and direct manipulation of an optical qubit <cit.>.A possible remedy to these problems exists in the synthetic A = 133 isotope of barium (τ_1/2 = 10.5 years), which combines the advantages of many different ion qubits into a single system.has nuclear spin I=1/2, allowing fast, robust state preparation and readout of the hyperfine qubit; metastable D states (τ ≈ 1 min), allowingultra-high fidelity readout <cit.>; and long-wavelength transitions enabling the use of photonic technologies developed for the visible and near infrared spectrum.Here, we demonstrate loading and laser-cooling ofatomic ions from a microgram source of barium atoms. We measure the previously unknownisotope shift of the↔transition and thehyperfine constant of thestate. Our measurements of the other spectroscopic features ofare in agreement with earlier measurements <cit.>. In addition, using the same techniques, we measure and report the isotope shifts of the↔transition in the rare ^130Ba^+ and ^132Ba^+ species.For this work, barium ions are confined using a linear radio frequency (rf) Paul trap. The minimum distance between the trap axis and the electrodes is 3 mm and the trap operates with a peak-to-peak rf voltage V_pp= 200 V at frequency Ω≈ 2π×1 MHz. Each electrode can be independently DC biased allowing for the compensation of stray fields and the ejection of trapped ions into a laser-cooling-assisted mass spectrometer (LAMS) <cit.>. Laser cooling of barium ions is accomplished with wavelengths near 493 nm and 650 nm. These lasers enter separate fiber electro-optic modulators (EOMs) with 6 GHz bandwidth and are delivered to the experiment via single-mode optical fibers. The EOMs are used to provide frequency sidebands on the laser spectrum, which allow cooling and/or heating multiple isotopes simultaneously, as well as for addressing the necessary transitions due to hyperfine structure in I≠ 0 isotopes (see Table <ref>). An applied magnetic field of a few Gauss along withlaser beams linearly polarized≈ 45^∘ from the magnetic field direction are used to destabilize dark states that result from coherent population trapping <cit.>.A source of ^133Ba atoms is produced by drying a commercially availablesolution of neutron activated BaCl_2 dissolved in 0.1 M HCl on a platinum ribbon substrate. The vendor reports that approximately2% of the total barium atoms are ^133Ba <cit.>. Atomic ion fluorescence and a LAMS spectrum indicate a highly enriched source of ^132Ba atoms due to the manufacturing process. The platinum ribbon substrate is ≈ 4 mmfrom the edge of the trap in the radial direction, and near the center of the trap axially. A 532 nm, 0.4 mJ, 5-7 ns laser pulse produces ions by ablating the barium on the platinum ribbon substrate. Turning on the rf voltage 10 μs after laser ablation confines ions in the ion trap.Overlapped cooling and repumping beams enter the trap at an angle of 45^∘ and 0^∘ with respect to the axial direction of the ion trap.To Doppler cool , a laser near 493 nm is slightly red-detuned (≈ 30 MHz) from the , F = 0 ↔ , F = 1transition, denoted ν_0^b in Fig. <ref>a. Transitions between the , F = 0 ↔ , F = 0 are forbidden, but off-resonant scattering via the , F = 1 states leads to population trapping in the , F = 0 state. To depopulate this state, the 493 nm fiber EOM is driven at ν_0 = 5.872 GHz resulting in a second-order sideband resonant withthe , F = 1 ↔ , F = 0 transition. A re-pumping laser near 650 nmis slightly red-detuned of the, F = 0 ↔ , F = 1 transition, denoted ν_0^r (see Fig. <ref>a). Transitions between the , F = 0 ↔ , F = 2 are dipole-forbidden, but decay from the, F=1 states populates the , F = 2 states. The off-resonant scatter rate out of the , F=2 states, from the applied laser frequency ν_0^r,is greater than the decay rate into the state due to off-resonant scatter from the application of laser frequency ν_0^b. Therefore, only the three frequencies ν_0^b, ν_1^b, and ν_0^r are required to cool and crystallize . To improve cooling,the 650 nm fiber EOM is driven at 904 MHz resulting in a first order sideband red-detuned from the, F = 1 ↔ , F = 2 transition, denoted ν_1^r in Fig. <ref>a.During laser ablation, other ions (here, mainly ^132Ba^+ due to their high abundance in our source) tend to be co-trapped with . Because thehyperfine qubit splitting ofis much larger than the isotope shift of the↔ transition in all Ba^+ isotopes, we are able to utilize a single high bandwidth fiber EOM to simultaneously laser coolwhile laser-heating any even barium isotopes out of the ion trap (see Fig. <ref>c). Additional laser sidebands can be used to laser-heat the odd isotopes out of the ion trap using the↔transitions, although in practice infrequent loading rates of these species from the neutron activated BaCl_2 microgram source rarely require this. Other chemical species with significantly different charge to mass ratio can be ejected from the ion trap by ramping the trap voltages. ^133Ba decays to form ^133Cs with a half-life of 10.5 years. Since ^133Ba and ^133Cs have nearly identical masses, trap filtration based on charge to mass ratio cannot be used to separate them. By monitoring thermionic emission from a heated platinum filament, we find that ^133Cs can easily and regularly be preferentially removed from a Ba source in situ. The technique of isotopic purification via isotope-selective heating appears to be effective at removing unwanted ions without any observable loss of the desired species. Detailed molecular dynamics simulations of the process have not revealed anyloss of the target ion, even when co-trapped with 499 ions undergoing laser-heating. This is critical for working with radioactive isotopes as it allows the use of non-isotope-selective loading techniques, like laser ablation, to be used. As shown in Fig. <ref>a, the magnetic moment of the I=1/2nucleus splits each fine-structure state byℋ = 𝒜I⃗·J⃗, where 𝒜 is the magnetic hyperfine constant associated with each fine structure state. The hyperfine splittings of the , , andlevels ofwere measured with the same atomic ion and are shown in Fig. <ref>. These spectra were obtained by using a modular digital synthesis platform <cit.> to rapidly alternate between Doppler cooling and weak optical excitation for fluorescence spectroscopy to prevent laser-induced lineshape distortions <cit.>. All measurements have a ±20 MHz systematic uncertainty primarily due to drift of the wavemeter used to stabilize the lasers. To measure thehyperfine splitting (Fig. <ref>a), a laser sideband frequency near the , F = 1 ↔ , F = 1 transition is scanned. When this frequency is near resonance, and without laser frequency ν_1^r, the population of the , F = 2 states is increased. We utilize the resulting decrease in fluorescence to measure thehyperfine splitting Δ_2 = 1840(2)_stat MHz (see Fig. <ref>b). To measure thehyperfine qubit splitting,the laser sideband ν_1^b near the , F = 1 ↔ , F=0transition is scanned. The fluorescence is maximized when 2ν_0 = Δ_1 + Δ_2 (see Fig. <ref>a). We measure the hyperfine qubit splitting Δ_1 = 9931(2)_stat MHz. In order to measure thehyperfine splitting, we increase the population of the , F = 2 manifold by applying a laser sideband at frequency ν_0^b - Δ_2. The fluorescence is maximized when the laser sideband ν_1^r = ν_0^r+Δ_3 - Δ_2 (see Fig. <ref>c). We measure Δ_3 = 937(3)_stat MHz.Efficient laser cooling of the ion also requires knowledge of the electronic transition frequencies. We measure these transitions inusing the values of the measured hyperfine splitting and scanning ν_0^b and ν_0^r. Defining the isotope shift of the i-th electronic transition δν_A,138^i≡ν_A^i - ν_138^i, with i = b (r) for the transitions near 493 nm (650 nm), we measure the isotope shifts in to obtain δν^b = 355(4)_stat MHzand δν^r =198(4)_stat MHz.With these data, the transition frequencies necessary for laser cooling and hyperfine qubit operation are now characterized for all isotopes of Ba^+ with half-life greater than a few weeks, and are shown in Table <ref>. Since all of these transitions are resolved and are simultaneously addressable using a broadband, fiber-coupled EOM,isotopic purification is possible in situ through laser heating. This allows for the production of single-species Coulomb crystals, even for trace species, as shown in Fig. <ref>b.Finally, the isotope shifts can be decomposed into two termsδν_A,A'^i = k_MS^i(1/A-1/A' ) + F_i λ_A,A'where k_MS is the sum of the normal and specific mass shifts, F_i is the field shift <cit.>, and λ_A,A' is the Seltzer moment of the nuclei of isotopes A and A'<cit.>. To lowest order, the Seltzer moment λ_A,A' is equal to the difference in the mean of the squared nuclear charge radii of an isotope pair: δ⟨ r^2 ⟩_A,A'=⟨ r_A^2 ⟩ - ⟨ r_A'^2 ⟩<cit.>. FollowingEqn. <ref>, a King plot, shown in Fig. <ref>,summarizes spectroscopic data for barium atomic ionsalong with our measurements of δν_130,138^r, δν_132,138^r, and δν_133,138^r. Using previous spectroscopic data <cit.>, the fitted slope of -0.26is close to a theoretical calculation of the slope -0.288 <cit.>. The fitted slope, field and specific mass shifts of 988 MHz/fm^2 and 360 MHz respectively <cit.>, and the new measurement of δν_133,138^r are combined to determine δ⟨ r^2 ⟩_133,138 = -0.104 fm^2.In summary, we have demonstrated trapping ofatomic ions produced via laser ablation of a microgram source.By leveraging the frequency selectivity of laser heating and cooling, we isotopically purify the trap sample to achieve efficient laser cooling of trappedions. Using the same, single trappedion we have measured the previously unknownhyperfine splitting Δ_3=937(3)_statMHz and isotope shift of the↔transition δν_133,138^r = 198 (4)_stat MHz. These measurements all have a ±20 MHz systematic uncertainty. The determination of these spectroscopic values along with the methods we have presented for trap loading from micrograms of radioactive material should enable the use offor trapped ion QIP.The advantages thation qubits promise over other species used for QIP are largely due to a unique combination of a nearly ideal atomic structure and the wavelength constraints of practical optical systems. First, unlike other ions hosting M = 0 clock-state qubits, the optical transitions that must be addressed to useare all in the visible and near IR, allowing the integration of photonic technologies – such as the fiber EOMs used in this work and very long optical fibers for quantum communication – that do not perform well in the UV. Second, the spin-1/2 nucleus ofproduces a hyperfine clock-state qubit that can be initialized and detected quickly using frequency-selective optical transitions <cit.>. Third, the unusually long-livedstate inshould allow both ultra-high fidelity state-selective shelving detection <cit.> and clock-state optical qubit operation. Therefore,ions provide robust hyperfine and optical frequency qubits in the same system, allowing use of the full suite of trapped ion entangling gates in a single species, and represent an attractive candidate for future trapped ion QIP. This work was supported by the US Army Research Office under award W911NF-15-1-0273. We thank Rainer Blatt, Jungsang Kim, Michael Mills, Chris Monroe, and Prateek Puri for helpful discussions. We thank Tyler Jackson, Saed Mirzadeh, Anthony Ransford, Christian Schneider, Calvin Ye, and Peter Yu for technical assistance. | http://arxiv.org/abs/1705.09736v1 | {
"authors": [
"David Hucul",
"Justin E. Christensen",
"Eric R. Hudson",
"Wesley C. Campbell"
],
"categories": [
"quant-ph",
"physics.atom-ph"
],
"primary_category": "quant-ph",
"published": "20170526230726",
"title": "Spectroscopy of a synthetic trapped ion qubit"
} |
IEEEexample:BSTcontrol Stochastic Feedback Control of Systemswith Unknown Nonlinear Dynamics^* Dan Yu^1, Mohammadhussein Rafieisakhaei^2 and Suman Chakravorty^1 *This material is based upon work partially supported by NSF under Contract Nos. CNS-1646449 and Science & Technology Center Grant CCF-0939370, the U.S. Army Research Office under Contract No. W911NF-15-1-0279, and NPRP grant NPRP 8-1531-2-651 from the Qatar National Research Fund, a member of Qatar Foundation, AFOSR contract Dynamic Data Driven Application Systems (DDDAS) contract FA9550-17-1-0068 and NSF NRI project ECCS-1637889. ^1D. Yu and S. Chakravorty are with the Department of Aerospace Engineering, and ^2M. Rafieisakhaei Electrical and Computer Engineering, Texas A&M University, College Station, Texas, 77840 USA. {[email protected], mrafieis, [email protected]}Version December 30, 2023 =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== This paper studies the stochastic optimal control problem for systems with unknown dynamics. First, an open-loop deterministic trajectory optimization problem is solved without knowing the explicit form of the dynamical system. Next, a Linear Quadratic Gaussian (LQG) controller is designed for the nominal trajectory-dependent linearized system, such that under a small noise assumption, the actual states remain close to the optimal trajectory. The trajectory-dependent linearized system is identified using input-output experimental data consisting of the impulse responses of the nominal system. A computational example is given to illustrate the performance of the proposed approach. § INTRODUCTIONStochastic optimal control problems, also known as Markov Decision Processes (MDPs) have found numerous applications in the Sciences and Engineering. In general, the goal isto control a stochastic system so as to minimize the expected running cost of the system.It is well known that the global optimal solution for MDPs can be found by solving the Hamilton-Jacobi-Bellman (HJB) equations <cit.>. The solution techniques can be further divided into model-based and model-free techniques according to whether the method uses an analytical model of the system, or it uses a black box simulation model or real experiments.In model-based techniques, many methods rely on a discretization of the underlying state and action space <cit.>, and hence, run into the curse of dimensionality, the fact that the computational complexity grows exponentially with the dimension of the state space of the problem. The most computationally efficient approach among these techniques are trajectory-based methods, first described in <cit.>. These methods linearize the nonlinear system equations about a deterministic nominal trajectory and perform a localized version of policy iteration to iteratively improve the policy. For instance, the Differential Dynamic Programming (DDP) <cit.> linearizes the dynamics and the cost-to-go function around a given nominal trajectory, and designs a local feedback controller using DP. The iterative LQG (iLQG) <cit.>, which is closely related to DDP, considers the first order expansion of the dynamics (in DDP, a second order expansion is considered), and designs the feedback controller using Riccati-like equations, which is shown to be computationally more efficient. In both approaches, the control policy is executed to compute a new nominal trajectory, and the procedure is repeated until convergence. Alternatively, the Trajectory-optimized LQG (T-LQG) approach <cit.> was recently proposed by us, which shows that under a first order approximation of the dynamics and cost-to-go function, a near optimal solution can be found by first solving a deterministic trajectory optimization problem, followed by a linear time-varying closed-loop controller design problem. This separated approach can also be extended to the model-free situation, which is the subject of this paper: we use a gradient descent algorithm, and a Linear Time-Varying (LTV) system identification technique, in conjunction with a black box simulation model of the process in order to accomplish the “separated" design.In the model-free case, the most popular approaches in the community are the Adaptive Dynamic Programming (ADP) <cit.> andReinforcement Learning (RL) paradigms <cit.>. They are essentially the same in spirit and seek to improve the control policy for a given black box system by repeated interactions with the environment while observing the system's responses. The repeated interactions, or learning trials, allow these algorithms to construct a solution to the DP equation, in terms of the cost-to-go function, in an online and recursive fashion. Another variant of the RL techniques is the so-called Q-learning method, where the basic idea is to estimate a real-valued Q(x, a) function of states, x, and actions, a, instead of the cost-to-go function, V(x). For continuous state and control space problems, the cost-to-go functions and the Q-functions are usually represented in a functionally parameterized form; for instance, in the linearly parametrized form Q(x,a) = θ^T ϕ(x,a), where θ is the unknown parameter vector, and ϕ isa pre-defined basis function. Multi-layer neural networks may also be used as nonlinearly parameterized approximators instead of the linear architecture above. The ultimate goal of these techniques is the estimation/learning of the parameters θ from learning trials/repeated simulations of the underlying system. However, the size of the parameter θ grows exponentially in the size of the state space of the problem without a compact parametrization of the cost-to-go or Q function in terms of the a priori chosen basis functions for the approximation, and hence, these techniques are typically subject to the curse of dimensionality. Albeit a compact parametrization may exist, a priori, it is usually never known.In the past several years, techniques based on the DDP/iLQG approach <cit.>, such as the RL techniques <cit.> have shown the potential for RL algorithms to scale to higher dimensional continuous state and control space problems, in particular, high dimensional robotic task planning and learning problems. These methods are a localized version of the policy gradient <cit.> technique that seek to directly estimate the feedback policy via a compact parameterization. For continuous state and control space problems, the method of choice is to wrap an LQR feedback policy around a nominal trajectory and then perform a recursive optimization of the feedback law, along with the underlying trajectory, via repeated simulations/iterations. However, the parametrization can still be very large for partially observed problems (O(d^2)) where d is the dimension of the state space) or large motion planning problems such as systems governed by partial differential equations wherein the (discretized) state is very high dimensional (millions of states). Furthermore, there are convergence problems with these techniques that can lead to policy chatter <cit.>.Fundamentally, rather than solve the derived “Dynamic Programming" problem as in the majority of the approaches abovethat requires the simultaneous optimization of the feedback law and the underlying trajectory, our approach is to directly solve the original stochastic optimization problem in the “separated open loop/closed loop" fashion wherein:1) we solve an open loop deterministic optimization problem to obtain an optimal nominal trajectory in a model-free fashion, and then 2) we design a closed loop controller for the linearized time-varying system around the nominal trajectory, again in a model-free fashion.This “divide and conquer" strategy is nonetheless theoretically sound as shown in the companion paper <cit.>.The primary contributions of the proposed approach are:1) Compared to other RL and ADP techniques, implementation using the proposed approach is simple. The stochastic optimal control problem is separated into two sub-problems: deterministic open-loop trajectory optimization problem and a linear time-varying system identification problem, and in each sub-problem, standard approaches can be used. The open-loop optimization problem is solved using gradient descent and input perturbations. The linearized system is identified via the time-varying ERA <cit.>, using the impulse responses of the nominal system, and an LQG controller is designed for the resulting linearized system. All of the above is accomplished by only considering a sequence of open loop impulse responses of the unknown system.2) Unlike other ADP and RL techniques, we specify a detailed set of experiments to accomplish the closed loop controller design for the unknown nonlinear system. This series of experiments consists of a sequence of input perturbations to collect the impulse response of the system, first to find an optimized nominal trajectory, and then to recover the LTV system corresponding to the perturbations of the nominal system in order to design the LQG controller corresponding to the LTV system.3) Albeit not covered in this paper due to a lack of space, in general, for large scale systems with partially observed states, the time-varying ERA constructs a reduced order model (ROM) of the LTV system, and hence, results in a reduced order estimator and controller. For example, the computational complexity of designing the LQG controller using the proposed approach is O(n_r^3), while it is O(n_x^3) for the original system, where n_r is the order of the reduced model,and n_r ≪ n_x. Therefore, for large scale systems such as partially observed systems and systems governed by PDEs, the online implementation of the LQG policy using the proposed approach is still computationally tractable. The rest of the paper is organized as follows. Section <ref> formulates the problem. In Section <ref>, we propose a separation-based stochastic optimal control algorithm with discussions of implementation. Last, in Section <ref>, wetest the proposed approach using the inverted pendulum problem. § PROBLEM FORMULATION Consider a discrete time nonlinear dynamical system:x_k + 1 = f(x_k, u_k, w_k), y_k= h(x_k ,v_k),where x_k ∈ℝ^n_x,y_k ∈ℝ^n_y, u_k ∈ℝ^n_u are the state, measurement, and the control vectors at time k, respectively, process model f(·) and measurement model h(·) are nonlinear, the process noise w_k and measurement noise v_k are assumed to be zero-mean, uncorrelated Gaussian white noise with covariances W and V, respectively.We assume the system is fully observed: y_k = h(x_k, v_k) = x_k . We make Assumption 1 to simplify the treatment of the problem; since otherwise, the stochastic control problem needs to be treated as a partially observed MDP (POMDP). This generalization may be done in a reasonably straightforward fashion (later discussed in Remark <ref>). Stochastic Control Problem: For the system with unknown nonlinear dynamics, f(·) , the optimal control problem is to find the control policies π = {π_0, π_1, ⋯, π_N -1} in a finite time horizon [0, N], where π_k is the control policy at time k, i.e., u_k = π_k(x_k),to minimize the cost functionJ_s = E(∑_k = 0^N - 1 (x_k^T Q_k x_k + u_k^T R_k u_k) + x_N^T Q_N x_N),where Q_k, Q_N≻ 0 and R_k≽ 0. § STOCHASTIC FEEDBACK CONTROL ALGORITHMWe compute a locally optimal solution to the stochastic control problem in a separated open loop/closed loop (SOC) fashion, i.e., we first solve a noiseless open-loop optimization problem to find a nominal optimal trajectory. Next,we design a linearized closed-loop controller around the nominal trajectory, such that, with existence of stochastic perturbations, the state stays close to the optimal open-loop trajectory. The separation-based approach has always been used by Control Engineers in Aerospace Guidance and Robotics problemsin a heuristic fashion <cit.>. However, our recent companion work <cit.>, using the theory of Large Deviations, shows that this separation, results in a near-optimal policy in the small noise case. Moreover, experimental results confirm its validity for moderate noise levels. The open loop optimization problem could be solved using a general nonlinear programming solver without knowing the explicit form of the underlying dynamics, i.e., it only accesses a black box simulation model of the dynamics. Next, we perform small input perturbations about the nominal trajectory in order to obtain the impulse responses of the LTV system governing the perturbations form the nominal trajectory, and identify the resulting linear time-varying derivation system from these impulse responses using time-varying Eigensystem Realization Algorithm (ERA) <cit.>. We consider quadratic cost functions, and design an LQR controller which results in an optimal linear control policy around the nominal trajectory. We discuss each of the above steps in the following section.§.§ Open Loop OptimizationConsider the noiseless nonlinear system:x_k+ 1 = f(x_k, u_k, 0), y_k = x_k,with known initial state x_0, and let the N-step cost functionJ_d (x_0, {u_k}_k = 0^N -1) =∑_k = 0^N - 1(x_k^T Q_k x_k + u_k^T R_k u_k)+ x_N^T Q_N x_N.The open loop optimization problem is to find the control sequence {u̅_k }_k = 0^N - 1, such that for a given initial state x_0,{u̅_k }_k = 0^N - 1 = min_{u_k}_k =0^N - 1 J_d ( x_0, {u_k}_k = 0^N -1),s. t.x_k + 1 = f(x_k, u_k, 0),y_k= x_k. Theproblem is solved using the gradient descent approach <cit.>,and the procedure is illustrated as follows. Starting from an initial guess of the control sequence U^(0)={u_k^(0)}_k = 0^N - 1, the control policy is updated iteratively viaU^(n + 1) = U^(n) - α∇_U J_d(x_0, U^(n)),until a convergence criterion is met, where U^(n) = {u_k^(n)}_k = 0^N - 1 denotes the control sequence in the n^th iteration, and α is the step size parameter. The gradient vector is defined as:∇_U J_d(x_0, U^(n)) =[ ∂ J_d/∂ u_0, ∂ J_d/∂ u_1, ⋯, ∂ J_d/∂ u_N - 1 ]|_x_0, {u_k ^(n)}_k = 0^N - 1,and without knowing the explicit form of the cost function, each partial derivative with respect to the i^th control variable u_i is calculated as:∂ J_d/∂ u_i|_x_0, U^(n)= 1/h(J_d(x_0, u_0^(n), u_1^(n),⋯, u_i^(n)+ h,⋯, u_N - 1^(n)) - J_d(x_0, u_0^(n), u_1^(n),⋯, u_i^(n),⋯, u_N - 1^(n))),where h is a small constant perturbation. Algorithm <ref> summarizes the gradient descent approach. The open loop optimization problem is solved using a black box model of the underlying dynamics, with sequence of input-output tests. Higher order approaches other than gradient descent are possible <cit.>, however, for a general system with complex cost functions, the gradient descent approach is easy to implement and is amenable to very large scale parallelization.§.§ Linear Time-varying System IdentificationWe linearize the system (<ref>) around the optimal nominal control and its corresponding state trajectory denoted by {u̅_k}_k = 0^N - 1 and {x̅_k}_k = 0^N, respectively, as: δ x_k+ 1 = A_k δ x_k + B_k δ u_k + G_k w_k,δ y_k= C_k δ x_k + F_k v_k,where δ x_k = x_k- x̅_k describes the state deviations from the nominal trajectory, δ u_k = u_k - u̅_k describes the control deviations,δ y_k = y_k - h(x̅_k, 0) describes the measurement deviations, andA_k =∂ f(x, u, w)/∂ x|_x̅_k, u̅_k, 0, B_k =∂ f(x, u, w)/∂ u|_x̅_k, u̅_k, 0, G_k =∂ f(x, u, w)/∂ w|_x̅_k, u̅_k, 0,C_k =∂ h(x, v)/∂ x|_x̅_k, 0,F_k =∂ h(x, v)/∂ v|_x̅_k, 0. Given the LTV system { A_k, B_k, C_k, G_k, F_k }between time [0, N-1],an LQR controller to track the nominal trajectory could be designed. However, since the dynamics are unknown, we first need to identify the LTV system.The time-varying ERA is used to construct a state space realization (Â_k, B̂_k, Ĉ_k , Ĝ_k, F̂_k } of system (<ref>). The state space realization is constructed using input and output experimental data and is shown to be minimal and balanced. The details of the time-varying ERA can be found in <cit.> and is briefly summarized next. Define the generalized Markov parameters h_k, j as:h_k, j=C_k A_k - 1 A_k - 2⋯ A_j+1 B_j, ifj < k - 1,C_k B_k -1,ifj = k - 1, 0, ifj > k - 1, and the generalized Hankel matrix as: H_k^(p, q)_pn_y × qn_u=[h_k, k-1h_k, k-2 ⋯h_k, k - q;h_k + 1, k - 1 h_k + 1, k -2 ⋯h_k + 1, k-q; ⋮ ⋮ ⋯ ⋮; h_k+ p- 1, k -1 h_k + p - 1, k -2 ⋯ h_k + p - 1, k -q ],with design parameters p and q. The time-varying ERA starts with an estimation of the generalized Markov parameters from input-output data using least squares solution as follows. Consider system (<ref>) with zero noise and for simplicity, assume δ x_0 = 0. Run M simulations and in the i^th simulation, choose input sequence {δ u_t^i }_t = 0^k, and collect the output δ y_k^i. The superscript (·)^i denotes the experiment number.From the input-output map, the generalized Markov parameters { h_k, j}_j = 0^k could be recovered via solving the least squares problem: [ δ y_k^1 δ y_k^2 ⋯ δ y_k^M ]= [ 0h_k, k-1 h_k, k- 2 ⋯h_k, 0 ]×[ δ u_k^1 δ u_k^2 ⋯ δ u_k^M;δ u_k -1^1δ u_k -1^2 ⋯ δ u_k - 1^M; ⋮ ⋮ ⋮; δ u_0^1 δ u_0^2 ⋯ δ u_0^M ], where M is a design parameter and is chosen such that the least squares solution is possible. After recovering the generalized Markov parameters, two Hankel matrices H_k^(p, q) and H_k + 1^(p, q) are constructed using (<ref>), and here, the design parameters p and q are chosen such that min{ p n_y, q n_u }≥ n_x, and could be tuned for best performance. Then we solve the singular value decomposition problem: H_k^(p, q) = U_k Σ_k^1/2_O_k^(p)Σ_k^1/2 V_k^T_R_k-1^(q).Suppose the rank of the Hankel matrix H_k^(p, q) is n_r, where n_r ≤ n_x. ThenΣ_k ∈ℝ^n_r × n_r is the collection of all non-zero singular values,and U_k ∈ℝ^p n_y × n_r, V_k ∈ℝ^q n_u × n_r are the corresponding left and right singular vectors. Similarly, H_k + 1^(p, q) = O_k + 1^(p) R_k^(q).Thus, the identified system using time-varying ERA is:Â_k_n_r × n_r = (O_k + 1^(p) ↓)^+ O_k^(p)↑ , B̂_k_n_r × n_u = R_k^(q)(:, 1: n_u), Ĉ_k_n_y × n_r = O_k^(p)(1: n_y, :),where (·)^+ denotes the pseudo inverse of (·), O_k + 1^(p) ↓ contains the first (p- 1) n_y rows of O_k + 1^(p), and O_k^(p)↑ contains the last (p-1) n_y rows of O_k^(p). Here, we assume that n_r is constant through the time period of interest, which could also be relaxed.The uncontrollable or unobservable eigenmodes of the dynamical system are not present in the input-output map, and hence, the state space realization using time-varying ERA is balanced in the sense that only the controllable and observable eigenmodes are preserved.Hence, for systems with high dimensions, such as systems discretized from partial differential equations (PDEs), and with partially observed states, we have n_r ≪ n_x. Therefore, one major contribution of this work is that we design a reduced order estimator and controller using the identified system, which implies that the computational complexity is reduced significantly. In comparison,the computational complexity of designing an LQG controller using the identified system is O((n_r/n_x)^3) using the full order system.Note that we cannot perturb the system (<ref>) directly. Instead, we identify the generalized Markov parameters as follows. Run M parallel simulations with the noise-free system:x_k + 1^i= f(x_k^i, u̅_k + δ u_k^i, 0), y_k^i= h(x_k^i, 0),and therefore,δ y_k^i = y_k^i - h(x̅_k, 0).where (u̅_k, x̅_k) is the open loop optimal trajectory. Then solve the same least squares problem with (<ref>).For simplicity, we assume that the process noise is independent of the state and control variables, and G_k = I_n_x × n_x, F_k = I_n_y × n_y, while the proposed algorithm is extendable to identify G_k and F_k.In general, the identified deviation system is: δ a_k+1 = Â_k δ a_k + B̂_k δ u_k + Ĝ_k w_k,δ y_k= Ĉ_k δ a_k + F̂_k v_k,where δ a_k ∈^n_r denotes the reduced order deviation states. Algorithm <ref> summarizes the time-varying ERA. §.§ Closed-loop Controller DesignGiven the identified deviation system (<ref>), we design the closed-loop controller totrack the optimal nominal trajectory, which is to minimize the cost functionJ_f = ∑_k = 0^N - 1 (δâ_k^T Q_k δâ_k+ δ u_k^T R_k δ u_k ) + δâ_N^T Q_N δâ_N,where δâ_k denotes the estimates of the deviation state δ a_k. For the linear system (<ref>), the separation principle of control theory is used to separate the design of an estimator and a fully observed controller. The feedback controller is:δ u_k = -L_k δâ_k,where δâ_k denotes the estimates from a Kalman observer, and the feedback gain L_k is computed by solving two decoupled Riccati equations: L_k = (B̂_k^T S_k + 1B̂_k + R_k)^-1B̂_k^T S_k + 1Â_k,where S_k is determined by running the following Riccati equation backward in time: S_k = Â_k^T S_k + 1Â_k+ Q_k- Â_k S_k + 1B̂_k(B̂_k^T S_k + 1B̂_k + R_k)^-1B̂_k^T S_k + 1Â_k,with terminal condition S_N = Q_N. The Kalman filter observer is designed as follows: δâ_k + 1 = Â_k δâ_k + B̂_k δ u_k+ K_k + 1(δ y_k + 1 - Ĉ_k + 1 (Â_k δâ_k + B̂_k δ u_k)),with δ y_k = h(x_k, v_k) - h(x̅_k, 0), and the covariance of the estimation is:P_k + 1 = Â_k(P_k - P_k Ĉ_k^T (Ĉ_k P_k Ĉ_k^T + F̂_k V F̂_k^T)^-1Ĉ_k P_k) Â_k^T + Ĝ_k W Ĝ_k^T,where the Kalman gain is:K_k = P_k Ĉ_k^T (Ĉ_k P_k Ĉ_k^T + F̂_k V F̂_k^T)^-1.Albeit we have only considered the fully observed problem in this paper, any implementation will have noisy measurements, and thus, an observer will be required to implement a feedback controller. In our case, it is simply the LQG controller as outlined above which can be conveniently designed using the identified LTV system . §.§ Stochastic Feedback Control AlgorithmAlgorithm <ref> summarizes the Stochastic Control Algorithm. Extension to Data-Driven Controller Design. As mentioned in data-based LQG <cit.> and data-driven MPC control <cit.>, the linear system (Â_k, B̂_k, Ĉ_k, Ê_k, F̂_k) need not be identified to design the LQG controller. Once the generalized Markov parameters are recovered, then together with other input-output data matrices, the controller can also be directly designed. The extension to designing such a “direct" data-driven controller is not covered in this paper.Discussion of Implementation Issues. * The proposed approach is implemented in an offine-online fashion. Starting from an given initial state, the open-loop trajectory optimization problem and the LTV system identification problem are solved offline, using the input-output experimental data. Then the LQG policy is implemented online.* An initial estimate of the optimal trajectory is required in the proposed approach, which poses the challenge, and results in a sub-optimal path. * The proposed approach is valid under small noise assumption. In practice, with the presence of noise, non-linearities, and unknown perturbations, the actual state deviates from the nominal trajectory during the execution, and if the actual system deviates too much from the nominal trajectory, the linearization becomes invalid. Therefore, once the deviation is greater than some predefined threshold, a replanning starting from the current location can be performed.Extension to Partially Observed MDP (POMDP). The proposed approach can also be extended to solve the stochastic optimal control problem for a general nonlinear system with unknown dynamics f(·) and h(·). The only difference is that for the partially observed case, we need to solve the open loop optimization problem in belief space instead of state space. Assume that the belief b_k = (μ_k, Σ_k) is approximately Gaussian, where μ_k and Σ_k represent the mean and covariance of the belief state.The challenge is that we do not have access to the covariance evolution equation. An effective way is to simulate the covariance evolution using an Ensemble Kalman Filter <cit.>. Then, thegradient descent approach can be used for solving the open loop belief space optimization problem. We linearize the system around the optimal control sequence {u̅_k}, with the associated nominal belief state (μ̅_k, Σ̅_k), and follow the same system identification and closed loop controller design procedure.The extension to POMDPs will be considered in future work.§ COMPUTATIONAL RESULTSWe test the method using the inverted pendulum problem <cit.>. The dynamics of the inverted pendulum mounted on a motor driven cart are given as: ẋ_1 = x_2, ẋ_2 =u cos x_1 - (M + m) g sin x_1 + m l (cos x_1 sin x_1) x_2^2/ml cos^2 x_1 - (M + m)l, ẋ_3= x_4, ẋ_4 = u + ml (sin x_1) x_2^2 - m g cos x_1 sin x_1/M + m - m cos^2 x_1with state variablesx_1 = θ, x_2 = θ̇, x_3 = x, x_4 = ẋ,where θ is the tilt angle referenced to the vertically upward direction andx represents the cart position, u is the x-directed external force, the cart mass is M = 2.4 kg, the pendulum point mass is m = 0.23 kg, the rod length is l = 0.36 m, and the standard gravity is g = 9.81 m/s^2. We use a time step of 0.1s to discretize the system in time:x_k + 1 = f(x_k, u_k) +w_k, y_k= x_k + v_k,where f is the unknown dynamics, the process noise and measurement noise w_k, v_k are Gaussian white noise with zero mean and covariances W = 0.01 I_4 × 4 and V = 0.01 I_4 × 4, respectively. The total simulation time is 5 seconds. The control objective is to find the control sequence { u_k }_k = 0^N - 1 to make the pendulum swing up within 3.5 seconds and minimize the cost function J_s in (<ref>), and maintain the pendulum to the upright vertical position. The initial state x_0 = [π; 0; 0; 0] is known, and the final state is x_N = [0; 0; x_f; 0], where the cart position x_f is not restricted.The open loop optimization problem is solved using Matlab nonlinear optimization solver 𝚏𝚖𝚒𝚗𝚞𝚗𝚌. The initial guess and optimal control are shown in Fig. <ref>. The corresponding nominal trajectory {x_k}_k = 0^N are plotted in Fig. <ref>.The implementation of Algorithm <ref> to identify the linearized system is performed as follows.Thesize of the generalized Hankel matrix H_k^(p, q) is pn_y × q n_u, and as discussed before, the design parameters p and q should be chosen such that min{pn_y, qn_u }≥ n_x, which for the current problem, n_x = 4, n_u = 1, and n_y = 4. Design parameters p, q are selected by trial and error. We start with some initial guess of p, q, compare the impulse responses of the original system and identified system, and check if the accuracy of theidentified system is acceptable.Here, we choose p = q = 5.We run M parallel simulations to estimate the generalized Markov parameters {h_k, j}_j = 0^k, k = 1, 2, ⋯, N -1. We perturb the nominal control {u̅_k }_k = 0^N - 1 with impulse, i.e., {δ u_k^i }_k = 0^N - 1 = (0, 0, ⋯, 0.01, ⋯, 0) is the input perturbation sequence in the i^th simulation, where only the i^th element is nonzero. Therefore,we choose design parameter M = N. In each simulation, we collect the outputs {δ y_k^i }_k = 0^N-1in (<ref>) corresponding to the control input {u̅_k + δ u_k^i }_k= 0^N - 1, and solve the least squares problem using (<ref>). We construct the generalized Hankel matrix, and solve the singular value decomposition problem. The rank of the Hankel matrix n_r = 4, and hence, the identified system Â_k ∈^4 × 4. To test the accuracy of the identified system, we calculate the identified Markov parameters using Â_k, B̂_k, Ĉ_k, and compare with the actual generalized Markov parameters (calculated using impulse responses as mentioned above). Since the generalized Markov parameter h_k, j∈^4 × 1, all four elements are shown in Fig. <ref> for k = 33.With the identified linearized system, we design the closed-loop controller. We run 1000 individual simulations, where performance of the closed-loop controller is shown in Fig. <ref>. There are two observations: 1) the averaged state estimates over 1000 Monte-Carlo simulations runs (plotted in solid green lines) are close to the open-loop optimal trajectory (plotted in red), implying that the control objective to minimize the expected cost function could be achieved using the proposed approach; 2) in this problem, the closed-loop controller has better control in the angle and angular velocity, so that the corresponding 2 σ bound is tight. However, the uncertainties of position and velocity rise especially after the pendulum reaches the upright position. This is due to the fact that with the presence of process noise in all states, it is not possible to stabilize all the states simultaneously. § CONCLUSIONIn this paper, we have proposed a separation-based design of the stochastic optimal control problem for systems with unknown nonlinear dynamics and fully observed states in a separated open loop-closed loop fashion. First, we design a deterministic open-loop optimal trajectory.Then we identify the nominal linearized system using time-varying ERA. The open-loop optimization and system identification are implemented offline, using the impulse responses of the system, and an LQG controller based on the ROM is implemented online.The offline learning procedure is simple, and the online implementation is fast. We tested the proposed approach on the inverted pendulum problem, and showed the performance of the proposed approach. Future work will generalize the proposed approach to large-scale, partially observed systems. IEEEtran | http://arxiv.org/abs/1705.09761v1 | {
"authors": [
"Dan Yu",
"Mohammadhussein Rafieisakhaei",
"Suman Chakravorty"
],
"categories": [
"cs.SY",
"cs.LG"
],
"primary_category": "cs.SY",
"published": "20170527035741",
"title": "Stochastic Feedback Control of Systems with Unknown Nonlinear Dynamics"
} |
http://arxiv.org/abs/1705.09528v2 | {
"authors": [
"Hang Deng",
"Cun-Hui Zhang"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170526110101",
"title": "Beyond Gaussian Approximation: Bootstrap for Maxima of Sums of Independent Random Vectors"
} |
|
[Correspondence to: ][email protected] of Physics and Astronomy, Seoul National University, Seoul 08826, Korea Superradiance is a quantum phenomenon emerging in macroscopic systems whereby correlated single atoms cooperatively emit photons. Demonstration of controlled collective atom-field interactions has resulted from the ability to directly imprint correlations with an atomic ensemble. Here, we report cavity-mediated coherent single-atom superradiance: single atoms with predefined correlation traverse a high-Q cavity one by one, emitting photons cooperatively with the N atoms already gone through the cavity.Enhanced collective photoemission of N-squared dependence was observed even when the intracavity atom number was less than unity. The correlation among single atoms was achieved by nanometer-precision position control and phase-aligned state manipulation of atoms by using a nanohole-array aperture.Our results demonstrate a platform for phase-controlled atom-field interactions.One-sentence summary:Enhanced collective photoemission is observed from correlated atoms traversing an optical cavity.Coherent single-atom superradiance Kyungwon An December 30, 2023 ================================== Superradianceis a collective radiation phenomenon by a number of quantum emitters<cit.>.In the original prediction, exchange symmetry is present in closely packed emitters whose inter-particle distance is much smaller than the transition wavelength, and therefore dipole-dipole correlation emerges during their spontaneous decay process. The correlation makes the ensemble behave collectively and induces enhanced interaction with the vacuum fields,leading to stronger and faster radiation emission compared to the ordinary spontaneous emission. Early experiments performed with a large number of emitters (as in a dense atomic vapor or in a beam) reported observations consistent with the prediction<cit.>. Recent technical advanceshaveenabled the realization of superradiance in various systems such as a Bose-Einstein condensate<cit.>, quantum dots<cit.> and trapped atoms coupled to a cavity<cit.>.The mutual phase correlation among atoms is the key to superradiance. It can make the ensemble behave as a single macro dipole.Moreover, direct control of atomic phases enables controllable collective atom-field interactions.In recent experiments, the phase of atoms in an ensemble was imprinted by a single photon pulse<cit.> or a frequency-swept laser pulse<cit.>. The ensemble then started superradiant emission without a threshold or an initial time delay. The output field in this case follows the given imprinted phase and thus its spatial mode overlaps with the input mode, making it hard to distinguish the input and output fields spatially.This approach works only in the pulsed regime. Observation of a superradiant state in a Bose-Einstein condensate couple to a cavity was another notable work<cit.>. However, it relied on self-organization of atoms based on a thermodynamic principle, and thus further tunability could not be attained.Another approach to achieve controllable superradiance is to prepare emitters in a cavity and to manipulate the quantum state of individual emitters. Ions<cit.>, neutral atoms<cit.> and artificial atoms based on superconducting circuits<cit.>have been used in this approach. The results includeimmediate strong and fast radiation emissionand controllability between superradiance and subradiance.However, technical difficulties have limited the number of emitters involved in the superradiance only up to two.We presentan approach to realize phase-controlled superradiancewhereby single atoms are prepared in the same quantum superposition of ground and excited states traverse a cavity one by one.The long-lived cavity field then mediates collective interaction among the phase-correlated single atoms separated in time, leading to superradiance. The collective interaction is one-sidedin that the emission of a particular atom in the cavity is cooperative only with the preceding atoms. Even when at most only one atom is present in the cavity, tens of atoms participate in the superradiance and the emission intensity is proportional to the square of the number of the participating atoms.Our system, adapted from Ref. <cit.>, consists of a supersonic barium atomic beam and a high-Q optical cavity which the atoms resonantly interact with (Fig. 1A).The barium-138 atoms are prepared in a superposition state of the ground and excited states just before they enter the cavity mode by a pump laser propagating perpendicular to the cavity axis as well as to the atomic beam direction. The atomic phase imprinted by the pump laser depends on the position at which the atom traverses the pump laser. The phase of the atom-cavity coupling also alternates 0 and π radian following the standing wave structure of the cavity mode. A checkerboard-pattern nanohole arrayis used as an atomic beam aperture in order to localize and control the atomic position. The localized atoms then selectively pick up the phase of the pump laser as well as the cavity field corresponding to their positions prescribed by the array structure.As a result, the atom-field relative phase is the same for every atom traversing the cavity(Fig. 1B).The desired atomic internal state is prepared by the pump laser with a pulse area of Θ = ∫[Ω_p (x)/v] dx, where Ω_p(x) is the Rabi frequency due to the pump laser and v is velocity of the atom. The atomic state can then be expressed as |ψ_atom⟩ = sin(Θ/2)|e⟩ + cos(Θ/2)e^iϕ|g⟩, where ϕ is the atomic phase imprinted by the pump laser. Atomic correlation between any two of the injected atoms is then given by ⟨σ_i^†σ_j⟩= 1/4sin^2 Θ, where σ_i=| g⟩⟨ e| is the lowering operator of the i-th atom, showingthat the atom-atom correlation is maximized when Θ = π/2.Injected atoms then emit photons into the cavity mode and build up the cavity field.A previous study assuming a lossless cavity expected enhanced collective emission by consecutively injected N atomic dipoles to show explicit N^2 dependence<cit.>.The longlasting cavity field links the atoms together and the expected photon number is exactly the same as that of simultaneously injected N dipoles (see Fig. S1).When a cavity has a finite decay, the gain (emission by atoms) and the loss (absorption by atoms as well as the cavity decay) of the cavity field would be balanced in its steady state. The averaged cavity photon number ⟨ n ⟩ in the steady state can be obtained from the quantum master equation (see Supplementary Text Section 2.2 for details) and it is approximately given by⟨ n⟩≈⟨ N_c⟩ρ_ee (gτ)^2/2-(2ρ_ee-1)⟨ N_c⟩ (gτ)^2 + (⟨ N_c⟩ |ρ_eg|gτ)^2. where ⟨ N_c⟩≡ r/γ_c is the mean number of atoms injected into the cavity during the cavity-field decay time 1/γ_c with r the atomic injection rate, ρ_ee and ρ_eg are the density matrix elements of atomic state with the subscripts `e' and `g' represent excited and ground states, respectively, g is the atom-cavity coupling constant and τ is the atom-cavity interaction time.The first term, approximately proportional to 1/2⟨ N_c⟩ when ⟨ N_c⟩ (gτ)^2≪ 1, is due to the non-collective emission of atoms, including spontaneous and stimulated emission as well as the cavity-QED effect. The second term, exhibiting a quadratic dependence on ⟨ N_c ⟩, is due to collective emission, i.e. the superradiance. Compared to the case with a lossless cavity<cit.>, the number of atoms participating in the superradiance is identified to be ⟨ N_c⟩ in our case (see Supplementary Text Section 2.1). When ⟨ N_c⟩≫ 1, the second term dominates the emission and the field stateapproximately becomes a coherent state |α⟩ with α = -i ⟨ N_c⟩ρ_ eg gτ.The mean intracavity atom number ⟨ N⟩ is related to ⟨ N_c⟩ as ⟨ N_c⟩ = ⟨ N ⟩ /(γ_cτ).If the cavity-field decay time 1/γ_c is much larger than τ (γ_cτ≪1), ⟨ N_c⟩ can be much greater than unity even when the mean intracavity atom number ⟨ N⟩ is less than unity and thus the collective effect can take place.The cavity field mediates the collective behavior among the time-separated ⟨ N_c⟩ atoms that are going through the cavity individually during the cavity-field decay time,leading to the single-atom superradiance.Around 22 atoms are involved in the collective emission when a single atom is present in the cavity mode on average. In our experiment, the phase-aligned atomic dipoles prepared with the aforementioned nanohole-array were injected into the cavityand the mean intracavity photon number in the steady state was measured with a single-photon-counting module. The atom-cavity interaction was in the strong coupling regime with (g,γ, γ_c) = 2π× (290, 25, 75)kHz, where g is the atom-cavity coupling averaged over the atomic distribution centered around the antinodes of the cavity and γ (γ_c) is the atomic polarization (cavity-field) decay rate.The single-atom cooperativity was C = g^2/γγ_c = 44. The mean travel time of atoms from the pump to the cavity field was about 200ns whereas the mean atom-cavity interaction time τ = 101ns.As a comparative counterpart, we also performed the experiment with a 250μm×25μm-sized rectangular atomic beam aperture for the case of atoms with random phases. In the latter case, the atomic beam was injected into the cavity mode with a small tilt angle in order to induce Doppler shifts so as to achieve a uniform atom-field coupling<cit.>, whose strength is a half of the maximum coupling strength.The collective emission described by the second term in Eq. (1)is expected to have the quadratic dependence on two parameters, the induced atomic dipole moment ∝|ρ_eg| and the atom number ⟨ N_c ⟩. First, we investigated |ρ_eg| dependence of collective emission by varying the pump pulse area (Fig. 2).Due to the relation |ρ_eg| = |1/2sinΘ | for the prepared superposition state, the atomic dipole moment would be maximized with equal ground- and excited-state populations (Θ = 0.5πor1.5π), and so would be the collective emission. Clear enhancement was observed when the atoms are prepared in the phase-aligned superposition states. The enhancement was more than ten-fold for Θ<0.3π (also see Fig. S2). Combined contributions by ρ_ ee (non-collective) and |ρ_ eg| (collective) make ⟨ n ⟩ maximized near Θ≃ 0.7π. Due to the small overlap between the pump laser field and the cavity mode (both are Gaussian),the collective emission process is somewhat disturbed by the stray pump field in the cavity when the pump intensity is strong, resulting in the enhancement reduction for Θ > π. On the other hand, in the case of random phase, the photon number is given by the non-collective emission only, and thus it is maximized with fully inverted atomic states (Θ = π). The enhancement is strongly dependent on the atomic phase purity. In reality, there are several sources of phase noise.Finite atomic localization sets the lower bound of atomic phase variance.Atomic spontaneous emission into free space also contributes to phase diffusion of atoms, reducing |ρ_eg| by 6%. In addition, the pump laser has a phase uncertainty: the laser phase diffuses in time with a finite laser linewidth. If we intentionally make the pump laser linewidth larger, the superradiant enhancement becomes smaller (see Fig. S3).We performed quantum-trajectory simulation as well as quantum master equation calculation with the experimental parameters and our data well agree with the numerical results <cit.> (also see Fig. S4). Figure 3shows the mean intracavity photon number ⟨ n ⟩ versus the excited state atom number ⟨ N ⟩ρ_ ee. When the atoms have no dipole moment (Fig. 3B),only the non-collective emission is present. With a small number of atoms, the cavity field is mainly made by spontaneous emission of atoms (dashed line) and its photon number increases linearly to the atom number. As the accumulated photon number gets larger, stimulated emission and absorption become dominant over the spontaneous emission and the system lases for positive inversion (ρ_ee - ρ_gg > 0) or the photon number plateaus for negative inversion (ρ_ee - ρ_gg <0). Especially for positive inversion, a rapid growth of the photon number starts to occur at ⟨ n ⟩≃ 1, which is the well-known lasing threshold in the conventional lasers<cit.>.However, when atoms have the same phase (Fig. 3A),photon emission is enhanced nonlinearly with its log-log slope getting steeper than unity. The measured intracavity photon numbers are consistently larger than the photon number made only by the cavity-enhanced spontaneous emission (dashed line). When the pump pulse area is 0.5π, corresponding to | ψ_atom⟩≃ (|e⟩ + e^iϕ|g⟩)/√(2), the observed log-log slope is 1.66±0.01.After subtracting the contribution by the non-collective emission corresponding to the dashed line,the recalculated log-log slope becomes 1.94±0.04 (see the inset of Fig. 3), which indicates the observed emission is dominantly superradiance proportional to the square of the number of atoms.A near-quadratic growth appears even in the negative inversion case of Θ = 0.3, in which only 21% of atoms are in the excited state with the rest in the ground state. When Θ >0.5π, the atoms have positive population inversion and thus the photon number grows further by stimulated emission beyond the level by the collective emission. In this case, it is impossible to isolate the collective emission effect clearly in the log-log plot. It is also notable that the log-log slope is almost invariant for a large range of ⟨ N_c ⟩ for Θ≤ 0.5π. The theory expects that the quadratic dependence on ⟨ N_c ⟩would be dominantin the region of (1+cosΘ)^-1 < ⟨ N_c ⟩<(gτ)^-2 for the perfectly phase-aligned atoms although the practical phase noise would make the domain somewhat reduced. Such a broad-range quadratic growth, occurring independently of ⟨ n ⟩ values, including ⟨ n ⟩≪ 1 as in Ref.<cit.>, is a distinctive feature of the present superradiance compared to the drastic slope change occurring near the threshold condition of ⟨ n ⟩≃ 1 in the ordinary lasing case.The absence of the usual lasing threshold or thresholdless lasing in the present superradiance cannot be explained in terms of theso-called β-factor in ordinary lasers based on non-collective emission<cit.>. In our case β=(gτ)^2≃ 0.034 in the nanohole-array-aperture case (Fig. 2A and 3A) and 0.011 in the rectangular-aperture case (Fig. 2B and 3B) (see Supplementary Text Section 2.3). The latter is consistent with the large mean photon number change occurring at the threshold in Fig. 3B (Θ>π/2).Note also that the range of superradiance or the maximum number of atoms participating in the collective emission can be easily scaled up by choosing smaller gτ values (see Fig. S5). This feature may provide a new approach in building thresholdless lasers.The present single-atom superradiance can be viewed as a consequence of one-sidedinteraction among a series of atoms separated by tens of meters. Note that the photon emitted by a preceding atom interacts with the next atom after traveling cτ/⟨ N ⟩ (about 30m for ⟨ N ⟩ = 1) when we unfold mirror refections although their average distance in real space is only hundreds of micrometers. Due to causality, only the preceding atoms can then affect the quantum states of the following atoms.Thisinteractioninduces the emission rate of the atom in the cavity to be twice larger than the emission rate per atom in the usual superradiance (see Supplementary Text as well as Fig. S6). The time-separated atoms linked by such one-sided interaction can formatom-atom interaction systems, which can serve as a testbed for various quantum many-body physics<cit.>. The present study deepens our understanding on matter-light collective interaction and provides a new insight on the field-mediated long-range<cit.>interactions.In addition, the phase-controlled many-atom-field interaction based on the nanohole-array technique can be used in non-classical field generation such as optical Schrödinger cat states and highly-squeezed vacuum states<cit.>, even in a lossy cavity contrary to the previous studies in the microwave region<cit.>,as well as in realizing superabsorption<cit.>. 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This work was supported by a grant from Samsung Science and Technology Foundation under Project No. SSTF-BA1502-05. The authors declare no competing financial interests. All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.List of Supplementary MaterialsMaterials and MethodsSupplementary TextFigs. S1 to S6References (32-35)Supplementary Materials1 Materials and methods 1.1 A nanohole-array aperture, a supersonic atomic beam and a high-Q cavityOur nanohole-array is fabricated on a silicon nitride membrane (Si_3Ni_4) of 10nm thickness by using the focused-ion-beam technique. The distance between holes in the pump beam direction as well as in the cavity axis directions is equal to the atomic transition wavelength λ=791nm.The nanohole diameter is 0.24λ.Actual atomic localization width would be slightly larger than the nanohole diameter because of a finite atomic beam divergence and a small angular mismatch between the array and the cavity axis.The atomic distribution in the cavity is estimated to have a full-width at half-maximum (FWHM) of 0.29λ for each nanohole,which is obtained from the observed contrast of the modulation of the cavity photon number as the array position along the cavity axiswhile the excited atoms are injected into the cavity through the nanohole-array. The nanohole array dimension is 25λ by25λ, spanning 19.8μm both horizontally and vertically. The vertical dimension is much smaller than the cavity-mode full width (83.8μm), ensuring near constant atom-cavity coupling constant in that direction. To investigate the single-atom superradiance effectively, we need to satisfy the following conditions: negligible frees space decay of atoms during the atom-field interaction (γτ≪1), a small single-atom vacuum-Rabi angle to suppress the gain saturation effect (gτ≪1) (18,19)and a sufficiently long cavity-field decay time (γ_cτ≪1).We use barium-138 having a narrow spin-forbidden transition (^1S_0 – ^3P_1) of γ = 2π× 25kHz natural linewidth (half width) and zero nuclear spin. We treat the atoms as a two-level system with ^1S_0 being the ground state and ^3P_1 the excited state. We use a supersonic beam of barium atoms with a fast mean velocity of v_mean = 755 m/s and a narrow velocity distribution with a FWHM of 0.24v_mean.A bias magnetic field is applied along the atomic beam direction so as to define the quantization axis of the atoms and to lift the magnetic sublevel degeneracy. The polarizations of the pump field and the cavity probe field are also set to parallel to the atomic beam direction in order to access the magnetic-field-insensitive π transition (Δ m = 0) of atoms. The cavity is made of supermirrors with a finesse of 0.92×10^6 and a radius of curvature of 10.0cm. The cavity length is 1.09mm and the mode waist is 42.5μm in the direction of the atomic beam and 41.9μm in the direction of the pump beam. 1.2 Success rate of atomic state manipulationAtomic state is prepared by a pump laser. However, the velocity distribution of the atomic beam and the free space spontaneous emission decay of the excited state induce small imperfection in state preparation. To verify resulting atomic state after passing through the pump laser, we measured the ground state population from the fluorescence of the 553nm cycling transition (^1S_0↔ ^1P_1). Under the π-pulse condition (Θ = π) of the pump, the observed population in the excited ^3P_1 state was 93±1%, well-agreeing with the numerical simulation result of 93.6%.1.3 Photon number calibration and atom number measurement The mean photon number <n> in the cavity is calibrated by using the photon number clamping effect. When the cavity is pumped by the atoms with a negative inversion and random phases, <n> would converge to ρ_ee/(1-2ρ_ee) as <N_c>→∞ (detailed in Section 2.2.1),only depending on the initial excited state population, which can be independently obtained with the aforementioned fluorescence measurement technique. We found that a single intracavity photon in the steady state corresponds to 1.1×10^5 counts per second measured by our single photon counting module.Once the photon number is calibrated, we can extract the mean intracavity atom number <N> by comparing the measured <n> in the case of fully excited atoms (Θ = π) with the result of the master equation calculation. The calibration result was verified to be reasonable by performing a linear regression analysis between the estimated <N> and the directly measured atomic fluorescence (at 553nm) in the large atom number region (with the adjusted coefficient of determination of R^2 = 0.94).1.4 Numerical simulations of experiment To compare the experimental results with theoretical expectations quantitatively, we performed two different numerical studies: the master equation calculation and the quantum trajectory simulation (QTS) (32,33).QTS appropriately treats transient fluctuations including phase diffusion and atom number fluctuation but requires high computational power. Therefore, we performed QTS for the phase-aligned atom case, which has less intracavity atoms, and the master equation approach for the random phase case. The master equation for our experiment is fully derived in Section 2.2. The master equation was solved for the steady state. With the same simulation parameters, two methods perfectly agree with each other. 2Supplementary text2.1 Qualitative description of the single-atom superradiance A two-level atom can be represented as a spin-1/2 particle and any quantum state of their ensemble can be expressed in terms of |J,M ⟩, eigenstates of total angular momentum operators σ_Σ^2 = (∑_i σ⃗_⃗i⃗^2) and σ_Σ,z= ∑_i σ_i,z. The spontaneous radiation rate of the state is then given by Γ_a <σ_Σ^†σ_Σ> = Γ_a ∑_i<σ_i^†σ_i> + Γ_a ∑_i≠ j<σ_i^†σ_j>where Γ_a is atomic natural linewidth,σ_Σ= ∑_iσ_i and σ_i(σ_i^†) is the lowering (raising) operator for the i-th atomic state.The first term of the right hand side represents the sum of excited state probabilities, corresponding to the total energy quanta carried by the atomic ensemble, and the second term is related with the mutual correlation among atoms. When the atoms have no correlation at all, atoms decay independently and no collective effect emerges.In the original idea of superradiance, an atomic ensemble has a perfect exchange symmetry and it is thus expressed in terms of the eigenstates with the maximum total angular momentum J=N/2. Their spontaneous decay rates are given by Γ_a<σ_Σ^†σ_Σ>= Γ_a (N/2 + M) (N/2 - M + 1), exhibiting the most enhanced decay rate of Γ_a N/2(N/2 + 1) when M=0, corresponding to the brightest superradiant state.On the other hands, when the atoms are in the same superposition state of C_e | e⟩ + C_g | g⟩, the total ensemble can be expressed as |Ψ_ atom⟩ = ∏_i(C_e | e_i ⟩ + C_g | g_i ⟩). This state also satisfies the exchange symmetry and thus can be decomposed into the sum of the eigenstates, |Ψ_ atom⟩ = ∑_k=0^N C_ e^N-k C_ g^k ([ N; k ]) |J=N/2, M=N/2 - k⟩.The spontaneous decay rate of this state is then given by Γ_a∑_k|C_ e|^2(N-k) |C_ g|^2k( [ N; k ]) (N-k)(k+1) =Γ_a N(N-1)|C_ e|^2 |C_ g|^2 ∑_k |C_ e|^2(N-k-1) |C_ g|^2(k-1)( [ N-2; k-1 ]) +Γ_a N|C_ e|^2 ∑_k |C_ e|^2(N-k-1) |C_ g|^2k( [ N-1; k ]) =N(N-1) Γ_a |ρ_eg|^2 + N Γ_a ρ_ee,also exhibiting N^2-dependence of the superradiance.The photon emission rate per atom is ρ_eeΓ_a+ (N-1) |ρ_eg|^2 Γ_a, interpreted as the rate sum ofnon-collective emission and collective emission with other N-1 atoms. When the ensemble is strongly coupled to a cavity with a coupling constant g during a definite interaction time τ, the emission rate becomes ρ_ee g^2τ + (N-1) |ρ_eg|^2 g^2τ,where the free-space decay rate Γ_a is replaced with the cavity-assisted decay rate g^2τ.Such collective emission can also appear when phase-aligned atoms do not interact with a field simultaneously, as in our single-atom superradiance. From the numerical calculation results on the consecutive injection of numerous atoms into a lossless cavity field, we have inductively found that the emission rate of the N-th atom is ρ_ee g^2τ + 2(N-1) |ρ_eg|^2 g^2τ. Here the preceding N-1 atoms influence the emission of N-th atom and the photons already emitted by the former ones induce the stronger collective emission, leading to a twice larger collective emission rate than that in the ensemble case (see also Fig. S1).When the cavity field dissipation is introduced, the field-mediated interaction with the preceding atoms is also reduced by exp(-γ_c Δ t), where γ_c is the cavity field decay rate and Δ t is the time elapsed since the preceding atom left the cavity. The effective number of atoms participating in the collective emission with the atom introduced in the cavity at time t_0 is then given byN_ eff = ∫_-∞^t_0 P(t) exp[-γ_c (t_0-t)] dtwhere P(t) is the injection rate of atoms at time t. The resulting emission rate per atom is now ρ_ee g^2τ + 2 N_ eff |ρ_eg|^2 g^2τ.When the atoms are randomly injected into the cavity field, P(t) is equivalent to the mean atomic injection rate r and the corresponding N_ eff is r/γ_c≡<N_c>. For regular injection of atoms with a constant time interval between them, P(t) = ∑_i δ(t-i/r) and N_ eff = ∑_i=1^∞exp(-iγ_c/r) = 1/exp(1/<N_c>) - 1 ,which can be also approximated by <N_c> for <N_c>≫ 1.In the steady state of the cavity field, the loss 2γ_c<n> by cavity dissipation and the gain r P_e from atomic emission should be balanced. Here, P_e is the photo-emission per atom in the cavity, given by the emission rate per atom times the interaction time. Therefore, the cavity photon number in the steady state is given by <n>= r P_e/2γ_c= r/2γ_c[ρ_ee (gτ)^2 + 2 N_ eff |ρ_eg|^2 (gτ)^2 ]= 1/2<N_c>ρ_ee (gτ)^2 + <N_c> N_ eff |ρ_eg|^2 (gτ)^2≈ 1/2<N_c>ρ_ee (gτ)^2 + <N_c>^2 |ρ_eg|^2 (gτ)^2. 2.2 Quantum master equation approachThe time evolution of the cavity field can be fully described by the quantum master equation. The equation is based on the atom-field interaction governed by the Jaynes-Cummings Hamiltonian, Ĥ = ħω_câ^†â + ħω_aσ̂_z+ħ g (âσ̂^†+â^†σ̂), and the stochastic decay described by the Lindblad equation.Here â (â^†) is the annihilation(creation) operator for the cavity mode photon and σ̂ (σ̂^†) is the lowering(raising) operator for the atomic state. The state of injected atoms is described by three parameters, ρ_ee, |ρ_eg| and arg(ρ_eg). For simplicity, we assume an uniform atom-field coupling constant g and a fixed interaction time τ.Adapting the approach in Chapter 13 of Ref. (34) , we assume that each atom traversing the cavity contributes to the cavity field state independently so that the gross change of the cavity density matrix ρ_field is just the product of an atomic injection rate r and a single-atom contribution δρ_field, i.e.ρ̇_field,interaction(t)= r ×δρ_field(t). Each atom interacts with the cavity during the given interaction time τ and the single-atom contribution can be derived by tracing the density matrix after the interaction over the atomic stateδρ_field(t_0)= ρ_field(t_0+τ) - ρ_field(t_0)=tr_atom[ρ_system(t_0+τ)] - ρ_field(t_0).The stochastic dissipation by the cavity decay can be described by the Lindblad equation,ρ̇_field,decay = γ_c [2âρ_fieldâ^† - (ρ_fieldâ^†â + â^†âρ_field)]Total change in the density matrix would be the sum of these two effects. With identification of <N_c> =rγ_c^-1, the resulting master equation is thenQ̇_n,m/γ_c = <N_c> { Q_n,m ( ρ_ee C_n C_m + ρ_gg C_n-1 C_m-1 -1 ) +Q_n+1,m+1ρ_ggS_n S_m + Q_n-1, m-1ρ_ee S_n-1 S_m-1+[i Q_n, m+1ρ_eg C_n S_m - Q_n+1, mρ_ge S_n C_m +Q_n, m-1ρ_ge C_n-1 S_m-1 - Q_n-1, mρ_eg S_n-1 C_m-1]}+ [2√((n+1)(m+1))Q_n+1, m+1 - (n+m) Q_n,m],where Q_n,m = ⟨ n |ρ_field|m⟩, C_n = cos(√(n+1)gτ) and S_n = sin(√(n+1)gτ).The steady state solution of the cavity field can be found by letting Q̇_n,m = 0 in Eq. (11). 2.2.1 The contribution of non-collective emission When ρ_eg = 0, i.e. atoms have no phase preference, the cavity field is affected only by the ground and excited state populations, ρ_gg and ρ_ee. Equation (11) with the steady state condition Q̇_n,m = 0 is reduced to<N_c> [ Q_n,m ( ρ_ee C_n C_m + ρ_gg C_n-1 C_m-1 -1 )+ Q_n+1,m+1ρ_ggS_n S_m + Q_n-1, m-1ρ_ee S_n-1 S_m-1]+ [2√((n+1)(m+1))Q_n+1, m+1 - (n+m) Q_n,m] = 0.It can be easily shown that if an initial field does not have non-zero off-diagonal density matrix elements (e.g. vacuum state) the resulting field state also does not have non-zero off-diagonal density matrix elements. So, Q_nn = P_n, the probability of having n photons in the cavity, are the only remaining non-zero density matrix elements. Using the relation C^2_n = 1 - S^2_n and ρ_gg = 1- ρ_ee, the recursion relation for P_n can be obtained as {<N_c> [ P_n+1(1-ρ_ee)S^2_n - P_n ρ_ee S^2_n ]+2(n+1)P_n+1}- {<N_c> [ P_n(1-ρ_ee)S^2_n-1 - P_n-1ρ_ee S^2_n-1]+2n P_n }=0. Equation (7) is satisfied if the quantities in both curly brackets are zero for any n, or ∀ n:P_n+1 =P_n <N_c> ρ_ee S^2_n/ 2(n+1) + <N_c>( 1-ρ_ee)S^2_n .So, the general expression for P_n is P_n = P_0 ∏_k=1^n <N_c> ρ_eesin^2(√(k)gτ)/ 2k + <N_c>( 1-ρ_ee) sin^2(√(k)gτ)and the probability P_0 of occupying the vacuum state can be found from the normalization condition ∑_k P_k = 1.When the intracavity mean photon number <n> is small enough to satisfy √(<n>)gτ≪1, the Rabi oscillation angle of each atom is not large and thus the gain saturation (see Sec. 2.2.4)does not occur. In this regime, we can approximate sin^2(√(k)gτ)≃k (gτ)^2. Then Eq. (15) is now simplified as P_n = P_0 [ p ρ_ee/2 + p (1-ρ_ee)]^n,where the pumping parameter p is defined as p ≡<N_c>(gτ)^2 and P_0 = 1 - ρ_eep/[2+(1-ρ_ee)p]. The intracavity mean photon number <n> in the steady state can then be derived as <n> = ∑_n n P_n = 1/2ρ_eep/1+(1-2ρ_ee)1/2p. It is noteworthy that the solution can be categorized in three cases depending on ρ_ee values.First, when ρ_ee>0.5, i.e. population inverted, the mean photon number diverges when p = (ρ_ee-1/2)^-1 under the approximation of √(<n>)gτ≪1, due to the stimulated emission amplification of the cavity field. In other words, the cavity field undergoes lasing. However, when ρ_ee < 0.5, the absorption is larger than the stimulated emission, and therefore the photon number plateaus at <n> = ρ_ee/(1-2ρ_ee) as p→∞. Lastly, when ρ_ee = 0.5, which means the excited and ground state populations are equal, the absorption and stimulated emission are balanced. As a result, only the spontaneous emission effect remains and the photon number is given by <n> = 1/4p = 1/4 <N_c>(gτ)^2, growing linearly until the gain saturation occurs. 2.2.2 The contribution of collective emission When the cavity is pumped by atoms in a superposition state (ρ_eg≠ 0), the equation to solve becomes a bit complicated. Under the assumption of √(<n>)gτ≪1, we can replace S_k and C_k with √(k+1)gτ and 1, respectively, in the approximation up to the first order of gτ. After such substitution to Eq. (11) and keeping the terms up to the first order of gτ, the equation for the steady state solution isi <N_c>(Q_n, m+1ρ_eg√(m+1)gτ - Q_n+1, mρ_ge√(n+1)gτ+ Q_n, m-1ρ_ge√(m)gτ - Q_n-1, mρ_eg√(n)gτ ) + [2√((n+1)(m+1))Q_n+1, m+1 - (n+m) Q_n,m] = 0,which can be rearranged as√(n+1) ( √(m+1) Q_n+1,m+1 - i ρ_ge<N_c> gτQ_n+1,m)+ √(m+1) ( √(n+1) Q_n+1,m+1 + i ρ_eg<N_c> gτQ_n,m+1) - √(m) ( √(m) Q_n,m - i ρ_ge<N_c> gτQ_n,m-1) + √(n) ( √(n) Q_n,m + i ρ_eg<N_c> gτQ_n-1,m)=0,which is satisfied if the quantity in each round bracket is zero for any n. So, the general expression for Q_n,m is thenQ_n,m =i ρ_ge<N_c> gτ Q_n,m-1/√(m)=-i ρ_eg<N_c> gτ Q_n-1,m/√(n)=(-i ρ_eg<N_c> gτ)^n (i ρ_ge<N_c> gτ)^m/√(n!m!) Q_0,0,which is exactly the density matrix element of the coherent state | α = -i <N_c> ρ_eggτ⟩. The mean photon number of the resulting coherent state is <n> = |α|^2=(<N_c> |ρ_eg| gτ)^2, exhibiting explicit quadratic dependence on <N_c>, characteristics of collective emission or the superradiance. Similar results were presented for different configurations by Refs. (17) and (35). Especially, Le Kien et al. considered a configuration with a lossless cavity. They found that the consecutive injection of N atoms in the same superposition state generates a coherent state of the cavity field, |α = -i N ρ_ eg gτ⟩. Comparing it with the result we derived above, we then confirm that the effective number of atoms involved in the collective emission in the single-atom superradiance is indeed <N_c>=r/γ_c.2.2.3 Competition of two contributionsIn general, both effects contribute to the cavity field and the steady state mean photon number can be approximately expressed as a sum of two terms:<n>≃ <N_c> ρ_ee(gτ)^2/2-(2ρ_ee-1)<N_c> (gτ)^2 + (<N_c>|ρ_eg| gτ)^2≈ 1/2<N_c> ρ_ee (gτ)^2 +<N_c>^2 |ρ_eg|^2(gτ)^2, when<N_c>≪ (gτ)^-2.We have verified the solution of Eq. (21) to be valid by comparing it with the numerical calculation results using the full density matrix equation (see Fig. S5).The competition between two terms is independent of gτ, which corresponds to the single-atom vacuum-Rabi oscillation angle, but only depends on the initial atomic state (ρ_ee, |ρ_eg|) and the atom number <N_c>. The second term becomes dominant when<N_c>>2ρ_ee / (2|ρ_eg|)^2 >1 (formaximumdipolecase,|ρ_eg| = 1/2) 2.2.4 Saturation due to coherent Rabi oscillation As in the cavity-QED microlaser, when the accumulated cavity photon number is sufficiently large, the Rabi angle √(<n>)gτ becomes large and thus the coherent Rabi oscillation of each atom gives a feedback to the photon number. As a result, the photon number becomes stabilized (16,17).This process starts when the Rabi angle due to the cavity field becomes comparable to the inversion angle of atoms (given by the pump pulse area Θ in the experiment).√(<n>+1)gτ∼ΘAssuming that the dipole pumping is dominant [Eq. (22)] , we substitute <n> with the result from Sec. 2.2.3. The saturation condition is then<N_c> ∼ (gτ)^-2Θ/sinΘ∼ (gτ)^-2. 2.3 Connection to thresholdless lasing Conventionally, the thresholdless lasers are studied in terms of the so-called β factor, which is the ratio of the spontaneous emission rate into the lasing cavity mode to the total decay rate (radiative as well as non-radiative) of the gain medium (of atoms or molecules).Specially designed micro/nano-cavities are used to maximize β factor by means of the Purcell effect in order to lower the lasing threshold. Thresholdless lasing is achieved when β≃ 1, for which almost all energy supplied to the excited state of the gain medium is transferred to the cavity mode per unit time via spontaneous emission of photons (24). Let us consider β factor in the non-collective emission case (usual lasing) in our experiment.From Eq. (21), with ρ_ eg=0 , we get the output flux of photons as 2γ_c ⟨ n⟩≃γ_c⟨ N_c⟩ρ_ ee (g τ)^2, which should be balanced by the emission from the atom to the cavity mode. Therefore, the spontaneous emission rate into the cavity mode is γ_c⟨ N_c⟩ρ_ ee (g τ)^2. Total decay rate of the excited state energy must be equal to the total pumping rate of energy in the excited state, ⟨ N⟩ρ_ ee/τ=γ_c⟨ N_c⟩ρ_ ee using the relation ⟨ N⟩=⟨ N_c ⟩γ_cτ. From this consideration, we then obtain β=(gτ)^2.In the non-collective emission experiment of Fig. 3B, β≃ 0.011. Despite the large single-atom cooperativity of C≃44, most of the excited atoms exit the cavity without emitting a photon into the cavity because of the short interaction time τ. This is consistent with the large mean photon number change occurring at the threshold in Fig. 3B. On the other hand, if we just apply this definition to the case of Fig. 3A, β≃ 0.034, which certainly cannot explain the thresholdless growth of the mean photon number observed there.One may extend the definition of β factor to the collective emission case by interpreting the spontaneous emission into the cavity mode to include the collective emission also. The collective emission rate into the cavity mode is obtained from Eq. (21) as 2γ_c⟨ N_c⟩^2 |ρ_ eg|^2 (gτ)^2, and thus for collective emission we get β_ coll=2⟨ N_c⟩ (|ρ_ eg|^2/ρ_ ee) (gτ)^2.As the atom number ⟨ N_c ⟩ increases, β_ coll also increases and finally converges to unity as the saturation condition, Eq. (24), sets in. So, it is tempting to use this effective β factor for explaining the thresholdless lasing in Fig. 3A.However, this effective β is not constant. It depends on the atom number ⟨ N_c⟩ or pumping rate, and thus no longer a geometric factor, loosing its usefulness in analysis. This consideration led us to conclude that the thresholdless lasing coming from the collective emission or superradiance in Fig. 3A cannot be explained in terms of the usual β factor in ordinary lasers. | http://arxiv.org/abs/1705.09136v3 | {
"authors": [
"Junki Kim",
"Daeho Yang",
"Seung-hoon Oh",
"Kyungwon An"
],
"categories": [
"quant-ph",
"physics.atom-ph",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170525115529",
"title": "Coherent single-atom superradiance"
} |
Term Models of Horn Clauses over Rational Pavelka Predicate Logic [ 30 July 2017 =================================================================A variant of the Parareal method for highly oscillatory systems of PDEs was proposed by <cit.>. In that work they proved superlinear convergence of the method in the limit of infinite time scale separation. Their coarse solver features a coordinate transformation and a fast-wave averaging method inspired by analysis of multiple scales PDEs and is integrated using an HMM-type method. However, for many physical applications the timescale separation is finite, not infinite.In this paper we prove convergence for finite timescale separaration by extending the error bound on the coarse propagator to this case. We show that convergence requires the solution of an optimization problem that involves the averaging window interval, the time step, and the parameters in the problem. We also propose a method for choosing the averaging window relative to the time step based as a function of the finite frequencies inherent in the problem. § INTRODUCTION A variation of the Parareal method<cit.><cit.> for highly oscillatory systems of equations was proposed in <cit.>. The method constructs the coarse solver based on a coordinate transformation and fast-wave averaging motivated by multiscale analysis of PDEs <cit.> and performs the integration using techniques from the Heterogeneous Multiscale Method <cit.>. They proved that the method provides parallel speedups<cit.> in the limit of infinite time scale separation. Finite time scale separation is an important case to understand for physical applications because many physical phenomena, such as those occurring in numerical weather prediction, have finite frequencies inherent in the problem, e.g. Earth's rotation rate is finite. In this paper we extend the work of <cit.> by showing that rapid convergence of the method is also possible for finite timescale separation by proving error bounds on the coarse solver in the case of finite timescale separation.Examples of applications of the Parareal algorithm being applied to parabolic PDEs include simulations of financial markets (i.e. the Black-Scholes equation for an American put <cit.> and a nonlinear parabolic evolutionary equation via the finite element method <cit.>. Hyperbolic systems solved with Parareal include simulation of molecular dynamics <cit.>, fluid/structure interaction <cit.>, solution of the Navier-Stokes equations <cit.>, and reservoir modelling <cit.>. In all of these applications, the degree of oscillatory stiffness was not sufficient to impede convergence, but it is known (cf. <cit.>) to be an issue for the Parareal method.There have been several modifications to the Parareal method which apply to highly oscillatory systems which assume that the system may be separated into fast and slow variables. In terms of ODEs, <cit.> have proposed a multiscale method for singularly perturbed ODEs where the fast dynamics are dissipative. Ariel et al (2016)<cit.> propose a method for highly oscillatory ODEs which is multiscale in nature but does not require explicit knowledge of the fast and slow variables. Gander and Hairer (2014) <cit.> suggest Parareal methods for Hamiltonian dynamics. Approaches using symplectic integrators with applications to molecular dynamics are presented in, for example, <cit.> and <cit.>. Finally, <cit.> proposed a method which is motivated by a asymptotic solutions for fast singular limits of nonlinear evolutionary PDEs. It is an extension of this method which we study here and which we refer to as Asymptotic Parallel-in-Time (APinT). It takes its name from the modified coarse solver which is inspired by methods used in the asymptotic analysis of PDEs.In this work we are primarily interested in oscillatory stiffness. Oscillatory stiffness places a restriction on the convergence of the Parareal method due to accuracy and stability limitations it places on the timestep size. We consider oscillatory stiffness to be a phenomenon arising from the presence of rapid oscillations which restricts the coarse timestep, with the degree of oscillatory stiffness being the degree to which the timestep is restricted. As discussed by Higham and Trefethen<cit.>, stiffness is a transient phenomenon involving finite time intervals. This is important here as in this paper we are concerned with mitigating oscillatory stiffness over the interval of the coarse timestep.We consider as a model equation a PDE of the form: ∂𝐮/∂ t + 1/εℒ𝐮 + 𝒩(𝐮,𝐮) = 0,where 𝐮 is the vector of unknowns, ℒ is a skew-Hermitian linear operator with purely imaginary eigenvalues, and 𝒩(·,·) is a nonlinear operator which is assumed to be of quadratic type. We further assume that the solution, 𝐮∈ L^2 and that we may approximate <ref> as a finite system of ODEs. The linear term then induces temporal oscillations on an 𝒪(ε) timescale, which can require the use of prohibitively small timesteps for standard numerical integrators if ε is small. If accuracy is required, as is necessary for the Parareal method, implicit methods must also use a small time step.The contribution of this work lies in an improved error estimate of the asymptotically motivated coarse solver which permits a mathematical description of the relationship between the time stepping error and the time-averaging of the the nonlinear operator. This understanding leads to an improved mathematical understanding of the convergence of the APinT method. With this improved notion of accuracy, we are then able to prove that the APinT algorithm converges for finite time scale separation. This is an advance on the previously shown case in the limit of ε→ 0.The slow solution relies on an averaged version of <ref>, with the average taken over an infinite window in the limit as ε→ 0. This is estimated numerically by a finite sum over a sufficiently large window. It has been shown<cit.><cit.><cit.> that the length required for this window may be reduced through the use of a smooth kernel of integration. In the small-ε limit, we find that this method provides a convergent algorithm. We have also found that for finite ε the averaging window may be chosen such that the slow solution is sufficiently accurate that the Parareal method remains convergent, as shown in <ref> and discussed in <ref>.In the next section we give an overview of the Parareal method to set the context of the work. In <ref> we discuss the slow solution which is found by fast-wave averaging and which forms the coarse propagator of the APinT method. We will discuss the implications of the existence of near resonances on the slow solution. With that in mind, we proceed in <ref> to prove the error bounds and therefore the convergence of the APinT method. Finally, in <ref> we will show numerical experiments on the one-dimensional rotating shallow water equations and discuss some particularities of solving these equations with the APinT method.§ THE PARAREAL ALGORITHM In this section we briefly review the Parareal method proposed by Lions et al.<cit.>, and further expanded upon by Maday and Turinici<cit.>. They proposed a generalisation of the concept of domain decomposition to the temporal domain, in which a `coarse' approximation to the solution is computed which is then refined, parallel in time, by the `fine' timestep. The solution has been shown to iteratively converge to the fine solution<cit.>. In practice, this method requires that the coarse timestepping method permits large timesteps, that it be inexpensive to compute and sufficiently accurate that the method converges quickly. In general, the maximum timestep is 𝒪(ε), so in the case of ε=𝒪(1), i.e. the less-stiff case, Parareal may be applied without any modifications. The insight of <cit.> was that a slow solution based on a coordinate transformation and a time average over the fast waves in the nonlinear operator provides a convergent and efficiently-computable coarse approximation. In fact, they showed that under suitable assumptions of smoothness, superlinear convergence is obtained as ε→ 0. Also related to this work is that of <cit.> who paid particular attention to the parallel implementation. They note that the coarse solver may employ a coarser timestep <cit.>, a coarser discretisation <cit.>, and/or a simpler physical model <cit.>. The APinT coarse solver as presented here us a combination of the first and third of these. A sketch of the algorithm is in <ref>. We assume for the sake of simplicity that we are interested in solving <ref> on the interval t∈[0,1]. Let φ_t(𝐮_0) denote the evolution operator associated with <ref> such that 𝐮(t) = φ_t(𝐮_0) solves the full equation. Similarly φ_t(𝐮_0) solves the averaged equations.We then divide the time domain into N finite subintervals, [nΔ T, (n+1)Δ T], where n=0,… N-1. The Parareal algorithm begins with a coarse solve and then proceeds by computing approximations to the solution, 𝐔_n^k, iteratively, following: 𝐔^k_n = φ_Δ T(𝐔^k_n-1) + (φ_Δ T(𝐔^k-1_n-1 - φ_Δ T(𝐔^k-1_n-1)), k=1,2,… Here, since the quantities 𝐔_n-1^k-1 in the difference (φ_Δ T(𝐔^k-1_n-1 - φ_Δ T(𝐔^k-1_n-1)) are already computed at iteration k, the difference can be computed in parallel for all n. Since the computation of φ_Δ T(𝐔_n-1^k) is cheap, the overall computation is quick in a parallel sense if the iterates converge quickly.§ THE SLOW SOLUTIONOur interest in solving general PDEs which arise in physical modelling, in particular those of weather and climate, requires that we confront the problem of oscillatory stiffness over the interval of a coarse time step. In this section we shall describe the mathematical roots of the slow solution, provide a short description of its historical context, and review its discretisation as a coarse solver for the Parareal algorithm. As the convergence of the coarse solver for the APinT algorithm depends on the quality of the approximation, we will then investigate the numerical behaviour of this solver before providing an improved proof of the performance of the slow solver.As an example of an application where the actual physics was thought to be asymptotic, but later shown not to be we look to the field of numerical weather prediction.Historically, the physical notion of `slow' dynamics, called Quasi-Geostrophic (QG) equations was a major advance in understanding weather. This insight was due to Charney<cit.> who derived the `slow' equations. These reduced equations allowed the fast waves, which cause the oscillatory stiffness, to be filtered while still resolving the large-scale motions of the fluid. The work was later expanded upon, leading to what is generally recognised as the first successful numerical weather prediction <cit.> <cit.>. Since those early days, the reduced equations have been rigorously shown to hold asymptotically in the limit of ε→ 0<cit.>. In contrast to these results, modern weather prediction has found that the reduced equations are not accurate enough to be predictive and therefore they rely on numerical approximations of the full equations of motion<cit.>. As such, we are confronted with the problem that at least some of the oscillations matter even for the large scale flow and so we must find some way to resolve the fast waves in order to capture the full dynamics.To address this problem for the Parareal method, <cit.> constructed a numerical approximation to the governing equations <ref> based on the consideration of fast singular limits<cit.>. One of the conclusions of this theory for when the linear operator, ℒ is skew-Hermitian is that though the leading order dynamics is not slow itself, the slow dynamics evolve independently of the fast, while the fast dynamics are `swept' by the slow <cit.>. For realistic weather, we expect ε∼ 0.01 to ε∼ 0.1 <cit.>. Since ε is finite for realistic cases, the timescale separation is also finite. Working in the limit of small ε and applying the method of multiple scales, an averaged equation for equations of the form <ref> was found by <cit.>. For a slow timescale, t, and a fast timescale, τ, they showed that averaged equations in the asymptotic limit ofε→ 0must satisfy: ∂𝐮(𝐱,t)/∂ t + lim_τ→∞1/τ∫_0^τe^sℒ𝒩(e^-sℒ𝐮(𝐱,t),e^-sℒ𝐮(𝐱,t)) ds=0,.𝐮(𝐱,t)|_t=0 = 𝐮^0(𝐱),where 𝐮̅ denotes the averaged 𝐮 and where the integral is taken over the nonlinear operator, not the solution itself, and where there is a mapping by the exponential of the linear operator between the averaged and `full' solutions:𝐮^0(x,t,τ) = e^-τℒ𝐮̅(𝐱,t).The above condition, studied in detail by <cit.>, <cit.>, and <cit.>, motivates our averaging, described below. We follow <cit.> and write the averaged equation in the following form:∂𝐮/∂ t + e^tℒ/ε𝒩(e^-tℒ/ε𝐮, e^-tℒ/ε𝐮) = 0,from which the full solution may be obtained through the application of the matrix exponential as in <ref>. The averaged equation, <ref>, has lost the factor of 1/ε from the full equation, <ref>, the main source of the oscillations, although another derivative will regain this term and so the oscillations have not been entirely eliminated.In order to use this equation as a coarse solver for Parareal with finite timescale separation, <cit.> retreated from the asymptotic limit by taking the integral over the nonlinear operator in <ref> over a finite time averaging window, rather than the infinite limit associated with ϵ→ 0. This integral is approximated numerically by using a smooth kernel, ρ(s), 0≤ s ≤ 1 which is chosen such that the length T_0 of the time window for the averaging is as small as possible, and approximate the averaged nonlinear operator <ref> as:𝒩(𝐮(t))≈1/T_0∫_0^T_0ρ(s/T_0)e^sℒ𝒩(e^-sℒ𝐮(t)) ds ≈1/M∑_m=0^M-1ρ(s_m/T_0)e^s_mℒ𝒩(e^-s_mℒ𝐮(t)) The Heterogeneous Multiscale Method (HMM) <cit.> is then applied to the slow equation by computing the averages numerically, as in <ref>. In practice, the width of the averaging window may be freely chosen, as illustrated in <ref>. The choice of this window has a significant effect on the convergence of the method, as illustrated in <ref>. As the computational cost of Parareal is proportional to the number of iterations required, an optimally-chosen window is necessary. In addition to a new error estimate our insight into the coarse error allows us to choose the optimal averaging window length as a function of the timescale separation. §.§ Triad ResonancesIn order to permit a long coarse timestep, the coarse solver proposed by <cit.> and described in <ref> filters the nonlinear operator. The effect of this is a change in the content of the nonlinear triad interactions which is a function of both the degree of near resonance of the interaction and the length of the averaging window. As we discuss in <ref>, the extent to which near-resonant sets are retained or rejected has an important impact on the convergence of the Parareal method.Therefore, in this section we review nonlinear triad resonances for the model problem that we use in our tests in <ref>, noting that a similar approach applies to all systems of the form <ref>.Systems governed by a quadratic nonlinearity with dispersive waves exhibit triad resonances<cit.><cit.><cit.>. Since we are motivated by geophysical modelling, we consider the general averaged equation for the rotating shallow water equations, which are commonly used as a test case for geophysical solvers. Following the notation of <cit.>, we decompose the right-hand side of <ref> in terms of its basis of eigenvectors and write: / t = -lim_τ→∞1/τ∫_0^τ∑_∈ℤ^2∑_α = -1^1[∑_=_1 + _2∑_α_1,α_2σ__1^α_1σ__2^α_2C_,_1,_2^α,α_1α_2e^i(·) - iΩ_,_1,_2^α,α_1,α_2s/ε]𝐫_^α ds.where Ω_,1,2^α,α_1,α_2 = ω__1^α_1 + ω__2^α_2 - ω_^α, α = -1,0,1 refers to the different branches of the eigenvalues, , _1, and _2 are the wavenumbers, ω_^α is the dispersion relation at a given α and wavenumber, σ denotes the Fourier coefficient in this basis, 𝐫_^α is the right eigenvector of the linear operator, and C__1,_2,^α_1,α_2,α is an interaction coefficient <cit.>.In the asymptotic case, the limit ε→ 0, by the orthogonality of the Fourier series, the only waves which remain after the wave averaging procedure (i.e. where τ→∞) are the direct three-wave resonances (cf. <cit.>, <cit.>, <cit.>), i.e. the elements of the resonant set, 𝒮_,α i.e.: 𝒮_,α = {(_1,_2,α_1,α_2): = _1 + _2,ω_^α = ω__1^α_1 + ω__2^α_2}. The wave-averaged solution then follows: σ_^α/ t + ∑_𝒮_,ασ__1^α_1σ__2^α_2C__1,_2,^α_1,α_2,α = 0, It is this three-wave resonance condition from which the behaviour of the averaging kernel can be understood. In the limit as τ→∞, only the direct resonances should remain. Because we have finite timescale separation,this integral is approximated over a finite averaging window which must be large enough to filter the non-resonant triad.As shown in <cit.>, using a finite time-averaging window and numerically integrating with respect to a smooth, finitely supported kernel permits this algorithm to result in a convergent Parareal algorithm in the limit of small ε.For finite ε we take a finite average and so the solution set is larger than the direct resonant set and this has an important effect on the convergence of Parareal. To better explain the finite-ε case, we define concentric shells of near-resonances, i.e. we rewrite the triad-based form <ref> as: e^sℒ/ε𝒩(e^-sℒ/ε𝐮(t),e^-sℒ/ε𝐮(t))= ∑_λ_ne^iλ_ns𝒩_n(𝐮(t)), = ∑_𝒮_,α𝒩_n(𝐮(t)) + ∑_β=1^∞(∑_𝒮_,α^ϵ_βe^iλ_ns𝒩_n(𝐮(t)))where 𝒮_,α^ϵ_β, β = 1, 2, … refers to a near-resonant set, i.e.: 𝒮_,α^ϵ_β = {(_1,_2,α_1,α_2): = _1 + _2, ϵ_β-1 < 1/ε|ω_^α - ω__1^α_1 + ω__2^α_2| ≤ϵ_β},where ϵ_0=0 by definition. The direct-resonant set results in a solution consisting of only the slow dynamics of the system – which was shown in <cit.> to be equivalent to the reduced equations for this system. Again, and as we will see in <ref>, the extent to which the near-resonant sets are retained and rejected by the averaging procedure is fundamental to the convergence of the APinT variation of the Parareal method.§ ERROR BOUNDS Now that we have discussed the key elements of the algorithm, we are in a position to discuss and prove convergence for the case when ε is finite. We shall first construct an improved error estimate for the coarse solution, and then use that result to prove the convergence of the Parareal method.§.§ A bound on the errors due to time-stepping and time-averaging in the coarse solverIn this section we employ the idea of near-resonant sets to extend the existing proof of APinT convergence<cit.> to the case of finite ε.As has been demonstrated above (cf. <ref>), the choice of the averaging window width, η, has a profound effect on the convergence of the method. While the choice of η is well-understood for the limit of small ε<cit.>, we show here that η may be similarly chosen to provide convergence for ε up to 𝒪(1) for an appropriate coarse timestep. We first reduce <ref> to a standard form for ODEs. Following <ref>, we write: 𝐯_t(t) = e^t/εN(e^-t/ε𝐯, e^-t/ε𝐯),t ∈ [0,Δ T],i.e. we are interested in the solution over a Δ T timescale. Let τ=t/(εΔ T), and so 𝐯̃(τ), defined on the interval [0,1/ε],𝐯̃(τ)=𝐯(t). Then differentiation gives, ∂_t𝐯(t)=∂_t𝐯̃(t/(εΔ T))=1/εΔ T∂_τ𝐯̃(t/(εΔ T))=1/εΔ T∂_τ𝐯̃(τ). Upon this substitution into the coarse solver <ref> over the discrete time interval <ref>, we arrive at the desired form which permits us to use the framework given in <cit.> where they have derived bounds for averaging methods. The aim of this is to modify and reapply their result for the error bound due to averaging, which holds on a general dynamical system of finite ODEs. This averaging error is one of the two major sources of error in the timestepping of the coarse solver. We then write the coarse solver in the form: ∂_τ𝐯̃(τ)=εΔ T e^τΔ T N(e^-τΔ T 𝐯̃(τ), e^-τΔ T 𝐯̃(τ)). While our interest is in solving PDEs describing physical systems, in practice we employ a Fourier spectral method, which has the effect of treating the PDE as a finite-dimensional system of ODEs. In this paper we show that the APinT method is convergent for finite systems of ODEs. This gives us access to the machinery of the numerical analysis of ODEs and averaging methods, following <cit.>. Let 𝐱 solve the governing equations when they are written as a system of ODEs, i.e. in the form shown in <ref>. For example, in the numerical experiments given in <ref>, 𝐱 is the Fourier solution. Then we may write: 𝐱_t=ε𝐟(𝐱,t). Similarly, we consider the coarse solver <ref> written as a system of ODEs. Let 𝐲 solve this averaged form of <ref>, i.e.: 𝐲_t=ε𝐟(𝐲,t),where the averaging follows directly from the averaged equation, <ref> and is written: 𝐟(𝐱,t)=1/η∫_0^ηρ(s/η)𝐟(𝐲,t+s)ds,where η denotes the finite length of the averaging window. Considering the initial value problems in 𝐱 and 𝐲 as stated above where 𝐟 is ℝ^n×ℝ Lipschitz continuous with constant β in 𝐱 on D ⊂ℝ^n and t on an 𝒪(1) timescale, i.e. for all 𝐱_1,𝐱_2∈ D, β is such that: ‖𝐟(𝐱_1,t)-𝐟(𝐱_2,t)‖≤β‖𝐱_1-𝐱_2‖. Let: M = sup_𝐱∈ Dsup_0≤ t≤ L‖𝐟(𝐱,t)‖. Then we can bound the difference between the exact solution 𝐱 and the averaged solution 𝐲 as: ‖𝐱-𝐲‖≤ M(1+1/2βε)εΔ Tη, The above lemma follows from a modification of Lemma 4.2.8 in <cit.> in order to include the kernel of integration (cf. appendix <ref>). We have here bounded the error over an 𝒪(1) time interval instead of 𝒪(1/ε) so that the rate of convergence at different degrees of scale separation may be more easily compared, as in practice we are interested in simulations over fixed timescales. Taking the unmodified lemma provides a slightly different result as it gives the averaging error over a simulation time which scales with ε. Due to the numerical nature of the proof here, the appropriate timescale is over a coarse timestep.<ref> places a bound on the error committed by averaging over the fast waves, independent of the numerical methods used for spatial or temporal discretisation. Next, we consider the error arising from the numerical approximation of <ref>. In doing so, we will need to assume bounds on 𝐟 in the region of phase space where 𝐲 exists. Then assume that ∂_𝐲𝐟(𝐲,t)≤ M_1,𝐲(t) ∈ D⊂ℝ^n. We assume that such a bound exists for higher spatial derivatives of 𝐟, such that max_j∂^j𝐟/∂𝐲_k^j≤ M,1 ≤ k ≤ n,0≤ j≤ p.Denote the numerical approximation to the averaged solution 𝐲(t) with timestep Δ T by a second-order timestepping method as 𝐲_Δ T(t). Assume that 𝐲(t) = ε𝐟(𝐱,t) and that 𝐟∈ D as in <ref>. Assume that integration is performed with respect to a smooth kernel, ρ(·), and let λ_n denote the n-th near resonant triad (cf. <ref>). Then the local time-stepping error of a second order time-stepping scheme applied to <ref> satisfies: 𝐲(t) - 𝐲_Δ T(t)≤ CMεΔ T^3max_x∈ℝ(λ_n^21/η∫_0^ηρ(s/η)e^iλ_nsds),for some constant, C ∈ℝ < ∞ and where M is the bound over the nonlinear operator as given in <ref>. The timestepping error of a p-th order scheme is bounded by <cit.>: 𝐲(t) - 𝐲_Δ T(t)≤ C_t(Δ T)^p+1max_t‖d^p+1𝐲/dt^p+1(t)‖ _2,First, decompose 𝐟 in terms of its basis of eigenvectors as discussed in <ref>. As with <ref>, we may write the solution as a sum of ODEs, each for a specific resonant nearness, λ_n. Then for the j-th component of 𝐲, we write dy_j/dt = ε1/η∫_0^ηρ(s/η)∑_nΔ T𝐍_n,j() ds,where the nearness of the resonances in any particular ODE is exposed through the eigenvalue sum, λ_n, in the exponent and where the subscript ,j denotes the j-th component and not a derivative, as it would with Einstein's notation. We then seek the third time derivative, which is found to be d^3y_j(t)/dt^3 = ε(i^2Δ T^3∑_nλ_n^2𝐍_n,j +.. 2iΔ T^2∑_n∑_kλ_n∂𝐍_n,j()/∂ y_kdy_k(t)/dt+. .Δ T∑_n∑_k,l∂^2𝐍_n,j()/∂ y_k∂ y_ldy_k(t)/dtdy_l(t)/dt+ ..Δ T∑_n∑_k∂𝐍_n,j()/∂ y_kd^2y_k(t)/dt^2) ds. This is then the right-hand side which is integrated with respect to the smooth kernel. The magnitude of the near-resonant triad, λ_n, now presents itself as a multiplier on the complex exponential. It is then clear that it is this value, which is zero for direct resonances but becomes large in general, which is the source of numerical stiffness. For convenience, we introduce P(η) =ds;e_n(t) = e^iΔ Tλ_nt,then d^3y_j(t)/dt^3 = εΔ T^3 P(η)∑_n[-λ_n^2e_n𝐍_n,j +(2iε∑_kλ_ne_n∂𝐍_n,j/∂ y_k)(∑_n'e_n'𝐍_n',j) + .. ε^2∑_k,le_n∂^2𝐍_n,j/∂ y_k∂ y_l(∑_n'e_n'𝐍_n',j)(∑_n” e_n”𝐍_n”,j) + ..∑_k'e_n∂𝐍_n,j/∂ y_k'( ε∑_n'λ_n'e_n'𝐍_n',j + ε^2∑_n”e_n”𝐍_n”,j∑_n”'∑_l'e_n”'∂𝐍_n”',j/∂ y_l') ], In bounding the timestepping error, we are interested in the norm of this quantity. Recalling that we are working with a finite-dimensional system of ODEs and applying the triangle and Cauchy-Schwarz inequalities we find that d^3y_j(t)/dt^3≤εΔ T^3P(η)∑_n|𝐍_n,j|(λ_n^2 + 2λ_nε|∑_k∂𝐍_n,j/∂ y_k| +.. ε^2|𝐍_n,j||∑_k,l∂^2𝐍_n,j/∂ y_k∂ y_l| ελ_n|∑_k∂𝐍_n,j/∂ y_k| + ε^2|∂𝐍_n,j/∂ y_k|^2). Now, as 𝐍 and all of its spatial derivatives up to and including p=2 are bounded by M by <ref>, we write d^3y(t)/dt^3 ≤εΔ T^3Mmax_x∈ℝP(η)λ_n^2 + 3λ_nε M + ε^2M^2≤εΔ T^3Mmax_x∈ℝP(η)λ_n + C_fε M^2, where C_f is a positive constant. We will now assume that |λ_n|≠ 0 as we are interested in the sup-norm of these values, which is nonzero when near-resonances are included. The directly resonant case has been treated by <cit.>. Then we must consider two possibilities. Firstly, if |λ_n|≤ 1, then we define some constant, K_1, K_1 = (1+C_fε M)^2. If |λ_n|>1, the binomial theorem yields: (|λ_n|+C_fε M)^p = ∑_j=0^ppj(|λ_n|)^p-j(C_fε M)^j,≤∑_j=0^ppj(|λ_n|)^p(C_fε M)^j,=|λ_n|^p∑_j=0^ppj(C_fε M)^j, =|λ_n|^pK_2. And then we may write (|λ_n| + εΔ TM)^2≤max(K_1, |λ_n|^2K_2). As for the Rotating Shallow Water Equations there must always be a value of λ_n which is strictly greater than one, we shall assume that it is the second value which is the maximum. We now let C = C_tK. Finally, we bound the nonlinear term in the same fashion as <ref>, where the fact that: M =sup_𝐱∈ Dsup_0≤ t≤ L‖𝐟(𝐱,t)‖ =sup_𝐱∈ Dsup_0≤ t≤ L‖∑_nΔ T e^iΔ T λ_nt𝐍_n(𝐲)‖≤ sup_𝐱∈ Dsup_0≤ t≤ L(∑_n‖𝐍_n(𝐲)‖) < ∞,completes the proof by providing an upper bound for the nonlinear operator as in <ref>. This provides a bound for the error due to timestepping which does not depend directly on the solution, but rather on the general properties of the nonlinearity, in particular the triadic interactions. With these two lemmas describing our primary sources of error, we will seek a bound on the error in the APinT algorithm and use this to show convergence.From <ref>, it follows that the timestepping error depends on: E(ε,η,λ_n,Δ T) ∼εΔ Tλ_n^2(1/η∫_0^ηρ(s/η)e^iλ_nΔ Tsds). We define the following term which describes the filtering, independent of the gain due to the scale separation and the coarse timestep, and which is the key insight into understanding how to regularise an oscillatory problem over a finite time interval. Λ(η) = max_x∈ℝλ_n^2∫_0^1ρ(s)e^iλ_nηΔ Ts ds. Λ(η) provides a measure of the extent to which the averaging integral mitigates the numerical stiffness. Recall that when the maximum λ_n is large, as it is for highly oscillatory problems, it contributes to large gradients on the right-hand side requiring a small numerical timestep. In contrast, the integral component tends to zero as λ_n gets large, and does so superlinearly because of the smooth kernel, ρ(s) <cit.>. This term is then where we see precisely how the averaging procedure filters the fast oscillations, causing Λ(η) to achieve a lower magnitude than λ_n^2 does on its own and therefore reducing the numerical stiffness.In seeking a bound on the error in the timestepping, it is necessary to bound this term for some particular averaging kernel, ρ(s). The choice of averaging kernel affects the error bounds through this function. Λ(η) is bounded and tends rapidly to zero as η→∞ (q.v. <ref>).With this in mind, we now prove <ref> which bounds the error committed by the coarse timestepping as compared to the fine. This will later allow us to prove error bounds on APinT subject to finite timescale separation. Let Δ T denote the coarse timestep for a second order numerical method. We assume a finite scale separation on the order of ε. For an averaging window of length η, the total error in the coarse timestepping for the APinT algorithm is bounded by: 𝐱(t) - 𝐲_Δ T(t)≤ MεΔ T((C_0 + C_1ε)η + D_1(Δ T)^3Λ(η)),where M is the sup-norm over the nonlinear operator as in lemmas <ref> and <ref> and C_0, C_1, and D_1 are finite constants. By the triangle inequality, we may write: ‖𝐱(t)-𝐲_Δ T(t)‖=‖𝐱(t)-𝐲(t)+𝐲(t)-𝐲_Δ T(t)‖, ≤ ‖𝐱(t)-𝐲(t)‖ +‖𝐲(t)-𝐲_Δ T(t)‖. <ref> is used to bound the first term, i.e.: 𝐱(t) - 𝐲(t)≤ M(C_0 + C_1ε)εΔ T η. Applying <ref> and <ref> to the second term yields: 𝐲(t) - 𝐲_Δ T(t) ≤ MCC_1(Δ T)^3εΛ(η),≤ MD_1(Δ T)^3εΛ(η),where Λ(η) is bounded independently of λ_n for any averaging window length, η. Combining the bounds in equations <ref> and <ref> gives the theorem as desired. §.§ Proof of Parareal ConvergenceWe may now derive error bounds for the Parareal iteration on finite systems of ODEs, given in <ref>. Using our improved error bound for the coarse solver which holds for finite ε, we modify the proof given in <cit.>, which held only as ε→ 0. For consistency we define several operators following <cit.>. Let φ̃_Δ T(·) be the evolution operator associated with numerically solving the slow equation using an 𝒪(p) method, such that φ̃_Δ T(·) is a numerical approximation of φ_Δ T(·). Furthermore, let φ_Δ T(·) denote the evolution operator for the fine solution. We then define: ℰ_φ,φ(·) = φ_Δ T(·) - φ_Δ T(·);ℰ_φ,φ̃(·) = φ_Δ T(·) - φ̃_Δ T(·), Then, as in <cit.>, <cit.>, and <ref> we make the following assumptions:* The operators φ(·) and φ(·) are uniformly bounded for 0≤ t ≤ 1:φ_t(𝐮_0)≤ C𝐮_0, φ_t(𝐮_0)≤ C𝐮_0 * The averaging method is accurate in the sense that:φ_t(𝐮_0) - φ_t(𝐮_0)≤εΔ Tη M(C_1 + C_2ε)𝐮_0 * The averaged evolution operator satisfies:φ_Δ T(𝐮_1) - φ_Δ T(𝐮_2)≤ (1 + CΔ T)𝐮_1 - 𝐮_2,and the numerical approximation to the evolution equation satisfies:φ̃_Δ T(𝐮_1) - φ̃_Δ T(𝐮_2)≤ (1 + CΔ T)𝐮_1 - 𝐮_2, * Following <ref> and <ref> and <ref>, the error operators satisfy: ℰ_φ,φ(𝐮_1) - ℰ_φ,φ(𝐮_2)≤εΔ T η M (C_1 + C_2ε) 𝐮_1 - 𝐮_2,and:ℰ_φ,φ̃(𝐮_1) - ℰ_φ,φ̃(𝐮_2)≤Δ T^3εΛ(η) M C 𝐮_1 - 𝐮_2,p ≥ 1. We have now quantified the major sources of error in the coarse timestepping which will affect the convergence of Parareal. The following proof of the convergence follows directly from these bounds. Subject to the above assumptions, the error, 𝐮(T_n)-𝐔_n^k, after the k-th Parareal iteration is bounded by: 𝐮(T_n) - 𝐔_n^k≤ MC_g(C_1Δ T^3εΛ(η) + (C_2 + C_3ε)εη)^k+1𝐮_0.The proof is by induction on k. When k=0: 𝐮(T_n)-𝐔_n^n = φ_Δ T(𝐮_0) - φ̃_Δ T(𝐮_0)≤φ_Δ T(𝐮_0) - φ_Δ T(𝐮_0) + φ_Δ T(𝐮_0) - φ̃_Δ T(𝐮_0)≤ M((C_1 + C_2ε)εΔ T η + C_3Δ T^2)𝐮_0, where we have used <ref>, which bounds the error induced by the averaging procedure, to bound the first term and <ref>, which governs the timestepping error, for the second. Now assume that: 𝐮(T_n)-𝐔_n^k-1≤ (Δ T + ε)(C_1Δ T^3εΛ(η) + (C_2+C_3ε)εη) 𝐮_0. We may then write the Parareal iteration, <ref> in the following form, using <ref> and <ref>: 𝐮(T_n) - 𝐔_n^k = (φ̃_Δ T(𝐮(T_n-1)) - φ̃_Δ T(𝐔_n-1^k)) + (ℰ_φ,φ(𝐮(T_n-1)) - ℰ_φ,φ(𝐔_n-1^k-1)) + (ℰ_φ,φ̃(𝐮(T_n-1)) - ℰ_φ,φ̃(𝐔_n-1^k-1)) By directly substituting equations <ref>, <ref>, and <ref>, we have: 𝐮(T_n) - 𝐔_n^k ≤ (1+CΔ T)𝐮(T_n-1)-𝐔_n-1^k +M(C_1Δ T^3εΛ(η) + (C_2 + C_3ε)εΔ Tη)𝐮(T_n-1)-𝐔_n-1^k-1≤ (1+CΔ T)𝐮(T_n-1)-𝐔_n-1^k +MΔ T(C_1Δ T^2εΛ(η) + (C_2 + C_3ε)εη)^k+1𝐮_0. Finally, application of the discrete Gronwall inequality gives: 𝐮(T_n) - 𝐔_n^k ≤(e^C(T_n-T_0)-1)M (C_1Δ T^2εΛ(η) + (C_2 + C_3ε)εη)^k+1𝐮_0≤ MC_g(C_1Δ T^2εΛ(η) + (C_2 + C_3ε)εη)^k+1𝐮_0.<ref> is one of the key contributions of this work. Using the understanding of near-resonance and the result of <ref>, it generalises the proof given by<cit.> of convergence for the asymptotic limit as ε→ 0 to finite ε. This is a significant improvement as for many physical applications such as weather and climate modelling ε remains finite. As the averaging window length, η, may be freely chosen we may select an optimal η for a wide range of ε subject to the other constants and choice of Δ T such that the method is convergent. We discuss this in the next section. §.§ Convergence for any εGiven <ref> we are in finally in a position to discuss convergence for any timescale separation. For the APinT algorithm to converge, we require that: C_1Δ T^3εΛ(η) + C_2εη + C_3ε^2η < 1, We are then left with the problem of choosing an appropriate averaging window length, η, depending on the degree of scale separation, ϵ, and the filtered contribution of the triads, Λ(η). In the interest of demonstrating that one exists, we assume the scaling (for example): η = Δ T/ε^s, 0<s<1. We then have: C_1Δ T^3εΛ(Δ T/ε^s) + C_2ε^1-sΔ T + C_3ε^2-sΔ T < 1,as ε→ 0, our error also decreases for any value of the power s. Λ(Δ T/ε^s) is bounded, so as ε→ 1, all terms remain bounded and we may choose our coarse timestep accordingly to ensure convergence. This means that the method proposed here may be applied across the full range of ε∈ (0, 1] with only a change of averaging window length, which allows convergence for physical problems where the time scale separation may change throughout the computation. This is in contrast to the proof in the limit <cit.> which proved convergence only for ε→ 0.§ THE ONE-DIMENSIONAL ROTATING SHALLOW WATER EQUATIONSWe now consider an example, using the one-dimensional rotating shallow water equation as a test-case, as did <cit.>. Let the unknown vector be: 𝐮(t,x) = (v_1(t,x), v_2(t,x), h(t,x))^T. We then write the linear and nonlinear operators in the full model <ref> as: ℒ = ([ 0-1 F^-1/2∂_x; 1 0 0; F^-1/2∂_x 0 0 ]); 𝒩(𝐮,𝐮) = ([ v_1(v_1)_x; v_1(v_2)_x; (hv_1)_x ]).for some constant, F∈ℝ. The corresponding eigenvalues are: ω_k^α = α√(1+F^-1k^2),α=-1,0,+1. In general, as ε→ 0 we expect that Λ(η) → 0 as well due to cancellation of oscillations in the integral<cit.>. As discussed in <ref>, oscillatory stiffness arises due to the magnitude of the gain term outside of the integral, which is large for highly oscillatory systems. The integral itself, however, is bounded from above by one, and achieves this value only for directly resonant triads (cf. <ref>), where the gain is zero. As the distance of resonance (i.e. the magnitude of |ω_k^α-ω_k_1^α_1 - ω_k_2^α_2|) increases, the integral tends to zero as well (and does so faster with larger η).The choice is then for a given degree of scale separation, ε, to choose an η which mitigates the stiffness sufficiently to allow the necessary coarse timestep, while retaining as much fidelity to the full equations as possible (cf. <ref>, where the averaging error is proportional to η). We shall deal with the practical implications of the form of Λ(η) in <ref>. §.§ The Optimally-Averaged Slow and Fast Solutions In order to illustrate the slow averaged solution over which the timestepping is performed and its relation to the full solution, <ref> compares the slow, full and true solutions for the stiff case where ε=0.01. The spatio-temporal oscillations are very rapid in the stiff case, which is the source of the timestep limitation by the CFL condition. However, the slow solution over which the timestepping is performed lacks these rapid oscillations and so permits the large timestep.Spatio-temporal oscillations from an initially stationary Gaussian height field are shown on a domain which is spatially periodic, i.e. the top and bottom boundaries of the plots wrap around. The decay of the height field into waves travelling in opposite directions is visible in <ref>. The optimal averaging window (q.v. <ref>) was applied, and convergence to single precision was obtained in six iterations. §.§ Numerical Results on the 1-D RSWEIn this section we present numerical results for the one-dimensional rotating shallow water equations which build on those presented in <cit.>. <Ref> shows the norm of the coarse error, i.e. 𝐱(t) - 𝐲(t)_2 computed relative to the fine timestep versus the width of the averaging window, T_0, which denotes the numerical choice of η, where 0≤ t≤ 1, δ t = 2e-4, and the spatial resolution is N_x = 64. This spatial resolution and fine timestepping regime were found to be within the asymptotic range of the timestepping. A second-order Strang splitting method was used for both the coarse and fine solves. The initial flow was stationary with a Gaussian height field.For the smallest ε the asymptotic behaviour is well approximated, as the fidelity of the coarse timestepping increases as the averaging window increases. This is consistent with the behaviour described in <ref>, where the theory predicts that η→∞ as ε→ 0. However, for larger ε such as the two cases shown, there is a clear optimal size for the averaging window to take, i.e. the minimum in the red and green curves in <ref>. The location but not the magnitude of this point is predicted by <ref>. <Ref> states that we should expect that outside of the small-ε limit the iterative error should decrease with k as in <ref> and exhibit a minimum where the sum of the timestepping and averaging errors is smallest. In the case of ε = 0.01, which is near the limit as ε→ 0, we expect that for a large enough averaging window we will have optimal convergence, with no improvement in solution quality for a larger averaging window. The numerical results are then consistent with the theory developed in this paper.In practice, we seek a choice for T_0 for which the solution is non-stiff on an 𝒪(Δ T) interval, and therefore as Δ T increases, so must the averaging window. Similarly, the oscillatory stiffness is proportional to 1/ε, and so as ε→ 0 it is necessary to choose a longer averaging window, and therefore to apply stronger smoothing to the solution.Comparing <ref> to <ref>, which shows the iterative error in the APinT method after three iterations for the same parameters, the direct computation of the coarse timestepping error provides good qualitative agreement with the optimal choice of η for the different values of ε. This is in direct agreement with the prediction of <ref>.As T_0 is taken smaller, instability is observed for all ϵ. This corresponds to the explicit CFL limit being violated, as reducing the length of the averaging window increases the maximum wave speed in the solution. For large T_0, the iterative error roughly stabilises for finite ε. In the limit as η is taken very large, the coarse timestepping corresponds to an incorrect equation (e.g. as in the QG equations for our example) being solved in a numerically stable fashion. The difference in the coarse and fine equations is sufficient to inhibit convergence, but does not violate the timestepping limit. §.§ Optimal Averaging for the 1-D RSWEIt was shown in <ref> that it is possible to choose the averaging window in such a way as to ensure convergence. Beyond doing this, we may choose the window optimally to obtain the fastest possible convergence (cf. <ref>).The reason we are able to describe the qualitative behaviour this way is a direct result of the Parareal algorithm and the sources of error present in it. The Parareal method consists of an initial approximation to the solution performed by the coarse solver which is accurate to within the coarse error predicted by <ref>. This is followed by a series of parallel-in-time corrections which converge to the full solution, derived from the difference between the coarse and fine solutions. The closer the initial approximation is to the solution, the less correction is required to converge.The optimal choice of η may be written as an optimisation problem: min_η∈ℝ^+(C_1Δ T^3εΛ(η) + (C_2 + C_3ε)εη),for some as-yet unknown constants C_1, C_2, and C_3. It is here that the fact that both the timestepping and averaging errors are bounded proportionally to the norm of the nonlinear term, M, becomes serendipitous, as this constant may be somewhat non-optimal in practice. In seeking the location, but not the magnitude, of the minimum coarse error, the bound on the norm of the nonlinear operator plays no role. Seeking stationary points with respect to η, this then requires us to find η such that: d/dηmax_ϵ_βmax_𝒮_k,α^ϵ_β|ω_k^α-ω_k_1^α_1 - ω_k_2^α_2|^p/ε∫_0^1ρ(s)e^i|ω_k^α-ω_k_1^α_1 - ω_k_2^α_2|ηΔ T/εs ds + C_2 + C_3ε/C_1Δ T^3 = 0. The result of <ref> is used to choose the optimal averaging window. This result captures the relationship of parameters such as the timestep and the scale separation on the optimal averaging, but relies on several unknown constants. If these constants C_n were known, the optimal averaging window could be determined computationally. <Ref> would then provide an approximation to the optimal window. Given some initial data of the type shown in <ref>, these constants may be fit by least-squares. Doing so fits the known trend to the known data, and permits the optimal averaging window to be recomputed `on the fly' in a computation.Certain practical issues arise in the computation of η. Firstly, the computation of dΛ/dη requires all triads to be investigated, i.e. the maximum is taken over the set of all near-resonant sets. Doing so is computationally expensive, although if this computation were to be performed infrequently the cost could be negligible compared to the simulation cost. Additionally, finding η requires solving a transcendental equation in at least two variables (η, ε), both for the initial fitting of constants, and for the optimisation on the fly. We therefore propose a simpler model based on the behaviour of Λ(s).Restricting ourselves for this example to a Gaussian kernel, we may consider the asymptotic behaviour of the kernel as λ_n is large. This gives: 1/η∫_0^ηρ(s/η)e^iλ_nΔ Tsds=∫_0^1ρ(s)e^iλ_nΔ Tη sds∼ C_0e^-C_1(|λ_n|Δ Tη)^2. We then multiply our approximation by 1=η^2/η^2, to obtain: η^2/η^2C_1Δ T^3εΛ(η) ≈D_1Δ Tε/η^2, for some constant, D_1, since x^2e^-x^2 is bounded independently of x (cf. <ref>). We then replace our first term in <ref> and seek fixed points corresponding to the minimum error. This yields: η_optimal = √(D_1Δ T/C_2 + C_3ε). This equation provides an estimate for the optimal averaging window length, η_opt, in terms of the computational parameters and the empirically-fit constants. This result is consistent with <ref>, as it exhibits a clear minimum for 𝒪(1) values of ε, with the optimal averaging window increasing as ε→ 0, as the asymptotic theory predicts. Both approximations are shown in <ref> for a set of minima extracted from a series of runs of the algorithm.The full model given in <ref> provides a much closer approximation both to the behaviour for ε = 1 and as ε→ 0, and as an actual fit to the points. It does this, however, at the cost of several orders of magnitude more computational difficulty. The simple model of <ref>, on the other hand, provides a reasonable approximation to the error as a function of ε, but has the disadvantage of poorly resolving the trend in the limit as ε→ 0. While the simple prediction underestimates the optimal as ε→ 0, the behaviour in this range is well-understood (cf. <cit.>, <cit.>) and so a hybrid model may easily be applied in practice.§ CONCLUSION We have investigated the convergence of a Parareal method using the APinT coarse solver, which provides a technique by which oscillatory-stiff equations may be solved with the Parareal method. The convergence of this method is due to the averaging applied to the coarse solution, which filters the fast waves and mitigates the oscillatory stiffness present in many of the equations of mathematical physics. This averaging must be performed over the entirety of the nonlinear operator due to the role the direct and near-resonances play in the oscillatory stiffness of the system.By describing the error of the coarse solver in terms of the interplay between the average over the rapid oscillations and the timestepping, we show the method converges for finite scale separation, significantly extending the domain of applicability for this method.We have shown here that this method is convergent across a wide range of scale separation, which is an improvement on the prior result<cit.> which held only in the small-ε limit. Further, in <ref> we considered both a full and a reduced model to predict the optimal averaging window in practical codes. § PROOF OF <REF>Consider d𝐮/dt(t)=𝐟(t/ε,𝐮(t)),0≤ t≤ h,and its averaged version d𝐮/dt(t)=𝐟_η'(t/ε,𝐮(t)),0≤ t≤ h,where with η'=ε hη,𝐟_η'(t/ε,𝐮(t))=1/η'∫_0^η'ρ(s/η')𝐟(t+s/ε,𝐮(t))ds.Then ‖𝐮(t)-𝐮(t)‖ =𝒪(εη h),0≤ t≤ h.To prove this, change variables: τ=t/(hε) and 𝐯(τ)=𝐯(t/hε)=𝐮(t), 𝐯(τ)=𝐯(t/hε)=𝐮(t).Then d/dt𝐮(t) =d/dt𝐯(t/hε) =1/hεd𝐯/dτ(τ).Thus,d𝐯/dτ(τ)=hε𝐟(hτ,𝐯(τ))≡ hε𝐠_h(τ,𝐯(τ)),0≤τ≤1/ε,and, with hτ'=s/ε, ds=ε hdτ',𝐟_η(t/ε,𝐯(τ)) =1/ε hη∫_0^ε hηρ(s/ε hη)𝐟(t+s/ε,𝐯(τ))ds =1/ε hη∫_0^ε hηρ(s/ε hη)𝐟(s/ε+hτ,𝐯(τ))ds =ε h/ε hη∫_0^ε hη/(ε h)ρ(ε hτ'/ε hη)𝐟(hτ'+hτ,𝐯(τ))dτ' =1/η∫_0^ηρ(τ'/η)𝐟(h(τ'+τ),𝐯(τ))dτ' =1/η∫_0^η𝐠_h(τ,+τ',𝐯(τ))dτ' =(𝐠_h)_η(τ,𝐯(τ)).Thus, d𝐯/dτ(τ)=hεd𝐮/dt(t)=hε𝐟_η(t/ε,𝐯(τ))=hε(𝐠_h)_η(τ,𝐯(τ)),0≤τ≤1/ε.Define 𝐟_η(t,𝐱)=1/η∫_0^ηρ(s/η)𝐟(t+s,𝐱)ds.If ϕ(t) is Lipschitz-continuous with Lipschitz-constant λ. Then|ϕ(t)-ϕ_η(t)|≤ C_0λη,whereC_0=∫_0^1ρ(s)sds. Using that1/η∫_0^ηρ(s/η)ds=∫_0^1ρ(s)ds=1,we have that |ϕ(t)-ϕ_η(t)| =|ϕ(t)-1/η∫_0^ηρ(s/η)ϕ(s+t)ds| =1/η∫_0^ηρ(s/η)|ϕ(t)-ϕ(s+t)|ds≤ 1/η∫_0^ηρ(s/η)sλ ds =ηλ∫_0^1ρ(s)sds.Consider d𝐯/dτ(t)=hε𝐟(ht,𝐯(t)), 0≤ t≤ε^-1,with 𝐟 continuous in each argument. Also assume that ‖𝐟(ht,𝐮)-𝐟(ht,𝐰)‖≤λ‖𝐮-𝐰‖ ,and M=sup_x∈ Dsup_0≤ t≤ε^-1‖𝐟(ht,𝐰)‖ <∞.Then defining ϕ(t)=∫_0^t𝐟(hτ,𝐯(τ))dτ,we have that |ϕ_η(t)-∫_0^t𝐟_η(hτ,𝐯(τ))dτ|≤ C_0(1+λ h)Mη.We calculate that ϕ_η(t) =1/η∫_0^ηρ(s/η)ϕ(s+t)ds =1/η∫_0^ηρ(s/η)(∫_0^t+s𝐟(hτ,𝐯(τ))dτ)ds =1/η∫_0^ηρ(s/η)(∫_s^t+s𝐟(hτ,𝐯(τ))dτ)ds+R_1 =1/η∫_0^ηρ(s/η)(∫_0^t𝐟(h(τ+s),𝐯(τ+s))dτ)ds+R_1 =1/η∫_0^ηρ(s/η)(∫_0^t𝐟(h(τ+s),𝐯(τ))dτ)ds+R_1+R_2 =∫_0^t(1/η∫_0^ηρ(s/η)𝐟(h(τ+s),𝐯(τ))ds)dτ+R_1+R_2 =∫_0^t∫_0^t𝐟_η(hτ,𝐯(τ))dτ+R_1+R_2,where ‖ R_1‖=‖1/η∫_0^ηρ(s/η)(∫_0^s𝐟(hτ,𝐯(τ))dτ)ds‖≤ 1/η∫_0^ηρ(s/η)∫_0^s‖𝐟(hτ,𝐯(τ))‖ dτ ds≤ 1/η∫_0^ηρ(s/η)∫_0^sMdτ ds = M1/η∫_0^ηρ(s/η)sds = Mη∫_0^1ρ(s)sds = C_0Mη,and ‖ R_2‖=‖1/η∫_0^ηρ(s/η)∫_0^t(𝐟(h(τ+s),𝐯(τ+s))-𝐟(h(τ+s),𝐯(τ)))dτ ds‖≤ 1/η∫_0^ηρ(s/η)∫_0^t‖𝐟(h(τ+s),𝐯(τ+s))-𝐟(h(τ+s),𝐯(τ))‖ dτ ds≤ 1/ηλ∫_0^ηρ(s/η)∫_0^t‖𝐯(τ+s)-𝐯(τ)‖ dτ ds =1/ηλ∫_0^ηρ(s/η)∫_0^t‖∫_τ^s+τd𝐯/dσ(σ)dσ‖ dτ ds =1/ηλ∫_0^ηρ(s/η)∫_0^t‖∫_τ^s+τhε𝐟(hσ,𝐯(σ))dσ‖ dτ ds≤ 1/ηhελ∫_0^ηρ(s/η)∫_0^t∫_τ^s+τ‖𝐟(hσ,𝐯(σ))‖ dσ dτ ds≤ 1/ηhελ M∫_0^ηρ(s/η)∫_0^t∫_τ^s+τdσ dτ ds =1/ηhελ M∫_0^ηρ(s/η)∫_0^tsdτ ds = C_0η hλ Mε t≤C_0hηλ M.In the last inequality, we used that 0≤ t≤ε^-1. Consider d𝐯/dτ(t)=hε𝐟(ht,𝐯(t)), 0≤ t≤ε^-1,with the same assumptions as in the previous lemmas. Let d𝐯/dτ(t)=hε𝐟_η(ht,𝐯(t)), 0≤ t≤ε^-1.Then ‖𝐯(t)-𝐯(t)‖≤ C_1hεη,0≤ ht≤ε^-1.Note that 𝐯(t)=𝐯(0)+hε∫_0^t𝐟(hτ,𝐯(τ))dτ.By <ref>,∫_0^t𝐟(hτ,𝐯(τ))dτ=∫_0^t𝐟_η(hτ,𝐯(τ))dτ+E_0,where ‖ E_0‖≤ C_0(1+λ h)Mη.Therefore,𝐯(t)=𝐯(0)+hε∫_0^t𝐟_η(hτ,𝐯(τ))dτ+E_1,where ‖ E_1‖ =‖ hε E_0‖≤ C_0(1+λ h)Mη hε.Also, since 𝐯(t)=𝐯(0)+hε∫_0^t𝐟_η(hτ,𝐯(t))dτ,we have that ‖𝐯(t)-𝐯(t)‖ ≤hε∫_0^t‖𝐟_η(hτ,𝐯(τ))-𝐟_η(hτ,𝐯(t))‖ dτ+C_0(1+λ h)Mη hε≤hελ∫_0^t‖𝐯(τ)-𝐯(t)‖ dτ+C_0(1+λ h)Mη hε.Finally, by Gronwall's inequality, ‖𝐯(t)-𝐯(t)‖≤ C_0(1+λ h)Mη hε e^hελ t. § ACKNOWLEDGMENTSWe would like the acknowledge the support of the University of Exeter and Los Alamos National Laboratory.siamplain | http://arxiv.org/abs/1705.09565v1 | {
"authors": [
"Adam Peddle",
"Terry Haut",
"Beth Wingate"
],
"categories": [
"math.NA",
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"primary_category": "math.NA",
"published": "20170526130012",
"title": "Parareal Convergence for Oscillatory PDEs with Finite Time-scale Separation"
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Topological defects (kinks) in a relativistic ϕ^4 scalar field theory in D=(1+1) are studied using the matrix product state tensor network. The one kink state is approximated as a matrix product state and the kink mass is calculated. The approach used is quite general and can be applied to a variety of theories and tensor networks. Additionally, the contribution of kink-antikink excitations to the ground state is examined and a general method to estimate the scalar mass from equal time ground state observables is provided. The scalar and kink mass are compared at strong coupling and behave as expected from universality arguments. This suggests that the matrix product state can adequately capture the physics of defect-antidefect excitations and thus provides a promising technique to study challenging non-equilibrium physics such as the Kibble-Zurek mechanism of defect [email protected]@imperial.ac.uk Department of Physics, Imperial College London, SW7 2AZ, UKTopological Defects in Quantum Field Theory with Matrix Product States Arttu Rajantie 31 October 2017 ======================================================================§ INTRODUCTION : TOPOLOGICAL DEFECTS AND TENSOR NETWORKS A significant problem within quantum field theory (QFT) is the calculation of non-perturbative and non-equilibrium problems. Standard perturbation theory can be used for near-equilibrium weak coupling problems but fails in other cases. For equilibrium calculations, lattice theory provides a powerful approach but has limited applicability to non-equilibrium problems <cit.>. On the non-equilibrium side, the use of 2-particle irreducible (2PI) or more generally nPI effective actions is common <cit.> and can be used in conjunction with large N expansions for a power series approach while stochastic quantisation allows for the extension of non-perturbative lattice techniques to the case of real-time <cit.>. Recently, Hamiltonian truncation techniques have also emerged as an alternative to the available functional/lattice techniques and have been applied to both equilibrium <cit.> and non-equilibrium calculations <cit.>. A particularly good test for the various available techniques is offered by the study of topological defects in QFT. Topological defects are naturally non-perturbative and cannot be included by perturbative expansions about classical vacuum solutions <cit.>. Instead, the QFT is split into different topological sectors, associated to a particular topological charge, and each sector must be treated separately in perturbative expansions. For non-perturbative equilibrium calculations involving defects, lattice techniques have been successful <cit.> and more recently Hamiltonian truncation has also been used <cit.>. However, non-equilibrium calculations can be more problematic as highlighted in the study of defect formation during phase transitions. Despite having a simple and general description via the Kibble-Zurek mechanism <cit.>, naive applications of the 2PI effective action techniques fail to include the contribution of defects <cit.> requiring the use of less standard techniques <cit.>.Additionally, simple observables such as the quasi-particle excitation density in general fail to capture the relevant physics in such problems <cit.> .In this paper we apply a recent approach based on the use of matrix product states (MPS) to the study of topological defects. These techniques deal with states directly and are most commonly used to approximate the ground states of strongly-coupled quantum systems in D=(1+1). While originally specialised to the study of gapped, finite size lattice systems with open boundary conditions <cit.>, these techniques have seen significant developments in recent years and have been applied to e.g. periodic systems <cit.>, infinite volume systems <cit.> and non-equilbrium calculations <cit.>. More generally, matrix product states are part of a broader class of tensor networks (TN) which allow further extensions to e.g. critical/gapless systems <cit.> and the equilbrium study of excited states (including kink excitations) <cit.>. Matrix product states have also been applied previously to the study of field theories in a number of settings. For example, they have been applied to lattice regularised relativistic field theories in D = (1+1) with both global symmetries <cit.> and gauge symmetries with substantial progress being made in the Schwinger model (U(1)) <cit.>, including an explcit example overcoming the sign problem <cit.>, in addition to studies in the SU(2) gauge theory <cit.>. More recently, there has also been a focus on the description of field theories using MPS and TN in the continuum, providing an alternative regularisation scheme to the lattice <cit.>. Finally, while many of the studies involving TN have been for D = (1+1), there have been a number of recent developments towards the application of tensor network methods to higher dimensions, particularly to D = (2+1) using the projected-entangled-pairs states (PEPS) TN <cit.>. These developments include work towards the application of TN to gauge theories in D=(2+1), <cit.>, making TN a promising tool for the future study of high energy physics.In the following, we apply matrix product states to the study of topological defects (kinks) in the ϕ^4 theory in D=(1+1). We study the equilibrium physics by obtaining MPS approximations to the ground state and one kink state in the lattice regularised setting. The main goal is to assess to what degree the MPS can capture the physics of defects non-perturbatively, both in the sense of providing a direct approximation to the one kink state and in capturing the contribution of kink-antikink excitations to the ground state observables. To achieve this, we focus on studying the kink mass M_K and the scalar mass m_S at strong and weak couplings where we can compare with analytic results. Confirming that the MPS can capture the physics of defects, particularly of the kink-antikink excitations, is essential if they are to be used to study non-equilibrium phenomena such as defect formation. The structure of our discussion will be as follows : Firstly, we will introduce the ϕ^4 theory in Section <ref> and review some weak coupling and strong coupling results before introducing the lattice regularisation used throughout. Secondly, we review the construction of matrix product states in Section <ref> and discuss their relationship to entanglement and tensor networks more generally. Following this, we provide an overview of the standard methods used to find MPS approximations to the ground state and show how they apply directly to one kink state in Section <ref>. More details on the information outlined in Sections <ref> and <ref> , along with a useful guide to implementation,can be found in the review <cit.>. In Section <ref> we provide a discussion of how to estimate the scalar mass from the equal time two point functions before examining some weak and strong coupling results in Section <ref> and concluding in Section <ref>. § Φ^4 SCALAR FIELD THEORY IN D = (1+1) §.§ Classical and Semi-Classical ResultsIn this section, we review some classical and semi-classical results for the ϕ^4 scalar field theory in D = (1+1) as defined by the actionS[ϕ] = ∫ dx dt [ 12(∂_tϕ)^2 - 12(∂_xϕ)^2 - μ_0^22ϕ^2 -λ_04!ϕ^4] .When μ_0^2 < 0, the classical potential densityU(ϕ) = μ_0^2/2ϕ^2 + λ_0/4!ϕ^4has two degenerate minima (vacua) ± v = ±√(-6 μ_0^2λ_0) corresponding to the spontaneous breaking of the global Z_2 symmetry ϕ(x) → -ϕ(x). Such static, uniform field configurations solve the classical equations of motion but there are additional topologically non-trivial static solutions called kinks ϕ_K(x) given byϕ_K(x) = v tanh( μ_0( x - x_0)/√(2))with the corresponding antikink given by -ϕ_K(x). Such solutions interpolate between the two minima and are not spatially uniform. Additionally, they have a degree of freedom (zero-mode) x_0 such that they form a continuous family of solutions. These kink solutions are the simplest possible example of a topological defect in quantum field theory. To relate these solutions to a topological charge, all the finite energy field configurations of the theory can be classified according to their homotopy. Configurations in distinct homotopy classes cannot be continuously deformed into one another (e.g. by time evolution) and the theory is split into distinct topological sectors. These sectors are labelled by a topological charge Q determined by the boundary conditions of the configurations asQ = 1/2 v( ϕ(∞) - ϕ(-∞) ) .The minima ± v have Q=0 while the kink and antikink have Q = 1 , -1 respectively and they provide the lowest energy configurations for each of these sectors. The existence of distinct topological sectors can be confirmed from a few general features of a theory so that one can check easily for the possibility of topological defects in a wide variety of cases <cit.>. The classical kink mass M_K can be calculated from the classical energyE = ∫ dx [ 12( ∂ϕ/∂ x )^2 + U(ϕ) ]by subtracting the classical vacuum energy to giveM_K = 4 √(2)μ_0^3/λ_0 . In the quantum theory, the kink and anti-kink will appear as charged particles. Since they lie outside the Q=0 vacuum sector, they are non-perturbative in the sense one cannot include their contribution by using perturbation theory starting from the vacuum sector. Instead, to gain semi-classical information about the defects, one must begin from the appropriate sector e.g. with classical kink configurations. Results in these sectors tends to require more work than the topologically trivial Q=0 sectorboth due to the fact the classical kink is not spatially uniform and due to the presence of the zero-mode x_0. The one-loop order calculation of the kink mass is a classic result known as the `DHN' result following the work of Dashen, Hasslacher and Neveu<cit.>. This can be written in terms of the scalar mass m_S^2 = 2 μ_0^2 + 𝒪(λ_0) up to 𝒪(λ_0) to giveM_K = 2 m_S^3/λ_0 + m_S/2( 1/6√(3/2) - 3/π√(2)) + 𝒪( λ_0) .From this expression one can see that in the semi-classical region the kink appears as a heavy particle such that M_K = 𝒪(m_S^3). As such, in this region kink-antikink excitations will provide a negligible contribution to ground state observables. More recently,zeta-function regularisation has allowed for one-loop results in finite size systems <cit.> and dimensional regularisation has provided a more systematic approach to one-loop corrections allowing for results at finite temperatures and in higher dimensions <cit.>. While giving equivalent results to one-loop order, a rigorous treatment of the zero-mode requires more work through e.g. the use of canonical coordinates <cit.>, which also allow for the computation of the scalar field n-point functions in the presence of the kink <cit.>. While semi-classical expansions can provide information about the weak-coupling region μ_0^2≫λ_0, at stronger couplings they will breakdown. This is particularly important for the ϕ^4 theory in D = (1+1) since the phase transition corresponds to strong couplings. Therefore, near the critical point perturbation theory can no longer be used and one must turn to alternative methods.§.§ Strong-Coupling Results : UniversalityOn approach to the critical point, the correlation length of the ϕ^4 theory diverges. In this regime, the microscopic (lattice) details are irrelevant and the critical behaviour is described by a simple field theory. The nature of this critical field theory depends only a few general features of the underlying theory, e.g. symmetry or dimensionality, so that many different theories have the same critical description and can be separated into universality classes. In D=2 the ϕ^4 theory falls into the same universality class as the classical D=2 Ising model. The critical field theory can be determined by the fixed points of a suitable renormalisation group (RG) flow and in this case corresponds to a Wilson-Fisher fixed point <cit.>. Since the corresponding critical field theory is not a free scalar field theory, standard perturbation theory cannot be used and we can say the phase transition is strongly coupled. Analytic results can instead be gained for the universality class by using RG or studying the Ising Model which is integrable.The most familiar universal results are the critical exponents and for the D=2 Ising universality class the correlation length ξ in the symmetric and symmetry broken phases is given by ξ ≈ξ_0 |τ|^ν τ >0 ξ ≈ξ'_0 |τ|^ν' τ < 0where τ is the reduced temperature such that τ >0 indicates the symmetric (high temperature) phase , τ < 0 the symmetry broken (low temperature) phase and τ = 0 the critical point. Additionally, universal amplitude ratios <cit.> can be derived and in particular ξ_0/ξ'_0 ≈ 2 .These results can be related to the topological defects in the ϕ^4 theory through the Kramers-Wannier duality present in the Ising Model. While explicit kink creation operators cannot be constructed for the ϕ^4 theory, the corresponding disorder operators μ(x) can be introduced in the classical Ising model <cit.>. The Kramers-Wannier duality relates the disorder operator two point correlation function on the dual lattice at coupling (temperature) K to the spin operator two point function at a coupling K^* as⟨μ(x) μ(x')|_⟩K = ⟨σ(x̃) σ(x̃')|_⟩K^* .This duality establishes the relation μ' = ν between the critical exponents where μ' is the critical exponent associated to the diverging correlation length ξ_K of the disorder two point function ⟨μ(x) μ(x')|$⟩ in the symmetry broken phase. When combined with the universal amplitude ratio Equation (<ref>) one can establish ξ_K/ξ ≈ξ_0 |τ|^ν/ξ_0'|τ|^ν'≈ 2which uses the hyperscaling relationν= ν'. This result corresponds in theϕ^4theory to the relationshipm_S ≈2 M_Kbetween the scalar massm_Sand the kink massM_K. While universality establishes this result rigorously in the vicinity of the critical point, physically it should hold from the point where firstm_S ≈2 M_Kdown to the critical point, since in this region the scalar excitation will decay into a kink-antikink pair excitation which is the lightest excitation for theQ=0sector in this region. Since this result incorporates a simple linear scalingν= 1with strong coupling physics and topological defects it provides a good test for non-perturbative calculation methods. §.§ Lattice Methods for Non-Perturbative Calculations For general non-perturbative calculations it is common to use lattice methods. Essentially, one approximates observables of the full theory by observables of a theory defined on a lattice. An appropriate lattice theory can be obtained for a particular QFT by discretising the continuum action. Usually, this discretisation takes place in both time and space with Euclidean spacetime being used to allow for sampling, effectively transforming the problem of quantum field theory into statistical mechanics. Since analytically continuing back to real time requires further assumptions in the discrete case and tends to dramatically increase the errors from sampling, this method is essentially restricted to equilibrium physics and has been highly successful in this regard, for an introduction to these techniques see <cit.>. Here, we will be interested in working with states at a particular time so that the lattice discretisation need only be applied in the spatial dimension leaving time continuous. In this case, the full QFT Hilbert space is truncated down to a lattice quantum theory and we can write schematicallyℋ_QFT →ℋ(a,L)where the lattice spacingaand sizeLact as truncation parameters. For spatial discretisation, the Hamiltonian formalism can be used and the continuum theory can be written asH[ϕ] = ∫ dx [ 12(∂_tϕ)^2 + 12(∂_xϕ )^2 + μ_0^22ϕ^2 + λ_04!ϕ^4] .An appropriate lattice theory can then be constructed by discretisation of this Hamiltonian. A good option is to simply replace the spatial derivatives by first order finite difference approximations leading to the HamiltonianH̃[ϕ] = ∑_x[ 12(π_x)^2 + 12(ϕ_x+a - ϕ_x )^2 + μ̃_0^22ϕ_x^2 + λ̃_04!ϕ_x^4]which is given in lattice units withH̃ = a H , μ̃_0^2 = a^2 μ_0^2 , λ̃_0 =a^2 λ_0 . Additionally, the time-derivatives have been replaced byπ_x = ∂ℒ/∂(∂_tϕ) = a (∂_tϕ) which obeys the canonical commutation relation[π_x,ϕ_y] = i δ_x,y. The finite lattice spacing, which we will set to one throughout, serves to restrict the possible momentum modes-π/a ≤p ≤π/aso that low momentum (long distance) observables of the full theory can be well approximated in the lattice theory, while higher momentum (short distance) behaviour will be modified significantly by the lattice. We will be particularly interested in the equal-time observables of the ground state⟨Ω|𝒪[ϕ]|Ω|$⟩ and the one kink state ⟨K|𝒪[ϕ]|K|$⟩. These are approximated in the lattice theory by the corresponding lattice observables⟨Ω(a,L)|𝒪[ϕ]|Ω(a,L)|$⟩ and ⟨K(a,L)|𝒪[ϕ]|K(a,L)|$⟩ where the state|Ω(a,L)⟩is simply the minimum energy state of the lattice theory and|K(a,L)⟩is the lattice one kink state. To approximate the one kink state on the lattice it is typical to write it explicitly as the minimum energy state of theQ=1topological sector which can be selected out by enforcing twisted periodic boundary conditions (TPBC) in the spatial dimension of the theoryϕ_x + L = -ϕ_x<cit.>. Both states of interest can then be written in terms of energy minimisation problems as |Ω(a,L)⟩ = min_|ψ⟩( ⟨ψ|H̃|ψ|-⟩λ[⟨ψ|ψ|-⟩ 1 ] )for the lattice ground state and |K(a,L)⟩ = min_|ψ⟩( ⟨ψ|H̃_(TPBC)|ψ|-⟩λ[⟨ψ|ψ|-⟩ 1] )for the one kink lattice state (note that higher kink number excitations are suppressed exponentially with the lattice sizeL). Specifying states further requires the choice of a basis. The field eigenbasis is a natural choice for mean field theory type approximations but to go beyond these it is better to pick a numerically stable basis of real space harmonic oscillators <cit.>. Introducing real space creation and annihilation operators viaϕ_x= 1√(2) ( a^†_x+ a_x)and[a_x , a^†_y] = δ_x,y,a natural basis set is then the tensor products|𝐧_𝐱⟩ = |n_1⟩⊗|n_2⟩⊗...⊗|n_L⟩where|n_x⟩are the eigenstates of the number operatorN_x = a^†_xa_xat each site. All basis states are now labelled by aL-tuple𝐧_𝐱 = ( n_1 , n_2 , n_3 , ... ,n_L)with n_x ∈Zand a general state|ψ⟩can be written as|ψ⟩ = ∑_n_1=0^∞ ... ∑_n_L=0^∞⟨ n_1 n_2 ... n_L | ψ||%s⟩⟩ n_1 n_2 ... n_L = ∑_𝐧_𝐱ψ_𝐧_𝐱|𝐧_𝐱⟩where the state coefficientψ_𝐧_𝐱(wavefunction) now specifies the state in this basis. § TENSOR NETWORKS AND MATRIX PRODUCT STATES§.§ Matrix Product States To study the ground state and one kink state observables in theϕ^4theory, we would like to solve the minimisation problems Equations (<ref>) and (<ref>) directly. Of course, this is not possible in general since even a finite dimensional Hilbert space grows exponentially with the number of lattice sites. However, matrix product states provide a method to truncate the Hilbert space further down to a tiny subset where states can be specified efficiently i.e. with a cost that rises at most polynomially in the number of lattice sites. Within this subset, the minimisation problems can be solved numerically and direct approximations of the ground state and one kink state can be obtained.To proceed, one first rewrites the expression for a generic state in the theory Equation (<ref>) in the matrix product form. For the lattice MPS used here, this first requires an additional truncation in the local Hilbert space dimension such that the resulting total state space is finite. This can be achieved by simply keeping the firstn = 0 , 1 , ... , n_max basis states at each site. A state can then be expressed as|ψ⟩ = ∑_n_1=0^n_max ... ∑_n_L=0^n_max⟨ n_1 n_2 ... n_L | ψ||%s⟩⟩ n_1 n_2 ... n_L .When using the field eigenbasis, the error associated with a similar truncation can be rigourously bound by the magnitude of local expectation values⟨ψ| ϕ_x^2|ψ|$⟩ and ⟨ψ | π_x^2|ψ|$⟩ or alternatively the energy expectation value E = ⟨ψ| H | ψ|$⟩ <cit.>. In practice, since the limit n_max→∞ tends to be well behaved, it is usually possible to simply increase the value of n_max in calculations until convergence in the desired observables is reached. To keep notation standard, we will let n_max = d-1 so that the total dimension of the regularised theory is now given by d^L. The matrix product form can now be introduced by considering the wavefunction ψ_𝐧_𝐱 as a rank-L tensor. Such a tensor can always be decomposed using a tensor train decomposition into a matrix product state form so thatψ_𝐧_𝐱 = ∑_α_1,α_2,..,α_L M^n_1_α_1,α_2(1) M^n_2_α_2,α_3(2) ...M^n_L_α_L,α_1(L)= (^n_1(1) ^n_2(2) ... ^n_L(L))= ( ∏_x^n_x(x) ) .Here, the state coefficient ψ_𝐧_𝐱 has been re-expressed as the nearest-neighbour contraction of a set of rank-3 tensors, one per site. The size of the tensors can vary site-to-site but we fix them for simplicity to all be (d,χ,χ). The first index provides the physical index corresponding to the local basis state |n_x⟩= |0⟩,|1⟩,...,|d-1⟩ while the latter two provide the internal or virtual degrees of freedom. All the internal indices are contracted over while the physical indices remain uncontracted. The MPS form is made more intuitive by the convenient diagrammatic notation available for tensor networks. In this notation, tensors are represented by shapes with legs corresponding to the indices of the tensor. Contractions between indices are then indicated by the joining of two legs. For example, the rank-3 tensor M^i_α,β is represented by [A/.style = shape=circle,draw=blue!100,fill=gray!50,empty/.style = shape=circle,draw=blue!0,node distance = 10pt] (M1) at (0,0) [A] ; [-] (M1) to (0,-0.5); [-] (M1) – (-0.5,0); [-] (M1) – (0.5,0); [empty] (L1) [ left=of M1] M^i_α , β =;[empty] (R1) [ right=of M1] ; such that the PBC lattice MPS with L = 7 sites is represented by [node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 , empty/.style = shape=circle,draw=blue!0];[A] (M3) at (0,0); [A] (M2) [left=of M3] ; [A] (M1) [left=of M2] ; [A] (M4) [right=of M3] ; [A] (M5) [right=of M4] ; [A] (BL) [left=of M1] ; [A] (BR) [right=of M5] ;[empty] (M1B) [below=of M1] ; [empty] (M2B) [below=of M2] ; [empty] (M3B) [below=of M3] ; [empty] (M4B) [below=of M4] ; [empty] (M5B) [below=of M5] ;[empty] (BLB) [below=of BL] ; [empty] (BRB) [below=of BR] ; [circle split] (M1A) [above=of M1] ; [empty] (M2A) [above=of M2] ; [empty] (M3A) [above=of M3] ; [empty] (M4A) [above=of M4] ; [circle split] (M5A) [above=of M5] ;[empty] (L1) [left = of BL]ψ_n =; (M1) – (M2); (M2) – (M3); ( M3) – (M4); (M4) – (M5);(M1) – (M1B); (M2) – (M2B); ( M3) – (M3B); (M4) – (M4B); (M5) – (M5B);(M1) – (BL); (M5) – (BR);(BLB) – (BL); (BRB) – (BR);(BL) to[in=180,out=180] (M1A);(M1A) to (M5A);(BR) to[in=0,out=0] (M5A); [empty] (R1) [ right=of BR ] .; Often, as in the original DMRG algorithm, MPS with open boundary conditions (OBC) are used. These can be written by considering the first and last tensors as rank-2 tensors so that ψ_𝐧_𝐱= (m_L^n_1)^T(∏_x = 2^L-1^n_x(x) ) m_R^n_Lwhere m_L^n_1, m_R^n_L are rank-2 tensors of size (d,χ) and the trace is no longer needed. The diagrams for OBC MPS are somewhat simpler than their PBC counterparts e.g. for L=5[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 , empty/.style = shape=circle,draw=blue!0]; [A] (M3) at (0,0); [A] (M2) [left=of M3] ; [A] (M1) [left=of M2] ; [A] (M4) [right=of M3] ; [A] (M5) [right=of M4] ;[empty] (M1B) [below=of M1] ; [empty] (M2B) [below=of M2] ; [empty] (M3B) [below=of M3] ; [empty] (M4B) [below=of M4] ; [empty] (M5B) [below=of M5] ;[empty] (L1) [left = of M1]ψ_n =; (M1) – (M2); (M2) – (M3); ( M3) – (M4); (M4) – (M5);(M1) – (M1B); (M2) – (M2B); ( M3) – (M3B); (M4) – (M4B); (M5) – (M5B);[empty] (R1) [ right=of M5] .; Due to this simplification, for convenience we will often use OBC in diagrams, though calculations will tend to involve PBC.We emphasise that this MPS form is complete in the sense that all states can be expressed in this manner. However, the real power of MPS is revealed by considering the subset of states that can be efficiently parametrised by the MPS form. The number of parameters required to specify a MPS is simply given by the lattice size multiplied by the size of the tensor i.e Ldχ^2. A general state requires an exponentially large number of parameters to specify so that at most χ∼ d^L/2. However, some states, often simply called “matrix product states" require only χ∼ const and these states make up the natural efficiently parametrised subset of the MPS form. By working with the MPS form and picking some value of χ, we can truncate the Hilbert space sufficiently so that we can deal with states directly, i.e. without the need for sampling. This allows one to approximate the states of interest by solving minimisation problems within this subset. Schematically, the series of truncations can then be shown as ℋ_QFT→ℋ(a,L) →(a,L,d,χ) where the subset of states (a,L,d,χ) no longer retains the structure of ℋ but can be shown in many cases to form a smooth manifold, for details see <cit.>.While states in the subset (a,L,d,χ) can be represented efficiently by a tensor network, the central question is which states actually belong to the subset and what physics is well approximated within it. The answer to this is given by considering the entanglement entropy of a reduced state ρ_𝒜 for a state in the subset, defined over a spatial subregion 𝒜. The entanglement entropy is then defined asS(ρ_𝒜) = - ( ρ_𝒜logρ_𝒜) . A generic, random state in the Hilbert space displays an entanglement entropy that is extensive S(ρ_𝒜) ∼ V_𝒜. However, states in the MPS subset have an entanglement entropy that is bound by a constant S(ρ_𝒜) = 𝒪(logχ). This is an example of a low entanglement law and is extremely restrictive making matrix product states highly atypical. However, perhaps surprisingly at first, many of the physical states of interest are also low entanglement states. This can be seen somewhat intuitively from the more familiar fact that often physical states are actually quite special in that they do not have arbitrarily long distance correlations. In particular, the ground states of gapped, local systems tend to have exponentially decaying correlations while the ground states of gapless/critical systems tend to have algebraically decaying correlations. As such, one would not expect the entire subregion 𝒜 of a system to be correlated with the rest of the system but for the dominant contribution to come from the boundary of the subsystem ∂𝒜. This intuition is indeed correct in many cases and e.g. it has been shown that states with exponentially decaying correlations defined on a ring obey entanglement area laws S(ρ_𝒜) ∼∂𝒜 <cit.>, see <cit.> for a review of entanglement area laws. The understanding that a lot of relevant physics is indeed low entanglement physics has been a driving force behind the development of tensor networks generally, see the review <cit.>. By using a tensor network form to parametrise the Hilbert space of interest, one can truncate down to a tiny low entanglement subset throwing out the vast majority of (highly entangled) states. In this way tensor networks provide a low entanglement effective theory with low entanglement observables being well approximated while high entanglement observables/contributions are lost. In the MPS case, the fact that S(ρ_𝒜) = 𝒪(logχ) means that they are suited to the description of area law states in D=(1+1), since in that case the area law is simply a constant. In fact, one can show that all ground states of local, gapped, lattice Hamiltonians in D=(1+1) can be efficiently represented as MPS and that MPS always have exponentially decaying correlations asymptotically, as expected of an area law state <cit.>. In terms of the observables of interest here, the effect of the entanglement cutoff can be seen easily in the connected equal-time two-point function G_2(r) = ⟨Ω | ϕ(x) ϕ(x+r) | Ω|-⟩⟨Ω | ϕ(x)|Ω|^⟩2. In the simple lattice truncation, such an observable is approximated by ⟨Ω(a,L) | ϕ_xϕ_x+r | Ω(a,L)|-⟩⟨Ω(a,L) | ϕ_x|Ω(a,L)|^⟩2 which will agree with the full observable in the region a ≪ r ≪ L. Outside this region the lattice effects will be significant and the approximation poor. Since the relevant physics of the full theory takes place on a scale determined by the correlation length ξ, this means that the physics of the full theory will be well approximated by the lattice theory whenever a ≪ξ≪ L. The additional truncation to the low entanglement subset (a,L,d,χ) then restricts this region of validly further and one can think of an additional infrared length scale ξ_χ being introduced after which the inevitable exponential decay of the MPS will dominate and the observable G_2(r) will be heavily modified by the entanglement restriction. Additionally, the truncation parameter χ provides a short distance ultraviolet cutoff which, while essential when studying MPS formulated in the continuum <cit.>, is made irrelevant by the lattice spacing a. Similarly, the lattice size L will tend to be less relevant that the long distance scale ξ_χ and so we can summarise that G_2(r) will be well approximated by the MPS theory within the range a ≪ r ≪ξ_χ so that to capture the relevant physics of the full theory we require the hierarchy a ≪ξ≪ξ_χ. §.§ Representation of Observables as Tensor Networks Once a MPS representation for a state has been obtained, one would like to be able to calculate observables in an efficient way. This can often be achieved by representing the observable as a tensor network. The simplest exampleof this is the representation of the overlap between two states ⟨ψ̃|ψ|$⟩ which can be found explicitly from the MPS form as⟨ψ̃ | ψ| ⟩ = ∑_𝐧_𝐱ψ̃_𝐧_𝐱^*ψ_𝐧_𝐱 = ∑_𝐧_𝐱( ∏_x(^n_x)^*(x) ) ( ∏_x^n_x(x) ) = ∑_𝐧_𝐱( ∏_x(^n_x)^*(x) ⊗^n_x(x) ) = ( ∏_x[ ∑_n_x (^n_x)^*(x) ⊗^n_x(x) ] ) .This expression is clearer in the diagrammatic representation and can be written by introducing the convention that the conjugation of a tensor is represented by flipping the vertical, physical index such that [A/.style = shape=circle,draw=blue!100,fill=gray!50,node distance = 10pt](M1) at (0,0) [A] ;[-] (M1) to (0,0.5);[-] (M1) – (-0.5,0);[-] (M1) – (0.5,0);(L1) [left= of M1] (M^i_α , β)^* =;(R1) [right= of M1] .; The overlap can then be represented as[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 , empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M3) at (0,0); [A] (M2) [left=of M3] ; [A] (M1) [left=of M2] ; [A] (M4) [right=of M3] ; [A] (M5) [right=of M4] ;[B] (M1B) [below=of M1] ; [B] (M2B) [below=of M2] ; [B] (M3B) [below=of M3] ; [B] (M4B) [below=of M4] ; [B] (M5B) [below=of M5] ; (M1) – (M2);(M2) – (M3);( M3) – (M4);(M4) – (M5); (M1B) – (M2B);(M2B) – (M3B);( M3B) – (M4B);(M4B) – (M5B); (M1) – (M1B);(M2) – (M2B);( M3) – (M3B);(M4) – (M4B);(M5) – (M5B); (L1) [left= of M1,yshift=-10pt] ⟨ψ̃|ψ|=⟩; (R1) [right= of M5,yshift=-10pt] .; In words, one simply contracts the physical indices of the states together site-by-site. This tensor network representation for the overlap contains no uncontracted indices so that contracting it fully will produce a single number i.e. the value of the overlap. To represent more general operator matrix elements⟨ψ̃|Ô|ψ|$⟩ as tensor networks it is standard to introduce a form for operators that corresponds to the MPS form. Such a representation for lattice systems is called a matrix product operator (MPO) form which can be found for a particular operator using some tricks (see Section <ref>) or using more generic construction methods <cit.>. To specify an operator, a natural basis choice is the transition basis consisting of tensor products of local operators T̂_n_x,m_x = |n_x⟩⟨m_x| so that the basis element is T̂_𝐧_𝐱,𝐦_𝐱 = T̂_n_1,m_1⊗T̂_n_2,m_2⊗ ... ⊗T̂_n_L,m_L which is labelled by 2 L-tuples (𝐧_𝐱,𝐦_𝐱). In this basis an operator can be expanded asÔ = ∑_𝐧_𝐱,𝐦_𝐱⟨𝐧_𝐱|Ô|𝐦_𝐱| ̂⟩T_𝐧_𝐱,𝐦_𝐱= ∑_𝐧_𝐱,𝐦_𝐱 O_𝐧_𝐱,𝐦_𝐱T̂_𝐧_𝐱,𝐦_𝐱 .A matrix operator form is then given by rewriting the coefficient O_𝐧_𝐱,𝐦_𝐱 as a set of rank-4 tensors W_α_x,α_x+1^n_x , m_x(x) of size (d,d,χ_W,χ_W) contracted in nearest-neighbour fashion asO_𝐧_𝐱,𝐦_𝐱 = ∑_α_1,α_2,...,α_L W_α_1,α_2^n_1 , m_1(1) W_α_2,α_3^n_2 , m_2(2) ... W_α_L,α_1^n_L , m_L(L)= ( 𝐖^n_1 , m_1(1) 𝐖^n_2 , m_2(2)... 𝐖^n_L , m_L(L) )= ( ∏_x𝐖^n_x , m_x(x))A diagrammatic expression for operators in (OBC) MPO form is then[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[empty] (M3) at (0,0) ; [empty] (M2) [left=of M3] ; [empty] (M1) [left=of M2] ; [empty] (M4) [right=of M3] ; [empty] (M5) [right=of M4] ;[O] (O1) [below=of M1] ; [O] (O2) [below=of M2] ; [O] (O3) [below=of M3] ; [O] (O4) [below=of M4] ; [O] (O5) [below=of M5] ;[empty] (M1B) [below=of O1] ; [empty] (M2B) [below=of O2] ; [empty] (M3B) [below=of O3] ; [empty] (M4B) [below=of O4] ; [empty] (M5B) [below=of O5] ;(O1) – (O2); (O2) – (O3); ( O3) – (O4); (O4) – (O5);(M1) – (O1); (M2) – (O2); (M3) – (O3); (M4) – (O4); (M5) – (O5);(O1) – (M1B); (O2) – (M2B); (O3) – (M3B); (O4) – (M4B); (O5) – (M5B);(L1) [left= of O1] O_𝐧_𝐱,𝐦_𝐱 =; such that operator matrix elements can be represented as a tensor network by sandwiching the MPO between two MPS as[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M3) at (0,0) ; [A] (M2) [left=of M3] ; [A] (M1) [left=of M2] ; [A] (M4) [right=of M3] ; [A] (M5) [right=of M4] ;[O] (O1) [below=of M1] ; [O] (O2) [below=of M2] ; [O] (O3) [below=of M3] ; [O] (O4) [below=of M4] ; [O] (O5) [below=of M5] ;[B] (M1B) [below=of O1] ; [B] (M2B) [below=of O2] ; [B] (M3B) [below=of O3] ; [B] (M4B) [below=of O4] ; [B] (M5B) [below=of O5] ;(M1) – (M2); (M2) – (M3); ( M3) – (M4); (M4) – (M5);(M1B) – (M2B); (M2B) – (M3B); ( M3B) – (M4B); (M4B) – (M5B);(O1) – (O2); (O2) – (O3); ( O3) – (O4); (O4) – (O5);(M1) – (O1); (M2) – (O2); (M3) – (O3); (M4) – (O4); (M5) – (O5);(O1) – (M1B); (O2) – (M2B); (O3) – (M3B); (O4) – (M4B); (O5) – (M5B);(L1) [left= of O1] ⟨ψ̃|Ô|ψ|=⟩;(R1) [right= of O5] .; Once an observable has been expressed as a tensor network in terms of MPS and MPO, it must be evaluated by contracting the tensors together in the pattern indicated. However, not all patterns will be equally efficient and can be exponentially expensive in the number of sites. In the present case, an efficient contraction ordering is given by proceeding horizontally from the left or right boundary. This can be expressed in terms of transfer matrices by defining the object E^A_B[O], or E^A_B if no operator is included, where A and B label the rank-3 tensors placed in the upper and lower positions respectively. Diagrammatically we have[node distance = 5pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M1); [empty] (O1) [below = of M1] ; [B] (M1B) [below = of O1];[empty] (M1L) [left = of M1] ; [empty] (O1L) [left = of O1] ; [empty] (M1BL) [left = of M1B] ; [empty] (M1R) [right = of M1] ; [empty] (O1R) [right = of O1] ; [empty] (M1BR) [right = of M1B] ; [-] (M1) – (M1B); [-] (M1) – (M1L); [-] (M1) – (M1R);[-] (M1B) – (M1BL); [-] (M1B) – (M1BR);[empty] (L2) [left=of O1] E^A_B =;and[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M1); [O] (O1) [below = of M1] ; [B] (M1B) [below = of O1];[empty] (M1L) [left = of M1] ; [empty] (O1L) [left = of O1] ; [empty] (M1BL) [left = of M1B] ; [empty] (M1R) [right = of M1] ; [empty] (O1R) [right = of O1] ; [empty] (M1BR) [right = of M1B] ; [-] (M1) – (O1); [-] (O1) – (M1B); [-] (M1) – (M1L); [-] (M1) – (M1R);[-] (O1) – (O1L); [-] (O1) – (O1R);[-] (M1B) – (M1BL); [-] (M1B) – (M1BR);[empty] (L1) [left=of O1] E^A_B[O] =;[empty] (R1) [right=of O1] .;In this notation, the matrix element ⟨ψ̃|Ô|ψ|$⟩ is given by⟨ψ̃|Ô|ψ| ⟩= ( E^A_1_B_1[O_1] E^A_2_B_2[O_2] ... E^A_L_B_L[O_L]) = ( 𝐄_1𝐄_2 ... 𝐄_L)= ( ∏_x𝐄_x) .Viewing𝐄_xas matrix of size(χ^2χ_W,χ^2χ_W)the cost of this matrix multiplication would naively be𝒪( χ^6 χ_W^3 )though this can be lowered in a variety of ways which we discuss further in Section <ref>.Since any state can be expressed as an MPS and any operator as an MPO this representation for observables can always be used. However, we will only be interested in those observables which can be represented efficiently. For the operators this means finding which can be represented with sufficiently smallχ_W. The set of operators that can be represented trivially, i.e. withχ_W = 1, are simply operators of the tensor product form e.g. all equal-timen-point functions. It is not so obvious that other important operators such as the lattice Hamiltonian can also be expressed in an efficient MPO form. However, this is indeed possible and we review the construction in the next section. §.§ Representation of Lattice Hamiltonian as MPOTo represent the lattice HamiltonianH̃[ϕ]as an MPO it is helpful to first simplify the expression by collecting all one site terms into a single operatorh_xso that the Hamiltonian takes the formH̃[ϕ] = ∑_x -ϕ_xϕ_x+1 + h_xwith h_x = 2+μ̃_0^22ϕ^2_x + λ̃_04!ϕ^4_x .Clearly, this has the same nearest neighbour structure as the Ising model Hamiltonian and we can use the same MPO construction methods as in that case <cit.>. A correct MPO representation can then be obtained by building the Hamiltonian up iteratively from the rightmost site to the left. To see this, consider the matrix of operators𝐖_x = [100; -ϕ_x00;h_xϕ_x1 ] .The matrix is tri-diagonal so that multiplying𝐖_x 𝐖_x+1gives𝐖_x𝐖_x+1 = [100; -ϕ_x00; h_x+h_x+1-ϕ_xϕ_x+1ϕ_x+11 ].The general structure remains the same while the Hamiltonian is built up in the bottom-left corner. The bulk terms of the Hamiltonian can be built up iteratively in this way. It then only remains to pick a single boundary term (e.g. at sitex=1) to correctly select out the bottom-left corner, where the bulk of the Hamiltonian has been built up, and encode any remaining boundaries using the trace appearing in the MPO definitions. For PBC a consistent choice is 𝐖_1 = [0ϕ_11;00 -ϕ_1;00h_1 ] .Open boundary conditions can also be encoded by setting𝐖_1and𝐖_Lto be vectors𝐖_1 = w_1 , 𝐖_L = w_Le.g. by the choicew_1^T = [ h_1 ϕ_1 1 ] , w_L = [1; -ϕ_L;h_L ].To specialise to TPBC such thatϕ(x+L) = -ϕ(x), we can introduce the field variableϕ̃(x) =ϕ(x)ifx ≤ L-ϕ(x)ifx > Lforx = 1,2,..,2L. This means we can treat the finite size Hamiltonian as having an impurity (the twist) at a particular lattice site and observables such as correlations functions must be transformed back to the original variables to keep the periodicity intact. The boundary matrix then takes the form𝐖_1 = [ 0 -ϕ̃_1 1; 0 0 -ϕ̃_1; 0 0 h_1 ] .Through the above constructions we have found that indeed the nearest-neighbour HamiltonianH̃[ϕ]can be expressed efficiently as a MPO withχ_W = 3, reflecting its local nature. Longer range interactions can also be considered, as would be the case if higher-order finite difference approximations were used. However, in that case larger values ofχ_Wwould be required reflecting the less local nature of the operator. Having an efficient representation of the lattice Hamiltonian as an MPO allows for the efficient computation of observables such as the energy expectation value or energy variance, which are central to obtaining approximations of the ground state and one kink state as MPS.§ APPROXIMATION OF Φ^4 GROUND STATE AND ONE KINK STATE VIA VARIATIONAL SEARCH.§.§ Variational Energy Minimisation for Matrix Product States We now turn to the approximation of the ground state and one kink state using MPS. While the minimisation problems given by Equations (<ref>) and (<ref>) are too difficult to solve in the full lattice Hilbert space, we can restrict them to the MPS subset. The approximations to the states then become|Ω(a,χ)⟩ = min_|ψ⟩∈(a,χ)( ⟨ψ|H̃|ψ|-⟩λ[⟨ψ|ψ|-⟩ 1])for the ground state and|K(a,χ)⟩ = min_|ψ⟩∈(a,χ)( ⟨ψ|H̃_(TPBC)|ψ|-⟩λ[⟨ψ|ψ|-⟩ 1] )for the one kink state, where only the most important truncation parametersaandχhave been kept explicit. Despite the restriction, such a problem is still too hard to solve globally but one can try an iterative procedure that minimises the energy at each step, converging to a best estimate for the global solution. This is now a standard procedure for approximating ground states with MPS and has been highly successful in a variety of cases, see <cit.> for a detailed guide to implementation. The tensor network structure of the MPS provides a natural way to proceed : one can minimise the energy with respect to just a single tensor (i.e. at a single site) while keeping all other tensors fixed. One then proceeds tensor by tensor minimising the energy iteratively. This is most efficiently performed in a sweeping pattern moving from site to site in a given direction until some convergence criteria are met. In the tensor network representation of observables this local minimisation has a useful form in terms of a generalised eigenvalue problem. To see this, consider the tensor network representation of the energy expectation value withH̃in MPO form withL=5[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M3) at (0,0) ; [A] (M2) [left=of M3] ; [A] (M1) [left=of M2] ; [A] (M4) [right=of M3] ; [A] (M5) [right=of M4] ;[O] (O1) [below=of M1] ; [O] (O2) [below=of M2] ; [O] (O3) [below=of M3] ; [O] (O4) [below=of M4] ; [O] (O5) [below=of M5] ;[A] (M1B) [below=of O1] ; [A] (M2B) [below=of O2] ; [A] (M3B) [below=of O3] ; [A] (M4B) [below=of O4] ; [A] (M5B) [below=of O5] ; (M1) – (M2);(M2) – (M3);( M3) – (M4);(M4) – (M5); (M1B) – (M2B);(M2B) – (M3B);( M3B) – (M4B);(M4B) – (M5B); (O1) – (O2);(O2) – (O3);( O3) – (O4);(O4) – (O5); (M1) – (O1);(M2) – (O2);(M3) – (O3);(M4) – (O4);(M5) – (O5); (O1) – (M1B);(O2) – (M2B);(O3) – (M3B);(O4) – (M4B);(O5) – (M5B); (L1) [left= of O1] ⟨ψ|H̃|ψ|=⟩; (R1) [right= of O5] .;If we are only interested in varying a single tensor at a particular site, e.g. the central sitex=3, all other tensors can be contracted together, giving [node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,OB/.style = shape=rectangle,draw=blue!100,fill=blue!30, empty/.style = shape=circle,draw=blue!0,B/.style = shape=circle,draw=blue!100,fill=gray!0];[A] (M1); [O] (O1) [below = of M1] ; [A] (M1B) [below = of O1]; [OB] (vL) [left = of O1]; [-] (O1) – (O1L); [OB] (v) [right = of O1]; [-] (O1) – (M1);[-] (O1) – (M1B);(vL) to (O1);(vL) to [bend left = 45] (M1) ;(vL) to [bend right = 45] (M1B); (v) to (O1);(v) to [bend right = 45] (M1) ;(v) to [bend left = 45] (M1B); (L1) [left= of vL] ⟨ψ|H̃|ψ|=⟩;for OBC while in PBC the boundary tensors would include additional indices to be traced over. This expression can then be thought of as the action of an effective Hamiltonian on the tensor at the uncontracted site. In terms of the tensor network, the effective Hamiltonian then takes the form[node distance = 10pt ,A/.style = shape=circle,draw=blue!100,fill=gray!50 ,O/.style = shape=rectangle,draw=blue!100,fill=black!50, empty/.style = shape=circle,draw=blue!0,OB/.style = shape=rectangle,draw=blue!100,fill=blue!30,B/.style = shape=circle,draw=blue!100,fill=gray!0];[empty] (M1); [O] (O1) [below = of M1] ; [empty] (M1B) [below = of O1]; [OB] (vL) [left = of O1]; [-] (O1) – (O1L); [OB] (v) [right = of O1]; [-] (O1) – (M1);[-] (O1) – (M1B); (vL) to (O1);(vL) to [bend left = 45] (M1) ;(vL) to [bend right = 45] (M1B); (v) to (O1);(v) to [bend right = 45] (M1) ;(v) to [bend left = 45] (M1B); (L1) [left= of vL] H_eff =;which is a linear operator on the space of rank-3 tensors. As such, it can be considered a matrix of size(d χ^2 , d χ^2)that acts on vectorsof size(d χ^2)(i.e. the rank-3 tensors). In a similar way, an effective normalisation matrix can be constructed by replacing the MPO representing the Hamiltonian by the identity operator via⟨ψ| ψ|=⟩ ⟨ψ| 1 | ψ|$⟩ where 1 = 1_1⊗1_2⊗ ... ⊗1_L. To emphasise this structure we can use the notation v_M = 𝐌^n_x_α_x,α_x+1 and write the two effective operators as matrices on this space, 𝐇_eff and 𝐍_eff. In this notation the (ground state) minimisation problem at this site can be written as min_v_M∈C^dχ^2( v_M^†𝐇_effv_M - λ[ v_M^†𝐍_effv_M -1] )which can be solved by finding the minimum eigenvector of the generalised eigenvalue problem 𝐇_effv_M = λ𝐍_effv_M .To find an approximation to the ground state one then initiates a (random) MPS, chooses a site i, forms the effective operators 𝐇_eff , 𝐍_eff, finds the minimum eigenvector ṽ_M of the generalised eigenvalue problem Equation (<ref>) and updates the current MPS by replacing the rank-3 tensor at the site i with the rank-3 tensor corresponding to the minimum eigenvector ṽ_M . The updated MPS will have a lower energy and so by proceeding to the next site the energy can be lowered iteratively until convergence is achieved.The above method can also be used to find an MPS approximation to the one kink state by simply exchanging the PBC Hamiltonian with the TPBC Hamiltonian. However, there are important differences to consider that make the approximation of the one kink state more difficult than the ground state. In particular, the matrix product state techniques described here are naturally inhomogeneous and during the minimisation procedure translational invariance will be broken numerically leading to spatial dependence of the tensors ^n_x(x). As such, the translational invariance of observables is only approximated. In the case of the ground state, this is no problem since the MPO representation of the lattice Hamiltonian is reasonably homogeneous and translational invariance can be easily approximated with a low χ MPS.However, for the one kink state, the Hamiltonian appears quite inhomogeneous with a particular location being selected for the twist. This makes it much harder to approximate translational invariance and the kink must be “delocalised" by using a sufficiently high χ. The approximation of translational invariance tends to happen quickly so that one can think of a threshold χ̃(d,L) after which the spatial variance of local observables drops dramatically. The value of χ̃(d,L) will depend on the observable in question as well as the values of d and L with higher d and larger L leading to an increased χ̃(d,L). The dependence on d is particularly important since it means that in regions of parameter space requiring high d it will become impossible to approximate translational invariance for the kink state with this method. Since the size of d is determined by the value of the field expectation value ⟨ϕ|$⟩ (or⟨ϕ^2|$⟩ to allow for Z_2 invariant cases) we see that it is the semi-classical region μ_0^2≫λ_0 that will be hard to approximate in this sense, while the strong coupling region will be less problematic. Of course, this is not too much of an issue since the semi-classical region can be treated perturbatively and observables that include contributions from the entire lattice, e.g. the kink mass, do not depend strongly on the translational invariance of the state. Similar issues are present when approximating excited states with MPS more generally and have led to the development of various tensor network excitation ansatz, including one for kinks, that enforce translational invariance explicitly <cit.>. However, there is some evidence that an inhomogeneous representation could better capture certain aspects of the kinks, which are naturally non-linear field configurations <cit.>. This would likely make little difference when considering an observable such as the kink mass, but might be important with observables that have significant contributions from the kink “width" e.g. the form factor or the equal time two point functions in the presence of a kink near the critical point. We do not explore this further but the excitation ansatz is at least more efficient when determining the mass of excitations near the critical point and we will use them for comparison with the methods here. §.§ Computational efficiency, numerical stability and uMPS minimisationThe computational efficiency and numerical stability of the minimisation procedure must be considered at two main stages. Firstly, observables must be calculated efficiently corresponding to the correct choice of contraction ordering. This also covers the construction of the effective operators H_eff , N_eff which are built up by the partial contraction of a similar tensor network. Secondly, the generalised eigenvalue problem Equation (<ref>) must be solved. In the first case, an efficient contraction pattern is given by simply multiplying the transfer matrices 𝐄_x as matrices for a computational cost 𝒪(χ^6). A more efficient way to multiply the transfer matrices can be found by making use of their tensor network structure, reducing the cost to 𝒪(χ^4). In the case of OBC MPS and MPO, a significant speed up is possible since the boundaries act as vectors and only matrix-vector multiplications are needed. In this case, the naive scaling (i.e. without taking advantage of the tensor network structure) is simply 𝒪(χ^4), which can be reduced to 𝒪(χ^3) when exploiting the tensor network structure. When considering sufficiently long chains of transfer matrices, the cost of PBC contractions can be reduced as the boundaries become less relevant (becoming completely irrelevant in the infinite distance limit), see <cit.> for details and implementation. In principle this reduces the cost to 𝒪(χ^3), equal to OBC, but numerical stability tends to require 𝒪(χ^4).The second case, that of solving the generalised eigenvalue problem, also demonstrates the significant computational advantage of OBC vs PBC. Naively, the cost of solving a generalised eigenvalue problem scales as matrix-matrix multiplication 𝒪(χ^6). However, if a sparse implementation is possible then only matrix-vector multiplications are required for a naive cost of 𝒪(χ^4). For OBC this can again be reduced to 𝒪(χ^3). Unfortunately, solving a generalised eigenvalue problem tends to be ill-conditioned and additional stabilisation steps must be taken. In MPS and tensor networks more generally, stabilisation is often achieved by exploiting the significant gauge freedom in the MPS representation. This freedom can be easily seen since any MPS can be equivalently rewritten by inserting identity matrices 1 of size (χ, χ) between any of the rank-3 tensors. Decomposing the identities as 1 = 𝐆(x)^-1𝐆(x) then leads to an equivalent MPS form of the state asψ_𝐧_𝐱=( ∏_x𝐆(x)^n_x(x)𝐆^-1(x+1) ) =( ∏_x^n_x(x)) .Equivalently, and more commonly in practice, one can think of performing a matrix decomposition on the tensors which can be chosen so that the eigenvalue problem to be solved is better conditioned, see <cit.> for details. In the case of OBC it is possible to choose the gauge such that the effective normalisation matrix simply becomes equal to the identity matrix i.e. the generalised eigenvalue problem is transformed into a standard eigenvalue problem which is considerably more stable. For PBC, this transformation is not possible and we have instead followed the stabilisation strategy outlined in <cit.>. Unfortunately, in the case of TPBC this was not sufficient to be able to solve the generalised eigenvalue problem with sparse methods and we have used dense methods at a cost of 𝒪(χ^6). Despite the relative expense, we find that the reachable χ≈ 20 are sufficient for studying the kink mass at strong couplings though for studying other observables e.g. the two point function in the presence of the kink, higher χ would be needed and an alternative stabilisation strategy would be useful such as the one in <cit.> which was applied to aspin system with TPBC.For the approximation of the ground state, since TPBC are not required, it is possible to gain the computational advantage of OBC by using a MPS with an explicitly translationally invariant representation which can be achieved by simply requiring all tensors to be identical i.e. 𝐌^n_x(x) = 𝐀^n_x for all x = (1,...,L). Since there is no spatial variation this MPS can be defined in the infinite size limit L →∞ such that the boundaries are irrelevant and the computational savings of OBC can be taken advantage of. Such MPS are called uniform matrix product state (uMPS) and, due to the greatly decreased number of free parameters, more standard minimisation procedures can be used to obtain approximations to the ground state <cit.>. Due to the greatly increased efficiency we use uMPS to approximate the ground state following the conjugate gradient procedure outlined in <cit.>. The use of uMPS allows for much higher values of χ which is essential for capturing the relevant physics of the ground state near the critical point due to the diverging correlation length. Additionally, we note there is an efficient time evolution procedure associated to uMPS known as the time dependent variational principle (TDVP) <cit.> making uMPS a good candidate for the study of non-equilibrium physics. For an open source uMPS code, which aided the development of the code used here, see <cit.>.§ SCALAR MASS§.§ Long-Distance Behaviour of G_2(r) Since tensor networks represent the state of the quantum system directly, a possible method for obtaining the scalar mass is to try and directly approximate the one particle excitation in the system. However, since tensor networks are often particularly suited to the description of ground states, it is useful to have a general procedure for extracting the scalar mass from ground state observables alone. While it is possible to extract the scalar mass from the long time behaviour of two point functions, for MPS it is much easier to consider the ground state equal time two point functions G_2(r) which can be directly calculated once a ground state approximation is found.To see how the scalar mass can be extracted, we can consider the Källén-Lehmann spectral representation of the time ordered ground state two point function which can be constructed quite generally for a Lorentz invariant theory <cit.>. This representation relates the full two point function to the two point function of the non-interacting theory, specifically the Feynman propagator D_F(x-y ; M^2) via⟨Ω|T ϕ(x)ϕ(y)|Ω|=⟩∫_0^∞dM^22 πρ(M^2) D_F(x-y;M^2)where x and y are space-time coordinates, ρ(M^2) is the spectral density given byρ(M^2)= ∑_λ (2 π) δ( M^2 - m_λ^2) |⟨Ω|ϕ(0)|λ_0||⟩^2= ∑_λ (2 π) δ( M^2 - m_λ^2) Zand |λ_0⟩ is a zero-momentum energy eigenstate. We can evaluate the Feynman propagator easily in D = (1+1) at equal times to giveD_F(r ; M^2) = 12π K_0(M r)where K_0(z) is a modified Bessel function of the second kind. The equal time two point function can then be written asG_2(r) = ∫_0^∞dM^24 πρ(M^2) K_0(M r) .If the spectrum contains an isolated pole then we can extract this contribution and write schematicallyG_2(r)=Z(2 π)^2 K_0(m_S r) + ∫_4 M^2^∞dM^24 πρ(M^2) K_0(M r) .This suggests that at sufficiently long distances, the non-interacting form of the two-point function will be dominant and depend on the dimensionless combination m_S r.§.§ Extracting m_S from G_2(r ; a , χ) The above arguments motivate the use of the ansatz G_2(r) = A K_0(m_S r)to extract the scalar mass. This can be achieved by taking the appropriate ratios (finite differences) to cancel overall factors as G_2(r+1)G_2(r)= K_0 (m_s (r+1))K_0 (m_s r) .This equation can then be solved numerically to extract m_S(r) which depends on r due the fact that G_2(r) is not a pure Bessel function. Following the previous arguments we can expect for some initial r ≲ξ the value of m_S(r) will vary due to the lattice effects and higher M^2 eigenstate contributions before becoming uniform such that m_S can be extracted (in practice, the uniform region must be selected by some criteria e.g. the gradient of m_S(r) falling below some specified tolerance, with m_S then estimated by averaging over the selected region). As discussed previously, the use of MPS will modify the long-distance behaviour of the observable G_2(r) due to finite entanglement effects ultimately leading to a pure exponential decay. The distance where this occurs is determined by the truncation parameter χ and we denote the length scale associated to this as ξ_χ. We can therefore expect a constant region of m_S(r) to occur in some intermediate distance region above the scale of higher mass contributions and below the scale of finite entanglement corrections. A method to diagnose the finite entanglement effects is to repeat the procedure used to extract m_S(r) but now use an exponentially decaying form as δ G_2 (r) G_2(r)= (G_2(r+1)- G_2(r) ) G_2(r)= G_2(r+1)G_2(r) - 1 =e^-m_D - 1so thatm_D(r) = -log( G_2(r+1)G_2(r)) .One again, m_D(r) will vary with r since only at long distances where the finite entanglement effects dominate will G_2(r) become a true exponential. This expression can be used to quickly get a sense of the distance where the correlation function becomes strongly modified by the finite entanglement effects of the MPS.§.§ Extracting m_S for Lattice MPS at Strong Couplings At strong couplings, the lightest excitations will become kink-antikink pairs. This motivates the use of a Bessel-squared function ansatz for the two point function G_2(r) = A K_0(m_S r)^2corresponding to the form for two non-interacting excitations. The behaviour of G_2(r) can be established more rigorously in the critical region by considering the critical behaviour of the classical D=2 Ising model which is described by a field theory of free massive Majorana fermions <cit.>. At strong couplings, the Bessel-squared ansatz can then be used in a similar way to the Bessel function form. In fact, both K_0 (z) and K_0 (z)^2 have similar exponentially decaying asymptotic forms K_0 (z)→ e^-z√(1z)[K_0 (z)]^2 → e^-2 z12z .As such, in principle either form can be used to estimate m_S. However, unlike K_0 (z), we can expect [K_0 (z)]^2 to be valid outside the asymptotic regime which is essential since then shorter distances of G_2(r) will need to be approximated, requiring smaller χ.§ RESULTSWe study the ϕ^4 QFT using MPS in the lattice regularised setting. An approximation to the ground state is obtained using uMPS while a finite size lattice MPS is used to approximate the one kink state. We study both weak and strong coupling behaviour by fixing the value of λ_0 = 0.1 or 2 while using a range of μ_0^2. When the effective coupling is small we compare the MPS approximation to the classical continuum. The lowest order mass renormalisation is included numerically by fitting the classical forms to the MPS data, replacing the bare mass with μ^2 = μ_0^2 - m_C^2 where m_C^2 is treated as a free parameter. In the region ξ≫ 1 the lattice effects will be small and the comparison with the classical continuum is appropriate. In the strong coupling region the MPS approximations can be compared with universal results and we will focus on a comparison of the mass ratio m_S/M_K to the universal result m_S/M_K≈ 2. The accuracy of our approximations will depend on the observable in question. Essentially, the important features are the observation distance and to what degree the observable represents an average over the system. In the first case, only the truncation parameter χ is important and increasing χ will allow longer distances to be better approximated. This can be seen clearly in ground state connected two point function G_2(r) where larger χ are required to approximate the observable at larger distances r. We can associate this behaviour to a length scale ξ_χ corresponding to the distance at which the approximation of G_2(r) is dominated by finite entanglement effects and decays as a pure exponential. In the second case, one can think of a particular threshold χ≈χ̃(d,L) being required before the translational invariance of observables is well approximated. This will depend strongly on the observable/state in question and on the truncation parameters d and L.The issues surrounding the approximation of translational invariance can be seen clearly when calculating the field expectation values ⟨Ω|ϕ|Ω|$⟩ and⟨K|ϕ|K|$⟩. Since both observables are local, they will converge quickly in χ so long as the threshold χ≈χ̃(d,L) is met. In the ground state case, χ̃(d,L) is essentially negligible and the approximation of ⟨Ω|ϕ|Ω|$⟩ converges rapidly. However, in the one kink stateχ̃(d,L)is important and, for sufficiently large values ofdorL, local expectation values such as⟨K|ϕ|K|$⟩ will show significant spatial variations and cannot be accurately approximated. We note that, even in the ground-state case, the field expectation value is in princible zero, respecting the Z_2 symmetry but is broken numerically during the approximation and must be enforced explicitly if desired <cit.>. The uMPS approximation of the vacuum expectation value ⟨Ω|ϕ|Ω|$⟩ is shown in Figure <ref> for perturbative and non-perturbative bare couplingsg_0 = λ_0/μ_0^2. In the first case, the results (blue triangles) can be compared with the classical continuum result (red line) v = √(-6μ^2/λ_0). In the stronger coupling case, there is no analytic comparison but the expected symmetry breaking pattern can be seen. In principle, such plots can be used to determine the location of the critical point e.g. by using the critical exponent associated with the vanishing field expectation value. However, as discussed in <cit.>, such fits are highly sensitive and it is much better to use observables with a simpler scaling, e.g. the kink mass, to determine the location of the critical point.The more problematic behaviour of⟨K|ϕ|K|$⟩ is shown in Figure <ref>. In the semi-classical case with bare coupling g_0≈ -0.33, the high field expectation value requires a relatively high value of d to converge and the threshold χ̃(d,L) is higher than the shown χ = 6, 10, 16. This means that a classical-like kink profile can be seen and increasing χ achieves only very slight changes to the width such that the correct ⟨ϕ|_⟩K = 0 value is not obtained. Moreover, the zero-mode means that the point at which ⟨ϕ(x)|_⟩K crosses zero is independent of the energy making convergence in χ or d difficult to quantify.However, at stronger couplings the field expectation value is much lower, corresponding to a lower d, which makes it easy to approximate translational invariance and obtain ⟨ϕ|_⟩K≈ 0 even for the modest values of χ shown. Observables that average over the whole system can be much less sensitive to spatial variations in the MPS representation than observables evaluated at a particular point. For example, the behaviour of ⟨K|ϕ^2(x)|K|=⟩⟨ϕ^2|_⟩K displays similar spatial variations at weak coupling as for the case of ⟨K|ϕ(x)|K|$⟩, see Figure <ref> . However, the spatial average of this expectation value⟨⟨ϕ^2|⟩|_⟩Khas a much weaker dependence onχ. This is shown in Figure <ref> where the spatial variation of the expectation values of both⟨⟨ϕ|⟩|_⟩Kand⟨⟨ϕ^2|⟩|_⟩Kare shown by error bars corresponding to their standard deviation withx. Despite the strong spatial variation in⟨ϕ^2|_⟩K, the spatial average changes only very weakly withχin both cases indicating that this observable can be well approximated even in the weak coupling case. This behaviour can be compared with that of⟨ϕ|_⟩K; while the spatial average does not display much variation in the weak coupling case, since the operator isZ_2anti-symmetric, it still gives the incorrect non-zero value and is only correctly approximated in the stronger coupling region where translational invariance is approximated. In general, observables corresponding toZ_2anti-symmetric operators cannot be reliably approximated outside the translational invariant region, while the spatial average of those corresponding toZ_2symmetric operators can be. The kink massM_Kis calculated from the difference of the one kink energy expectation value⟨K|H̃_TPBC|K|$⟩, obtained from the finite size lattice MPS, and the ground state energy density, obtained from the uMPS. The latter converges quickly in χ and the only potential issue is the approximation of ⟨K|H̃_TPBC|K|$⟩. However, since the kink mass both includes contributions from the whole system, is fairly local andZ_2symmetric, we can expect a reasonable convergence withχeven in the weak coupling case. In the case of the scalar mass, since it is estimated from the ground state observableG_2(r), approximating translational invariance should not be an issue and we can focus on the need to increaseχso that the region up tor ≈ξis well approximated. At weak couplings,ξis relatively small so that the requiredχshould not be too high allowing for an estimate ofm_Sto be extracted relatively easily. The kink mass and scalar mass are shown for a variety of weak couplings in Figure <ref> along with the classical continuum results for comparison. At stronger couplings the correlation lengthξincreases so that longer distances ofG_2(r)need to be approximated requiring largerχ. Ultimately, this means that this method cannot be used with MPS arbitrarily close to the critical point where the scalar mass vanishes. This is reflected in the fact that at the critical point the correlation length diverges leading to algebraically decaying correlations which correspond to a logarithmic violation of the entanglement area law i.e.S_𝒜 ∼log(L_𝒜) ∂𝒜. While an MPS can still be used to approximate short distance observables in the critical region <cit.> an alternative tensor network , e.g. the multi-scale entanglement renormalisation ansatz (MERA) <cit.>, that obeys the correct low entanglement law will also allow the approximation of the long range physics. Of course, this means that MPS are not especially suited to the study of universal physics and we can expect difficulty when trying to reproduce the strong coupling behaviour e.g. the universal mass ratio.The scalar mass at strong couplings is plotted along with the kink mass in Figure <ref> and the qualitative change in the scaling can been seen in the left-hand plot at the point when2 M_K ≈m_Sas expected. However, the estimate ofm_Swithχ= 32in the critical region extracted from the Bessel function tends to be somewhat higher than the value of2 M_Ksuggesting that, as might be expected, it is inaccurate in this region. The scalar mass extracted from the excitation ansatz withχ= 32is also plotted (red dashed line) and is somewhat closer to the value of2 M_Ksuggesting that it can provide a more efficient and accurate method to extract the scalar mass in the critical region. To increase the accuracy of the uMPS method one can simply increase the value ofχbut it is also possible to use the Bessel-squared ansatz Equation (<ref>). A comparison of these methods is shown for the strong coupling region in the right-hand plot. The estimate of the scalar mass is closer to the expected behaviour whenχis increased (red dots and diamonds) but the use of the Bessel-squared ansatz improves the estimate again (black squares) agreeing fairly well with the excitation ansatz. The significant improvement of the Bessel-squared method over the single Bessel method suggests that the uMPS is able to capture the contributions coming from the kink-antikink excitations in this observable. To achieve higher accuracies than obtained here, largerχcan be used by following more recently developed algorithms than the conjugate gradient minimisation used here<cit.>. Alternatively, one can also turn to better suited tensor networks such as MERA and both the methods to obtain the kink mass and scalar mass should be easily adaptable to this case. Of course, if one is only interested in the equilibrium physics chosen here, the excitation ansatz provides good accuracy and efficiency. However, it is less flexible and cannot be readily applied to different tensor networks and instead must be built explicitly for each case. § CONCLUSION We have studied the topological defects (kinks) of the relativisticϕ^4quantum field theory (QFT) inD=(1+1)using matrix product states (MPS). We have shown how a finite size lattice MPS approximation to the one kink state can be obtained by making use of twisted periodic boundary conditions (TPBC) and that the resulting kink mass agrees with expectations. While alternative specific excitation ansatz provide a more efficient method to calculate the kink mass, the TPBC method can be easily adapted to other theories and tensor networks while also allowing for the easy calculation of a wide variety of observables e.g. equal time field n-point functions in the presence of the kink. We have also outlined a general method to extract the scalar mass from the ground state equal time two point functions. A comparison of the one kink and ground state approximations with universal results suggests that the MPS (specifically the uMPS) is able to capture the contribution of kink-antikink excitations to observables making it an interesting candidate to study challenging non-equilibrium phenomena such as defect formation via the Kibble Zurek mechanism during quantum phase transitions. § ACKNOWLEDGEMENTS We have made use of the Imperial College London High Performance Computing Service. E.G. was supported by the EPSRC Centre for Doctoral Training in Controlled Quantum Dynamics and A.R. by STFC grant ST/L00044X/1. | http://arxiv.org/abs/1705.09802v3 | {
"authors": [
"Edward Gillman",
"Arttu Rajantie"
],
"categories": [
"quant-ph",
"hep-lat"
],
"primary_category": "quant-ph",
"published": "20170527104722",
"title": "Topological Defects in Quantum Field Theory with Matrix Product States"
} |
http://arxiv.org/abs/1705.09166v2 | {
"authors": [
"Michael I. Eides",
"Valery A. Shelyuto"
],
"categories": [
"hep-ph",
"physics.atom-ph"
],
"primary_category": "hep-ph",
"published": "20170525132425",
"title": "One More Hard Three-Loop Correction to Parapositronium Energy Levels"
} |
|
Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation Fei Liu^♯ and Lexing Ying^†♯† Department of Mathematics, Stanford University♯ Institute for Computational and Mathematical Engineering, Stanford University ======================================================================================================================================================================== This paper presents an efficient preconditioner for the Lippmann-Schwinger equation that combines the ideas of the sparsifying and the sweeping preconditioners. Following first the idea of the sparsifying preconditioner, this new preconditioner starts by transforming the dense linear system of the Lippmann-Schwinger equation into a nearly sparse system. The key novelty is a newly designed perfectly matched layer (PML) stencil for the boundary degrees of freedoms. The resulting sparse system gives rise to fairly accurate solutions and hence can be viewed as an accurate discretization of the Helmholtz equation. This new PML stencil also paves the way for applying the moving PML sweeping preconditioner to invert the resulting sparse system approximately. When combined with the standard GMRES solver, this new preconditioner for the Lippmann-Schwinger equation takes only a few iterations to converge for both 2D and 3D problems, where the iteration numbers are almost independent of the frequency. To the best of our knowledge, this is the first method that achieves near-linear cost to solve the 3D Lippmann-Schwinger equation in high frequency cases. Keywords. Lippmann-Schwinger equation, acoustic and electromagnetic scattering, quantum scattering, preconditionerAMS subject classifications. 65F08, 65F50, 65N22, 65R20, 78A45§ INTRODUCTION This paper concerns the time-harmonic scattering problem(-Δ-ω^2c(x)^2)(u(x)+u_I(x)) = 0,x ∈^d, lim_r→∞ r^(d-1)/2(r-ω) u(x) = 0,where u_I(x) is the given incoming wave, u(x) is the scattered field to solve, ω is the angular frequency and c(x)=Θ(1) is the velocity field such that c(x) = 1 outside some bounded region Ω. See Figure <ref> for an example. The incoming wave u_I(x) satisfies the homogeneous Helmholtz equation(-Δ-ω^2)u_I(x) = 0,x∈Ω. Let m(x) = 1 - 1/c(x)^2 be the perturbation field. Rewriting (<ref>) in terms of m(x) we have(-Δ-ω^2 + ω^2 m(x))u(x) = -ω^2m(x)u_I(x),x∈^d.Let G(x) be the Green's function of the free space Helmholtz equationG(x)= 4H_0^(1)(ω |x|),d = 2, exp(ω |x|)4π |x|,d = 3.Convolving both sides of (<ref>) with G(x) givesu(x) + ω^2 ∫_Ω G(x-y) m(y) u(y)y = -ω^2 ∫_Ω G(x-y) m(y) u_I(y)y ,which is known as the Lippmann-Schwinger equation written in terms of the scattered field u(x).Solving the integral equation (<ref>) has several advantages compared to solving (<ref>). First, since m(x) is compactly supported, we only need to solve (<ref>) in Ω. The scattered field u(x) for x∈Ω^c is explicitly given by (<ref>) once u(x) in Ω is known. More importantly, the resulting wave field u(x) in ^d automatically satisfies the Sommerfeld radiation condition. On the contrary, for (<ref>) one has to truncate the domain ^d to some bounded region and impose appropriate boundary conditions to simulate the radiation condition. Second, most local discretizations of (<ref>) suffer from the pollution effect <cit.> due to inaccurate dispersion relations. (<ref>) avoids this problem by leveraging the Green's function explicitly in the equation.However, discretizing (<ref>) also raises several issues. First, the resulting linear system is dense. By the Nyquist theorem, a constant number of points per wavelength is needed to capture the oscillations, thus the number of degrees of freedom N is at least Θ(ω^d). In high frequency cases, N can be rather large where it is impractical to solve general dense linear systems with direct method. Second, the discretized system can have very large condition number for non-negligible perturbations m(x) due to multiple scattering when ω is large. As a result, most standard iterative solvers require a large number of iterations to converge.Recently, several progresses have been made to solve the Lippmann-Schwinger equation <cit.>. <cit.> proposes a numerical scheme that has spectral accuracy for smooth media by truncating the interactions on the physical domain. <cit.> presents an adaptive method for the Lippmann-Schwinger equation in 2D. <cit.> solves the 2D Lippmann-Schwinger equation with a technique which is now often referred to as recursive interpolative factorization or recursive skeletonization, where the setup cost is O(N^3/2) and the solve cost is O(N log N). <cit.> approximates the discretized dense system by a sparse system, and applies the nested dissection factorization <cit.> to the sparse system as a preconditioner to the original dense system. The costs are dominated by merely the nested dissection solver, which are O(N^3/2) and O(N log N) for setup and solve in 2D, O(N^2) and O(N^4/3) for setup and solve in 3D respectively. <cit.> combines the sparsifying preconditioner <cit.> with the method of polarized traces <cit.> to design a preconditioner for the Lippmann-Schwinger equation in 2D, which achieves O(N) setup and O(N log N) solve costs. As far as we know, <cit.> is the first to achieve near-linear cost in 2D high frequency cases.Meanwhile, a series of domain decomposition methods were developed to solve the Helmholtz equation with Sommerfeld radiation condition <cit.>.The idea is to divide the domain into slices and impose suitable transmission conditions between these slices. These methods reduce the computational costs to O(N) for setup and O(N) for solve in 2D, and O(N^4/3) for setup and O(N log N) for solve in 3D, which is a notable improvement over the nested dissection method. A recursive technique <cit.> further reduces both the setup and solve costs in 3D to O(N).This work combines the sparsifying preconditioner in <cit.> with the sweeping preconditioner in <cit.> to develop a new preconditioner which solves the Lippmann-Schwinger equation in near-linear cost. The sketch of the method is as follows. We first construct two types of compact stencil schemes to approximate the discretized dense system by a sparse system, and then apply the sweeping factorization to the sparse system. The solving process of the sweeping factorization induces an approximating solution, which defines a preconditioner to the original system. The setup and application costs are O(N) and O(N) in 2D and O(N^4/3) and O(N log N) in 3D respectively. Furthermore, the costs in 3D can be reduced to O(N) for setup and O(N) for application by a recursive sweep similar to <cit.>. When combined with the standard GMRES solver, the preconditioner only needs a few iterations to converge, where the iteration number is almost independent of the angular frequency ω as shown by the numerical results. To the best of our knowledge, this is the first algorithm to solve the Lippmann-Schwinger equation in near-linear cost in 3D high frequency cases.Another highlight of this work is the newly designed compact stencil introduced for the preconditioner. The design approach focuses on fitting the stencils to the wave data given by the analytic expressions such as the Green's function. This approach is quite different from the state-of-the-art methods <cit.> to design compact stencils, which focus more on the analytic property of the underlying differential operator. Numerical results show that, when used as a method for solving the Helmholtz equation, this scheme is comparably as accurate as the Quasi-Stabilized FEM (QSFEM) method in <cit.> in terms of the phase error.The rest of the paper is organized as follows. Sections <ref> and <ref> present the preconditioners and the numerical results in 2D and 3D respectively, where the detailed approach is explained in Section <ref> for the 2D case, and Section <ref> generalizes it to 3D with necessary modifications. Section <ref> presents numerical results to show the validity of the compact stencil sparsifying scheme presented in this work when used as a direct method. Conclusions and future work are given in Section <ref>. § PRECONDITIONER IN 2D This section describes the preconditioner for the 2D Lippmann-Schwinger equation. Starting by formalizing the dense linear system obtained from discretization, we transform it into an approximately sparse one by introducing two types of compact stencils. After that, the sweeping factorization is used to solve the truncated sparse system approximately. The whole process can then be treated as a preconditioner for the original dense system of the Lippmann-Schwinger equation. §.§ Problem formulationWithout loss of generality, we assume that Ω = (0,1)^2 and that m(x) is supported in Ω. The task is to discretize the Lippmann-Schwinger equation (<ref>) and solve for u(x) in Ω.The domain Ω is discretized by a uniform Cartesian grid, which allows for the rapid evaluation of the convolution in (<ref>) by FFT. Let n be the number of grid points per unit length, h1/(n+1) be the step size, and Nn^2 be the number of degrees of freedom.Denote i as the 2D index point and p_i as the grid point with step size h byi(i_1, i_2),i_1,i_2 ∈,p_ii h = (i_1 h, i_2 h).Letbe the index set of the grid points in Ω andbe the set of the corresponding grid points, given by{ i = (i_1,i_2) : 1 ≤ i_1, i_2 ≤ n},{ p_i : i ∈}.We also introduceas the index set for Ω̅ and ∂ as the boundary index set by{i = (i_1,i_2) : 0 ≤ i_1, i_2 ≤ n+1},∂∖,and correspondingly we haveand ∂ as{ p_i : i ∈}, ∂{ p_i : i ∈∂}. Let u_i be the numerical solution of (<ref>) at p_i for i∈. To compute the integral in (<ref>), we use the Nyström method∫_Ω G(p_i-y) m(y) u(y)y ≈∑_j ∈ k_i - j m_j u_j,wherem_im(p_i),i ∈,k_iG(p_i)h^2,i(0,0),and k_(0,0) is the weight given by a quadrature correction at the singular point of G(x) at x=0, which achieves O(h^4 log(1/h)^2) accuracy when m(x) is smooth <cit.>. This gives the discretized equationu_i + ω ^2 ∑_j ∈ k_i - j m_j u_j = g_i, i ∈,whereg_i- ω^2 ∑_j ∈ k_i - j m_j [u_I]_j,i∈,and [u_I]_ju_I(p_j) is the discrete value of the incoming wave. Higher order quadrature can be achieved by using more extended local quadrature correction <cit.>.With a slight abuse of the notations, we extend the discrete vectors m and g to the whole 2D grid by zero paddingm_i0, i ∈^2 ∖, g_i0, i ∈^2 ∖.Introducing matrix K with K_i,j k_i-j, (<ref>) can be written into a more compact form(I + ω^2 K M) u = g,where M (m).A subtle difference between (<ref>) and (<ref>) is that, (<ref>) is a set of equations for the unknowns with indices i ∈, while (<ref>) can be regarded as an equation set defined on the infinite index set ^2, where the unknown vector u is also extended to the whole 2D grid with the extension value determined by the equation implicitly. We have two observations for (<ref>) * The solution of (<ref>) agrees with the one of (<ref>) in . To get the numerical solution of (<ref>) in Ω, we can solve (<ref>) and then restrict the solution toinstead of solving (<ref>).* The solution of (<ref>) does not match the numerical solution of (<ref>) outside Ω since the zero padding of g differs from the discretized value of the right-hand side of (<ref>) in Ω^c. Nonetheless, this is not an issue as we only care about the solution of (<ref>) in Ω. One may ask: why do we extend the discrete domain to the infinite grid and consider a problem with infinite size? Besides, the zero padding pattern of g seems rather irrational as it creates discontinuities at ∂. The answer is that, we are not going to actually solve the ^2-size problem. The purpose of extending the unknown to a larger domain is to introduce the wave attenuation by PML on the extended grid to simulate the Sommerfeld radiation condition as we shall see in Section <ref>. The zero padding of g is to ensure that there is no source outside Ω such that the PML approximation holds.The reader may notice that, if we just use the discretized value of the right-hand side of (<ref>) defined on the whole plane, the solution will also satisfy the Sommerfeld condition, so it seems meaningless to introduce the zero padding. It is true that the right-hand side of (<ref>) on the whole plane will induce a solution satisfying the radiation condition. However, in some cases, when solving (<ref>), we are only given g defined inwithout knowing the actual incoming wave u_I, and it's computationally impractical to get the extension of g determined by (<ref>). This is especially true when we develop preconditioners where the input only involves the right-hand side in the domain of interest.With the extended problem (<ref>), we are now ready to build a sparse system to approximate (<ref>). §.§ Sparsification In this section, we adopt the idea of the sparsifying preconditioner <cit.> to build a sparse system which serves as an approximation to (<ref>). The sparse system to be constructed has the same sparsity pattern as a compact stencil scheme, i.e., each equation only involves the unknowns at one grid point and its neighbor points, unlike (<ref>) where each equation is dense in .To be specific, we define μ_i as the neighborhood for the index iμ_i {j : j - i_∞≤ 1}.Now the task is to build for each point i a local stencil supported only in μ_i.We shall build two types of stencils in what follows. The first type is for the interior points, while the second type is for the points near the boundary which are inside what we call “the PML region”.The perfectly matched layer (PML) <cit.> is a technique to attenuate the waves exponentially near the boundary of the domain so that the zero Dirichlet boundary conditions can be imposed directly to simulate the radiation boundary condition without bringing in too much error. We will explain the PML usage during the construction of the second type of the stencils.Let's start with extending our domain Ω. Denote Ω^h as Ω with an h-size extension, Ω^h+η as the PML extension of Ω^h with width η, given byΩ^h(-h, 1+h)^2, Ω^h+η (-h -η, 1+h + η)^2.Here η = bh is the PML width, where b = O(1) is the number of discrete layers in each side of the PML region. η is typically around one wavelength. The PML region Ω^h+η∖Ω^h is where we will attenuate the scattered field u(x) in Section <ref>. Note that there is a small h-distance between the domain of interest Ω and the PML region Ω^h+η∖Ω^h. This small distance is introduced on purpose and the reason will be clear later.The corresponding index sets in these regions are^h {i : 0 ≤ i_1, i_2 ≤ n + 1}, ^h+η{ i : -b ≤ i_1, i_2 ≤ n + 1 + b}.Similar to the notations , ∂, , ∂ and , we introduce ^h, ∂^h, ^h, ^h+η, ∂^h, etc, as the corresponding grid point sets, boundary sets, closures and so on. The meanings are straightforward and we omit the formal definitions. See Figure <ref> for an illustration.We now describe how to design two types of stencils for the unknowns indexed by ^h+η: first for the ones in ^h and then for the ones in ^h+η∖^h. At the end, we assemble them together to form our sparse system.§.§.§ Stencils for the interior points in ^h Following the approach in <cit.>, we design the first type of the stencils for the neighborhood μ_i where i ∈^h (see Figure <ref>, the 3× 3 green grids). Taking out the equations in (<ref>) indexed by μ_i we haveu_i + ω ^2 ∑_j ∈ K_i,j m_j u_j = g_i, i ∈μ_i,which can be written asu_μ_i + ω^2 (K_μ_i,μ_i [m u]_μ_i + K_μ_i,μ_i^c [mu]_μ_i^c) = g_μ_i.Here are some explanations for the notations in (<ref>): * The subscript μ_i stands for the corresponding vector restricted to the index set μ_i, for example, [mu]_μ_i is the vector of the elementwise multiplication of m and u restricted to μ_i.* μ_i^c ∖μ_i, which is the complement of μ_i with respect to .* K_μ_i, μ_i^c is the sub-matrix of K with row index set μ_i and column index set μ_i^c. Let's consider a linear combination of the equations in (<ref>). Suppose α is a column vector supported on μ_i. Multiplying both sides of (<ref>) by α^* givesα^* u_μ_i + ω^2 (α^* K_μ_i,μ_i [m u]_μ_i + α^* K_μ_i,μ_i^c [mu]_μ_i^c) = α^* g_μ_i,where α^* is the conjugate transpose of α.To design a local stencil, we hope that the resulting equation (<ref>) only involves unknowns indexed by μ_i. Observing the left-hand side of (<ref>), we found if α^* K_μ_i,μ_i^c≈ 0, then we can truncate the terms involving u_μ_i^c and the resulting equation will be local. But does there exist an α such that α^* K_μ_i,μ_i^c≈ 0? The answer is yes. The reason is that the elements of K are defined by the Green's function G(x), which satisfies(-Δ - ω^2) G(x) = 0,x ∈^2 ∖{(0,0)}.Each column of the matrix K_μ_i,μ_i^c can be treated as the Green's function centered at some grid point indexed by j ∈μ_i^c and evaluated at the points indexed by the neighborhood μ_i, which does not involve the singular point of G(x) at x= 0. Thus it's reasonable to expect some local stencil α, which can be thought of as a discretization of the local operator (-Δ - ω^2), such that α^* K_μ_i,μ_i^c≈ 0. By the translational invariance of the Green's function, to find such α, it suffices to require that α^* K_μ, μ^c≈ 0, whereμμ_0 = {j : j_∞≤ 1 }, μ^c {i : -n ≤ i_1, i_2 ≤ n}∖μ,which means that we can translate the index i to the origin and consider an equivalent problem. Here the complement of μ is taken with respect to a larger index set. The reason is that, when we translate different indices i to the origin, the corresponding complement μ_i^c will also be translated. The larger set is taken as the union of all those translated complements to ensure that the condition is sufficient.To minimize α^* K_μ, μ^c, we consider the optimization problemmin_α : α_2α^* K_μ, μ^c_2.The solution is the left singular vector corresponding to the smallest singular value of K_μ, μ^c, which can be solved in O(N).Once we have α, we compute β by settingβ^* α^* K_μ, μ.Then (<ref>) can be approximated asα^* u_μ_i + ω^2 β^* [m u]_μ_i≈α^* g_μ_i.This defines the local stencil for each i ∈^h.Note that, if we do the same thing for i ∉^h, the right-hand side α^* g_μ_i will be 0 due to the zero padding of g. If we build the stencils for all i ∈^2 ∖^h and combine them with the Sommerfeld radiation condition at infinity, it will induce a discrete DtN map at ∂^h. This linear map, though existing in theory, is dense and expensive to compute. Section <ref> circumvents this issue by exploiting PML on the extended domain and introducing the second type of stencils to approximate this dense map efficiently.Now why do we introduce the h-size padded domain Ω^h and build the first type of stencils for i ∈^h rather than just for ? The reason is that, for i∈∂, α^* g_μ_i is not necessarily zero, thus we cannot assign ∂ to the second type where the corresponding right-hand side is zero. So we enlarge Ω by h-size and build stencils of the first type for ^h =. Figure <ref> shows this subtlety in 1D as an illustration.§.§.§ Stencils for the PML points in ^h+η∖^h Next, we design the stencils for i ∈^h+η∖^h (see Figure <ref>, the 3× 3 blue grids). Define the auxiliary functionσ(x)- Cω(x + hη)^2, -h - η <x ≤ -h,0, -h < x < 1 + h, Cω(x - 1 -h η)^2, 1+h ≤ x < 1 + h + η,where C ∼Θ(1) is some positive constant. We attenuate the scattered field u(x) in the PML region Ω^h+η∖Ω^h by introducing the complex stretchingx^σ (x^σ_1,x^σ_2) = (x_1 + σ(x_1), x_2 + σ(x_2)),u^σ (x)u(x^σ) = u(x_1 + σ(x_1), x_2 + σ(x_2)),u^σ_iu^σ (p_i) = u(p^σ_i). By changing variable from x to x^σ, we know that the function u^σ(x) satisfies the modified Helmholtz equation in the PML region (-∑_d = 1 ^ 2 (∂_d1+ σ'(x_d))^2 - ω^2 ) u^σ (x) = 0, x ∈Ω^h+η∖Ω^h, A simple way to build local stencils for ^h+η∖^h is to discretize (<ref>) explicitly with some local scheme such as the central difference scheme. Unfortunately, it turns out to be not accurate enough to do so. We adopt a different approach. The idea is similar to what we did in the previous section, where we aim to find some local stencil to annihilate a set of given functions evaluated at the points indexed by μ_i. In what above, we used the Greens function G(x) to design the stencil α. Here we use a set of “modified plane waves” to achieve the same goal.Specifically, we first note that the plane wave functionF(x) exp(ω (r· x)), r_2 = 1,satisfies the free space Helmholtz equation(-Δ - ω^2 ) F(x) = 0, x ∈^2.Let F^σ(x)F(x^σ) be the complex stretching of F(x). We immediately have that F^σ(x) satisfies (<ref>) by definition. If we were to design a local stencil γ for μ_i where i∈^h+η∖^h, we would hope that γ^* F^σ_μ_i≈ 0, where F^σ_μ_i is the function F^σ(x) evaluated at the grid points indexed by μ_i. Note that any direction r such that r_2=1 induces a “modified plane wave” F^σ(x). We hope to solve γ by annihilating as many r as possible. To be precise, Let R be a set of directions where the elements are sampled uniformly from the unit circle {r: r_2=1}, and F^σ_μ_i, R be a matrix of size |μ_i| × |R|, each column of which is a modified plane wave function F^σ(x) with a direction r∈ R, evaluated at the grid points indexed by μ_i. Then we solve γ bymin_γ:γ_2 = 1γ^* F^σ_μ_i, R_2. Intuitively, it's better to increase the sample size |R| to improve the reliability of the stencil. However, larger sample size also leads to more computational cost. Fortunately, it turns out that not too many samples are needed for a reliable result. It suffices to use only the eight most common directions – north, south, west, east, northwest, northeast, southwest and southeast – to form R, and γ is given by the vector perpendicular to the eight corresponding vectors on μ_i. Note that the solution is unique up to a coefficient ± 1 since we have 8 independent modified plain waves and the size of the neighborhood μ_i is 9.In the PML region, we need to compute different stencils for different neighborhoods due to the lack of translational invariance as a result of the complex stretching. Nevertheless, by the symmetry of the stretching, we only need to compute the stencils near a corner of ^h+η∖^h, which takes only O(b^2) work in total. See Figure <ref> for an illustration.We denote γ_i as the stencil for μ_i, then the corresponding approximating equation isγ_i^* u^σ_μ_i≈ 0.This defines the local stencil for each i ∈^h+η∖^h.§.§.§ Assemble togetherAssembling (<ref>) and (<ref>) together and noting that u^σ_i = u_i for i∈^h, we haveα^* u^σ_μ_i + ω^2 β^* [m u^σ]_μ_i≈α^* g_μ_i,i ∈^h, γ_i^* u^σ_μ_i≈ 0, i ∈^h+η∖^h,where α, β, and γ are given in (<ref>), (<ref>) and (<ref>) respectively. Noticing also that u^σ almost satisfies the zero Dirichlet boundary conditionsu^σ_i ≈ 0,i ∈∂^h+η,we can introduce the sparse linear systemα^* ũ_μ_i + ω^2 β^* [m ũ]_μ_i = α^* g_μ_i,i ∈^h, γ_i^* ũ_μ_i = 0, i ∈^h+η∖^h, ũ_i = 0, i ∈∂^h+η.for the unknown ũ defined on ^h+η that serves as an approximation to u^σ. In what follows, we write this system conveniently as H ũ = f,where the right-hand side f is given byf_i α^* g_μ_i,i ∈^h,0,i ∈^h+η∖^h. The system (<ref>) is defined on ^h+η. To get the unknowns on , we simply solve (<ref>) and extract the solutions on . The result is an approximation to the true solution of (<ref>), and this process can serve as a preconditioner for the linear system (<ref>). In the next section, we present an approach for approximating the solution of (<ref>) efficiently by leveraging the idea of the sweeping preconditioner. §.§ Sweeping factorizationIn this section, we adopt the sweeping factorization to solve the sparse system (<ref>) approximately. The main idea of the sweeping factorization is to divide the domain into slices and eliminate the unknowns slice by slice. An auxiliary PML region is introduced for each slice to build a subproblem to approximate the inverse of the Schur complement during the Gaussian elimination to save computational cost.To be specific, we first divide the 2D grid into ℓ slices along the x_1 direction. Each slice contains only a few layers. The leftmost slice contains the left PML region and the rightmost one contains the right PML region (see Figure <ref>). For simplicity, we assume that each of the middle slices contains b layers and each of the two boundary slices contains 2b layers – b normal layers plus b attenuating layers in the PML region. Let _1,…, _ℓ be the discrete points in each slice correspondingly, and define ũ_[i] and f_[i] as the restrictions of ũ and f on _i respectively. The sparse system (<ref>) can be written as the block tridiagonal form[H_[1,1]H_[1,2];H_[2,1]H_[2,2]⋱; ⋱⋱⋱;⋱⋱ H_[ℓ-1, ℓ];H_[ℓ, ℓ-1] H_[ℓ, ℓ] ][ ũ_[1]; ũ_[2]; ⋮; ũ_[ℓ-1]; ũ_[ℓ] ] = [ f_[1]; f_[2]; ⋮; f_[ℓ-1]; f_[ℓ] ],where H_[i,j]'s are the corresponding sparse blocks. Note that we use the bracket subscripts [ · ] to emphasize that the corresponding unknowns are grouped together in each slice.We introduce the Schur complement S_[i] and its inverse T_[i] slice by slice recursivelyS_[1] = H_[1,1],T_[1] = S_[1]^-1,S_[i] = H_[i,i] - H_[i,i-1] T_[i-1] H_[i-1,i],T_[i] = S_[i]^-1, i=2,…, ℓ.Then we can solve ũ by the Gaussian eliminationũ_[1] = T_[1] f_[1], ũ_[i] = T_[i] (f_[i] - H_[i,i-1]ũ_[i-1]), i = 2,…, ℓ, ũ_[i] = ũ_[i] - T_[i] (H_[i,i+1]ũ_[i+1]), i= ℓ-1,…, 1. The expensive part of the above process is to compute T_[i] and apply it to the vectors on _i. If say we formed T_[i] directly, the computation would take O(b^3n^3) steps and the application O(b^2n^2) steps. The sweeping factorization reduces the cost by approximating T_[i] with a subproblem. To introduce the approximation, we first make a key observation of the operator T_[i]: inverting the top left i× i block of H, one notices that T_[i] appears at the bottom right block of the resulting matrix. In other wordsH_[1:i,1:i]^-1= [ H_[1,1] H_[1,2]; H_[2,1] H_[2,2] ⋱; ⋱ ⋱ ⋱; ⋱ ⋱ H_[i-1,i]; H_[i,i-1] H_[i,i] ]^-1 = [ * * … * *; * … * *; ⋮ ⋮ ⋱ ⋮ ⋮; * … * *; * … * T_[i] ].This means T_[i] is the restriction of H_[1:i,1:i]^-1 to _i. Think of T_[i] as an operator from some input vector v to T_[i] v on the grid _i. Then given v, we can compute T_[i] v by solving the equation[ H_[1,1] H_[1,2]; H_[2,1] H_[2,2] ⋱; ⋱ ⋱ ⋱; ⋱ ⋱ H_[i-1,i]; H_[i,i-1] H_[i,i] ][ *;; ⋮;; w ] = [ 0; 0; ⋮; 0; v ]where w is exactly equal to T_[i] v. That is to say, given v, we can find T_[i] v by padding v with zeros on _1:(i-1), solving the unknowns on _1:i by (<ref>) and then extracting the solution on _i.Note that the right-hand side of (<ref>) is zero on _1:(i-1), thus the only role of the first i-1 blocks of equations in (<ref>) is to induce the radiation condition at the left boundary of _i implicitly. To simulate this radiation condition, one can directly put the PML region to the left side of _i instead of putting it far away on _1. That's the key idea of the sweeping factorization: move the PML region adjacent to the domain of interest _i and approximate the operator T_[i] by solving a much smaller system compared to (<ref>) (see Figure <ref>).By introducing the modified plain waves, we can build the local stencils for points in the auxiliary PML region on the left of _i similar to what was done in Section <ref>. A subtle difference is that, the local spacial frequency is perturbed to ω√(1 - m(x)) instead of ω at location x, and we need to use this local frequency to build the local stencil for each point.To save computational cost of the stencil construction, we do not use the exact value of the local frequency. Though building local stencil in the PML region with the exact local frequency takes only constant steps per point in theory, the constant is not small since it involves finding the kernel of a 8× 9 matrix. Instead, we consider the square frequency range:[ω^2(1 - max{m(x)}) ,ω^2(1 - min{m(x)})].We choose some samples uniformly from this range interval, and build local stencils only for those samples. Then for each point in the PML region, we assign the stencil to be the one from the samples with the closest local square frequency value, a technique introduced earlier in <cit.>. In practice, only n samples will be enough for an accurate approximation. So it only takes O(b n) steps to build these stencils, which is negligible compared to the problem size O(n^2). An intuition of why we only need n samples is that, O(ω^2 / n) is the size of the variation in one neighborhood μ_i on average, so there's no need to make the sampling scale smaller than that.With the auxiliary PML region on the left of _i, we can solve a much smaller system instead of solving (<ref>). In our setting, the set of the auxiliary PML points for _i is just _i-1 since the width of the PML region is the same as _i-1. The auxiliary system can be written as[ H̃_[i-1,i-1] H̃_[i-1,i];H_[i,i-1]H_[i,i] ][ *; w ] = [ 0; v ],where the bottom block of equations is inherited from (<ref>), and the top block is defined by the local stencils of the second type in the auxiliary PML region, the role of which is to simulate the radiation boundary condition on the left of _i.A minor problem here is that the auxiliary PML region for _2 consists only the normal layers in _1 rather than all the layers, so (<ref>) needs a slight modification for i = 2: we restrict the columns of H_[2,1] to the normal layers in _1 so that the two blocks are compatible. This problem is inessential and the patch here is only to make the discussion strictly correct. In practice, the width of the slices and the PML regions can be rather flexible.Equation (<ref>) defines an approximating operator T̃_[i]: v → w for i∈ 1…, ℓ by restricting the system (<ref>) on _(i-1):i to _i. Note that for i = 1, T̃_[1] is exactly equal to T_[1] if we treat _0 as ∅ naturally. Compared to (<ref>), Equation (<ref>) is a much smaller quasi-1D problem, which can be solved efficiently with the LU factorization. §.§ Putting togetherWe now have all the tools needed to design a linear-complexity preconditioner for the discretized Lippmann-Schwinger equation (<ref>). The setup and application processes of the preconditioner are given by Algorithms <ref> and <ref> respectively. The slice width b is typically a small integer less than 10, thus both the setup and the application costs are linear.We would like to make some comments below for the actual implementation of the algorithm. * The algorithm presented above constructs the sweeping factorization along the x_1 direction from left to right. Indeed, since we have radiation conditions on all sides of the domain, we can construct the factorization from both sides and sweep toward the middle slice. The two sweeping fronts can be processed independently until they meet in the middle, where they exchange some local information in the middle slice and then sweep back to the boundaries independently. This is potentially helpful for the parallelization of the algorithm.* The widths of the slices and the auxiliary PML regions are completely arbitrary. There are two reasons why we set them to be b uniformly in what above. The first is for the simplicity of discussion. The second is that, given the PML width b, it is optimal to set the width of each slice to be also b to minimize the setup and application costs of the preconditioner. In practice, it may not be possible to uniformly divide the domain where each slice contains b layers exactly. In that case, we change the widths of one or two slices accordingly, which has negligible effect to the cost and efficiency of the preconditioner.* The constructions of the stencils α, β and γ, though depending on n and ω, are essentially independent of the velocity field c(x). First, the computation of α and β only involves the free space Green's function G(x), where the velocity field is completely irrelevant. Next, for the local PML stencils γ, they might depend on c(x) slightly, but only on the range as we see from the sampling process. In practice c(x)=Θ(1), so the range is actually bounded for fixed ω. Thus we can precompute the stencils without given the velocity field. This means that the stencil construction only needs a fixed cost for given problem size, which can be eliminated from the setup process of the algorithm for the input c(x).§.§ Numerical resultsIn this section we present the numerical results in 2D. The algorithm is implemented in MATLAB and the tests are performed on a 2.4-GHz server. We force MATLAB to use only one computational thread to test the sequential time cost. The preconditioner is combined with the standard GMRES solver with relative tolerance 10^-6 and restart value 20. The domain is discretized with h = λ / 8 where λ = 2π / ω is the typical wavelength.We choose b = 8 as the width of the slices and the auxiliary PML regions. This corresponds to about one wavelength width for the PML regions and the slices used in the sweeping preconditioner. The sweeping factorization is built with two fronts sweeping toward the middle slice, and the middle slice is padded with auxiliary PMLs on both sides for the corresponding quasi-1D subproblem.Four velocity fields are tested in 2D, which are *A converging Gaussian centered at (0.5,0.5).*A diverging Gaussian centered at (0.5,0.5).*32 randomly placed converging Gaussians with narrow width.*A random velocity field that is equal to 1 at ∂Ω.The incoming wave u_I(x) for each test is a plane wave shooting downward at frequency ω. The test results are given in Tables <ref>, <ref>, <ref> and <ref> respectively. The notations in the tables are listed as follows. * ω is the angular frequency.* N is the number of unknowns.* T_setup is the setup cost of the preconditioner in seconds.* T_apply is the application cost of the preconditioner in seconds.* N_iter is the iteration number.* T_solve is the solve cost of the preconditioner in seconds. From the numerical tests we observe that both the setup time and the application time scale linearly in N, which are in accordance with the complexity analyses. More importantly, the iteration numbers change only slightly as the problem size grows, almost independent of ω.We notice that the iteration number also depends on the velocity field. For simple fields such as the diverging Gaussian, it requires less iterations compared to more complicated fields such as the narrower converging Gaussians. This makes sense intuitively since converging lenses and velocity fields with drastic local variations increase the oscillations and refractions of the wave field, thus the corresponding systems are harder to solve. In addition, for the sweeping factorization to work well, we need to assume that there are no strong reflections and refractions during the transmission of the waves so that the auxiliary PMLs in the intermediate slices can make correct approximations to the true underlying DtN maps. In practice, moderate amount of wave-ray bendings can be taken care of by a few more iterations as we see in the tests for the multiple diverging Gaussians and the random field. If the velocity field is even worse, for example, if the field has large region of strong discontinuities, then neither will the Nyström method be able to give an accurate discretization scheme, nor can the sweeping factorization provide an accurate approximating solution due to the strong reflections caused by the discontinuities. Thus for our preconditioner to work, we require certain smoothness from the velocity fields. Nonetheless, as we can tell from the numerical examples, the preconditioner works well even when the fields have drastic transitions in narrow regions. So this approach can be widely applied to many use cases.§ PRECONDITIONER IN 3DThis section presents the preconditioner in 3D. As we see from Section <ref>, the approach is essentially dimension independent and it can be easily generalized to 3D. We will keep the description short, mainly emphasizing the differences compared to the 2D case so that the reader can get the central idea effortlessly. The 3D numerical results for both the recursive approach and the non-recursive approach of the sweeping factorization are provided in the second part of this section. §.§ Problem formulation, sparsification and sweeping factorizationIn this section we formulate the approach in 3D. All the notations in 2D can be easily reused without causing any ambiguities. We will keep them unless otherwise stated.We assume Ω=(0,1)^3 contains the support of m(x). The domain is discretized with step size h = O(1/ω) in each dimension. A similar quadrature correction formula is used for the central weight of the Green's function, which gives an accuracy of O(h^4).For the sparsification process, the first type of stencils α and β can be constructed similarly, where now each neighborhood μ_i has 27 points. For the second type of stencils in the PML region, we use the modified plain waves in 3D, defined similarly asF^σ(x) exp(ω (r · x^σ)), r_2 = 1,where now x^σ and r are in ^3, and x^σ is stretched to the complex plane from x for all three coordinates. In 2D, the stencil γ is defined as the kernel vector which annihilates the independent waves shooting toward the eight most common directions. This can be done similarly in 3D. We now need a set of 26 directions, which is defined asR {(r_1,r_2,r_3)√(r_1^2+r_2^2+r_3^2) : (r_1,r_2,r_3) ∈{-1,0, 1}^3 ∖{(0, 0, 0)}}.In other words, these are the directions shooting from the center of a neighborhood to the 26 boundary neighbor points.The computational cost of constructing the stencils in 3D seems higher due to more degrees of freedom and larger size of the neighborhoods. But indeed, the relative cost compared to the sweeping factorization is lower than the 2D case, let alone that the stencil computations are independent of the velocity field and they can be done by a once-in-a-life-time preprocessing.For the sweeping factorization, the domain are now divided into ℓ quasi-2D slices. The auxiliary PMLs are padded to each slice similarly. Each subproblem is quasi-2D, which can be solved efficiently by the nested dissection algorithm with O(b^3n^3) setup cost and O(b^2n^2log n) application cost. Consisting of ℓ≈ n / b subproblems, the whole process has a total setup cost O(b^2n^4) = O(b^2 N^4/3) and application cost O(b n^3 log n) = O(b N log N). Note that the direct use of the nested dissection algorithm to the 3D sparse system costs O(N^2) for setup and O(N^4/3) for solve. The sweeping factorization drastically reduces the costs by dimension reduction.For each of the quasi-2D problem, we can sweep similarly along the x_2 direction, reducing it to ℓ quasi-1D subproblems. This reduces the setup cost to O(b^4 N) and the application cost to O(b^2 N), which are both linear in N, but more sensitive to the slice width b. We call this the recursive approach <cit.> while the one in the previous paragraph as non-recursive. §.§ Numerical resultsIn this section we present the numerical results in 3D. The test configurations are the same as Section <ref> unless otherwise stated. In the 3D tests, we set b = 4 for the slice width and PML width.The four velocity fields tested are *A converging Gaussian centered at (0.5,0.5,0.5).*A diverging Gaussian centered at (0.5,0.5,0.5).*256 randomly placed converging Gaussians ofnarrow width.*A random velocity field that is equal to 1 at ∂Ω.The right-hand side is a plain wave shooting downward at frequency ω.The tests of the non-recursive approach are given in Tables <ref>,<ref>, <ref> and <ref>, and the ones of the recursive approach are in Tables <ref>, <ref>, <ref>, <ref>, where the relative costs compared to the non-recursive approach are also listed as percentages, together with the iteration numbers of the non-recursive ones in the parentheses for the convenience of comparison.From the numerical tests we see that, same as the 2D cases, the iteration numbers remain essentially independent of the problem size. The preconditioner converges in a few iterations for all the test cases. Another highlight is that, the recursive approach requires only zero or one more iteration compared to the non-recursive approach, which means that the recursive sweeping factorization for the quasi-2D linear systems keeps the total approximation error almost at the same level.§ SPARSIFYING SCHEME AS A DIRECT METHODIn this section, we show that the compact stencils acquired by the sparsifying scheme can be viewed as accurate discretizations of the Helmholtz equation. Specifically, we will solve the 2D homogeneous Helmholtz equation with the compact scheme introduced in the sparsification process, and compare it with the Quasi-Stabilized FEM (QSFEM) method in <cit.>. As we shall see from the numerical tests, both methods did comparably well at minimizing the pollution error with only a small number of points per wavelength.Let's consider(-Δ - ω^2) u(x) = f(x),x ∈^2,where f(x) is a delta source centered at (0.5, 0.5). The exact solution is given by the Green's function with a shift of the center (see Figure <ref> for an example).For the sparsifying scheme, we have the discrete equationα^* u_μ_i = β^* f_μ_ifor each of the interior point i, where α and β are 9-point stencils given by (<ref>) and (<ref>) respectively, and f is the discrete delta function.For the QSFEM method, the 9-point stencil for u is given byA = [ A_2 A_1 A_2; A_1 A_0 A_1; A_2 A_1 A_2 ],whereA_0 = 4,A_1 = 2 c_1(κ)s_1(κ) - c_2(κ) s2(κ)c_2(κ) s_2(κ)(c_1(κ)+s_1(κ)) - c_1(κ) s_1(κ) (c_2(κ) + s_2(κ)),A_2 = 2 c_2(κ) + s_2(κ) - c_1(κ) - s1(κ)c_2(κ) s_2(κ)(c_1(κ)+s_1(κ)) - c_1(κ) s_1(κ) (c_2(κ) + s_2(κ)),c_1(κ)cos(κcosπ16),s_1(κ) cos(κsinπ16), c_2(κ)cos(κcos3π16),s_2(κ) cos(κsin3π16),κ ω h,and h is the step size. The right-hand side is the discrete delta function with a scaling.We solve the 2D homogeneous Helmholtz equation (<ref>) and compare the phase errors against the true solution. Specifically, we write the solutions u(x) as u(x) = A(x) e^2πϕ(x) and compare the phase ϕ(x) with the one acquired by the Green's function. The boundary points are discretized by a slowly turning-up PML such that the reflection error is negligible compared to the phase error.Figure <ref> shows the phase errors for a large test case (1024 waves across each dimension) with a small number of points (3 to 5) per wavelength. From the tests we see that the phase error of the sparsifying scheme is comparable to the one of the QSFEM method in <cit.>.We would like to comment that, as pointed out in <cit.>, 2D compact stencils can be optimized to reduce the pollution error, but cannot completely eliminate it. For example, in Figure <ref>, the phase shifts near the four corners are about 1/6 for the 3 p.p.w. test cases, which is not negligible for practical usage. Hence for large problems, one would eventually have to increase the stencil width, or use more points per wavelength.§ CONCLUSIONS AND FUTURE WORKThis paper presents the sparsify-and-sweep preconditioner for the Lippmann-Schwinger equation in 2D and 3D. The preconditioner involves two steps. The first step is to sparsify the system by introducing the compact stencil sparsifying scheme. The second step is to apply the sweeping factorization to the sparsified system. Numerical results show that the iteration number is essentially independent of the angular frequency ω.Though the cost is reduced to linear, potential improvements can be made regarding parallelizations. First, the factorization of the auxiliary subproblems are completely independent, thus can be done in parallel, especially when the recursive approach in 3D is adopted where there are O(n^2) quasi-1D subproblems that can be processed at the same time. Second, the setup and application processes of the nested dissection algorithm can also be parallelized for independent skeleton fronts (see <cit.> for example). Third, the two sweeping fronts during the application process are also independent and can be processed in a parallel way.Another future work is on the sparsification of dense systems by the data-fitting approach. This approach was first proposed by Ying in <cit.> for solving highly indefinite systems including time-independent high frequency wave propagations with radiation conditions or periodic boundary conditions. This paper generalizes it to incorporate the PML approach. There have been some explorations and applications of this sparsification method, such as solving the nonlinear eigenvalue problems in soliton systems <cit.>. This data-fitting approach to design local schemes is quite different from most classical approaches, and could be potentially generalized to other types of integral equations and dense systems.§ ACKNOWLEDGMENTSThe authors are partially supported by the National Science Foundation under award DMS-1521830 and the U.S. Department of Energy's Advanced Scientific Computing Research program under award DE-FC02-13ER26134/DE-SC0009409.abbrv | http://arxiv.org/abs/1705.09443v2 | {
"authors": [
"Fei Liu",
"Lexing Ying"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170526062329",
"title": "Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation"
} |
Center-Excised Luminosity as a Cluster Mass Proxy]Center-Excised X-ray Luminosity as an Efficient Mass Proxy for Future Galaxy Cluster Surveys A. B. Mantz et al.]Adam B. Mantz,^1,2E-mail: mailto:[email protected]@slac.stanford.eduSteven W. Allen,^1,2,3 R. Glenn Morris,^1,3 Anja von der Linden^4^1Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA^2Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA^3SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA94025, USA^4Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA firstpage–lastpage 2017 [ [ Submitted 25 May 2017. Accepted 26 September 2017. ====================================================== The cosmological constraining power of modern galaxy cluster catalogs can be improved by obtaining low-scatter mass proxy measurements for even a small fraction of sources. In the context of large upcoming surveys that will reveal the cluster population down to the group scale and out to high redshifts, efficient strategies for obtaining such mass proxies will be valuable. In this work, we use high-quality weak lensing and X-ray mass estimates for massive clusters in current X-ray selected catalogs to revisit the scaling relations of the projected, center-excised X-ray luminosity (), which previous work suggests correlates tightly with total mass. Our data confirm that this is the case, withhaving an intrinsic scatter at fixed mass comparable to that of gas mass, temperature or . Compared to these other proxies, however,is less susceptible to systematic uncertainties due to background modeling, and can be measured precisely with shorter exposures. This opens up the possibility of usingto estimate masses for large numbers of clusters discovered by new X-ray surveys (e.g. eROSITA) directly from the survey data, as well as for clusters discovered at other wavelengths with relatively short follow-up observations. We describe a simple procedure for making such estimates from X-ray surface brightness data, and comment on the spatial resolution required to apply this method as a function of cluster mass and redshift. We also explore the potential impact ofand XMM-Newton follow-up observations over the next decade on dark energy constraints from new cluster surveys.galaxies: clusters: intracluster medium – X-rays: galaxies: clusters § INTRODUCTIONMeasurements of observable quantities that correlate tightly with total mass, i.e. that function as effective low-scatter proxies, can significantly boost the cosmological constraining power of galaxy cluster surveys (; ). Key considerations are the intrinsic scatter in the mass proxy at fixed true mass and the observing resources required to measure the proxy.Among X-ray mass proxies, total X-ray luminosity is the most straightforward to measure, requiring (in addition to the cluster redshift) only tens of source counts, and comes “for free” from surveys which identify clusters through their X-ray emission. Other proxies such as gas mass and temperature require hundreds to thousands of X-ray source counts, but have a smaller intrinsic scatter than total luminosity. While the relatively large (∼40 per cent) scatter in total X-ray luminosity at fixed mass limits its utility as a mass proxy, it has been recognized that X-ray luminosity measured in an annulus, excluding the cluster center, has a significantly smaller intrinsic scatter, while retaining the attractive simplicity of a luminosity measurement (e.g. ).Physically, the reason why excising cluster centers reduces the intrinsic scatter is easy to understand. The intracluster medium shows great variation in cluster centers, primarily driven by the development of bright, dense “cool cores” of gas, associated with sharp density and surface brightness peaks and reduced temperatures, in a fraction of the cluster population (e.g. ). In contrast, the gas density and temperature profiles outside of cluster centers are remarkably similar (e.g. ).For massive (hot) clusters, the soft X-ray bremsstrahlung emissivity, and the associated K corrections, are nearly independent of temperature. For concreteness, we define the “soft” X-ray band to be 0.1–2.4 keV (rest frame), and “hot” temperatures to mean 4 keV; the important general features here are that bremsstrahlung continuum emission dominates over line emission for a “hot” source, and that the (redshifted) exponential cut-off of the bremsstrahlung spectrum falls outside of the “soft” energy band (see more detailed discussion in Section <ref>). In the appropriate regime, the soft-band surface brightness can thus be considered a relatively simple function of the gas density, with similarity in density profiles outside cluster centers translating into a small intrinsic scatter in both center-excised soft-band luminosity (hereafter ) and integrated gas mass (). In this context, a key difference betweenandis that the former is a straightforward integral of the projected, K-corrected surface brightness in a given annulus, which is dominated by the region where the brightness is highest, usually the smallest radii in the annulus of integration. In contrast, the gas mass within a sphere isa volume-weighted integral, and is thus dominated by the largest radii in the integration, where the signal-to-noise is typically lowest. Furthermore, estimatingwithin a sphere requires knowledge of the gas profile at yet larger radii, so that projected emission can be accounted for.While center-excised temperature measurements have become common for mass estimation (e.g. ), the same cannot be said for . Yet, the existing evidence points tobeing potentially a useful, not to mention relatively cheap, mass proxy. To the extent that density profiles outside of cluster centers are self-similar, the intrinsic scatter inshould be comparable to that ofand temperature (∼10–15 per cent). For perfectly similar profiles measured with high signal-to-noise, it would provide identical information to . In practice, using a signal that is most sensitive to relatively small radii (while still excluding the core) is an advantage, because the impact of the X-ray background and the likelihood of having limited azimuthal coverage of the cluster (at relatively low redshifts) both increase with radius. Similar comments apply to the comparison ofand temperature, the latter being more sensitive to background modeling than either luminosity or gas mass. In addition, cross-calibration studies indicate good agreement in soft X-ray flux measurements betweenand XMM-Newton, whereas temperature measurements for hot clusters with these two workhorse telescopes remain discrepant <cit.>.Motivated by these considerations, we revisit in this work the –mass scaling relation, using a relatively large sample of massive clusters for which we have high quality X-ray and/or weak gravitational lensing mass estimates. Section <ref> describes these data, while Section <ref> presents the scaling relations linkingto gas mass and total mass, as probed by weak lensing. In Section <ref>, we outline the procedure for applying the –mass relation to obtain mass estimates of new clusters, address the applicability of(and center-excised measurements in general) in the context of the spatial resolution provided by current and planned X-ray observatories, and discuss the potential ofobserving programs to boost cosmological constraints from clusters. We conclude in Section <ref>.We define the characteristic radius and mass of a cluster in the conventional way, with respect to the critical density at its redshift,M_Δ = 4π/3Δ(z) r_Δ^3,with Δ=500. For brevity of notation, we will forgo subscripts “500” for mass, , etc., but all quantities measured in this work are referenced to r_500. We assume a flatcosmology with parameters H_0=70^-1^-1 and =0.3 throughout.§ DATAOur data set consists of 139 clusters withX-ray follow-up observations, originally selected from the ROSAT All-Sky Survey (RASS). Thedata, occasionally supplemented by ROSAT PSPC surface brightness information, are used to extract integrated observables such as gas mass, average temperature, and total X-ray luminosity, which are known to correlate with total mass, with different amounts of intrinsic scatter. The analysis of these X-ray data, and their scaling relations with mass, is described in detail in <cit.>. Here we consider an additional mass proxy, . To be precise, this refers to the intrinsic (unabsorbed), rest-frame 0.1–2.4 keV band[The exact choice of energy band is somewhat arbitrary. While the 0.1–2.4 keV band provides continuity with luminosity measurements going back to the RASS, similar bands that have been used in the literature are in principle equally useful for our purposes. In our results (Table <ref>), we also provide scaling relations appropriate for the commonly used 0.5–2.0 keV band.] X-ray luminosity of a cluster, projected within an annular aperture with an inner radius of 0.15 r_500 and an outer radius of r_500. The particular choice of inner radius is intended to comfortably exclude the variability observed in cluster centers (e.g. ); we comment on the use of larger excluded regions in Section <ref>.In addition, we use mass estimates from weak gravitational lensing data for 50 clusters from the Weighing the Giants project. The analysis of the lensing data, extraction of mass estimates, and accounting for systematic uncertainties are discussed by <cit.>, <cit.> and <cit.>.To determine r_500 for each cluster, we adopt a fiducial value of the integrated gas mass fraction at this radius, (<r_500)/M(<r_500) = 0.125, based on the –M relation from . Statistical uncertainties on the gas mass profiles, and measurement correlations among the various observables, are propagated through this procedure to our final results. This includes the correlation induced directly by the determination of and uncertainty in r_500, which determines the measurement aperture for each quantity. § SCALING RELATIONS OF CENTER-EXCISED LUMINOSITYOur data allow us two routes to constraining the –mass relation. On one hand, we have bothandmeasurements for >100 clusters.showed thatand M (calibrated from weak lensing) are tightly correlated, with a power-law slope very near unity, M∝^1.007±0.012, for the clusters in this sample. Simulations predict the intrinsic scatter in the –M relation to be small, 10 per cent <cit.>, a feature which has been explicitly confirmed for massive and dynamically relaxed clusters, where precise total masses can also be derived from X-ray data <cit.>. We can, therefore, use the abundant gas mass measurements to fit an – relation (accounting for the correlation in measurement uncertainties), and then straightforwardly translate these constraints to an estimate of the –M relation. One non-trivial aspect of this approach is the indication, from relaxed clusters, of a strong correlation in the intrinsic scatters ofandat fixed mass <cit.>; this is a direct consequence of the near equivalence ofandfor self-similar clusters. While this will not affect the slope of the inferred –M relation, it does mean that any measured intrinsic scatter in | may underestimate the intrinsic scatter in |M.On the other hand, we have, for a smaller number of clusters, measurements of both the total mass from weak lensing and . The statistical uncertainties on individual weak lensing mass estimates are relatively large compared to those of , as is the expected intrinsic scatter between lensing-derived mass and true mass <cit.>. Nevertheless, careful weak lensing mass estimates represent the current gold standard of absolute cluster mass calibration (i.e., accuracy in the average). Our approach in this section is, therefore, to fit the – scaling relation, as described above, and then to demonstrate the consistency of this relation with the weak lensing mass estimates. The expected self-similar scaling of , following <cit.>, has the form∝ E(z)^2+α β_tM^1+α β_t∝ E(z)^2+α β_t ^1+α β_t,where E(z)=H(z)/H_0. Here β_t is the power-law slope of the temperature–mass relation (from , β_t=0.62±0.04, consistent with the self-similar value of 2/3), and α≈-0.13 is the intrinsic temperature dependence of bremsstrahlung emissivity, accurate for temperatures 3, accounting for the limited energy band in our definition of luminosity (rest-frame 0.1–2.4 keV). Our baseline expectation can thus be written in a simpler form,E(z)^-1∝[E(z) M]^γ,with γ=1+α β_t≈ 0.92 for β_t=2/3. The center-excised luminosity, gas mass and lensing mass data are shown in terms of this scaling in Figures <ref> and <ref>, although we consider a more general dependence on redshift and mass below.We fit scaling relations to the data using a model that includes a log-normal intrinsic scatter (i.e. a Gaussian distribution in ln), accounts for the correlation between measurement uncertainties, and marginalizes over a log-normal distribution of the covariates <cit.>. This methodology does not explicitly account for selection effects (e.g. Malmquist bias), but because the underlying data set is X-ray flux limited, with the majority of the intrinsic scatter in total X-ray luminosity being due to the core region that is excluded from<cit.>, we do not expect selection effects to significantly impact our results. Fitting the simpler form of the – relation from Equation <ref>, we findE(z)^-1/10^45^-1 = e^(-1.081±0.010)[E(z) /10^14]^0.979±0.019.Note that the departure of the power-law slope from the self-similar baseline is in the same sense (steeper), but smaller, than that of the center-included luminosity scaling derived from the same cluster sample (). The log-normal intrinsic scatter is consistent with zero, with a 95.4 per cent confidence upper limit of 0.04. Figure <ref> compares this best fit, and its associated 68.3 per cent confidence predictive interval, with the data, and highlights the dramatic reduction in scatter resulting from the center excision. Allowing separate power-law dependences ofon E(z) and , ∝ E(z)^γ_1^γ_2, we find/10^45^-1 =e^(-0.878±0.011)[E(z)/E(0.2)]^(2.03±0.19) ×[/10^14]^0.977±0.022.This is consistent with the simpler scaling in Equation <ref>; in particular, γ_1-γ_2 = 1.06±0.20.Using the –M relation from , we can translate Equation <ref> to a scaling relation betweenand total mass:E(z)^-1/10^45^-1 = e^(-0.86±0.04)[E(z) M_500/10^15]^0.99±0.02.Figure <ref> compares this relation with X-ray measurements ofand weak lensing mass estimates; note that the predictive interval shown accounts for a log-normal intrinsic scatter of 0.17 in lensing masses (). To maximize the number of clusters in the comparison, we include those which have mass estimates from <cit.>, even if they are not in the X-ray flux limited sample used to fit the – relation. Whether or not one makes this distinction, the scaling relation in Equation <ref> is in excellent agreement with the measured weak lensing masses.While we can directly fit for the intrinsic scatter inat fixed , inferring the scatter inat fixed total mass is more complex. For massive, relaxed clusters where total masses can be estimated from hydrostatic equilibrium, <cit.> found a strong correlation in the joint scatters ofandat fixed mass (assuming a bivariate log-normal form of the scatter), with ρ=0.88^+0.06_-0.16; the same study found an –M power-law slope of 1.02±0.09, in excellent agreement with our current results. If the strongly correlated scatter holds for the entire cluster population, our measured scatter in | will be smaller than the scatter in |M. Indeed, the same study of relaxed clusters found a marginal log-normal intrinsic scatter in |M of 0.17±0.05. If we fit for the intrinsic scatter about the best-fitting –M relation using the weak lensing data (approximating the statistical uncertainties in the lensing masses as log-normal) the resulting scatter estimate is 0.15 ± 0.04. Both of these scatter estimates contain contributions from other sources: the former from the unknown scatter in hydrostatic mass estimates for relaxed clusters, and the latter from scatter in weak lensing mass estimates due to correlated large scale structure (estimated to be 0.17±0.06; , see also ). Taken together, these lines of evidence cannot straightforwardly put a direct constraint on the intrinsic scatter in |M, but support a value in the range 0.1–0.2, with the lower bound given approximately by the scatter in |M and the upper bound by the estimates from hydrostatic and weak-lensing masses.Note that the simple power-law behavior expected for(Equation <ref>) depends on the cluster temperature being 3 keV. At cooler temperatures (lower masses), as the luminosity in soft energy band progressively approaches the bolometric luminosity, we would expect a change in slope of the –M relation, although not necessarily an increase in scatter. Use of the above scaling relations in this regime, which is in any case an extrapolation beyond the data set employed here, is therefore discouraged. At even lower temperatures of 2 keV, we might also expect the scaling relations to become more complex due to the appearance of strong emission lines in the soft band, in addition to the relatively greater role of complex feedback processes in these less massive systems. Given the challenge of obtaining individual mass measurements at low masses, the calibration of(and mass proxies in general) in this regime remains an subject for future exploration. We discuss these and other practical considerations further in Section <ref>.For completeness, we also consider luminosity measured in an annulus spanning radii of (0.15–1) E(z)^-2/3 Mpc. We denote this luminosity , since <cit.> suggested this annulus definition as one that approximates the redshift dependence of r_500 but lacks the dependence on mass. <cit.> found that using this annulus, as opposed to (0.15–1)r_500, had negligible effect on the inferred scatter in the center-excised luminosity– relation. In the case of the – relation, we find a scatter of 0.106±0.015. The difference between this result and the much smaller scatter obtained for the – annulus is consistent with our expectation of a strong correlation betweenandat fixed mass (see above) due to the similarity of gas density profiles at the relevant radii, which the definition ofdoes not take advantage of (though it does eliminate the scatter in luminosity due to cool cores). Note that this scatter constraint is not at odds with the results of <cit.>, given the relative sizes of the scatters inandfound by .§ DISCUSSION §.§ Usingto Estimate Mass In this section, we describe a straightforward procedure for estimating r_500 and M from X-ray data, given an –M scaling relation. We do not explicitly address statistical uncertainties in the scaling relation or the measured luminosities, but instead recommend propagating these by repeating the procedure for many monte-carlo realizations of each.The starting point for a mass estimate is an X-ray surface brightness profile, ideally in a relatively soft energy band. We use the observer-frame 0.6–2.0 keV band, which is typical in the literature. To use Equation <ref>, the surface brightness must be corrected for Galactic absorption and converted to intrinsic (rest-frame) 0.1–2.4 keV band luminosity per unit solid angle. Both of these steps are, strictly speaking, temperature-dependent spectral corrections. However, for clusters in the mass and redshift range of our sample (i.e. where the –M relation has been calibrated), the temperature dependence is small enough to ignore, typically. In a more thorough treatment, one could also straightforwardly make use of an empirical temperature estimate (if available), or consistently incorporate the temperature–mass relation from .At this point, it is straightforward to generate a projected aperture luminosity profile like the example shown in Figure <ref>. As a function of radius, r, this profile is nothing more than the surface brightness (in luminosity units) integrated between radii of 0.15 r and r. While the qualitative shape of the profile in Figure <ref> is typical, we note that there is great variety in the aperture luminosity profiles across our sample, mostly driven by variety in the central surface brightness of the clusters. In other words, despite the self similarity of density profiles (outside cluster centers) and of the –M relation, there is not a widely applicable scaled profile of aperture luminosity as a function of r/r_500, as there is for, e.g., . In extreme cases, for very diffuse clusters with shallow central surface brightness profiles, the aperture luminosity can even be flat or increasing at r_500.An estimate for r_500 (and hence M) is obtained by finding the intersection between the aperture luminosity profile and aperture luminosity predicted by the fiducial –M scaling relation (the dashed line shown in Figure <ref>). Combining Equations <ref> and <ref>, we arrive at the implicit equation=L(r_500)-L(0.15 r_500)=A E(z) [2π E(z)(z)/312r_500^3]^B,where A=(4.23±0.17)44^-1 and B=0.99±0.02.The original r_500 estimates for our sample were arrived at by following an analogous procedure, using the measured gas mass profiles and the nominal –M scaling relation from . Comparing the nominal r_500 estimates from the two techniques (that is, using the mean scaling relations and measured profiles, without worrying about statistical or parameter uncertainties, or intrinsic scatter), we find a ratio of 0.99±0.03 (average and standard deviation). Radius and mass estimates fromare thus highly consistent with those from , as the tight scaling relation in Figure <ref> requires. §.§ Impact of the Telescope ResolutionFrom a practical standpoint, one of the essential differences between usingandas mass proxies is that the former is mostly determined by emission at relatively small projected radii, while the latter is most heavily influenced by the measured surface brightness at larger radii (∼ r_500) and requires relatively complex modeling (deprojection). This is an advantage forin some respects, namely the exposure time required for a simple mass estimate and the ability to access the most important radii at all azimuths for relatively low-redshift, massive clusters in a single field of view. For distant or less massive clusters, however, we need to consider whether a given telescope's point spread function (PSF) can reliably permit the central portion of the cluster to be excised. If not, photons leaked from cluster centers can be expected to increase the intrinsic scatter inmass estimates. There are other practical implications of PSF smoothing that we do not explicitly consider here, most notably the greater challenge of identifying and masking contaminating point sources in lower resolution images.To estimate the impact of the PSF, we consider as an extreme case the cool core cluster hosting 3C 186, an X-ray bright AGN in the central galaxy <cit.>. At a redshift of 1.06, this is one of the most centrally X-ray peaked massive clusters known at high redshifts. Based on the originalimage, we simulate a suite ofmeasurements of morphologically equivalent clusters with different redshifts and masses, observed with various PSF sizes, and with various fractions of r_500 excised. In general, the PSF was assumed to be Gaussian in shape; however, we verified that a Gaussian+beta model approximation to the real XMM PSF produces very similar results in this case to a Gaussian with the same half energy width (HEW). For centrally peaked sources such as 3C 186, we find, intuitively, that the fractional contamination of the center-excised luminosity depends on the ratio of the PSF width to the size of the excised region, and not on the cluster redshift or mass explicitly. For this test case, we find that the ratio HEW/(xθ_500) must be <0.35 (0.40) to limit the contamination of themeasurement to 5 (10) per cent, where xθ_500 is the inner angular radius of themeasurement annulus (i.e., x is the fraction of r_500 excised). Note that these limiting values are a function of cluster morphology, and in particular are larger (less restrictive) for less extremely peaked systems.At redshifts 1 z2, where the angular diameter distance changes relatively slowly with z, a maximum allowed level of contamination translates to a maximum PSF size, with little residual dependence on cluster redshift. If we adopt a limit of 5 per cent contamination for a 3C 186-like cluster (small compared with the expected intrinsic scatter ofwith mass), then our chosen excision size of 0.15 r_500 requires a HEW 5”. Currently, such fine resolution isonly provided by(HEW<0.5”). To meet our contamination requirement with XMM (HEW≈15”) over this redshift range would require us to raise the excision radius to x=0.5. This comes at the cost of excluding a significant fraction (∼75 per cent) of the flux in the 0.15–1 r_500 annulus for our toy model, although the larger effective area of XMM compared toeffectively compensates for this; for more typical (less strongly peaked) clusters, XMM would have an advantage in terms of the required observing time even using this larger excision region. In the context of future X-ray observatories, the most relevant PSF sizes are those of eROSITA (HEW∼15” for pointed observations, ∼30” when scanning),(HEW∼5”), and(HEW<1”).Figure <ref> shows the angular radii corresponding to r_500 for our sample, which comprises the most massive clusters at redshifts <0.5, as well as for the most massive South Pole Telescope (SPT) clusters out to higher z <cit.>. Also shown are curves corresponding to masses of M_500=10^13, 10^14 and 10^15. Dashed lines correspond to the minimum cluster θ_500 accesible (according to the requirement above) to a telescope with a HEW of 5” or 15” (-like versus XMM/eROSITA-like) with an excision of x=0.3, while dot-dashed lines show the same for an excision of x=0.5. In the context of z>1 clusters specifically, XMM and eROSITA will be able to measurefor massive clusters by employing anexcision radius of x∼0.5.will have access to a larger range in mass, even with a more modest excised region. Very high resolution observatories such asandare essentially unrestricted. At lower redshifts, especially z0.5, all the observatories considered here can access a wide range in mass. Given the significant practical advantages of being able to use XMM, and eventually eROSITA and , formeasurements at high redshifts, we list in Table <ref> scaling relation parameters corresponding to excision regions of x=0.3 and 0.5, in addition to 0.15. The table also includes fits where luminosity corresponds to the alternative but commonly used rest-frame band of 0.5–2.0 keV. §.§ Regime of ApplicabilitySeveral considerations impact the regime of redshift, temperature, and mass where we can expect the scaling relations given in Section <ref> and Table <ref> to provide a good description of clusters. To begin with, the scaling relations reported here are calibrated using high quality X-ray and weak lensing observations of clusters with masses M_500 314 (temperatures 4) at redshifts <0.5. The good agreement with relaxed clusters of similar masses out to z∼1.1, for which we have reliable hydrostatic mass estimates <cit.>, suggests that the scaling relations remain valid for massive clusters at least out to these redshifts. To extend their use to even higher redshifts and/or lower masses, additional gas mass and/or total mass measurements should be obtained in order to verify that the scaling relations remain valid.More broadly, we can ask in what regime we theoretically expect the –M relation to remain a simple power law with small intrinsic scatter. One key requirement is the approximate self-similarity of gas density profiles within the luminosity measurement aperture. This property has been repeatedly verified for massive clusters over a wide range in redshifts. Going down to the group scale, we would generally expect the sphere of influence of a central AGN to be larger relative to r_500, potentially altering the mean scaling relation and sourcing additional intrinsic scatter; however, measurements of a constant gas mass fraction between r_2500 and r_500 (≈0.5–1 r_500) by <cit.> suggest thatmay remain well behaved for a sufficiently large excision radius.Another requirement for the scaling relation to remain simple is that the emission in the band whereis defined be dominated by bremsstrahlung. At temperatures 2, line emission plays an increasingly important role. In principle, this contribution is predictable, and one could simply alter the power law expectation accordingly. However, the dependence on a potentially complex metallicity structure in poor clusters and groups likely introduces a significant scatter.A related consideration at high redshifts and/or low masses (temperatures) is the ease of converting measured count rates in the observer-frame to intrinsic luminosity in the cluster rest frame. For kT/(1+z)>3, the temperature dependence of K corrections in the soft X-ray bands generally used is small, making relatively shallow observations to obtaina viable option for mass estimation. At sufficiently low temperatures and/or high redshifts, however, the bremsstrahlung spectral cut-off will be present within the observer-frame energy band used to measure fluxes, at which point the temperature dependence of the K corrections can no longer be neglected. (At low temperatures, line emission provides another source of temperature dependence in the K correction.) Calculation ofin this regime thus requires observations to be deep enough to measure temperature, which provides a mass proxy in its own right. Depending on the cluster redshift and the instrumental background, the required observations could also potentially provide gas mass measurements. It remains to be seen whetheror one of the other mass proxies, or some combination of them <cit.> is most efficient in this regime. §.§ Exploiting New Cluster SurveysIn the coming decade, new cluster surveys at X-ray, optical/NIR and mm wavelengths will vastly expand the population of clusters known. To obtain the tightest possible cosmological constraints from these new catalogs, additional data will be required to set the absolute cluster mass calibration (e.g. through weak lensing) and also to provide information on the evolving shape of the mass function through precise relative mass measurements. The latter task is where mass proxies such ascan contribute, provided that their scaling relations with mass and redshift are understood. In this context, a low-scatter mass proxy that is straightforward to measure from relatively short X-ray observations is potentially invaluable. For new X-ray surveys such as eROSITA,can be estimated directly from survey data for many more clusters than other X-ray mass proxies such asand temperature; by the same token, follow-up observations of clusters discovered in mm-wavelength and optical surveys can be shorter if the target is to measurerather than the other proxies. For Sunyaev-Zel'dovich (SZ) effect surveys especially, newly discovered clusters will tend to have high temperatures, making them well suited to mass estimates based on(Section <ref>).To provide a rough quantitative estimate of the impact that follow-up observations with , XMM and/or future X-ray telescopes might have, we use the Fisher forecasting code of <cit.>. The fiducial survey we consider has a mass limit of 214 and a redshift range of 0.0<z<1.5 and covers 2000 sq. deg., finding ∼5500 clusters. This design is approximately modelled after the union of RASS and the SPT-3G surveys, but since the forecasted improvement due to follow-up data is not very sensitive to the details, the results are more broadly applicable. A limitation that should be noted, however, is that this forecasting code assumes that the survey observable scaling relation is modeled by a power-law in mass and redshift with constant intrinsic scatter. In practice, more flexible scaling models are likely to be applied to future large surveys, in which case our simple forecasts will underestimate the value of auxiliary mass proxy information.Figure <ref> shows predicted improvements in the dark energy figure of merit <cit.> as a function of the number of follow-up mass proxy measurements with 15 per cent intrinsic scatter. The upper edge of the shaded region corresponds to follow-up targets that are chosen to optimize the FoM (assuming a power-law form for the scaling relation and its evolution; see ), while the lower edge corresponds to representative follow-up of the survey. In the optimized case, roughly half of the targets are relatively low-redshift, high-mass clusters of the kind that already populate theand XMM archives and will have high signal-to-noise in the eROSITA survey; hence, new observations would focus on the relatively high-redshift and low-mass clusters. Adopting a target of 100 source counts in the 0.15–1 r_500 aperture, and accounting for the mass and redshift distribution of follow-up targets, the -equivalent exposure time required is approximately 1 Ms per 50 new cluster observations, with most individual exposures in the 10–30 ks range.Arguably the most beneficial follow-up strategy in terms of the potential for discovery is not one that is optimized assuming a particular model for the cosmology and scaling relations, as above, but rather one that spreads follow-up observations throughout the interesting redshift and mass range. The FoM improvement per target for such an “evenly sampled” program would lie between the extremes represented by the optimal and representative follow-up cases, i.e. order of magnitude improvements in the FoM for ∼100–500 mass proxy measurements in total. In this context, we note that the exposure time required per 50 clusters uniformly distributed in redshift and log(M), for 0.3<z<1.5 and 10^14<M/M_⊙<614 (i.e., the regime not well represented in archival data), is again ∼1 Ms with , similar to the optimized case above. The rough scale of a follow-up program like the one outlined here is thus not too sensitive to the choice of targets, with-equivalent investments of 3–5 Ms potentially producing order of magitude improvements in the FoM with respect to no follow-up for the fiducial SPT-3G case, scaling to ∼10 Ms spread over the next decade to enable more ambitious surveys such as CMB-S4.Inevitably, relatively short observations that are optimized to measurewill produce less auxiliary astrophysics per observation than the deeper exposures that have been the norm to date. Nevertheless, there is science beyond cosmology that such observations would enable, such as the evolution of cool cores (as identified by surface brightness), cluster morphologies, and the identification of active galactic nuclei (AGN) in and around clusters (given sufficient spatial resolution e.g. ). Given that the clusters that new surveys will uncover are, naturally, X-ray fainter than those that have already been studied, it is reasonable that the first systematic forays into this new regime be focussed primarily on such basic measurements, with interesting candidates for deeper exposures being identified from these initial observations. In practice, deeper observations of a subset of clusters would be desirable in any case, to verify that scaling relations among various X-ray observables remain well behaved, as well as to exploit additional cosmological constraints from the gas mass fractions of the most dynamically relaxed clusters identified <cit.>. Together, these considerations suggest that, in the near term, the most efficient follow-up strategy will utilize the complementary strengths of bothand XMM. Relatively shortobservations can efficiently constrain AGN populations and cluster morphologies, and measure(all requiring roughly equivalent exposure time). For clusters that are especially interesting in their own right, and for a subset where additional mass proxies are used to test thescaling relations, XMM observations (benefitting from theconstraints on AGN contamination) can then provide deeper imaging and more detailed spectroscopy.In this section, we have focussed on the potential application ofas a mass proxy for upcoming SZ surveys, but it can potentially also benefit cosmology with optically selected cluster samples, such as the Dark Energy Survey. However, because the bulk of optically selected clusters are lower in mass than the sample considered in this work, the first step in this case is to extend the calibration of the –M relation to somewhat lower masses. In the longer term, follow-up programs analogous the one described above, but targeting the much larger sample of fainter clusters discovered by LSST, Euclid and CMB-S4, could be enabled by upcoming X-ray facilities such as Athena and Lynx. § CONCLUSIONWe present constraints on the scaling relations of the projected, center-excised X-ray luminosity, using a large sample of massive galaxy clusters with X-ray and weak lensing mass estimates. Our analysis confirms earlier indications thatcorrelates tightly with mass in the mass and redshift range probed by our data set (M_500≥314, z≤0.5), with an intrinsic scatter of 15 per cent. We outline a straightforward procedure for estimating masses using this scaling relation, which requires only a cluster redshift, standard manipulations of X-ray surface brightness measurements, and the solution of an implicit equation. We comment on the spatial resolution required to take advantage ofas a mass proxy for particularly high-redshift clusters. This is especially apt in the context of new X-ray surveys such as eROSITA, sincecan be estimated directly from survey data for many more clusters than more expensive low-scatter mass proxies such asand temperature, and for upcoming mm-wavelength surveys that will discover large numbers of high-redshift clusters. While conventionallyhas been measured in an aperture of 0.15–1 r_500, the scatter of center-excised luminosity at fixed mass remains small with significantly more generous center excisions (e.g. 0.3–1 or 0.5–1 r_500), making these observations feasible for telescopes with HEWs of >10” such as XMM and eROSITA.The relative inexpensiveness ofcompared with other X-ray mass proxies opens up the possibility of following up large numbers of clusters discovered by upcoming X-ray, optical and mm-wavelength surveys, with higher redshifts and/or lower masses than the majority of clusters in current archival data. From Fisher matrix calculations, we estimate that order of magnitude improvements in dark energy constraints from upcoming cluster catalogs are possible by investing the equivalent of ∼3–5 Ms oftime to follow up hundreds of clusters, ideally with a comparable investment by XMM providing deeper exposures for a subset of targets. On the scale of the next decade, this represents a significant but reasonable and practical investment; indeed, our most recent cluster cosmology work, using the most massive X-ray selected clusters at z<0.5 (; ), employed almost 10 Ms of archivaldata, all of which has also been used in multiple astrophysical studies (and would be leveraged again in the work described here). While relatively shallow observations aimed at measuringdo not provide very detailed thermodynamic information, they do nevertheless enable a subset of interesting and important astrophysical investigations, and could be used as a first pass to identify targets for deeper observations. 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H., 2010, http://adsabs.harvard.edu/abs/2010ApJ...713.1207W, 713, 1207[Zhang et al.Zhang et al.2007]Zhang0702739 Zhang Y.-Y., Finoguenov A., Böhringer H., Kneib J.-P., Smith G. P., Czoske O.,Soucail G., 2007, http://adsabs.harvard.edu/abs/2007A 467, 437 | http://arxiv.org/abs/1705.09329v2 | {
"authors": [
"Adam B. Mantz",
"Steven W. Allen",
"R. Glenn Morris",
"Anja von der Linden"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170525185947",
"title": "Center-Excised X-ray Luminosity as an Efficient Mass Proxy for Future Galaxy Cluster Surveys"
} |
Unsupervised Feature Learning for Writer Identification and Writer Retrieval Vincent Christlein1, Martin Gropp1, Stefan Fiel2, and Andreas Maier1 1 Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg,91058 Erlangen, Germany 2 Computer Vision Lab, TU Wien, 1040 Vienna, Austria, [email protected], [email protected], [email protected], [email protected] ========================================================================================================================================================================================================================================================================================================================================In the present paper, we consider large-scaledifferential Lyapunov matrixequations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is based on the integral expression of the exact solution and an approximation method forthe computation of the exponential of a matrix times a block of vectors. In the second approach, we firstproject the initial problem onto a block (or extended block)Krylov subspace and get a low-dimensional differential Lyapunov matrixequation. The latter differential matrix problem is then solved by the Backward Differentiation Formula method (BDF) and the obtained solution is used to build the low rank approximate solution of the original problem. The process being repeated until some prescribed accuracy is achieved. We give some newtheoretical results and present some numerical experiments.Extended block Krylov; Low rank; Differential Lyapunov equations.65F10, 65F30 myheadings plain M. Hached and K. JbilouLow rank approximate solutions ...§ INTRODUCTION In the present paper, we consider thedifferential Lyapunov matrixequation (DLE in short) of the form {[ Ẋ(t)=A(t) X(t)+X(t) A^T(t)+B(t)B(t)^T;(DLE); X(t_0)=X_0, t ∈ [t_0, T_f], ].where the matrix A(t) ∈ℝ^n × n is assumed to be nonsingularandB(t) ∈ℝ^n × sisa full rank matrix, with s ≪ n.The initial condition X_0 is assumed to be a symmetric and positive low-rank given matrix. Differential Lyapunovequations play a fundamental role in many areas such as control, filter design theory, model reduction problems, differential equations and robust control problems <cit.>. For thoseapplications, the matrixA is generally sparseand very large. For such problems, onlya few attempts have been made to solve (<ref>).Let us first recall the following theoretical result which gives an expression of the exact solution of (<ref>).<cit.> The unique solution of thegeneral Lyapunov differential equationẊ(t)=A(t) X+X A(t)^T+M(t); X(t_0)=X_0is defined byX(t) = Φ_A(t,t_0)X_0Φ^T_A(t,t_0)+∫_t_0^t Φ_A(t,τ)M(τ)Φ^T_A(t,τ)dτ.where the transition matrix Φ_A(t,t_0) is the unique solution to the problemΦ̇_A(t,t_0)=A(t) Φ_A(t,t_0),Φ_A(t_0,t_0)=I.Futhermore, if A is assumed to be a constant matrix, then we haveX(t)=e^(t-t_0)AX_0e^(t-t_0)A^T+∫_t_0^t e^(t-τ)AM(τ)e^(t-τ)A^Tdτ. We notice that the problem (<ref>) is equivalent to the linear ordinary differential equation{[ẋ(t) = 𝒜(t)x(t)+b(t); x_0 =vec(X_0) ].where 𝒜= I ⊗ A(t) + A(t) ⊗ I, x(t)=vec(X(t)) and b(t) = vec(B(t)B(t)^T), where vec(Z) is the longvector obtained by stacking the columns of the matrix Z. For moderate size problems, it is then possible to use an integration method to solve(<ref>). However, this approach is not adapted to large problems. In the present paper, we will consider projection methods onto extended block Krylov (or block Krylov if A is not invertible) subspacesassociated to the pair (A,B). These subspaces are defined as follows_m(A,B)= range(B,AB,…,A^m-1B)for block Krylov subspaces, orK_m(A,B)= range(A^-m,…,A^-1B,B,AB,…,A^m-1B)for extended block Krylov subspaces. Notice thatthe extended Krylov subspace K_k(A,B) is a sum of two block Krylov subspaces K_m(A,B)=_m(A,B)+_m(A^-1,A^-1B).To compute an orthonormal basis {V_1,…,V_m},where V_i is of dimension n× s for the block Krylov and n × 2s in the extendedblock Krylov case, twoalgorithms have been defined: the first one is the well known block Arnoldi algorithm and the second one is the extended block Arnoldi algorithm <cit.>. These algorithms also generate block Hessenberg matrices T̅_m =V_m+1^T AV_m satisfying the following algebraic relations AV_m = V_m+1 T̅_m,= V_mT_m + V_m+1 T_m+1,m E_m^T,where T_m = T̅_m (1:d,:)= V_m^T AV_m andwhere T_i,j is the(i,j) block of T̅_m of size d × d, and E_m = [ O_d × (m-1)d, I_d ]^T is the matrix of the last d columns of the md × md identity matrix I_md with d=s for the block Arnoldi and d=2s for the extended block Arnoldi. When the matrix A is nonsingular and when the computation of W=A^-1V is not difficult (which is the case for sparse and structured matrices), the use of the extended block Arnoldi is to be preferred. The paper is organized as follows: In Section 2, we present a first approach based on the approximation of the exponential of a matrix times a block using a Krylov projection method. We give some theoretical results such as an upper bound for the norm of the error and an expression of the exact residual.A second approach,presented in Section 3, forwhich the initial differential Lyapunov matrix equation is projected onto a block (or extended block) Krylov subspace. Then, the obtained low dimensional differential Lyapunov equation is solved by using the well known Backward Differentiation Formula (BDF). In Section 4, an application to balanced truncation method for large scale linear-time varying dynamical systems is presented. The last section is devoted to some numerical experiments. § THE FIRSTAPPROACH: USING AN APPROXIMATION OF THE MATRIX EXPONENTIAL In this section, we give a newapproach for computing approximate solutions to large differential equations (<ref>).The expression of the exact solution asX(t)=e^(t-t_0)AX_0e^(t-t_0)A^T+∫_t_0^te^(t-τ)AB B^Te^(t-τ)A^Tdτ,suggests the idea of computing X(t)by approximating the factor e^(t-τ)AB and thenusing a quadrature method to compute the desired approximate solution. As computing the exponential of a small matrix is straightforward , this is not the case for large scale problems, as e^(t-τ)A could be dense even though A is sparse. However, in our problem, the computation of e^(t-τ)A is not needed as we will rather consider the product e^(t-τ)AB, for which approximations via projection methods onto block or extended block Krylov subspaces are well suited.Krylov subspace projection methods generate a sequence ofnested subspaces (Krylov or extended Krylov subspaces).Let 𝒱_m=[V_1,…,V_m] be the orthogonalmatrix whosecolumns form an orthonormalbasis of the subspace 𝐾_m, Following <cit.>, an approximation to Z=e^(t-τ)AB can be obtained asZ_m(t) = 𝒱_m e^(t-τ)𝒯_m 𝒱_m^T Bwhere 𝒯_m=𝒱^T_m A 𝒱_m.Therefore, the term appearing in the integral expression (<ref>) can be approximated as e^(t-τ)ABB^Te^(t-τ)A^T≈ Z_m(t) Z_m(t)^T.If for simplicity, we assume X_t_0=0, an approximation to the solution of the differential Lyapunov equation (<ref>) can be expressed as X_m(t) = 𝒱_m G_m(t) 𝒱_m^T,where G_m(t) = ∫_t_0^tG_m(τ) G_m^T(τ) dτ, and G_m(τ)= e^(t-τ)𝒯_mB_m. The next result shows that the matrix function G_m is the solution of a low-order differential Lyapunov matrix equation.Let G_m(t) be the matrix functiondefined by (<ref>), then it satisfies the following low-order differential Lyapunov matrix equationĠ_m(t) = 𝒯_m G_m(t) + G_m(t)𝒯_m^T+B_mB_m^T,t ∈ [t_0, T_f]The proof can be easily derived from the expression (<ref>) and the result of Theorem <ref>. As a consequence, intruducingthe residual R_m(t) = Ẋ_m(t)-A X_m-X_m A^T- BB^T associated to the approximation X_m,we have the following relationV^T_m R_m(t)V_m= V^T_m (Ẋ -AX_m(t)-X_m(t)A^T-BB^T)V_m =Ġ_m(t) -𝒯_m G_m(t) - G_m(t)𝒯_m^T-B_mB_m^T = 0,which shows that the residual satisfies a Petrov-Galerkin condition. As mentioned earlier,onceG_m(τ) is computed, we use a quadrature method to approximate the integral (<ref>) in order to approximate G_m(t).We now briefly discusssome practical aspects of the computation of e^(t-τ)𝒯_m B_m where B_m= V_m^TB, when m is small and 𝒯_m is a an upper block Hessenberg matrix. In the last decade, many approximation techniques such as the use of partial fraction expansions or Padé approximation have been proposed, see for example<cit.>. However, it was remarkedthat a good way for evaluating the exponential of matrix times by a vector by using rational approximation to the exponential function. One of the main advantages ofrational approximations as compared to polynomial approximations isthe better stability of theirintegration schemes.Let us consider the rational function F(z)= a_0 + ∑_i=1^p a_i/z-θ_i,where the θ_i's are the poles of the rational function F. Then, the approximation to G_m(τ)=e^(t-τ)𝒯_m is given byG_m(τ) ≈ a_0B_m +∑_i=1^pa_i [(t-τ)𝒯_m- θ_i I]^-1B_m.One of the possible choices for the rational function F isbased on Chebychev approximationof the function e^x on [0, ∞[, see <cit.>.Wenotice that for small values of m,one can also directly compute the matrix exponentiale^(t-τ)𝒯_m by using the well-known 'scaling and squaring method for the matrix exponential' method, <cit.>. This method wasassociated toa Padé approximation and is implemented in the expm Matlab routine. From now on, we assume that the basis formed by the orthonormalcolumns of V_mis obtained by applyingthe block Arnoldi or the extended block Arnoldi algorithmto the pair (A,B). The computation of X_m(t) (and of R_m(t)) becomes expensive as m increases. So, in order to stop the iterations, one has to test if ∥ R_m ∥ < ϵ without having to compute extra products involving the matrix A. The next result shows how to compute the residual norm of R_m(t) without forming the approximation X_m(t) which is computed in a factored form only when convergence is achieved.Let X_m(t) =V_mG_m(t) V_m^T be the approximation obtained at step m by theblock (or extended block)Arnoldimethod. Then the residual R_m(t) satisfies ∥ R_m(t) ∥ = ∥ T_m+1,mG̅_m(t) ∥, where G̅_m is the d × mdmatrix corresponding to the last d rows of G_m where d=s when using the block Arnoldi and d=2s for the extended block Arnoldi.The proof of this theorem comes directly from (<ref>) and the fact that G_m solves the low dimensional problem (<ref>). The result of Theorem <ref> is veryimportant in practice, as it allows us to stop the iterations when convergence is achieved without computing the approximate solution X_m(t). The following resultshowsthat the approximation X_m is an exact solution of a perturbeddifferentialLyapunovequation. Let X_m(t) be the approximate solution given by (<ref>). Then we have Ẋ_m(t)=(A-F_m) X_m+X_m (A-F_m)^T+BB^T.where F_m = V_m T_m+1,m^T V_m+1^T.The proof is easily obtainedfrom (<ref>) andthe expression (<ref>) of theapproximate solution X_m(t). The solution X_m(t) can be given as a product of two low rank matrices. Consider the eigen-decomposition of the symmetric and positive matrixmd × md G_m(t)=UDU^T where D is the diagonal matrix of theeigenvalues of G_m(t) sorted in decreasing order and d=s for the block Arnoldi or d=2s for the extended block Arnoldi. Let U_l be the md × l matrixof the first l columns ofUcorresponding to the l eigenvalues of magnitude greater thansome tolerance dtol. We obtain the truncated eigen-decompositionG_m(t) ≈ U_lD_lU_l^T where D_l =diag[λ_1, …, λ_l].Setting Z_m(t)= V_m U_lD_l^1/2, it follows thatX_m(t) ≈Z_m(t) Z_m(t)^T.Therefore, one has to compute and to store only the matrix Z_m(t) which is usually the required factorin some control problems such as in the balanced truncation method for model reduction in large scale dynamical systems. This possibility is very important for storage limitations in thelarge scale problems. The next result states that the error matrix X(t)-X_m(t) satisfies a differential Lyapunov matrix equation.Let X(t) be the exact solution of (<ref>) and let X_m(t) be the approximate solution obtained at step m. The error E_m(t)=X(t)-X_m(t)satisfies the following equationĖ_m(t)=AE_m(t)+E_m(t)A^T-R_m(t),andE_m(t)=e^(t-t_0)AE_m,0e^(t-t_0)A^T+∫_t_0^t e^(t-τ)AR_m(τ)e^(t-τ)A^Tdτ,t ∈ [t_0,T_f].whereE_m,0=E_m(0).The result is easily obtained by subtracting the residual equation from the initial differential Lyapunov equation(<ref>). Next, we givean upper bound for the norm of the error in the case where A is a stable matrix. Assume that A is astable matrix and X(t_0)=X_m(t_0). Then we have the following upper bound∥ E_m(t) ∥≤∥T_m+1,m∥ ∥G̅_m ∥_∞e^2(t-t_0)μ_2(A)-1/2 μ_2(A),where μ_2(A)=1/2λ_max(A+A^T)<0is the 2-logarithmic norm and ∥G̅_m ∥_∞=max_τ∈ [t_0,t]∥G̅_m(τ) ∥. The matrixG̅_m is the d × mdmatrix corresponding to the last d rows of G_m where d=s when using the block Arnoldi and d=2s for the extended block Arnoldi. We first remind that if A is a stable matrix, then the logarithmic norm provides the following bound∥ e^tA∥≤ e^μ_2(A)t. Therefore, using the expression (<ref>), weobtain the following relation ∥ E_m(t) ∥≤∫_t_0^t ∥ e^(t-τ)A∥^2∥ R_m(τ) ∥ d τ.Therefore, using (<ref>) and the fact that ∥ e^(t-τ)A∥≤ e^(t-τ) μ_2(A), we get ∥ E_m(t) ∥ ≤ ∥T_m+1,mG̅_m ∥_∞∫_t_0^t e^2(t-τ) μ_2(A) dτ≤ ∥T_m+1,m∥ ∥G̅_m ∥_∞ e^2tμ_2(A)∫_t_0^t e^-2τμ_2(A) dτ≤ ∥T_m+1,m∥∥G̅_m ∥_∞ e^2tμ_2(A) ×e^-2μ_2(A)t-e^-2μ_2(A)t_0/-2 μ_2(A) =∥T_m+1,m∥ ∥G̅_m ∥_∞e^2(t-t_0)μ_2(A)-1/2 μ_2(A),which gives the desired result.Notice that if ∥ T_m+1,m∥ is close to zero, which is the case when m is close to the degree of the minimal polynomial of A for B, then Theorem <ref> shows that the error E_m(t) tends to zero. Next, we give anothererror bound for the norm of the error for every matrixA.Let X(t) be the exact solution to (<ref>) and let X_m(t) be the approximate solution obtained at step m.Then we have‖ X(t)- X_m(t) ‖ ≤e^tμ_2(A) (‖ B ‖ +‖ B_m ‖ )∫_t_0^te^-τμ_2(A)‖ e^(t-τ)AB- V_m e^(t-τ) 𝒯_m B_m‖ d τwhere μ_2(A)= λ_max((A+A^T)/2), Z(τ) = e^(t-τ)A B and Z_m(τ)= V_m e^(t-τ)𝒯_m B_m with B_m= V_m^T B. From the expressions of X(t) and X_m(t), we have ‖ X(t) - X_m(t) ‖=‖∫_t_0^t (Z(τ)Z(τ)^T -Z_m(τ)Z_m(τ)^T) d τ‖=‖∫_t_0^t [Z(τ)(Z(τ)-Z_m(τ))^T +(Z(τ)-Z_m(τ))Z^T_m(τ)] d τ‖≤ ∫_t_0^t(‖ Z(τ) ‖ + ‖ Z_m(τ) ‖) ‖Z(τ) -Z_m(τ) ‖d τ,Therefore, using the fact that μ_2(𝒯_m)=λ_max((𝒯_m+𝒯_m^T)/2) ≤λ_max((A+A^T)/2)=μ_2(A), where 𝒯_m= 𝒱_m^T A 𝒱_m, it follows that ‖ X(t) - X_m(t) ‖ ≤e^tμ_2(A) (‖ B ‖ +‖ B_m ‖ ) ∫_t_0^te^-τμ_2(A)‖ Z(τ) -Z_m(τ) ‖ d τ≤e^tμ_2(A) (‖ B ‖ +‖ B_m ‖ )∫_t_0^te^-τμ_2(A)‖ e^(t-τ)AB- V_m e^(t-τ) 𝒯_m B_m‖ d τ,When using a block Krylov subspace method such as the block Arnoldi method, then one can generalize to the block case the results already statedin many papers; see <cit.>. In particular, we can easily generalize the result given in <cit.> for the case s=1 to the case s >1. In this case, we have the following upper bound.‖ e^AB- V_m e^𝒯_m B_m ‖≤ 2 ∥ B ∥ ρ^m e^ρ/m!,where ρ= ‖ A ‖The rupper bound (<ref>) could be used in Theorem <ref> to obtain a new upper bound for the norm of the error. In that case, we obtain the following upper bound‖ X(t) - X_m(t) ‖≤ 2 ∥ B ∥ρ^m/m! e^t(μ_2(A)+ρ) (‖ B ‖ +‖ B_m ‖ )∫_t_0^te^-τ (μ_2(A)+ρ) (t-τ)^m d τ, We summarize the steps of our proposed first approach (usingthe extended block Arnoldi) in the following algorithm§ A SECOND APPROACH: PROJECTINGAND SOLVING WITH BDF §.§ Low-rank approximate solutions via BDFIn this section, we show how to obtain low rank approximate solutions to the differential Lyapunov equation (<ref>) by projecting directly the initial problem onto small block Krylov or extended block Krylov subspaces. We first apply theblock Arnoldialgorithm (or the extended block Arnoldi)to the pair (A,B) to get the matrices V_m and T_m= V^T_m AV_m. Let X_m(t) be the desired low rank approximate solutiongiven as X_m(t) =V_m Y_m(t)V_m^T,satisfying the Petrov-Galerkin orthogonality conditionV_m^T R_m(t)V_m =0,t ∈ [t_0,T_f],where R_m(t) is the residual R_m(t) = Ẋ_m(t)-A X_m(t)-X_m(t) A^T- BB^T.Then, from (<ref>) and (<ref>), we obtain the low dimensional differential LyapunovequationẎ_m(t)-T_m Y_m(t)-Y_m(t)T_m^T- B_mB_m^T=0,withT_m=V^T_m AV_m andB_m=V_m^T B. The obtained low dimensional differential Lyapunov equation (<ref>) isthesame as the one given by (<ref>). For this second approach, we have to solve the latter low dimensional differential Lyapunov equation by some integration method such as the well knownBackward DifferentiationFormula (BDF).Notice that we can also compute the norm of the residual without computing the approximation X_m(t) which is also given, when convergence is achieved, in a factored form as in (<ref>). The norm of the residual is given as∥ R_m(t) ∥ = ∥ T_m+1,mY̅_m(t) ∥, where Y̅_m is the d × mdmatrix corresponding to the last d rows of Y_m where d=s when using the block Arnoldi and d=2s for the extended block Arnoldi. §.§ BDF for solving the low order differential Lyapunov equation (<ref>)In this subsection, we will apply theBackward Differentiation Formula (BDF) method for solving, at each step m of the block (or extended) blockArnoldi process,the low dimensional differential Lyapunov matrix equation (<ref>).We notice that BDF isespecially used for the solution of stiff differential equations. At each time t_k, letY_m,k of the approximation of Y_m(t_k), where Y_m is asolution of(<ref>).Then, the new approximation Y_m,k+1 ofY_m(t_k+1) obtained at step k+1 by BDF is definedby the implicit relation Y_m,k+1 = ∑_i=0^p-1α_i Y_m,k-i +h_k βℱ(Y_m,k+1), where h_k=t_k+1-t_k is the step size, α_i and β_i are the coefficients of the BDF method as listedin Table <ref> and ℱ(X) isgiven by ℱ(Y)=T_m Y+YT_m^T+ B_m B_m^T.The approximate Y_m,k+1 solves the following matrix equation-Y_m,k+1 +h_kβ ( T_m Y_m,k+1 + Y_m,k+1 T_m^T)+ B B^T + ∑_i=0^p-1α_i Y_m,k-i = 0,which can be written as the followingLyapunov matrix equation 𝕋_mY_m,k+1+Y_m,k+1𝕋_m^T+ 𝔹_m,k 𝔹_m,k^T =0.We assumethat at each time t_k, the approximation Y_m,k isfactorizedas a low rank productY_m,k≈ Z_m,kZ_m,k^T, where Z_m,k∈ℝ^n × m_k, with m_k ≪ n. In that case, the coefficient matrices appearing in (<ref>) are given by𝕋_m= h_kβ T_m -1/2I 𝔹_m,k+1=[√(h_kβ) B^T, √(α_0)Z_m,k^T,…,√(α_p-1) Z_m,k+1-p^T]^T.TheLyapunov matrixequation (<ref>) can be solved by applying direct methods based on Schur decomposition such as the Bartels-Stewart algorithm <cit.>. We notice that for large problems, many Krylov subspace type methods have been proposed to solve (<ref>); <cit.>. The main difference between Approach 1 and Approach 2 is the fact that in the first case, we compute an approximation of an integral using a quadrature formulae while in the second case, we have to solve a low dimensional differential Lyapunov equation using the BDF method. Mathematically, the two approaches are equivalent and they differ only in the way of computing numerically the low-order approximations:G_m in the first approach and Y_m in the second one.`We summarize the steps of our proposed first approach (usingthe extended block Arnoldi) in the following algorithm§ APPLICATION: BALANCED TRUNCATION FOR LINEAR TIME-VARYING DYNAMICALSYSTEMSIn this section, we assume that the coefficient matrices A and B are time-dependent. It is the case for example when we are dealing with Multi-Input Multi-Output (MIMO)linear-time varying (LTV) dynamical systems{[ẋ(t) = A(t)x(t)+B(t)u(t),x(t_0)=0,;y(t) =C(t) x(t), ] .where x(t) ∈ℝ^n is the state vector, u(t) ∈ℝ^pis the control and y(t) ∈ℝ^p is the output. The matrices A(t) ∈ℝ^n × n, B(t) ∈ℝ^n × p and C(t) ∈ℝ^p × n are assumed to becontinuous and bounded for all t ∈ [t_0,T_f].The LTV dynamical system (<ref>) can also be denoted asΣ(t)≡ [ [ A(t) B(t); C(t)0 ] ].In many applications, such as circuit simulation, or time dependent PDE control problems, the dimension n of Σis quite large, while the number of inputsand outputsis small p ≪ n. In these large-scale settings, the system dimension makes the computation infeasible due to memory, time limitations and ill-conditioning. To overcome these drawbacks, one approach consists in reducing the model. The goal is to produce a low order system that has similar response characteristics as the original system withlower storage requirements and evaluation time. Thereduced orderdynamical system can be expressed as follows Σ_m {[ ẋ_m(t) = A_m(t) x_m(t) +B_m(t)u(t); ;y_m(t) = C_m(t)x_m(t);]. wherex_m∈ℝ^m,y_m ∈ℝ^p, A_m ∈ℝ^m × m, B ∈ℝ^m × p and C_m ∈ℝ^p × m with m ≪ n. The reduced dynamical system (<ref>) is also represented as Σ_m(t)≡ [[ A_m(t) B_m(t); C_m(t)0 ] ]. The reduced order dynamical system should be constructed in order that * Theoutput y_m(t) of the reduced system approachesthe output y(t) of the original system. * Someproperties of the original system such as passivity and stability arepreserved. * The computationmethods aresteady and efficient. One of the well known methods for constructing such reduced-order dynamical systems is the balanced truncation method for LTV systems <cit.>; see also <cit.> for the linear time-independent case.This method requires the LTV controllabilityand observabilityGramians P(t) and Q(t) defined as the solutions of the differential Lyapunov matrix equationsṖ(t) = A(t)P(t)+P(t)A(t)^T +B(t)B(t)^T,P(t_0)=0,andQ̇(t) = A^T(t)P(t)+P(t)A(t) +C(t)^TC(t),Q(T_f)=0.Using the formulae (<ref>), the differential Lyapunov equation (<ref>) has the unique symmetric and positive solution P(t) given byP(t)=∫_t_0^t Φ_A(t,τ)B(τ)B^T(τ)Φ^T_A(t,τ)dτ,where the transition matrix Φ_A(t,τ) is the unique solution of the problemΦ̇_A(t,τ)=A(t) Φ_A(t,τ),Φ_A(t,t)=I.The observability Gramian is given byQ(t)=∫_t^T_fΦ^T_A(τ,t)C^T(τ)C(τ)Φ_A(τ,t) dτ.The twoLTV controllabilityand observability Gramians P(t) and Q(t) are then used to construct a new balanced systemsuch that P̃(t)=Q̃(t)=diag(σ_1(t),…,σ_n(t)) where the Hankel singular values are given byσ_i(t)= √(λ_i(P(t)Q(t)), i=1,…,nand order in decreasing order.The concept of balancingis to a transform the original LTV system to an equivalent onein which the states that are difficult to reach are also difficult to observe, which is finding an equivalent new LTV system such that the new Gramians P and Q are such that P(t) = Q(t)=diag(σ_1,…,σ_n)where σ_i is the i-th Hankel singular value of the LTV system; i.e.σ_i = √(λ_i(P(t)Q(t))).Considerthe Cholesky decompositions of the Gramians P and Q:P(t)=L_c(t)L_c(t)^T, Q(t)=L_o(t)L_o(t)^T,and consider also the singular value decomposition of L_c(t)^T L_o(t) asL_c(t)^T L_o(t) = Z(t) Σ(t) Y(t)^T,where Z(t) and Y(t) are unitary n × n matrices and Σ is a diagonal matrix containing thesingular values.The balanced truncation consists in determining a reduced order modelby truncating the states corresponding to the small Hankel singular values. Under certain conditions stated in <cit.>, one can construct the low order model Σ_m(t) as follows:We set V_m(t) =L_o(t) Y_m(t) Σ_m(t)^-1/2and W_m(t) = L_c(t) Z_m(t) Σ_m(t)^-1/2,where Σ_m(t)=diag(σ_1(t),…,σ_m(t)); Z_m(t) and Y_m(t) correspond to the leading m columns of the matrices Z(t) and Y(t) given by the singular value decomposition (<ref>). Thematrices of the reduced LTV system W_m(t)^TV_m(t)A_m(t) = V_m(t)^T A(t) W_m(t)-V_m(t)^T Ẇ_m(t),B_m(t)=V_m(t)^T B(t),C_m(t) = C(t)W_m(t).The use of Cholesky factors in the Gramians P(t) and Q(t) is not applicable for large-scale problems. Instead,one can compute low rank approximations of P(t) an Q(t) as given by (<ref>) and use them to construct an approximate balanced truncation model. As A, B and C are time-dependent, the direct application of the two approaches we developed is too expensive. Instead, we can apply directly an integration method such as BDF to the differential Lyapunov matrix equations (<ref>) and (<ref>). Then, at each iteration of the BDF method, we obtain a large Lyapunov matrix equation that can be numerically solvedby using the extended block Arnoldi algorithm. Consider the differential matrix equation (<ref>), then,at each iteration of the BDF method, the approximation P_k+1 ofP(t_k+1) where P is the exact solution of (<ref>),is given by the implicit relation P_k+1 = ∑_i=0^p-1α_i P_k-i +h_k β𝒢(G_k+1), where h_k=t_k+1-t_k is the step size, α_i and β_i are the coefficients of the BDF method as listedin Table <ref> and 𝒢(X) isgiven by 𝒢(X)= A^T X+X A+ B B^T. The approximate solution P_k+1 solves the following matrix equation-P_k+1 +h_kβ (A^T P_k+1 + P_k+1 A+ B B^T) + ∑_i=0^p-1α_i P_k-i = 0,which can be written as the followingcontinuous-time algebraic Riccati equation𝒜_k^TP_k+1+ P_k+1 𝒜_k+ ℬ_k ℬ_k^T=0.Assuming that at each timestep, P_k can be approximated as a product oflow rank factorsP_k≈Z̃_kZ̃_k^T, Z̃_k ∈ℝ^n × m_k, with m_k ≪ n, the coefficient matrices are given by𝒜_k= h_kβ A -1/2I, ℬ_k+1=[√(h_kβ) B, √(α_0)Z̃_k^T,…,√(α_p-1)Z̃_k+1-p^T]^T. A good way for solving the Lyapunov matrix equation (<ref>) is byusing the block or extended block Arnoldi algorithm applied to the pair (𝒜_k,ℬ_k). This allows usto obtain low rankapproximate solutions in factored forms. The procedure is as follows: applying for example theblock Arnoldi to the pair (𝒜_k,ℬ_k) we get, at step m of the Arnoldi process,an orthonormal basis of the extended block Krylov subspace formed by the columns of the matrices: {V_1,k,…,V_m,k} and also a block upper Hessenberg matrix ℍ_m,k. Let𝕍_m,k=[V_1,k,…,V_m,k] and ℍ_m,k = 𝕍_m,k^T 𝒜_k 𝕍_m,k. Then the obtained low rank approximate solution to the solution P_k+1 of(<ref>) is given as P_m,k=𝕍_m,k𝕐_m,k𝕍_m,k^T where 𝕐_m,k is solution of the followinglow order Lyapunov equation ℍ_m,k𝕐_m,k+𝕐_m,kℍ_m,k^T+ ℬ̃_k ℬ̃_k^T=0,where ℬ̃_k=𝕍_m,k^Tℬ_k. As stated in Remark 1, the approximate solution can be given in a factored form. § NUMERICAL EXAMPLES In this section, we compare the two approaches presented in this paper. The exponential approach (EBA-exp) summarized in Algorithm <ref>, which is based on the approximation of the solution to (<ref>) applying a quadrature method to compute the projected exponential form solution (<ref>).We used a scaling and squaring strategy, implemented in the MATLABexpm function; see<cit.> for more details.The second method (Algorithm <ref>) is based on the BDFintegration method applied to the projected Lyapunov equation (<ref>). The basis of the projection subspaces were generated by the extended block Arnoldi algorithm for both methods. All the experiments were performed on a laptop with anIntel Core i7 processor and 8GB of RAM. The algorithms were coded in Matlab R2014b.Example 1.The matrix Awas obtained from the 5-point discretization of the operators L_A=Δ u-f_1(x,y)∂ u/∂ x+ f_2(x,y)∂ u/∂ y+g_1(x,y),on the unit square [0,1]× [0,1] with homogeneous Dirichlet boundary conditions.The number of inner grid points in each direction isn_0= and the dimension of the matrix A was n = n_0^2=. Here we set f_1(x,y) = 10xy, f_2(x,y)= e^x^2y, f_3(x,y) = 100y, f_4(x,y)= x^2y ,g_1(x,y) = 20y and g_2(x,y)=x y.The time interval considered was [0, 2] and the initial condition X_0=X(0) was choosen as the low rank product X_0=Z_0Z_0^T, where Z_0=0_n × 2.For both methods, we usedprojections onto the Extended Block Krylov subspaces𝒦_k(A,B) =Range(B,A B,…,A^m-1 B,A^-1 B,…,(A^-1)^m B)and thetolerance was set to 10^-10 for the stop test on the residual.For the EBA-BDF method,we used a 2-step BDF scheme with a constant timestep h. The entries of the matrix B were random values uniformly distributed on the interval [0, 1] and the number of the columns in Bwas s=2. literature. In order to check if our approaches produce reliable results, we began comparing our results to the one given by Matlab's ode23s solver which is designed for stiff differential equations. This was done by vectorizing our DLE, stacking the columns of X one on top of each other. This method, basedon Rosenbrock integration scheme, is not suited to large-scale problems. Due to the memory limitation of our computer when running the ode23s routine,we chose asize of 100× 100 for the matrix A.In Figure <ref>, we compared the component X_11 of the solution obtained by the methods tested in this section, to the solution provided by the ode23s method from Matlab,on the time interval [0, 2], for size(A)=100× 100 and a constant timestep h=10^-3.We observe that all the considered methods give similar results in terms of accuracy. The relative error norms X_EBA-exp(t_f)-X_ode23s(t_f)/X_ode23s(t_f) and X_EBA-BDF(2)(t_f)-X_ode23s(t_f)/X_ode23s(t_f) at final time t_f=2 were equal to 1.8× 10^-10 and9.1× 10^-11 respectively.The runtimes were respectively 0.59s, 5.1s for the EBA-exp and EBA-BDF(2) methods and 1001s for the ode23s routine. In Table <ref>, we givethe obtained runtimes in seconds, for the resolution of Equation (<ref>)for t ∈ [0,2], with a timestep h=0.001 and the Frobenius norm of the residual at the final time.The resultsin Table <ref> illustrate that the EBA-exp method clearly outperforms the EBA-BDF(2) method in terms of computation time even though both methods are equally accurate. In Figure <ref>, we featured the norm of the residual at final time t=2 for both EBA-exp and EBA-BDF(2) methodsfor size(A)=6400 × 6400 in function of the number m of extended Arnoldi iterations. We observe that the plots coincide for both methods. Example 2.This examplecomes from the autonomous linear-quadratic optimal control problem of one dimensional heat flow∂/∂ t x(t,η) =∂^2/∂η^2 x(t,η)+b(η) u(t) x(t,0) = x(t,1)=0, t>0 x(0,η) = x_0(η), η∈ [0,1] y(x) =∫_0^1 c(η) x(t,η) d η, x>0.Using a standard finite element approach based on the first order B-splines, we obtain the following ordinary differential equation M ẋ(t)= Kx(t) + F u(t)y(t) =Cx(t), where the matrices M and K are given by:M=1/6n( [ 4 1; 1 4 1; ⋱ ⋱ ⋱; 1 4 1; 1 4 ]),K=-α n ( [2 -1; -12 -1; ⋱⋱⋱;-12 -1;-12 ]).Using the semi-implicit Euler method, we getthe following discrete dynamical system(M-Δ t K) ẋ(t)= Mx(t)+ Δ tF u_k.We set A=(M-Δ t K)^-1M and B=Δ t(M-Δ t K)^-1F. The entries of the n × smatrix F and the s × n matrixC were random values uniformly distributed on [0, 1]. In our experiments we usedn=, s=2, Δ t = 0.01 andα=0.05.In Table <ref>, we givethe obtained runtimes in seconds, for the resolution of Equation (<ref>)for t ∈ [0,2], with a timestep h=0.001 and the Frobenius norm of the residual at the final time. The figures in Table <ref> illustrate the gain of speed provided by the EBA-exp method. Again, both methods performed similarly in terms of accuracy. In figure <ref>, we considered the case size(A)=100× 100 andplotted the upper bound of the error norms as stated in Formula (<ref>) at the final time T_f against the computed norm of the errors, taking the solution given by the integral formula (<ref>) as a reference, in function of the number m of Arnoldi iterations for the EBA-exp method.Example 3In this last example, we applied the EBA-BDF(1) method to the well-known problem Optimal Cooling of Steel Profiles. The matrices were extracted from the IMTEK collection [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark]. We compared the EBA-BDF(2) method to the EBA-exp method for problem sizes n=1357 and n=5177, on the time interval [0 ,1000]. The initial value X_0 was chosen as X_0=0 andthe timestep was set to h=0.01. The tolerance for the Arnoldi stop test was set to 10^-7 for both methods and the projected low dimensional Lyapunov equations were numerically solved by thesolver (lyap from Matlab) at each iteration of the extended block Arnoldialgorithm for the EBA-BDF(2) method.In Table <ref>,we listed the obtained runtimes which again showed the advantage of the EBA-exp method in terms of execution time and similar accuracy for both methods.§ CONCLUSIONWe presented in the present paper two new approaches for computing approximate solutions to large scale differentialLyapunov matrix equations. The first one comes naturally from the exponential expression of the exact solution and the use of approximation techniques of the exponential of a matrix times a block of vectors. The second approach is obtained by first projecting the initial problem onto a block Krylov (or extended Krylov) subspace, obtain a low dimensional differential Lyapunov equation which is solved by using the well known BDF integration method. We gave some theoretical results such as the exact expression of the residual norm and also upper bounds for the norm of the errors. An application in model reduction forlinear time-varying dynamicalsystems is also given. Numerical experiments show that both methods are promising for large-scale problems, with a clear advantage for the EBA-exp method in terms of computation time.plain99abou03 H. Abou-Kandil, G. Freiling, V. Ionescu, G. Jank, Matrix Riccati Equations in Control and Sytems Theory, in Systems & Control Foundations & Applications, Birkhauser, (2003).anderson71 B.D.O. Anderson, J.B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, (1971).bartels R.H. Bartels, G.W. 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Jbilou and A. J. Riquet, Projection methods for large Lyapunov matrix equations,Lin. Alg. Appl.,415 (2006), 344–358. moler03 C.B. Moler, C.F. Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review 20, 1978, pp. 801–836. Reprinted and updated as "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later," SIAM Review 45(2003), 3–49.moore B.C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Contr., AC-26(1981), 17–32.mullis C. T. Mullis and R. A. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans. Acoust. Speec Signal Process, 24, 1976.saad1 Y. Saad, Numerical solution of large Lyapunov equations,in Signal Processing, Scattering, Operator Theory and Numerical Methods. Proceedings of the international symposium MTNS-89, vol. 3, M.A. Kaashoek, J.H. van Schuppen and A.C. Ran, eds., Boston, 1990, Birkhauser, pp. 503–511.saad2 Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), 209–228.sandberg H. Sandberg, Linear Time-Varying Systems: Modeling and Reduction, Ph.D. thesis, Department of Automatic Control, Lund Institute of Technology, Lund, 2002.shokoohi S. Shokoohi, L. Silverman, and P. Van Dooren, Linear time-variable systems: balancing and model reduction, IEEE Trans. Automat. Control, 28 (1983),810–822.simoncini1 V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comp., 29(3) (2007), 1268–1288.verries E. I. Verriest and T. Kailath, On generalized balanced realizations, IEEE Trans. Automat. Control, 28 (1983), 833–844.vorst1 H. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, Cambridge, 2003. | http://arxiv.org/abs/1705.09362v2 | {
"authors": [
"M. Hached",
"K. Jbilou"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170525210751",
"title": "Numerical solutions to large-scale differential Lyapunov matrix equations"
} |
Entanglement properties of the time periodic Kitaev Chain Aditi Mitra December 30, 2023 ========================================================= At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks and convolutional LSTMs. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their real-valued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech Spectrum Prediction using the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks. § INTRODUCTION Recent research advances have made significant progress in addressing the difficulties involved in learning deep neural network architectures. Key innovations include normalization techniques <cit.> and the emergence of gating-based feed-forward neural networks like Highway Networks <cit.>. Residual networks <cit.> have emerged as one of the most popular and effective strategies for training very deep convolutional neural networks (CNNs). Both highway networks and residual networks facilitate the training of deep networks by providing shortcut paths for easy gradient flow to lower network layers thereby diminishing the effects of vanishing gradients <cit.>. <cit.> show that learning explicit residuals of layers helps in avoiding the vanishing gradient problem and provides the network with an easier optimization problem. Batch normalization <cit.> demonstrates that standardizing the activations of intermediate layers in a network across a minibatch acts as a powerful regularizer as well as providing faster training and better convergence properties. Further, such techniques that standardize layer outputs become critical in deep architectures due to the vanishing and exploding gradient problems.The role of representations based on complex numbers has started to receive increased attention, due to their potential to enable easier optimization <cit.>, better generalization characteristics <cit.>, faster learning <cit.> and to allow for noise-robust memory mechanisms <cit.>. <cit.> and <cit.> show that using complex numbers in recurrent neural networks (RNNs) allows the network to have a richer representational capacity. <cit.> present an LSTM <cit.> architecture augmented with associative memory with complex-valued internal representations. Their work highlights the advantages of using complex-valued representations with respect to retrieval and insertion into an associative memory. In residual networks, the output of each block is added to the output history accumulated by summation until that point. An efficient retrieval mechanism could help to extract useful information and process it within the block. In order to exploit the advantages offered by complex representations, we present a general formulation for the building components of complex-valued deep neural networks and apply it to the context of feed-forward convolutional networks and convolutional LSTMs. Our contributions in this paper are as follows: * A formulation of complex batch normalization, which is described in Section <ref>;* Complex weight initialization, which is presented in Section <ref>;* A comparison of different complex-valued ReLU-based activation functions presented in Section <ref>;* A state of the art result on the MusicNet multi-instrument music transcription dataset, presented in Section <ref>;* A state of the art result in the Speech Spectrum Prediction task on the TIMIT dataset, presented in Section <ref>. We perform a sanity check of our deep complex network and demonstrate its effectiveness on standard image classification benchmarks, specifically, CIFAR-10, CIFAR-100. We also use a reduced-training set of SVHN that we call SVHN*. For audio-related tasks, we perform a music transcription task on the MusicNet dataset and a Speech Spectrum prediction task on TIMIT. The results obtained for vision classification tasks show that learning complex-valued representations results in performance that is competitive with the respective real-valued architectures. Our promising results in music transcription and speech spectrum prediction underscore the potential of deep complex-valued neural networks applied to acoustic related tasks[The source code is located at <http://github.com/ChihebTrabelsi/deep_complex_networks>] – We continue this paper with discussion of motivation for using complex operations and related work.§ MOTIVATION AND RELATED WORKUsing complex parameters has numerous advantages from computational, biological, and signal processing perspectives. From a computational point of view, <cit.> has shownthat Holographic Reduced Representations <cit.>, which use complex numbers, are numerically efficient and stable in the context of information retrieval from an associative memory. <cit.> insert key-value pairs in the associative memory by addition into a memory trace. Although not typically viewed as such, residual networks <cit.> and Highway Networks <cit.> have a similar architecture to associative memories: each ResNet residual path computes a residual that is then inserted – by summing into the “memory” provided by the identity connection. Given residual networks' resounding success on several benchmarks and their functional similarity to associative memories, it seems interesting to marry both together. This motivates us to incorporate complex weights and activations in residual networks. Together, they offer a mechanism by which useful information may be retrieved, processed and inserted in each residual block.Orthogonal weight matrices provide a novel angle of attack on the well-known vanishing and exploding gradient problems in RNNs. Unitary RNNs <cit.> are based on unitary weight matrices, which are a complex generalization of orthogonal weight matrices. Compared to their orthogonal counterparts, unitary matrices provide a richer representation, for instance being capable of implementing the discrete Fourier transform, and thus of discovering spectral representations. <cit.> show the potential of this type of recurrent neural networks on toy tasks. <cit.> provided a more general framework for learning unitary matrices and they applied their method on toy tasks and on a real-world speech task.Using complex weights in neural networks also has biological motivation. <cit.> have proposed a biologically plausible deep network that allows one to construct richer and more versatile representations using complex-valued neuronal units. The complex-valued formulation allows one to express the neuron’s output in terms of its firing rate and the relative timing of its activity. The amplitude of the complex neuron represents the former and its phase the latter. Input neurons that have similar phases are called synchronous as they add constructively, whereas asynchronous neurons add destructively and thus interfere with each other. This is related to the gating mechanism used in both deep feed-forward neural networks <cit.> and recurrent neural networks <cit.> as this mechanism learns to synchronize inputs that the network propagates at a given feed-forward layer or time step. In the context of deep gating-based networks, synchronization means the propagation of inputs whose controlling gates simultaneously hold high values. These controlling gates are usually the activations of a sigmoid function. This ability to take into account phase information might explain the effectiveness of incorporating complex-valued representations in the context of recurrent neural networks.The phase component is not only important from a biological point of view but also from a signal processing perspective. It has been shown that the phase information in speech signals affects their intelligibility <cit.>. Also <cit.> show that the amount of information present in the phase of an image is sufficient to recover the majority of the information encoded in its magnitude. In fact, phase provides a detailed description of objects as it encodes shapes, edges, and orientations.Recently, <cit.> leveraged the Fourier spectral representation for convolutional neural networks, providing a technique for parameterizing convolution kernel weights in the spectral domain, and performing pooling on the spectral representation of the signal. However, the authors avoid performing complex-valued convolutions, instead building from real-valued kernels in the spatial domain. In order to ensure that a complex parametrization in the spectral domain maps onto real-valued kernels, the authors impose a conjugate symmetry constraint on the spectral-domain weights, such that when the inverse Fourier transform is applied to them, it only yields real-valued kernels.As pointed out in <cit.>, the use of complex-valued neural networks <cit.> has been investigated long before the earliest deep learning breakthroughs <cit.>. Recently <cit.> have tried to bring more attention to the usefulness of deep complex neural networks by providing theoretical and mathematical motivation for using complex-valued deep networks. However, to the best of our knowledge, most of the recent works using complex valued networks have been applied on toy tasks, with the exception of some attempts. In fact, <cit.> have used complex representation in vision tasks. <cit.> have also performed a real-world speech task consisting of predicting the log magnitude of the future short time Fourier transform frames. In Natural Language Processing, <cit.> have used complex-valued embeddings. Much remains to be done to develop proper tools and a general framework for training deep neural networks with complex-valued parameters.Given the compelling reasons for using complex-valued representations, the absence of such frameworks represents a gap in machine learning tooling, which we fill by providing a set of building blocks for deep complex-valued neural networks that enable them to achieve competitive results with their real-valued counterparts on real-world tasks.§ COMPLEX BUILDING BLOCKSIn this section, we present the core of our work, laying down the mathematical framework for implementing complex-valued building blocks of a deep neural network. §.§ Representation of Complex NumbersWe start by outlining the way in which complex numbers are represented in our framework. A complex number z = a + ib has a real component a and an imaginary component b. We represent the real part a and the imaginary part b of a complex number as logically distinct real valued entities and simulate complex arithmetic using real-valued arithmetic internally. Consider a typical real-valued 2D convolution layer that has N feature maps such that N is divisible by 2; to represent these as complex numbers, we allocate the first N/2 feature maps to represent the real components and the remaining N/2 to represent the imaginary ones. Thus, for a four dimensional weight tensor W that links N_in input feature maps to N_out output feature maps and whose kernel size is m× m we would have a weight tensor of size (N_out× N_in× m× m )/2 complex weights. §.§ Complex ConvolutionIn order to perform the equivalent of a traditional real-valued 2D convolution in the complex domain, we convolve a complex filter matrix W = A + iB by a complex vector h = x + i y where A and B are real matrices and x and y are real vectors since we are simulating complex arithmetic using real-valued entities. As the convolution operator is distributive, convolving the vector h by the filter W we obtain:W * h = (A * x - B * y) + i (B * x + A * y).As illustrated in Figure <ref>, if we use matrix notation to represent real and imaginary parts of the convolution operation we have:[ (W * h); (W * h) ] = [A -B;BA ] * [ x; y ].§.§ Complex DifferentiabilityIn order to perform backpropagation in a complex-valued neural network, a sufficient condition is to have a cost function and activations that are differentiable with respect to the real and imaginary parts of each complex parameter in the network. See Section <ref> in the Appendix for the complex chain rule.By constraining activation functions to be complex differentiable or holomorphic, we restrict the use of possible activation functions for a complex valued neural networks (For further details about holomorphism please refer to Section <ref> in the appendix). <cit.> shows that it is unnecessarily restrictive to limit oneself only to holomorphic activation functions; Those functions that are differentiable with respect to the real part and the imaginary part of each parameter are also compatible with backpropagation. <cit.> have used non-holomorphic activation functions and optimized the network using regular, real-valued backpropagation to compute partial derivatives of the cost with respect to the real and imaginary parts.Even though their use greatly restricts the set of potential activations, it is worth mentioning that holomorphic functions can be leveraged for computational efficiency purposes. As pointed out in <cit.>, using holomorphic functions allows one to share gradient values (because the activation satisfies the Cauchy-Riemann equations <ref> and <ref> in the appendix). So, instead of computing and backpropagating 4 different gradients, only 2 are required. §.§ Complex-Valued Activations§.§.§ ModReLUNumerous activation functions have been proposed in the literature in order to deal with complex-valued representations. <cit.> have proposed modReLU, which is defined as follows:modReLU(z) = ReLU(|z| + b) e^iθ_z=(|z| + b)z/|z| if|z| + b ≥0, 0otherwise,where z ∈ℂ, θ_z is the phase of z, and b ∈ℝ is a learnable parameter. As |z| is always positive, a bias b is introduced in order to create a “dead zone” of radius b around the origin 0 where the neuron is inactive, and outside of which it is active. The authors have used modReLU in the context of unitary RNNs. Their design of modReLU is motivated by the fact that applying separate ReLUs on both real and imaginary parts of a neuron performs poorly on toy tasks. The intuition behind the design of modReLU is to preserve the pre-activated phase θ_z, as altering it with an activation function severely impacts the complex-valued representation. modReLU does not satisfy the Cauchy-Riemann equations, and thus is not holomorphic. We have tested modReLU in deep feed-forward complex networks and the results are given in Table <ref>.§.§.§ ℂReLU and zReLUWe call Complex ReLU (or ℂReLU) the complex activation that applies separate ReLUs on both of the real and the imaginary part of a neuron, i.e:ℂReLU(z) = ReLU((z)) + iReLU((z)).ℂReLU satisfies the Cauchy-Riemann equations when both the real and imaginary parts are at the same time either strictly positive or strictly negative. This means that ℂReLU satisfies the Cauchy-Riemann equations when θ_z∈]0,π / 2[ or θ_z∈]π,3 π / 2[. We have tested ℂReLU in deep feed-forward neural networks and the results are given in Table <ref>.It is also worthwhile to mention the work done by <cit.> where a ReLU-based complex activation which satisfies the Cauchy-Riemann equations everywhere except for the set of points {(z) > 0, (z)=0}∪{(z)=0, (z)>0} ias used. The activation function has similarities to ℂReLU. We call <cit.> activation as zReLU and is defined as follows:zReLU(z) =zif θ_z∈ [0, π / 2], 0otherwise,We have tested zReLU in deep feed-forward complex networks and the results are given in Table <ref>. §.§ Complex Batch NormalizationDeep networks generally rely upon Batch Normalization <cit.> to accelerate learning. In some cases batch normalization is essential to optimize the model. The standard formulation of Batch Normalization applies only to real values. In this section, we propose a batch normalization formulation that can be applied for complex values.To standardize an array of complex numbers to the standard normal complex distribution, it is not sufficient to translate and scale them such that their mean is 0 and their variance 1. This type of normalization does not ensure equal variance in both the real and imaginary components, and the resulting distribution is not guaranteed to be circular; It will be elliptical, potentially with high eccentricity.We instead choose to treat this problem as one of whitening 2D vectors, which implies scaling the data by the square root of their variances along each of the two principal components. This can be done by multiplying the 0-centered data (x - 𝔼[ x] ) by the inverse square root of the 2 × 2 covariance matrix V:x̃ = ( V)^-1/2(x - 𝔼[ x] ),where the covariance matrix V isV= ( [ V_rr V_ri; V_ir V_ii ]) = ( [ Cov({ x}, { x}) Cov({ x}, { x}); Cov({ x}, { x}) Cov({ x}, { x}) ]).The square root and inverse of 2 × 2 matrices has an inexpensive, analytical solution, and its existence is guaranteed by the positive (semi-)definiteness of V. Positive definiteness of V is ensured by the addition of ϵ I to V (Tikhonov regularization). The mean subtraction and multiplication by the inverse square root of the variance ensures that x̃ has standard complex distribution with mean μ = 0, covariance Γ = 1 and pseudo-covariance (also called relation) C = 0. The mean, the covariance and the pseudo-covariance are given by:μ = 𝔼[ x̃]Γ = 𝔼[ (x̃ - μ) (x̃ - μ)^*] = V_rr + V_ii + i (V_ir - V_ri)C= 𝔼[ (x̃ - μ) (x̃ - μ) ] = V_rr - V_ii + i (V_ir + V_ri).The normalization procedure allows one to decorrelate the imaginary and real parts of a unit. This has the advantage of avoiding co-adaptation between the two components which reduces the risk of overfitting <cit.>.Analogously to the real-valued batch normalization algorithm, we use two parameters, β and γ. The shift parameter β is a complex parameter with two learnable components (the real and imaginary means). The scaling parameter γ is a 2 × 2 positive semi-definite matrix with only three degrees of freedom, and thus only three learnable components. In much the same way that the matrix ( V)^-1/2 normalized the variance of the input to 1 along both of its original principal components, so does γ scale the input along desired new principal components to achieve a desired variance. The scaling parameter γ is given by:γ =( [ γ_rr γ_ri; γ_ri γ_ii ]). As the normalized input x̃ has real and imaginary variance 1, we initialize both γ_rr and γ_ii to 1 / √(2) in order to obtain a modulus of 1 for the variance of the normalized value. γ_ri, {β} and {β} are initialized to 0.The complex batch normalization is defined as:BN( x̃) = γ x̃ + β.We use running averages with momentum to maintain an estimate of the complex batch normalization statistics during training and testing. The moving averages of V_ri and β are initialized to 0. The moving averages of V_rr and V_ii are initialized to 1 /√(2). The momentum for the moving averages is set to 0.9. §.§ Complex Weight Initialization In a general case, particularly when batch normalization is not performed, proper initialization is critical in reducing the risks of vanishing or exploding gradients. To do this, we follow the same steps as in <cit.> and <cit.> to derive the variance of the complex weight parameters.A complex weight has a polar form as well as a rectangular formW= |W| e^iθ = {W} + i {W},where θ and |W| are respectively the argument (phase) and magnitude of W.Variance is the difference between the expectation of the squared magnitude and the square of the expectation:Var(W)= 𝔼[ WW^*] - (𝔼[ W ])^2 = 𝔼[ |W|^2 ] - (𝔼[ W ])^2,which reduces, in the case of W symmetrically distributed around 0, to 𝔼[ |W|^2 ]. We do not know yet the value of Var(W) = 𝔼[ |W|^2 ]. However, we do know a related quantity, Var(|W|), because the magnitude of complex normal values, |W|, follows the Rayleigh distribution (Chi-distributed with two degrees of freedom (DOFs)).This quantity isVar(|W|)= 𝔼[ |W||W|^*] - (𝔼[ |W| ])^2 = 𝔼[ |W|^2] - (𝔼[ |W| ])^2.Putting them together:Var(|W|)= Var(W) - (𝔼[ |W| ])^2, and Var(W) = Var(|W|) + (𝔼[ |W| ])^2.We now have a formulation for the variance of W in terms of the variance and expectation of its magnitude, both properties analytically computable from the Rayleigh distribution's single parameter, σ, indicating the mode. These are:𝔼[ |W| ]= σ√(π/2),Var(|W|) = 4-π/2σ^2.The variance of W can thus be expressed in terms of its generating Rayleigh distribution's single parameter, σ, thus:Var(W) = 4-π/2σ^2 + ( σ√(π/2))^2= 2σ^2. If we want to respect the <cit.> criterion which ensures that the variances of the input, the output and their gradients are the same, then we would have Var(W) = 2 / (n_in + n_out), where n_in and n_out are the number of input and output units respectively. In such case, σ = 1 / √(n_in + n_out). If we want to respect the <cit.> initialization that presents an initialization criterion that is specific to ReLUs, then Var(W) = 2 / n_in which σ = 1/√(n_in).The magnitude of the complex parameter W is then initialized using the Rayleigh distribution with the appropriate mode σ.We can see from equation <ref>, that the variance of W depends on on its magnitude and not on its phase. We then initialize the phase using the uniform distribution between -π and π. By performing the multiplication of the magnitude by the phasor as is detailed in equation <ref>, we perform the complete initialization of the complex parameter.In all the experiments that we report, we use variant of this initialization which leverages the independence property of unitary matrices. As it is stated in <cit.>, <cit.>, and <cit.>, learning decorrelated features is beneficial for learning as it allows to perform better generalization and faster learning. This motivates us to achieve initialization by considering a (semi-)unitary matrix which is reshaped to the size of the weight tensor. Once this is done, the weight tensor is mutiplied by √(He_var/Var(𝑊)) or √(Glorot_var/Var(𝑊)) where Glorot_var and He_var are respectively equal to 2 / (n_in + n_out) and 2 / n_in. In such a way we allow kernels to be independent from each other as much as possible while respecting the desired criterion. Note that we perform the analogous initialization for real-valued models by leveraging the independence property of orthogonal matrices in order to build kernels that are as much independent from each other as possible while respecting a given criterion.§.§ Complex Convolutional Residual NetworkA deep convolutional residual network of the nature presented in <cit.> consists of 3 stages within which feature maps maintain the same shape. At the end of a stage, the feature maps are downsampled by a factor of 2 and the number of convolution filters are doubled. The sizes of the convolution kernels are always set to 3 x 3. Within a stage, there are several residual blocks which comprise 2 convolution layers each. The contents of one such residual block in the real and complex setting is illustrated in Appendix Figure <ref>.In the complex valued setting, the majority of the architecture remains identical to the one presented in <cit.> with a few subtle differences.Since all datasets that we work with have real-valued inputs, we present a way to learn their imaginary components to let the rest of the network operate in the complex plane. We learn the initial imaginary component of our input by performing the operations present within a single real-valued residual blockBN → ReLU → Conv → BN → ReLU → ConvUsing this learning block yielded better emprical results than assuming that the input image has a null imaginary part. The parameters of this real-valued residual block are trained by backpropagating errors from the task specific loss function. Secondly, we perform a Conv → BN → Activation operation on the obtained complex input before feeding it to the first residual block. We also perform the same operation on the real-valued network input instead of Conv → Max pooling as in <cit.>. Inside, residual blocks, we subtly alter the way in which we perform a projection at the end of a stage in our network. We concatenate the output of the last residual block with the output of a 1x1 convolution applied on it with the same number of filters used throughout the stage and subsample by a factor of 2. In contrast, <cit.> perform a similar 1x1 convolution with twice the number of feature filters in the current stage to both downsample the feature maps spatially and double them in number.§ EXPERIMENTAL RESULTSIn this section, we present empirical results from using our model to perform image, music classification and spectrum prediction. First, we present our model's architecture followed by the results we obtained on CIFAR-10, CIFAR-100, and SVHN^* as well as the results on automatic music transcription on the MusicNet benchmark and speech spectrum prediction on TIMIT. §.§ Image RecognitionWe adopt an architecture inspired by <cit.>. The latter will also serve as a baseline to compare against. We train comparable real-valued Neural Networks using the standard ReLU activation function. We have tested our complex models with the ℂReLU, zReLU and modRelu activation functions. We use a cross entropy loss for both real and complex models. A global average pooling layer followed by a single fully connected layer with a softmax function is used to classify the input as belonging to one of 10 classes in the CIFAR-10 and SVHN datasets and 100 classes for CIFAR-100.We consider architectures that trade-off model depth (number of residual blocks per stage) and width (number of convolutional filters in each layer) given a fixed parameter budget. Specifically, we build three different models - wide and shallow (WS), deep and narrow (DN) and in-between (IB). In a model that has roughly 1.7 million parameters, our WS architecture for a complex network starts with 12 complex filters (24 real filters) per convolution layer in the initial stage and 16 residual blocks per stage. The DN architecture starts with 10 complex filters and 23 blocks per stage while the IB variant starts with 11 complex filters and 19 blocks per stage. The real-valued counterpart has also 1.7 million parameters. Its WS architecture starts with 18 real filters per convolutional layer and 14 blocks per stage. The DN architecture starts with 14 real filters and 23 blocks per stage and the IB architecture starts with 16 real filters and 18 blocks per stage.All models (real and complex) were trained using the backpropagation algorithm with Stochastic Gradient Descent with Nesterov momentum <cit.> set at 0.9. We also clip the norm of our gradients to 1. We tweaked the learning rate schedule used in <cit.> in both the real and complex residual networks to extract small performance improvements in both. We start our learning rate at 0.01 for the first 10 epochs to warm up the training and then set it at 0.1 from epoch 10-100 and then anneal the learning rates by a factor of 10 at epochs 120 and 150. We end the training at epoch 200.Table <ref> presents our results on performing image classification on CIFAR-10, CIFAR-100. In addition, we also consider a truncated version of the Street View House Numbers (SVHN) dataset which we call SVHN*. For computational reasons, we use the required 73,257 training images of Street View House Numbers (SVHN). We still test on all 26,032 images. For all the tasks and for both the real- and complex-valued models, The WS architecture has yielded the best performances. This is in concordance with <cit.> who observed that wider and shallower residual networks perform better than their deeper and narrower counterpart. On CIFAR-10 and SVHN^*, the real-valued representation performs slightly better than its complex counterpart. On CIFAR-100, the complex representation outperforms the real one. In general, the obtained results for both representation are quite comparable. To understand the effect of using either real or complex representation for a given task, we designed hybrid models that combine both. Table <ref> contains the results for hybrid models. We can observe in the Table <ref> that in cases where complex representation outperformed the real one (wide and shallow on CIFAR-100), combining a real-valued convolutional filter with a complex batch normalization improves the accuracy of the real-valued convolutional model. However, the complex-valued one is still outperforming it. In cases, where real-valued representation outperformed the complex one (wide and shallow on CIFAR-10 and SVHN^*), replacing a complex batch normalization by a regular one increased the accuracy of the complex convolutional model. Despite that replacement, the real-valued model performs better in terms of accuracy for such tasks. In general, these experiments show that the difference in efficiency between the real and complex models varies according to the dataset, to the task and to the architecture.Ablation studies were performed in order to investigate the importance of the 2D whitening operation that occurs in the complex batch normalization. We replaced the complex batch normalization layers with a naive variant (NCBN) which, instead of left multiplying the centred unit by the inverse square root of its covariance matrix, just divides it by its complex variance. Here, this naive variant of CBN is Mimicking the regular BN by not taking into account correlation between the elements in the complex unit. The Naive variant of the Complex Batch Normalization performed very poorly; In 5 out of 6 experiments, training failed with the appearance of NaNs (See Section <ref> for the explanation). By way of contrast, all 6 complex-valued Batch Normalization experiments converged. Results are given in Table <ref>.Another ablation study was undertaken to compare ℂReLU, modReLU and zRELU. Again the differences were stark: All ℂReLU experiments converged and outperformed both modReLU and zReLU, both which variously failed to converge or fared substantially worse. We think that modRelu didn't perform as well as ℂReLU due to the fact that consecutive layers in a feed-forward net do not represent time-sequential patterns, and so, they might need to drop some phase information. Results are reported in Table <ref>. More discussion about phase information encoding is presented in section <ref>. §.§ Automatic Music TranscriptionIn this section we present results for the automatic music transcription (AMT) task. The nature of an audio signal allows one to exploit complex operations as presented earlier in the paper. The experiments were performed on the MusicNet dataset <cit.>. For computational efficiency we resampled the original input from 44.1kHz to 11kHz using the algorithm described in <cit.>. This sampling rate is sufficient torecognize frequencies presented in the dataset while reducing computational cost dramatically. We modeled each of the 84 notes that are present in the dataset with independent sigmoids (due to the fact that notes can fire simultaneously). We initialized the bias of the last layer to the value of -5 to reflect the distribution of silent/non-silent notes. As in the baseline, we performed experiments on the raw signal and the frequency spectrum. For complex experiments with the raw signal, we considered its imaginary part equal to zero. When using the spectrum input we used its complex representation (instead of only the magnitudes, as usual for AMT) for both real and complex models. For the real model, we considered the real and imaginary components of the spectrum as separate channels. The model we used for raw signals is a shallow convolutional network similar to the model used in the baseline, with the size reduced by a factor of 4 (corresponding to the reduction of the sampling rate). The filter size was 512 samples (about 12ms) with a stride of 16. The model for the spectral input is similar to the VGG model <cit.>. The first layer has filter with size of 7 and is followed by 5 convolutional layers with filters of size 3. The final convolution block is followed by a fully connected layer with 2048 units. The latter is followed, in its turn, by another fully connected layer with 84 sigmoidal units. In all of our experiments we use an input window of 4096 samples or its corresponding FFT (which corresponds to the 16,384 window used in the baseline) and predicted notes in the center of the window. All networks were optimized with Adam. We start our learning rate at 10^-3 for the first 10 epochs and then anneal it by a factor of 10 at each of the epochs 100, 120 and 150. We end the training at epoch 200. For the real-valued models, we have used ReLU as activation. ℂReLU has been used as activation for the complex-valued models.The complex network was initialized using the unitary initialization scheme respecting the He criterion as described in Section <ref>. For the real-valued network, we have used the analogue initialization of the weight tensor. It consists of performing an orthogonal initialization with a gain of √(2). The complex batch normalization was applied according to Section <ref>.Following <cit.> we used recordings with ids '2303', '2382', '1819' as the test subset and additionally wecreated a validation subset using recording ids '2131', '2384', '1792', '2514', '2567', '1876' (randomly chosen from the training set). The validation subset was used for model selection and early stopping. The remaining 321 files were used for training.The results are summarized on Table <ref>. We achieve a performance comparable to the baseline with the shallow convolutional network. our VGG-based deep real-valued model reaches 69.6% average precision on the downsampled data. With significantly fewer parameters than its real counterpart, the VGG-based deep complex model, achieves 72.9% average precision which is the state of the art to the best of our knowledge. See Figures <ref> and <ref> in the Appendix for precision-recall curves and a sample of the output of the model.§.§ Speech Spectrum Prediction We apply both a real Convolutional LSTM <cit.> and a complex Convolutional LSTM on speech spectrum prediction task (See section <ref> in the Appendix for the details of the real and complex Convolutional LSTMs). In this task, the model predicts the magnitude spectrum. It implicitly infers the real and imaginary components of the spectrum at time t+1, given all the spectrum (imaginary part and real components) up to time t. This is slightly different from <cit.>. The real and imaginary components are considered as separate channels in both model. We evaluate the model with mean-square-error (MSE) on log-magnitude to compare with the others <cit.>. The experiments are conducted on a downsampled (8kHz) version of the TIMIT dataset. By following the steps in <cit.>, raw audio waves are transformed into frequency domain via short-time Fourier transform (STFT) with a Hann analysis window of 256 samples and a window hop of 128 samples (50% overlap). We use a training set with 3690 utterances, a validation set with 400 utterances and a standard test set with 192 utterance. To match the number of parameters for both model, the Convolutional LSTM has 84 feature maps while the complex model has 60 complex feature maps (120 feature maps in total). Adam <cit.> with a fixed learning rate of 1e-4 is used in both experiments. We initialize the complex model with the unitary initialization scheme and the real model with orthogonal initialization respecting the Glorot criterion. The result is shown in Table <ref> and the learning curve is shown in Figure <ref>. Our baseline model has achieved the state of the art and the complex convolutional LSTM model performs better over the baseline in terms of MSE and convergence. § CONCLUSIONS We have presented key building blocks required to train complex valued neural networks, such as complex batch normalization and complex weight initialization. We have also explored a wide variety of complex convolutional network architectures, including some yielding competitive results for image classification and state of the art results for a music transcription task and speech spectrum prediction. We hope that our work will stimulate further investigation of complex valued networks for deep learning models and their application to more challenging tasks such as generative models for audio and images. § ACKNOWLEDGEMENTSWe are grateful to Roderick Murray-Smith, Jörn-Henrik Jacobsen, Jesse Engel and all the students at MILA, especially Jason Jo, Anna Huang and Akram Erraqabi for helpful feedback and discussions. We also thank the developers of Theano <cit.> and Keras <cit.>. We are grateful to Samsung and the Fonds de Recherche du Québec – Nature et Technologie for their financial support. We would also like to acknowledge NVIDIA for donating a DGX-1 computer used in this work.unsrtnat § APPENDIX In practice, the complex convolution operation is implemented as illustrated in Fig.<ref> where M_I, M_R refer to imaginary and real feature maps and K_I and K_R refer to imaginary and real kernels. M_IK_I refers to result of a real-valued convolution between the imaginary kernels K_I and the imaginary feature maps M_I.§.§ MusicNet illustrations§.§ Holomorphism and Cauchy–Riemann Equations Holomorphism, also called analyticity, ensures that a complex-valued function is complex differentiable in the neighborhood of every point in its domain. This means that the derivative, f'(z_0) ≡lim_Δ z → 0 [(f(z_0) + Δ z) - f(z_0)/Δ z] of f, exists at every point z_0 in the domain of f where f is a complex-valued function of a complex variable z = x + i y such that f(z) = u(x, y) + i v(x, y). u and v are real-valued functions. One possible way of expressing Δ z is to have Δ z = Δ x + iΔ y. Δ z can approach 0 from multiple directions (along the real axis, imaginary axis or in-between). However, in order to be complex differentiable, f'(z_0) must be the same complex quantity regardless of direction of approach. When Δ z approaches 0 along the real axis, f'(z_0) could be written as:f'(z_0) ≡lim_Δ z → 0[(f(z_0) + Δ z) - f(z_0)/Δ z] =lim_Δ x → 0lim_Δ y → 0[Δ u(x_0, y_0) + iΔ v(x_0, y_0)/Δ x + iΔ y] = lim_Δ x → 0[Δ u(x_0, y_0) + iΔ v(x_0, y_0)/Δ x + i 0].When Δ z approaches 0 along the imaginary axis, f'(z_0) could be written as:= lim_Δ y → 0lim_Δ x → 0[Δ u(x_0, y_0) + iΔ v(x_0, y_0)/Δ x + iΔ y]= lim_Δ y → 0[Δ u(x_0, y_0) + iΔ v(x_0, y_0)/0 + iΔ y] Satisfying equations <ref> and <ref> is equivalent of having ∂ f/∂ z = ∂ u/∂ x + i∂ v/∂ x = -i∂ u/∂ y + ∂ v/∂ y. So, in order to be complex differentiable, f should satisfy ∂ u/∂ x = ∂ v/∂ y and ∂ u/∂ y = -∂ v/∂ x. These are called the Cauchy–Riemann equations and they give a necessary condition for f to be complex differentiable or "holomorphic". Given that u and v have continuous first partial derivatives, the Cauchy-Riemann equations become a sufficient condition for f to be holomorphic. §.§ The Genralized Complex Chain Rule for a Real-Valued Loss FunctionIf L is a real-valued loss function and z is a complex variable such that z = x + i y where x, y ∈ℝ, then:∇_L(z) = ∂ L/∂ z = ∂ L/∂ x + i ∂ L/∂ y = ∂ L/∂(z) + i ∂ L/∂(z) = (∇_L(z)) + i (∇_L(z)).Now if we have another complex variable t = r + i s where z could be expressed in terms of t and r, s ∈ℝ, we would then have:∇_L(t) = ∂ L/∂ t = ∂ L/∂ r + i ∂ L/∂ s = ∂ L/∂ x∂ x/∂ r + ∂ L/∂ y∂ y/∂ r + i (∂ L/∂ x∂ x/∂ s + ∂ L/∂ y∂ y/∂ s)= ∂ L/∂ x(∂ x/∂ r + i ∂ x/∂ s) + ∂ L/∂ y(∂ y/∂ r + i ∂ y/∂ s)= ∂ L/∂(z)(∂ x/∂ r + i ∂ x/∂ s) + ∂ L/∂(z)(∂ y/∂ r + i ∂ y/∂ s) = (∇_L(z)) (∂ x/∂ r + i ∂ x/∂ s) + (∇_L(z)) (∂ y/∂ r + i ∂ y/∂ s).§.§ Computational Complexity and FLOPSIn terms of computational complexity, the convolutional operation and the complex batchnorm are of the same order as their real counterparts. However, as a complex multiplication is 4 times more expensive than its real counterpart, all complex convolutions are 4 times more expensive as well.Additionally, the complex BatchNorm is not implemented in cuDNN and therefore had to be simulated with a sizeable sequence of elementwise operations. This leads to a ballooning of the number of nodes in the compute graph and to inefficiencies due to lack of effective operation fusion. A dedicated cuDNN kernel will, however, reduce the cost to little more than that of the real-valued BatchNorm.Ignoring elementwise operations, which constitute a negligible fraction of the floating-point operations in the neural network, we find that for all architectures inand for all of CIFAR10, CIFAR100 or SVHN, the inference cost in real FLOPS per example is roughly identical. It is ∼ 265 MFLOPS for the ℝ-valued variant and ∼ 1030 MFLOPS for the ℂ-valued variant of the architecture, approximately quadruple.§.§ Convolutional LSTMA Convolutional LSTM is similar to a fully connected LSTM. The only difference is that, instead of using matrix multiplications to perform computation, we use convolutional operations. The computation in a real-valued Convolutional LSTM is defined as follows: i_t = σ( W_xi * x_t + W_hi * W_t-1 + b_i) f_t = σ(W_xf * x_t + W_hf * h_t-1 + b_f) c_t = f_t ∘c_t-1 + i_t ∘tanh(W_xc * x_t + W_hc * h_t-1 + b_c) o_t = σ(W_xo * x_t + W_ho * h_t-1 + b_o) h_t = o_t ∘tanh(c_t)Where σ denotes the sigmoidal activation function, ∘ the elementwise multiplication and * the real-valued convolution. i_t, f_t, o_t represent the vector notation of the input, forget and output gates respectively. c_t and h_t represent the vector notation of the cell and hidden states respectively. the gates and states in a ConvLSTM are tensors whose last two dimensions are spatial dimensions. For each of the gates, W_xgate and W_hgate are respectively the input and hidden kernels.For the Complex Convolutional LSTM, we just replace the real-valued convolutional operation by its complex counterpart. We maintain the real-valued elementwise multiplication. The sigmoid and tanh are both performed separately on the real and the imaginary parts. §.§ Complex Standardization and Internal Covariate Shift §.§ Phase Information Encoding | http://arxiv.org/abs/1705.09792v4 | {
"authors": [
"Chiheb Trabelsi",
"Olexa Bilaniuk",
"Ying Zhang",
"Dmitriy Serdyuk",
"Sandeep Subramanian",
"João Felipe Santos",
"Soroush Mehri",
"Negar Rostamzadeh",
"Yoshua Bengio",
"Christopher J Pal"
],
"categories": [
"cs.NE",
"cs.LG"
],
"primary_category": "cs.NE",
"published": "20170527090455",
"title": "Deep Complex Networks"
} |
Dynamical Generation of Topological Masses in Dirac Fermions Zhong-Yi Lu December 30, 2023 ============================================================New system for i-vector speaker recognition based on variational autoencoder (VAE) is investigated. VAE is a promising approach for developing accurate deep nonlinear generative models of complex data. Experiments show that VAE provides speaker embedding and can be effectively trained in an unsupervised manner. LLR estimate for VAE is developed. Experiments on NIST SRE 2010 data demonstrate its correctness. Additionally, we show that the performance of VAE-based system in the i-vectors space is close to that of the diagonal PLDA. Several interesting results are also observed in the experiments with β-VAE. In particular, we found that for β≪ 1, VAE can be trained to capture the features of complex input data distributions in an effective way, which is hard to obtain in the standard VAE (β=1). Index Terms: speaker verification, i-vectors, PLDA, β-VAE§ INTRODUCTIONIn recent years the promising deep generative model VAE (variational autoencoder) was developed <cit.> which has the following properties:(i) it can be made sufficiently deep to capture the complex data structures; (ii) it provides fast sampling of data from the inference model and (iii) it is computationally feasible and scalable. This paper presents the attempt to apply this VAE model in i-vectors space <cit.> for the speaker verification task. We deliberately chose these features in spite of the fact that they are highly Gaussian after i-vector extractor <cit.> and length normalization <cit.> and are ideal for subsequent modeling with Gaussian PLDA (Probabilistic Linear Discriminant Analysis) <cit.>. So it should be expected that for these features the performance of VAE will be limited by that of PLDA.The main goal of this paper is to develop and assay the verification backend for VAE.It is convenient to solve this task in the i-vectors space and then to extend the solution to other features.§ VERIFICATION SYSTEM BASED ON VAEIn this paper we confine to the investigation of the simplest diagonal version of VAE with a single hidden stochastic layer. §.§ VAEIt is more convenient to consider VAE as a model of deep nonlinear factor analysis (FA), though its original name <cit.> suggests the obvious relation to conventional autoencoders <cit.>. In autoencoders all hidden layers consist of only deterministic neurons whereas for the factor analysis latent variable we need at least one hidden layer consisting of stochastic neurons (see Figure <ref> where it is denoted as 𝐡).Similar to the classic factor analysis we should be able to perform the following actions: (i) to make inference for the latent variable posterior and (ii) to sample observed data vectors X. To meet these requirements VAE comprises two neural nets, namely inference net and generative net shown at the right and left parts of Figure <ref> respectively. Both of them involve the Gaussian assumptions and have identical structure.In addition to the the input layer X of size D_x and stochastic layer 𝐡 of size D_h, this structure contains the layers 𝐳 and 𝐲 of deterministic neurons of size D_d, shown on Figure <ref> as rhombs. These layers in VAE are responsible for the additional depth and for the nonlinearity:𝐳 = tanh[ W_v^(θ) + b_v^(θ)] =tanh[ W_v^(θ)], 𝐲 = tanh[ W_v^(ϕ) + b_v^(ϕ)] =tanh[ 𝐱W_v^(ϕ)],while parameters of the mean vectors and precision matrices of both generation and inference nets are computed with the linear connections only:μ_g(𝐡,θ) = 𝐳W_μ^(θ) ,τ_g(𝐡,θ) =exp[ 𝐳W_μ^(θ)], μ_r(𝐱,ϕ) = 𝐲W_μ^(ϕ), τ_r(𝐱,ϕ) = exp[ 𝐲W_μ^(ϕ)],where indices r and g are used for the inference and generation nets respectively. Hereinafter all vectors are treated as row vectors. In the expressions (<ref>–<ref>) the entire set of the generated net's parameters is denoted as θ and that of the inference net is denoted as ϕ, following the original paper <cit.>, and the additional notations like ≡ [1] andW_v^(θ)≡[W_v^(θ)^Tb_v^(θ)^T]^T are used. We consider only diagonal precision matrices τ_g and τ_r also treated as vectors. This is what we mean by diagonality of VAE. §.§ Learning VAELet 𝐗={𝐱^(i)}_i=1^N be a training set of i-vectors of the dimension D_x. Due to the nonlinearity and depth it is difficult to maximize likelihood directly via analytical EM-algorithm. That is why the authors of <cit.> have to use the VBA approximation. In the context of analogy to factor analysis we are comparing VAE to FA-VBA. Following <cit.>, let us separate the lower bound ℒ(𝐱) from the evidencelog p(): log p() = ℒ() + D_KL[q( | ) ∥ p( | )],where lower bound isℒ()=𝔼_q(|)[ logp(,)/q(|)] =𝔼_q(|)[ logp( | )p() /q(|)] = 𝔼_q(|)[ p( | ) ]- D_KL[q(|) || p()]The true posterior p(|) in (<ref>) is intractable, therefore it is approximated by the variational posterior q(|).Like in the conventional Gaussian FA-VBA, the hidden variable prior is assumed to be p()=𝒩(0|𝐈), the posteriors q(|,ϕ) and p(|(ϕ), θ) are Gaussian and the KL-divergence can found analyticallyand does not depend on . As in FA-VBA we need to maximize the lower bound ℒ() to solve the optimization task for the VAE parameters Ψ={θ, ϕ}. Due to nonlinearity and depth we are now unable to find this VBA-solution analytically. So we have to resort to the search for the stationary point Ψ_0 using the numerical stochastic gradient ∇_Ψℒ() ascent to update parameters. We also should be able to sample from q(|) during the inference stage.The computation of gradient ∇_θℒ(^(i)) does not reveal any difficulties so the standard deterministic backpropagation can be used. However the gradient ∇_ϕℒ(^(i)) looks more problematic. It is known that the naïve Monte Carlo approximation of expectation in (<ref>) which uses K samples directly from the inference net ^(k)∼ q(|^(i),ϕ) results in very high variance <cit.>. In this case the training is slow because the gradients oflog p(|(ϕ),θ) with respect to latent variableare not used <cit.>. In the papers <cit.> the reparametrization trick was proposed, according to which the vectors ^(i,k) for the Monte-Carlo estimation are not sampled from q(|^(i),ϕ) but instead generated from the deterministic transform^(i,k) = μ_r(𝐱^(i),ϕ) + [τ_r(𝐱^(i),ϕ)]^-1/2⊙ϵ^(k),where ϵ^(k) are sampled from the fixed distribution ϵ^(k)∼𝒩(0,𝐈). Using the reparametrization trick makes it possible to push gradient ∇_ϕ inside the expectation in (<ref>) because it is now taken over the fixed distribution of ϵ which is independent of ϕ. As a result, the final expression for the gradient of (<ref>) is as follows:∇_Ψℒ(^(i)) ≈1/K∑_k=1^K [∇_Ψlog p(^(i)|^(i,k))]- -∇_ΨD_KL[q(|^(i)) || p()],where p(^(i)|^(i,k)) is Gaussian with parameters μ_g and τ_g. Ultimately, we have the following expressions for the gradients of ℒ(^(i)) with respect to θ:∂ℒ(^(i))/∂W_μ^(θ) =^TA,∂ℒ(^(i))/∂W_μ^(θ) =^TB, ∂ℒ(^(i))/∂W_v^(θ) =^TG,and with respect to ϕ:∂ℒ(^(i))/∂W_μ^(ϕ) =^T(S-μ_r),∂ℒ(^(i))/∂W_τ^(ϕ) = ^T(S⊙ F+R), ∂ℒ(^(i))/∂W_v^(ϕ) = ^T{([ S⊙ F ]W_τ^(ϕ)^T + SW_μ^(ϕ)^T)⊙ T+. +.( RW_τ^(ϕ)^T - μ_rW_μ^(ϕ)^T )⊙ T},whereA = (^(i)-μ_g)⊙τ_g,B=1/2[E_x - (x^(i)-μ_g)⊙ A ], R = 1/2[τ_r^-1-E_h] , F = -1/2[τ_r^-1/2⊙ϵ^(k)], C = 1-tanh^2^(i,k), T = 1-tanh^2^(i), G = C ⊙( BW_τ^(θ)^T + AW_μ^(θ)^T), S = G W_v^(θ)^T, E_h = [diag^-1(I_D_h)]^T,E_x = [diag^-1(I_D_x)]^T.We found that K=1 and minibatch size 100 provided the best results. Moreover, when we trained VAEwith K=10 and K=100 it ceased to capture a complex structure of data and was able to generate data only from adistribution like a single Gaussian (which corresponds to the classical FA). The similar situation was observed when using a naïve Monte Carlo estimate of expectation in (<ref>) instead of reparametrization trick. §.§ RMS-prop optimizerThe choice of the optimizer like SGD or AdaGrad <cit.> is crucial for training VAE. In this work we used the RMS-prop optimizer <cit.>:MS_j^(new) = γ MS_j^(old) +(1-γ)[∂ℒ(x^(i))/∂Ψ_j]^2,where 0⩽γ⩽ 1. We divide gradient with respect to the parameter Ψ_j by square root of the smoothed value MS_j^(new). §.§ LLR scoring for VAESince VAE is a discriminative model we can only use evidence or marginal likelihood to obtain the speaker verification scores. Thus our verification score for the pair of i-vectors {_1=_test, _2=_enroll} is a Likelihood Ratio:LR(_1,_2)=P(_1,_2| H_tar)/P(_1,_2| H_imp)= P(_1,_2| θ)/P(_1| θ)P(_2| θ),where H_tar, H_imp — are the hypotheses about the facts that _1, _2 are related to the same or different speakers respectively.If only a single latent variableis used then we can estimate the marginal likelihood under impostor hypothesis with the help of the importance sampling which uses q(|,ϕ) as a proposal distribution:P(|θ) =∫p(|,θ)p()/q(|,ϕ) q(|,ϕ)d≈ ≈1/K∑_k=1^K p(|^(k),θ)p(^(k))/q(^(k)|,ϕ),where K samples ^(k) are obtained from the inference net^(k)∼ q(|,ϕ)via reparametrization trick.For the target hypothesis the situation is more complicated. To make computation feasible we assume that _1 and _2 are conditionally independent given :P(_1,_2|θ) = ∫ p(_1, _2|,θ)p() d = = ∫ p(_1|,θ)p(_2|,θ)p() d.Since such an assumption is specific for training the conventional Gaussian PLDA analyzer model <cit.>, it is natural for testing PLDA model as well <cit.>. However it is not the case for VAE, where training vectors are fed into the model without speaker labels, in fully unsupervised manner. However, our experiments (see Section <ref>) demonstrated that this assumption is highly reasonable, so VAE performs a speaker embedding which is discussed in <ref>. With using this assumption we can use the importance sampling once again to compute marginal likelihood:P(_1, _2 |θ) ≈1/K∑_k=1^K p(_1|^(k),θ)p(_2|^(k),θ)p(^(k))/q(^(k)|_2,ϕ).This expression is asymmetric, because samples are taken from q(^(k)|_2,ϕ) when i-vector _2 is an enrollment one. However, under the target hypotheses they could be taken from q(^(k)|_1,ϕ) as well. Therefore one can use the symmetric LR estimate, where P(_1, _2 | θ) = (P(_1, _2 | θ)+ P(_2, _1 | θ)/2 takes both these sampling variants into account. However we found no significant difference between P(_1, _2 |θ) and P(_1, _2 |θ) in our experiments on NIST-2010 (DET-5) <cit.>. That is why all results shown below were obtained with the use ofP(_1, _2 |θ) in log-LR estimate, i.e. on the assumption of feeding enrollment vector into inference net. §.§ β-VAEIn the recent paper <cit.> on β-VAE the empirical deviation from the exact lower bound was used:ℒ() = 𝔼_q(|)[ p( | ) ]- β D_KL[q(|) || p()].The KL-divergence term in (<ref>) can be treated as a natural regularizer (which follows from the variational Bayes) for the lower bound. It was observed in <cit.> that if VAE is trained with β>1 (i.e. with high penalty on the likelihood term) then it can better disentangle factors than with the theoretical value β=1. In the speaker recognition domain the factors are, for example, eigenvoices in PLDA model. In Section <ref> we demonstrate the results of our experiments on investigating β-VAE in both “hard” (β>1) an “soft” (0<β<1) modes.§ EXPERIMENTS AND DISCUSSIONAll our experiments were carried out for two homogeneous cellular corpora, namely NIST and RusTelecom. Train part for the NIST corpus consists of 17486 sessions from 1763 male speakers taken from NIST 1998-2008. Tests were carried out on the male part of NIST 2010 (DET-5 extended protocol) <cit.>. Train part of RusTelecom database consists of 116678 sessions from 6508 male speakers and test part consists of 235 male speakers. The details of the extraction of 400-dimensional i-vectors for the NIST corpus with using English ASR DNN and 600-dimensional i-vectors for the RusTelecom corpus with using Russian ASR DNN are described in <cit.>.For the correct comparison of VAE and PLDA the latter should have diagonal covariances for both noise and posterior components. Here we moved from PLDA with latent variable to a simple diagonal two-covariance model <cit.>. All input vectors for both PLDA and VAE experiments were centered, whitened and length-normalized in both training and testing. Hereinafter we denote a whitening matrix as U. We used full matrix U_full and diagonal matrix U_diag for the full-covariance PLDA and diagonal PLDA respectively. §.§ Speaker Embedding VAE The fact that we selected i-vectors features and thus limited the effectiveness of VAE by that of PLDA is very convenient. By carrying out extensive comparison of VAE and PLDA for β=1 (see Tables <ref> and <ref>) we can obtain two conclusions at once: * the correctness of LLR-score (<ref>),* the confirmation of the assumption (<ref>).The second conclusion states that VAE performs speaker embedding in space of latent variable . In other words, similar to PLDA,for the target hypothesis VAE is able to sample _1 and _2 from the likelihood p(|, θ) conditioned onof a single speaker. §.§ Exploring β-VAE in low-dimensional space The second effect was found during β-VAE experiments, when we explored the “soft” training mode (0<β<1).Carrying out the experiments on synthetic data we found that when 0<β<1 diagonal VAE model starts to behave like full-covariance (in posterior) VAE model being able to capture the observed training data from Gaussian clusters with non-diagonal covariance. In order to investigate this property in real-life speaker verification task we selected 11119 files of 660 male speakers having at least 10 sessions. We used PCA projections of 400-dimensional i-vectors in order to operate with a wide range of VAE's number of parameters under comparatively small training dataset.Figure <ref> shows two modes of β-VAE training for PCA=2. In Figure <ref>a the obvious capture (red points) of 660 speakers training data (green points) is observed for the weak regularization mode (β<1). This differs from the standard mode (β=1) shown in <ref>b. We found that the necessary condition for such a behavior is not only weak regularization but also the sufficient number of neurons in both stochastic and deterministic layers. For instance we unable to achieve this capture for the configuration {D_x=2; D_d=2; D_h=2}, the minimum configuration required is {D_x=2; D_d=4; D_h=4}. Our explanation is as follows. Since the expressiveness of the VAE model depends a lot on a posterior power then when increasing a number of posterior diagonal covariance elements to D_h=4 we can expect that the capabilities of the diagonal covariance VAE will be strengthened up to those of the full-covariance VAE. Anyway we can assert that making hidden layerwider than deterministic ones is necessary for such behavior.In order to find out if this effect is only a result of overfitting or not and if it may be useful in speaker verification, we carried out a number of verification experiments at PCA=10 for the different VAE configurations. To avoid a strong overfitting we performed a verification tests on the rest 6367 files out of total 17486 in parallel to training. And we stopped training when EER and minDCF metrics computed on this development set started to degrade. Then we tested the obtained VAE model on the male part of the NIST 2010 (DET-5 extended protocol) <cit.> using K=100 for estimating LR-score (<ref>). The results are shown in Table <ref>.It can be seen from Table <ref> that, contrary to out prior expectations, in both β modes VAE is able to exceed the plateau of diagonal PLDA with respect to EER and minDCF. For the “soft” β-VAE there is a conspicuous extremum on the stochastic layer sizes between D_h=40 and D_h=100. It is not the case for the standard VAE which provides good results starting from the minimal configuration {D_x=10; D_d=10; D_h=5} and right up to the maximal number of parameters which is reasonable to use when training on our small training set of 11119 files. We carried out such experiments for several values of PCA dimensions and in all cases we observed the same above behavior for two β-VAE modes. “Soft” β-VAE is better than standard one and they both are superior to the diagonal PLDA up to the dimension PCA=15 inclusive. However, starting from PCA=20 the standard VAE becomes worse than diagonal PLDA with respect to EER, though comparable to it with respect to minDCF. One can observe this behavior up to maximal PCA dimensions (limited by i-vector dimension). In these cases only minimal configurations like one shown in the first line of Table <ref> can be trained because of the small training set size.§.§ β-VAE in homogeneous corpusExperiments on the original cellular corpus NIST (17486 training i-vectors) without PCA dimensionality reduction, i.e. for D_x=400, represent the extreme case of our above observations for large PCA dimensions. The results are shown in Table <ref>. Here for all β-VAE modes best results are achieved for the configuration{D_x=400; D_d=100; D_h=50} and switched off RMS-prop (γ=1). There were 160 iterations of VAE training for it to saturate. Here we also tested “hard” β-VAE (β=4) and found that its behavior doesn't differ significantly fromthat of standard VAE (β=1).It is interesting that LLR estimate depends only marginally on a number of samples used for both β=1 and β=4. It seems, one might expect that VAE performance improves when K increases, however this behavior is observed for only “soft” β-VAE.As our experiments show such a behavior of LLR estimate is determined not by a difference in β modes. The main factor here is a tightness of the lower boundduring the training. Indeed the situation similar to that for the “soft” β-VAE in Table <ref> is observed if the tested model is underfit. In order to improve conditions for training “soft” β-VAE we moved to a larger training corpus of Russian speech, RusTelecom database <cit.>. In these experiments the optimal configuration was {D_x=600; D_d=400; D_h=200} and RMS-prop was switched off. The learning rate was piecewise-constant starting from 1e-6 and decreasing once in the middle of training. The number of iterations was 220. The results shown in Table <ref> demonstrate that we managed to slightly improve the “soft” β-VAE situation. However we should have even larger training datasets to achieve the results comparable to those of full-covariance PLDA with “soft” β-VAE. § CONCLUSIONSThe VAE-based speaker verification system in i-vector space is proposed. The LLR estimate for VAE is developed which demonstrates high effectiveness in all experiments with VAE. We showed that VAE performs a speaker embedding during training and thus, contrary to PLDA, can be trained in a fully unsupervised manner on large unlabeled datasets. We found that β-VAE can be trained in a “soft” mode which results in that its properties are close to those of full-covariance VAE model. Last, we demonstrated that in i-vectors space the effectiveness of standard diagonal VAE tends to the plateau corresponding to diagonal PLDA. Therefore we conclude that application of VAE in other features space is of interest.IEEEtran | http://arxiv.org/abs/1705.09185v1 | {
"authors": [
"Timur Pekhovsky",
"Maxim Korenevsky"
],
"categories": [
"cs.SD",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.SD",
"published": "20170525135918",
"title": "Investigation of Using VAE for i-Vector Speaker Verification"
} |
^1Applied Physics, University of Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany^[email protected]^[email protected] Efficient sources of many-partite non-classical states are key for the advancement of quantum technologies and for the fundamental testing of quantum mechanics. We demonstrate the generation of time-correlated photon triplets at telecom wavelengths via pulsed cascaded parametric down-conversion in a monolithically integrated source. By detecting the generated states with success probabilities of (6.25±1.09)×10^-11 per pump pulse at injected powers as low as 10 μW, we benchmark the efficiency of the complete system and deduce its high potential for scalability. Our source is unprecedentedly long-term stable, it overcomes interface losses intrinsically due to its monolithic architecture, and the photon-triplet states dominate uncorrelated noise significantly. 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The creation of high-dimensional (entangled) photon states, such as tripartite Greenberger-Horne-Zeilinger-states (GHZ) <cit.>, is desirable for proving deterministically the non-classical nature of quantum physics as a complete theory <cit.>, but it requires sophisticated quantum technologies to do so <cit.>. Many recent developments in quantum optics build on the benefits of robust and compact integrated circuits as parts of complex quantum networks. Integrated devices with multiple functionalities have been successfully demonstrated, e. g., in the fields of photon entanglement <cit.>, quantum interference <cit.> and boson sampling <cit.>. The technological challenge to combine multiple functionalities in a mutually compatible manner remains. To date, photonic tripartite states have been generated successfully via simultaneous <cit.> and cascaded parametric down-conversion (PDC) <cit.>, by cascading three- and/or four-wave mixing processes <cit.> or by using tri-exciton decays in coupled solid-state quantum dot sources <cit.>. Also, the generation of photon triplets using cascaded superlattices in nonlinear crystals has been studied in detail in <cit.>. Due to the chosen architectures, most of the experimental approaches inherently suffer from loss at the interfaces of the involved elements, they are space-consuming and can be susceptible to long-term stability issues. Additionally, the required pump powers for photon-triplet generation are typically of the order of several milliwatts, whereas more energy-efficient systems are favorable for real-world applications.Based on the idea of cascaded PDC <cit.>, we pursue a fully monolithic approach to generate photon-triplet states on a second-order nonlinear waveguide chip. We use lithium niobate with diffused waveguide structures <cit.>, since they offer low loss <cit.>, high source brightness <cit.> and fast electro-optical switching capabilities <cit.> for reconfigurable quantum optical applications. By contrast to schemes that utilize continuous-wave pump lasers, we deploy pump pulses, which makes our source compatible to synchronized quantum network architectures.§ DEVICE DESIGN AND THEORETICAL IMPLICATIONSWe designed and fabricated an integrated device, which is illustrated and explained in Fig. <ref>. A coupled-waveguide structure of constant waveguide width and based on titanium-diffusion <cit.> has been introduced to a 76 mm long lithium niobate chip. In front of and behind the integrated wavelength division multiplexer (WDM) we have implemented two differently poled structures, which act as guided-wave PDC sources, by periodically inverting the nonlinear susceptibility using a pulsed electric-field-poling technique. Although our waveguides in principle support both polarizations, we restrict ourselves here to PDC processes, where only TM modes are involved, since the highest nonlinear coefficient d_33 can be deployed for both down-conversions. The choice of only one polarization also implies that the designed integrated WDM has to act only as a wavelength demultiplexer, but not as a polarization-wavelength-splitting element.The actual photon-triplet generation process is considered as follows: picosecond pulses at λ_p=532 nm serve as the pump and drive the first type-0 PDC process in order to produce pairs comprising signal 1 photons (s1) at λ_s1=λ_p2=790.3 nm and idler 1 photons (i1) at λ_i1=1625 nm. The WDM separates the generated primary photons pairs with high probability in a spatio-spectral manner. The signal 1 photons remain in the original arm, whereas the idler 1 photons are transferred to the adjacent waveguide. By deploying the signal 1 photons as the pump (p2) in the second type-0 PDC stage we generate secondary signal photons (s2) at λ_s2=(1551±25) nm and idler photons (i2) at λ_i2=(1611±25) nm. In summa, we are able to create three time-correlated photons in the telecom wavelength regime with the cascaded PDC processes, while the energy conservation, ħω_p=ħω_i1+ħω_s2+ħω_i2, must be fulfilled. Likewise, the wavelength-dependent conservations of momenta in both PDC processes (commonly called phase-matchings) have to be made mutually compatible as described in Appendix A. Note that our compact monolithic approach provides intrinsic spatial mode matching and conveniently tunable spectral mode matching of the intermediate signal 1/pump 2 photons along the coupled-waveguide structure. Additionally, our titanium-diffused waveguides exhibit very low propagation loss of ∼0.08 dB/cm on average at telecom wavelengths.The first PDC stage in the cascade is pumped with a classical field. It is known that parametric down-conversion processes, which are not pumped by single-photons, produce not only single, but also higher-order photon pairs <cit.>. Thus, we must expect to generate a statistical mixture of genuine photon triplets, |ψ_triplet⟩=|1 1 1⟩, and states including higher-order photon contributions in our process. This means that we also generate states of the form |ψ_m-plet⟩=|m 1 1⟩ with certain probabilities. The mean photon number per optical pulse behind the primary PDC is denoted by ⟨ m⟩. We write for the vector containing the photon-number-occupation probabilities of the first PDC process:ρ=[ ρ_0; ⋮; ρ_n ],where the vector components ρ_m∈{ρ_0,...,ρ_n} are the probabilities to generate m PDC photon pairs, and 0≤ m≤ n is an integer number. For our case of spectrally multi-mode PDC, the photon probabilities obey Poisson statistics and are, thus, given byρ_m=e^-⟨ m ⟩⟨ m⟩^m/m!,m∈𝐍_0.Because the mean photon pair number per optical pulse can be written as⟨ m⟩=∑_m=0^∞ mρ_m,a reasonable photon-triplet generation approach must provide ⟨ m⟩≪1 for the primary PDC process. Given that case, the higher-order photon contributions are significantly reduced such that ∑_m≥2^∞ mρ_m≪ρ_1, meaning that we pump the secondary PDC stage almost exclusively with single photons. Thus, at low pump powers, we will measure mainly genuine photon triplets in the overall process. Reference <cit.> provides an in-depth theoretical analysis of quantitative measures for state preparation fidelities based on observed experimental parameters. § LONG-TERM STABLE EXPERIMENTAL SETUP For testing our device, we implement the setup shown in Fig. <ref>. Thermal stabilization of our integrated chip at temperatures of θ=164.8^∘C ensures that the intermediate signal 1 wavelength is λ_s1=λ_p2=790.3 nm, which is required for non-degenerate secondary PDC generation (see Appendix A for detailed explanation). This setting allows for optimum mutual compatibility of the two PDC processes, and it improves the spatio-spectral separability of secondary photon pairs by using fiber-based coarse wavelength division multiplexers (CWDM). These standard components also serve as high-performance filters for noise events as well as for parasitic photons from the primary PDC stage. Otherwise, those photons could affect the detection of secondary PDC photons, because the first PDC process happens around ten million times more often than the second one.We measure the photon-triplet events using two superconducting nanowire single photon detectors (SNSPDs) with η_det,i1=0.6 and η_det,i2=0.7 of detection efficiency (Opus One, Quantum Opus/PicoQuant Photonics North America Inc.), as well as one InGaAs avalanche photo diode with η_det,s1=0.25 (ID230-SMF-FR, ID Quantique SA) in conjunction with a time-tagging module. Optical and electronic path differences between the three detectors have been compensated for, such that the expected, time-correlated detection events can be registered at around zero delay with respect to each other. The measurements have been performed for 11.5 hours with very high stability. This is indicated by the plot in Fig. <ref>, where we show the relative change of the idler 1 count rate with respect to the average value. We chose the idler 1 detection events for monitoring the stability, because they occur orders of magnitudes more often and with significantly better signal-to-noise ratios than secondary PDC detection events. The single event rates of signal 2 and idler 2 are orders of magnitudes lower than the respective detector noise count rates and have not been considered as stability indicators.The continuous-wave-equivalent pump power of P_pump=(10.0±0.1) μW at a repetition rate of 10 MHz corresponds to a mean photon pair number per pulse of only ⟨ m⟩=0.215±0.02 behind the first PDC stage. Thus, we expect that predominantly (∼88 %) genuine photon-triplet states |ψ_triplet⟩=|1,1,1⟩ are generated. Likewise, we deduce that a high conversion efficiency in our primary PDC stage limits the cleanliness of the generated photon-triplets by generating higher-order photon pairs <cit.>.We chose the signal 2 photon detection as the reference events for our data analysis, because these occur at the lowest detection rates due to the InGaAs detector efficiency. A three-fold coincidence is given, if a signal 2 detection event announces the detection of its idler 2 twin photon, and if the corresponding idler 1 photon is also registered. Hence, the relative arrival times for idler 1 events are labeled τ_1-τ_2, whereas τ_3-τ_2 denotes the relative arrival time of idler 2 photons with respect to the signal 2 photons. This pseudo-heralding method significantly reduces the computational effort for the post-selection. Additionally, we merge the bins of our time-tagging module sixteen-fold in order to include the joint timing jitter of our apparatus. Thus, the time-bins for the data analysis have widths of (1.317±0.002) ns in both temporal directions.§ RESULTS AND DISCUSSION Our data analysis benefits from the pulsed pump, which allows us to distinguish between time-correlated photon triplets and noise-related three-fold coincidences. The latter appear due to dark counts of the detector and the blackbody radiation emitted by our heated integrated device. The effect of noise is shown in Fig. <ref> (left) for a time window of about 40 ns×40 ns. At relative arrival time delays between individual detection events of τ_3-τ_2=τ_1-τ_2=(-0.165±0.001) ns, we notice time-correlations as an indicator for photon-triplet detection. However, due to the pulsed operation, this result alone does not prove the generation of genuine photon-triplet states. For the verification we have to make sure that the influence of accidental three-fold-coincidences is negligible.By analogy to conventional pulsed parametric down-conversion <cit.>, we can deduce the impact of accidentals by extracting three-fold coincidences, where neighboring pulses are involved. For their identification we analyze a larger, 600 ns×600 ns-wide, time window. This corresponds to ∼210000 bins of (1.317±0.002) ns width, surrounding the signal 2 detection events. It also implies that, due to the repetition time of our pump laser system of 100 ns and the sixteen-fold merging of the time-bins, we have access to 41 neighboring pump pulses within the time window for the estimation of higher-order photons and other accidental contributions to the three-fold coincidence rate.Indeed, we find accidental three-fold coincidences at multiple integers of the pump repetition time for both temporal directions. We illustrate this in Fig. <ref> (right), where only a fraction of the analyzed time window is shown for clarity. The graph shows that higher-order photons contribute to the three-fold coincidences along the (τ_1-τ_2)-axis, indicating an increase on the primary idler detection probability. The appearance of three-fold coincidences in neighboring pulses along the (τ_3-τ_2)-axis, where only secondary photons should reside, indicates other parasitic influences: either higher-order idler 1 photons survive the demultiplexing on-chip and the subsequent CWDM-filtering, or primary PDC processes involving higher-order mode combinations produce idler 1 photons at secondary PDC photon wavelengths. It is also possible that nonlinear Cherenkov-type PDC <cit.> is generated in the primary PDC stage with a broad idler distribution in the secondary output arms.We quantify the impact of accidentals by comparing the number of three-fold coincidences in the center spot of the graph in Fig. <ref>, where we suspect our photon-triplets to reside, with the average number of accidental three-folds in the 41 bins, where neighboring pulses are involved. By division of the two results we infer a coincidences-to-accidentals-ratio of CAR=9.4±1.9. This means that the influence of higher-order photon contributions and other accidentals from the primary PDC stage is not negligible, but very low.Additionally, we perform a statistical analysis of the 210000, 1.317ns-wide, time-bins in order to answer the question: how many three-fold coincidences occur how often. Our intention for this analysis method is to identify noise-related contributions, genuine photon triplets, and also accidental three-fold coincidences. Besides our expectation, that the photon triplets overcome the noise background significantly, the distribution of the accidentals should also deviate from the noise statistics, because those pseudo-time-correlated accidental events are generated by the same pulsed pump that generates the photon triplets.In the histogram in Fig. <ref>, we show the result of our statistical analysis. On the x-axis, we plot the number of three-fold coincidences per time-bin during 11.5 hours of measurement time. The y-axis shows the absolute frequencies of these events' occurrences. We measured 33 three-fold coincidences only once with standard deviation of σ_triplet=5.7. The noise-related background events are visibly separated and average to ⟨ N_3-fold⟩=0.048 three-fold coincidences with a standard deviation of σ_3-fold=⟨ N_3-fold⟩^1/2=0.218. The blue line is a Poisson fit of the overall measurement data, the vast majority of which are time bins containing noise. Our result indicates that we are able to detect time-correlated three-fold coincidences with a signal-to-noise-ratio of SNR>680. Assuming for now that the measured rate per time-bin of 33 three-folds stems solely from noise contributions would mean, that it was 150 standard deviations away from the average noise-related three-fold coincidence rate per bin. In other words: the probability of measuring a noise-related rate of 33 three-folds per bin in 11.5 hours is around p^noise_3-fold(33)≈3.3×10^-81 and can be considered impossible. Thus, our statistical analysis underlines the strong evidence for time-correlated photon triplets in only one temporal measurement bin. We also notice in the histogram that the accidentals, stemming from neighboring pulses along τ_3-τ_2 and τ_1-τ_2, also deviate significantly from the noise-dominated fit curve. This behavior indicates the suspected pseudo-time-correlations of the accidentals due to generation in neighboring pulses. We refer the kind readership to Appendix B, where we provide arguments for the validity of Poisson statistics of noise-related three-fold coincidences.Our findings verify the generation of 33±5.7 time-correlated photon triplets per 11.5 hours at the expected relative arrival times. From the absolute number of triplets within the whole measurement duration, we deduce a success probability for the detection of photon triplets P^exp_triplet=(6.25±1.3)×10^-11 per pump pulse. In order to compare this value with the success probability expected from our experimental circumstances, we take the set pump power and wavelength, 𝐏_p and λ_p, the pump laser repetition rate R_rep and the efficiencies of the three measurement arms, η_i1, η_s2 and η_i2, into account. From separate measurements we inferred the individual PDC conversion efficiencies, P_PDC,1=(8.1±0.1)×10^-8 and P_PDC,2=(2.7±0.1)×10^-7 pairs per pump photon. Additionally, the injection efficiency of the pump into the waveguide structure is around η_p^in=0.5±0.1. By calculatingP^th_triplet=η_i1η_s2η_i2· P_PDC,1 P_PDC,2𝐏_pη_p^inλ_p/h c_vac R_repwith the vacuum speed of light c_vac, we get a theoretical success probability of P^th_triplet=(6.35±1.5)×10^-11 per pulse. This is in excellent agreement with the experimentally derived benchmark. Note that the scalability of our source is inherent to Eq. (<ref>), because for identical pulse energies the ratio 𝐏_p/R_rep and P^th_triplet do not change, whereas the absolute number of successfully detected photon triplets increases with higher repetition rates of the pump laser. The pulsed excitation in general and in conjunction with the emission wavelengths of the triplet states make our source fully compatible with existing synchronized telecom infrastructure.In comparison to other approaches on tripartite-state generation <cit.>, our integrated source offers around two to four orders of magnitude less photon-triplets per unit time. The main limiting factors on the detected photon triplets rates in our setup are given by the necessity of spectral filtering in the secondary PDC output and by the individual conversion efficiencies of the two PDC stages. Further improvements of our waveguide technology can reduce their impact on the success probability in the future. We expect an increase of detected photon-triplet rates of at least one order of magnitude solely by accessing the full spectral bandwidth of the secondary PDC outcome. The resulting, spectrally multimode, secondary photon pairs can be deployed for example for absolute calibration of broad-band-sensitive single photon detectors in the telecommunication bands. Besides, the strong temporal correlations of our spectrally multi-mode secondary PDC photons offer energy-time-entanglement. This could be combined in future work with time-bin-entanglement schemes and would allow for the heralded generation of hyper-entangled Bell-states.The implementation of reverse proton-exchanged waveguide structures <cit.> could also increase the PDC conversion efficiencies, each by at least one additional order of magnitude. Finally, higher repetition rates of the pump laser, e. g. by temporal multiplexing <cit.> will lead to increasing numbers of detectable photon triplets and indicate the scalability of our integrated device. Note that state-of-the-art detector recovery times of around 75 ns for highly efficient MoSi-based SNSPDs <cit.>, in conjunction with detection efficiencies of η_det∼87%, are still the limiting factor to date rather than available repetition rates for the pump laser. By contrast, increasing the pump pulse energy will not improve the output of photon triplets at high CAR-values, because of the growing impact of higher-order photon contributions. Assuming an identical performance of our source in combination with three high-efficiency MoSi detectors, we are actually limited to pump repetition rates of around 62 MHz without losing photons due to detector recovery effects. The corresponding gain in terms of photon triplet detection rate would be about 6.2 as compared to this work. Another factor of 6 can be achieved due to the increased efficiency of those novel detectors. Summarizing these sources for improvement we expect to increase the photon-triplet verification rates by around four to five orders of magnitude in future work. This will let the application of pulsed and integrated cascaded parametric down-conversion sources in the field of quantum communication get into reach. Note also that our device does not yet represent a source for multi-partite entanglement. Slight technical variations can be made in order to generate polarization-entangled GHZ-states on-chip, such as replacing our type-0 PDC sources with cascaded type-II PDC stages, each having interlaced poling structures <cit.>. Additional guided-wave polarizing beam splitters and electro-optical polarization controllers could support these integrated devices. § CONCLUSION In conclusion, our monolithic photon-triplet source demonstrates the strengths of integrated quantum optics in second-order nonlinear materials in terms of robustness to environmental influences and state preparation with high signal-to-noise ratios and coincidences-to-accidentals-ratios. Our integrated device marks important progress towards scalable, miniaturized and reconfigurable quantum circuits with high integration densities, long-term stability and the mutual compatibility with the infrastructure of existing and future quantum networks.The fundamental dependence of the cascaded triplet generation on higher-order photon contributions also offers new ways for studying decoherence at the transition between the micro- and the macro-world. Seeding our primary PDC process with synchronized weak coherent light at idler 1 wavelengths, for example, provides the generation of single-photon-added coherent states <cit.> paired with two single photons. This lies at the heart of micro-macro-entanglement and allows an integrated approach, e.g. for the generation of Schroedinger-cat-like states <cit.>. Likewise, the monolithic generation of heralded exotic quantum states, in conjunction with the opportunity to add fast optical switches to the very same chip, paves the way for future prospects of quantum communication and quantum network technology.§ ACKNOWLEDGMENTS The authors thank R. Ricken, H. Suche, K. Shalm and T. J. Bartley for fruitful discussions and A. J. Miller, V. Ansari and G. Harder for their help with the detection apparatus. This work has been supported by the German Research Foundation (DFG) (Graduiertenkolleg 1464 "Mikro- und Nanostrukturen in Optoelektronik und Photonik"). § APPENDIX A: TESTING THE MUTUAL COMPATIBILITY OF TWO PARAMETRIC DOWN-CONVERSION PROCESSES ON-CHIP Parametric down-conversion (PDC) processes require energy conservation of pump (p), signal (s) and idler (i) photons. This can be expressed in the frequency notationħω_p=ħω_s+ħω_i,and in the wavelength notation1/λ_p=1/λ_s+1/λ_i.Likewise, the conservation of momentum must be fulfilled, which is commonly referred to as phase-matching and expressed byΔk⃗=k⃗_p-k⃗_s-k⃗_i.In waveguides, which are usually dispersive, the co-linear propagation reduces the vectorial notation to a wave-number representation, k⃗_i→ k_i=2π n_eff,i/λ_i. The effective refractive indices of the guided waves are given by n_eff,i. Guided-wave PDC requires the compensation of a phase-mismatch Δ k between the three involved photons at wavelengths λ_p, λ_s and λ_i. By periodically inverting the nonlinear susceptibility with period Λ_G, we create a rectangular grating with the corresponding wave-number Δ k=2π/Λ_G. We include this to the momentum conservation condition:k_p-k_s-k_i±2π/Λ_G=Δ k=0.This expression facilitates quasi-phasematching of almost arbitrary wavelength combinations. The effective refractive indices are typically dependent on the temperature of the waveguide material. Thus, the PDC emission wavelengths can be tuned by thermal manipulation.In our integrated lithium niobate chip, we aim for TM_00 mode-conversion of green picosecond pump photons (p) to a pair of TM_00 photons at around λ_s1=790.5 nm (signal 1) and λ_i1=1625 nm (idler 1). The signal 1 photon can subsequently decay to TM_00 “granddaughter” photon pairs, signal 2 and idler 2, with a spectral distribution of about Δλ≈±35 nm around the degeneracy wavelength λ_s2/i2=1581 nm. The overall cascaded PDC process is described by the formula[ p →i1 +s2 + i2,;532 nm → 1625 nm +(1581∓35) nm + (1581±35) nm. ] Before setting up our cascaded PDC process, the individual PDC sections have been characterized thoroughly, because the intermediate signal photons will serve as the pump for the secondary PDC process. This means that we have to make both processes mutually compatible, since the chosen poling periods are fixed and allow only for raw setting of the quasi-phase-matching conditions. Thus, we acquired the spectra of the signal photons of the primary PDC process at different temperatures using a commercial fiber-coupled spectrometer system and deduced the temperature tuning curve of the first PDC process. Additionally, we characterize the secondary PDC stages in two different ways. First, we make use of the fact that second harmonic generation (SHG) represents the reverse three-wave mixing process of degenerate PDC. Therefore, we inject coherent fundamental light from a tunable external cavity laser at wavelengths 1570 nm≤λ_F≤1610 nm to our waveguide structures and measure the SHG with a photo-diode. In order to fine-tune the secondary quasi-phasematching condition, we also focus on the temperature-dependent behavior of the SHG peak wavelength.With the temperature tuning curves of both down-conversion stages stages at hand, we extract a principle operation temperature (POT) for the cascaded parametric down-conversion process. At θ^POT=163.5^∘C, we observe a signal 1 wavelength of λ_s1=λ_p2=(790.47±0.35) nm, which is shown in Fig. <ref> (left). However, at this temperature-wavelength-combination the secondary PDC emission will be degenerate with broad spectral distribution. But for our photon-triplet detection we want to split the secondary photons quasi-deterministically. A non-degenerate operation is beneficial to achieve this. As the second characterization method we therefore performed the direct generation of secondary PDC photon pairs at the fixed device temperature θ=(163.5±0.1)^∘C. Picosecond pulsed laser light is deployed in the range of 785.84 nm≤λ_p2≤791.88 nm and in steps of Δλ_p2=0.23 nm. We use a highly dispersive fiber in conjunction with a superconducting nanowire single photon detector (Opus One, Quantum Opus) and a time-tagging module (TTM8000, Austrian Institute of Technology) in order to build a calibrated spectrometer<cit.>. This system stretches the unfiltered signal 2/idler 2 pulses in time, according to their spectral components<cit.>.We acquire the numbers of click events from the secondary PDC source for 30 seconds and plot the outcomes pump-wavelength-dependent and color-coded in Fig. <ref> (right). When pumping at λ_p2=(790.49±0.23) nm, we identify signal 2 and idler 2 at degenerate wavelengths of λ_s2=λ_i2=(1581.0±0.5) nm, as we expected it. We also infer from the graph that, with decreasing pump wavelengths, the PDC emission splits into two arms of non-degenerate signal and idler wavelengths. The spectral bandwidth of the signal arm narrows down at shorter pump wavelengths. The same holds true for idler photons due to energy conservation. The graph does not provide this feature, because the idler photons tend to be weakly guided at wavelengths higher than λ_i2≥1635 nm. This effect could be reduced by dispersion engineering of our waveguides.In order to prevent idler photon scattering to the lithium niobate substrate, we inferred λ_s1=λ_p2=790.3 nm as the optimum pump wavelength for the secondary PDC process. This has also the advantage, that signal and idler emission are concentrated in the wavelength regions around λ_s2=(1551±25) nm and λ_i2=(1611±25) nm, respectively. That choice allows us to use fiber-based coarse wavelength division multiplexers (CWDM) with very good filtering properties for unwanted wavelengths. Note, however, that these filterws have a narrower transmission bandwidth than our PDC emission, andwe will reduce the detectable event rates in turn.Additionally, we estimate the pump acceptance bandwidth of our secondary PDC stage, which should spectrally match with the primary signal wavelength in the cascaded process. Integrating the individual 2^nd-stage PDC emission spectra over time and subtracting the integral noise background results in the graph in Fig. <ref>, where we plot the dependency on the pump wavelength. The accumulated emission shows a maximum at the degeneracy point λ_p2=790.5 nm. The data points at short pump wavelengths reflect non-degenerate PDC. By contrast, the steep drop above the degeneracy pump wavelength indicates the tendency to non-phase-matched cases. From the Gaussian fit, we deduce a spectral acceptance bandwidth of Δλ^FWHM_p2=(0.749±0.054) nm. This value is narrower than what we measured for the emission of the primary PDC signal photons, which means that the spectral overlap of the two processes was limited to η_λ_s1-λ_p2=0.88.Further technological improvement will allow for the exact matching of the two bandwidths, e. g. by adapting the effective lengths of the two involved periodically poled areas.As the final characterization result, we extract the optimum operating temperature of θ^opt=163.8^∘C for the cascaded PDC process. We take the temperature tuning curves of each PDC stage into account and consider the desired non-degenerate emission of secondary PDC photons for optimized filtering. At this operating temperature, the intermediate primary signal wavelength is stabilized at λ_s1=λ_p2=(790.3±0.032) nm, which provides very good spectral overlap of the two individual PDC processes. § APPENDIX B: HISTOGRAM OF THE ABSOLUTE FREQUENCIES OF THREE-FOLD COINCIDENCES In Fig. <ref> of the main text, we fit the absolute frequencies of the detected three-fold coincidences per time bin in terms of a Poisson distribution. Because our data analysis software does not provide factorials, we calculate them by using the gamma function N!=Γ(N+1), n∈ℕ. The Poisson fit method is reasonable here, but requires additional explanation at this point.We are aware that each detector might show an individual (probably thermal) dark count statistics. Additional contributions to our noise floor, i. e. the emitted blackbody radiation of our source, is also expected to be distributed with thermal statistics. However we expect that our actual three-fold coincidence statistics involve several modes and detectors such that their convolution results in a Poisson distribution. Moreover, in the limit of small expectation values or mean values of the absolute frequency of three-fold coincidence per time bin, both distributions tend to behave identical. Thus we consider the description of our rates by means of Poisson statistics to be adequate. The three-fold coincidence rate for noise and accidentals is hence modeled by the convolution of three Poisson distributions. By contrast, the generation of the overall photon triplets is also assumed to follow Poisson distribution, since they stem from cascaded multi-mode PDC processes. But the triplets manifest themselves by strong correlations for the three-fold coincidence rates, because three photons will be generated for every cascaded PDC event. We note that it is worth to model our system in terms of the exact statistical behavior. But since the description of the individual detector responses is typically non-trivial, it should be given elsewhere in more detail. | http://arxiv.org/abs/1705.09734v1 | {
"authors": [
"Stephan Krapick",
"Benjamin Brecht",
"Harald Herrmann",
"Viktor Quiring",
"Christine Silberhorn"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170526225025",
"title": "On-chip generation of photon-triplet states"
} |
Term Models of Horn Clauses over Rational Pavelka Predicate Logic [================================================================= Many data producers seek to provide users access to confidential data without unduly compromising data subjects' privacy and confidentiality.One general strategy is to require users to do analyseswithout seeing the confidential data; for example, analysts only get access to synthetic data or query systems that provide disclosure-protected outputs of statistical models.With synthetic data or redacted outputs, the analyst never really knows how much to trust the resulting findings. In particular, if the user did the same analysis on the confidential data, would regression coefficients of interest be statistically significant or not?We presentalgorithms for assessing this question that satisfy differential privacy.We describe conditions under which the algorithms should give accurate answers about statistical significance.We illustrate the properties of the proposed methods using artificial and genuine data.Keywords: Confidentiality, Disclosure, Laplace, Query, Synthetic, Verification§ INTRODUCTION In many settings, data producers such as national statistical agencies, survey organizations, health systems, and private sector companies—henceforth all called agencies—seek to provide researchers and the broader public access to data on individual records. However, these agencies are ethically and often legally obligated to protect the confidentiality of data subjects' identities and sensitive attributes.Research has shown that stripping obvious identifiers, like names and addresses, may not suffice to protect confidentiality<cit.>.Ill-intentioned users—henceforth called intruders—may be able to learn sensitive information by linking released data files to records in external databases by matching on fields common to both datasets, thereby breaking the protection from de-identification.In recognition of this threat, agencies have developed and deployed techniques that allow users to do analyses without seeing the actual data.One approach is to use remote access query systems <cit.> in which the user submits a queryto a server that holds the data for output from some statistical model.The server runs the query and reports back the analysis results to the user, e.g., estimated regression coefficients and their standard errors, without ever allowing the user to see the individual-level data.To further reduce disclosure risks, the outputs usually are coarsened or perturbed <cit.>; for example, the server can add noise to outputs that satisfies the risk criterion differential privacy <cit.>. Query system approaches are used by many government agencies, such as the Census Bureau and the Australian Bureau of Statistics,and are being implemented for general social science data access, for example in DataVerse <cit.> and in the Private data Sharing Interface <cit.>.A second approach is to release fully synthetic data <cit.>. Here, the agency generates new values for every confidential datum by sampling from a predictive distribution estimated with the confidential data.Since all values are simulated, it is nonsensical for intruders to match released cases to external records.Synthetic data have been used in several public use data products, including the Longitudinal Business Database<cit.>,the Survey of Income and Program Participation<cit.>, and the OnTheMap application <cit.>.While query systems and synthetic data are appealing options for data release, they have a significant drawback: it is difficult for analysts to know how much they should trust the results of their analyses.For example, in a query system, the user might ask for outputs from a regression model that, in actuality, fits poorly on the confidential data. This lack of fit cannot be easily detected from the coefficients and standard errors alone, whether they are perturbed or not.Additionally, the steps taken to perturb the outputs could infuse substantial error into the reported coefficients.Similar dilemmas arise for synthetic data.By default, the synthetic data reflect only those distributional features and relationships encoded in the synthesis models<cit.>. The synthesis models may fail to describe the data in ways that lead the analyst to findings that are not supported by the confidential data.Further, even when the synthesis models adequately describe the distributions in the confidential data, the process of generating synthetic data tends to increase standard errors, which could obscure important relationships.The literature on privacy-preserving data analysis has begun to address aspects of this problem. <cit.>suggests that linear regression output from query systems be accompanied by synthetic plots of residuals versus predicted values.Related residual diagnostics for logistic regressions are proposed in <cit.> and <cit.>. <cit.>present an algorithm for releasing residual plots (and also an algorithm for ROC curves for logistic regression) that satisfies differential privacy. Their algorithm takes as input the privately-estimated coefficients, which could come from noisy outputs or a synthetic data analysis.While useful diagnostic tools, these plots do not provide analysts with means to compare inferences obtained via the privacy-preserving mechanism to those that would be obtained from the confidential data.It may be, for example, that a particular regression coefficient of substantive interest has a large p-value in the noisy output or synthetic data, even though it has a small one in the confidential data, or vice versa.In this article, we present algorithms for comparing the sign and significance level of privately-computed regression coefficients, i.e., those computed via output perturbation or via synthetic data, with those computed from the confidential data.We envision the outputs of these algorithms being delivered to users via a verification server <cit.>.This is a query system that allows users to ask for measures indicating how similar privately-computed results are to those based on the confidential data without allowing users to see the confidential data.The algorithms satisfy differential privacy, which has important benefits in this interactive context.As shown in <cit.> and <cit.>, when verification servers provide exact (unperturbed) answers to queries about similarity of results, intruders can query the server repeatedly to gather information that, in combination, provides unacceptably tight ranges for individual confidential values. Differential privacy provides provable bounds for the amount of information leaked by the server over repeated queries, regardless of their nature. The basic idea of the algorithms is built on the subsample and aggregate mechanism of <cit.>.We randomly partition the confidential data into M disjoint subsets. In each subset, we estimate the regression using only the data in that subset, from which we compute the univariate t-statistic for the regression coefficient(s) of interest to the user.Wetruncate each t-statistic at some user-defined threshold a; this facilitatesdifferentially private algorithm design, as we discuss later. We add noise to the average of the truncated t-statistics, sampled from a Laplace distribution with variance tuned to satisfy differential privacy.We refer the resulting noisy statistic to an appropriate reference distribution under the null hypothesis that the coefficient equals zero, resulting in calibrated p-values.The p-value can be used directly as evidence of the significance of the coefficient, or it can be compared with the corresponding, privately-computed p-value for purposes of verification.The sign of the noisyt-statistic also provides a differentially private estimate of the sign of the coefficient. We are not aware of algorithms for differentially private significance tests for linear regression coefficients, although the literature on differential privacy includes significance tests for models appropriate for other settings.Several authors have developed differentially private significance tests for categorical data and contingency tables<cit.>.These tests cannot be sensibly used for linear regression, where the outcome variable, as well as potentially some of the explanatory variables, are assumed to be continuous rather than categorical.<cit.> and <cit.> propose differentially private significance tests for the mean and the difference of means of Gaussian random variables, respectively.These authors assume that bounds for the means or the data values are known, whereas we work in multivariate regression settings where bounds on the variables need not be known. <cit.> propose a differentially private algorithm for analysis of variance, which is a special case of linear regression with only categorical explanatory variables.They restrict results to outcomes that lie on the unit interval, do not consider continuous predictors, and report only the result of the omnibus significance test that all coefficients simultaneously equal zero. <cit.> propose differentially private algorithms to obtain confidence intervals for single means of Gaussian random variables, which can be inverted to significance tests.Their approach relies on spending some privacy budget to bound the range and standard deviation of the data values for the single variable.Their approach does not apply in a straightforward manner to regression modeling with multiple explanatory variables, as the expression for the variance of any estimated regression coefficient has a numerator and denominator that are non-linear functions of all the variables used in the regression model.Multiple authors have developed differentially private algorithms for estimating pieces of the outputs needed for significance testing of coefficients in linear regressions or other predictive models<cit.>. None of these works includes procedures to compute standard errors, making it impossible to conduct significance tests.<cit.> presents an algorithm for estimating regressions that does provide standard errors and hence significance tests; however, the algorithm sometimes returns output associated with a variant of ridge regression rather than strictly linear regression. The algorithm also requires all data values to be bounded, which we do not require in our algorithms. Finally, the algorithm appears to require (ϵ, δ > 0)-differential privacy to give useful outputs, whereas our algorithm allows for ϵ-differential privacy,The remainder of the article is organized as follows. In Section <ref>, we review differential privacy and some of the techniques used to design algorithms that satisfy it. In Section <ref>, we present the algorithm for the differentially private t-statistic, including its reference distribution. In Section <ref>, we discuss some theoretical aspects of the differentially private t-statistic, focusing on approximation properties and type II error rates. In Section <ref>, we present results of simulation studies that illustrate the performance of the differentially private t-statistic in finite samples.In Section <ref>, we present an approach for choosing the number of partitions and the threshold level.These drive the accuracy and usefulness of the inferences.In Section <ref>, we conclude with suggestions for implementation of these techniques, as well as discuss future research topics around verification of privately-computed regression quantities.§ REVIEW OF DIFFERENTIAL PRIVACY Before reviewing differential privacy, we motivate why one should not simply release verification measures without redaction. Suppose that an intruder asks a verification server to return the value of the t-statistic for the slope in a regression of an outcome y on a single predictor x, and that the verification server provides this value from the regression estimated with the confidential data. Consider a worst case scenario: the intruder knows the values of (x_i, y_i) for all but one record in the confidential data.If the intruder submits a regression involving all records and then requests the t-statistic, the user can try various combinations of (x, y) for that unknown record until finding the set of values that yield the reported t-statistic.More generally, similar attacks work for an intruder who knows the values of (x_i, y_i) for any r records in the confidential data when the intruder can request output from a regression estimated with those r records plus one additional record.As these examples illustrate, it is desirable to redact the verification measures before releasing them, which we do using differential privacy.Let 𝒜 be an algorithm that takes as input a database 𝐃 and outputs some quantity o, i.e., 𝒜(𝐃) = o.In our context, these outputs are used to form verification measures for the t-statistic and sign.Define neighboring databases, 𝐃 and 𝐃', as databases that differ in one row and are identical for all other rows. Specifically, 𝐃 and 𝐃' are neighboring databases if there exists only one record d ∈𝐃 and one record d' ∈𝐃' such that d ≠ d' and 𝐃 - {d} = 𝐃' - {d'}.An algorithm 𝒜 satisfies ϵ-differential privacy if for any pair of neighboring databases (𝐃, 𝐃'), and any non-negligible measurable set S ⊆ range(𝒜),the Pr(𝒜(𝐃) ∈ S) ≤exp(ϵ) Pr(𝒜(𝐃') ∈ S). Intuitively, 𝒜 satisfies ϵ-DP when the distributions of itsoutputs are similar for any two neighboring databases, where similarity is defined by the factor exp(ϵ). The ϵ, also known as the privacy budget, controls the degree of the privacy offered by 𝒜, with lower values implying greater privacy guarantees. ϵ-DP is a strong criterion, since even an intruder who has access to all of 𝐃 except any one row learns little from 𝒜(𝐃)about the values in that unknown row when ϵ is small.Differential privacy has three other properties that are appealing for verification measures.Let 𝒜_1(·) and 𝒜_2(·) be ϵ_1-DP and ϵ_2-DP algorithms.First,for any database 𝐃, releasing the outputs of both 𝒜_1(𝐃) and 𝒜_2(𝐃) ensures (ϵ_1 + ϵ_2)-DP.Thus, we can quantify and track the total privacy leakage from releasing verification measures.Second, releasing the outputs of both 𝒜_1(𝐃_1) and 𝒜_2(𝐃_2), where 𝐃_1 ∩𝐃_2 = ∅, satisfies max{ϵ_1, ϵ_2}-DP.Third, for any algorithm 𝒜_3(·), releasing 𝒜_3(𝒜_1(𝐃)) for any 𝐃 still ensures ϵ_1-DP.Thus, post-processing the output of ϵ-DP algorithms does not incur extra loss of privacy.A common method for ensuring ϵ-DP is the Laplace Mechanism <cit.>. For any function f : 𝐃→ℝ^d, let Δ(f) = max_(𝐃_1,𝐃_2) ||f(𝐃_1) - f(𝐃_2)||_1, where (𝐃_1, 𝐃_2) areneighboring databases.This quantity, known as the global sensitivity of f, is the maximum L_1 distance of the outputs of the function f between any two neighboring databases.The Laplace Mechanism is 𝐋𝐌(𝐃) = f(𝐃) + η,where η is a d × 1 vector ofindependent draws from a Laplace distribution with density p(x |λ) = (1/(2λ)) exp(-|x| / λ), where λ = Δ(f) / ϵ. We use the Laplace Mechanism to design verification measures that satisfy ϵ-differential privacy, which we refer to as ϵ-DP verification measures. We also use the subsample and aggregate technique <cit.>. This technique allows us to reduce the global sensitivity of f, thereby reducing the variance in the noise distribution. To implement this technique, we randomly partition the datasetinto M disjoint subsets, _1,…,_M. We then compute f(_1),…,f(_M) and their average M^-1∑_l=1^M f(_l).The global sensitivity of M^-1∑_l=1^M f(_l) is 1/M times that of f(), since any single observation appears in at most one of the partitions.Finally, we use the Laplace mechanism to release a noisy version of M^-1∑_l=1^M f(_l). <cit.> - Johnson-Lindenstrauss projection of A=[X;Y]_n × d - Output 1: RA = A'=[X';Y']_r × d, where R is random and r is an input that reduces the dimension of A.- Output 2: A'=RA^*=[X';Y']_r × d where A^* is an augmented version of A.- If the smallest singular value of A is large, then Output 1 is return with high probability.- If the smallest singular value of A is small, then Output 2 is return with high probability.- β̃ = ((RX')^T RX)^-1 (RX')^T RY- When output 2 is return, β̃ is a good approximation to the ridge estimator of β with penalty term w = f(B,ϵ,δ,r) ... B is a bound on the l_2 norm of any row in A, which is also an input.- If there is a predictor in X that predicts very well Y, then the singular value of A is small and output 2 will be return. <cit.> * They project the outliers in the private dataset. To do so, they have to fix two bounds B_y and B_x and truncate Y and X using these bounds.* Let Y' and X' be the truncated versions of Y and X.* X^TX and X^TY are released using the Wishart and Laplace mechanism ...this not enough to compute standard errors* They do not use measurement error models to make inferences. <cit.> * They use the data to compute the sensitivity of a function. I do not know how illegal is to do that. * They trimmed the data.* Inputs are ϵ and m, where m is the percentile.<cit.> - They use objective pertubation.- The minimization is made over a closed convex set ...“The setting we are interested in is where each row of the design matrix X has L_2 norm at most √(p) and the parameter vector θ^∗ has L_2 norm at most √(p).” ... As mentioned above, the privacy proof of CMS required that r be differentiable and θ be unconstrained.- Point estimation.- Lasso regression.<cit.> - Point estimate.- The algorithm first randomly partitions the n inputs into disjoint blocks of 2 data points each. Compute the coefficients in each group and returns a trimmed median of them. <cit.>- Point estimate.§ THE DIFFERENTIALLY PRIVATE TEST STATISTIC We begin by laying out relevant notation and formally specifying our objectives. Letbe a confidential dataset comprising n individuals. For each individual i=1, …, n, let y_i∈ℝ be its univariate response variable and x_i = (1,x_i,1,…,x_i,p)^⊤∈ℝ^p+1 be its (p+1) × 1 vector of predictors.Hence,={(x_i, y_i)}_i=1^n.An analyst seeks to estimate the parameters in the regression, y_i =β^⊤ x_i + e_i, where β = (β_0,…,β_p)^⊤∈ℝ^p+1 and e_i are i.i.d. random errors with E(e_i)=0 and Var(e_i)=σ^2. In linear regression, we typically assume that e_i ∼ N(0, σ^2) for all i, although our algorithm can be used with other error distributions.We assume that, if the analyst had direct access to , he or she would make inferences about each β_j based on the maximum likelihood estimator (MLE), β̂_̂ĵ, and its corresponding sampling distribution.However, the analyst does not get direct access to ; instead, the analyst can learn only the privately-computed estimates β̃_̃j̃, which could arise from perturbed versions of β̂_j or from synthetic data. Our key question is whether or not inferences about β_j are similar when using β̂_j or β̃_̃j̃.To address this question, we develop differentially private significance tests.Let T() be the standardized estimator of β_j obtained from , that is, T() = β̂_̂ĵ/ √(Σ̂_j,j),where (Σ̂_j,j) is the (j,j)th element of the matrix, Σ̂ = σ̂^2(_^⊤_)^-1. Here,σ̂^2 = (_-β̂^⊤_)^⊤(_-β̂^⊤_)/(n-p-1), where _=(y_1,…,y_n)^⊤, and _=[x_1^⊤,…,x_n^⊤]^⊤ is the design matrix associated with the regression when estimated with .Under certain conditions on _, β̂_̂ĵis asymptotically normally distributed for a large range of error distributions (, , page 21;, , section 6.8). The key condition on the design is that the maximum among the diagonal elements of the matrix _(_^⊤_)^-1_^⊤ goes to zero as n →∞.Hence, for a large enough n, the distribution of T() can be suitably approximated by a standard Gaussian distribution. In the remainder of the article, we refer to T() simply as the t-statistic.T() provides all the information needed for inferences about the sign and significance of β_j.Hence, we can address our key question and account for privacy by developing algorithms for releasing differentially private versions of T(), along with deriving reference distributions for the private t-statistics.Taken together, these algorithms enable the analyst to assess the significance level for β_j directly from the private output.Unfortunately, we cannot simply apply the Laplace mechanism in (<ref>) to create the differentially private test statistic, as the global sensitivity of T() is unbounded.A possible remedyis to work with some bounded statistic instead of T().For statistical significance, two obvious candidates include (i) the p-value associated with T() and (ii) a truncated version of T().We next describe some of the pros and cons of each approach.Let p^T be the p-value associated with T() for a two-tailed significance test of the null hypothesis β_j=0. Any p^T has global sensitivity equal to one. Let p^T,ϵ be the ϵ-differentially private p-value obtained after adding Laplace noise to p^T based on the global sensitivity of one. With high probability, adding this noise to small values of p^T could inflate them so much as to change our opinion of the significance of β_j. This is less problematic when adding noise to large values of p^T. In other words, for a given ϵ, the probability that an analyst reaches the same decisions about statistical significance when using p^T,ϵ or p^T is higher when p^T falls in an acceptance region for H_0 than when p^T falls in a rejection region. Regarding the second approach, let the truncated t-statistic be given by T^t(D)= {[-a T() < -a,; T() -a ≤ T() ≤ a,; aT() > a,; ].where a>0 is a user-defined parameter. Here,a has to be large enough to ensure that T(D) and T^t(D) lead to the same conclusion regarding the null hypothesis with high probability.Because of the truncation, the global sensitivity of T^t(D) equals 2a.LetT^t,ϵ(D)= T^t(D)+ η be a noisy version of T^t(D), where η∼Lap(0,2a/ϵ) and Lap(l,s) denotes the Laplace distribution with location l and scale s.The problematic situations for T^t,ϵ(D) are the reverse of those for p^T. With undesirably high probability, adding noise to values of T^t(D) near zero could make an insignificant effect appear significant, whereas the noise is not likely to change our opinion about significance when T(D) is large. Put another way,for a given ϵ, the probability that an analyst reaches the same decisions about statistical significance when using T^t,ϵ(D) or T^t(D) is higher when T^t(D) falls in a rejection region for H_0 than when T^t(D) falls in an acceptance region. The arguments above suggest that neither approach always outperforms the other.We opt for the second approach because T^t() is more analytically tractable than p^T. As a result, we find it easier to develop a properly calibrated significance test, and understand its theoretical properties, for T^t() than for p^T.Using T^t() also allows the release of a noisy estimate of the sign of β_j without additional expenditure of ϵ, which improves the overall utility of the data release without sacrificing privacy.Since the length of the range of T^t() coincides with its global sensitivity,we need to use a large ϵ to ensure that T^t,ϵ(D) is practically useful, perhaps larger than what we would like from the perspective of protecting privacy.Put another way, for a small ϵ the noise introduced by the Laplace mechanism may be so large compared to T^t() that the statistic has little ability to detect any deviations from the null hypothesis.Hence, we need to adapt the truncated t-statistic to reduce the global sensitivity.We consider a way to do so based on the subsample and aggregate method described in Section <ref>. We first randomly partitioninto M disjoint subsets, 𝒫 = {_1,…,_M}, of equal size (or as close to equal as possible when n/M is not an integer).In each _l, we estimate the regression model of interest using only _l. For the regression coefficient of interest, we then compute the set of Mt-statistics, {T(_1), …, T(_M)}, and truncate each T(_l) at [-a,a], akin to (<ref>). Let {T^t(_1), …, T^t(_M)} be the set of M truncated t-statistics.We then computeT̅^t(𝒫) = ∑_l=1^M T^t(_l)/M.The sensitivity of T̅^t(𝒫) is 1/M times the sensitivity of T^t(), which apparently achieves our goal. However, we do not create the differentially private measure by adding Laplace noise to T̅^t(𝒫), for reasons we now describe.Because of the random partitioning, it is reasonable to consider each _l as a random sample from a population (with infinite sample size) and, thus, each T^t(_l) as a random draw from its sampling distribution.We would like the sampling distributions of T^t() and T̅^t(𝒫) to be approximately the same, so that the significance test based on T̅^t(𝒫) would have approximately the same power function as that based on T^t().When this is the case,analysts should have high probability of reaching similar conclusionswhen using the adapted t-statistic or T^t().However, the variance of T̅^t(𝒫) is roughly M times smaller than the variance of T^t().Thus, instead of using T̅^t(𝒫) directly,we equate the variances by multiplying T̅^t(𝒫)by √(M); that is, we use T̅^t,R(𝒫)= √(M)T̅^t(𝒫).This changes the global sensitivity, as it increases from 2a/M to 2a/√(M).Hence, the differentially private version of the t-statistic is T̅^t,ϵ(𝒫)= T̅^t,R(𝒫)+ η, where η∼Lap(0,2a/√(M)ϵ). Figure <ref> displays the steps that agencies can use to release T̅^t, ϵ(𝒫) (Algorithm 1). The figure also describes simple Monte Carlo algorithms that can be used to approximate the sampling distribution ofT̅^t, ϵ(𝒫) (Algorithm 2).This reference distribution can be used to obtain approximate p-values corresponding to the test statistic. The Monte Carlo simulations are needed to properly account for all sources of randomness, including the noise from the Laplace Mechanism. Taken together, Algorithm 1 and 2 provide a means to perform differentially private significance tests. Inferences about the sign of β_j can be obtained from the sign of T̅^t, ϵ(𝒫).Finally, we conclude this section with a formal theorem and proof that Algorithm 1 is differentially private. Algorithm 1 satisfies ϵ-differential privacy. T̅^t,Rhas global sensitivity equal to 2a/√(M). Hence, definingT̅^t,ϵ(𝒫)= T̅^t,R(𝒫) + Lap(2a/√(M)ϵ) implies that, by Definition <ref>, Algorithm 1 satisfies ϵ-differential privacy. § THEORETICAL PROPERTIES In this section, we discuss some of the theoretical properties of T̅^t,ϵ(𝒫). First, we derive conditions that characterize the distance between T̅^t,ϵ(𝒫) and T(). We then study the asymptotic probability of type II errors for the statistical test defined from T̅^t,ϵ(𝒫). §.§ Distance between T̅^t,ϵ(𝒫) and T() To characterize the distance between T̅^t,ϵ(𝒫) and T(), we focus on the probability P{|T̅^t,ϵ(𝒫)- T()|>c |_,ℛ} for all c>0, where ℛ denotes the random mechanism used to partitionand to create 𝒫. We note thatand 𝒫 are functions of (_,_) and (_,_,ℛ), respectively.Since we are conditioning on _ and ℛ, the randomness in T() and T̅^t,ϵ(𝒫) comes from treating _ as a random variable. We use the triangle inequality to bound this probability and, in this way, focus on characterizing each of the terms defining the right hand-side of P{|T̅^t,ϵ(𝒫)- T()|>c |_,ℛ} ≤ P{|T̅^t,ϵ(𝒫)-T̅^t,R(𝒫)|>c |_,ℛ}+ P{|√(M)T̅(𝒫)-T()|>c |_, ℛ}+ P{|T̅^t,R(𝒫)-√(M)T̅(𝒫)|>c |_, ℛ},where T̅(𝒫) = M^-1∑_l=1^M T(_l). The first term is relatively straightforward to understand theoretically and corresponds to computing a probability under the Laplace distribution. Thus, P{|T̅^t,ϵ(𝒫)-T̅^t,R(𝒫)|>c |_,ℛ} = exp(-cϵ√(M)/2a).As expected, the bound in (<ref>) indicates that T̅^t,ϵ(𝒫) gets closer to T̅^t,R(𝒫) as the scale of the underlying Laplace distribution, 2a/ϵ√(M), goes to zero, that is, as M and ϵ increase and a decreases.Turning to the second term in (<ref>), we now provide conditions that ensure √(M)T̅(𝒫) is a reasonable approximation of T(). To begin, we treat each _l as an independent sample from an infinite population.We make inferences conditional on each __l. Throughout, we rely on the following assumption. Assumption A1. The distribution of β̂_jl, i.e., the MLE of β_j estimated with _l treating __l as fixed, can be suitably approximated by a Gaussian distribution with mean β_j and variance Σ_j,j(_l), where Σ_j,j(_l) denotes the jth diagonal element of Σ(_l) = σ^2(__l^T__l)^-1 and __l is the design matrix associated with_l.A1 is widely assumed in practice in regression modeling (see , , page 21;, , section 6.8).Under A1, treating each _l as independent samples and conditioning on __l implies that √(M)T̅(𝒫) and T() are both Gaussian-distributed with variance equal to one.However, it is not necessarily the case that the means of each T(_l) are equal just because the conditional means of each β̂_jl are equal, nor that these means equal the mean of T().In particular, when β_j ≠ 0,T(_l) is based on a smaller sample size and a different design matrix than T(), which results in different (typically smaller)expected values.Thus, we need to derive conditions on the means of each √(M)T(_l), where l=1, …, M, that guarantee the mean of √(M)T̅(𝒫) is close to the mean of T(). Since each _l in actuality is a random sample from , as long as the sample size in each partition is large it is reasonable to assume that (__l^⊤__l) ≈ (__k^⊤__k) for all pairs of datasets (l,k).Moreover, it also is reasonable to make the following assumption. Assumption A2. M(__l^⊤__l) ≈ (_^⊤_) for all l = 1, …, M. With A2, we haveM^-1Σ(_l) ≈Σ().Using this approximation, we take expectations of √(M)T̅(𝒫) conditional on the realized (_, ℛ). We have E{√(M)T̅(𝒫)|_, ℛ}= M^-1∑_l=1^M E{√(M) T(_l)|_, ℛ} =M^-1∑_l=1^M √(M)β_j/√(Σ_j,j(_l))≈ M^-1∑_l=1^M β_j/√(Σ_j,j())=E{T()|_}.Additionally, using the approximation in A2, for any _l theCov{(_^⊤_)^-1_^⊤_,(__l^⊤__l)^-1__l^⊤__l|_, ℛ} = σ^2(_^⊤_)^-1 = Σ().Therefore, we have Cov{.T(),√(M)T̅(𝒫)|_, ℛ}= √(M)/M∑^M_l=1Cov{.T(),T(_l)|_, ℛ} =1/M∑^M_l=1√(Σ_j,j())/√(M^-1Σ_j,j(_l))>0.Thus, the smaller the distance is between _^⊤_ and M(__l^⊤__l), the higher is the correlation between √(M)T̅(𝒫) and T(). A direct application of the Markov inequality implies thatc^2P{|√(M)T̅(𝒫) - T()|>c |_,ℛ} ≤ Var{√(M)T̅(𝒫)|_,ℛ} + Var{T()|_,ℛ} - 2Cov{.T(),√(M)T̅(𝒫)|_,ℛ}+ [E{.T()|_,ℛ} - E{.√(M)T̅(𝒫)|_,ℛ}]^2.Thus, under A2,by A1 and (<ref>) we have Var{√(M)T̅(𝒫)|_,ℛ}≈ Var{T()|_,ℛ}≈ 1and E{.T()|_,ℛ}≈ E{.√(M)T̅(𝒫)|_,ℛ}.By (<ref>), we have Cov{.T(),√(M)T̅(𝒫)|_,ℛ}≈ 1.Therefore, the probability in (<ref>) is near zero, implying that the distance between √(M)T̅(𝒫) and T() has high probability of being small. Finally, we provide an upper bound for the last term in (<ref>). This bound allows us to understand how choices of M and a affect the distance between √(M)T̅(𝒫) and T̅^t,R(𝒫). Specifically, it follows thatP{|√(M)T̅(𝒫) - T̅^t,R(𝒫)|>c |_,ℛ} ≤P{|√(M)T̅(𝒫) - T̅^t,R(𝒫)|>0 |_,ℛ} = 1-P{√(M)T̅(𝒫) = T̅^t,R(𝒫) |_,ℛ}≤1-∏_l=1^M P{T(_l) = T^t(_l)|_,ℛ} = 1-(Φ(a-μ) - Φ(-a-μ))^M,where μ = E{T(_l)|__l, ℛ} and Φ denotes the cumulative distribution function of the standard Gaussian distribution. The probability in (<ref>) reveals that T̅^t,R(𝒫) gets closer to √(M)T̅(𝒫) as a increases and M decreases. Together, (<ref>)–(<ref>) provide a full characterization of (<ref>). §.§ Asymptotic power properties ofT̅^t,ϵ(𝒫)We next study the type II error rates for the significance test defined from T̅^t,ϵ(𝒫).For given values of (M, a, ϵ), and under H_0: β_j=0 (i.e., μ=0), let r be a positive constant such that P_H_0{|T̅^t, ϵ(𝒫)| < r | _,ℛ. }= 1-α,where P_H_0 denotes the probability computed under H_0. Here,r corresponds to the critical value that ensures a significance level of α for the test 𝕋_t,ϵ(𝒫) = 𝕀_(-r,r)(T̅^t, ϵ(𝒫)), where 𝕀_B(b)=1 if b∈ B and 𝕀_B(b)=0 otherwise. Thus, the type II error probability associated with this test is given byE_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.}, where E_H_1 denotes expectation under H_1: β_j ≠ 0. The strategy used to define T̅^t, ϵ(𝒫) makes it difficult to derive analytical expressions for r and E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.} in terms of (α,M,a,ϵ). However, sinceT̅^t,ϵ(𝒫) is a function of random variables that are easy to generate numerically, it is trivial to use Monte Carlo simulation to provide accurate approximations of r and E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.}. This allows us assess type II error rates for the test both asymptotically, which we study in this section, and in finite samples, which we study in Section <ref>.As n goes to infinity, T̅^t,R(𝒫) converges in probability to √(M)a.Hence, the asymptotic probability of type II error is approximately equal to the probability of the acceptance region (-r,r) under a Laplace distribution with location and scale equal to √(M)a and 2a/√(M)ϵ, respectively. This characterization immediately reveals that the probability of type II error for this test never equals zero.Figure <ref> displays a Monte Carlo approximation of the asymptotic probability of type II error associated with the test 𝕋_t,ϵ(𝒫) for different values of (α,a,M,ϵ). For most of the combinations of (α,M,a,ϵ) studied here, it is possible to obtain an asymptotic probability of type II error that is close to zero, with (α=0.01, ϵ = 0.1) being the lone exception. For any given (α,M,ϵ), the asymptotic probability of type II error is almost constant as a function of a, that is, lim_n→∞E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.} decreases at a slow rate as a increases.The type II error probabilities also decrease as any one of M, α, or ϵ increases, holding the others constant. We note that analysts can use this Monte Carlo approach to approximate the asymptotic probability of type II error for any combination of (α,M,a,ϵ).Thus, for example, for a fixed (α,ϵ), analysts can determine values of (M,a) that lead to a specified asymptotic probability of type II error. For example, for α= 0.05, ϵ=1, and lim_n→∞E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.}< 0.001, we find that M needs to be greater than 25 provided that a is greater than one. Although the Monte Carlo approach provides accurate and straightforward approximations, it is also instructive to characterize the asymptotic behavior of E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.} mathematically under arbitrary choices of (α,a,M,ϵ). Theorem <ref> provides an upper bound for lim_n→∞E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.}. Analogous to Figure <ref>, in the supplementary material we present a graphical representation of the upper bound. We observe that (<ref>) is a sharp bound when a>2. For 1 ≤ a ≤ 2 and some values of M, the bound in (<ref>) is a moderately less precise, but still valid, bound for lim_n→∞E_H_1{𝕋_t,ϵ(𝒫) | _,ℛ.}.The proof of Theorem <ref> is in the supplementary material. Under H_1: β_j ≠ 0 and assumption A1,lim_n →∞ E{𝕋_t,ϵ| _,𝒫.}<𝕀_{r^*<√(M)a}1/2(α^-1-ϵ√(M)/2a-α^1+ϵ√(M)/2a)exp(-ϵ M/2)+𝕀_{r^*>√(M)a}( 1-1/2α^1+ϵ√(M)/2a[exp(ϵ M/2)- exp(-ϵ M/2)]),where r^*=-log(α)(2a/ϵ√(M)+1).§ EMPIRICAL ILLUSTRATIONS The results in Section <ref> suggest we should make a as large as possible and M as small as possible to ensure the proximity between T̅^t,R(𝒫) and T(). However, the accuracy of the approximation of T() is only part of the story. We need to add Laplace noise to protect privacy.To maximize the usefulness of the privately-computed t-statistic T̅^t,ϵ(𝒫), we seek to add as little noise as possible while still satisfying differential privacy. This pushes us to make a smaller rather than larger, and to make M larger rather than smaller. How do we trade off accuracy in T̅^t, R(𝒫) for reductions in variance of the Laplace noise? We have a partial answer this question from the asymptotic results in Section <ref>, but, in practice, we have to analyze finite samples. Which choices of (M,a) tend to offer higher accuracy for a given risk level ϵ with finite samples? In this section, we address this question using simulation studies. In all simulations, we assume that β̂_j ∼ N(β_j, Σ_j,j()) and β̂_j,l∼ N(β_j, Σ_j,j(_l)) for any j and l, i.e., we assume that A1 holds. We consider two scenarios, one where M__l^⊤__l≈ (_^⊤_), where l=1,…,M, i.e., where A2 holds, and the other where this is not necessarily the case. The scenarios are generated as follows.In Scenario I, we work directly with the theoretical distributions of the t-statistics, without simulating and partitioning values of . For an arbitrary regression coefficient β, let μ_T be the number of standard deviations that its value is from zero.We consider multiple values of μ_T in the simulation.For any μ_T, we generateT andT̅^t,Rfrom their sampling distributions as follows,T∼ N(μ_T,1) T̅^t,R= 1/M∑_l=1^M √(M)(-a𝕀_(-∞,-a)(Z_l) + Z_l𝕀_[-a,a](Z_l) + a𝕀_(a,∞)(Z_l)).We let Z_l i.i.d.∼ N(√(M)μ_T,1). We use the Laplace mechanism based on the appropriate global sensitivity values to generate T̅^t, ϵ.Each T and T̅^t,R is generated independently.Generally, one would expect their values to be positively correlated when computed on some .However, generating them independently guarantees that A2 holds. Scenario Iprovides lower bounds for cases where the t-statistics are positively correlated, since thestatistics should be more similar when positively correlated than when independent.In Scenario II, we work with a subset of the March 2000 Current Population Survey (CPS) public use file comprising n = 49,436 heads of households with non-negative incomes.This dataset was used by <cit.> and <cit.>, among others.In order to use realistic predictor distributions, we set _ to be an n × 25 matrix of values derived from the CPS data.Its columns include age in years, age squared, education (16 levels), marital status (7 levels), and sex (2 levels). We generate multiple sets of the response variable Y from linear regressions on _, each using a different, pre-specified set of β= (β_0,…,β_24)^⊤, so as to control the importance of the regression coefficients.For j=0, …, 24, let Σ_j,j() be the jth diagonal element of σ^2(_^⊤_)^-1,where σ = 0.82. This value of σ corresponds to the residual standard error of a linear regression fitted using the logarithm of income—one of the variables in the CPS data—as the response variable and _ as predictors.To derive any one β= (β_0,…,β_24)^⊤, we set each β_j = μ_T √(Σ_j,j()) for some specified number of standard deviations μ_T from zero.Using this β, we simulate realizations of T and T̅^t,R via the following steps. i) For i=1, …, n, generate y_i from N(x_i^⊤β,σ^2), where x_i denotes the ith row-vector of _ and σ =0.82. ii) Set ={(x_i, y_i)}_i=1^n, and generate a random partition 𝒫={_l}_l=1^M.iii) Get a realization of T = T() by computing the t-statistic of the jth regression coefficient estimated from . iv) For l=1, …, M, set Z_l = T(_l) as the t-statistic of the jth regression coefficient obtained from the regression of y on __l.Get realizations of T̅^t,R from (<ref>). We then add Laplace noise to generate T̅^t, ϵ.For both scenarios, we let μ_T ∈{0,0.1,…,1,2,…,10}, let M ∈{10, 25, 50, 75, 100}, let a ∈{1, …, 10}, and let ϵ∈{0.5,1,2.5,5}.We generate simulations for all possible combinations of (μ_T, M, a, ϵ). For each combination (μ_T, M, a, ϵ), we generate 100,000 realizations of (T, T̅^t,R) for Scenario I and1,000 realizations for Scenario II.We evaluate the significance tests based on T̅^t, ϵ by comparing the powerat each value of μ_T to the power of the test based on T at each corresponding value of μ_T.We evaluate the properties of the differentially private sign measures by comparing how often one can infer the correct sign of each β_j from the corresponding T and T̅^t,ϵ. We also compute the probability that a user makes the same decision about the significance level or sign when using T and T̅^t,ϵ; we call these matching probabilities.§.§ Assessing inferences aboutsignificance We study the significance properties of the t-statistics using the following quantities,p_0( t,γ) = P{|T| <t| μ_T = γ, _. },p_t,ϵ( t,γ,M,a,ϵ)=P{|T̅^t, ϵ| <t| μ_T= γ, M, a, ϵ, _, ℛ. }.As a slight abuse of notation, we condition these probabilities on (γ,M,a,ϵ) to highlight that they parametrize the probabilities.For a given significance level α and type II error rate λ_0, let r_0 and q_0 be positive constants such that p_0(r_0,0)= 1-α and p_0(r_0,q_0)= λ_0. Notice that r_0 is the (1-α/2)th quantile of the distribution of T when μ_T = 0, i.e., r_0 is the critical value under the null hypothesis H_0: β_j=0 that ensures a confidence level of 1-α for the test 𝕋 = 𝕀_(-r_0,r_0) (T). The value q_0 is the number of standard deviations from zero at which the test 𝕋 reaches the desired Type II error λ_0, i.e., under H_1: β_j = q_0 √(Σ_j,j()), the power of this test is equal to 1-λ_0.Let r and λ(M, a, ϵ) be positive constants such that p_t,ϵ(r,0,M,a,ϵ)= 1-α and λ(M,a,ϵ) = p_t,ϵ(r,q_0,M,a,ϵ). For given values of (M, a, ϵ), r is the critical value under H_0 that ensures a confidence level of 1-α for the test 𝕋_t,ϵ = 𝕀_(-r,r)(T̅^t, ϵ). The value λ(M,a,ϵ) corresponds to the type II error rate of 𝕋_t,ϵ under H_1: β_j = q_0 √(Σ_j,j()). To assess how similar the test 𝕋_t,ϵ is to 𝕋, we use the loss function, L_t,ϵ^sig(M,a,ϵ) = max(0,λ(M,a,ϵ)-λ_0).For a given significance level α, we say there is zero loss of power from using 𝕋_t,ϵ when the type II error rate of 𝕋_t,ϵ at q_0 is less than λ_0. Otherwise, we record the corresponding loss of power, λ(M,a,ϵ)-λ_0. For all results in this subsection, we set α = 0.05 and λ_0 = 0.2.We begin by examining thevalues of L_t,ϵ^sig(M,a,ϵ) at the different combinations of (M,a) when ϵ = ∞, i.e., no noise is added to T̅^t,ϵ.To save space, we present these results in the supplementary material. In Scenario I, the power for 𝕋_t,ϵ and𝕋 are almost identical for a ≥ 2for all values of M. This finding conforms with the theory in Section <ref>. The results with the CPS data in Scenario II follow a similar pattern except for M=100. This is expected since large values of M weaken the validity of A2.We next consider ϵ < ∞, as needed to satisfy differential privacy. Figure <ref> displays values of L_t,ϵ^sig(M,a,ϵ) for ϵ∈{0.5, 1.0, 2.5, 5.0}.In Scenario I, L_t,ϵ^sig(M,a,ϵ) tends to be smaller when M is large and a is small, and largest when M is small and a is large. In other words, the discrepancy between the power of 𝕋_t,ϵand 𝕋 is smaller when the global sensitivity is smallest.For values of ϵ > 1, there is at least one combination of (M,a) for 𝕋_t,ϵ that provides almost no loss inpower.For ϵ = 1, with 𝕋_t,ϵ we need M ≥ 50 and a ∈{1,2}to experience only a small power loss. For ϵ = 0.5, with 𝕋_t,ϵwe still can achieve only modest power losses by using(M ≥ 75, a = 1). As expected, the power of 𝕋_t,ϵ is strongly influenced by ϵ. As ϵ gets smaller, so does the power.The reduction in power for small values of ϵ is the price to pay for strengthening the privacy guarantee. However, we emphasize that 𝕋_t,ϵ is still a valid test when ϵ is small, in that it gives the correct type I error rates regardless of the value of ϵ. Finally, the result patterns obtained under Scenario II generally match those obtained under Scenario I except for M=100. For this value of M, we notice a loss of power for some of the regression coefficients. We attribute this discrepancy to the fact that large values of M weaken the validity of A2. §.§ Assessing inferences about signs of coefficientsSince the sign of μ_T and β_j is the same, we restrict our analysis to the sign of μ_T only. Because the Laplace and Student-t distributions are symmetric, we only consider the case where μ_T ≥ 0.We study the sign of the statistics using the following quantities,s_0(γ) = P{ sign(T) =sign(μ_T) | μ_T = γ, _. }, s_t,ϵ(γ,M,a,ϵ)= P{ sign(T̅^t,ϵ) = sign(μ_T) | μ_T= γ, M, a, ϵ, _, ℛ. }.These representprobabilities that the t-statistics have the same sign as β_jwhen the value of β_j is γ standard deviations from zero. For a given probability α_0, let μ_0 be a positive constant such that s_0(μ_0) = α_0, i.e., μ_0 is the number of standard deviations at which T() has the same sign of β_j with a probability equal to α_0. To assess how similar α_0 is to s_t,ϵ(μ_0,M,a,ϵ), we use the loss function L_t,ϵ^sgn(M,a,ϵ) = max(0,α_0-s_t,ϵ(μ_0,M,a,ϵ)).We say that T̅^t,ϵ and T() result in similar inferences about the sign of β_j when α_0>s_t,ϵ(μ_0,M,a,ϵ). For all results in this subsection, we set α_0 = 0.95. We again begin with the values of L_t,ϵ^sgn(M,a,ϵ) for ϵ=∞; results are displayed in the supplementary material.Across scenarios, inferences about the sign of β_j based on T̅^t,ϵ=∞ are quite accuratefor all combinations of (M,a).Figure <ref> displays results for L_t,ϵ^sgn(M,a,ϵ) when ϵ∈{0.5, 1.0, 2.5, 5.0}.In general, when ϵ≤ 2.5,T̅^t,ϵ offers many combinations of (M, a) that result in accurate inferences about the sign of β_j, especially when a is small and M is large. We also find combinations of (M,a) that result in accurate inferences about the sign even when ϵ = 0.5, in particular when M ≥ 50 and a ∈{1,2}.When ϵ = 5, the results forT̅^t,ϵ are practically indistinguishable for almost all combinations of (M,a). These general findings hold for both scenarios, so thatT̅^t,ϵ provides a differentially private mechanism to release the sign of β_j that seems robust against violations of A2.§.§ Matching probabilities We define matching probabilities as the probability that sign(T) =sign(T̅^t, ϵ) and the probability that 𝕋_t,ϵ=𝕋. We assess these probabilities using the quantities,m_t, ϵ^sig(μ_T,ϵ)= min_M,aP{𝕋_t, ϵ=𝕋|μ_T,M,a,ϵ, _, ℛ}, m_t, ϵ^sgn(μ_T,ϵ)= min_M,aP{ sign(T) =sign(T̅^t, ϵ)|μ_T,M,a,ϵ, _, ℛ},where each minimum is over all possible combinations ofM ∈{10,25,50,75,100} and a ∈{1,2,…,10}.As we minimize over (M,a), these metrics represent the worst case matching probabilities for these simulations. In the supplementary material, we present analogous figures showing the maximum values of the matching probabilities, which represent the best case matching probabilities for these simulations.Figure <ref> displays the values of m_t, ϵ^sig(μ_T,ϵ) for different values of μ_T and ϵ. The results in Scenario I and II follow similar patterns so we describe them simultaneously. Under the null hypothesis, i.e., μ_T=0, values of m_t, ϵ^sig are greater than 0.85.When μ_T is large enough, values of m_t, ϵ^sig are close to one. The value of μ_T at which m_t, ϵ^sig is close to one is inversely related to the value of ϵ. For example, when ϵ =5 and μ_T>5, then m_t, ϵ^sig≈ 1. However, when ϵ =0.5, m_t, ϵ^sig < 0.5 for all values of μ_T.Values of m_t, ϵ^sig tend to be small when μ_T ∈ [r_0,r], where r_0 and r are the critical values associated with 𝕋 and 𝕋_t, ϵ, respectively. Because r increases as ϵ decreases, the range of values of μ_T where m_t, ϵ^sig is small becomes wider as ϵ decreases. Figure <ref> also summarizes the results for m_t,ϵ^sgn(μ_T,ϵ). As expected, in both scenarios, increases in μ_T correspond to increases in the matching probability. The rate at whichm_t,ϵ^sgn increases as a function of μ_T depends on ϵ: the larger the ϵ, the faster the rate. In both scenarios, we observe a high matching probability (m_t,ϵ^sgn>0.9) when μ_T ≥ 3 and ϵ≥ 2.5. When μ_T<3,m_t,ϵ^sgn ranges from 0.5 to 0.9 for almost all ϵ. It reaches a minimum value when μ_T=0, regardless of the value of ϵ; however, when μ_T=0, matching the sign ofT andT̅^t,ϵ arguably is not important for interpretations. § CHOOSING M AND A WITHOUT ADDITIONAL PRIVACY LOSS To use these differentially private test statistics, the data producer or, when permitted in a verification server, the analyst first fixes the desired privacy level ϵ and then must select values for M and a. Here, we consider an analyst who does not get to choose ϵ but does get to choose (M,a).Analysts who get to choose ϵ, e.g., when allocating a total privacy budget across multiple queries, could repeat the approach described here with different values of ϵ. In this section, we present a three step approach for selecting values of (M, a) that does not incur additional privacy loss.We illustrate these steps with a regression analysis of the CPS data, using the same _ as before and the reported values of household income on a logarithmic scale as the response variable. This regression fits reasonably well without obvious violations of the assumption ofi.i.d. Gaussian errors. We present the methodology for T̅^t,ϵ with ϵ = 1.5. The three-step approach also can be used for the sign.Step 1: Fix an upper bound for L_t,ϵ^sig(M,a,ϵ). To begin, the user specifies an upper bound for L_t,ϵ^sig(M,a,ϵ) for their desired significance level α and type II error rate λ_0. By fixing α and λ_0, it is implied that there exists q_0>0 such that the power of T equals (1-λ_0)when β_j is q_0 standard deviations from zero. Thus, the bound for L_t,ϵ^sig(M,a,ϵ) represents the loss of power that a user is willing to accept by using T̅^t, ϵ instead of using T when β_j isq_0 standard deviations from zero.In our illustrative example, we set α = 0.05 and λ_0 = 0.2, and fix an upper bound of 0.1.Step 2: Simulate values of L_t,ϵ^sig(M,a,ϵ) for choices of M and a. The user can simulate values ofL_t,ϵ^sig(M,a,ϵ) for different combinations of (M, a) using the strategy described in Scenario I of Section <ref>.As an example, Table <ref> displays the values of L_t,ϵ^sig(M,a,ϵ) for different combinations of (M, a).Of course, this table is not comprehensive; users can create different tables with different combinations of (M,a).Importantly, we do not compute the entries in Table <ref> using the CPS data; otherwise, the results would leak information about the confidential data. When auxiliary data are available, such as synthetic data, analysts could base the tables off these auxiliary data using the strategy described in Scenario II of Section <ref>. Step 3: Choose (M, a). The user considers all values of (M,a) corresponding to values of L_t,ϵ^sig(M,a,ϵ)below the fixed upper bound. When no combination of (M, a) satisfies this condition, the user has to sacrifice accuracy and increase their error tolerance. Once possible solutions exist, we recommend that the user choose the smallest value of M from these solutions. Smaller values of M result in larger sample sizes within the partitions, lessening the possibility that A1 or A2 are invalid. In our example, based on Table <ref>, the user should choose the smallest values of M for which L_t,ϵ^sig(M,a,ϵ=1.5) is below 0.1, which is M = 25.The user then chooses the value of a that minimizes L_t,ϵ^sig(M=25,a,ϵ=1.5).From Table <ref>, L_t,ϵ^sig(M=25,a,ϵ=1.5) reaches its minimum at 0.05 for 𝕋_t,ϵ and a ∈{1,2}. In theory, any of these values of a should work. In this case, we recommend a=2 since this truncation level has the least effect on the approximation to T(). We now illustrate this method of choosing (M, a) on the CPS data.Using (M=25, a = 2), we compute the differentially private p-value and sign using Algorithms 1 and 2; results for all 25 coefficients are displayed in the supplementary material.The p-values from the differentially private significance test and the test based on the confidential data agree substantially, resulting in essentially the same conclusions about the significance for most of the regression coefficients. We observe discrepancies when the p-values based onprovide weak evidence against the null. Regarding the sign, the outputs from the differentially private algorithm agree with the observed signs for almost all coefficients, except some with large p-values.When p-values for the test of H_0: β_j =0are quite large, changes in sign arguably are inconsequential.As an additional illustration, we repeat the model selection and estimation using ϵ= 0.5 and upper boundfor L_t,ϵ^sig(M,a,ϵ) equal to 0.2; detailed results are in the supplementary material. The (M, a) selection algorithm suggests that we set (M=100, a=1).For estimating the signs, the differentially private algorithms continue to be effective: the privately-computed and observed signs systematically agree on all but three of the coefficients with significant p-values.For p-values, we seedifferent interpretations of the significance for 3 out of 25 coefficients.We conjecture why this occurs when discussing the results in the supplementary material. Of course, interpretations about the quality of the results for different (M, a) and ϵ are data-specific, and we expect lower reductions in data quality forwith larger sample size. § CONCLUDING REMARKS The methods described here allow data producers to provide ϵ-differentially private answers to queries about the statistical significance and signs of coefficients in linear regression models.Further, the strategy from Section <ref> provides users with a principled way to choose (M, a).We expect that these methods can be applied to significance testing with other regression models.The key assumptions are A1 and A2, whichare reasonable for many models and data settings.Themethods extend trivially to two contexts that are not discussed in previous sections. First, we can test general null hypotheses, H_0: β_j = b with b ∈ℝ. We simply define the t-statistic as (β̂_j-b)/ √(Σ̂_j,j), and apply the algorithms without modification.One application of this extension, which we leave to future investigation, is to set b equal to the output of a privately computed regression coefficient β̃_̃j̃.The test statistic then could be interpreted as a noisy, truncated estimate of the number of standard errorsβ̃_̃j̃ is from β̂_j.Second, we can test the significance of the intercept, i.e., H_0: β_0 = b. This is equivalent to a ϵ-DP, one sample t-test for a single mean. Unlike other tests for means, this significance test does not presume known bounds on data values or population means.Nonetheless, it would be interesting to compare the power of the various tests for single means to see when each offers the best performance. The algorithm described here provides results for one β_j at a time. If analysts are given a finite privacy budget, they must spend part of that budget for each coefficient they wish to verify.Thus, a key area for research is to develop differentially private algorithms that allow queries for multiple test statistics without burning through the privacy budget too quickly. § SUPPLEMENTARY MATERIAL The online supplementary materials include a graphical representation of the upper bound provided in Theorem <ref>, and the proof of this theorem. They also include additional plots from simulations in Section <ref> and Section <ref> when ϵ=∞, and in Section <ref> when the maximum of the matching probabilities is taken over (M,a) for fixed values of ϵ. Finally, the include the p-values and signs for the coefficients in the two examples considered in Section <ref>. § ACKNOWLEDGMENTSThis work is supported by grants from the National Science Foundation (ACI 1443014 and SES 1131897) and the Alfred P. 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"authors": [
"Andrés F. Barrientos",
"Jerome P. Reiter",
"Ashwin Machanavajjhala",
"Yan Chen"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170526125536",
"title": "Differentially private significance tests for regression coefficients"
} |
firstpage–lastpageBuilding crystalline topological phases from lower-dimensional states Michael Hermele December 30, 2023 =====================================================================We perform controlled N-body simulations of disc galaxies growing within live dark matter (DM) haloes to present-day galaxies that contain both thin and thick discs. We consider two types of models: a) thick-disc initial conditions to which stars on near-circular orbits are continuously added over ∼10, and b) models in which the birth velocity dispersion of stars decreases continuously over the same time-scale. We show that both schemes produce double-exponential vertical profiles similar to that of the Milky Way (MW). We indicate how the spatial age structure of galaxies can be used to discriminate between scenarios. We show that the presence of a thick disc significantly alters and delays bar formation and thus makes possible models with a realistic bar and a high baryon-to-DM mass ratio in the central regions, as required by microlensing constraints. We examine how the radial mass distribution in stars and DM is affected by disc growth and non-axisymmetries. We discuss how bar buckling shapes the vertical age distribution of thin- and thick-disc stars in the bar region. The extent to which the combination of observationally motivated inside-out growth histories and cosmologically motivated dark halo properties leads to the spontaneous formation of non-axisymmetries that steer the models towards present-day MW-like galaxies is noteworthy. methods: numerical - galaxies:evolution - galaxies:spiral -Galaxy: disc - Galaxy: kinematics and dynamics - Galaxy: structure; § INTRODUCTION The vertical density profile of stars in the Milky Way (MW) is fitted well by a sum of two exponentials <cit.>. <cit.> find scaleheights of ∼900 for the geometrical thick disc and ∼300 for the thin disc. Similarly, the verticalsurface brightness profiles of the majority of bright edge-on spiral galaxies show thin and thickcomponents <cit.>. Studies of the chemical abundances of solar neighbourhood (Snhd) starsreveal that populations with hotter vertical kinematics and thus larger scaleheights have abundancesof α elements relative to iron () that are larger than those of populations with smallscaleheights and comparable iron abundances<cit.>. In the - plane, the low- and high-α populations are generally found to separate into fairly distinct sequences. Higherindicates shorter chemical enrichment time-scales and age determinations find systematically older ages for stars of the α enhancedchemical thick disc <cit.>: they are generally found to be older than 8 Gyr. The distinct sequences have motivated models in which the two componentsformed in two temporally separated phases in very different conditions (e.g. ), but can also be explained as the result of continuous star formation and chemical enrichment<cit.>.It has now become clear that chemical and geometrical definitions of the thick disc yield different results. Whereas <cit.>, who determined the density of all stars in the Snhd independent of chemistry, found that the geometrically thick disc has a longer radial scalelength than the thin disc, high- stars are found to form a thicker, but more centrally concentrated component than low- stars<cit.>. A scenario in which the disc forms inside-out and each mono-age population flares, i.e. is thicker at outer than at inner radii, could potentially explain these observations <cit.>. Flaring can be caused by radial migration of stars <cit.> or by vertical heating of the outer disc through satellite interactions <cit.> or misalignedgas infall <cit.>.<cit.> (hereafter Papers 1 and 2) presented ∼100 idealized N-body models of disc galaxies growing within live dark matter (DM) haloes over ∼ 10. These models covered a large variety of star formation and radial growth histories and most of them followed the assumption that all stars are born on near-circular orbits as in the MW today. Structural and kinematical properties of the MW's thin disc, such as an exponential profile with scaleheight ∼300 or the local age-velocity dispersion relations could be reproduced if giant molecular clouds (GMCs) were included. Additionally, bars of comparable size and structure to that of the MW formed in these models. However, none of the models produced a realistic thick disc. The conclusion was that additional sources of heating were required early in the disc's life, prior to the onset of thin-discformation.In this paper, we create models similar to those of Paper 1 that, in addition to a realisticthin disc, also contain an appropriate ∼ 10 old thick disc.Demanding the presence of an old thick component makes it much harder to steer a model to a configuration consistentwith current data, in part because the number of observations that need to be explainedsimultaneously increases roughly twofold. Moreover, the thick disc must be steered into itspresent form by adjusting the conditions at the onset of disc formation, which will modify the subsequent formation of the thin-disc component by altering spiral and bar structures and their interaction with the dark halo. These changes to the thin disc and dark halo will themselves modify the appearance of the current thick disc. Moreover, the old chemical thick disc of the MW is centrally concentrated and should be important in the central ∼5, a regiondominated by the Galactic bar <cit.>, a structure that is supposed to have formedfrom a rather cold disc.To create a thick disc we follow two approaches. In one scheme, we assume that the velocity dispersions of newborn stars decline with time. This is motivated by observations of redshift z_ rs∼2 galaxies, which show a high fraction of galaxies with Hα kinematicsconsistent with ordered rotation, but significantly higher velocity dispersions than today's discgalaxies <cit.> and of galaxies at lower redshifts that indicate a continuous decline of gasvelocity dispersion with decreasing redshift (, but seefor a different conclusion). Such a declining birth dispersion has also been found in hydrodynamical cosmological simulations of disc galaxy formation <cit.>, and can be understood in terms of galaxies that gradually become less gas-rich and are characterizedby gravitationally driven turbulence that decays <cit.>.In an alternative scheme, we model thin+thick disc systems by creating thick initial conditions and growing thin discs within them. These models are thus representations of two-phase formation scenarios. We here do not model the formation of the thick disc prior to redshift z_ rs∼2, but several overlapping scenarios envisage the production of such an object: heating of an initially thinner disc by a merger (e.g. ); formation in early gas-rich mergers <cit.>; the formation in a clumpy, turbulent disc <cit.>.It should be noted that the latter two scenarios could also be fitted into the picture of declining birth dispersions that constitutes our alternative modelling scheme.In this paper, we concentrate on the setup of our models and on their structural evolution. We explore how models with double-exponential vertical profiles, circular speed curves like that of the MW and bars of appropriate structure can be constructed.We examine how the presence of a thick disc changes the preferred density of the dark halo and influences the evolution of the thin disc and the formation of a bar. We show that different scenarios for the formation of the thick disc leave signatures in the current distribution of age with the (R,z) plane. A companion paper (, hereafter Paper 4) focuses on disc heating and radial migration in the models.Our paper is structured as follows.Section <ref> describes the setup and parameters of our simulations. Section <ref> discusses the evolution of vertical density profiles and howthis shapes the final age structure of the disc. Section <ref> analysesthe radial distribution of dark and baryonic mass components and Section <ref> illustrates the evolution and structure of bars in the presence of thick discs. Section <ref> discusses the successes and problems of our models. Section <ref> concludes.§ SIMULATIONSThe simulations analysed in this paper are similar to the models presented in Paper 1. These are simulations of growing disc galaxies within non-growing live DM haloes. They are run with the Tree code GADGET-3, last described in <cit.>. We focus on standard-resolution models, which contain N=5×10^6 particles in theDM halo and a similar number of particles in the final stellar system. We here rely on collisionless simulations; models that contain an isothermal gas component were discussed in Papers 1 and 2 and shown to differ only mildly from collisionless models in terms of the structure and kinematics of thestellar component. In addition, and crucially, all simulations contain a population ofshort-lived, massive particles representing GMCs. Papers 1 and 2 demonstrated the importance of GMC heating in creating thin-disc components with realisticvertical structure as non-axisymmetric structure contributes little to vertical disc heating (see also ). §.§ Initial conditions Table <ref> gives an overview of the initial conditions (ICs) of our models. The ICs were created using the GALIC code <cit.>.We refer the reader to Paper 1 for a full description of the IC creation process and here focus on how the ICs used in this paper differ from each other. All modelsdiscussed start with a spherical DM halo with a <cit.> profileρ_DM(r)=M_DM2πar(r+a)^3.The total mass of all ICs is M_ tot=10^12, of which M_ b,i=M_ disc,i+M_ bulge,i is in a stellar component and the rest M_DM is in DM. The inner density profile is adjusted so that it is similar to an NFW profile with concentration c_ halo=6-9 and thus scale radii are in the range a=30-40. The kinematics of the DM particles are initially isotropic (i.e. have equal velocity dispersions σ_r=σ_ϕ=σ_θ). Each DM halo studied here is resolved with N_ DM=5 000 000 particles.ICs contain either a disc or a bulge component or both. IC discs have a mass in the rangeM_disc,i=5-25× 10^9 and a density profile of the formρ_disc,i(R,z) = M_disc,i4πz_0, disch_R, disc^2^2 (zz_0, disc)exp(-Rh_R, disc),where h_R, disc=1.5-2.5 is the exponential disc scalelength. Radially constant isothermal vertical profiles with scaleheights z_0,disc=0.1-1.7 are assumed. The ratios of the radial to vertical velocity dispersions are also radially constant in the IC discs and have values in the range σ_R^2/σ_z^2=1.0-2.0.IC bulge components are set up with Hernquist density profiles with scalelengths a_ bulge=0.4-1.5 that are distorted to be oblate spheroids with axis ratios in the range s=1-3 asρ_ bulge(R,z)=s ρ_ Hernquist(√(R^2+s^2z^2)).They are supposed to model either compact, non-rotating bulges or rotating spheroidal components. Following section 3.2 of <cit.>, the rotation of the axisymmetric bulge components iscontrolled via the <cit.> parametrization<v_ϕ>^2 = k^2 ( < v_ϕ^2> -σ_R^2).Non-rotating bulges assume k=0, whereas rotating bulges are modelled as isotropic rotators with k=1.Stellar particles in the ICs have a particle mass m=1×10^4 andour ICs thus contain N_ b,i=5-25× 10^5 stellar particles. As in Paper 1, the applied force softening lengths for the given resolutionsare ϵ_ b=30 for baryonic particles (including GMCs) andϵ_ DM=134 for DM particles.The first letter of a model's name specifies its ICs according to the scheme laid out in Table <ref>. In summary: * Y ICs have a compact and thin baryonic disc;* P, Q, R, U and T ICscontain more extended and thicker discs, with varying size,thickness, mass and DM halo concentration;* C ICs have a compact non-rotating bulge;* W and X ICs combine a compact non-rotating bulge with a thick disc;* K, M, O and V ICs contain rotating oblate spheroids that crudely represent elliptical galaxies.§.§ Growing the disc To simulate the growth of galaxies over cosmological time-scales, stellar particles with a particle mass m=1×10^4 are continuously added to the simulations. As we are interested in the coevolution of thin and thick discs, we pursue two different ideas, which were already touched upon in Paper 1. a) We assume that a thick and rotating stellar component was formed early on in a galaxy's history (e.g. through a merger), represent this component with a thick disc or rotating spheroid in our ICs and over the timespan of the simulation add star particles to the system on near-circular orbits. b) We assume that thebirth velocity dispersions of stars have been continuously declining over thehistory of a galaxy and add stellar populations with continuously decreasing dispersions.§.§.§ Evolution of input velocity dispersion In case a), the young stellar populations are assigned low birth velocity dispersions in the range σ_0=σ_1=6-10 in all three directions R, ϕ and z as observedin the MW (see blue stars in ). The mean rotation velocity v_ϕ(R) at radius R is set to the circular velocity v_ circ=√(a_R(R) R), where a_R(R) is the azimuthal average of the radial gravitational acceleration, ∂Φ/∂ R. As was shown in Papers 1 and 2, the in-plane dispersions quickly adjust to higher valuesand an appropriate ratio σ_ϕ/σ_R due to spiral and bar heating. In case b) we choose a value of σ_0 that is declining with time. Paper 1 had appliedσ_0(t)=(6+30^-t/1.5),which provided significantly too few hot stars. So, here we consider different functional forms:σ_0(t)=σ_1 [ arctan( t_1-t1) + π2]+σ_2 (type atan)orσ_0(t)=σ_1 ( t+t_1t_2)^-ι -σ_2 (type plaw) In observations of high-redshift galaxies, it is common to study kinematics of the Hα emission line and to assign a single characteristic velocity dispersion σ_Hα to each galaxy. Applying such a procedure, <cit.> find a dependence of observed Hα dispersionsσ_Hα∝ (1+z_ rs), where z_ rs is redshift (see also ). The decline of our input velocity dispersion parameter σ_0(t) for simulations of type `plaw' (Equation <ref>) resulted from a rough approximation to this proportionality under the crude assumption that the kinematics of young stars follow Hα kinematics. Note that this trend is observed for populations of galaxies at varying redshifts z_ rs. The spread in σ_Hα for each given redshift range in <cit.> is significant and the expected formation histories for galaxies are diverse, so that it is reasonable to assume that individual galaxies can show evolutions of velocity dispersions very different from σ_Hα∝ (1+z_ rs). We thus also test decline histories of type `atan' (Equation <ref>), which represents a scenario in which the transitionfrom hot formation to cold formation is faster, yielding rather distinct hot and cold phases. Figure <ref> visualizes the difference between assumed σ_0 histories.We assume that σ_0 is radially constant and always set σ_z(R,t)=σ_0(t). Resolution effects prevent high-z_ rs observations from constraining the radial dependence of σ_Hα reliably, but the following two observational findings motivate this simple modelling approach. a) <cit.> find that, if one characteristic value is assigned to each galaxy, σ_Hαdepends little on the size of the galaxy. b)<cit.> show radial σ_Hα(R) profiles of lensed galaxies and these show various shapes and high dispersions at outer radii.In the case of high input dispersions, it is not desirable to use σ_ϕ/σ_R=1 for an input, as this creates velocity distributions out of equilibrium. When σ_0<10, heating by non-axisymmetries is very efficient for young stars (see Paper 2) and the dispersions will quickly adjust to an appropriate ratio σ_ϕ/σ_R, but heating after birth will playa minor role for high σ_0. Since σ_ϕ/σ_R depends on the shape of the rotation curve and the radial profile σ_R(R) (see e.g. , section 4.4.3), we apply a simple approach: We assume σ_ϕ=√(0.5)σ_R and an asymmetric drift correction for <v_ϕ> derived from Equation 4.228 in <cit.>. We find that σ_ϕ/σ_R for these assumptions only adjusts mildly after insertion.=4.pt The choice for the ratio σ_z/σ_R is also unclear. For high-z_ rs discs, <cit.> studied σ_Hα as a function of disc inclination. Edge-on galaxies would thus be dominated by in-plane dispersions, whereas face-on galaxies would be dominated by vertical dispersions. They found mild indications for σ_R>σ_z, but <cit.> could not confirm this trend with a larger sample of galaxies. From the perspective of the thick-disc stars in the MW, <cit.> findσ_z≈σ_R. We test values σ_R=λσ_0 with λ=1-1.3.As we place all newly added star particles in the midplane z=0, the measured vertical dispersionσ_z of a young stellar component is smaller than σ_0, as the midplane is at the bottom of the vertical potential well and particles quickly lose kinetic energy moving away from it.§.§.§ Star formation history The star formation rate (SFR) is either constant (type 0) orSFR(t)= SFR_0 ×exp(-t/t_ SFR) (type1),or SFR(t)= SFR_0 ×exp(-t/t_ SFR-0.5/t) (type2),with t_ SFR=6-12. Our simulations run for a total time of t_ f=10-12and the constant SFR_0 is adjusted to produce at t_ f a target final baryonic mass M_ f in the range 5-6×10^10, including the mass of the stars in the ICs.Paper 1 showed that for the range of DM density profiles adopted for our models, galaxy massesM_ f in this range provide the right level ofself-gravity to explain the age-velocitydispersion relation of the Snhd. <cit.> favours similar Galaxy masses for his MW mass models.§.§.§ Radial growth history Particles are added randomly distributed in azimuth every five Myrwith an exponential radial density profile Σ_ SF(R)∝exp(-R/h_R(t)). The scalelength h_R(t) of the newly added particles grows in time ash_R(t)=h_R, i+(h_R, f-h_R, i)(t/t_ f)^ξ.To avoid inserting particles in the bar region, where near-circular orbits do not exist, particles are not added inside the cutoff radius R_ cut, which is determined by the current bar length (`adaptive cutoff'; see Paper 1 for details). §.§ GMCs GMCs are modelled as a population of massive collisionless particles drawn from a mass function of the form dN/ dM∝ M^γ with lower and upper masslimits M_ low=10^5 and M_ up=10^7 and an exponent γ=-1.6. Their radial density is proportional to the star formation surface density Σ_ SF(R), and their azimuthal surface density is given byΣ_ GMC(ϕ)∝[Σ_ ys(ϕ)]^α,where Σ_ ys(ϕ, R) is the surface density of young stars withages 200-400 Myr and α=1. The mass in GMCs is determined by the SFR efficiencyζ.Specifically, for each Δ m_ stars of stars formed, a totalGMC mass Δ m_ GMC = Δ m_ stars/ζ is created. GMC particles live for only 50: for 25 their masses grow with time as m∝ t^2, and for the final 25 of their lives their masses are constant, before they disappear instantaneously. GMCs are added on orbits with σ_0=6.See Paper 1 for more details. §.§ Overview of Models and their Naming Table <ref> gives an overview of the models discussed. We use an extended version of the naming convention described in Paper 1. All model names start with a capital letter identifying their IC according to the scheme definedby Table <ref>. A Greek letter α or β following the initial capital letter indicates that the model has declining birth velocity dispersion σ_0 rather than a thick disc in its ICs.In α models the decline of σ_0 follows an `atan' shape (Equation <ref>) while in β models it follows a `plaw' shape (Equation <ref>). For simplicity the model names do not reflect the specific choices for the parameters of Equations (<ref>) and (<ref>), as they are of minor importance for the analyses in this paper.These capital and, if present, Greek letters are followed by a number between 1 and 9 describing the radial growth history of the model, determined by parameters h_R,i, h_R,f and ξ. Growth histories `1', `2' and `5' were already used in Paper 1, `8' (h_R,i=1.0, h_R,f=4.3 and ξ=0.6) and `9' (h_R,i=1.0, h_R,f=3.5 and ξ=0.6) are new and represent inside-out growth from a very compact disc into an extended disc.The final baryonic masses M_ f of all models lie in the rather narrow range 5-6×10^10 and we do not include the variations in the naming convention. For all other parameters we define standard values and additional digits added to the model name only when a model deviates in one or more parameters from the standard. The meanings of the additional digits are: * The overall star formation history (SFH) of a model is described by theSFR type, the final time t_ f and the SF time-scale t_ SFR. Our standard choice is a type 1 SFR with t_ f=10 and t_ SFR=8. We have applied four additional SFHs, which are labelled by `s5',...,`s8'. * The standard input velocity dispersion in models without declining σ_0is σ_0=6 . Models with σ_0=10 are labelled as `σ'. * The GMC star formation efficiency ζ has a standard value of 0.08. Models with ζ=0.04 are labelled as `ζ-', and models with ζ=0.06are labelled as `ζ*'. * The standard choice for the radial-to-vertical dispersion ratio for insertedparticles is λ=1. Models with λ=1.25 are labelled as `λ'.§ VERTICAL PROFILES<cit.> studied the vertical stellar density profile in the Snhd and presented a bias-corrected model fit to Sloan Digital Sky Survey data of the form ρ(z, R=8)=ρ_0 [ exp(-|z|/h_ thin)+fexp(-|z|/h_ thick) ].They found h_ thin=300 and h_ thick=900 with20 per cent uncertainty each and f=0.12 with 10 per cent uncertainty. f here is the ratio of local densities, whereas f_Σ=f h_ thick / h_ thin=0.36 is the ratio of local surface densities. <cit.> show that while most studies in the literature find similar scaleheights h_ thick and h_ thin to <cit.>, this determination ranks at the upper end in terms of f_Σ for photometric surveys,the average literature value of which is significantly lower, f_Σ=0.12. This strong variation in f_Σ between different studies is likely caused by significant differences in the survey selection functions and degeneracies between f_Σ andthe scaleheights. §.§ Vertical profile shape In the models of Paper 1, GMC heating created remarkably exponential vertical profiles with scaleheights h_ thin=200-350. Thus these models can reproduce thescaleheight of the MW's thin disc but fail to create a realistic thick disc. In Figure <ref> we examine the vertical profiles of the present models at t=t_ f and R=8±0.5 as black symbols. We overplot in red fits of Equation (<ref>) to these profiles with the scaleheights given in the top-left corner of each panel. The values of f and f_Σ are given in the top-right corner of each panel. The figure illustrates that both approaches to producing thin+thick disc systems presented here produce double-exponential profiles similar to the one observed in the MW.In the upper row of Figure <ref> we show profiles for models with thick-disc ICs.We find that using disc-like ICs with z_0∼1.7 yields values of h_ thick in therange 815-1133 at t_ f similar to the one inferred for the MW. As was alreadyshown in Paper 1, z_0∼ 1 yields final thick discs that are too thin. For elliptical ICs we find that setting the axis ratio s=2 in model K2 yields h_ thick≈2.4 and s=3 in model O2 yields h_ thick≈1.4, which are both too thick. We thusdecided to focus on disc-like thick ICs. In terms of thin-disc scaleheights, we find values h_ thin=200-339. As was already discussed in Paper 1, for a given GMC mass function, lower values of the star formation efficiency ζ∼0.05 and thus ahigher total mass in GMCs per mass of formed stars are required to obtain h_ thin∼ 300 as in the MW. Lowering the DM halo concentration reduces the vertical force contribution from the halo and thus also mildly increases h_ thin. The highest values for both h_ thick and h_ thin for disc IC modelsare found for model U1 that at t_ f features an overly extended and thick bar as is discussed in Section <ref>. As was discussed in Paper 1, bars that extend beyond R=5 can significantly thicken vertical profiles at R=8.For the density and surface density ratios in models with thick-disc ICs, we find ranges of f=0.04-0.167 and f_Σ=0.14-0.53, which include the <cit.> values and are in the upper half of thevalues of the <cit.> literature compilation. Naturally, increasing the thick-disc mass M_ disc, iat fixed thick-disc scalelength h_R,disc increases these ratios (Q versus P models) and decreasingh_R,disc at fixed M_ disc, i lowers them (U versus Q models). As far as other model parameters are concerned there are no clear patterns apparent. This is likely connected to competing effects. For example, few GMCs produce less vertical heating, but lead to stronger bars, which, if long enough, can thicken the vertical profile.The lower row of Figure <ref> shows models with declining birth dispersions. We find values for the scaleheights in the ranges h_ thin=214-284 and h_ thick=863-1234, very similar to the ranges found for thick IC disc models. Model Cα2 has the thinnest thin disc as at late times it has a very low value ofσ_0<5 and it also has a high star formation efficiency ζ=0.08. 6 has ζ=0.06and σ_0∼10 at late times and thus the highest h_ thin among these models. Vα8s5 and Vβ8s5 only differ in the shape of the declining σ_0 curve:Vα8s5 is of type `atan', whereas Vβ8s5 is of type `plaw' (see Equations <ref> and <ref>).Their final vertical profiles are rather similar, so both types of decline are acceptable. The value of h_ thickis also mildly influenced by the ICs, which for V and M models is a low-mass s=2 elliptical. As modelsVα8s5/Vβ8s5 have a higher final mass than Mα1 and thus a lower fraction of IC stars,their thick discs appear mildly thinner. The corresponding thin-to-thick disc ratios are f=0.025-0.055 and f_Σ=0.11-0.22, which are lowerthan the <cit.> values, but comfortably within the range found in the literature. For models with decliningσ_0(t) the thick-disc mass fraction depends on the fraction of mass formed during earlyformation stages with high σ_0 and thus on the detailed forms of σ_0(t) and the SFH. Moreover, the radial growth history determines how many stars are formed at a certain radius duringthis period. Dynamical heating and migration processes also influence the number of old stars found at t_ f, so the final value of f is not easily predicted. §.§ Vertical profile evolution with time In Figure <ref> we analyse the temporal evolution of the shapes of vertical profiles in two different types of simulations. In the models of Paper 1 that included GMC heating, thin-disc vertical profiles are at all times exponential and their scaleheights change very little with time. This finding reflects balance between mass growth, which continuously supplies cold particles and deepens the vertical potential well, and GMC heating which efficiently increases the vertical velocity dispersions of young stars.The thick-disc IC model 5 shows a double-exponential profile from early on. The thin-disc part of its profile is very constant, just like the thin-disc-only models of Paper 1. What changes are the surface density ratio f_Σ, which by construction becomes more and more thin-disc-dominated, and the scaleheight of the thick disc, which becomes smallerwith time, because the growth in the thin disc's mass deepens the vertical potential well and thethick disc is not heated significantly.Mα1 represents models with declining σ_0(t). Its profile at early times is closer to a single- than a double-exponential, as there are no cold, thin-disc populations present. Only as σ_0(t) falls below ∼ 20 does a thin disc build up. The scaleheight of the thin disc h_ thin becomes smaller with time as the decline in σ_0(t) cools the thin-disc population as a whole despite the vertical heating due to the GMCs. The deepening of the vertical potential well adds to that effect and reduces h_ thick in the same way as in 5.§.§ Radial dependence of vertical profiles The vertical profiles of observed disc galaxies are very constant radially <cit.>. Paper 1 showed that in thin-disc-only models that include GMC heating, vertical profiles are almost independent of radius R, unless there is a buckled bar, which thickens only the central region. Figure <ref> shows the radial variations of the vertical profiles in five of our thin+thick models. The inner parts of models 1o and O2 show thickening by a bar. 1o otherwise shows a profile that is almost constant in radius. Model 5 shows a thin disc that becomes mildly thicker and has a higher mass fraction towards larger R, whereas h_ thick stays roughly constant. This is caused by inside-out formation, on account of which the young thin disc has a longer scalelength than the thick disc, and higher σ_0=10 than the models studied in Paper 1. Combined with a shallower potential well at outer radii, where heating is limited, this value of σ_0 yields thicker thin discs.1 shows a stronger fading and a mild thickening of the thick populations towards the outskirts.Compared to P ICs, the T ICs have a more compact and more massive thick disc and also a bulge component. The effective vertical profile of the two components thus varies with radius alreadyin the ICs. O2 has elliptical ICs, which lead to a thick population that becomes thicker with increasing R and also attains a higher mass fraction in the outskirts.Model Mα1 with declining σ_0 shows a thin disc that thickens mildly with R, because atlate times it has σ_0∼10 and thus behaves similarly to model 5 discussed above.The thick disc becomes thicker with R. This is characteristic for models of this type. It is a consequence of our assumption that σ_0 is constant with radius. This leads to a flaring of the hot component, which is hardly influenced by vertical heating. Note that in contrast to the thick discs in models withdeclining σ_0, the thick-disc ICsof our alternative modelling scheme were set up with a radially constant scaleheight and thus avertical velocity dispersion that declines with R. §.§ Flaring of mono-age components In Figure <ref> we examine for a selection of models how the median distance from the midplane |z|_ med varies with radius R for populations ofdifferent ages. It has been suggested that the lack of significant changes with discradius R in the double-exponential vertical mass profiles of disc galaxies is aconsequence of inside-out growth combined with |z|_ med being an increasingfunction of R for all mono-age populations (`flaring'; ).Flaring can be caused by satellite interactions or misaligned infall of gas, which are not present in our models. However, Paper 1 showed that disc galaxies formed in isolation also have flaring mono-age components. The amount of flaring is determined by theradial mass profile of the disc and the radial profile of the vertical velocitydispersion of mono-age components σ_z(R,τ), which is determined by the birth dispersions of stars and the vertical heating mechanism(s) at work. Radial migration of stars can also influence σ_z(R,τ) <cit.>.The left most panel of Figure <ref> depicts |z|_ med(R) for the mono-age populations of model 5, which has thick-disc ICs. The thin-disc populations are at all radii substantially thinner than the old thick-disc stars. These mono-age populations all show flaring, and for the youngest populations |z|_ med increases from R=1 to 15 by up to a factor of η≡|z|_ med(15)/|z|_ med(1)∼ 5.In fact, in 5 the structure of the thin-disc populations is similar to that in thin-disc-only model Y1 examined in Paper 1. The relative increase in |z|_ med with R becomes smaller with increasing age, and we find η∼ 2.5 for the oldest inserted stars. In model 1o (second panel in Figure <ref>) the situation is altered by a vertically extended bar. On account of bar buckling, |z|_ med for intermediate-age populations now peaks around R∼3, then declines slightly to R∼ 8, and gradually increases further outward. The youngest populations have not been affected by bar buckling and have η∼ 3.In all our models, non-IC stars at any given time t are inserted with a radiallyconstant birth dispersion σ_0(t). On account of the outward decrease insurface density, this results in flaring. GMC heating increases σ_z morestrongly in the centre than in the outskirts and thus flattens the increase in|z|_ med(R). For models without buckled bars and with ζ=0.04 and thusmore GMCs per unit mass of inserted stars, η can be as low as ∼ 1.5 forold inserted stars.Bars can additionally heat the central regions of galaxies(see e.g. ) and thus cause even flatter runs of |z|_ med(R). The three panels on the right of Figure <ref> show models with declining σ_0, and in these models the structure of |z|_ med(R) is quite different from what it is in the models with thick IC discs. Now the low-mass elliptical ICs are unimportant because the thick disc is formed mainly by old added stars. Due to the continuous decline in σ_0(t), the curves form a continuum rather than a bimodal grouping. The young, thin-disc populations behave very similarly to those in the models with thick-disc ICs, the local maxima in |z|_ med at R∼ 2 in model Vα8s5 being caused by a buckled bar. The thick-disc stars in models with declining σ_0 (red and orange curves) yield a similar value of η to the youngest thin-disc stars in all models. Again this reflects our decision to make the declining birth dispersion σ_0 independent of R.Given that vertical GMC heating has little influence on the thick-disc populations, and that the depth of the vertical potential well declines with R, strong flaring is an inevitable consequence. The flattest curves are thus found for the intermediate-age populations, which were already born on relatively cold orbits and have been significantly affected by GMC and bar heating. The differences between models Mα1 and 2 can be explained by the different shapes of σ_0(t) applied (see Figure <ref>). Whereas the oldest component of each model flares in a similar way, the two next-oldest populations show stronger flaring in the atan model Mα1 than in the plaw model 2, in which σ_0 declines more gradually.Also, in the atan model |z|_ med declines with decreasing age faster than in the plaw model.The flaring of all disc populations discussed so far can be qualitatively explained by birth-dispersion profiles and disc heating mechanisms. A comparison between the thick-disc IC stars in models 5 and 1o indicates that radial migration plays a role as well. The IC stars in 5 only show a strong outward increase in |z|_ med in the centre and hardly any flaring at larger radii, whereas in 1o, the flaring of the IC stars in the outer disc is stronger. The thick-disc ICs were created with a radially constant scaleheight and are hardly affected by vertical heating mechanisms, so their curve of |z|_ med(R) at t_ f is determined by the change in the vertical potential well together with extent to which their stars migrate radially. Bar formation funnels a lot of mass to the centre and consequently the disc's thickness decreases there. The stronger flaring in 1o is likely explained by higher levels of radial migration.When stars migrate radially, their vertical actions J_z are conserved <cit.>.Populations of stars that are born with a radially constant scaleheight in a MW-like disc have their mean J_z decreasing with R. Consequently, outward migrators at a given R have higher J_z (and thus higher σ_z) than non- or inward migrators. If there are more outward than inward migrators, as expected for the outer disc regions, such populations are expected to flare <cit.>. This principle is complicated by the finding that stars migrate less if they have high J_z <cit.>, but Paper 4 shows that for our thin-disc populations and for stars from thick-disc ICs at t=t_ f, outward migrators are indeed more numerous in the outer disc and have higher σ_z than inward migrators (see also ). Model 1o has both a more compact IC disc and a lower density dark halo than model 5, which leads to stronger non-axisymmetries and a higher fraction of outward migrators in the outer disc (Paper 4), which in turn explains the stronger flaring of IC stars in 1o.Compared to the oldest inserted stars in models Mα1 and 2, in model Vα8s5 this population shows a much flatter curve |z|_ med(R) at R>5. Model Vα8s5 has a lower density dark halo and a more compact disc at early times than the M models. Hence in this model the oldest inserted stars in the outer disc have a higher fraction of outward migrators.For models with declining σ_0, thick-disc stars are born with radially constant σ_z and thus their mean J_z increasing with R. As they are hardly affected by disc heating, in these models old outward migrators at a given R have lower J_z and thus lower σ_z than inward migrators. This likely explains the flatter curve |z|_ med(R) for the oldest inserted stars in model Vα8s5. So depending on the shape of σ_z(R) at birth, radial migration can both strengthen and weaken the flaring of a mono-age population. A detailed analysis of radial migration in our models is presented in Paper 4. §.§ Radial and vertical age structure The combination of recent and ongoing astrometric and spectroscopic surveys of MW stars is about to increase vastly our knowledge of the age structure in the Galactic disc(s) (e.g. ). In the top row of Figure <ref> we therefore show maps of median age τ_ med as a function ofR and |z| for various models. To do so we assume that the oldest stars in all models are 13 old and thus randomly assign ages in the range [t_f,13] to IC star particles.The leftmost panel shows the thin-disc-only model Y1 from Paper 1. The innermost region R<4at all altitudes and high altitudes |z|>1.5 at all radii are dominated by old stars. The youngest τ_ med are found at high R and low |z|. Intermediate τ_ med are confined to |z|<200 at R<6, to |z|<500 at R<10, but can populate regions up to |z|∼1500 at R∼15. This characteristic τ_ med pattern is caused by a combination of inside-out formation and disc heating. The oldest population is more compact and thicker, whereas the younger population lives closer to the midplane and preferentially at larger radii, where it extends to higher |z|.Model 5 has an old thick disc, which is more massive, much thicker and more extendedthan the oldest disc population in Y1. This changes the τ_ med map only marginally. The high-|z| populations are by construction older than in Y1 and the young, outermost populations are also mildly older than their counterparts in Y1. The τ_ med pattern is however rather similar. P2 differs from 5 in lacking inside-out formation, which makes the outer populations older. It also has lowerσ_0, which reduces the flaring of the outer populations. Consequently, at all radii R, all altitudes|z|>500 are dominated by old stars and the zone populated by intermediate τ_ med is confinedto |z|>300 at R<10 and to |z|>500 at all R. Model U1 is an inside-out model, but compared to 5 has a lower concentration halo, a more massive and more compact IC thick disc and lower σ_0. It has a thicker and longer bar than the other depicted models. The buckled bar causes an area at R<5, which is populated by stars with old τ_ med and shows no noticeable vertical age gradient. The bar also heats the disc at R∼5-10 vertically and thus increases the altitudes at which youngerstars are found, so the radial increase of the maximum |z| at which intermediate τ_ med are found is flatter in this model.Model 2 is a model with declining σ_0. It shows a τ_ med pattern that differs from those of the other models shown in Figure <ref>. At R<5, its vertical age structure is similar to that in the P models, but at R>5 the intermediate-age stars reach higher altitudes, which leads to declining radial age gradients at all |z|. This is caused by our assumption of radially constant σ_0(t), as was also discussed in relation to the flaring of mono-age populations shown in Figure <ref>. Moreover, vertical heating is inefficient for stars with high σ_0 and for stars at large radii.<cit.> have recently presented measurements of radial gradients in τ_ med at various altitudes |z| in the Snhd. At |z|>500, they find significant declines in τ_ med with radius R at all radii. However, the value of ∂τ_ med / ∂ R is still very uncertain. Close to the plane near the solar radius R_0, τ_ med(R) is rather flat. In the middle row of Figure <ref> we show τ_ med versus R at four ranges in |z|.As model Y1 lacks a thick disc, we will not discuss it in detail. The P and U models with thick IC discs clearly show no decline in τ_ med with R at |z|∼ 1.5 and only 5 shows a negative τ_ med /R at |z|∼ 1.0 and R>10 due to inside-out formation and a stronger flaring in the young disc dueto higher σ_0. By contrast, 2 shows clear negative gradients throughout the whole disc at all latitudes and is thus more in agreement with the age determinations of <cit.> .At |z|∼ 0.5, all models show a negative radial τ_ med gradient. This gradient is weaker in U1 due to the influence of the unrealistically long and thick bar. In the midplane, the inside-out models also show clear negative age gradients, whereas P2, which has a constant radial feeding scalelength, shows a rather flat age profile, as does U1, again strongly affected by the long bar. As the age gradients in 2 and 5 are flatter closer to the midplane and the observations are still very uncertain, little can be deduced yet about inside-out growth.<cit.> have presented evidence for a vertical age gradient in the Snhd from asteroseismology. They find a decline by ∼4± 2/, but have little knowledge of the shape of the decline. In the lower row of Figure <ref> we present τ_ med as a function of |z| at three radii, the cyan line representing a solar-like radius and the dashed line showing a constant vertical gradient of 4/.Due to the dominance of the thick disc at |z|>1 and the weak radial age gradient, at all radii in P2, age increases more strongly than at 4/ up to |z|∼ 1 and then flattens out. In 5 at R=13, the enhanced presence of younger stars away from the plane results in aflatter increase of τ_ med with |z|, but at R=8 the situation is similar to that in P2. In U1 the long bar causes an almost flat τ_ med versus |z| plot in the bar region and a flatter gradient at R=8. In 2, τ_ med(|z|) at R=8 flattens more gradually and at R=13 has an almost constant slope due to the stronger flaring of mono-age populations.Averaged over the studied vertical extent of 2, all models show a vertical gradient consistent with the findings of <cit.>. As they observe very few stars above |z|>1.2, model 2 shows the best agreement with the still very uncertain data. § RADIAL MASS DISTRIBUTIONIn this section we investigate the radial distribution of baryonic and dark matter in our models and the circular speed curves v_ circ(R) that result from them. §.§ Dark matter density As discussed in Paper 1 the parameters for our DM haloes as set up in the ICs are motivated by what Λ cold dark matter (ΛCDM) predicts for haloes associated with MW mass galaxies. The DM profile ρ_ DM(r) will be modified by growing a massive baryonic disc within the DM halo and by interaction with non-axisymmetric disc structures such as the bar and spirals. As the halo is always spherical in the ICs, but the disc mass fraction in the ICs varies strongly between ICs and the various galaxy models evolve differently, the final haloes differ even if two models share the same DM IC parameters.Our best constraints on the DM content in the MW come from dynamical measurements of the total matter surface density in the Snhd. Subtracting the baryonic components, <cit.> find a local DM density of ρ_ DM=0.013±0.003^-3. In Figure <ref> we plot in red the DM density profiles ρ_ DM(R) as measured in the midplane of the galaxy at t=t_ f and compare them to the Snhd constraints assuming that the solar Galactic radius is R_0=8.3±0.3 <cit.>.We find thatat t_ f all models fall within the constraints of <cit.>. The four shown inFigure <ref> are a representative selection. The main drivers for ρ_ DM(R_0) at t_ f are, as expected, initial halo concentration c and the added disc massM_ add=M_ f-M_ disc, i-M_ bulge, i. Consequently 2, which has c=6.5 and M_ add=3.5×10^10 has the lowest ρ_ DM(R_0), whereas Mα1 with c=9 and M_ add=4.5×10^10 has the highest ρ_ DM(R_0) among themodels shown. P2 with c=9 and M_ add=3.5×10^10 and Vα8s5 withP2 with c=6.5 and M_ add=5.5×10^10 show intermediate ρ_ DM(R_0). It is worth noting that the models with c=4 presented in Paper 1 and discarded because of overly strong bars indeed show too low ρ_ DM(R_0).We also note that at t_ f none of our models shows a cored DM profile in the centre, as was recently favoured by <cit.>. Our IC DM profiles do not contain a core as is indicated by the pink dashed lines. The DM densities ρ_ DM(R_0) of the ICs are significantly lower thanin the final models. Initial profiles with c=9 lie at the lower allowed limit for today's Snhd and models with c=6.5 are clearly below this limit. During the simulations they are altered bycompression due to the added mass in stars and by angular momentum transfer from stars toDM due to bars and spirals. As all models have declining SFR(t) and bars form in the later evolution stages as shown in Section <ref>, the increase in the DM density at R<15, where the disc grows, is strong up to t=0.3t_ f as indicated by the green dashed lines and rather weak afterwards. In the four models shown, the relative increase in ρ_ DM(R<15) is strongest in models Vα8s5 as it has the largest baryonic mass fraction and the largest M_ add, and the increase is weakest in P2. Angular momentum transfer to the halo by spirals and the bar is not strong enough to create cores, as was alsoshown by <cit.>. §.§ Solar Neighbourhood surface densityPaper 1 showed that despite having control over the evolution of the input scalelength h_R (t), there was little control over the final surface density profile of the models. The more compactly a disc was fed, the earlier it grew a bar, which redistributed matter and, as we avoid inserting particles into the bar region, shifted the inner cutoff radius outwards. In the end, the surface density profiles of a range of models with different radial growth histories were thus rather similar. Our surface density was thus decided by the total mass of the final model, which we justified from the reasonable agreement of our models with a) the vertical scaleheightof the thin disc, b) an appropriate local circular speed, c) an appropriate amount of radial migration to R_0 and d) appropriate vertical and radial velocity dispersions.In Figure <ref> we examine how well our models fulfil constraints on the Snhd baryonic surfacedensity Σ_ b(R_0). Table 3 of <cit.> gives an overview of determinations ofΣ_ b(R_0) including gas of all phases, stars and stellar remnants. Σ_ b is consistently found to be in the range Σ_ b(R_0)=40-60^-2, where the given errors are included in the interval. Gas is found to contribute 25-30 per cent. As in our simulations gas is only represented by GMCs, our gas fractions are much lower. We choose Σ_ b(R) over the stellar surface density Σ_⋆(R), as we are interested in theconnection between disc structure and kinematics, and the strength of non-axisymmetries and thus the levels of radial disc heating and radial migration are determined by Σ_ b(R). Moreover, the interplay between vertical profiles and vertical velocity dispersions depends on the total mass surface density and not only on Σ_⋆(R)and it is therefore appropriate to compare Σ_ b in models and observations, although the divisionof mass between gas and stars is very different. It should be noted that the neutral hydrogen component,which is missing in our models, will have a smaller scaleheight than the stars and thus a model that has theright vertical profile, kinematics and DM halo is not expected to agree with the Snhd Σ_ b(R_0).The local radial exponential scalelength h_R (R_0) of the MW is rather uncertain. <cit.> recently compiled a variety of measurements in the optical and infrared, the vast majority of which fall in the range 2-4. Their meta-analysis of 29 previous measurements yields an estimate of h_R(R_0)∼2.65. <cit.> showed that populations of stars with different chemical abundances show widely varying scalelength, the most compact of which have h_R∼ 1.5 and the most extended of which are consistent with locally flat profiles. As we are plotting Σ_ b(R_0) that includes GMCs, the comparison is not exact, but because at t=t_ f the GMC mass fractions are 2-3 per cent as for molecular gas in the MW today, the correction is negligible for our purposes. A more relevant question is whether the missing neutral gas mass, which is a highly relevant mass component in the outer MW disc, is properly represented in our models. In the upper row of Figure <ref> we plot Σ_ b(R) for various models and overplot a blue box indicating Σ_ b(R_0)=40-60^-2 at R_0=8.3±0.3 and a dashed line indicating an exponential with h_R (R_0)=2.65. Due to the connection of Σ_ b(R) to bars, we also plot the m=2 FourieramplitudeA_2(R)≡1N(R)∑_j=1^N(R)e^2ıϕ_jin the lower row of Figure <ref>. The dashed line marks lnA_2=-1.5, which is used for determining the adaptive cutoff region, within which no particles are inserted in our models. Model P2 has a constant feeding scalelength h_R=2.5, an IC disc scalelengthh_R,disc=2.5 and lives in a c=9 halo. Despite the constant input scalelength the final profile is very different from a simple exponential. At R<5, the profile is shaped by the bar,which at t_f has a length of ∼ 5, similar to that of the MW bar. The bar steepens the profile in the centre and flattens it at radii similar to those of the bar tips. At R=5-10 the surface density profile is mildly flatter than the dashed h_R (R_0)=2.65 line, whereas at R>10 the profile is steeper. The Snhd surface density is close to the upper limit of theobserved range =40-60^-2.Model 1 has a more compact and more massive IC thick disc than P2, inside-out formation in the range 1.5-3.5, a mildly higher final mass and a higher GMC mass fraction. Its thick disc also has a higher-than-average ratio of radial to vertical velocity dispersions σ_R^2/σ_z^2=1.8. The outcome is a model witha weaker bar and thus a Σ_ b(R) profile that is well-fit by an exponential at R=3-15. Σ_ b(R_0)agrees well with the Snhd constraints. This is one of the models which comes closest to the inferred local profile of the MW.Model 2 has declining σ_0. It lives in a c=9 halo and has inside-out growth in the range1.5-4.5, which generates a final exponential mildly flatter than h_R (R_0)=2.65. Its value of Σ_ b(R_0) is in agreement with the Snhd constraints. It has a rather weak bar, which influences the profile only at the inner radii. Model U1 lives in a c=7.5 halo and has a massive and compact thick IC disc and inside-out formation in the range1.5-4.5. It has a stronger and longer bar compared to the two previous models with c=9 haloes due to a higher baryon fraction as discussed in Paper 1. Its profile shows a steep bar region out to R∼5, a flat region at R∼5-8 and a shallow exponential decline at R>8. Like for the previous models,Σ_ b(R_0) agrees well with Snhd constraints. The declining σ_0 model 6 has a c=6.5 halo, a high final mass M_ f=6×10^10 andgrows inside out in the range 1.0-3.5. Its Σ_ b(R_0) is too high for MW constraints andalthough its bar is weaker than in U1 and also P2, its Σ_ b(R) profile is significantly flattened at R∼4-7. At R∼7-12 the profile agrees well with an exponential with h_R (R_0)=2.65.There is a clear tendency of A_2 being lower in models with declining σ_0 compared to models withthick IC discs. We will discuss this further in Section <ref>. §.§ Radial profile evolution of mono-age components Paper 1 and Section <ref> have shown that the output scalelengths are somewhat independent of the input scalelengths, as bars and spirals redistribute matter. We know from observations of stars in the Snhd that the old thick populations are more compact than the young thin ones <cit.>. To test, how different populations of stars are affected by changes in the radial distribution, inFigure <ref> we plot local scalelengths h_R(R_0) as a function of time for populations of different ages. The pink line is for IC stars and the other colours are for 10 equally spaced age bins of all stars inserted during the simulations. h_R(R_0) is determined by a single-exponential fit to the surface density profile Σ(R) at R=6-10, irrespective of how good the fit is.Model P1s6 has little bar activity at any stage of its evolution and is thus well suited to understand the plots. As it has inside-out growth from 1.5 to 4.3 paired with a thick IC disc with h_R,disc=2.5, different age components are rather well separated in size. The IC component has a shorter scalelength than in the setup, because it is compressed by the disc's gravitational field. Apart from a mild shrinking of all populations due to compression and a mild level of noise, which is likely caused by spiral activity, the output scalelengths are essentially set by the input scalelengths.P2 has a constant input scalelength h_R=2.5 and the same ICs as P1s6 and thus the populations of different ages are only mildly separated in size due to continuous compression. At t∼ 8 bar formation causes an increase in h_R(R_0), which is stronger for younger populations, so the measured scalelengths increase to h_R(R_0)∼2.5 for the oldest and ∼ 4 for the youngest components. As alreadyshown in Figure <ref>, U1 is more strongly affected by a bar. U1 has inside-out growth as in P1s6 and a shorter IC disc scalelength h_R,disc=2.0. Bar formation at t∼4 causes a strongincrease in h_R(R_0) for all age groups and at t∼9bar growth causes another increase,so at t=t_f, the oldest population has h_R(R_0)∼2.5 and the youngest populations have an essentially flat profile. The declining σ_0 models Mα1 and 6 have low-mass elliptical ICs, for which we find h_R(R_0)∼2.5 fits at early times. They both grow inside out, Mα1 from 1.5 to 4.3 and 6 from 1.0 to 3.5. As the initial mass of the baryonic ICs is much lower than in P and U models, the amount of compression for the oldest components is stronger. This is especially true for 6, which has more compact feeding scalelengths at early times and also a shorter-than-average SFR time-scale t_ SFR=6. The latter increases the mass instars added at early times and thus also the mass in GMCs present at these formation stages. Mα1 has a bar from t∼ 6, which causes a mild increase for all h_R(R_0), whereas bar activity is measurable from t∼ 3 onwards in 6.Irrespective of how long a bar is and how strongly it affects disc evolution and how high is the level of compression, the final ordering of h_R(R_0) always reflects the ordering of scalelengths at input. Additionally, in combination with the results of Section <ref>, it is clear that all models with a bar similar to that of the MW show an increase for h_R(R_0) of all age components with time due to bar formation and growth. §.§ Circular speed curvesRecently, various surveys of bulge/bar stars and microlensing data have enabled more detailed mass models of the centre of the MW <cit.>. These models agree in the following points: 1) The centre of the MW is baryon dominated; 2) The baryonic contribution to the rotation curve at R∼3 is v_ circ, b∼185 [althoughfind an uncertainty ∼± 25]; 3) The contributions of DM and baryons to v_ circ are roughly equal at R_0. Further constraints on v_ circ(R) come from the motion of stars in the Snhd: <cit.> finds v_ circ(R_0)=238±9 and R_0=8.3±0.3.<cit.> (hereafter AS15) presented an inside-out growing model in a c=9 halo. Its rotation curve fulfilled the <cit.> constraints, but has too few baryons in the centre to match any of the constraints from microlensing. In Figure <ref> we present circular speed curves for a selection of our models: black is total v_ circ(R), red is the baryonic contribution v_ circ, b(R) and blue is the DM contribution v_ circ, DM(R). The pink boxes mark the <cit.> constraints. v_ circ(R) is measured in the midplane of the disc and averaged azimuthally.Figure <ref> shows two c=9 models: P2 and 2. P2 has h_R=2.5 both in the ICs and at all times through the simulation. Its value of v_ circ(R_0) is at the upper end of allowed values. Its central baryonic contribution is higher than that of AS15 but still too low for the microlensing constraints. 2 grows inside out in the range 1.5-4.3 and has a higher added mass M_ add. The former leads to a weaker central baryonic contribution than in P2 and the latter causes a stronger compression of the halo and thus an unacceptably high v_ circ(R_0).The microlensing constraints suggest shifting mass from the halo to the discs. As was shown above, the Snhd DM density allows concentrations as low as c=6.5 given a constant IC halo mass of M_ tot=10^12. Models U1 and 1 have haloes with c=7.5 and 6 has a c=6.5 halo. U1 has a massive thick IC disc with M_ disc,i=2.5×10^10 and h_R,disc=2.0. It grows inside out from 1.5 to 4.3 reaching a final mass of M_ f=6×10^10. Its baryonic contribution to the rotation curve peaks at v_ circ, b∼190 and falls below the DM contribution at R∼8 and thus fulfils all microlensing constraints. Its Snhd v_ circ(R_0) is lower than that found by <cit.>.Model 1 has a thick IC disc with M_ disc,i=2.0×10^10 and h_R,disc=2.0 and in addition an IC bulge with a_ bulge=0.7 and M_ bulge,i=0.5×10^10. It has inside-out growth in the range 1.5-3.5and the same final mass as U1. Consequently, its peak v_ circ, b is higher at ∼200, but it still fulfils all constraints from microlensing, as well as the Snhd v_ circ(R_0) constraints. The declining σ_0 model 6 also fulfils all constraints but in a different way. It starts from a low-mass elliptical IC and grows a disc with M_ f=6×10^10 like those of two previous models. Its inside-out growth is from 1.0 to 3.5 and its final v_ circ, b(R) is rather constant at 170-180 in the range R=2-10. Unlike 1, which has a flat total circular speed with 240 for R=3-10, 6 hasv_ circ(R) increasing in this radial range from 220 to 250.§ BAR FORMATION AND EVOLUTIONPaper 1 showed that models starting with a thin-disc IC and having no GMC heating undergo strong bar activity from early times. GMC heating can delay and weaken bar formation andevolution and in extreme cases prevent the formation of a strong bar over cosmological time-scales. In this section, we examine how this picture is modified by an old thick-disc component.§.§ Bar strengths Figure <ref> displays for 10 models the evolution of the m=2 Fourier amplitude A_2 (see Equation <ref>) for all the stars within R=3. Model Y2 represents thin-disc-only models and we see that ln(A_2) instantaneously increases to -2.5 as the addition of mass to the thin andcompact IC disc makes the system develop non-axisymmetries. As discussed in Paper 2, the radial heating by GMCsat low disc mass and high SFR is important and delays the formation of a strong bar during the first 3 of evolution in Y2. This effect is enhanced in model Y1ζ-, which has ζ=0.04 and thustwice as many GMCs per unit mass of added stars: ln(A_2) is kept at ∼ -3 until t∼7, when itincreases to ∼ -1.7, indicating a rather weak bar.The curve for model P2 is very different. P2 shares with Y2, the constant radial growth history h_R(t)=2.5,the final mass M_ f=5×10^10 and the shape of the SFH. Its IC disc is thicker, more extended and more massive than the one in Y2 and the normalization of its SFH is thus lower. The existence of a hot disc, which is stable against bar formation prevents the growth of A_2, although stars are continuously added on cold orbits throughout the simulation. Only at t∼7 has enough thin disc been accumulated to make the composite system unstable to bar formation.Paper 1 showed that, in the absence of a thick disc, lowering the halo concentration from c=9 isproblematic, as the system becomes more self-gravitating and bar unstable. U1 has a more massive andmore compact IC thick disc than P2, lives in a c=7.5 halo and has an inside-out growth history.Despite the higher thick-disc central surface density and the lower halo concentration than P2,the thick disc still suppresses bar formation for 4. Another ingredient here is the radial-to-vertical velocity dispersion ratio in the IC thick disc. Model 1 has a more massive and more compact IC thick disc and a slightly higher final mass than P2 and inside-out formation from 1.5 to 3.5. Despite the higher surface densities, σ_R^2/σ_z^2=1.8 for the ICs in 1 compared to a value of 1 in P2 makes the systemmore stable against bar formation and thus weakens bar formation more strongly than in P2.To understand bar evolution in models with declining σ_0, which start with low-mass elliptical ICs, we first examine model K2 that grows a thin disc inside a higher mass elliptical IC. K2 and P2 differ only in that P2 has a thick-disc IC of the same mass. During the early evolution phases of K2, A_2 is suppressed due to the elliptical ICs. However, in K2 A_2 increases at an earlier time than in P2 and from t∼ 3 on shows a significant bar. By construction, the in-plane velocity dispersions of the IC stars at R 2are similar in these models. However, the surface densities are higher and thus the rotation velocities of the IC stars are faster in K2. Consequently, K2 is more unstableto bar formation than P2.Compared to the K ICs, the M ICs contain an elliptical, which is three times less massive. So a cold disc model in M would have high A_2 at an earlier time than K2. The hot input dispersions for the old populationsin Mα1, however, act in the same way as the thick IC disc in the P models and delay bar formationuntil t∼6. We find that the specific shape of declining σ_0(t) does not significantlyinfluence bar formation.Vα1 is the equivalent model to Mα1, but it lives in a lower concentration c=6.5 halo. Still, bar formation is delayed until t∼4. 6 has a higher mass, a more compact feeding history and a shorter SFR time-scale t_ SFR than Vα1. All three factors lead to a much fasterincrease in surface density at early times, which outweighs the fact that at feeding the radial-to-vertical input dispersion ratio σ_R/σ_z=1.25 is higher than in Vα1. Consequently, A_2 is higherat early times.The specifics of the SFH, the radial growth history, the dispersions of the old components and GMC heating thus determine the bar formation history of an individual model. Figures <ref> and <ref> however show that halo concentrations c=6-7 allow models with reasonable final bars in the presence of hot disc components. The fact that in the lower row of Figure <ref>, the m=2 amplitudes at final times and radii R>5 are lower in models with declining σ_0 than in the thick IC disc models is connected to the gradients of σ_R. In models with thick-disc ICs, the oldest stars have radially constant scaleheights and σ_R/σ_z and thus declining σ_R(R),whereas models with declining σ_0 assume a radially constant input dispersion σ_R. Thus in the end, the outer thick components are radially hotter in models with declining σ_0 and thus less unstable to m=2 modes. §.§ Bar morphology The central region of our Galaxy is dominated by a bar, the inner part of which consists of a boxy/peanut-shaped bulge at R<2with an X-shape at |z|>500 <cit.> surrounded by a vertically thin part, the long bar,extending to R∼ 4-5 <cit.>. AS15 demonstrated that an inside-out growing model without GMCs in a c=9 halo produces a bar with X-shaped structure with the tips of this structure at (x,z)∼(2,1.3), very similar to the structure of the MW bulge/bar region inferred by<cit.>. Paper 1 showed that some thin-disc-only models with GMCs also displayed bars very similar to the one in the MW, but also noted that not all of these bars are buckled and that lower concentration haloes favoured unrealistically long bars.Here we test how well our thick-disc models can reproduce the MW bar. This is interesting as the chemically defined thick disc is concentrated and should thus have a high mass fraction in the bar region, but can only form a bar if the thin-disc fraction is high enough, as discussed above. Figure <ref> shows edge-on and face-on surface density maps of several galaxies at t_ f.Like the standard Y models in Paper 1 and the model in AS15, P2 lives in c=9 halo and shows a bar that is ∼ 5 long. Model P1s6, which has a more radially extended feeding history shows only a small, weak bar in the central R<2. The vertical structure of P1s6 is indistinguishable from a pure disc galaxy, whereas P2 has a boxy shape with a lateral extent of ±∼ 2 anda vertical extent of ±∼ 1 and a mild hint of an X structure. As in the MW bar, the outer regions are thinner. 1 has a more compact and more massive, but radially hotterthick disc and does not show a bar, just mildly elliptical surface density contours.The galaxy in model O2 evolves in a c=9 halo from an elliptical IC. At t_ f, its bar is ∼ 4 longand is currently buckling as indicated by the broken mirror symmetry relative to the x-axisas first observed in a simulation by <cit.>.This event will eventually produce an edge-on peanut bulge with a characteristic X-shape (see ). U1 has a compact, massive IC thick disc, a higher-than-average final mass and lives in a c=7.5 halo.Its bar grows to a length ∼ 6 and is thus only mildly longer than the P2 bar. However, bar buckling has created a significantly more extended X-shape with a lateral extent of ±∼ 4 anda vertical extent of ±∼ 2.The lower row displays bars in models with declining σ_0. Model 3b grows a disc around a compact bulge IC in a c=9 halo. At t_ f, it clearly shows an X-shaped edge-on structure in the central ∼ 2, stronger than the similarly sized one in P2. However, its face-on image reveals an almost axisymmetric image. Going back in time, we find that at t=8.5 3b exhibited a strong bar that had not yet buckled. From the evolution of A_2(R<3) depicted in Figure <ref>, we learn that the bar formed around t=6. Between t=8.5 and t_ f 3b undergoes buckling, but in the final ∼ 2 of the simulation its bar becomes continuously weaker.Models Mα1ζ* (c=9 halo) and Vβ8s5 and 6 (c=6.5 halo) are models with declining σ_0 starting from low-mass elliptical ICs. They display bars of reasonable sizes (3.5-5.5) andvarying strengths. Their vertical profiles all show boxy edge-on shapes with vertical extents±∼1.0-1.5 and lateral extents that are wider for longer bars.In summary, it is possible to create models that show reasonable agreement with the vertical profile of the MW, its circular speed curve and at the same time contain a bar, which in length, strength and vertical extent agrees reasonably with that of the MW. The problem is that the details depend on mass and size growth history, DM halo density and GMC heating and that, additionally, the evolution ofbar length and strength is to some degree stochastic <cit.>, so that, not even within the limits of our methods, it is possible to determine which model best represents our Galaxy. §.§ Bar age structure In Figure <ref>, we examine the age structure of the bar. For several models we plot a map of median age τ_ med in the x-z plane, where x is measured along the major axis of the bar and z is perpendicular to the disc. All stars with |y|<1.5 are considered for the map. If τ_ med at a certain position corresponds to a star particle from the ICs, we apply a distinctyellow colour, whereas inserted star ages range from black (young) to orange (old). We overplot in white edge-on density contours.We start by analysing model 1, which does not show a noticeable bar, in the left-hand panelof the middle row. Clearly, above |z|∼600 the IC thick disc dominates at all x. The thin disc is youngest in the plane and has a vertical age gradient, as an effect of stars being borncold and being vertically heated by GMCs. It also has a radial age gradient, due to inside-out formation.In contrast to model 1, the three models displayed in the the top row of Figure <ref> at t_ fshow X-shaped edge-on structures. These three models all grow thin discs within thick IC discs. The X-shape is beautifully visible in the τ_ med maps. The cones of relatively lower surface density above and below the galactic centres are dominated by old, thick-disc stars, whereas younger stars fill diamonds that lie at both sides of the cones in the x-direction. These diamond-shaped regions have vertical age gradients of different strengths. This is connected to the fact that we do not feed new stars into regions of strong bars. The bar in U1 forms already at t=4, whereas the bar in P2 only forms at t=7.5. As shown in AS15, stars can be captured by the bar, but the rates of this process are low, so the age structure is only mildly affected. Consequently, the number of relatively young stars present in P2 is high and vertical and radial age gradients are present in the bar region. In U1, the region at |x|<4, which is part of the diamonds, is well mixed in τ_ med. The bar in W1 forms at t=6 and the model is thus intermediate to P2 and U1 in terms of age gradients.The vertical mixing of stars of various ages happens during bar buckling, as is illustrated by model O2, which is undergoing the process at the depicted moment. We see how the younger stars are spread vertically at around |x|∼2 and the model thus transitions from a disc-like to an X-shaped age structure. Note that, as bars can vary in length and strength with time, cold, young stars can be added in the plane after buckling events and there can be multiple buckling events, which complicates the age structure.The remaining four models in Figure <ref> feature declining σ_0. As the thick discs of these models comprise fed-in stars, which have radial and vertical density distributions that vary continuously with age, the age structure is more complicated to interpret, as in real galaxies. At t_ f, Vα1, like 1, has no X-shaped edge-on structure. The thin discs of these galaxies exhibit qualitatively similar vertical andradial age gradients. The thick disc of Vα1 shows a continuation of these gradients at higher |z|; the gradients in these regions are, however, much shallower. Mα1 and 6 have edge-on peanut-shaped density structures similar to that of P2 and, indeed, the age structures of the three bars are qualitatively similar. As the thick disc background, however, is, again, much less distinct,the structure combining old cones plus younger diamonds is much less evident and might be hard to measure in a real galaxy.Model 3b differs from the other models with declining σ_0 in having compact bulge ICs and thus the density of IC stars near the galactic centre is higher and the τ_ med gradients away from the centre are steeper than in models with low-mass elliptical ICs. A relatively recent and strong buckling event before thedisappearance of the bar and a radial growth history with constant h_R=2.5 has led to relatively youngareas at |x|∼2.§ DISCUSSION §.§ Vertical profiles and age gradients In all final models the vertical profile of the Snhd is fitted well by the sum of two exponentials similar to that of the MW. When the IC contains a thick disc, the vertical profile is double-exponential from the outset, with a growing thin-to-thick density ratio, whereas when σ_0 declines, the vertical profile gradually evolves a double-exponential structure as a sufficiently massive, cold and thin population forms, and the scaleheights of both components change continuously. We cannot directly measure the evolution of scaleheights over cosmic time, but in nearby galaxies we can probe this evolution through observations of the radial variation of the vertical profile and age structure of the disc. The radial variation of the vertical profile is dominated by the evolution of the thick disc because our thin discs have scaleheights that are almost independent of both R and t. Our thick-disc ICs are set up with declining σ_z(R) and radially constant scaleheights, and the latter property is roughly conserved to the present epoch, as radial migration causes only mild levels of flaring. Models with declining σ_0 have, byconstruction, radially constant σ_z at birth. As the oldest and hottest stars are barely affected by vertical heating, their thick discs flare strongly and their scaleheights increase with radius. In these models, radial migration weakens flaring, but the effect is not strong enough to balance the outward increase in scaleheight imprinted at birth.This has important consequences on the age structure of the disc. At radii R=5-12 and altitudes |z|>1 models with thick-disc ICs are dominated by old IC disc stars; they thus show no radial variation in median age τ_ med. At lower altitudes these discs become younger with increasing R.The age structure is markedly different in models with declining σ_0.On account of the strongly flaring old and intermediate-age components, they show negative τ_ med gradients at all |z|. Models with declining σ_0 thus agree better with recent measurements of the radial age structure at various altitudes in the MW by <cit.>. They also show somewhat flatter vertical age gradients at R_0, in rough agreement with measurements by <cit.>. For thick IC disc models, strong inside-out growth improves the agreement with these measurements.Our discs evolve in isolation, whereas at least the low-density outskirts of discs are likely to be affected by processes capable of significant vertical thickening, such as disc-satellite interactions <cit.> or infall of gas with misaligned angular momentum <cit.>.Including minor mergers or adding stars in tilted outer discs would thus be a valuable extension of our models. §.§ The interplay between the dark halo, the thick disc and the bar The evolution of a disc depends on the local density of the dark halo because increasing the latter reduces the extent to which the disc controls the gravitational field in which it moves. In particular, decreasing the local DM density increases the amplitude of non-axisymmetric structure.We varied the local DM density by varying the initial concentration parameter c at a fixed halo mass, M_ tot=10^12.Concentrations in the range c=6-9 work well. Indeed, after the DM has been compressed by the disc, the final DM density in the Snhd then agrees with observational constraints, and in many models a bar similar to that of the MW emerges before the current epoch. Although non-axisymmetric structures in the disc transfer angular momentum to the DM, the dark halo does not acquire a core like that favoured in the MW <cit.>. If such a core exists in a ΛCDM context, it thus probably formed in the very early evolution stages of the Galaxy that are not modelled here. Observational constraints on the circular speed, v_ circ(R_0), near the Sun <cit.> and on the microlensing optical depth towards the MW bar/bulge <cit.> indicate a baryon-dominated central MW and roughly equal contributions of DM and baryons to v_ circ(R_0). To achieve this in our models, initial concentrations c∼6-7 are favoured as higher values do not allow for enough baryonic mass in the central regions.Paper 1 favoured c∼9 because thin-disc-only models with lower values of c showed unrealistically long and strong bars.The presence of old thick, and thus kinematically hot, disc components alleviates this problem. By shifting mass from the cold thin disc to a thick, radially hot component, the formation of a bar can be delayed by several Gyr. It is immaterial whether the hot component is included in the ICs or arises from declining σ_0, and the ratio of radial to vertical dispersions can be σ_R/σ_z∼1, so smaller than the values σ_R/σ_z∼2 in thin discs. As a consequence, the time available for the bar to grow in strength and length is limited. Indeed,models with c=9 haloes tend to show an unrealistically weak final bar. Models with c∼6-7 generally have bars with lengths similar to that of the MW's bar. The delaying of bar formation by old thick discs also explains the observationthat the fraction of barred disc galaxies decreases with increasing redshift <cit.>. §.§ The edge-on structure of bars Several models have bars that are morphologically similar to that of the MW: a boxy bulge in the central R2 with an X-shape up to |z|∼1-1.5 at the centre of a thinner outer bar that extends to R∼5 <cit.>. In one of these models the bar dissolved at the very end of the simulation. However, notwithstanding the face-on surface density being almost axisymmetric, the edge-on peanut shape survived. Thus not all observed boxy edge-on bulges need be bars.In edge-on density projections of these models, the X-shapes are not as striking as in the model of AS15, which lacks a thick disc and GMCs, but features an isothermal gas component. This is not surprising as the MW bulge does not show an X-shape in all stellar components. <cit.> found that old and metal-poor RR Lyrae stars appear to have a more spheroidal shape and <cit.> found that low-metallicity stars (<-0.5)in the bar/bulge contain a much lower fraction of stars on bar-supporting orbitsthan stars with higher metallicities. Moreover, the X is possibly absent inyounger populations as well <cit.>. Analysis on the edge-on age structureof our bars offers insight into why the shape should vary with stellar population.The X structures are particularly pronounced in models with thick-disc ICs that have final boxy bulge/bar regions that are more extended than in the MW. Here bar buckling spreads stars from the thin disc vertically, and because they mainly populate the 2:2:1 resonant orbit family <cit.>, thesestars form a structure that, seen edge-on, resembles two diamonds overlappingat the galactic centre. The cones above and below the centre, which are notpopulated by the 2:2:1 orbits, are dominated by thick-disc stars. Young starsthat were captured by the bar after the buckling event will be found in theplane. Consequently, a distinct age pattern should be observed in bucklededge-on bars.A similar separation in edge-on morphology between the oldest,the intermediate-age and the youngest stars in a barred galaxy has also beenfound in the simulations of <cit.>. <cit.> recently studied bar formation in galaxies that contain disc populations with differing randommotions. They showed that radially cooler populations form stronger bars, the edge-onprofiles of which are vertically thinner and peanut-shaped, whereas the hotterpopulations form a weaker bar with a vertically thicker edge-on box shape (see also).The thick and thin discs of models with declining σ_0 are not strictly separated in age, as they are in models with thick-disc ICs, and in consequence their characteristic age structure is less clear in an edge-on age map. Thus while observations of the edge-on age structures of bars have the potential to betray the formation history of the bulge, central thick and thin discs and the timing of the bar buckling event, the constraints will be less tight if the scenario with declining σ_0 is more appropriate than that in which the thick disc is included in the ICs. §.§ Radial redistribution and inside-out growth Bars and spiral structure make it hard to steer the disc's radial scalelengthh_R at the solar radius R_0 to a preferred value, h_R(R_0)∼ 2.6. As was already discussed in Paper 1, making the disc more compact results in strongernon-axisymmetries, which in turn leads to more mass redistribution and largerh_R(R_0). We have shown that this is the case for all age components and forall models with appropriate bars, so in the past h_R(R_0) would likely havebeen smaller than it is today. Assigning higher values of σ_R/σ_zto the old thick disc can yield steeper-than-average profiles, but doingso weakens bars inappropriately. Still, for models that grow inside out, h_R(R_0) always increases with decreasing age, just as observations of the Snhd suggest <cit.>. As these observations show scalelengths h_R(R_0)∼1.5-2.0 for the most compact and oldest mono-abundance populations (see also ), h_R=2 can be regarded as an upper limit on the birth scalelength at the earliest times, but model 6 demonstrates that the scalelength at birth could have been as small as h_R=1, a conclusion similar to that of <cit.>.The flat age-metallicity relation of the Snhd and the radial metallicity gradient in the MW make it hard to infer the scalelength of current star formation by studying mono-abundance populations. In our models input scalelengths at late times in the range h_R∼3-4 give reasonable results. §.§ Thick-disc formation scenarios The age structure of the MW disc points towards a model with declining σ_0. Most hydrodynamical cosmological simulations of disc galaxiessupport a picture in which birth dispersions and gas fractions declinecontinuously with time (e.g.). Such a scenario has also been inferred from observations of Hα kinematics (e.g.). However, <cit.> argue that there is nosubstantial difference between the gas kinematics of galaxies at redshiftsz_ rs∼1 and today. Moreover, <cit.> find that models thathave kinematics in line with those found in the Snhd favour a two-phaseformation scenario, in which the thick-disc stars are born in a turbulent,merger-dominated phase and the thin-disc stars are born cold and heatedsubsequently.Such a two-phase scenario motivated our models with thick-disc ICs, but in setting up an equilibrium thick stellar disc with a radially constant scaleheight we have ignored correlations between stellar ages, kinematics and density profiles that would naturally arise during formation of the proto-thick disc.Declining σ_0 inherently produces such correlations. However, an inappropriate thick disc is still liable to emerge through poor choices for σ_0(t) or h_R(t), or the choice of a radially constant σ_0.The main limiting factor of our models is thus the lack of a self-consistent heating mechanism for thick-disc stars: in both scenarios thick-disc stars are created ad hoc. The heating mechanism will affect non-axisymmetries and the radial distribution of matter, which are crucial for disc evolution. As measurements of gas fractions at redshifts z_ rs∼2, a time consistent with the formation of the chemically defined thick disc of the MW, indicate that molecular gas makes up 50 per cent or more of the baryonic masses of galaxies (e.g. ), the lack of a realistic gas component is a connected problem. Although at early times models such as 6 have as much as 45 per cent of their baryonic mass in GMCs, the GMCs have the same mass function as a present-day spiral galaxy, in which the gas fraction is lower and stars form cold. In a picture in which turbulence driven by gravitational disc instabilities causes stars to form with large dispersions <cit.>, molecular complexes would be expected to be more massive.Scattering of stars by massive clumps could contribute to thick-disc formation <cit.>.§CONCLUSIONS We have presented a new set of idealized N-body simulations of disc galaxies with both thin and thick discs within live dark haloes. These models are grown over 10-12 by continuously adding new stellar particles with specified age-dependent velocity dispersions. Short-lived massive particles represent GMCs. Thin discs grow by the addition ofstars on near-circular orbits, whereas for thick-disc components we relyon two different concepts: a) create an appropriate thick disc in theICs and only add thin-disc stars during the simulation, or b) start withlow-mass, diffuse elliptical or compact bulge ICs and add stars withcontinuously declining input velocity dispersion σ_0(t). Hence inscenario b) we form kinematically hot thick-disc stars at early timesand cold thin populations at late times, whereas in scenario a) thesimulation starts after the structure that will morph into the thickdisc is fully formed. To understand the evolution of our models, wesimulate a variety of histories of star formation, dark halo densities and thick disc properties.Both types of models can produce at final time t_ f models that are similar in structure to the MW. We find: * Both scenarios create double-exponential vertical profiles. The scaleheight of the thin disc is governed by GMC heating. To achieve a MW-like thick-discexponential scaleheight h_ thick∼ 1 at t_ f, thick-disc ICs withisothermal vertical scaleheights z_0∼1.7 are suitable. For decliningσ_0 models, the input velocity dispersions should be σ_0∼40-50 at the earliest formation stages. Thick-disc scaleheights are not affected by GMC heating.* Models need to undergo inside-out growth to reproduce the observed dependence of radial scalelength h_R on chemical composition of disc stars. We find that models that grow from h_R∼1-2 at early times to h_R∼3-4 today are suitable.* To explain the baryon dominance of the Galactic Centre, the circular speed curve of the MW and the structure of the bar in the presence of a thick disc, we favour DMhaloes that at mass M_DM=10^12 have an initial concentration parameterc∼7. It is essential that thick-disc stars are already hot when the thin disc starts forming because Paper 1 showed that heating by GMCs and non-axisymmetries is incapable of producing the thick disc, although it explains the properties of the thin disc. The presence of the thick disc modifies the evolution of the thin disc, but the final properties of the thin discs in our thin+thick disc models are similar to those of thin-disc-only models in slightly more concentrated dark haloes. Crucially, this change in halo density and the presence of a hot and thick disc make it possible to bring models with appropriate bars into agreement with the baryon fractions inferred for the central MW.Regarding the non-axisymmetric structures of the disc, we find: * Bars with a structure similar to that of the MW bar, i.e. a boxy/peanut-shaped bulge at R<2 with an X-shape surrounded by a vertically thin part extending to R∼ 4-5, can be found in some of our viable models. Stochasticity in the evolution of bar lengths and strengths complicates the comparison. * The presence of a hot, thick-disc stellar population at the start of thin-disc formation suppresses non-axisymmetries and delays the formation of the bar. * In models with an appropriate bar, the local exponential scalelengths h_R(R_0) of all mono-age populations are increased by the radial redistribution of matter that the bar and spirals generate. Populations measured in the Snhd today have thus likely had lower h_R(R_0) in the past. * The dark halo's density profile is modified by the growth of the disc and the non-axisymmetric structures that form in the disc, but the profile does not develop a central core as is currently favoured for the centre of the MW. To distinguish between formation scenarios, it is helpful to study the radial andvertical age structure of disc galaxies. We find: * Our two types of models for the creation of the thick disc differ significantly in the predicted age maps of the discs. The observed Snhd radial age gradient at |z|>1 and the vertical age gradient both favour models with declining σ_0. However, the measurements are still rather uncertain and models with thick IC discs by construction ignore any internal structure of the thick disc. Such structure could be added to these models. * Bar buckling in thin+thick disc systems creates characteristic age patterns in edge-on views of the bar region. Buckling predominantly affects thin-disc stars and causes them to form a structure in the (R,z) plane resembling two diamonds overlapping at the galactic centre. The cones above and below the centre are dominated by thick-disc stars. Considering the wealth of data on the structure of the MW that will soon become available from surveys such as Gaia <cit.>, evolutionary models of disc galaxies that grow over cosmological time-scales and contain both thick and thin discs will be essential to connect the data to the formation history of the MW. Our models allow for a relatively controlled and flexible setup, can be produced in large numbers and capture a wealth of important dynamical processes.We have demonstrated that our models can reasonably reproduce a variety of observations of the structure of the MW. No model sticks out as particularly similar to the MW in allaspects, but this isto be expected given the remaining shortcomings inmodelling. In a companion paper (Paper 4) we examine the models presented here inlight of Snhd kinematics and constraints on radial migration.§ ACKNOWLEDGEMENTSWe thank the referee for comments that helped improve the paper. It is a pleasure to thank Ralph Schönrich for valuable discussions and comments on the manuscript.This work was supported by the UK Science and Technology Facilities Council (STFC) through grant ST/K00106X/1 and by the European Research Council under the EuropeanUnion's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 321067. This work used the following compute clusters of the STFC DiRAC HPC Facility(www.dirac.ac.uk): i) The COSMA Data Centric system at Durham University, operated by the Institute for Computational Cosmology. This equipment was funded by a BISNational E-infrastructure capital grant ST/K00042X/1, STFC capital grantST/K00087X/1, DiRAC Operations grant ST/K003267/1 and Durham University.ii) The DiRAC Complexity system, operated by the University of LeicesterIT Services. 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"authors": [
"Michael Aumer",
"James Binney"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170525155641",
"title": "The structural evolution of galaxies with both thin and thick discs"
} |
[t1]A preliminary version of this paper has appeared in theSpringer LNCS Proceedings of the 21st International Conference on Developments in Language Theory (DLT 2017), pp. 235–246.[t2]©2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license <http://creativecommons.org/licenses/by-nc-nd/4.0/>label1]Oscar H. Ibarrafn1 [label1]Department of Computer ScienceUniversity of California, Santa Barbara, CA 93106, USA [label1][email protected] [fn1]Supported, in part, by NSF Grant CCF-1117708 (Oscar H. Ibarra).label2]Ian McQuillanfn2 [label2]Department of Computer Science, University of SaskatchewanSaskatoon, SK S7N 5A9, Canada [label2][email protected][fn2]Supported, in part, by Natural Sciences and Engineering Research Council of Canada Grant 2016-06172 (Ian McQuillan). We introduce a model of one-way language acceptors(a variant of a checking stack automaton) andshow the following decidability properties: * The deterministic version has a decidable membership problem but has an undecidable emptiness problem.* The nondeterministic version has an undecidable membershipproblem and emptiness problem.There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature.We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.checking stack automata pushdown automata decidability reversal-bounded counters § INTRODUCTIONThe deterministic and nondeterministic versions of most known models of language acceptors exhibit the same decidability properties for each of the membership and emptiness problems.In fact, it is possible to define machine models in a general fashion by varying the allowed store types, such as withAbstract Families of Acceptors (AFAs) from <cit.>, or a similar type of machine model with abstract store types used in <cit.> and in this paper.Studying machine models defined in such a general fashion is advantageous as certain decidability problems are equivalently decidable for arbitrary machine models defined using such a framework, and thereforeit is possible to see which problems could conceivably differ in terms of decidability. For arbitrary one-way machine models defined in the way used here, the emptiness problem for the nondeterministic machines of this class, the membership problem for nondeterministic machines of this class, and the emptiness problem for the deterministic machines in this class, must all be either decidable or undecidable. Membership for deterministic machines could conceivably differ from the other three decidability problems. However, as far as we know, no one-way model has been shown to exhibit different decidability properties for deterministic and nondeterministic versions.The question arises of whether there is a model where membership for deterministic machines is decidable while it is undecidable for nondeterministic machines?A second topic of interest here is that of studying decidability properties of different classes of machines when adding additional data stores. In <cit.>, it was shown that for any one-way machine model (defined using another method used there), if the languages accepted by these machines are all semilinear[See <cit.> for the formal definition. Equivalently, a language is semilinear if it has the same commutative closure as some regular language.], then augmenting these machines with additional reversal-bounded counters[A counter stores a non-negative integer that can be tested for zero, and it is reversal-bounded if there is a bound on the number of changes between non-decreasing and non-increasing.] produces only semilinear languages. And, if semilinearity is effective with the original model, then it is also effective after adding counters, and therefore the emptiness problem is decidable. However, it is unknown what can occur when augmenting a class of machines that accepts non-semilinear languages with reversal-bounded counters. Can adding such counters change decidability properties? These two topics are both simultaneously studied in this paper. Of primary importanceis the one-way checking stack automaton <cit.>, which is similar to a pushdown automaton that cannot erase its stack, but can enter and read the stack in two-way read-only mode, but once this mode is entered, the stack cannot change. This model accepts non-semilinear languages, but has decidable emptiness and membership problems. Here, we introduce a newmodel of one-way language acceptors by augmenting a checking stack automaton with reversal-bounded counters, and the deterministic and nondeterministic versions are denoted by and , respectively. The models with two-way input (with end-markers) are called 2 and 2. These are generalized further to models with k checking stacks: k-stack 2 and k-stack 2. These models can be defined within the general machine model framework mentioned above. We show the following results concerning membership and emptiness:* The membership and emptiness problems for s are undecidable, even when there are only two reversal-bounded counters.* The emptiness problem foris decidable when there is only one reversal-bounded counter but undecidable whenthere are two reversal-bounded counters.* The membership problem for k-stack 2s is decidable for any k.Therefore, this machine model provides the first known model where membership is decidable for deterministic machines, while the other decidability properties are undecidable, which is the only property that can conceivably differ. It also shows one possible scenario that can occur when augmenting a machine model accepting non-semilinear languages with reversal-bounded counters: it can change the emptiness problem for both nondeterministic and deterministic machines to be undecidable, as with the membership problem for nondeterministic machines, but membership for deterministic machines can remain decidable (and therefore, all such languages accepted by deterministic machines are recursive).In addition, we define another family of one-way acceptors where the deterministic version has a decidable emptiness problem, but the nondeterministic version has an undecidable emptiness problem. This model must necessarily not be defined using the general machine model framework, as emptiness for deterministic and nondeterministic machine models are always equivalently decidable. But the model is still natural and demonstrates what must occur to obtain unusual decidable properties. Further, we introduce a new family with decidable emptiness, containment, and equivalence problems, which is one of the most powerful families to have these properties (one-way deterministic machines with one reversal-bounded counter and achecking stack that can only read from the stack at the end of the input). We also investigate the decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.).§ PRELIMINARIES This paper requires basic knowledge of automata and formal languages, including finite automata, pushdown automata, and Turing machines <cit.>. An alphabet Σ is a (usually finite unless stated otherwise) set of symbols. The set Σ^* is the set of all words over Σ, which contains the empty word λ. A language is any set L ⊆Σ^*. Given a word w ∈Σ^*, |w| is the length of w. A language L is bounded if there exists words w_1, …, w_k such that L ⊆ w_1^* ⋯ w_k ^*, and L is letter-bounded if w_1, …, w_k are letters. We use a variety of machine models here, mostly built on top of the checking stack. It is possible to define each machine model directly.As discussed in Section <ref>, an alternate approach isto define “store types” first, which describes just the behavior of the store, including instructions that can change the store, and the manner in which the store can be read. This can capture standard types of stores studied in the literature, such as a pushdown, or a counter. Defined generally enough, it can also define a checking stack, or a reversal-bounded counter. Then, machines using one or more store types can be defined, in a standard fashion. A (Ω_1, …, Ω_k)-machine is a machine with k stores, where Ω_i describes each store. This is the approach taken here, in a similar fashion to the one taken in<cit.> or <cit.> to define these same types of automata. This generality will also help in illustrating what is required to obtain certain decidability properties; see e.g.Lemma <ref> and Proposition <ref> which are proven generally for arbitrary store types. Furthermore, these two results are used many times within other proofs rather than having many ad hoc proofs. Hence, this generality in defining machines serves several purposes for this work. First, store types, and machines using store types are defined formally using the same framework used by the authors in <cit.>.A store type is a tuple Ω = (Γ, I,f,g,c_0, L_I), where Γ is the store alphabet (potentially infinite available to all machines using this type of store), I is theset of allowable instructions, c_0 is the initial configuration which is a word in Γ^*, and L_I ⊆ I^* is the instruction language (over possibly an infinite alphabet) of allowable sequences of instructions, f is the read function, a partial function from Γ^* to Γ, and g is the write function, a partial function fromΓ^* × I to Γ^*.We will study a few examples of store types. First, a pushdown store typeis a tupleΩ = (Γ, I,f,g,c_0, L_I), whereΓ is an infinite set of store symbols available to pushdowns, where special symbol Z_b∈Γ is the bottom-of-stack marker, andΓ_0 = Γ - {Z_b}, I = {(y) | y ∈Γ_0}∪{,} is the set of instructions of the pushdown, where the first set are called the push instructions, and the second set contains the pop and stay instruction,L_I = I^*, c_0 = Z_b, f( x a) = a, a ∈Γ, x∈Γ^* withxa ∈ Z_b Γ_0^*, and g is defined as: * g(Z_b x, (y)) = Z_b x y for x ∈Γ_0^*, y ∈Γ_0,* g(Z_b x a, ) = Z_b x, for x ∈Γ_0^*, a ∈Γ_0,* g(Z_b x , ) = Z_b x, for x ∈Γ_0^*.A counter store tape can be obtained by restricting the pushdown store type to only having a single symbol c ∈Γ_0 (plus the bottom-of-stack marker).The instruction language L_I in the definition of Ω restricts the allowable sequences of instructions available to the store type Ω (that is, a computation can only accept if its sequence of instructions is in the instruction language). This restriction does not exist in the definition of AFAs, but can be used to define many classically studied machine models, while still preserving many useful properties. For example, an l-reversal-bounded counter store type is a counter store type with L_I equal to the alternating concatenation of{(c),}^* and { , }^* with l applications of concatenation (this is more classically stated as, there are at most l alternations between non-decreasing and non-increasing).Next, the more complicated stack store type is a tupleΩ = (Γ, I,f,g,c_0, L_I), where* Γ is an infinite set of store symbols available to stacks, where special symbols ↓∈Γ are the position of the read/write head in the stack, Z_b∈Γ is the bottom-of-stack marker, and Z_t ∈Γ is the top-of-stack marker, with Γ_0 = Γ - {↓, Z_b, Z_t},* I = {(y) | y ∈Γ_0}∪{,}∪{ D,S,U} is the set of instructions of the stack, where the first set are called the push instructions, the second set is the pop and stay instruction, and the third set are the move instructions (down, stay, or up),* L_I = I^*, c_0 = Z_b ↓Z_t, f( x a ↓ x') = a,a ∈Γ_0 ∪{Z_t, Z_b}, x,x'∈Γ^* withxax' ∈ Z_b Γ_0^* Z_t,* and g is defined as: * g(Z_b x ↓ Z_t, (y)) = Z_b x y ↓ Z_t for x ∈Γ_0^*, y ∈Γ_0,* g(Z_b x a ↓ Z_t, ) = Z_b x ↓ Z_t, for x ∈Γ_0^*, a ∈Γ_0,* g(Z_b x ↓ Z_t, ) = Z_b x ↓ Z_t, for x ∈Γ_0^*,* g(Z_b x a ↓ x' ,D) = Z_b x ↓ a x', for x, x' ∈Γ^*, a ∈Γ_0 ∪{Z_t}, with xax' ∈Γ_0^* Z_t,* g(Z_b x ↓ x',S) = Z_b x↓ x', for x,x' ∈Γ^*, x x' ∈Γ_0^* Z_t,* g(Z_b x ↓ a x',U) = Z_b x a ↓ x', for x,x' ∈Γ^*, a ∈Γ_0 ∪{Z_t}, xax' ∈Γ_0^* Z_t.That is, a stack is just like a pushdown with the additional ability to read from the “inside” of the stack (but not change the inside) in two-way read-only mode. Also, the checking stack store type is a restriction of stack store type above where L_I is restricted to be in {(y), | y ∈Γ_0}^* { D,S,U}^*. That is, a checking stack has two phases, a “writing phase”,where it can push or stay (no pop), and then a “reading phase”, where itenters the stack in read-only mode. But once it starts reading, it cannot change the stack again.Given store types (Ω_1, …, Ω_k), with Ω_i = (Γ_i, I_i,f_i,g_i,c_0,i, L_I_i), a two-way r-head k-tape(Ω_1, …, Ω_k)-machine is a tuple M = (Q,Σ, Γ, δ,,, q_0, F) where Q is the finite set of states, q_0 ∈ Q is the initial state, F ⊆ Q is the set of final states, Σ is the finite input alphabet,Γ is a finite subset of the store alphabets of Γ_1 ∪⋯∪Γ_k, δ is the finite transition relation from Q× [Σ]^r ×Γ_1 ×⋯×Γ_kto Q × I_1 ×⋯× I_k × [{-1,0,+1}]^r.An instantaneous description (ID) is a tuple (q, w , α_1, …, α_r, x_1, …, x_k), where q ∈ Q is the current state, w is the current input word (surrounded by left input end-marker and right input end-marker), 0 ≤α_j ≤ |w|+1 is the current position of tape head j (this can be thought of as 0 scanning, and |w|+1 scanning ), for 1 ≤ j ≤ r, and x_i ∈Γ_i^* is the current word in the Ω_i store, for 1 ≤ i ≤ k. Then M is deterministic if δ is a partial function (i.e. it only maps each element to at most one element). Then (q, w , α_1, …, α_r, x_1, …, x_k) ⊢_M (q', w , α_1', …, α_r', x_1', …, x_k'), (two IDs) if there exists a transition (q', ι_1, …, ι_k, γ_1, …, γ_r) ∈δ(q, a_1, …, a_r, b_1, …, b_k), where a_j is character α_j+1 of w (1 is added sinceis the letter at position 0), and α_j' = α_j + γ_j, for 1 ≤ j ≤ r, b_i = f_i(x_i), and g_i(x_i, ι_i) = x_i' for 1 ≤ i ≤ k.Instead of ⊢_M, we can also write ⊢_M^(ι_1, …, ι_k) to show the instructions applied to each store on the transition. We let ⊢_M^* be the reflexive and transitive closure of ⊢_M, and let ⊢_M^(γ_1, …, γ_k), where γ_i ∈ I_i^* is the sequence of instructions applied to store i, 1 ≤ i ≤ k, in the sequence of transitions applied, and |γ_1| = ⋯ = |γ_k|. The language accepted by M, L(M), is equal to { w |(q_0,w , 1, …, 1, c_0,1, …, c_0,k) ⊢_M^(γ_1, …, γ_k) (q_f,w , α_1, …, α_r, x_1, …, x_k),q_f ∈ F, γ_i ∈ L_I_i, 1 ≤ i ≤ k}.Thus, the sequence of instructions applied to each store must satisfy its instruction language, and they must each be of the same length.The different machine modes are combinations of either one-way or two-way,deterministic or nondeterministic, and r-head for some r ≥ 1. For example, one-way, 1-head, deterministic, is a machine mode. Given a sequence of store types Ω_1, …, Ω_k and a machine mode, one can study the set of all (Ω_1, …, Ω_k)-machines with this mode. The set of all such machines with a mode is said to be complete. Any strict subset is said to be incomplete. Given a set of (complete or incomplete) machines M of this type, the family of languages accepted by these machines is denoted L( M). For example, the set of all one-way deterministic pushdown automata is complete as it contains all one-way deterministic machines that use the pushdown store.But consider the set of all one-way deterministic pushdown automata that can only decrease the size of the stack when scanning the right end-marker. This is a strict subset of all one-way deterministic machines that use the pushdown store, since the instructions available to such machines depend on the location of the input (whether it has reached the end of the input or not). Therefore, this is an incomplete set of machines. The instruction language of a store does allow a complete class of machines to restrict the allowable sequences of instructions, but it has to apply to all machines using the store. Later in the paper, we will consider variations of checking stack automata such as one called no-read, which means that they do not read from the inside of the checking stack before hitting the right input end-marker. This is similarly an incomplete set of automata since the instructions allowed differs depending on the input position.The class of one-way deterministic (resp. nondeterministic) checking stack automata is denoted by(resp., ) <cit.>. The class of deterministic (resp. nondeterministic), finite automata is denoted by(resp., ) <cit.>. For k,l ≥ 1, the class of one-way deterministic (resp. nondeterministic) l-reversal-bounded k-counter machines is denoted by (k,l) (resp. (k,l)). If only one integer is used, e.g. (k), this class contains all l-reversal-bounded k counter machines, for some l, and if the integers is omitted, e.g.,and , they contain all l-reversal-bounded k counter machines, for some k,l. Note that a counter that makes l reversals can be simulated by ⌈l+1/2⌉ 1-reversal-bounded counters <cit.>. Closure and decidable properties of various machines augmented with reversal-bounded counters have been studied in the literature (see, e.g., <cit.>). For example, it is known that the membership and emptiness problems are decidable for<cit.>.Also, here we will study the following new classes of machines that have not been studied in the literature: one-way deterministic (resp. nondeterministic)machines defined by stores consisting of one checking stack and k l-reversal-bounded counters, denoted by (k,l) (resp. (k,l)), those with k-reversal-bounded counters, denoted by (k)(resp. (k)), and those with some number of reversal-bounded counters, denoted by(resp. ).All models above also have two-way versions of the machines defined, denoted by preceding them with 2, e.g.2, 2, 2(1), 2, 2, etc. We will also define models with k checking stacks for some k, which we will precede with the phrase “k-stack”, e.g.k-stack 2, k-stack 2, k-stack 2, k-stack 2, etc.When k=1, then this corresponds with omitting the phrase “k-stack”. § A CHECKING STACK WITH REVERSAL-BOUNDED COUNTERSBefore studying the new types of stores and machine models, we determine several properties that are equivalent for any complete set of machines. This helps to demonstrate what is required to potentially have a machine model where the deterministic version has a decidable membership problem with an undecidable emptiness problem, while both problems are undecidable for the nondeterministic version.First, we examine a machine's behavior on one word.Let M be a one- or two-way, r-head, for some r≥ 1, (Ω_1, …, Ω_k)-machine,and let w∈Σ^*. We can effectively construct another (Ω_1, …, Ω_k)-machine M_w that is one-way and 1-head which accepts λ if and only if M accepts w. Furthermore, M_w is deterministic if M is deterministic. The input w is encoded in the state of M_w, and M_w on input λ, simulates the computation of Mand accepts λ if and only if M accepts w. This uses a subset of the sequence of transitions used by M (and thereby would satisfy the instruction language of each store). Since M_w is only on λ input, two-way input is not needed in M_w, and the r-heads are simulated completely in the finite control.Then, for all machines with the same store types, the following decidability problems are equivalent: Consider store types (Ω_1, …, Ω_k). The following problems are equivalently decidable, for the stated complete sets of automata: * the emptiness problem for one-way deterministic(Ω_1, …, Ω_k)-machines,* the emptiness problem for one-way nondeterministic(Ω_1, …, Ω_k)-machines,* the membership problem for one-way nondeterministic(Ω_1, …, Ω_k)-machines,* acceptance of λ, for one-way nondeterministic(Ω_1, …, Ω_k)-machines,* the membership problem for two-way r-head (for r ≥ 1) nondeterministic (Ω_1, …, Ω_k)-machines. The equivalence of 1) and 2) can be seen by taking a nondeterministic machine M. Let T = {t_1, …, t_m} be labels in bijective correspondence with the transitions of M. Then construct M' which operates over alphabet T. Then M' reads each input symbol t and simulates t of M on the store, while always moving right on the input. However, if it is a stay transition on the input of M, then M' also checks that the next input symbol read (if any), t', is defined on the same letter of Σ in M. Then M' is deterministic, and changes its stores identically in sequence to M (thereby still satisfying the same instruction language), and L(M') is therefore empty if and only if L(M) is empty.It is immediate that 5) implies 4), and it follows that 4) implies 5) from Lemma <ref>. Similarly, 3) implies 4), and 4) implies 3) from Lemma <ref>.To show that 4) implies 2), notice that any complete set of nondeterministic one-way automata are closed under homomorphisms h where h(a) ≤ 1, for all letters a. Considering the homomorphism that erases all letters, the resulting language is empty if and only if λ is accepted by the original machine. To see that 2) implies 4), take a one-way nondeterministic machine and make a new one that cannot accept if there is aninput letter. This new machine is non-empty if and only if λ is accepted in the original machine.It is important to note that this proposition is not necessarily true for incomplete sets of automata, as the machines constructed in the proof need to be present in the set. We will see some natural restrictions later where this is not the case, such as sets of machines where there is a restriction on what instructions can be performed on the store based on the position of the input. And indeed, to prove the equivalence of 1) and 2) above, the deterministic machine created reads a letter for every transition of the nondeterministic machine applied. Hence, consider a set of machines that is only allowed to apply a strict subset of store instructions before the end-marker. Let M be a nondeterministic machine of this type, and say that M applies some instruction on the end-marker that is not available to the machine before the end-marker. But the deterministic machine M' created from M in Proposition <ref> reads an input letter when every instruction is applied, even including those applied on the end-marker of M. But since M' is reading an input letter during this operation, it would violate the instructions allowed by M' before the end-marker. The above proposition indicates that for every complete set of one-way machines, membership for nondeterminism, emptiness for nondeterminism, and emptiness for determinism are equivalent. Thus, the only problem that can potentially differ is membership for deterministic machines. Yet we know of no existing model where it differs from the other three properties. We examine one next.We will study s and s, which are s and s (nondeterministic and deterministic checking stack automata)respectively, augmented by reversal-bounded counters. First, two examples will be shown, demonstrating a language that can be accepted by a . Consider the languageL = {(a^n#)^n | n ≥ 1}. AM with one 1-reversal-bounded counter can accept L as follows:M when given an input w (we may assume that the input is of the form w = a^n_1#⋯ a^n_k# for some k ≥ 1 and n_i ≥ 1 for 1 ≤ i ≤ k, since the finite control can check this), copies the first segment a^n_1 to the stack while also storing number n_1 in the counter. Then M goes up and down the stack comparing n_1 to the rest of the inputto check that n_1 = ⋯ = n_k while decrementing the counter by 1 for each segment it processes.Clearly, L(M) = L and M makes only 1 reversal on the counter. We will show in Proposition <ref> that L cannot be accepted by an(or an ).Let L = {a^i b^j c^k | i, j ≥ 1, k = i · j }. We can construct a (1) M to accept L as follows. M reads a^i and stores a^i in the stack.Then it readsb^j and increments the counter by j. Finally, M reads c^k while moving up and down the stack containing a^i and decrementing the counter by 1 every time the stack has moved i cells, to verify thatk is divisible by i and k/i = j.Then M accepts L, and M needs only one 1-reversal counter. We will see in Proposition <ref> that L cannot be accepted by a 2(1).The following shows that, in general,s and sare computationally more powerful than s and s, respectively. There are languages in ((1,1)) - (()∪()). Hence, () ⊊((1,1)), and () ⊊((1,1)). Consider the languageL = {(a^n#)^n | n ≥ 1} from Example <ref>. L cannot be accepted by an ; otherwise, L' = {a^n^2 | n ≥ 1} can also be accepted by an(sincelanguages are closed under homomorphism), but it was shown in <cit.> that L' cannot be accepted by any .However, Example <ref> showed that L can be accepted by a (1,1). Furthermore, L is not semilinear, butonly accepts semilinear languages <cit.>. We now proceed to show that the membership problem for s is decidable. In view of Lemma <ref>, our problem reduces to deciding, given aM, whether it accepts λ. For acceptance of λ, the next lemma provides a normal form. Let M be a . We can effectively constructaM' such that: * all counters of M' are 1-reversal-bounded and each must return to zero before accepting,* M' always writes on the stack ateach step during the writing phase,* the stack head returns to the left end of the stack before accepting,whereby M' accepts λ if and only if M accepts λ.It is evident that all counters can be assumed to be 1-reversal-bounded as with<cit.>, and that each counter can be forced to return to zero before accepting. Similarly, the checking stack can be forced to return to the left end before accepting. We introduce a dummy symbol $ to the stack alphabet so that if in a step, M does not write on the stack, then M' writes $.When M' enters the reading phase, M' simulatesM but ignores (i.e., skips over) the $'s. ThenM' accepts λ if and only if M accepts λ. In view of Lemma <ref>, wemay assume that awrites a symbol at the end of the stack at each step during the writing phase. This is important for deciding the following problem. Let M be asatisfying the assumptions of Lemma <ref>.We can effectively decide whether or not M, on λ input,has an infinite writing phase (i.e., will keep on writing).Let s be the number of states of M.We constructanM' which, when given an input w over the stack alphabet of M, does the following: simulatesthe computation of M on `stay' transitions while checking thatw could be written by M on the stack at some point during the computation of the writing phase of w, while also verifying that there is a subwordx of w of length s +1 such that x was written by M without:* incrementing a counter that has so far been at zero, and* decrementinga non-zero counter.If so, M' accepts w. Next, it will be argued that L(M') is not empty if and only if M has an infinite writing phase on λ, and indeed this is decidable since emptiness foris decidable <cit.>.If L(M') is not empty,then there is a sequence of s+1 transitions during the writing phase where no counter during this sequence is increased from zero, and no counter is decreased. Thus, there must be some state q hit twice by the pigeonhole principle, and the sequence of transitions between q and itself must repeat indefinitely in M. Thus, M has an infinite writing phase on λ input.Conversely, assume M has an infinite writing phase. Then there must be a sequence of s+1 transitions where no counter is decreased, and no counter is increased from zero. Thus, L(M') must be non-empty. From this, decidability of acceptance of λ is straightforward. It is decidable, given aM satisfying the assumptions of Lemma <ref>, whether or not M accepts λ.From Lemma <ref>, we can decide if M has an infinite writing phase.If so, M will not accept λ (as the stack must return to the bottom before accepting).IfM does not have an infinite writing phase, the (final) word w written in the stack is unique andhence has a unique length d.In this case, wecan simulate faithfully the computation of M (on λ input) and determine d.We then construct a DCM M_d, which onλ input, encodes the stack in the state and simulates M. Thus, M_d needs a buffer of size d to simulate the operation of the stack, and M_d accepts if and only if M accepts. The result follows, sincethe membership problem foris decidable <cit.>. From Lemmas <ref>, <ref>, and <ref>: For r ≥ 1, the membership problem for r-head 2 is decidable. We now give some undecidability results. The proofs will use thefollowing result in <cit.>: <cit.>It is undecidable, given a 2(2) M over a letter-bounded language, whether L(M) is empty. The membership problem for (2) is undecidable.Let M be a 2(2) machine over a letter-bounded language. Construct from M anM' which, on λ input (i.e. the input is fixed),guesses an input w to M andwrites it on its stack. Then M' simulates the computation of M by usingthe stack and two reversal-bounded counters and accepts if and only if M accepts.Clearly, M'accepts λ if and only if L(M) is not empty which is undecidable by Proposition <ref>. By Propositions <ref> and <ref>, the following is true: The emptiness problem for (2) is undecidable.Let M be a 2with two reversal-bounded counters with input alphabet Σ.Construct from M, aM' whichreads its (one-way) input(over Σ^*) into its stack.Then, as in the proof ofProposition <ref>, M' simulates M usingthe stack and two reversal-bounded counters. Then L(M')is empty if and only if L(M) is empty,which is undecidable. Combining together the results thus far demonstrates thatis a model where, * the deterministic version has a decidable membership problem,* the deterministic version has an undecidable emptiness problem,* the nondeterministic version has an undecidable membership problem,* the nondeterministic version has an undecidable emptiness problem.Moreover, this is the first (to our knowledge) model where these properties hold.The next restriction serves to contrast this undecidability result. Consider anwhere during the reading phase, the stack head crosses the boundary of any two adjacent cells on the stack at mostd times for some given d ≥ 1.Call this machine a d-crossing . Then we have:It is decidable, given a d-crossingM,whether or not L(M) = ∅. Let M have k reversal-bounded counters, k ≥ 1. As in the proof of Proposition <ref>, we construct fromM a 2 M' with k reversal-bounded counters whose two-way input is over the two-track alphabet Δ. M' simulates M as described in the proof of Proposition <ref>. Clearly, M' also crosses the boundary of any two adjacennt cells on of its input at most d times. Then L(M) = ∅ if and only if L(M') = ∅, wgich is decidable by Proposition <ref>, part 3.Define a d-crossingto be an nondeterministic Turing machinewith a one-way read-only input tape and a d-crossing read/write worktape (i.e., the worktape head crosses the boundary between any two adjacent worktape cells at most d times) augmented withreversal-bounded counters. Note that a d-crossingcan be simulated by a d-crossing .It was shown in <cit.> that it is decidable, given a d-crossingM, whether L(M) = ∅. The proposition follows.Although we have been unable to resolve the open problem as to whether the emptiness problem is decidable for bothandwith one reversal-bounded counter, as with membership for the nondeterministic version, we show they are all equivalent to an open problem in the literature. The following are equivalent: * the emptiness problemis decidable for 2(1),* the emptiness problem is decidable for (1),* the emptiness problem is decidable for (1),* the membership problem is decidable for r-head 2(1),* it is decidable if λ is accepted by a (1).The last four properties are equivalent by Proposition <ref>.It can be seen that 2) implies 1) because a (1) machine can simulate a 2(1) machine by taking the input, copying it to the stack, then simulating the 2(1) machine with the two-way stack instead of the two-way input.Furthermore, it can be seen that 1) implies 5) as follows: given a(1) machine M,assume without loss of generality, that M immediately and nondeterministically sets the stack and returns to the bottom of the stack in read-only mode in some special state q before changing any counter (as it can verify that M would have pushed the stack contents). Then, build a 2(1) machine M' that on some input over the stack alphabet, simulates the stack using the input, and the counter using the counter starting at state q.Then L(M') is non-empty if andonly if λ is accepted by M.It is indeed a longstanding open problem as to whether the emptiness problem for 2(1) is decidable <cit.>. Now consider the following three restricted models, with k counters: For k ≥ 1, a (k) (or a (k)) machine is said to be: * no-read/no-counter if it does not read the checking stack nor use any counter before hitting the right input end-marker,* no-read/no-decrease if it does not read the checking stack nor decrease any counter before hitting the right input end-marker,* no-read if it does not read the checking stack before hitting the right input end-marker.We will consider the families of (k) ((k)) machines satisfying each of these three conditions.For any k ≥ 1, every 2(k) machine can be effectively converted to an equivalent no-read/no-decrease (k) machine, and vice-versa.First, a 2(k) machine M can be simulated by a no-read/no-decrease (k) machine M' that first copies the input to the stack, and simulates the input of M using the checking stack, while simulating the counters faithfully. Indeed, the checking stack is not read and counters are not decreased until M' reads the entire input. Next we will prove the converse.Let M be a no-read/no-decrease (k) machine with input alphabet Σ and stack alphabet Γ.A two-way deterministic gsm, 2 DGSM, is a deterministic generalized sequential machine with a two-way input (surrounded by end-markers), accepting states, and output. It is known that if L is a language accepted by a two-wayk-head deterministic machine augmented with some storage/memory structure (such as a pushdown, checking stack, k checking stacks, etc.), then T^-1(L)= { x|T(x) = y,y ∈ L} is also accepted by the same type of machine <cit.>. Let T be 2 DGSM which, on input x ∈Σ^*, first outputsx #. Then it moves to the left end-marker and on the second sweep of input x, simulates M and outputs the string z written on the stack during the writing phase of M. Note that T can successfully do this as M generates the checking stack contents from left-to-right, and does not read the contents during the writing phase; and becausethe counters of M are not decreased during the writing phase of M, the counters can never empty during the writing phase, thereby affecting the checking stack contents created. When T reachesits right end-marker, it outputs the state s of M at that time, andthen T enters an accepting state.Thus, T(x) = x # z s.Now construct a 2(k) M' which when given a string x #z s, reads x, and while doing so, M' simulates the writing phase of M on x by only changing the counters as M would do. Then, M' moves to the right and stores the state s in the finite control.Then M' simulates the reading phase of M on string z (which only happens after the end of the input has been reached), starting in state s and the current counter contents, and accepts if and only if M accepts.It is straightforward to see that T^-1(L(M')) = L, which can therefore be accepted by a 2(k) machine. From this, the following is immediate, since emptiness for 2(1) is known to be decidable <cit.>. The emptiness problem for no-read/no-decrease (1) is decidable. In the first part of the proof of Proposition <ref>, the (k) machine created from a 2(k) machine was also no-read/no-counter. Therefore, the following is immediate:For k ≥ 1, the family of languages accepted by the following three sets of machines coincide: * all no-read/no-decrease (k) machines,* all no-read/no-counter (k) machines,* 2(k). One particularly interesting corollary of this result is the following: * The family of languages accepted by no-read/no-decrease (respectively no-read/no-counter) (1) is effectively closed under union, intersection, and complementation.* Containment and equivalence are decidable for languages accepted by no-read/no-decrease (1) machines. This follows since this family is equal to 2(1), and these results hold for 2(1) <cit.>. Something particularly noteworthy about closure of languages accepted by no-read/no-decrease 2(1) under intersection, is that, the proof does not follow the usual approach for one-way machines. Indeed, it would be usual to simulate two machines in parallel, each requiring its own counter (and checking stack). But here, only one counter is needed to establish intersection, by using a result on two-way machines. Later, we will show that Corollary <ref>, part 2 also holds for no-read (1)s. Also, since emptiness is undecidable for 2(2), even over letter-bounded languages <cit.>, the following is true: The emptiness problem for languages accepted by no-read/no-counter (2) is undecidable, even over letter-bounded languages. Turning now to the nondeterministic versions, from the first part of Proposition <ref>, it is immediate that for any k ≥ 1, every 2(k) can be effectively converted to an equivalent no-read/no-decrease (k). But, the converse is not true combining together the following two facts: * For every k ≥ 1, the emptiness problem for languages accepted by 2(k) over a unary alphabet is decidable.* The emptiness problem for languages accepted by no-read/no-counter (also for no-read/no-decrease) (2) over a unary alphabet is undecidable.The first partwas shown in <cit.>. For the second part, it is known that the emptiness problem for 2(2) M (even over a letter-bounded language) is undecidable by Proposition <ref>. We construct a no-read/no-counter (2) M' which, on a unary input, nondeterministically writes some string w on the stack.Then M' simulates M using w.The result follows since L(M') = ∅ if and only if L(M) = ∅.In contrast to part 2 of Proposition <ref>:For any k ≥ 1, the emptiness problem for languages accepted by no-read/no-decrease (k) machines over a unary alphabet, is decidable. If M is a no-read/no-decrease (k) over a unary alphabet, we can effectively construct an equivalent 2(k) M (over a unary language) from Proposition <ref>.The result follows since the emptiness problem for 2(k) over unary languages is decidable <cit.>. Combining these two results yields the following somewhat strange contrast: Over a unary input alphabet and for all k ≥ 2,the emptiness problem forno-read/no-counter (k)s is undecidable, but decidable for no-read/no-counter (k)s. As far as we know, this demonstrates the first knownexample of a family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. This presents an interesting contrast to Proposition <ref>, where it was shown that for complete sets of automata for any store types, the emptiness problem of the deterministic version is decidable if and only if it is decidable for the nondeterministic version. However, the set of unaryno-read/no-counter (k) machines can be seen to not be a complete set of machines, as a complete set of machines contains every possible machine involving a store type. This includes those machines that read input letters while performing read instructions on the checking stack.And indeed, to prove the equivalence of 1) and 2) in Proposition <ref>, the deterministic machine created reads a letter for every transition applied, which can produce machines that are not of the restriction no-read/no-counter. With only one counter,decidability of the emptiness problem for no-read/no-decrease (1), and for no-read/no-counter (1) can be shown to be equivalent to all problems listed in Proposition <ref>. This is because 2) of Proposition <ref> implies each immediately, and each implies 1) of Proposition <ref>, as a2(1) machine M can be converted to a no-read/no-decrease, or no-read/no-counter (1) machine where the input is copied to the stack, and then the 2(1) machine simulated. Therefore, it is open as to whether the emptiness problem for no-read/no-decrease (or no-read/no-counter) (1)is decidable, as this is equivalent to the emptiness problem for 2(1).One might again suspect that decidability of emptiness for no-read/no-decrease (1) implies decidability of emptiness for no-read/no-decrease (1) by Proposition <ref>. However, it is again important to note that Proposition <ref> only applies to complete sets of machines, including those machines that read input letters while performing read instructions on the checking stack, again violating the `no-read/no-decrease' condition.Even though it is open as to whether the emptiness problem is decidable for no-read/no-decrease (1)s, we have the following result, which contrasts Corollary <ref>, part 2: The universe problem is undecidable for no-read/no-counter (1)s.(Thus, containment and equivalence are undecidable.) It is known that the universe problem for a one-way nondeterministic 1-reversal-bounded one-counter automaton M is undecidable <cit.>.Clearly, we can construct ano-read/no-counter (1) M' to simulate M. In the definition of a no-read/no-decrease , we imposed the condition that the counters can only decrement when the input head reaches the end-marker. Consider the weaker condition no-read, i.e., the only requirement is that the machine can only enter the stack when the input head reaches the end-marker, but there is no constraint on the reversal-bounded counters. It is an interesting open question about whether no-read (k) languages are also equivalent to a 2(k) (we conjecture that they are equivalent). However, the following stronger version of Corollary <ref> can be proven. The emptiness problem is decidable for no-read (1)s. Let M be a no-read (1).Let T = {t_1, …, t_m} be symbols in bijective correspondence with transitions of M that can occur in the writing phase. Then, build a 2(1) machine M' that, on input w over T, reads w while changing states as w does, and changing the counter as the transitions do. Let q be the state where the last transition symbol ends. Then, at the end of the input, M' simulates the reading phase of M starting in q by scanning w, and interpreting a letter t of w as being the stack letter written by t in M (while always skipping over a letter t if t does not write to the stack in M). Then L(M') is empty if and only if L(M) is empty. We can further strengthen Proposition <ref> somewhat.Define a restricted no-read (1) to be a no-read (1) which is only nondeterministic during the writing phase.Then the proof of Proposition <ref> applies to the following, as the sequence of transition symbols used in the proof can be simulated deterministically: The emptiness problem is decidable for languages accepted by restricted no-read (1) machines. As was mentioned previously, it is anopen problem whether the membership problem (hence, also the emptiness problem) for (1) is decidable. However, for a special case, it is decidable: The membership problem is decidable for (1) whose stackcan only write a string from a bounded language (i.e., from w_1^* ⋯ w_k^*, for some given k and strings w_1,…, w_k). This follows from the fact that emptiness for 2(1) is decidable over bounded languages <cit.>.While we are unable to show that the intersection of two no-read (1) languages is a no-read (1)language, we can prove:It is decidable,giventwo no-read (1)s M_1 and M_2, whether L(M_1) ∩ L(M_2) = ∅. Let M_1 and M_2 be no-read (1) over input alphabet Σ. Let T_i be symbols in bijective correspondence with transitions of M_i that can occur in the writing phase, for each i ∈{1,2}. Let T' be the set of all pairs of symbols (r,s), where r is a transition of M_1, s is a transition of M_2, and where both r and s read the same input letter of Σ. Let T” be all those symbols (r,$) where r is a transition of M_1 that stays on the input, and let T”' be all those symbols ($,s) where s is a transition of M_2 that stays on the input.Build a 2(1) machine M' operating over alphabet T' ∪ T”∪ T”'. On input w, M' verifies that the first component changes states as M_1 does (skipping over any $ symbol) and that if a stay transition is read, the next letter has a first component on the same input letter, and changing the counter as M_1 does. Let q be the state where the last transition symbol ends. Then, at the end of the input, M' simulates the reading phase of M_1 starting in q by scanning w, and interpreting a letter t ≠$ in the first component of w as being the stack letter written by t in M, and skipping over $ or any t that does not write to the stack. After completion, then M' does the same thing with M_2 using the second component. Notice that the alphabet is structured such that a transition of M_1 on a letter a ∈Σ is used exactly when a transition of M_2 using a ∈Σ is used, since M_1 and M_2 are both no-read (so their entire input is used before the reading phases starts). For example, a word w = (s_1,r_1) (s_2, $) (s_3,$) ($, r_2) (s_4,r_3) implies s_1 reads the same input letter in M_1 as does r_1 in M_2, similarly with s_4 and r_3, s_2 and s_3 are stay transitions in M_1, and r_2 is a stay transition in M_2. Hence, L(M') is empty if and only if L(M_1) ∩ L(M_2) is empty. One can show that no-read(1) languages are effectively closed under complementation.Thus,from Proposition <ref>:The containment and equivalence problems are decidable for no-read (1)s. No-read (1) is indeed quite a large family for which emptiness, equality, and containment are decidable. The proof of Proposition <ref> also applies to the following:It is decidable,giventwo restricted no-read (1)s M_1 and M_2, whether L(M_1) ∩ L(M_2) = ∅. Finally, consider the general model (1) (i.e., unrestricted). While it is open whether no-read (1) is equivalent to 2(1), we can prove:(2(1)) ⊊((1)). It is obvious that any 2(1) can be simulated bya (1) (in fact by a no-read/no-counter (1)). Now let L = {a^i b^j c^k | i, j ≥ 1, k = i · j }. We can construct a (1) M to accept L by Example<ref>. However, it was shown in <cit.> that L cannot be accepted by a 2(1) by a proof that shows that if L can be accepted by a 2(1), then one can use the decidability of the emptiness problem for 2(1)s to show that Hilbert's Tenth Problem is decidable.§ MULTIPLE CHECKING-STACKS WITH REVERSAL-BOUNDED COUNTERS In this section, we will study deterministic and nondeterministic k-checking-stack machines. These are defined by usingmultiple checking stack stores.Implied from this definition is that each stack has a “writing phase” followed by a “reading phase”, but these phases are independent of each letter for each stack.A k-stack( respectively) is the deterministic (nondeterministic) version of this type of machine. The two-way versions (with input end-markers) are called k-stack 2 and k-stack 2, respectively. These k-stack models can also be augmented with reversal-bounded counters and are called k-stack , k-stack , k-stack 2, and k-stack 2. Consider a k-stackM. By Lemma <ref>, for the membership problem, we need only investigate whether λ is accepted.Also, as in Lemma <ref>, we may assume that each stack pushes a symbol at each move during its writing phase, and that allcounters are 1-reversal-bounded. We say that M has an infinite writing phase (on λ input) if no stack enters a reading phase. Thus, all stacks will keep on writing a symbol at each step.If M has a finite writing phase, then directly before a first such stack enters its reading phase, all the stacks would have written strings of the same length.Let k ≥ 1 and M be a (k+1)-stackM satisfying the assumption of Lemma <ref>. * We can determine if M has an infinite writing phase. If so, M does not accept λ. * If M has a finite writing phase, we can construct a k-stackM” satisfying the assumption of Lemma <ref> such that M” accepts λ if and only if M acceptsλ.Let M have s states and stack alphabets Γ_1, …, Γ_k+1 for the k+1 stacks. Let Γ = {[a_1, …, a_k+1] | a_i ∈Γ_i, 1 ≤ i ≤ k+1 }. By assumption, each stack of M writes a symbol during its writing phase.We can determine if M has a finite writing phase as follows: Asin Lemma <ref>, we constructanM' which, when given an input w ∈Γ^*, does the following: simulates the computation of M on `stay' transitions such that the input w was written by M (in a component-wise fashion on each checking stack) and there is a subwordx of w of length s+1 such that the subword was written by M without:* incrementing a counter that has so far been at zero, and* decrementinga non-zero counter.If so, M' accepts w. So we need only check if L(M') is not empty, which is decidable since emptiness is decidable for<cit.>. Then, M does not accept λ if and only if M has an infinite writing phase, and if and only if L(M') is not empty, which is decidable.If L(M') is empty, we then simulate M faithfully to determine the unique word w ∈Γ^* and its length d just before the reading phase of at least one of the stacks, say S_i, isentered. Note that by construction, no stack entered its stack earlier. We then construct a k-stackM” which, onλ input, encodes the operation of stack S_i in the state and simulates M (also converted into satisfying the assumptions of Lemma <ref>). Thus, M” needs a buffer of size d to simulate the operation of stack S_i.M” accepts if and only if M accepts, and has one less stack than M. Notice that M” has fewer stacks than M. Then, from Proposition <ref> (the result for a single stack)and using Lemma <ref> recursively: The membership problem for k-stack s is decidable.Then, by Lemma <ref>:The membership problem for r-head k-stack 2 is decidable.This is one of the most general machine models known with a decidable membership problem. Although space complexity classes of Turing machines are also very general, the membership problem for both deterministic and nondeterministic Turing machines satisfying some space complexity function are both decidable. However, for s, membership is undecidable but is decidable for deterministic machines. Moreover, unlike space-bounded Turing machines, r-headk-stack 2s do not have a space restriction on their stacks. § CONCLUSIONSWe introduced several variants of checking stack automata and showed the difference between the deterministic and nondeterministic models with respect to the decidability of the membership and emptiness problems. The main decision problems are summarized in Table <ref>. We believe the contrasting results obtained are the first of its kind.An interesting open question is the status of the emptiness problem for nondeterministic checking stack automataaugmented with one reversal-bounded counter which can only read the stack and decrease the counter at the end of the input. As shown in the paper,this problem is equivalent to a long-standing open problem of whether emptiness for two-way nondeterministic finite automata augmented with one reversal-bounded counter is decidable. Furthermore, we investigated possible scenarios that can occur when augmentinga machine model accepting non-semilinear languages with reversal-bounded counters. This contrasts known results on models accepting only semilinear languages. § ACKNOWLEDGEMENTSWe thank the Editor and the referees for their expeditious handling of our paper and, in particular, the referees for their comments that improved the presentation of our results. elsarticle-num | http://arxiv.org/abs/1705.09732v2 | {
"authors": [
"Oscar H. Ibarra",
"Ian McQuillan"
],
"categories": [
"cs.FL"
],
"primary_category": "cs.FL",
"published": "20170526222201",
"title": "Variations of Checking Stack Automata: Obtaining Unexpected Decidability Properties"
} |
RWT estimation via ResRNNWufeng Xue et al.Department of Medical Imaging, Western University, London, ON, [email protected], [email protected] Direct Estimation of Regional Wall Thicknesses via Residual Recurrent Neural Network Wufeng Xue, Ilanit Ben Nachum, Sachin Pandey, James Warrington, Stephanie Leung, and Shuo Li December 30, 2023 ================================================================================================Accurate estimation of regional wall thicknesses (RWT) of left ventricular (LV) myocardium from cardiac MR sequences is of significant importance for identification and diagnosis of cardiac disease. Existing RWT estimation still relies on segmentation of LV myocardium, which requires strong prior information and user interaction. No work has been devoted into direct estimation of RWT from cardiac MR images due to the diverse shapes and structures for various subjects and cardiac diseases, as well as the complex regional deformation of LV myocardium during the systole and diastole phases of the cardiac cycle. In this paper, we present a newly proposed Residual Recurrent Neural Network (ResRNN) that fully leverages the spatial and temporal dynamics of LV myocardium to achieve accurate frame-wise RWT estimation. Our ResRNN comprises two paths: 1) a feed forward convolution neural network (CNN) for effective and robust CNN embedding learning of various cardiac images and preliminary estimation of RWT from each frame itself independently, and 2) a recurrent neural network (RNN) for further improving the estimation by modeling spatial and temporal dynamics of LV myocardium. For the RNN path, we design for cardiac sequences a Circle-RNN to eliminate the effect of null hidden input for the first time-step. Our ResRNN is capable of obtaining accurate estimation of cardiac RWT with Mean Absolute Error of 1.44mm (less than 1-pixel error) when validated on cardiac MR sequences of 145 subjects, evidencing its great potential in clinical cardiac function assessment. § INTRODUCTION Estimation of regional wall thicknesses (RWT) of left ventricle (LV) myocardium is of significant importance for early identification and diagnosis of cardiac disease <cit.>. Fig. <ref> demonstrates the RWT to be estimated for a short-axis view cardiac image. A traditional way for RWT estimation is to segment the LV myocardium from other structures first and then measure the corresponding RWT of each region. However, existing segmentation methods for cardiac images <cit.> require strong prior information and user interaction to obtain reliable results, which may hinder them from efficient clinical application. An alternative way is to circumvent this segmentation step and estimate RWT from cardiac images directly. Direct estimation of cardiac volumes <cit.> have achieved great success in recent years, while direct estimation of RWT has never been explored due to the diversity of cardiac shape and structures for various subjects and various diseases, as well as the complication of regional myocardium deformation through the cardiac cycle.In this work, we provide a method to estimate the frame-wise RWT from cardiac MR sequences through a newly proposed Residual Recurrent Neural Network (ResRNN). This ResRNN contains two paths: 1) a CNN path for low dimension embedding to robustly represent cardiac images of diverse structures, and preliminary estimation of RWT independently from the embedding of each frame itself, and 2) an RNN path for residual estimation from neighboring frames by leveraging the temporal and spatial dynamic deformation in cardiac sequences simultaneously. In the RNN path, a temporal RNN is applied to the features of temporally neighboring frames for modeling the complex long-range temporal dependencies, and another spatial RNN is applied to the predicted results of spatially neighboring wall thicknesses for modeling the mutual dependencies among these wall thicknesses. For the RNN module, a new Circle-RNN is designed to eliminate the effect of null initial hidden states by characterizing the periodicity of cardiac sequences and the circular spatial layout of cardiac RWT. With image represented by the CNN embeddings, the dynamic deformation of myocardium and the diversity of cardiac shape are well captured by the temporal and spatial RNN, thus leading to accurate estimation of RWT. §.§ Related work §.§.§ Segmentation-based and direct methods for cardiac volumes estimation.Currently, the most related work to RWT estimation is cardiac volumes estimation, which falls into two categories: segmentation-based methods <cit.> and direct methods <cit.>. Segmentation-based methods rely on the premise of cardiac segmentation, which is still a great challenge due to the diverse structure of cardiac image and therefore requires strong prior information and user interaction <cit.>. To circumvent these limitations, direct methods without segmentation have grown in popularity in cardiac volumes estimation <cit.>. In these methods, hand-crafted features extracted from cardiac images are fed into regression models such as random forest (RF), adaptive K-clustering RF (AKRF),Bayesian model, and neural networks, to predict cardiac volumes. The employed features include Bhattacharyya coefficient between image distributions <cit.>, appearance features <cit.>, multiple low level image features <cit.>, as well as features from multiscale convolutional deep belief network (MCDNB) <cit.> and supervised descriptor learning (SDL) <cit.>.Although these methods obtained effective performance, two limitations still exist: 1) they followed two separated phases, i.e., feature extraction+ volumes regression, and no feedback exists between them to make them compatible with each other; 2) neither the temporal dependencies nor the spatial dependencies are taken into account, while the dependencies are important for dynamic modeling of cardiac sequence. In the present work, we provide an elegant solution for direct RWT estimation with an end-to-end architecture that respects both temporal and spatial dependencies. §.§.§ Recurrent neural network.Recurrent neural network, especially when the long short-term memory units (LSTM) are deployed, is specialized in long-range temporal dynamic modeling and spatial context modeling. Promising results have been achieved by RNN in a wide spectrum of applications including language modeling <cit.>, object recognition <cit.>, visual recognition and description <cit.>, and also medical image analysis <cit.>. In the work of cardiac image segmentation <cit.>, an RNN was employed to capture the spatial changes of cardiac structure (represented as low dimensional CNN embeddings) in cardiac sequences. In <cit.>, an RNN with LSTM was employed to model the temporal dependencies in cardiac MR sequences to identify the end-diastole and end-systole frames across a cardiac cycle. In <cit.>, an RNN was trained to describe the contexts of detected disease in Chest X-Rays. These methods only explored one of the spatial or temporal dependencies while in cardiac sequences, both are important for robust dynamic modeling.To fully explore the dependencies that exist in cardiac sequences during RWT estimation, two RNN modules are deployed in our work accounting for the temporal and spatial dependencies simultaneously. Besides, we propose a Circle-RNN for periodic cardiac sequences to better serve this aim, avoiding the effect of the null initial hidden input in existing RNN. §.§ ContributionsThe contributions of our work include: * An effective end-to-end method that has great potential in clinical cardiac function assessment is proposed for direct cardiac RWT estimation, which has never been investigated previously. * The newly proposed two-path ResRNN endows the network with abilities to robustly represent complex cardiac structure, and effectively model the capricious spatial and temporal dynamic deformation. * The temporal RNN that accounts for the temporal deformation of the cardiac shape, and the spatial RNN that accounts for the smoothness of the LV myocardium shape, enable ResRNN to estimate collaboratively RWT of all frames and all regions by leveraging the temporal and spatial dependencies in cardiac MR sequences, rather than to estimate independently each cardiac RWT from a single image.* A Circle-RNN designed for characterizing the periodicity of cardiac sequences and the circular spatial layout of cardiac RWT is proposed to eliminate the effect of the null hidden input for the first time step, to incorporate both the future and the past information in the dynamic modeling, and to treat every frame in the sequence with equally long-term dependencies. § RWT ESTIMATION VIA RESRNN§.§ Problem formulationFor a set of cardiac MR sequences 𝒳={X^s_f}, where s=1⋯ S indexes the subject and f=1⋯ F indexes the frame in a cardiac cycle, we aim to estimate the frame-wise values of RWT 𝒴={y^s_f,l} for all the frames, where l=1⋯ 6 indexes the spatial location of each RWT (see Fig.<ref>, from IS to AS in counter-clockwise order). We consider in this work the mid-cavity of LV myocardium in short axis view, which is divided into six segments according to the AHA 17 segments model <cit.>. The objective function can be formulated as:min_θ1/2S× F∑_f∑_sy^s_f-𝐐(X^s_f|θ)^2_2+λℛ(θ)where 𝐐 is the network, θ is the parameter vector to be learned, and ℛ(θ) regularizes the parameter vector. §.§ Overview of the proposed methodTo estimate the frame-wise RWT from cardiac MR sequence, we build a new architecture of network: ResRNN. As shown in Fig. <ref>, two paths are included in ResRNN: with the CNN path 𝐐_𝐂𝐍𝐍, each frame in the sequence is independently processed by the CNN network, forming a low dimensional embedding of the cardiac images, from which the RWT is preliminarily estimated with another fully-connected layer; with the RNN path 𝐐_𝐑𝐍𝐍, two RNN modules are deployed to model the temporal dependencies between neighboring frames and the spatial dependencies between neighboring RWTs, so as to further correct the residual of the preliminary estimation. The RNN path shares the same CNN embedding with the CNN path. The final RWT estimation is computed as:𝐐(X^s_f|θ)=𝐐_CNN(X^s_f|θ)+𝐐_RNN(X^s_f|θ)§.§ Preliminary estimation with the CNN pathThe diagram of our CNN path is shown in Fig. <ref>. Three convolution (conv1∼3) and one fully-connected (fc1) layers are deployed to obtain the low dimensional CNN embedding e^s_f of cardiac images. The second fully-connected layer (fc2) estimates a preliminary results y^s,CNN_f of RWT from the CNN embeddings. y^s,CNN_f=θ_fc2:we^s_f+θ_fc2:bwhere θ_fc2:w and θ_fc2:b are the weight matrix and bias of the fc2 layer.As a feed forward neural network, the CNN path bears a notable limitation that it relies on the assumption of independence among samples, which does not hold for cardiac sequence. The dependencies among cardiac sequences have to be modeled to further reduce the residual of the CNN estimation.§.§ Residual estimation with the RNN path The diagram of the RNN path is shown in Fig. <ref>. Based on the CNN embedding obtained with the CNN path, temporal and spatial RNN are employed to effectively model the dependencies existing among the RWT of all frames. In this section, we first introduce the memory unit LSTM that we use in the RNN path, and then describe the temporal and spatial RNN. §.§.§ LSTMIn order to learn the long-term dynamics in sequential data and avoid the gradient vanishing/exploding problem in traditional RNN, LSTM unit <cit.>, as shown in Fig. <ref>, was introduced into RNN. The input gate, output gate, forget gate and the memory cell allow the network to learn when to forget previous hidden states and when to update current hidden states given current input. This strategy enables LSTM to adaptively memorize and access information long term ago. The LSTM computations for time step t given the current input x_t, the previous hidden states h_t-1 and memory states c_t-1, are as follows <cit.>:i_t =σ(W_xix_t+W_hih_t-1+b_i)f_t =σ(W_xfx_t+W_hfh_t-1+b_f)o_t =σ(W_xox_t+W_hoh_t-1+b_o)g_t =φ(W_xcx_t+W_hch_t-1+b_c)c_t =f_t⊙ c_t-1+i_t⊙ g_th_t =o_t⊙φ(c_t)where σ(·) and φ(·) are element-wise sigmoid and hyperbolic tangent non-linearity functions, ⊙ are element-wise product. The first three equations map the current input and previous hidden states to the input gate i_t, the forget gate f_t and the output gate o_t, to adaptively control the information flow. Ws are the weight matrices to be learned and bs are the corresponding bias terms.§.§.§ Temporal RNN and Spatial RNNWith CNN embedding e^s_f, we first deploy a temporal RNN with the frame index in a cardiac sequences as time step to predict the values of RWT h^s_temp,f∈ℛ^6 for each frame f taking account of the dependencies between neighboring frames (See Temporal RNN in Fig. <ref>). h^s_temp,f = LSTM(e^s_f, h^s_temp,f-1), f=1… FBased on the prediction of temporal RNN, we again deploy a spatial RNN with spatial location as time step to predict RWTh^s_spa,l∈ℛ^F of one specific location l for all the frames in the sequences (See Spatial RNN in Fig. <ref>). We rearrange column vectors [h^s_temp,1, h^s_temp,2, …, h^s_temp,F] to row vectors [ĥ^s,T_spa,1; ĥ^s,T_spa,2,…, ĥ^s,T_spa,6]. Then we haveh^s_spa,l = LSTM(ĥ^s_spa,l, h^s_spa,l-1), l=1… 6By rearranging these row vectors [h^s,T_spa,1;h^s,T_spa,2;…,h^s,T_spa,6] back to column vectors [y^s,RNN_1, y^s,RNN_2, …, y^s,RNN_F], we get the RNN predictions for frame-wise RWT. §.§.§ Circle-RNNAs can be seen from Eqs. <ref> and <ref>, three limitations exist for RNN: 1) for the first time step (f=1 or l=1), there is no value for previous hidden units, which influences the prediction of the first frame; 2) only past information can be used to determine the output of current time step, while future information is also equally important; 3) for the first few time steps, long-term dependency model can not be well built from the limited past information, leading to unfair treatment of different frames.We provide an elegant solution to overcome these limitations for cardiac MR sequence: Circle-RNN, which connects the output of the last frame to the hidden input of the first frame, as the red arrows show in Fig. <ref>. Within Circle-RNN, Eqs. <ref> and <ref> become: h^s_temp,f = LSTM(e^s_f, h^s_temp,mod(f-1-1,F)+1), f=1… F h^s_spa,l = LSTM(ĥ^s_spa,l, h^s_spa,mod(l-1-1,6)+1), l=1… 6where mod(a,b) computes the modulus. Circle-RNN can be easily optimized with the BPTT algorithm <cit.>. To avoid infinite information loop within this Circle-RNN, we introduce a parameter depth to control how many rounds the information flow in our Circle-RNN. Fig. <ref> shows the error reduction of Circle-RNN over RNN when predicting cardiac RWT from the CNN embedding with only temporal RNN employed. § EXPERIMENTS§.§ Dataset and Implementations A dataset of short-axis cine MR images paired with manually obtained ground truth values of RWT is constructed to evaluate the performance of our method, which includes 2900 images from 145 subjects. These subjects are collected from 3 hospitals affiliated with two health care centers and 2 vendors (GE and Siemens). Each subject contains 20 frames throughout a cardiac cycle. In each frame, the middle slice is selected following the standard AHA prescriptions <cit.> for validation of the proposed cardiac RWT estimation method. The ground truth values of RWT are manually obtained for each image.In our experiments, two landmarks, i.e, junctions of the right ventricular wall with the left ventricular, are manually marked for each cardiac image to provide reference for ROI cropping and the LV myocardial segments division. The cropped images are resized to dimension of 80×80. All values of RWT are normalized to the range of [0,1] according to the image dimension (80). Five-fold cross validation is employed for performance evaluation and comparison. Mean absolute error (MAE) between estimation and the ground truth is computed to evaluate the performance. The network is implemented by Caffe <cit.> with SGD solver. Learning rate and weight decay are set to (0.05, 0.0005). ‘step’ learning policy is employed with gamma and step size being (0.5, 2500) and momentum 0.9. The depth of Circle-RNN is set as the number of time steps, i.e, 20 for the temporal RNN and 6 for the spatial RNN. Data augmentation is conducted by randomly cropping images of size 75× 75 from the original image. §.§ Performance comparison: ResRNN vs. CNN and RNN The advantages of the proposed ResRNN are firstly demonstrated by comparing performance of three different network architectures: 1) CNN, i.e. the CNN path as shown in Fig. <ref>; 2) RNN, i.e. the RNN path as shown in Fig. <ref>), and 3) the proposed ResRNN. For RNN and ResRNN, both the original RNN (the plain RNN) and Circle-RNN are employed for comparison. From the average estimation error shown in Table <ref>, we can observe the followings. 1) The two-path ResRNN outperforms CNN and RNN, with either plain or circle RNN deployed, which can be ascribed to the complementarity of the preliminary estimation from each frame itself and the residual estimation that modeling the dependencies of cardiac sequence. 2) When Circle-RNN, rather than the plain RNN, is deployed, lower estimation error can be obtained by RNN or ResRNN, due to the fact that Circle-RNN is capable of memorizing useful cardiac dynamic information for the first time step. In the following experiments, Circle-RNN is deployed in ResRNN. §.§ Performance Comparison: ResRNN vs. state-of-the-art To demonstrate the advantages of our proposed method over segmentation based <cit.> and two-phase direct methods <cit.>, we apply these methods to our database for cardiac RWT estimation. For the direct methods, the same five-fold cross-validation protocol is used. As can be observed in Table <ref> and Fig. <ref>, the proposed ResRNN demonstrates great advantages over existing segmentation-based and two-phase direct methods for cardiac RWT estimation. From Table <ref>, we can see that the proposed ResRNN estimates cardiac RWT with high accuracy (average MAE of 1.44mm) and outperforms all competitors. Specifically, it outperforms the Max Flow method with 55.14% MAE reduction. Note that Max Flow obtained high dice metric of 0.9182 for LV cavity segmentation when applied to our database. In fact, the dependency on manual segmentation of the first frame makes the estimation error of Max Flow increase as the estimated frame is far from the first frame within the cardiac cycle (see the frame-wise estimation error of Max Flow in Fig. <ref>). ResRNN outperforms the best of the direct methods (MCDBN+RF <cit.>) with a clear error reduction: 12.73%. Within the two-phase framework, the hand-crafted multifeatures, the features obtained by supervised learning, and MCDBN features employed in existing methods all fail to beat ResRNN, which evidences the benefits of the network architecture in ResRNN over the two-phase direct methods. Fig. <ref> also shows that ResRNN estimates each frame with consistently lower MAE.Besides, we can draw that all the deep neural network based methods in Table <ref> outperform existing two-phase direct methods in Table <ref>. This further confirms the argument that independent feature extraction and regression cannot make the two phases maximumly compatible with each other. The end-to-end learning procedure of neural network integrates both phases together and leads to clearly improved performance.§ CONCLUSIONSIn this paper, we propose an effective network architecture ResRNN for the task of cardiac RWT estimation, which has never been explored before. In ResRNN, a CNN path is employed to estimate from each cardiac image independently the preliminary results of RWT, and an RNN path is employed to compensate the residual of CNN estimations with temporal and spatial dependencies being accounted by Circle-RNN. Validation on a data set of cardiac MR sequences from 145 subjects demonstrates that the proposed ResRNN is capable of estimating cardiac RWT values with performance better than state-of-the-art methods, and is of great potential in clinical cardiac function assessment.splncs03 | http://arxiv.org/abs/1705.09728v1 | {
"authors": [
"Wufeng Xue",
"Ilanit Ben Nachum",
"Sachin Pandey",
"James Warrington",
"Stephanie Leung",
"Shuo Li"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170526220446",
"title": "Direct Estimation of Regional Wall Thicknesses via Residual Recurrent Neural Network"
} |
^1Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, Hampshire, PO1 3FX, UK We extend our previous redshift space power spectrum code to the redshift space correlation function. Here we focus on the Gaussian Streaming Model (GSM). Again, the code accommodates a wide range of modified gravity and dark energy models. For the non-linear real space correlation function used in the GSM we use the Fourier transform of the RegPT 1-loop matter power spectrum. We compare predictions of the GSM for a Vainshtein screened and Chameleon screened model as well as GR. These predictions are compared to the Fourier transform of the Taruya, Nishimichi and Saito (TNS) redshift space power spectrum model which is fit to N-body data. We find very good agreement between the Fourier transform of the TNS model and the GSM predictions, with ≤ 6% deviations in the first two correlation function multipoles for all models for redshift space separations in 50Mpc/h ≤ s ≤ 180Mpc/h. Excellent agreement is found in the differences between the modified gravity and GR multipole predictions for both approaches to the redshift space correlation function, highlighting their matched ability in picking up deviations from GR. We elucidate the timeliness of such non-standard templates at the dawn of stage-IV surveys and discuss necessary preparations and extensions needed for upcoming high quality data. 98.80.-k A Perturbative Approach to the Redshift Space Correlation Function: Beyond the Standard Model Benjamin Bose^1, Kazuya Koyama^1 today =============================================================================================§ INTRODUCTIONSince its conception, Einstein's general relativity (GR) has enjoyed numerous successes, most recently the detection of long predicted gravitational waves <cit.>. Despite this, with the discovery of cosmic acceleration <cit.>, this model of gravity has needed to dramatically expand its `dark' sector to include dark energy in the form of a cosmological constant - Λ. This joins the long present dark matter (DM) component, putting the dark sector at around 95% of the energy content of the universe <cit.>. Besides being non-interacting with light, making it very difficult to detect through conventional methods, the tiny value of the cosmological constant leads to the infamous fine tuning problem e.g. <cit.>. By adopting GR, one is also lead to tensions between low redshift large scale structure data sets and the CMB <cit.>. Motivated by these issues, alternatives to a pure Λ-CDM description of our universe have been investigated in great depth e.g.<cit.>.A basic perquisite of such modifications is to satisfy the many solar system tests that GR has passed over the years. But by modifying GR one generally introduces an additional degree of freedom usually cast in the form of a scalar field, which leads to an additional force on top of the GR-predicted one. This force is not observed in the local universe and so this naturally leads to the idea of screening, where the so called fifth force is shut off in areas of high density (See <cit.> for reviews). Screening generally comes in two flavours, environmental screening where either the local density distribution effectively kills scalar field gradients sourcing the fifth force or it suppresses the coupling between the scalar and matter, and screening via new derivative interactions which modify the derivative structure of the Klein Gordon equation. The former can come in two forms, the Chameleon mechanism <cit.> and the symmetron mechanism <cit.>, while the latter is known as Vainshtein screening<cit.>.It is because of screening that the large scale structure (LSS) of the universe becomes such an attractive laboratory in which to test deviations from GR. At the relevant scales, the unscreened fifth force can work to produce a detectable signal of modified gravity (MG) <cit.>. In particular the so called Redshift Space Distortions (RSD)<cit.> in galaxy clustering promises to be a great candidate for testing MG. RSD is a matter clustering anisotropy thatcomes from the non-linear mapping between real position and redshift space position. This mapping must account for the peculiar velocities of the clustering galaxies. In the presence of a fifth force, these peculiar velocities will be boosted giving us a signal in the observed anisotropy<cit.>. By comparing clustering statistics predicted by theory with the observed statistics coming from spectroscopic surveys such as the Baryon Oscillation Spectroscopic Survey within SDSS III (BOSS) [http://www.sdss3.org], one can get a measurement of the growth of structure <cit.>, typically coming in the form of the parameter combination fσ_8, where f=d ln(F_1)/ dln(a), with F_1 being the linear growth of structure and a being the scale factor, while σ_8 normalises the linear power spectrum (see eg. <cit.>). Such a measurement can be used to constrain modified gravity parameters <cit.>, with the caveat that the limitations of the theoretical template applied to data are known. On this point, work has been done in determining the importance of using correctly modelled theoretical templates when comparing to data in order to obtain an unbiased measurement of growth and consequently MG parameters <cit.>. A GR based modelling of RSD with additional degrees of freedom quantifying our ignorance (eg. tracer bias or velocity dispersions), has seemed to be a good approximation when placing general constraints on gravity. But as the precision of surveys increases, modelling gravity consistently becomes more and more important as any MG signal loses room within the error bars to hide. The same is also true as theory pushes its applicability to smaller scales, where the strongest statistical power comes from.In <cit.> we presented a means for consistently constructing the redshift space power spectrum for an MG theory within the Horndeski class with a generalised potential assuming quasi static perturbations. The model for the redshift space power spectrum considered was the TNS model <cit.>built upon standard perturbation theory (SPT). This model has been shown to perform very well when compared to N-body simulations <cit.> and has consequently been employed in deriving constraints using survey data <cit.>. Part of its success at smaller scales is due to the introduction of a velocity dispersion parameter which is fit to data. The aim of <cit.> was to provide a means of applying a wide class of MG templates to data and in doing so bypass the question of model bias discussed in <cit.>. This work moves in a similar vein.In this paper we move out of Fourier space and into configuration space by applying a similar approach as in <cit.> to the redshift space correlation function. On smaller scales, where the so called fingers-of-god effect and survey systematics become more dominant, a Fourier analysis of data becomes less ideal because -modes will not evolve independently. Large modes mixing with small ones affect our measurements/predictions at larger scales. Traditionally the correlation function measured from surveys has been used for theory-data comparisons. Here we focus on the Gaussian streaming model (GSM) described in <cit.> which assumes a Gaussian form for the velocity probability distribution function. The GSM has been shown to fit N-body data to percent level on and below the Baryon Acoustic Oscillation (BAO) scale in the case of GR and has been widely applied to data <cit.>. Work has also been done in extending and modifying the model to achieve a better range of applicability <cit.>. When considering the first two multipole moments, the GSM can accurately probe down to scales s ≥ 30 /h in the GR case when compared to N-body simulations for halos and matter and does very well in comparison to other redshift space correlation function models <cit.>.The TNS and GSM models are two of the most widely used theoretical templates applied to observational data. By having these two templates at one's disposal it is possible to make independent comparisons to data where observational systematics are different for each statistic. This will help beat down systematic uncertainty in growth measurements. Further, we compare the GSM predictions with the Fourier transform of the TNS power spectra. We expect the TNS transform to be the more accurate of the two because it makes use of the free parameter σ_v to capture small scale dispersion effects whereas the pure GSM model contains no phenomenological ingredients. To merit the GSM approach, the TNS approach requires resummation techniques such as regularised perturbation theory (RegPT) <cit.> to perform the Fourier transform. These techniques are currently under scrutiny in light of the effective field theory of large scale structure (EFToLSS)<cit.>. Suffice to say it is not well defined how one should treat small scale SPT divergences. These techniques are not needed in the GSM modelling. Of course one needs simulation measurements to see which approach does better, but their comparison has merit as a consistency check in their ability to model the observations as well as deviations from GR. This consistency is explored at and around the baryon acoustic oscillation (BAO) scale (50/h ≤ s ≤ 180 /h) in this paper.This paper is organised as follows: In Sec.II we review the SPT building blocks for the construction of the GSM for generalised gravitational theory templates.This is followed by a review of the TNS power spectrum which we will use to compare with our GSM predictions. We conclude this section by going over the GSM and its ingredients in terms of the generalised perturbation kernels.In Sec.III we present the GSM predictions. We validate the predictionsby comparing them with the Fourier transform of TNS predictions fitted to three sets of N-body data, each run under a different model of gravity. Finally, we summarise our results, review our pipeline, its successes and shortcomings, and highlight future work in Sec.IV. § THEORY §.§ Evolution equations for the perturbationsWe begin by considering perturbations in homogeneous and isotropic expanding background. This is strongly supported by various experiments, most notably the Planck mission [http://sci.esa.int/planck/]. These can be described by the perturbed Friedman-Robertson-Walker (FRW) metric. In the Newtonian gauge this is given byds^2=-(1+2Φ)dt^2+a^2(1-2Ψ)δ_ijdx^idx^j. We will assume the matter content of the universe is well described by a perfect fluid and will work on the evolution of density fluctuations inside the Hubble horizon. We work on sub-horizon scales where we can use the quasi-static approximation and neglect time derivatives of the perturbed quantities compared with the spatial derivatives. It is worth pointing out that the validity of this approximation weakens in certain cases.The large distance modification of gravity which is invoked to explain cosmic acceleration introduces a new scalar degree of freedom. The quasi static approximation generally holds as long as the sound speed of this degree of freedom is marginally larger than the speed of light <cit.> within the scales considered. This is model and scale dependant and one should be aware of its validity when applying the treatment described here.In this paper we work in the Jordan frame. The evolution equations for matter perturbations are obtained from the conservation of the energy momentum tensor. Before shell crossing and assuming no vorticity in the velocity field, a safe assumption at large scales and late times, the evolution equations can be expressed in Fourier space asa ∂δ()/∂ a+θ() =- ∫d^3_1d^3_2/(2π)^3δ_ D(-_1-_2) α(_1,_2) θ(_1)δ(_2), a ∂θ()/∂ a+ (2+a H'/H)θ() -(k/a H)^2 Φ()= -1/2∫d^3_1d^3_2/(2π)^3δ_ D(-_1-_2) β(_1,_2) θ(_1)θ(_2), where the prime denotes a scale factor derivative, δ is the density contrast representing an over or under density in the background density field and θ is the velocity divergence expressed in terms of the peculiar velocity field _p() as θ( x)= ∇·_p()/aH(a) f. H(a) is the Hubble factor and f=d ln(F_1)/ dln(a), with F_1 being the linear growth of structure defined below. The kernels in the Fourier integrals, αand β, are given by α(_1,_2)=1+_1·_2/|_1|^2, β(_1,_2)= (_1·_2)|_1+_2|^2/|_1|^2|_2|^2. The assumption of perturbation theory is that the non-linear density and velocity perturbations can be written out as a perturbative expansion of increasing order in the linear perturbations (See Ref.<cit.> for a review).Once we assume this, we can solve eq.(<ref>) and eq.(<ref>) order by order. For our needs we will expand δ and θ up to the third order.Gravity enters the evolution via the Newtonian potential Φ in the Euler equation (eq.(<ref>)) and its non-linear relation to the matter perturbationsis governed by the Poisson equation <cit.>. -(k/a H)^2Φ= 3 Ω_m(a)/2μ(k,a) δ() + S(), where Ω_m(a) = 8 π G ρ_m/3 H^2 and the function S() is the non-linear source, term up to the third order given by S() = ∫d^3_1d^3_2/(2π)^3 δ_ D(-_12) γ_2(, _1, _2;a) δ(_1) δ(_2), +∫d^3_1d^3_2d^3_3/(2π)^6δ_ D(-_123) γ_3( , _1, _2, _3;a) δ(_1) δ(_2) δ(_3), where γ_2( , _1, _2; a)and γ_3(, _1, _2, _3;a) are symmetric under the exchange of _i. Expressions for these functions are derived under the quasi static Klein Gordon equation for the additional scalar degree of freedom typical in MG theories. In the case of GR γ_2( , _1, _2; a) = γ_3(, _1, _2, _3;a) = 0 andμ(k,a)=1.In <cit.> we specify the form of these functions for the Vainshtein screened DGP model of gravity <cit.> and for the chameleon screened Hu-Sawicki form of f(R) gravity <cit.>.As said, we solve eq.(<ref>) and eq.(<ref>) perturbatively for n-th order kernels F_n and G_n which give the n-th order solutions δ_n(k ; a)= ∫ d^3k_1...d^3 k_n δ_D(k-k_1...n) F_n(k_1,...,k_n ; a) δ_0(k_1)...δ_0(k_n),θ_n(k; a)= ∫ d^3k_1...d^3 k_n δ_D(k-k_1...n) G_n(k_1,...,k_n; a) δ_0(k_1)...δ_0(k_2),where k_1...n = k_1 + ...+ k_n. In the following we will omit the scale factor dependence of the kernels to simplify the expressions. Using the numerical algorithm described in <cit.> we can do this for all orders in a general way. Using the perturbations up to third order we can construct the so called 1-loop power spectrumP^1- loop_ij(k) = P_0(k) + P^22_ij(k) + P^13_ij(k),where P_0(k) is the linear power spectrum defined as ⟨δ_0() δ_0(')⟩ = (2π)^3δ_ D(+') P_0(k), ⟨ ... ⟩ denoting an ensemble average. The higher order terms are given by⟨ g_i^2() g_j^2(')⟩ = (2π)^3δ_ D(+') P_ij^22(k),⟨ g_i^1() g_j^3(') +g_i^3() g_j^1(') ⟩ = (2π)^3δ_ D(+') P_ij^13(k),where g^i_1 = δ_i and g^i_2= θ_i.The power spectrum is a Fourier space measure of the correlation between the fields and the higher order or 1-loop terms give the first order in non-linearity on top of the linear power spectrum. The inclusion of loop terms has been shown to improve the prediction of theory <cit.>, an improvement more pronounced at higher redshift <cit.>. Despite this, the loop expansion of the power spectrum is known to have divergent behaviour at small scales making the benefits of the 1-loop terms only enjoyable within a restricted range of scales. Further this bad behaviour makes it difficult to move out of Fourier space which involves an integral over UV scales. In the next section we describe the RegPT treatment which allows us to do the Fourier transform safely. This treatment will be employed to Fourier transform the best fit TNS power spectrum in Sec.III which we compare to the GSM. We proceed by reviewing the TNS model of RSD. §.§ Modelling RSD in Fourier Space : TNSAs we mentioned in the introduction, the anisotropy of galaxy clustering provides a very promising means of testing gravity.The anisotropy arises from the non-linear mapping between real and redshift space and it is because of the mapping's non-linear nature that makes modelling RSD complex. A model of the effect was first given by Kaiser <cit.> which captures the coherent, linear peculiar infalling motion of galaxies in a clusterP_K^S(k,μ;a) = (1+fμ^2)^2P_δδ(k;a),where μ [The use of μ here should not be confused with the function μ(k;a) which will always include its arguments.] is the cosine of the angle between the line of sight and , f = d lnF_1/dlna is the logarithmic growth rate and P_δδ(k) is the linear matter power spectrum. The model does not account for the small scale damping effect of incoherent virialised motion - the fingers-of-god effect. Many authors accounted for this effect via a phenomenological pre factor term, usually taking the form of an exponential or Gaussian <cit.>.For robust tests of gravity a move beyond linear models is needed. As mentioned in the introduction, the non-linear TNS model of RSD has proved its merit against N-body and survey data. It is derived partly perturbatively with the small scale fingers-of-god effect being treated phenomenologically through a damping factor. The derivation can be followed in <cit.>, but we simply present it here P^S(k,μ) = _ (kμσ_v) { P^1- loop_δδ (k) - 2μ^2 P^1- loop_δθ(k) + μ^4 P^1- loop_θθ (k) + A(k,μ) + B(k,μ) }.The A and B terms account for higher-order interactions between the density and velocity fields and are given byA(k,μ)=-(k μ) ∫ d^3 k'[k_z '/k'^2 B_σ(k',k-k',-k) +kμ-k_z'/|-'^2| B_σ(k-k', k',-k) ],B(k,μ)= (k μ)^2 ∫ d^3k' F(k') F(k-k'), where F(k) = k_z/k^2[P_δθ (k) - k_z^2/k^2P_θθ (k) ],where the power spectra here are calculated at linear order. The cross bispectrum B_σ is given by δ_D(k_1+ k_2+ k_3)B_σ( k_1,k_2,k_3) = ⟨θ(k_1){δ(k_2) - k_2z^2/k_2^2θ(k_2)}{δ(k_3) - k_3z^2/k_3^2θ(k_3)}⟩,We choose an exponential form forthe fingers-of-god damping factor D_FoG(kμσ_v)= exp(-k^2 μ^2 σ_v^2), where σ_v is treated as a free parameter quantifying the dispersion in velocities (expressed in units Mpc/h) <cit.>.eq.(<ref>) gives very good predictions in the mildly non-linear regime but due to convergence problems of SPT a Fourier transform of the expression is difficult. Since we are concerned with configuration space statistics in this paper we will adopt the RegPT treatment for the power spectra components as well as for the A and B correction terms. This treatment essentially damps the divergences of the loop terms allowing an easy transform to configuration space. §.§.§ RegPTWe go over the terms used to construct eq.(<ref>) using the RegPT treatment described in e.g <cit.>. This treatment is based on expanding the multipoint propagators which contain the entire non-perturbative nature of the field's evolution <cit.>. These propagators can be analytically described in terms of the perturbative kernels and allow for the construction of a quasi non-linear power spectrum and correlation function which show excellent agreement with N-body data in real and redshift space as well as for models other than GR <cit.>.In terms of the multipoint propagators Γ_a^(n) the RegPT 1-loop power spectrum is given by P_bc(k;a) = Γ^(1)_b(k;a)Γ_c^(1)(k;a)P_0(k)+ 2 ∫d^3 /(2π)^3Γ_b^(2)(,-;a)Γ_c^(2)(,-;a)P_0(q)P_0(|-|),where P_0 is the initial linear matter power spectrum and b,c ∈{δ ,θ}.The propagators are given in terms of the perturbative kernels by Γ^(1)_b(k;a) =[ J_b^(1)(k;a){ 1+ k^2σ_d^2/2}. . + 3∫ d^3/(2π)^3 J_b^(3)(,,-;a)P_0(q)] e^-k^2σ_d^2/2,Γ^(2)_b(,-;a)= J_b^(2)(,-;a)e^-k^2σ_d^2/2,where J_b^(n) = (F_n,G_n). σ_d^2 is the dispersion of the linear displacement field given by σ_d^2(k) = ∫^k/2_0 dq/6π^2 F_1(q;a)^2P_0(q).This accounts for the loop terms in eq.(<ref>). The A and B correction terms are evaluated at tree-level since they are treated as next to leading order and are given in terms of the propagators as P_bc,tree(k;a)= Γ_b^(1)(k;a)Γ_c^(1)(k;a)P_0(k), B_bcd,tree(_1,_2,_3)= 2 Γ_b^(2)(_2,_3;a)Γ^(1)_c(k_2;a),Γ_d^(1)(k_3;a) P_0(k_2)P_0(k_3) + ( cyc. perm)where now the propagators are evaluated at tree level too Γ_b,tree^(n)(_1, ... _n;a) = J_a^(n)(_1 ... _n;a)e^-k_1...n^2σ_d^2/2,where k_1...n = |_1 + ... + _n|. We end by noting that the generality of these expressions applies to all MG models within the framework described in <cit.> although corrections may be incurred in some models as discussed in <cit.>. For the models treated in this paper the corrections are negligible at the scales of interest. Finally, we will conclude this section by reviewing the GSM model for the redshift space correlation function. §.§ Modelling RSD in Configuration Space: GSM In <cit.> the first study of the relation between eq.(<ref>) and the redshift space correlation function was made. Working in the linear regime, the authors developed the so called linear streaming model (LSM) which uses the mean infall velocity between pairs (v_12)and the velocity dispersion along the line of sight (LOS) (σ_12^2) as ingredients connecting theory to the RSD phenomenon in the linear regime. Specifically, it is the scale dependence of v_12 and σ_12^2 which drives the distribution of galaxies away from isotropy. By assuming a Gaussian form for the joint density-velocity distribution, the LSM for biased tracers is given by1+ξ^s_ LSM (r_σ, r_π) = ∫ G(r,y) e^-[r_π - y]^2/2σ_ 12,lin^2(r,μ)dy/√(2πσ_ 12,lin^2(r,μ)),where G(r,y) = [1 + ξ^r_L(r) + y/r(r_π -y)v_ 12,lin(r)/σ_ 12,lin^2(y) - 1/4y^2/r^2v_ 12,lin^2(r)/σ_ 12,lin^2(y)(1-(r_π-y)^2/σ_ 12,lin^2(y)) ], The variables are as follows: r_π and y are the separations parallel to the LOS of matter particles in redshift and real space respectively, r_σis the separation perpendicular to the LOS. ξ^r_L is the linear real space galaxy correlation function determined by Fourier transforming the linear power spectrum, v_ 12,lin = ⟨δ() (')⟩is the linear mean infall velocity of a particle pair with real space separation r = √(y^2 + r_σ^2) and σ_ 12,lin^2(r) = ⟨ (_ LOS() -_ LOS(') )^2 ⟩ is the linear velocity dispersion. The linear predictions for these are given below in terms of the generalised 1st order perturbative kernels (F_1,G_1)_ 12,lin(r) = v_ 12,lin = b/π^2∫ dk kj_1(kr) G_1(k;a) F_1(k;a) P_0(k), where j_1(k) is the 1st order spherical Bessel function and b is the linear bias factor. σ_ 12,lin^2(r, μ^2) = 2[σ_1^2 - 1/2π^2∫ dk G_1(k;a) 𝒥(kr,μ^2) P_0(k)], where σ_1^2 =1/3⟨() ·() ⟩ is the 1-dimensional velocity dispersion and 𝒥(kr,μ^2) = μ^2(j_0(kr)-2j_1(kr)/kr) + (1-μ^2)j_1(kr)/kr.Moving away from the linear regime, we consider the non-linear redshift space correlation function developed in <cit.>, known as the Gaussian streaming model from its core assumption that the matter's pairwise velocity probability distribution is of a Gaussian form. This is given by1+ξ^s_ GSM(r_σ, r_π) = ∫ [1+ξ^r(r)] e^-[r_π - y - μ v_12(r)]^2/2σ_12^2(r,μ)dy/√(2πσ_12^2(r,μ)). ξ^r is the non-linear real space correlation function, v_12(r) is the non-linear mean infall velocity of a particle pair and σ_12^2(r,μ) is the non-linear, non-isotropic velocity dispersion. Note for biased tracers we must include tracer bias in ξ^r. At linear bias level, this is given as b^2ξ^r, where b is the linear bias factor. In <cit.> the authors use the Lagrangian perturbation theory (LPT) model of <cit.> for the real space correlation function. Here we use a RegPT prescription where our real space correlation function is produced by Fourier transforming the RegPT 1-loop matter power spectrumξ^r(r) = ∫d^3k/(2π)^3 e^i · P^ 1-loop,RegPT_δδ (k).Because of RegPT's damping of the power spectrum at small scales, we can do the above integral without having to worry about SPT divergences.As discussed in a previous section, P^1-loop_ab(k) can be readily constructed for general models of gravity.Finally, the mean infall velocity and velocity dispersion are given by correlations between the density field and the velocity field. Using a perturbative treatment of the fields we can derive expressions for these ingredients for general models of gravity in the linear and quasi non-linear regime. Here we give expressions for v_12 and σ_12^2 appearing in eq.(<ref>) in terms of the generalised kernels (F_n,G_n) (see eq.(<ref>) and eq.(<ref>)). In the case of GR, using the Einstein de Sitter approximation for the kernels, one can follow Appendices A1 and A2 of <cit.>.§.§.§ Mean Infall Velocity v_12(r)The mean infall velocity arrises from correlating the density field with the velocity. In terms of these correlations we can write (eq.(27) of <cit.>) [1+b^2ξ^r(r)] v_12(r)r̂ = 2b⟨δ_1()_1(+)⟩ + 2b∑_i>0⟨δ_i()_4-i(+)⟩ +2b^2∑_i,j>0⟨δ_i() δ_j(+)_4-i-j(+) ⟩,where we have included b, the linear bias factor. ξ^r is the matter correlation function andis the velocity field perturbation. The correlations in the above expression up to 2nd order in the linear power spectrum are given below. 2b ( ⟨δ_1()_1(+)⟩ + ∑_i>0⟨δ_i()_4-i(+)⟩) = b/π^2∫ dk k P_δθ^ 1- loop(k; a) j_1(kr),where P_δθ^ 1- loop is given by eq.(<ref>). The last term has three contributions at 1-loop order, i,j=(1,1),(1,2),(2,1).The contribution from the first two of these (A5 of <cit.>) is given as2b^2 ( ⟨δ_1() δ_1(+)_2(+) ⟩ + ⟨δ_1() δ_2( ..+)_1(+) ⟩) =b^2/2π^4∫_0^∞ dk dy ∫_-1^1 dx k^4 y j_1(kr) P_0(k)P_0(ky)(F_1(k;a)F_1(ky;a) G_2(ky,k,-x;a) y(1-yx)/1+y^2-2yx. + G_1(ky;a)F_1(k;a)F_2(ky,k,-x;a)x ),and the (2,1) contribution (A6 of <cit.>) is given as 2b^2⟨δ_2() δ_1(+)_1(+) ⟩ =b^2/2π^4∫_0^∞ dk dy∫_-1^1 dx k^4 y x j_1(kr) P_0(ky)P_0(k√(1+y^2-2yx)) ×F_1(k√(1+y^2-2yx);a)G_1(ky;a)F_2(ky,k√(1+y^2-2yx), u ;a),where we have written the kernels in terms of the integrated vector's magnitudes and angle between them:|| = k, || =ky and | - | = k√(1+y^2-2yx) with x = · and u = (̂-̂)̂·. This notation is used for the velocity dispersion expressions below.§.§.§ Velocity Dispersion σ_12^2(r,μ^2)The velocity dispersion depends on both the separation of the pair r and the angle the separation vectormakes with the LOS, ϕ_lr, expressed through the argument μ^2 =cos^2 (ϕ_lr). One can combine the perpendicular and parallel components of σ_12^2 to get the expression(eq.(29) to eq.(32) of <cit.>) [1+b^2ξ^r(r)] σ_12^2(r,μ^2)= 2 ( ⟨ (v^ℓ())^2⟩ - ⟨ v^ℓ()v^ℓ(+) ⟩)+ 2b ⟨δ()( v^ℓ())^2⟩ + 2b [⟨δ() (v^ℓ(+))^2 - 2 ⟨δ()v^ℓ() v^ℓ(+) ⟩]+ 2b^2 [ ⟨δ() δ( + )(v^ℓ())^2 ⟩ - ⟨δ() δ( + )v^ℓ()v^ℓ( + ) ⟩], where ℓ denotes the component ofalong the LOS. We give these component by component below. 2 ⟨ (v^ℓ())^2⟩ = 6 σ_1^2 =1/3π^2∫ dk j_1(kr)/kr G_1(k;a)^2 P_0(k), - 2 ⟨ v^ℓ()v^ℓ(+) ⟩ = -1/π^2∫ dk P_θθ^ 1-loop (k;a) 𝒥(kr,μ^2), where again P_θθ^ 1-loop (k;a) is evaluated using eq.(<ref>) and 𝒥(kr,μ^2) is given in eq.(<ref>). The third term contributes a constant to σ_12^2(r,μ^2). This is treated as a free parameter (σ^2_ iso) in our analysis in Sec.III (see <cit.> for example) but we give the PT prediction for this isotropic contribution below2b ⟨δ()( v^ℓ())^2⟩ = b/6π^4∫ dk dy ∫_-1^1 dx k^3 y^2 P_0(k) P_0(ky) G_1(k;a), ×(2G_2(ky,k,x;a) F_1(ky;a)(1+yx)/√(1+y^2+2yx)-xG_1(ky;a)F_2(ky,k,x;a)/y). We can expand the 2nd line of eq.(<ref>) as 2b [⟨δ() (v^ℓ(+))^2⟩ - 2 ⟨δ()v^ℓ() v^ℓ(+) ⟩] = 4b ⟨δ_1()v^ℓ_1(+) v^ℓ_2(+) ⟩ + 2b ⟨δ_2() (v^ℓ_1(+))^2⟩ -4b⟨δ_1()v^ℓ_1() v^ℓ_2(+) ⟩ -4b⟨δ_1()v^ℓ_2() v^ℓ_1(+) ⟩- 4b⟨δ_2()v^ℓ_1() v^ℓ_1(+) ⟩. The integrals of these terms are given below4b ⟨δ_1()v^ℓ_1(+) v^ℓ_2(+) ⟩ = b/2π^4∫ dk dy ∫_-1^1 dx k^3 y P_0(k)P_0(ky)F_1(k;a) G_1(ky;a) G_2(ky,k,x;a)/√(1+y^2+2yx)×( j_0(kr)(y-2x-3x^2y) - 𝒥(kr,μ^2) y(1-x^2) ), 2b ⟨δ_2() (v^ℓ_1(+))^2⟩ = -1/16π^6∫ dk dy k^3 y G_1(ky;a)G_1(k;a) P_0(k) P_0(ky) ∫_-1^1 dx_1 dx_2cos(kyrx_1 + krx_2)∫_0^2π dϕ_1 dϕ_2F_2(k,y,x̅;a)[ μ^2(2x_1x_2 - x̅) + x̅-x_1 x_2 ], where x̅ = x_1x_2 + √((1-x_1^2)(1-x_2^2))sinϕ_1sinϕ_2. The 4 dimensional angular integration in this expression is performed using the Monte Carlo integration algorithm Cuba <cit.>. -4b⟨δ_1()v^ℓ_1() v^ℓ_2(+) ⟩ = -b/π^4∫ dk dy ∫_-1^1 dx k^3 y x P_0(ky) P_0(k√(1+y^2-2yx)) 𝒥(kr,μ^2)G_1(ky;a) F_1(k√(1+y^2-2yx);a) G_2(ky,k√(1+y^2-2yx),u;a), -4b⟨δ_1()v^ℓ_2() v^ℓ_1(+) ⟩- 4b⟨δ_2()v^ℓ_1() v^ℓ_1(+) ⟩= -b/π^4∫ dk dy ∫_-1^1 dx k^3 y P_0(k) P_0(ky) 𝒥(kr,μ^2) × G_1(k;a)(G_2(ky,k,x;a)F_1(ky;a)y(1+yx)/1+y^2+2yx - x G_1(ky;a)F_2(ky,k,x;a)).Finally, the last term in eq.(<ref>) evaluates to 2b^2 [ ⟨δ() δ( + )(v^ℓ())^2 ⟩ - ⟨δ() δ( + )v^ℓ()v^ℓ( + ) ⟩]= b^2ξ^r(r) σ_12, lin^2(r,μ^2) + 1/2v_12, lin^2(r) μ^2, where σ_12, lin^2(r,μ^2) and v_12, lin^2(r) are the linear predictions for the velocity dispersion eq.(<ref>) and mean infall velocity eq.(<ref>).At leading order the first term in eq.(<ref>) cancels with the 2nd term on the LHS of eq.(<ref>) and so we omit in our calculations and simply include the linear mean infall velocity term. For the calculations in the next section we have set b=1 and so only dark matter particles are considered. Perturbation theory predicts a constant contribution to the velocity dispersion, σ^2_ iso (eq.(<ref>)) given in units of (Mpc/h)^2. As mentioned, this is treated as a free parameter allowing us to describe deviations to the predicted scale dependance on small scales where non-linear fingers of god effects are strong and unable to be treated perturbatively. § RESULTSIn this section we will present predictions using eq.(<ref>) for three models of gravity, namely the Vainshtein screened normal branch of DGP gravity (nDGP) <cit.>, the Chameleon screened Hu-Sawicki form of f(R) gravity <cit.> and GR. We will compare these results with the FT of eq.(<ref>) which is fit to N-body simulations. This is done for dark matter only and no tracer bias is included.The Fourier space comparisons for nDGP can be found in Appendix A while for GR and f(R) we use the best fit σ_v found in <cit.>. Our background cosmology is taken from WMAP9 <cit.>: Ω_m = 0.281, h=0.697, and n_s=0.971. The box width is 1024 /h with 1024^3 dark matter particles used and a starting redshift of 49. The linear theory power spectrum normalisation was set to be σ_8=0.844. The nDGP simulation uses Ω_rc=1/4r_c^2H_0^2=0.438 while the f(R) simulation uses |f_R0|=10^-4. We consider the redshift of z=0.5 where SPT benefits from a good range of validity while still being relevant for upcoming surveys such as Euclid [<www.euclid-ec.org>] and DESI[<http://desi.lbl.gov/>]. We start with a comparison of linear and non-linear predictions for the real space correlation function followed by comparing different predictions for the non-linear redshift space correlation function predictions: the GSM using RegPT and the FT of TNS multipoles.We compare the FT of the RegPT 1-loop expressions with the LPT model of <cit.>. This model has been tested against N-body simulations in the GR case and has shown to be percent level accurate at scales of r≥25 Mpc/h <cit.>. It has been employed in spectroscopic survey analyses with BOSS <cit.>. Although we only do this for GR, it gives us a handle on the accuracy of our FT approach to the multipoles. The transform of the RegPT power spectrum was compared to N-body results for GR and f(R) in <cit.> showinggood agreement above and around the BAO scale. Fig.<ref> shows the real space correlation function as predicted by eq.(<ref>) and the LPT prediction of<cit.> for dark matter. The FT of the linear power spectrum is also shown as the linear prediction. We see both RegPT and LPT give a smoothing of the BAO bump - a well known non-linear effect - and that they agree on small and large scales at the percent level while around the BAO bump they show up to a 4% difference with the RegPT treatment showing slightly more damping around this scale. For completeness wealso show the RegPT predictions against the linear predictions for the other models of gravity considered (Fig.<ref>). We notice that the non-linear RegPT predictions for these models show more damping of the BAO bump when compared to the GR case, an expected effect of enhanced structure growth as well as enhanced 2nd and 3rd order non-linearities.Moving to redshift space, we will use the FT of the best fit multipoles shown on the left of Fig.<ref> in the Appendix for nDGP and Table. II of <cit.> for f(R) and GR. Because of the TNS's extra degree of freedom (σ_v), the model should have an advantage in goodness of fit when compared to the GSM, which can be completely determined by SPT. In general the correlation function needs to be measured many times from N-body simulations and averaged because of the small imprint of the acoustic features which can be greatly hidden by scatter. MG simulations are more computationally expensive than GR ones and so only a few are available. Thus, a clean configuration space measurement in MG theories is not readily available. This makes the TNS transform a good and practical benchmark to compare the GSM predictions to in the absence of averaged N-body correlation function measurements. The configuration space multipoles are given by <cit.> ξ_ℓ^(S)(s)= i^ℓ/2π^2∫ dk k^2 P_ℓ^(S)(k)j_ℓ(ks),where j_ℓ is the ℓ^ th order spherical Bessel function and P_ℓ^(S) is given byP_ℓ^(S)(k)=2ℓ+1/2∫^1_-1dμ P_ TNS^(S)(k,μ)𝒫_ℓ(μ),where 𝒫_l(μ) denote the Legendre polynomials. Again we will only consider the first two multipoles, ℓ = 0,2. The top panel of Fig.<ref> shows the monopole (left) and quadrupole (right) predictions for the redshift space correlation function within GR. We have plotted the TNS transform with σ_v = 4.75Mpc/h in black against the GSM predictions for three different values of the parameter σ_ iso defined in Sec.II C. The blue curve is the GSM prediction where σ_ iso takes the PT predicted value. The predictions look very reasonable with significant smearing of the BAO due to non-linear effects, mostly seen in the monopole.The bottom panels of Fig.<ref> show the fractional differences between the TNS transform and the GSM predictions. Fractional differences go up to 4% in the monopole around the BAO scale and slightly less for the quadrupole, with slightly more damping of the BAO bump by the GSM predictions. We find that around this scale the PT prediction (eq.(<ref>)) for the isotropic contribution to the velocity dispersion does well for the monopole, whereas for the quadrupole the higher valued green curve (σ_ iso = 5Mpc/h) does better, a value consistent with the TNS best fit velocity dispersion. Similar results are found for the nDGP model of gravity, shown in Fig.<ref>. The deviations of the GSM predictions from the TNS transform are only slightly larger than in the GR case, going up to 6% in the monopole at the BAO scale. The PT prediction for σ_ iso (σ_ iso = 3.9Mpc/h) does the best over both multipoles at smaller scales with the green (σ_ iso = 5.5Mpc/h) doing a bit better around the BAO bump. Both these values are consistent with the TNS best-fit value.The f(R) predictions are shown in Fig.<ref>. In this case the monopole's fractional differences are significantly larger with up to 8% more damping in the GSM model. The quadrupole differences remain ≤ 3% around the BAO scale. In this case the PT predicted value for σ_ iso (5.2Mpc/h) seems to underestimate the value withσ_ iso = 7.5Mpc/h being more consistent with the TNS transform. This being said, to really tell which treatment performs better we wait for comparisons with simulation data. As mentioned earlier, many realisations are needed to get a converged measurement of the correlation function. This can be done for GR but for MG theories simulations are expensive computationally. By using COmoving Lagrangian Acceleration (COLA) approaches such as those described in <cit.>, this problem becomes tractable and we leave this to a future work.Fig.<ref> and Fig.<ref> show the differences between the modified gravity predictions and the GR ones for both theoretical predictions for the correlation function as well as the linear prediction. We see that in both the FT of TNS and GSM the effect of modifying gravity is very similar indicating both approaches give comparable signals of deviations from GR.In the monopole, around the acoustic bump, both non-linear approaches reduce the MG-Signal with a larger difference seen in linear modelling. The LSM also shows larger differences at scales below the BAO in the quadrupole. One other feature is that f(R) gravity shows a suppression compared to GR around40 Mpc/h ≤ s ≤100 Mpc/hwhile nDGP shows an enhancement over GR for the monopole.§ SUMMARYThis work has extended the code described in <cit.> to calculate the non-linear redshift space correlation function as modelled by <cit.> for a general class of gravity and dark energy models. We have also extended the code to calculate the non-linear redshift space correlation function as described by the TNS model using the RegPT treatment as done in <cit.>. To make comparisons between the two predictions the TNS power spectrum monopole and quadrupole were first compared to N-body data in order to obtain the best fit σ_v (See Fig.<ref> and Table.II of <cit.>). This required finding a realm of validity for the SPT predictions which was found by comparing the real space power spectra (See Appendix A). We then found fair agreement between these two treatments to within 4% for GR and nDGP with Ω_rc = 0.438 and up to a 8% deviation in the treatments for the chameleon screened f(R) model with |f_R0|=10^-4 around the BAO scale (Fig.<ref>, Fig.<ref> and Fig.<ref>). We have also compared the LPT correlation function <cit.> in real space with that obtained using a FT of the RegPT 1-loop spectrum (Fig.<ref>). The RegPT treatment gives up to 4% more damping around the BAO scale. Recently a LPT prediction for MG models has been developed <cit.> allowing the extension of such comparisons.We observe large damping in the GSM and FT of TNS treatments over the linear predictions with more damping observed in the f(R) and nDGP cases. This is due to gravity being boosted by additional non-linearities encoded in the extra γ functions for these theories. The difference between the GSM and FT of TNS predictions comes from their treatment of the RSD. While the GSM is completely perturbative in making the non-linear mapping to redshift space within configuration space, the TNS is partly phenomenological and further, a resummation technique such as RegPT is needed to make the transform to configuration space. Despite its added degree of freedom, σ_v, it is unclear how best to treat small scale SPT divergences and further how robust and consistent the methods on the market are (examples of such treatments include RegPT <cit.>, renormalised perturbation theory <cit.> and EFToLSS prescriptions <cit.>). This issue has yet to be investigated thoroughly. In light of this, one cannot say with certainty which approach to the redshift space correlation function will perform better when matching simulation or observational data. This will be the focus of a future work. To give the GSM model extra freedom, we promote the isotropic velocity dispersion contribution to the GSM's pairwise dispersion σ_12^2as a free parameter σ_ iso, which is physically equivalent to TNS's σ_v parameter. By doing this we can enhance the PT prediction, given in eq.(<ref>), and better match the TNS on small scales. We find that the PT prediction for σ_ iso = 3.9 Mpc/h does well for the nDGP model and we are able to match the FT of TNS prediction at scales s ≤ 100Mpc/h to within 2% (Fig.<ref>).For f(R) and GR we find the PT prediction underestimates the small scale velocity dispersion, and we find the larger values of σ_ iso^ GR = 5 (3.16) Mpc/h and σ_ iso^f(R) = 7.5(5.2) Mpc/h (PT prediction in brackets) better match the FT of TNS at smaller scales, specifically in the quadrupole prediction (Fig.<ref> and Fig.<ref>). Around the scales 100 Mpc/h ≤ s ≤ 180 Mpc/h σ_ iso has a marginal effect. The preferred values of σ_ isoin the modified gravity theories both differ by around 30 % when compared with the best fit values of σ_v of the TNS model. The GR value of σ_ iso is within ∼ 5% of its TNS equivalent. In summary, we find that both approaches model the RSD consistently in the range 50/h ≤ s ≤ 180 /h withthe GSM requiring the promotion of σ_12^2 to a free parameter to be consistent with the TNS approach, particularly for the quadrupole.Using the best fit values for σ_ iso we find that the differences between GR and MG-GSM predictions for the correlation function multipoles accurately follow those using the FT of TNS indicating that both approaches to modelling the RSD consistently treat modifications to gravity, with neither giving an enhanced MG signal over the other (Fig.<ref> and Fig.<ref>). The non-linear differences follow the LSM differences in all cases with the LSM generally picking up larger deviations from GR consistently in both multipoles. This may be because of MG's enhanced non-linear gravitation which suppresses enhancements in the multipoles. The survey comparisons done in <cit.> imply the GSM treatment overdamps the BAO wiggle in redshift space. This suggests a preference of the RegPT treatment to the real and redshift space correlation function although marginally. Again, we wait for the availability of simulation data to make this conclusion. In any case, the ability to compute the redshift space correlation function for generalised models should prove to be very useful when performing statistical analyses on survey data and obtaining gravitational parameter constraints. The importance of correctly modelling gravity has been investigated in a number of works<cit.> and has been shown to be of growing importance as we enter the era of stage IV surveys. Using the pipeline described here we can perform consistent analyses of the high quality upcoming data from surveys such asthe Dark Energy Spectroscopic Instrument (DESI) [<http://desi.lbl.gov/>] and the ESA/Euclid survey[<www.euclid-ec.org>].Further, by moving to smaller scales and using a fuller shape of the correlation function we expect any deviations from GR to become less able to hide in nuisance degrees of freedom such as σ_v, σ_ iso or tracer bias. Fig.<ref> and Fig.<ref> show the difference between the MG and GR predictions for the correlation function. We see that at the BAO scale down to the scales valid for the GSM treatment, we have a significant MG signal.By pushing into these scales we enter regions as yet unused for constraining models beyond GR <cit.>. This work primes the consistent probing of parameter space in this regime by using currently available spectroscopic data such as BOSS and further the possibility of using a combination of configuration and Fourier space measurements which will be very useful in overcoming systematics. Finally we comment on the preparation of the code for such statistical analyses. Currently optimisation needs to be made in the computation of eq.(<ref>) which on top of the 2 spatial integrals and 1 angular for the multipoles, 4 additional angular integrals need to be performed. The current method is to use an Monte Carlo integration technique to evaluate the integral which is slow when looking to achieve the desired accuracy. For statistical analyses of data a lower time cost is essential.Further, for scale dependent models of gravity, the perturbation kernels need to be initialised many times which also incurs a significant time cost. These issues have been relievedto some extent through parallelisation. One can also perform an interpolation technique in gravitational parameter space as done for the BOSS analysis in <cit.> which reduces the number of model computations significantly. We aim to optimise the computation of eq.(<ref>) and make use of the code to perform analyses of MG models with currently available data in a future work. § ACKNOWLEDGMENTSThe authors would like to thank Yuting Wang for useful discussions. We would like to thank Gong-bo Zhao and Wojciech Hellwing for supplying us with the N-body data used in Appendix A. BB is supported by the University of Portsmouth. KK is supported by the European Research Council through 646702 (CosTesGrav). KK is also supported by the UK Science and Technologies Facilities Council grants ST/N000668/1.§ FOURIER SPACE COMPARISONS : NDGP To get a good benchmark for the accuracy of the GSM predictions we consider the FT of eq.(<ref>). By fitting σ_v to N-body simulations we are able to accurately reproduce quasi non-linear effects which are then transferred to the correlation function. Using the RegPT prescription we are not punished by divergences in the integration over highermodes. We begin by finding the best fit σ_v and to do this we first must determine the range of validity of SPT. The left pane of Fig.<ref> shows the real space matter-matter (blue), matter-velocity divergence (green) and velocity divergence(red) power spectra modelled using SPT (dashed) and RegPT (solid) against N-body data for the nDGP model of gravity.The k_ maxh/Mpcwe use for the fitting of σ_v in the multipoles is given by the solid arrow which delimits the 1% deviation region. We have fitted Gaussian error bars to the data assuming a survey volume of 1 ^3/h^3. With a range of validity we can now fit the TNS free parameter σ_v. We consider the multipoles of eq.(<ref>) given by eq.(<ref>). The monopole and quadrupole, ℓ = 0,2 respectively, are then fit up to the k_ max found previously. Higher order multipoles have a very low signal to noise ratio making them problematic to measure in practice and so we will not consider them in our results.The right pane of Fig.<ref> shows the monopole (magenta) and quadrupole (cyan) N-body measurements against the RegPT-TNS predictions for three different values of σ_v. The fractional difference of the best fit σ_v with N-body is shown in the bottom panels. The best fit values for σ_v is found to be 5.1Mpc/h. The best fit value for f(R) and GR were found to be 6Mpc/h and 4.75Mpc/h respecitvely in <cit.>. | http://arxiv.org/abs/1705.09181v2 | {
"authors": [
"Benjamin Bose",
"Kazuya Koyama"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170525135431",
"title": "A Perturbative Approach to the Redshift Space Correlation Function: Beyond the Standard Model"
} |
Anisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions Michael Strickland December 30, 2023 ==============================================================empty § INTRODUCTION Synthetic biology is an interdisciplinary field of science and engineering that aims to construct biochemical systems with prescribed behaviors <cit.>. At the theoretical level, the synthetic systems may significantly enhance our understanding of biology. At the practical level, they may have broad applications, e.g. in medicine <cit.>, industry <cit.>, and nanotechnology <cit.>. The systems may also be of interest to NASA for optimizing extraterrestrial explorations <cit.>. A proof-of-concept for synthetic biology is a synthetic oscillator called the repressilator, which was implemented in vivo <cit.>. The experimental advances since the repressilator range from isolated synthetic biochemical networks, to microorganisms containing partially, or even fully, synthetic DNA molecules (synthetic life) <cit.>. Examples include microorganismscontaining a synthetic bistable switch <cit.>, and a cell-density controlling quorum sensor <cit.>, microorganisms producing antimalarial drugs <cit.>, and synthetic systems designed for tumor detection, diagnosis and adaptive drug-response <cit.>.The construction of biochemical networks in synthetic biology may be broken down into two steps: firstly, an abstract system is constructed, displaying prescribed properties, and taking the form of a chemical reaction network <cit.>. Secondly, the abstract network is mapped to a suitable physical network, which may then be integrated into a desired environment (e.g. a test-tube, or a living cell) <cit.>. In the first step of network construction, the goal is to obtain an abstract network with desired dynamics. In this paper, we consider two dynamical models of reaction networks under mass-action kinetics <cit.>: the deterministic model, and the stochastic model (see Methods for more details). The deterministic model takes the form of the reaction rate equations, which are ordinary-differential equations governing the time-evolution of the species concentrations <cit.>. The stochastic model takes the form of a Markov chain, which may be simulated using the Gillespie stochastic simulation algorithm <cit.>. The Gillespie algorithm generates noisy copy-number time-series, with the copy-number distribution matching that obtained from the underlying chemical master equation <cit.>. The stochastic model is more-detailed, taking into an account the discreteness of the species counts, and the stochastic nature of the dynamics, which may be particularly important in biochemistry, where reaction networks may contain low-abundance species <cit.>. On the other hand, the deterministic model is less-detailed, and more appropriate when the species are in high-abundance, and the discreteness and stochasticity are negligible <cit.>.In the second step of network construction, the goal is to engineer a physical network whose dynamics match well the dynamics of a given abstract network, over a suitable time-interval. Engineering an appropriate physical network may proceed indirectly, by altering a preexisting physical network, or directly, by engineering anetwork, which involves a given set of physical species, from scratch. The advantage of the former approach is that a preexisting network may display (partially) desirable dynamical properties. However, such a network may involve DNA and RNA molecules, proteins, and metabolites <cit.>, some of which may have complex biophysical properties. Consequently, the disadvantage is that the structure (and, thus, the dynamics) of such a network cannot generally be modified in an arbitrary manner. In the latter approach, one may choose the physical species, at the expense of having to build a network from scratch. In the subfield of DNA computing, the latter approach is followed, and physical networks are engineered with chemical species consisting exclusively of DNA molecules, interacting via the toehold-mediated DNA strand-displacement mechanism <cit.>. DNA production is systematic and cost-effective, and, due to the fact that DNA biophysics is relatively well-understood, one has more freedom in controlling the structure of corresponding physical networks. More precisely, an abstract network under mass-action kinetics may be mapped to a DNA-based physical network provided it consists of up to second-order reactions, with rate coefficients varying over up to six orders of magnitude. The resulting physical network has identical deterministic dynamics as the abstract network (in the asymptotic limit of some of the kinetic parameters <cit.>), up to a scaling of the dependent variables.A proof-of-concept for DNA computing is a synthetic oscillator called the displacillator, which was implemented in vitro <cit.>.While the deterministic model of reaction networks is less-detailed, it is also simpler than the stochastic model, making it attractive for guiding the construction of networks, predicting accurately their mean-field behavior <cit.>. However, when noise is an important part of the dynamics, the stochastic model has to be considered. The intrinsic noise, often arising in biochemistry, may be controlled in two ways: it may be decreased (e.g. as in <cit.>), in order to reduce the differences between the stochastic and deterministic dynamics. On the other hand, it may be increased, in a state-dependent manner, in order to favorably change the stochastic dynamics. In the language of molecular computing, the latter approach corresponds to exploiting the proven computational power of the stochastic reaction networks <cit.>, by reprogramming the underlying intrinsic noise. Let us note that exploitations of the noise for enhancing biological functions have been reported in applications <cit.>. In this paper, we follow the latter approach, and present the noise-control algorithm (given as Algorithm <ref>) which maps an input reaction network to output networks whose stochastic dynamics have an additional controllable state-dependent noise. Importantly, the input and output networks have identical deterministic model in appropriate limits of some of the parameters introduced by the algorithm. The algorithm may play a significant role in the biochemical network synthesis, allowing for a deterministic-stochastic hybrid approach. More precisely, when constructing abstract and physical networks, one may use the deterministic model to guide the construction, and then apply the algorithm to favorably modify the intrinsic noise in the stochastic model, while preserving the desired deterministic dynamics. The algorithm may also be used to adjust the intrinsic noise to favorably interact with environment-induced effects (e.g. extrinsic noise).The rest of the paper is organized as follows: we introduce Algorithm <ref> by applying it to the test network (<ref>), which at the deterministic level displays a globally attracting equilibrium point. We show that the algorithm can favorably modify the stationary probability distribution underlying (<ref>) at arbitrary points of the state-space, without influencing the deterministic dynamics. For example, it is shown that the algorithm may be used to redesign (<ref>) to achieve noise-induced multimodality (multistability). We then apply Algorithm <ref> to the exotic network (<ref>), which at the deterministic level displays a bistability involving an equilibrium point and a limit cycle. The algorithm is used to redesign (<ref>) to increase the stochastic switching between the two attractors, and to achieve noise-induced oscillations.§ A ONE-SPECIES REGULAR SYSTEM Consider the one-species production-decay reaction network ℛ̂(s), given by (<ref>)..5ℛ̂(s): ∅s, s ∅, .4dx̂/d t=k_1 - k_2 x̂,x̂(0) = x̂_0. Species s from network (<ref>) reacts according to the two reactions with rate coefficients k_1,k_2 ∈ℝ_≥,where ℝ_≥ is the set of nonnegative real numbers, and ∅ is the zero-species (denoting species which are not of interest). In this paper, we assume reaction networks are under mass-action kinetics, with the reactions taking place in unit-volume reactors. Let us denote the concentration of species s from (<ref>) at time t ∈ℝ_≥ by x̂ = x̂(t) ∈ℝ_≥. The initial value problem for the deterministic model (also called the drift) for network (<ref>) is given by system (<ref>), with x̂_0 ≥ 0 (see also Methods). Since the deterministic model (<ref>) has a globally attracting equilibrium point, given by k_1/k_2, network (<ref>) is said to be regular <cit.>.Let us denote the copy-number of species s from (<ref>) at time t ≥ 0 by X̂(t) ∈ℕ_0, where ℕ_0 is the set of integers. Under the stochastic model, X̂(t) is modelled as a continuous-time, discrete-space Markov chain (see also Methods), which can be generated by using the Gillespie stochastic simulation algorithm <cit.>. Given X̂(t), there will be a mean interevent time until one of the reactions from (<ref>) fires. The mean interevent time is given by 1/α̂(X̂(t)), and when the event takes place, the probability that the i-th reaction from (<ref>) fires is equal to α̂_i(X̂(t))/α̂(X̂(t)), for i ∈{1,2}. Here, α̂_1 = k_1, and α̂_2(x) = k_2 x, are the so-called propensity functions of the first, and second, reactions from (<ref>), respectively. Function α̂(x) = k_1 + k_2 x is the total propensity function of network (<ref>), i.e. the sum of propensity functions of all the underlying reactions.We now wish to structurally modify network (<ref>) in such a way that the deterministic model from (<ref>) is preserved, while an arbitrary nonnegative function, defined on a bounded discrete domain, is added to the total propensity function of (<ref>). The latter requirement implies that the interevent time would be controllably decreased in a state-dependent manner. Equivalently, the two requirements imply that a controllable state-dependent noise would be introduced into the stochastic dynamics. We have designed a three-step algorithm, given as Algorithm <ref>, which achieves such goals for arbitrary reaction networks under mass-action kinetics. Let us describe properties of the algorithm by applying it on network (<ref>). Firstly, we wish to introduce an additional species s̅ into network (<ref>), in such a way that species s and s̅ satisfy a pairwise stoichiometric conservation law. Secondly, we require that the enlarged network has the same deterministic model as network (<ref>), despite the added species s̅, which may be achieved by adding another auxiliary species. More precisely, let us consider network ℛ̂^1(s,s̅) ∪ℛ_1^2(s̅), given by:ℛ̂^1(s,s̅):s̅ + I^1s + I^1, ss̅,ℛ_1^2(s̅):∅ I^1,s̅ + I^1s̅.Species s, s̅, I^1 from (<ref>) react according to the four reactions with rate coefficients k_1, k_2, 1/μ∈ℝ_≥. Network ℛ̂^1 = ℛ̂^1(s,s̅), given in (<ref>), is obtained from network ℛ̂ = ℛ̂(s), given by (<ref>), in the following way: since the first reaction in ℛ̂ increases copy-number of s by one, s̅ and I^1 are added to the reactants of the reaction, and I^1 is added to the products, leading to the first reaction in ℛ̂^1. Since the second reaction in ℛ̂ decreases copy-number of s by one, s̅ is added to the products, leading to the second reaction in ℛ̂^1. This ensures that the desired conservation law holds. The superscript in I^1 indicates that species I^1 is involved as a catalyst in a reaction of ℛ̂^1 in which s is increased by one. The subscript in ℛ_1^2 = ℛ_1^2(s̅) indicates that the network describes production and decay of I^1.The initial value problem for the deterministic model of (<ref>) is given byd x/d t=k_1 (c - x) y - k_2 x,d y/d t= 1/μ( 1 - (c - x) y ), x(0) = x_0, y(0) = y_0,where x = x(t) ∈ [0,c] ∩ℝ_≥, and y = y(t) ∈ℝ_≥, are the concentrations of species s, and I^1, from (<ref>), respectively, with x_0, y_0, c ∈ℝ_≥. We have used the kinetic conservation law x̅(t) = c - x(t), where x̅(t) is the concentration of species s̅, and c < ∞ is a time-independent conservation constant. Note that the conservation law truncates x-state-space. Let us now describe relationships between systems (<ref>) and (<ref>), starting with the weak statement: for c > k_1/k_2, and for any μ≥ 0, solutions of (<ref>) and (<ref>) are the same in the long-time limit t →∞. More precisely, the x-component of the equilibrium point of (<ref>) is identical to the equilibrium point of (<ref>), and both are stable. In Supplementary Information (SI) Text, we justify the strong statement: for sufficiently large c, and for μ≪ 1, solutions of (<ref>) and (<ref>), with the same initial conditions, are approximately the same at each time t ≥ 0. For these reasons, we call ℛ_1^2 a drift-corrector network.§.§ Zero-Drift Network ℛ_1,1^3Having completed the first two steps, let us focus on the third (and final) step, in which we introduce arbitrary noise into the stochastic model of (<ref>), without influencing the deterministic model (<ref>). Let us start our consideration by embedding into (<ref>) network ℛ_1,1^3 = ℛ_1,1^3(s,s̅), which is given byℛ_1,1^3(s,s̅):s + s̅ 2 s,s + s̅ 2 s̅.The subscript in ℛ_1,1^3 indicates that the underlying reactions have one molecule of s, and one of s̅, as reactants. The two reactions in (<ref>) preserve the conservation law from (<ref>). Furthermore, they fire with the same rates, with the first reaction leading to a unit-production, while the second to a unit-decay, of species s. Consequently, embedding ℛ_1,1^3 into (<ref>) does not affect the underlying deterministic model (<ref>), and we call ℛ_1,1^3 a zero-drift network. However, ℛ_1,1^3 does affect the underlying stochastic model <cit.>. To illustrate this, let us consider network ℛ_1,1^3 in isolation: the reactions from (<ref>) fire when X(t) ∈ (0,C), but not when X(t) ∈{0,C}, so that ℛ_1,1^3 in isolation fires until X(t) takes one of the extreme values {0,C}. Here, X(t) ∈ℕ_0, and C ∈ℕ, C < ∞, are the copy-number of species s appearing in (<ref>) and (<ref>) at time t ≥ 0, and the conservation constant, respectively. Let us note that a possible biologically-relevant realization of network (<ref>), aside from DNA strand-displacement mechanism, is a dimer version of the bifunctional histidine kinase/phosphatase reported in <cit.>. In SI Text, we derive equation (SI7) which describes the effective behavior of the Markov chain X(t) from network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3 in the limit μ→ 0, and it follows that the effective total propensity function of the network, denoted α(x), satisfies α(x)≈α̂(x) + 2 K_1,1β_1,1(x),as μ→ 0,α̂(x) = k_1 + k_2 x.Function α̂(x) has the form of the total propensity of network (<ref>), and K_1,1β_1,1(x) is the propensity function of reactions in (<ref>), with the scaled factors given byK_1,1= (C/2)^2 k_1,1,β_1,1(x) = (C/2)^-2 x (C-x). Function β_1,1(x) is displayed in Figure <ref>(a), where one can notice its parabolic shape, arising from the underlying conservation law X(t) + X̅(t) = C, which holds for all t ≥ 0, where X̅(t) ∈ℕ_0 is the copy-number of s̅ at time t ≥ 0. Comparing (<ref>) and (<ref>), it follows that, as μ→ 0, the mean interevent time for X(t), from network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3, is lower than that of X̂(t), from network (<ref>), in the regions of the common state-space where β_1,1(x)0, i.e. for x ∈ (0,C). Coefficient K_1,1 controls by how much the interevent time is reduced. Equivalently, β_1,1(x), and K_1,1, determine the support, and magnitude, respectively, of the state-dependent intrinsic noise which network (<ref>) introduces into the dynamics of network (<ref>). To study this further, in SI Text we derive the following two equations (given as (SI9), and (SI13), respectively)lim_K_1,1→ 0 p(x)≈1/x!( k_1/k_2)^xexp(- k_1/k_2),ifx ∈ [0,C],0,otherwise, lim_K_1,1→∞ p(x)≈1 - 1/Ck_1/k_2,ifx = 0,1/Ck_1/k_2,ifx = C, 0,otherwise,where p(x) is the stationary probability mass function (PMF) corresponding to network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3 in the limit μ→ 0, i.e. the probability that there are x molecules of species s as μ→ 0 in the long-time limit t →∞. Let us interpret analytical results (<ref>) and (<ref>), and compare them with the numerically obtained counterparts. In Figure <ref>(b), we display numerically obtained stationary x-marginal PMFs for different values of K_1,1, with the rest of the (dimensionless) parameters fixed to k_1 = 2.5, k_2 = 0.5, μ = 10^-3, and C = 15. It can be seen that, for K_1,1 = 0, i.e. when the zero-drift network ℛ_1,1^3 does not fire, the PMF matches that of network (<ref>), i.e. it is a Poissonian, as predicted by (<ref>). Let us note that the matching of the PMFs of networks (<ref>) and ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3 relies on choosing sufficiently large rate coefficients 1/μ in the drift-corrector network ℛ_1^2. When K_1,1 = 5, the PMF appears closer to a uniform distribution, than does the PMF when K_1,1 = 0. Finally, for the larger value K_1,1 = 10^5, i.e. when zero-drift network ℛ_1,1^3 fires much faster than network ℛ̂^1, the PMF redistributes across the domain, accumulating at the boundary, and becoming bimodal. This is in qualitative agreement with (<ref>), and in quantitative agreement with (<ref>), which predicts p(0) ≈ 0.7 and p(15) ≈ 0.3. In Figure <ref>(c), a representative sample path is shown, obtained by applying the Gillespie algorithm on network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3, when K_1,1 = 10^5. Also shown is a trajectory obtained by numerically solving the deterministic model (<ref>). Consistent with Figure <ref>(b), the sample path switches between the boundary of the state-space, with a bias towards the left boundary point x = 0. This is in contrast to the deterministic trajectories, which are globally attracted to the equilibrium point x = 5.§.§ General Zero-Drift Networks ℛ_n,n̅^3Zero-drift network ℛ_1,1^3(s,s̅), given by (<ref>), involves a single molecule of s and s̅ as reactants, and adds the noise at x ∈ [1,C-1], i.e. in the interior of the state-space. Similar networks may be used to add the noise at any point in the state-space, without influencing the deterministic dynamics. In particular, in (<ref>) and (<ref>), we present general zero-drift networks ℛ_n,n̅^3(s,s̅), which involve n molecules of s, and n̅ of s̅, as reactants, and add the noise at x ∈ [n,C-n̅], where n,n̅∈ℕ_0, and (n + n̅) ≤ C (see also SI Text). Embedding a union of such networks, ∪_(n,n̅)ℛ_n,n̅^3(s,s̅), into (<ref>), we arrive at the result similar to (<ref>), with K_1,1β_1,1(x) replaced by the linear combination ∑_(n, n̅) K_n,n̅β_n,n̅(x). The scaled rate coefficient K_n,n̅, and function β_n,n̅(x), are given as (S14), and (S15), respectively, inSI Text, where we also justify that an arbitrary nonnegative function, with compact support, may be approximated by a suitable sum ∑_(n, n̅) K_n,n̅β_n,n̅(x). To illustrate general zero-drift networks, let us start with embedding into network (<ref>) zero-drift network ℛ_5,10^3(s,s̅), satisfying (<ref>) with n = 5 and n̅ = 10. In Figure <ref>(d), we show propensity function β_5,10(x), which is nonzero only at x = 5. In (e), we show the numerically approximated stationary x-marginal PMFs underlying network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_5,10^3 for different values of K_5,10, with the rest of the parameters as in Figure <ref>(b). One can notice that, under the action of network ℛ_5,10^3, the PMF is gradually decreased to nearly zero at x = 5 (the deterministic equilibrium), and becomes bimodal, with the two noise-induced maxima at x = 4 and x = 6. In (f), we show a corresponding representative sample path.In general, noise-induced multimodality may be achieved by a suitable combination of zero-drift networks. For example, let us synthetize noise such that the stationary PMF is trimodal, and nearly zero everywhere, except at x ∈{1,7,11}. Such a task may always be achieved by a suitable combination of the basis zero-drift networks, i.e. those zero-networks that induce noise only at a single point in the state-space (e.g. subnetwork ℛ_5,10^3 with propensity function shown in Figure <ref>(d), see also SI Text). In the present case, one could construct the thirteen basis zero-drift networks which add large enough noise at x ∈ [0,15] ∖{1,7,11}.Here, for simplicity, we achieve the task with only four zero-drift networks. In Figures <ref>(g)–(i), we consider network ℛ̂^1 ∪ℛ_1^2 ∪ (ℛ_0,15^3 ∪ℛ_2,9^3 ∪ℛ_8,5^3 ∪ℛ_12,0^3). We denote β(x) ≡β_0,15(x) + β_2,9(x) + β_8,5(x) + β_12,0(x), and, for simplicity, take K ≡ K_0,15 = K_2,9 = K_8,5 = K_12,0. The resultant propensity function β(x) is shown in (g), while in (h) it can be seen that the PMF becomes trimodal for sufficiently large K, with the maxima at x = {1,7,11}. This is consistent with the corresponding representative sample path shown in blue in panel (i), which display tristability. Let us note that, while the stochastic dynamics display multistability in (c), (f) and (i), the corresponding deterministic dynamics, also shown in the plots, remain monostable. § A TWO-SPECIES EXOTIC SYSTEM Consider the two-species network ℛ̃(s_1,s_2), given byℛ̃(s_1,s_2): ∅s_1,∅s_2,s_12 s_1, s_2 ∅,2 s_13 s_1,s_1 + s_2s_1 + 2 s_2, s_1 + s_2s_2,2 s_23 s_2,2 s_1 + s_2s_1 + s_2, 3 s_22 s_2, s_1 + 2 s_22 s_1 + 2 s_2, where species s_1 and s_2 react according to the eleven reactions with rate coefficients k_1, k_2, …, k_11≥ 0. We denote the copy-numbers of species s_1, and s_2, at time t by X_1(t), and X_2(t), respectively. It was established in <cit.> that, for particular choices of the rate coefficients, the deterministic model of reaction network (<ref>), given as equation (SI17) in SI Text, exhibits exotic dynamics: it undergoes a homoclinic bifurcation, and displays a bistability involving a limit cycle and an equilibrium point. On the other hand, it is demonstrated in <cit.> that the stochastic model of (<ref>) is not necessarily sensitive to the deterministic bifurcation, and may effectively behave in a monostable manner. The latter point is demonstrated in Figure <ref>(c), where we show in red numerically approximated x_1-solutions of (SI17), one initiated in the region of attraction of the equilibrium point, while the other of the limit cycle. For a comparison, we also show in blue a representative sample path generated by applying the Gillespie algorithm on (<ref>). It can be seen that the stochastic solution spends significantly more time near the deterministic equilibrium point. To gain a clearer picture, we display in Figures <ref>(a), and (b), the joint, and the x_1-marginal, stationary PMFs, respectively, underlying network (<ref>), which have been obtained numerically for the same parameter values as in Figure <ref>(c). In (b), one can notice that the PMF is bimodal, but the left peak, corresponding to the limit cycle, is significantly smaller than the right peak, which corresponds to the stable equilibrium point.We now apply Algorithm <ref> on network (<ref>) to achieve two goals. Firstly, we balance the sizes of the two peaks of the stationary PMF from Figure <ref>(b), thereby forcing the stochastic system to spend comparable amounts of time at the two deterministic attractors. Secondly, we reverse the situation shown in Figure <ref>(b), by making the left PMF peak significantly larger than the right one, thereby forcing the stochastic system to spend most of the time near the limit cycle. We could achieve the goals by introducing species s̅_1, s̅_2 into (<ref>), and using suitable basis zero-drift networks. We take a simpler approach, by mapping (<ref>) to ℛ̃^1(s_1,s_2,s̅_2) ∪ℛ_1^2(s̅_2) ∪ (ℛ_0,C_2 - 10^3(s_2,s̅_2) ∪ℛ_30,0^3(s_2,s̅_2)), which is given by equation (SI18) in SI Text. For our purposes, only one of s̅_1, s̅_2 is sufficient, since the stochastic dynamics of s_1 and s_2 are coupled. We have chosen s̅_2 for convenience, since x_2-state-space may be truncated at a lower value, C_2 = 180, than x_1-state-space (see also Figure <ref> (a)). The x_2-component of the deterministic limit cycle satisfies x_2 ∈ (10,30). Correspondingly, we introduce two zero-drift networks: ℛ_0,C_2 - 10^3(s_2,s̅_2), and ℛ_30,0^3(s_2,s̅_2), which redistribute the PMF from x_2 ∈ [0,10], and from x_2 ∈ [30,C_2], respectively, to the limit cycle region, x_2 ∈ (10,30). We fix the scaled rate coefficient K_0,C_2-10^2 to a large value (so that the PMF is nearly zero for x_2 ∈ [0,10]), and vary the coefficient K_30,0^2, which redistributes the PMF from the deterministic equilibrium point to the limit cycle. Network ℛ_1^2(s̅_2) is necessary for the preservation of the deterministic dynamics of (<ref>) under the application of Algorithm <ref>. In Figures <ref>(d), and (e), we show the joint, and x_1-marginal, stationary PMFs for an intermediate value of K_30,0^2, when the PMF is partially redistributed from x_2 ∈ [30,C_2] to x_2 ∈ (10,30), so that the two peaks in (e) are of comparable sizes. In Figure <ref>(f), we show a representative sample path, obtained by applying the Gillespie algorithm on network (SI18) from SI Text, together with the deterministic trajectories obtained by solving (SI17). One can notice that the stochastic system now spends significantly more time near the limit cycle, when compared to (c). In Figures <ref>(f)–(g), we show analogous plots, but for a sufficiently large value of K_30,0^2, when the PMF is almost completely redistributed from x_2 ∈ [30,C_2] to x_2 ∈ (10,30). Now, in contrast to Figures <ref>(a)–(c), the PMF becomes essentially unimodal, and concentrated around the limit cycle. Let us note that the red trajectories from Figures <ref>(f) and (i) were generated by numerically solving the deterministic model of network (<ref>), given by (SI17). For our purposes, it is not necessary to solve the corresponding (stiff) deterministic model of network (SI18). The reason is that Algorithm <ref> does not influence the deterministic equilibrium points of a given reaction network, regardless of the choice of the kinetic algorithm parameters. For example, while the deterministic limit cycle is not necessarily preserved for the algorithm parameters chosen in Figure <ref>(i), the enclosed deterministic unstable focus is necessarily preserved. Thus, the blue sample path corresponds to noise-induced oscillations either near a deterministic limit cycle, or near a deterministic unstable focus.§ SUMMARYIn this paper, we have presented the noise-control algorithm, which is given as Algorithm <ref>. The algorithm maps an input chemical reaction network to output networks, all under mass-action kinetics, by introducing appropriate additional species and reactions, such that the output networks satisfy the following two properties.Firstly, the output networks have the same deterministic model as the input network, in appropriate limits of some of the parameters (rate coefficients) introduced by the algorithm. Secondly, controllable state-dependent noise is introduced into the stochastic model of the output networks. Thus, Algorithm <ref> may be used to control the intrinsic noise of a given reaction network under mass-action kinetics, while preserving the deterministic dynamics. Let us note that the asymptotic conditions for the algorithm parameters are necessary for preservation of the time-dependent deterministic solutions. However, the time-independent deterministic solutions (the deterministic equilibrium points), which capture important features of the deterministic dynamics, are preserved under the algorithm even if the asymptotic conditions are not satisfied.The algorithm has been applied to a test problem, taking the form of the one-species production-decay system given by (<ref>). Using analytical and numerical methods, we have shown that the additional intrinsic noise, introduced by the algorithm, may be used to favorably modify the stationary probability mass function at arbitrary points in the state-space, as demonstrated in Figure <ref>. For example, in Figure <ref>(b), the noise is added to the whole interior of the state-space, while in (e) only at a single point, in both cases resulting in noise-induce bimodality. On the other hand, in Figure <ref>(h), by adding the noise to specific points in the state-space, the network is redesigned to display noise-induced trimodality. As shown in Figures <ref>(c), (f), (i), the blue stochastic trajectories display multistability, while the red deterministic ones remain monostable. The algorithm has also been applied to a more challenging problem, taking the form of the two-species system given by (<ref>), which, for the parameters taken in this paper, at the deterministic level displays a bistability involving an equilibrium point and a limit cycle <cit.>. At the stochastic level, the system is significantly more likely to be found near the equilibrium point, as demonstrated in Figures <ref>(a)–(c). We have used the algorithm to redesign network (<ref>), so that the stochastic system spends comparable amounts of time near the two attractors, as demonstrated in Figures <ref>(d)–(f). The network was also redesigned to display noise-induced oscillations, which is shown in Figures <ref>(g)–(i).The controllable state-dependent noise is generated by Algorithm <ref> using the zero-drift networks (<ref>) and (<ref>). Any nonnegative function, defined on a bounded discrete domain, may be represented by a linear combination of propensity functions induced by an appropriate union of the zero-drift networks. Thus, choosing suitable zero-drift networks, the algorithm may control the intrinsic noise at arbitrary points in the state-space of the stochastic dynamics of reaction networks. The cost of such a precision in nose-control is a larger number of reactants in the underlying zero-drift networks. However, while the high-molecular reactions introduced by the algorithm are more expensive to synthetize, they do not limit applicability of Algorithm <ref> to synthetic biology. The reason for this is that such reactions may always be broken down into sets of up-to bi-molecular reactions, with asymptotically equivalent deterministic and stochastic dynamics <cit.>. In particular, a zero-drift network, involving reactions of order (n + n̅), may be broken down into 2 (n + n̅) - 2 reactions of up-to second-order, which may be readily mapped to DNA-based physical networks.Algorithm <ref> may constitute a qualitatively novel finding which will facilitate the progress of DNA computing <cit.>. In particular, a hybrid approach for constructing DNA-based reaction networks may be used: the deterministic model may be used to guide the construction of reaction networks, and then Algorithm <ref> may be applied to favorably reprogram the intrinsic noise in the stochastic model, while preserving the mean-field behavior. The algorithm may be of critical importance when the synthetic networks involve species at low copy-numbers, since then the stochastic effects may play a significant role <cit.>, uncontrollably contaminating the performance of the synthetic networks. In such circumstances, Algorithm <ref> may be used for controlling the stochastic effects, enriching the DNA-based synthetic systems with novel, noise-induced functionalities.§ METHODSLet us consider the mass-action reaction network ℛ given byℛ(s_1, …, s_N): ∑_i = 1^N c_i j s_i∑_i = 1^N c_i j' s_i, j ∈{1, …, M},where s_1, …, s_N are the reacting species, k_j the reaction rate coefficients, and c_i j, c_i j' the stoichiometric coefficients. Let us denote by 𝐜_j, 𝐜_j' ∈ℕ_0^N the vectors of the stoichiometric coefficients of reaction j, and Δ𝐱_j = 𝐜_j' - 𝐜_j. The deterministic model of reaction network (<ref>) is given by the following system of ordinary-differential equations (ODEs), also known as the reaction rate equations <cit.>:d𝐱/d t=∑_j = 1^M k_j𝐱^𝐜_jΔ𝐱_j,i ∈{1, …, N}.Here, 𝐱 = 𝐱(t) ∈ℝ_≥^N is the vector of species concentrations, i.e. x_i(t) is the concentration of species s_i at time t, and 𝐱^𝐜_j≡∏_l = 1^N x_l^c_l j, with the convention that 0^0 ≡ 1.The stochastic model of reaction network (<ref>) is given by the following system of difference-differential equations, also known as the chemical master equation (CME) <cit.>:∂/∂ t p(𝐱,t) =ℒ p(𝐱,t) = ∑_j (E_𝐱^-Δ𝐱_j - 1) (α_j(𝐱) p(𝐱,t) ).Here, p(𝐱,t) is the probability mass function (PMF), i.e. the probability that the vector of copy-numbers 𝐗 = 𝐗(t) ∈ℕ_0^N of species s_1, …, s_N at time t is given by 𝐱. Linear operator ℒ is called the forward operator, and step operator E_𝐱^-Δ𝐱_j is such that E_𝐱^-Δ𝐱_j p(𝐱,t) = p(𝐱 - Δ𝐱_j,t). Function α_j(𝐱) is the propensity function <cit.> of the j-th reaction from (<ref>), and is given byα_j(𝐱) =k_j 𝐱^𝐜_j= k_j ∏_l = 1^N x_l^c_l j,where x_l^c_l j denotes a falling factorial of x_l, i.e. x_l^c_l j≡ x_l (x_l - 1) … (x_l - c_l j + 1).§ ACKNOWLEDGMENTS The authors would like to thank the IsaacNewton Institute for Mathematical Sciences, Cambridge, for supportand hospitality during the programme “Stochastic Dynamical Systemsin Biology: Numerical Methods and Applications”, where work on thispaper was undertaken. The authors would also like to thank John J. Tyson (Department of Biology, Virginia Polytechnic Institute and State University, USA) for a discussion on a possible realization of network (<ref>) via a bifunctional histidine kinase/phosphatase from <cit.>. This work was supported by EPSRC grant noEP/K032208/1. This work was partially supported by a grant fromthe Simons Foundation. Konstantinos C. Zygalakis was supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1. David F. Anderson would like to acknowledge the NSF grant NSF-DMS-1318832, and Army Research Office grant W911NF-14-1-0401. Radek Erban would alsolike to thank the Royal Society for a University Research Fellowship. 10 SynthBio1 Endy, D., 2005. Foundations for Engineering Biology. Nature, 483: 449–453.SynthBio2 Andrianantoandro, E., Basu, S., Karig, D. K., Weiss, R., 2006. Synthetic biology: new engineering rules for an emerging discipline. Molecular Systems Biology, 2: 2006.0028.SynthBioMed1 Abil, Z., Xiong, X., Zhao, H., 2015. Synthetic Biology for Therapeutic Applications. Molecular Pharmaceutics, 12(2): 322–331.SynthBioMed2 Anderson, J. C., Clarke, E. J., Arkin, A. P., Voigt, C. A., 2006. Environmentally Controlled Invasion of Cancer Cells by Engineered Bacteria. Journal of Molecular Biology, 355(4): 619–627.SynthBioMed3 Benenson, Y., Gil, B., Ben-Dor, U., Adar, R., and Shapiro, E., 2004. 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Programming Chemical Kinetics: Engineering Dynamic Reaction Networks with DNA Strand Displacement. PhD Thesis, California Institute of Technology, Pasadena, California, 2015.Computing5 Soloveichik, D., Cook, M., Winfree, E., Bruck, J., 2008. Computation with Finite Stochastic Chemical Reaction Networks. Natural Computing, 7(4): 615–633.ZeroDrift1 Ohkubo, J., Shnerb, N., and Kessler, D. A., 2008. Transition Phenomena Induced by Internal Noise and Quasi-Absorbing State. Journal of the Physical Society of Japan, 77, 044002.ZeroDrift2 Biancalani, T., Dyson, L., and McKane, A. J., 2014. Noise-Induced Bistable States and Their Mean Switching Time in Foraging Colonies. Physical Review Letters, 112, 038101.ZeroDrift3 Saito, N., and Kaneko, K., 2015. Theoretical Analysis of Discreteness-Induced Transition in Autocatalytic Reaction Dynamics. Physical Review Letters, 91, 022707.Tyson Subramanian, K., Paul, M. R., Tyson, J. J., 2013. Potential Role of a Bistable Histidine Kinase Switch in the Asymmetric Division Cycle of Caulobacter crescentus. PLOS Computational Biology, 9, e1003221.UNI Wilhelm, T., 2000. Chemical systems consisting only of elementary steps - a paradigma for nonlinear behavior. Journal of Mathematical Chemistry, 27: 71–88.Me3 Plesa, T., 2017. Stochastic Approximation of High-molecular by Bi-molecular Reactions. In preparation.QSA1 Klonowski, W., 1983. Simplifying principles for chemical and enzyme reaction kinetics. Biophysical Chemistry, 18(3): 73–87.§ SUPPLEMENTARY INFORMATION (SI) TEXT§.§ The Deterministic Dynamics of Network ℛ̂^1 ∪ℛ_1^2 in the Limit μ→ 0Let us analyse system (4) in the asymptotic limit μ→ 0. It follows from the Tikhonov theorem [42] that the ODE for y, given by second equation in (4), reduces to the algebraic equation y = (c - x)^-1 as μ→ 0. Substituting the algebraic equation into (4) results in d x/d t=k_1 - k_2 x, x(0) = x_0,as μ→ 0. SI1Initial value problems (2) and (<ref>) have the same form, and let us denote their solutions by x̂(t;x̂_0) and x(t; x_0), respectively. Then, choosing c ≥max_t ≥ 0x̂(t;x̂_0) < ∞, and x_0 = x̂_0, ensures that concentration of auxiliary species s̅ is nonnegative, x̅(t) = c - x(t) ≥ 0, and that the solutions of (2) and (4) are asymptotically equivalent in the limit μ→ 0.§.§ The Stochastic Dynamics of Network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3 in the Limit μ→ 0The chemical master equation (CME) [27] induced by network ℛ̂^1 ∪ℛ_1^2 ∪ℛ_1,1^3 is given by∂/∂ t p(x,y,t) = (ℒ^1 + 1/μℒ_1^2 + K_1,1ℒ_1,1^3 ) p(x,y,t), SI2where x(t),y(t) ∈ℕ_0 are copy-numbers of species s,I^1 from (3), respectively, with ℒ^1 = k_1 (E_x^-1 - 1) ( (C-x) y ) + k_2 (E_x^+1 - 1) x,ℒ_1^2 = (E_y^-1 - 1)+ (C-x) (E_y^+1 - 1) y,ℒ_1,1^3 =(E_x^-1 + E_x^+1 - 2) β_1,1(x), SI3and K_1,1, β_1,1(x) given in (8). Operators ℒ^1, ℒ_1^2, ℒ_1,1^3 are induced by subnetworks ℛ̂^1, ℛ_1^2, ℛ_1,1^3, respectively. Let us analyse system (<ref>) in the limit μ→ 0, and consider the following power-series expansion:p(x,y,t) = p_0(x,y,t)+ μ p_1(x,y,t)+ … + μ^i p_i(x,y,t) + …, SI4with i ≥ 2. Substituting (<ref>) into (<ref>), and equating terms of equal powers in μ, the following system of equations is obtained:𝒪(1/μ): - ℒ_1^2 p_0(x,y,t)= 0,𝒪(1): - ℒ_1^2 p_1(x,y,t)= (ℒ^1 + K_1,1ℒ_1,1^3 - ∂/∂ t) p_0(x,y,t) . SI5Order 1/μ equation. A suitable form of the zero-order approximation of the PMF follows from the Bayes theorem: p_0(x,y,t) =p_0(y|x) p_0(x,t), where p_0(y|x) is the stationary PMF of y conditional on x, while p_0(x,t) is the marginal PMF of x. Substituting p_0(x,y,t) =p_0(y|x) p_0(x,t) into the first equation in (<ref>), with t, x fixed, leads to - ℒ_1^2 p_0(y|x) = 0. It follows that p_0(y|x) is a Poisson distribution with parameter (C-x)^-1, so that the zero-order PMF is given byp_0(x,y,t) = (1/y!(1/(C-x))^yexp(-1/(C-x)) ) p_0(x,t). SI6 Order 1 equation. Substituting (<ref>) into the second equation in (<ref>),summing over all the possible states y ∈ℕ_0, using (<ref>), and equalities ∑_y y p_0(y|x) = (C-x)^-1 and ∑_y p_0(y|x)= 1, one obtains the effective CME, given by∂/∂ t p_0(x,t) = (ℒ +K_1,1ℒ_1,1^3) p_0(x,t), SI7where ℒ is the forward operator corresponding to network (1), and has the following formℒ= k_1 (E_x^-1 - 1) + k_2 (E_x^+1 - 1) x. SI8 §.§.§ Limit K_1,1→ 0Setting the left-hand side (LHS) to zero, and taking K_1,1 = 0 in (<ref>), and assuming C is fixed to a sufficiently large value, it follows that the stationary PMF is a Poisson distribution with parameter k_1/k_2 [27]:p_0(x) = 1/x!( k_1/k_2)^xexp(- k_1/k_2),ifx ∈ [0,C], 0,otherwiseSI9. §.§.§ Limit K_1,1→∞Let us substitute the power-series expansionp_0(x) = f_0(x)+ 1/K_1,1 f_1(x)+ … + (1/K_1,1)^i f_i(x) + …, SI10with i ≥ 2, into (<ref>) with the LHS set to zero, and consider the limit K_1,1→∞. Then, equating terms of equal powers in 1/K_1,1, one obtains:𝒪(1 ): - ℒ^3 f_0(x)= 0,𝒪(1/K_1,1): - ℒ_1,1^3 f_1(x) = ℒ f_0(x) . SI11Order 1 equation. The solution to the first equation in (<ref>) is given byf_0(x) =1 - a/C,ifx = 0,a/C,ifx = C, 0,otherwise,SI12where a ∈ℝ_≥ is an arbitrary constant.Order 1/K_1,1 equation. Multiplying the second equation in (<ref>) by x, and summing over x ∈ℕ_0, with the convention that f_0(x) = 0 and β_1,1(x) = 0 for x ∉ [0,C], one obtains the solvability condition 0 = ∑_x = 0^∞ x ℒ f_0(x), which implies a = k_1/k_2. Substituting a into (<ref>) leads to the zero-order approximation of the stationary PMF:f_0(x) =1 - 1/Ck_1/k_2,ifx = 0,1/Ck_1/k_2,ifx = C, 0,otherwise. SI13§.§ Zero-Drift Networks ℛ_n,n̅^3The propensity function of reactions underlying ℛ_n,n̅^3(s,s̅),n,n̅∈ℕ_0, and (n + n̅) ≤ C, is given by K_n,n̅β_n,n̅ : [0,C] →ℝ_≥, withK_n,n̅ = M_n,n̅ k_n,n̅, SI14andβ_n,n̅(x) = (M_n,n̅)^-1∏_i = 0^n - 1( x - i )∏_i = 0^n̅ - 1((C - i) - x ), SI15where the scaling factor M_n,n̅ is introduced to approximately normalize β_n,n̅(x), and is given byM_n,n̅= ∏_i = 0^n - 1( n/n + n̅ C - i )∏_l = 0^n̅ - 1(n̅/n + n̅ C - i ). SI16Here, we take the convention ∏_i = 0^N f(i) = 1 if N < 0, where f(i) is an arbitrary function of i. Function β_n,n̅(x) is nonzero on the interval [n,C-n̅], with the single maximum approximately at C n/(n + n̅).Interior zero-drift networks. Zero-drift network ℛ_n,n̅^3(s,s̅), with n, n̅ 0, satisfies (19), and the propensity function of its reactions, which is proportional to (<ref>), is nonzero only in the interior of the state-space. Since the propensity function of ℛ_n,n̅^3(s,s̅), with n, n̅ 0, attains its maximum in the interior of the domain, we call the network an interior zero-drift network.Boundary zero-drift networks. Network ℛ_0,n̅^3(s,s̅), satisfying (20), is a zero-drift network in the limit μ_0,n̅→ 0. Furthermore, in the same limit, the first two reactions from (20) have the same propensity function, which is proportional to (<ref>) with n = 0, and which is nonzero at the left boundary point, x = 0. Similarly, network ℛ_n,0^3 = ℛ_0,n^3(s̅, s;B̅, k_n,0, μ_n,0) is a zero-drift network as μ_n,0→ 0, and its first two reactions have the same propensity function, which is nonzero at the right boundary point, x = C. Since networks with n = 0 (respectively, n̅ = 0) generate propensity functions with the maximum values at the left (respectively, right) boundary point, we call such networks left (respectively, right) boundary zero-drift networks.Basis zero-drift networks. Stoichiometric coefficients n, n̅ control the support of the intrinsic noise, which network ℛ_n,n̅^3 introduces into the stochastic dynamics, via the control of support of the compact function (<ref>). The larger the sum (n + n̅) is, with (n + n̅) ≤ C, the smaller the support of (<ref>), and, hence, one obtains a more precise noise-control. In the special case when n + n̅ = C, the propensity function (<ref>) is nonzero only at a single point in the state-space, x = n. We call networks ℛ_n,n̅^3(s,s̅), with n + n̅ = C, basis zero-drift networks, and the corresponding propensity functions basis propensity functions. Any nonnegative function, defined on a bounded discrete domain, may be represented by a suitable linear combination of the basis propensity functions.§.§ The Deterministic Model for Network ℛ̃The deterministic model of network (11) is given byd x_1/d t=k_1 + k_2 x_1 + k_3 x_1^2 - k_4 x_1 x_2 - k_5 x_1^2 x_2 + k_6 x_1 x_2^2,d x_2/d t=k_7- k_8 x_2 + k_9 x_1 x_2 + k_10 x_2^2 - k_11 x_2^3, SI17where x_1 = x_1(t), x_2 = x_2(t) are the concentrations of species s_1, s_2, respectively, at time t. §.§ Applying Algorithm 1 on Network ℛ̃Network ℛ̃^1(s_1,s_2,s̅_2) ∪ℛ_1^2(s̅_2) ∪ (ℛ_0,C_2 - 10^3(s_2,s̅_2) ∪ℛ_30,0^3(s_2,s̅_2)) is given byℛ̃^1(s_1,s_2,s̅_2):∅s_1, s_12 s_1, 2 s_13 s_1, s_1 + s_2s_2, 2 s_1 + s_2s_1 + s_2, s_1 + 2 s_22 s_1 + 2 s_2,s̅_2 + I_2^1 s_2 + I_2^1, s_2 s̅_2, s_1 + s_2 + s̅_2 + I_2^1 s_1 + 2 s_2 + I_2^1, 2 s_2 + s̅_2 + I_2^13 s_2 + I_2^1, 3 s_22 s_2 + s̅_2,ℛ_1^2(s̅_2):∅ I_2^1,s̅_2 + I_2^1s̅_2,ℛ_0,C_2-10^3(s_2,s̅_2): (C_2-10) s̅_2 s_2 + (C_2 - 11) s̅_2, C_2 s_2 + B_2 (C_2 - 1) s_2 + s̅_2 + B_2, (C_2-10) s̅_2 (C_2-10) s̅_2 + B_2, C_2 s_2 + B_2 C_2 s_2,ℛ_30,0^3(s_2,s̅_2): 30 s_2 29 s_2 + s̅_2, C_2 s̅_2 + B̅_2 s_2 + (C_2 - 1) s̅_2 + B̅_2, 30 s_2 30 s_2 + B̅_2, C_2 s̅_2 + B̅_2 C_2 s̅_2. SI18 | http://arxiv.org/abs/1705.09392v5 | {
"authors": [
"Tomislav Plesa",
"Konstantinos C. Zygalakis",
"David F. Anderson",
"Radek Erban"
],
"categories": [
"q-bio.MN"
],
"primary_category": "q-bio.MN",
"published": "20170525230146",
"title": "Noise Control for DNA Computing"
} |
=1JHEP Kähler | http://arxiv.org/abs/1705.09247v3 | {
"authors": [
"Renata Kallosh",
"Andrei Linde",
"Diederik Roest",
"Yusuke Yamada"
],
"categories": [
"hep-th",
"astro-ph.CO",
"gr-qc",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170525162123",
"title": "$\\overline{D3}$ Induced Geometric Inflation"
} |
A greedy approximation algorithm for the minimum (2,2)-connected dominating set problem Yash P. Aneja Odette School of Business University of Windsor Windsor, Canada Asish Mukhopadhyay School of Computer Science University of Windsor Windsor Canada Md. Zamilur Rahman School of Computer Science University of Windsor Windsor Canada============================================================================================================================================================================================================================================================================= Abstract Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless sensor network (WSN) is an effective way to save energy and reduce the impact of broadcasting storms. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. This could be modeled as a k-connected, m-fold dominating set ((k,m)-CDS). Given a virtual undirected network G=(V,E), a subset C⊂ V is a (k,m)-CDS of G if (i) G[C], the subgraph of G induced by C, is k-connected, and (ii) each node in V\ C has at least m neighbors in C. We present a two-phase greedy algorithm for computing a (2,2)-CDS that achieves an asymptotic approximation factor of (3+ln(Δ+2)), where Δ is the maximum degree of G. This result improves on the previous best known performance factor of (4+lnΔ+2ln(2+lnΔ)) for this problem.§ INTRODUCTIONSuppose G=(V,E) is a connected graph. A subset C of V is a said to be a connected dominating set (CDS) of G if G[C], the induced graph on C, is connected and every vertex v in V∖ C is a neighbor of C (connected by an edge to some vertex u∈ C). Nodes in C are called dominators, and nodes in V\ C are called dominatees. To save energy and reduce interference, it is desirable that the CDS size is as small as possible. Computing a minimum CDS is a well known NP-hard problem <cit.>. By showing that finding a minimum set cover is a special case of finding a minimum CDS, Guha and Khullar <cit.> established that a minimum CDS can not be approximated within ρln n for any 0 <ρ<1 unless NP⊂ DTIME(N^O(loglog n)). In the same paper Guha and Khullar <cit.> proposed a two-phase greedy algorithm, with an approximation factor of (3+lnΔ) for fining a minimum sized CDS. Subsequently, Ruan et. al. <cit.> used a potential function approach to come up with a single phase greedy algorithm improving the approximation ratio to (2+lnΔ). There are, in the literature, several approximation algorithms for finding a minimum CDS for a general graph <cit.>.To make a virtual backbone more robust to deal with frequent node failures in WSNs, researchers have suggested using a (k,m)-CDS. As mentioned in the abstract, C⊂ V is a (k,m)-CDS if every node in V\ C is adjacent to at least m nodes in C, and G[C], the subgraph induced by C, is k-connected. The k-connectedness means that |C|>k and G[C\ X] is connected for any X⊂ C with |X|<k. In other words, no two vertices of G[C] are separated by removal of fewer than k other vertices of C. With such a C, messages can be shared by the whole network, where every node in V\ C can tolerate up to m-1 faults (node failures) on its dominators, and the virtual backbone G[C] can tolerate up to k-1 faults.Zhou et al. <cit.>, using a more complex potential function than the one in Ruan et al. <cit.>, provide a single phase (2+ln(Δ+m-2))-approximation algorithm for the minimum (1,m)-CDS problem in a general graph.Shi et al. <cit.>, using a two-phase approach, provide a (α+2(1+lnα))-approximation algorithm for the minimum (2,m)-CDS, where m≥2 and α is the approximation ratio for the computation of a (1,m)-CDS. Using the solution obtained for the minimum (1,m)-CDS problem, they augment the connectivity of G[C] by merging blocks (a block is defined as a maximal connected subgraph without a cut-vertex) of G[C] recursively. When m=2, this approximation ratio becomes (4+lnΔ+2ln(2+lnΔ)).In this paper, we present a different two-phase approach to the (2,2)-CDS problem. The first phase ends up obtaining a C such that it is a 2-fold dominating set, and all connected components of G[C] are biconnected (2-connected). The second phase, at each iteration, needs two nodes from V\ C to reduce the number of these biconnected components by at least one. This results in an algorithm with an asymptotic approximation factor of (3+ln(Δ+2)). By a simple modification of the potential function, our approach provides a (3+ln(Δ+m))-approximation algorithm for computing a (2,m)-CDS.For related and earlier work, the reader may refer to the papers by <cit.> and <cit.>.§ MAIN RESULTS Let G=(V,E) be a biconnected graph. For a C⊂ V, define p(C) to be the number of (connected) components of G[C], the subgraph induced by C. Define G⟨ C⟩ to be the spanning subgraph of G, with vertex set V, and edge set {e∈ E:e has at least one end in C}. Let q(C) represents the number of components of G⟨ C⟩. For each node v∈ V, define m_C(v) as:m_C(v)=0,if v∈ C,or adjacent to at least 2 nodes in C1,if v∈ V∖ C, and adjacent to at most 1 node in C.Let m(C)=∑_v∈ Vm_C(v). Thus m(C) represents the number of nodes in V\ C which have at most one neighbor in C. Note that for m = 2, q(C) and m(C) are defined exactly as in <cit.> . Again, as in <cit.>, we assign a color to each node in V relative to a given C as follows. All nodes in C are colored black, nodes in V\ C which have at least two neighbors in C are colored gray, nodes in V\ C that have exactly one neighbor in C are colored red, and all other nodes are colored white. Given C, we define p̂(C) to bep̂(C) =max_x∈ Cp(C∖{x})=p(C∖{x_C})in an attempt to capture the bi-connectivity deficit of G[C].A node x_C for which the maximum in (<ref>) is attained is called a critical node of C.The component of G[C] that contains x_C is called the critical component of C. Note that if G[C] is biconnected then p̂(C)=1, in which case every node in C can be viewed as a critical node. For a given C, its potential function, f(C), is defined by:Finally, we use the functions, p̂(C), q(C) and m(C) to define a potential function, f(C), on C as:f(C)=p̂(C)+q(C)+m(C)and the difference function Δ_yf(C) byΔ_yf(C)=f(C)-f(C ∪{y}),where y∈ V. We can also, equivalently, writeΔ_yf(C)=Δ_yp̂(C)+Δ_yq(C)+Δ_ym(C) Result. Function f(C) is monotonically non-increasing. That is, Δ_yf(C)≥0 for every v in V. We need to consider three cases:Proof: Several cases arise. * Suppose y is gray. This means that p̂(C∪{y})≤p̂ (C). Clearly m(C∪{y})≤ m(C), and q(C∪{y})≤ q(C). Thus, f(C∪{y})≤ f(C), and hence Δ_yf(C)≥0. * Suppose y is red. It is then connected to only one node in C. As y is added to C, its m-value goes down by one and its q-value cannot increase. Its p̂ value may increase by 1. Thus, f(C∪{y})≤ f(C). * Suppose y is white. As y is added to C, its m-value goes down by one, q-value goes down by at least one, and p̂-value goes up by one. Hence, f(C∪{y})≤ f(C).We now give a definition and a proposition from Diestel <cit.> to exploit the structure of a biconnected graph. The following characterization of the structure of a biconnected graph <cit.> is useful for us. Given a graph H, we call a path P an H-path if P meets H exactly in its ends.[Asish, may be reove it.In particular, the edge of anyH-path of length 1 (has only one edge) is never an edge of H.]For example consider a biconnected graph that is a cycle H of three nodes: x_1,x_2, and x_3. Then a path P of three nodes x_1,x_4, and x_3 is an H-path of H. Adding this H-path to cycle H, keeps it biconnected. The following proposition formalizes this observation and is illustrated in Fig. <ref> <cit.> A graph H is biconnected if and only if it can be constructed from a cycle by successively adding H-paths to graphs H already constructed. Suppose C^∗ is a minimum (2,2)-CDS of G. Since it is biconnected, using the above proposition we can list the nodes in an order such that each sublist starting from the beginning is essentially a “path", where the first node of this “path" might correspond to a biconnected subgraph of C^∗. Let us illustrate this with the following example <ref> of G[C^∗].We can list 8 nodes of this graph as the following list with sublists: ((1,2,3,4),5,6,7),8). Node 2 is adjacent to node 1, node 3 is adjacent to only node 1. Node 4, however, is adjacent to both nodes 2 and node 3. So (1,2,3,4) corresponds to a biconnected graph (cycle), and is now designated as a “single meta-node" in our list. Next, the H-path (4,5,6,7,2) is added to this subgraph, resulting in another biconnected subgraph. Finally, adding the H-path (7,8,4) results in G[C^∗]. The next lemma exploits this interpretation of a biconnected graph as a “path". For any two subsets A,B⊆ V and any node y∈ V, if B is a“path" thenΔ_yf(A∪ B)≤Δ_yf(A)+1 Proof:The result is obvious if y∈ A. Suppose y∈ B\ A. Then the above result follows as Δ_yf(A)≥0 for all y∈ V. Thus we assume from here on that y∉ A∪ B. Define μ(f)=Δ_yf(A∪ B)-Δ_yf(A). It is useful to write μ(f) as:μ(f)=μ(p̂)+μ(q)+μ(m).We first look at μ(m). Define S to be set of nodes which are neighbors of y which are white with respect to A and red with respect to B. Let |S|=s. We first want to show that:μ(m)={[ s-1, if y is gray for A∪ B,but not gray for A; s, otherwise. ].It is easy to formalize and establish this result by looking at the following two example figures: case (i): μ(m)=s-1, case (ii): μ(m)=s.In both 3(a) and 3(b) of Figure <ref>, |S|=s=1. In Fig. 3(a), m(A)=4, m(A∪{y})=3. Hence Δ_ym(A)=1. m(A∪ B)=2, m(A∪ B∪{y})=1. Hence Δ_ym(A∪ B)=1. So μ(m)=Δ _ym(A∪ B)-Δ_ym(A)=0.In Fig. 3(b), m(A)=4, m(A∪{y})=3. Hence Δ_ym(A)=1. m(A∪ B)=3,m(A∪ B∪{y})=1, Δ_ym(A∪ B)=2. Hence μ (m)=Δ_ym(A∪ B)-Δ_ym(A)=1.We now look at μ(q). We want to show that:μ(q)≤{[ -s, if y is adjacent to B;-(s-1), otherwise. ].Let N_A(y) be the set of components of G⟨ A⟩ that are adjacent with node y in G (the component of G⟨ A⟩ containing node y, if any, is not counted). Then Δ_yq(A)=|N_A(y)|. Hence μ(q)=|N_A∪ B(y)|-|N_A(y)|. Again, it is easy to formalize and establish the above result by looking at the above two example figures, in Figure 3,covering the cases: 3(a), μ(q)≤-s, and 3(b): μ(q)≤-(s-1).In Figure 3(a), Δ_yq(A)=|N_A(y)|=3, Δ_yq(A∪ B)=N_A∪ B(y)=1. Hence μ(q)=1-3=-2.In Figure 3(b), Δ_yq(A)=N_A(y)=2, Δ_yq(A∪ B)=|N_A∪ B(y)|=1. Hence μ(q)=-1. Zhou et al. <cit.> have established the above two results in a more general setting. Now we focus on μ(p̂)=Δ_yp̂(A∪ B)-Δ_yp̂ (A).If G[A] is not biconnected but G[A∪{y}] is biconnected, then Δ_yp̂(A)=p̂(A)-p̂(A∪{y})=p̂(A)-1.Then we claim that:Δ_yp̂(A ∪ B) ≤Δ_yp̂(A), whenyis not adjacent toB; otherwise,Δ_yp̂(A ∪ B) ≤Δ_yp̂(A) + 1.Proof: By definition,Δ_yp̂(A ∪ B) = p̂(A ∪ B) - p̂(A ∪ B ∪{y})Assume that y is not adjacent to B. Now p̂(A ∪ B ∪{y}) ≥2, as 2 is the fewest number of components that can be generated by the removal of a vertex from the set A ∪ B ∪{y}. On the other hand, p̂(A ∪ B) ≤p̂(A) + 2 as p̂(B) ≤2 and the number of components of A is bounded above by p̂(A). ThusΔ_yp̂(A ∪ B) = p̂(A ∪ B) - p̂(A ∪ B ∪{y})≤p̂(A) + 2 - 2≤p̂(A) In the case that y is adjacent to B, the only change is that p̂(A ∪ B ∪{y}) ≥1. HenceΔ_yp̂(A ∪ B) = p̂(A ∪ B) - p̂(A ∪ B ∪{y})≤p̂(A) + 2 - 1≤p̂(A) + 1So assume that adding y to A does not make it biconnected.Let r be a critical node of G[A]. Let A_r be the set of nodes in the component of G[A] containing node r. [Note that if G[A] is connected then A_r=A.] The component of G[A] that contains the critical node is called the critical component.We define three constants α, β, and γ in G[A] as follows. Letα = p(A_r\{r}), the number of components in G[A_r\{r}].Call these components α-components.β = The number of components G[A\ A_r] which are adjacent to node y in G[A].γ = The number of components of G[A_r\{r}] which are are adjacent to node y in G[A]. (γ≤α).Refer to Fig. <ref> for an illustration.Result-1: Δ_yp̂(A) = min{α, β +γ}-1.Proof: Referring to the figure above, note that p̂(A)=α+β. Now let us calculate p̂(A∪{y}). Whichever of the two nodes, node r or node y, whose removal results in the higher number of components in G[A∪{y}] is the critical node . Now if we remove node r, the resulting number of components will be (α -γ)+1. If we remove node y then this number is β+1. Hencep̂(A∪{y})=max{α-γ,β}+1Hence,Δ_yp̂(A) =p̂(A)-p̂(A∪{y})=α+β-[max{α-γ,β}]+1=α+β+min{γ-α,-β}-1=min{β+γ,α}-1Returning to μ(p̂)=Δ_yp̂(A∪ B)-Δ_yp̂(A), we use result-1 to make some assertions about μ(p̂). As we mentioned earlier, we can assume that B is a set of nodes which form a “path". Since B is a “path", adding B to A does not create a new critical node in G[A∪ B]. Result-2: Suppose y is not adjacent to B, then μ (p̂)=0.Proof: Since y is not adjacent to B, adding B to A does not change β and γ values. α value may increase. Hence, min{β+γ,α} does not change, implying μ(p̂)=0. Result-3: If y is adjacent to B, then μ(p̂ )≤1.Proof: If B is not adjacent to r, then β goes up by 1, α and γ do not change. Hence μ≤1. If B is adjacent to r, then both α and γ go up by 1, but β does not change. Hence μ(p̂)=1. Hence we have the thirdinequality:μ(p̂) {[ =0, if y is not adjacent to B,; ≤1, otherwise. ].Combining the three inequalities (4), (5), and (6), proves our Lemma 2.3. Let G=(V,E) be a biconnected graph. Then, C is a 2-fold dominating set if Δ_yf(C)=0 for every y∈ V. Proof: The following claims establish the proof.Claim-1. C≠∅.Suppose C=∅. We have p̂(∅)=0, q(∅ )=|V|, m(∅)=|V|. Since G is biconnected, every node in G has degree at least 2. Pick any node y. So C={y}, and p̂({y})=0, q({y})=|V|-|N_G(y)|, m({y})=|V|. Hence Δ_yf(C)>0, a contradiction.Claim-2. |C|≥3. Its proof is straightforward.Claim-3. m(C)=0. This claim would imply that C is a 2-fold CDS.Suppose m(C)>0. This means that there is at least one node y which is red or white with respect to C. Suppose that y is a white node. This means that y is an isolated node in G⟨ C⟩, and hence accounts for one component in computing q(C). Adding y to C implies Δ _yq(C)≥1, and Δ_ym(C)=1. Since Δ_yp̂(C)≥-1, we have Δ_yf(C)≥1, a contradiction. So assume that there are no white nodes. Suppose y is red. This mean that y is adjacent to only one node in C. Since G is biconnected, y is adjacent to another node y_1∉ C. So y_1 is either red or gray. Suppose y_1 is red. Adding y to C makes y_1 gray. Hence Δ_ym(C)=2. Since Δ_yq(C)≥0, and Δ_yp̂(C)≥-1, we have Δ_yf(C)≥1, a contradiction. So assume y_1 is gray. Then Δ_y_1p̂ (C)≥0, Δ_y_1m(C)=1, since y is now gray in G[C∪{y_1}], Δ_y_1q(C)≥0, implying Δ_y_1f(C)>0, a contradiction. This proves the claim. Claim-4. Every (connected) component in G[C] is biconnected.To prove this, suppose C_1 is a component of C which is not biconnected. Hence C_1 has a critical vertex x such that p̂(C_1 )=p(C_1∖{x})=t≥2. Since G is biconnected, there exists a gray node y that is connected to two different components of G[C_1∖{x}]. Hence p̂(C_1∪{y})≤ t-1, implying Δ_yp̂(C)≥1, Δ_yq(C)≥0, and hence Δ _yf(C)>0, a contradiction.When Δ_yf(C)=0, ∀ y∈ V∖ C, we say that phase-I of the algorithm has ended.A formal description of the Phase I algorithm is given below.(<ref>). Now the question is how much effort is needed to connect these, say r, biconnected components of C.At the end of Phase I, G[C] has t biconnected components, t≥1. If t=1, there is nothing more to do. Again, since G is biconnected, if C_1 and C_2 are any two components(of course biconnected)of G[C], there must exist at least two nodes y_1 and y_2 in V∖ C such that both y_1 and y_2 are connected to both C_1 and C_2, making G[C∪{y_1,y_2}] having one less component than G[C].So, if at the end of phase-I, we have t components in G[C], we need to add at most 2t nodes to C to obtain a (2,2)-CDS. The greedy algorithm with potential function f for (2,2)-CDS is bounded by the approximation ratio (3+ln(Δ+2)), where Δ is the maximum degree of G. Proof: Assume |V|=n. Let C_G={x_1,x_2,….x_g}, in the order of nodes selected by the algorithm (phase-I). For 0≤ i≤ g, let C_i={x_1,….,x_i}. In particular, C_g is the output of the algorithm. Suppose C^∗ is a minimum (2,2)-CDS with θ=|C^∗|. Since G[C^∗] is biconnected, we can arrange the elements C^∗ as y_1,….,y_θ such that for each j≥2, C_j-1^∗={y_1,….,y_j-1} can be written as a “path", such that y_j is connected to y_j-1, and perhaps to the first node (or meta-node) of this path. If y_j is also connected to the first “node", then G[y_1,….,y_j] is biconnected, and considered as a single meta-node. Let C_0 =C_0^∗=∅. Since f(C^∗)=2, we havef(C_i-1)-2 =f(C_i-1)-f(C_i-1∪ C^∗)=∑_j=1^θΔ_y_j(C_i-1∪ C_j-1^∗)≤∑_j=1^θ(Δ_y_j(C_i-1)+1)By the pigeonhole principle, there exists a node y_j in C^∗ such thatΔ_y_jf(C_i-1)+1≥f(C_i-1)-2/θSince phase-I follows greedy strategy,Δ_x_if(C_i-1)≥Δ_yf(C_i-1)≥f(C_i-1)-2/θ-1oror f(C_i)≤ f(C_i-1)-f(C_i-1)-2/θ+1Denote a_i=f(C_i)-2. Then, we can equivalently writea_i≤ a_i-1-a_i-1/θ+1Since all a_i's are integers, we havea_i≤ a_i-1-⌈a_i-1/θ⌉ +1Now a_i>θ implies ⌈a_i-1/θ⌉≥2, which means a_i<a_i-1. So long as a_i>2+θ, phase-I continues. Now,we can write inequality <ref> as:a_i≤ a_i-1(1-1/θ)+1, whose solution, as in <cit.>, is a_i≤ a_0(1-1/θ)^i +∑_j=0^i-1(1-1/θ)^jSo after θln(a_0/θ) iteration, as in <cit.>, a_i<2θ. Since phase-I continues as long as a_i>θ, after at most θ iterations a_i≤θ since each iteration of phase-I reduces a_i^' by at least one unit. Suppose phase-I ends at this stage. At this stage f(C_i)≤θ+2. Thus C has at most θ+2 biconnected components, and needs at most 2θ+4 additional nodes in C to obtain a (2,2)-CDS, resulting in a bound ofθln(a_0/θ)+θ+2θ+4=θ[ lna_0/θ+3+4/θ]Asymptotically, 4/θ can be ignored. So the asymptotic approximation factor is 3+ln(a_0/θ)=3+ln(2n/θ).To bound 2n/θ, we proceed as follows.Taking i=1, C_0=∅. Then p̂(∅)=0, q(∅ )=n, m(∅)=n. So f(∅)=2n. f(x_1)=p̂({x_1})+q({x_1})+m({x_1}), p̂({x_1})=0, q({x_1})=n-|N_G(x_1)|-1, m({x_1})=n-1. This implies that f(x_1 )=2n-2-|N_G(x_1)|=2n-2-Δ. HenceΔ_x_1f(∅)=|N_G(x_1)|+2=Δ+2Now f(C_1) ≤ f(C_0)-f(C_0)-2/θ+1 or2n-2/θ≤Δ+2 or2n/θ≤Δ+2+2/θ So the approximation ratio asymptotically becomes 3+ln(Δ+2).§ CONCLUSION In this paper, we proposed a (3+ln(Δ+2))-approximation algorithm for the (2,2)-connected dominating set for a general graph. This algorithm can easily be generalized for the (2,m)-CDS problem, for m≥2, resulting in a (3+ln(Δ+m))-approximation algorithm. 1Diestel_2000 Reinhard Diestel. Graph theory; 2nd ed. Graduate texts in mathematicals. Springer, Heidelberg, 2000. Record from the Electronic Library of Mathematics/European Mathematical Society.Du_et_al-2012 Ding-Zhu Du, Ker-I Ko, and Xiaodong Hu. Design and Analysis of Approximation Algorithms. Springer Publishing Company, Incorporated, 2011.Garey-Johnson Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979.Guha_Khuller_1998 Sudipto Guha and Samir Khuller. Approximation algorithms for connected dominating sets. Algorithmica, 20(4):374–387, 1998.Ruan_Du_Jia_Wu_Li_Ko_2004 Lu Ruan, Hongwei Du, Xiaohua Jia, Weili Wu, Yingshu Li, and Ker-I Ko. A greedy approximation for minimum connected dominating sets. Theor. Comput. Sci., 329(1-3):325–330, 2004.Shi_Zhang_Zhang_Wu_2016 Yishuo Shi, Yaping Zhang, Zhao Zhang, and Weili Wu. A greedy algorithm for the minimum 2-connected m-fold dominating set problem. J. Comb. Optim., 31(1):136–151, 2016.Zhou_Zhang_Wu_Xing_2014 Jiao Zhou, Zhao Zhang, Weili Wu, and Kai Xing. A greedy algorithm for the fault-tolerant connected dominating set in a general graph. J. Comb. Optim., 28(1):310–319, 2014. | http://arxiv.org/abs/1705.09643v1 | {
"authors": [
"Yash P. Aneja",
"Asish Mukhopadhyay",
"Md. Zamilur Rahman"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170526163345",
"title": "A greedy approximation algorithm for the minimum (2,2)-connected dominating set problem"
} |
C.C.]Carolyn Chun United States Naval Academy, Annapolis, MD, 21402 USA. [email protected].]Rhiannon Hall Department of Mathematics, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom. [email protected] C.M.]Criel Merino Instituto de Matemáticas,Universidad nacional Autónoma de México,Ciudad de México, 04510 México. Investigación realizada gracias al Programa UNAM-DGAPA-PAPIIT IN102315 [email protected]. Moffatt]Iain Moffatt Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom. [email protected].]Steven Noble Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London, WC1E 7HX, United Kingdom. [email protected][2010]05B35 The width of a delta-matroid is the difference in size between amaximal and minimal feasible set.We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. Weapply this theorem to give an excluded minor characterisation of delta-matroids that admit a twist of width at most one.The structure of delta-matroids with width one twists [ December 30, 2023 =====================================================§ INTRODUCTION, RESULTS AND NOTATION Delta-matroids are a generalisation of matroids introduced by A. Bouchet in <cit.>. They can be thought of as generalising topological graph theory in the same way that matroids can be thought of as generalising graph theory (see, e.g.,<cit.>). Roughly speaking, delta-matroids arise by dropping the requirement that bases are of the same size in the standard definition of a matroid in terms of its bases. (Formal definitions are provided below.) In the context of delta-matroids these generalised “bases” are called “feasible sets”. A basic parameter of a delta-matroid is its “width”, which is the difference between the sizes of a largest and a smallest of its feasible sets. One of the most fundamental operations in delta-matroid theory is the “twist”. In this paper we examine how the structure of a delta-matroid determines the width of the delta-matroids that are in its equivalence class under twists. Formally, a delta-matroid D=(E,) consists of a finite set E and a non-empty set ℱ of subsets of E thatsatisfies the Symmetric Exchange Axiom: for all X,Y∈ℱ, if there is an element u∈ X Y, then there is an element v∈ X Y such that X{u,v}∈ℱ.Here X Y denotes the symmetric difference of sets X and Y. Note that it may be the case that u=v in the Symmetric Exchange Axiom. Elements of ℱ are called feasible sets and E is the ground set. We often use ℱ(D) and E(D) to denote the set of feasible sets and the ground set, respectively, of D. A matroid is a delta-matroid whose feasible sets are all of the same size. In this case the feasible sets are calledbases. This definition of a matroid is a straightforward reformulation of the standard one in terms of bases. In general a delta-matroid has feasible sets of different sizes. The width of a delta-matroid, denoted w(D), is the difference between the sizes of its largest and smallest feasible sets: w(D):=max_F∈ |F| - min_F∈ |F|. Twists, introduced by Bouchet in <cit.>, are one of the fundamental operations of delta-matroid theory. Given a delta-matroid D=(E,ℱ) and some subset A⊆ E, thetwist of D with respect to A, denoted by D* A, is the delta-matroid given by (E,{A F :F∈ℱ}). (At times we write D∗ e for D∗{e}.) Note that the “empty twist” is D*∅ = D. The dual of D, written D^*, is equal to D*E. Moreover, in general, the twist can be thought of as a “partial dual” operation on delta-matroids.Forming the twist of a delta-matroid usually changes the sizes of its feasible sets and its width. Here we are interested in the problem of recognising when a delta-matroid has a twist of small width. Our results are a Rough Structure Theorem for delta-matroids that have a twist of width one, and an excluded minor characterisation of delta-matroids that have a twist of width at most one. To state the Rough Structure Theorem we need the following. Let D=(E,ℱ) be a delta-matroid and let ℱ_min be the set of feasible sets of minimum size. ThenD_min:=(E,ℱ_min) is a matroid. Fora matroid M with ground set E, a subset A of E is said to be a separator of M if A is a union of components of M. Note that both ∅ and E are always separators. In terms of the matroid rank function, where the rank r(X) of a set X⊆ E is defined to be the size of the largest intersection of X with a basis of M, the set A is a separator if and only if r(A) + r(E-A) = r(M). Throughout the paper we usefor the complement E- A of A, and D|X denotes the restriction of D to X⊆ E (see the beginning of Section <ref> for its definition).We now state the first of our two main results: a Rough Structure Theorem for delta-matroids admitting a twist of width one. Let D=(E,) be a delta-matroid. Then D has a twist of width one if and only if there is some A⊆ E such that* A is a separator of D_min,* D|A is a matroid, and* D| is of width one. Weactually prove a result that is stronger than Theorem <ref>. This stronger result appears below as Theorem <ref> andthe present theorem follows immediately from it. As an application of Theorem <ref>, we find an excluded minor characterisation of the class of delta-matroids that have a twist of width one as our second main result, Theorem <ref>.This class of delta-matroids is shown to be minor closed in Proposition <ref>, and its set of excluded minors comprises the delta-matroids in the following definition together with their twists.Let D_1 denote the delta-matroid on the elements a,b with feasible sets(D_1)={∅, {a}, {b}, {a,b}}.For i=2,…, 5 letD_i denote the delta-matroid on the elements a,b,c with feasible sets given by(D_2) ={∅, {a}, {b}, {c},{a,b,c}},(D_3) ={∅, {a,b}, {b,c}, {a,c}},(D_4) ={∅, {a,b}, {b,c}, {a,c},{a,b,c}},(D_5) ={∅, {a}, {a,b}, {b,c}, {a,c}}.Throughout this paper D_1, …,D_5refer exclusively to these delta-matroids. Let 𝒟_[5] be the set of all twists of these delta-matroids.Note that D_i∈𝒟_[5] for all i∈{1,2,… ,5} via the empty twist.A delta-matroid has a twist of width at most one if and only if it has no minor isomorphic to a member of 𝒟_[5]. The proof of this theorem appears at the end of Section <ref>.We note that the excluded minors of twists of matroids (i.e., twists of width zero delta-matroids) has been shown, but not explicitly stated, to be( {a}, {∅ , {a}} ), D_3, and D_3∗{a} by A. Duchamp in <cit.>. This result can be recovered from Theorem <ref> by restricting to even delta-matroids, where an even delta-matroid is a delta-matroid in which the difference in size between any two feasible sets is even.Above wementioned the close connection between delta-matroids and graphs in surfaces. The width of a delta-matroid can be viewed as the analogue of the genus (or more precisely the Euler genus) of an embedded graph, while twisting is the analogue of S. Chmutov's partial duality of <cit.>. Thus characterising twists of width one is the analogue of characterising partial duals of graphs in the real projective plane. The topological graph theoretical analogues of Theorems <ref> and <ref> can be found in <cit.>. § THE PROOF OF THE ROUGH STRUCTURE THEOREM For the convenience of the reader, we recall some standard matroid and delta-matroid terminology. Given a delta-matroid D=(E,ℱ) and elemente∈ E, if e is in every feasible set of D then we say that e is a coloop of D. If e is in no feasible set of D, then we say that e is a loop of D. If e∈ E is not a coloop, then D delete e, denoted by D e, is the delta-matroid (E-e, {F : F∈ℱ andF⊆ E-e}). If e∈ E is not a loop, then D contract e, denoted by D/e, is the delta-matroid (E-e, {F-e : F∈ℱ ande∈ F}). If e∈ E is a loop or coloop, then D/e=D e. Useful identities that we use frequently are D/e = (D∗e)e and D e = (D∗e) / e. If D' is a delta-matroid obtained from D by a sequence of deletions and contractions, then D' is independent of the order of thedeletions and contractions used in its construction, so we can define D X/Y for disjoint subsets X and Y of E, as the result of deleting each element in X and contracting each element in Y in some order. A minor of D is any delta-matroid that is obtained from it by deleting or contracting some of its elements. The restriction of D to a subset A of E, written D|A, is equal to D. Note that if ∅∈(D) then F is feasible in D|A if and only if F⊆ A and F∈(D).The connectivity function λ_M of a matroid M on ground set E with rank function ris defined on all subsets A of E byλ_M(A) = r(A) + r() - r(E). Recall thatA is said to be a separator of M if A is a union of components of M. This happens if and only if λ_M(A)=0. Moreover, A is a separator if and only ifis a separator.We will use Bouchet's analogue of the rank function for delta-matroids from <cit.>. For a delta-matroid D=(E,ℱ), it is denoted by ρ_D or simply ρ when D is clear from the context. Its value on a subset A of E is given byρ(A):=|E|-min{|A△ F| : F∈ℱ}.The following theorem determines the width of a twist of a delta-matroid.Let D=(E,) be a delta-matroid and A⊆ E. Then the width, w(D*A), of the twist of D by A is given byw(D*A)=w(D|A)+w(D|) +2 λ _D_min(A). The largest feasible set in D*A has size max{|F A|:F∈ (D)}. Take F'∈ such that |F' A| is maximal. Then |F' | is minimal. As ρ ()=|E|-min{|F |:F∈}, we see that ρ ( )=|E|-|F' |=|F' A|. Hence the largest feasible set in D*A has size equal to ρ ( ). Next, the size of the smallest feasible set in D*A is |E| minus the size of the largest feasible set in (D*A)^*=D*. By an application of the above, it follows that the size ofthe smallest feasible set in D*A is |E|-ρ (A). Hence w(D*A)=ρ ( )-|E|+ρ (A).We let r and n be the rank and nullity functions, respectively, of D_min. From <cit.>, we know that w(D|A)=ρ (A)-r(A)-n(E)+n(A). As n(A)=|A|-r(A) and n(E)=|E|-r(E), w(D|A) +w(D| ) =ρ (A)-r(A)-|E|+r(E)+|A|-r(A) + ρ ( )-r( )-|E|+r(E)+| |-r( )= ρ ( )-|E|+ρ (A) -2(r(A)+r( )-r(E))=w(D*A)-2 (λ _D_min(A)),giving the result.The following two theorems are immediate consequences of Theorem <ref>. The Rough Structure Theorem, Theorem <ref>, follows immediately from the second of them. Let D=(E,) be a delta-matroid, A⊆ E, and =E-A. Then D*A is a matroid if and only ifA is a separator of D_min,and both D|A and D| are matroids.Let D=(E,) be a delta-matroid, A⊆ E, and =E-A. Then D*A has width one if and only ifA is a separator of D_min, and one of D|A and D| is a matroid and the other has width one. For convenience, we write down the following straightforward corollary. It provides the form of the Rough Structure Theorem that we use to find excluded minors in the next section.LetD=(E,) be a delta-matroid in which ∅ is feasible. Then the following hold. * D has a twist of width zeroif and only if there exists A⊆ E such that D|A and D| are both of width zero.* D has a twist of width one if and only if there exists A⊆ E such that D|A is a matroid, andD| is of width one.This is a straightforward consequence of the fact that if ∅ is feasible in D, then D_min is the matroid on E(D) where each element is a loop, thus every set A⊆ E is a separator of D_min.§ THE PROOF OF THE EXCLUDED MINOR CHARACTERISATIONWe begin this section by verifying that the class of delta-matroids in question is indeed minor-closed. For each k∈ℕ_0, the set of delta-matroids with a twist of width at most k is minor-closed. Let D=(E,ℱ) and suppose w(D∗ A) ≤ k for some A⊆ E. If E is empty the result is trivial, so assume not and let e∈ E.If e∉ A then (D e)∗ A = (D∗ A) e, and (D/e)*A=((D*e) e)*A= ((D*e)*A) e = ((D*A)∗e)e = (D*A) / e. Similarly, if e∈ A then e∉ A-e, so using and extending the previous argument, (D/e)∗ (A-e)= (D*(A-e))/e = ((D*A)* e)/e = (D*A) e, and(D e) ∗ (A-e) = (D∗ (A-e))e=((D∗ A)∗ e) e = (D∗ A)/e.In each case we see thatD/eand D e have a twist that can be written as (D∗ A)/e or(D∗ A)e.Since deletion and contraction never increase width it follows thatD/eand D e have twists of width at most w(D∗ A) ≤ k. The result follows. LetD=(E,) be a delta-matroidand A⊆ E. Then { H : His a minor ofD∗ A }={ J∗ (A∩ E(J)) : Jis a minor ofD }.In the proof of Proposition <ref> it was shown thatif e∉ Athen (D*A) / e=(D/e)*A and (D∗ A) e = (D e)∗ A, whereas if e∈ A then(D*A) e = (D/e)∗ (A-e) and(D∗ A)/e = (D e) ∗ (A-e). The result follows immediately from this.LetD be a delta-matroid in which the empty setis feasible. Then D has a twist of width at most 1, or contains a minor isomorphic to one ofD_1, …,D_5.For any delta-matroidD in which the empty set is feasible, setL:= {x∈ E(D) : {x}∈(D)}and=E(D)-L.(Technically we should record the fact that L depends upon D in the notation, however we avoid doing this for notational simplicity. This should cause no confusion.) Note that L may be empty.Construct a (simple) graph G_D as follows.Take one vertex v_x for each element x∈, and add one other vertex v_L.The edges ofG_D arise from certain two-element feasible sets of D.Add an edge v_xv_y to G_D for each pair x,y∈ with{x,y}∈(D); add an edge v_xv_L to G_D if {x,z}∈(D) for some z∈ L. We consider two cases: when G_D is bipartite, and when it is not. We will show that if G_D is bipartite thenD must have a twist of width at most one or a minor isomorphic to D_1 or D_2; if G_D is not bipartite then it must have a minor isomorphic to D_1, D_3, D_4, or D_5.Case 1. LetD be a delta-matroid in which the empty set is feasible, and such that G_D is bipartite. Fix a 2-colouring of G_D. LetA be the set of elements in E(D) that correspond to the vertices in the colour class containing v_L together with the elements in L, and let⊆ E(D) be the set of elementscorresponding to the vertices in the colour class not containing v_L. We start by showingD|≅ U_0, ||,where U_0, || denotes the uniform matroid with rank zero and || elements.To see why (<ref>) holds, note that (D|)={F : F⊆ andF∈(D)}. Since the elements incorrespond to vertices in , no feasible sets of D| have size one. Furthermore,(D|) cannot contain any sets of size two since, by the construction of G_D, whenever {x,y}∈(D) the corresponding vertices v_x and v_y are in different colour classes.Since ∅∈(D|), the Symmetric Exchange Axiom ensures that there are no other feasible sets.(If F∈(D|) with F≠∅, take x∈∅ F. Then by the Symmetric Exchange Axiom ∅{x, y} must be in (D|) for some y, but there are no feasible sets of size one or two.) This completes the justification of (<ref>). Next we examine the feasible sets in D|A. Trivially ∅∈(D|A). The set of feasible sets of D|A of size one is{ F∈(D|A) :|F|= 1} ={ F∈(D) :|F|= 1}= {{x} : x∈ L }. If (D|A) contains a set {x,y} of size two then x,y∈ L as otherwise there would be an edge v_xv_y in G_D whose ends are in the same colour class.It follows in this case that D|A and hence D contains a minor isomorphic to D_1.Now assume that (D|A) does not contain a set of size two.If (D|A) has no sets of size one then, arguing via the Symmetric Exchange Axiom as in the justification of (<ref>), wehave D|A ≅ U_0,|A|. Taken together with (<ref>), this implies that A satisfies the conditions of the first part ofCorollary <ref>, so D has a twist of width zero.Suppose that (D|A) does contain a set of size one.If it contains no sets of size greater than one thenD|A is of width one, and by combining this with (<ref>), itfollows from Corollary <ref> that D has a twist of width one(D∗ A and D∗ are such twists). On the other hand,if(D|A) does contain a set of size greater than one, then, as it does not containa set of size two, the Symmetric Exchange Axiom guarantees there is a setin(D|A) of size exactly three. (If not, let F be a minimum sized feasible set with |F|>3.Then F {x,y} is feasible and of size at least two for some x,y∈∅ Fcontradicting the minimality of |F|>3.) Let {x,y,z}∈(D|A). Then after possibly relabelling its elements, the collection of feasible sets of D|{x,y,z} is one of {∅ , {x}, {y}, {z}, {x,y,z}}, {∅ , {x}, {y},{x,y,z}}, {∅ , {x} , {x,y,z}}.Only the first of the three cases is possible asthe Symmetric Exchange Axiom fails for the other two showing that neither is the collection of feasible sets of a delta-matroid. Hence, restricting D to {x,y,z} results in a minor isomorphic to D_2. Thus we have shown that if G_D is bipartite then D has a twist of width at most one or contains a minor isomorphic to D_1 or D_2. This completes the proof of Case 1.Case 2. LetD be a delta-matroid in which the empty set is feasible, and such that G_D is non-bipartite. We will show that D contains a minor isomorphic to one of D_1, D_3, D_4 or D_5 by induction on the length of a shortest odd cycle in G_D. For the base of the induction suppose that G_D has an odd cycle C of length three. There are two sub-cases, when v_L is not in C and when it is. Note that the former sub-case includes the situation where L=∅.Sub-case 2.1. Suppose that v_L is not in C. Let x,y,z∈ E(D) be the elements corresponding to the three verticesof C. We have x,y,z∈, so {x}, {y}, {z}∉(D).From the three edges of C we have {x,y}, {y,z}, {z,x}∈(D). It follows that D|{x,y,z} is isomorphic to either D_3 or D_4 giving the required minor.Sub-case 2.2. Suppose that v_L is in C. Let v_x, v_y, v_L be the vertices in C. The edges of C give that {x,y}∈(D),and since x,y∈ we have {x}, {y}∉(D). We also know that there are elements , ∈ L such that {}, {}, {x,},{y,}∈(D), where possibly =.If = then D|{x,y,} must have feasible sets {∅ , {} , {x,},{y,} ,{x,y}}or{∅ , {} , {x,},{y,} ,{x,y}, {, x,y}}.The first case gives a minor of D isomorphic to D_5; in the second case, (D|{x,y,}) / is aminor of D isomorphic to D_1.If ≠ then the feasible setsof D|{x,y,,} of size zero or one are exactly∅, {}, and {}.From G_D, the feasible sets of size two include{x,},{y,} ,{x,y}.If { y,} is also feasible then D|{x,y,} is isomorphic to one of the delta-matroids arising from (<ref>), so D has a minor isomorphic to D_1 or D_5. The case when { x,} is feasible is similar. If {,} is feasible thenD|{,} is isomorphic to D_1.The case that remains is when the feasible sets of D|{x,y,,} of size at most two are exactly∅ , {}, {}, {x,},{y,} ,{x,y}.By applying the Symmetric Exchange Axiom to each of the pairs of feasible sets ({},{x,y}),({},{x,y}), ({},{x,}) and ({},{y,}), one can show that each of the three element sets,{,x,y}, {,x,y}, {,,x},{,,y},is feasible in D|{x,y,,}. Finally, {,, x,y} may or may not be feasible. If {,, x,y} is feasible then (D|{x,y,,})/{x,y} is isomorphic to D_1; if {,, x,y} is not feasible then (D|{x,y,,})/{} is isomorphic to D_5.This completes the base of the induction.For the inductive hypothesis, we assume that, for some n>3, if D is a delta-matroid such that ∅∈ℱ(D) and G_D has an odd cycle of length less than n, then D has a minor isomorphic to D_1,D_3,D_4, or D_5.Suppose that ∅∈ℱ(D) and a shortest odd cycle C of G_D has length n. Again there are two sub-cases: when v_L is not in C and when it is.Sub-case 2.3. Suppose that v_L is not in C. Let C=v_x_1v_x_2… v_x_nv_x_1. Since each x_i∈ and C is the shortest odd cycle in G_D, ∅ , { x_1,x_2 }, { x_2,x_3 }, …,{ x_n,x_1 }is a complete list of the feasible sets of size at most two in D|{x_1,…, x_n}. Next, we show{x_i,x_j,x_k}∉(D|{x_1,…, x_n}),for any distinct1≤ i,j,k≤ n.To see why (<ref>) holds, first note that, since n>3, every set of three distinct vertices in the cycle includes a non-adjacent pair.If {x_i,x_j,x_k} were feasible in D|{x_1,…, x_n}, then, without loss of generality, {x_j,x_k}∉ℱ(D|{x_1,…, x_n}).As x_i∈{x_i,x_j,x_k}∅, an application of the Symmetric Exchange Axiom would imply that {x_i,x_j,x_k}{x_i,z} is feasible for some z∈{x_i,x_j,x_k}.Thus {x_j,x_k},{x_j}, or {x_k} would be feasible, a contradiction to (<ref>).Thus (<ref>) holds. Next we show that, taking indices modulo n,{x_i,x_i+1,x_j,x_j+1}∈(D|{x_1,…, x_n}),for any i and j such that 1 ≤ i,j ≤ n and i,i+1,j,j+1 are pairwise distinct.For this, first suppose that neither x_i+1 and x_j nor x_j+1 and x_i are adjacent in C.Then by (<ref>), { x_i,x_i+1} and{ x_j,x_j+1} are feasible.As x_j is in their symmetric difference, by the Symmetric Exchange Axiom, {x_i,x_i+1}{x_j,y} is feasible for some y∈{x_i,x_i+1,x_j,x_j+1}.Thus {x_i,x_i+1,x_j}, {x_i,x_j}, {x_i+1,x_j} or {x_i,x_i+1,x_j,x_j+1} is feasible.By (<ref>) and (<ref>), {x_i,x_i+1,x_j,x_j+1} is feasible. If x_i+1 and x_j are adjacent then the Symmetric Exchange Axiom implies that {x_i,x_i+1}{x_i+3,z} is feasible for some z∈{x_i,x_i+1,x_i+2,x_i+3}.Again, (<ref>) and (<ref>) imply that {x_i,x_i+1,x_i+2,x_i+3} must be feasible. The other case is identical. This completes the justification of (<ref>).Combining (<ref>)–(<ref>) gives that all of∅ , { x_1,x_2 }, { x_2,x_3 }, …,{ x_n-2,x_1 },but none of {x_1}, …, {x_n-2}, arefeasible in (D|{x_1,…, x_n})/{ x_n-1,x_n}. Hence the graph G_(D|{x_1,…, x_n})/{ x_n-1,x_n}has a shorter odd cycle than G_D. By the inductive hypothesis, (D|{x_1,…, x_n})/{ x_n-1,x_n} and hence D has a minor isomorphic to one of D_1, D_3, D_4 or D_5. Sub-case 2.4. Suppose that v_L is in C. Let C=v_Lv_x_2v_x_3… v_x_nv_L. The edges of the cycle give that, for each 2≤ i≤ n-1,{x_i,x_i+1}∈(D).Also, for 2≤ i≤ n, since x_i∈ we have {x_i}∉(D).We also know that there are elements , ∈ L such that {}, {}, {, x_2},{,x_n}∈(D) where possibly =. (This possibility is covered in the following analysis.)When ≠, if {,}∈ℱ(D), then D|{,} is isomorphic to D_1,therefore we assume {,}∉ℱ(D). Using that C is a shortest odd cycle, the feasible sets of D|{, , x_2,…, x_n} of size at most two are exactly∅ , {} , {}, {,x_2},{ x_2,x_3 }, { x_3,x_4 }, …,{ x_n-1,x_n }, {, x_n}.An argument similar to the justification of (<ref>) gives that {x_i,x_j,x_k}∉(D|{,, x_2,…, x_n}),for any distinct2≤ i,j,k≤ n.However {, x_n-1, x_n},{, x_n-1, x_n}∈(D|{,,x_2,…, x_n}).To see this note that x_n-1∈{}{ x_n-1,x_n }, so the Symmetric Exchange Axiom gives that one of {,x_n-1}, {x_n-1}, or {,x_n-1, x_n } is feasible, andwe know from (<ref>) that the feasible set must be the third option.That {, x_n-1, x_n} is feasible follows from a similar argument.We next show that for each 2≤ i < n-2,{, x_2 , x_n-1, x_n}, {x_i, x_i+1 , x_n-1, x_n} , {, x_n-2 , x_n-1, x_n}∈(D|{,,x_2,…, x_n}).For this, first consider x_2 ∈{ x_n-1, x_n}{, x_2}. The Symmetric Exchange Axiom implies that {x_n-1,x_n}{x_2,z} is feasible for some z∈{,x_2,x_n-1,x_n}.By (<ref>) and (<ref>), z=, thus {,x_2,x_n-1,x_n} is feasible.Next, to show that {x_i,x_i+1,x_n-1,x_n} is feasible, we take x_i∈{x_n-1,x_n}{x_i,x_i+1} and apply the Symmetric Exchange Axiom as above to see that {x_n-1,x_n}{x_i,z} is feasible, where z must equal x_i+1.Lastly, to show that {,x_n-2,x_n-1,x_n} is feasible, we first show that {,x_n-2,x_n}∉(D|{,,x_2,…, x_n}). If {,x_n-2,x_n} were feasible, then since x_n-2∈∅{,x_n-2,x_n}, the Symmetric Exchange Axiom would give {x_n-2}, {,x_n-2} or {x_n-2,x_n} as feasible, a contradiction. Now showing that {,x_n-2,x_n-1,x_n} is feasible comes from taking x_n-2∈{,x_n}{x_n-2,x_n-1}. The Symmetric Exchange Axiom gives that {,x_n}{x_n-2,z} is feasible for some z∈{,x_n-2,x_n-1,x_n}, of which z=x_n-1 is the only possibility.From (<ref>)–(<ref>) it follows that all of ∅ , {}, {}, {,x_2}, { x_2,x_3 }, …, {, x_n-2}, but none of {x_2}, …, {x_n-2},arefeasible in (D|{, x_2,…, x_n,})/{ x_n-1,x_n}. Hence the graph G_(D|{, x_2,…, x_n,})/{ x_n-1,x_n}has a shorter odd cycle than G_D. The inductive hypothesis gives that (D|{, x_2,…, x_n,})/{ x_n-1,x_n} and hence D has a minor isomorphic to one of D_1, D_3, D_4 or D_5.This completes the proof of the sub-case, and the lemma.We now apply Lemma <ref> to prove our excluded minor characterisation of the family of delta-matroids admitting a twist of width at most one. All twists of the delta-matroids D_1, …,D_5 are of width at least two. Since the set of delta-matroids with a twist of width at most one is minor-closed it follows thatno minor of a delta-matroid with a twist of width at most one is isomorphic toa member of 𝒟_[5]. This proves one direction of the theorem.Conversely suppose that every twist of a delta-matroid D=(E,) is of width at least two. Let A∈. Then D*A is a delta-matroid in which ∅ is feasible and in which every twist is of width at least two. By Lemma <ref>, D*A has a minor isomorphic to one of D_1, …,D_5. It follows from Lemma <ref> that D has a minor isomorphic to a member of 𝒟_[5].plain | http://arxiv.org/abs/1705.09129v1 | {
"authors": [
"Carolyn Chun",
"Rhiannon Hall",
"Criel Merino",
"Iain Moffatt",
"Steven Noble"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170525113404",
"title": "The structure of delta-matroids with width one twists"
} |
Alibaba [email protected] [email protected] [email protected] [email protected] [email protected] In recent years, RTB(Real Time Bidding) becomes a popular online advertisement trading method. During the auction, each DSP(Demand Side Platform) is supposed to evaluate current opportunity and respond with an ad and corresponding bid price. It's essential for DSP to find an optimal ad selection and bid price determination strategy which maximizes revenue or performance under budget and ROI(Return On Investment) constraints in P4P(Pay For Performance) or P4U(Pay For Usage) mode. We solve this problem by 1) formalizing the DSP problem as a constrained optimization problem, 2) proposing the augmented MMKP(Multi-choice Multi-dimensional Knapsack Problem) with general solution, 3) and demonstrating the DSP problem is a special case of the augmented MMKP and deriving specialized strategy. Our strategy is verified through simulation and outperforms state-of-the-art strategies in real application. To the best of our knowledge, our solution is the first dual based DSP bidding framework that is derived from strict second price auction assumption and generally applicable to the multiple ads scenario with various objectives and constraints. Dual Based DSP Bidding Strategy and its Application Hao Wang Received: date / Accepted: date ===================================================§ INTRODUCTIONIn recent years, RTB(Real Time Bidding) becomes a popular online advertisement trading method. There are three major roles in the market, namely SSP(Supply Side Platform), DSP(Demand Side Platform), and AdX(Ad Exchange). SSP controls huge amount of websites and earns money by supplying impressions. DSP holds a lot of advertisers and makes profit through fulfilling their demands. AdX, an online advertisement exchange, docks SSPs and DSPs and holds auctions.In a typical scenario, an audience visits one of the SSP's websites, then the AdX is informed and an auction is initiated. The AdX broadcasts bid request to DSPs and waits for a short time(e.g. 100ms). Each DSP is supposed to evaluate current opportunity and respond with an ad and corresponding bid price. The AdX gathers bid responses arriving before deadline and determines the winner and its bidding cost. Finally, the AdX notifies the SSP about the auction result and the SSP serves the winner's ad to the audience.There are two popular payment modes for advertisers * P4P(Pay For Performance): the advertiser sets a CPP(Cost Per Performance) and pays DSP the CPP times the units of performance delivered by DSP(e.g. 1$/click*10clicks=10$).* P4U(Pay For Usage): the advertiser sets a CR(Commission Rate) and pays DSP the total bidding cost plus the fraction of it as commission(e.g. (1+10%)*100$=110$). DSP is interested in optimizing one of the following objectives * Revenue: the total amount of money(e.g. 50$) earned from advertisers through either payment mode mentioned above.* Performance: the total units of performance(e.g. 20 clicks) delivered to advertisers. During the optimization, several constraints must be satisfied * Budget Upper Bound: the maximum amount of money the advertiser is willing to spend in DSP for a certain period of time (e.g. 100$/day).* ROI Lower Bound: the minimum value of ROI(Return On Investment) which is defined as, for DSP, the revenue earned from advertisers over the bidding cost payed to AdX (e.g. DSP ROI is 1.1 when DSP earns 110$ and pays 100$) and, for advertiser, the performance delivered by DSP over the money spent in DSP (e.g. advertiser ROI is 0.8 when advertiser spends 100$ for 80 clicks). It's essential for DSP to find an optimal ad selection and bid price determination strategy which maximizes revenue or performance under budget and ROI constraints in P4P or P4U mode. We solve this problem by* formalizing the DSP problem as a constrained optimization problem(Section <ref>),* proposing the augmented MMKP(Multi-choice Multi-dim-ensional Knapsack Problem) with general solution(Section <ref>),* and demonstrating the DSP problem is a special case of the augmented MMKP and deriving specialized strategy(Section <ref>). Our strategy is verified through simulation(Section <ref>) and outperforms state-of-the-art strategies in real application(Section <ref>). To the best of our knowledge, our solution is the first dual based DSP bidding framework that is derived from strict second price auction assumption and generally applicable to the multiple ads scenario with various objectives and constraints. These are the main contributions of this document.Before further discussion, it's worth to mention several points about our problem configuration. First, PPI(Performance Per Impression) is defined as the expected performance of one impression with certain ad and its accurate prediction is of great importance in performance estimation. However, PPI prediction is beyond the scope of this document and we assume that the PPI is always explicitly provided in the rest of our discussion. Second, it is assumed that all advertisers agree to the same payment mode and performance metric and DSP prefers to optimize a pure objective rather than a hybrid one. Third, the CPP in P4P mode or CR in P4U mode are set on the ad level, i.e. the advertiser is able to set different CPP or CR for his ads. And the constraints are set on ad group level, e.g. the budget might be shared by ads of the same advertiser and DSP might be interested in controlling its global ROI. At last, the ROI lower bound for advertiser could also be interpreted as the CPP upper bound which might be more familiar to some readers.§ RELATED WORKS <cit.> suggests a linear bidding strategy which, given base price, bids in proportion to the relative quality of impression. However, their method is a heuristic one and lacks theoretical foundations.Based on calculus of variations, <cit.> suggests a non-linear relationship between optimal bid price and KPIs. However, their strategy is derived from first price auction assumption which doesn't hold in RTB. Besides, winning rate is explicitly modeled as a function of bid price in <cit.>. To find the analytical solution of the optimal bid price, the winning rate function must be of specific forms, which makes their method inflexible.Both win rate and winning price are estimated in <cit.>, and the corresponding bidding strategy is provided. However, their strategy doesn't consider any constraints(i.e. budget) which are common in real DSP applications.While all above researches consider only one campaign, <cit.> extends <cit.> and proposes bidding strategy for multiple campaigns. However, <cit.> also shares the drawbacks of <cit.> as listed above.<cit.> studies the joint optimization of multiple objectives with priorities. <cit.> argues that the bid price should be decided based on the performance lift rather than absolute performance value. Risk management of RTB is discussed and risk-aware bidding strategy is proposed in <cit.>. By modeling the state transition of auction competition, the optimal bidding policy is derived in <cit.> based on reinforcement learning theory.The probability estimation of interested feedbacks plays a central role in performance based advertising. CTR(Click Through Rate) prediction is of great importance and extensively studied by researchers. FTRL-Proximal, an online learning algorithm, is proposed in <cit.> and sparse model is learned for CTR prediction. In <cit.>, a hybrid model which combines decision trees with logistic regression outperforms either of these methods on their own. In <cit.>, field-aware factorization machines are used to predict CTR. Compared with clicks, the conversions are even more rare and harder to predict. To tackle the data sparseness, a hierarchical method is proposed in <cit.> for CVR(Conversion Rate) prediction. Feedbacks are usually delayed in practice and <cit.> tries to distinguish negative training samples without feedbacks eventually from those with delayed ones.Bidding landscape is studied in <cit.> and log normal is used to model the distribution of winning price. <cit.> predicts win price with censored data, which utilizes both winning and losing samples in the sealed auction. Traffic prediction for DSP is discussed in <cit.>. Budget pacing is achieved through throttling in <cit.> and bid price adjustment in <cit.>.Our work is mainly inspired by <cit.> in which compact allocation strategy, after modeling its problem as linear programming, is derived from complementary slackness. Sealed second price auction is studied in <cit.>. After all, DSP problem is a sort of online matching problem and <cit.> is an informative survey of this area.§ FORMALIZATION §.§ Primal The DSP problem could be formalized as follows. Once we bid Impression with Ad, it results in gain V and resource consumptions W, both of which are functions of BidPrice. Our total gain should be maximized under resource constraints B with x and BidPrice as variables. In addition, each Impression should be distributed to no more than one Ad. To conquer the computational hardness, indicator variable x is relaxed from {0, 1} to [0, 1]. Although most kinds of resources(e.g. budget) are sort of private and only accessible to very limited number of Ads in practice, we assume, without loss of generality, that all resources are public and shared by all Ads in this formalization. max_,() ∀k ∀i ≥0 ∀i,j i N is the index of Impressionj M is the index of Adk K is the index of Constraint is a relaxed variable, indicating whether Impression_i should be given to Ad_j, short for BidPrice_ij, is a variable(bp) is the gain function of BidPrice with support [0, ∞)(bp) is the k-th resource consumption function of BidPrice with support [0, ∞) is a resource limit constantThe above formalization might seem too abstract to capture the details of those practical objectives and constraints discussed in Section <ref>. To make things clearer, we * derive the expected winning probability and bidding cost under second price auction assumption(Section <ref>),* define the utility function familybased on previous derivation(Section <ref>),* and show how to systematically encode those practical objectives and constraints into above formalization by setting B and choosing V(bp) and W(bp) from (Section <ref>).§.§ Second Price AuctionMost AdXes adopt sealed second price auction mechanism in which the DSP with the highest bid price wins and pays the second highest bid price. For example, three DSPs bid 2$, 1$, 3$ respectively, so the third DSP wins and pays 2$.Furthermore, only the winner has access to the second highest bid price while the others observe nothing except the fact that they lose.Due to the dynamic nature of auction, the outcome is random. To model this uncertainty, p_i(x) is defined as the distribution of the highest bid price among all other DSPs' bid prices for Impression_i with support [0, ∞). In another word, the most competitive DSP will bid x for Impression_i with probability p_i(x) d x.To win Impression_i, our BidPrice must be higher than x, but we will only pay x eventually. Then the expected winning probability and bidding cost for our DSP could be defined as follows. As our BidPrice goes infinite, we'll win Impression_i with probability 1 and our bidding cost must be the mean of p_i(x). Prob_i(BidPrice)= ∫_0^BidPrice p_i(x) d xCost_i(BidPrice)= ∫_0^BidPrice x p_i(x) d x It is the non-negativity property of p_i(x) and the integral forms of Prob_i(BidPrice) and Cost_i(BidPrice) which play a central role in our theory(Section <ref>). Except that, we make little, if any, assumption about the distribution family of p_i(x). In addition, in some special cases, even explicit modeling of p_i(x) is unnecessary (Section <ref> & <ref>), which simplifies the implementation of our strategy.Whenever it's mandatory, p_i(x) could be modeled with method proposed by <cit.>. We could pick a distribution family p(x; θ)(e.g. log normal) with parameter θ and learn a parameter predictor θ̂(i) from historical bidding data which maximizes the following likelihood. ∏_i ∈Win p_i(Cost_i; θ̂(i)) ∏_i ∈Lose ∫_BidPrice_i^∞ p_i(x; θ̂(i)) d x The likelihood could be separated into two parts, i.e. one for the impressions we won and the other for those we lost. For any Impression_i that we won, the bid price of the most competitive DSP must be equal to our Cost_i, which suggests the first part. Otherwise, the only thing for sure is that it must be higher than our BidPrice_i, which suggests the second part. §.§ Utility Function FamilyThe practical V(bp) and W(bp) in DSP problem come from a certain family which is defined here and whose properties are shown without proof.is the function family that ∀ f ∈ is of the form =Prob(bp) +Cost(bp) = ∫_0^bp ( +x)p(x)dx. Given ∀ f ∈, we have f'=( + bp)p(bp). Given ∀ f ∈, we have _bp f = -/. Given ∀ g,h ∈ with shared p(x) and , we have max_bp g ≥max_bp f if and only if _g ≥_f. Given ∀ g,h ∈ with shared p(x), we have d^2h/dg^2 = _g _h - _g _h/(_g + _g bp)^3 p(bp). All above theorems are listed here for summarization purpose and will be referenced when actually used. It's safe to skip them for now and come back later. §.§ Objectives and ConstraintsThere are 4 practical objectives as listed in Table <ref>, i.e. revenue and performance objectives in P4P and P4U modes. It's straightforward to encode those objectives into standard form by definition.There are 6 practical constraints as listed in Table <ref>, i.e. budget, DSP ROI and advertiser ROI constraints in P4P and P4U modes. Constraints like budget could be expressed in standard form naturally. Others, though not so obvious at the first glance, could be rewritten into standard form as well.Take DSP ROI constraint in P4P mode for example. By definition, we have/≥ After multiplying both sides with the denominator, subtracting both sides with the nominator, and combining items by , we have{-}≤0It's easy to encode this constraint into standard form with =-, = and =0.§ AUGMENTED MMKP §.§ Primal Now we propose the augmented MMKP which could be formalized as follows and seen as an extension of MMKP with infinitely many sub-choices. In the original MMKP, both V and W are constants, while, in the augmented MMKP, V is variable and W becomes function of V. Our main-choice and sub-choice are indicated by x and corresponding V respectively. max_,∀k ∀i ≥0 ∀i,j i N is the index of Itemj M is the index of Userk K is the index of Constraint is a relaxed variable, indicating whether Item_i should be given to User_j is a gain variable(V) is the k-th resource consumption function of V with support [0, ∞) is a resource limit constant §.§ Dual We define several basic functions and show the dual of augmented MMKP based on them. (V; ) = V - (V)() = _V (V; )() = max_V (V; )min_,+ ∀i,j ≥0 ∀k ≥0 ∀i (V; ) serves as sort of score function which is used to estimate the utility of distributing Item_i to User_j with sub-choice V. It could be interpreted as the compromised gain function in which our original gain V is degenerated by resource consumption W with opportunity price α. §.§ Strong Duality The strong duality of augmented MMKP is provable under mild assumption. A brief proof is provided here and more details are revealed in the appendix.If W(V) is convex function of V, strong duality of augmented MMKP holds.Several auxiliary problems are defined in Table <ref>. P is the primal and could be separated into 2 nested steps. The inner step, given , maximizes objective withas variables. The outer step, maximizes objective withas variables.Since W(V) is convex function of V, inner is a strong duality problem, inner = dualize(inner), P = d. Since dualize(inner) is a strong duality problem, dualize(inner) = dualize(dualize(inner)), d = dd. Since D is a strong duality problem, D = DD. dd and DD happen to have the same form, dd = DD. As a result, P = D, strong duality of augmented MMKP holds.§.§ Dual Based Strategy With strong duality satisfied, several important properties are claimed about the optimal solution of both primal and dual problems(i.e. ^*, ^*, ^* and ^*), based on which we propose the dual based strategy. ^* = (^*).^*(^* - (^*)) = 0.If (^*) < 0, we have ^* = 0.Since (^*) < 0 and ^* ≥ 0, we have ^* > (^*). Now that ^* - (^*) > 0 and ^* ≥ 0, taking above theorem into consideration, ^* must be 0, that is Item_i should not be distributed to User_j.If S_ij_1(^*) < S_ij_2(^*), we have x_ij_1^* = 0.Similarly, since S_ij_1(^*) < S_ij_2(^*) and ^* ≥ S_ij_2(^*), we have ^* > S_ij_1(^*). Now that ^* - S_ij_1(^*) > 0 and x_ij_1^* ≥ 0,taking above theorem into consideration, x_ij_1^* must be 0, that is Item_i should not be distributed to dominated User_j_1.^*(∑_j ^* - 1) = 0.If ∃ j that (^*) > 0, we have ∑_j ^* = 1.Since (^*) > 0 and ^* ≥(^*), we have ^* > 0. Now that ^* > 0 and ∑_j ^* - 1 ≤ 0, taking above theorem into consideration, ∑_j ^* must be 1, which means Item_i should not be discarded. In summary, for each Item_i, every User_j should propose its own best score ^* achieved by ^*. Item_i should be awarded to the dominating User_j^* if its best score S_ij^*^* is positive and discarded if that is negative. Theoretically speaking, while most of which are determined by above corollaries, behaviors remain undefined in two special cases. First, there might be multiple dominating users with the same best score. Second, the best score of dominating user might be exactly zero. In practice, however, both cases are probably rare due to the high resolution of items and users, and prone to cause relatively limited damage. Ties could be broken by random or heuristics. §.§ Numeric Optimization Note that, during the execution of the dual base strategy, only the ^* is mandatory while the others(i.e. ^*, ^* and ^*) could be recovered with ^*, which makes our strategy storage efficient. Next, we propose the numeric method to solve ^*. () = max{ 0, () ∀ j }() = ∑_k /N + () By fixingin the dual problem, ^* could be calculated as ^* = (). Then the dual problem could be rewritten as min_≥0 ∑_i ()and solved by SGD(Stochastic Gradient Descent). Due to the convexity of dual problem, it must converge to the global optimal ^*.§ SOLUTION §.§ Dual We define corresponding basic functions and show the dual of DSP problem based on them. (bp; ) = (bp) - (bp)() = _bp(bp; )() = max_bp(bp; )min_,+ ∀i,j ≥0 ∀k ≥0 ∀i Note that, in DSP problem, our sub-choice is indicated by bp rather than V. Sinceis the linear combination of (bp) and (bp) fromwith shared p_i(x), it must belong totoo with its _ and _ as follows._= _ - ∑_k _ _= _ - ∑_k _In practice, each ad is usually subjected to very limited number of constraints, which makes the calculation of _ and _ light-weighted. §.§ Strong Duality Due to the nice property of , it's easy to check that, as to practical objectives and constraints(Section <ref>), W(bp) is indeed convex function of V(bp), which immediately justifies the strong duality of DSP problem.Strong duality of DSP problem holds.According to Theorem <ref>, V'(bp)>0 and W(bp) must be function of V(bp). According to Theorem <ref>, d^2W/dV^2≥0 and W(bp) must be convex function of V(bp). As a result, according to Theorem <ref>, strong duality of DSP problem holds.§.§ Dual Based StrategyWith strong duality satisfied, the dual based strategy developed for augmented MMKP is also applicable to DSP problem. Generally speaking, ^* could be determined without p_i(x) according to Theorem <ref>. In certain applications, _ is the same for given i and all j, then j^* could also be determined independent of p_i(x) according to Theorem <ref>. By disposing of p_i(x) completely from deciding process, it not only simplifies the computation, but also encourages p_i(x) free training process. §.§ Numeric OptimizationThe numeric method developed for augmented MMKP is also applicable to DSP problem. It's easy to prove that d()/d must be either -(()) if the best score of the dominating Ad_j is positive or 0 otherwise. This optimization method, though generally applicable, requires explicit modeling of p_i(x).Through executing our strategy in production environment, the randomized version of (()) is revealed and the gradients could be approximated with these feedbacks. This optimization method is p_i(x) free and much easier to implement.§ SIMULATION §.§ Methodology To eliminate the uncertainty, our strategy is verified in P4P and P4U modes through simulation. Due to the limited space, we focus on the P4P mode in the rest of Section <ref>.There are two simulation cases, i.e. one for revenue maximization and the other for performance maximization. Two mocked ads Ad_1 and Ad_2 are created with CPP_1=1 and CPP_2=2. Four mocked constraints are listed in Table <ref>. Budget of Ad_1 and Ad_2 are 20 and 10 respectively. The global DSP ROI lower bound is 2, while the global advertiser ROI lower bound is 0.5. As suggested by <cit.>, p(x) is assumed to be log normal distribution p(x;μ,σ) with mean μ and standard deviation σ as parameters. To mock the impressions, 200 tuples <μ_i, σ_i, PPI_i1, PPI_i2> are drawn randomly.Once the configurations are ready, ^* are solved by SGD(Section <ref>). After that, the dual based strategy is applied on the same cases and the consequent statistics are collected. §.§ Results and Analysis The statistics and ^* are listed in Table <ref>. In both cases, all resources have non-negative surplus and no constraint is violated. In addition, the gap between primal and dual objective values is negligible(Theorem <ref>). As mentioned earlier, the α^* serves as so called opportunity price of the resource. Intuitively speaking, waste of resource with positive surplus shouldn't lead to any opportunity cost. As a result, the corresponding α^* tends to be 0.§ APPLICATION §.§ Scenario We also deploy our strategy in the DSP platform of Alibaba. In our application, advertisers set budgets and pay for clicks, while DSP is willing to maximize revenue under daily global DSP ROI constraint. There are so many ads in our inventory that it's impossible to go through each ad before auction deadline. Although these budgets are quite large totally, they are relatively small on average.With well calibrated CPP and PPI predictors, the problem could be transformed equivalently into one in P4P mode. And to meet the latency requirement, the whole deciding process is decomposed into two stages with so called logical ad.Logical ad should be seen as proxy of physical ads and binded with specific ad retrieval algorithm. In the first stage, DSP is supposed to make decisions among just a few logical ads and respond in time. In the second stage, once the chosen logical ad wins the auction, physical ad is lazily retrieved with corresponding algorithm.Our logical ads are actually based on 4 heterogeneous ad retrieval algorithms whose details are beyond the scope of this document. These algorithms are sorted by their historical performance in descending orders and 4 logical ads are constructed correspondingly.In summary, our problem could be approximately modeled as, given 4 logical ads with literally unlimited budget, maximizing revenue under daily global DSP ROI constraint in P4P mode. Since there is only one resource constraint, superscript k is omitted and ROI is short for global DSP ROI in the rest of Section <ref>.According to our theory, we have _=(1+α) and _=-αROI. Since _ is always -αROI, no p_i(x) is required in deciding process as discussed in Section <ref>. To take full advantage of that, we adopt a simplified version of the p_i(x) free optimization method suggested in Section <ref>, i.e. . §.§ Experiment Groups We compare our strategy with a variation of linear bidding strategy. In <cit.>, it's suggested that =ActualCTR_ij/CTRBid with Bid set by operation team. However, unlike ActualCTR_ij which is independent of , ActualROI_ij indeed varies with it. As a result, we iteratively updatewith .We also apply optimal RTB theory to our application for comparison. According to <cit.>, we model the win probability as w(bp;c)=bp/c+bp and bid with =, in which c is fitted with method proposed by <cit.> and λ is iteratively tuned with .Four experiment groups are shown in Table <ref>. The first three groups are designed to compare different strategies with single logical ad, while the last group is used to test our strategy with multiple logical ads.To eliminate potential bias, the experiment lasts for a whole ordinary week. Bidding opportunities are distributed to each group randomly with equal probability. For fairness, the same CPP and PPI predictors are shared by all groups. The lower bound of daily ROI is set to 3.5.Strategy parameters(i.e. , λ and α) are randomly initialized and periodically adjusted with actual ROI since last update. The period is set to 24 hours for the LIN group due to the data sparseness and 10 minutes as to the others for robustness and faster convergence. Note that the more frequent update introduces inexplicitly a 10 minutes ROI constraint which is stricter than the daily one and might degenerate the theoretical optimal. §.§ Results and Analysis For each group, the daily statistics of four metrics are plotted in Figure <ref>, namely revenue, actual ROI, number of winning impressions and revenue per winning impression.The LIN, though with theoretical optimal intact, tends to earn less revenue than the others in practice. In addition, it usually violates the daily ROI constraint seriously, so it's an inferior strategy.Compared with the DB_s who claims a linear relationship between bid price and expected revenue, the ORTB, derived from first price auction assumption, suggests a non-linear one. It is biased towards the impressions with low expected revenue and against those with high expected revenue, which leads to more impressions and lower averaged quality. While the daily ROI constraint is satisfied by both strategies, the DB_s earns more revenue than the ORTB. As a result, the DB_s is superior theoretically and practically.The DB_m, as an ensemble of four ad retrieval algorithms, achieves the most revenue without violation of the daily ROI constraint and becomes the best strategy.§ CONCLUSIONS AND FUTURE WORKS In this document, we propose a dual based DSP bidding strategy derived from second price auction assumption according to convex optimization theory. Our strategy is verified through simulation and outperforms state-of-the-art strategies in real application. It's a theoretically solid and practically effective strategy with simple implementation and various applications.Three problems remain unsolved and deserve further study. First, is there a better way to solve ^* of large scale in dynamic environment? On the one hand, in a typical DSP, there will be millions of constraints shared by similar number of ads. Each of the constraints deserves a α, which makes the vectorvery large. On the other hand, billions of impressions are broadcast by AdX every day and bid by hundreds of DSPs simultaneously. The bidding strategies are interactively adjusted by DSPs and the inventories are frequently updated by advertisers, which makes the bidding landscape unstable. Both properties make the ^* hard to solve.Second, how to construct and index logical ads automatically in massive ads applications, balancing latency and performance? It's obvious that both deciding and training processes share the same ad evaluation and maximum determination style, which makes their computational complexities linearly related with the number of candidate logical ads. At one extreme, each ad is represented by exactly one logical ad, and the consequent latency is unacceptable. At the other extreme, all ads are represented by the only logical ad, while the performance might be seriously degenerated. A proper compromise combined with efficient indexing tricks will accelerate both processes by orders of magnitude.Third, how to optimally break ties when they are common and critical? Take an imaginary scenario for example. DSP is willing to maximize its revenue in P4P mode. There are two identical ads with the same CPP and PPI, but they are targeted to overlapped sets of impressions and subjected to different budget constraints. In this circumstance, resolution of impressions and ads is extremely low and ties are very prevalent. To tackle the tie breaking problem, we might try randomized soft-max instead of hard-max during ad selection. However, the theoretical soundness and practical effectiveness of this tie breaking strategy are to be verified.§ STRONG DUALITY PROOF Here we give the detailed proof of the strong duality. We first prove that P ≤ D by dualizing P. P = - min_,≥0{ - } = - min_, { max_,,≥0 { - + [() - ] + (- 1) + (-) } } = - min_, { max_,,≥0 { - - + [-+ () + - ] } } ≤- max_,,≥0 { min_, { - - + [-+ () + - ] } } = - max_ ,≥0{ -- } = min_ ,≥0{ + } = DNext, we prove that D = DD by dualizing DD. D = min_, { max_,,≥0{ + + [() - ] + (-) + (-) } } = min_, { max_,,≥0{ (- ) + (1 - - ) + () } } = max_,,≥0{ min_, { (- ) + (1 - - ) + () } } = max_,≥0{ min_ { (- ) + () } } = DDAfter that, we prove that P = d by dualizing the inner step of P with the outer step unchanged. P = max_ ≥0{ max_{ } } = max_ ≥0{ - min_{ - } } = max_ ≥0{ - min_ { max_≥0 { - + [() - B^(k)] } } } = max_ ≥0{ - min_ { max_≥0 { - B^(k) + [-+ ()] } } } = max_ ≥0{ - max_≥0 { min_ { - B^(k) + [-+ ()] } } } = max_ ≥0{ - max_≥0 { - - () } } = max_ ≥0{ min_≥0 { + () } } = dAt last, we prove that d = dd by dualizing the inner step of d with the outer step unchanged. d = max_ ≥0{ min_ { max_≥0 { + () + (-)} } } = max_ ≥0{ min_ { max_≥0 { (- ) + () } } } = max_ ≥0{ max_≥0 { min_ { (- ) + () } } } = max_ ,≥0{ min_ { (- ) + () } } = ddIt's obvious that DD and dd are of the same form, so DD = dd. As a result, we have P = D and strong duality holds.ACM-Reference-Format | http://arxiv.org/abs/1705.09416v2 | {
"authors": [
"Huahui Liu",
"Mingrui Zhu",
"Xiaonan Meng",
"Yi Hu",
"Hao Wang"
],
"categories": [
"stat.ML",
"cs.GT"
],
"primary_category": "stat.ML",
"published": "20170526024308",
"title": "Dual Based DSP Bidding Strategy and its Application"
} |
Instituto de Astrofísica de Canarias, 38205, C/ Vía Láctea, s/n, La Laguna, Tenerife, Spain Departamento de Astrofísica, Universidad de La Laguna, 38205, La Laguna, Tenerife, Spain NorthWest Research Associates, Boulder, CO 80301, USAMax-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, GermanyImproving methods for determining the subsurface structure of sunspots from their seismic signature requires a better understanding of the interaction of waves with magnetic field concentrations. We aim to quantify the impact of changes in the internal structure of sunspots on local helioseismic signals. We have numerically simulated the propagation of a stochastic wave field through sunspot models with different properties, accounting for changes in the Wilson depression between 250 and 550 km and in the photospheric umbral magnetic field between 1500 and 3500 G. The results show that travel-time shifts at frequencies above approximately 3.50 mHz (depending on the phase-speed filter) are insensitive to the magnetic field strength. The travel time of these waves is determined exclusively by the Wilson depression and sound-speed perturbation. The travel time of waves with lower frequencies is affected by the direct effect of the magnetic field, although photospheric field strengths below 1500 G do not leave a significant trace on the travel-time measurements. These results could potentially be used to develop simplified travel-time inversion methods.Helioseismic Holography of Simulated Sunspots: dependence of the travel time on magnetic field strength and Wilson depression T. Felipe<ref>,<ref> D. C. Braun<ref> A. C. Birch<ref>December 30, 2023 ============================================================================================================================= § INTRODUCTION Solar active regions, such as sunspots, are the most remarkable manifestations of solar magnetism and have a key role in the coupling between the interior and the atmosphere, which makes the understanding of sunspot formation, stability, and decay one of the outstanding problems in solar physics. Local helioseismic studies have been focused on solar active regions for more than two decades, but still significant uncertainty exists regarding their interpretation due to the complexity of wave interactions with magnetic fields.Several helioseismic techniques have been developed to probe the subsurface structure of active regions. The most commonly used measurement is the phase change of waves traveling through the active region, usually described as a change in travel time, which can be obtained from time-distance helioseismology <cit.>, Fourier-Hankel analysis <cit.>, or helioseismic holography <cit.>. The first attempts to interpret the seismic signals assumed that sunspots can be characterized as small perturbations to the thermal structure of a quiet Sun model and neglected the direct effects of the magnetic field on helioseismic waves. From this approach two different sunspot scenarios were inferred from observations: a shallow positive perturbation of the sound speed <cit.> and a two-layer model with a reduction of the wave-speed in the top layer and an increase in the wave-speed down to 10 Mm below the surface <cit.>. Subsequent observational <cit.> and numerical <cit.> studies have shown that surface magnetic fields strongly alter travel times. In addition to the changes in the phase speed of the waves introduced by the magnetic field, mode conversion <cit.> can also affect the photospheric seismic measurements through the returning of atmospheric fast waves <cit.> or Alfvén waves <cit.>. <cit.> quantified the contribution of these returned waves to the travel-time shifts obtained from the analysis of the photospheric wave field. They found a significant travel-time shift which depends on the propagation direction of the waves and the magnetic field inclination. Recently, <cit.> evaluated the individual contributions of the direct and indirect magnetic effects on the travel-time perturbations measured using helioseismic holography. The term “direct magnetic effects” refers to the changes in wave propagation due to the wave interaction with the magnetic field through mode conversion and fast wave refraction, which produces variations in the wave speed and ray paths. The “indirect magnetic effects” account for the changes in the thermal structure of the sunspot imposed by the presence of the magnetic field. Their results show that waves filtered for certain phase speeds and frequencies are less sensitive to the magnetic field. For those combinations of filters, the travel-time shifts are accurately predicted from the thermal structure of the sunspot, accounting for the Wilson depression and the changes in the sound speed but neglecting the direct effect of the magnetic field. In this work, we aim to further explore the dependence of the helioseismic signals on sunspot properties by analyzing a larger sample of sunspot models which differ in their magnetic field strength and Wilson depression. The numerical methods are described in Sect. <ref> and the comparison of the results obtained for different models is presented in Sect. <ref>. Finally, the conclusions and the potential applications of our results are discussed in Sect. <ref>. § NUMERICAL PROCEDURESWe carried out twelve numerical simulations of wave propagation through magnetohydrostatic (MHS) sunspot models using the code MANCHA <cit.>. The numerical set up of all the simulations is the same described in <cit.>. Here we include only a brief description of the numerical configuration. We address the interested reader to <cit.> for a detailed discussion of the numerical methods. The vertical computational domain spans from z=-25 Mm to z=1 Mm, with z=0 defined at the quiet-Sun photosphere and a uniform vertical grid size of 50 km. The dimension of the horizontal directions is 102.4 Mm and they are sampled with a resolution of 200 km. The axes of the sunspots are located at the middle of the domain. The Alfvén speed has been artificially limited to 80 km s^-1 following <cit.>. The simulations differ in the properties of the MHS models used as background. All the sunspot models were constructed with the same method <cit.>, but changing two of their properties: magnetic field strength and Wilson depression. The method combines a self-similar solution <cit.> in the deep layerswith a current-distributed solution <cit.> in the upper layers. This class of models allows the freedom to select the values of several parameters which control the properties of the resultant sunspot model. The magnetic configuration is given by the parameters B_0^L, η, and a. The former produces changes in the magnetic field strength, while the magnetic field inclination, the curvature of the field lines, and the radius of the structure depend on the other two parameters. The model used as a boundary condition at the axis of the sunspot can be shifted up or down on the axis in order to account for the Wilson depression z_ WD. The last parameter that can be modified is the height z_0 where the self-similar and current-distributed solutions are merged. The space of parameters explored includes umbral photospheric field strengths of B_ u= 1500, 2500, and 3500 G and Wilson depressions of z_ WD= 250, 350, 450, and 550 km. All the possible combinations of these two parameters were computed, producing a total of twelve independent simulations. The parameters η, a, and z_0 are the same for all the models. The radial variation of the photospheric vertical magnetic field approximately follows a Gaussian distribution with a FWHM of 19 Mm (a Gaussian is set at the height z_0, but it is modified through an iterative process to concatenate the self-similar and current-distributed models and obtain a MHS solution). The use of models with different magnetic field strength but otherwise the same field geometry allow us to evaluate the effect of field strength on the seismic signal independently of the magnetic field inclination. Figure <ref> shows the vertical stratification of the isothermal cut-off frequency for all the models at the center of the sunspot. As expected, an increase in the Wilson depression increases the depth of the upper turning point. In the following, we will call the “upper turning point” the depth where the wave frequency is equal to the local cut-off frequency, that is, to the upper turning point of the radial mode. The depth of the β=c^2/v_ A^2=1 layer, where c is the sound speed and v_ A is the Alfvén speed, increases with the magnetic field. This layer is located at a similar depth for all the models with B_u=3500 G, but for lower field strengths the depth of the β=1 layer depends on the Wilson depression. Figure <ref> illustrates the sound speed. Models with the same Wilson depression but different field strengths present differences in their sound speed, since their gas pressure stratifications differ due to the pressure deficit imposed by the magnetic field. A quiet Sun simulation with the same set up of the sunspot cases was performed for applying the method of noise subtraction <cit.>. Waves are excited by sources introduced in the equations. They are located at 150 km below the photosphere in quiet-Sun regions and their depth increases toward the axis of the spot following a constant temperature surface from the simulation with B_u= 2500 G and z_ WD= 450 km. Wave sources are randomly distributed in the horizontal directions, but all the simulations use the same random distribution. As a result, the same quiet-Sun simulation can be used for applying noise subtraction to all the sunspot cases. The duration of the simulations with B_ u=1500 G and B_ u=2500 G is 8 h, while the simulations with B_ u=3500 G span 7 h of solar time. Travel times are computed by applying the procedures for surface-focused helioseismic holography <cit.> to the vertical velocity maps at constant optical depth τ=0.01. Filtering has been applied in k-ω space following many previous studies <cit.>. We use a set of phase-speed filters described in <cit.>. Their mean phase speed spans from 14.87 km s^-1 (TD2) to 35.46 km s^-1 (TD5). A specific pupil function is associated with each of the filters, with larger pupils for higher phase speeds. See <cit.> for details of the form of the filters and pupils. The wavefield has also been filtered in temporal frequency, isolating bandpasses with widths of 0.5 mHz centered at 2.75, 3.25, 3.75, 4.25, 4.75, and 5.25 mHz. Finally, the travel-time shifts are obtained by subtracting the quiet-Sun travel time from the sunspot travel time. The quiet-Sun travel time was obtained from the analysis of 7 or 8 h temporal series as reference for the B_ u=3500 G cases or lower field strength cases, respectively. Figure <ref> illustrates the power spectra of the quiet-Sun simulation, with the model S <cit.> eigenfrequencies of the f mode and p_1 to p_4 modes overplotted as a reference. The regions in the k-ω domain spanned by the frequency and phase-speed filters are indicated. § TRAVEL-TIME SHIFT MEASUREMENTS §.§ MapsThe dependence of the travel-time shifts with frequency and phase speed (positive values for low frequencies and low phase speeds and negative values for high frequencies and high phase speeds) is in qualitative agreement with the results obtained from actual observations <cit.> and previous numerical simulations <cit.>. Figures <ref>-<ref> illustrate the travel-time shift maps of the three simulations with a Wilson depression of 550 km. We do not show the maps for the other sunspot cases, since their qualitative behavior is similar and the quantitative differences are better appreciated in the radial-average plots in the next sections. §.§.§ TD4 and TD5 phase-speed filtersNegative travel-time shifts are obtained for the phase-speed filters TD4 and TD5 for all frequencies in all of our sunspot models. The magnitude depends on the phase speed and frequency. In the case of the sunspot with B_ u=3500 G (Fig. <ref>), the maximum travel-time perturbation is found for the phase-speed filter TD4 and frequency of 3.25 mHz. Higher frequencies show a lower magnitude in the travel time perturbation. On the contrary, the sunspot with the lowest magnetic field strength (Fig. <ref>) shows stronger perturbations for high frequencies. For those filter combinations (high phase speed and high frequency) the travel-time perturbation is reduced in amplitude with increasing magnetic field (considering the same Wilson depression).§.§.§ TD2 and TD3 phase-speed filters The travel-time shifts for low phase-speed filters (TD2 and TD3) changes from positive values (waves apparently slower in the sunspot than in the quiet Sun) to negative values (faster waves in the sunspot) as the frequency increases. The frequency at which this sign reversal is produced is lower for the TD3 filter than for the TD2 filter, but it also depends on the sunspot model. For the sunspot with B_ u=3500 G and z_ WD=550 km (Fig. <ref>) the travel-time shift measured at 2.75 mHz and TD3 is mainly positive, while a 3.25 mHz frequency shows a mix of positive and negative shifts. In the other two sunspots with lower field strengths (Figs. <ref> and <ref>) this combination of filters produces a negative travel-time shift, which is specially prominent in the case with B_ u=1500 G. For the TD3 case, higher frequencies are clearly negative in the three sunspots shown in Figs. <ref>-<ref>. With regards to TD2 phase-speed filter, in the sunspot with stronger magnetic field negative travel-time shifts are found for frequencies higher than 4 mHz. For the sunspots with B_ u=1500,2500 G, the travel-time shift of waves with 3.75 mHz frequency and phase speed given by TD2 filter shows a negative value surrounded by an annular region with positive shift. For higher frequencies the positive ring vanishes, and only the negative signal remains.§.§ Sensitivity to the Wilson depression We computed azimuthally averaged travel-time shifts for all of the sunspot models. Figure <ref> shows the radial variation of the averaged travel-time shifts for some selected frequencies for all the simulations with B_ u=1500 G. To assess the significance of differences between measurements of the travel-time shifts among the models, we compute errors in the following manner. We first take the difference between the travel-time shift maps for two models with different values of the Wilson depression. Although the same distribution of acoustic sources are used to reduce the realization noise (see Sect. <ref>), a residual amount of noise remains. The map of the travel-time shift difference is divided into four quadrants centered on the sunspot and the error is estimated as the total spread (maximum minus minimum value) of the four azimuthal averages within each individual quadrant (all of the models considered here are cylindrically symmetric, so any variations between quadrants is due to noise). In order to simplify the figure, only the errors of the difference between the z_ WD=350 km and 450 km cases are shown, superimposed on the z_ WD=350km measurements. Errors computed using other pairs of models are similar. As expected, the measured signal increases in amplitude with the Wilson depression. An increase of the Wilson depression produces a shift in the depth of the upper turning point of the waves toward deeper layers. As a result, the path traveled by wave rays hitting the center of the sunspot is shorter than that of waves reaching the photosphere in quiet-Sun regions. This effect leads to shorter travel times in the sunspot, and the magnitude of this reduction is stronger for more depressed sunspot atmospheres. §.§.§ Frequency dependenceMost of the averaged travel-time shifts illustrated in Fig. <ref> show a negative travel-time shift. Only the case with TD2 and 3.25 mHz frequency (and lower frequencies with phase speed given by TD2 and TD3, as seen in Fig. <ref>) produces a positive perturbation in the travel time. It is interesting to note that this positive perturbation increases with the Wilson depression. In this case, the variation of the travel time cannot be understood as the result of the changes in the depth of the upper turning point. The sensitivity of the travel-time shift to the Wilson depression depends on the combination of filters used for the measurement. The travel-time shift measured for waves with 3.25 mHz frequency at the central part of the umbra shows differences around 7 s at most between the results obtained for the case with z_ WD=250 km and the case with z_ WD=550 km. On the contrary, for waves with 5.25 mHz frequency this variation can be larger than 25 s, as found for the phase-speed filter TD3. The top panel of Fig. <ref> shows the vertical stratification of the cut-off frequency from the B_ u=1500 G cases, whose travel-time shifts are plotted in Fig. <ref>. A wave with 3.25 mHz frequency (bottom horizontal dashed line) propagating upward from the interior will reach the turning point of the z_ WD=550 km model (red line) earlier than that of the smaller Wilson depression models. Waves propagating in sunspot models with small Wilson depression will travel a longer path, closer to that traveled in a quiet-Sun region (thick black dashed line). The difference in the depth where the cut-off frequency is 3.25 mHz between the z_ WD=250 km and z_ WD=550 km cases is around 220 km, in the sense that the path of the later is shorter. For waves with 5.25 mHz, the reduction of the travel path is more than 300 km, producing a larger difference in the travel-time perturbation signal. This effect can qualitatively explain the dependence of the travel-time shift with the frequency for models with the same field strength. The variation of the travel-time signal with the Wilson depression at constant field strength shows a similar trend for the case with B_ u=3500 G (Fig. <ref>), although filters TD3 and TD4 show a weaker dependence at the high frequencies. The results for the model with B_ u=2500 G (not shown in the figures) are somewhat between the other two cases. §.§ Sensitivity to the magnetic field strengthFigure <ref> shows the azimuthally averaged travel-time shift measured for the simulations with a Wilson depression of 450 km, but with different magnetic field strengths. Errors are estimated for differences in measurements using models with different magnetic field strength in a manner similar to that discussed in Sect. <ref>. Shown are errors of the difference between measurements from the B_ u=2500 G and the B_ u=1500 G models, superimposed on the B_ u=2500 G measurements. Errors computed using other pairs of models are similar. The largest differences between models are found for the TD2 filter and frequencies between 3.25 and 4.00 mHz. In that region the magnetic field causes changes in the travel-time shift of around 30 s, and the perturbations can even show an opposite sign, as seen in Sect. <ref>. For example, TD2 waves with 3.50 mHz frequency show a negative travel-time shift for the sunspot with B_ u=1500 G but a positive shift for the other two cases. The dashed green lines illustrate the signal obtained from a background model with the density, pressure, and coefficient of specific heats from the sunspot with B_ u=2500 G and z_ WD=450 km but with the magnetic field set to zero. These data correspond to the “thermal sunspot” analyzed in <cit.>.§.§.§ Low frequencies The travel-time shifts at lower frequencies (2.75 mHz for all the phase-speed filters and also 3.00 mHz for TD2) show a weak dependence on the magnetic field strength. For most phase speed filters and radial positions it is below the error of the measurement procedure. The change in the travel-time shift between models with B_ u=1500 G and 3500 G is only measurable for the phase speed filter TD2 near the center of the sunspots and for TD4. In all the sunspot models, the upper turning point for low frequencies is deeper than the depth where β=1 (Fig. <ref>). Upward propagating waves with low frequencies reach the turning point before the magnetic effects are relevant. They neither reach the mode conversion layer nor propagate at depths where the fast speed is significantly modified by the increase of the Alfvén velocity.§.§.§ Intermediate frequencies For frequencies between 3.00 and 3.25 mHz, the negative travel-time perturbations measured with the higher phase-speed filters (TD4 and TD5) increase with the strength of the magnetic field. The case with B_ u=3500 G shows shifts around -40 s for the TD4 filter at the central part of the umbra. The magnitude of the shift is lower for the B_ u=2500 G and B_ u=1500 G, but a comparison of the later with the thermal sunspot reveals a perfect match. Interestingly, although the travel-time perturbations for frequencies between 3.00 mHz and 3.75 mHz are sensitive to the field strength, the thermal sunspot produces signals in quantitative agreement with the sunspot with B_ u=1500 G (except for TD2 filter). That is, the travel times are nearly insensitive to changes in the magnetic field for strengths below 1500 G. As can be seen in Fig. <ref>, the distance between the turning point and the β=1 depth decreases with the magnetic field strength. Waves with frequency around 3.00 mHz propagating through the B_ u=2500 G and B_ u=3500 G models can be affected by the magnetic effects, but in the B_ u=1500 G model they do not reach the region where these effects are significant. The shorter travel times for the strongly magnetized sunspots agrees with the faster propagation velocity in those models, since their fast magnetoacoustic speed is higher than for the low field strength model. This situation differs from the TD2 and TD3 cases with frequencies between 3.25 mHz and 4.00 mHz, where the models with strong magnetic field show longer travel times. In those cases the interaction with the magnetic field is more complex and must be associated with the larger angle of incidence of those waves, since the lower phase speed waves propagate in a shallower cavity.§.§.§ High frequencies The travel-time shifts measured with all phase-speed filters show a slight variation with magnetic field for high frequencies (generally around 10 s). In these cases the magnitude of the perturbation decreases with the field strength, opposite to the results found for low frequency and high phase speed waves. The differences between the B_ u=1500 G and B_ u=2500 G cases are small but higher than the value of the measurement error. The travel time obtained for the stronger magnetic field simulation is clearly distinguishable from the other two cases. However, the “thermal sunspot” shows a quantitative agreement with the sunspot model with B_ u=2500 G and z_ WD=450 km. These two simulations have exactly the same thermal structure, they only differ in the absence of magnetic field in the former. Their comparison implies that a 2500 G field strength change does not produce significant changes in the travel time. Instead, the variations identified between the three models with different field strength must be due to thedifferences in their thermal structure, even though they have the same Wilson depression. The construction of the MHS models requires the imposition of force balance. The magnetic force is different in the three models and, thus, their thermal structure must necessarily differ. The three models have the same Wilson depression, but show differences in the stratification of pressure, density and coefficient of specific heats which lead to differences in their sound speed, as shown in Fig. <ref>. The sound speed in the layers between z=-1.7 Mm and z=-0.7 Mm decreases with increasing field strength. p-mode waves propagating through the B_ u=1500 G sunspot will be faster than waves in the other sunspots, leading to the reduced travel time measured for high frequencies in this model. The frequency above which the travel-time perturbations can be interpreted as a result of the changes in the Wilson depression and sound speed rather than magnetic field depend of the phase-speed filter. For the TD2 case the threshold is at 4.25 mHz, for TD3 at 4.00 mHz, and for TD4 and TD5 filters at 3.75 mHz. Models with other value for the Wilson depression show a similar dependence with the magnetic field strength. The results of these cases are in qualitative agreement with those illustrated in Fig. <ref>. Figure <ref> shows the travel-time shift between models with z_ WD=450 km and models with z_ WD=250 km. That is, it is equivalent to Fig. <ref> except that the travel times of the z_ WD=250 km cases were subtracted instead of the travel times obtained from the quiet Sun simulation. This representation highlights the differences between pairs of sunspot models. A negative travel-time shift indicates that the travel time of the waves in the simulation with z_ WD=450 km is shorter and vice versa. In the cases with small Wilson depression, the dependence of the high frequency travel times with the magnetic field strength is lower, since the difference in the sound speed is smaller than in the z_ WD=450 km cases (compare the first and third panels of Fig. <ref>). § DISCUSSION AND CONCLUSIONSWe have performed a parametric study of the sensitivity of the travel-time shifts measured using helioseismic holography to sunspot models with different Wilson depressions and magnetic field strengths. This study is a continuation of the paper <cit.>. The results confirm those from the previous work and reveal new findings. Our main conclusions are:* The Wilson depression has a strong effect on the measured travel-time shifts. The depression of the atmosphere lowers the height of the upper turning point of the waves, and the path they travel beneath the sunspot is shorter than that of waves in quiet-Sun regions. This causes a reduction in the travel time. The magnitude of the reduction increases with the Wilson depression (Figs. <ref> and <ref>). Low frequencies are less sensitive to this effect, since the change in the depth of their upper turning point is smaller than for high frequency waves (Fig. <ref>).* Waves with frequencies below 3.00 mHz show a weak dependence on the magnetic field strength. Their cut-off height is below the region where magnetic field effects are significant. In the case of the TD4 and TD5 phase-speed filters, they show a negative travel-time perturbation, which is in agreement with the deeper turning point of those waves in the sunspot models with respect to the quiet-Sun atmosphere. This is opposite to the positive shifts obtained for the low phase-speed filters (TD2 and TD3). <cit.> and <cit.> have argued that this sign change is a property of the chosen data analysis filters. * The direct effects of the magnetic field on the travel time are apparent for some combinations of phase speed and frequency filters. For the phase-speed filter TD2, the travel-time perturbation measured for frequencies between 3.25 and 4.00 mHz is clearly modified by the magnetic field strength. For the filter TD3, waves with frequencies between 3.00 and 3.75 mHz are sensitive to the magnetic field, while the filters TD4 and TD5 show variation of the travel-time shift with magnetic field for frequencies between 3.00 and 3.50 mHz. These conclusions are based on the analysis of the sunspots with a Wilson depression of 450 km (Fig. <ref>), but the results are similar for the other models. * For all phase-speed filters analyzed, the travel-time perturbation at high frequencies is likely caused by the changes in the thermal model. The Wilson depression shortens the path of the waves, reducing the travel times. However, the Wilson depression on its own cannot account for the obtained travel-time shifts. The variation of the sound speed (among models with the same Wilson depression) at depths between z=-1.7 Mm and z=-0.7 Mm is also measurable, and it is responsible of changes around 10 s in the travel time (see Fig. <ref>).* The travel-time shift is insensitive to the direct effects of photospheric magnetic field strengths below 1500 G. As indicated in the previous bullet point, travel-times at high frequencies are insensitive to the magnetic field. Only low frequencies depend on the field strength, but for those filters the travel-time perturbation measured from the sunspot with B_ u=1500 G shows a perfect match with the “thermal sunspot”, whose magnetic field vanishes (Fig. <ref>). For small sunspots and pores, the only contribution to the travel time comes from the indirect effect of the magnetic field on the thermal structure, through changes in the sound speed and Wilson depression.We have used a set of forward calculations to show examples of how travel times depend on the properties of various sunspot models. We emphasise that the model sunspots described in this paper are not intendeded to represent any particular observed sunspot (for example, by matching the observed sunspot radius or radial profile of magnetic field) and it is therefore premature to compare the modeled travel times with observed sunspot travel times. It is, however, important to know which of these travel-time variations are detectable above the observational noise levels. Figure 9 from <cit.> shows the umbral averages (and errors) of the travel-time shifts measured in the sunspot in AR11092 using 24 hr of Helioseismic and Magnetic Imager <cit.> data. The error they retrieved is below 10 s for all the phase-speed and frequency filters, while for some filter combinations (especially for TD4) the measured error is significantly lower (below 5 s). For photospheric umbral magnetic field around or below 2500 G the change in the travel-time perturbation produced by 100 km shifts in the Wilson depression could be detected above the noise level in the TD3 and TD4 filters for frequencies around 5 mHz (Fig. <ref>). Models with B_ u=3500 G do not show such a strong travel time variation with Wilson depression at high frequencies (Fig. <ref>), and the precision of the Wilson depression estimation would be lower. For other filter combinations the observational noise is comparable or above the travel-time shift associated to changes in the Wilson depression.The highest sensitivity of the travel-time shift to the Wilson depression is obtained for the phase-speed filter TD3 at 5.25 mHz. This combination of phase-speed and frequency filters encompasses the ridge of the p_ 2 mode (Fig. <ref>). This is in general agreement with <cit.>, who concluded that p_ 2 waves are a good candidate for constraining the depth of the Wilson depression. We also find that the phase-speed filter TD4 at 5.25 mHz (p_ 3 mode) is another good candidate for measuring the Wilson depression. This mode was not considered by <cit.>.The travel-time shifts caused mainly by the changes in the sound speed among models with the same Wilson depression but different field strengths (high frequencies, Fig. <ref>) are also above the observational noise level when the sound speed perturbation is high enough (sound speed changes between models with B_ u=3500 G and B_ u=1500 G). <cit.> evaluated the sensitivity of the travel time of p_1 waves to sound-speed perturbations located at z=-1.5 Mm (close to the depth of the sound-speed perturbation in our models). They found that the p_1 travel-time shift is below the observed noise level, even for cases with a sound speed perturbation much higher than that introduced in our models. Our results are more sensitive to the changes in the sound-speed because they include higher order p-modes (according to the phase-speed and frequency filters used in this work, we have measured up to p_4). Interestingly, we found that the lowest frequency for which the travel-time shifts depend on the sound-speed perturbation decreases with increasing phase speed, which agrees with the fact that higher phase-speed filters are sensitive to higher order p-modes at lower frequencies (see Fig. <ref>). The depth sensitivity of p_2 to p_4 modes provides a better sampling of the region where the sound-speed perturbation is located in our models. <cit.> suggested a path toward simplified travel-time inversion methods by selecting some combinations of phase-speed and frequency filters that are less sensitive to the magnetic field. The inversion should account for the changes in the Wilson depression and sound speed. This approach would eliminate the need to compute the sensitivity of wave travel times to the strength and geometry of themagnetic field <cit.>. Our results support this suggestion.We acknowledge the financial support by the Spanish Ministry of Economy and Competitiveness (MINECO) through projects AYA2014-55078-P, AYA2014-60476-P and AYA2014-60833-P. D.C.B acknowledges support from the NASA Living With a Star Program through grant NNX14AD42G awarded to NWRA. A.C.B. acknowledges the EU FP7 Collaborative Project “Exploitation of Space Data for Innovative Helio- and Asteroseismology” (SPACEINN). This work used the NASA's Pleiades supercomputer at Ames Research Center, Teide High-Performance Computing facilities at Instituto Tecnológico y de Energías Renovables (ITER, SA), and MareNostrum supercomputer at Barcelona Supercomputing Center. aa | http://arxiv.org/abs/1705.09135v1 | {
"authors": [
"T. Felipe",
"D. C. Braun",
"A. C. Birch"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170525114617",
"title": "Helioseismic Holography of Simulated Sunspots: dependence of the travel time on magnetic field strength and Wilson depression"
} |
A feature-rich transmission spectrum for WASP-127b E. Palle1,2 G. Chen1,2,3 J. Prieto-Arranz1,2 G. Nowak1,2 F. Murgas1,2 L. Nortmann1,2 D. Pollacco4 K. Lam4 P. Montanes-Rodriguez1,2 H. Parviainen1,2 N. Casasayas-Barris1,2Received Month 00, 2017; accepted Month 00, 2017 ============================================================================================================================================================================================================================================================================================= In this paper, we focus on learning structure-aware document representations from data without recourse to a discourse parser or additional annotations. Drawing inspiration from recent efforts to empower neural networks with a structural bias <cit.>, we propose a model that can encode a document while automatically inducing rich structural dependencies. Specifically, we embed a differentiable non-projective parsing algorithm into a neural model and use attention mechanisms to incorporate the structural biases. Experimental evaluations across different tasks and datasets show that the proposed model achieves state-of-the-art results on document modeling tasks while inducing intermediate structures which are both interpretable and meaningful.§ INTRODUCTIONDocument modeling is a fundamental task in Natural Language Processing useful to various downstream applications including topic labeling <cit.>, summarization <cit.>, sentiment analysis <cit.>, question answering <cit.>, and machine translation <cit.>.Recent work provides strong evidence that better document representations can be obtained by incorporating structural knowledge <cit.>. Inspired by existing theories of discourse, representations of document structure have assumed several guises in the literature, such as trees in the style of Rhetorical Structure Theory <cit.>, graphs <cit.>, entity transitions <cit.>, or combinations thereof <cit.>.The availability of discourse annotated corpora <cit.> has led to the development of off-the-shelf discourse parsers <cit.>, and the common use of trees as representations of document structure. For example, <cit.> improve document-level sentiment analysis by reweighing discourse units based on the depth of RST trees, whereas <cit.> show that a recursive neural network built on the output of an RST parser benefits text categorization in learning representations that focus on salient content. Linguistically motivated representations of document structure rely on the availability of annotated corpora as well as a wider range of standard NLP tools (e.g., tokenizers, pos-taggers, syntactic parsers). Unfortunately, the reliance on labeled data, which is both difficult and highly expensive to produce, presents a major obstacle to the widespread use of discourse structure for document modeling.Moreover, despite recent advances in discourse processing, the use of an external parser often leads to pipeline-style architectures where errors propagate to later processing stages, affecting model performance. It is therefore not surprising that there have been attempts to induce document representations directly from data without recourse to a discourse parser or additional annotations.The main idea is to obtain hierarchical representations by first building representations of sentences, and then aggregating those into a document representation <cit.>. <cit.> further demonstrate how to implicitly inject structural knowledge onto the representation using an attention mechanism <cit.> which acknowledges that sentences are differentially important in different contexts. Their model learns to pay more or less attention to individual sentences when constructing the representation of the document.Our work focus on learning deeper structure-aware document representations, drawing inspiration from recent efforts to empower neural networks with a structural bias <cit.>. <cit.> introduce structured attention networks which are generalizations of the basic attention procedure, allowing to learn sentential representations while attending to partial segmentations or subtrees. Specifically, they take into account the dependency structure of a sentence by viewing the attention mechanism as a graphical model over latent variables. They first calculate unnormalized pairwise attention scores for all tokens in a sentence and then use the inside-outside algorithm to normalize the scores with the marginal probabilities of a dependency tree.Without recourse to an external parser, their model learns meaningful task-specific dependency structures, achieving competitive results in several sentence-level tasks. However, for document modeling, this approach has two drawbacks. Firstly, it does not consider non-projective dependency structures, which are common in document-level discourse analysis <cit.>. As illustrated in Figure <ref>, the tree structure of a document can be flexible and the dependency edges may cross. Secondly, the inside-outside algorithm involves a dynamic programming process which is difficult to parallelize, making it impractical for modeling long documents.[In our experiments, adding the inside-outside pass increases training time by a factor of 10.] In this paper, we propose a new model for representing documents while automatically learning richer structural dependencies. Using a variant of Kirchhoff's Matrix-Tree Theorem <cit.>, our model implicitly considers non-projective dependency tree structures. We keep each step of the learning process differentiable, so the model can be trained in an end-to-end fashion and induce discourse information that is helpful to specific tasks without an external parser. The inside-outside model of <cit.> and our model both have a O(n^3) worst case complexity. However, major operations in our approach can be parallelized efficiently on GPU computing hardware. Although our primary focus is on document modeling, there is nothing inherent in our model that prevents its application to individual sentences. Advantageously, it can induce non-projective structures which are required for representing languages with free or flexible word order <cit.>.Our contributions in this work are threefold: a model for learning document representations whilst taking structural information into account; an efficient training procedure which allows to compute document level representations of arbitrary length; and a large scale evaluation study showing that the proposed model performs competitively against strong baselines while inducing intermediate structures which are both interpretable and meaningful.§ BACKGROUNDIn this section, we describe how previous work uses the attention mechanism for representing individual sentences.The key idea is to capture the interaction between tokens within a sentence, generating a context representation for each word with weak structural information. This type of intra-sentence attention encodes relationships between words within each sentence and differs from inter-sentence attention which has been widely applied to sequence transduction tasks like machine translation <cit.> and learns the latent alignment between source and target sequences.Figure <ref> provides a schematic view of the intra-sentential attention mechanism.Given a sentence represented as a sequence of n word vectors [u_1,u_2,⋯,u_n], for each word pair ⟨u_i, u_j⟩, the attention score a_ij is estimated as:f_ij = F(u_i, u_j) a_ij = exp(f_ij)/∑_k=1^nexp(f_ik)where F() is a function for computing the unnormalized score f_ij which is then normalized by calculating a probability distribution a_ij.Individual words collect information from their context based ona_ij and obtain a context representation:r_i = ∑_j=1^na_iju_jwhere attention score a_ij indicates the (dependency) relation between the i-th and theand how information from u_j should be fed into u_i.Despite successful application of the above attention mechanism in sentiment analysis <cit.> and entailment recognition <cit.>, the structural information under consideration is shallow, limited to word-word dependencies. Since attention is computed as a simple probability distribution, it cannot capture more elaborate structural dependencies such as trees (or graphs).<cit.> induce richer internal structure by imposing structural constraints on the probability distribution computed by the attention mechanism. Specifically, they normalize f_ij with a projective dependency tree using the inside-outside algorithm <cit.>:f_ij = F(u_i, u_j) a = inside-outside(f) r_i = ∑_j=1^na_iju_jThis process is differentiable, so the model can be trained end-to-end and learn structural information without relying on a parser. However, efficiency is a major issue, since the inside-outside algorithm has time complexity O(n^3) (where n represents the number of tokens) and does not lend itself to easy parallelization. The high order complexity renders the approach impractical for real-world applications. § ENCODING TEXT REPRESENTATIONSIn this section we present our document representation model. We follow previous work <cit.> in modeling documents hierarchically by first obtaining representations for sentences and then composing those into a document representation. Structural information is taken into account while learning representations for both sentences and documents and an attention mechanism is applied on both words within a sentence and sentences within a document. The general idea is to force pair-wise attention between text units to form a non-projective dependency tree, and automatically induce this tree for different natural language processing tasks in a differentiable way. In the following, we first describe how the attention mechanism is applied to sentences, and then move on to present our document-level model. §.§ Sentence ModelLet T=[u_1, u_2, ⋯, u_n] denote a sentence containing a sequence of words, each represented by a vector u, which can be pre-trained on a large corpus.Long Short-Term Memory Neural Networks <cit.> have been successfully applied to various sequence modeling tasks ranging from machine translation <cit.>, to speech recognition <cit.>, and image caption generation <cit.>.In this paper we use bidirectional LSTMs as a way of representing elements in a sequence (i.e., words or sentences) together with their contexts, capturing the element and an “infinite” window around it. Specifically, we run a bidirectional LSTM over sentence T, and take the output vectorsas the representations of words in T, where h_t ∈ℝ^k is the output vector for word u_t based on its context.We then exploit the structure of T which we induce based on an attention mechanism detailed below to obtain more precise representations.Inspired by recent work <cit.>, which shows that the conventional way of using LSTM output vectors for calculating both attention and encoding word semantics is overloaded and likely to cause performance deficiencies, we decompose the LSTM output vector in two parts:[e_t, d_t] = h_twhere e_t ∈ℝ^k_t, the semantic vector, encodes semantic information for specific tasks, and d_t∈ℝ^k_s, the structure vector, is used to calculate structured attention. We use a series of operations based on the Matrix-Tree Theorem <cit.> to incorporate the structural bias of non-projective dependency trees into the attention weights.We constrain the probability distributions a_ij (see Equation (<ref>)) to be the posterior marginals of a dependency tree structure. We then use the normalized structured attention, to build a context vector for updating the semantic vector of each word, obtaining new representations [r_1, r_2, ⋯, r_n].An overview of the model is presented in Figure <ref>. We describe the attention mechanism in detail in the following section. §.§ Structured Attention Mechanism Dependency representations of natural language are a simple yet flexible mechanism for encoding words and their syntactic relations through directed graphs. Much work in descriptive linguistics <cit.> has advocated their suitability for representing syntactic structure across languages. A primary advantage of dependency representations is that they have a natural mechanism for representing discontinuous constructions arising from long distance dependencies or free word order through non-projective dependency edges. More formally, building a dependency tree amounts to finding latent variables z_ij for all i≠ j, where word i is the parent node of word j, under some global constraints, amongst which theconstraint is the most important, since it forces the structure to be a rooted tree. We use a variant of Kirchhoff's Matrix-Tree Theorem <cit.> to calculate the marginal probability of each dependency edge P(z_ij=1) of a non-projective dependency tree, and this probability is used as the attention weight that decides how much information is collected from child unit j to the parent unit i. We first calculate unnormalized attention scores f_ij with structure vector d (see Equation (<ref>)) via a bilinear function:t_p = tanh(W_pd_i) t_c = tanh(W_cd_j) f_ij = t_p^TW_at_cwhere W_p ∈ℝ^k_s*k_s and W_c ∈ℝ^k_s*k_s are the weights for building the representation of parent and child nodes.W_a ∈ℝ^k_s*k_s is the weight for the bilinear transformation.f∈ℝ^n*n can be viewed as a weighted adjacency matrix for a graph G with n nodes where each node corresponds to a word in a sentence.We also calculate the root score f^r_i, indicating the unnormalized possibility of a node being the root:f^r_i = W_rd_iwhere W_r ∈ℝ^1*k_s. We calculate P(z_ij=1), the marginal probability of the dependency edge, following <cit.>:A_ij = 0if i=jexp(f_ij)otherwise L_ij =∑_i'=1^n A_i'j if i=j - A_ij otherwise L̅_ij = exp(f^r_i) i=1 L_iji>1 P(z _ij=1) = (1-δ_1,j)A_ij[L̅^-1]_jj -(1-δ_i,1)A_ij[L̅^-1]_jiP(r oot(i)) = exp(f^i_r)[L̅^-1]_i1 where 1≤ i≤ n, 1≤ j≤ n. L∈ℝ^n*n is the Laplacian matrix forgraph G and L̅∈ℝ^n*n is a variant of L that takes theroot node into consideration, and δ is the Kronecker delta.The key for the calculation to hold is for L^ii, the minorof the Laplacian matrix L with respect to row i and columni, to be equal to the sum of the weights of all directed spanningtrees of G which are rooted at i.P(z_ij=1) is the marginalprobability of the dependency edge between the i-th andj-th words. P(root(i)=1) is the marginal probability of thei-th word headed by the root of the tree.Details of the proof canbe found in <cit.>. We denote the marginal probabilities P(z_ij=1) as a_ijand P(root(i)) as a^r_i. This can be interpreted asattention scores which are constrained to converge to a structuredobject, a non-projective dependency tree, in our case.We update thesemantic vector e_i of each word with structured attention:p_i= ∑_k=1^na_kie_k+a^r_ie_root c_i= ∑_k=1^na_ike_i r_i= tanh(W_r[e_i, p_i, c_i])where p_i ∈ℝ^k_e is the context vector gathered from possible parents of u_i and c_i ∈ℝ^k_e the context vector gathered from possible children, and e_root is a special embedding for the root node. The context vectors are concatenated with e_i and transformed with weights W_r ∈ℝ^k_e*3k_e to obtain the updated semantic vector r_i ∈ℝ^k_e with rich structural information (see Figure <ref>). §.§ Document ModelWe build document representations hierarchically: sentences are composed of words and documents are composed of sentences. Composition on the document level also makes use of structured attention in the form of a dependency graph. Dependency-based representations have been previously used for developing discourse parsers <cit.> and in applications such as summarization <cit.>.As illustrated in Figure <ref>, given a document with n sentences [s_1,s_2,⋯,s_n], for each sentence s_i, the input is a sequence of word embeddings [u_i1,u_i2,⋯,u_im], where m is the number of tokens in s_i.By feeding the embeddings into a sentence-level bi-LSTM and applying the proposed structured attention mechanism, we obtain the updated semantic vector [r_i1,r_i2,⋯,r_im].Then a pooling operation produces a fixed-length vector v_i for each sentence. Analogously, we view the document as a sequence of sentence vectors [v_1, v_2,⋯, v_n] whose embeddings are fed to a document-level bi-LSTM. Application of the structured attention mechanism creates new semantic vectors [q_1,q_2,⋯,q_n] and another pooling operation yields the final document representation y.§.§ End-to-End TrainingOur model can be trained in an end-to-end fashion since all operations required for computing structured attention and using it to update the semantic vectors are differentiable.In contrast to in <cit.>, training can be done efficiently. The major complexity of our model lies in the computation of the gradients of the the inverse matrix. Let A denote a matrix depending on a real parameter x; assuming all component functions in A are differentiable, and A is invertible for all possible values, the gradient of A with respect respect to x is:dA^-1/dx = -A^-1dA/dxA^-1Multiplication of the three matrices and matrix inversion can be computed efficiently on modern parallel hardware architectures such as GPUs. In our experiments, computation of structured attention takes only 1/10 of training time. § EXPERIMENTS In this section we present our experiments for evaluating the performance of our model. Since sentence representations constitute the basic building blocks of our document model, we first evaluate the performance of structured attention on a sentence-level task, namely natural language inference. We then assess the document-level representations obtained by our model on a variety of classification tasks representing documents of different length, subject matter, and language. Our code is available at <https://github.com/nlpyang/structured>.§.§ Natural Language InferenceThe ability to reason about the semantic relationship between two sentences is an integral part of text understanding. We therefore evaluate our model on recognizing textual entailment, i.e., whether two premise-hypothesis pairs are entailing, contradictory, or neutral. For this task we used the Stanford Natural Language Inference (SNLI) dataset <cit.>, which contains premise-hypothesis pairs and target labels indicating their relation. After removing sentences with unknown labels, we obtained 549,367 pairs for training, 9,842 for development and 9,824 for testing. Sentence-level representations obtained by our model (with structured attention) were used to encode the premise and hypothesis by modifying the model of <cit.> as follows.Let [x^p_1, ⋯, x^p_n] and [x^h_1, ⋯, x^h_m] be the input vectors for the premise and hypothesis, respectively. Application of structured attention yields new vector representations [r^p_1, ⋯, r^p_n] and [r^h_1, ⋯, r^h_m]. Then we combine these two vectors with inter-sentential attention, and apply an average pooling operation:o_ij = MLP(r^p_i)^TMLP(r^h_j) r̅^p_i = [r^p_i, ∑_j=1^mexp(o_ij)/∑_k=1^mexp(o_ik)] r̅^h_i = [r^h_i, ∑_i=1^mexp(o_ij)/∑_k=1^mexp(o_kj)] r^p = ∑_i=1^ng(r̅^p_i),r^h = ∑_i=1^mg(r̅^h_i)where MLP() is a two-layer perceptron with a ReLU activation function. The new representations r^p, r^h are then concatenated and fed into another two-layer perceptron with a softmax layer to obtain the predicted distribution over the labels. The hidden size of the LSTM was set to 150. The dimensions of the semantic vector were 100 and the dimensions of structure vector were 50.We used pretrained 300-D Glove 840B <cit.> vectors to initialize the word embeddings. All parameters (including word embeddings) were updated with Adagrad <cit.>, and the learning rate was set to 0.05. The hidden size of the two-layer perceptron was set to 200 and dropout was used with ratio 0.2.The mini-batch size was 32. We compared our model (and variants thereof) against several related systems. Results (in terms of 3-class accuracy) are shown in Table <ref>. Most previous systems employ LSTMs and do not incorporate a structured attention component. Exceptions include <cit.> and <cit.> whose models include intra-attention encoding relationships between words within each sentence (see Equation (<ref>)). It is also worth noting that some models take structural information into account in the form of parse trees <cit.>. The second block of Table <ref> presents a version of our model without an intra-sentential attention mechanism as well as three variants with attention, assuming the structure of word-to-word relations and dependency trees. In the latter case we compare our matrix inversion based model against Kim et al.'s () inside-outside attention model. Consistent with previous work <cit.>, we observe that simple attention brings performance improvements over no attention. Structured attention further enhances performance. Our own model with tree matrix inversion slightly outperforms the inside-outside model of <cit.>, overall achieving results in the same ballpark with related LSTM-based models <cit.>.Table <ref> compares the running speed of the models shown in the second block of Table <ref>.As can be seen matrix inversion does not increase running speed over the simpler attention mechanism and is considerably faster compared to inside-outside. The latter is 10–20 times slower than our model on the same platform. §.§ Document ClassificationIn this section, we evaluate our document-level model on a variety of classification tasks. We selected four datasets which we describe below. Table <ref> summarizes some statistics for each dataset.Yelp reviews were obtained from the 2013 Yelp Dataset Challenge.This dataset contains restaurant reviews, each associated with human ratings on a scale from 1 (negative) to 5 (positive) which we used as gold labels for sentiment classification; we followed the preprocessing introduced in <cit.> and report experiments on their training, development, and testing partitions (80/10/10). IMDB reviews were obtained from <cit.>, who randomly crawledreviews for 50K movies. Each review is associated with user ratings ranging from 1 to 10.Czech reviews were obtained from <cit.>. The dataset contains reviews from the Czech Movie Database[<http://www.csfd.cz/>] each labeled as positive, neutral, or negative. We include Czech in our experiments since it has more flexible word order compared to English, with non-projective dependency structures being more frequent. Experiments on this dataset perform 10-fold cross-validation following previous work <cit.>.Congressional floor debates were obtained from a corpus originally created by <cit.> which contains transcripts of U.S. floor debates in the House of Representatives for the year 2005. Each debate consists of a series of speech segments, each labeled by the vote (“yea” or “nay”) cast for the proposed bill by the the speaker of each segment. We used the pre-processed corpus from <cit.>.[<http://www.cs.cornell.edu/ ainur/data.html>]Following previous work <cit.>, we only retained words appearing more than five times in building the vocabulary and replaced words with lesser frequencies with a special UNK token. Word embeddings were initialized by training word2vec <cit.> on the training and validation splits of each dataset.In our experiments, we set the word embedding dimension to be 200 and the hidden size for the sentence-level and document-level LSTMs to 100 (the dimensions of the semantic and structure vectors were set to 75 and 25, respectively).We used a mini-batch size of 32 during training and documents of similar length were grouped in one batch.Parameters were optimized with Adagrad <cit.>, the learning rate was set to 0.05.We used L_2 regularization for all parameters except word embeddings with regularization constant set to 1e^-4. Dropout was applied on the input and output layers with dropout rate 0.3. Our results are summarized in Table <ref>.We compared our model against several related models covering a wide spectrum of representations including word-based ones (e.g., paragraph vector and CNN models) as well as hierarchically composed ones (e.g., a CNN or LSTM provides a sentence vector and then a recurrent neural network combines the sentence vectors to form a document level representation for classification).Previous state-of-the-art results on the three review datasets were achieved by the hierarchical attention network of <cit.>, which models the document hierarchically with two GRUs and uses an attention mechanism to weigh the importance of each word and sentence.On the debates corpus, <cit.> obtained best results with a recursive neural network model operating on the output of an RST parser. Table <ref> presents three variants[We do not report comparisons with the inside-outside approach on document classification tasks due to its prohibitive computation cost leading to 5 hours of training for one epoch.] of our model, one with structured attention on the sentence level, another one with structured attention on the document level and a third model which employs attention on both levels. As can be seen, the combination is beneficial achieving best results on three out of four datasets. Furthermore, structured attention is superior to the simpler word-to-word attention mechanism, and both types of attention bring improvements over no attention. The structured attention approach is also very efficient, taking only 20 minutes for one training epoch on the largest dataset.§.§ Analysis of Induced StructuresTo gain further insight on structured attention, we inspected the dependency trees it produces. Specifically, we used the Chu-Liu-Edmonds algorithm <cit.> to extract the maximum spanning tree from the attention scores. We report various statistics on the characteristics of the induced trees across different tasks and datasets. We also provide examples of tree output, in an attempt to explain how our model uses dependency structures to model text.Sentence Trees We compared the dependency trees obtained from our model with those produced by a state-of-the-art dependency parser trained on the English Penn Treebank. Table <ref> presents various statistics on the depth of the trees produced by our model on the SNLI test set and the Stanford dependency parser <cit.>. As can be seen, the induced dependency structures are simpler compared to those obtained from the Stanford parser. The trees are generally less deep (their height is 5.78 compared to 8.99 for the Stanford parser), with the majority being of depth 2–4.Almost half of the induced trees have a projective structure, although there is nothing in the model to enforce this constraint. We also calculated the percentage of head-dependency edges that are identical between the two sets of trees. Although our model is not exposed to annotated trees during training, a large number of edges agree with the output of the Stanford parser.Figure <ref> shows examples of dependency trees induced on the SNLI dataset.Although the model is trained without ever being exposed to a parse tree, it is able to learnplausible dependency structures via the attention mechanism.Overall we observe that the induced trees differ from linguistically motivated ones in the types of dependencies they create which tend to be of shorter length. The dependencies obtained from structured attention are more direct as shown in the first premise sentence in Figure <ref> where words at and bar are directly connected to the verb drink. This is perhaps to be expected since the attention mechanism uses the dependency structures to collect information from other words, and the direct links will be more effective.Document Trees We also used the Chu-Liu-Edmonds algorithms to obtain document-level dependency trees.Table <ref> summarizes various characteristics of these trees. For most datasets, document-level trees are not very deep, they mostly contain up to nodes of depth 3. This is not surprising as the documents are relatively short (see Table <ref>) with the exception of debates which are longer and the induced trees more complex. The fact that most documents exhibit simple discourse structures is further corroborated by the large number (over 70%) of projective trees induced on Yelp, IMBD, and CZ Movies datasets. Unfortunately, our trees cannot be directly compared with the output of a discourse parser which typically involves a segmentation process splitting sentences into smaller units. Our trees are constructed over entire sentences, and there is no mechanism currently in the model to split sentences into discourse units.Figure <ref> shows examples of document-level trees taken from Yelp and the Czech Movie dataset.In the first tree, most edges are examples of the “elaboration” discourse relation, i.e., the child presents additional information about the parent. The second tree is non-projective, the edges connecting sentences 1 and 4 and 3 and 5 cross.The third review, perhaps due to its colloquial nature, is not entirely coherent. However, the model manages to link sentences 1 and 3 to sentence 2, i.e., the movie being discussed; it also relates sentence 6 to 4, both of which express highly positive sentiment. § CONCLUSIONSIn this paper we proposed a new model for representing documents while automatically learning rich structural dependencies. Our model normalizes intra-attention scores with the marginal probabilities of a non-projective dependency tree based on a matrix inversion process. Each operation in this process is differentiable and the model can be trained efficiently end-to-end, while inducing structural information. We applied this approach to model documents hierarchically, incorporating both sentence- and document-level structure. Experiments on sentence and document modeling tasks show that the representations learned by our model achieve competitive performance against strong comparison systems. Analysis of the induced tree structures revealed that they are meaningful, albeit different from linguistics ones, without ever exposing the model to linguistic annotations or an external parser.Directions for future work are many and varied. Given appropriate training objectives <cit.>, it should be possible to induce linguistically meaningful dependency trees using the proposed attention mechanism. We also plan to explore how document-level trees can be usefully employed in summarization, e.g., as a means to represent or even extract important content.Acknowledgments The authors gratefully acknowledge the support of the European Research Council (award number 681760). We also thank the anonymous TACL reviewers and the action editor whose feedback helped improve the present paper,members of EdinburghNLP for helpful discussions and suggestions, and Barbora Skarabela for translating the Czech document for us.acl_natbib | http://arxiv.org/abs/1705.09207v4 | {
"authors": [
"Yang Liu",
"Mirella Lapata"
],
"categories": [
"cs.CL",
"cs.AI"
],
"primary_category": "cs.CL",
"published": "20170525145407",
"title": "Learning Structured Text Representations"
} |
a1]Samir Karasuljić c1 [email protected]]Enes Duvnjaković [email protected]]Vedad Pasic [email protected]]Elvis Barakovic [email protected][a1]Department of Mathematics, Faculty of Natural Sciences and Mathematics, University of Tuzla, Univerzitetska 4, Tuzla, Bosnia and Herzegovina 0.1cm[c1]Corresponding authorS. Karasuljić, E. Duvnjaković, V. Pasic, E. Barakovic We consider an approximate solution for the one–dimensional semilinear singularly–perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an ε–uniform convergence of such gained the approximate solutions, in the maximum norm of the order 𝒪(N^-1) on the observed domain.After that, the constructed approximate solution is repaired and we obtain a solution, which also has ε–uniform convergence, but now of order 𝒪(ln^2N/N^2) on [0,1]. In the end a numerical experiment is presented to confirm previously shown theoretical results. Singular perturbationnonlinearboundary layerBakhvalov meshlayer-adapted meshuniform convergence.65L1065L1165L50.Construction of a global solution for the one dimensional singularly-perturbed boundary value problem [ December 30, 2023 ======================================================================================================definition theoremTheorem[section] lemma[theorem]Lemma proposition[theorem]Proposition corollary[theorem]Corollary definition[theorem]Definition example[theorem]Example xca[theorem]Exercise problem[theorem]Problem remark remark[theorem]Remark equationsection§ INTRODUCTION We will consider the singularly–perturbed boundary value problemε^2y” =f(x,y), x∈ I=[0,1], y(0)=0, y(1)=0,with the condition∂ f(x,y)∂ y:=f_y⩾ m>0, ∀(x,y)∈ I×ℝ,where 0<ε<1 is a perturbation parameter, f is a nonlinear function f∈ C^k(I×ℝ), k⩾ 2 and m is a real constant.The boundary value problem (<ref>)–(<ref>), with the condition (<ref>), has a unique solution, see <cit.>. Contributions tonumerical solutions of the problem (<ref>)–(<ref>) with different assumptions on the function f and similar problems were obtained by many authors, see for example Flaherty and O'Malley <cit.>, Cvetković and Herceg <cit.>, Herceg <cit.>, Herceg, Surla and Rapajić <cit.>, Kopteva <cit.>, Linß and Vulanović <cit.>, Niijima <cit.>, Stynes and O'Riordan<cit.>, Vulanović <cit.> etc. The method that will be used in this paper in order to obtain a discrete approximate solution, i.e. values of the approximate solution in the mesh points, of the problem (<ref>)–(<ref>) was first developed by Boglaev <cit.>, who constructed a difference scheme and showed convergence of order 1 on the modified Bakhvalov mesh. Using the method of <cit.>, we constructed new difference schemes in <cit.> and we carried out numerical experiments.In <cit.> we constructed new difference schemes and we proved uniqueness of the numerical solution and ε–uniform convergence on the modified Shishkin mesh and at the end presented numerical experiments. In this paper we will use the difference scheme from <cit.> in order to calculate values of the approximate solution of the problem on the mesh points and then construct an approximate solution. § THEORETICAL BACKGROUND AND KNOWN RESULTS Let us set up an arbitrary mesh on [0,1]0=x_0<x_1<…<x_N=1.A construction of a difference scheme, which will be used for calculation of the approximate solution of the problem (<ref>)–(<ref>) in the mesh points, is based on the representation of the exact solution on the interval [x_i,x_i+1], i=0,…,N-1y_i(x)=y_iu_i^I(x)+y_i+1u_i^II(x)+∫_x_i^x_i+1G_i(x,s)ψ(s,y(s)) s,where G_i(x,s) is the Green's functionG_i(x,s)=1/ε^2w_i(s){[ u_i^II(x)u_i^I(s),x_i⩽ x⩽ s⩽ x_i+1,; ; u_i^I(x)u_i^II(s),x_i⩽ s⩽ x⩽ x_i+1, ]. ψ(s,y(s))=f(s,y(s))-γ y(s),and w_i(s)=-β/sinh(β h_i), s∈[x_i,x_i+1], u_i^I(x)=sinh(β(x_i+1-x))/sinh(β h_i), u_i^II(x)=sinh(β(x-x_i))/sinh(β h_i), h_i=x_i+1-x_i, β=√(γ)/ε, y_i:=y(x_i) andγ is a constant for whichγ⩾ f_y, (details can be found in <cit.>). The difference scheme constructed in <cit.>, which we will use, has the following form a_i+d_i2y_i-1-( a_i+d_i2+a_i+1+d_i+12)y_i+ a_i+1+d_i+12y_i+1 = d_iγf_i-1+ d_i+1γf_i,where y_k, k∈{i-1,i,i+1} are values of the approximate solution in the mesh points, d_i=d_i-a_i, d_i=1/tanh(β h_i), a_i=1/sinh(β h_i) and f_i=f((x_i+x_i+1)/2,(y_i+y_i+1)/2), i=1,…,N-1. The difference scheme generates a system of nonlinear equations and the solutions of this system are values of the approximate solution in the mesh points. An answer to the question of existence and uniqueness will be given in the next theorem, however before that, it is necessary to define the operator (or discrete problem) F:ℝ^N+1↦ℝ^N+1 and a corresponding norm that is necessary in formulation of the theorem. Therefore, we will now use the difference scheme (<ref>) in order to obtain a discrete problem of the problem (<ref>)–(<ref>). We have thatFy=( (Fy)_0,(Fy)_1,…,( Fy)_N )^T=0,where(Fy)_0 := 0, (Fy)_i := γ/ d_i+ d_i+1[ a_i+d_i2y_i-1-( a_i+d_i2+a_i+1+d_i+12)y_i. +. a_i+1+d_i+12y_i+1 - d_iγf_i-1- d_i+1γf_i], i=1,…,N-1 (Fy)_N := 0. Here we use the maximum norm u_∞=max_0⩽ i⩽ N|u_i|,for any vector u=(u_0,u_1,…,u_n)^T∈ℝ^N+1 and the corresponding matrix norm. <cit.> The discrete problem (<ref>) for γ≥ f_y, hasthe unique solutiony=(y_0, y_1, y_2, …, y_N-1, y_N)^T, with y_0=y_N=0. Moreover, the following stability inequality holdsw-v_∞⩽1/mFw-Fv_∞,for any vectors v=(v_0,v_1,…,v_N)^T∈ℝ^N+1, w=( w_0,w_1,…,w_N)^T∈ℝ^N+1.The mesh that will be used here is a modified Shishkin mesh from <cit.>, which has a greater smoothness compared to the generating function. Before the construction of the mesh, we are stating a theorem about the decomposition and estimates of the derivatives, which is necessary for the construction and further analysis. <cit.> The solution y to the problem (<ref>)–(<ref>) can be represented in the following wayy=r+s,where for i=0,1,…,k and x∈[0,1] we have that |r^(i)(x)|⩽ C, |s^(i)(x)|⩽ Cε^-i(e^-x/ε√(m)+e^-1-x/ε√(m)). LetN+1 be the number of mesh points, q∈(0,1/2) and σ>0 be the mesh parameter. We will define the transition point of the Shishkin mesh withλ:=min{σε/√(m)ln N,q}.Let σ=2.Forthe sake of simplicity in representation, we assume that λ=2ε (√(m))^-1ln N, as otherwise the problem can be analysed in the classical way. We shall also assume that q N is an integer. This is easily achieved by choosing q=1/4 and N divisible by 4 for example. The mesh :x_0<x_1<...<x_N is generated by x_i=φ(i/N) with the mesh generating functionφ(t):={[λqt t∈[0,q],; p(t-q)^3+λ t∈[q,1/2],; 1-φ(1-t) t∈[1/2,1], ].where pis chosen so that φ(1/2)=1/2, i.e. p=12(1-λq)(12-q)^-3. Note that φ∈ C^1[0,1] with φ'_∞,φ”_∞≤ C. Therefore the mesh sizes h_i=x_i+1-x_i, i=0,1,...,N-1 satisfyh_i⩽C/N and|h_i+1-h_i|⩽C/N^2,see <cit.> for details.<cit.> The difference scheme (<ref>) on the mesh generated by the function (<ref>)is uniformly convergent with respect to ε andmax_0≤ i≤ N|y(x_i)-y_i|≤ Cln^2 NN^2,where y(x) is the solution of the problem (<ref>)–(<ref>), y is the corresponding numerical solution of (<ref>), and C>0 is a constant independent of N and ε. § MAIN RESULTS On the interval[x_i,x_i+1] using the representation (<ref>), we look for an approximate solution in the following formỹ_i(x)= y_iu_i^I(x)+y_i+1u_i^II(x)+ψ_i∫_x_i^x_i+1G_i(x,s) s, i=0,…,N-1,whereψ_i=ψ((x_i+x_i+1)/2,(y_i+y_i+1)/2), i=0,1,…,N-1.We obtain that it is∫_x_i^x_i+1G_i(x,s) s=-sinh(β(x_i+1-x))/γsinh(β h_i)[ cosh(β(x-x_i))-1] - sinh(β(x-x_i))/γsinh(β h_i)[ cosh(β(x_i+1-x))-1], i=0,…,N-1. We are looking for an approximate solution on [0,1] in the form .Y(x)|_[x_i,x_i+1]=ỹ_i(x), i=0,…,N-1.Using the maximum norm, we estimate the difference between the exact solution of the problem (<ref>)–(<ref>) and approximate solutions given by (<ref>). This difference will be estimated on each interval [x_i,x_i+1], i=0,…,N-1. Taking into account (<ref>), (<ref>) and(<ref>), we have that|y_i(x)-ỹ_i(x)|⩽|y_i-y_i||u^I_i(x)|+|y_i+1-y_i+1||u^II_i(x)|+|∫_x_i^x_i+1G_i(x,s)( ψ(s,y(s))-ψ_i)s|, i=0,…,N-1. An estimate of the value of difference |y(x)-Y(x)|, ∀ x∈[0,1], or estimate of the error will be done for [0,1/2]. An analogue estimate would hold on [1/2,1].Note that e^-x√(m)/ε⩾ e^-(1-x)√(m)/ε and h_i+1⩾ h_i for x∈[0,1/2] and e^-x√(m)/ε⩽ e^-(1-x)√(m)/ε and h_i+1⩽ h_i for x∈[1/2,1].Let us first estimate ∫_x_i^x_i+1G_i(x,s) s for x∈[0,λ].Forx∈[x_i,x_i+1], i=0,…,N/4-1, we have the following estimate| sinh(β(x_i+1-x))/γsinh(β h_i)[ cosh(β(x-x_i))-1].. + sinh(β(x-x_i))/γsinh(β h_i)[ cosh(β(x_i+1-x))-1]|⩽Cln^2N/N^2. sinh(β(x_i+1-x))/γsinh(β h_i)[ cosh(β(x-x_i))-1] + sinh(β(x-x_i))/γsinh(β h_i)[ cosh(β(x_i+1-x))-1]=sinh(β (x_i+1-x_i))-sinh(β(x_i+1-x))-sinh(β(x-x_i))/γsinh(β h_i)=β h_i+β^3h_i^3/6+𝒪_1(β^5h^5_i )-β(x_i+1-x)-β^3(x_i+1-x)^3/6-𝒪_2(β^5(x_i+1-x)^5 )/γ[ β h_i+β^3h^3_i/6+𝒪_1(β^5h^5_i)] -β(x-x_i)+β^3(x-x_i)^3/6+𝒪_3(β^5(x-x_i)^5 )/γ[ β h_i+β^3h^3_i/6+𝒪_ 1(β^5h^5_i)] =1/2β^3(x-x_i)(x-x_i+1)(x_i-x_i+1)/γ[ β h_i+β^3h^3_i/6+𝒪_1(β^5h^5_i)]+𝒪_1(β^5h^5_i )-𝒪_2(β^5(x_i+1-x)^5 )-𝒪_3(β^5(x-x_i)^5 )/γ[ β h_i+β^3h^3_i/6+𝒪_1(β^5h^5_i)] .Furthermore, based on the value of parameter β and the properties of the mesh, we have that |1/2β^3(x-x_i)(x-x_i+1)(x_i-x_i+1)/γ[ β h_i+β^3h^3_i/6+𝒪_1(β^5h^5_i)].+.𝒪_1(β^5h^5_i )-𝒪_2(β^5(x_i+1-x)^5 )-𝒪_3(β^5(x-x_i)^5 )/γ[ β h_i+β^3h^3_i/6+𝒪_1(β^5h^5_i)]|⩽ C_1ln^3N/N^3+ln^5N/N^5/ln N/N⩽Cln^2 N/N^2 .Now, using (<ref>), we obtain (<ref>). For x∈[x_i,x_i+1], i=N/4,…,N/2-1, we have the following estimate| sinh(β(x_i+1-x))/γsinh(β h_i)[ cosh(β(x-x_i))-1].. + sinh(β(x-x_i))/γsinh(β h_i)[ cosh(β(x_i+1-x))-1]|⩽ C.In the proof of the Lemma <ref>, it is shown thatsinh(β(x_i+1-x))/γsinh(β h_i)[ cosh(β(x-x_i))-1] + sinh(β(x-x_i))/γsinh(β h_i)[ cosh(β(x_i+1-x))-1]=sinh(β (x_i+1-x_i))-sinh(β(x_i+1-x))-sinh(β(x-x_i))/γsinh(β h_i) .We get that| sinh(β (x_i+1-x_i))-sinh(β(x_i+1-x))-sinh(β(x-x_i))/γsinh(β h_i)| ⩽1/γ(1+|sinh(β(x_i+1-x))/sinh(β h_i)|+| sinh(β(x-x_i))/sinh(β h_i)| ) ⩽ C. Lety be the exact solution of the problem (<ref>)–(<ref>),and Y be the appropriate approximate solution given in (<ref>). We have the following estimatemax_x|y(x)-Y(x)|⩽ C {[ ln^2NN^2,x∈[0,λ],; ; 1N, x ∈[λ,1-λ],; ; ln^2NN^2, x∈ [1-λ,1], ].where the constant C does not depend on the perturbation parameter ε nor N. We divide [0,1] by the mesh points x_i, i=1,…,N-1 into subintervals [x_i,x_i+1], i=0,…,N. Since Y(x)=ỹ_i(x) on [x_i,x_i+1], we estimate the difference|y(x)-ỹ_i(x)| on each subinterval [x_i,x_i+1]. Based on representations of the exact solution (<ref>) and the approximate solution (<ref>) on the interval [x_i,x_i+1], we have that the estimate (<ref>) holds and| y_i(x)-ỹ_i(x)| ⩽|y_i-y_i|u^I_i(x)+|y_i+1-y_i+1|u^II_i(x) +|∫_x_i^x_i+1G_i(x,s)[ψ(s,y(s))- ψ_i] s|.Let us first estimate the differenceψ(x,y(x))- ψ_i on the interval [x_i,x_i+1], i=0,…,N/4-1, which appears in the integrand in (<ref>). Using Lagrange's theorem we obtain|ψ(x,y(x))- ψ_i|= |f(x,y(x))-f(x_i+x_i+12,y_i+y_i+12) -γ(y(x)-y_i+y_i+1/2)|= |(∂ f(ξ,η)/∂ y-γ)(y(x)-y_i+y_i+1/2)+∂ f(ξ,η)/∂ x(x-x_i+x_i+1/2)| ⩽ Cln N/N.Let now i=N/4+1,…,N/2-1.We have that|ψ(x,y(x))- ψ_i|= |f(x,y(x))-f(x_i+x_i+12,y_i+y_i+12)-γ(y(x)-y_i+y_i+1/2)| = |(∂ f(ξ,η)/∂ y-γ)(y(x)-y_i+y_i+1/2)+∂ f(ξ,η)/∂ x(x-x_i+x_i+1/2)| ⩽ C/N,where ξ∈(x,(x_i+x_i+1)/2) or ξ∈((x_i+x_i+1)/2,x) in (<ref>), andη∈ (y,(y_i+y_i+1)/2) orη∈((y_i+y_i+1)/2,y) in (<ref>). Let us estimate another difference ψ(x,y(x))- ψ_i on the interval [ N/4,N/4+1]. Since ε^2y”(x)=f(x,y(x)), we get the estimate|f(x,y(x))-f(x_i+x_i+12,y_i+y_i+12)|⩽|f(x,y(x)|+|f(x_i+x_i+12,y_i+y_i+12)| ⩽C/N^2.Now, from |y(x_i)-y_i|⩽Cln^2 N/N^2, i=0,…,N,and decomposition and estimates from Theorem <ref>, we get the following estimate|y(x)-y_i+y_i+1/2|⩽ |y(x)-y(x_i)+y(x_i+1)/2|+C_1ln^2 N/N^2 ⩽ | s(x)-s(x_i)+s(x_i+1)/2| +|r(x)-r(x_i)+r(x_i+1)/2|+C_1ln^2 N/N^2 ⩽ C_2/N^2+|r'(μ)|(x-x_i+x_i+1/2)+C_1ln^2 N/N^2⩽C/N,where μ∈(x,(x_i+x_i+1)/2) or μ∈((x_i+x_i+1)/2,x). Now from(<ref>), Lemma <ref>, Lemma <ref>, and the estimates (<ref>), (<ref>), (<ref>) and(<ref>)the assertion of the theorem follows. According the proof of the previous theorem it isshown thatthe difference between the exact and approximate solution| y(x)-Y(x)| on [0,λ] is of the order 𝒪(ln^2 N/N^2), while on [λ,1-λ] that order of the error is𝒪(1/N). Based on the Theorem <ref>, the difference between the exact and the approximate solution on the mesh points is of order𝒪(ln^2N/N^2). In order to get the approximatesolution with a satisfactory value of the error, we must conduct the correction of the approximatesolutions given in (<ref>). Namely, since this constructed approximate solution performs well at the layer, which is the most problematic part of the analysis, we will take on this part the approximate solution which was given in (<ref>). In the remaining part of the observed domain, i.e. for x∈[λ,1-λ] we will use a piecewise linear function. Therefore, for x∈[0,λ]∪[1-λ,1], we useỹ_i(x)=y_iu^I_i(x)+y_i+1u^II_i(x)+∫_x_i^x_i+1G_i(x,s)ψ(x_i,y) s,while for x∈[λ,1-λ],we use the following interpolation polynomialp(x)={[p_N/4(x)x∈[x_N/4,x_N/4+1],; ⋮;p_i(x)x∈[x_i,x_i+1],; ⋮; p_3N/4-1(x) x∈[x_3N/4-1,x_3N/4] , ].wherep_i(x)={y_i+1-y_ix_i+1-x_i (x-x_i)+y_i x∈[x_i,x_i+1], 0x ∉[x_i,x_i+1].and y_i, i=N/4,…,3N/4-1 are the already calculated values of the approximate solutions in the mesh points. Now, the approximate solution to the problem(<ref>)–(<ref>), has the following formY(x)={[ ỹ_i(x) x∈[0,λ],; ; p(x) x∈[λ,1-λ],; ; ỹ_i(x) x∈[1-λ,1]. ].In the following theorem, the estimate of the error will be calculated only forx∈[λ,1/2], i.e. for the value of the indexes i=N/4,…,N/2. We use the same assumptions as previously listed in Remark <ref>. The following estimate of the error between the exact and approximate solution (<ref>)–(<ref>) holds:max_x∈[0,1]|y(x)-Y(x)|⩽Cln^2N/N^2. The case of x∈[0,λ] has already been proved in the Theorem <ref>.Let us show now (<ref>) on [λ,1/2]. Let us denote byp a polynomial which is defined in the same way as the polynomial p in (<ref>)–(<ref>). The polynomial p will pass through the points with coordinates(x_i,y_i) and (x_i+1,y_i+1), (y_i andy_i+1 are values of the exact solution in the mesh points, i.e. y_i=y(x_i), y_i+1=y(x_i+1)). We have that|y(x)-p(x)|=|y(x)-p(x)+p(x)-p(x)|⩽|y(x)-p(x)|+|p(x)-p(x)|.On every interval [x_i,x_i+1], i=N/4,…,N/2, we get thatp(x)-p(x)= y_i+1-y_ix_i+1-x_i(x-x_i)+y_i-y_i+1-y_ix_i+1-x_i(x-x_i)-y_i = y_i+1-y_i+1-(y_i-y_i)/x_i+1-x_i(x-x_i)-(y_i-y_i),therefore in view of the Theorem<ref> we obtain the estimate|p(x)-p(x)|⩽Cln^2N/N^2, i=N/4,…,N/2.In the part of the mesh when i=N/4+1,…,N/2,on basis of <cit.>, (<ref>), (<ref>) and (<ref>), we obtain |y-p_i(x)|⩽h^2/8max_η∈[x_i,x_i+1]|y”(η)| ⩽C/N^2. For i=N/4, according to thedecomposition(<ref>) from Theorem <ref>,we obtainy-p_i(x)= y- y_i+1-y_ix_i+1-x_i(x-x_i)+y_i= s- s_i+1-s_ix_i+1-x_i(x-x_i)+s_i+r- r_i+1-r_ix_i+1-x_i(x-x_i)+r_i.For the layer component, on the basis of the estimate(<ref>) we obtain|s- s_i+1-s_ix_i+1-x_i(x-x_i)+s_i|⩽ |s|+|s_i+1-s_i|+|s_i| ⩽ C_1[ e^-x/ε√(m)+e^-1-x/ε√(m) +(e^-x_i+1/ε√(m)+e^-1-x_i+1/ε√(m)).+.2( e^-x_i/ε√(m)+e^-1-x_i/ε√(m)) ]⩽C/N^2.For the regular component we apply again the estimate from<cit.>, and on the basis of (<ref>) we get that|r- r_i+1-r_ix_i+1-x_i(x-x_i)+r_i|⩽h^2/8max_η∈[x_i,x_i+1]| y”(η)|⩽C/N^2.Now, from (<ref>), (<ref>), (<ref>) and (<ref>), and the part of the proof of Theorem <ref>, which is related to x∈[0,λ], we obtain (<ref>). § NUMERICAL EXPERIMENTSIn this section the theoretical results of the previous section will be checked on the following exampleε^2y”=y+(1-2x)^2-8ε^2, x∈(0,1), y(0)=0, y(1)=0.The exact solution of the test example (<ref>) isy(x)=e^-x/ε+e^-(1-x)/ε/1+e^-1/ε+4x(1-x)-1.First we will calculate a discrete approximate solution, i.e. the value of approximate solutions in the mesh points, using the difference scheme (<ref>) and then based on those results we will construct approximate solutions (<ref>) and (<ref>). Plots of exact and approximate solutions (<ref>) and(<ref>) are represented by Figure <ref> and Figure <ref>, while the values of errors are presented in na Figure <ref>. The system of equations is solved by Newton's method with initial guess y_0=-0.5. The value of the constant γ=1 has been chosen so that the condition γ≥ f_y(x,y), ∀(x, y)∈[0,1]×ℝ is satisfied. Because of the fact that we know the exact solution, we definethe computed error E_N and the computed rate of convergence Ord in the usual way E_N=max_0≤ i≤ N|y(x_i)-y^N(x_i) |, Ord=ln E_N-ln E_2Nln2k/k+1,whereN=2^k, k=5,6,…,11, y^N(x_i) is the numerical solution on a mesh with N subintervals. Values E_N andOrd are represented in the following table.The explanations about the figures. In Figure (<ref>), (<ref>) and(<ref>), the plotsof the exact solution of the problem (<ref>)–(<ref>) and the approximate solutions (<ref>)are presented, for the values of the parameters N=32 and ε=2^-4, 2^-6, 2^-10, respectively, while in figures(<ref>), (<ref>)and (<ref>) graphics of exact and numerical solution(<ref>) were given for the values of the parameters N=64, 128, 256 and ε=2^-10, respectively. In figure (<ref>), (<ref>) and (<ref>) one can notice an increase of the error value, or differences in the graphs between the exact and numerical solutions, while in Figure (<ref>) it is very difficult to distinguish between the exact and numerical solutions (<ref>), in Figure(<ref>) the deviation between the numerical and exact solution can be seen. From the presented graphs it is evident that there is a decrease of the error value due to an increase in the number of pointsN.In Figures (<ref>), (<ref>) and (<ref>) the plots of the exact (<ref>)–(<ref>) and approximate solution (<ref>) are given. For the calculation of the approximate solutions we used N=32 points, while the value of the perturbation parameter was ε=2^-4, 2^-6, 2^-10, respectively. From the presented graphics it can be seen a decrease of perturbation parameter ε, with a constant value of the number of points N a value of the error is slightlyincreasing. However, this increase is smaller than in the case of use of approximate solutions(<ref>).Inthe Figure (<ref>), (<ref>) and (<ref>) thereare graphs of the correct solution of the problems (<ref>)–(<ref>) and approximate solutions. Graphs on all three figures are obtained for a fixed value of parameter ε, while approximate solution is obtained by using N=64, 128,256 number of points, respectively.In Figures (<ref>), (<ref>) and(<ref>) the plots of the error of the approximate solutions(<ref>) are represented, while in Figures(<ref>), (<ref>) and(<ref>) are graphs of the error of the approximate solution (<ref>). Side by side are graphs of the errors of the approximate solution, to the left is (<ref>), while on the right are approximate solution (<ref>) for the same values of the parameter ε and N. From the graph we can see that values of the error agree with the theoretical results. In the graph, on the right side is a value of the error from the order 𝒪(N^-1), while on the graphs from the right side is a value of the error from the order 𝒪(N^-2ln^2N), and therefore in this way we have a confirmation of the theoretical results.§ CONCLUSIONIn this paper we performed a construction of approximate solutions for singularly–perturbed boundary value problem (<ref>)–(<ref>). First, we calculated a discrete approximate solution, i.e. the value of approximate solution in points of the mesh, and then we constructed an approximate solution by using a representation of the exact solution via Green's functions. Order of the value of the error is𝒪(N^-1) in the maximum norm. The basis functions are exponential.From Theorem <ref> we can see that the value of errors in this way constructed approximate solution is in the part of the domain where lies boundary layer of order 𝒪(ln^2N/N^2), while out of the layer are of order 𝒪(1/N). In order to gain the approximate solution with the smallest error, basis function of the exponential type of the outer boundary layer is replaced with linear functions. Error in this case is in the order 𝒪(ln^2N/N^2), also in the maximum norm.amsplain | http://arxiv.org/abs/1705.09608v1 | {
"authors": [
"Samir Karasuljić",
"Enes Duvnjaković",
"Vedad Pasic",
"Elvis Barakovic"
],
"categories": [
"math.NA",
"65L10, 65L11, 65L50"
],
"primary_category": "math.NA",
"published": "20170526150544",
"title": "Construction of a global solution for the one dimensional singularly-perturbed boundary value problem"
} |
Learning Local Feature Aggregation Functions with Backpropagation Angelos Katharopoulos*, Despoina Paschalidou*, Christos Diou, Anastasios Delopoulos * These two authors contributed equally Multimedia Understanding GroupECE Department, Aristotle University of Thessaloniki, Greece{katharas, pdespoin}@auth.gr; [email protected]; [email protected] December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================ This paper introduces a family of local feature aggregation functions and a novel method to estimate their parameters, such that they generate optimal representations for classification (or any task that can be expressed as a cost function minimization problem). To achieve that, we compose the local feature aggregation function with the classifier cost function and we backpropagate the gradient of this cost function in order to update the local feature aggregation function parameters. Experiments on synthetic datasets indicate that our method discovers parameters that model the class-relevant information in addition to the local feature space.Further experiments on a variety of motion and visual descriptors, both on image and video datasets, show that our method outperforms other state-of-the-art local feature aggregation functions, such as Bag of Words, Fisher Vectors and VLAD, by a large margin.§ INTRODUCTION A typical image or video classification pipeline, which uses handcrafted features, consists of the following components: local feature extraction (e.g. Improved Dense Trajectories <cit.>, SIFT <cit.>), local feature aggregation (e.g. Bag of Words <cit.>, Fisher Vectors <cit.>) and classification of the final aggregated representation. This work focuses on the second component of the classification pipeline, namely the generation of discriminative global representations from the local image or video features.The majority of existing local feature aggregation functions <cit.> rely on a visual codebook learned in an unsupervised manner. For instance, Bag of Words <cit.> quantizes every local feature according to a codebook, most commonly learned with K-Means, and represents the image as a histogram of codewords. Fisher Vectors <cit.>, on the other hand, capture the average first and second order differences between the local feature descriptor and the centres of a GMM. Furthermore, the Kernel Codebook encoding <cit.> is analogous to the Bag of Words, with the only difference that it uses soft assignments, which are functions of the distances between the local features and the codewords.There have been several attempts to improve the feature aggregation step by improving the codebook. For example, the authors of <cit.> propose a K-Means alternative that improves modelling of sparse regions of the local feature space. Other researchers focus on the indirect use of the class information in order to influence the codebook generation. For instance, in <cit.> Lazebnik et al. propose a technique for codebook learning that aims to minimize the loss of the classification-relevant information. Finally, in <cit.> and <cit.> the authors make direct use of the class labels in order to improve the Bag of Words representation using a classifier.In this paper, we define a family of local feature aggregation functions and we propose a method for the efficient estimation of their parameters in order to generate optimal representations for classification. In contrast to former research, our method: * Can be used to estimate any type of parameters and not only codebooks.* Can be used to create representations optimal for any task that can be expressed as a differentiable cost function minimization problem, not just classification.To demonstrate these properties, we introduce two feature aggregation functions that outperform state-of-the-art local feature aggregation functions in terms of classification accuracy in various descriptors for both image and video datasets.The rest of the paper is structured as follows. In Section <ref> we introduce and explain the proposed method. Experimental results are reported in Section <ref>, followed by conclusions in Section <ref>.§ LEARNING LOCAL FEATURE AGGREGATION FUNCTIONS Let F = {f_1, f_2, …, f_N_F} be the set of N_F local descriptors extracted from an image or video. In order to derive a global representation for this feature set, we consider feature aggregation functions that can be expressed in the form of equation <ref>, where T(·Θ): ℝ^D↦ℝ^K is a differentiable function with respect to the parameters Θ. R(F Θ) = 1/N_F∑_n=1^N_F T(f_n Θ) By appropriately defining the T(·Θ) function, in the above formulation, we are able to express many local feature aggregation functions. For instance, the soft-assignment Bag of Words <cit.> can be expressed with the T(·Θ) function given in equation <ref> T_BOW(f_nC)= 1/∑_k=1^K D(f_n, C_k)[ D(f_n, C_1); ⋮; D(f_n, C_K) ]where D(f_n, C_k)= exp-γ (f_n - C_k)^T(f_n - C_k)is a Gaussian-shaped kernel with Euclidean distance and C ∈ℝ^D × K is the codebook. In the following sections, we propose a generic method to estimate the parameters Θ^* of the local feature aggregation functions, such that they generate representations that are optimal for classification. To do that, we backpropagate the gradient of a classifier's cost function in order to update the parameters Θ using gradient descent. §.§ Parameter estimation Most approaches for parameter estimation of local feature aggregation functions do not take into consideration the subsequent usage of the global feature representation. For instance, in the case of the classification task, the extensively used K-Means and GMM methods, ignore the class labels of the feature vectors in the training set. In this work, we propose a supervised method for the parameter estimation of any local feature aggregation function that belongs in the family of functions of equation <ref>. Even though our method can be used for any task that can be expressed as a differentiable cost function minimization problem, in the rest of this paper we focus on the classification task.In particular, we estimate the values of the parameters Θ by minimizing the cost function J(·) of a classifier.Let J(x, yW) be the cost function of a classifier with parameters W that aims to predict the class label y from a global feature vector x. Training a classifier is equivalent to finding the W^* = _W1/N∑_i=1^N J(x^(i), y^(i) W), where x^(i) and y^(i) are the i-th training sample and its corresponding class label from a total of N samples. Instead of using traditional clustering methods, such as K-Means and GMM, to learn the parameters of the feature aggregation function, we compose J(· W) with R(·Θ). This allows us to jointly learn a classifier and a feature aggregation function by solving the optimization problem of equation <ref>. W^*, Θ^* = _W, Θ∑_i=1^N J(R(F^(i)Θ), y^(i) W) Due to the differentiability of T(·Θ), a straight-forward way to solve this optimization problem is to use Stochastic Gradient Descent (SGD). However, this optimization problem becomes computationally intensive in case of multimedia and especially for video datasets, due to the large number of local features F of each video (e.g. more than 20,000 local features in the case of Improved Dense Trajectories <cit.>). In order to address this problem, we approximate the gradient of R(·) with respect to the k-th parameter θ_k, of equation <ref>, by using a random sample of local features, S_F, instead of computing the gradient for every local feature. Jθ_k = JR(F Θ)R(F Θ)θ_k= JR(F Θ)1/N_F∑_n=1^N_FT(f_n Θ)θ_k≈JR(F Θ)1/N_S_F∑_n ∈ S_FT(f_n Θ)θ_k Empirical results indicate that this approximation has similar effects to the stochastic gradient approximation of SGD, namely efficiency and robustness. §.§ Aggregation functions In this section, we make use of the previous analysis in order to create two local feature aggregation functions that outperform other state-of-the-art methods such as Bag of Words <cit.> and Fisher Vectors <cit.> on a variety of descriptors, as shown in the Experiments section <ref>.Firstly, we consider the representation R_1(·), which is a generalization of the soft-assignment Bag of Words and employs the encoding function T_1(·) of equation <ref> T_1(f_nC, Σ)= 1/Z(f_n, C, Σ)[ D(f_n, C_1, Σ_1);⋮; D(f_n, C_K, Σ_K) ]whereD(f_n, C_k, Σ_k)= exp-γ (f_n - C_k)^TΣ_k^-1(f_n - C_k)andZ(f_n, C, Σ)= ∑_k=1^K D(f_n, C_k, Σ_k)involve the codebook C_k and the diagonal covariance matrix Σ_k used to compute the Mahalanobis distance between the n-th local feature and the k-th codeword.On the other hand, we consider the representation R_2(·), produced by the encoding function T_2(·) of equation <ref>, which is exactly the soft-assignment Vector of Locally Aggregated Descriptors (VLAD) <cit.> and thus the dimensionality of the resulting representation is D × K because f_n - C_k is a vector of size D. T_2(f_nC, Σ)= 1/Z(f_n, C, Σ)[ D(f_n, C_1, Σ_1)(f_n - C_1); ⋮; D(f_n, C_K, Σ_K)(f_n - C_K) ] In order to compute the optimal parameters C and Σ of the local feature aggregation functions, we optimize equation <ref> using a Logistic Regression classifier with a cross-entropy loss according to equation <ref>.While linear classifiers are very efficient, non-linear classifiers tend to yield better classification results, especially in the case of Bag of Words <cit.>. Therefore, we decided to adopt an approximate feature map of χ^2 <cit.> that is used in combination with T_1(·) and Logistic Regression to retain both the training efficiency of a linear classifier and the classification accuracy of a non-linear classifier. J(x, yW) = -logexpW_y^T x/∑_ŷexpW_ŷ^T x We could have used any classifier whose training is equivalent to minimizing a differentiable cost function, such as Neural Networks. Nevertheless, we use Logistic Regression and a χ^2 feature map in order to fairly compare our method to existing feature aggregation functions. §.§ Training procedure In Algorithm <ref>, we present the training procedure for the feature aggregation functions introduced in Section <ref>. The training process consists of three main parts, the initialization step, the optimization step and the classifier fine-tuning step.Regarding the initialization, we have experimented with three methods to initialize the codebook C and the covariance matrices Σ. In particular, we used:* Random sampling from the set of local features to initialize the codebook and the identity matrix to initialize the covariance matrices.* K-Means clustering to initialize the codebook and the identity matrix to initialize the covariance matrices.* GMM clustering to initialize both the codebook and the covariance matricesThe proposed method can be used in combination to any of the aforementioned initializations. However, we empirically observe that when initialized with K-Means it results in a smoother parameter space, hence it is easier to choose a suitable value for the SGD learning rate.Finally, the reason for adding the classifier fine-tuning step emerged from the need to alleviate the effects of gradient noise, produced by the sampling of local features in equation <ref>.§ EXPERIMENTS This section presents an experimental evaluation of the proposed method on real and artificial datasets in order to assess its effectiveness and provide insights into the resulting feature aggregation functions. In particular, we have conducted experiments on the CIFAR-10 <cit.> image classification dataset and the UCF-11 (YouTube) Action dataset <cit.>. In case of CIFAR-10, we have extracted local features with a pre-trained deep convolutional neural network.Specifically, we have used the conv3_3 layer from VGG-16 architecture <cit.>, pre-trained on Imagenet, which results in 25 local features in ℝ^256 for each image. In addition, in case of the video data, we have extracted Improved Dense Trajectories <cit.>, after removing videos that have less than 15 frames, which results on an average of approximately 22,000 local features per video.In Section <ref>, we present a comparative evaluation of the discovered codewords in two synthetic datasets, in order to acquire a better understanding of the way our method chooses the codebook, compared to unsupervised methods. Subsequently, in Section <ref>, we present the classification accuracy of various representations on CIFAR-10, with respect to the training epochs, and compare it to the corresponding results using Bag of Words. Finally in Section <ref>, we compare the proposed method on CIFAR-10 and UCF-11 with respect to the classification accuracy to Fisher Vectors, Bag of Words and VLAD on a variety of descriptors. §.§ Synthetic dataset Figure <ref> compares the generated codebooks by K-Means, GMM and the proposed method on two artificial two-class datasets. In both cases, we generate and visualize 10 codewords, especially in the case of GMM we visualize additionally the covariance matrices. For our method, we use the T_1(·) feature aggregation function, from equation <ref>, to learn the codebook with the covariance matrices being fixed and equal to the identity matrix. In contrast to K-Means and GMM, our method focuses on generating representations that can be separated by the classifier without necessarily retaining the structural information of the local features. It only suffices to observe Figure <ref> to note that K-Means and GMM do not respect the circular class boundary, while our method focuses mainly on generating a linearly separable representation. In addition, owing to the fact that our method does not try to describe the local features it results in a more separable representation with a smaller codebook. For instance, it only requires a single codeword to successfully separate the concentric dataset of Figure <ref>.§.§ Training evolution For this experiment, we generate codebooks using K-Means of sizes {64, 128, 256, 512, 1024, 2048}, which we subsequently use to create the corresponding Bag of Words representations. To classify the produced representations, we train a linear SVM with a χ^2 feature map. Moreover, we use the T_1(·) feature aggregation function, of equation <ref>, with Logistic Regression, a χ^2 feature map and K-Means as an initialization method according to Algorithm <ref>. In order to select a value for the hyper-parameter γ of the T_1(·) function, we perform cross-validation.By observing Figure <ref>, we conclude that the proposed method produces discriminative representations even with a small number of dimensions.In particular, it outperforms Bag of Words with 2048 dimensions by almost 4 percentage points with only 64 dimensions. Furthermore, we also notice that our method considerably improves the representation during the first epochs, thus we conclude that it can be used to fine-tune any differentiable feature aggregation function (e.g. Fisher Vectors) with little computational effort. Finally, we anticipate that increasing the number of training epochs will further increase the classification accuracy. §.§ Classification results In the current experiment, we assess the discriminativeness of the produced representations by evaluating their classification performance on a variety of descriptors and comparing it to several state-of-the-art feature aggregation methods. In the case of CIFAR-10, we use the provided train-test split while for UCF-11, we create three random 60/40 train-test splits and report both the mean classification accuracy and the standard error of the mean. Table <ref> summarizes the results. The experimental setup for CIFAR-10 is analysed in Section <ref>.Regarding UCF-11, we generate codebooks of sizes {1024, 2048} using K-Means, both to create Bag of Words representations and to initialize the codebooks for the T_1(·) function. In addition, we train a GMM with 64 Gaussians to generate Fisher Vectors representations and again K-Means with 64 centroids to generateVLAD and initialize T_2(·).For both datasets, we train an SVM with a χ^2 feature map for Bag of Words and T_1(·) and a linear SVM for the rest of the local feature aggregation functions in Table <ref>.Moreover, in case of CIFAR-10, T_1(·) is trained for only 10 epochs, while for UCF-11, both T_1(·) and T_2(·) are trained for 30 epochs. In the conducted experiments, we have observed that both T_1(·) and T_2(·) are very sensitive with respect to the hyper-parameter γ, which must be carefully selected using a validation set or cross-validation. In particular, the reported results are generated using γ=70 for UCF-11 “idt_traj”, γ=50 for UCF-11 “idt_hof” and γ=5 × 10^-8 for CIFAR-10. The large differences in the range of γ make intuitive sense upon observing the distribution of the pairwise distances of the local features.Furthermore, we additionally report the classification accuracy attained by T_1(·) and T_2(·), without learning the parameters using the proposed method; the results are reported in Table <ref> as “initial”. This allows us to quantify the improvement in terms of classification accuracy achieved using the proposed method. In particular, we observe an average improvement of approximately 3.5 percentage points in all cases.1.2 § CONCLUSIONS We have introduced a new method to learn the parameters of a family of local feature aggregation functions through optimization, which can be used to learn any type of parameters and is not limited to codebooks.Furthermore, it can be used to generate an optimal representation for any task that can be expressed as a cost function minimization problem. In particular, in the conducted experiments, we have demonstrated the effectiveness of the proposed method in the classification task.We observed that the proposed local feature aggregation functions outperform Bag of Words, Fisher Vectors and VLAD in a variety of descriptors on image and video data. Our method opens up a multitude of new research directions. Initially, we could use the proposed method to learn extra parameters, such as γ, in order to further improve the generated representation. Moreover, it would be interesting to conduct experiments on other large-scale video classification datasets, such as UCF101 <cit.> and compare the performance of our method to state-of-the-art Neural Network architectures, such as the hybrid deep learning framework, as it was introduced in <cit.>.Finally, we can explore the use of the proposed method for the generation of optimal representations for other types of tasks, such as regression or ranking. abbrv | http://arxiv.org/abs/1706.08580v1 | {
"authors": [
"Angelos Katharopoulos",
"Despoina Paschalidou",
"Christos Diou",
"Anastasios Delopoulos"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170626201341",
"title": "Learning Local Feature Aggregation Functions with Backpropagation"
} |
UAV Assisted Public Safety Communications with LTE-Advanced HetNets and FeICIC Abhaykumar Kumbhar^1,2, Simran Singh^3, and İsmail Güvenç^3^1Dept. Electrical and Computer Engineering, Florida International University, Miami, FL, 33174^2Motorola Solutions, Inc., Plantation, FL, 33322^3Dept. Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, 27606December 30, 2023 ======================================================================================================================================================================================================================================================================================================================== Establishing a reliable communication infrastructure at an emergency site is a crucial task for mission-critical and real-time public safety communications (PSC). To this end, use of unmanned aerial vehicles (UAVs) has recently received extensive interest for PSC to establish reliable connectivity in a heterogeneous network (HetNet) environment. These UAVs can be deployed as unmanned aerial base stations (UABSs) as part of the HetNet infrastructure. In this article, we explore the role of agile UABSs in LTE-Advanced HetNets by applying 3GPP Release-11 further-enhanced inter-cell interference coordination (FeICIC) and cell range expansion (CRE) techniques. Through simulations, we compare the system-wide 5th percentile spectral efficiency (SE) when UABSs are deployed in a hexagonal grid and when their locations are optimized using a genetic algorithm, while also jointly optimizing the CRE and the FeICIC parameters. Our simulation results show that at optimized UABS locations, the 3GPP Release-11 FeICIC with reduced power subframes can provide considerably better 5th percentile SE than the 3GPP Release-10 with almost blank subframes. Cell range expansion, drone, eICIC, FeICIC, FirstNet, genetic algorithm, interference coordination, public safety, quadcopter, unmanned aerial base station.§ INTRODUCTIONPublic safety communications (PSC) is considered to be the cornerstone of public safety response system and plays a critical role in saving lives, property, and national infrastructure during a natural or man-made emergency. The legacy PSC technologies are designed predominantly for delivering mission-critical voice communications over narrowband channels, which have been so far met by operating in the pre-defined channelized spectrum allocation. However, the evolution of data and video applications demands higher channel capacity and improved spectral efficiency (SE) <cit.>.To enhance the capabilities of next-gen broadband PSC networks, recently, FirstNet in the United States is building a 4G Long Term Evolution (LTE) based coast-to-coast public safety network deployed in the 700 MHz band <cit.>. Similarly, the United Kingdom plans to replace the TETRA system, which currently provides mission-critical communications for public safety agencies and other government organizations, with LTE by the year 2020 <cit.>. The 4G mobile networks as considered in these examples have great potential to revolutionize PSC during emergency situations by providing high-speed real-time video and multimedia services along with mission-critical communication. Furthermore, LTE-Advanced capabilities such as small cell deployment, interference coordination, and cell range extension can restore or extend coverage beyond the existing or damaged PSC networks. Unmanned aerial base stations (UABSs) such as balloons, quadcopters, and gliders equipped with LTE-Advanced capabilities can be utilized for emergency restoration and temporary expansion of public safety network in case of disaster recovery <cit.>. These UABSs can be deployed with minimum interdependencies, at low cost, and provide virtually omnipresent coverage which is essential for first-responders to be efficient and save lives. The FirstNet PSC network requires 95% geographical coverage of the country, which will be difficult to achieve using only dedicated cell towers. The UABSs can be deployed when necessary to assist in achieving this coverage goal. On the other hand, the UABSs may also introduce significant interference problems with the ground network <cit.>. Recent studies in the literature <cit.> have addressed the application of UABSs for rendering mission-critical communication. However, in a heterogeneous network (HetNet) environment,use of UABSs introduce only limited performance gains due to high inter-cell interference. Deployment of unmanned aerial vehicles (UAVs) as mobile LTE relays to offload traffic in a HetNet scenario while also considering inter-cell interference has been studied in <cit.>. The effectiveness of 3GPP Release-10/11 inter-cell interference coordination (ICIC) techniques for fixed HetNet deployments has been explored in <cit.>, while the use of cell range expansion (CRE) techniques for offloading users from MBSs to UABSs has been analyzed in <cit.> without considering ICIC. To our best knowledge, merits of 3GPP Release-10/11 techniques along with CRE and UABS mobility have not been evaluated in the literature, and such an evaluation is the main goal of this paper. We consider an LTE band class 14 PSC network <cit.> as shown in Fig. <ref>; by randomly removing macro base stations (MBSs), we simulate a mock emergency situation to study the impact of interference and CRE when the UABSs are deployed. Subsequently, we explore potential gains in 5th percentile SE (5pSE) from the use of Release-10/11 ICIC techniques for a UABS based PSC network.The rest of this paper is organized as follows. In Section <ref>, we provide the UABS-based HetNet model, assumptions, and definition of 5pSE as a function of network parameters. The UABSs deployment and ICIC parameter configurations using the genetic algorithm and hexagonal grid UABS model are described in Section <ref>. In Section <ref>, we analyze and compare the 5pSE of the HetNet using extensive computer simulations for various ICIC techniques, and finally, the last section provides some concluding remarks.§ SYSTEM MODELWe consider a two-tier HetNet deployment with MBSs and UABSs as shown in Fig. <ref>, where all the MBSs and UABS locations are captured in matrices X_ mbs∈ℝ^N_ mbs× 3and X_ uabs∈ℝ^N_ uabs× 3, respectively, where N_ mbs and N_ uabs denote the number of MBSs and UABSs within the simulation area, and UABSs are deployed at a fixed height. The MBS and user equipment (UE) locations are each modeled using a two-dimensional Poisson point process (PPP) with intensities λ_ mbs and λ_ ue, respectively <cit.>. The UABSs are deployed either at fixed locations in a hexagonal grid, or the locations are optimized using the genetic algorithm. We assume that the MBSs and the UABSs share a common transmission bandwidth, round robin scheduling is used in all downlink transmissions, and full buffer traffic is used in every cell. The transmit power of the MBS and UABS are P_ mbs and P_ uabs, respectively, while K and K^' are the attenuation factors due to geometrical parameters of antennas for the MBS and the UABS, respectively. Then, the effective transmit power of the MBS is P^'_ mbs = KP_ mbs, while the effective transmit power of the UABS is P^'_ uabs = K^' P_ uabs.An arbitrary UE n is always assumed to connect to the nearest MBS or UABS, where n∈{1,2,...,N_ ue}.Let the nearest macro-cell of interest (MOI) be at a distance d_mn and the nearest UAV-cell of interest (UOI) be at a distance d_un.Then, for the nth UE the reference symbol received power from the mth MOI and the uth UOI are given by <cit.>S_ mbs(d_mn) = P^'_ mbs/d_mn^δ,S_ uabs(d_un) = P^'_ uabs/d_un^δ,where δ is the path-loss exponent, and d_un depends on the locations of the UABSs that will be dynamically optimized. §.§ 3GPP Release-10/11 Inter-Cell Interference Coordination Due to their low transmission power, the UABSs are unable to associate a larger number of UEs compared to that of MBSs. However, by using the cell range expansion (CRE) technique defined in 3GPP Release 8, UABSs can associate a large number of UEs by offloading traffic from MBSs. A negative effect of CRE includes increased interference in the downlink on cell-edge UEs or the UEs in CRE region of the UABS, which is addressed by using ICIC techniques in LTE and LTE-Advanced <cit.>. 3GPP Release-10 introduced a time-domain based enhanced ICIC (eICIC). It uses almost blank subframes (ABS) which require the MBS to completely blank the transmit power on the physical downlink shared channel (PDSCH) resource elements as shown in Fig. <ref>(a). This separates the radio frames into coordinated subframes (CSF) and uncoordinated subframes (USF). 3GPP Release-11 defines further-enhanced ICIC (FeICIC), where the data on PDSCH is still transmitted but at a reduced power level as shown in Fig. <ref>(b). We assume that the ABS and reduced power pattern are shared via the X2 interface, which is a logical interface between the base stations. Implementation of the X2 interface for UABSs is left as a future consideration.The MBSs can schedule their UEs either in USF or in CSF based on the scheduling threshold ρ. Similarly, the UABSs can schedule their UEs either in USF or in CSF based on the scheduling threshold ρ^'. Let β denote the USF duty cycle, defined as the ratio of number of USF subframes to the total number of subframes in a radio frame. Then, the duty cycle of CSFs is (1-β). For ease of simulation, the USF duty cycle β is fixed at 50% in this paper for all the MBSs, which is shown in <cit.> to have limited effect on system performance when ρ and ρ' are optimized. Finally, let 0≤α≤ 1 denote the power reduction factor in coordinated subframes of the MBS for the FeICIC technique. As two special cases,α=0 corresponds to Release-10 eICIC, while α=1 corresponds to no ICIC.Given the eICIC and FeICIC framework in 3GPP LTE-Advanced as in Fig. <ref>, and following an approach similar to that in <cit.> for a HetNet scenario, the signal-to-interference ratio (SIR) experienced by an arbitrary nth UE can be defined for CSFs and USFs for the mth MOI and the uth UOI as follows:Γ = S_ mbs(d_mn)/S_ uabs(d_un) + Z→ USF SIR from MOI,Γ_ csf = α S_ mbs(d_mn)/S_ uabs(d_un) + Z→ CSF SIR from MOI,Γ^' = S_ uabs(d_un)/ S_ mbs(d_mn) + Z→ USF SIR from UOI,Γ^'_ csf = S_ uabs(d_un)/α S_ mbs(d_mn)+ Z→ CSF SIR from UOI, where Z is the total interference power at a UE during USF or CSF from all the MBSs and UABSs, excluding the MOI and the UOI. In hexagonal grid UABS deployment model (and in <cit.>), locations of the UABSs (and small cells) are fixed. To maximize the 5pSE of the network, we actively consider the SIRs in (<ref>)–(<ref>) while optimizing the locations of the UABSs using the genetic algorithm. §.§ UE Association and SchedulingThe cell selection process relies on Γ and Γ^' in (<ref>) and (<ref>), respectively, for the MOI and UOI SIRs, as well as the CRE τ. If τΓ^' is less than Γ, then the UE is associated with the MOI; otherwise, it is associated with the UOI. After cell selection, the MBS-UE (MUE) and UABS-UE (UUE) can be scheduled either in USF or in CSF radio subframes as: If ΓτΓ^' and Γ≤ ρ→ USF-MUE,If ΓτΓ^' and Γρ→ CSF-MUE,If Γ≤τΓ^' and Γ^'ρ^'→ USF-UUE, If Γ≤τΓ^' and Γ^'≤ ρ^'→ CSF-UUE.Once a UE is assigned to an MOI/UOI, and it is scheduled within a USF/CSF, then the SE for this UE can be expressed for the four different scenarios in (<ref>)-(<ref>) as follows:C_ usf^ mbs = β log_2(1+Γ)/N_ usf^ mbs, C_ csf^ mbs = (1-β) log_2(1+Γ_ csf)/N_ csf^ mbs,C_ usf^ uabs =log_2(1+Γ^')/N^ uabs_ usf, C_ csf^ uabs =log_2(1+Γ^'_ csf)/N^ uabs_ csf,where N_ usf^ mbs, N_ csf^ mbs, N^ uabs_ usf, and N^ uabs_ csf are the number of MUEs and UUEs scheduled in USF and CSF radio subframes, and Γ, Γ_ csf, Γ^', Γ^'_ csf are as in (<ref>)-(<ref>). In this paper, we consider the use of 5pSE which corresponds to the worst fifth percentile UE capacity among the capacities of all the N_ ue UEs (calculated based on (<ref>)-(<ref>)) within the simulation area. We believe it is a critical metric particularly for PSC scenarios to maintain a minimum QoS level at all the UEs in the environment. We define the dependency of the 5pSE to UABS locations and ICIC parameters as C_ 5th( X_ uabs, S_ mbs^ ICIC, S_ uabs^ ICIC) ,where X_ uabs∈ℝ^N_ uabs× 3 captures the UABS locations as defined earlier, S_ mbs^ ICIC = [α,ρ] ∈ℝ^N_ mbs× 2 is a matrix that captures individual ICIC parameters for each MBS, while S_ uabs^ ICIC = [τ,ρ'] ∈ℝ^N_ uabs× 2 is a matrix that captures individual ICIC parameters for each UABS. In particular,α=[α_1,...,α_N_ mbs]^T,ρ=[ρ_1,...,ρ_N_ mbs]^Tare N_ mbs× 1 vectors that include the power reduction factor and MUE scheduling threshold parameters for each MBS. On the other hand, τ=[τ_1,...,τ_N_ uabs]^T, ρ^'=[ρ_1^',...,ρ_N_ uabs^']^Tare N_ uabs× 1 vectors that involve the CRE bias and UUE scheduling threshold at each UABS. As noted in Section <ref>, the duty cycle β of ABS and reduced power subframes is assumed to be set to 0.5 at all MBSs to reduce search space and complexity. Considering that the optimum values of the vectors α, ρ, ρ', and τ are to be searched over a multi-dimensional space, computational complexity of finding the optimum parameters is prohibitively high. Hence, to reduce system complexity (and simulation runtime) significantly, we consider that the same ICIC parameters are used for all MBSs and all UABSs. In particular, we consider that for i=1,...,N_ mbs we have α_i=α and ρ_i=ρ, while for j=1,...,N_ uabs we have τ_j=τ and ρ_j^'=ρ^'. Therefore, the dependence of the 5pSE on the UABS locations and ICIC parameters can be simplified asC_ 5th( X_ uabs,α,ρ,τ,ρ^') ,which we will seek ways to maximize in the next section. We leave the problem of individually optimizing ICIC parameters for the MBSs and UABSs as a future work due to the high computational complexity of the problem.§ UABS DEPLOYMENT OPTIMIZATION We consider that the cell-edge user SE is captured by the 5pSE of the cumulative distribution function of the user throughput, which we will use as a metric to measure the overall network performance. In this section, we discuss the UABS deployment using the genetic algorithm (GA) and the hexagonal grid model, where we use the 5pSE as an optimization metric to maximize for both scenarios.§.§ Genetic Algorithm based UABS Deployment Optimization The GA is a population-based optimization technique that can search a large environment simultaneously to reach an optimal solution <cit.>. In this paper, the UABS coordinates and the ICIC parameters constitute the GA population, and a subsequent chromosome is illustrated in Fig. <ref>. Listing <ref> describes the main steps used to optimize the UABS locations and ICIC parameters while computing the 5pSE.We apply the GA to simultaneously optimize the UABS locations and ICIC parameters to maximize the 5pSE of the network over a given geographical area of interest. The location of each UABS within a rectangular simulation area is given by (x_i,y_i) where i ∈{ 1,2,...,N_ uabs}. The UABS locations and the ICIC parameters that maximize the 5pSE objective function can be calculated as [X̂_ uabs,α̂,ρ̂, τ̂,ρ̂^̂'̂]=X_ uabs,α,ρ,τ,ρ^'max C_ 5th( X_ uabs,α,ρ,τ,ρ^').Since searching for optimal X_ uabs and ICIC parameters through a brute force approach is computationally intensive, in this paper, we use the GA to find optimum UABS locations and the best-fit ICIC parameters τ, α, ρ, and ρ^'.[caption=Steps for optimizing population using GA., basicstyle=, language=R, breaklines=true, numbers=none, frame=single, showstringspaces=false, xleftmargin=0.2cm, linewidth=8.7cm,label=GaListing] Input: Population: set of UABS locations and ICIC parameters FITNESS function: 5pSE of the network Output: Args: Best individuals of ICIC parameters and highest 5pSE Method: NewPopulation <- empty set StopCondition: Number of iterations = 6 SELECTION: Roulette wheel selection method while(! StopCondition)for i = 1 to Size doParent1 <- SELECTION(NewPopulation,FITNESS function) Parent2 <- SELECTION(NewPopulation,FITNESS function) Child <- Reproduce(Parent1, Parent2) if(small random probability) child <- MUTATE(Child)add child to NewPopulation EVALUATE(NewPopulation, FITNESS function); Args <- GetBestSolution(NewPopulation) Population <- Replace(Population, NewPopulation) §.§ UABS Deployment on a Hexagonal Grid As a lower complexity alternative to optimizing UABS locations, we consider deploying the UABSs on a hexagonal grid, where the position of the UABSs are deterministic. We assume that the UABSs are placed within the rectangular simulation area regardless of the existing MBS locations. The 5pSE for this network is determined by using a brute force technique as described in the pseudo-code <ref> which only considers optimization of the ICIC parameters captured through the matrix S_ ICIC. The optimized ICIC parameters that maximize the 5pSE can then be calculated as:[α̂,ρ̂,τ̂,ρ̂^̂'̂]=α,ρ,τ,ρ^'max C_ 5th( X^ (hex)_ uabs,α,ρ,τ,ρ^'),where X_ uabs^ (hex) are the fixed and known hexagonal locations of the deployed UABSs within the simulation area.[caption=Steps for computing 5pSE for hexagonal grid deployment., basicstyle=, language=R, breaklines=true, numbers=none, frame=single, showstringspaces=false, xleftmargin=0.2cm, linewidth=8.7cm, label=HexListing] Input: set of UABS locations and ICIC parameters Output: SE: 5pSE for the network Method: StopCondition: Number of iterations = 100 while(! StopCondition)Generate UABS locations for t = 1 to ICICParms.tau[t] dofor a = 1 to ICICParms.alpha[a] dofor r = 1 to ICICParms.rho[r] dofor p = 1 to ICICParms.rhoprime[p] doSE = Calc5thPercentileSE(nodal locations, nodal Tx powers, path-loss, tau, beta, alpha, rho, rhoprime)§ SIMULATION RESULTS In this section, using Matlab based computer simulations, we compare the 5pSE with and without ICIC techniques while considering different UABS deployment strategies. Unless otherwise specified, the system parameters for the simulations are set to the values in Table <ref>. §.§ 5pSE with UABSs Deployed on a Hexagonal GridThe variations in 5pSE with respect to CRE, when the UABSs are deployed on a hexagonal grid and utilizing optimized ICIC parameters (see (<ref>) and Listing <ref>) are shown in Fig. <ref>. With no CRE, the number of UEs associated with the UABSs and the interference experienced by these UEs is minimal. With the no-ICIC mechanism (NIM) the peak value for the 5pSE is observed at around 0 dB CRE as seen in Fig. <ref>(a). On the other hand, the 5pSE for ICIC techniques at 0 dB CRE are relatively lower as seen in Fig. <ref>(b) and Fig. <ref>(c), due to blank subframes at the MBSs for eICIC, and power reduction of the CSFs at the MBSs for FeICIC. As the CRE increases, the number of UEs associated with the UABSs increases and so does the interference experienced by these UEs. Hence, with NIM the 5pSE decreases with increasing CRE as seen in Fig. <ref>(a). On the other hand, the ICIC techniques observe improvement in SE performance. The peak values of the 5pSE for the ICIC techniques is observed when the CRE is between 6-9 dB. This influence of CRE on the 5pSE for NIM and ICIC is summarized in Fig. <ref>. Overall, the 5pSE for the network is higher when larger numbers of UABSs are deployed and when fewer MBSs are destroyed. Also, the 5pSE decreases with the increasing number of destroyed MBSs as seen in Fig. <ref>. §.§ 5pSE with GA Based UABS Deployment OptimizationUsing the UABS locations and ICIC parameters optimized through the GA as in (<ref>) and Listing <ref>, we plot the peak 5pSE for the network with respect to the optimized CRE value in Fig. <ref>. In the GA based simulations, the optimum CRE value is directly related to the locations of the UABSs with respect to the MBSs, the number of UEs offloaded to the UABSs, and the amount of interference observed by the UEs.Consider first that the 50% of the MBSs are destroyed, which implies that there are still a large number of MBSs present and the interference from these MBSs is substantial. Hence, offloading a large number of UEs from MBSs to UABSs with higher values of CRE and ICIC is necessary for achieving better 5pSE gains as shown in Fig. <ref>(a) and Fig. <ref>(b) for eICIC and FeICIC, respectively. When most of the infrastructure is destroyed (i.e., when 97.5% of the MBSs destroyed), the interference observed from the MBSs is limited and larger number of UEs need to be served by the UABSs. Therefore, with fewer UABSs deployed, higher CRE are required to serve a larger number of UEs and achieve better 5pSE. On the other hand, when a larger number of UABSs are deployed, smaller values of CRE will result in better 5pSE gains. We record these behavior in Fig. <ref>(a) and Fig. <ref>(b) for eICIC and FeICIC, respectively. §.§ Performance Comparison Between Fixed (Hexagonal) and Optimized UABS Deployment with eICIC and FeICIC We summarize our key results from earlier simulations in Fig. <ref> to compare the key trade-offs between fixed (hexagonal) deployment and GA based deployment of UABSs. The comparison of Fig. <ref> and Fig. <ref> show that the optimized deployment of UABSs provides a better 5pSE than the UABSs deployed on a fixed hexagonal grid, which are also reflected in the comparative analysis in Fig. <ref> which considers an optimized CRE. Moreover, Fig. <ref> shows that the 5pSE gains from the optimization of UABS locations are more significant when 50% of the MBSs are destroyed and less significant when 97.5% of the MBSs are destroyed. When 50% MBSs are destroyed, there are still a large number of MBSs present which causes substantial interference. Hence, in such interference driven scenario it is important to optimize the locations of the UABSs, and use of larger number of UABSs provide only marginal gains in the 5pSE.On the other hand, with 97.5% of the MBSs destroyed, the interference from the MBSs is small, and deploying the UABSs on a hexagonal grid will perform close to optimum UABS deployment. The difference between the hexagonal deployment and optimized deployment is especially small for the FeICIC scenario where power reduction factor α in the MBS CSFs provides an additional optimization dimension for improving the 5pSE. Use of a larger number of UABSs when 97.5% of the MBSs are destroyed is also shown to provide significant gains in the 5pSE, in contrast to modest gains in the 5pSE when 50% of the MBSs are destroyed. § CONCLUDING REMARKSIn this article, we show that the mission-critical communications could be maintained by deploying UABSs in the event of any damage to the public safety infrastructure. Through simulations, we compare and analyze the 5pSE of the network when the UABSs are deployed on a hexagonal grid and when placed optimally using the GA. Our analysis shows that the deployment of the UABSs on a hexagonal grid is close to optimal when the observed interference is limited. In the presence of substantial interference, the GA approach is more effective for deploying UABSs. Finally, we observe that the HetNets, with reduced power subframes (FeICIC) yields better 5pSE than that with almost blank subframes (eICIC). Future research directions include optimizing the UABS locations using learning techniques and developing a path-planning algorithm for UABS placement.§ ACKNOWLEDGMENTThis research was supported in part by NSF under the grants AST-1443999 and CNS-1453678. The authors would like to thank A. Merwaday for his helpful feedback. IEEEtran | http://arxiv.org/abs/1706.08997v2 | {
"authors": [
"Abhaykumar Kumbhar",
"Simran Singh",
"Ismail Guvenc"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170627182139",
"title": "UAV Assisted Public Safety Communications with LTE-Advanced HetNets and FeICIC"
} |
Managing a Fleet of Autonomous Mobile Robots (AMR) using Cloud Robotics Platform Aniruddha Singhal, Nishant Kejriwal, Prasun Pallav, Soumyadeep Choudhury, Rajesh Sinha and Swagat KumarEmail IDs:The authors are with TCS Research, Tata Consultancy Services, New Delhi, India 201309. Accepted 24/05/2017 ================================================================================================================================================================================================================ § INTRODUCTIONString theory methods have had a remarkable impact on the calculation of field theory scattering amplitudes.In particular, in string theory, gravitational amplitudes are naturally related to the square of those for Yang-Mills. This leads to corresponding statements in field theory that are now well established at tree-level in the form of the KLT relations <cit.>.These relations have been extended to a notion of colour-kinematic duality or more simply double copy, in which gravity amplitudes can be obtained from Yang Mills by replacing the colour structures for Yang-Mills with their associated kinematic numerators in a specific class of representations <cit.>. In this form, the double copy has been applied at increasingly high loop order, where it is instrumental in rendering the calculations feasible (e.g., <cit.>). These computations have demonstrated the inadequacy of standard techniques for determining the onset of UV divergences in supergravity <cit.>, and have even fueled speculations that four-dimensional =8 supergravity could be perturbatively ultraviolet finite <cit.>.The double copy is a precise conjecture about how, in a specific class of representations, momentum space formulae for gravity scattering amplitudes are related to those of gauge theory.Suppose there exist representations for which the kinematic numerators of a gauge theory scattering amplitude (expressed as a sum over cubic Feynman graphs) obey the same Jacobi-like relations as the colour factors of the amplitude. If such a set of numerators can be found, then the corresponding gravity amplitude is given by simply replacing the colour factors in the gauge theory amplitude by another copy of the kinematic factors in this gauge. At tree-level, the double copy conjecture has been proven in a number of different ways <cit.>, and is equivalent to the KLT relations <cit.> between open and closed string amplitudes in the low-energy limit. While there is currently no general proof at higher loop orders in perturbation theory, a growing body of evidence suggests that the double copy also holds at loop level, at the time of writing to 5 loops. The success of the double copy prescription has led to an oft-repeated slogan in the amplitudes community: Gravity = ()^2.Yet, despite this array of evidence, the geometric and fully non-linear origins of the double copy remain mysterious. Most clear proofs thus far are expressed in momentum space for perturbations around a flat background.A body of recent work has explored how to manifest the double copy at the level of classical non-linear solutions in gauge theory and gravity <cit.>. However, these studies have been restricted to algebraically special solutions (in particular those of Kerr-Schild type), and do not probe dynamics in the same way as scattering amplitudes. In this paper, we address the question as to whether the double copy relationship between gauge theory and gravity holds for perturbation theory on curved backgrounds. To do this, we consider the simplest curved backgrounds for which there is a well-defined notion of S-matrix: sandwich plane waves <cit.>. These are metric or gauge field backgrounds which are flat in the asymptotic past and future in generic directions but contain a compactly supported region of curvature. This curvature can be thought of as a burst of unidirectional radiation (gravitational or electromagnetic) which is turned on and then switched off at some finite retarded times. The possibility of scattering on a plane wave background may seem controversial in light of the fact that such space-times are not in general globally hyperbolic <cit.>. Nevertheless, we will see that the evolution of massless fields is unitary without leakage, so the S-matrix does indeed make sense. The relationship Gravity =(Yang Mills)^2 is already nicely manifest in the underlying gravitational and electromagnetic plane waves, written in Brinkmann coordinates. With coordinates X^μ=(u,v,x^a), a=1,… , d-2, the Brinkmann form of the metric is Kerr-Schild, given by ṣ^2=ṣ^2_ flat-H_ab(u) x^a x^b ụ^2, ṣ^2_ flat=2 ụ ṿ- δ_ab x̣^a x̣^b,whereas the corresponding electromagnetic potential is=F(u)_ax^a ụ,so that the metric perturbation from flat space is naturally a sum of terms of the form A⊙ A.Here H_ab(u) and F_a(u) are curvatures and are freely prescribable functions of u subject to H_ab being trace-free for the Einstein equations to be satisfied (this restriction disappears if a dilaton is allowed).[Note that this classical double copy differs from that for the more general Kerr-Schild pp-waves considered in <cit.>. There, if the Maxwell field is ϕ k_μ, the metric is ṣ^2_flat+ϕ k_μ k_ν where k_μ is a null vector and ϕ a solution to the transverse wave equation. Such solutions can often be considered to be longitudinal with ϕ playing the role of a Coulomb-like source term that is analogous to a propagator and therefore not squared. We consider plane waves with a radiative Maxwell term, so the whole Maxwell field must be squared to obtain a gravitational field.] For a sandwich wave, H_ab and F_a are supported in some interval u∈[u_1,u_2] so that space-time and connection are flat for u→±∞.For both types of plane wave we will see that it is possible to find complete sets of polarization states for in and out momentum eigenstates for linear massless fields of integral spins.The flat `in' and `out' regions of sandwich plane waves allow us to define the S-matrix. We focus on the special case of 3-point amplitudes; in flat space, this is where the slogan Gravity = ()^2 of the double copy is literally <cit.>:_3^flat = (^flat_3)^2,where _3^flat and ^flat_3 are the 3-point gravity and gauge-theory amplitudes in Minkowski space, stripped of overall momentum conserving delta functions and coupling constants. Hence, we expect that if there is a notion of double copy which holds in curved backgrounds, it should be most easily found at the level of 3-point amplitudes for which propagators are not yet required.We consider such 3-point amplitudes for scalars, gauge theory and gravity on a gravitational plane wave background, and for charged scalars and gauge theory on a Yang-Mills plane wave background in any number of space-time dimensions. In each case, the computation reduces to an integral which depends on the background field; it turns out that the integrand[This `tree-level integrand' is the equivalent of `stripping off momentum conserving delta functions' in the flat space amplitudes.] of the resulting expression carries sufficient information to determine if there is a double copy. We find that the 3-point amplitudes for gluons on a plane wave gauge background and for gravitons on plane wave space-times have two parts written symbolically as^pw_3=F+C, ℳ_3^pw=ℱ^2 - 𝒞.Here, F is precisely the flat space-time integrand for three gluon scattering, whereas ℱ is the 3-gluon integrand on the gravitational plane wave background.Thus, there is a correction term between the square of the gluon 3-point amplitude and the graviton 3-point amplitude on a plane wave metric. The flat space F can be mapped to ℱ after some replacements of momenta and polarization vectors by their curved (and non-constant) counterparts. These replacements are non-local on space-time and are fixed by finding solutions to the Hamilton-Jacobi equations that allow one to bring momentum eigenstates into the interior of space-time from future or past infinity in the curved case.That it is non-local on a curved space-time is not a surprise as the double copy is onlyexpressed locallyon momentum space. The correction terms C and 𝒞 arise from the `tails' formed by the linearized free fields backscattering off the background. Scalar waves propagate cleanly on a plane wave background subject to Huygens' principle <cit.>, but spin one and spin two do not <cit.>. The tails of momentum eigenstates in the past pick up terms encoding the `memory' of the field through which they have passed (i.e., the integral of the field strength in the electromagnetic case). Remarkably, we find that C^2→𝒞 with an extension of the same replacements used to relate F and ℱ. Define _3=F-C to be the gluon 3-point integrand on a gauge background with flipped sign (or colour charge) for the background gauge field, and let ρ to be the replacement maps from flat to curved kinematics and gauge to gravitational background fields.Then our double copy can be written as_3 =ρ(_3_3).This is strong evidence that a notion of double copy persists more generally in the presence of background curvature.Our formulae therefore also allow a study of the memory effect for plane waves on the amplitude. The key ingredient in the integrand is a vielbein whose non-trivial change from past to future exemplifies the memory effect <cit.>, which has been studied in detail for sandwich plane waves (e.g., <cit.>). For a charged field on a gauge background, it gives a momentum shift from past to future infinity proportional to the integral of the field.On a gravitational background, the linear planes that are wave fronts of a standard momentum eigenstate in the past becomediverging quartic surfaces, Dupin cyclides, in the future <cit.>.This memory effect will also give rise to new infrared divergences that have been studied in the case of a charged field on an electromagnetic plane wave background <cit.>.We review the non-linear plane wave backgrounds for both gravity and gauge theory in Section <ref>. Free fields on these backgrounds are constructed in Section <ref>, where we also confirm that (for scalars, gauge theory and gravity) the S-matrix for these states is well-defined in the sense that scattering is unitary and there is no particle creation. We close this section with a brief discussion of Huygens' principle and tails. Section <ref> contains the calculation of 3-point amplitudes and integrands for scalars, gauge theory and gravity on the gravitational plane wave background; Section <ref> contains the analogous calculations for charged scalars and Yang-Mills theory on a background plane wave gauge field. In Section <ref>, these two calculations are mapped onto each other; this map defines the double copy for 3-point amplitudes on plane wave backgrounds. We also show how the gauge theory 3-point functions on the two backgrounds are related by a double copy map which acts only on the background. Section <ref> concludes. In Appendix <ref>, we provide explicit amplitude formulae for the special case of the impulsive plane wave background. Appendix <ref> contains the operational definitions of tree-level amplitude and integrand used throughout the paper. § PLANE WAVE BACKGROUNDSWe begin with a brief review of plane wave backgrounds in both the gravitational and gauge theoretic contexts. More thorough treatments can be found in the literature; the focus is on those features relevant to our calculations. §.§ Gravitational plane waves Non-linear plane waves are among the oldest exact solutions to the field equations of general relativity, and have many fascinating properties (c.f., <cit.>). These metrics describe space-times composed of pure radiation of the gravitational field itself or a Maxwell field, propagating from past to future null infinity along a given constant null direction. Our focus will be on purely gravitational plane wave metrics, which can be interpreted as a coherent superposition of gravitons. There are two standard coordinate systems: the Einstein-Rosen <cit.> and the Brinkmann <cit.> coordinates.In Einstein-Rosen coordinates, the metric is given by:ṣ^2 = 2 Ụ Ṿ - γ_ij(U) ỵ^i ỵ^j , where the indices i,j,…=1,…,d-2 and the only non-trivial metric components, γ_ij, depend on U. These coordinates are useful because they manifest many of the symmetries of the space-time which are `hidden' in the other coordinates. The metric(<ref>) clearly has Killing vectors ∂/∂ V, ∂/∂ y^i, and the vectors𝒳^i = y^i∂/∂V + F^ij(U)∂/∂y^j , F^ij(U):= ∫^Uṣ γ^ij(s) , are also Killing. The vectors ∂_V, ∂_i and 𝒳^i form a Heisenberg algebra,[^i,^j]=0 , [∂/∂y^i, ^j]=δ^j_i∂/∂V , so plane wave metrics are endowed with an abelian isometry group generated by translations of the constant U planes as well as this (solvable) Heisenberg symmetry. We will also see that massless field equations are most easily solved in these coordinates.The main drawback of Einstein-Rosen coordinates is that they are essentially never global coordinates: the metric will develop coordinate singularities due to the focusing of the null geodesic congruence tangent to _U <cit.>. Furthermore, the curvature and field equations are given by somewhat complicated expressions in terms of γ_ij. For instance, the Ricci curvature isR_UU=-γ^ij/2(γ̈_ij+1/2γ̇_ikγ^klγ̇_lj) ,where ḟ=∂_Uf for any function f(U). Thus the vacuum equations impose conditions on γ_ij in the form of a second-order ODE.The Brinkmann coordinates have the advantage that they are global, and the curvature is easily identified. In the Brinkmann chart, the metric is:ṣ^2=2 ụ ṿ - H(u,𝐱) ụ^2 - x̣_a x̣^a , with indices a,b,…=1,…,d-2. In these coordinates, the u=const. metric is completely flat. For pp-waves H(u,x) can have general x-dependence, but for plane waves it is constrained to be quadratic in x^a:H(u,𝐱)=H_ab(u) x^a x^b . The non-vanishing Christoffel symbols in these coordinates are:Γ^a_uu=-H_ab(u) x^b , Γ^v_ua=-H_ab(u) x^b , Γ^v_uu=-Ḣ(u,𝐱)/2 , and the non-vanishing curvature components areR^a_ubu=-H^a_b(u) , so the vacuum equations in Brinkmann coordinates simply impose that H_ab be trace-free: H^a_a=0.The sandwich plane wave setup is one for which H_ab(u) is compactly supported in u <cit.>. Without loss of generality, we assume that H_ab(u)≠0 only for u_1≤ u≤ u_2≤0; for u<u_1 or u> u_2, the space-time is a flat. The flat region u<u_1 is referred to as the in-region, while u>u_2 is the out-region. See Figure 1 for a schematic of this setup.Although we work mostly in Brinkmann coordinates, the relationship between the Brinkmann and Einstein-Rosen coordinate systems will be important. It can be understood in terms of the solutions to the equation:ë_a=H_ab e^b , for some functions e^a(u) . Setting e^a(u)=Δ x^a, (<ref>) is the geodesic deviation equation in Brinkmann coordinates; this follows from the fact that the connecting vectors between the geodesics,e^a∂/∂ x^a - ė_a x^a∂/∂ v ,areKilling vectors. A set of (d-2) Killing vectors is obtained by choosing a full (d-2)× (d-2) matrix of solutions to (<ref>), E^a_i(u) (and its inverse E^i_a(u)), subject toĖ^a_[i E_|a| j]=0 . The Killing vectors are then:𝒟^i=E^a i∂/∂ x^a - Ė^i_a x^a∂/∂ v .The commutation relations between the 𝒟^i and the 𝒳^i (transformed to Brinkmann coordinates) give the Heisenberg algebra which was more manifest in Einstein-Rosen coordinates.By comparing the line elements (<ref>), (<ref>), the diffeomorphism linking Einstein-Rosen and Brinkmann coordinates is identified as: U = u , V = v +1/2Ė^i_a E_b i x^ax^b , y^i= E^i_a x^a . The array E^a_i and its inverse will be referred to as vielbeins since they give the d-2 orthonormal 1-forms x̣^a=E^a_i ỵ^i in terms of the Einstein-Rosen coordinates. They obeyË_a i=H_ab E^b_i , γ_ij=E^a_(i E_|a| j) . As part of the geometry of the Einstein-Rosen waves, the hypersurfaces V=constant are null and transverse to the geodesic shear-free null congruence _v that rules the u=constant null hypersurfaces.The _U null congruence has a deformation tensor, measured in Brinkmann coordinates byσ_ab=Ė^i_a E_b i ,whose trace is the expansion and trace-free part is the shear.Note that any other choice of vielbein, say f^a_i, is related to E^a_i byf^a_i=E^a_j( F^jk b_ki+c^j_i) , for constant matrices b_ij, c^i_j, and F^ij(u) defined as:F^ij(u):=∫^u ṣ γ^ij(s) = ∫^u ṣ E^a (i(s) E^j)_a(s) . In particular, given some initial value for the vielbein on the in-region of a sandwich plane wave, (<ref>) encodes how the vielbein changes after passing through the curved interior to the out-region. For the sandwich wave, two natural initial values are given by requiring the vielbein to become trivial in the past or future:lim_u→±∞E^i ±_a(u) = δ_a^i . Since solutions to (<ref>) are simply linear in flat regions, we haveE^a -_i(u)=b^a +_i u+c^a +_iu→ +∞,E^a +_i(u)=b^a -_i u+c^a-_iu→ -∞.From (<ref>) and the conservation of the Wronskian between E^+ and E^-, it follows that b_[i^a ±c_j] a^±=0, b^a +_i=δ^aj δ_bi b^b -_jand we can use a rotation of the Brinkmann coordinates to make b symmetric if desired.Note that it is essentially impossible to have E invertible for all u for non-trivial b, so the Einstein-Rosen coordinates are generically singular. This is the inevitable consequence of null geodesic focusing of the V=constant null hypersurfaces as emphasized by Penrose <cit.>. Both E^a +_i and E^a -_i will describe the same flat metric in the asymptotic regions but with different Einstein-Rosen forms. In particular, if the deformation tensor σ_ab vanishes in one asymptotic region, it will generically be nontrivial in the other, albeit falling off as 1/u. This non-trivial change in σ_abis an example of the memory effect <cit.>, which has been studied in detail for sandwich plane waves (e.g., <cit.>). §.§ Gauge theory plane waves An `Einstein-Rosen' plane wave in gauge theory is a gauge potential which satisfies properties similar to a plane wave metric in Einstein-Rosen coordinates. It is often used to model the electromagnetic fields of lasers (c.f., <cit.>). In particular, we demand that– a priori taking values in the adjoint of some Lie algebra 𝔤 – manifests the symmetries generated by ∂/∂ v and ∂/∂ x^a. The most general connection satisfying these conditions has the form:=_0(u) ṿ + _a(u) x̣^a , where we write the potential in the coordinatesṣ^2 =2 ụ ṿ - x̣_a x̣^a , of Minkowski space. We want (<ref>) to be preserved under the same Heisenberg symmetry algebra (<ref>) that generated the isometries of the plane wave metrics in Einstein-Rosen coordinates. This requires there to be a vector field_φ^a=x^a ∂/∂v + u ∂/∂x_a +φ^a , with φ^a a Lie algebra-valued function for which[^a_φ,^b_φ]=0 , [∂/∂x^a, ^b_φ]=δ^b_a∂/∂v . These conditions imply that φ^a=φ^a(u) and [φ^a,φ^b]=0. Furthermore, we require that ^a_φ generates a further symmetry of the gauge connection; namely, that =+̣ is covariantly Lie-dragged along the ^a_φ. This imposes further constraints on :_a=-φ̇_a , [_0, φ^a]=0 , [_a, φ^b]=δ^b_a _0 . For simplicity, we restrict our attention to the special case where φ^a is valued in the Cartan subalgebra 𝔥⊂𝔤. With this choice, consistency of the symmetry algebra reduces to_0=0 , φ^a(u)=-∫^uṣ ^a(s) , and the functional form of ^a_φ closely resembles that of its gravitational counterpart (<ref>).To summarize, our definition of an `Einstein-Rosen' plane wave gauge field (valued in the Cartan of the gauge group) results in a gauge potential of the form:=-_a(u) x̣^a , where an overall negative sign has been included for convenience. Just as the Brinkmann form of a plane wave metric can be obtained by the diffeomorphism (<ref>) from Einstein-Rosen form, a gauge transformation of (<ref>) gives the plane wave gauge potential in `Brinkmann' form. In particular, taking →+(̣x^a_a) gives =x^a _a ụ . The fact thatis a linear polynomial in x^a, rather than a quadratic function as in the gravitational setting (<ref>), is a first glimpse of the double copy. It has already been noted that plane wave background geometries (for gauge theory and gravity) exhibit the double copy structure <cit.>, although the distinction between linear and quadratic functions does not seem to have been noticed previously. Although we obtained (<ref>) from the Einstein-Rosen gauge by working in the Cartan subalgebra of the gauge group, general non-abelian plane waves also take this functional form <cit.>.The field strength isF=_a x̣^a∧ụ . As for the Brinkmann metric, the gauge field (<ref>) directly encodes the field strength; (<ref>) obeys the Maxwell equations, and hence the Yang-Mills equations when valued in the Cartan subalgebra of the gauge group. The sandwich gauge field plane wave is analogous to that for gravity; the field strengthF_a=_a(u) is taken to be compactly supported for u_1≤ u≤ u_2≤0, so that it is flat in the in-region (u<u_1) and out-region (u>u_2). The memory effect here is associated with the fact that ifis taken to vanish in the past, it will be constant and non-zero in the future_a|_out-_a|_in=∫_u_1^u_2 F_aụ , By analogy with the gravitational case, (<ref>) can be viewed as encoding the electromagnetic memory effect <cit.> for plane wave gauge theory backgrounds. § FREE FIELDS ON PLANE WAVE BACKGROUNDS AND INNER PRODUCTSAmplitudes in flat space are functionals of free fields and are usually expressed as functions of momenta after being evaluated on momentum eigenstates. In curved space, such solutions are not so obviously available and it is here that we must use the special structure of plane waves. Friedlander showed that Huygens' principle remains valid for the scalar wave equation in plane wave space-times: there exist solutions with delta-function support on null hypersurfaces through every null direction <cit.>. These null hypersurfaces are level surfaces of solutions to the Hamilton-Jacobi equation, which provide curved space analogues of the function k· X for null vectors k in Minkowski space. Such functions provide analogues of momentum eigenstates, and also lead to integral formulae for general solutions to the wave equation <cit.>. Generalizing <cit.>, we can raise the spin to obtain free fields of spin one and two with arbitrary polarizations, but Huygens' principle no longer holds and tails appear. Furthermore, a consequence of the memory effect will be that, unlike flat space-time, a momentum eigenstate in the past will not evolve into one in the future. Nevertheless,we can show that, despite the lack of global hyperbolicity of plane waves <cit.>, the scattering problem is well-defined on a plane wave background, featuring unitary evolution without leakage or particle creation.§.§ Scalar wave equation The plane progressing waves of Friedlander are obtained from solutions to the Hamilton-Jacobi equation for null geodesics g^μν(_μϕ)(_νϕ)=0,such that arbitrary functions of ϕ satisfy the wave equation (when multiplied by a fixed pre-factor). Solutions are most easily obtained in Einstein-Rosen coordinates where they can be separated using the explicit symmetries leading to ϕ_k=k_0v+ k_i y^i +k_ik_jF^ij(U)/2 k_0,where (k_0,k_i) are constants and F^ij=∫γ^ij(s)ṣ as in (<ref>). The wave equation in Einstein-Rosen coordinates is1/√(-|g|)∂_μ(√(-|g|) g^μν ∂_ν Φ)=(2∂_U ∂_V +(_U √(γ))_V - γ^ij∂_i ∂_j)Φ=0 , and it can be seen directly that this is solved by <cit.>Φ(X)=Ω(U) ^ ϕ_k , Ω(U):=|γ^-1(U)|^1/4 = |E(u)|^-1/2 , In Brinkmann coordinates, the wave equation is (2∂_u ∂_v+H(u,𝐱) ∂^2_v - ∂_a ∂^a)Φ=0 , and of course this is solved by the same Φ. Using (<ref>), it can be expressed in Brinkmann coordinates as: ϕ_k:= k_0/2σ_ab x^ax^b+k_iE^i_a x^a + k_0 v + k_i k_j/2 k_0 F^ij , with F^ij(u) and (k_0,k_i) as before, and σ_ab=Ė^i_a E_b i the deformation tensor defined by (<ref>). The natural momentum associated with ϕ_k is:K_μ X̣^μ := ϕ̣_k= k_0 ṿ+( k_0/2 σ̇_bc x^bx^c+k_iĖ^i_bx^b+k_ik_j/2k_0γ^ij)ụ+(k_iE^i_a+k_0 σ_abx^b)x̣^a .Although K_μ is a (u,x^a)-dependent generalization of the constant momentum familiar from flat space, it is nevertheless null by construction from the Hamilton-Jacobi equation. To see this explicitly, note that σ̇_bc=Ė^i_bĖ_c i-H_bc.The solutions Φ=Ω^ϕ_k clearly reduce to on-shell momentum eigenstates when the background is Minkowski space, and hence can be chosen to do so in one or other asymptotic region. We can use this to characterize in and out scattering states in terms of ϕ_k: an in state Φ^- is one which looks like a plane wave ^ k· X in the in-region (u<u_1), while an out state Φ^+ looks like a plane wave in the out-region (u>u_2). This comes down to requiring the vielbein to become trivial in the past or the future:lim_u→±∞E^a ±_i(u) = δ^a_i . In terms of the solution to the Hamilton-Jacobi equations, ϕ_k, the distinction becomes:ϕ^-_k|_in= k_0 v+k_iδ^i_a x^a+u δ^ijk_ik_j/2k_0=ϕ^+_k|_out . The positive frequency condition on these states is simply that k_0≥0.Even at the level of the free theory, some interesting facts about the S-matrix on a plane wave space-time can be derived by making use of the natural inner product between two solutions to the free equation of motion. This uses complex conjugation to turn the standard symplectic form on the space of solutions of the wave equation into an inner product: Φ_1 | Φ_2= ∫_Σ (Φ_1∧* ̣̅Φ_2 - Φ̅_2∧* Φ̣_1) , where Σ is an arbitrary Cauchy surface. Plane wave space-times do not admit a Cauchy hypersurface <cit.>, but one can instead choose the foliation by hypersurfaces Σ_u of constant u. In this case, the inner product gives:Φ_1 | Φ_2= ∫_Σ_u ṿ ^̣d-2x (Φ_1 ∂_vΦ̅_2 - Φ̅_2 ∂_vΦ_1) ,evaluated at some fixed u. Consider the inner product between two positive frequency in states, say Φ^-_1 and Φ^-_2 with constant momentum components {k_0,k_i} and {l_0,l_i} respectively. Using (<ref>), this givesΦ^-_1|Φ^-_2=2 k_0 δ(k_0-l_0) δ^d-2(k_i-l_i) , with all u-dependence dropping out. As desired, the evolution problem underlying the scattering theory is unitary, since there is no `leakage' of momentum – at any value of u – between the two in states.Similarly, the inner product between a positive frequency in state and a negative frequency out state (namely Φ^+_1|Φ̅^-_2) encodes the presence of `particle creation' in the plane wave background. Without loss of generality, the inner product can be evaluated at u=0>u_2, leading to:Φ^+_1|Φ̅^-_2= δ(k_0+l_0) (k_0-l_0) Ω^-(0)∫^̣d-2x exp[(l_0/2σ_ab^-(0) x^ax^b....+(k_a+l_iE^- i_a(0))x^a+l_i l_j/2l_0F^ij_-(0))] .However, the assumption of positive frequency means that k_0+l_0≥0, so on the support of the overall delta function this inner product vanishes:Φ^+_1|Φ̅^-_2=0 , confirming the well-known result that there is no particle creation for scalar QFT in plane wave space-times <cit.>. Equivalently: positive frequency in states do not develop a negative frequency part in the out-region.The final independent inner product is between positive frequency in and out states, Φ^+_1|Φ^-_2. This quantity encodes the amplitude for in-to-out scattering in the plane wave space-time <cit.>. The inner product can again be evaluated at u=0:Φ^+_1|Φ^-_2=2 k_0 δ(k_0-l_0) ^-s_l Ω^-(0)×∫^̣d-2x exp[((k_a-l_i E^- i_a(0)) x^a -l_0/2σ_ab^-(0) x^ax^b)] ,where the (constant) phase s_l is defined ass_l:=l_il_j/2l_0F^ij_-(0) .Now, by (<ref>) it follows thatE^-_ia(u)=u b_ia+c_ia , ∀u>u_2 , where b, c are constant, invertible (d-2)×(d-2) matrices. This leaves a Gaussian integral to do in (<ref>), with the result:Φ^+_1|Φ^-_2=2 k_0(2π/l_0)^d-2/2 δ(k_0-l_0) ^-(s_l+r_k,l)/√(|b|) , after using the fact that Ω^-(0)=√(|c^-1|) and defining another phaser_k,l:=-1/2 l_0(k_a-l_ic^i_a)c^ak (b^-1)^b_k (k_b-l_jc^j_b) .As expected, this matches the result in the literature <cit.>. §.§ Spin one Theaction for free gauge fields propagating on a plane wave space-time isS^free[A]=1/g^2∫_Mụ ṿ ^̣d-2x(∇_[μ A_ν] ∇^μA^ν) , where A_μ is the gauge field and ∇ the Levi-Civita connection. We will see that on a plane wave it is consistent to simultaneously impose both a Lorenz gauge ∇_μA^μ=0 and a light-cone gauge A_v=0, since _v is Killing.With this, the linearized equations of motion for the gauge connection areg^ρσ∇_ρ∇_σ A_μ=0 , ∂_μ A^μ=0=A_v . These can be solved using the d-2 spin-raising operatorsℛ^a:=ụ δ^ab∂/∂x^b +x̣^a ∂/∂v , where the free index labels different possible polarization states. As tensors, the ℛ^a are covariantly constant. Acting on a solution to the wave equation, Φ, it is easily checked that ℛ^aΦ satisfies (<ref>), so ℛ^a is naturally a spin-raising operator (this generalizes the four-dimensional approach in <cit.>).Thus with Φ the scalar wave (<ref>) we construct the free gauge fieldA_μ X̣^μ=1/k_0ϵ_a ℛ^aΦ= 1/k_0ϵ_a ℛ^a (Ω ^ ϕ_k) , where ϕ_k and Ω are as before and the polarization vector ϵ^a is constant. We can also define a `curved' ε_μ so thatA_μ= ε_μΦ , ε_μX̣^μ=ϵ^a(k_j/k_0E^j_a+σ_ab x^b) ụ +ϵ_ax̣^a . This satisfies the free equation of motion and gauge-fixing conditions. Similarly to its flat space counterpart, the curved polarization vector obeys·K=g^μν _μ K_ν=0 , where K is as defined in (<ref>). In the flat space limit,A_μ reduces to a standard linearized plane wave ε^flat_μ^ k· X, with the non-trivial constant components of ε^flat_μ being ϵ_a.In and out states are defined in the same way as for the scalar: an in state looks like a Minkowski plane wave in the in-region, while an out state looks like a Minkowski plane wave in the out-region.As in the scalar case, an inner product on free gauge fields is induced by the boundary term of the action <cit.>. Restricted to a constant u hypersurface, this inner product is:A_1|A_2:= ∫_Σ_uṿ ^̣d-2x (A_1^μ F̅_2 vμ-A̅^μ_2 F_1 vμ) , which is easily used to compute the three cases of interest. Assuming positive frequency for all (un-conjugated) fields, one finds: A^-_1 | A^-_2 =2 k_0 ϵ_1 ·ϵ_2 δ(k_0-l_0) δ^d-2(k_i-l_i) ,A^+_1 | A̅^-_2 =0 ,A^+_1 | A^-_2 =2k_0 (2π/l_0)^d-2/2 ϵ_1 ·ϵ_2 δ(k_0-l_0) ^-(s_l+r_k,l)/√(|b|) , where ϵ_1 ·ϵ_2 = ϵ_1^a ϵ_2^b δ_ab and the phases s_l, r_k,l are the same as the scalar case. Unsurprisingly, (<ref>) indicate that the evolution problem is unitary and that there is no particle creation for gauge fields propagating on the plane wave space-time. §.§ Spin two Finally, consider linearized metric fluctuations h_μν on the plane wave background. Assuming that the background is a solution to the vacuum Einstein equations and choosing a transverse-traceless gauge for the perturbations∇_μh^μ_σ=0=h_μ^μ , the linearized Einstein equation is:∇_σ∇^σh_μν-2 R^ρ_ μνσ h_ρ^σ=0 , with R^ρ_ μνσ the background curvature tensor. For a vacuum plane wave in Brinkmann coordinates (i.e., H_a^a=0), the gauge for h_μν can be further fixed by requiring the vanishing of the v-components h_v μ=0. With these conditions, the linearized equation is:g^μν∂_μ∂_νh_ρσ+4 δ^u_(ρ ∂_|v| h_σ)a H^a_b x^b-2 δ^u_ρδ^u_σH^abh_ab=0 , where all Christoffel symbols have been written out explicitly in Brinkmann coordinates.Solutions to (<ref>) can be constructed by acting on the massless scalar twice with the spin-raising operator (<ref>). This leads to:h_μν X̣^μ X̣^ν = 1/k_0^2ϵ_a ℛ^a(ϵ_b ℛ^b Φ)= ((ε·X̣)^2 - /k_0 ϵ_a ϵ_b σ^ab ụ^2) Φ , where ϵ_a is chosen to be null with respect to δ^ab to ensure that the gauge condition h_μ^μ=0 is obeyed. Note in particular the `tail' term proportional to ϵ_a ϵ_bσ^ab: unlike in Minkowski space-time, metric perturbations on a plane wave background do not carry a polarization which is simply the `square' of a gauge field's polarization. The reason for this is that the second spin raising operator in (<ref>) acts not only on the scalar solution (which contributes a second copy of ε_μ) but also on the first spin raising operator (or equivalently, on the first copy of _μ, which – unlike in Minkowski space – is not a constant vector).Thus the perturbative double copy for plane wave backgrounds involves subtleties not present in Minkowski space. For linear perturbations around flat space,h_μν∼ A_μ⊙ A_ν formomentum eigenstates, whereas in plane wave space-times we have h_μν∼ A_μ⊙ A_ν +C_μν, withcorrectionC_μν given by the last term proportional to σ^ab in (<ref>). The boundary term in the linearized Einstein-Hilbert action induces an inner product on metric fluctuations <cit.>:h_1|h_2=∫_Σ_u ṿ ^̣d-2x (h_1^μσ ∂_vh̅_2 μσ-h̅_2^μσ ∂_vh_1 μσ) . Once again calculating the inner products between incoming and outgoing states gives:h^-_1 | h^-_2=2 k_0(ϵ_1 ·ϵ_2)^2 δ(k_0-l_0) δ^d-2(k_i-l_i) ,h^+_1 | h̅^-_2 =0 ,h^+_1 | h^-_2 =2k_0 (2π/ l_0)^d-2/2 (ϵ_1 ·ϵ_2)^2 δ(k_0-l_0) ^-(s_l+r_k,l)/√(|b|) .So despite the `correction' term in h_μν, the physical properties of unitary evolution and no particle creation are preserved. §.§ Charged free fields in plane wave gauge fields Although we assume that the background gauge potential in (<ref>) is valued in the Cartan algebra, it couples non-trivially to free fields which are charged under the gauge group. Consider a free, charged scalar: S^free[Φ]= 1/2∫ụ ṿ ^̣d-2x D_μΦD^μΦ , where D_μ=∂_μ- e_μ, with _μ the background gauge field (<ref>) and e the charge of Φ.In the first instance, we will take e to be a standard (1) charge, but more generally,takes values in the Cartan subalgebra of some gauge group, Φ in some root space, and e will then be the corresponding root and e the corresponding contraction withencoding the commutator.The free equation of motion for the charged scalar is thusD_μD^μΦ(X)=(2∂_u ∂_v -∂_a ∂^a-2 x^ae _a ∂_v)Φ(X)=0 . Solutions to this `charged' wave equation are given by:Φ(X)=^ ϕ̃_k , whereϕ̃_k=k_0 v +(k_a+e_a) x^a+1/2 k_0 f(u) . The function f(u) is the analogue of the F^ij(u) which appeared in the gravitational case:f(u):=∫^uṣ (k_a+e_a(s)) (k^a+e^a(s)) . When the background gauge field is turned off, it is easy to see that these solutions become the usual momentum eigenstates of Minkowski space.The natural momentum associated with these scalars is defined by _μ X̣^μ:=-^-ϕ̃_k D_μ ^ϕ̃_k X̣^μ = k_0 ṿ+ 1/2 k_0(k_a+e_a)(k^a+e^a)ụ +(k_a+e_a)x̣^a .The components of _μ are functions of u, but it is easy to see that this momentum is null. The distinction between in and out states for the charged scalar is in direct analogy with the definitions on the gravitational background. An incoming state is one which looks like a Minkowski plane wave in the in-region, while an outgoing state looks like a Minkowski plane wave in the out-region. This distinction manifests itself in the boundary conditions on :lim_u→±∞^±_a(u)=0 . Note that unlike the massless scalar in the gravitational background, the exponential dependence on x^a for the charged scalar is at most linear in any region.The inner product on the charged scalars is given byΦ_1|Φ_2=∫_Σ_uṿ ^̣d-2x (Φ_1 ∂_vΦ̅_2- Φ̅_2 ∂_vΦ_1) , and once again there are three inner products of physical interest. These are: Φ^-_1|Φ^-_2 = 2k_0 δ(k_0-l_0) δ^d-2(k_a-l_a) ,Φ^+_1|Φ̅^-_2 = 0 , Φ^+_1|Φ^-_2 = 2k_0 δ(k_0-l_0) δ^d-2(k_a-l_a+c_a) ^ s̃_l ,where c_a is the inner product of ^-_a(0) in the Cartan subalgebra with the charge of the field.The momentum conservation then indicates the `kick' received by the field from the memory effect. The phase s̃_l is defined bys̃_l:=f_-(0)/2 l_0 .The equations (<ref>) indicate that the classical S-matrix associated with this charged scalar is unitary with no particle production. §.§ Spin one on a gauge background The linearized equation of motion for a gauge field a_μ charged under the same gauge group as the background _μ is:D_μ(D^μa^ν-D^νa^μ)+a_μ(∂^μ^ν-∂^ν^μ)=0 . Solutions to this equation are simplified by choosing a Lorenz gauge D_μa^μ=0 along with[This is of course not possible on a general background, but is possible here because _v is a symmetry.] a_v=0; the latter condition actually reduces the Lorenz condition to ∂_μa^μ=0. Solutions are then found by acting on the charged scalar solution with ℛ^a as before in the gravitational case. This leads toa_μ X̣^μ=ϵ̃_a( x̣^a + 1/k_0(k^a+e^a) ụ) ^ϕ̃_k , where ϵ̃_a is a (constant) (d-2)-dimensional vector which we will take to be null. As in the gravitational case, we define a polarisation d-vector _μ as _μX̣^μ= ϵ̃_a (x̣^a + 1/k_0(k^a+e^a) ụ).This polarization is on-shell in the sense that ·=0.With these gauge choices, the inner product is essentially equivalent to (<ref>) giving:a^-_1|a^-_2 = 2k_0 ϵ̃_1·ϵ̃_2 δ(k_0-l_0) δ^d-2(k_a-l_a) ,a^+_1|a̅^-_2 = 0 ,a^+_1|a^-_2 = 2 k_0 ϵ̃_1·ϵ̃_2 δ(k_0-l_0) δ^d-2(k_a-l_a+c_a) ^ s̃_l .So we again have a unitary classical S-matrix with no particle creation, as before. §.§ Huygens' principle and tails The wave equation in flat and plane wave space-times satisfies Huygens' principle <cit.>.In intuitive terms, the principle states that waves can propagate in all directions withoutscattering off the background metric and generating a tail. The sharp definition is that there should exist solutions to the wave equation with delta-function support along null hypersurfaces tangent to every null direction through every point. These are simply given in the above by Ω δ(ϕ_k -c) where c is a constant. This principle fails for linear fields of spin one and spin two <cit.>, however.We can construct these fields by spin raising as above. At spin one, to get a field with delta function support along ϕ_k=0, we must start by raising the spin of a solution to the scalar wave equation of the form Ω ϕ_k Θ(ϕ_k) where Θ is the Heaviside step function. With this, the corresponding spin-one potential is A =Θ(ϕ_k) ϵ_a/k_0 ℛ^a (Ω ϕ_k Θ(ϕ_k))= Ω ϵ^a (x̣_a +(k_j/k_0 E^j_a + σ_abx^b)ụ) Θ(ϕ_k),and the field strength isF =Ạ= δ(ϕ_k) Ω ϵ^a (x̣_a +(k_j/k_0E^j_a +σ_abx^b) ụ)∧ϕ̣_k +Θ(ϕ_k) Ω ϵ^a (σ_ab x̣^b∧ụ -σ^b_b x̣_a∧ụ).We see that the field strength has developed a tail in the second line, which is not supported at ϕ_k=0. This tail can be thought of as the consequence of the interaction between the impulsive electromagnetic field and the gravitational background. There is a similar story for the spin-two field where one starts with Φ= ϕ_k^3 Θ(ϕ_k).In these examples, the tail is proportional to the shear of the ∂_U null geodesic congruence (i.e., trace-free part of σ_ab). So tails are generally identified by the part of the field in which the shear appears explicitly. In the free solutions constructed above, terms contributing to the tails are readily identified: σ_ab x^bụ from ε·X̣ at spin one and two, and the spin two correction term C=-/ k_0ϵ_a ϵ_bσ^abụ^2. However, we will see that the contributions to the tail from ε_μ alone actually drop out of amplitude calculations. So for spin one fields on a plane wave space-time, the tail terms do not effect the amplitude – even though they appear explicitly in the scattering states.This observation is perhaps related to a different definition of tails for the propagation of gauge fields on a plane wave space-times, in terms of a Green's function in <cit.>. That discussion does not give tails for gauge fields but does for graviton propagation <cit.>, and indeed we will see that it is the extra correction term C that is important forgraviton amplitudes. Note that this treatment of tails does not simply extend to fields propagating on the gauge theory plane wave background because we cannot simply obtain solutions fromarbitrary functions of Φ̃ as it now has charge. So, in the gauge background case, we will simply take the tail to be those terms in a curved polarization vector that depend explicitly on the potential . This is consistent with the fact that such potential terms encode the memory in the asymptotic regions via (<ref>), just as the deformation tensor σ_ab does on a gravitational background. § 3-POINT AMPLITUDES ON THE GRAVITATIONAL BACKGROUNDWe now consider the 3-point amplitudes of scalars, gauge fields and gravitons on the gravitational sandwich plane wave background. In each case, this calculation is performed by evaluating the cubic part of the action on solutions to the linearized equations of motion on the background. For each theory, the amplitude formulae are presented in terms of an integral over the u variable (in Brinkmann coordinates), which cannot be done explicitly for general space-times. Stripping off the integration underlying the action integral, together with the three Φs associated with the three on-shell fields, we are left with a tree-level integrand expression which is sufficient for exploring the double copy structure of the amplitudes. See appendix <ref> for further discussion of the scattering amplitudes and tree-level integrand. §.§ Scalars Consider the cubic scalar theoryS[Φ]=1/2∫_M ụ ṿ ^̣d-2x (g^μν∂_μΦ ∂_νΦ- λ/3Φ^3) , where g^μν is the inverse of the plane wave metric (<ref>) in Brinkmann coordinates. The 3-point amplitudes of interest are given by evaluating the cubic portion of the action[A similar calculation has been done for scalar contact interactions of arbitrary valence in certain homogeneous plane wave backgrounds <cit.>.]-λ/6∫_Mụ ṿ ^̣d-2x Φ_1(X) Φ_2(X) Φ_3(X) , where Φ_r(X) are solutions to the linearized equations of motion of (<ref>) for r=1,2,3. When evaluating (<ref>), there are basically two distinct configurations which need to be considered: three in states, or one out and two in states (the other configurations are easily related to these). The case when all three states are incoming is the easiest. This gives-λ/6∫_Mụ ṿ ^̣d-2x Φ^-_1(X) Φ^-_2(X) Φ^-_3(X)=-λ/6 δ^d-1(∑_r=1^3k_r)∫ụ |E^-| (Ω^-)^3 exp( F^ij∑_s=1^3k_s ik_s j/2k_s 0) =-λ/6 δ^d-1(∑_r=1^3k_r) ∫ụ/√(|E^-|) exp( F^ij∑_s=1^3k_s ik_s j/2k_s 0) .whereδ^d-1(∑_r=1^3k_r):= δ(∑_r=1^3 k_r 0) δ^d-2(∑_r=1^3 k_r i) .The delta functions arise from performing the integrations in ṿ and ^̣d-2x, with |E^-| an overall Jacobian factor appearing in the second line. Using the relationship (<ref>) between Ω(u) and |E|, the various u-dependent factors left inside the integral can be slightly simplified in passing to the third line.The other configuration is a bit more complicated. In this case one has-λ/6∫_Mụ ṿ ^̣d-2x Φ^-_1(X) Φ^-_2(X) Φ^+_3(X) =-λ/6 δ(∑_r=1^3k_r 0) ∫ụ ^̣d-2x(Ω^-)^2Ω^+× exp(k_3 0/2(σ_ab^--σ_ab^+) x^a x^b.. + (k_1 i+k_2 i)E^i -_a x^a + k_3 iE^i +_a x^a + ∑_s=1^3k_s ik_s j/2k_s 0 F^ij_s) .Due to the mixed asymptotic conditions, momentum conservation in the v-direction no longer eliminates the quadratic x-dependence from the exponential, leaving a (d-2)-dimensional Gaussian integral. Performing this integral leaves:-λ/6 (k_3 0)^d-2/2 δ(∑_r=1^3k_r 0) ∫ụ(Ω^-)^2Ω^+ √((2π)^d-2/|A|) ×exp(-/2 k_3 0J_a J_b (A^-1)^ab +∑_s=1^3k_s ik_s j/2k_s 0 F^ij_s) ,whereA_ab:= σ_ab^--σ_ab^+ ,J_a:=(k_1 i+k_2 i) E^i -_a + k_3 i E^i +_a .Nevertheless, applying the definition of the tree-level integrand to these results (see earlier or appendix <ref>), somewhat tautologically gives the extremely simple answer_3(Φ^-_1, Φ^-_2, Φ^±_3)=1 , after stripping off a power of the coupling, overall delta-functions, and `universal' u-dependent functions that depend on the choice of Φ's.This is a general feature. Although the precise form of the amplitude will vary significantly between different configurations of incoming and outgoing states – as in (<ref>) versus (<ref>), the integrands will be the same. This is the closest thing to CPT symmetry in flat space-time – interpreted here as the ability to exchange incoming and outgoing states while simultaneously conjugating polarizations and charges – which survives on a sandwich plane wave background. §.§ Gauge theoryThe Yang-Mills action on a curved background is:S[A]=1/g^2∫_M (F∧* F) , where * is the Hodge star and F=[D,D] is the curvature of the connection D=∇ + A, for ∇ the Levi-Civita connection. The 3-point amplitude is given by the cubic portion of the action (<ref>) evaluated on linearized states of the form (<ref>). In the Lorenz gauge of section <ref>, the 3-point amplitude reads:g f^_1_2_3 ∫ụ ṿ ^̣d-2x(A_3^b A^μ_2 ∂_μA_1 b - A_2^b A_3^μ ∂_μA_1 b + cyclic) , where f^_1_2_3 are the structure constants of the gauge group. As before, there are essentially two independent configurations in which this amplitude can be evaluated: three in states or two in states and one out state. However, some simplifications occur in the amplitude even before the asymptotic behaviour of the states has been specified. Evaluated on general linearized free fields, (<ref>) becomes g f^_1_2_3 ∫ụ ṿ ^̣d-2x (_1·_3 (K_1·_2-K_3·_2) +cyclic ) ∏_r=1^3Ω_r ^ϕ_r , where the Ω_r and ϕ_r (r=1,2,3) depend on whether the state is incoming or outgoing. Since the functional form of the integrand (i.e., the portion of this expression in the parentheses) is independent of the state configuration, it suffices to identify the integrand in the simplest configuration. As in the scalar example, this will be the all incoming configuration, since there are more delta functions in this case.Even for the three-incoming configuration, the integrand of (<ref>) is a priori a function of the x^a through the polarizations (<ref>) and momenta (<ref>). However, thanks to the identities: K_r·ε_s = {[0ifr=s; E^i a(k_r 0k_s i/k_s0ϵ_s a-k_r iϵ_s a)otherwise ].,_r·ε_s = {[ 0 ifr=s;-ϵ_r·ϵ_s otherwise ]. ,it follows that the integrand is actually independent of the x^a.This allows the ṿ and ^̣d-2x integrals to be done as the only dependence on these variables is in the exponential:g f^_1_2_3 δ^d-1(∑_r=1^3k_r) ∫ụ/√(|E^-|) (ε_1·_3 (K_1·_2-K_3·_2) + cyclic)× exp( F^ij∑_s=1^3k_s ik_s j/2k_s 0) .On the support of the momentum conserving delta functions, this simplifies to2g f^_1_2_3 δ^d-1(∑_r=1^3k_r) ∫ụ/√(|E^-|) (_1·_3 K_1·_2 +cyclic)exp(F^ij∑_s=1^3k_s ik_s j/2k_s 0) . As we saw for the scalar, the amplitude boils down to a u-integration which depends on the particulars of the background plane wave geometry. The integrand, though, is easily identified as:_3(A_1,A_2,A_3)=_1·_3 K_1·_2+ cyclic. Note that although this has the same functional form as the flat space 3-point integrand for Yang-Mills theory, it is not equal to the flat space result. Indeed, the integrand in this case is a function of u, given explicitly by_3(A_1,A_2,A_3)= - ϵ_1 ·ϵ_3 E^i_a(k_1 0/k_20 k_2 i ϵ^a_2-k_1 i ϵ^a_2) +cyclicafter using (<ref>)–(<ref>). Note that the tails associated with the asymptotic states do not contribute to the amplitude, as a result of the identities (<ref>)–(<ref>).The other configuration – two incoming states and one outgoing state – is more complicated. The primary reason for this is that the x-dependence of the integrand does not drop out. Assuming that the scattering states are A^-_1, A^-_2 and A_3^+ we now have_r·_3 = -ϵ_r·ϵ_3 , K_r·_3 = ϵ_3^a (k_r 0k_3 i/k_30E^+ i_a-k_r iE^- i_a)+k_r 0ϵ_3^a x^b (σ^+_ab-σ^-_ab) ,K_3·_r = ϵ_r^a (k_3 0k_r i/k_r 0E^- i_a-k_3 iE^+ i_a)+k_3 0ϵ^a_rx^b (σ^-_ab-σ^+_ab) ,for r=1,2. The integration over ^̣d-2x is now a rather involved Gaussian integral, which has the rough structure of (<ref>) plus a derivative of this result. Since the integrand is the primary object of interest here, we will onlyconsider (<ref>). §.§ Gravity The 3-point amplitude for gravitons on the plane wave background is encoded by extracting the cubic portion of the Einstein-Hilbert action,S[g]=1/κ^2∫_M ^̣dX √(-|g|) R , perturbed around the plane wave background metric. To do this, a recent perturbative re-writing of the Einstein-Hilbert action is useful <cit.>. For perturbations h_μν around a fixed background geometry g_μν, this action takes the form:S[h]=1/4 κ^2∫^̣dX √(-|g|)[∇_μσ_νρ ∇_λσ^κρ(σ^μλδ^ν_κ - 2 σ^νλδ^μ_κ) +σ^μν R_μν] , where the perturbations are encoded inσ_μν=g_μν+κ h_μν+κ^2/2h^2_μν +⋯ , σ^μν=g^μν-κ h^μν+κ^2/2h^μν-⋯ ,and indices are raised and lowered with the background metric (e.g., h^2_μν=h_μρg^ρσh_σν). On the vacuum plane wave background in Brinkmann coordinates, |g|=-1 and R_μν=0 so expanding (<ref>) to cubic order is straightforward. This leads to the cubic term:κ/4∫ụ ṿ ^̣d-2x(h^μν∇_μh_ρσ∇_νh^ρσ-2 h^ρν∇_μh_ρσ∇_νh^μσ) . We have checked that this matches the cubic contribution from expanding the standard Einstein-Hilbert action around a plane wave background.The 3-point amplitude is given by evaluating (<ref>) on three of the linearized perturbations (<ref>). With the transverse-traceless gauge conditions on h_μν, the covariant derivatives in (<ref>) reduce to partial derivatives, leaving:κ/4∫ụ ṿ ^̣d-2x(h_1^μν∂_μh_2 ρσ∂_νh_3^ρσ-2 h_1^ρν∂_μh_2 ρσ∂_νh_3^μσ) + allpermutations . A computation gives a typical term in the sum over permutations of external states to be:h_1^μν∂_μh_2 ρσ∂_νh_3^ρσ-2 h_1^ρν∂_μh_2 ρσ∂_νh_3^μσ=((2_3· K_2 _1· K_3 _1·_2-_1· K_2 _1· K_3 _2·_3) (_2·_3)- _2·_3 σ^ab(k_2 0k_3 0/k_1 0_2·_3 ϵ_1 aϵ_1 b -2k_2 0 _1·_2 ϵ_1 bϵ_3 a)) ^(ϕ_1+ϕ_2+ϕ_3) .To proceed further, the configuration of the external states must be specified. Building on the scalar and gauge theory calculations, it is clear that the easiest configuration to treat is the one with all three states incoming.In this configuration, identities of the form (<ref>)–(<ref>) ensure that the only x-dependence in termslike (<ref>) is in the overall exponential. This allows the ṿ and ^̣d-2x integrations to be done explicitly, resulting in momentum conserving delta functions. On the support of these delta functions, the 3-point amplitude for incoming states reads:κ/2 δ^d-1(∑_r=1^3k_r) ∫ụ/√(|E^-|) [(_1·_3 K_1·_2+ cyclic )^2.-k_1 0k_2 0k_3 0 σ^ab𝒞_a𝒞_b ] × exp( F^ij∑_s=1^3k_s ik_s j/2k_s 0) .where the quantity 𝒞_a is defined as𝒞_a:= _2·_3 ϵ_1 a/k_10 + _1·_3ϵ_2 a/k_20+_1·_2ϵ_3 a/k_30 . The upshot is that the 3-point integrand for gravity on a plane wave space-time is given by_3(h_1,h_2,h_3)=(_1·_3 K_1·_2+_1·_2 K_2·_3 +_2·_3 K_3·_1)^2-k_1 0k_2 0k_3 0 σ^ab 𝒞_a 𝒞_b ,This structure mirrors what one might have guessed based solely on the structure of the linearized perturbations (<ref>). So it seems that 3-point amplitudes on a plane wave space-time do not simply obey double copy as they do in flat space. Indeed, we find that_3(h_1,h_2,h_3)=(_3(A_1,A_2,A_3))^2-k_1 0k_2 0k_3 0 σ^ab 𝒞_a 𝒞_b . Unlike the gluon amplitudes, the tails associated to graviton perturbations do contribute to the amplitude. Note that they do so in an intrinsically geometric way: the tail contribution couples via the deformation tensor associated with the background geometry. To find the `square root' of perturbative gravity on a plane wave background, one must instead turn to Yang-Mills theory in the presence of a background plane wave gauge field. § 3-POINT AMPLITUDES ON THE GAUGE FIELD BACKGROUNDThe 3-point amplitudes for charged scalars and Yang-Mills theory in a plane wave background gauge field are now computed. As in the gravitational setting, these amplitudes reduce to an integral over the u-coordinate which depends on the particulars of the background, but the tree-level integrands are easily identified. §.§ Charged scalarsTo obtain a gauge invariant cubic scalar interaction that carries charge with respect to the background gauge field, the charges of the three fields must add up to zero.S_int[Φ]= ∫ụ ṿ ^̣d-2x ( Φ_1Φ_2Φ_3) , where D_μΦ_r=(∂_μ- e_r_μ)Φ_r, with _μ the background gauge field (<ref>). The charges e_r as roots encode the commutators.Armed with the linearized solutions (<ref>), we can compute the 3-point amplitudes by evaluating the cubic portion of the action (<ref>). This means that the amplitude can be reduced to a u-integration fairly straightforwardly in an arbitrary configuration:δ^d-1(∑_r=1^3k_r) ∫ụ exp(∑_s=1^3f_s/2k_s 0) . Note that the translation action of the gauge field on the total momentum has cancelled because the charges must add up to zero by gauge invariance.From this expression it is easy to read off the tree-level integrand for the 3-point scattering of charged scalars on the plane wave gauge field background:_3(Φ_1,Φ_2,Φ_3)=1 . This is independent of the specifics of the configuration as forthe gravitational background. §.§ Gauge theory Now consider a dynamical gauge field a on the fixed plane wave background . Although the background gauge fieldis valued in the Cartan of the gauge group, the dynamical gauge field carries arbitrary colour structure. The dynamical gauge field is governed by the actionS[a]=1/g^2∫(∧*-∧*) , whereis the curvature of +a and the kinetic term for the non-dynamical background field is subtracted. The cubic term in the action (<ref>) is∫ụ ṿ ^̣d-2x (a_μ a_ν(∂^μa^ν-∂^νa^μ+[^μ,a^ν])). We must choose the colour structure so as to obtain a non-trivial trace. All non-trivial examples are essentially the same and are equivalent to taking the (2) case with a_3 in the Cartan, and a_1, a_2 respectively of charge ± 1 with respect to the Cartan generator. In particular the three charges add up to zero. Together with the gauge choices made in (<ref>), the 3-point amplitude reduces tog f^_1_2_3 ∫ụ ṿ ^̣d-2x (a_2^μ a^ν_3∂_μa_1 ν-a_2^μ a_3^ν∂_νa_1 μ + cyclic) . Evaluating on the states (<ref>) (with arbitrary asymptotics) leads tog f^_1_2_3 δ^d-1(∑_r=1^3k_r) ∫ụ [_1·_3 (_1·_2-_3·_2)+ cyclic ] exp(∑_s=1^3f_s/2 k_s 0) . On the support of these delta functions, the result further reduces to:2 g f^_1_2_3 δ^d-1(∑_r=1^3k_r) ∫ụ[_1·_3 _1·_2+cyclic ] exp(∑_s=1^3f_s/2 k_s 0) . Thus the integrand can be written in terms of on-shell data:_3(a_1,a_2,a_3)=_1·_3 _1·_2+cyclic , as expected. This formula hides explicit dependence on the potential.Using (<ref>) and (<ref>), it follows that:_r·_s = {[0ifr=s; ϵ̃_s^a/k_s 0(k_r 0k_s a-k_s0k_r a+ k_r 0 e_s_a- k_s0e_r_a)otherwise ].,_r·_s = {[0ifr=s; -ϵ̃_r·ϵ̃_sotherwise ]. .In particular, the background gauge field does enter into the functional form of the integrand (<ref>). The explicit form of the integrand is:_3(a_1,a_2,a_3)=-ϵ̃_1·ϵ̃_3/k_2 0[(k_1 0 k_2·ϵ̃_2-k_2 0k_1·ϵ̃_2)+·ϵ̃_2(k_1 0e_2- k_20e_1)]+cyclic .Crucially, the terms linear ingive a background-dependent correction to the flat space result analogous to the tail terms involving σ_ab appearing in the gravity integrand (<ref>).In both cases, they encode the memory. § THE DOUBLE COPYArmed with explicit formulae for the 3-point integrands on both gravitational and gauge theory plane wave backgrounds, a precise statement of double copy can now be made. From (<ref>), the 3-point integrand for gluons on the gauge theory plane wave background can be written compactly as: _3(a_1,a_2,a_3)= F({k_r 0,k_r a,ϵ̃_r}) +𝖢({k_r 0,k_r a,ϵ̃_r}|) , where the functionF({k_r 0,k_r a,ϵ̃_r}):=-ϵ̃_1·ϵ̃_3/k_20(k_1 0 k_2·ϵ̃_2-k_2 0 k_1·ϵ̃_2) +cyclicis the `flat' contribution to the integrand.[The spurious poles in k_0 are associated with our projection of the polarization vectors ϵ_a to be orthogonal to both _u and _v.] The tail-dependent correction term is𝖢({k_r 0,k_r a,ϵ̃_r}| ):= _1·_3/k_20 ·ϵ̃_2(k_1 0e_2-k_2 0e_1)+ cyclic Note that both F and 𝖢 are real functions, in the sense that they take real values provided the kinematic data is real-valued.To double copy the integrand (<ref>), one performs a sequence of simple steps: * Flip the charge (i.e., the sign of the colour factor of ) to define 𝒜_3=F-𝖢 and regard this as the conjugate of _3: |_3|^2:= _3 _3= F^2({k_r 0,k_r a,ϵ̃_r}) - 𝖢^2({k_r 0,k_r a,ϵ̃_r}|) * Replace every spatial (d-2)-momentum by a curved version using the vielbein of the gravitational plane wave background (e.g., k_1 a→ k_1 i E^i_a). Replace the gauge background polarisations ϵ̃_a with gravitational background polarisations ϵ_a. This yields[The latter operation is just a relabelling by removing all tildes. In particular, this replacement implies _r·_s→_r·_s.]F^2({k_r 0,k_r i E^i_a,ϵ_r}) - 𝖢^2({k_r 0,k_r i E^i_a,ϵ_r}|) . * Replace the remaining (quadratic) dependence on the background gauge field with dependence on the background gravitational field using the rule:e_re_s ^a ^b →{[ k_r 0 σ^abr=s; (k_r 0+k_s 0) σ^ab]. ,where e_r is the charge under the background gauge field associated with external state r=1,2,3. The final step is motivated by dimensional considerations and suggested by the fact that _a encodes the gauge theory memory effect; if it is set to vanish in the in-region it will generically be a non-zero constant in the out-region remembering an integral of the field. Thus the quadratic combination _a _b is where the memory effect can be seen in the amplitude.In the gravitational case, the deformation tensor σ_ab can be chosen to vanish in the past, but is then non-trivial in the future, although nowgenerically falling off asymptotically asu^-1, by (<ref>).Therefore, the replacement (<ref>) identifies the fields responsible for memories, albeit with different functional dependence on u.An additional power of momenta is needed on the gravitational side to ensure that the two combinations have the same mass dimension.Steps 1-3 result in an expression of the formF^2({k_r 0,k_r i E^i_a,ϵ_r}) - 𝖢^2({k_r 0,k_r i E^i_a,ϵ_r}|σ) . Working on the support of momentum conservation in the v-direction – which holds regardless of the asymptotic configuration of the three external states – a bit of algebra reveals that𝖢^2({k_r 0,k_r i E^i_a,ϵ_r}|σ)=k_1 0k_2 0k_3 0 σ^ab 𝒞_a 𝒞_b , and therefore that the expression (<ref>) is in fact equal to the 3-point integrand for gravitons on the gravitational plane wave background. There is also a canonical way to map the 3-point integrand for gluons on a gauge theory background to the 3-point integrand for gluons on a gravity background. This entails a `classical' double copy of the background (in the sense of <cit.>) while leaving the functional form of the integrand unchanged. To see how this works,use the integrand expression:_3(a_1,a_2,a_3)=_1·_3 _1·_2+_1·_2 _2·_3 +_2·_3 _3·_1 , where _r a and _r a are given by (<ref>), (<ref>) for r=1,2,3. Now perform the following replacements everywhere in (<ref>):k_r a→k_r i E^i_a , ϵ̃_r a→ϵ_r a, e_r _a→k_r 0 σ_ab x^b . The last of these replacements is motivated by the observation that the non-trivial component of the plane wave gauge field, namely x^a _a is a linear function of x while the non-trivial component of the plane wave metric, namely -Ë^i_aE_b i x^a x^b, is quadratic. After making the replacements (<ref>), the polarization vectors in the gauge field background are mapped directly onto the polarization vectors in the gravitational background: _r μ→_r μ. Although _r μ is not quite mapped onto K_r μ, it is easy to see that_r·_s→ K_r·_s .Calling this substitution map ψ, it follows immediately thatψ(_3(a_1,a_2,a_3))=_3(A_1, A_2, A_3) , where the two integrands have the same kinematic data but are defined on different backgrounds.§ DISCUSSIONIn this paper we have made a preliminary investigation of how the notion of double copy generalizes to curved scattering backgrounds starting with the three point amplitude on sandwich plane waves.We find new features, but see that the double copy nevertheless does extend to this curved setting: 3-point graviton amplitudes on a plane wave space-time can be obtained by taking the double copy of 3-point gluon amplitudes on a gauge theory plane wave background. This statement can be expressed succinctly by encoding steps 2 and 3 of the double copy procedure ina `replacement map' ρ, that acts on the spaces of (d-2)-kinematics and background gauge fields. The double copy for 3-point integrands on plane wave backgrounds is then simply:_3(h_1, h_2, h_3)= ρ(|_3(a_1, a_2, a_3)|^2) . This is consistent with the usual double copy on flat backgrounds expressed inthe KLT relations. In a flat background, ρ acts trivially and this is the usual squaring relation. We have only investigated the simplest scattering amplitudes (i.e., 3-point amplitudes), which are generated by contact interactions in the space-time action. Higher-point amplitudes will involve propagator contributions; although explicit forms for propagators on plane wave backgrounds are known (e.g., <cit.>), these are significantly more complicated that those arising from flat space. Nevertheless, the prescription given in section <ref> seems universal: it dictates how to double copy the data for any n-point scattering amplitude. Steps 1-3 do not depend on the number of external particles being three. So one canoptimistically conjecture a heuristic form of the double copy for n-point integrands on plane wave backgrounds:_n(h_1,…,h_n) = ρ(∑_α,β∈S_n/_n _n(α) 𝒮^[α|β] _n(β)) , where the sum is over distinct colour-orderings for the n-point integrands on the gauge theory background, ρ is the replacement map defined by steps 2 and 3 of the double copy, _n is the integrand with opposite charges for the background and 𝒮^[α|β] is a plane wave analogue of the KLT matrix (perhaps obtained from the same replacement algorithm for the momenta).However, now theandmust incorporate the non-trivial propagators on those backgrounds, and it is likely that these must also be subject to some replacement to work correctly on a gravitational background. Our procedure is not a straightforward local identification of integrands.It requires the replacement of certain structural functions appropriate for propagation on a gauge theory background by those for a gravitational background.Indeed, colour/kinematics duality is usually expressed locally in momentum space, and so should not be expected to be local in space-time. Here we see evidence that a non-local procedure based onHamilton-Jacobi functions for propagation of momentum eigenstates from null infinity will do the trick.Thus, the most optimistic message from this for the general curved colour-kinematic duality is that although a space-time procedure cannot be local, it can workby referring to null infinity, using Hamilton-Jacobi generating functions to create the identifications. It would also be desirable to extend the double copy to other curved backgrounds. Although plane waves are a very special example of such backgrounds, there is some sense in which they are universal limits of all space-times <cit.>. It would be interesting to see in what sense the results found here inform those for more general space-times.Finally, we note that our original motivation for considering scattering on plane wave backgrounds was to provide a space-time result to compare with an alternative calculation of these amplitudes using ambitwistor string theory <cit.> adapted to a curved background <cit.>. As we will show in <cit.>, ambitwistor strings provide an alternative `stringy' approach to calculating amplitudes on curved backgrounds which gives pure field theory amplitudes without α' corrections, in a way that cleanly manifests the double copy found here.The use of Hamilton-Jacobi functions to bring in momenta and polarization vectors from null infinity should then tie in with the work in <cit.> where ambitwistor strings are formulated at null infinity.We would like to thank Pedro Vieira, Kai Röhrig, David Skinner and Piotr Tourkine for useful conversations. TA, EC and LM thank the Kavli Institute for Theoretical Physics for hospitality while this work was completed; this research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915. TA is supported by an Imperial College Junior Research Fellowship; EC and LM are supported by EPSRC grant EP/M018911/1; SN is supported by EPSRC grant EP/M50659X/1 and a Studienstiftung des deutschen Volkes scholarship.§ THE IMPULSIVE PLANE WAVEFor both gauge theory and gravitational sandwich plane waves, the computation of 3-point amplitudes (rather than integrands) boils down to performing integrations that depend on the particulars of the background geometry. In this appendix, we consider the simplest concrete example of a sandwich plane wave: the impulsive plane wave <cit.>. Impulsive plane waves correspond to gluing two flat regions together along an infinitesimal burst of radiation; in other words, the radiation region of the sandwich plane wave has delta function support. In the case of the impulsive gauge theory background, the scalar and gluon 3-point amplitudes can be computed in closed form. For the impulsive gravitational background, the 3-point amplitudes can be written in terms of integrals which are suitable to numerical approximation. §.§ Gauge theory background For an impulsive gauge theory plane wave, we have_a(u)=δ(u) _a , for _a a set of d-2 constants which characterize the impulsive wave. Using the asymptotic conditions (<ref>), it follows that^-_a(u)=Θ(u) _a , ^+_a(u)=-Θ(-u) _a , where Θ(u) is the Heaviside step function. Proceeding from (<ref>) it is a straightforward calculation to obtain the 3-point amplitudes of charged scalars on this background. The results for the two independent configurations – all incoming or two incoming and one outgoing – are given by:M_3(Φ^-_1,Φ^-_2,Φ^-_3)=λ/6 δ^d-1(∑_r=1^3 k_r)[(∑_s=1^3𝐤_s^2/2 k_s 0)^-1..-(∑_s=1^3𝐤_s^2+2 e_s k_s^a_a+e_s^2^2/2 k_s 0)^-1] ,andM_3(Φ^-_1,Φ^-_2,Φ^+_3)=λ/6 δ^d-1(∑_r=1^3 k_r) [(𝐤_3^2-2e_3 k_3^a_a+e_3^2^2/2 k_3 0+∑_s=1,2𝐤_s^2/2 k_s 0)^-1..-(𝐤_3^2/2 k_0 3+∑_s=1,2𝐤_s^2+2e_s k_s^a_a+e_s^2^2/2 k_s 0)^-1] ,where 𝐤_s^2:=k_s ak^a_s for any s=1,2,3.The 3-point amplitudes for gluons on the impulsive gauge theory background follow similarly from (<ref>). A calculation leads to:M_3(a^-_1,a^-_2,a^-_3)=2 g δ^d-1(∑_r=1^3 k_r)[(∑_s=1^3𝐤_s^2/2 k_s 0)^-1 F({k_t, _t}) .-(∑_s=1^3𝐤_s^2+2e_s k_s^a_a+e_s^2^2/2 k_s 0)^-1(F({k_t,_t}) -^a(_1·_3/k_2 0 _2 a(e_2k_1 0-e_1k_2 0) ..+..._1·_2/k_3 0 _3 a(e_3k_2 0-e_2k_3 0) +_2·_3/k_1 0 _1 a (e_1k_3 0-e_3k_1 0)))] ,andM_3(a^-_1,a^-_2,a^+_3)=2 g δ^d-1(∑_r=1^3 k_r) [(𝐤_3^2-2e_3 k_3^a_a+e_3^2^2/2 k_3 0+∑_s=1,2𝐤_s^2/2 k_s 0)^-1.×(F({k_t,_t})+e_3 ^a(k_2 0/k_3 0 _1·_2 _3 a-_2·_3 _1 a))-(𝐤_3^2/2 k_0 3+∑_s=1,2𝐤_s^2+2e_s k_s^a_a+e_s^2^2/2 k_s 0)^-1 (F({k_t,_t})-^a(_1·_3/k_2 0 _2 a(e_2k_1 0-e_1k_2 0).....-e_2 _1·_2 _3 a+e_1 k_3 0/k_1 0 _2·_3 _1 a))] ,where the function F of the kinematic data is defined by (<ref>).In each of these expressions a Hartle-Hawking contour deformation is used to dampen rapidly oscillating contributions to the u-integrations near u=±∞. This is the same as the prescription on Minkowski space, and corresponds to selecting the physical vacuum. §.§ Gravitational background For an impulsive gravitational wave, the non-trivial metric component H(u,𝐱) in Brink– mann coordinates has delta function support:H(u,𝐱)=δ(u) H_ab x^a x^b , with H_ab a trace-free and constant (d-2)×(d-2) matrix. Assuming that H_ab is corank zero with distinct eigenvalues, it can be diagonalized using rotations in the x^a-plane. So without loss of generality, we takeH_ab=λ_(a) δ_ab , ∑_a=1^d-2λ_(a)=0 . The vielbein E^a_i must solve the equationË_a i=λ_(a) δ_ab δ(u) E^b_i , subject to incoming or outgoing boundary conditions (<ref>). In each case, one findsE^-_a i=δ_ai(1+u λ_(a) Θ(u)) , E^+_a i=δ_ai(1-u λ_(a) Θ(-u)) , so the transverse metric γ_ij(u) is given in incoming or outgoing coordinates by:γ_ij^-(u)=δ_ij(1+u λ_(i) Θ(u))^2 , γ_ij^+(u)=δ_ij(1-u λ_(i) Θ(-u))^2 , where λ_(i) is identified with λ_(a) using δ_a^i. This demonstrates that the impulsive gravitational wave is two copies of Minkowski space glued together along a single pulse of gravitational radiation. While the metrics (<ref>) are continuous across the pulse at u=0, they have discontinuous first derivatives.To compute 3-point amplitudes, it is also important to have the inverse vielbeins:E^i -_a=δ^i_a(1+u λ_(a) Θ(u))^-1 , E^i +_a=δ^i_a(1-u λ_(a) Θ(-u))^-1 , leading to expressions for F^ij_±(u): F^ij_-(u)=u δ^ij/1+u λ_(i) Θ(u),F^ij_+(u)=u δ^ij/1-u λ_(i) Θ(-u) . So in both cases F^ij(u) gets an infinite series of O(u^2) corrections upon crossing the pulse at u=0.Even at the level of scalar amplitudes, the situation on the gravitational background is more complicated than on the gauge theory background. Unlike (<ref>)–(<ref>), on the impulsive gravitational wave (relatively) compact expressions for the u-integrations are not available. Instead, we find explicit expressions which could be evaluated (numerically or possibly analytically) when the momenta and eigenvalues {λ_(a)} are specified. For instance, one finds:M_3(Φ^-_1,Φ_2^-,Φ^-_3)=λ /6 δ^d-1(∑_r=1^3k_r) [(∑_s=1^3𝐤_s^2/2 k_s 0)^-1.. +∫_0^∞+iϵụ ∏_a=1^d-2(1+u λ_(a))^-1/2 exp( u ∑_s=1^3∑_i=1^d-2k_s i^2/2k_s 0 (1+uλ_(i)))] ,for the all-incoming configuration.The expression for the two-incoming, one-outgoing configuration is similarly given in terms of u-integrals over the in- and out-regions:M_3(Φ^-_1,Φ_2^-,Φ^+_3)=-λ/6 √((2 π)^d-2/k_3 0^d-2) δ(∑_r=1^3k_r 0)×[∫_-∞-ϵ^0ụ/∏_a=1^d-2√(λ_(a)) exp(-/2 k_3 0J_aJ_b (A^-1)^ab+∑_s=1^3k_s ik_s j/2 k_s 0 F^ij_s)..+∫^∞+ϵ_0ụ ∏_a=1^d-2(λ_(a)+u λ^2_(a))^-1/2 exp(-/2 k_3 0J_aJ_b (A^-1)^ab+∑_s=1^3k_s ik_s j/2 k_s 0 F^ij_s)] .Here, the F^ij_s(u) are given by (<ref>), whileA_ab(u)=-λ_(a) δ_ab/1+|u| λ_(a) , andJ_a(u)=k_1 a+k_2 a+k_3 a+u λ_(a)(k_3 a Θ(u)-(k_1 a+k_2 a) Θ(-u))/1+|u| λ_(a) . § CLASSICAL S-MATRIX & TREE-LEVEL INTEGRANDSThis appendix reviews the notion of classical S-matrix which is used throughout the paper, as well as providing a precise definition for the tree-level integrand. On a sandwich plane wave background (for either gauge theory or gravity), the tree-level S-matrix for a theory encodes the evolution of asymptotic free states from the in-region of the space-time (i.e., u<u_1) through the non-trivial, or radiation region (u_1≤ u≤ u_2), to the out-region (u>u_2) as governed by the classical theory. Rather than work out the curved space Feynman rules, we use a definition of the classical S-matrix in which tree-level amplitudes are given by extracting certain multi-linear pieces of the classical action evaluated on a perturbative solution to the non-linear equations <cit.>.In general this has the interpretation of the field-theoretic Hamilton-Jacobi generating function for the evolution and gives the tree-level contribution to the S-matrix. For the 3-point calculations in the body of the paper, there is no need to iterate the perturbative solution, but here we present the general framework. Let S[Φ] be the classical action, a functional of some fields Φ which is defined on the sandwich plane wave background (gravitational or gauge theoretic – at this stage it makes no difference). We assume that this action takes the generic form:S[Φ]=∫^̣dX (_kin+_int) , where _kin is the kinetic portion of the action, which is quadratic in Φ and governs the free theory, and _int contains all higher-point interactions.Define the following object:Φ^[n](X):=∑_i=1^nϵ_i φ_i(X)+∫^̣dY Δ(X,Y) .δ_int/δΦ|_Φ=∑_j=1^nϵ_jφ_j(Y). This is essentially an integral form of the full non-linear equations from the action S with data given by the first term on the right hand side. Here, the {ϵ_i} are n parameters that will eventuallybe thought of as infinitesimal; {φ_i} are n solutions to the free equations of motion of _kin with specified asymptotic behaviour; and Δ(X,Y) is a Green's function defined by _kin. There are precise formulae for various useful definitions of this Δ(X,Y) (e.g., advanced, retarded, Feynman) in scalar, gauge, and gravitational theories on plane wave backgrounds <cit.>, though we will not make explicit use of them here. Specifying the asymptotic behaviour of the free solution φ_i boils down to saying whether it looks like an `in' or `out' state.Both the in- and out-regions are flat, so asymptotically free states φ_i should look like free states in Minkowski space in at least one of these regions. In a momentum space representation, such free states in Minkowski space are modelled on massless plane wave momentum eigenstates, ^ k· X for k^2=0. Unlike Minkowski space, in the sandwich plane wave a state which looks like ^ k· X in the in-region will not look like ^ k· X in the out-region. This is a consequence of the `memory' relations (<ref>), (<ref>). Hence, the specification of asymptotic behaviour for φ_i boils down to stating whether it is an incoming or outgoing state, denoted respectively as φ_i^- or φ_i^+. An incoming state is one which looks like a free solution in Minkowski space the in-region; an out state looks like a free solution in Minkowski space in the out-region. More precisely, φ^-_i|_in∼^ k·X∼φ^+_i|_out , for both the gravitational and gauge theory backgrounds.The n-point tree-level scattering amplitude for the states {φ_i} – with their given asymptotic configuration of in and out states – is then a multi-linear piece of the classical action:M^(0)_n(φ_1,…,φ_n)=.∂^nS[Φ^[n]]/∂ϵ_1⋯∂ϵ_n|_ϵ_1=⋯=ϵ_n=0 . For flat backgrounds, this agrees with the usual definition of the S-matrix and would also correspond with a Feynman diagram definition for sandwich plane waves.For the purposes of investigating the double copy, a notion of tree-level integrand closely related to the tree-level amplitude is useful. Indeed, it is actually this tree-level integrand that appears in the KLT relations of the standard double copy. From the definition (<ref>) it is straightforward to see that the tree-level scattering amplitude will always take the form:M^(0)_n = ∫^̣dX _n(X) ∏_i=1^n f_i(X) , where each of the f_i(X) is a solution to the free scalar wave equation on the plane wave background. The object _n is defined to be the tree-level integrand; generically, it will be formed of polarizations, momenta and propagators and depends on the background geometry. It captures everything that is encoded by the kinematic numerators and denominators which would result from a conventional Feynman diagram approach. In more heuristic terms, the tree-level integrand is what remains after removing the final integral that forms the action functional in (<ref>), along with `universal' spin-independent functions.In Minkowski space, it is easy to see that∏_i=1^n f_i(X)=^(k_1+⋯+k_n)· X ,so the effect of isolating _n is to strip off an overall momentum conserving delta function. On non-trivial backgrounds such as the sandwich plane wave, the result of the final ^̣dX integrals is more complicated, but the principle is the same: _n contains all of the information which one could expect to be `squared' in taking the double copy. Another interesting property of the integrand is that it is functionally independent of the asymptotic conditions of the states being scattered. This enables the investigation of double copy by considering the computationally simplest configuration of incoming and outgoing states.Clearly, there is a sense in which the tree-level integrand is not a gauge-invariant object, just as one can add boundary terms to an action. This lack of gauge invariance is analogous to the statement that individual Feynman diagrams – or individual terms contributing to (<ref>) – are not gauge invariant. However, once a gauge for performing perturbative calculations has been fixed (i.e., specific linearized solutions {φ_i} and a Green's function Δ(X,Y) have been consistently chosen), the object _n is well-defined. In our calculations, we always work in a Lorenz or de Donder gauge, so the resulting expressions for the integrand should be viewed as expressions in these particular gauges. Their integrals, however, do not depend on the gauge choice.Throughout the paper, the tree-level integrand for theories on the gravitational plane wave background is denoted by _n, and the tree-level integrand for theories on the gauge theory plane wave background by _n. JHEP | http://arxiv.org/abs/1706.08925v2 | {
"authors": [
"Tim Adamo",
"Eduardo Casali",
"Lionel Mason",
"Stefan Nekovar"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170627161440",
"title": "Scattering on plane waves and the double copy"
} |
NOMA based Random Access with Multichannel ALOHA Jinho ChoiThe author is with School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology (GIST), Gwangju, 61005, Korea (Email: [email protected]). This work was supported by the “Climate Technology Development and Application" research project (K07732) through a grant provided by GIST in 2017. today ==================================================================================================================================================================================================================================================================================================================================================== Matthieu GendulpheDipartimento di MatematicaUniversità di Pisa A celebrated result of Mirzakhani states that, if (S,m) is a finite area orientable hyperbolic surface, then the number of simple closed geodesics of length less than L on (S,m) is asymptotically equivalent to a positive constant times L^(S), where (S) denotes the space of measured laminations on S. We observed on some explicit examples that this result does not hold for nonorientable hyperbolic surfaces. The aim of this article is to explain this surprising phenomenon. Let (S,m) be a finite area nonorientable hyperbolic surface. We show that the set of measured laminations with a closed one–sided leaf has a peculiar structure. As a consequence, the action of the mapping class group on the projective space of measured laminations is not minimal. We determine a partial classification of its orbit closures, and we deduce that the number of simple closed geodesics of length less than L on (S,m) is negligible compared to L^(S). We extend this result to general multicurves. Then we focus on the geometry of the moduli space. We prove that its Teichmüller volume is infinite, and that the Teichmüller flow is not ergodic. We also consider a volume form introduced by Norbury. We show that it is the right generalization of the Weil–Petersson volume form. The volume of the moduli space with respect to this volume form is again infinite (as shown by Norbury), but the subset of hyperbolic surfaces whose one–sided geodesics have length at least >0 has finite volume.These results suggest that the moduli space of a nonorientable surface looks like an infinite volume geometrically finite orbifold. We discuss this analogy and formulate some conjectures. § INTRODUCTION In this article we are interested in three aspects of the geometry of hyperbolic surfaces and their moduli spaces:* the growth of the number of closed geodesics of a given topological type,* the mapping class group action on the space of measured laminations,* the Teichmüller and Weil–Petersson volumes of the moduli space.The recent work of Mirzakhani brought to light many connections between these topics, in addition to solving important problems. However her work only deals with orientable surfaces.The purpose of this article is to point out some interesting phenomena that occur in the case of nonorientable surfaces. They are interesting for two reasons. Firstly they show a difference between the orientable and the nonorientable cases. Secondly they suggest that the moduli space of a nonorientable surface looks like an infinite volume geometrically finite orbifold. Before going to this conclusion let us describe our results and emphasize the differences with the orientable setting.In this introduction (S,m) is a finite area hyperbolic surface without boundary. Given a homotopy class of closed curves , we define its m–length ℓ_m() as the infimum of the lengths of its representatives. Ifis nontrivial and nonperipheral, then ℓ_m() is realized by its unique geodesic representative. The mapping class group (S) acts on the set of homotopy classes, we denote by __0 =(S)·_0 the orbit of a homotopy class _0. §.§ Growth of simple closed geodesicsLet us start with the result that motivated our study. In <cit.> Mirzakhani established:[Mirzakhani]Let (S,m) be a finite area orientable hyperbolic surface. For any simple closed geodesic _0 of (S,m) there exists c(m,_0)>0 such thatlim_L→ +∞| {∈__0 ; ℓ_m()≤ L }| /L^(S) =c(m,_0).The case of the punctured torus is due to McShane and Rivin (<cit.>).We denote by (S) the space of measured laminations of S, its dimension is given by: (S) = {[ 6g-6+2rif S is orientable; 3g-6+2r if S is nonorientable ]. ,where g is the genus of the surface and r its number of punctures.In contrast to Mirzakhani's result we show:Let (S,m) be a finite area nonorientable hyperbolic surface. For any simple closed geodesic _0 of (S,m) we havelim_L→ +∞| {∈__0 ; ℓ_m()≤ L }| /L^(S) =0. This phenomenon was already known in a few cases.In <cit.> we treated the case of nonorientable surfaces with Euler characteristic-1. Based on Mirzakhani's result, we showed that | {∈__0 ; ℓ_m()≤ L }| is asymptotically equivalent to a monomial whose degree is an integer less than (S). In <cit.> Huang and Norbury studied the case of the thrice–punctured projective plane denoted by N_1,3. They related the growth of the number of one–sided simple closed geodesics to the growth of the number of Markoff quadruples. In the recent work <cit.>, Gamburd, Magee and Ronan determined the asymptotic of the growth of the number of integral Markoff tuples. Let m_0 be the most symmetric hyperbolic metric (up to isometry) on the thrice–punctured projective plane, and let __0 be the orbit of the one–sided simple closed geodesics. A combination of both works entails the existence of ∈̱(2,3) such that the limitlim_L→ +∞| {∈__0 ; ℓ_m_0()≤ L }| /L^exists and is positive. This result has been extend to any hyperbolic metric on N_1,3 by Magee (<cit.>). §.§ Counting measuresTo prove her theorem, Mirzakhani introduced a family of counting measures (ν__0^L)_L>0 on (S), and showed its weak^∗ convergence towards a positive multiple of the Thurston measure . This convergence can be interpreted as the equidistribution of __0 in the projective space of measured laminations (S).The framework of Mirzakhani's proof is avaible for nonorientable surfaces as well as for orientable ones. Actually the difference between the two theorems reflects a difference in the dynamics of the action of (S) on (S). In the nonorientable case, the orbit __0 accumulates on the subset ^+(S)⊂(S)of measured laminations without one–sided closed leaves, which is negligible with respect to the Thurston measure(Danthony and Nogueira <cit.>). This implies that the family of counting measures (ν^L__0)_L>0 weak^∗ converges towards the zero measure (Proposition <ref>).§.§ Dynamics of the mapping class group action Let us focus on the action of (S) on (S). When S is orientable this action is rather well–understood: * the action of (S) on (S) is minimal (Thurston, see <cit.>),* the action of (S) on (S) is ergodic with respect to(Masur <cit.>),* there is a classification of (S)–invariant locally finite ergodic measures on (S) (Hamenstädt <cit.>, Lindenstrauss–Mirzakhani <cit.>),* there is a classification of the orbit closures of (S) in (S) (ibid.). In contrast to these results we show that: * the action of (S) on (S) is not minimal (Proposition <ref>),* the action of (S) on (S) is topologically transitive if and only if S is of genus one (Proposition <ref>),* we have a partial classification of the orbit closures of (S) on (S), which is complete when S has genus one (Theorem <ref>).* the action of (S) on (S) is not topologically transitive, thus is not ergodic with respect to(Proposition <ref>), We also determine the unique minimal invariant closed subset of (S) (§<ref>).§.§ Structure of (S)These results are consequences of the peculiar structure of (S) when S is nonorientable. As mentioned above, the set ^-(S)⊂(S) of measured laminations with a one–sided closed leaf has full Thurston measure (Danthony and Nogueira <cit.>). In addition it admits a canonical cover by topological open balls of dimension (S) (§<ref>). Each ball () is associated to a collection of disjoint one–sided simple closed geodesics ={_1,…,_k}, and consists in the set of measured laminations that admit _1,…,_k as leaves. The projection of () into (S) is a topological open ball. These balls form a packing (i.e. are pairwise disjoint) when S has genus one. §.§ Volumes of moduli spacesThe particular structure of (S) influences the geometry of the moduli space (S). This is logical since (S) is the Thurston boundary of the Teichmüller space. When S is nonorientable, we show that the Teichmüller flow is not ergodic and that the Teichmüller volume of the moduli space is infinite (§<ref>). We also consider a kind of volume form introduced by Norbury (<cit.>) as a generalization of the Weil–Petersson volume form to the case of nonorientable surfaces. We show (§<ref>) that it is indeed the right generalization from the point of view of the twist flow. Then we provide a simpler proof of the fact — due to Norbury (ibid.) — that (S) has infinite volume with respect to this volume form. We also show that the subset {^-≥} of points in (S) whose one–sided simple closed geodesics have length at least >0 is a finite volume deformation retract of (S) (§<ref>).§.§ Counting multicurves Mirzakhani (<cit.>) has extended her theorem on the growth of simple closed geodesics to general multicurves. The particular case of the punctured torus has been solved independently by Erlandsson and Souto (<cit.>), who also established some other interesting results. We first need few lines to precise the terminology. A multicurve is a formal _+^∗–linear combination =a_1_1+… +a_n_n of distinct homotopy classes of noncontractible and nonperipheral closed curves. For technical reasons we assume that each _i is primitive, that is corresponds to the conjugacy class of primitive elements of π_1(S). We say thatis an integral (resp. rational) multicurve if moreover a_i∈ (resp. a_i∈) for any i=1,…,n. We denote bythe set of integral multicurves of S. The mapping class group acts on , we denote by _=(S)· the orbit of . Given a hyperbolic metric m on S, we define the m–length ofby ℓ_m()=a_1ℓ_m(_1)+…+a_nℓ_m(_n). We say that a multicurve is simple if its components are simple and disjoint.For sake of clarity, we stated Mirzakhani's theorem in the case of a simple closed geodesic, but it applies to any simple integral multicurve. As we mentioned above, she has extended her theorem to all multicurves (<cit.>). We do the same way with our:Let (S,m) be a finite area nonorientable hyperbolic surface. For any integral multicurve _0 we have lim_L→ 0| {∈__0 ; ℓ_m()≤ L }| /L^(S) =0.We prove Theorem <ref> in the same manner as Theorem <ref> by showing that the family of counting measures (ν__0^L)_L>0 converges to the zero measure. These measures are not defined on (S) anymore, but on the space(S) of geodesic currents. We show that any limit point of (ν__0^L)_L>0 is supported on ^+(S), and we conclude using a powerful result of Erlandsson and Souto (<cit.>) which states that limit point of (ν__0^L)_L>0 is absolutely continuous with respect to .The method developed in <cit.> is very different, and does not make use of geodesic currents. Working with geodesic currents present the advantage that Theorem <ref> extends immediately to all geometric structures that define a filling geodesic currents, like complete negatively curved metrics (see Theorem <ref> for a precise statement).Let us mention a result which explains why an accumulation point in (S) of an orbit __0 can not have a one–sided leaf: Let ł∈^-(S) be a measured lamination with a one–sided closed leaf . For any k≥ 1, there exists a neighborhood U_k of ł in (S) such that for any geodesic current c∈ U_k there exists a geodesic ∈̣(c) that projects either oneither on a geodesic with k self–intersections.§ CONCLUSION The analogy between moduli spaces and locally symmetric orbifolds has guided the study of Teichmüller spaces and mapping class groups since many years. So far, moduli spaces were compared to finite volume locally symmetric orbifolds. The results established in this paper suggest that, in the case of a nonorientable surface, the moduli space looks like a geometrically finite locally symmetric orbifold of infinite volume. The aim of this section is to defend this point of view.For sake of simplicity, we compare the mapping class group (S) with a geometrically finite Kleinian group . We mostly focus on the dynamical properties of their actions. In the first paragraphs we recall some basic results and definitions. Then we describe the analogy and state some conjectures. Such an analogy has been particularly fruitful in the orientable setting (see <cit.>). §.§ Geometrically finite Kleinian groupsWe recall that a Kleinian groupis a discrete subgroup of (^n). Its limit set Λ is the set of points in ∂^n that are adherent to any orbit ofin ^n. We assume thatis non elementary, which means that Λ is infinite. The convex core of the orbifold ^n/ is the projection convex(Λ)/ of the convex hull of Λ in ^n.Following <cit.>, we say thatis geometrically finite if there exists >0 such that the –tubular neighborhood of the convex core has finite volume. For example, a Kleinian group which admits a convex fundamental domain bounded by finitely many geodesic faces is geometrically finite, but the converse is not true. Note that a geometrically finite Kleinian group is a lattice (i.e. has finite covolume) if and only if Λ=∂^n. There are various characterizations of geometrical finiteness, let us mention the following: a Kleinian groupis geometrically finite if and only if there exists {B_i}_i∈ I a –invariant collection of disjoint horoballs centered at fixed points of parabolic subgroups ofsuch that (convex(Λ)-∪_i B_i)/is compact(see <cit.> or <cit.>). §.§ Patterson–Sullivan theoryLetbe a Kleinian group. To study the dynamics of the action ofon ∂^n one naturally looks for an invariant Radon measure supported on Λ. Such a measure is necessarily trivial whenis non elementary, therefore one considers a more general object called conformal density.A conformal density of dimension >̣0 is a family {μ_x}_x∈^n of Radon measures on ∂^n such that any two measures μ_x and μ_y are absolutely continuous one with respect to the other, and are related by the following equality:d μ_y/ d μ_x(ξ) = e^-̱̣_ξ(y,x)(∀ξ∈∂^n),where _̱ξ(y,x) is the Busemann cocycle that gives the signed distance between the horospheres centered at ξ passing through y and x. We say that the conformal density is –invariant if _∗μ_x = μ_ x for any ∈ and any x∈^n. The Lebesgue measure on ^n-1 induces an invariant conformal density of dimension n-1. More interestingly, Patterson and Sullivan (<cit.>) constructed a non zero invariant conformal density supported on Λ, and they showed that its dimension is equal to the critical exponent of : _̣ =inf{>̣0 ; ∑_ e^-ḍ_^n(o, o)<+∞}, where o is any point in ^n.To any –invariant conformal density corresponds a measure on the unit tangent bundle of ^n which is invariant under the geodesic flow and the action of , this is the associated Bowen–Margulis–Sullivan measure (<cit.>). In the case of the conformal density induced by the Lebesgue measure on ^n-1, the associated Bowen–Margulis–Sullivan measure is simply the Liouville measure. The properties of a conformal density and its Bowen–Margulis–Sullivan measure are related by the Hopf–Tsuji–Sullivan dichotomy (see <cit.>).Let us assume thatadmits a convex fundamental domain bounded by finitely many geodesic faces. Then the –invariant conformal density of dimension _̣ is unique up to scaling, and has finite total mass. Moreover _̣ is equal to the Hausdorff dimension of Λ⊂∂^n with respect to the angular distance at any point of ^n (<cit.>). The associated Bowen–Margulis–Sullivan measure on ^n/ is ergodic. This measure is a crucial tool to establish growth and equidistribution results (see <cit.>).§.§ The analogy Let S be a surface of finite type with χ(S)<0. The Teichmüller space equipped with the Teichmüller metric plays the role of ^n. The mapping class group acts properly and discontinuously by isometry on (S), like a Kleinian group on ^n. The Thurston boundary replaces ∂^n, even though (S) is not the visual boundary of the Teichmüller metric (Kerckhoff <cit.>). The Thurston measure induces a conformal density {μ_X}_X∈(S) analogous to the conformal density induced by the Lebesgue measure, it is given byμ_X(U) =(B_X∩π^-1(U)),for any U⊂(S) measurable, where π:(S)→(S) is the canonical projection and B_X={ł∈(S) ; ℓ_X(ł)≤ 1}. The Thurston measure is also related to the Teichmüller volume defined onthe space of quadratic differentials (see §<ref>). This space identifies canonically with the cotangent bundle of (S), so that the Teichmüller volume appears as an analogue of the Liouville measure. We refer to <cit.> for more details. Now we assume that S is nonorientable. The mapping class group shouldn't be considered as a lattice since it has infinite covolume. We suggest to compare it with a geometrically finite Kleinian group. To motivate this analogy we first look at its limit set. In §<ref> we prove the following conjecture in the case of genus one surfaces: The limit set of (S) in (S) is the set of projective measured laminations without one–sided closed leaves, it is denoted by ^+(S).We have already mentioned that ^+(S) is negligible with respect to the Thurston measure class (Danthony and Nogueira <cit.>). Similarly, in dimension three, the limit set of an infinite covolume geometrically finite Kleinan group is negligible with respect to the Lebesgue measure class (Ahlfors <cit.>). We observe that ^+(S) coincides with the set of points in the Thurston boundary that are adherent to {^-≥}⊂(S) for >0 small enough, where ^- denotes the length of the shortest closed one–sided geodesic. We believe that {^-≥}⊂(S) is the analogue of an infinite convex polyhedron of ^n.Let us write {^-≥} = ∩_∈^-{ℓ_≥} where we denote by ^- the set of isotopy classes of one–sided simple closed curves. Whenis a two–sided simple closed geodesic, the hypersurface {ℓ_=} is a kind of horosphere. Whenis a one–sided simple closed geodesic, the boundary of {ℓ_=} in the Thurston compactification is a polyhedral sphere (see §<ref>), like the boundary of a geodesic hyperplane in ^n is a conformal sphere in ∂^n. So it seems natural to think about {ℓ_=} as a geodesic hyperplane (or maybe as a hypersurface made of points at a given distance from a geodesic hyperplane).Then, the fact that {^-≥}⊂(S) has finite Weil–Petersson volume (§<ref>) is analogous to geometrical finiteness. Note that the tubular neighborhood of {^-≥} with respect to the Teichmüller metric is contained in {^-≥_1} for some _1>0 small enough (use the main theorem of <cit.>).Let us examine the characterization of geometrical finiteness we mentioned in a previous paragraph. We have {^- ≥} - ⋃_∈^+{ℓ_<} ={≥} where ^+ is the set of isotopy classes of two–sided simple closed curves. As well–known, the subset {≥}⊂(S) is compact, and any {ℓ_<}⊂(S) with ∈^+ is a kind of horoball. These horoballs are not pairwise disjoint, but any two of them are disjoint if their corresponding geodesics intersect (Collar Lemma). So the characterization of geometrical finiteness is somehow satisfied. We believe that (S) should be compared to a Kleinian group that admits a convex fundamental domain bounded by finitely many faces. Such fundamental domains have been constructed when S has a small Euler characteristic (the author <cit.>, Huang and Norbury <cit.>).We now formulate some optimistic conjectures. The following conjecture is the analogue of Sullivan's theorem (<cit.>):There exists a unique (up to scaling) ergodic (S)–invariant Radon measure whose support is ^+(S), it is $̣–homogeneous where$̣ is the Hausdorff dimension of ^+(S). The Hausdorff dimension of ^+(S) is the one defined by any Euclidean norm in any train–track chart. The transition maps between the train–track charts are piecewise linear— in particular Lipschitz — therefore the Hausdorff dimension does not depend on the choice of a chart. Let us now consider the growth of simple closed geodesics. The conjecture below is true when the Euler characteristic of S is -1 (see <cit.>), and seems to be true when S is the thrice–punctured projective plane (Magee <cit.>):For any hyperbolic metric m on S, and for any simple closed geodesic _0, there exists a constant c>0 such that |{∈__0 ; ℓ_m()≤ L}|≃ c L^$̣ asLtends to infinity, where$̣ is the Hausdorff dimension of ^+(S). §.§ A remarkable exampleLet N_1,3 denote the thrice–punctured projective plane. In a forthcoming paper (<cit.>) we show that the action of (N_1,3) on the Thurston compactification of (N_1,3) is topologically conjugate to the action of a geometrically finite Kleinian groupon ^3∪∂^3. A finite index subgroup ofis the so–called Apollonian group, that is the group of Möbius transformations that preserve the Apollonian packing. Many counting problems related to the Apollonian packing have been studied recently (see <cit.>). § ORGANIZATION OF THE PAPERThe rest of paper falls into three parts. The first part is devoted to the space of measured laminations. We start with some explicit examples (§<ref>) before describing the peculiar structure of ^-(S) in the general case (§<ref>).Then we study the dynamics of the mapping class group action (§<ref>) and prove Theorem <ref> (§<ref>). In the second part we extend Theorem <ref> to multicurves. The main difficulty is to show that the orbit of a multicurve in (S) accumulates on ^+(S) (§<ref>). In the third part we consider various aspects of the geometry of Teichmüller spaces. We show that the Teichmüller volume of the moduli space of unit area quadratic differentials is infinite, and that the Teichmüller flow is not ergodic (§<ref>). Then we discuss the definition of the Weil–Petersson volume (§<ref>), and we show that the subset {^-≥} of the moduli space has finite volume (§<ref>).§ ACKNOWLEDGEMENTS I would like to thank Juan–Carlos Álvarez Paiva and Gautier Berck for an original idea of title (see <cit.>). I would like to thank Gabriele Mondello who gave me interesting questions and ideas, a part of this work was realized when I was in Rome. I would like to thank the members of the Mathematics department of the university of Pisa for their friendly welcome. Finally I thank the Italian Republic for the last four years of financial support.§ NOTATIONS For the convenience of the reader we list some notations introduced in the paper: B_X(L) set of measured laminations ł∈(S) with ℓ_X(ł)≤ L (§<ref>)b_X(L) Thurston measure of B_X(L) (§<ref>)()⊂(S) set of measured laminations that admit the components of the simple multicurve =_1+…+_k as closed leaves (§<ref>)(S) curve complex of S^-(S) subcomplex of one–sided curves (§<ref>)S smooth connected surface of finite type with negative Euler characteristicN_g,n compact nonorientable surface of genus g with n boundary components (§<ref>) set of isotopy classes of simple closed curves that do not bound a disk, a punctured disk, an annulus or a Möbius strip (§<ref>)^-⊂ set of isotopy classes of one–sided simple closed curves (§<ref>)^-_b⊂^- set of isotopy classes of one–sided simple closed curves whose complement is orientable (§<ref>)^-_nb⊂^- set of isotopy classes of one–sided simple closed curves whose complement is nonorientable (§<ref>) set of integral multicurves (§<ref>)_ (S)–orbit of an integral multicurve ∈ (§<ref>)_k set of integral multicurves with exactly k self–intersections (Part <ref>)S̃ universal cover of SS̃_∞ visual boundary of S̃(S̃) space of geodesics of S̃(S) space of geodesic currents on S(S) space of measured laminations on S (§<ref>)^-(S) subspace of measured laminations with a one–sided closed leaf (§<ref>)^+(S) subspace of measured laminations without one–sided closed leaves (§<ref>)(S,) set of integral simple multicurves (§<ref>)(S,) set of rational multiples of elements of (S,) (§<ref>)_N^-(S,) set of simple multicurves in (S,) whose one–sided components have weight bounded by N (§<ref>)(S) projective space of measured laminations (§<ref>)(S) Teichmüller space of S (§<ref>)(S) space of quadratic differentials on S (§<ref>)(S) moduli space of S (§<ref>)(S) mapping class group of S (§<ref>)^∗(S) extended mapping class group of S (§<ref>) or ^S Thurston measure on (S) (§<ref>)μ^L counting measure of (S,) (see (<ref>) in §<ref>)μ^L__0 counting measure of the orbit __0^- the length of the shortest closed one–sided geodesic. § PRELIMINARIESIn this section we recall some classical definitions and results. Most of them deal with measured laminations. We refer to <cit.> for a more detailed exposition.In all this text S is a smooth connected surface of finite type with negative Euler characteristic. A hyperbolic metric on S is a finite area complete metric of constant curvature -1. We assume that the boundary ∂ S is geodesic if it is nonempty. Given a hyperbolic metric m on S, each homotopy classof noncontractible and nonperipheral closed curves admits a unique geodesic representative. We abusively use the same notation for a geodesic and its homotopy class. In the sequel we implicitly assume that a closed curve is not homotopic to a point nor a puncture. §.§ Teichmüller spaces and mapping class groupsThe Teichmüller space (S) is the space of isotopy classes of hyperbolic metrics on S. We assume that the lengths of the boundary components are fixed. The Teichmüller space is a smooth manifold diffeomorphic to an open ball of dimension -3χ(S)-n where n is the sum of the number of boundary components plus the number of punctures of S.The extended mapping class group ^∗(S) is the group of diffeomorphisms of S up to isotopy. The mapping class group (S) is the subgroup of ^∗(S) that preserves each boundary component, each puncture, and the orientation. The mapping class group acts properly and discontinuously by diffeomorphisms on the Teichmüller space. The orbifold (S)=(S)/(S) is the moduli space of S. §.§ Measured laminationsLet m be a hyperbolic metric on S. A m–geodesic lamination on S is a compact subset of S foliated by simple m–geodesics. The boundary components can not be leaves of the lamination. A transverse measure μ for ł is the data of a Radon measure μ_k on each arc k transverse to ł such that: * the restriction of μ_k to a transverse subarc k'⊂ k is μ_k',* an isotopy preserving ł between two transverse arcs k,k' sends μ_k on μ_k',* the support of μ_k is k∩ł.A measured lamination is a m–geodesic lamination equipped with a transverse measure. The transverse measure determines the m–geodesic lamination. We denote by (S) the space of measured laminations on S.Letbe the set of isotopy classes of simple closed curves that do not bound a disk, a punctured disk, an annulus or a Möbius strip. The map (ł,μ)↦ (μ())_∈ is a proper topological embedding of (S) into the affine space ^ endowed with the product topology. The space (S)∪{0} is homeomorphic to an open ball of the same dimension as (S). The extended mapping class group acts by homeomorphisms on (S)∪{0}.§.§ The Thurston compactificationThe space (S) is a cone since one can multiply a transverse measure by a positive number. We denote by (S) the projective space of measured laminations. It is homeomorphic to a sphere and embedds topologicallyinto (^) through the map [ł,μ]↦ [μ()]_∈.The map [m]↦[ℓ_(m)]_∈ is a topological embedding of (S) into (^). The closure of its image is a closed ball whose boundary is exactly the image of (S) in (^). The union (S)∪(S) endowed with the topology induced by (^) is called the Thurston compactification of (S). The mapping class group acts by homeomorphism on the Thurston compactification.§.§ Train–tracks and piecewise integral linear structureThe space (S) has no canonical smooth structure, however it admits a piecewise integral linear structure. This means that it has an atlas whose transition maps are piecewise linear and coincide with integral linear isomorphisms over pieces defined by integral linear inequalities. In particular, there is a well–defined notion of integral point and a canonical measure on (S). This atlas is canonical and explictly defined through train–tracks. In this article we do not use directly train–tracks, we refer to<cit.> for more details. These references consider both orientable and nonorientable surfaces.§.§ Integral points and Thurston measure We denote by (S,) the set of integral points of (S), they correspond to integral simple multicurves, that is to linear combinations of the form a_1_1+…+a_k_k where a_1,…,a_k∈^∗ and _1,…, _k are disjoint elements of . In the same way we denote by (S,) the set of rational multiples of elements in (S,).The Thurston measureon (S) (also denoted ^S) corresponds to the Lebesgue measure in any train–track chart. Thus it is also the weak^∗ limit of the counting measures of integral points. To be more precise, let μ^L (L>0) be the measure defined byμ^L=1/L^(S)∑_∈(S,) 1_1/L.The measure μ^L tends toin the weak^∗ topology as L tends to infinity. Integral points and the Thurston measure are invariant under the action of (S). §.§ The intersection formThe intersection number i(,)̣ between two simple closed geodesicsand $̣ is well–defined. It extends to(S,)by linearity, and then to(S)by density and uniform continuity ofi(·,·). §.§ How to look at nonorientable surfacesIn order to work with nonorientable surfaces it is convenient to look at them as orientable surfaces with boundary submitted to some identifications. There are two kinds of identifications, but we are going to use only one in this article. Letcbe a boundary component of a hyperbolic surface(S,m), one can identify the points ofcas follows: two points are identified if they dividecinto two segments of equal length. The quotient of(S,m)under this identification is a nonorientable hyperbolic surface, and the boundary componentcprojects onto a one–sided geodesic. Note that this identification does not change the Euler characteristic. Let us consider two examples. Let(,m)be a one–holed torus equipped with a hyperbolic metric. The surface obtained by identification of∂is homeomorphic to the connected sum of three real projective planes. Let(P,m)be a hyperbolic pair of pants, the identification of each boundary component ofPproduces again a hyperbolic surface homeomorphic to the connected sum of three projective planes. §.§ One–sided simple closed curves We denote by^-the subset of one–sided elements of. There are exactly two topological types of one–sided simple closed curves : bounding and non bounding. We say that a one–sided simple closed curve is bounding if its complement is orientable, otherwise we say that it is non bounding. A simple closed curve is bounding if and only if it intersects any other one–sided simple closed curve. We denote by^-_b(resp.^-_nb) the subset of bounding (resp. non bounding) elements of^-, it forms an orbit under the mapping class group action. Let us note that a genus one nonorientable surface has only bounding one–sided simple closed curves, and an even genus nonorientable surface has only non bounding one–sided simple closed curves.PART:Spaces of measured laminations§ NONORIENTABLE SURFACES OF SMALL COMPLEXITYIn this section we provide some examples that illustrate the results of the next sections. We quickly describe the projective space of measured laminations(S)in some particular cases. We consider the three compact nonorientable surfaces with Euler characteristic-1and the three–holed projective plane.The curve complex of these surfaces has been studied by Scharlemann (<cit.>). We studied the nonorientable surfaces with Euler characteristic-1in <cit.>, and the three–holed projective plane in <cit.>. All the statements made in this section can be found in these articles. We also refer to <cit.> for the particular case of the three–holed projective plane. The nonorientable surfaces with Euler characteristic-1are very peculiar, and they appear as exceptions of some theorems. For instance, they do not admit pseudo–Anosov mapping classes, and they are hyperelliptic in the sense that there is a unique element in the mapping class group which acts as-Idon the first homology group. This element is central and acts trivially on the space of measured laminations. In the sequel we denote byN_g,nthe compact nonorientable surface of genusgwithnboundary components. §.§ The two–holed projective planeThis surface has only two simple closed geodesics (that are one–sided), and any other simple geodesic spirals around one of these closed geodesics or along a boundary component. So(N_1,2)consists in only two points, that are exchanged by the mapping class group(N_1,2)which is isomorphic to/2×/2. §.§ The one–holed Klein bottleThis surface has exactly one two–sided simple closed geodesic, that we call_∞. It has infinitely many one–sided simple closed geodesics, that we denote by(_n)_n∈. Any other simple geodesic spirals around a closed geodesic or along the boundary component. So(N_2,1)is a circle with a marked point_∞, which is the limit of the points(_n)_n∈asntends to±∞. Each arc[_n,_,n+1]⊂(N_2,1)consists in the set of projective measured laminations of the form[t_n+(1-t)_n+1]witht∈[0,1]. The mapping class group(N_2,1)is isomorphic toD_∞×/2, whereD_∞is the infinite dihedral group. The index two subgroup⩽D_∞is generated by the Dehn twist along_∞and acts transitively on{_n ; n∈}.For any hyperbolic metric, the number of one–sided simple closed geodesics of length less thanLis asymptotically equivalent to a positive constant timesL, the constant depending on the metric (<cit.>). §.§ The connected sum of three projective planesThis surface is very close to be a one–holed torus. Indeed it has a unique simple closed geodesicwhose complement is a one–holed torus(Scharlemann <cit.>, see also <cit.>). As a consequence there is a canonical embedding()⊂(N_3), and also a canonical isomorphism(N_3)≃^∗(). We recall that the symplectic representation induces an isomorphism between^∗()andGL(2,).We showed in <cit.> that any two–sided simple closed curve is homotopic to a curve contained in, and more generally any measured laminationłwithout one–sided closed leaf is contained in. The circle()divides(N_3)into two open disks. One open disk consists in all projective measured laminations of the form[+ł]withł∈(). The other, given by the inequalityi(,·)>0, consists in all measured laminations with a one–sided closed leaf distinct from. We refer to <cit.> for more details and a nice picture of(N_3). For any hyperbolic metric, the number of two–sided simple closed geodesics of length less thanLis asymptotically equivalent to a positive constant timesL^2, the constant depending on the metric. The same is true for the number of one–sided simple closed geodesics of length less thanL.§.§ The three–holed projective planeThe complementN_1,3-of any one–sided simple closed geodesicis a four–holed sphere. Let us denote by()the set of measured laminations of the formt+łwitht>0andł∈(N_1,3-). It is an open ball invariant under multiplication by a positive scalar, it projects onto an open disk in(N_1,3)which is a2–sphere. Any other one–sided simple closed geodesic$̣ intersects , therefore the balls () are ()̣ are disjoint. One easily observes that () and ()̣ are tangent (i.e. their boundaries havenontrivial intersection) if and only if i(,)̣=1. The projections on (N_1,3) of the balls () with ∈^- form a packing of disks. There exists a homeomorphism :(N_1,3)→^2 that sends this packing of disks on the so–called Apollonian packing (<cit.>). Moreoveris equivariant with respect to the actions of (N_3) and of a discrete subgroupof (^3)≃Conf(^2). The complement of the packing of disks is a minimal (N_1,3)–invariant closed subset of (N_1,3). Huang and Norbury <cit.> showed that (N_1,3)≃(/2⊕/2)∗/2.§ A DECOMPOSITION OF THE SPACE OF MEASURED LAMINATIONS§.§ The ball of a simple multicurve Given a simple multicurve =_1+…+_k, we call ball ofthe set () of measured laminations on S that admit _1,…,_k as closed leaves:() =_+^∗_1+… +_+^∗_k +(S-)∪{0}.It is homeomorphic to an open ball of dimension ()=(S)-k^+, where k^+ is the number of two–sided components of . If all the components _i are one–sided, then () is an open subset of (S). The ball () is convex in any train–track chart, and stable under multiplication by a positive scalar. Its projection on (S) is homeomorphic to an open ball of dimension (S)-k^+-1 bounded by a polyhedral sphere.Inside (), the Thurston measure ^S decomposes as a product:^S =_1 ⊗…⊗_n⊗^S-.We use abusively the notation _i for the differential of the weight of _i. Let us consider V=I_1_1+…+I_n_n+U where I_i⊂_+^∗ is an open interval and U is an open subset of (S-)∪{0}. Open subsets like V generate the Borel σ–algebra of (S). The number of integral points in V is the product of the numbers of integral points in the I_i's and in U. We conclude that (<ref>) is true since the Thurston measure can be defined in terms of integral points.The combinatorics of the intersection of balls is very simple:{[ ()∩(δ)=∅ if i(,δ)≠ 0,;()∩(δ)=(+)̣ if i(,δ)= 0. ].In particular, an intersection of balls is contractible whenever it is nonempty. §.§ Laminations with a closed one–sided leafWe denote by ^-(S) the set of measured laminations with a closed one–sided leaf, and by ^+(S) the set of measured laminations without closed one–sided leaves. We have^-(S) =⋃_∈^-().The collection of balls () such thatis a simple multicurve whose components are one–sided forms an open cover of ^-(S) which is stable under intersection. As each () is contractible, the nerve of this cover has same homotopy type as ^-(S). From the combinatorics of the intersection of balls we deduce that the nerve is the complex of one–sided closed curves, denoted by ^-(S). A n–simplex of ^-(S) is a family of n+1 disjoint isotopy classes of one–sided simple closed curves. Alternatively, one can show that ^-(S) is a deformation retract of ^-(S). Any ł in ^-(S) is uniquely written as ł=ł^+ + ł^- where ł^- is a weighted sum of one–sided closed curves, and ł^+ has no one–sided closed leaf. Let us define H:[0,1]×^-(S)→^-(S) by H(t,ł)=(1-t)ł^+ + ł^-. This is clearly a homotopy between the identity of ^-(S) and the retraction ł↦ł^-. As H commutes with the multiplication by a scalar, it induces a deformation retraction of ^-(S) onto ^-(S). Note that these two constructions are (S)-invariant.Let us look at the connectivity of ^-(S):The connected components of ^-(S) are ∪_∈^-_nb() and the balls () with ∈^-_b. Each ball () with ∈^-_b is obviously a connected component of ^-(S). So it remains to show that the subcomplex ^-_nb(S)⊂^-(S) spanned by _nb^- is connected. This comes directly from the following lemma. For any non disjoint ,∈̣^-_nb we have d(,)̣≤ 2 i(,)̣ where d is the distance on the 1-skeleton of ^-(S) obtained by fixing the length edge to 1.We endow S with a hyperbolic metric, and we work with the geodesic representatives of the isotopy classes.We proceed by induction on i(,)̣. We first consider the case i(,)̣=1.Let P be a sufficiently small tubular neighborhood of ∪$̣. ThenPis a projective plane with two boundary components embedded inS. The complementS-Pis a (possibly non connected) nonorientable surface. OtherwiseSwould be of genus one, and any one–sided simple closed curve ofSwould be bounding, which is impossible sinceis non bounding. AsS-Pis nonorientable, it contains a one–sided simple closed curve, necessarily disjoint formand$̣. This shows that d(,)̣=2.Now we assume i(,)̣≥ 2. We cut , this produces a boundary componentof length 2ℓ(). The trace of $̣ inS-consists in a collections of disjoint arcs_̣1,…,_̣mwithm=i(,)̣. We assume that a transverse orientation of_̣1induces distinct orientations ofat the endpoints of_̣1, this is possible for$̣ is one–sided. We denote by p and q the endpoints of _̣1 on . Letbe the shortest arc ofthat joins q to the antipodal point of p. This arc intersects _̣1∪…∪_̣m in at most i(,)̣ points, because its length is less than ℓ(). From an arc parallel to _̣1 and a subegment ofone can make a one–sided curve c with i(c,_̣1)=0 and i(c,)̣<i(,)̣. Moreover we have i(c,)=1. So d(,c)=2 and by induction d(c,)̣≤ 2i(c,)̣≤ 2(i(,)̣-1). We conclude with the triangular inequality§.§ Genericity of laminations with a one–sided leafThe following theorem plays a crucial role in the proofs of our main theorems :^-(S) is a dense open subset of (S) of full Thurston measure. We say that a subset of (S) has full Thurston measure if its complement is –negligible. The measurable part of the statement is difficult, it involves the Rauzy induction for linear involutions (a generalization of interval exchanges). The topological part is relatively easy (Proposition 1.2 in <cit.> is rather similar). In the following two corollaries we precise the structure of a generic lamination:The set of measured laminations ł∈(S) such that ł^- bounds an orientable subsurface is a dense open subset of full Thurston measure.A collection of disjoint non isotopic simple closed curves bounds an orientable subsurface if and only if it is maximal among such collections. We proceed by induction on the genus g≥ 1 of S.(Initialization) If g=1, then the corollary comes directly from Theorem <ref>.(Induction) We assume the property true for all nonorientable surfaces of genus less than a given g≥ 1. From Theorem <ref> it suffices to show that, for any one–sided simple closed curve , the set of ł∈() such that ł^- bounds an orientable subsurface is open and has full measure in () (note thatis automatically a component of ł^-). If S- is orientable, then the assertion is obvious. If S- is nonorientable, then we use the induction hypothesis and the decomposition of the Thurston measure (<ref>). From Corollary <ref> and <cit.> we immediately get: Almost every measured lamination ł∈(S) is of the form ł=ł^-+ł^+ where ł^- bounds an orientable subsurface, and ł^+ is a maximal measured lamination of S-ł^-.§.§ An identityHere we present another consequence of Theorem <ref> in the form of an identity. Following <cit.> we set X=(S,m) and, for any L>0,B_X(L) ={ł∈(S) ; ℓ_m(ł)≤ L},b_X(L) =(B_X(L)).For any simple closed geodesicof X, we abusively denote by X- the hyperbolic surface with boundary which is the metric completion of X-. In particular, b_X-(1) is the Thurston measure of {ł∈(S-) ; ℓ_X-(ł)≤ 1} where we denote by S- the underlying topological surface of X-. Let X be a finite area nonorientable hyperbolic surface. Then b_X(1) = ∑_(d-n)!/d!b_X-(1)/ℓ_X(_1)⋯ℓ_X(_n), where d=(S) and =_1+…+_n (n∈^∗) runs over the set of maximal families of disjoint non isotopic one–sided simple closed curves.We don't have any application of this formula which shows a relation between the volume b_X-(1) and the lengths ℓ_X(_1),…,ℓ_X(_n). It would be interesting to bound b_X-(1) in terms of these lengths. From Corollary <ref> we have:(B_X(L)) =∑_(B_X(L) ∩()) (∀ L>0),where =_1+…+_n (n∈^∗) runs over the set of maximal families disjoint non isotopic one–sided simple closed curves. Using (<ref>) we find:(B_X(L) ∩()) =∫_L≥ℓ_X( x·) b_X-(L-ℓ_X(x·))x,=b_X-(1)∫_L≥ℓ_X( x·) (L-x_1ℓ_X(_1)-… -x_nℓ_X(_n))^d-n x , =b_X-(1)/ℓ_X(_1)⋯ℓ_X(_n)∫_L≥ y_1+…+y_n (L-y_1-… -y_n)^d-n y,= b_X-(1)/ℓ_X(_1)⋯ℓ_X(_n)(d-n)!/d!L^d ,where d=(S) and x·=x_1_1+…+ x_n_n for any x∈(_+^∗)^n.§.§ A (S)–invariant continuous functionWe have seen that any measured lamination ł∈(S) is uniquely written asł =ł^-+ł^+ where ł^+∈^+(S) and ł^- is a simple multicurve whose components are one–sided:ł^- = a_1_1+…+a_k_kwhere a_1≥…≥ a_k>0 and _1,…,_k are disjoint (non isotopic) one–sided simple closed curves. We set a_k+1=…=a_g=0 if k is less than the genus g of S. With the above notations, we define a function[ w^-:(S)⟶ _+^g; ł⟼ (a_1,…, a_g) ].The function w^- is continuous and (S)–invariant. It induces a continuous and (S)–invariant function from ^-(S) to (^g)The function w^- is obviously nonconstant. The induced function from ^-(S) to (^g) is nonconstant if g≥ 2.The only difficulty is the continuity, but w^- gives in decreasing order the values of the weight functions of the one–sided simple closed geodesics, so the continuity of w^- follows directly from the continuity of the weight functions (Lemma <ref>).Given an isotopy classof one–sided simple closed curves, the weight function w_:(S)→ is defined as follows: for any ł∈(S), the weight w_(ł) is the unique nonnegative real number such that ł=w_(ł)·+ł' where ł'∈(S) has no leaf isotopic to . The weight function w_ of a one–sided curveis continuous.This function is identically equal to zero outside the open subset (). So it suffices to show that w_ is continuous on the closure of () in (S).We fix a hyperbolic metric on S, and we denote by C_r() the collar of width r>0 around . As well–known, for r>0 sufficiently small, any simple geodesic disjoint fromeither is asymptotic to , either does not penetrate C_r(). In particular, a measured lamination in the closure of () has no leaf that penetrate C_r() except . As a consequence, for a sufficiently small geodesic arc k that intersectstransversely, we have w_(ł)=i(k,ł) for any ł in the closure of (). We conclude that w_ is continuous on the closure of () by continuity of i(·,·). §.§ Consequences on the dynamics of the (S) actionThe action of (S) on (S) is not minimal, and is not topologically transitive if the genus of the surface is at least 2.The action of (S) on (S) is not topologically transitive, in particular it is not ergodic with respect to the Thurston measure. In the case of an orientable surface, the action of the mapping class group on the space of measured laminations is ergodic with respect to the Thurston measure (<cit.>). Moreover, the orbit of any measured lamination without closed leaves is dense (see <cit.>).The orbit of any [ł]∈^+(S) is disjoint from the open set ^-(S), so the action of (S) on (S) is not minimal. The other statements come from the existence of nonconstant, continuous and (S)–invariant functions (see Proposition <ref> and Remark <ref>). § PARTIAL CLASSIFICATION OF ORBIT CLOSURESIn this section we study the topological dynamics of the (S)–actions on (S) and (S). We give a partial classification of its orbit closures. The idea is to use the (S)–invariance of the decomposition of ^-(S) into open balls () wherea simple multicurve whose component are one–sided. In the orientable case, Hamenstädt (<cit.>) and Lindenstrauss–Mirzakhani (<cit.>) gave a complete classification of the closed invariant subsets of the space of measured laminations. Their works rely on the properties of the Teichmüller geodesic and horocyclicflows, for instance they both use the nondivergence of the Teichmüller horocyclic flow established by Minsky and Weiss (see <cit.> and <cit.>). It seems difficult to adapt their arguments since the Teichmüller horocylic flow is not well–defined in the nonorientable case (see §<ref>).We recall that S is a finite type nonorientable surface with χ(S)<0. We denote by ^+(S,) the set of elements in (S,) whose components are two–sided. The closure of ^+(S,) is included in ^+(S), we prove that there is equality when S has genus one (Lemma <ref>). §.§ Orbit closures in (S) Following <cit.> we define a complete pair as a couple R=(R,) where =x_1_1+…+x_n_n (x_i>0) is a simple multicurve, and R is a union of connected components of the complement S-. As in <cit.> we use the following notations: G^ R=+(^+(R)∪{0})and G^[ R] =∪_f∈(S)G^f· R.To any measured lamination ł we associate a complete pair R_ł=(R,) as follows: the multicurvecorresponds to the atomic part of the transverse measure of ł, and the subsurface R is the union of the connected components of S- that contain a noncompact leaf of ł. For any measured lamination ł∈(S) we have(S)·ł⊂ G^[ R_ł].The inclusion is an equality if R is orientable.1) We show that (S)·ł⊂ G^[ R_ł].Let ł_∞ be a measured lamination in the closure of (S)·ł. We consider a sequence (g_n)_n in (S) such that (g_nł)_n converges to ł_∞ in (S).The measured lamination ł is uniquely written in the form ł=+ł' where =x_1_1+…+x_m_m (x_i>0) is a simple multicurve, and ł' is a measured lamination without compact leaves. We clearly have g_n≤ g_nł and g_nł'≤ g_nł for any n, here we consider a measured lamination as a geodesic current (see §<ref>). As (g_nł)_n converges, it comes that the sequences (g_n)_n and (g_nł')_n are bounded in the space of Radon measures. Therefore they admit convergent subsequences with respect to the weak* topology (Tychonoff theorem). So we assume that (g_n)_n and (g_nł')_n are convergent with respective limits _∞ and ł'_∞. Using the same argument, we assume that each sequence (g_n_i)_n converges. Actually this implies that each sequence (g_n_i)_n stablizes, for a compact subset of (S) contains only a finite number of simple closed geodesics. We fix an integer N such that g_n_i=g_N_i for any n≥ N and any i=1,…,m.Up to the choice of a subsequence and a bigger N, we assume that the mapping classes g_N^-1g_n (n≥ N) do not permute the connected components of S-_∞. Then the lamination ł'_∞=lim_n g_nł' is clearly contained in the subsurface g_N R, where R is the subsurface defined before the statement of the theorem.It remains to show that ł'_∞ has no closed one–sided leaf. This comes directly from the fact that the set +(S) of measured laminations without one–sided leaf is closed in (S), or equivalently that the set ^-(S) of measured laminations with a one–sided leaf is open in (S) (see §<ref>). 2) If R is orientable. Then ^+(R)=(R) and, according to Theorem 1.2 of <cit.>, the orbit (S-)·ł' is dense in (R).§.§ Orbit closures in (S)As well–known, for a finite type orientable surface with negative Euler characteristic, the action of the mapping class group on the projective space of measured laminations is minimal (see <cit.>). The following theorem shows that the mapping class group action has a very different topological dynamics in the nonorientable case.Let PG^[ R_ł] denote the image of G^[ R_ł] in (S). For any projective measured lamination [ł]∈(S) we have (S)· [ł]⊂PG^[ R_ł]∪^+(S).The inclusion is an equality if S has genus one.The theorem gives a complete classification of orbit closures in genus one.The proof falls into two parts.1) The inclusion. Let ł_∞ be a measured lamination such that [ł_∞] is in (S)· [ł]. We want to show that [ł_∞]∈PG^[ R_ł]∪^+(S). If ł_∞ does not have a one–sided leaf, then ł_∞∈^+(S) and we are done. So we assume that ł_∞ has a one–sided leaf. Then There exists a number >0 and a sequence (g_n)_n⊂(S) such that (g_nł)_n converges to ł_∞ in (S).There exists two sequences (_n)_n⊂_+^∗ and (g_n)_n⊂(S) such that _n(g_nł) tends to ł_∞ in (S) as n tends to infinity. Clearly ł_∞ belongs to the open ball (ł_∞^-), therefore g_nł∈(ł_∞^-) for n big enough. In particular, for n big enough, any component of ł^-_∞ is a component of g_nł^-.Let us write ł^-=x_1_1+…+x_m_m where _1,…,_m are disjoint one–sided geodesics and x_1,…,x_m>0. Up to the choice of a subsequence of (g_n)_n we have ł_∞^-=y_1(g_n_1)+…+y_l (g_n_m) for n big enough and for some y_1,…,y_m≥ 0not all equal to zero and independent of n. The role of the subsequence is to avoid any permutation of the _i's. From lim_n _n (g_nł)=ł_∞ we deduce lim_n _n x_i=y_i and consequently lim_n _n=y_i/x_i for any i=1,…,m. We conclude that (g_nł)_n converges to ł_∞ with =x_i/y_i (i=1,…,m). As a direct consequence of the above lemma we have ł_∞∈(S)·ł.Then Theorem <ref> implies ł_∞∈ G^[ R_ł] and [ł_∞]∈PG^[ R_ł]. This prove the first part of the statement.2) The equality. From Lemmas <ref> and <ref> we have ^+(S)⊂(S)·[ł]. From the equality case of Theorem <ref> we have PG^[ R_ł]⊂(S)·[ł].This concludes the proof of the theorem. If S has genus one then ^+(S) is the closure of ^+(S,).Let ł be a measured lamination without one–sided leaf: ł∈^+(S). We show that ł is a limit of elements in ^+(S,). To do this we consider a sequence (_n)_n⊂(S,) that converges to ł in (S). If infinitely many _n have no one–sided leaf, then we are done. So, up to the choice of a subsequence, we assume that each _n has a one–sided leaf _n. Thus _n belongs to the ball (_n). Our problem is to construct from (_n)_n a sequence of two–sided simple multicurves (_̣n)_n⊂^+(S,) which has same limit as _n.Let U⊂^m be a convex set which is a neighborhood of ł in some train–track chart. For n big enough we have _n∈ U, and the segment [_n,ł] is contained in U. It intersects the boundary ∂(_n)=(S-_n) in a point p_n. This point corresponds to a lamination contained in S-_n. Note that S-_n is a sphere with some punctures and holes, because S has genus one. As S-_n is orientable there exists a two–sided rational simple multicurve _̣n on S-_n which is at distance at most 1/n from p_n in U⊂^m with respect to the canonical Euclidean norm of ^m. We have ł-_̣n ≤ł-p_n+1/n≤ł-_n +1/n, and we conclude that the sequence of rational two–sided simple multicurves (_̣n)_n converges to ł in (S).For any [ł]∈(S) the orbit closure (S)· [ł] contains ^+(S,), except if [ł] is the projective class of the unique bounding one–sided simple closed geodesic of the connected sum of three projective planes.If S is the projective plane with two boundary components then ^+(S) is empty. If S is the Klein bottle with one boundary component then ^+(S) has only one point that represents the unique two–sided simple closed geodesic. If S is the connected sum of three projective planes, then the bounding one–sided simple closed geodesic is the unique geodesic lamination that does not intersect any two–sided simple closed geodesic.We assume that S is not the projective plane with two holes or the Klein bottle with one hole (in both cases the lemma is trivially true). Let us consider a measured lamination ł∈(S) which is not the bounding geodesic of the connected sum of three real projective planes. Then there exists a two–sided simple closed geodesicthat intersects ł. Using the Dehn twist alongwe find that [] is in the closure of (S)· [ł] (see <cit.>).Given ∈̣^+(S,) there exists a two–sided simple closed geodesic _1 that intersectsand each leaf of $̣. Using Dehn twists, we first show that[_1]belongs to the closure of(S)·[], and then that[]̣belongs to the closure of(S)·[_1]. We conclude by transitivity of the relation to belong to the orbit closure of.§.§ Topological transitivity in (S)We recall that an action is topologically transitive if it has a dense orbit. The action of (S) on (S) is topologically transitive if and only if S has genus one.A direct application of Theorem <ref> shows that, if S has genus one, then an orbit (S)·[ł] is dense in (S) if and only if [ł]=[+ł'] whereis a one–sided geodesic and ł' has no closed leaf. We distinguish the two cases:1) S has genus one.We recall that ^-(S)=∪_() whereruns over the set of one–sided simple closed geodesics. From the hypothesis that S has genus one we deduce that (S) acts transitively on the set of components () (there is only one topological type of one–sided simple closed geodesic). Using Lemma <ref> we conclude that the (S)–action on (S) is topologically transitive.2) S has genus at least two. See Proposition <ref>.Letbe a simple closed geodesic such that S- is orientable. Then the action of (S) on the projective ball PB() is topologically transitive.Let us consider the subset +(S-)⊂(). The action of (S-) on (S-) is ergodic with respect to the Thurston measure on (S-) (Masur <cit.>), in particular it is topologically transitive (see also <cit.> for more precise results). We conclude as the projection (S)→(S) induces a (S)-invariant homeomorphism between +(S-) and PB().§.§ Minimal invariant in (S)We have studied in §<ref> the case of surfaces withχ(S)=-1, so we assumeχ(S)<-1.Then the above lemma says that ^+(S,) is the unique minimal (S)–invariant closed subset of (S). This minimal invariant can alternatively be described in terms of pseudo–Anosov laminations (Papadopoulos and McCarthy (<cit.>). Indeed, the closure of the set of pseudo–Anosov laminations is(S)–invariant and contained in the closure of any(S)–orbit. The only problem is to show that this set is nonempty (this happens whenχ(S)=-1). Thurston explained how to construct pseudo–Anosov mapping classes provided thatShas a pair of two–sided simple closed geodesics that fill upS(<cit.>, see also <cit.>). Such a pair does not exist whenχ(S)=-1, but it does exist whenχ(S)<-1(Lemma <ref>). §.§ ConjecturesIn view of our partial classifications it is natural to formulate the following conjectures:We have ^+(S)=^+(S,). We have seen that this is true whenShas genus one (Lemma <ref>). It would be a first step in the direction of the following more general conjecture: The inclusions in Theorems <ref> and <ref> are equalities, except if ł is the bounding geodesic of the genus three closed surface. We have seen in Remark <ref> that the bounding geodesic of the connected sum of three real projective planes is an exception.§ GROWTH OF THE NUMBER OF SIMPLE CLOSED GEODESICS We now prove Theorem <ref> on the growth of__0=(S)·_0. Actually we are going to prove a more general result (Theorem <ref>). We first need to introduce some notations.For anyN∈^∗, we denote by^-_N(S,)the set of integral simple multicurves whose one–sided components have weight at mostN. In other words, a simple multicurve=n_1_1+…+n_k_k(n_i∈^∗) belongs to^-_N(S,)if and only ifn_i≤Nfor every one–sided component_i. This set is clearly(S)–invariant, and it contains infinitely many(S)–orbits since the weights of the two–sided components are not bounded. Note also that, for any_0∈(S,), there existsNsuch that__0⊂^-_N(S,). The following theorem is the generalization of Theorem <ref>:Any nonorientable hyperbolic surface of finite area (S,m) satisfieslim_L→∞{∈^-_N(S,) ; ℓ_m()≤ L }/L^(S) =0.In Part <ref> we extend this result to all multicurves (not necessarily simple) and to all filling geodesic currents (not necessarily Liouville currents of hyperbolic metrics).The global scheme of the proof is similar to the one of <cit.>: we introduce a family of counting measures(ν^L)_L>0on(S)and show its weak^∗convergence. We use the theorem of Danthony and Nogueira (§<ref>) instead of Masur's ergodic theorem. We do not need to integrate over the moduli space, which is the most difficult part of Mirzakhani's proof (<cit.>). Let us consider the sequence of counting measures (ν^L)_L>0 defined byν^L=1/L^(S)∑_∈^-_N1_1/L,where for short we set ^-_N=^-_N(S,). We have ν^L(B_m(1))= |B_m(1)∩1/L^-_N|/L^(S)= |B_m(L)∩^-_N|/L^(S)=| {∈^-_N ; ℓ_m()≤ L} |/L^(S). Thus the conclusion of the theorem is equivalent tolim_L→∞ν^L(B_m(1))= 0,which is a direct consequence of the weak^∗ convergence of (ν^L)_L towards the zero measure (Proposition <ref> and Lemma <ref>).The measure ν^L weak^∗ converges to the zero measure as L tends to infinity.Let us recall that (μ^L)_L is the sequence of counting measures associated to the set of all integral points of (S) (see (<ref>) for an explicit formula). We have ν^L≤μ^L for any L>0, and we have seen (§<ref>) that μ^L weak^∗ converges toas L tends to infinity. We deduce that (ν^L)_L>0 is relatively compact in the space of Radon measures (Tychonoff's theorem). Thus, to prove its convergence, it suffices to show that it has a unique limit point.Let ν^∞ be the weak^∗ limit of a sequence (ν^L_n)_n where (L_n)_n is a sequence of positive real numbers that tends to infinity as n tends to infinity. From the inequality ν^L_n≤μ^L_n we get that ν^∞ is absolutely continuous with respect to . But its support supp(ν^∞) is –negligible (Lemma <ref>), so we conclude that ν^∞ is the zero measure. We have supp( ν^∞)⊂^+(S) which is –negligible.Let us consider a measured lamination ł∈^-(S). Our aim is to show that ł∉supp(ν^∞). We write ł=ł^-+ł^+ with ł^+∈^+(S)∪{0} and ł^-=x_1_1+… +x_k_k (k≥ 1) where x_1,…,x_k>0 and _1,…,_k are disjoint (non isotopic) one–sided simple closed geodesics. We consider a neighborhood U=I_1_1+…+I_k_k+^+(S-ł^-)∪{0} of ł where each I_i⊂_+^∗ is a compact interval containing x_i in its interior. Clearly ν^∞(U)=0 implies ł∉supp(ν^∞). Therefore our aim is to show that ν^∞(U)=0. We claim that ν^∞(U)=lim_nν^L_n(U). As (ν^L_n)_n weak^∗ converges to ν^∞ we just have to check that ∂ U is ν^∞–negligible. But ν^∞ is absolutely continuous with respect to , anddecomposes as a product on U (see §<ref>), hence ∂ U is ν^∞–negligible. Now we show that ν^L(U)=0 for L big enough. Let η be an element of ^-_N(S,)∩ (L· U) for some L>0. From the definition of ^-_N(S,) we have η=n_1_1+…+ n_k_k+η^+ with N≥ n_i≥ 0 and η^+∈^+(S-ł^-)∪{0}. And from η∈ L· U we have n_i∈ L· I_i. The conditions N≥ n_i and n_i∈ L· I_i imply that ^-_N(S,)∩ (L· U)=∅ for L big enough. This is precisely equivalent to ν^L(U)=0 for L big enough. Let us recall the following obvious lemma (see <cit.>): The boundary ∂ B_m(1) is –negligible.We have B_m(1)⊂ B_m(1+)-B_m(1-) for any >0 small enough. By multiplicativity of the Thurston measure we find (for any >0 small enough)(∂ B_m(1))≤ (B_m(1+))-(B_m(1-)), (∂ B_m(1))≤ (B_m(1)) · ((1+)^d-(1-)^d), (∂ B_m(1))≤ (B_m(1)) ·· P(),where d=(S) and P() is a polynomial in . We conclude by taking the limit astends to zero.§ PAIRS OF FILLING CURVES LetSbe a compact surface withχ(S)<0. Following Thurston (<cit.>) we say that a pair of simple closed geodesics{,}̣fills upSif each component ofS-(∪)̣is an open disk or a half–open annulus whose boundary lies in∂S. Equivalently{,}̣is filling if every simple closed geodesic ofSintersects∪$̣. If S is nonorientable, then it has a filling pair of two–sided simple closed geodesics if and only if χ(S)<-1.See <cit.> for a proof in the orientable case.If χ(S)=-1 then one easily sees that S has no pair of filling two–sided simple closed geodesics. So we assume χ(S)<-1.We consider a simple closed geodesicwhose complement is orientable. Let ł be a filling measured lamination of S-, that is a measured lamination that intersects any simple closed geodesic of S-. Let $̣ be a two–sided simple closed geodesic ofSthat intersects(it exists thanks to the assumptionχ(S)<-1). The functioni(,̣·)+i(ł,·)is positive on(S). Let(_n)_nbe a sequence of two–sided simple closed geodesics that converges to[ł]in(S). The existence of such a sequence is obvious forłis contained in an orientable subsurface. Let(a_n)_nbe a sequence of positive real numbers such that(a_n_n)converges tołin(S). Let us show that there exists N such that $̣ and_Nfill upS. By contradiction we assume that for eachnthere exists_̱n∈such thati(,̣_̱n)+i(_n,_̱n)=0. Up to the choice of a subsequence, we assume that(_̱n)_nconverges in(S)to a point[]̱. Let(b_n)_nbe a sequence of positive real numbers such that(b_n_̱n)converges to$̱ in (S). By continuity of i(·,·) on (S)×(S) it comes i(,̣)̱+i(ł,)̱=lim_n→∞ i(,̣b_n_̱n)+i(a_n_n,b_n_̱n)=0. This contradicts the positivity of i(,̣·)+i(ł,·) on (S).If S is nonorientable, then it has a filling pair of one–sided simple closed geodesics.We first assume χ(S)<-1 and following the previous lemma we consider{,}̱ a pair of filling two–sided simple closed geodesics. Let (_n)_n and (_̱n)_n be two sequences of one–sided simple closed geodesics that converge respectively toand $̱ in(S). One easily shows the existence of such sequences using Dehn twists. We choose two sequences(a_n)_nand(b_n)_nof positive real numbers such that(a_n_n)and(b_n_̱n)converge respectively toand$̱ in (S). We conclude by contradiction as in the proof of the previous lemma.For any nonorientable surface S with χ(S)=-1 it is not difficult to find an explicit filling pair of one–sided simple closed geodesics.PART:Growth of geodesics with self–intersections The aim of this part is to establish the following theorem:Let (S,m) be a nonorientable hyperbolic surface of finite area (possibly with geodesic boundary). For any integral multicurve _0 we havelim_L→∞|{∈__0 ; ℓ_m()≤ L}|/L^(S)= 0. Actually we prove a more general result (Theorem <ref>) that deals with geodesic currents.This result contrasts with the result of Mirzakhani (<cit.>) which states that, for orientable surfaces, the above limit exists and is positive. It contrasts also with the fact that for any k≥ 1 there exists C_k>0 such that 1/C_k· L^(S)≤ |{∈_k ; ℓ_m()≤ L}| ≤ C_k · L^(S), for any L large enough. Here we denote by _k the set of integral multicurves with exactly k self–intersections. We refer to <cit.> for a proof of this fact that works for both orientable and nonorientable surfaces of finite type. This kind of estimates was first obtained by J. Sapir (<cit.>). As in the case of simple closed curves, the theorem comes with the convergence of a family of counting measures (ν__0^L)_L>0 defined by ν__0^L =1/L^(S) ∑_∈__01_1/L.These are (S)–invariant measures over the space of geodesic currents (S) (see §<ref>). We show that the accumulation points of __0 in (S) are contained in ^+(S) (§<ref>). This implies that the support of any limit point of (ν^L__0)_L is contained in ^+(S). Then we conclude that (ν__0^L)_L converges to the zero measure (§<ref>) using a theorem of Erlandsson and Souto (<cit.>). Along the way we show that, if (_n)_n is a sequence of closed geodesics that converges to a one–sided simple closed geodesic in the projective space of geodesic currents (S), then either (_n)_n stabilizes either the number of self–intersections of _n tends to infinity with n (Proposition <ref>). § GEODESIC CURRENTS In this section we recall some basic facts about Bonahon's geodesic currents. We refer to the article <cit.>, or to the textbook <cit.>, for more details and complete proofs. For sake of simplicity, we restrict our attention to closed surfaces, but it is explained in <cit.> how to deal with punctured surfaces. §.§ Space of geodesics Let (S,m) be a closed hyperbolic surface. We denote by S̃_∞ the boundary at infinity of the universal cover S̃. A geodesic of (S̃,m̃) is encoded by its endpoints in S̃_∞, that is why we call space of geodesics of S̃ the quotient (S̃)=(S̃_∞×S̃_∞-Δ)/(/2) where Δ is the diagonal of S̃_∞×S̃_∞ and /2 acts by transposition. Given another hyperbolic metric m' on S, the identity id_S̃:(S̃,m̃)→ (S̃,m̃') is a quasi–isometry, therefore it extends to a π_1(S)–invariant homeomorphism between the visual boundaries. This shows that S̃_∞ and (S̃) are of topological nature, they do not depend on m. Similarly, the objects introduced below are topological and independent of m (except ł_m). §.§ Geodesic currentsA geodesic current on S is a π_1(S)–invariant (positive) Radon measure on (S̃). We denote by (S) the space of geodesic currents on S endowed with the weak^∗ topology. It is stable under addition and multiplication by a positive scalar. The projective space of geodesic currents (S) is compact with respect to the quotient topology (<cit.>). §.§ Multicurves as geodesic currents To a (primitive) closed geodesicof (S,m) corresponds the geodesic current ∑_ 1_ whereruns over the set of lifts of . By linearity, this map extends to a canonical injection of the set of integral multicurvesinto the space of geodesic currents (S).The set of integral multicurves is discrete in (S), but the set of closed geodesics is dense in (S) (<cit.>). The geometric intersection between closed geodesics extends to a bilinear continuous function i:(S)×(S)→_+.§.§ Hyperbolic metrics as geodesic currentsAny hyperbolic metric m on S determines a geodesic current ł_m, called the Liouille current of m. As indicated by its name, it is related to the Liouville form on T^∗ S. For our purpose, it is enough to mention the following properties: the map m↦ł_m induces a topological embedding (S)→(S) and, for any multicurve ∈, we have the following nice relation i(ł_m,)=ℓ_m(). §.§ Measured laminationsThe injective map (S;)⊂↪(S) extends to a topological embedding of (S) into (S). The image of (S) coincide with the light cone {c∈(S) ; i(c,c)=0}. § ACCUMULATION POINTS OF __0 IN (S)For any _0∈, the accumulation points of __0 in (S) are contained ^+(S).One easily shows that any point in ^+(S,) is an accumulation point of __0 in (S), as well as any projective pseudo–Anosov lamination.Combine the Lemma <ref> with Corollary <ref>.The accumulation points of _k in (S) are contained in (S).Let [ł] be an accumulation point of _k in (S). There exist a sequence (_̣n)_n of distinct elements of _k, and a sequence (d_n)_n of positive real numbers such that (d_n_̣n)_n converges to ł in (S). The geodesic current ł is a measured lamination if and only if i(ł,ł)=0 (<cit.>).We fix a hyperbolic metric m on S. On one hand we have ℓ_m(_̣n)→∞ as n tends to infinity (the _̣n are distinct). On another hand ℓ_m(d_n _̣n)→ℓ_m(ł) as n tends to infinity (ℓ_m is continuous). So d_n→ 0 as n tends to infinity. We conclude thati(ł,ł) = lim_n i(d_n_̣n ,d_n_̣n ) =k lim_n d_n^2=0 by continuity of i(·,·). §.§ Neighborhood of a lamination with a one–sided leafLet ł∈^-(S) be a measured lamination with a one–sided closed leaf . For any k≥ 1, there exists a neighborhood U_k of ł in (S) such that for any geodesic current c∈ U_k there exists a geodesic ∈̣(c) that projects either oneither on a geodesic with at least k self–intersections.*We do not say that the projection of $̣ onSis closed.*The property satisfied byU_kis invariant by multiplication by a scalar. As a consequence the cone_+^∗ U_ksatisfies this property.Let [ł]∈(S) be an accumulation point of __0 in (S). Then ł has no one–sided leaf.Let [ł]∈(S) be the limit in (S) of a sequence (_̣n)_n of elements of __0. There exists a sequence (d_n)_n of positive real numbers such that (d_n_̣n)_n converges to ł in (S). As noted in the proof of Lemma <ref> we have lim_n d_n=0.By contradiction we assume that ł has a one–sided leaf . We denote by U the neighborhood of ł given by Proposition <ref> for k=i(_0,_0)+1. For n big enough we haved_n_̣n∈ U, which implies thatis a component of _̣n, and consequently can be written in the form _̣n= c_n+'̣_n with c_n∈^∗, and _̣n'∈_≤ i(_0,_0) such thatis not a component of '̣.The weight c_n is uniformly bounded by the number of components of _0 counted with multiplicity. Therefore lim_n d_nc_n=0 and lim_n d_n'̣_n=lim_n d_n_̣n=ł. So d_n '̣_n belongs to U for n big enough. This contradicts Proposition <ref>.We fix a hyperbolic metric m on S. Let C be a collar neighborhood ofwhich is homeomorphic to a Möbius band. We denote bya lift ofto the universal cover, and by C̃ the lift of C that contains .We first choose U_k. According to Lemma <ref> there exists L_k>0 such that any geodesic arcin C of length ℓ_m()≥ L_k has at least k self–intersections. We take U_k to be the neighborhood of ł given by Lemma <ref> for L=L_k. Let us show that U_k satisfies the expected property. For any c∈ U_k there is a geodesic ∈̣(c) such that ∩̣C̃ has length at least L_k (Lemma <ref>). If $̣ is asymptotic to, then we conclude that∈(c)because(c)is a closedπ_1(S)–invariant subset ofG(S̃). If$̣ is not asymptotic to , then ∩̣C̃ is a closed segment that projects onto a geodesic arcin C of length ℓ_m()≥ L_k. In that case we conclude that the projection of $̣ onShas at leastkself–intersections (by choice ofL_kandU_k). With the notations introduced in the proof of Proposition <ref> we have: For any L>0 there exists a neighborhood U⊂(S) of ł such that for any c∈ U there is a geodesic ∈̣(c) such that ∩̣C̃ has length at least L.In the proof of the lemma, we only use the fact that ł has a closed leaf , we do not use the hypothesisone–sided.We denote by w the width of C. Let p,q be two points onat distance L+2w from each other. We denote by V⊂ G(S) the set of geodesics that intersect the balls of radius w>0 centered at p and q. This is an open subset of G(S̃). We denote by U the set of geodesic currents c∈(S) such that c(V)>0, or equivalently U={c∈(S) ; (c)∩ V≠∅}. It is an open neighborhood of ł.By construction, for any c∈ U there is a geodesic ∈̣(c) that intersects the balls of radius w centered at p and q, thus ∩̣C̃ has length at least L. Letbe a one–sided simple closed geodesic of a hyperbolic surface, and C be a collar neighborhood ofhomeomorphic to a Möbius band. Any geodesic arcin C satisfies | i(,)-ℓ()/2ℓ()| ≤ℓ(∂ C)+2w/2ℓ()+1, where i(,) is the number of self–intersections of , and w is the width of C.This lemma says that, for a geodesic arccontained in a collar neighborhood of a one–sided geodesic, the lengthℓ()is quasi–proportional to the intersection numberi(,). We fix two distinct points p and q on ∂ C. We first enumerate the geodesic arcs whose endpoints are p and q. Let $̱ be a loop based atpwhose homotopy class generatesπ_1(C,p)≃, and let$̣ be an arc of ∂ C that goes from p to q. We assume that the orientation of $̱ and$̣ are compatible (note that ^̱2 is homotopic to ∂ C). The collection {∗̣^̱k}_k∈ is a set of representatives of the homotopy classes of paths from p to q. We denote by _k the unique geodesic arc homotopic to ∗̣^̱k(Figure <ref>).C at 17 300 p at 180 345 q at 180 30at 195 120 $̱ at 235 252$̣ at 285 300 _3 at 481 275 _2 at 841 260< g r a p h i c s > Examples of geodesic arcs _k Now we show that |i(_k,_k)-|k|/2| ≤ 1.The universal cover C̃ of C is an infinite band isometric to the w–tubular neighborhood of a geodesic of the hyperbolic plane. The Deck transformation given by $̱ acts by translation–reflection along the geodesic. A lift_kof_kdividesC̃into two connected components. The numberi(_k,_k)is half the number of lifts of_kwhose endpoints do not belong to the same component ofC̃-_k(Figure <ref>). One easily finds thati(_k,_k)={[k/2 if k≥0is even;|k|/2-1 if k<0is even;(k+1)/2 ifk>0is odd;(|k|-1)/2 ifk<0is odd ]. .In the casep=qwe find thati(^̱k,^̱k)is the integer part of|k|/2.C̃ at 20 89 p̃ at90 -10 q̃ at445 180 _3 at30065< g r a p h i c s > The lift _3 in the universal cover C̃ By comparing the lengths of_kand^kwith the lengths of some piecewise geodesic loops in the same homotopy classes we getℓ(^k)+ℓ()̣+2w>ℓ(_k)andℓ(_k)+ℓ()̣ > ℓ(^k), wherewis the width of the collarC. This impliesℓ(_k)+ℓ()̣/ℓ()>|k|> ℓ(_k)-ℓ()̣-2w/ℓ().We conclude using the above expression ofi(_k,_k).§ CONVERGENCE OF COUNTING MEASURES Let_0∈be an integral multicurve. For anyL>0we setν__0^L =1/L^(S) ∑_∈__01_1/L.This is a locally finite(S)–invariant Borel measure on the space(S). The measure ν^L__0 tends to the zero measure (with respect to the weak^∗ topology) as L tends to infinity.According to the Proposition <ref> of Erlandsson and Souto it suffices to show that the only limit point of (ν__0^L)_L>0 is the zero measure. Such limit point is absolutely continuous with respect to(same proposition of Erlandsson and Souto) and supported on ^+(S) (Lemma <ref>). The conclusion comes from the fact that ^+(S) is –negligible (Theorem <ref> of Danthony andNogueira).Any limit point of the family (ν^L__0)_L>0 is supported on ^+(S).Let (L_n)_n be a sequence of positive numbers that converges to infinity, and let us assume that (ν^L_n__0)_n converges to a measure μ in the weak^∗ topology. For any open set U we have lim inf_n ν__0^L_n(U)≥μ(U), which implies |1/L_n·__0∩ U|≥μ(U)/2L_n^(S) for any n big enough. We deduce that if U∩(μ)≠∅ then the projection of U in (S) contains infinitely many points ofthe projection of __0. It follows that the projection of any point c∈(μ) is an accumulation point of __0 in (S). We conclude that c∈^+(S) by applying Proposition <ref>. The proposition below is the nonorientable analogue of <cit.>,it has the same proof until the use of Masur ergodic theorem (as mentioned in the remark below). Let (L_n)_n be a sequence of positive real numbers that tends to infinity with n. There exists a subsequence (L_n_i)_i such that the sequence of measures (ν^L_n_i__0)_i converges in the weak^∗ topology to a measure which is absolutely continuous with respect to the Thurston measure .We recall that, if S is orientable, then the action of (S) on (S) is ergodic with respect to(Masur <cit.>). This explains the difference between Theorem <ref> and <cit.>. The proof of <cit.> involves all the preceding results of the article. It might not be clear that they all apply to nonorientable surfaces. Let us say few words about it to convince the reader. Some results extend directly to nonorientable surfaces by lifting the situation to the orientation cover (see for instance <cit.>), but in general this does not work. The main problem occurs when looking at some limit of the formlim_L→∞1/L^ S |{∈∗ ; ℓ_m()≤ L }|(see Corollary 3.6, Theorem 1.2 and Lemma 3.5 in <cit.>). Indeed the quantitiy(S)looses its meaning when we pass to the orientation cover (it does not represent anymore the dimension of the space of measured laminations of the surface we are dealing with). Still the results concerned extend to the nonorientable setting. The idea behind them is to count multicurves using some specific graphs (train–tracks or a generalization of them called radalla). To a multicurveimmersed on such a graphτwe associate the vectorω_∈^E(τ)that gives the number of times each edge of the graph is followed. This correspondance is bounded–to–one (<cit.>) so that counting multicurves is almost counting the vectors in^E(τ)that satisfy some integral equations (the switch equations). These equations define a linear subspace whose dimension is exactly(S). This explains that the number of vectorsω_of norm less thanLis bounded by a constant timesL^(S). This kind of argument works perfectly well for nonorientable surfaces.§ PROOF OF THE THEOREMWe prove the following theorem which is more general than Theorem <ref>: Let S be a nonorientable surface of finite type with χ(S)<0. For any _0∈, and for any filling geodesic current c∈(S), we havelim_L→∞| {∈__0 ; i(c,)≤ L }|/L^(S)= 0.It is the nonorientable analogue of <cit.>. We say that a geodesic current fills the surface if every geodesic ofS̃intersects transversely a geodesic in its support. The Liouville current of a hyperbolic metric has full support, therefore it is a filling geodesic current. For our purpose, the important property is the following: ifcis a filling geodesic current, then the unit ballB_c(1)={ł∈(S) ; i(c,ł)≤ 1}is compact (see <cit.>). We havelim_L→∞| {∈__0 ; i(c,)≤ L }|/L^(S)= lim_L→∞ν__0^L(B_c(1)) We conclude by Theorem <ref> and by compacity of B_c(1). PART:Geometry of Teichmüller spaces In this last part, we study the Teichmüller and Weil–Petersson geometry of Teichmüller spaces. We focus on the volume of the moduli space, which plays a key role in the work <cit.> of Mirzakhani. We first recall the definitions of the Teichmüller metric, the Teichmüller flow and the Teichmüller volume ; we note that they extend to the nonorientable setting.§ PRELIMINARIES ON TEICHMÜLLER GEOMETRY§.§ The Teichmüller metric in the orientable settingLetTbe a smooth closed oriented surface of genusg≥ 2. §.§.§ Quadratic differentialsGiven a complex structureXonT, a quadratic differential is a tensorqwhich is locally of the formq_z=f(z)z^2withfholomorphic. The Riemannian metric|q|is flat with holonomy in{±Id}and conical singularities at the zeros ofq. Equivalently(T,|q|)is isometric to the quotient of a polygon of(,| z|^2)by a pairing of the sides realized by isometries of the formz↦± z+c(c∈). There are two preferred orthogonal geodesic foliations on(T,|q|): the horizontal and the vertical ones which are respectively given by the real1–forms(q)and(q). Each1–form defines a transverse measure on the corresponding foliation. We abusively denote by(q)and(q)these transverse measured foliations. The couple(X,q)is completely determined by the pair((q), (q))or by the polygonP⊂ (, z^2)and the pairing of its sides.§.§.§ Bundle of quadratic differentialsThe bundle(T)→(T)of isotopy classes of pairs(X,q)is called the bundle of quadratic differentials. We denote by(T;a_1,…,a_k)the subset of(T)that consists in quadratic differentials whose zeros have multiplicitiesa_1,…,a_k. From the Gauss–Bonnet formula we have-2χ(T)=a_1+…+a_k. The subset(T;a_1,…,a_k)is a smooth submanifold of dimension-2χ(T)+2k. In particular(T;1,…,1)is a dense open subset of full measure (with respect to the Lebesgue class). Let us denote byZthe set of zeros of a quadratic differentialq∈(T;1,…,1). The flat surface(T,|q|)admits a geodesic triangulation whose set of vertices isZ. Let us choose a collection of edges of the triangulation(e_1,…,e_-2χ(T))which realizes a basis of_1(T,Z;). We callq–holonomy ofe_ithe quantityhol_q(e_i)=∫_e_i√(q),which depends on the choice of a branch of√(q)and of an orientation ofe_i. Any quadratic differentialq'sufficiently close toqadmits a geodesic triangulation in the same isotopy class (relative toZ). So that one can define a continuous mapq'↦ (_q'(e_i))_ion a neighborhood ofqin(T). This map is actually a local diffeomorphism. Moreover the set of such maps forms an atlas for a piecewise linear integral structure on(T;1,…,1). Note thatqcorresponds to an integral point if and only if_q(e_i)∈⊕ ifor anyi. The induced notion of volume, called Teichmüller volume, is locally given by⋀_i (_q(e_i))∧(_q(e_i)).§.§.§ From quadratic differentials to measured laminationsThe following map is a(S)–invariant homeomorphism[(T)⟶(T)×(T)-Δ;q⟼ ((q), (q)) ]whereΔ={(ł,η) ; i(,ł)+i(,η)=0 for some ∈(T)}.Mirzakhani (<cit.>) showed that the restriction of this map to(T;1,…,1)is a piecewise integral linear isomorphism, in particular it preserves the volume. Note that|_q(e_i)|=i(e_i,(q))and|_q(e_i)|=i(e_i,(q)). In the sequel we adopt the point of view of measured laminations jusitified by the above isomorphism. One advantage of this point of view is that all notions extends immediately to all surfaces of finite type with negative Euler characteristic. §.§.§ The Teichmüller metric There is a canonical identification between(T)and the cotangent bundle of(T), whereas the bundle of Beltrami differentials onTidentifies with the tangent bundle of(T). The pairing between Beltrami and quadratic differentials induces an isomorphism between the corresponding bundles. Therefore one can define a Finsler metric on(T)through(T). The Teichmüller metric is the Finsler metric on(T)defined by the normq =∫_T |q|=area(q)=i((q),(q)).We set(T)={q∈(T) ; q=1}.TheTeichmüller flow is the geodesic flow of the Teichmüller metric. In terms of polygons, the Teichmüller geodesic passing throughq∈(T)att=0is given byt↦[e^t0;0 e^-t ]· Pfor all t∈,wherePis a polygon that representsq. In terms of measured foliations, the same Teichmüller geodesic is given byt⟼ (e^t(q),e^-t(q))for allt∈.The trajectory of the Teichmüller horocyclic flow passing throughq∈(T)att=0is given byt↦[ 1 t; 0 1 ]· Pfor all t∈,In terms of measured laminations, the Teichmüller horocyclic flow corresponds to the earthquake flow (see <cit.>). §.§ Definitions in the nonorientable setting LetSbe a closed nonorientable surface withχ(S)<0. Then the bundle of quadratic differentials(S)is the set of isotopy classes of pairs(X,q)whereXis a dianalytic structure onSandqa quadratic differential with respect toX. We recall that a dianalytic structure is given by an atlas whose changes of charts are holomorphic or anti–holomorphic.§.§.§ The flat surface (X,|q|)It is isometric to the quotient of a polygonP⊂(,|dz|^2)by a pairing of the sides realized by isometries of the formz↦± z+corz↦±z̅+cwithc∈. Thus the holonomy is not in{±Id}anymore, and there are geodesics (possibly closed) with self–intersections outside the singularities. However the holonomy preserves the horizontal and vertical directions, thus the horizontal and vertical geodesics do not self–intersect outside the singularities, and there are two measured foliations(q)and(q).§.§.§ The Teichmüller flow The linear part of the isometryz↦±z̅+cis diagonal, thus it commutes with the linear map(x,y)↦ (e^t,e^-t). It follows that the identifications of the sides of the polygon[e^t0;0 e^-t ]· P⊂ (, z^2)are of the formz↦± z+corz↦±z̅+c. Therefore the polygon and the pairing of the sides determine a quadratic differential onS. This shows that the Teichmüller flow is well–defined on(S). It is still given byt↦ (e^t(q),e^-t(q))in terms of measured laminations.§.§.§ The horocyclic flow On the contrary there is no Teichmüller horocyclic flow on(S). Indeed, the conjugate ofz↦±z̅ +cby(x,y)↦ (x+ty,y)is not an isometry whenevert≠ 0. As well–known, the Teichmüller space of the Klein bottle can be identified with the geodesici^∗_+of, wherehas to be understood as the Teichmüller space of the torus equipped with its Teichmüller metric. Clearly the horocyclic flow ofdoes not stabilizei_+^∗.§.§.§ Volume and piecewise integral linear structureThese structures extends readily to the nonorientable setting. As in the orientable case, it is possible to define them through the holonomy of quadratic differentials, or through the real and imaginary measured foliations. §.§.§ The orientation cover point of viewLet us denote bySthe orientation cover ofS, and byits automorphism. Any quadratic differential(X,q)onSlifts to a quadratic differential(X,q̂)onS. The automorphismchangesq̂into its conjugate. The mapq↦q̂identifies(S)with the fixed–point locus()⊂(S)of the mapping class[]. The bundle structure of(S)is the one induced by(S)on(). §.§ Volume on (S) The moduli space (S) of quadratic differentials on S is the quotient(S)/(S). Similarly the moduli space of unit area quadratic differentials on S is(S)=(S)/(S). We define a Borel measure_1on(S)as follows: we set_1(U)=((0,1)· U)for any measurableU⊂^1(S)whereis the Teichmüller volume on(S). Since the action of(S)on(S)is proper and discontinuous, the measure_1induces a measure on(S)still denoted by_1. § TEICHMÜLLER VOLUMEIn this section we prove the following theorem: Let S be a nonorientable surface of finite type with χ(S)<0. The moduli space (S) of unit area quadratic differentials on S has infinite volume. Actually we are going to prove a more precise result:Ifand $̣ are two maximal one–sided simple multicurves that fillS. Then the projection of()×()̣in(S)has infinite volume. A multicurve is one–sided if each of its components is one–sided. A one–sided simple multicurve=_1+…+_nis maximal if its complementS-is orientable. Two simple multicurvesand$̣fillS if i(,η)+i(,̣η)>0 for any simple closed geodesic η. Note thatand $̣ can not share a component. The projection π:(S)→(S) is a covering with negligible ramification locus. Its restriction ()×()̣→π(()×()̣) is a ramified covering of finite degree d≥ 1 (Lemma <ref>). Thus the volume of the projection of ()×()̣ on (S) is 1/d·({(ł,μ)∈()×()̣ ; i(,)̣≤ 1}). But this quantity is infinite (Lemma <ref>).Let us denote byπthe projectionπ:(S)→(S). This is a ramified covering since the action of(S)on(S)is a proper and discontinuous. Here ramified means that, in the neighborhood of a singular point, the projectionπis an étale covering up to the action of a finite group.The restriction of π to ()×()̣→π(()×()̣) is a ramified covering of finite degree.Two elements in ()×()̣ have same image in (S) if and only if they belong to the same (S)–orbit. Thus it suffices to show that the number of f∈(S) such that f(())∩()≠∅ and f(()̣)∩()̣≠∅ is finite.Any f∈(S) satisfies the alternative f()= or i(f(),)≠ 0, and the similar alternative with $̣. This is an easy consequence of the fact thatand$̣ are maximal among families of disjoint one–sided simple closed geodesics. We deduce that if f∈(S) identifies two elements in ()×()̣, then f fixesand $̣.The complementS-(∪)̣consists in a finite number of finite sided polygons that may have a puncture or a hole. If there is a hole then the boundary of the polygon is a component of∂ S. The number of such polygons is finite. We deduce that the number off∈(S)fixingand$̣ is finite. In view of the above alternative, this is equivalent to say that the number of f∈(S) such that f(())∩()≠∅ and f(()̣)∩()̣≠∅ is finite.The volume of {(ł,μ)∈()×()̣ ; i(,)̣≤ 1} is infinite.We setV ={v ∈()̣ ; i(,v)< 1/2},U ={u ∈(S-) ; sup_v∈ V i(u,v)<1/2},and W_t = (t· (+U))× (t^-1· V),W =⋃_t>0 W_t.We note that W⊂()×()̣, and that the union is disjoint (consider the weight ofon the first component).By construction, any (ł,μ)∈ W satisfies i(ł,μ)<1. Thus it suffices to show that W has infinite volume to prove the lemma. Using the splitting of the Thurston measure (§<ref>) we find(W) =∫_0^+∞(W_t)dt=(U) (V) ∫_0^+∞ t^-n d t =+∞, where n>0 is the number of components of . We still have to show that W is measurable, to do this it suffices to prove that U and V are open. The set V is open by continuity of i(,·). It is also relatively compact in (S)∪{0} sinceand $̣ fill upS(use <cit.>). Now we prove thatUis a nonempty relatively compact open subset of(S-)∪{0}. Let us picku∈(S-), the supremumt_u=sup_v∈ V i(u,v)is finite by relative compactness ofV, so that(3t_u)^-1u ∈ U. This shows thatUis nonempty. Clearly1/3i(,)̣∈̣V, thusi(u,)̣/3i(,)̣≤sup_v∈ V i(u,v)andi(u,)+i(u,)̣=i(u,)̣< 3i(,)̣/2for anyu∈ U. This implies thatUis relatively compact in(S-)∪{0}sinceand$̣ fill up S. It remains to show that U is open. Let K be a compact subset of (S)∪{0} whose interior contains U. We consider the family {u↦ i(u,v)}_v∈ V of continuous functions from K to . These functions are L–Lipschitz continuous for some constant L that depends on V. Indeed the intersection function i:(S)×(S)→_+ is Lipschitz (Rees <cit.>, Luo and Stong <cit.>) with respect to some natural metrics (for instance they use the norm of the Dehn–Thurston coordinates in <cit.>). According to Ascoli's theorem, the family {u↦ i(u,v)}_v∈ V is relatively compact in C(K,) equipped with the uniform norm, therefore u↦sup_v∈ V i(u,v) is continuous over K. We conclude immediately that U is open.Let S be a closed nonorientable surface with χ(S)<0. Then the Teichmüller geodesic flow on (S) is not topologically transitive, in particular is not ergodic with respect to any Borel measure of full support.We identifiy (S) with {(ł,μ)∈(S)×(S) ; i(ł,μ)=1}. We write ł and μ in the form ł=ł^-+ł^+ and μ=μ^-+μ^+ where ł^+,μ^+∈^+(S) and ł^-,μ^- are one–sided simple closed multicurves. It suffices to remark that the four continuous functions (ł,μ)↦ i(ł^±,μ^±) over (S)×(S) are invariant under the Teichmüller flow but not globally constant.§ WEIL–PETERSSON VOLUME§.§ The orientable case LetTbe a closed oriented surface withχ(T)<0. Its Teichmüller space(T)admits a(T)–invariant Kählerian metric called the Weil–Petersson metric. Wolpert showed that the Fenchel–Nielsen coordinates are Darboux coordinates for the Weil–Petersson symplectic formω_WP. More precisely, given any pants decomposition={_1,…,_n}ofS, we have (see <cit.>):ω_WP=∑_i=1^nd τ_i∧ d ℓ_i,where(τ_1,ℓ_1,…,τ_n,ℓ_n)are the Fenchel–Nielsen coordinates associated to. More generally the formula (<ref>) defines a(T)–invariant symplectic form on(T)for any compact oriented surfaceTwithχ(T)<0.Letν_WPbe the(T)–invariant volume form which is then–th exterior power ofω_WP. In the Fenchel–Nielsen coordinates we have:ν_WP =⋀_i=1,…,n d τ_i ∧ d ℓ_i.The Weil–Petersson volume of the moduli space(T)is finite.Mirzakhani (<cit.>) found a recursive formula forν_WP((T))when∂ T≠∅, and deduced thatν_WP((T))is a polynomial of degree(T)in the lengths of the boundary components. She also gave a formula for the average∫_(T) F(x) ν_WPof a function of the formF=∑_f∈(T)ℓ_f·_0where_0is a simple closed geodesic (<cit.>) . This is a key ingredient in the determination of the asymptotic of the number of simple closed geodesic of length less thanL(Step 3 in the proof of <cit.>). §.§ The Teichmüller space as a Lagragian submanifoldLetSbe a compact nonorientable surface withχ(S)<0. We denote bySits orientation cover whose automorphism is:S→S. Note thatis an orientation reversing involution. The mapm↦m̂, that consists in lifting a hyperbolic metric onStoS, induces a diffeomorphism between the Teichmüller space(S)and()⊂(S), the fixed–point locus of the mapping class[]. Letbe a pants decomposition which is stabilized by[](such that there is a permutationof{1,… , n}satisfying[]·_i=_(i)for anyi=1,…, n). Asreverses orientation we have[]^∗τ_i=-τ_(i), therefore[]^∗ω_WP=-ω_WP. So the fixed–point locus()is a Lagrangian submanifold of((S),ω_WP). Let us denote byπthe fundamental group ofS. As well–known, there is a(S)–equivariant diffeomorphism between(S)and(π,(2,)), the space of faithfull and discrete representationsρ:π→(2,)up to conjugacy. Moreover, the tangent spaceT_[ρ](π,(2,))identifies with^1(π,𝗌𝗅(2,)), where𝗌𝗅(2,)is the Lie algebra of(2,)seen as aπ–module with respect to the action∘ρ. Goldman (<cit.>) showed that the image ofωthrough the diffeomorphism(S)→(π,(2,))is a multiple of the symplectic form^1(π,𝗌𝗅(2,))×^1(π,𝗌𝗅(2,))→^2(π,)≃,induced by the cup product together with the Killing form of𝗌𝗅(2,). The set of representations invariant under the action of[]forms a submanifold of(π,(2,))whose tangent space at[ρ]is the set of fixed points of_∗in^1(π,𝗌𝗅(2,)). By naturality of the cup product, and because_∗sends the fundamental class to its opposite, it comes that_∗is an anti–symplectomorphism, so its set of fixed points is a Lagrangian submanifold. §.§ Norbury's volume formFrom the above paragraph, it seems rather difficult to find a(S)–invariant symplectic form on(S)(or maybe another structure) that generalizesω_WP. Note also that the dimension of(S)can be odd. However it might possible to find a volume form that generalizesν_WP. Indeed, Norbury (<cit.>) suggested the following:ν_N =(⋀__ione–sided(ℓ_i) ℓ_i)∧( ⋀__itwo–sidedτ_i ∧ d ℓ_i ).where{_1,…,_n}is a pants decomposition ofS. He showed that the Norbury's volume formν_Nis(S)–invariant up to sign, in particular its absolute value is a(S)–invariant measure.One can wonder in what respectν_nis a generalization ofν_WP. What guided Norbury is the formula (<ref>), and he was looking for a measure given by a similar formula. In the next paragraph we provide a more conceptual justification.§.§ A characterization by mean of the twist flowLetTbe compact oriented surface withχ(T)<0. The twist flow is simply defined by twisting a hyperbolic metric along a simple closed geodesic. It comes directly from the formula (<ref>) of Wolpert thatω_WP(∂/∂τ_,·) =ℓ_for any simple closed geodesic. Hereτ_is the vector field associated to the twist flow in the direction. By definition, this means that the twist flow is the Hamiltonian flow ofℓ_. In particular the twist flow is volume preserving. All these considerations extend to the earthquake flow, and also to the shearing flow (<cit.>). It is worth mentioning that the twist flow can be defined on spaces of representations (see <cit.> and <cit.>). LetSbe a nonorientable surface of finite type withχ(S)<0. In that case, the twist flow is only defined for two–sided simple closed curves, and the earthquake flow is only defined for measured laminations that are contained in some orientable subsurface. Moreover these flows are well–defined up to sign. From the discussion above, it appears that the twist flow — and more generally the earthquake flow — is a canonical volume preserving flow. So we logically look for a characterization of Norbury and Weil–Petersson volume forms in terms of the twist flow. Let S be a surface of finite type with χ(S)<0. If S is orientable, then ν_WPis the unique (up to a multiplicative constant) volume form on (S) invariant under the twist flow. If S is nonorientable with χ(S)<-1, then ν_N is the unique (up to a multiplicative constant) volume form on (S) invariant under the twist flow.Let S be a nonorientable surface with χ(S)=-1. Then there exists a simple closed geodesicsuch that i(,)̣=0 for any two–sided simple closed geodesic $̣. In particularℓ_$̣ is constant along the trajectories of the twist flow. This shows that the statement is false for these surfaces. From the defining formulas, we see that ν_WP and ν_N are invariant under the twist flow. So the problem is to show their uniqueness.We consider a volume form ν on (S) invariant under the twist flow. We write ν=fν_WP or ν=fν_N, where f:(S)→ is smooth and invariant under the twist flow. We first treat the case S orientable. We note that f is invariant under the earthquake flow. This comes directly from the following facts: f is continuous, the earthquake flow (S)×(S)→(S) is continuous, and multiples of simple closed geodesics are dense in (S).Then, we apply a classical theorem of Thurston (see <cit.>) which states that any two points in (S) are related by an earthquake path, and we conclude that f is constant on (S). Now we assume S nonorientable. We prove below that any two points in (S) are related by a combination of at most two earthquake paths (with respect to measured laminations contained in orientable subsurfaces), this finishes the proof.Letbe a simple closed geodesic of S whose complement is orientable. Let $̣ be a two–sided simple closed geodesic that intersects(such a geodesic exists becauseχ(S)<-2). By twisting along$̣ we can increase arbitrarily the length of , and thus join any two fibers of ℓ_. Any fiber of ℓ_ is canonically identified to the Teichmüller space of S-, thus we can apply the theorem of Thurston mentioned above to join any two points on the same fiber. It would be interesting to have other characterizations ofν_N, for instance in terms of spaces of representations (see <cit.> for representations of nonorientable surface groups into compact Lie groups) or shearing coordinates (see <cit.> for the relations between various notions of volumes on moduli spaces).§.§ Infinite volumeThe aim of this paragraph is to explain the following theorem due to Norbury (<cit.>). The ideas contained in this paragraph are also known to Yi Huang.[Norbury] If S is a nonorientable surface of finite type with χ(S)<0, then ν_N((S)) is infinite. Norbury's proof mimic Mirzakhani's computation of Weil–Petersson volumes, in particular it relies on an identity à la McShane. So it does not really explain why the volume is infinite. Here we determine rather precisely which part of(S)has infinite volume. The first step of the proof is similar to Lemma <ref>, that is we work with a finite covering. Our proof goes as follows: we consider the subset of(S)that consists in hyperbolic surfaces having a short one–sided multicurve=_1+…+_nwhose complementS-is orientable, we show that this subset fibers in Teichmüller spaces(S-)of orientable subsurfaces. We know the volume of the fibers thanks to Mirzakhani's results, and the transverse measure is obviously given by(ℓ_1)⋯(ℓ_n)ℓ_1⋯ℓ_n, so that we can explicitely compute the volume.Let =_1+…+_n be a maximal family of disjoint one–sided simple closed geodesics. Note that S- is orientable. For any >0 we denote by U_() the projection in (S) of the open subsetU_() ={[m]∈(S) ; ℓ__i(m)< fori=1,…,n }.By Lemma <ref> the volume ν_N( U_()) is infinite.For >0 small enough, U_()/(S-) is a finite ramified covering of U_().By the Collar Lemma (see <cit.>), for any point in U_() the set of one–sided simple closed geodesics of length less thanis precisely . This implies that an element of (S) that preserves U_() stabilizes . Thus the subgroup of (S) that preserves U_() is ^∗(S-) and U_()=U_()/^∗(S-). But (S-) is a subgroup of finite index of ^∗(S-). So U_()/(S-) is a finite covering of U_().The ν_N–volume of U_() is infinite.According to the above lemma, it suffices to show that U_()/(S-) has infinite volume.As (S-) acts trivially on , the family of length functions (ℓ__1,…,ℓ__n) induces a map L:U_()/(S-)→(0,)^n. Each fiber L^-1(x) identifies canonically with (S-,x) whose volume V_S-(x) is a polynomial of degree (S-) in x (Mirzakhani <cit.>). We find (U_())/(S-)) = ∫_(0,)^n(x_1)⋯(x_n) V_S-(x)dx= +∞,which concludes the proof. It is interesting to note that if=_1+…+_nand=̣_̣1+…+_̣mare maximal families of disjoint one–sided simple closed geodesics, then for anysmall enough either U_()=U_(̣)or U_()∩ U_(̣)=∅, the first case occurring if and only ifand$̣ have same topological type (i.e. are in the same (S)–orbit). This fact is an obvious consequence of the Collar Lemma. One can easily compute the number of such topological types which is the integer part of (g+1)/2 where g is the genus of S. § A FINITE VOLUME DEFORMATION RETRACT We call systole of one–sided geodesics the length of the shortest one–sided simple closed geodesic. This metric invariant defines a continuous and (S)–invariant function ^-:(S)→_+^∗. In this section we study the (S)–invariant subset^-_(S) ={[m]∈(S) ; ^-(m)≥}and its quotient _^-(S)=_^-(S)/(S). §.§ Noncompact subsets of finite volumeFor any >0 the subset _^-(S) has finite ν_N–volume. It is noncompact ifis sufficiently small and if S is not the two–holed projective plane.Using standard methods (see <cit.>), one easily shows that there exists a constant B(S), called the Bers' constant, such that any point in (S) admits a pants decomposition whose components have length at most by B(S). Let us recall that, in our definition of (S), we assume that the lengths of the boundary components of S are fixed.Given a pants decomposition ={_1,…,_n}, we denote by A_()⊂(S) the subset defined by the following inequalities in the Fenchel–Nielsen coordinates associated to : {[0≤τ_i≤ℓ_iand ℓ_i≤ B(S) for alli=1,…, n; ℓ_i≥ for alli=1,…,n such that _iis one–sided. ]. We observe immediately that A_() has finite ν_N–volume.Let us denote by A_() the projection of A_() in (S). The existence of the Bers' constant implies that any point in _^-(S) belongs to some A_() for some pants decomposition . But A_() depends only on the topological type of , and there are finitely many topological types of pants decomposition. So we can cover _^-(S) by a finite number of subsets of the form A_(). We conclude that _^-(S) has finite ν_N–volume since every A_() has finite ν_N–volume.If S is not the two–holed projective plane, then S has a pants decomposition with a two–sided component. By pinching the two–sided component while keeping the lengths of the other components equal to 2 we leave any compact in (S) (Mumford's compactness criterion) but remain in _^-(S) (Collar Lemma). This proves the second assertion.§.§ Quasi–convexityIn this paragraph we just ask the following natural question (see §<ref> for some motivations): Is _^-(S) quasi–convex with respect to the Teichmüller metric ? We believe that Minsky's product theorem (<cit.>) gives some evidence to a positive answer. It provides an approximation of the Teichmüller distance when the systole is small, but not an approximation of the Finsler metric, so we were not able to conclude. §.§ A retraction Let us fix >0 small enough in the sense that two closed geodesics of length at mostcan not intersect in any hyperbolic surface (independently of the topology). For >0 small enough there is a (S)–invariant (strong) deformation retraction of (S) onto ^-_(S).It is well–known that forsmall enough {≥} is a (S)–invariant deformation retract of (S) (see <cit.> for a proof and a discussion of its relation with the well–rounded retract). The statement above is a little bit different since we deal with ^- and not . Our deformation retraction follows the flow of a well–chosen vector field that increases the length of the geodesics realizing ^-. This is rather classical, the additional difficulty is to make sure that the surface does not degenerate before we reach ^-_(S). To do so we need to control also the length of the two–sided geodesics. This is done using strip deformations. Let us introduce some notations, for short we drop the (S) in all notations involving (S). We denote by ^≤ n_ the set of X∈ which have at most n≥ 0 one–sided geodesics of length less than . We have ^≤ 0_=^-_ and ^≤ n_= for any n greater or equal to the genus g of S. Given a one–sided simple multicurve =_1+…+_n we denote by ^_ the set of X∈ whose one–sided geodesics of length less thanare exactly _1,…,_n. For short we denote by ℓ_i the length function ℓ__i. The subsets ^_ are pairwise disjoint.The strategy of the proof is to decrease the maximal number of one–sided simple closed geodesics of length less than . We successively retract =^≤ g_⟶_^≤ g-1⟶…⟶^≤ 1_⟶^≤ 0_=^-_The construction of the retraction ^≤ n_⟶_^≤ n-1 is based on the following statement (Lemma <ref>) which is the heart of the proof: given a one–sided multicurve =_1+…+_n there exists a ^∗(S-)–invariant retraction R^:^≤ n_⟶^≤ n_-^_.Once we have R^ we get for free a (S)–invariant retractionR^[]:^≤ n_→^≤ n_-(∪_f∈(S)^f()_),defined by {[ R^[](X)=f(R^(X)) if X∈_^f() for some f∈(S),;R^[](X)=X otherwise. ]. Note that R^[] does not depend on the particular f∈(S) such that X∈^f()_ since R^ is ^∗(S-)–invariant. The continuity of R^[] follows directly from the continuity of R^ and the relative positions of the subsets ^f()_, namely they are pairwise disjoint and relatively open in ^≤ n (Lemma <ref>). Now we conclude easily: we perform the retraction R^[] for each topological type [] of one–sided multicurve with n components (there are finitely many such topological types), this gives the expected retraction ^≤ n_⟶_^≤ n-1. It can be checked that it is a deformation retraction like R^ (Lemma <ref>).Let =_1+…+_n be a one–sided multicurve. The set ^_ is open in _^≤ n.We want to show that any X_0∈^_ has a neighborhood U insuch that U∩^_=U∩^≤ n_. If X_0 belongs to the interior of ^_, then one obviously takes U=^_. So we assume that X_0∈^_ belongs to the frontier ∂^_ of ^_ in . Let U be a neighborhood of X_0 insuch that ℓ_i< on U for any i. Then U-^_ is the set of X∈ U such that ℓ_(X)< for some one–sided geodesic $̣ which is not a component of. In particularU-^_⊂ U-^≤ n_. We deduce thatU∩^_=U∩^≤ n_, which is what we wanted to show.Given a one–sided multicurve =_1+…+_n there exists a ^∗(S-)–invariant (strong) deformation retraction R^:^≤ n_⟶^≤ n_-^_. Let us completeinto a pants decomposition . Let v^ be the vector field on (S) given by v^=∂/∂ℓ_1+…+∂/∂ℓ_n in the Fenchel–Nielsen coordinates associated to . Actually the twist coordinates are not canonically associated to a pants decomposition, but this does not matter here. It obviously satisfiesℓ_i(v^) = 1(i=1,…,n).Its flow ϕ is explicitely given by ϕ(X;t)=X+(t,…,t,0,…,0) with respect to the linear structure of the Fenchel–Nielsen coordinates. Note that ϕ is complete in positive time. The flow ϕ can be realized as a strip deformation. This is a construction of Thurston which has been studied in details in <cit.> (see also <cit.>). Given a surface with boundary, a strip deformation is parametrized by a system of arcs {_1,…,_k} together with apoint p_j∈_j and a width w_j>0 for any 1≤ j≤ k. For any hyperbolic metric m, the time 1 strip deformation of m in the direction (,p,w) consists in cutting each _j, and inserting a strip of width w_j in such a way that the geodesic segment contained in the strip and joining the points identified to p_j is orthogonal to the boundary components of the strip. Actually this construction is realized in the Nielsen extension of the surface. We refer to <cit.> for a more precise description. In our case, the surface with boundary is S-, the arcs _j's are the common perpendicular between the _i's with respect to the pants decomposition , the points p_j's are the midpoints of the common perpendiculars, the widths w_j's are chosen so that (<ref>) is realized (w_j depends on the metric). This point to view has the advantage that one can easily control the variation of the length of any closed geodesic $̣ contained inS-. Looking at the trace of$̣ in each pair of pants of S- we find (see also <cit.>) 0 ≤ ℓ_(̣v^)≤i(,̣_1)+…+i(,̣_k).We observe that ℓ_(v^) is uniformly controlled over . This gives another proof of the completeness of ϕ in positive time. We will use this argument later.Unfortunately ϕ is not (S-)^∗–invariant. To fix this problem we endowwith the Teichmüller distance d_, and choose a smooth function ψ:→ [0,1] with ψ(0)=1 and ψ≡ 0 outside [0,1]. Then we set V_X =∑_f∈^∗(S-)ψ(d_(X,f(X))) · v^f()_X.The sum is finite at any point X∈(S) since d_ si complete (closed balls are then compact) and (S) acts properly. So the vector field V is well–defined and locally Lipschitz, in particular integrable. It is moreover ^∗(S-)–invariant:V_g(X)=∑_f∈^∗(S-)ψ(d_(g(X),f(X)))v^f()_g(X), =∑_f∈^∗(S-)ψ(d_(X,g^-1f(X))) g_X(v^g^-1f()_X), =∑_h∈^∗(S-)ψ(d_(X,h(X))) g_X(v^h()_X), = g_X(V_X).We have used the equality g_X(v^g^-1f()_X)=v^f()_g(X) which comes from the fact that the Fenchel–Nielsen coordinates of g(X) with respect to f() are equal (up to some additive constants) to the Fenchel–Nielsen of X with respect to g^-1f(). The additive constants are due to the fact that the twist coordinates are not canonically associated to a pants decomposition.Let Φ be the flow of the vector field V, we claim that Φ is complete in positive time.This is equivalent to say that, for any simple closed geodesic $̣ and anyX∈, the lengthℓ_(̣Φ(X;t))can not tend to zero or infinity in a positive finite time. From equations (<ref>) and (<ref>) and1≥ψ≥ 0we deduce thatΦ(X,t)can not degenerate in a positive finite time. Using the flowΦwe construct the retractionR^of the statement. For anyX∈we sett_X=inf{t≥ 0 ; Φ(X;t)∉^_}. This number is finite since the lengths of the_i's increase at least linearly (equation (<ref>) andψ(0)=1). MoreoverX↦ t_Xis continuous sincet_X=inf{t≥ 0 ; ℓ_i(Φ(X;t))≥ for any i=1,… n}(the lengths of the geodesics disjoint fromincrease). We conclude that the mapX↦Φ(X;t_X)gives the expected deformation retraction. §.§ The frontier of ^-_(S)For >0 small enough, the frontier of ^-_(S) in (S) is homotopy equivalent to the geometric realization of ^-(X).*The frontier of ^-_(S) is simply {^-=}.*As well–known {=} is homotopy equivalent to the complex of curves (S) (see <cit.>).We use the standard idea, that is we construct a cover of {^-=} by contractible open subsets(i.e a good cover) whose nerve is isomorphic to ^-(S). Then we conclude (using <cit.>) that {^-=} is homotopy equivalent to ^-(S). Note that {^-=} is homeomorphic to a CW–complex.Given a one–sided multicurve =_1+…+_n we denote by U_ the set of X∈{^-=} such that ℓ_(̣X)> for any one–sided geodesic $̣ disjoint from. AlternativelyU_={^-=}∩{min_ℓ_>̣}where$̣ runs over the set of one–sided geodesics disjoint from . In this form we see that U_ is relatively open in {-=} since min_ℓ̣_$̣ is continuous. Let us show thatU_is also contractible. The open subset{min_ℓ̣_>̣}⊂(where$̣ should be taken as previously) is contractible. Actually one can prove that {min_ℓ̣_≥̣2} is a deformation retract of bothand {min_ℓ̣_>̣}. We realize the deformation retractions through the flow of a well–chosen vector field, as in the proof of Lemma <ref>, but in a much simpler way since we do not require the (S)–invariance.To prove that U_ is contractible we show that it is a deformation retract of {min_ℓ̣_>̣}. We proceed as in beginning of the proof of Lemma <ref>. We completeinto a pants decomposition . Then we consider the flow ϕ of the vector field v^=∂/∂ℓ_1+…+∂/∂ℓ_n. As we have seen, this flow increases the length of all geodesics disjoint from , in particular it preserves the open set{min_ℓ̣_>̣}. We sett_X=min{t≥ 0 ; ℓ_i(ϕ(X,t))≥ fori=1,…, n}. Clearly X↦ t_X is a well–defined and continuous function, so (X,s) ↦ϕ(X, st_X) is a deformation retraction. This shows that U_ is contractible. The family {U_ ; is a one–sided muticurve} is a good open cover of ^-_, its nerve is isomorphic to ^-(S) due to the equality U_∩ U_'=U_∩' for any simple one–sided multicurvesand '. So ^-_ is homotopy equivalent to ^- (S).*alpha | http://arxiv.org/abs/1706.08798v1 | {
"authors": [
"Matthieu Gendulphe"
],
"categories": [
"math.GT"
],
"primary_category": "math.GT",
"published": "20170627115448",
"title": "What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces"
} |
]Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat spaceG. Khakimzyanov]Gayaz Khakimzyanov G. Khakimzyanov: Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia [email protected]. Dutykh]Denys Dutykh^* D. Dutykh: LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, F-73376 Le Bourget-du-Lac Cedex, France [email protected] http://www.denys-dutykh.com/ ^* Corresponding authorZ. I. Fedotova]Zinaida Fedotova Z. I. Fedotova: Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia [email protected]. Mitsotakis]Dimitrios Mitsotakis D. Mitsotakis: Victoria University of Wellington, School of Mathematics, Statistics and Operations Research, PO Box 600, Wellington 6140, New Zealand [email protected] http://dmitsot.googlepages.com/ emptyGayaz KhakimzyanovInstitute of Computational Technologies, Novosibirsk, Russia Denys DutykhCNRS–LAMA, Université Savoie Mont Blanc, France Zinaida FedotovaInstitute of Computational Technologies, Novosibirsk, Russia Dimitrios MitsotakisVictoria University of Wellington, New Zealand[t]1.0Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat spaceemptyLast modified: December 30, 2023[ [ December 30, 2023 =====================emptyIn this paper we review the history and current state-of-the-art in modelling of long nonlinear dispersive waves. For the sake of conciseness of this review we omit the unidirectional models and focus especially on some classical and improved Boussinesq-type and Serre–Green–Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.: long wave approximation; nonlinear dispersive waves; shallow water equations; solitary wavesMSC: [2010] 76B15 (primary), 76B25 (secondary) PACS: [2010] 47.35.Bb (primary), 47.35.Fg (secondary)empty§ INTRODUCTION The history of nonlinear dispersive modelling goes back to the end of the XIXth century <cit.>. At that time J. Boussinesq (1877) <cit.> proposed (in a footnote on page 360) the celebrated Korteweg–de Vries equation, re-derived later by D. Korteweg & G. de Vries (1895) <cit.>. Of course, J. Boussinesq proposed also the first Boussinesq-type equation <cit.> as a theoretical explanation of solitary waves observed earlier by J. Russell (1845) <cit.>. After this initial active period there was a break in this field until 1950's. The silence was interrupted by the new generation of `pioneers' — F. Serre (1953) <cit.>, C.C. Mei & Le Méhauté (1966) <cit.> and D. Peregrine (1967) <cit.> who derived modern nonlinear dispersive wave models. After this time the modern period started, which can be characterized by the proliferation of journal publications and it is much more difficult to keep track of these records. Subsequent developments can be conventionally divided in two classes: * Application and critical analysis of existing models in new (and often more complex) situations* Development of new high-fidelity physical approximate modelsSometimes both points can be improved in the same publication. We would like to mention that according to our knowledge the first applications of Peregrine's model <cit.> to three-dimensional practical problems were reported in <cit.>.In parallel, scalar model equations have been developed. They describe the unidirectional wave propagation <cit.>. For instance, after the above-mentioned KdV equation, its regularized version was proposed first by Peregrine (1966) <cit.>, then by Benjamin, Bona & Mahony (1972) <cit.>. Now this equation is referred to as the Regularized Long Wave (RLW) or Benjamin–Bona–Mahony (BBM) equation. In <cit.> the well-posedness of RLW/BBM equation in the sense of J. Hadamard was proven as well. Even earlier Whitham (1967) <cit.> proposed a model equation which possesses the dispersion relation of the full Euler equations (it was constructed in an ad-hoc manner to possess this property). It turned out to be an excellent approximation to the Euler equations in certain regimes <cit.>. Between unidirectional and bi-directional models there is an intermediate level of scalar equations with second order derivatives in time. Such an intermediate model was proposed, for example, in <cit.>. Historically, the first Boussinesq-type equation proposed by J. Boussinesq <cit.> was in this form as well. The main advantage of these models is their simplicity on one hand, and the ability of providing good quantitative predictions on the other hand.One possible classification of existing nonlinear dispersive wave models can be made upon the choice of the horizontal velocity variable. Two popular choices were suggested in <cit.>. Namely, one can use the depth-averaged velocity variable (see <cit.>). Usually, such models enjoy nice mathematical properties such as the exact mass conservation equation. The second choice consists in taking the trace of the velocity on a surface defined in the fluid bulk y = (, t). Notice, that surface (, t) may eventually coincide with the free surface <cit.> or with the bottom <cit.>. This technique was used for the derivation of several Boussinesq type systems with flat bottom, initially in <cit.> and later in <cit.> and analysed thoroughly theoretically and numerically in <cit.>. Sometime the choice of the surface is made in order to obtain a model with improved dispersion characteristics <cit.>. One of the most popular model of this class is due to O. Nwogu (1993) <cit.> who proposed to use the horizontal velocity defined at y = ()βh() with β ≈ 0.531. This result was improved in <cit.> to β ≈ 0.555 (taking into consideration the shoaling effects as well). However, it was shown later that this theoretical `improvement' is immaterial when it comes to the description of real sea states <cit.>.Later, other choices of surface (, t) were proposed. For example, in <cit.> the surface (, t) was chosen to be genuinely unsteady (due to the free surface and/or bottom motion). This choice was motivated by improving also the nonlinear characteristics of the model. Some other attempts can be found in <cit.>. On the good side of these models we can mention accurate approximation of the dispersion relation up to intermediate depths and in some cases good well-posedness results. On the other side, equations are often cumbersome with unclear mathematical properties (well-posedness, existence of travelling waves, ). Below we shall discuss more closely some of the models of this type.For another recent complementary review of Boussinesq-type and other nonlinear dispersive models, which discusses also applications and some numerical approaches we refer to <cit.> and for a detailed analysis of the theory and asymptotics for the water-wave problem we refer to <cit.>.This manuscript is the first part in a series of four papers (the other parts are <cit.>). Here we attempt to make a literature review on the topic of nonlinear weakly dispersive wave modelling in shallow water environments. This topic is so broad that we apologize in advance if we forgot to mention someone's work. It was not made on purpose. Moreover, we propose a unified modeling framework which encompasses some more or less known models in this field. Namely, we show how several well-known models can be derived from the base model by making judicious choices of dynamic variables and/or their fluxes. We also try to point out some important properties of some model equations that have not attracted so much the attention of the researchers. The second part will be devoted to some numerical questions <cit.>. More precisely, we shall propose an adaptive finite volume discretization of a particular widely used dispersive wave model. The numerical method adaptivity is achieved by moving grid points to the locations where it is needed. The title of the first two parts include the wording `on a globally flat space'. It means basically that we consider a fluid flow with free surface on a Cartesian space, even if some bathymetry variations[The amount of bathymetry variations allowed in our modelling will be discussed in the second part <cit.> of this series.] are allowed, the bottom is not necessarily flat. The (globally) spherical geometries will be discussed in some detail in Parts III & IV <cit.>.The present article is organized as follows: In Section <ref> we derive the base model. However, the derivation procedure is quite general and it can be used to derive many other particular models, some of them being well-known and some possibly new. In Section <ref> we propose also a weakly nonlinear version of the base model. Finally, in Section <ref> we outline the main conclusions and perspectives of the present study.§ BASE MODEL DERIVATIONFirst of all we describe the physical problem formulation along with underlying constitutive assumptions. Later on this formulation will be further simplified using the asymptotic (or perturbation) expansions methods <cit.>.Consider the flow of an ideal incompressible liquid in a physical three-dimensional space. We assume additionally that the fluid is homogeneous ( the density ρ =) and the gravity acceleration g is constant everywhere[This assumption is quite realistic since the variation of this parameter around the Earth is less than 1%.]. Without any loss of generality from now on we can set ρ ≡ 1. For the sake of simplicity, in this study we neglect all other forces (such as the Coriolis force and friction). Hence, we deal with pure gravity waves.In order to describe the mathematical model, we introduce a Cartesian coordinate system O x_1 x_2y. The horizontal plane O x_1 x_2 coincides with the still water level y = 0 and the axis O y points vertically upwards. By vector = (x_1, x_2) we denote the horizontal coordinates. The fluid layer is bounded below by the solid (impenetrable) bottom y = -h (, t) and above by the free surface y = η (, t). The sketch of the fluid domain is schematically shown in Figure <ref>.The flow is considered to be completely determined if we find the velocity field (, y, t) = ((̆, y, t), v(, y, t)) (=̆ (u_1, u_2) being the horizontal velocity components) along with the pressure field p(, y, t) and the free surface elevation η(, t), which satisfy the system of Euler equations:÷+ v_y= 0 ,_̆t + ()+ v _̆y +p= 0 ,v_t +v + vv_y + p_y= -g ,where = (∂_x_1, ∂_x_2) denotes the horizontal gradient operator. The Euler equations are completed with free surface kinematic and dynamic boundary conditionsη_t + η= v ,y = η(, t) , p= 0 ,y = η(, t) .Finally, on the bottom we impose the impermeability condition ( the fluid particles cannot penetrate the solid boundary), which states that the normal velocity on the bottom vanishes:h_t +h + v = 0 ,y = -h(, t) .Below we shall discuss also the components of the vorticity vector =, which are given byω_1= v_x_2 - u_2, y ,ω_2= -v_x_1 + u_1, y ,ω_3= u_2, x_1 - u_1, x_2 .§.§ Dimensionless variables In order to study the propagation of long gravity waves, we have to scale the governing equations (<ref>)–(<ref>) along with the boundary conditions (<ref>)–(<ref>). For this purpose we choose characteristic scales of the flow. Let ℓ, d and α be the typical (wave or basin) length, water depth and wave amplitude correspondingly (they are depicted in Figure <ref>). Then, dimensionless independent variables can be introduced as followsx_1,2^∗ = x_1,2/ℓ ,y^∗ = y/d , t^∗ = t/ℓ/√(g d) .The dependent variables are scaled[We would like to make a comment about the pressure scaling. For dimensional reasons we added in parentheses the fluid density ρ. However, it is not present in governing equations since for an incompressible flow of a homogeneous liquid ρ can be set to the constant 1 without loss of generality.] ash^∗ = h/d , η^∗ = η/α ,p^∗ = p/(ρ) g d , ^̆∗ = /√(gd) ,v^∗ = v/d√(gd)/ℓ .The components of vorticityare scaled asω_1,2^∗ = ω_1,2/√(gd)/d , ω_3^∗ = ω_3/√(gd)/ℓ .The scaled version of the Euler equations (<ref>)–(<ref>) read now÷+ v_y= 0 ,_̆t + ()+ v _̆y +p= 0 , μ^2 (v_t +v + vv_y) + p_y= -1 ,where we drop the asterisk symbol ∗ for the sake of notation compactness. Boundary conditions at the free surface similarly become(η_t + η)= v ,y =η(, t) , p= 0 ,y =η(, t) .It can be easily checked that the bottom boundary condition (<ref>) remains invariant under this scaling. Finally, the scaled components of vorticity ω^∗ areω_1= μ^2 v_x_2 - u_2, y ,ω_2= -μ^2 v_x_1 + u_1, y ,ω_3= u_2, x_1 - u_1, x_2 . Above we introduced two important dimensionless parameters: Nonlinearity αd, which measures the deviation of waves with respect to the unperturbed water levelDispersion μdℓ, which indicates how long the waves are comparing to the mean depth (or equivalently how shallow is the water)§.§ Long wave approximation In approximate shallow water systems the dynamic variables are the total water depth (̋, t)h(, t) +η(, t) and some vector (, t) which is supposed to approximate the horizontal velocity vector of the full model (̆, y, t). In many works (, t) is chosen as the trace of the horizontal velocity $̆ at certain surfacey = _σ(, t)in the fluid bulk <cit.>,(, t)(, _σ(, t),t) .Another popular choice for the velocity variable consists in taking the depth-averaged velocity <cit.>:(, t)1/(̋, t) ∫_-h(, t)^η(, t) (̆, y, t)y .By applying the mean value theorem <cit.> to the last integral, we obtain that two approaches are mathematically formally equivalent:(, t) ≡ (, _ξ(, t),t) .However, this time the surfacey = _ξ(, t)remains unknown, while above it was explicitly specified. We only know that such surface exists.Below we shall consider only long wave approximation to the full Euler equations. Namely, we assume that(, t)approximates the true horizontal velocity(̆, y, t)to the orderØ(μ^2),(̆, y, t) = (, t) + μ^2 (, y, t) .By integrating the continuity equation (<ref>) over the total depth and taking into account boundary conditions (<ref>), (<ref>) we obtain the mass conservation equation_̋t + ÷() = -μ^2 ÷() ,where(, t)1/(̋, t) ∫_-h(, t)^ η (, t) (, y, t)y .If we choose the variableto be depth-averaged, then(, t) ≡ 0and the mass conservation equation (<ref>) takes the very familiar form_̋t + ÷() = 0 .Integration of equation (<ref>) over the vertical coordinate in the limits from-h(, t)toyand taking into account the bottom boundary condition (<ref>) leads to the following representation for the vertical velocity in the fluid column:v(, y, t) = - h - (y + h)÷ + Ø(μ^2) ,where for the sake of simplicity we introduced the material (or total, or convective) derivative operator:[ · ][ · ]_ t +[ · ] .Below the powers of this operator will appear in our computations:^k [ · ]··…·_k[ · ] ,k ≥ 1 . We have to express asymptotically also the pressure fieldp(, y, t)in terms of the dynamic variables((̋, t),(, t)). Thus, we integrate the vertical momentum equation (<ref>) over the vertical coordinate in the limits fromyto the free surface:p(, y, t) = μ^2 ∫_y^ η(, t)[ v + v v_y + Ø(μ^2) ]y - y +η(, t) .The integrand can be expressed in term of(̋, t)and(, t)using representation (<ref>):v + v v_y = -(y + h) _1 - _2 + Ø(μ^2) ,where we defined_1(, t)÷ - (÷)^2 , _2(, t) ^2h .Substituting the last result into the integral representation (<ref>) and integrating it exactly inyleads the following expression of the pressure field in the fluid layer:p = - (y + h) - μ^2 [ (- (y + h)) _2 + (^̋ 2/2 - (y + h)^2/2) _1 ] + Ø(μ^4) .Notice that this representation does not depend on the expression of the velocity correction(, t). If in the last formula we neglect terms ofØ(μ^4)and return to physical variables, we can obtain the pressure reconstruction formula in the fluid bulk:p/ρ = g [ - (y + h) ] - [ (- (y + h)) _2 + (^̋ 2/2 - (y + h)^2/2) _1 ] .We underline the fact that the last formula is accurate to the orderØ(μ^4). This formula will be used in <cit.> in order to reconstruct the pressure field under a solitary wave, which undergoes some nonlinear transformations.In order to obtain an evolution equation for the approximate horizontal velocity(, t)we integrate over the vertical coordinate equation (<ref>):∫_-h^ η[ _̆t + ()+ v _̆y ]y + ∫_-h^ η py - .p|_y = -h· h = 0 .The pressure variable can be easily eliminated from the last equation using the representation formula (<ref>):∫_-h^ η py - .p|_y = -h· h ==h - μ^2[ (^̋ 3_1 + ^̋ 2_2) -h (_1 + _2) ] + Ø(μ^4) .Then, using the representation (<ref>) for the vertical velocityv, we can write∫_-h^ η v _̆yy= -μ^2∫_-h^ η[h + (y + h)÷ ] _yy + Ø(μ^4) = -.μ^2( h)·|_y=-h^y= η - μ^2 ÷∫_-h^ η(y + h)_yy_(∗) + Ø(μ^4) .The integral (∗) can be computed using integration by parts∫_-h^ η(y + h) _yy = .·̋|^y= η -.Combining together these results, we obtain the following asymptotic formula1/μ^2∫_-h^ η v _̆yy = .( h)·|_y=-h - [h + ÷ ].|^y= η + ÷ + Ø(μ^2) .Finally, we take care of convective terms∫_-h^ η[ _̆t + ()]y = ∫_-h^ ηy + μ^2∫_-h^ ηy + μ^2∫_-h^ η()y + Ø(μ^4)=+ μ^2[ [] - [ η ]·.|^y =η -h·.|_y = -h + (̋)] + Ø(μ^4) .Finally, we obtain∫_-h^ η[ _̆t + ()+ v _̆y ]y =- μ^2[ + ÷̋ ]_(∗∗)·.|^y = η + + μ^2 [ [] + (̋)+ ÷ ] .From the mass conservation equation (<ref>) we have+ ÷ = -μ^2 ÷() = Ø(μ^2) .Thus, the term (∗∗) can be asymptotically neglected. As a result we have∫_-h^ η[ _̆t + ()+ v _̆y ]y =+ μ^2 [ [] + (̋)+ ÷ ] + Ø(μ^4) .Substituting all these intermediate results into depth-integrated horizontal momentum equation (<ref>), we obtain the required evolution equation for:_t + () +η = μ^2/ [ (^̋ 3_1 + ^̋ 2_2) -h (_1 + _2) ]- μ^2/ [ [] + (̋)+ ÷ ] .The last equation may look complicated. However, it can be rewritten in a clearer way by pointing out explicitly the non-hydrostatic pressure effects. It turns out that it is advantageous to introduce the depth-integrated (but not depth-averaged) pressure:(,̋ )∫_-h^ ηpy = ^̋ 2/2 - μ^2 (^̋ 3_1 + ^̋ 2_2) .We introduce also the pressure traceat the bottom:(x, t).p|_y = -h = - μ^2 (^̋ 2_1 + _2) .Using these new variables equation (<ref>) becomes_t + () + / = h/ - μ^2/ [ ()_t + ()() + (̋)+ ÷ ] . The derived system of equations admits an elegant conservative form[This form becomes truly conservative (in the sense of hyperbolic conservation laws) only on the flat bottom, h(, t) = h_0 =⇒h ≡.]:_̋t + ÷[]= 0 , ()_t + ÷[+ (,̋ )· + μ^2]= h ,where we introduced a new velocity variable + μ^2 and ∈ _2 × 2()is the identity matrix. Operatoris the tensorial product, for two vectors∈ ^mand∈ ^n(u_i· v_j)^1 ≤i ≤ m_1 ≤j ≤n ∈ _m × n() .From now on equations (<ref>), (<ref>) will be referred to as the base model of our study. In order to close the last system of equations (<ref>), (<ref>), we have to express the variablein terms of other dynamic variables(̋, t)and(, t). Several popular choices will be discussed below. Notice also that nowhere in the derivation above the flow irrotationality was assumed. Notice that taking formally the limit μ → 0 in equations (<ref>), (<ref>) yields straightforwardly the well-known Nonlinear Shallow Water (NSW or Saint-Venant) Equations <cit.>. Thus, our base model satisfies the Bohrcorrespondence principle[This principle was formulated by Niels Bohr (1920) <cit.>. Loosely speaking, this principle states that Quantum Mechanics reproduces Classical Mechanics in the limit of large quantum numbers. Correspondingly, a nonlinear dispersive model should describe correctly the propagation of non-dispersive waves in the limit when the dispersion vanishes.]. This property is crucial for robust physical wave modelling in coastal environments. Indeed, a wave approaching continental shelf undergoes nonlinear transformations: the water depth is decreasing and the wave amplitude grows, which often leads to the formation of undular bores. The model has to follow these transformations. Mathematically it means that the model equations should encompass a range of physical regimes varying from fairly shallow water to intermediate depths <cit.>. There exists an option of coupling different hydrodynamic models as it was donein <cit.>. However, the coupling represents a certain number of difficulties,*Boundary conditions at artificial interfaces?*How to determine automatically the physical regime?*Dynamic evolution and handling of model applicability areas… Consequently, in this study we let the physical model to do this work for us. §.§.§ Energy conservation We would like to raise the question of energy conservation in nonlinear dispersive wave models. The full Euler equations naturally have this property. So, it is a priori natural to require that a good approximation to Euler equations conserves the energy as well <cit.>. An energy conservation equation can be established for the base model (<ref>), (<ref>) for some choices of the variable(,̋ ). For instance, the classical SGN model discussed in the following section enjoys this property (it corresponds to the choice ≡ ). On moving bottoms this property was discussed in <cit.>. Here we provide only the final result,the total energy equation for SGN model on a general moving bottom[Of course, this equation becomes a conservation law only when the bottom is static (but not necessarily flat).]:()_t + ÷[( + /) ] = -h_t ,where the total energyis defined as^2 +^̋ 2 (÷)^2 +( h) (÷) +( h)^2 + g/2 (- 2h) .For other choices of the closure(,̋ )this question of energy conservation has to be studied separately. Recently, Clamond, Dutykh&Mitsotakis (2015) <cit.> proposed a dispersion-improved SGN-type model which enjoys the energy conservation property. The method employed in that study is the variational approach: the preservation of the variational structure is crucial for the preservation of several invariants. §.§.§ Galilean invariance The same questions can be raised about the Galilean invariance property as well. This property is of fundamental importance for any mathematical model that provide a physically sound description of water waves (stemming from Classical Mechanics and Classical Physics). Some thoughts and tentative corrections can be found in <cit.>. The base model (<ref>), (<ref>) is Galilean invariant under reasonable assumptions on the closure velocity vector.Galilean invariance principle states that all mechanical laws are the same in any inertial frame of reference <cit.>. Consequently, the mathematical form of governing equations should be the same as well. It was proposed by Galileo Galilei in 1632 <cit.>. Consider the horizontal Galilean boost transformation between two inertial frames of reference:^ ' =+ t ,y^ ' = y ,t^ ' = t ,whereis a constant motion speed of the new coordinate system (with primes) relatively to the initial one (without primes). Notice that scalar quantities such as(̋, t)andh(, t)remain invariant since they are defined as distances between two points and distances are preserved by the Galilean transformation (<ref>). Let us see how the horizontal velocity variable changes under the Galilean transformation:(̆, y, t)t = ^ 't - ^̆ '(^ ', y^ ', t^ ') -.It is not difficult to understand that the same transformation rule applies to(, t)regardless if it is defined as a trace or depth-averaged velocity:= ^ ' -.Indeed, the last claim is obvious for the case of the trace operator. Let us check it for the depth-averaging operator:(, t)1/ ∫_-h^η y = 1/^̋ ' ∫_-h^ '^η^ '(^̆ ' - )y^ ' = = 1/^̋ ' ∫_-h^ '^η^ '^̆ 'y^ ' - ^ '(^ ', t^ ') -.If the velocity(, t)is defined in a different way, its transformation rule has to be studied separately. From the definition (<ref>) it follows that the velocity correctionshould remain invariant under the Galilean boost (<ref>) (since it is defined as a difference of two velocities):^ ' ≡.In the following we shall assume that the chosen closure(,̋ )satisfy the last transformation rule.Finally, let us discuss the invariance of the base model (<ref>), (<ref>). Basically, this property follows from the transformation rule (<ref>), from the fact that = and the following observation[Let us prove, for example, the first identity:≡ _̋t + () = ^̋ '_t^' +^̋ ' + ((^ ' - )) ^̋ ' = ^̋ '_t^' + (^ ') ^̋ ' ≡ ^ '^̋ ' . ]:≡ ^ '^̋ ' , ≡ ^ '^ ' .The pressure variablesandremain invariant as well, since they depend on velocity through_1and_2, which depend in their term only on the full derivative and divergence of the velocity. Thus, the base model (<ref>), (<ref>) is Galilean invariant under not very restrictive assumptions made above. Many Boussinesq-type equations derived and published in the literature are not Galilean invariant. As a classical such example we can mention Peregrine's (1967) system <cit.>. In <cit.> it was shown how to derive a weakly nonlinear model from the fully nonlinear one in such a way that the reduced Boussinesq-type model has the Galilean invariance and energy conservation properties.§.§ Serre–Green–Naghdi equations The celebrated Serre–Green–Naghdi (SGN) equations can be obtained by choosing the simplest possible closure,≡.This closure follows from the fact that the velocity variablechosen in SGN equations is precisely the depth-averaged velocity. Thus,(, t) ≡ and from (<ref>) we have that(, t) ≡ . By substituting the proposed closure into equations (<ref>), (<ref>), we obtain the SGN equations:_̋t + ÷[]= 0 , ()_t + ÷[+ (,̋ )· ]= h ,where(,̋ )was defined in (<ref>). The last equation can be written in a non-conservative form as well:_t + () + / = h/ .The SGN equations have been rediscovered independently by a number of authors. The steady version of these equations can be already found in Rayleigh (1876) <cit.>. Then, this model in 1D was derived by Serre (1953) <cit.> and by Su&Gardner (1969) <cit.>. A modern derivation was done by Green, Laws&Naghdi (1974) <cit.>. Later, in Soviet Union this system was derived also by Pelinovsky&Zheleznyak (1985) <cit.>. More recently, modern derivations of these equations based on variational principles have been proposed. Namely, Miles&Salmon (1985) <cit.> gave a derivation in Lagrangian ( particle) description. The variational derivation in Eulerian description was given by Fedotova&Karepova (1996) <cit.> and later by Kim (2001) <cit.> and Clamond&Dutykh (2012) <cit.>. Recently the multi-symplectic structure for SGN equations was proposed in <cit.>. §.§ Other particular cases The scope of the present section is slightly broader than its title may suggest. More precisely, we consider the whole class of models where the velocity variable is defined on a certain surface inside the fluid, see equation (<ref>) for the definition. We show in this section that the base model (<ref>), (<ref>) can be closed using the partial irrotationality condition. Namely, we assume that only two horizontal components of vorticity vanish,_̆y = μ^2v .Integration of this identity overyand using representations (<ref>), (<ref>) leads(, y, t) = -(y + h) [ ( h) +h (÷) ] - (y + h)^2/2(÷) + .|_y=-h + Ø(μ^2) .Consequently, from (<ref>) we obtain(̆, y, t) =+ μ^2 [(y + h)+ 1/2 (y + h)^2+ ] + Ø(μ^4) ,where we introduced for simplicity the following notation:(, t) -( h) -h (÷) ,(, t) -(÷) ,(, t) .|_y=-h .Let us evaluate both sides of equation (<ref>) aty_σ = _σ(, t). According to (<ref>) we must have(, _σ(, t), t) ≡ (, t) .Consequently, we have(, t) ≡ -(y_σ + h)- 1/2 (y_σ + h)^2.Thus, coefficientcan be eliminated from (<ref>) to give the following representation(̆, y, t) =+ μ^2 [(y - y_σ)+ 1/2 [ (y + h)^2 - (y_σ + h)^2 ] ] + Ø(μ^4) .Substituting the last result into equation (<ref>) yields the required closure relation:(,̋ ) = [ /2 - (y_σ + h)]+ [1/6 ^̋ 2 - 1/2 (y_σ + h)^2 ]+ Ø(μ^2) .To summarize, under the assumption (<ref>) that the first two components of the vorticity field vanish, we can propose a closure to the base model, after neglecting the terms of orderØ(μ^2)in (<ref>). *Depth-averaged velocity.It is interesting to obtain also the 3D velocity reconstruction formula in the case, where(, t)is defined as the depth-averaged velocity (<ref>). To do it, we average the equation (<ref>) over the depth:1/ ∫_-h^ η(̆, y, t)y = (, t) + μ^2 [ /2+ ^̋ 2/6+] + Ø(μ^2) .Using the definition (<ref>) of the depth-averaged velocity, we conclude that= -/2- ^̋ 2/6+ Ø(μ^2) .By substituting the last expression into (<ref>) we obtain the desired representation:(̆, y, t) = (, t) + μ^2 [ (/2 - y - h)·( h + (÷)h) + (^̋ 2/6 - (y + h)^2/2) (÷) ] + Ø(μ^4) .The last formula will be used in <cit.> in order to reconstruct the 3D field under a propagating wave, which undergoes some nonlinear transformations. Formula (<ref>) shows also that in shallow water flows the velocity distribution in the vertical coordinateyis nearly quadratic. We underline that formula (<ref>) is obtained under the assumption that the flow is irrotational. Without this assumption, in the most general case we can only use formula (<ref>) by neglecting terms of the order Ø(μ^2). In other words, the velocity variable (, t) approximates the 3D velocity field (̆, y, t) throughout the fluid to the order Ø(μ^2). However, in many applications this accuracy is not enough. §.§.§ Lynett–Liu's model It can be shown that the base model (<ref>), (<ref>) supplemented by the proposed closure (<ref>) is asymptotically equivalent to the well-known Lynett–Liu (2002) model derived in <cit.> under an additional assumption that the initial 3D flow is irrotational. This claim is true only up to the approximation orderØ(μ^4)and it can be checked by straightforward but tedious calculations.Various choices of the levely_σ, where the horizontal velocity is defined, allow to obtain in a straightforward manner the fully nonlinear analogues of various existing models. Some of popular choices are discussed below.§.§.§ Mei–Le Méhauté's model Consider the horizontal velocity variable defined at the bottom,y_σ = -h(, t) .Substituting this value into (<ref>) we obtain straightforwardly the following closure:(,̋ ) = 1/2+ 1/6 ^̋ 2+ Ø(μ^2) .In this way, the base model (<ref>), (<ref>) with the last closure becomes the celebrated Mei–Le Méhauté (1966) model <cit.>.§.§.§ Peregrine's model and its generalizations In 1967 Peregrine<cit.> considered a weakly nonlinear model withy_σ = 0. The fully nonlinear analogue of Peregrine's model can be obtained if we takey_σ =η (, t) .Closure relation (<ref>) then becomes:(,̋ ) = -1/2- 1/3 ^̋ 2+ Ø(μ^2) ,and base model (<ref>), (<ref>) becomes the fully nonlinear Peregrine's system. The momentum equation of this model takes a very simple form, when the Boussinesq regime is considered:_t + () +η =.In other words, if initially the vertical component of vorticity is zero, then it is so for all times,_2, x_1 - _1, x_2 = 0, ∀ t ≥ 0 .The last assertion is true only in Boussinesq approximation in for the Cauchy problem. The irrotationality can break when boundary conditions are applied on finite domains, <cit.>.§.§.§ Nwogu's model and its generalizations In 1993 Nwogu proposed the following choice <cit.>:y_σ ≈ -β· h(, t) , β ≈ 0.531 .This choice was motivated by linear dispersion relation considerations (optimization of dispersive characteristics). The nonlinearity of Nwogu's model was improved in <cit.>. The idea consists in finding surface between the bottomy = -h(, t)and free surfacey =η(, t)(instead of the bottom andy = 0in weakly nonlinear considerations). In this way, a free parameterβ ∈ [0, 1]at our disposal:y_σ(, t) = -βh(, t) + (1 - β)η (, t) .In this case the closure relation becomes:(,̋ ) = ( β - 1/2 )- ^̋ 2/6 ( 3β^2 - 6β + 2 )+ Ø(μ^2) .The `optimal' value ofβwill coincide with that given by Nwogu<cit.> since linearizations of both models coincide.§.§.§ Aleshkov's model As the last example, we show here how to obtain Aleshkov's (1996) model <cit.>, which was generalized later to include moving bottom effects in <cit.>. Aleshkov's model (with moving bottom) can be obtained from the base model (<ref>), (<ref>) if we adopt the following closure:(,̋ ) = -( h)h + 1/2+ 1/6 ^̋ 2+ Ø(μ^2) .This closure is similar to Mei–Le Méhauté closure (<ref>) except for the first term. The horizontal velocity in Aleshkov's model does not coincide with the horizontal fluid velocity at any surface inside fluid bulk. Instead, Aleshkov's velocity variable is given by the gradient of the velocity potential evaluated at solid bottom. For non-flat bottoms it does not coincide with.|_y=-h. These subtle differences are discussed in some detail in Appendix <ref>. Since this model is not widely known, we give here the governing equations:_̋t + ÷[] = μ^2 ÷[ (̋ h)h + ^̋ 2/2 [ ( h) + (÷)h ] + ^̋ 3/6 (÷) ] ,_t + ()+η = μ^2[ _2 + ^̋ 2/2 _1 + 1/2 ( h)^2 ] + Ø(μ^4) .One big advantage of equations above is that the irrotational flow is preserved by its dynamics of equations (<ref>), (<ref>) in the sense of definition given in equation (<ref>). The proof of this fact is given in Appendix <ref>.§ WEAKLY-NONLINEAR MODELS We considered the fully nonlinear version of the base model (<ref>), (<ref>) previously since the small amplitude assumption was never used (even if we introduced formally the nonlinearity parameter). The only constitutive assumption employed was the long wave hypothesis or, in other words, the waves are only weakly dispersive. In the present section we derive a weakly nonlinear variant of the base model (<ref>), (<ref>). In this way we achieve a further simplification of governing equations. Moreover, we shall work in the so-called Boussinesq regime:= Ø(μ^2)⟺= Ø(1) ,whereμ^2 ≡ α ℓ^ 2d^ 3is the so-called Stokes–Ursell number <cit.>. In other words, we assume that the nonlinearity and dispersion parameters have approximatively the same order of magnitude. It is under this assumption that one can obtain numerous Boussinesq-type models <cit.>. Sometimes the simplifying Boussinesq assumption (<ref>) is accompanied also by explicitly (or implicitly) stated assumptions on the bottom variations,h ∼ Ø() ≃ Ø(μ^2) , as it is the case for the base model.The most difficult task here is to keep as many good properties of the base model as possible, while simplifying the governing equations. It is not always possible and some illustrations will be given below. §.§ Weakly nonlinear base model In the present Section we derive the Weakly Nonlinear Base Model (WNBM) starting from the base model equations (<ref>), (<ref>). The first goal here is to preserve at least the conservative form of the equations when simplifying the base model.First of all, we notice that the vectoralways enters into governing equations with coefficientμ^2,μ^2. Consequently, under the assumption (<ref>), the vectorcan be formally split as= _0 + _1_Ø(1) + Ø(μ^2) ,where_0contains all the terms independent of the system solution and_1contains everything else involvingη, . For instance, to illustrate this idea for the closure relation (<ref>), which gives the Lynett–Liu's model, we have the following decomposition:_0= (h/2 + y_σ)h_t ,_1= (h/2 + y_σ)·(( h) + (÷) h) - (h^2/6 - (y_σ + h)^2/2) (÷) .From now on we use the following notation for the main part of vector:^♭_0 + _1 .The mass conservation equation for WNBM model is directly obtained from (<ref>):_̋t + ÷() = -μ^2 ÷(h ^♭) .We notice that the last equation is in the conservative form as well. In a similar way, we obtain the weakly nonlinear analogue of the momentum conservation equation:()_t + ÷(⊗) + ^ ♭ = ^ ♭h - μ^2[ (h ^♭)_t + ÷(h _0⊗ + h ⊗_0) ] .In some cases, it is useful to have also a non-conservative form of the momentum conservation equation (<ref>), which can be obtained using the weakly nonlinear form of the mass conservation (<ref>):_t + () + ^ ♭/ = ^ ♭h/- μ^2/ [ (h ^♭)_t -÷(h ^♭) + ÷(h _0⊗ + h ⊗_0) ] .To complete the description of the WNBM, we have to explain how to compute the non-hydrostatic pressure in this model:^♭ ^̋ 2/2 - μ^2[ h^3/3 _1^ ♭ + h^2/2 _2^ ♭ ] ,^♭ - μ^2 [ h^2/2 _1^ ♭ + h _2^ ♭ ] ,where_1^ ♭(÷)_t , _2^ ♭h_tt + 2h_t + _t h .We underline that the non-conservative form (<ref>) contains one nonlinear dispersive term ÷(h _1)while in the conservative form (<ref>) all dispersive terms are linear. Equations (<ref>), (<ref>) constitute the WNBM. Below we derive some important particular cases of WNBM.§.§.§ Depth-averaged WNBM Consider a particular case of the WNBM when the velocity variable is chosen to be depth-averaged. In this case we showed above that ≡. Consequently,^♭ ≡ as well. WNBM equations (<ref>), (<ref>) take the simplest form in this particular case:_̋t + ÷()= 0 , ()_t + ÷(⊗) + ^ ♭= ^ ♭h .On the flat bottom the last equation becomes even simpler:()_t + ÷(⊗) + ^ ♭ =.The equivalent non-conservative form of the momentum conservation equation (on uneven bottoms) is_t + ()+ α η = μ^2/ {[ h^3/3 _1^ ♭ + h^2/2 _2^ ♭ ] - [ h^2/2 _1^ ♭ + h _2^ ♭ ]h}_() . §.§.§ Peregrine's system In the pioneering work <cit.>Peregrine derived a weakly-nonlinear model over a stationary bottom,h_t ≡ 0 . The mass conservation in Peregrine's system coincides exactly with the mass conservation equation from the previous Section <ref>. We show below that the Peregrine's momentum conservation can be obtained from the non-conservative equation (<ref>) under the Boussinesq assumption (<ref>). The right-hand side()can be rewritten as()/μ^2 = 1/h [ h^3/3 (÷)_t + h^2/2 (_t h) ] - [ h/2 (÷)_t + _t h ]h == [ h/2(÷)h + h^2/3 (÷) + h/2 ( h) ]_t .Then, we use the relationh = ÷(h ) - h ÷ .Finally, we obtain the right-hand side of Peregrine's model:()/μ^2 = [ h/2 (÷(h )) - h^2/6 (÷) ]_t .Hence, the non-conservative momentum equation reads_t + ()+ α η = μ^2 [ h/2 (÷(h )) - h^2/6 (÷) ]_t_≃ () .However, the simplifications we made above were drastic in some sense. For instance, the Peregrine's model cannot be recast in a conservative form even on a flat bottom. It goes without saying that the energy equation cannot be established for this model either. These are the main drawbacks of the weakly nonlinear Peregrine's system. Moreover, the numerical schemes based on non-conservative equations may be divergent <cit.>. Despite all this critics, the Peregrine's system supplemented with moving bottom effects (h_t ≠ 0) was successfully used to model wave generation in closed basins <cit.>.It is interesting to note that the depth-averaged WNBM and Peregrine's system give the same linearisation over the flat bottomh(x) ≡ d:η_t + d ÷= 0 ,_t + α η= μ^2 d^ 2/3 (÷_t) .In particular, it implies that dispersive properties are the same.§.§.§ WNBM with the velocity given on a surface When the velocity variable is defined on a surface in the fluid bulk as in (<ref>), WNBM equations are (<ref>), (<ref>) and the closure relation for variable^ ♭is given by formulas (<ref>), (<ref>). Consequently, the dispersive terms are present in both mass and momentum conservation equations. Moreover, in the case of the stationary bottom (h_t ≡ 0) we have automatically that_0 ≡. Consequently, the WNBM equations with this choice of the velocity variable read:_̋t + ÷()= -μ^2 ÷(h _1) , ()_t + ÷(⊗) + ^ ♭= ^ ♭h - μ^2 (h _1)_t .The last equation can be recast in the non-conservative form:_t + ()+ α η = μ^2/ {[ h^3/3 _1^ ♭ + h^2/2 _2^ ♭ ] - [ h^2/2 _1^ ♭ + h _2^ ♭ ]h - (h _1)_ t +÷(h _1)}_(ƪ) .Below we show an important application of this variant of the WNBM.§.§.§ Nwogu's system Nwogu's model was derived in <cit.> under the assumption of the stationary bottom (h_t ≡ 0) that we adopt here as well. First of all, the expression (<ref>) can be transformed using the relation (<ref>):_1 = (h/2 + y_σ) (÷(h )) + (y_σ^2/2 - h^2/2) (÷) .In this way we obtain straightforwardly the mass conservation equation of Nwogu's system <cit.>. In order to obtain the momentum equation of Nwogu's system, first we neglect in(ƪ)the nonlinear dispersive term ÷(h _1) . Then, the non-hydrostatic pressure terms are transformed similarly to Peregrine's system case studied above in Section <ref>. So, the right-hand side(ƪ)of WNBM becomes:(ƪ)/μ^2 = h/2 (÷(h )_t) - h^2/6 (÷)_t- (h/2 + y_σ) (÷(h )_t) - (y_σ^2/2 - h^2/6) (÷)_t ≡ -[ y_σ (÷(h )_t) + y_σ^2/2 (÷)_t ] .As a result, we obtain the momentum equation of Nwogu's system <cit.>:_t + ()+ α η = -μ^2 [ y_σ (÷(h )_t) + y_σ^2/2 (÷)_t ]_≃ (ƪ) .Using the low-order linear terms in the dispersive terms again, other asymptotic equivalent models can also be derived, <cit.>. The WNBM equations and Nwogu's system linearize on the flat bottomh(x) ≡ dto the same equations:η_t + d ÷= -μ^2 d^ 3(β + 1/3) ÷((÷)) ,_t + α η= -μ^2 d^ 2 β (÷_t) ,where we introduced the following parameter:βy_σ/d + y_σ^2/2 d^ 2 .However, in the nonlinear case the WNBM system has the advantage of admitting the conservative form on general (unsteady and uneven) bottoms. This fact can be used to develop efficient numerical algorithms to solve nonlinear dispersive equations numerically. For instance, this conservative property will be exploited in <cit.> in order to construct adaptive and efficient numerical discretizations.§ DISCUSSION We presented a certain number of developments going from the derivation of the base model (<ref>), (<ref>) to obtaining some particular models as particular cases. The main conclusions and perspectives of this study are outlined below. §.§ Conclusions In the present manuscript we attempted to meet two main goals. First of all, we tried to make a review of the continuously growing field of long wave modelling. In particular, we focused on nonlinear dispersive wave models such as some improved Boussinesq-type and Serre–Green–Naghdi (SGN) equations <cit.>, which were not covered in previously published review papers. We apologize in advance if we forgot to mention somebody's contribution to this field. The topic being so broad that it is practically impossible to referred to all the published literature.Then, we attempted to present a unified approach which incorporates some well-known and some less known models in the same modelling framework. The derivation procedure is based on the minimal set of assumptions. Various models can be obtained as particular cases of the so-called base model presented in our study. In the same time, the base model allows to obtain fully nonlinear analogues of previously derived weakly-nonlinear models. The linearizations of old and new models will coincide exactly, hence leaving dispersive characteristics unchanged. Moreover, the resulting models admit an elegant conservative form by construction. The improvement of dispersive characteristics can be achieved by a judicious choice of the closure relation(,̋ )as it was illustrated, for example, in Section <ref>. §.§ Perspectives In the present study we discussed modeling and derivation of models for shallow water waves flowing over uneven bottoms, but the whole system was defined on a flat domainΩof the Euclidean space^d, with dimensiond = 1, 2. The bottom represents only a deformation (not necessarily small) of the mean water depth. Among the main perspectives of this study we would like to mention the derivation of fully nonlinear shallow water models defined on more general geometries. In particular, the spherical geometry represents a lot of interest in view of applications to atmospheric sciences. The first steps in this direction have already been made in <cit.>. The derivation of shallow water equations on a sphere will be discussed in Part III<cit.>.The numerical discretization of the derived above equations on moving adaptive grids will be considered in details in the companion paper <cit.> (Part II), while the numerical simulation of shallow water waves on a sphere will be considered in Part IV of this series of papers <cit.>. §.§ Acknowledgments tocsubsectionAcknowledgmentsThis research was supported by RSCF project No 14–17–00219. D. Mitsotakis was supported by the Marsden Fund administered by the Royal Society of New Zealand.§ ALESHKOV'S MODEL VS. MEI–LE MÉHAUTÉ'S MODEL In this Appendix we assume the flow to be irrotational. Consider the fluid velocity potential expansion around the bottom:ϕ(, y, t) =- μ^2(y + h)(h_t +h) - μ^2 (y + h)^2/2 ^2 + Ø(μ^4) ,whereis the velocity potential trace at the bottom,(, t).ϕ(, y, t)|_y=-h .A similar formula can be found in <cit.> for the stationary bottom and in <cit.> for moving bottoms. The horizontal fluid velocity can be readily obtained by differentiating equation (<ref>):(̆, y, t) ≡ ϕ =- μ^2(h_t +h)h - μ^2(y + h) (h_t +h)- μ^2(y + h) (^2)h - μ^2 (y + h)^2/2 (^2) + Ø(μ^4) .Then, the whole family of models can be obtained by choosing the velocity variable(, t)at different levels in the fluid. Here we take the velocity at solid bottom:(, t).(̆, y, t)|_y = -h =- μ^2(h_t +h)h .Hence, from definition (<ref>) we can compute the expression for:(, t) = -(y + h) (h_t +h) - (y + h) (^2)h - (y + h)^2/2 (^2) + Ø(μ^2) ,and taking into account the fact that =+ Ø(μ^2)we have(, t) = -(y + h)h - (y + h) (÷)h - (y + h)^2/2 (÷) + Ø(μ^2) ≡ (y + h)+ (y + h)^2/2+ Ø(μ^2) .After applying the depth-averaging operator we obtain the corresponding closure variable:(, t)1/H ∫_-h^η(, y, t)y = H/2+ H^2/6+ Ø(μ^2) .It coincides exactly with the closure relation (<ref>) given above. This concludes our clarifications regarding Mei–Le Méhauté's model <cit.>.In Aleshkov's model the velocity variable(, t)is defined in a different way:(, t)(, t) .Then, the fluid horizontal velocity takes the form(̆, y, t) =- μ^2(h_t +h)h - μ^2(y + h) (h_t +h)- μ^2(y + h) (^2)h - μ^2 (y + h)^2/2 (^2) + Ø(μ^4) =- μ^2(h_t +h)h - μ^2(y + h) (h_t +h)- μ^2(y + h) (÷)h - μ^2 (y + h)^2/2 (÷) + Ø(μ^4) = + μ^2[ - h( h) + (y + h)+ (y + h)^2/2] + Ø(μ^4) .From the last formula it is straightforward to obtain the closure relation (<ref>) which yields Aleshkov's model <cit.>. It explains also the differences between between Aleshkov's and Mei–Le Méhauté's models.§ VORTICITY IN ALESHKOV'S MODEL In this Appendix we study how the vertical component of vorticity evolves under the dynamics of Aleshkov's model (<ref>), (<ref>). Consequently, we rewrite equations (<ref>) in the following equivalent form:_1, t + _1 _1, x_1 + _2 _1, x_2 + _x_1= 0 ,_2, t + _1 _2, x_1 + _2 _2, x_2 + _x_2= 0 ,whereis a scalar function defined asη - μ^2 [ H_2 +H^2 _1 +( h)^2 ] .The same equations (<ref>) can be rewritten also as_1 t - _2 ω + [+ _1^ 2 + _2^ 2/2 ]_x_1= 0 ,_2 t + _2 ω + [+ _1^ 2 + _2^ 2/2 ]_x_2= 0 ,where we introduced the vertical vorticity functionω_2, x_1 - _1, x_2. Making a cross differentiation of two last equations and subtracting them yields the following vorticity equation:ω_t + [ ω _1 ]_x_1 + [ ω _2 ]_x_2 = 0 .Let us assume that initially we haveω(, 0) ≡ 0and equation (<ref>) admits a unique solution. By noticing thatω(, t) ≡ 0solves equation (<ref>) and satisfies the initial condition, we obtain the required result.There is a much shorter (but less insightful) proof of the same result. 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"authors": [
"Gayaz Khakimzyanov",
"Denys Dutykh",
"Zinaida Fedotova",
"Dimitrios Mitsotakis"
],
"categories": [
"physics.flu-dyn",
"math.AP",
"physics.class-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20170627122811",
"title": "Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space"
} |
Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USADepartment of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China Institut Laue Langevin, 71 Avenue des Martyrs, 38042 Grenoble, France [email protected] Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China 74.70.Xa, 75.30.Gw, 78.70.Nx We use neutron polarization analysis to study temperature dependence of the spin excitation anisotropy in BaFe_2As_2, which has a tetragonal-to-orthorhombic structural distortion at T_sand antiferromagnetic (AF) phase transition at T_N with ordered moments along the orthorhombic a-axisbelow T_s≈ T_N≈ 136 K. In the paramagnetic tetragonal state at 160 K, spin excitations are isotropic in spin space with M_a=M_b=M_c, where M_a, M_b, and M_c are spin excitations polarized along the a, b, and c-axis directions of the orthorhombic lattice, respectively. On cooling towards T_N, significant spin excitation anisotropy with M_a>M_b≈ M_c develops below 3 meVwith a diverging M_a at T_N.The in-plane spin excitation anisotropy in the tetragonal phase of BaFe_2As_2 is similar to those seen in the tetragonal phase of its electron and hole-doped superconductors, suggesting that spin excitation anisotropy is a direct probe of doping dependence of spin-orbit coupling and its connection to superconductivity in iron pnictides.Spin excitation anisotropy in the paramagnetic tetragonal phase of BaFe_2As_2 Pengcheng Dai December 30, 2023 =============================================================================The iron pnictide superconductors have a rich phase diagram including an orthorhombic lattice distortion associated with ferro-orbital order and nematic phase,antiferromagnetic (AF) order, andsuperconductivity <cit.>.In the undoped state, a parent compound of iron pnictide superconductors BaFe_2As_2forms stripe AF order at T_N near a tetragonal-to-orthorhombic structural transition temperature T_s [Fig. 1(a)] <cit.>. Superconductivity can be induced by partially replacing Ba by K in BaFe_2As_2 to form hole-doped Ba_1-xK_xFe_2As_2 or by partially replacing Fe by TM (TM=Co, Ni) to form electron-doped BaFe_2-xTM_xAs_2 <cit.>. Although much attention has been focused on understanding theinterplay between magnetism and superconductivity in these materials <cit.>, a more subtle and much less explored facet involvesthe effect of spin-orbit coupling (SOC) <cit.>, which translates anisotropies in real space into anisotropies in spin space and determines the easy axis of the magnetic ordered moment [Fig. 1(b)], and its connection with the electronic nematic phase and superconductivity <cit.>. Since a nematic quantum critical point is believed to occur near optimal superconductivity in electron and hole-doped iron pncitides <cit.>, it is important to determine the temperature and electron/hole doping evolution of SOC and its association with the nematic phase and superconductivity.One way to achieve this in iron pnictides is to studythe energy, wave vector, temperature, and doping dependence of thespin excitation anisotropy using neutron polarization analysis. Compared with angle resolved photoemission experiments <cit.>, polarized neutron scattering experiments typically have much better energy and momentum resolution <cit.>.In previous work on electron-doped BaFe_2-xTM_xAs_2 <cit.> and hole-doped Ba_1-xK_xFe_2As_2 iron pnictides <cit.>,there are clear evidence for spin excitation anisotropy in the paramagnetic tetragonal phasewith M_a≈ M_c>M_b, where M_a, M_b, and M_c are spin excitations polarized along the a, b, and c-axis directions of the AF orthorhombic lattice, respectively,at temperatures well above T_N and T_s <cit.>.Although low-energy spin waves in the parent compound BaFe_2As_2 are also anisotropic in the orthorhombic AF ordered state with M_c>M_b>M_a <cit.>, temperature dependence of the inelastic magnetic scattering at theAF ordering wave vector Q_AF= Q_1=(1,0,1) [Figs. 1(b) and 1(c)] and an energy transfer of E=10 meV changes from isotropic to anisotropic on cooling below T_N <cit.>. However, the energy scale of isotropic paramagnetic scattering at E=10 meV in BaFe_2As_2 is considerably larger than that of the anisotropicparamagnetic spin excitation in doped superconductors(E<6 meV)<cit.>.Since the SOC-induced spin space anisotropy is present in the paramagnetic tetragonal phase of doped iron pnictide superconductors and is alsoexpected to be present in undoped BaFe_2As_2, it is possible that paramagnetic spin excitations in BaFe_2As_2 are also anisotropic,but with an energy scale smaller than E=10 meV. To test if this is indeed the case, we carried out polarized neutron scattering experiments on BaFe_2As_2 with T_N≈ T_s≈ 136 K to study thetemperature dependence of the spin excitation anisotropy [Fig. 1(d)].In the AF ordered state at T=135 K, we find M_c>M_b>M_a at Q_AF=(1,0,1) [Figs. 2(a), 2(b), and 3(a)],confirming the earlier results at 10 K <cit.>. On warming to T=138 K (>T_N,T_s) in the paramagnetic tetragonal state, spin excitations at Q_AF=(1,0,1) are still anisotropic below E=4 meV but with M_a>M_b≈ M_c [Figs. 2(c), 2(d), and 3(b)]. For comparison, spin excitations at the AF zone boundary (ZB)Q_ZB=(1,0,0) are isotropic for energies above E=2 meV [Fig. 3(d)].Upon further warming to T=160 K, paramagnetic scattering becomes isotropicat all energies probed (8≥ E≥ 2 meV) [Fig. 3(c)].While temperature dependence of the spin excitations at E=8 meV and Q_AF=(1,0,1) transforms from isotropic to anisotropicbelow T_N with no evidence of critical scattering consistent with earlier measurements at E=10 meV <cit.>, paramagnetic scattering at E=2 meVstarts to develop spin space anisotropy below about 160 K with enhanced M_a (>M_b≈ M_c) on approaching T_N due to condensationof the longitudinal component of the magnetic critical scattering into a-axis aligned AF Bragg peak below T_N [Fig. 4(a)-4(f)] <cit.>.On the other hand, paramagnetic scattering at E=2 meVand Q_ZB=(1,0,0) is isotropicat all temperatures above T_N [Fig. 4(g)-4(h)]. By comparing these results with spin excitation anisotropy in the paramagnetic tetragonal phase of electron/hole doped iron pnictide superconductors <cit.>, we conclude that electron/hole doping inBaFe_2As_2 necessary to induce superconductivity also enhances thec-axis polarized spin excitations associated with superconductivity. These results arealso in line with the tetragonal C4 magnetic phase with spins aligned along the c-axis in near optimally hole doped superconducting Ba_1-xK_xFe_2As_2 <cit.>.Our polarized neutron scattering experiments were carried out using the IN22 triple-axis spectrometers at the Institut Laue-Langevin, Grenoble, France. Polarized neutrons were produced using a focusing Heusler monochromator and analyzed with a focusing Heusler analyzer with a final wave vector ofk_f=2.662 Å^-1. About 12-g single crystals of BaFe_2As_2 used in previous work <cit.> are used in the present experiment.Figure 1(a) shows the collinear AF structure of BaFe_2As_2 withordered moments along the a-axis <cit.>. The orthorhombic lattice parameters of the AF unit cell are a≈ b≈ 5.549 Å, and c=12.622 Å. The wave vector transfer Q in three-dimensional reciprocal space in Å^-1 is defined as Q=H a^*+K b^*+L c^*, with a^*=2π/aâ, b^*=2π/bb̂ and c^*=2π/cĉ, where H, K and L are Miller indices. The samples were co-aligned in the [H,0,L] scattering plane [Figs. 1(b) and 1(c)]. In this notation, the AF Bragg peaks occur at [1,0,L] with L=1,3,…, while the AF zone boundaries along the c-axis occur at L=0,2,….The magnetic responses at a particular Q along the a-, b-, and c-axis directions are marked as M_a, M_b, and M_c, respectively as shown in Fig. 1(b). In the paramagnetic tetragonal state, these correspond to magnetic excitations polarized along the in-plane longitudinal, in-plane transverse, and out-of-plane directions, respectively.The neutron polarization directions x, y, and z are defined as along Q, perpendicular to Q but in the scattering plane,and perpendicular to both Q and the scattering plane, respectively [Fig. 1(c)]. From the observed neutron spin-flip (SF) scattering cross sections σ^SF_x, σ^SF_y, and σ^SF_z, we can calculate thecomponents M_a, M_b, and M_c via σ _x^SF = R/R+1 (sin ^2 θ M_a + cos ^2 θ M_c) + R/R+1 M_b +B,σ _y^SF = 1/R+1 (sin ^2 θ M_a + cos ^2 θ M_c) + R/R+1 M_b +B, andσ _z^SF = R/R+1 (sin ^2 θ M_a + cos ^2 θ M_c) + 1/R+1 M_b +B, where R is the flipping ratio(R=σ _Bragg^NSF/σ _Bragg^SF≈ 13) and B is the background scattering. By measuring σ_x,y,z^SF at two equivalent AF zone center wave vectors Q_AF= Q_1 = (1,0,1) and Q_2 = (1,0,3),one can determine all three components of the magnetic response M_a, M_b, and M_c <cit.>.For the zone boundary position at Q_0=(1,0,0) with θ=0, one can determine M_b and M_c using σ_x,y,z^SF at this position.To determine the magnetic ordering temperature of BaFe_2As_2, we show in Fig. 1(d) background subtracted elastic SF cross section σ _x^SF measured at Q_1 = (1,0,1).The solid line is a fit of the magnetic order parameter withGaussian convolved power-law M(T)^2=B^2∫(1-T/T_N)^2βe^-(T-T_N)^2/2σ^2 <cit.>. Although this formula is used to account for sample inhomogeneities and a distribution of Néel temperaturesin Co-doped Ba(Fe_1-xCo_x)_2As_2<cit.>, we use it for pure BaFe_2As_2, where disorder is not expected to be important,to compare with β and σ in lightly Co-doped samples.We find T_N=135.9±0.4 K, σ = 0.51 ± 0.07, and β = 0.1 ± 0.02 for BaFe_2As_2. While the value of σ in BaFe_2As_2 is very similar to that of x=0.021 suggesting a small distribution of T_N <cit.>,the β value is considerably smaller than the Co-doped samples but similar to previous value of β=0.103 for pure BaFe_2As_2 <cit.>. Figure 2 shows energy scans at the AF wave vectors Q_1=(1,0,1) and Q_2=(1,0,3) attemperatures below and above T_N.In an isotropic paramagnet with negligible background scattering and R→∞, we would expect σ _x^SF /2≈σ _z^SF≈σ _y^SF. At T=135 K below T_N,magnetic scattering at Q_1=(1,0,1) shows strong anisotropy with σ _z^SF > σ _y^SF [Fig. 2(a)]. Figure 2(b) plots similar scan at Q_2=(1,0,3) with σ _z^SF≈σ _y^SF. Since Q_1=(1,0,1) and Q_2=(1,0,3) correspond to angles of θ_1=23.4^∘ and θ_1=52.4^∘, respectively [Fig. 1(c)],we can use σ_x,y,z^SF at these two wave vectors to completely determine M_a, M_b, and M_c <cit.>.Figure 3(a) shows our calculated M_c, M_b, and M_a (M_c>M_b>M_a), and the outcome is similar to spin excitationsof BaFe_2As_2 <cit.> and BaFe_1.91Co_0.09As_2 <cit.> in the low-temperature AF ordered phase. In previous work, it was found that paramagnetic spin excitations of BaFe_2As_2 above T_N and T_s are isotropic at E=10 meV and Q_1=(1,0,1) <cit.>. To see if spin excitation anisotropy ispresent at T=138 K (>T_N,T_s) in the paramagnetic tetragonal state, we carried out constant-Q measurements at Q_1 [Fig. 2(c)] and Q_2 [Fig. 2(d)]. Inspection of the figures finds clear difference in spin excitations (σ _z^SF > σ _y^SF) below about E≈ 6 meV at Q_2.Figure 3(b) shows the energy dependence of M_a, M_b, and M_c obtained by using the data in Figs. 2(c) and 2(d), revealing M_a>M_b≈ M_c for energies below 6 meV. Upon further warming the system to 160 K (>T_N,T_s), magnetic signal at Q_1[Figs. 2(e)] and Q_2[Figs. 2(f)]becomes purely paramagneticisotropic scattering in the energy region probed satisfying(σ _x^SF-B)/2≈ (σ _y^SF-B)≈ (σ _z^SF-B). The energy dependence of M_a, M_b, and M_c shown in Fig. 3(c) confirmthe isotropic paramagnetic nature of the scattering. Figure 3(d) shows the energy dependence of M_b and M_c as obtained from constant-Q scanat the zone boundary Q_0=(1,0,0), indicating isotropic paramagnetic scattering at energies probed.Figures 3(a)-3(c) summarize temperature evolution of the estimated M_a, M_b, and M_c at the AF zone center Q_AF, obtained by using data inFig. 2 after taking into account the magnetic form factor differences at Q_1 and Q_2 and other effects as shown in Ref. <cit.>.In the AF ordered state at T=135 K (≈ T_N-1 K), the M_c component dominates the spin excitation spectrum below 10 meV, followed by M_b and M_a [Fig. 3(a)].For comparison, the M_a component of the spin waves is completely gapped out below ∼10 meV at 2 K [dashed line in Fig. 3(a)].When warming the system to T=160 K (≈ T_N+24 K), paramagnetic scattering is isotropic in spin space at all probed energieswith M_a=M_b=M_c. At a temperature T=138 K (≈ T_N+2 K) slightly above T_N, paramagnetic spin excitations are anisotropic below ∼5 meV with M_a>M_b≈ M_c.In previous unpolarized neutron scattering experiments on BaFe_2As_2 <cit.>, two-dimensional (2D) magnetic critical scattering has been observed at temperatures far aboveT_N. Upon cooling, the longitudinal component of the critical scattering above T_N (M_a) is expected to increase with decreasing temperature and condense into the 3D AF Bragg positions at the 2D-3D dimensional crossover temperature T_ 3D near T_N <cit.>. The transverse components of spin excitations (M_b and M_c) are the spin wave contributions not expected to diverge at T_N <cit.>.To test if this is indeed the case, we measured temperature dependence ofσ_x,y,z^SF at E=2 meV and 8 meV at the AF zone center Q_1 [Figs. 4(a) and 4(b)] and Q_2 [Figs. 4(c) and 4(d)]. With decreasing temperature, σ_x,y,z^SF increases in intensity with the differences between σ_z^SF and σ_y^SF most obvious near T_N at Q_2 [Fig. 4(c)]. Using data in Fig. 4(a) and 4(c), we estimate the temperature dependence of M_a, M_b, and M_c in Fig. 4(e).Consistent with the expectations from the magnetic critical scattering measurements <cit.>, we see a diverging longitudinal spin excitations M_a at E=2 meV while transverse spin excitations show nocritical scattering around T_N.On cooling below T_N, all three polarizations of spin excitations are suppressed due to the formation of spin gaps <cit.>.Similar measurements at E=8 meV show isotropic paramagnetic scattering behavior (M_a≈ M_b≈ M_c) down to T_N before splitting into M_c>M_b>M_a seen in the AF ordered state [Fig. 4(f)]. Figure 4(g) shows temperature dependence of the spin excitations σ_x,y,z^SF at E=2 meV and zone boundary position Q_0.We see that magnetic scattering is isotropic at all measured temperatures with no evidence of spin anisotropy. The diverging M_a near T_N in BaFe_2As_2 may arise from the longitudinally polarized spin excitation in the critical scattering regime of a Heisenberg antiferromagnet with Ising spin anisotropy [Fig. 4] <cit.>.This means that the effect of critical scattering in BaFe_2As_2 can force the fluctuating moment along the longitudinal (a-axis) direction in the paramagnetic critical regime without the need for orthorhombic lattice distortion and associated ferro-orbital (nematic) ordering. Although this scenario is interesting, we note thattemperature dependence of spin excitation anisotropy in the paramagnetic state of AF ordered NaFeAs <cit.> and electronunderdoped BaFe_1.904Ni_0.096As_2 <cit.> behave differently.In previous polarized neutron scattering experiments on NaFeAs,which has a collinear AF order at T_N=45 K and an orthorhombic-to-tetragonal lattice distortion at T_s≈ 58 K <cit.>,M_a≈ M_c is larger than M_b in the paramagnetic orthorhombic phase below T_s and the in-plane anisotropy M_a-M_b enhances on approaching T_N from T_s <cit.>.When warming up to above T_s, the statistics of the data in NaFeAs is insufficient to establish possible spin anisotropy <cit.>. Since one of the key differences between BaFe_2As_2 and NaFeAs is the coupled structural and magnetic phase transitions in BaFe_2As_2, our datasuggest that the orthorhombic lattice distortion lifting the degeneracy of the Fe d_xz and d_yz orbitals also induces the M_c and M_b anisotropy. This is consistent withthe observation that M_c has the lowest energy in spin waves of the AF ordered BaFe_2As_2 <cit.> and NaFeAs <cit.>, suggesting that it costs less energy for the a-axis ordered moment to rotate out of the plane than to rotate within the plane.For electron doped BaFe_1.904Ni_0.096As_2superconductor with T_c=19.8 K and T_N≈ T_s=33± 2 K, spin excitation anisotropy at E=3 meV and zone center Q_AF with M_a≈ M_c>M_b first appears below ∼70 K and shows no anomaly across T_s/T_N before changing dramatically below T_c <cit.>.For hole-doped Ba_0.67K_0.33Fe_2As_2 superconductor with T_c=38 K and no structural/magnetic order, spin excitation anisotropy at E=3 meV and Q_AF with M_a≈ M_c>M_b appears below ∼100 K, and also decreases abruptly T_c <cit.>.The similarities of these results to those of NaFeAs in the nematic temperature regime (T_s>T>T_N) suggest that the ferro-orbital order or fluctuations <cit.> in electron and hole-doped BaFe_2As_2 first appear in theparamagnetic tetragonal phase at temperatures well above T_s <cit.>. Since SOC in iron pnictides is a single iron effect not expected to change dramatically as a function of electron and hole doping <cit.>,the weak/absence of M_c and M_b spin excitation anisotropy in the tetragonal phase of BaFe_2As_2 is difficult to understand. One possibility is that the nearly coupled structural and magnetic phase transitions in BaFe_2As_2 <cit.> suppress the role of the SOC inducedferro-orbital fluctuations above T_s.Although hole-doped Ba_1-xK_xFe_2As_2 also has coupled structural and magnetic phase transitions in the underdoped regime <cit.>,it changes to a double-Q tetragonal magnetic structure with ordered moments along the c-axis near optimal superconductivity <cit.>. When hole and electron doping in BaFe_2As_2 reduces the structural and magnetic ordering temperatures, the SOC inducedferro-orbital fluctuations start to appear at temperatures above T_s.In this picture, the spin excitation anisotropy in the superconducting iron pnictides originates from similar anisotropy already present in their parent compounds below T_s. The dramatic change in spin excitation anisotropy across T_c seen in electron and hole-doped BaFe_2As_2 suggests a direct coupling of the SOC to superconductivity.The systematic polarized neutron scattering measurements present here and in previous work on doped BaFe_2As_2 familyof materials <cit.>call for quantitative calculations on how SOC is associated with spin excitation anisotropy in iron pnictides.The neutron scattering work at Rice is supported by the U.S. NSF-DMR-1436006 and NSF-DMR-1362219 (P.D.). The materials synthesis efforts at Rice are supported by the Robert A. Welch Foundation Grant No. C-1839 (P.D.).hosono H. 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"authors": [
"Yu Li",
"Weiyi Wang",
"Yu Song",
"Haoran Man",
"Xingye Lu",
"Frederic Bourdarot",
"Pengcheng Dai"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.str-el"
],
"primary_category": "cond-mat.supr-con",
"published": "20170626072521",
"title": "Spin excitation anisotropy in the paramagnetic tetragonal phase of BaFe2 As2"
} |
Tiziano Zingales University College London INAF- Osservatorio Astronomico di Palermo Giovanna Tinetti University College London Ignazio Pillitteri INAF - Osservatorio Astronomico di Palermo Jérémy Leconte CNRS, Université de Bordeaux Laboratoire d'Astrophysique de BordeauxGiuseppina Micela INAF - Osservatorio Astronomico di PalermoSubhajit Sarkar Cardiff University, Cardiff, UK The ARIEL Mission Reference Sample Tiziano Zingales Giovanna Tinetti Ignazio PillitteriJérémy Leconte Giuseppina Micela Subhajit SarkarReceived: date / Accepted: date ============================================================================================================ The ARIEL (Atmospheric Remote-sensing Exoplanet Large-survey) mission concept is one of the three M4 mission candidates selected by the European Space Agency (ESA) for a Phase A study, competing for a launch in 2026.ARIEL has been designed to study the physical and chemical properties of a large and diverse sample of exoplanets, and through those understand how planets form and evolve in our galaxy.Here we describe the assumptions made to estimate an optimal sample of exoplanets – including both the already known exoplanets and the "expected” ones yet to be discovered – observable by ARIELand define a realistic mission scenario.To achieve the mission objectives, the sample should include gaseous and rocky planets with a range of temperatures around stars of different spectral type and metallicity. The current ARIEL design enables the observation of ∼1000 planets, covering a broad range of planetary and stellar parameters, during its four year mission lifetime. This nominal list of planets is expected to evolve over the years depending on the new exoplanet discoveries.§ INTRODUCTION §.§ Mission overviewToday we know over 3500 exoplanets and more than one third are transiting (http://exoplanets.eu/). Theseinclude Earths, super-Earths, Neptunes and Giant planets around a variety of stellar types. The Kepler space mission has discovered alone more than 1000 new transiting exoplanets between 2009 and 2015 and more than 3000 still unconfirmed planetary candidates. The number ofknown exoplanets is expected to increase in the next decade thanks to current and future space missions (K2, GAIA, TESS, CHEOPS, PLATO) and a long list of ground-based surveys (e.g. HAT-NET, HARPS, WASP, MEarth, NGTS, TRAPPIST, Espresso, Carmenes). They will detect thousands of new transiting exoplanets. ARIEL (Atmospheric Remote-sensing Exoplanet Large-survey) is one of the three candidate missions selected by the European Space Agency (ESA) for its next medium-class science mission due for launch in 2026. The goal of the ARIEL mission is to investigate the atmospheres of several hundreds planets orbiting distant stars in order to address the fundamental questions on how planetary systems form and evolve. Key objective of the mission is to find out whether the chemical composition of exoplanetary atmospheres correlate with basic parameters such as the planetary size, density, temperature, and stellar type and metallicity. During its four-year mission, ARIEL aims at observing a statistically significant sample of exoplanets,ranging from Jupiter- and Neptune-size down to super-Earth and Earth-size in the visible and the infrared with its meter-class telescope. The analysis of ARIEL spectra and photometric data will allow to extract the chemical fingerprints of gases and condensates in the planets' atmospheres, including the elemental composition for the most favorable targets. It will also enable the study of thermal and scattering properties of the atmosphere as the planet orbit around the star.The main purpose of this paper is to estimate an optimallist of targets observable by ARIEL or a similar mission in ten years time and quantify a realistic mission scenario to be completed in 4 year nominal mission lifetime, including the commissioning phase.To achieve the mission objectives, the sample should include gaseous and rocky planets with a range of temperatures around stars of different spectral type and metallicity. With this aim, it is necessary to consider both the already known exoplanets and the “expected” ones yet to be discovered.The data collected by Kepler allow to estimate the occurrence rate of exoplanets according to their size and orbital periods. Using this planetary occurrence rate and the number density of stars in the Solar neighbourhood, we can estimate the number of exoplanets expected to exist with a particular size, orbital period range and orbitinga star of a particular spectral type and metallicity. Here we describe the assumptions made to estimate an optimal sample of exoplanets observable by ARIEL and define the Mission Reference Sample (MRS). It is clear thatthis nominal list of planets will change over the years depending on the new exoplanetary discoveries. In Section <ref> we explain the method used to estimate the number and the parameters of the planetary systems yet to be discovered. All the potential ARIEL targets will be presented in Section <ref>, where we show all the planets that can be observed individually during the mission lifetime, and out of which we want to select the optimal sample. Section <ref> is dedicated to the selection and description of an ARIEL MRSfulfillingthe mission requirements, we compare the proposed ARIEL MRS to the sample expected to be discovered by TESS, confirming that TESS could provide a large fraction of the ARIEL targets. A sample including only planets known today is identified. In Section <ref> we show a possible MRS which maximises the coverage of the planetary and stellar physical parameters. §.§ Description of the models We use the ESA Radiometric Model <cit.> to estimate the performances of the ARIEL mission given the planetary, stellar and orbital characteristics: namely the stellar type and brightness, the planetary size, mass, equilibrium temperature and atmospheric composition, the orbital period and eccentricity. This tool takes into account the mission instrumental parameters and planetary system characteristics to calculate: * The SNR (Signal to Noise Ratio) that can be achieved in a single transit;* The SNR that can be achieved in a single occultation;* The number of transit/occultation revisits necessary to achieve a specified SNR;* The total number and types of targets that can be included in the mission lifetime. In this work, the input planet list for the radiometric model is a combination of known and simulated exoplanets, as detailed in the following sections. We used the instrument parameters of the ARIEL payload as designed during the phase A study.To increase the efficiency of our simulations we used a Python tool as a wrap of the ESA Radiometric Model, so we could test different mission configurations that fulfil the mission science objectives. The results were validated with ExoSim, a time domain simulator used for the ARIEL space mission, but thanks to its modularity it can be used to study any transit spectroscopy instrument from space or ground. ExoSim has been developed by <cit.> (see App <ref>).§ SIMULATIONS OF PLANETARY SYSTEMS EXPECTED TO BE DISCOVERED IN THE NEXT DECADE§.§ Star count estimateWe used the stellar mass function as obtained from the 10-pc RECONS (REsearch Consortium On Nearby Stars) to estimate the number of stars as a function of the K magnitude. We assume mass-luminosity-K magnitude conversions from <cit.>. The same procedure was adopted by <cit.>. The number of main sequence stars with limit K-mag m_K = 7 used to infer the number density of stars in the Solar neighbourhood is shown in Tab <ref>. The number density and the number of stars are related through Eq <ref>:ρ_* = N_*(K < 7)/4/3π d^3where the distance d has been calculated in the ARIEL Radiometric Model <cit.> using the relation between K magnitude m_K and the distance d:m_K = -2.5 logR_*^2S_s(Δλ)/d^2S_0^K(Δλ)In Eq <ref>, R_* is the stellar radius, S_0^K (Δλ) is the zero point flux for the standard K-band filter profile, Δλ is the filter band pass given in <cit.> and S_s (Δλ) the stellar flux density evaluated over the same bandwidth. We neglect the interstellar absorption since our stars are at a relatively short distance. §.§ Planetary population and occurrence rateIn this section webriefly review the current knowledge about the occurrence rate ofplanets, i.e. the average expected number of planets per star. <cit.> used the Kepler statistics to publish the planetary occurrence rates around F, G, K main sequence stars ordered by orbital periods and planetary types.An accurate planetary occurrence rate is pivotal to the reliability of the estimate of the existing planets in the Solar neighbourhood. We used the planetary occurrence rate values for F,G,K and M stars from <cit.>, being the most conservative and currently the most complete, i.e. covering all planetary types and stars. Therefore, our estimates for the ARIEL sample are very conservative.<cit.>updated the planetary occurrence rate for planets between 0.5R_⊕ and 4R_⊕ and orbital period < 50 days, using a more recent list of planets discovered by the Kepler satellite. Fig <ref> shows the comparison between <cit.> and <cit.>. The differences between the two occurrence ratescan be up to an order of magnitude.<cit.> show that M stars have 3.5 times more small planets (1.0-2.8 R_⊕) thanFGK stars, but two times fewer Neptune-sized and larger (>2.8 R_⊕) planets. The fraction of M-stars considered in our work is only ∼ 7% of the total stellar sample, so we are significantly underestimating the number of small planets around M-dwarfs, which are optimal targets for transit spectroscopy. More recent and completeresults from Mulders and collaborators are expected to be published in the next months and they are not yet available for our simulations. Given the discrepancy between Mulders and Fressin's statistics we expect a substantial improvement in our estimates when the most recent Kepler statistics will become available.<cit.> provided the following statistics fordifferent planetary classes:* Jupiters: 6R_⊕ < R_p ≤ 22R_⊕* Neptunes: 4R_⊕ < R_p ≤ 6R_⊕* Small Neptunes: 2R_⊕ < R_p ≤ 4R_⊕* Super Earths: 1.25R_⊕ < R_p ≤ 2R_⊕* Earths: 0.8R_⊕ < R_p ≤ 1.25R_⊕We adopted a size resolution of 1R_⊕ in each of these classes.The number of planets can be estimated as:N_p = 4/3π d^3 ρ_* P_t, p P_geomwhere d is the radius of a spherewith the Sun at the centre, ρ_* is the number density of the stars, P_t, p is the probability of having a t-type planet orbiting with an orbital period p (See Fig <ref>). P_geom =R_* / a is the geometrical probability of a transit.We simulated all the transiting planets in the solar system neighbourhood up to m_K = 14, all these planets described by N_p constitute the “Mission Reference Population”.To include in the population sample the exoplanets known today, every time we predict a system with the same physical properties of a known system we replace it with the known one. In Sec <ref> we show that in the solar system neighbourhood there are ∼ 9500 planets for which the ARIEL science requirements can be reached in less that 6 transits or eclipses.The equilibrium temperature (Eq <ref>) of the planet can be evaluated assumingthe incoming and outgoing radiation at the planetary surface are in equilibrium: T_p = T_*( R_*/2a)^1/2( 1-A/ε)^1/4 Here T_* and R_* are the stellar temperature and radius,a the semi-major axis of the orbit, A is the planetary albedoand ε is the atmospheric emissivity. The ARIEL space mission will focus on planets with an orbital period shorter than 50 days. As expected, shorter periods mean shorter semi-major axis and, therefore, from Eq <ref>, typically higher effective temperature. §.§ Planetary masses and densitiesTo simulate a realistic planetary population we need to consider a distribution of masses given a planetary radius. The planetary mass affects directly the surface gravity and therefore the scale height (H) of the atmosphere: H = k T/μg The mass estimate is not a trivial task, given the range of planetary densities observed today. We used a Python tool written by <cit.> to estimate the mass of all the planets in our simulated sample. In the ARIEL planetary sample there are both known and simulated planets. <cit.> use the currently known planets to derive the statistical distribution of the mass of a given planet when its radius is known. Thus, for each planet in our initial sample, the mass is randomly drawn following this distribution except for known systems. In Fig <ref> we show the mass distribution for all the planets in our simulations. Moreover, as a very few planets have a radius larger than 20R_⊕, we use that radius as an upper limit. There is already a well known degeneracy in the 7-20R_⊕ range: objects with a radius within that range can be planets as well as very cool stars. However, this should not be too concerning, as observations have shown that very short-period, low-mass stellar companions are much less frequent than hot giant planets <cit.>. § ARIEL SCIENCE GOALS AND MISSION REFERENCE POPULATION §.§ The 3 tier approachThe ARIEL primary science objectives call for atmospheric spectra or photometric lightcurves of a large and diverse sample of known exoplanets covering a wide range of masses, densities, equilibrium temperatures, orbital properties and host-stars. Other science objectives require, by contrast, the very deep knowledge of a select sub-sample of objects. To maximise the science return of ARIEL and take full advantage of its unique characteristics, a three-tiered approach has been considered, where three different samples are observed at optimised spectral resolutions, wavelength intervals and signal-to-noise ratios. (a summary of the three-tiers and observational methods is given below in table <ref>). In this section we present the pool of potential targets that could reach the specifications for each tier in a reasonable number of observations. The number of targets for the various Tiers are shown as a function of planetary radius in Fig<ref>, <ref> and <ref> and as a function of effective temperature in<ref>, <ref> and <ref>. Note that the planets shown in these figures do not represent the final sample, as it would take too long to observe all of them. They are the pool from which the MRS can be selected to best address the scientific questions summarized below. The fact that the number of potential targets is much larger than the number that can be observed illustrates that ARIEL can choose the final sample among a great variety of observable planets, providing a lot a flexibility.The key questions and objectives of each tier can be summarised as follows (see Tinetti et al., in prep. for further details):Survey:* What fraction of planets are covered by clouds?– Tier 1 mode is particularly useful for discriminating between planets that are likely to have clear atmospheres, versus those that are so cloudy that no molecular absorption features are visible in transmission. Extremely cloudy planets may be identified simply from low-resolution observations over a broad wavelength range. This preliminary information will therefore allow us to take an informed decision about whether to continue the spectral characterization of the planet at higher spectral resolution, and therefore include or not the planet in the Tier 2 sample.* What fraction of small planets have still hydrogen and helium retained from the protoplanetary disk? – Primordial (primary atmosphere) atmospheres are expected to be mainly made of hydrogen and helium, i.e. the gaseous composition of the protoplanetary nebula. If an atmosphere is made of heavier elements, then the atmosphere has probably evolved (secondary atmosphere). An easy way to distinguish between primordial (hydrogen-rich) and evolved atmospheres (metal-rich), is to examine the transit spectra of the planet: the main atmospheric component will influence the atmospheric scale height, thus changing noticeably the amplitude of the spectral features. This question is essential to understand how super-Earths formed and evolved.* Can we classify planets through colour-colour diagrams or colour-magnitude diagrams? – Colour-colour or colour-magnitude diagrams are a traditional way of comparing and categorising luminous objects in astronomy. Similarly to the Herzsprung-Russell diagram, which led to a breakthrough in understanding stellar formation and evolution, the compilation of similar diagrams for exoplanets might lead to similar developments <cit.>.* What is the bulk composition of the terrestrial exoplanets? – The planetary density may constrain the composition of the planet interior. However this measurement alone may lead to non-unique interpretations <cit.>. A robust determination of the composition of the upper atmosphere of transiting planets will reveal the extent of compositional segregation between the atmosphere and the interior, removing the degeneracy originating from the uncertainty in the presence and mass of their (inflated?) atmospheres. * What is the energy balance of the planet? – Eclipse photometric measurements in the optical and infraredcan provide the bulk temperature and albedo of the planet, thereby allowing the estimation of the planetary energy balance and whether the planet has an internal heat source or not. Deep:A key objective of ARIEL is to understand whether there is a correlation between the chemistry of the planet and basic parameters such as planetary size, density, temperature and stellar type and metallicity. Spectroscopic measurements at higher resolution will allow in particular to measure: * The main atmospheric component for small planets;* The chemical abundances of trace gases, which is pivotal to understand the type of chemistry (equilibrum/non equilibrium). * The atmospheric thermal structure, both vertical and horizontal;* The cloud properties, i.e. cloud particles size and distribution, * The elemental composition in gaseous planets. This information can be used to constrain formation scenarios <cit.>. Benchmark:A fraction ofplanets around very bright stars will be observed repeatedly through time to obtain: * A very detailed knowledge of the planetary chemistry and dynamics;* An understanding of the weather, and the spatial and temporal variability of the atmosphere. Benchmark planets are the best candidates for phase-curve spectroscopic measurements.§.§ Key science questionsIn this section we show a full list of potential targets for ARIEL: these are expected to evolveuntil launch, and will be updated regularly to include newplanet discoveries.ARIEL Tier 1 (Survey) will analyse a large sample of exoplanets to address science questions where a statistically significant population of objects needs to be observed. ARIEL Tier 1 will also allow a rapid, broad characterisation of planets permitting a more informed selection of Tier 2 and Tier 3 planetary candidates. For most Tier 1 planetary candidates, Tier 1 performances can be reached between 1 and 2 transits/eclipses. In Fig <ref> and <ref> we show that in the solar system neighbourhood there are ∼ 9500 observable by ARIEL for which the science requirements can be reached in less than 6 transits or eclipses.ARIEL Tier 2 (Deep, the core of the mission) will analyse a sub-sample of Tier 1 planets with a higher spectral resolution, allowing an optimal characterisation of the atmospheres, including information on the thermal structure, abundance of trace gases, clouds and elemental composition. In Fig <ref> and <ref> we show the properties of all the planetary candidates that could be studied by ARIEL in the Deep mode with a small/moderate number of transit or eclipse events. The third ARIEL Tier (Benchmark, the reference planets) will study the best planets (section <ref>), i.e. the ones orbiting very bright stars which can be studied in full spectral resolution with a relatively small number of transits/eclipses.For the planets observed in benchmark mode in 1 or 2 events, it is possible to study the spatial and temporal variability (i.e. study the weather and evaluate its impact when observations are averaged over time). In Fig <ref> and <ref> we show the properties of the Tier 3 planetary candidates. § A POSSIBLE SCENARIO FOR THE ARIEL SPACE MISSIONIn Section <ref> we presented a comprehensive list of planet candidates which could be observed with the ARIEL space mission.Here we discuss possible optimisations of the Mission Reference Sample, which ideally should include a large and diverse sample of planets, have the right balance among the three Tiers and, most importantly, must be completed during the nominal mission lifetime (4 years including the commissioning phase). In Fig <ref> we show a possible MRS with all the three tiers nestedtogether. The first MRS shows the maximum number of targets, taking into account the nominal mission lifetime. It has been built starting from all the targets feasible within one transit/eclipse, and adding all the targets that can be done within 2, 3, 4 and so on transits/eclipses in ascending order. This is just one of the possible configurations for the MRS, and one would expect the ARIEL MRS to evolve in response of new exoplanetary discoveries in the next decade. §.§ MRS Tier 1: SurveyOur simulations indicate that the current ARIEL design as presented at the end of the Phase A study, allows to observe 1002 planets in Tier 1. All the planets can be observed in 1538 satellite visits i.e. 37% of the mission lifetime. Most giant planets and Neptunes fulfil the Tier 1 science objectives in 1 transit/eclipse, the smaller planets require up to 6 events (fig. <ref> and <ref> ). Fig. <ref> and <ref> illustrate how the 1002 planets are distributed in terms of planetary size, temperature, density and stellar type. §.§ MRS Tier 2: DeepThe Deep is the core of the mission. Our simulations indicate that the current ARIEL design as presented at the end of the Phase A study, allows to observe ∼ 500 planets in Tier 2 assuming 60% of the mission lifetime. Fig. <ref> and <ref> illustrate how the 500 planets are distributed in terms of planetary size, temperature, density and stellar type. Most gaseous planets fulfil the Tier 2 science objectives in less than five transits/eclipses, the small planets require up to twenty events (fig. <ref> and <ref> ). We included a variety of planets from cold (300 K) to very hot (2500 K) as shown in Fig <ref>. We scheduled also ∼ 50 planets that will be studied with both transit and eclipse methods, indicated by stripes in Fig <ref>). These are the best candidates for phase-curves observations, which can be included at the expenses ofthe number ofTier 2 planets observed.§.§ MRS Tier 3: BenchmarkIn the current MRS, we have selected as Tier 3, 67 gaseous planets for weather studies. Fig. (<ref>) shows the temperature distribution covered by the Tier 3 sample. Only 3% of the mission lifetime is required to achieve the Tier 3 science objectives for this sample.§.§ Compliance with TESS expected yieldsThe Transiting Exoplanet Survey Satellite (TESS) is expected to provide a large fraction of the targets observable by ARIEL. The numbers of targets envisioned in the sample presented here are perfectly in line with the expected yield from The Transiting Exoplanet Survey Satellite (TESS), as shown in Fig <ref> where we compare the expected TESS discoveries and the ARIEL MRS. We see that the ARIEL MRS is well within the TESS sample <cit.>. The success of the TESS mission will allow the characterisation of hundreds of planets by ARIEL. §.§ ARIEL MRS with currently known targets In February 2017 ∼210 transiting planets fulfill the ARIEL previous criteria. It means that, even if ARIEL were launched tomorrow, it would observe at least 210 relevant targets.Using the planets known today, we could organise the MRS into the following three tiers: * Survey: 210 planets using 30% of the mission lifetime (Fig <ref>);* Deep: 158 planets using 60% of the mission lifetime (Fig <ref>);* Benchmark: 67 planets using 10% of the mission lifetime (Fig <ref>).In Fig <ref>, <ref> and <ref> we showthe key physical parameters of the known planets defining the current observable MRS. In Fig <ref> and <ref> we show the properties of the stellar hosts. As mentioned previously, the number of known planets is expected to increase dramatically in the future.Pictorial representation (M. Ollivier, private comm.) of the known planets sky coordinates and their sky visibility all over the year is given in Fig <ref>. It shows that objects far away from the ecliptic plane will be visible longer than the planet close to this plane. § MRS OPTIMISATION FOR STELLAR HOSTS In this section we show another possible selection of the Tier 1 sample that maximises also the diversity of stellar hosts, additionally to other planet parameters.In particular, the stellar metallicity is expected to play an important role in the planet formation process and type of chemistry of the planet <cit.>. ARIEL will also collect important data to understand better the relationship between stellar metallicity and planetary characteristics. §.§ MethodWe will limit our analysis to those systems which can be studied in up to six visits for each planet (either a transit or an occultation).We chose four physical quantities that define a 4D space to distribute the ARIEL targets. The quantities are: stellar effective temperature (T_eff ), metallicity ([Fe/H]), planetary radius (R_pl) and planetary theoretical equilibrium temperature (T_pl). For the metallicity we use the values observed in the solar neighbourhood and reported by <cit.>. We adopt three bins for stellar T_eff, [Fe/H] and planetary R_pl, while for the T_pl we use five bins, as detailed in Table <ref>. The three T_eff bins correspond approximately to the ranges of spectral types M-Late / K stars, Early K-G stars and F-G stars, respectively, as indicated in the labels in Fig <ref> to <ref>. Analogously, we separated the sample in low metallicity, solar metallicity and high metallicity, according to individual temperature values. The binning into 3 intervals of T_eff, [Fe/H] and R_pl is a reasonable trade-off between a detailed representation of the sample and a simple visualization of the richness and diversity of the physical configurations of the sample. We inferred from <cit.> that the metallicities of stars in the solar neighbourhood are consistent with a normal distribution with mean -0.1 and standard deviation sd=0.2. Using such model distribution we simulated the values of [Fe/H] for each star in the ARIEL sample.The 4D space of T_eff , [Fe/H], R_pl and T_pl is composed by a total of 3×3×3×5=135 cells. We assume that 10 systems are sufficiently reliable to determine the properties of the atmospheres of planets in each cell. §.§ Results The population of 9545 planets is distributed in the 4-D bins as in Fig <ref>.From this distribution we selected 1002 exoplanets, requiring altogether 1538 satellite visits. These 1002 planets are distributed in the 4D space as shown in Fig. <ref>. The 3×3 panel grid distributes the sample along the 3 spectral types and the metallicity ranges reported in Table <ref>. Each panel is a matrix with planetary radii along x-axis and (calculated) equilibrium temperatures along y-axis, as specified in Table <ref> and discussed above. The numbers in each box identify the numbers of systems with the corresponding R_pl, T_pl, spectral type, and [Fe/H] values. The 1002 systems in Fig. <ref> tend to populate the cells corresponding to F-G-early and K stars orbited by Neptunes/Jupiters size planets (with a number of planets per cell N>20), as these systems are the easiest to be observed with high signal to noise and, on average, with one or two visits. At the same time, planets around M or late K stars are much less represented in this distribution, especially planets smaller than Neptunes.This issue is addressed by prioritising these targets over the rest of the population. We found that planets around M stars require on average more visits than the analogues around early K, G, and F stars. We managed to select 908 planets and, in particular, 594 of them require only 1 visit (65.4%), 151 planets require 2 visits (16.6%), 83 planets require 3 visits (9.1%), 41 planets require 4 visits (4.5%), and 39 planets require 5 visits (4.4%). The corrected sample is shown in Fig <ref>, where now ∼ 19% of the population are Earths/Super Earth or Neptunes around M or K stars observable with less than 6 visits.Assuming a total number of visits as in the 1002 planets configuration (approximately 1500 visits), we fixed the maximum number of systems (10 planets in our choice) in each 4D space cell. This choice implies that any additional targets in an “already full" cell will be discarded. In this way we can include planets in the empty or poorly populated parts of the parameter space. The goal is to verify that we can cover with enough statistics most of the 4D parameter space. The distribution of systems selected with such criteria is shown in Fig. <ref>. Compared to Fig. <ref>, we see that we can efficiently cover most of the 4D space in planetary sizes, planetary temperatures, host temperatures and metallicities, apart from those combination of parameters corresponding to not physical or rare systems (e.g., very hot planets around very cool stars). Our selection is composed by 908 unique planets requiring a total of 1504 visits. Among already known systems, 92 of the initial 211 systems are in this new list. This selection is not unique, and depends on our choices, but our exercise shows that we have great freedom on the final choice on how to spend ARIEL observing time, as it can be easily tuned on specific needs.Fig. <ref> shows the average number of visits required to cover each cell of the 4D space. The number of visits needed for Jupiters and Neptunes is, typically, one or two, while Earths/Super Earths require from 3 to 5 visits each. To summarise, out of the 908 planets in our selection there are 594 planets requiring only 1 visit (65.4%), 151 planets requiring 2 visits (16.6%), 83 planets requiring 3 visits (9.1%), 41 planets requiring 4 visits (4.5%), and 39 planets requiring 5 visits (4.4%).As a final comment, we have verified that, by increasing the maximum number of systems per 4D cell while keeping fixed the total number of visits to ∼1500, we obtain that the number of observed planets increases (for example assuming N=15 as maximum systems per cell, we can observe up to 1000 systems), but at the same time the 4D cells of systems with cold/warm Earths/Super Earths would tend to be left empty and thus unexplored. This exercise shows the degree of flexibility offered by ARIEL in the choice of the target sample.§ CONCLUSIONSIn this paper we demonstrated that the current ARIEL design enables the observation of 900-1000 planets during its four-year lifetime, depending on the physical parameters of the planet/star systems which one wants to optimise. The optimal sample of targets fulfils all the science objectives of the mission. While we currently know only ∼200 transiting exoplanets which could be part of the mission reference sample, new space missions and ground-based observatories are expected to discover thousands of new planets in the next decade. NASA-TESS alone is expected to deliver most ARIEL targets.§ ACKOWLEDGEMENTS T. Z. is supported by the European Research CouncilERC projects ExoLights (617119) and from INAF trough the "Progetti Premiali” funding scheme of the Italian Ministry of Education, University, and Research. I.P and G.M. are supported by Ariel ASI-INAF agreement No. 2015-038-R.0. G.T. is supported by a Royal Society URF.We thank Enzo Pascale and Ludovig Puig for their help in setting up the ESA's Radiometric model.§ ESA RADIOMETRIC MODEL VALIDATION WITH EXOSIMWe compare the out-of-transit signal and noise from ESA Radiometric Model (ERM) with that from ExoSim. An early version of ARIEL with a grating design was used for the instrument model in each.We model 55 Cancri and GJ 1214 with the same PHOENIX spectra in each simulator and include only photon noise and the noise floor, N_min(λ), which is dominated by dark current noise. All the calculations are done per unit time and per spectral bin (R=30 in Ch1 and R=100 in Ch0). The noise variance was compared assuming an aperture mask on the final images, and the noiseless signal per unit time was compared assuming no aperture. In the ERM, we use the following expression for N_min giving the noise variance:N_min(λ) = 2.44 fλ^2/mRΔ_pix^2I_dcwhere I_dc is the dark current per pixel, m is the reciprocal linear dispersion of the spectrum in μm wavelength per μm distance, R is the spectral resolving power and Δ_pix is the pixel pitch. The noise variance from ExoSim is given as the average of 50 realizations with a standard deviation (shown as error bars in the following figures). For 55 Cancri e case (Fig <ref>), over all wavelength bins, the ERM signal is always within 2% of ExoSim, and the averaged noise variance within 5% of the ERM.In 94% of the bins, the ERM noise variance is within the standard deviation from ExoSim.For GJ 1214 (Fig <ref>), the ERM signal is within 4% of ExoSim over all bins and the averaged noise variance within 6% of ExoSim over all bins. The ERM noise variance is always within the standard deviation from ExoSim over all bins.There is therefore good agreement between the two simulators. § KNOWN PLANETS OBSERVABLE BY ARIELapalike | http://arxiv.org/abs/1706.08444v3 | {
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Dahlem Center for Complex Quantum Systems and Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, GermanyDepartment of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USAWe present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions f in the lowest Landau level, with Coulomb interactions. At f=1/3,1/5, and 2/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at f=1/3. 03.67.Mn, 73.43.-f, 71.23.An Evolution of quantum entanglement with disorder in fractional quantum Hall liquids R. N. Bhatt December 30, 2023 ==================================================================================§ INTRODUCTIONThe rich physics of two-dimensional (2D) electron systems subject to a strong perpendicular magnetic field in the fractional quantum Hall (FQH) regime has attracted intensive interest. In particular, studying the FQH effect<cit.>, namely the quantized Hall conductance plateaus near various fractional Landau level filling fractions f<cit.>, is not only fundamentally crucial for understanding various types of exotic topological order<cit.>, but is also closely relevant to the technological development of quantum computers<cit.>.The FQH plateaus, unlike plateaus near integer fillings whose explanation requires only single-particle physics<cit.>, are a consequence of the strong electron-electron interaction combined with (the ever-present) weak disorder in the system. The interaction opens an energy gap to protect the topologically ordered FQH ground state at filling f with a quantized Hall conductance, and the weak disorder localizes excitations to provide a finite Hall conductance plateau near f. However, if disorder is much stronger than the interaction scale, the topological FQH ground state and its corresponding plateau will be eventually destroyed, which is consistent with the experimental fact that the FQH effect only exists in samples with high mobility<cit.>. A few numerical studies<cit.> of the disorder-driven transition from a topological FQH state to an insulating phase exist, tracking the closing of the energy gap and mobility gap and the collapse of Hall conductance quantization with increasing disorder strength.Tremendous effort has also gone into understanding FQH physics at fillings where Hall conductance plateaus are absent<cit.>. A representative example is f=1/2, where both experimental<cit.> and theoretical<cit.> work has suggested that composite fermions (CFs)<cit.> form a Fermi sea in clean systems. Although the precise nature of the CF sea is still a central topic of current research<cit.>, early studies already predicted that a transition to an insulator occurs at strong disorder<cit.>.Following the extensive applications of quantum entanglement spectroscopy methods to clean FQH systems<cit.>, corresponding studies in disordered FQH liquids started recently<cit.>. In particular, the ground-state entanglement entropy has been demonstrated as a new and powerful diagnostic of disorder-driven transitions in FQH liquids<cit.>. In Ref. zhao2016, we first applied this diagnostic to electrons with Haldane's pseudopotential interaction<cit.> at f=1/3 in the lowest Landau level (LLL), by tracking the entanglement evolution with increasing disorder. It was shown that the phase transition point can be precisely identified by a sharp increase in the magnitude of the ground-state entanglement entropy derivative with respect to disorder, and a finite-size scaling analysis of the entropy derivative can be used to extract the critical exponent ν of the diverging correlation length at the transition point. Moreover, it was found that the nearest-neighbor level repulsion statistics of the ground-state entanglement spectrum (for the same system sizes as the entanglement entropy) does not dramatically change at the critical point, and is thus not sensitive to the phase transition.In this paper, we extend our research of the disorder-driven entanglement evolution to other fractional filling fractions, where the ground states at zero disorder are either incompressible topological FQH states or gapless CF seas. We consider Coulomb interaction between electrons, which is more realistic than the Haldane's pseudopotential used in Ref. zhao2016. The structure of this paper is as follows. We first introduce our model in detail in Sec. <ref>, including the many-body Hamiltonian, the disorder models, the underlying symmetries, and the definitions of the ground-state entanglement entropy. Then in Secs. <ref> and <ref>, we focus on the Laughlin fillings f=1/3,1/5 and the hierarchy filling f=2/5, where the ground states at zero disorder are incompressible topological FQH states. At all of these fillings, we observe a similar evolution of the ground-state entanglement entropy to that reported in Ref. zhao2016, and the entropy derivative with respect to disorder provides a clear signal for the expected transition from a topological FQH state to an insulator. We extract the length exponent ν of this transition by a finite-size scaling analysis. In Sec. <ref>, we consider f=1/2 where the ground state at zero disorder is a gapless CF sea. We examine two disorder models, for both of which we observe a very chaotic behavior of the entropy derivative without a convincing phase transition signal for all probed system sizes within our exact diagonalization (ED) capability. However, the scaling of the entanglement entropy itself versus the system size changes with increasing disorder, which is consistent with the predicted transition from a CF sea to an insulator. In Sec. <ref>, we summarize our results, and list some open questions for future work. Finally, we also discuss the entropy evolution at f=1/7, and demonstrate the ground-state entanglement spectrum (ES) level statistics at f=1/3 for completeness in the Appendix. Similar to the pseudopotential case in Ref. zhao2016, we also observe the localization of the low-energy part of Coulomb ES with increasing disorder, which, however, again only occurs deeply in the insulating phase.§ MODEL We consider N interacting electrons in a 2D random potential U( r) on an L_1× L_2 rectangular torus penetrated by a uniform perpendicular magnetic field. We suppose Coulomb interaction V( r_i, r_j)=e^2/ϵ1/| r_i- r_j| between electrons, where -e is the electron charge, ϵ is the dielectric constant, and r_i is the coordinate of the ith electron. For convenience, we set e^2/(ϵℓ)=1 and the magnetic length ℓ=1 as the energy and length units throughout the paper. In a strong magnetic field, the energy scales of both interaction and disorder are small compared with the Landau level spacing, so we project the many-body Hamiltonian H=∑_i<j^N V( r_i- r_j)+∑_i=1^N U( r_i) to the LLL, which can be written in the LLL orbital basis asH = ∑_m_1,m_2,m_3,m_4=0^N_ϕ-1V_m_1,m_2,m_3,m_4c_m_1^† c_m_2^† c_m_3c_m_4+ ∑_m_1,m_2=0^N_ϕ-1U_m_1,m_2c_m_1^† c_m_2.Here N_ϕ=L_1 L_2/(2π) is the number of magnetic flux quanta penetrating the torus, and c_m^† (c_m) creates (annihilates) an electron in the LLL orbital m. After choosing the single-particle wave function of orbital m as ψ_m(x,y)=(1/√(π)L_2)^1/2∑_n=-∞^+∞ e^i2π/L_2(m+nN_ϕ)ye^-1/2[x-2π/L_2(m+nN_ϕ)]^2, we can compute the interaction matrix elements V_{m_i} and the disorder matrix elements U_{m_i} by the standard second-quantization procedure, which givesV_{m_i}=1/2δ_m_1+m_2,m_3+m_4^ mod N_ϕ∑_s,t=-∞^+∞δ_t,m_1-m_4^ mod N_ϕV_ q× e^-1/2| q|^2e^ i2π s/N_ϕ(m_1-m_3)andU_{m_i}=∑_s,t=-∞^+∞δ_t,m_1-m_2^ mod N_ϕU_ q e^-1/4| q|^2 e^ iπ s/N_ϕ(2m_1-t).Here δ^ modN_ϕ_i,j is the periodic Kronecker delta function with period N_ϕ, q=(q_x,q_y)=(2π s/L_1,2π t/L_2) with | q|^2=q_x^2+q_y^2, V_ q=1/N_ϕ1/| q| is the Fourier transform of Coulomb interaction, and U_ q=1/2π N_ϕ∫ U( r) e^- i q· rd r is the Fourier transform of U( r). s=t=0 must be excluded from the sum in Eq. (<ref>) to remove the artificial divergence (caused by the lack of a positive countercharge in the above model, which is always present in experiments).We model disorder for the most part using Gaussian white noise, which satisfies ⟨ U( r)⟩=0,⟨ U( r)U( r')⟩=W^2 δ( r- r') and ⟨ U_ q⟩=0,⟨ U_ qU_ q'⟩=W^2/2π N_ϕδ_ q,- q', where W is the strength and ⟨⋯⟩ represents the sample average. In each sample, we generate real U_ q=0 from a Gaussian distribution with zero mean and variance W^2/2π N_ϕ. The real part and imaginary part of U_ q≠ 0 are separately produced from a Gaussian distribution with zero mean and variance W^2/4π N_ϕ. Because U_ q^*=U_- q, the above generation procedures are only implemented for independent U_ q's with q∈{ q|q_x=0,q_y≥ 0}∪{ q|q_x>0}. For filling factor f = 1/2, for reasons that will become clear later, we consider, in addition to Gaussian white noise, an ensemble of short-range scatterers corresponding to U( r)=∑_n W_n e^-| r- R_n|^2/λ^2 and U_ q=λ^2/2N_ϕ∑_n W_n e^-1/4λ^2 | q|^2 e^- i q· R_n. Here λ is the range of scatterers, and W_n is the strength of the nth scatterer and R_n its position, the latter being randomly chosen in each sample. When averaging over N_s samples, we estimate the error bar of quantity A by √((⟨ A^2⟩-⟨ A⟩ ^2)/(N_s-1)).In the absence of disorder, Eq. (<ref>) is invariant under the particle-hole (PH) transform c_m^†↔ c_m up to a constant shift<cit.>. A single disorder configuration breaks this symmetry, because the PH transform replaces U_{m_i} by -U_{m_i}^* in Eq. (<ref>) up to a constant shift, which is equivalent to U_q_x,q_y→ -U_q_x,-q_y in Eq. (<ref>). However, the PH symmetry is statistically preserved by Gaussian white noise, because -U_q_x,-q_y still satisfies the Gaussian white noise conditions. Moreover, if the distributions of W_n and R_n^y for an ensemble of scatterers are symmetric with respect to zero, the PH symmetry is also statistically preserved, otherwise it is broken. On the other hand, the magnetic translation invariance conserved in the interaction term (<ref>) is always broken by the disorder (<ref>), making the numerical simulation significantly more challenging.In the following, we choose the isotropic limit with L_1=L_2=√(2π N_ϕ). In order to study the entanglement properties, we divide the whole system by two cuts at orbital m=0 and m=⌈ N_ϕ/2⌋-1, respectively, where ⌈ x⌋ is the integer part of x. This procedure gives two subsystems A and B with boundary length L=2√(2π N_ϕ), consisting of orbital m=0,...,⌈ N_ϕ/2⌋-1 and m=⌈ N_ϕ/2⌋,...,N_ϕ-1, respectively. We have checked that different positions of the cuts provide the same results statistically for averaged quantities. The entanglement entropy between A and B can be defined as the von-Neumann entropy S(ρ)=- Trρ_Alnρ_A, where ρ_A= Tr_Bρ is the reduced density matrix of part A, and ρ is the density matrix describing a suitably chosen ground-state manifold.For a partially filled LLL at filling f=N/N_ϕ=p/q with coprime p and q, the ground states of any translation invariant Hamiltonian in clean samples are exactly D-fold degenerate with D≥ q, guaranteed by the magnetic translation invariance<cit.>. If D is independent of the system size, such a degeneracy motivates us to consider a ground-state manifold containing the lowest D eigenstates |Ψ_i=1,⋯,D⟩ of the Hamiltonian (<ref>) at any W for consistency, rather than a single eigenstate. In that case, we will use three choices of ρ:(i)ρ=1/D∑_i=1^D|Ψ_i⟩⟨Ψ_i|;(ii)ρ_i=|Ψ_i⟩⟨Ψ_i|;and (iii)ρ_i^min=|Ψ_min^i⟩⟨Ψ_min^i|,where |Ψ_min^i⟩ is the minimally entangled state (MES)<cit.> in the ground-state manifold.Depending on different choices, we correspondingly measure the entanglement entropy by S(ρ), S=1/D∑_i=1^D S(ρ_i), and S_min=1/M∑_i=1^M S(ρ_i^min) respectively<cit.>, where M is the number of MESs. The sum over states in S and S_min is to minimize the effect of statistical fluctuations for a finite number of samples of finite size. However, if D depends on the system size, we just choose a ground-state manifold only containing the lowest eigenstate |Ψ_1⟩ of the Hamiltonian (<ref>), i.e. ρ=|Ψ_1⟩⟨Ψ_1|, so the ground-state entanglement entropy is measured by S(ρ)=S(|Ψ_1⟩).§ LAUGHLIN FILLINGS We first consider fillings f=1/q with q=3 and 5. In clean systems, we have numerically confirmed for various system sizes that the Coulomb ground states at these fillings are always exactly q-fold degenerate, so we choose the lowest q eigenstates of the Hamiltonian (<ref>) as the ground-state manifold. The overlap and energy gap calculations at these fillings suggest that the Coulomb ground states in clean systems are gapped and well captured by the Laughlin model states, although the deviation from the model states increases for larger q (Table <ref>). In the non-interacting limit with W=∞ for PH-symmetric disorder, because extended single-particle states only exist at the center of the LLL band<cit.>, all occupied single-particle states below the Fermi level at f=1/3 and 1/5 are localized, which means that the ground state is an Anderson insulator. Therefore, we expect a transition from the topological Laughlin phase to an insulating phase with increasing disorder at these fillings. In the following, we will characterize this transition by the entanglement entropy of the ground-state manifold, with disorder modeled by Gaussian white noise. §.§ f=1/3For q=3, the Coulomb ground states at zero disorder are very well described by the f=1/3 Laughlin model states, as indicated by the extremely high overlaps that are stable against increasing system size (Table <ref>). The Hall conductance plateau at f=1/3 is the first reported FQH effect in experiments<cit.>. Strong disorder closes the energy gap and the mobility gap, leading to a phase transition to an insulating phase<cit.>. We compute the ground-state manifold by ED for N≤ 10 electrons with Hilbert space dimension up to 30045015, then monitor the evolution of its entanglement entropy with increasing disorder. We first measure the entanglement entropy by S(ρ). Similar to the case of f=1/3 with Haldane's pseudopotential<cit.>, we find that ⟨ S(ρ) ⟩ decreases with W for a fixed system size [Fig. <ref>(a)]. However, it increases with the system size at a fixed W, always agreeing with an area law ⟨ S(ρ) ⟩∝ L [Fig. <ref>(b)]. We further compute the derivative of S(ρ) with respect to the disorder strength, dS(ρ)/dW, approximated in each sample by [S(ρ)|_W+Δ W-S(ρ)|_W]/Δ W with Δ W=0.001W, where only the magnitude of W is changed by a small percentage but the disorder configuration is kept fixed. One can see that all ⟨ dS(ρ)/dW⟩ curves exhibit a pronounced minimum that gets deeper for larger systems [Fig. <ref>(c)]. Except the smallest N=4, this minimum is located at W_c≈ 0.09, which is almost independent of the system size. As demonstrated in a double logarithmic plot [Fig. <ref>(d)], the magnitude of the minimum h=|min_W⟨ dS(ρ)/dW⟩| grows with N as h∝ N^1.10, which is consistent with a divergence in the thermodynamic limit. Informed by the fact that thermal phase transitions are very often characterized by a singularity in the specific heat (which is proportional to the temperature derivative of thermal entropy), we consider this divergence of the disorder derivative of entanglement entropy as a convincing signature of the expected quantum phase transition from the f=1/3 Laughlin phase to an insulating phase. A sharp drop in the entanglement entropy and a pronounced peak in the entanglement derivative were also used to identify a first-order transition in clean bilayer quantum Hall systems as a function of layer separation<cit.>.For a continuous phase transition in our case, the area law shown in Fig. <ref>(b) suggests a scaling behaviorS(ρ) ∝ N^1/2f[N^1/2ν(W-W_c)],for large N (here we have used L=2√(2π N_ϕ)=2√(2π N/f)∝√(N)), leading todS(ρ)/dW∝ N^1/2+1/2νf'[N^1/2ν(W-W_c)]and, in particular,dS(ρ)/dW|_W=W_c∝ N^1/2+1/2ν,where f' means derivative. Thus h∝ N^1.10 in Fig. <ref>(d) implies that ν≈ 0.9. By plotting the rescaled variable ⟨ dS(ρ)/dW⟩/N^1/2+1/2ν versus N^1/2ν(W-W_c) for W_c≈ 0.09 and ν≈ 0.9, we indeed find that besides the smallest size N=4, all data in Fig. <ref>(c) collapse onto a single curve [Fig. <ref>(c) inset].W_c obtained from Coulomb interaction is very different from its counterpart obtained from Haldane's pseudopotential<cit.>. For Coulomb interaction, we find W_c≈ 0.09, which is seven times smaller than the reported value ≈ 0.6 for Haldane's pseudopotential, reflecting the fact that Coulomb ground states are protected by a smaller gap and are more fragile against disorder. Since ν is a critical exponent, however, we would expect that Coulomb interaction and pseudopotential interaction should give roughly the same ν. However, we numerically get ν≈ 0.9 for Coulomb interaction, which is 50% larger than the reported value ≈ 0.6 for Haldane's pseudopotential and almost reaches the conventional ν≥ 2/d bound for d-dimensional disordered systems<cit.>. We have further examined an interpolation between Coulomb and Haldane's pseudopotential, and observed a continuous varying of ν. Such an apparent dependence of ν on the interaction suggests that corrections to finite-size scaling are still significant in the system sizes reached by ED.An alternative measure of the entanglement entropy is given by S. Because the ground states in clean systems are exactly degenerate and different choices of ground states may lead to very different S<cit.>, this quantity is not well defined at W=0. This also causes the larger error bars of ⟨ dS/dW⟩ compared to ⟨ dS(ρ)/dW⟩ at very small W. However, once the disorder is not too weak, the results of S (Fig. <ref>) are very similar to those of S(ρ). Remarkably, the minimum of ⟨ dS/dW⟩ is located at W_c≈ 0.085, which is almost the same as that of ⟨ dS(ρ)/dW⟩ [Fig. <ref>(c)]. The finite-size scaling analysis also gives a similar ν≈ 0.9 [Fig. <ref>(c) inset].Finally, we study the entanglement entropy of the MES<cit.> in the ground-state manifold. We consider all superpositions |Ψ⟩=sinθ_1sinθ_2|Ψ_1⟩+sinθ_1cosθ_2 e^ iϕ_1|Ψ_2⟩+cosθ_1 e^ iϕ_2|Ψ_3⟩ with θ_1,θ_2∈[0,π/2] and ϕ_1,ϕ_2∈[0,2π), then numerically search for the local minima of S(|Ψ⟩) in the parameter space spanned by (θ_1,θ_2,ϕ_1,ϕ_2). |Ψ⟩'s at these local minima correspond to the MES |Ψ_min^i=1,⋯,M⟩. Since searching for these MESs is a complicated four-dimensional minimization problem, we can only reach N=8 electrons with less samples, and do not perform the calculation of ⟨ dS_min/dW⟩. Instead, in Fig. <ref>, we show ⟨S_min⟩ as a function of W. At small disorder, an almost constant ⟨S̅_min⟩ suggests that the ground-state topological properties are the same as those in clean systems. However, the plateau of ⟨S̅_min⟩ is not as good as that for Haldane's pseudopotential<cit.>. We attribute this as being due to the larger finite-size effect in Coulomb ground states. ⟨S̅_min⟩ starts to significantly drop at W≈0.08, signifying a transition point consistent with those indicated by ⟨ dS(ρ)/dW⟩ and ⟨ dS/dW⟩.In summary, at f=1/3, all of the three entanglement measurements give consistent identifications of the transition from the Laughlin phase to an insulating phase. We have examined that this consistency also holds for f=1/5 and 2/5. Therefore, we will only demonstrate the results of dS(ρ)/dW in the remainder of this section as well as in the following section. §.§ f=1/5For q=5, the overlaps between the ground states at zero disorder and the f=1/5 Laughlin model states are still high (Table <ref>), which is consistent with the experimental observation of a robust FQH effect in high-quality samples at f=1/5<cit.>. However, compared with the f=1/3 case, the deviation from the model states is larger, and the energy gaps are smaller, implying that the topological ground states at f=1/5 are more fragile against disorder than those at f=1/3. In the following, we compute the ground-state manifold by ED for N≤ 7 electrons with Hilbert space dimension up to 6724520, and track their entanglement entropy evolution. The reason why we can only reach a smaller N at f=1/5 than at f=1/3 is that the Hilbert space for a fixed N is larger at lower fillings.We show ⟨ dS(ρ)/dW⟩ in Fig. <ref>. Similar to the f=1/3 case, we observe a pronounced minimum in all ⟨ dS(ρ)/dW⟩ curves at f=1/5, whose magnitude increases with the system size [Fig. <ref>(a)]. However, as expected from the overlap and energy gap calculations at zero disorder, this minimum is located at a much smaller W_c≈ 0.012 (except for the smallest system size N=4) than that ≈ 0.09 at f=1/3. The minimum magnitude h also shows a larger finite-size effect than the f=1/3 case. At f=1/5, the data point of N=4 in the ln h-ln N plot obviously deviates from the linear growth of other three points [Fig. <ref>(b)]. With the point of N=4 neglected, we obtain h∝ N^0.90, which suggests ν≈ 1.3 according to the finite-size scaling. Indeed, if we set W_c≈ 0.012 and ν≈ 1.3, all data in Fig. <ref>(a) collapse to a single rescaled curve for N=5-7 electrons [Fig. <ref>(a) inset]. We notice that ν obtained at f=1/5 is larger than that ≈ 0.9 at f=1/3, and is consistent with the ν≥ 2/d bound. § F=2/5 We next go beyond the Laughlin fillings and consider f=2/5. A robust FQH effect was experimentally observed at this filling<cit.>, which can be interpreted as two fully filled effective Landau levels of CFs<cit.>, or the daughter of the f=1/3 FQH effect in the hierarchy scenario<cit.>. At zero disorder, the Coulomb ground states at f=2/5 are incompressible FQH states with large overlaps with the CF ansatz wave functions<cit.>. Here again, for PH symmetric disorder, in the non-interacting limit, all single-particle states below the Fermi level at f=2/5 are localized, leading to an Anderson insulator. Therefore, we again expect a transition from a topological FQH phase to an insulator with increasing disorder at this filling. Since the Coulomb ground states at f=2/5 are fivefold degenerate at zero disorder for all system sizes, we choose the ground-state manifold containing the lowest five eigenstates of the Hamiltonian (<ref>), which is obtained by ED for N≤ 10 electrons with Hilbert space dimension up to 3268760. We model disorder by Gaussian white noise in this section.The evolution of entanglement entropy at f=2/5 is similar to those at Laughlin fillings (Fig. <ref>). We observe a pronounced minimum in all ⟨ dS(ρ)/dW⟩ curves. The position of this minimum stays around W_c≈ 0.06 for N≥ 8 electrons [Fig. <ref>(a)], which means that the FQH phase with Coulomb interaction at f=2/5 is more robust against disorder than that at f=1/5. There is a large finite-size effect in the minimum magnitude h: the data points of N=4 and N=6 electrons in the ln h-ln N plot deviate from the linear growth of other two points [Fig. <ref>(b)]. Based on the finite-size scaling analysis of the data for N=8 and N=10 electrons, we obtain ν≈ 0.6 [Fig. <ref>(a) inset]. This value is smaller than those at f=1/3 and f=1/5, and again violates the ν≥ 2/d bound.§ F=1/2 Having considered several filling fractions where incompressible FQH phases exist in clean systems, we now extend our discussion to f=1/2. For the half-filled Landau level without disorder, in a mean-field picture CFs feel a zero effective magnetic field and consequently form a gapless CF sea instead of an incompressible FQH state<cit.>. For finite systems, the shape of such a CF sea depends on the system size, leading to a variable ground-state degeneracy D, as shown in Table <ref>. Because the averaging method used in Secs. <ref> and <ref> is not appropriate for a manifold with a degeneracy depending on the system size, here we just focus on the entanglement entropy of the lowest eigenstate |Ψ_1⟩ of the Hamiltonian (<ref>), which is obtained by ED for N≤ 13 electrons with Hilbert space dimension up to 10400600.Naively one may expect that the metallic CF sea in clean systems will be destroyed by an arbitrarily small disorder, according to the scaling theory of localization, which excludes the metallic behavior in 2D non-interacting systems with random disorder if the magnetic field is absent<cit.>. However, being different from ordinary fermions, CFs carry magnetic flux. Early arguments suggested<cit.> that the disorder-induced inhomogeneous CF density produces random local fluctuations of the effective magnetic field (although the net effective magnetic field is still zero), thus suppressing the localization of CFs and stabilizing the metallic CF sea at weak disorder<cit.>. Then, at strong disorder, the transition to an insulator occurs, consistent with the conventional localization theory. In the following, we will look for the clue of this phase transition in the entanglement entropy.We first consider Gaussian white noise, which preserves the PH symmetry. S(|Ψ_1⟩) and ⟨ dS(|Ψ_1⟩)/dW⟩ are shown in Figs. <ref>(a) and <ref>(b), respectively. Strikingly, we observe a chaotic behavior of the entropy derivative at f=1/2, which is very different from those at f=1/3,1/5, and 2/5. For some system sizes such as N=6 and N=7 electrons, it is difficult to identify a minimum in ⟨ dS(|Ψ_1⟩)/dW⟩. For other system sizes where an obvious minimum in ⟨ dS(|Ψ_1⟩)/dW⟩ exists, its position is still changing significantly with the system size, and the magnitude does not nicely scale with N. Therefore, at least for the system sizes that we can reach by ED, the disorder derivative of the entanglement entropy with Gaussian white noise does not provide a convincing signal of the phase transition at f=1/2.As a result of the PH symmetry, the f=1/2 ground state in the W=∞ limit for Gaussian white noise is not an insulator, but a metallic critical phase with the same Hall and longitudinal conductance σ_xy=σ_xx=0.5e^2/h<cit.>, which may make Gaussian white noise inappropriate for the study of a transition at f=1/2 from a CF sea to an insulator. In order to understand whether the chaotic ⟨ dS(|Ψ_1⟩)/dW⟩ observed above is due to the absence of an insulator in the non-interacting limit, we then consider a different disorder model that breaks the PH symmetry. We choose an ensemble of scatterers<cit.>, and assume that (i) the range of scatterers is λ=1/√(2), which is comparable with the magnetic length ℓ=1; (ii) the number of scatterers in the ensemble is 3N_ϕ, which is significantly more than one per flux quantum; (iii) W is negative, and W_n=10W for 20% of the scatterers and W_n=W for the remaining in each sample; (iv) the distribution of R_n is symmetric with respect to zero. In the non-interacting limit, these settings significantly skew the density of states and shift the position of extended single-particle states from LLL filling f_c=1/2 to f_c≈ 0.6, thus breaking the PH symmetry. The f=1/2 ground state at W=∞ for such an ensemble of scatterers is hence an Anderson insulator with σ_xy=σ_xx=0. However, even in this case, we still observe a very chaotic ⟨ dS(|Ψ_1⟩)/dW⟩ [Fig. <ref>(b)]. S(|Ψ_1⟩) [Fig. <ref>(a)] is also similar to that for Gaussian white noise. Therefore, our numerical data suggest that, for the system sizes that we can reach by ED, the significant size-dependence in the disorder derivative of ground-state entanglement entropy at f=1/2 is not significantly affected by the choice of the disorder model, or the state in the infinite disorder limit, but is more likely due to the absence of a gap at zero disorder.It is known that a logarithmic correction to the area law of entanglement entropy is expected if a Fermi surface is present, like in the case of CF sea<cit.>. However, if disorder really induces the collapse of the CF sea, the area law should be recovered at strong disorder. Therefore we further examine the scaling of S(|Ψ_1⟩) versus the boundary length L of the subsystem A at various disorder. For both of the Gaussian white noise and the scatterer ensemble, we indeed find that S(|Ψ_1⟩) grows with L faster than a linear scaling at small disorder, then gradually evolves to a linear scaling with increasing disorder [Figs. <ref>(c) and <ref>(c)]. This tendency is consistent with the predicted transition from a CF sea to an insulator. Although our system sizes are still too small to manifest S/L∝ln L at small disorder [Figs. <ref>(d) and <ref>(d)] (a similar finite-size effect at small L can also be seen in Ref. shao15), we can estimate an upper bound of the critical point as W_c∼ 0.1 for Gaussian white noise and W_c∼ 0.05 for the scatterer ensemble, after which an unambiguous linear scaling of S∝ L starts to show.Considering that the CF picture conceptually does not apply to the non-interacting case, the f=1/2 metallic critical phase for Gaussian white noise at W=∞, which only exists in the W=∞ limit, is not contradictory with the insulating phase at strong disorder.§ DISCUSSION In this paper, we have tracked the evolution of ground-state entanglement entropy with increasing disorder for electrons with Coulomb interactions at various LLL filling fractions, and estimated the critical points and the length exponents of pertinent disorder-driven phase transitions using a finite-size scaling analysis of the ground state entanglement entropy. Our main results are summarized in Table <ref>. At f=1/3,1/5, and 2/5, we observe the same feature in the derivative of the entropy with respect to disorder: there is always a pronounced minimum whose position is size independent, but whose magnitude increases markedly with the system size, and is consistent with a divergence in the thermodynamic limit. We consider the location of this minimum as the critical point W_c of the expected transition from a topological FQH state to an insulator. A finite-size scaling analysis of the magnitude of the minimum gives us an estimation of the critical length exponent ν at these fillings. The values of ν that we obtain by this method vary by as much as 50% depending on the filling fraction. Moreover, our estimates lie on either side of the conventional bound ν≥ 2/d for the length exponent for non-topological transitions in disordered systems. This suggests that while our data for the transition from the gapped FQH states (especially for the larger sizes) are quite consistent with finite-size scaling, there are likely corrections to finite-size scaling. These corrections are likely largest for the FQH states with the smallest gaps (like f = 1/5). The larger finite-size effects observed at f=1/5 and 2/5 than that at f=1/3 may be related to the larger size of composite fermions at these fillings<cit.>.However, it is noteworthy that all our estimates are very different from ν≈ 2.5 for the localization length at integer plateau transitions<cit.>. It would therefore be of great interest to have experimental estimates of these exponents to see whether they are the same as that for integer plateau transitions. Previous experiments have studied integer quantum Hall plateau transitions by tuning the magnetic field; this may not work for FQH-insulator transition because of the possibility of intervening FQH states. Tuning the disorder, while significantly more challenging, is not out of the question, e.g., by gating a sample, and thereby changing the disorder potential felt by the 2D electron gas. On the numerical side, clearly the best possibilities for improvement remain for FQH states with the largest gaps, which also have the largest W_c, as indicated in Table <ref> by a nearly constant Δ/W_c that is close to the value found at f=1/3 for the Haldane pseudopotential<cit.>.At f=1/2, the entropy derivative with respect to disorder has a behavior that is quite size-dependent, varying in a somewhat chaotic manner for the system sizes we are able to study. At this filling, the effective magnetic field for composite fermions vanishes; as a result, there is no gap and the effective length scale (magnetic length) diverges. Consequently, finite-size effects are much more prominent. A similar chaotic behavior is also observed at f=1/4 – another filling fraction where compressible CF liquid is formed at zero disorder. Access to larger systems, probably with the help of more advanced numerical techniques, is needed to verify whether the derivative will become regular when the system is large enough. Even so, extracting W_c and ν through a finite-size scaling technique could be more complicated because of the different size-dependence of the entanglement entropy for the CF Fermi liquid and the disordered insulator.Several future directions are suggested by our present work. One is to study the entanglement evolution at f>1/2 driven by PH symmetric disorder, and compare the result with that of the PH conjugate at filling 1-f. It should be noted, however, that because the orbital partition used here cannot distinguish f and 1-f in the presence of PH symmetry<cit.>, the entanglement computed from a real-space partition<cit.> will be necessary in that case. Another, more interesting topic is to investigate the role of disorder for non-Abelian FQH states, such as those at f=5/2 and 12/5. However, even at zero disorder, these states are more difficult to stabilize than the Abelian states studied in this work. Before considering the disorder effect, we first need to modify the bare Coulomb interaction, for example, by tuning its pseudopotential components or sample thickness<cit.>, to reach robust non-Abelian phases. We thank Scott Geraedts, Zlatko Papic, and Kun Yang for useful discussions, and an anonymous referee for useful comments. This work was supported by the Department of Energy, Office of Basic Energy Sciences through Grant No. DE-SC0002140. Z. L. was also supported by an Alexander von Humboldt Research Fellowship for Postdoctoral Researchers. R. N. B. thanks the Aspen Center for Physics for hospitality while this paper was being completed.§ F=1/7 Previous studies in clean systems have shown that the proximity to compressible Wigner crystals makes the Coulomb ground states at f=1/7 deviate more substantially from the Laughlin model states than the f=1/3 and 1/5 cases<cit.>. At zero disorder, we indeed find that both the overlap between the Coulomb ground state and the Laughlin model state and the energy gap drop significantly at N=8 electrons (Table <ref>). The high overlaps for smaller N are then probably because the formation of Wigner crystal, which is sensitive to the sample geometry, is frustrated in these smaller systems on the square torus. In experiments, the f=1/7 FQH effect was also observed only at relatively high temperature<cit.> compared with the f=1/3 and 1/5 cases, which can be explained as the melting of Wigner crystals. Therefore, both numerics and experiments suggest that the f=1/7 Coulomb ground state in a large clean system at zero temperature is a compressible Wigner crystal instead of a FQH phase.In the presence of Gaussian white noise, we track the evolution of the entanglement entropy at f=1/7 in a manifold containing the lowest seven eigenstates of the Hamiltonian (<ref>). Due to the very fast growth of the Hilbert space with increasing electron numbers, we can only efficiently study up to N=5 electrons by ED. We find that the behavior of entropy derivative for these very small systems at f=1/7 (Fig. <ref>) are similar to that at f=1/3 and 1/5. There is a pronounced minimum in all ⟨ dS(ρ)/dW⟩ curves. Except for the smallest system size N=3, the position of this minimum is around a very small value W_c≈ 0.003 [Fig. <ref>(a)], but a reasonable linear fitting of the points in the ln h-ln N plot [Fig. <ref>(b)] is not possible. Nevertheless, this similarity with the f=1/3 and 1/5 cases is probably just a result of the non-vanishing finite-size gaps at zero disorder in these very small systems (Table <ref>). Once the compressible Wigner crystal dominates at zero disorder for large enough systems, we expect that the behavior of entanglement entropy would likely become strikingly different.§ ENTANGLEMENT SPECTRUM LEVEL STATISTICS Finally, we discuss the level statistics of ground-state ES, i.e., the spectrum of -lnρ_A, in the presence of Gaussian white noise. In clean systems, each ES level can be labeled by the number of electrons N_A and the total momentum K_A in part A<cit.>. Disorder breaks the conservation of K_A. However, N_A remains a good quantum number, which still allows us to decompose the ES into various N_A sectors. ES levels in different N_A sectors are independent, so putting them together will hide the true level statistics in each N_A sector. Therefore, in the following we will focus on a specific N_A sector, the one with N_A=⌈ N/2⌋, to study the ES level statistics therein.In Ref. zhao2016, the ES level statistics were diagnosed using the distribution P(s) of the normalized level spacing s_n/⟨ s_n⟩, where s_n=ξ_n-ξ_n-1 with ξ_n's the unfolded<cit.> ES levels with N_A=⌈ N/2⌋ sorted in ascending order in each sample. Here we consider the ratio between two consecutive level spacings, i.e., ⟨ r_n⟩ with r_n=min(s_n,s_n+1)/max(s_n,s_n+1), as another indicator of the ES level statistics. We first compute ⟨ r_n⟩ for the ES of each eigenstate of the Hamiltonian (<ref>) in the ground-state manifold, then further average it over the whole manifold to get the mean ⟨r_n⟩. In Fig. <ref>, we show the evolution of ⟨r_n⟩ at f=1/3 in two windows ξ∈(0,10] and ξ∈(0,15]. The results at other fillings are similar.We indeed observe a transition of ⟨r_n⟩ from the Gaussian unitary ensemble (GUE) value ⟨ r⟩≈ 0.603 to the Poisson value ⟨ r⟩≈ 0.386, which becomes more obvious for larger system sizes. However, contrary to the naive expectation that the ES level statistics might have a dramatic change at the same disorder strength as W_c≈ 0.09 where dS/dW diverges, the transition of ⟨r_n⟩ occurs at a much larger W_c^ξ. By assuming that ⟨r_n⟩ crosses with ⟨ r⟩≈ 0.495 (i.e., the middle value between GUE and Poisson) at W=W_c^ξ, we find W_c^ξ≈ 3 for ξ∈(0,10] [Fig. <ref>(a)] and W_c^ξ≈ 6 for ξ∈(0,15] [Fig. <ref>(b)], respectively, which are more than an order of magnitude larger than W_c≈ 0.09 indicated by dS/dW. In both cases, W_c^ξ almost does not move with increasing system sizes. The smaller W_c^ξ in the ξ∈(0,10] window is consistent with our previous observation for the pseudopotential interaction in Ref. zhao2016 that the localization in the ES, reflected by P(s) changing from Gaussian unitary ensemble (GUE) to semi-Poisson and finally to Poisson, is first activated among low levels, then propagates towards higher-ξ region with increasing disorder strength.Thus, like in Ref. zhao2016, instead of being at (or near) the same disorder strength where dS/dW diverges, the transition point of ES level-spacing statistics from GUE to Poisson is at a very different value of W; further, it depends on the choice of the ξ window. Therefore it is not feasible to use this measure of the entanglement spectrum to precisely locate the ground-state phase transition. One may wonder whether using a narrower window above ξ=0 can approach W_c≈ 0.09. 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"authors": [
"Zhao Liu",
"R. N. Bhatt"
],
"categories": [
"cond-mat.str-el",
"cond-mat.dis-nn"
],
"primary_category": "cond-mat.str-el",
"published": "20170627124112",
"title": "Evolution of quantum entanglement with disorder in fractional quantum Hall liquids"
} |
1Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA; mailto:[email protected]@lco.global2Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA3Department of Astronomy/Steward Observatory, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA4Department of Physics & Astronomy, Texas Tech University, Box 41051, Lubbock, TX 79109-1051, USA5Department of Physics, University of California, 1 Shields Avenue, Davis, CA 95616-5270, USA6Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA7Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-8169, USA8Departments of Physics and Astronomy, University of California, Berkeley, CA 94720-7300, USA9Department of Physics, Florida State University, 77 Chieftain Way, Tallahassee, FL 32306-4350, USA10Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark2017 June 27 2017 July 31 2017 August 2 2017 August 14 The Astrophysical Journal Letters, 845:L11 (8pp), 2017 August 20 Early Blue Excess from the Type Ia SN 2017cbv Hosseinzadeh et al. We present very early, high-cadence photometric observations of the nearby Type Ia SN 2017cbv.The light curve is unique in that it has a blue bump during the first five days of observations in the U, B, and g bands, which is clearly resolved given our photometric cadence of 5.7 hr during that time span.We model the light curve as the combination of early shocking of the supernova ejecta against a nondegenerate companion star plus a standard SN Ia component. Our best-fit model suggests the presence of a subgiant star 56 R_ from the exploding white dwarf, although this number is highly model-dependent.While this model matches the optical light curve well, it overpredicts the observed flux in the ultraviolet bands. This may indicate that the shock is not a blackbody, perhaps because of line blanketing in the UV. Alternatively, it could point to another physical explanation for the optical blue bump, such as interaction with circumstellar material or an unusual nickel distribution.Early optical spectra of SN 2017cbv show strong carbon (C2 λ6580) absorption up through day -13 with respect to maximum light, suggesting that the progenitor system contains a significant amount of unburned material. These early results on SN 2017cbv illustrate the power of early discovery and intense follow-up of nearby supernovae to resolve standing questions about the progenitor systems and explosion mechanisms of SNe Ia.§ INTRODUCTION Einstein FellowType Ia supernovae (SNe Ia) are the thermonuclear explosions of carbon–oxygen white dwarfs and are standardizable candles vital for cosmological distance measurements. Despite intense study, the progenitor scenarios and explosion mechanisms for these events are still not understood, and may have multiple pathways <cit.>.The two main progenitor pictures are the single-degenerate (SD) scenario, where the white dwarf accretes material from a nondegenerate secondary star <cit.>, and the double-degenerate scenario (DD), where two white dwarfs are present in the pre-supernova system <cit.>. However, the details of both scenarios are still under investigation <cit.>.The early light curves of SNe Ia are promising ways to constrain the progenitor systems and the physics of the explosion.For instance, the collision of the SN ejecta with a nondegenerate companion star may manifest as an early blue or ultraviolet (UV) bump in the light curve, depending on the viewing angle <cit.>. Early temperature or luminosity measurements can directly constrain the radius of the progenitor <cit.>; observed limits have confirmed that the exploding star must be a white dwarf <cit.>. Recently, <cit.> explored how early SN Ia light curve behavior depends on the amount and extent of circumstellar material (CSM) and the distribution of ^56Ni in the ejecta, which is expected to vary considerably with the location(s) of ignition in the progenitor.Finally, different explosion mechanisms may also produce distinct early light curves <cit.>.Very early SN Ia light curve observations are becoming more common, and recent studies have shown that they display a range of early behaviors <cit.>.iPTF14atg <cit.> and SN 2012cg <cit.> both showed early, UV/blue excesses in their light curves.iPTF14atg was an SN 2002es-like event, which are subluminous, do not follow the <cit.> relation, have low velocities, and show Ti2 absorption <cit.>. <cit.> found that the UV excess in iPTF14atg was consistent with the SN ejecta interacting with a companion star, although <cit.> find the SD scenario incompatible with the observed spectral evolution.In SN 2012cg, a normal SN Ia, the early blue bump was again interpreted as a signature of ejecta–companion interaction, consistent with a 6 M_ main-sequence companion 29 R_ from the white dwarf, using the <cit.> formulation.[Note that the models of <cit.> directly constrain the binary separation, not the companion mass. Masses can be inferred by assuming the companion is in Roche lobe overflow and applying a mass-radius relationship.] However, <cit.> find that other probes of the progenitor system do not support the SD interpretation for SN 2012cg.Here, we present the early light curve and spectra of SN 2017cbv, which show a clear blue excess during the first several days of observations. This excess may be a more subtle version of that seen in SN 2012cg and iPTF14atg, but seen more clearly here thanks to denser sampling.§ OBSERVATIONS AND DATA REDUCTIONSN 2017cbv (a.k.a. DLT17u) was discovered on MJD 57822.14 (2017 March 10 UT) at a magnitude of R ≈ 16 by the Distance Less Than 40 Mpc survey (DLT40; L. Tartaglia et al. 2017, in preparation), a one-day cadence SN search using a PROMPT 0.4 m telescope <cit.>, and was confirmed by a second DLT40 image during the same night <cit.>. Our last nondetection was on MJD 57791. The SN is located on the outskirts of the nearby spiral galaxy NGC 5643. Within hours of discovery (MJD 57822.7), the transient was classified as a very young SN Ia with the robotic FLOYDS spectrograph mounted on the Las Cumbres Observatory <cit.> 2 m telescope in Siding Spring, Australia <cit.>. UBVgri follow-up observations were obtained with Sinistro cameras on LCO's network of 1 m telescopes. Using<cit.>, a PyRAF-based photometric reduction pipeline, we measured aperture photometry of the SN. Because the SN is bright and far from its host galaxy, image subtraction and PSF fitting are not required. Local sequence stars were calibrated to the L101 standard field observed on the same night at the same observatory site using UBV Vega magnitudes from <cit.> and gri AB magnitudes from the .The Swift satellite began observing SN 2017cbv on MJD 57822.52.Ultra-Violet Optical Telescope <cit.> photometry is given in the UVOT Vega photometry system using the pipeline for the Swift Optical Ultraviolet Supernova Archive <cit.> and the zeropoints of <cit.>.Smaller hardware windows were used in the optical near maximum brightness in order to reduce coincidence loss and measure brighter magnitudes without saturation <cit.>.A few U- and B-band observations on the rising branch were saturated and are excluded.Finally, we obtained absolute magnitudes by applying the distance modulus, μ = 31.14 ± 0.40 mag (16.9 ± 3.1 Mpc), of <cit.> and the Milky Way extinction corrections, E(B-V) = 0.15 mag, of <cit.>. We assume no additional host galaxy extinction, given the SN's position in the outskirts of NGC 5643 and a lack of narrow Na1 D absorption in high-resolution spectra (D.J. Sand et al. 2017, in preparation).Our photometry is shown in Figure <ref>. By fitting quadratic polynomials to the observed light curve, one around peak and one around +15 days, we find that SN 2017cbv reached a peak magnitude of B=11.72 mag (M_B=-20.04 mag) on MJD 57841.07, with Δ m_15(B)=1.06 mag. The decline rate of SN 2017cbv is near the average for normal SNe Ia <cit.>. The peak absolute magnitude appears to be on the bright end of SNe Ia <cit.>, but this is uncertain due to the poorly constrained distance to NGC 5643. All figures use MJD 57821 as the nominal explosion date (see <ref>).§ LIGHT CURVE MORPHOLOGY AND FITTINGThe very early light curve of SN 2017cbv shows a prominent blue bump in the U, B and g bands during the first five days of observation, also visible in its U-B and B-V colors (Figure <ref>). This indicates a high-temperature component of the early light curve in addition to the normal SN Ia behavior.There are several possibilities for the origin of the early light curve bump (see <ref> for a discussion), but here we fit the analytic models of <cit.> for a nondegenerate binary companion shocking the SN ejecta, as might be expected from the SD scenario. While in reality such a collision would be highly asymmetric, we use 's analytic expressions for the luminosity within an optimal viewing angle. By further assuming that the shock consists of a spherical blackbody,arrives at the following equations for the photospheric radius and effective temperature[Equation (<ref>) corrects the exponent on κ in 's Equation (25).]:R_phot = (2700 R_) x^1/9κ^1/9 t^7/9 T_eff = (25,000 K) a^1/4 x^1/144κ^-35/144 t^-37/72where a is the binary separation in units of 10^11 m (144 R_), κ is the opacity in units of the electron scattering opacity (we fix κ=1 in our fits), and t is the time since explosion in days.In addition, we definex ≡M/M_Ch(v/10,000 km s^-1)^7,where M is the ejected mass, M_Ch = 1.4 M_ is the Chandrasekhar mass, and v is the transition velocity between power laws in the density profile.Our light curve model is the sum of the SiFTO template <cit.> to account for the normal SN Ia emission, and 's shock model to account for the blue excess. We scale each band of the SiFTO template independently. In the U, B, V, and g bands, we fix the scaling factor to match the observed peak. In the r and i bands, we leave the scaling factor as a free parameter in order to account for any contribution from the shock in those bands around peak (predicted for some combinations of parameters). We also allow the time of B-band maximum light and the stretch <cit.> of the SiFTO template to vary.In total, we have eight parameters: * The explosion time;* The binary separation, a;* x ∝ M v^7 (Equation (<ref>));* A factor on the r SiFTO template;* A factor on the i SiFTO template;* The time of peak;* The stretch; and* A factor on the shock component in U (see below). We fit this combined model to our UBVgri light curve from LCO using a Markov Chain Monte Carlo routine based on thepackage <cit.>. We cannot include the Swift data in the fit due to a lack of early UV SN Ia templates. In addition to the photometric uncertainty, we add a 2% systematic (0.02 mag) uncertainty in quadrature as an estimate for our calibration uncertainties. For each of our observations, we find the expected R_phot and T_eff at that time, calculate the corresponding average L_ν in each filter, and compare that to our measured L_ν.One caveat to our approach is that the SiFTO template may not describe the early light curve behavior of normal SNe Ia correctly in all filters. Given the small number of events observed 10–20 days before peak, reliable templates do not exist at these phases. However, we are encouraged by the fact that our SiFTO-based results qualitatively agree with other template fitters that we experimented with—SALT2 <cit.>, MLCS2k2 <cit.>, and the observed light curve of SN 2011fe <cit.>—even at these early times.We find that the SiFTO+ model provides an excellent fit to our ground-based data (reduced χ^2 = 8.6 after including our 2% calibration uncertainty), correctly predicting the blue bump at early times (Figure <ref>, top). Although at first glance our light curves do not look peculiar in the redder bands, emission from the shock model several weeks after explosion contributes to the observed luminosity around peak. Specifically, our best-fit model indicates that 5% and 15% of the r- and i-band peak luminosities, respectively, come from the shock component.The best-fit binary separation is 56 R_, implying a stellar radius of ∼20 R_ <cit.>. 56 R_ is among the largest binary separations for SD SN Ia progenitors from binary population synthesis calculations <cit.>. However, this value is quite sensitive to the early color evolution of our SN Ia template. As these templates may not be valid 15–20 days before peak, this result should be treated with caution. Furthermore, our simplified spherical model ignores the degeneracy between binary separation (a) and viewing angle; a bright (large a) off-angle shock looks similar to a faint (small a) shock along the line of sight.Given the strong dependence of x on the transition velocity (∝ v^7), which is not observable, we cannot robustly estimate the ejecta mass. However, taking our best-fit value of x=3.84±0.19 and assuming a Chandrasekhar mass of ejecta, we find a reasonable transition velocity of v≈12000 km s^-1 (subject to uncertainty in the distance modulus). The best-fit explosion time is MJD 57821.9, about 7 hr before discovery, and the best-fit time of B-band peak for the SiFTO component is MJD 57840.2. The best-fit stretch from the SiFTO template is 1.04.Despite the success of the binary companion shock model in the optical, we required a scaling factor of 0.61 on the U-band shock component in order to fit the data. The UVW1, UVM2 and UVW2 emission are even further overpredicted by the shock model (Figure <ref>, bottom). We discuss the potential causes of this discrepancy in <ref>.§ EARLY SPECTRA We obtained several additional optical and near-infrared (NIR) spectra of SN 2017cbv with FLOYDS and the SpeX NIR spectrograph <cit.> at the NASA Infrared Telescope Facility, a selection of which are presented in Figure <ref>. As the event is ongoing at the time of publication, the full data set and analysis will be presented elsewhere (D. J. Sand et al. 2017, in preparation). In summary, the spectrum of SN 2017cbv greatly resembles the spectrum of SN 2013dy <cit.> during the two weeks before maximum light, with Si2 and Ca2 absorption features weaker than the prototypical Type Ia SN 2011fe but stronger than the somewhat overluminous SN 1999aa <cit.>. Despite the spectroscopic similarity to SN 2013dy, the light curve of SN 2017cbv declines slightly faster: Δ m_15(B)=1.06 mag for SN 2017cbv versus 0.92 mag for SN 2013dy <cit.>.We measure a Si2 λ6355 velocity of 22,800 km s^-1 in the initial spectrum of SN 2017cbv, 19 days before maximum light. The absorption feature just redward of Si2 could either be a lower-velocity component (9400 km s^-1) of the same line or C2 λ6580 at 19,800 km s^-1. We prefer the latter interpretation, as (1) it correctly predicts the tentative position of C2 λ7234 in the same spectrum and (2) 9400 km s^-1 would be uniquely low among very early SN Ia spectra <cit.>. The detection of unburned carbon can directly discriminate between the proposed SN Ia explosion mechanisms but is rarely seen in optical spectra even at these early times <cit.>, except for in super-Chandrasekhar SNe Ia <cit.>. In SN 2017cbv, this feature disappears by day -13, reinforcing the need for early spectroscopy to fully account for unburned carbon.The Si2 absorption feature at -13 days clearly shows signs of two velocity components at 16,500 and 10,500 km s^-1.It is likely that the earlier spectra of SN 2017cbv are dominated by the high-velocity component of Si2, but we cannot fully trace the transition to low-velocity Si2 due to our lack of FLOYDS spectra between -13 and -2 days.We note that -13 days was also the approximate epoch at which SN 2012fr, another SN Ia with a prominent high-velocity Si2 component, began showing low-velocity Si2 <cit.>.A similar multi-component velocity structure is evident for the Ca2 H&K feature in our pre-maximum spectra.We fit the early velocity evolution of Ca2 H&K and the Si2 high-velocity component (the only one we can clearly identify at early times) to a t^-0.22 power law, as suggested by <cit.> for finding the explosion time for SNe Ia.To do this, we mimicked the methodology of <cit.> and allowed the power-law dependence to vary between t^-0.20 and t^-0.24 to estimate our uncertainties. This fit implies an explosion on MJD 57821.0 ± 0.3, 1.1 days prior to discovery and 0.9 days before the implied explosion time from our binary shock + standard SN Ia model presented in <ref>.If SN 2017cbv had an SD progenitor, we might expect to see hydrogen in its late-time spectra <cit.>. However, we do not detect Paβ emission in an NIR spectrum taken 34 days after maximum light (Figure <ref>, bottom). Following the method of <cit.>, we calculate a rough limit of ≲0.1 M_ of hydrogen by comparing to a spectrum of ASASSN-14lp at a similar phase, although this limit depends on the viewing angle.§ DISCUSSIONThe companion-shocking models provide a good fit to our optical data, but not to our UV data. This discrepancy is not likely to be a reddening effect (unless the reddening varies very quickly with time) because the UV luminosities around peak are not unusual for SNe Ia. However, it could stem from several simplifying assumptions in our model:* Blackbody: The analytic models assume a blackbody spectrum for the shock component. However, the observed spectral energy distribution (SED) during the bump deviates significantly from a blackbody spectrum in the UV (Figure <ref>). A Swift grism spectrum taken during the bump is similar to UV spectra of other SNe Ia, showing significant absorption relative to a blackbody continuum (D.J. Sand et al. 2017, in preparation). This UV suppression is likely due to line blanketing (e.g., from iron lines). Any alternative model, companion shocking or otherwise, will need to account for this deviation from a blackbody spectrum.* Constant Opacity: We fixed the opacity to be that of electron scattering throughout the first 40 days of evolution, whereas in reality the opacity should change over time as the ejecta cool. Opacity and/or line blanketing that vary with time could potentially explain the discrepancy between the models and our UV data.* Density Profile: The shock models we quote here assume a broken power-law density profile for the ejecta: ρ_inner∝ r^-1 and ρ_outer∝ r^-10. The earliest emission, which should peak in the UV bands, depends strongly on the density of the outermost ejecta layers. In particular, a steeper density profile could suppress the early luminosity.* Spherical Symmetry: In order to make the problem analytically tractable, we have ignored the asymmetry that must be present in a binary system. The analytic predictions are roughly equivalent to numerical predictions for a favorable viewing angle <cit.>. If in reality we are viewing the collision off-axis, we might invoke a larger binary separation, ejecta mass, or ejecta velocity to match our observations. However, 3D numerical modeling of the ejecta–companion interaction would be needed to disentangle the early SN color diversity from the angular dependence of the shock component's color. If the companion-shocking scenario is correct, but the model is inaccurate in the UV, then the companion sizes found by <cit.> and <cit.> and the constraints of <cit.> would have to be reevaluated. However, if the UV overprediction is only due to line blanketing, which depends on temperature, the models may be more susceptible to failure in relatively low-temperature events. Alternatively, there could be another cause of early bumps in the UV or optical.The observed bump could also be interpreted as a collision with CSM, rather than a collision with a companion star, as has recently been modeled by <cit.>. <cit.> estimates that 0.01–0.1 M_ of CSM would be needed for a significant effect, and that the material would have to be located at distances comparable to the binary separation. Stellar winds would be unlikely to produce such a configuration, but a DD system could potentially eject enough mass to large enough distances during the pre-supernova accretion phase <cit.> or during the merger itself <cit.>. A large mass of CSM would likely have decelerated the ejecta below the velocities we measure in <ref>, although an unshocked high-velocity component could still persist.A third possibility is that the bump arises from a bubble of radioactive nickel that escaped most of the ejecta, allowing it to radiate light away faster than the typical diffusion timescale. <cit.> explored various nickel distributions and their effect on early SN light curves.Shallow nickel distributions result in steeper, bluer early light curves.The early light curves in <cit.> that included both CSM interaction and significant nickel mixing bear a qualitative resemblance to the light curve of SN 2017cbv. Likewise, <cit.> find an early blue bump in their sub-Chandrasekhar double-detonation model, in which the initial helium detonation on the surface produces a small amount of radioactive material. We cannot rule out these possibilities, but more detailed modeling of this event in particular would be necessary to distinguish them from the companion-shocking case.§ SUMMARYWe have presented early photometry and spectroscopy of the Type Ia SN 2017cbv, which was discovered within ∼1 day of explosion. Its light curve shows a conspicuous blue excess during the first five days of observations. We find a good fit between our UBVgri data and models of binary companion shocking from <cit.>, but the fit overpredicts the observed UV luminosity at early times. This discrepancy might be due to several simplifying assumptions in the models. Alternatively, the excess emission could be due to interaction with CSM or the presence of radioactive nickel in the outer ejecta.We observe no indication of ejecta interaction with hydrogen-rich material stripped from a companion star in the spectra of SN 2017cbv. However, more deep optical and near-infrared spectra out to the nebular phase are needed to confirm this finding.Intriguingly, we do detect unburned carbon in the earliest spectra at a level rarely seen in normal SNe Ia. A connection between an early light curve bump and the presence of unburned carbon could provide an important clue about SN Ia progenitors, but the scarcity of events with either of these observations prevents us from drawing any conclusions now.Our analysis demonstrates the importance of (1) discovering and announcing SNe as early as possible and (2) obtaining extremely well-sampled follow-up light curves and spectral series. The DLT40 survey and LCO follow-up network are uniquely suited to find very young SNe and follow them with sub-day cadence for long periods.Such observations, which will become increasingly common in the coming years, greatly enhance our ability to confront theoretical models.We thank Kyle Barbary for his help withand Alex Conley and Santiago González Gaitán for providing the SiFTO templates. This work makes use of observations from the LCO network as part of the Supernova Key Project. A portion of the Swift observations were obtained by time allocated to the Danish astronomy community via support from the Instrument Center for Danish Astrophysics (PI: Stritzinger). G.H., D.A.H., and C.M. are supported by the National Science Foundation (NSF) under grant No. 1313484.D.J.S. and L.T. are supported by the NSF under grant No. 1517649. Support for S.V. was provided by NASA through a grant (program number HST-GO-14925.007-A) from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Support for I.A. was provided by NASA through the Einstein Fellowship Program, grant PF6-170148. E.Y.H., S.D., and M.S. are supported by the NSF under grant No. AST-1613472 and by the Florida Space Grant Consortium. M.D.S. acknowledges support by a research grant (13261) from the Villum Fonden. D.J.S. is a visiting astronomer at the Infrared Telescope Facility, which is operated by the University of Hawai`i under contract NNH14CK55B with NASA. This research has made use of the NASA/IPAC Extragalactic Database, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. | http://arxiv.org/abs/1706.08990v3 | {
"authors": [
"Griffin Hosseinzadeh",
"David J. Sand",
"Stefano Valenti",
"Peter Brown",
"D. Andrew Howell",
"Curtis McCully",
"Daniel Kasen",
"Iair Arcavi",
"K. Azalee Bostroem",
"Leonardo Tartaglia",
"Eric Y. Hsiao",
"Scott Davis",
"Melissa Shahbandeh",
"Maximilian D. Stritzinger"
],
"categories": [
"astro-ph.HE",
"astro-ph.SR"
],
"primary_category": "astro-ph.HE",
"published": "20170627180951",
"title": "Early Blue Excess from the Type Ia Supernova 2017cbv and Implications for Its Progenitor"
} |
Laboratoire de Physique Théorique et Modélisation (CNRS Unité 8089), Université de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France We investigate the issue of single particle nonlocality in a quantum system subjected to time-dependent boundary conditions. We discuss earlier claims according to which the quantum state of a particle remaining localized at the center of an infinite well with moving walls would be specifically modified by the change in boundary conditions due to the wall's motion. We first prove that the evolution of an initially localized Gaussian state is not affected nonlocally by a linearly moving wall: as long as the quantum state has negligible amplitude near the wall, the boundary motion has no effect. This result is further extended to related confined time-dependent oscillators in which the boundary's motion is known to give rise to geometric phases: for a Gaussian state remaining localized far from the boundaries, the effect of the geometric phases is washed out and the particle dynamics shows no traces of a nonlocal influence that would beinduced by the moving boundaries. Single particle nonlocality, geometric phases and time-dependent boundary conditions A. Matzkin December 30, 2023 ===================================================================================== § INTRODUCTION Quantum systems with time-dependent boundary conditions are delicate to handle. Even the simplest system – a particle in a box with infinitely high but moving walls – remains the object of ongoing investigations. From a mathematical standpoint, a consistent and rigorous framework hinges on unifying the infinite number of Hilbert spaces (one for each time t), each endowed with its own domain of self-adjointness <cit.>. Explicit solutions have been found in specific cases, notably for the infinite well with linear expanding or contracting walls <cit.>, later generalized to a family of confined time-dependent linear oscillators whose frequency is related to the wall's motion <cit.>. However general methods, such as employing a covariant time derivative <cit.> in order to track the change in the boundary conditions or implementing a quantum canonical transformation <cit.> so as to map the time-dependent boundary conditions problem to a fixed boundary problem with another Hamiltonian can at best give perturbartive results. Explicit solutions call for numerical methods <cit.> but these are not well suited to investigate fundamental effects, in particular when controversial effects are discussed.This work precisely deals with a controversial effect, namely the existence of possible nonlocal effects induced by time-dependent boundary conditions on a quantum state remaining localized far from the boundaries. From a general standpoint, it is known that systems with a cyclic evolution may display geometric phases, a global property often said to be “nonlocal”or “holistic”. However it was initially suggested by Greenberger <cit.>, and subsequently mentioned by several authors, eg <cit.>, that time-dependent boundary conditions could give rise to a genuine form of nonlocality: a particle at rest and localized in the center of the box, remaining far from the moving walls, would be physically displaced by the changing boundary conditions induced by the walls motion. This claim was never shown rigorously to be exact (some arguments were given to support the idea of nonlocality, sometimes in a hand waving fashion), but to the best of our knowledge this claim was not shown to be incorrect either.In this work we show that the moving walls have no effect on the dynamics of a quantum state placed far from the wall. More precisely we prove that the dynamics of a particle with an initial Gaussian wavefunction (the most widely investigated case) does not depend on the boundary conditions as long as the wavefunction remains negligible at the boundaries. This will first shown to be the case in the infinite well with linearly moving walls, and we will then extend these results to a family of related systems in which the moving boundaries can give rise to geometric phases. The ingredients employed, combining a time-dependent unitary transformation with a property of the Jacobi theta functions, will be described in Sec. 2. Sec. 3 will deal with the infinite potential well with linearly expanding walls, including the periodic case with instantaneous reversal. Sec. 4 will investigate confined time-dependent oscillators with a specific relation between the oscillator frequency and the position of the confining walls; contrary to the infinite potential well, such systems admit cyclic states displaying geometric phases that will be seen to be induced by the wall's motion. We will nevertheless show that the effect of the geometric phases is washed out when the initial quantum state is localized inside the well. The results obtained will be discussed in Sec. 5.§ QUANTUM CANONICAL TRANSFORMATION§.§ Hamiltonian and boundary conditions The Hamiltonian for a particle of mass m placed in a potential v(x,t) inside a confined well of width L(t) with moving boundaries is given byH =P^2/2m+VV(x,t) ={[ v(x,t) for-L(t)/2≤ x≤L(t)/2;+∞ otherwise ]..The solutions of the Schrödinger equationiħ∂_tψ(x,t)=Hψ(x,t)must obey the boundary conditions ψ(± L(t)/2)=0 (if the well is embedded in a larger system more general boundary conditions can be considered <cit.>). The even and odd instantaneous eigenstates of H,ϕ_n(x,t)=√(2/L(t))cos[(2n+1)π x/L(t)]andφ_n(x,t)=√(2/L(t))sin[(2n)π x/L(t)]verify H|ϕ_n⟩=E_n(t)|ϕ_n⟩ and H|φ_n⟩=E_n (t)|φ_n⟩. The instantaneous eigenvalues are E_n(t)=( 2n+1)^2ħ^2π^2/2mL^2(t) (with n a positive integer) and E_n(t)=(2n)^2ħ^2π^2/2mL^2(t) (with n a strictly positive integer) for the even and odd instantaneous eigenstates resp. However the ϕ_n or φ_n are not solutions of the Schrödinger equation. Indeed, due to the time-varying boundary conditions, the problem is ill defined, eg the time derivative ∂_tψ(x,t) involves the difference of two vectors with different boundary conditions belonging to different Hilbert spaces <cit.>. Hence neither the difference ψ(x,t^')-ψ(x,t) nor inner products taken at different times ⟨ψ(t^')|. ψ(t)⟩ are defined.In the following we will restrict our discussion to symmetric boundary conditions as specified by Eq (<ref>) and to initial states of even parity in x (in practice, states initially located at the center of the box), so that only the even states ϕ_n(x,t) will come into play. The reason for this choice is that the derivations are technically simpler and the discussion more transparent. The extension to initial states with no definite parity and to non-symmetric boundary conditions is given in the Appendix. §.§ Unitary transformation To tackle this problem the most straightforward approach is to map the Hamiltonian H of the time-dependent boundary conditions to a new Hamiltonian H̃ of a fixed boundary problem. This is done by employing a time-dependent unitary transformation implementing a “canonical” change of variables <cit.>. Letℳ(t)=exp(iξ(t)/2ħ(XP+PX) )be a unitary operator with a time-dependent real function ξ(t) defining the canonical transformation <cit.>|ψ̃⟩ =ℳ(t)|ψ⟩ H̃(t) =ℳ(t)H(t)ℳ^† (t)+iħℳ(t)∂_tℳ^† (t) Ã =ℳ(t)Aℳ^†(t)the latter holding for time-independent observables A such as X or P. Note that ℳ(t) represents a dilation, ie any arbitrary function f(x)transforms as ℳ(t)f(x)=e^ξ(t)/2f(e^ξ(t)x). It is therefore natural to choose ξ(t) so that exp(ξ(t))= L(t)/L_0 where L_0≡ L(t=0) so as to map the original problem to the initial interval [-L_0/2,L_0/2], withψ(x,t)=⟨ x|ℳ ^†(t)|ψ̃⟩=√(L_0/L(t)) ψ̃(L_0/L(t)x,t).|ψ̃⟩ is the solution of the fixed boundary Hamiltonian (<ref>) whose explicit form isH̃(t)=P̃^2/2m+V(X̃)-∂_t L(t)/2L(t)(XP+PX).Eq. (<ref>) suggests to work with solutions of H̃(t). This is particularly handy when a set of complete solutions |ψ̃_n⟩obeying the canonically transformed Schrödinger equationiħ∂_t|ψ̃_n⟩ =H̃|ψ̃_n⟩are known: the initial state |ψ(t_0)⟩ is mapped to |ψ̃(t_0)⟩, which is evolved by expansion over the basis functions |ψ̃_n⟩ before being transformed by the inverse unitary transformation.§ EVOLUTION OF A LOCALIZED STATE IN AN INFINITE POTENTIAL WELL §.§ Moving walls at constant velocity: basis solutions Let us now consider the infinite potential well corresponding to v(x,t)=0 in Eq. (<ref>). We will assume throughout that the walls move at constant velocity q, so that the wall's motion followsL(t)=L_0+qt.q>0 (q<0) corresponds to linearly expanding (contracting) walls. The linear motion (<ref>) has been indeed the main case studied in the context of nonlocality induced by boundary conditions, due to the existence of a complete basis of exact solutions of the canonically transformed Schrödinger equation (<ref>). These solutions were originally obtained by inspection <cit.>, or later from a change of variables in the Schrödinger differential equation <cit.>. From these solutions it is straightforward to guess the basis functions |ψ̃_n⟩ of Eq (<ref>) that are found to be given byψ̃_n(x,t)=√(2/L_0)e^imx^2L(t)[ ∂_tL(t)]/2ħ L_0^2-iħπ^2(2n+1)^2∫ _0^tL(t^')^-2 dt^'/2mcos(π(2n+1)x/L_0 )where n=0,1,2... For the linear motion (<ref>), the integral immediately yields∫_0^t1/L(t^')^2 dt^'=t/L_0(L_0 +qt).As mentioned above the ψ̃_n(x,t) are not eigenfunctions of H̃, but they can be employed as a fundamental set of solutions in order to obtain the state |ψ̃(t)⟩ evolved from an arbitrary initial state |ψ̃(t=0)⟩ expressed as|ψ̃(t)⟩=∑_n ⟨ψ̃_̃ñ(t=0)|ψ̃(t=0)||%s⟩⟩ψ̃_̃ñ(t).The solution ψ(x,t) of the original problem with moving boundaries is recovered from ψ̃(x,t) through Eq. (<ref>). In particular, each solution ψ̃_n(x,t) is mapped intoψ_n(x,t)=√(2/L(t))e^imx^2[∂ _tL(t)]/2ħ L(t)-iħπ^2(2n+1)^2∫_0^t L(t^')^-2 dt^'/2mcos(π(2n+1)x/L(t)). §.§ Gaussian Evolution §.§.§ Initial Gaussian Assume the initial wavefunction is a Gaussian of width d,⟨ x|.G(t=0)⟩≡ G(x,0)=(1-i)e^-x^2/4d^2/2^3/4π^1/4√(-id)with a maximum at the center of the box (x=0) and with negligible amplitude at the box boundaries x=± L_0/2. We will consider in the Appendix the more general case of an initial Gaussian with arbitrary initial average position and momentum, given by Eq. (<ref>). | G(t=0)⟩ is expanded over the basis states |ψ̃_n(t=0)⟩ as per Eq. (<ref>) where g_n (q)=⟨ψ̃_n(t=0)|G(t=0)|$⟩ is readily obtained analytically fromg_n(q) =∫_-∞^+∞ψ_n^∗(x,0)G(x,0)dx =(1-i)2^3/4π^1/4/√(-idl_0)√(1/d^2 +2imq/ħ l_0)exp(-π^2d^2ħ(2n+1)^2 /l_0(ħ l_0+2id^2mq))The fact that the solutionsψ_n(x,t)stretch (in the expanding case) as time increases has been taken as an indication that the initial Gaussian would also stretch provided the expansion is done adiabatically so that the expansion coefficientsg_nremain unaltered <cit.>. Hence the physical state of the particle would be changed nonlocally by the expansion, although no force is acting on it.We show however that the evolution of the initial Gaussian can be solved exactly in the linear expanding or retracting cases by using Eqs. (<ref>) and (<ref>), displaying no dependence of the time-evolved Gaussian on the walls motion. The periodic case, in which the walls motion reverses and starts contracting atT/2so thatL(T)=L_0follows by connecting the solutions att=T/2.§.§.§ Sum in terms of Theta functions Our approach to this problem involves the use of special functions, the Jacobi Theta functions, and a well-known peculiar property of these functions (the Transformation theorem <cit.>). Let us introduce the Jacobi Theta function,ϑ_2(z,κ), defined here asϑ_2(z,κ)=2∑_n=0^∞e^iπκ(n+1/2) ^2cos[(2n+1)z]withIm(κ)>0. It can be verified that the time evolved solutionψ̃(x,t)=∑_ng_n(q)ψ̃_n(x,t)can be summed to yield a theta functionϑ_2, and that further applying Eq. (<ref>) gives the wavefunction evolved fromG(x,0)asψ(x,t)=(1-i)(2π)^1/4e^imx^2∂ _tL(t)/2hL(t)ϑ_2(z,κ)/√(-idL_0 L(t))√(1/d^2+2im/hL_0∂_tL(t)_t=0)withz=π x/L(t); κ=4πħ d^2/L_0( 2d^2m∂_tL(t)_t=0-iħ L_0)-2πħ/m ∫_0^t1/L(t^')^2dt^'In generalψas well aszandκdepend onq, the velocity of the walls motion. We will explicitly denote this functional dependence, iez(q),κ(q).Note that the particular caseq=0corresponds to static walls with fixed boundary conditions.§.§.§ Comparing the static and expanding walls cases In order to compare the time evolved wavefunction in the static and moving problems, let us computeψ(x,t;q=0)/ψ(x,t;q)which after some simple manipulations takes the formψ(x,t;q=0)/ψ(x,t;q)=e^iz^2(0)/πκ(0) -iz^2(q)/πκ(q)(κ(0)/κ(q)) ^1/2ϑ_2(z(0),κ(0))/ϑ_2( z(q),κ(q)).We now prove that for the physical values of the parameters corresponding to a localized Gaussian, this expression is unity. The first step is to use the Jacobi transformation <cit.> ϑ_2(z,κ)=e^-iz^2/κπ/( -iκ)^1/2ϑ_4(z/κ,-1/κ)for bothϑ_2functions of Eq. (<ref>).ϑ_4is the Jacobi Theta function defined byϑ_4(z,κ)=∑_n=-∞^∞( -1)^ne^iπκ n^2e^2inz.Eq. (<ref>) then becomesψ(x,t;q=0)/ψ(x,t;q)=ϑ_4(z(0)/κ(0),-1/κ(0))/ϑ_4(z(q)/κ(q),-1/κ(q)).We then note thatIm[-1/κ(q)]=d^2m^2L(t)^2/π( 4d^4m^2+h^2t^2).This is typically a very large quantity,Im-1/κ(q)≫1. This comes from the fact that the typical spatial extensionΔxof a Gaussian at timetis deduced from its variance( Δx)^2.Δxneeds to be much less than the spatial extension of the wellL(t)since by assumption the quantum state remains localized at the center of the box, far from the box boundaries.Recall indeed that for a Gaussian(Δx)^2=d^2+(ħt)^2/(2dm)^2, so for expanding walls the condition(Δx)(t)≪L(t)can be fulfilled even for largetprovidedqis sufficiently large. However, since we are comparing here the evolution for an arbitrary value ofqwith the fixed walls case (q=0), the stricter condition forq=0(Δ x)(t)≪ L_0 is the one that needs to hold. This condition will only hold for a limitedtime interval, given that the initially localized quantum state will spread and necessarily reach the walls. But then of course the question regarding nonlocal effects of the boundaries motion becomes moot, since a local contact with an infinite wall (be it fixed or moving) reflects the wavefunction and modifies its dynamics.This is why the investigation concerning nonlocal effects is only relevant for times such that Eq. (<ref>)holds, although it should be stressed that the time evolved expression forψ(x,t)that we have derived, given by Eq. (<ref>) remains valid for anyt. Now from the definition ofϑ_4we haveϑ_4(z(q)/κ(q),-1/κ(q)) =∑_n=-∞^∞(-1)^ne^i(π n^2-2nz(q))[Re(-1/κ (q))]e^-(π n^2-2nz(q))[ Im(-1/κ(q))].The last term of Eq. (<ref>) is negligible except forn=0, ieexp(πn^2-2nz(q))[Im( 1/κ(q))]≃0+δ_n,0becausez(q)is real, with|z(q)|≪1/2(since the spatial wavefunction is assumed to vanish outside the central part of the well), andIm(1/κ)<0.Therefore Eq. (<ref>) is reduced to the single termn=0yieldingϑ_4(z(q)/κ(q),-1/κ(q))≃1. This holds for any value ofqand in particular forq=0(fixed walls). Hence, according to Eq. (<ref>), we haveψ(x,t;q)=ψ(x,t;q=0)meaning that the dynamics of the wavefunction initially localized at the center of the box does not depend on the expanding motion of the walls at the boundaries of the box. In particular the adiabatic condition does not play any particular role, as Eq. (<ref>) holds for any value of the wall velocityq. While each individual stateψ_n(x,t)does stretch out as time increases, the sum (<ref>) forψ(x,t)ensures that the interferences cancel the stretching for the localized state. From a physical standpoint no motion is induced superluminally on a localized quantum state by the walls expansion.§.§.§ Contracting and periodic walls motion The same results hold for walls contracting linearly (with nowq<0), provided the wavefunction remains localized far from the walls throughout . The evolution in the periodic case follows by considering successively an expansion withL(t)=L_0+qtup tot=T/2followed by a contraction fromt=T/2toTwith the walls positions determined fromL^c(t)=L_0+q(T-t),now withq>0. The analytic solutions (<ref>) and (<ref>) do not verify the Schrödinger equation during the reversal. Assuming the walls motion is instantaneously reversed att=T/2,the continuity of the wavefunction imposes to match the expanding and contracting solutions at that time. Note in particular that an expanding basis stateψ_n (x,T/2-ε), whereεis small, does not evolve into the “ reversed” stateψ_n (x,T/2+ε)after the walls motion reversal. Indeed the basis solutions of the Schrödinger equation with the contracting boundary conditions given by Eq. (<ref>) areψ_n^c(x,t)=√(2/L_0+q(T-t))e^(-iπ^2 ħ(2n+1)^2(2t-T)/2m(2L_0+qT)(L_0+q(T-t)) -imqx^2/2ħ(L_0+q(T-t)))cos(π (2n+1)x/L_0+q(T-t)),and obviouslyψ_n(x,T/2)≠ψ_n^c(x,T/2).We have instead a diffusion process, in which a given basis functionψ_nof the expanding boundary condition is scattered into several outgoing channelsψ_j^cof the contracting boundary case. This holds for any nonvanishing value ofq; to first order, we haveψ_n(x,T/2)/ψ_n^c(x,T/2)=1+iqmx^2/ħ L_0 +o(q^2),so that even in the adiabatic limit the expanding basis wavefunction cannot be matched to a contracting one, as implicitly assumed in Ref. <cit.>.In order to obtain evolved localized Gaussian in the periodic case, we can proceed as follows. From Eq. (<ref>) (taken forq→∞), we know thatψ(x,T/2;q)is a freely evolved Gaussian. We can thus repeat the same steps leading to (<ref>), but starting from the time evolved GaussianG(x,T/2)=(1-i)e^imx^2/2(ħ T/2-2id^2m) /(2π)^1/4√(d(ħ T/2/d^2 m-2i))instead of Eq. (<ref>).G(x,T/2)is the expanded over the contracting basis functionsψ̃_n^c(x,t)[cf Eq. (<ref>)], the expansion coefficientsg_n^c(q)replacing the formerg_n(q)introduced above in Eq. (<ref>). The result isg_n^c(q)=(1-i)2^3/4π^1/4exp(-ħ(2π n+π)^2(4d^2m+iħ T)/2m(2L_0+qT)(ħ (L_0+qT)-2id^2mq))/√(2L_0+qT)√(ħ T/dm-4id)√(m(ħ(L_0+qT)-2id^2mq)/h(2L_0+qT)(4d^2m+iħ T)).The final step, as above, is to write the formal infinite sum in terms of the Theta functionϑ_2. Att=T, when the walls have recovered their initial positionL(T)=L_0, the time evolved Gaussian is given byψ^c(x,T;q)=(1-i)(2π)^1/4e^-imqx^2 /2ħ L_0ϑ_2(π x/L_0,κ^c(q)) /√(L_0(T/d-4idm/ħ))√(-2id^2mq+ħ L_0+ħ qT/4d^2m+iħ T)whereκ^c(q)(at timet=T) is given byκ^c(q)=-2πħ(ħ T-2id^2m)/L_0m( ħ(L_0+qT)-2id^2mq).We then use the same method that led us from Eq. (<ref>) to Eq. (<ref>) based on the Jacobi transformation theorem to show thatψ^c (x,T;q)=ψ^c(x,T;q=0),that is the walls motion after a full cycle has no consequence on the dynamics of a localized quantum state of the particle.§ EFFECT OF GEOMETRIC PHASES ON A LOCALIZED STATE EVOLUTION§.§ Geometric phases and nonlocality For the infinite potential well with moving boundaries, the fact that the basis functions are not cyclic states even in the case of periodic motion of the walls (as seen in Sec. <ref>) precludes the existence of a cyclic non-adiabatic geometric phase <cit.>. However geometric phases <cit.> could be relevant to the issue of nonlocality. Indeed, a geometric phase is a global quantity, affecting the quantum state globally even if the effect causing the geometric phase lies in a localized space-time region (we will see an explicit example below). Some authors even ascribe to geometric phases nonlocal properties <cit.> including in the context of time-dependent boundary conditions <cit.>.For these reasons it is relevant to see if the results obtained in Sec. <ref> for the infinite potential well could be affected in systems admitting geometric phases. It turns out that there are systems, a family of time-dependent linear oscillators (TDLO) confined by infinitely high moving walls, whose solutions are closely related to the ones of the infinite well with time-dependent boundary conditions, that admit cyclic states that pick up geometric phases. We will see by using a simple scaling property that the geometric phase in this system is caused by the walls motion, but that nevertheless the geometric phases have no consequence on the dynamics of a localized quantum state. §.§ Confined time-dependent oscillators: geometric phase and basis states §.§.§ Confined TDLO Let us start again from the Hamiltonian (<ref>) but now takev(x,t)of Eq. (<ref>) to be given byv(x,t)=-m/2∂_t^2L(t)/L(t)x^2.This is a TDLO confined in the interval-L(t)/2≤x≤L(t)/2,where as aboveL(t)represents the size of the box between infinitely high and moving walls. This TDLO is special in that the frequencyΩ^2(t)=-∂_t^2L(t)/L(t)depends on the walls motion[Note that when L(t) is linear in t,∂_t^2L(t) vanishes and the potential (<ref>) becomes that of the infinite well. Hence in general Eq. (<ref>) represents the solution of a confined TDLO, only the special case for which ∂_t^2L(t)=0 corresponds to the infinite well with moving walls.]. It is then known <cit.>, as can be checked directly by inspection, that the functionsψ̃_n(x,t)andψ_n(x,t)defined respectively by Eqs. (<ref>) and (<ref> ) still obey the Schrödinger equationsiħ∂_tψ̃=H̃ψ̃andiħ∂_tψ=Hψwhere the potential between the walls is now given by Eq. (<ref>).§.§.§ Cyclic Evolution Assume a confined TDLO with a real and periodic functionL(t)with periodTis initially in a stateψ_n(x,t=0), given by Eq. (<ref>). After a full cyclic evolutionψ_n(x,T)returns to the initialψ_n(x,0)but acquires a total phaseμ_n,ieψ_n(x,T)=e^-iμ_nψ_n(x,0).Following Aharonov and Anandan <cit.>,μ_ncan be parsed into a “ dynamical” partδ_nencapsulating the usual phase increment by the instantaneous expectation value of the Hamiltonian and a “ geometric” partγ_nreflecting the curve traced during the evolution in the projective Hilbert space (defined as the space comprising the rays, that is the states giving rise to the same density matrix <cit.>).μ_nis directly obtained from Eq. (<ref>) [with Eq. (<ref>)] and is seen to be proportional to∫_0^TL(t^')^-2 dt^'.The dynamical phaseδ_n=-ħ^-1∫_0^T⟨ψ_n(t^')| H|ψ_n(t^')⟩ dt^'is computed through a tedious but straightforward calculation. The nonadiabatic geometric phaseγ_nis then obtained asγ_n=μ_n-δ_n=m/24(1-6/(2π n+π)^2 )∫_0^T(∂_tL(t))^2-L(t)∂ _t^2L(t)dt.Note thatγ_nis nonzero for nontrivial choices ofL(t).§.§.§ Scaling In order to get a handle on the physical origin of the geometric phase on a stateψ_n, we use the following scaling property. By rescalingL(t),L̅(t)=kL(t), withk>1one changes the walls position while leaving the dynamics invariant: putL̅(t)=kL(t), withk>1.Then the frequencyΩ^2(t)=-∂_t^2L(t)/L(t)and therefore the Hamiltonian are not modified, by virtue of Eq. (<ref>). However as is apparent from Eq. (<ref>) the geometric phase scales asγ̅_n=k^2γ_n. Hence increasing the walls motion by a factorkinduces a change in the geometric phase on the basis statesψ̅_n(x,T)that can be detected at any pointxinside the confined oscillator.This is illustrated in Fig. <ref> featuring a TDLO withL(t)=L_0((1+q)/(1+qcosω t) )^1/2and the frequency in Eq. (<ref>) is given byΩ^2(t)=-∂_t^2L(t)/L(t)=qω^2(q(cos(2ω t)-5)-4cos(ω t))/8(qcos(ω t)+1)^2.This particular choice ofL(t)for the boundary motions has been previously investigated and is known in the infinite potential well case to lead to chaotic or regular behavior asL_0,qandωare varied <cit.>. Here instead we are looking at the confined TDLO, ie with the potential given by Eq. (<ref>): Fig. <ref> shows the geometric phaseγ_n,computed from Eq. (<ref>) for the first basis statesψ_nand for different values ofL̅_0=kL_0thus illustrating the dependence ofγ_non the walls motion. §.§ Confined time-dependent oscillators: Localized state §.§.§ Evolution of Gaussian state The time-dependent boundary conditions induce geometric phases on the basis statesψ_n. Although a Gaussian stateG(x,0)initially given by Eq. (<ref>) can be expanded at any time in terms of these basis statesψ_nthis does not imply of course that the evolved wavefunctionψ(x,t)will also pick up a phase after a full cycle.Actually, since Eqs. (<ref>) and (<ref>) still hold for the confined TDLO with moving walls, we can again write the time-evolved solutionψ(x,T), here after a periodTin terms of a Theta function. Formallyψ(x,t)is again given by Eq. (<ref>), the only difference relative to the infinite potential well of Sec. <ref> being thatL(t)is a periodic function and not linear int. To assess the relevance of geometric phases, we rescale the walls motion while leaving the Hamiltonian invariant as explained above by puttingL̅(t)=kL(t). We have seen that this rescaling modifies the geometric phases. Hence by comparing the rescaled wavefunctionψ̅(x,T)with the original solutionψ(x,T),evolved in both cases from the same initial stateG(x,0),we can infer whether the geometric phases modify the quantum state evolution.Writingψ̅(x,T)/ψ(x,T)in terms ofϑ_4functions as per Eq. (<ref>), and noting thatz̅=z/k,κ̅ =κ/k^2andL(T)=L_0,we apply the Jacobi transformation (<ref>) to find given by Eq. (<ref>). From Eq. (<ref>) we see thatz̅=z/kandκ̅=κ/k^2so that by using Eq. (<ref>) and the Jacobi transformation (<ref>) we are led toψ̅(x,T)/ψ(x,T)=ϑ_4(1/k z/κ,-k^2/κ)/ϑ_4(z/κ,-1/κ).The equality on the right handside holds only provided the conditions given above between Eqs. (<ref>) and (<ref>) hold (recall we havek>1). Under these circumstances we see, by following exactly the reasoning given above that bothϑ_4(1/kz/κ ,-k^2/κ)≃1andϑ_4(z/κ,-1/κ)≃1.Eq. (<ref>) proves that while rescaling the walls motion changes the geometric phase of the basis functions according toγ̅_n =k^2γ_n, no such change takes place when the initial state is the GaussianG(x,0)localized at the center of the confined time-dependent potential. The geometric phases picked up by each basis state over whichG(x,0)is expanded vanish by interference. Recall that an arbitrary initial Gaussian placed in a periodic (unconfined) potential is not cyclic unless specific conditions are verified <cit.>. Eq. (<ref>) does not depend on whether these conditions are met and suggests that the wavefunction in the time-dependent boundary problem follows the same evolution as the one of the unconfined problem with the time-dependent potential:ψ(x,T)can thus pick up a nonadiabatic cyclic geometric phase if the evolution in the unconfined potential leads to such a geometric phase, but there will be no additional effect due to the time-dependent boundaries. §.§.§ Approximate solution for an unconfined time-dependent oscillator Note that as a byproduct of the present treatment, we have obtained an interesting closed form expression for the evolution of an initial Gaussian in an unconfined time-dependent linear oscillator potential for which there is a functionL(t)such that the frequency can be put under the formΩ^2(t)=-∂_t^2L(t)/L(t). Indeed, Eq. (<ref>) along with the Jacobi transformation (<ref>) andϑ_4(z/κ,-1/κ)≃1give the evolved Gaussianψ(x,t)asψ(x,t)=(1-i)(2π)^1/4e^imx^2∂ _tL(t)/2hL(t)/√(-idL_0L(t))√(1/d^2+2im/hL_0∂_tL(t)_t=0)e^ix^2/[4ħ L(t)^2( τ(t)/2m+d^2/-2d^2mL_0∂_tL(t)_t=0+iħ L_0^2)]/√(4π d^2ħ/L_0(ħ L_0+2id^2m∂_tL(t)_t=0)+2iπħτ(t)/m) ,whereτ(t)≡∫_0^tL^-2(t^')dt^'. Contrary to the standard approaches for solving Gaussian problems in TDLOs, that involve nonlinear equations calling for numerical integration <cit.>, Eq. (<ref>) can be often obtained explicitly analytically, depending on whether the closed form integral ofτ(t)is known (of course the range of application of Eq. (<ref>) is very limited compared to standard methods).§.§.§ Example Let us look at the localized state evolution for the TDLO whose geometric phases in the basis states were shown in Fig. <ref>. We start with an initial Gaussian state and let it evolve up tot=Tfor the TDLO confined by infinitely high moving walls on the one hand, and for the same but unconfined TDLO on the other. Fig. <ref> shows the real part of the evolved wavefunction in both cases. The curves are identical, illustrating that the walls motion has no influence on the evolution of a localized state. Note that the wavefunction for the unconfined TDLO has been computed by employing an independent and totally different method, based on Gaussian propagation through the solutions of Ermakov systems (see Ref. <cit.> for details).§ CONCLUSION To sum up we have shown that contrary to earlier claims, time-dependent boundary conditions do not induce an effective or explicit form of nonlocality, as happens eg for Bell correlations. This was seen to be the case for the paradigmatic particle in a box with moving walls and also holds for systems in which the moving boundaries induce geometric phases. Although a moving wall changes the boundary conditions, this change modifies the entire quantum state of the system (instantaneously in the non-relativistic framework) only if the state has a non-negligible amplitude in the boundary region. This is clearly not the case for a localized state placed far from the moving walls.§ TIME EVOLVED STATE IN TERMS OF THETA FUNCTIONS We first consider the case of an initial state given by the Gaussian⟨ x|.𝒢(t=0)⟩≡𝒢(x,0)=e^-(x-x_0)^2/4d^2+ip_0x/h /(2π)^1/4√(d).Contrary to the initial Gaussian given by Eq. (<ref>),𝒢(x,0)has its maximum atx_0anywhere inside the box (but sufficiently far from the box boundaries, since by assumption the initial state has negligible amplitude at the boundaries), and a mean momentump_0. In addition to the even basis functions (<ref>) and (<ref>) derived from the even instantaneous eigenstatesϕ_n(x,t)given by Eq. (<ref>), we will also need odd basis functions derived in the same way from the odd eigenstatesφ_n(x,t)given by Eq. (<ref>); for example the odd counterpart toψ_n(x,t)defined by Eq. (<ref>) isζ_n(x,t)=√(2/L(t))e^imx^2[∂ _tL(t)]/2ħ L(t)-iħπ^2(2n)^2∫_0^tL(t^')^-2 dt^'/2msin(π(2n)x/L(t)).The expansion coefficientsg_nof Eq. (<ref>) now becomeh_n(q) =∫_-∞^+∞ψ_n^∗(x,0)𝒢(x,0)dx=(1-i)(-π/2)^1/4e^-x_0^2/4d^2 (e^i[2π d^2ħ(2n+1)+(2d^2p_0-iħ x_0)L_0]^2/4d^2ħ L_0(2d^2m∂ _tL(t)_t=0-iħ L_0)+e^i[2π d^2 ħ(2n+1)+(-2d^2p_0+iħ x_0)L_0]^2 /4d^2ħ L_0(2d^2m∂_tL(t)_t=0-iħ L_0) )/√(dL_0)√(1/d^2+2im∂ _tL(t)_t=0/ħ L_0)for the even basis functions andj_n(q) =∫_-∞^+∞ζ_n^∗(x,0)𝒢 (x,0)dx=i(2π)^1/4e^-x_0^2/4d^2 e^i[2πħ d^2(2n)+(-2d^2p_0+iħ x_0)L_0]^2/4d^2ħ L_0(2d^2m∂ _tL(t)_t=0-iħ L_0)-e^^i[2πħ d^2(2n)+(2d^2p_0-iħ x_0)L_0]^2/4d^2ħ L_0(2d^2m∂_tL(t)_t=0-iħ L_0) /√(dL_0)√(1/d^2+2im∂_tL(t)_t=0/ħ L_0)for the odd basis functions.It can be checked, after a tedious but straightforward calculation that the time evolved stateψ(x,t)=∑_n⩾0h_n(q)ψ_n (x,t)+∑_n>0j_n(q)ζ_n(x,t)can be written in terms of 8 Jacobi theta functions, half of them being theta functions of the second typeϑ_2(z,κ)introduced above [Eq. (<ref>)], the other half (for the odd part of the sum) being functionsϑ_3(z,κ)defined byϑ_3(z,κ)=∑_n=-∞^∞e^iπκ n^2e^2inz.PutA ≡exp(-x_0^2/4d^2+i(2d^2 p_0-iħ x_0)^2L_0/4d^2ħ(2d^2m∂ _tL(t)_t=0-iħ L_0)+imx^2∂_tL(t)_t=0 /2ħ L_0) B ≡√(dL_0L(t))√(1/d^2+2im∂ _tL(t)_t=0/ħ L_0)C ≡π2d^2p_0-iħ x_0/iħ L_0-2d^2 m∂_tL(t)_t=0.Note thatAandCdepend onx_0andp_0. Withκdefined by Eq. (<ref>) above, we introduce the functionsθ_1(x,t;q) =(1-i)(-π)^1/4Aϑ_2 (-π x/L(t)-C,κ)/B θ_2(x,t;q) =(1-i)(-π)^1/4Aϑ_2 (π x/L(t)-C,κ)/B θ_3(x,t;q) =(1-i)(-π)^1/4Aϑ_2 (-π x/L(t)+C,κ)/B θ_4(x,t;q) =(1-i)(-π)^1/4Aϑ_2 (π x/L(t)+C,κ)/B θ_5(x,t;q) =(π/2)^1/4Aϑ _3(-π x/L(t)-C,κ)/B θ_6 (x,t;q) =-(π/2)^1/4Aϑ _3(π x/L(t)-C,κ)/B θ_7(x,t;q) =-(π/2)^1/4Aϑ _3(-π x/L(t)+C,κ)/B θ_8(x,t;q) =(π/2)^1/4Aϑ _3(π x/L(t)+C,κ)/B.Then the time-evolved stateψ(x,t)analogous to the one obtained above [Eq. (<ref>)] but when the initial state is the general Gaussian given by Eq. (<ref>) is given in terms of the functionsθ_kasψ(x,t;q)=1/2∑_k=1^8θ_k(x,t;q). § MOVING WALLS AT CONSTANT VELOCITY Let us assess the effect of walls moving at constant velocity, discussed in Sec. <ref>, on the wavefunction evolving from𝒢(x,0). For each of the even functionsθ_k(k=1,..,4)the transformation (<ref>) leads to the analog of Eq. (<ref>) in the formθ_k(q=0)/θ_k(q)=ϑ_4(z_k(0)/κ(0),-1/κ(0))/ϑ_4(z_k(q)/κ(q),-1/κ(q)), k=1,..,4wherez_kis the relevant argument of the theta function in the expression ofθ_kgiven by Eqs. (<ref>)-(<ref>), that isz_k=±πx/L(t)±C.As explained in Sec. <ref> in the case of a single theta function, this leads here, under the same assumptions, toθ_k(x,t;q=0)=θ_k(x,t;q), so that the walls motion does not impinge on the evolution of each of these even functionsθ_k.For the odd functionsθ_k(k=5,..,8)involvingϑ_3,we use instead of Eq. (<ref>) the Jacobi transformation <cit.>ϑ_3(z,κ)=e^-iz^2/κπ/( -iκ)^1/2ϑ_3(z/κ,-1/κ).The resultθ_k(q=0)/θ_k(q)=ϑ_3(z_k(0)/κ(0),-1/κ(0))/ϑ_3(z_k(q)/κ(q),-1/κ(q))=1, k=5,..,8is shown to hold by following the same arguments given in Sec. <ref>, but by using the expansion (<ref>) instead of (<ref>). Thus Eq. (<ref>) above stating thatψ(x,t;q)=ψ(x,t;q=0)also holds when the initial state is the Gaussian (<ref>) andψ(x,t;q)is given by Eq. (<ref>).§ A SINGLE MOVING WALL In the main text we have considered the symmetric boundary conditions specified by Eq (<ref>), as this gives a simpler treatment. However in most of the works <cit.> dealing with the subject of nonlocality induced by time-dependent boundary conditions, the problem of an infinite well with a single moving wall was considered. In that case, the Hamiltonian has the following boundary conditions:H =P^2/2m+VV(x,t) ={[ 0 for0≤ x≤ L(t);+∞ otherwise ]..The instantaneous eigenstates ofHare similar to the odd functionsφ_n(x,t)introduced in Eq. (<ref>) and the basis functions to theζ_n(x,t)of Eq. (<ref>); they are obtained by replacing in these expressionsnbyn/2,yieldingf_n(x,t)=√(2/L(t))sin[nπ x/L(t)]for the instantaneous eigenstates andF_n(x,t)=√(2/L(t))e^imx^2[∂ _tL(t)]/2ħ L(t)-iħπ^2n^2∫_0^tL(t^')^-2 dt^'/2msin(π nx/L(t))for the basis functions. Therefore, provided we are willing to keep the-∞bound in Eq. (<ref>), a harmless approximation given the assumptions concerning the initial Gaussian, we can transpose the results obtained in the present Appendix [Eqs. (<ref>), (<ref>)-(<ref>) and (<ref>)] to the case of a single moving wall (note that relative to these expressions, the arguments ofϑ_3are rescaled asz→z/2andκ→κ/4). 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Lett. A 163 142.zou2000J. Zou and B. Shao 2000, Int. J. Mod. Phys. B 14 1059.yao2001Q.K. Yao Qian-Kai, G. W. Ma, X. F. Chen and Y. Yu 2001, Int. J. Theor. Phys. 40 551.wang2008Z. S. Wang, C. Wu, X. L Feng , L.C. Kwek, C.H. Lai, C.H. Oh and V. Vedral 2008, Phys. Lett. A 372 775mousavi2012S. V. Mousavi 2012, EPL 99 30002.mousavi2014S. V. Mousavi 2014, Phys. Scr. 89 065003.facchiP. Facchi, G. Garnero, G. Marmo and J. Samuel 2016, Ann. Phys. 372 201bellmanR. Bellman, A brief introduction to Theta functions (Dover: Mineola, NY, 2013).aharonov-anandanY. Aharonov and J. Anandan 1987, Phys. Rev. Lett. 58 1593.book-GPA. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, The Geometric Phase in Quantum Systems (Springer, Berlin, 2003).anandanJ. S. Anandan 1988, Ann. Inst. Henri Poincaré 49 271.matzkin2012A. Matzkin 2012, Phys.Rev. Lett. 109 150407child98Y. C. Ge and M. S. Child 1997, Phys. Rev. Lett. 78 2507.A2015JPAA. Matzkin 2015, J. Phys. A 48 305301 | http://arxiv.org/abs/1706.08617v3 | {
"authors": [
"A. Matzkin"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170626223252",
"title": "Single particle nonlocality, geometric phases and time-dependent boundary conditions"
} |
A free energy Lagrangian variational formulation of the Navier-Stokes-Fourier system François Gay-Balmaz^1 and Hiroaki Yoshimura^2====================================================================================emptyemptySynthetiser This article presents a newclass of Pseudorandom Number Generators.The generators are based on traversing a n-cube where a Balanced Hamiltonian Cycle has been removed. The construction of such generators is automatic for small number of bits, but remains an open problem when thisnumber becomes large. A running example is used throughout the paper. Finally, first statistical experiments of these generators are presented, they show how efficient and promising the proposed approach seems.§ INTRODUCTIONMany fields of research or applications like numerical simulations,stochastic optimization, or information security are highly dependent on the use of fast and unbiased random number generators. Depending on the targeted application, reproducibility must be either required, leading to deterministic algorithms that produce numbers as close as possible to random sequences, or refused, which implies to use an external physical noise. The former are called pseudorandom number generators (PRNGs) while the latter are designed by truly random number generators (TRNGs). TRNGs are used for instance in cypher keys generation, or in hardware based simulations or security devices. Such TRNGs are often based on a chaotic physical signal, may be quantized depending on the application. This quantization however raises the problem of the degradation of chaoticproperties.The use of PRNGs, for its part, is a necessity in a large varietyof numerical simulations, in which responses of devices under study must be compared using the same “random” stream. This reproducibility is required too for symmetric encryption like one-time pad, as sender and receiver must share the same pad. However, in that situation, security of the pseudorandom stream must be mathematically proven: an attacker must not be able to computationally distinguish a pseudorandom sequence generated by the considered PRNG with a really random one. Such cryptographically secure pseudorandom number generators are however only useful in cryptographic contexts, due to their slowness resulting from their security.Other kind of properties are desired for PRNGs used in numerical simulations or in programs that embed a Monte-Carlo algorithm. In these situations, required properties are speed and random-like profiles of the generatedsequences. The fact that a given PRNG is unbiased and behaves as a white noise is thus verified using batteries of statistical tests on a large amount of pseudorandom numbers. Reputed and up-to-date batteries are currently the NIST suite <cit.>, and DieHARD <cit.>.Finally, chaotic properties can be desired when simulating a chaotic physical phenomenon or in hardware security, inwhich cryptographical proofs are not realizable. In both truly and pseudorandom number generation, there is thus a need to mathematically guarantee the presence of chaos, and to show thata post-treatment on a given secure and/or unbiased generator can be realized, which adds chaos without deflating these desired properties. This work takes place in this domain with the desire of automaticallygenerating a large class of PRNGs with chaos and statistical properties. In a sense, it is close to <cit.> where the authors shown thatsome Boolean maps may be embedded into an algorithm to provide a PRNG that has both the theoretical Devaney's chaos property and the practicalproperty of succeeding NIST statistical battery of tests. To achieve this, it has been proven inthis article that it is sufficientfor the iteration graph to be strongly connected, and it is necessary and sufficient for its Markov probability matrix to be doubly stochastic. However, they do not purpose conditionsto provide such Boolean maps. Admittedly, sufficient conditions to retrieve Boolean maps whose graphs arestrongly connected are given, but it remains to further filter those whoseMarkov matrix is doubly stochastic. This approach suffers from delaying the second requirement to a final step whereas this is a necessary condition.In this position article, we provide a completely new approach to generate Boolean functions, whose Markov matrix is doubly stochastic and whose graph of iterations is strongly connected.Furthermore the rate of convergence is always taken into consideration to providePRNG with good statistical properties.This research work is organized as follows. It firstly recall some preliminaries that make the document self-contained (Section <ref>), The next section (Section <ref>) shows how theproblem of finding some kind of matrices is moved into the graph theory. Section <ref> is the strongest contribution of this work. It presents the main algorithm to generate Boolean maps with all the required properties andproves that such a construction is correct.Statistical evaluations are then summarized in Section <ref>.Conclusive remarks, open problems, and perspectives arefinally provided. § PRELIMINARIES In what follows, we consider the Boolean algebra on the set={0,1} with the classical operators of conjunction '.',of disjunction '+', of negation '', and ofdisjunctive union ⊕.Let n be a positive integer. ABoolean map f isa function from the Boolean domain to itselfsuch thatx=(x_1,…,x_n) maps to f(x)=(f_1(x),…,f_n(x)). Functions are iterated as follows.At the t^th iteration, only the s_t-th component is “iterated”, where s = (s_t)_t ∈N is a sequence of indices taken in 1;n called “strategy”. Formally, let F_f: 1;n×^n to ^n be defined byF_f(i,x)=(x_1,…,x_i-1,f_i(x),x_i+1,…,x_n).Then, let x^0∈^n be an initial configuration and s∈1;n^ be a strategy,the dynamics are described by the recurrencex^t+1=F_f(s_t,x^t).Let be given a Boolean map f. Its associatediteration graphΓ(f) is the directed graph such thatthe set of vertices is ^n, and for all x∈^n and i∈1;n, the graph Γ(f) contains an arc from x to F_f(i,x).It is easy to associate a Markov Matrix M to such a graph G(f) as follows:M_ij = 1/n if there is an edge from i to j in Γ(f) and i ≠ j;M_ii = 1 - ∑_j=1, j≠ i^n M_ij; and M_ij = 0 otherwise. Let us consider for instance n=3. Letf^*: ^3 →^3 be defined by f^*(x_1,x_2,x_3)=(x_2 ⊕ x_3, x_1 ⊕x_3,x_3). The iteration graph Γ(f^*) of this function is given inFigure <ref> and its Markov matrix is givenin Figure <ref>. The mixing time <cit.> is one of the usual metricsthat gives how far the rows of a Markov matrix converge to a specific distribution.It defines the smallest iteration numberthat is sufficient to obtain a deviation lesser than a given ε for each rows of such kind of matrices.Let us finally present the pseudorandom number generator χ_14Secrypt which is based on random walks in Γ(f).More precisely, let be given a Boolean map f:^n →^n, a PRNG Random, an integer b that corresponds to an awaited mixing time, andan initial configuration x^0.Starting from x^0, the algorithm repeats b timesa random choice of which edge to follow and traverses this edge. The final configuration is thus outputted. This PRNG is formalized in Algorithm <ref> further denotedas χ_14Secrypt.Let f: ^n→^n. It has been shown <cit.> thatif its iteration graph is strongly connected, thenthe output of χ_14Secrypt followsa law that tends to the uniform distributionif and only if its Markov matrix is a doubly stochastic matrix. The nextsection presentsan efficient method to generate Boolean functions with Doubly Stochastic matrix and Strongly Connected iteration graph, further (abusively) denoted as DSSC matrix. § GENERATION OF DSSC MATRICESFinding DSSC matrices can be theoretically handled by Constraint Logic Programming on Finite Domains (CLPFD): all the variables range into finite integer domains with sum andproduct operations.However, this approach suffers from not being efficient enough for large n due to a generate and test pattern.Intuitivelly, considering the n-cube andremoving one outgoing edge and one ongoing edge for each node should be a practical answerto the DSSC matrix finding problem.Moreover, the previous wish of exaclty removing exactly one outgoingand one ongoing edge for each node is solved by removing aHamiltonian cycle in the n-cube. The next section details this step.The iteration graph of f^*(given in Figure <ref>)is the 3-cube in which the Hamiltonian cycle000,100,101,001,011,111,110,010,000has been removed.§ REMOVING HAMILTONIAN CYCLESThe first theoretical section (Section <ref>) shows that this approach produces DSSC matrix, as wished. The motivation to focus on balanced Gray code is then given in Sec. <ref>. We end this section by giving some discussion about practical aspecctsof an existingalgorithm that aims at computing such codes (Section <ref>). §.§ Theoretical Aspects of Removing Hamiltonian Cycles We first have the following result on stochastic matrix and n-cube without Hamiltonian cycle.The Markov Matrix M resulting from the n-cube in which an Hamiltoniancycle is removed, is doubly stochastic.The proof is left as an exercise for the reader. The following result states that the n-cube without one Hamiltonian cyclehas the awaited property with regard to the connectivity. The iteration graph issued from the n-cube where an Hamiltoniancycle is removed is strongly connected.Again, the proof is left as an exercise for the reader. Removing an Hamiltonian cycle in the n-cube solves thus the DSSC constraint. We are then left to focus on the generation of Hamiltonian cycles in then-cube. Such a problem is equivalent to find cyclic Gray codes, i.e., to find a sequence of 2^n codewords (n-bits strings)where two successive elements differ in only one bit position and and where the last codeworddiffers in only one bit position from the first one. The next section is dedicated to these codes. §.§ Linking to Cyclic (Totally) Balanced Gray Codes Let n be a given integer. As far as we know,the exact number of Gray codes in ^n is not known buta lower bound, (n*log2/e loglog n*(1 - o(1)))^2^n has been given in <cit.>. For example, when n is 6, such a number is larger than 10^13. To avoid this combinatorial explosion, we want to restrict the generation to any Gray codesuch that the induced graph of iteration Γ(f) is“uniform”. In other words, if we count in Γ(f)the number of edges that modify the bit i, for 1 ≤ i ≤ n, all these values have to be close to each other. Such an approach is equivalent to restrict the search of cyclic Gray codes which are uniform too.This notion can be formalized as follows. LetL = w_1, w_2, …, w_2^n be the sequenceof a n-bits cyclic Gray code. Let S = s_1, s_2, …, s_2^n be thetransition sequence where s_i, 1 ≤ i ≤ 2^n indicates which bit position changes betweencodewords at index i and i+1 modulo 2^n.Let TC_n : {1,…, n}→{0, …, 2^n} the transition count function that counts the number of times i occurs in S, i.e., the number of timesthe bit i has been switched in L.The Gray code is totally balanced if TC_n is constant (and equal to 2^n/n). It is balanced if for any two bit indices i and j,|TC_n(i) - TC_n(j)| ≤2.Let L^*=000,100,101,001,011,111,110,010 be the Gray code that corresponds tothe Hamiltonian cycle that has been removed in f^*. Its transition sequence is S=3,1,3,2,3,1,3,2 and its transition count function isTC_3(1)= TC_3(2)=2 andTC_3(3)=4. Such a Gray code is balanced. Let now L^4=0000, 0010, 0110, 1110, 1111, 0111, 0011, 0001, 0101, 0100, 1100, 1101, 1001, 1011, 1010, 1000 be a cyclic Gray code. Since S=2,3,4,1,4,3,2,3,1,4,1,3,2,1,2,4, its transition count TC_4 is equal to 4 everywhere and this code is thus totally balanced. §.§ Induction-Based Generation ofBalanced Gray CodesThe article <cit.> proposed the “Construction B” algorithm to produce Balanced Gray Codes. This method inductively constructs n-bits Gray code given a n-2-bit Gray code.The authors have proventhatS_n is transition sequence of a cyclic n-bits Gray codeif S_n-2 is. It starts with the transition sequence S_n-2 of such codeand the following first step: Let l be an even positive integer. Findu_1, u_2, … , u_l-2, v (maybe empty) subsequences of S_n-2such that S_n-2 is the concatenation ofs_i_1, u_0, s_i_2, u_1, s_i_3, u_2, . . . , s_i_l-1, u_l-2, s_i_l, v where i_1 = 1, i_2 = 2, and u_0 = ∅ (the empty sequence).However, this first step is notconstructive:it does not precises how to select the subsequences which ensures thatyielded Gray code is balanced.Let us nowevaluate the number of subsequences u than can be produced. Since s_i_1 and s_i_2 are well defined, we have to chose the l-2 elements of s_3,s_4,…,s_2^n-2 that become s_i_3,…, s_i_l. Let l = 2l'. There are thus#_n = ∑_l'=1^2^n-32^n-2-22l'-2 distinct subsequences u. Numerical values of #_n are given in table <ref>. Even for small values of n, it is not reasonable to hope to evaluate the wholeset of subsequences. However, it isshown in the article that TC_n(n-1) and TC_n(n) are equal to l. Since this step aims at generating (totally) balanced Gray codes, we have set l to be the largest even integer less or equal than 2^n/n. This improvement allows to reduce the number of subsequences to study. Examples of such cardinalities are given in Table <ref> and are referred as#'_n.Finally, the table <ref> gives the number of non-equivalent functions issued from(totally) balanced Gray codes that can be generated with the approach presented in this article with respect to the number of bits. In other words, it corresponds to the size of the class of generators that can be produced. Notice that when n is 7 and 8, we only give lower bounds for2.5E5 distinct choices for the u subsequence. § EXPERIMENTSWe have directly implemented the algorithm given in Figure <ref>. For function f and our experiments b is set with the value given in the fourth column of Table <ref>.For each number n=4,5,6,7,8 of bits, we have generatedthe functions according the methodgiven in Section <ref> . For each n, we have then restricted this evaluation to the functionwhose Markov Matrix has the smallest mixing time. Such functions aregiven in Table <ref>. In this table, let us consider for instancethe function a from ^4 to ^4 defined by the following images :[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]. In other words,the image of 3 (0011) by a is 14 (1110): it is obtained asthebinaryvalueofthefourth elementinthesecondlist (namely 14).Experiments have shown that all the generators pass the NIST and the DieHARD batteries of tests.§ CONCLUSIONThis article has presented a method to compute a large class of truly chaotic PRNGs. First experiments through the batteries of NIST, and DieHardhave shown that the statistical properties are almost established for n=4,5,6,7,8. The iterated map inside the generator is built by removingfrom a n-cube an Hamiltonian path that corresponds to a (totally) balanced Gray code. The number of balanced gray code is large and each of them can beconsidered as a key of the PRNG. However, many problems still remain open, most important ones being listedthereafter. The first one involves the function to iterate. Producing a DSSC matrix is indeed necessary and sufficient but is not linked with the convergence rate to the uniform distribution.To solve this problem, we have proposed to remove from the n-cube an Hamiltonian path thatis a (totally) balanced Gray code. We do not have proven that thisproposal is the one that minimizes themixing time. This optimization task is an open problem we plan to study.Secondly, the approach depends on finding (totally) balanced Graycodes. Even if such codes exist for all even numbers, there is no constructive method to built them when n is large, as far as we know. These two open problems will be investigated in a future work.plain | http://arxiv.org/abs/1706.08923v1 | {
"authors": [
"Jean-François Couchot",
"Pierre-Cyrille Heam",
"Christophe Guyeux",
"Qianxue Wang",
"Jacques M. Bahi"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170626092052",
"title": "Traversing a n-cube without Balanced Hamiltonian Cycle to Generate Pseudorandom Numbers"
} |
Institute of High Energy Physics, [email protected] Double Calorimetry System in JUNO Miao He on behalf of the JUNO collaboration ================================================= The Jiangmen Underground Neutrino Observatory (JUNO) is a multipurpose neutrino-oscillation experiment, with a 20 kiloton liquid scintillator detector of unprecedented 3% energy resolution (at 1 MeV) at 700-meter deep underground. There are 18,000 20-inch photomultiplier tubes (PMTs) in the central detector with an optical coverage greater than 75%. Control of the systematics of the energy response is crucial to archive the designed energy resolution as well as to reach 1% precision of the absolute energy scale. The detected number of photoelectrons in each PMT differs by two orders of magnitude in the reactor antineutrino energy range in such a large detector, which is a challenge to the single channel charge measurement. JUNO has approved a new small-PMT system, including 25,000 3-inch PMTs, installed alternately with 20-inch PMTs. The individual 3-inch PMT receives mostly single photoelectrons, which provides a unique way to calibrate the energy response of the 20-inch PMT system by a photon-counting technology. Besides, the small-PMT system naturally extends the dynamic range of the energy measurement to help the high-energy physics, such as cosmic muons and atmospheric neutrinos. We will present the physics concept of this double calorimetry, the design and implementation of the 3-inch PMT and its readout electronics system. § INTRODUCTION JUNO <cit.> is a neutrino experiment under construction in southern China, ∼53 km from two powerful nuclear plants, Yangjiang and Taishan. The detector target contains 20 kiloton liquid scintillator with 3% energy resolution (at 1 MeV). It is located 700 m underground with a cosmic muon flux of 0.0030 Hz m^-2. The primary goal of JUNO is to determine the neutrino mass hierarchy using reactor antineutrinos by the interference between two oscillation frequency components driven by Δ m^2_31 and Δ m^2_32, respectively. On the other hand, the precise measurement of antineutrino spectra allows JUNO to be the first experiment to measure solar and atmospheric mass splitting simultaneously, with better than 1% precision of θ_12, Δ m^2_21 and Δ m^2_31 (or Δ m^2_32). In addition, the large detector volume, good energy resolution and very low radioactive background allow more physics possibilities, including supernova neutrinos, terrestrial neutrinos, solar neutrinos and exotic searches <cit.>.The schematic design is shown in Fig. <ref>. There is an acrylic sphere with an inner diameter of 35.4 m, which is the container of the liquid scintillator, and supported by a stainless steel latticed shell at a diameter of 40.1 m with 590 connecting bars. There are ∼18,000 20-inch photomultiplier tubes (PMTs) and ∼25,000 3-inch PMTs installed on the stainless steel shell, watching inward on the light generated by the interaction of neutrinos. All of the detector components are immersed in a large pool, filled with 35 kiloton pure water. The water pool also serves as a Cherenkov detector after equipping with 2,000 20-inch PMTs to tag cosmic muons. A redundant muon veto system made of plastic scintillators is deployed on top of the pool.§ DOUBLE CALORIMETRY Determination of the mass hierarchy requires precision measurement of the energy spectrum to separate Δ m^2_31 and Δ m^2_32, and the sensitivity of the mass hierarchy heavily depends on the detector energy resolution <cit.>. JUNO aims for 3%/√(E(MeV)) resolution, which needs very large number of detected photons and strict control of the systematics. High light yield, high transparency liquid scintillator and high quantum efficiency 20-inch PMTs with 75% optical coverage give >1,200 photoelectrons at 1 MeV with a statistical fluctuation of 2.9%, which leaves a room of <1% for the systematic uncertainty.Unfortunately, the 20-inch PMT is too large to see large variation of the detected number of photoelectrons by two orders of magnitude in the reactor antineutrino energy range in such a large detector. It is a challenge to calibrate the non-linear response of the single channel charge measurement to sub-percent level <cit.>. Moreover, when the event vertex differs in the detector volume, the charge range at a single PMT also changes. Therefore, the nonlinearity of the single channel contributes to a non-uniformity of the detector and thus deteriorates the energy resolution.A small-PMT system was proposed in 2014 to install up to 36,000 3-inch PMTs in the gap between 20-inch PMTs. The photocathode area is ∼1/50 compared to the large PMT, and more than 98% of small PMTs only detect single photoelectron in the reactor antineutrino energy range in the JUNO detector according a Monte Calor simulation. Therefore, the total charge of the small-PMT system can be obtained with a photon-counting technology and thus there is almost no charge nonlinearity. This feature provides a unique way to calibrate the energy response of the 20-inch PMT system. Combination of the large and the small PMTs system constitutes a double calorimetry to control both stochastic and non-stochastic effects. Besides, the small-PMT system naturally extends the dynamic range of the energy measurement to help the high-energy physics, such as cosmic muons and atmospheric neutrinos. The physics concept and the system design of the double calorimetry was approved sequentially by the JUNO collaboration in 2015 and 2016.§ SMALL PMT SYSTEM The small-PMT system consists of PMTs and the readout electronics. As shown in Fig. <ref>, a group of 128 small PMTs (SPMTs) with HV divider and water proofing is connected to an underwater box, which is the container of the readout electronics system. A ∼100 m cable transmits power and data to the surface.An international bidding of PMTs was organized in May, 2017. In the end, Hainan Zhanchuang (HZC) photonics, a Chinese industry who introduced the production line and technologies from PHOTONIS, was chosen to be the supplier. HZC will produce 25,000 3-inch PMTs (XP72B22) with 1,000 spares for JUNO in the next two years. XP72B22 is an upgrade of previous XP72B20, with dedicated R&D of better timing based on the requirements of JUNO. HZC will also collaborate with JUNO and produce HV dividers and do the water proofing for all PMTs.Performances of XP72B22 are listed in Table. <ref>. All of them meet JUNO's requirements. In particular, the single photon detection is critical for JUNO since the small PMT works mostly in the photon-counting mode, and the resolutions of the single photoelectron of 5 samples from HZC were measured to be (35±2)%, showing very good uniformity.The HV unit and the control unit <cit.> are expected to be the same as the 20-inch PMT system. A a multichannel front-end ASIC CATIROC was chosen to read out the charge and time information from the small PMT with almost no dead time. Details of CATIROC can be found in Ref. <cit.>. The preliminary design of the underwater box includes a commercial stainless steel tube with two caps. A multichannel connector between PMTs and the box is under consideration for easier installation with reasonable price.§ CONCLUSION Energy resolution is the key of the reactor antineutrino spectrum measurement, for the determination of the neutrino mass hierarchy. A small-PMT system was proposed to work with the large-PMT system as a double calorimetry, in order to control both the stochastic and non-stochastic effects, and thus to improve the energy resolution. JUNO has signed a contract with HZC in May 2017, and is going to install 25,000 3-inch PMTs in the central detector. The mass production and testing of PMTs are in preparation, and R&D of the readout electronics and the underwater box is ongoing. The system is expected to be ready in 2020, and starts taking data with the full detector.This work is supported by the National Natural Science Foundation of China No. 11575226, and the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDA10011200. 5juno.cdr T. Adam et al. (JUNO collaboration), JUNO Conceptual Design Report, arXiv:1508.07166 (2015)juno.yb F.P. An et al. (JUNO collaboration), Neutrino Physics with JUNO, J. Phys. G 43, 030401 (2016)liyf.mh Yu-Feng Li, Jun Cao, Yifang Wang, and Liang Zhan, Unambiguous determination of the neutrino mass hierarchy using reactor neutrinos, Phys. Rev. D 88, 013008 (2013)dyb.nl F.P. An et al. (Daya Bay collaboration), Measurement of electron antineutrino oscillation based on 1230 days of operation of the Daya Bay experiment, Phys. Rev. D 95, 072006 (2017)gcu Davide Pedretti, The Global Control Unit for the JUNO front-end electronics, talk at TIPP 2017, http://indico.ihep.ac.cn/event/6387/session/52/contribution/192/material/slides/0.pdfcatiroc Selma Conforti, CATIROC: a multichannel front-end ASIC to read out the SPMT system of the JUNO experiment, talk at TIPP 2017, http://indico.ihep.ac.cn/event/6387/session/39/contribution/113/material/slides/1.pdf | http://arxiv.org/abs/1706.08761v1 | {
"authors": [
"Miao He"
],
"categories": [
"physics.ins-det"
],
"primary_category": "physics.ins-det",
"published": "20170627100608",
"title": "Double Calorimetry System in JUNO"
} |
[email protected], [email protected]. Mikheev Institute of Metal Physics, Russian Academy of Sciences, 620108 Ekaterinburg, Russia Simple scaling consideration and NRG solution of the one- and two-channel Kondo model in the presence of a logarithmic Van Hove singularity at the Fermi level is given. The temperature dependences of local and impurity magnetic susceptibility and impurity entropy are calculated. The low-temperature behavior of the impurity susceptibility and impurity entropy turns out to be non-universal in the Kondo sense and independent of the s-d coupling J. The resonant level model solution in the strong coupling regimeconfirms the NRG results. In the two-channel case the localsusceptibility demonstrates a non-Fermi-liquid power-law behavior. Kondo modelVan Hove singularities strong correlationsAnomalous f- and d-systems possess highly unusual electronic properties and magnetism. Besides the heavy-fermion behavior, they demonstrate the non-Fermi-liquid (NFL) behavior: logarithmic or anomalous power-law temperature dependences of magnetic susceptibility and electronic specific heat <cit.>. Their magnetism has both localized and itinerant features, being determined by both Kondo effect and density of states (DOS) singularities which are especially important for magnetic ordering.The NFL behavior is related to peculiar features of electron and spin fluctuation spectra.In particular, the multichannel Kondo model is often used which assumes existence of degenerate electron bands. This model explains power-law or logarithmic behavior of electronic specific heat and magnetic susceptibility <cit.>. Recently, the one- and two-channel charge Kondo effect was extensively discussed for nanostructures, layer systems and quantum dots <cit.>.In the present Letter we treat the Kondo model with the electron spectrum containing a logarithmic DOS singularity. This van Hove singularity is typical, in particular, for the 2D case. This is present, e.g., in the electron spectrum of graphene where the Kondo effect is tunable with carrier density <cit.>. Recently, a possibility of graphene doping for the exploration of Van Hove physics was proposed <cit.>.Whereas the flat-band Kondo model permits exact Bethe ansatz solution, the model with singular DOS is a challenge for analytical field-theoretical consideration. At present, the case of empty conduction band was investigated by numerical renormalization group (NRG)method <cit.>. A NRG andresonant level model treatment of the power-law and 1/|E|ln^2|E| divergent bare DOS was performed in Refs.<cit.>. To investigate the case of singular DOS for the Kondo metal we use a simple scaling consideration and compare the results with more advanced NRG calculations.We start from the Hamiltonian of the one-center s-d(f) exchange (Kondo) modelH_sd=∑_𝐤 m αε _𝐤c_𝐤 m α^†c_𝐤 m α^-∑_𝐤𝐤^' mαβJ_𝐤𝐤^'𝐒 _αβc_𝐤mα^†c_ 𝐤^'mβ^ .Here ε _𝐤 is the band energy, 𝐒 are spin operators with spin value being S, σ are the Pauli matrices, in the case of contact coupling J_𝐤𝐤^'=J/N_s where J is the s-d(f) exchange parameter, N_s is the number of lattice sites, m=1...M is the orbital degeneracy index, α, β are spin indices.The density of states corresponding to the spectrum ε_𝐤 is supposed to contain a Van Hove singularity near the Fermi level. In particular, for the square lattice with next- and next-to-nearest neighbour transfer the spectrum readsε_𝐤=2t(cos k_x+cos k_y)+4t^'(cos k_xcos k_y+1)and we have the density of statesρ (E) ≃1/2π ^2√(t^2-4t^' 2)ln16 √(t^2-4t^' 2)/|E|wherethe bandwidth is determined by |E-8t^'|<4|t|. Below we use the approximate density of states fort^'=0ρ (E)=ϱ F(E),F(E)=ln4D/|E| ,ϱ=2/π^2 D, D= 4t. We apply the “poor man scaling” approach <cit.>. This considers the dependence of effective (renormalized) couplingJ_ef(C) on the flow cutoff parameter C→ -0 which occurs at picking out the singular Kondo contributions.To find the scaling equation we pick out in the sums for the Kondo terms the contribution of intermediate electron states near the Fermi level with C<ε_𝐤<C+δ C to obtain to next-leading order in J (see details in <cit.>)δ J_ef(C)=2ϱ J^2[F(C) + Jϱ MF(C/2)F(-C/2)]δ C/Cwhere the factor of F comes from the singularity of DOS. For the correction to the localized magnetic moment S_ef determining the Curie constant, as determined from the Kondo contribution to local magnetic susceptibility <cit.>, we have (cf. <cit.>)δ S_ef(C)/S=2ϱ ^2J^2MF(C/2)F(-C/2)δ C/C. The lowest-order scaling calculation according to (<ref>)(see also Refs. <cit.>) yields for the boundary of the strong-coupling regionT_K Dexp[ -( π ^2D/2|J|) ^1/2] . However to describe a possible NFL-type behavior (intermediate-coupling fixed point), we have to use the next-order scaling equations∂ g_ef(ξ )/∂ξ =[ξ-M/2(ξ +ln 2)^2 g_ef(ξ )]g_ef^2(ξ )where ξ =ln |4D/C| and we haveintroduced the dimensionless effectives-d coupling constantg_ef(C) = -2ϱ J_ef(C).In the flat-band case (where the singular factors ξ +ln 2 are replaced by unity) such equations give a finite fixed pointg_ef(ξ→∞)=2/M. It is known that this point is unphysical (unreachable) for M=1, but for M>2 the scaling consideration gives a qualitatively correct description, see review paper <cit.>. The case M=2 is marginal, so that logarithmic factors occur which are missed by simple approaches.Unlike the flat-band case, the equation (<ref>) cannot be solved analytically, but only asymptotic solution for G_ef(ξ )≡ g_ef(ξ )ξat large ξ can be obtained, which has the formG_ef(ξ ) =2/M+(1-4ln 2/M)1/ξ .The second term can change sign and is positive for large M, so that the derivative of G_ef(ξ) changes its sign. Thus the details of scaling behavior are rather sensitive to parameters. Besides that, the factors in (<ref>) is well determined only within the 1/M-expansion. Nevertheless, these results demonstrate existence of the “fixed point” G_ef(ξ )→ 2/M By analogy with Refs. <cit.> we can write down the scaling equation for the effective localized magnetic moment∂ln S_ef(ξ )/∂ξ =-M/2G_ef^2(ξ )so that to leading approximationS_ef(C)≃ (|C|/T_K)^Δ,Δ =2/Mand for the local magnetic susceptibility,χ _loc(T)=∫_0^1/T⟨ S_z(τ )S_z⟩ dτ ,we obtainthe power-law dependenceχ_loc (T) S_ef^2(T)/T (T/T_K)^2Δ -1where we have taken into account that G_ef(C) reaches the value about 2/M at |C| ∼ T_K. The dependence χ_loc(T) follows at high temperatures the Curie–Weiss law, has a maximum at T ∼ T_K and decreases with further increasing T for small M (however, in this case the scaling results for T <T_K are not reliable). For large M, χ_loc(T) is divergent at low temperatures. Note that the exponent in (<ref>) becomes modified in higher orders in 1/M, and in the flat-band case one has Δ =2/(M+2) according to the Bethe ansatz solution, see Refs. <cit.>. Below we take into account this replacement at comparison of analytical and NRG results.We see that simple analytical methods do not provide definite results, which is connected with insufficient information provided by 1/M expansion. In particular the result for susceptibility (<ref>) does not differ from the corresponding flat-band result and does not take into account the logarithmic factor at M=2 <cit.>.At the same time a simple “poor man” renormalization group treatment captures the differences in the perturbation expansion for the singular DOS and flat-band cases. Leaving the algebraical structure of the perturbation series the same, it leads to the expansion in terms of G_ef(ξ) for the singular DOS rather than g_ef(ξ) for the flat-band case. Moreover, it gives the inverse logarithmic contributions to the impurity entropy and the specific heat (see below).Therefore, we calculate impurity magnetic susceptibility, entropy and specific heat by using numerical renormalization group (NRG) approach <cit.> in the one- and two-channel cases.The NRG procedure starts from the solution of the isolated-impurity problem (sites “imp” and ϵ _0 in Fig. <ref>).At the initial step, we add a first conducting electronic site ϵ _1, and construct and diagonalize a Hamiltonian matrix on this Hilbert space (with a 4^M–fold higher dimensionality). This procedure is multiply repeated. However, since the dimensionality of the Hilbert space grows as 4^MN (N is the number of an iteration), it is impossible to store all the eigenstates during the calculation. Therefore, it is necessary to retain after each iteration only the states with the lowest energies. If we restrict ourselves to a certain maximum number of stored states (determined by the computational possibilities), it is necessary, starting from a certain iteration, to retain of the order of 1/4^M of states at each step. Practically, the number of the states is reasonable in the one- and two-channel cases. To take into account the disturbance introduced by the elimination of the high-lying states we use Wilson's logarithmic discretization of the conduction band <cit.> (see also Ref. <cit.>). In real calculations for M=2 we took into account of the order of 10^4 states at each NRG iteration with Wilson's logarithmic discretization factor being Λ=3. The agreement in the entropy value S(T=0) with the Bethe ansatz results for the flat-band DOS, which is ln 2/ 2, <cit.> (see below Fig. <ref>) demonstrates sufficient of NRG calculations. For the case M=1 we performed NRG calculations using Λ=2 and Λ=1.5 with subsequent extrapolation to Λ=1 which corresponds to non-discretized zone.When performing NRG calculations in the singular DOS case, we chose t=1/4, t^'=0 in approximation (<ref>) and calculated the half-width of the DOS support D^' from the normalization condition, which gave D^'=1.05977 D. The picture of energy levels depending on the NRG step N (multiplied by Λ^(N-1)/2) is presented in Figs. <ref>. The energies are referred to the energy of the ground state. For flat-band and singular DOS cases the values of the corresponding parameters J were chosen from the approximate equality of Kondo temperatures T_K for the cases addressed. An important difference between the flat-band and singular DOS situations is considerably slower tending of the curves E(N) to the asymptotic values in the latter case (cf. Ref. <cit.>). The slow fall off in the Van Hove case is somewhat similar to the situation for an underscreened Kondo model <cit.>. Because ofretaining only part of the energy spectrum at the N-th step of the NRG procedure, thermodynamic averages should be calculated at a temperature that depends on Λ, T_N = Λ^-N/2T_0 <cit.>. Here the starting temperature T_0 should benot too small to avoid the problem of discreteness of the energy spectrum. The total entropy andspecific heat read <cit.>𝒮_tot = ⟨ H⟩_ tot/T + ln Z_ tot , C_ tot = [⟨ H^2⟩_ tot-⟨ H⟩_ tot^2]/T^2 ,where Z is partition function. On differentiating ⟨ S_z⟩_ tot with respect to magnetic field one obtains <cit.>Tχ_ tot(T) = ⟨ S_z^2⟩_ tot - ⟨ S_z⟩^2_ tot ,The quantities Tχ_ band(T), 𝒮_ band and C_ bandare calculated in a similar way, and the corresponding impurity contributions are obtained by subtracting them from (<ref>)-(<ref>).First we consider the results for M=1 and discuss the local magnetic susceptibility χ_loc (<ref>). This is the susceptibility of a single impurity in a magnetic field that acts locally only on this impurity; this can be measured experimentally from the impurity spin correlation function and is obtained in simple perturbation calculations <cit.>.Instead of calculating (<ref>) directly we use the following procedure. One can apply small magnetic field h to impurity spin only and then calculate numerically derivative of the local magnetization induced, d⟨ S_z⟩/dh, in the limit h→ 0. To be sure that this limit with a linear dependence ⟨ S_z⟩∝ h has been reached we performed calculations for a series of h values.Numerical results for χ _loc are shown in Figs. <ref>–<ref>.They clearly demonstrate that there exists characteristic crossover temperature T_K^loc for χ_loc. Similar to Wilson <cit.> we use (somewhat ambiguously) the definition T_K^locχ_loc(T_K^loc)=0.0701 (see also Ref. <cit.> and discussion therein). The values of T_K^loc are presented in Table <ref>.However, unlike flat-band case, we did not observe exact universal behavior of T_K^locχ_loc (T) as a function of T/T_K^loc (see Fig. <ref>). The behavior of χ _loc(T) at low temperatures is shown in Fig. <ref>. The empirical linear dependenceχ _loc(T)/χ _loc(0) = 1 + 2/3χ _loc(0) Tdescribes the low-temperature behavior rather well, with χ_loc(0) presented in Table <ref>.Generally, the low temperature dependence of χ_loc israther unusual: instead of nearly constant value below T_K one observes a linear behavior. To confirm that thisbehavior is not an artifact, NRG calculations were performed for a series of values of Λ (see Fig. <ref>).Thus the situation in the singular DOS casediffers from that in the flat band case where typical parabolic Fermi liquid dependence T_K^locχ_loc (T)=0.097 - 0.07 (T/T_K^loc)^2 takes place (see Fig. <ref>).If the singularity is slightly shiftedfrom the Fermi level, with μ being the value of the shift, we should have the crossover to the typical for the flat-band temperature behavior. However at small μ the dependence of χ _loc(T) acquires at lowering temperaturea linear increase instead of a maximum (see inset in Fig. <ref>).For completeness we introducean alternatively defined susceptibility χ_imp, which can be expressed as a difference of magnetic susceptibilities of the whole system and the system without impurity:χ _imp(T)=χ _tot(T)-χ _band(T) ,where χ _tot is the total magnetic susceptibility, and χ _band is the susceptibility of non-interacting band electrons. Since in this definition magnetic field acts on the whole system, this quantity can be experimentally determined from magnetic measurements; it is also usually treated in Bethe ansatz solutions. In the flat-band case we have at low temperatures the standard Fermi-liquid behavior T_K^impχ _imp(T)= 0.103 - 0.11(T/T_K^imp)^2, where according to Wilson <cit.> we use the definition T_K^impχ_imp(T_K^imp)=0.0701.As discussed, e.g., in Refs. <cit.> and <cit.> (Sect.6.2.2), χ_imp and χ_loc can be different. In our case χ_imp(T) at not too low T demonstrates the usual universal Kondo behavior with T_K^imp being defined according to Wilson.However, with decreasing temperature χ_imp(T) deviates from the universal behavior and changes its sign.At very low T we have irrespective of J (Fig. <ref>)Tχ _imp(T)≈-0.115/ln(D/T) .The temperature of loss of universality for χ _imp corresponds to that for χ _loc(T): both the sign change in the former and maximum in the latter are connected with overcompensation of impurity spin by conduction electrons. For the case M=1, χ _imp was discussed in details in Ref. <cit.>.To compare these results with above perturbative consideration, we remember that in our case, as follows from the structure of perturbation theory, g_ef(T)→ G_ef(T)= g_ef(T)ln(D/T), andG^*- G_ef∼ 1/ln(D/T)according to (<ref>). Thus the terms, proportional to G^*- G_ef, should give 1/ln(D/T)-contribution to thermodynamic characteristics. In particular, the 1/ln(D/T) terms in χ_imp(T) are expected. However, the calculation of χ_imp(T) requires a careful collection of all the contributions to susceptibility.A similar behavior is obtained for the impurity entropy (Fig. <ref>) and for specific heat,S_imp(T)≈-1.3/ln(D/T), C_imp(T)≈-1.3/ln^2 (D/T) .According to Ref. <cit.> (Eq.44), the entropy in the flat band multichannel model contains the contribution (π^2/4) (M g_ef^3(T )- (3/8) M^2 g^4_ef(T)) with g^*- g_ef(T) ∼ T^Δ. To leading order in 1/M, this yields the contribution, proportional to [g^*- g_ef(T)]^2 ∼ T^2Δ. One can expect that in our case terms linear in g_ef(T)→ G_ef(T)= g_ef(T)ln(D/T) will occur in the entropy. However, to obtain a correct description of strong coupling regime, a more accurate analysis is required. These results correct somewhat our previous calculations <cit.>. The occurrence of the inverse-logarithm contributions is in a qualitative agreement with the above scaling consideration.The physical picture can be explained as follows. At low temperatures the impurity spin is completely screened and we come to the situation where χ_imp is determined by the contribution of conduction electrons which is independent of J. The effective bandwidth in the singular case decreases, and the situation is close to that in the model with a hole at the magnetic impurity site (J→ -∞): here the total magnetic susceptibility of the system is smaller than that of bare electrons, so that the contribution χ_imp is evidently negative.Thus in the strong coupling regime (at low temperatures)the problem is reduced to the resonant level model. A quantitative analytical solution in this model can be obtained similar to <cit.>. In terms of the impurity free energy one obtains S_imp = -∂ F_imp/∂ T= -∑_σ∫_-∞^∞dz/πz/cosh^2zln (-G_r,σ^-1(2Tz))Tχ_imp= -1/4∑_σ∫_-∞^∞dz/πsinh z/cosh^3 zln (-G_r,σ^-1(2Tz))where G_r,σ(z)is the retarded Green's function of the resonant level hybridized with the band.Calculating theGreen's function of the level hybridized with the singular DOS we haveG_r,σ^-1(z)= z+i0^+-v^2G_r,σ^0(z)where v is an effective hybridization matrix element andG_r,σ^0(z)= ρ[P∫_-D^Ddϵln|4D/ϵ|/z-ϵ -iπln|4D/z|θ (D^2-z^2) ]is the band Green's function for the singular DOS at the resonant level site, θ(x) is the Heaviside step function. Since the low T behavior is dominated by small |z|≪ Done hasln (-G_r,σ^-1(z))≈ -π/2 + sign(z)arctanπ/2ln|4D/z|and we derive for leading corrections irrespective of vS_imp≈ -ln 4/ln |D/T|, Tχ_imp≈ -1/8/ln|D/T|in a fair agreement with the NRG results (<ref>),(<ref>). The above consideration shows that the unusual low temperature behavior is solely determined by the logarithmic van Hove singularity of the band spectrum (cf. <cit.>), the singular contribution having essentially one-electron nature.Now we pass to the two-channel situation. In the flat-band caseχ_loc (T) is known to behave as ln(T_K/T) <cit.> (such a behavior was also reproduced by our test calculation, Fig. <ref>). However, for the logarithmic DOSthe local susceptibility χ _loc(T) demonstrates a power-lawnon-Fermi-liquid behavior (Fig. <ref>), which can be fitted at low T asT_K^locχ _loc(T) ∼ (T_K^loc/T)^αwhereαslightly decreases with decreasing |J| (see Table <ref>).Thus the power-law behavior of physical quantities occurs in the presence of the DOS singularity already for M=2 (in the flat-band case this takes place starting from M=3). One can suppose that the singularity leads to an effective increase of the number of scattering channels. Although it is often difficult to distinguish between the experimental logarithmic and power-law dependences, especially in the case of small exponents, this conclusion is important from the theoretical point of view.As demonstrated in Ref. <cit.> from the 1/M-expansion, in the flat-band case the change in the gyromagnetic ratio (which enters total magnetic susceptibility) leads to a change in numerical factors only, so thatboth χ_loc(T) andχ_imp(T) are positive and behave insimilar way. However, in our case the behavior of χ_imp(T) is again qualitatively different: we have the dependence (see Fig. <ref>)Tχ _imp(T)≈-0.075/ln(D/T) . For the M=2 flat-band Kondo model the ground state impurity entropy is known to be equal to ln2/2 <cit.>. Our NRG calculations confirm this value (see Fig. <ref>). For the singular DOS the temperature behavior is again unusual: at low T the entropy approaches this value from below according to the lawS_imp(T)≈ln2/2-1.3/ln(D/T)≈ S_imp^1channel + ln2/2 . Thus the temperature dependence in (<ref>) is the same as in the resonant level model (<ref>).We demonstrated that the behavior in the Kondo model with singular logarithmic DOS differs radically from that in the smooth DOS model. This has unusual behavior with negative impurity magnetic susceptibility and entropy at low T. On the other hand, the local magnetic susceptibility which is determined by the linear response remains positive. This has a shallow maximum in the one-channel case and demonstrates a power-law NFL behavior for two-channel case. As for the 1/M-expansion, this yields the results which differ from those for smooth DOS case by the replacement of the effective coupling constant by G_ef. However, the NRG method enables one to obtain a more detailed information (although the calculations are rather cumbersome because of slow convergence in the singular case). In particular, NRG calculations reproduce the Bethe ansatz results for theflat band M=2 case (logarithmic factors in local magnetic susceptibility). In the case of singular DOS, theM=2 results turn out to be different (weak power-law divergence). We note also absence of Wilson's self-similarity in terms of T/T_K. When shifting the singularity from the Fermi level, the system can demonstrate non-trivial crossovers at lowering T, which can be observed experimentally.The research was carried out within the state assignment of FASO of Russia (theme “Quantum” No. 01201463332) and supported in part by Ural Branch of Russian Academy of Science (project no. 15-8-2-10). 99 Stewart1l G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001); Rev. Mod. Phys. 78, 743 (2006).Wiegmann A. M. Tsvelick and P. B. Wiegmann, Adv. Phys. 32, 453 (1983).Tsv B. Andraka and A. M. Tsvelik, Phys. Rev. Lett. 67, 2886 (1991). Cox D. L. Cox and A. Zawadowski, Adv. Phys. 47, 599 (1998).Nature K. Le Hur, Nature 526, 203 (2015); Z. Iftikhar et al,Nature 526, 233 (2015); A. J. Keller et al,Nature 526, 237 (2015). Law1 K. T. Law, C. Y. Seng, P. A. Lee, and T. K. Ng,Phys. Rev. B 81, 041305 (2010).Khar M. Kharitonov and G. Kotliar,Phys. Rev. B 88, 201103(R) (2013).Pot1 R. M. Potok et al, Nature 446, 167 (2007).Chen J.-H. Chen, L. Li, W. G. Cullen, E. D. Williams, and M. S. Fuhrer, Nature Phys.7, 535 (2011).Takeshi T. Inoshita, M. Tsukada, S. Saito, H. Hosono, arXiv:1708.01048 [cond-mat.mtrl-sci]Zitko R. Zitko and A. Horvat, Phys. Rev. B 94, 125138 (2016).Mitchell A. K. Mitchell and L. Fritz, Phys. Rev. B 88, 075104 (2013); A. K. Mitchell, M. Vojta, R. Bulla and L. Fritz, Phys. Rev. B 88, 195119 (2013).And P. W. Anderson, J. Phys. C: Solid State Phys. 3, 2436 (1970).I11 V. Yu. Irkhin, J. Phys.: Condens. Matter 23, 065602 (2011); J. Exp. Theor. Phys. 125, 159 (2017).I16 V. Yu. Irkhin, Eur. Phys. J. B 89, 117 (2016).Kondo J. Kondo, Solid State Physics. Vol. 23. Eds. F. Seitz, D. Turnbull, H. Ehrenreich. N. Y.: Academic Press, 1969, p. 183.IK89 V. Yu. Irkhin, M. I. Katsnelson, Z. Phys. B 75, 67 (1989); Phys. Rev. B 56, 8109 (1997).Zhur2009 A. K. Zhuravlev, Phys. Met. Metallogr. 108, 107 (2009); A. K. Zhuravlev and V. Yu. Irkhin, Phys. Rev. B 84, 245111 (2011).Gan J. Gan, J. Phys.: Condens. Matter 6, 4547 (1994).Wilson K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980).NRGI. Affleck, A. W. W. Ludwig, H.-B. Pang, and D. L. Cox, Phys. Rev. B 45, 7918 (1992).Mehta P. Mehta, L. Borda, G. Zarand, N. Andrei, and P. Coleman, Phys. Rev. B 72, 014430 (2005)Pruschke R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys., 80, 395 (2008).ZhurKett A. Zhuravlev, I. Zharekeshev, E. Gorelov, A. I. Lichtenstein, E. R. Mucciolo, and S. Kettemann, Phys. Rev. Lett. 99, 247202 (2007).Santoro G. E. Santoro and G. F. Giuliani, Phys. Rev. B 44, 2209 (1991). | http://arxiv.org/abs/1706.08443v2 | {
"authors": [
"A. K. Zhuravlev",
"A. O. Anokhin",
"V. Yu. Irkhin"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170626154104",
"title": "One- and two-channel Kondo model with logarithmic Van Hove singularity: a numerical renormalization group solution"
} |
Validating a novel angular power spectrum estimator using simulated low frequency radio-interferometric data [2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/] [================================================================================================================================================================================================================================================== The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way.This paper gives a generalizated approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward.MSC 2010: 65N30. Keywords: Finite element method, basis function, pull-back.§ INTRODUCTIONAt the heart of any finite element implementation lies the evaluation of basis functions and their derivatives on each cell in a mesh. These values are used to compute local integral contributions to stiffness matrices and load vectors, which are assembled into a sparse matrix and then passed on to an algebraic solver.While it is fairly easy to parametrize local integration routines over basis functions, one must also provide an implementation of those basis functions.Frequently, finite element codes use a reference element, on which a set of basis functions is constructed once and mapped via coordinate change to each cell in a mesh.Alternately, many finite element bases can be expressed in terms of barycentric coordinates, in which case one must simply convert between the physical and barycentric coordinates on each cell in order evaluate basis functions.Although we refer the reader to recent results on Bernstein polynomials <cit.> for interesting algorithms in the latter case, the prevelance of the reference element paradigm in modern high-level finite element software <cit.> we shall restrict ourselves to the former.The development of FIAT <cit.> has had a significant impact on finite element software, especially through its adoption in high-level software projects such as FEniCS <cit.> and Firedrake <cit.>.FIAT provides tools to describe and construct reference bases for arbitrary-order instances of many common and unusual finite elements.Composed with a domain-specific language for variational problems like UFL <cit.> and a form compiler mapping UFL into efficient code for element integrals <cit.> gives a powerful, user-friendly tool chain. However, any code based on the reference element paradigm operates under the assumption that finite elements satisfy a certain kind of equivalence.Essentially, one must have a pull-back operation that puts basis functions on each cell into one-to-one correspondence with the reference basis functions. Hence, the original form of <cit.> used only (arbitrary order) Lagrange finite elements, although this was generalized to H(div) and H(curl) elements using Piola transforms in <cit.>. Current technology captures the full simplicial discrete de Rham complex and certain other elements, but many famous elements are not included. Although it is possible to construct reference elements in FIAT or some other way, current form compilers or other high-level libraries do not provide correct code for mapping them.Elements such as Hermite <cit.>, Argyris <cit.>, Morley <cit.>, and Bell <cit.>, shown alongside the Lagrange element in Figure <ref>, do not satisfy the proper equivalence properties to give a simple relationship between the reference basis and nodal basis on a general cell.Typically, implementations of such elements require special-purpose code for constructing the basis functions separately on each element, which can cost nearly as much in terms of work and storage as building the element stiffness matrix itself.It also requires a different internal workflow in the code.Although Domínguez and Sayas <cit.> give a technique for mappingbases for the Argyris element and a separate computer implementation is available (<https://github.com/VT-ICAM/ArgyrisPack>) and Jardin <cit.> gives a per-element construction technique for the Bell element, these represents the exception rather than the rule.The literature contains no general approach for constructing and mapping finite element bases in the absence of affine equivalence or a suitable generalization thereof.In this paper we provide such a general theory for transforming finite elements that supplements the theory on which FIAT is based for constructing those elements.Our focus is on the case of scalar-valued elements in affine spaces, although we indicate how the techniques generalize on both counts.We begin the rest of the paper by recalling definitions in <ref>.The bulk of the paper occurs in <ref>, where we show how to map finite element bases under affine equivalence, affine-interpolation equivalence, and when neither holds.We also sketch briefly how the theory is adapted to the case of more general pullbacks such as non-affine coordinate mappings or Piola transforms.All the theory in <ref> assumes that the natural pull-back operation ( composition with coordinate change) exactly preserves the function spaces between reference and physical space. However, in certain notable cases such as the Bell element, this condition fails to hold.In <ref>, we give a more general theory with application to the Bell element. Finally, in <ref>, we present some numerical results using these elements. § DEFINITIONS AND PRELIMINARIESThrougout, we let C_b^k(Ω) denote the space of functions with continuous and bounded derivatives up to and including order k over Ω, and C_b^k(Ω)^' its topological dual.A finite element is a triple (K, P, N) such that * K ⊂ℝ^d is a bounded domain.* P ⊂ C^k_b(K) for some integer k ≥ 0 is a finite-dimensional function space.* N = {n_i}_i=1^ν⊂ C^k_b(K)^' is a collection of linearly independent functionals whose actions restricted to P form a basis for P^'.The nodes in N are taken as objects in the full infinite-dimensional dual, although sometimes we will only require their restrictions to members of P. For any n ∈ C^k_b(K)^', define π n ∈ P^' by restriction.That is, define π n (p) = n(p) for anyp ∈ P. Further, with a slight abuse in notation, we will let N = [ n_1 n_2 … n_ν ]^T denote a functional on P^ν, or equivalently, a vector of ν members of the dual space.As shorthand, we define these spaces consisting of vectors of functions or functionals byX≡( P )^ν, X^† ≡( C_b^k( K )^')^ν. We can “vectorize” the restriction operator π, so thatfor anyN ∈ X^†, π N ∈ (P^ν)^' has (π N)_i = π(n_i).Galerkin methods work in terms of a basis for the approximating space, and these are typically built out of local bases for each element:Let (K, P, N) be a finite element with P = ν.The nodal basis for P is the set {ψ_i}_i=1^ν such that n_i(ψ_j) = δ_i,j for each 1 ≤ i,j ≤ν. The nodal basis also can be written as X ∋Ψ = [ ψ_1 ψ_2 … ψ_ν ].Traditionally, finite element codes construct the nodal basis for a reference finite element (K̂, P̂, N̂) and then map it into the basis for (K, P, N) for each K in the mesh.Let F:K →K̂ be the geometric mapping, as in Figure <ref>.We let J denote the Jacobian matrix of this transformation.Similarly to (<ref>), we define the vector spaces relative to the reference cell:X̂ ≡( P̂)^ν,X̂^† ≡( C_b^k( K̂ )^')^ν.As with π, we define π̂n̂ as the restriction of n̂ to P̂, and can vectorize it over X̂^† accordingly. This geometric mapping induces a mapping between spaces of functions over K and K̂ as well as between the dual spaces.These are called the pull-back, and push-forward operations, respectively: The pull-back operation mapping C^k_b(K̂) → C^k_b(K) is defined byF^*(f̂) = f̂∘ Ffor each f̂∈ C^k_b(K̂). The push-forward operation mapping the dual space C^k_b(K)^' into C^k_b(K̂)^' is defined byF_*(n) = n∘ F^*for each n ∈ C^k_b(K)^'. It is easy to verify that the pull-back and push-forward are linear operations preserving the vector space operations.Moreover, they are invertible iff F itself is.Therefore, we have Given finite elements (K,P,N) and (K̂,P̂,N̂) such that F(K)=K̂ and F^*(P̂) = P, F^*:P̂→ P and F_*:P^'→P̂^' are isomorphisms.The pull-back and push-forward operations are also defined over the vector spaces X, X^†, X̂, and X̂^†. If N is a vector of functionals and Φ a vector of functions, then the vector push-forward and pull-back are, respectively F_*(N) ∈X̂^†,( F_*(N) )_i = F_*(n_i), F^*(Φ̂) ∈ X,( F^*(Φ̂) )_i = F^*(ϕ̂_̂î).It will also be useful to consider vectors of functionals acting on vectors of functions.We define this to produce a matrix as follows. If N = [ n_1 n_2 … n_k ]^T is a collection of functionals and Φ = [ ϕ_1 ϕ_2 … ϕ_ℓ ]^T a collection of functions, then we define the (outer) product N(Φ) to be the k ×ℓ matrix( N(Φ) )_ij = n_i(ϕ_j).For example, if N is the vector of nodes of a finite element and Ψ contains the nodal basis functions, then the Kronecker delta property is expressed as N(Ψ) = I. If M is a matrix of numbers of appropriate shape and Φ∈ Xmembers of a function space P, then MΦ is just defined by (M Φ)_i = ∑_j=1^ν M_ijΦ_j, according to the usual rule for matrix-vector multiplication. Let N ∈ X^† and Φ∈ X and M ∈ℝ^ν×ν.ThenN(MΦ) = N(Φ) M^T.The proof is a simple calculation:( N(MΦ) )_ij = n_i ( ( M Φ)_j ) = n_i ( ∑_k=1^ν M_jkϕ_k ) = ∑_k=1^ν n_i ( ϕ_k) = ∑_k=1^ν(N(Φ))_ik M_jk.The relationship between pull-back and push-forward also leads to the vectorized relationLet N ∈ X^† and Φ̂∈X̂.ThenN(F^*(Φ̂)) = F_*(N)(Φ̂)Let (K,P,N) and (K̂, P̂, N̂) be finite elements and F an affine mapping on K. Then (K,P,N) and (K̂,P̂, N̂) are affine equivalent if * F(K) = K̂,* The pullback maps F^*(P̂) = P (in the sense of equality of vector spaces),* F_*(N) = N̂ (in the sense of equality of finite sets). Let (K, P, N) be a finite element of class C^k and Ψ∈ X its nodal basis.The nodal interpolant ℐ_N: C_b^k(K) → P is defined by ℐ(f) = ∑_i=1^ν n_i(f) ψ_i. This interpolant plays a fundamental role in establishing approximation properties of finite elements via the Bramble-Hilbert Lemma <cit.>. The homogeneity arguments in fact go through for the following generalized notion of element equivalence:Two finite elements (K, P, N) and (K, P, Ñ) are interpolation equivalent if ℐ_N = ℐ_Ñ.If (K, P, Ñ) is affine equivalent to (K̂, P̂, N̂) and interpolation equivalent to (K, P, N), then (K, P, N) and (K̂, P̂, N̂) are affine-interpolation equivalent. Brenner and Scott <cit.> give the following result, of which we shall make use:Finite elements (K, P, N) and (K, P, Ñ) are interpolation equivalent iff the spans of N and Ñ, (viewed as subsets of C_b^k(K)^'), are equal. For Lagrange and certain other finite elements, one simply has that F^*(Ψ̂) = Ψ, which allows for the traditional use of reference elements used in FEniCS, Firedrake, and countless other codes. However, for many other elements this is not the case.It is our goal in this paper to give a general approach that expresses Ψ as a linear transformation M applied to F^*(Ψ̂).Before proceeding, we note that approximation theory for Argyris and other families without affine-interpolation equivalence can proceed by means of establishing the almost-affine property <cit.>.Such proofs can involve embedding the inequivalent element family into an equivalent one with the requisite approximation properties. For example, the Argyris element is proved almost-affine by comparison to the “type (5)” quintic Hermite element. Although we see definite computational consequences of affine-equivalence, affine-interpolation equivalence, and neither among our element families, we our approach to transforming inequivalent families does not make use of any almost-affine properties.§ TRANSFORMATION THEORY WHEN F^*(P̂) = P For now, we assume that the pull-back operation (<ref>) appropriately converts the reference element function space into the physical function space and discuss the construction of nodal bases based on relationships between the reference nodes N̂ and the pushed-forward physical nodes F_*(N).We focus on the simplicial case, although generalizations do not have a major effect, as we note later.Throughout, we will use following convention, developed in <cit.> for handling facetorientation in mixed methods but also useful in order higher-order Lagrange degrees of freedom.Since our examples are triangles (2-simplices), it is not necessary to expand on the entire convention.Given a triangle with vertices (𝐯_1, 𝐯_2, 𝐯_3), we define edge γ_i of the triangle to connect the vertices other than 𝐯_i.The (unit) tangent vector 𝐭_i =[ t^𝐱_i t^𝐲_i ]^T, points in the direction from the lower- to the higher-numbered vertex. When triangles share an edge, then, they agree on its orientation.The normal to an edge is defined by rotating the tangent by applying the matrix R = [01; -10 ] so that 𝐧_i = R 𝐭_i = [ n^𝐱_i n^𝐲_i ]^T We also let 𝐞_i denote the midpoint of γ_i.Now, we fix some notation for describing nodes.First, we define δ_𝐱 acting on any continuous function by pointwise evaluation.That is:δ_𝐱(p) = p(𝐱).We let δ^𝐬_𝐱 denote the directional derivative in direction 𝐬 at a point 𝐱, so thatδ^𝐬_𝐱(p) = 𝐬^T ∇ p(𝐱).We use repeated superscripts to indicate higher-order derivatives, so that δ^𝐱𝐱_𝐱 defines the second directional derivative along the x-axis at point 𝐱.It will also be convenient to use block notation, with a single symbol representing two or items.For example, the gradient notation∇_𝐱 = [ δ^𝐱_𝐱 δ^𝐲_𝐱 ]^Tgives the pair of functionals evaluating the Cartesian derivatives at a point 𝐱.To denote a gradient in a different basis, we append the directions as superscripts so that∇^𝐧𝐭_𝐱 = [ δ^𝐧_𝐱 δ^𝐭_𝐱 ]^Tcontains the normal and tangential derivatives at a point 𝐱. Similarly, we let△_𝐯 = [ δ^𝐱𝐱_𝐱 δ^𝐱𝐲_𝐱 δ^𝐲𝐲_𝐱 ]^Tdenote the vector of three functionals evaluating the unique(supposing sufficient smoothness) second partials at 𝐱.Let Ψ = {ψ_i}_i=1^ν be the nodal basis for a finite element (K, P, N) and Ψ̂ = {ψ̂_i }_i=1^ν that for a reference element (K̂,P̂,N̂).We also assume that F(K) = K̂ and F^*(P̂)=P. Because the pull-back is invertible, it maps linearly independent sets to linearly independent sets.So, F^*(Ψ̂) must also be a basis for P. There exists an invertible ν×ν matrix M such that Ψ = M F^*(Ψ̂),or equivalently, that each nodal basis function is some linear combination of the pull-backs of the reference nodal basis functions.Our theory for transforming the basis functions ( computing the matrix M) will work via duality – relating the matrix M to how the nodes, or at least their restrictions to the finite-dimensional spaces, push forward.It will be useful to define as an intermediate ν×ν matrix B=F_*(N)(Ψ̂).Recall from (<ref>) that its entries for 1 ≤ i, j ≤ν areB_ij≡ F_*(n_i)(ψ̂_j) = n_i(F^*(ψ̂_j))This matrix, having nodes only applied to members of Pis indifferent to restrictions and so B = F_*(π N)(Ψ̂) as well.Because of Proposition <ref> and finite-dimensionality, thethe nodal sets π̂N̂ and F_*(π N) are both bases for P̂^', and so there exists an invertible ν×ν matrix V such thatπ̂N̂ = V F_*(π N) Frequently, it may be easier to express the pushed-forward nodes as a linear combination of the reference nodes.In this case, one obtains the matrix V^-1.At any rate, the matrices V and M are closely related.For finite elements (K,P,N) and (K̂,P̂,N̂) with F(K)=K̂ and F_*(P̂) = P, the matrices in (<ref>) and (<ref>) satisfyM = V^T. We proceed by relating both matrices to B defined in (<ref>) via the Kronecker property of nodal bases. First, we haveI = N(Ψ) = N(MF^*(Ψ̂)) = N(F^*(Ψ̂)) M^T = B M^T.so thatM = B^-T.Similarly,I = (VF_*(N))(Ψ̂) = V F_*(N)(Ψ̂) = V B,so that V=B^-1 and the result follows. That is, to relate the pullback of the reference element basis functions to any element's basis functions, it is sufficient to determine the relationship between the nodes. §.§ Affine equivalence: The Lagrange element When elements form affine-equivalent families, the matrix M has a particularly simple form. If (K,P,N) and (K̂,P̂,N̂) are affine-equivalent finite elements then the transformation matrix M is the identity. Suppose the two elements are affine-equivalent, so that F_*(N) = N̂.Then, a direct calculation givesN(F^*(Ψ̂)) = F_*(N)(Ψ̂) = N̂(Ψ̂) = Iso that M=I. The Lagrange elements are the most widely used finite elements and form the prototypical affine-equivalent family <cit.>. For a simplex K in dimension d and integer r ≥ 1, one defines P = P_r(K) to be the space of polynomials over K of total degree no greater than r, which has dimension r+dd.The nodes are taken to be pointwise evaluation at a lattice of r+dd points.Classically, these are taken to be regular and equispaced, although options with superior interpolation and conditioning properties for large r are also known <cit.>.One must ensure that nodal locations are chosen at the boundary to enable C^0 continuity between adjacent elements.A cubic Lagrange triangle (r=3 and d=2) is shown earlier in Figure <ref>.The practical effect of Theorem <ref> is that the reference element paradigm “works.”That is, a computer code contains a routine to evaluate the nodal basis Ψ̂ and its derivatives for a reference element (K̂, P̂, N̂).Then, this routine is called at a set of quadrature points in K̂.One obtains values of the nodal basis at quadrature points on each cell K by pull-back, so no additional work is required.To obtain the gradients of each basis function at each quadrature point, one simply multiplies each basis gradient at each point by J^T.On the other hand, when M ≠ I, the usage of tabulated reference values is more complex.Given a table_iq = ψ̂_i(ξ̂_q)of the reference basis at the reference quadrature points, one finds the nodal basis for (K, P, N) by constructing M for that element and then computing the matrix-vector product M so thatψ_i(ξ_q) = ∑_k=1^ν M_i,k_k,q Mapping gradients from the reference element requires both multiplication by M as well as application of J^T by the chain rule.We define D∈ℝ^ν× |ξ| × 2 byD_i,q,: = ∇̂ψ̂_i(ξ̂)_q. Then, the basis gradients requires contraction with MD^'_i,q,: := ∑_k=1^ν M_i,k D_k,q,:,followed by the chain ruleD_i,q,: := J^T D^'_i,q,:.In fact, the application of M and J^T can be performed in either order.Note that applying M requires an ν×νmatrix-vector multiplication and in principle couples all basis functions together, while applying J^T works pointwise on each basis function separately.When M is quite sparse, one expects this to be a small additional cost compared to the other required arithmetic.We present further details for this in the case of Hermite elements, to which we now turn.§.§ The Hermite element: affine-interpolation equivalenceThe Hermite triangle <cit.>, show in Figure <ref> is based cubic polynomials, although higher-order instances can also be defined <cit.>.In contrast to theLagrange element, its node set includes function values and derivatives at the nodes, as well as an interior function value. The resulting finite element spaces have C^0 continuity withC^1 continuity at vertices. They provide a classic example of elements that are not affine equivalent but instead give affine-interpolation equivalent families. We will let (K,P,N) be a cubic Hermite triangle, specifying the gradient at each vertex in terms of the Cartesian derivatives – see Figure <ref>.Let {𝐯_i}_i=1^3 be the three vertices of K and 𝐯_4 its barycenter.We order the nodes N byN = [ δ_𝐯_1 ∇_𝐯_1^T δ_𝐯_2 ∇_𝐯_2^T δ_𝐯_3 ∇_𝐯_3^T δ_𝐯_4 ]^T,using block notation.Now, we fix the reference element (K̂, P̂, N̂) with K̂ as the unit right triangle and express the gradient by the derivatives in the direction of the reference Cartesian coordinates, as in Figure <ref>. Let {𝐯̂_i}_i=1^3 be the three vertices of K̂ and 𝐯̂_4 its barycenter. We define N̂ analogously to N.Consider the relationship between the nodal basis functions Ψ and the pulled-back F^*(Ψ̂).For any ψ̂∈P̂, the chain rule leads to∇ (ψ̂∘ F) = J^T ∇̂ψ̂∘ F. Now, suppose that ψ̂ is a nodal basis function corresponding to evaluation at a vertex or the barycenter, so that δ_𝐯̂_iψ̂ = 1 for some 1 ≤ i ≤ 4, withthe remaining reference nodes vanishing on ψ̂. We compute thatδ_𝐯_i F^*(ψ̂) = ( ψ̂∘ F )( 𝐯_i ) = ψ̂(v̂_i) = 1,while δ_𝐯_j F^*(ψ̂) = 0 for 1 ≤ j ≤ 4 with j ≠ i. Also, since the reference gradient of ψ̂ vanishes at each vertex, (<ref>) implies that the physical gradient of F^*(ψ̂) must also vanish at each vertex.So, pulling back ψ̂ gives the corresponding nodal basis function for (K, P, N).The situation changes for the derivative basis functions.Now take ψ̂ to be the basis function with unit-valued derivative in, say, the 𝐱̂ direction at vertex 𝐯̂_i and other degrees of freedom vanishing.Since it vanishes at each vertex and the barycenter of K̂, F^*(ψ̂) will vanish at each vertexand the barycenter of K.The reference gradient of ψ̂ vanishes at the vertices other than i, so the physical gradient of its pullback must also vanish at the corresponding vertices of K. However, (<ref>) shows that ∇(ψ̂∘ F) will typically not yield [ 1 0 ]^T at 𝐯_i. Consequently, the pull-backs of the reference derivative basis functions do not produce the physical basis functions.Equivalently, we may express this failure in terms of the nodes – pushing forward N does not yield N̂.We demonstrate this pictorially in Figure <ref>, showing the images of the derivative nodes under push-forward do not correspond to the reference derivative nodes.Taking this view allows us to address the issue using Theorem <ref>.This discussion using the chain rule can be summarized by the matrix-valued equationF_*(N) = [ 1 0 0 0 0 0 0; 0 J^T 0 0 0 0 0; 0 0 1 0 0 0 0; 0 0 0 J^T 0 0 0; 0 0 0 0 1 0 0; 0 0 0 0 0 J^T 0; 0 0 0 0 0 0 1 ]N̂,noting that the second, fourth, and sixth rows and columns of this matrix are blocks of two, and each “0” is taken to be the zero matrix of appropriate size.This is exactly the inverse of V from Theorem <ref>.In this case, the transformation V is quite local – that is, only the push-forward of nodes at a given point are used to construct the reference nodes at the image of that point.This seems to be generally true for interpolation-equivalent elements, although functionals with broader support (e.g. integral moments over the cell or a facet thereof) would require a slight adaptation. We will see presently for Morley and Argyris elements that the transformation neeed not be block diagonal for elements without interpolation equivalence.At any rate, the following elementary observation from linear algebra suggests the sparsity of V: Let W be a vector space with sets of vectors W_1 = { w^1_i }_i=1^m ⊂ W and W_2 = { w^2_i }_i=1^n.Suppose that span W_1 ⊂span W_2 so that there exists a matrix A ∈ℝ^m × n such that w^1_i = ∑_k=1^n A_ik w^2_k.If we further have that some w^1_i ∈span{ w^2_j }_j∈𝒥 for some 𝒥⊂ [1, n], then A_ij = 0 for all j ∉𝒥. Our theory applies equally to the general family of Hermite triangles of degree k ≥ 3. In those cases, the nodes consist of gradients at vertices together with point-wise values at appropriate places.All higher-order cases generate C^0 families of elements with C^1-continuity at vertices.The V matrix remainsanalogous to the cubic case, with J^-T on the diagonal in three places corresponding to the vertex derivative nodes.No major differences appear for the tetradral Hermite elements, either.As we saw earlier, Hermite and other elements for which M ≠ I incur an additional cost in mapping from the reference element, as one must compute basis function values and gradients via (<ref>) and (<ref>).The key driver of this additional cost is the application of M.Since M is very sparse for Hermite elements – just 12 nonzeros counting the 1's on the diagonal – evaluating (<ref>) requires just 12 operations per column, so a 10-point quadrature rule requires 120 operations. Evaluating (<ref>) requires twice this, or 240 operations.Applying J^T in (<ref>) is required whether Hermite or Lagrange elements are used.It requires 4 × 10 times the number of quadrature points used – so a 10-point rule would require 400 operations.Hence, the chain rule costs more than the application of M in this situation. On the other hand, building an element stiffness matrix requires a double loop over these 10 basis functions nested with a loop over the, say, 10 quadrature points.Hence, the loop body requires 1000 iterations, and with even a handful of operations will easily dominatethe additional cost of multiplying by M.§.§ The Morley and Argyris elementsThe construction of C^1 finite elements, required for problems such as plate bending or the Cahn-Hilliard equations, is a long-standing difficulty.Although it is possible to work around this requirement by rewriting the fourth-order problem as a lower order system or by using C^0 elements in conjunction with variational form penalizing the jumps in derivatives <cit.>, this doesn't actually give a C^1 solution. The quadratic Morley triangle <cit.>, shown in Figure <ref>, finds application in plate-bending problems and also provides a relatively simple motivation for and application of the theory developed here. The six degrees of freedom, vertex values and the normal derivatives on each edge midpoint, lead to an assembled finite element space that is neither C^0 nor C^1, but it is still suitable as a convergent nonconforming approximation for fourth-order problems.The quintic Argyris triangle <cit.>, shown in Figure <ref>, with its 21 degrees, gives a proper C^1 finite element. Hence it can be used generically for fourth-order problems as well as second-order problems for which a continuously differentiable solution is desired.The Argyris elements use the values, gradients, and second derivatives at each triangle vertex plus the normal derivatives at edge midpoints as the twenty-one degrees of freedom.It has been suggested that the Bell element <cit.> represents a simpler C^1 element than the Argyris element, on the account that it has fewer degrees of freedom.Shown in Figure <ref>, we see that the edge normal derivatives have been removed from the Argyris element.However, this comes with a (smaller but) more complicated function space.Rather than full quintic polynomials, the Bell element uses quintic polynomials that have normal derivatives on each edge of only third degree.This constraint on the polynomial space turns out to complicate the transformation of Bell elements compared to Hermite or even Argyris. For the rest of this section, we focus on Morley and Argyris, returning to Bell later.It can readily be seen that, like the Hermite element, the standard affine mapping will not preserve nodal bases.Unlike the Hermite element, however, the Morley and Argyris elements do not form affine-interpolation equivalent families – the spans of the nodes are not preserved under push-forward thanks to the edge normal derivatives – see Figure <ref>. As the Morley and Aryris nodal sets do not contain a full gradient at edge midpoints, the technique used for Hermite elements cannot be directly applied. To work around this, we introduce the following idea:Let (K,P,N) and (K̂, P̂, N̂) be finite elements of class C^k with affine mapping F:K→K̂ and associated pull-back and push-forward F^* and F_*. Suppose also that F^*(P̂) = P. Let N^c = {n^c_i}_i=1^μ⊂ C_b^k(K)^' and N̂^c = {n̂^n_i }_i=1^μ⊂ C^k(K̂)^' be such that * N ⊂ N^c (taken as sets rather than vectors),* N̂⊂N̂^c (again as sets),* span(F_*(N^c)) = span(N̂^c) in C^k(K̂)^'.Then N^c and N̂^c form acompatible nodal completion of N and N̂.Let (K,P,N) and (K̂, P̂, N̂) be the Morley triangle and reference triangle.Take N^c to contain all the nodes of N together with the tangential derivatives at the midpoint of each edge of K and similarly for N̂^c. In this case, μ = 9.Then, both N^c and N̂^c contain complete gradients at each edge midpoint and function values at each vertex.The push-forward of N^c has the same span as N̂^c and so N^c and N̂^c form a compatible nodal completion of N and N̂.This is shown pictorially in Figure <ref>. A similar completion – supplementing the nodes with tangential derivatives at edge midpoints – exists for the Argyris nodes and reference nodes <cit.>.Now, since the spans of N̂^c and F_*(N^c) agree (even in C_b^k(K̂)^'), there exists a μ×μ matrix V^c, typically block diagonal, such thatN̂^c = V^c F_*(N^c).Let E ∈ℝ^ν×μ be the Boolean matrix with E_ij = 1 iff n̂_i = n̂_j^c so thatN̂ = E N̂^c,and it is clear thatN̂ = E V^c F_*(N^c).That is, the reference nodes are linear combinations of the pushed-forward nodes and the extended nodes, but we must have the linear combination in terms of the pushed-forward nodes alone.Recall that building the nodal basis only requires the action of the nodes on the polynomial space.Because μ > ν, the set of nodes π N^c must be linearly dependent.So, we seek a matrix D ∈ℝ^μ×ν such thatπ N^c = D π N.Since F_* is an isomorphism, such a D also givesπ̂ F_*(N^c) = D π̂ F_*(N). Rows i of the matrix D such that n^c_i = n_j forsome j will just have D_ik = δ_kj for 1 ≤ k ≤ν.The remaining rows must be constructed somehow via an interpolation argument, although the details will vary by element.This discussion suggests a three-stage process, each encoded by matrix multiplication, for converting the push-forwards of the physical nodes to the reference nodes, hence giving a factored form of V in (<ref>).Before working examples, we summarize this in the following theorem:Let (K,P,N) and (K̂, P̂, N̂) be finite elements with affine mapping F:K→K̂ and suppose that F^*(P̂) = P.Let N^c and N̂^c be a compatible nodal completion of N and N̂. Then given matrices E ∈ℝ^ν×μ from (<ref>), V^c ∈ℝ^μ×μ from (<ref>) and D ∈ℝ^μ×ν from (<ref>) that builds the (restrictions of) the extended nodes out of the given physical nodes, the nodal transformation matrix V satisfiesV = E V^C D.This gives a general outline for mapping finite elements, and we illustrate now by turning to the Morley element.§.§.§ The Morley elementFollowing our earlier notation for the geometry and nodes, we order the nodes of a Morley triangle byN =[ δ_𝐯_1 δ_𝐯_2 δ_𝐯_3 δ^𝐧_1_𝐞_1 δ^𝐧_2_𝐞_2 δ^𝐧_3_𝐞_3 ]^T Nodes N^C will also include tangential derivatives at the edge midpoint.We putN^c = [δ_𝐯_1δ_𝐯_2δ_𝐯_3 (∇^𝐧_1𝐭_1_𝐞_1)^T (∇^𝐧_2𝐭_2_𝐞_2)^T (∇^𝐧_3𝐭_3_𝐞_3)^T ]^T,Again, this is a block vector the last three entries each consist of two values.We give the same ordering of reference element nodes N̂ and N̂^c.The matrix E simply extracts the members of N^C that are also in N, so with η = [ 1 0 ], we have the block matrixE = [ 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 1 0 0 0; 0 0 0 η 0 0; 0 0 0 0 η 0; 0 0 0 0 0 η ]. Because the gradient nodes in N^c use normal and tangential coordinates, V^c will be slightly more more complicated than V for the Hermite element. For local edge γ_i, we define the (orthogonal) matrixG_i = [ 𝐧_i 𝐭_i ]^Twith the normal and tangent vector in the rows.Similarly, we letĜ_i = [ 𝐧̂_i 𝐭̂_i ]^Tcontain the unit normal and tangent to edge γ̂_i of the reference cell K̂.It is clear thatF_*(∇^𝐧_i 𝐭_i_𝐞_i) = F_*(G_i ∇_𝐞_i) = G_i F_*(∇_𝐞_i) = G_i J^T ∇̂_𝐞_i = G_i J^T Ĝ_i^T ∇̂^𝐧̂_i𝐭̂_i_𝐞̂_i,so, definingB^i = (G_i J^T Ĝ_i^T)^-1 = Ĝ_i J^-T G_i^T,we have thatV^C = [ 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 1 0 0 0; 0 0 0 B^1 0 0; 0 0 0 0 B^2 0; 0 0 0 0 0 B^3; ]. Now, we turn to the matrix D ∈ℝ^9 × 6, writing members of π N^c in terms of π N alone.The challenge is to express the tangential derivative nodes in terms of the remaining six nodes – vertex values and normal derivatives.In fact, only the vertex values are needed.Along any edge, any member of P is just a univariate quadratic polynomial, and so the tangential derivative is linear.Linear functions attain their average value over an interval at its midpoint.But the average value of the derivative over the edge is just the difference between vertex values divided by the edge length.The matrix D must beD = [ 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 -ℓ_1^-1ℓ_1^-1 0 0 0; 0 0 0 0 1 0; -ℓ_2^-1 0ℓ_2^-1 0 0 0; 0 0 0 0 0 1; -ℓ_3^-1ℓ_3^-1 0 0 0 0; ]We can also arrive at this formulation of D in another way, that sets up the discussion used for Argyris and later Bell elements. Consider the following univariate result:Let p(x) any quadratic polynomial on [-1, 1].Thenp^'(0) = 12( p(1) - p(-1) )Write p(x) = a + b x + c x^2.Then p^'(x) = b + 2 c x so that p^'(0) = b. Also note that p(1) = a + b + c and p(-1) = a - b + c.Wanting to write p^'(0) = d_1 p(1) + d_-1 p(-1) for constants d_1 and d_-1 leads to a 2 × 2 linear system, which is readily solved to give d_1 = -d_-1 = 12.Then, by a change of variables, this rule can be mapped to [ -ℓ2, ℓ2] so thatp^'(0) = 1ℓ( p(ℓ2) - p(-ℓ2) ).Finally, one can apply this rule on the edge of a triangle running from 𝐯_a to 𝐯_b to find thatπδ^𝐭_i = ℓ2( πδ_𝐯_b - πδ_𝐯_a).It is interesting to explicitly compute the product V = E V^C D, as giving a single formula rather than product of matrices is more useful in practice.Multiplying through gives:V = [100000;010000;001000;0 -B^1_12ℓ_1B^1_12ℓ_1 B^1_1100; -B^2_12ℓ_20B^2_12ℓ_20 B^2_110; -B^3_12ℓ_3B^3_12ℓ_3000 B^3_11 ] From the definition of B^i, it is possibly to explicitly calculate its entries in terms of the those of the Jacobian and the normal and tangent vectors for K and K̂.Only the first row of each B^i is neededB^i_11= n̂^𝐱_i ( n^𝐱_i ∂ x∂x̂ + t^𝐱_i ∂ y∂x̂) + t̂^𝐱_i ( n^𝐱_i ∂ x∂ŷ + t^𝐱_i ∂ y∂ŷ) B^i_12= n̂^𝐱_i ( n^𝐲_i ∂ x∂x̂ + t^𝐲_i ∂ y∂x̂) + t̂^𝐱_i ( n^𝐲_i ∂ x∂ŷ + t^𝐲_i ∂ y∂ŷ)We can also recall that the normal and tangent vectors are related by n^𝐱 = t^𝐲 and n^𝐲 = -t^𝐱 to express these entries purely in terms of either the normal or tangent vectors. Each entry of the Jacobian and normal and tangent vectors of K and K̂ enter into the transformation.In this form, V has 12 nonzero entries, although the formation of those entries, which depend on normal and tangent vectors and the Jacobian, from the vertex coordinates requires an additional amount of arithmetic.The Jacobian will typically be computed anyway in a typical code, and the cost of working with M = V^T will again be subdominant to the nested loops over basis functions and quadrature points required to form element matrices, much like Hermite.§.§.§ The Argyris elementBecause it is higher degree than Morley and contains second derivatives among the nodes, the Argyris transformation is more involved.However, it is a prime motivating example and also demonstrates thatthe general theory here reproduces the specific technique in <cit.>. The classical Argyris element has P as polynomials of degree 5 over a triangle K, a 21-dimensional space. The 21 associated nodes N are selected as the point values, gradients, and all three unique second derivatives at the vertices together with the normal derivatives evaluated at edge midpoints.These nodal choices lead to a proper C^1 element, and C^2 continuity is obtained at vertices.Since the Argyris elements do not form an affine-interpolation equivalent family, we will need to embed the physical nodes into a larger set.Much as with Morley elements, the edge normal derivatives will be augmented by the tangential derivatives.With this notation, N is a vector of 21 functionals and N^C a vector of 24 functions written asN = [ [ δ_𝐯_1 ∇_𝐯_1 △_𝐯_1 δ_𝐯_2 ∇_𝐯_2 △_𝐯_2 δ_𝐯_3 ∇_𝐯_3 △_𝐯_3 δ^𝐧_1_𝐞_1 δ^𝐧_2_𝐞_2 δ^𝐧_3_𝐞_3 ]]^T, N^C= [[δ_𝐯_1∇_𝐯_1△_𝐯_1δ_𝐯_2∇_𝐯_2△_𝐯_2δ_𝐯_3∇_𝐯_3△_𝐯_3 ∇^𝐧_1𝐭_1_𝐯_1 ∇^𝐧_2𝐭_2_𝐯_2 ∇^𝐧_3𝐭_3_𝐯_3 ]]^T,with corresponding ordering of reference nodes N̂ and N̂^c. The 21 × 24 matrix E just selects out the items in N^C that are also in N, so thatE_ij = 1,for1 ≤ i=j ≤ 19or(i,j) ∈{(20, 21), (21, 23)}0,otherwise. The matrix V^C relating the push-forward of the extended nodes to the extended reference nodes is block diagonal and similar to our earlier examples.We use (<ref>) to map the vertex gradient nodes as in the Hermite case.Mapping the three unique second derivatives by the chain rule requires the matrix:Θ = [ ( ∂x̂∂ x)^2 2 ∂x̂∂ x∂ŷ∂ x( ∂ŷ∂ x)^2;∂x̂∂ y∂x̂∂ x ∂x̂∂ y∂ŷ∂ x + ∂x̂∂ x∂ŷ∂ y∂ŷ∂ x∂ŷ∂ y;(∂x̂∂ y)^2 2 ∂x̂∂ y∂ŷ∂ y( ∂ŷ∂ y)^2 ]The edge midpoint nodes transform by B just as in (<ref>), so that the V^C isV^C = [ [100000000000;0 J^-T0000000000;00 Θ^-1000000000;000100000000;0000 J^-T0000000;00000 Θ^-1000000;000000100000;0000000 J^-T0000;00000000 Θ^-1000;000000000B^100;0000000000B^20;00000000000B^3;]]. Constructing D, like for Morley, is slightly more delicate.The additional nodes acting on quintic polynomials – tangential derivatives at edge midpoints – must be written in terms of the remaining nodes.The first aspect of this involves a univariate interpolation-theoretic question.On the biunit interval [-1, 1], we seek a rule of the formf'(0) ≈ a_1 f(-1) + a_2 f(1) + a_3 f^'(-1) + a_4 f^'(1) + a_5 f^''(-1) + a_6 f^''(1)that is exact when f is a quintic polynomial.The coefficients may be determined to by writing a 6× 6 linear system asserting correctness on the monomial basis.The answer, given in <cit.>, is thatAny quintic polynomial p defined on [-1, 1] satisfiesp^'(0) = 1516( p(1) - p(-1) ) - 716( p^'(1) + p^'(-1) ) + 116( p^''(1) - p^''(-1) ). This can be mapped to the interval [-ℓ2, ℓ2] by a change of variables:p^'(0) = 158ℓ( p(ℓ2) - p(-ℓ2) ) - 716( p^'(ℓ2) + p^'(-ℓ2)) + ℓ32( p^''(ℓ2) - p^''(-ℓ2) ). Now, we can use this to compute the tangential derivative at an edge midpoint, expanding the tangential first and second derivatives in terms of the Cartesian derivatives.If 𝐯_a and 𝐯_b are the beginning and ending vertex of edge γ_i with midpoint 𝐞_i and length ℓ_i, we write the tangential derivative acting on quintics asπδ^𝐭_i_𝐞_i =158ℓ_i( δ_𝐯_b - δ_𝐯_a) - 716( t^𝐱_i ( δ^𝐱_𝐯_b + δ^𝐱_𝐯_a) +t^𝐲_i ( δ^𝐲_𝐯_b + δ^𝐲_𝐯_a) )+ ℓ_i32( (t_i^𝐱)^2 ( δ^𝐱𝐱_𝐯_b - δ^𝐱𝐱_𝐯_a) + 2 t_i^𝐱 t_i^𝐲( δ^𝐱𝐲_𝐯_b - δ^𝐱𝐲_𝐯_a) +(t_i^𝐲)^2 ( δ^𝐲𝐲_𝐯_b - δ^𝐲𝐲_𝐯_a) ). For each edge γ_i, define the vector τ_i byτ_i = [ (t^𝐱_i)^2 2 t^𝐱_i t^𝐲_i (t^𝐲_i)^2 ]^T. The end result is thatD = [ [ 1 0 0 0 0 0 0 0 0 0 0 0; 0 I_2 0 0 0 0 0 0 0 0 0 0; 0 0 I_3 0 0 0 0 0 0 0 0 0; 0 0 0 1 0 0 0 0 0 0 0 0; 0 0 0 0 I_2 0 0 0 0 0 0 0; 0 0 0 0 0 I_3 0 0 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 0 0; 0 0 0 0 0 0 0 I_2 0 0 0 0; 0 0 0 0 0 0 0 0 I_3 0 0 0; 0 0 0 0 0 0 0 0 0 1 0 0; 0 0 0 -158ℓ_1716𝐭^T_1 -ℓ32τ^T_1158ℓ_1716𝐭^T_1ℓ32τ^T_1 0 0 0; 0 0 0 0 0 0 0 0 0 0 1 0; -158ℓ_2716𝐭^T_2 -ℓ32τ^T_2 0 0 0158ℓ_2716𝐭^T_2ℓ32τ^T_2 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1; -158ℓ_3716𝐭^T_3 -ℓ32τ^T_3158ℓ_3716𝐭^T_3ℓ32τ^T_3 0 0 0 0 0 0; ]]. If this transformation is kept in factored form, D contains 57 nonzero entries and V^c contains 54 nonzero entries. E is just a Boolean matrix and its application requires copies.So, application of M requires no more than 111 floating-point operations, besides the cost of forming the entries themselves.While this is about ten times the cost of the Hermite transformation, it isfor about twice the number of basis functions and still well-amortized over the cost of integration loops.Additionally, one can multiply out the product E V^c D symbolically and find only 81 nonzero entries, which reduces the cost of multiplication accordingly. §.§ Generalizations§.§.§ Non-affine mappingsNon-affine geometric transformations, whether for simplicial or other element shapes, present no major complications to the theory.In this case, K and K̂ are related by a non-affine map, and P is taken to be the image of P̂ under pull-backP = { F^*(p̂) : p̂∈P̂},although this space need not consist of polynomials for non-affine F.At any rate, one may define Hermite elements on curvilinear cells <cit.>.In this case, the Jacobian matrix varies spatially so that each instance of J^T in (<ref>) must be replaced by the particular value of J^T at each vertex.§.§.§ Generalized pullbacksMany vector-valued finite element spaces make use of pull-backs other than composition with affine maps.For example, the Raviart-Thomas and Nédélec elements use contravariant and covariant Piola maps, respectively.Because these preserve either normal or tangential components, one can put the nodal basis functions of a given element (K, P, N) and reference element (K̂, P̂, N̂) into one-to-one correspondence by means of the Piola transform, a fact used heavily in <cit.> possible.It would be straightforward to give a generalization of affine equivalence to equivalence under an arbitrary pull-back F^*, with push-forward defined in terms of F^*.In this case, the major structure of <ref> would be unchanged.However, not all H(div) elements form equivalent families under the contravariant Piola transform.For example, Mardal, Tai, and Winther <cit.> give an element that can be paired with discontinuous polynomials to give uniform inf-sup stability on a scale of spaces between H(div) and (H^1)^2, although it is H^1-nonconforming. The degrees of freedom include constant and linear moments of normal components on edges, which are preserved under Piola mapping. However, the nodes also include the constant moments of the tangential component on edges, which are not preserved under Piola transform.One could push-forward both the normal and tangential constant moments, then express them as a linear combination of the normal and tangential moments on the reference cell in a manner like (<ref>).One could see the Mardal–Tai–Winther element as satisfying a kind of “Piola-interpolation equivalence” and readily adapt the techniques for Hermite elements, §.§ A further note on computationWe have commented on the added cost of multiplying the set of basis functions by M during local integration.It also also possible to apply the transformation in a different way that perhaps more fully leverages pre-existing computer routines.With this approach, M can also be included in local matrix assembly by means means of a congruence transform acting on the “wrong” element matrix as follows.Given a finite element (K, P, N) with nodal basis Ψ = {ψ_i }_i=1^ν and bilinear form a_K(·, ·) over the domain K, we want to compute the matrixA^K_ij = a_K(ψ_j, ψ_i).Suppose that a computer routine existed for evaluating A^K via a reference mapping for affine-equivalent elements.That is, given the mapping F:K̂→ K, this routine maps all integration to the reference domain K̂ assuming that the integrand over K is just the affine pull-back of something on K̂.Consider the following computation: A^K_ij= a_K(ψ_j, ψ_i)= a_K ( ∑_ℓ_2=1^ν M_jℓ_2 F^*(ψ̂_ℓ_2) , ∑_ℓ_1=1^ν M_iℓ_1 F^*(ψ̂_ℓ_1) )= ∑_ℓ_1, ℓ_2 = 1^ν M_jℓ_2 M_iℓ_1 a_K( F^*(ψ̂_ℓ_2), F^*(ψ̂_ℓ_1))Now, this is just expressed in terms of the affine pullback of reference-element integrands and so could use the hypothesized computer routine. We then haveA^K_ij = ∑_ℓ_1, ℓ_2 = 1^ν M_jℓ_2 M_iℓ_1 a_K̂( ψ̂_ℓ_2, ψ̂_ℓ_1) = ∑_ℓ_1, ℓ_2 = 1^ν M_jℓ_1 M_iℓ_2Â^K_ℓ_1ℓ_2,or, more compactly,A^K = M Ã^K M^T,where Ã^K is the matrix one would obtain by using the pull-back of the reference element nodal basis functions instead of the actual nodal basis for (K, P, N).Hence, rather than applying M invasively at each quadrature point, one may use existing code for local integration and pre- and post-multiply the resulting matrix by the basis transformation.In the case of Hermite, for example,applying M to a vector costs 12 operations, so applying M to all 10 columns of Ã^K costs 120 operations, plus another 120 for the transpose.This adds 240 extra operations to the cost of building Ã^K, or just 2.4 extra FLOPs per entry of the matrix.One may also apply this idea in a “matrix-free” context.Given a routine for applying Ã^K to a vector, one may simply apply M^T to the input vector, apply Ã^K to the result, and post-multiply by M.Hence, one has the cost of muliplying by Ã^K plus the cost of applying M and its transpose to a single vector.In the case of Hermite, one has the cost of computing the “wrong” local matrix-vector product via an existing kernel plus 24 additional operations.Finally, we comment on evaluating discrete functions over elements requiring such transforms.Discrete function evaluation is frequently required in matrix-free computation, nonlinear residual evaluation, and in bilinear form evaluation when a coefficient is expressed in a finite element space. Suppose one has on a local element K a function expressed byu = ∑_j=1^ν c_j ψ_j,where c ∈ℝ^ν is the vector of coefficients and {ψ_j } is the nodal basis for (K, P, N). In terms of pulled-back reference basis functions, u is given byu = ∑_j=1^ν c_j ( ∑_k=1^ν M_jk F^*(ψ̂_k) ) = ∑_j, k=1^ν M_jk c_j F^*(ψ̂_k),which can also be written asu = ∑_k=1^ν (M^T c)_k F^*(ψ̂_k) = ∑_k=1^ν (V c)_k F^*(ψ̂_k).Just as one can build element matrices by means of the “wrong” basis functions and a patch-up operation, one can also evaluate functions by transforming the coefficients and then using the standard pullback of the reference basis functions.Such observations may make incorporating nonstandard element transformations into existing code more practical.§ WHAT IF P ≠ F^*(P̂)? The theory so far has been predicated on F^* providing an isomorphism between the reference and physical function spaces.In certain cases, however, this fails.Our main motivation here is to transform the Bell element, a near-relative of the quintic Argyris element.In this case, one takes P to be the subspace of P_5 that has cubic normal derivatives on edges rather than the typical quartic values. This reduction of P by three dimensions is accompaniedby removing the three edge normal derivatives at midpoints from N.In general, however, the pull-back F^*(P̂) does not coincide with P.Instead of cubicnormal derivatives on edges, F^*(P̂) has reduced degree in some other direction corresponding to the image of the normal under affine mapping.The theory developed earlier can be extended somewhat to resolve this situation. §.§ General theory: extending the finite elementAbstractly, one may view the Bell element or other spaces built by constraint as the intersection of the null spaces of a collection of functionals acting on some larger space as follows.Let (K, P, N) be a finite element.Suppose that P ⊂P̃ and that {λ_i }_i=1^κ⊂( C^k_b )^' are linearly independent functionals that when acting on P̃ satisfyP = ∩_i=1^κnull(λ_i). The following result is not difficult to prove:Let (K, P, N) be a finite element with ∩_i=1^κnull(λ_i) = P ⊂P̃ as per (<ref>).Similarly, let Let (K̂, P̂, N̂) be a reference element with ∩_i=1^κnull(λ̂_i) = P̂⊂P̃̂̃.Suppose that P̃ = F^*(P̃̂̃). Then P = F^*(P̂) iffspan{F_*(λ_i)}_i=1^κ = span{λ̂_i}_i=1^κ.In the case of the Bell element, the span condition (<ref>) fails and so that the function space is not preserved under affine mapping.Consequently, the theory of the previous section predicated on this preservation does not directly apply.Instead, we proceed by making the following observation. Let (K, P, N) be a finite element with P ⊂P̃ satisfying P = ∩_i=1^κnull(λ_i) for linearly independent functionals {λ_i}_i=1^κ. DefineÑ = [ N; L ]to include the nodes of N together with L = [ λ_1 λ_2 … λ_κ ]^T. Then (K, P̃, Ñ) is a finite element. Since we have a finite-dimensional function space, it remains to show that Ñ is linearly independent and hence spans P̃^'.Consider a linear combination in P̃^' ∑_i=1^ν c_i n_i + ∑_i=1^κ d_i λ_i = 0.Apply this linear combination to any p ∈ P to find∑_i=1^ν c_i n_i(p) = 0since λ_i(p) = 0 for p ∈ P. Because (K, P, N) is a finite element, the n_i are linearly independent in P^' so c_i = 0 for 1 ≤ i ≤ν. Applying the same linear combination to any ∈P̃\ P then gives that d_i=0 since the constraint functionals are also linearly independent.Given a nodal basis (K, P̃, Ñ), it is easy to obtain one for (K, P, N).Let (K, P, N), {λ_i}_i=1^κ, and (K, P̃, Ñ) be as in Proposition <ref>.Order the nodes in Ñ by Ñ = [ N; L ] with L_i = λ_i for 1 ≤ i ≤κ.Let {ψ̃_i}_i=1^ν + κ be the nodal basis for (K, P̃, Ñ). Then {ψ̃_i }_i=1^ν is the nodal basis for (K, P, N). Clearly, n_i(ψ̃_j) = δ_ij for 1 ≤ i, j ≤ν by the ordering of the nodes in Ñ.Moreover, {ψ̃_i}_i=1^ν⊂ P because λ_i(ψ̃_j) = 0 for each 1 ≤ i ≤κ.§.§ The Bell elementSo, we can obtain a nodal basis for the Bell element or others with similarly constrained function spaces by mapping the nodal basis for a slightly larger finite element and extracting a subset of the basis functions.Let(K, P, N) and (K̂, P̂, N̂) be the Bell elements over K and reference cell K̂.Recall that the Legendre polynomial of degree n is orthogonal to polynomials of degree n-1 or less. Let ℒ^n be the Legendre polynomial of degree n mapped from the biunit interval to edge γ_i of K. Define a functionalλ_i(p) = ∫_γ_iℒ^4(s) ( 𝐧_i ·∇ p )ds.For any p ∈ P_5(K), its normal derivative on edge iis cubic iff λ_i(p) = 0.So, the constraint functionals are given in L = [ λ_1 λ_2 λ_3 ]^T and Ñ = [ N; L ] as in Proposition <ref>.We defineλ̂_i(p) = ∫_γ̂_iℒ^4(s) ( 𝐧̂_i ·∇ p )dsand hence (K̂, P̂, N̂) as well as L̂ and Ñ̂̃ in a similar way.P and P̂ are the constrained spaces – quintic polynomials with cubic normal derivatives on edges, while P̃ and P̃̂̃ are the spaces of full quintic polynomials over K and K̂, respectively.We must construct a nodal basis for (K̂, P̃̂̃, Ñ̂̃), map it to a nodal basis for (K, P̃, Ñ) by the techniques in Section <ref>, and then take the subset of basis functions corresponding to the Bell basis.This is accomplished by specifying a compatible nodal extension of Ñ and Ñ̂̃ by including the edge moments of tangential derivatives against ℒ^4with those of Ñ and Ñ̂̃.We defineλ_i^'(p) = ∫_γ_iℒ^4(s) ( 𝐭_i ·∇ p ) ds,λ̂_i^'(p) = ∫_γ̂_iℒ^4(s) ( 𝐭̂_i ·∇ p ) ds. We must specify the E, V^c, and D matrices for this extended set of finite element nodes.We focus first on D, needing to compute each λ_i^' in terms of the remaining functionals. As with Morley and Argyris, we begin with univariate results.The following is readily confirmed, for example, by noting the right-hand side is a quintic polynomial and computing values and first and second derivatives at ± 1:Let p be any quintic polynomial on [-1, 1].Then16 p(x) =-( x-1 )^3 ( p^''(-1) ( x+1 )^2+ p^'(-1) ( x+1) ( 3x + 5 ) + p(-1) ( 3 x^2 + 9x + 8 ) )+ ( x + 1 )^3 ( p^''(1) ( x-1 )^2 - p^'(1)( x-1 )( 3x - 5 ) + p(1) ( 3 x^2 - 9 x + 8 ) ). The formula (<ref>) can be differentiated and then integrated against ℒ^4 to show that∫_-1^1 p^'(x) ℒ^4(x) dx = 121[p(1) - p(-1) - p^'(1) - p^'(-1) + 13(p^''(1) - p^''(-1)) ]. Then, this can be mapped to a general interval [-ℓ2, ℓ2] by a simple change of variables:∫_-ℓ2^ℓ2 p^'(x) ℒ^4(x) dx = 121[p(ℓ2) - p(-ℓ2) - ℓ2(p^'(ℓ2) + p^'(-ℓ2)) + ℓ^212( p^''(ℓ2) - p^''(-ℓ2)) ]. Now, we can use this to express the functionals λ_i^' from (<ref>) as linear combinations of the Bell nodes:Let K be a triangle and 𝐯_a and 𝐯_b are the beginning and ending vertex of edge γ_i with length ℓ_i. Let p be any bivariate quintic polynomial over K and λ_i^' defined in (<ref>).Then the restriction of λ_i^' to bivariate quintic polynomials satisfies πλ_i^' = 121[ πδ_𝐯_b - πδ_𝐯_a - ℓ_i2(πδ_𝐯_b^𝐭_i+ πδ_𝐯_b^𝐭_i) + ℓ_i^212( πδ_𝐯_b^𝐭_i𝐭_i - πδ_𝐯_b^𝐭_i𝐭_i) ],and henceπλ_i^' =121[ πδ_𝐯_b - πδ_𝐯_a]-ℓ_i42[ t_i^𝐱( πδ_𝐯_b^𝐱 + πδ_𝐯_a^𝐱) + t_i^𝐲( πδ_𝐯_b^𝐲 + πδ_𝐯_a^𝐲) ]+ℓ_i^2252( (t_i^𝐱)^2 ( πδ_𝐯_b^𝐱𝐱 -πδ_𝐯_a^𝐱𝐱) + 2 t_i^𝐱t_i^𝐲( πδ_𝐯_b^𝐱𝐲 -πδ_𝐯_a^𝐱𝐲) + (t_i^𝐲)^2 ( πδ_𝐯_b^𝐲𝐲 -πδ_𝐯_a^𝐲𝐲) ).Now, V^c is quite similar to that for the Argyris element.There is a slight difference in the handling the edge nodes, for we have an integral moment instead of a point value and must account for the edge length accordingly.By converting between normal/tangent and Cartesian coordinates via the matrix G_i and mapping to the reference element, we find that for any p,[ λ_i(p); λ_i^'(p) ]= ∫_γ_iℒ^4(s) ( G_i ∇ p ) ds= ∫_γ̂_i|dŝds|ℒ^4(ŝ) ( G_i J^T Ĝ_i^T ∇̂^𝐧̂_i𝐭̂_ip̂) dŝ = |dŝds|G_i J^T Ĝ_i^T [ λ̂_̂î(p); λ̂_̂î^'(p) ]This calculation shows that V^C for the Bell element is identical to (<ref>) for Argyris, except with a geometric scaling of the B matrices.The extraction matrix E for the extended Bell elements consisting of full quintics now is identical to that for Argyris. Then, when evaluating basis functions, one multiplies the affinely mapped set of basis values by V^T and then takes only the first 18 entries to obtain the local Bell basis. §.§ A remark on the Brezzi-Douglas-Fortin-Marini elementIn <cit.>, we describe a two-part process for computing the triangular Brezzi-Douglas-Fortin-Marini (BDFM) element <cit.>, an H(div) conforming finite element based on polynomials of degree k with normal components constrained to have degree k - 1.This is a reduction of the Brezzi-Douglas-Marini element <cit.> somewhat as Bell is of Argyris.However, as both elements form Piola-equivalent families, the transformation techniques developed here are not needed.Like the Bell element, one can define constraint functionals (integral moments of normal components against the degree k Legendre polynomial) for BDFM.In <cit.>, we formed a basis for the intersection of the null spaces of these functionals by means of a singular valuedecomposition.A nodal basis for the BDFM space then followed bybuilding and inverting a generalized Vandermonde matrix on the basis for this constrained space.In light of Propositions <ref> and <ref>, however, this process was rather inefficient.Instead, we could have merely extended the BDFM nodes by the constraint functionals, building and inverting a single Vandermonde-like matrix.If one takes the BDM edge degrees of freedom as moments of normal components against Legendre polynomials up to degree (k-1) instead of pointwise normal values, then one can even build a basis for BDM that includes a a basis for BDFM as a proper subset.§ NUMERICAL RESULTS Incorporation of these techniques into high-level software tools such as Firedrake is the subject of ongoing investigation.In the meantime, we provide some basic examples written in Python, with sparse matrix assemble and solvers using petsc4py <cit.>. §.§ Scaling degrees of freedomBefore considering the accuracy of the L^2 projection, achieved via the global mass matrix, we comment on the conditioning of the mass and other matrices when both derivative and point value degrees of freedom appear.The Hermite element is illustrative of the situation.On a cell of typical diameter h, consider a basis function corresponding to the point value at a given vertex.Since the vertex basis function has a size of 𝒪(1) on a triangle of size 𝒪(h^2), its L^2 norm should be 𝒪(h).Now, consider a basis function corresponding to a vertex derivative.Its derivative is now 𝒪(1) on the cell, so that the H^1 seminorm is 𝒪(h). Inverse inequalities suggest that the L^2 norm could then be as large as 𝒪(1).That is, the different kinds of nodes introduce multiple scales of basis function sizes under transformation, which manifests in ill-conditioning.Where one expects a mass matrix to have an 𝒪(1) condition number, one now obtains an 𝒪(h^-2) condition number.This is observed even on a unit square mesh, in Figure <ref>.All condition numbers are computed by converting the PETSc mass matrix to a dense matrix and using LAPACK via scipy <cit.>However, there is a simple solution.For the Hermite element, one can scale the derivative degrees of freedom locally by an “effective h”.All cells sharing a given vertex must agree on that h, which could be the average cell diameter among cells sharing a vertex.Scaling the nodes/basis functions (which amounts to multiplying V on the right by a diagonal matrix with 1's or h's) removes the scale separation among basis functions and leads again to an 𝒪(1) condition number for mass matrices, also seen in Figure <ref>.From here, we will assume that all degrees of freedom are appropriately scaled to give 𝒪(1) conditioning for the mass matrix. §.§ Accuracy of L^2 projectionNow, we demonstrate that optimal-order accuracy is obtained by performing L^2 projection of smooth functions into the Lagrange, Hermite, Morley, Argyris, and Bell finite element spaces.In each case we use an N × N mesh divided into right triangles. Defining u(x, y) = sin(π x) sin(2 π y) on [0,1]^2, we seek u_h such that( u_h , v_h ) = ( u , v_h )for each v_h ∈ V_h, where V_h is one of the the finite element spaces.Predicted asymptotic convergence rates – third for Morley, fourth for Hermite and Lagrange, fifth for Bell, and sixth for Argyris, are observed in Figure <ref>.Note that the Hermite and Lagrange elements have the same order of approximation, but the Lagrange element delivers a slightly lower error.This is to be expected, as the space spanned by cubic Hermite triangles is a proper subset of that spanned by Lagrange. §.§ The Laplace operatorAs a simple second-order elliptic operator, we consider the Dirichlet problem for the Laplace operator on the unit square Ω:-Δ u = f,equipped with homogeneous Dirichlet boundary conditions u = 0 on ∂Ω.We divide Ω into an N × N mesh of triangles and let V_h be one of the Lagrange, Hermite, Argyris, or Bell finite element spaces, all of which are H^1-conforming, over this mesh.The Morley element is not a suitable H^1 nonconforming element, so we do not use it here. We then seek u_h ∈ V_h such that( ∇ u_h , ∇ v_h ) = ( f , v_h )for all v_h ∈ V_h.Enforcing strong boundary conditions on elements with derivative degrees of freedom is delicate in general.However, with grid-aligned boundaries, it is less difficult.To force a function to be zero on a given boundary segment, we simply require the vertex values and all derivatives tangent to the edge vanish.This amounts to setting the x-derivatives on the top and bottom edges of the box and y-derivative on the left and right for Hermite, Argyris, and Bell elements.Dirichlet conditions for Lagrange are enforced in the standard way.By the method of manufactured solutions, we select f(x, y) = 8 π^2 sin(2 π x) sin(2 π y) so that u(x,y) = sin(2 π x) sin(2 π y).In Figure <ref>, we show the L^2 error in the computed solution for both element families.As the mesh is refined, both curves approach the expected order of convergence – fourth for Hermite and Lagrange, fifth for Bell, and sixth for Argyris.Again, the error for Lagrange is slightly smaller than for Hermite, albeit with more global degrees of freedom. §.§ The clamped plate problemWe now turn to a fourth-order problem for which the Argyris and Bell elements provide conforming H^2 discretizations and Morley a suitable nonconforming one. Following <cit.>, we take the bilinear form defined on H^2( Ω ) to bea(u, v) = ∫_ΩΔ u Δ v - ( 1 - ν) ( 2 u_xx v_yy + 2 u_yy v_xx - 4 u_xy v_xy) dx dy,where 0 < ν < 1 yields a coercive bilinear form for any closed subspace of H^2 that does not contain nontrivial linear polynomials.We fix ν = 0.5.Then, we consider the variational problema(u, v) = F(v) = ∫_Ω f vdx,posed over suitable subspaces of H^2.It is known <cit.> that solutions of (<ref>) that lie in H^4(Ω) satisfy the biharmonic equation Δ^2 u = f in an L^2 sense.We consider the clamped plate problem, in which both the function value and outward normal derivative are set to vanish, which removes nontrivial linear polynomials from the space.Again, we use the method of manufactured solutions on the unit square to select f(x,y) such that u(x,y) = ( x(1-x) y(1-y) )^2, which satifies clamped boundary conditions.We solve this problem with Argyris and Bell elements, and then also use the nonconforming Morley element in the bilinear form.Again, expected orders of convergence are observed in Figure <ref>.§ CONCLUSIONSMany users have wondered why FEniCS, Firedrake, and most other high-level finite element tools lack the full array of triangular elements, including Argyris and Hermite.One answer is that fundamental mathematical aspects of mapping such elements have remained relatively poorly understood.This work demonstrates the challenges involved with mapping such elements from a reference cell, but also proposes ageneral paradigm for overcoming those challenges by embedding the nodes into a larger set that transforms more cleanly and using interpolation techniques to relate the additional nodes back to original ones. In the future, we hope to incorporate these techniques in FInAT (<https://github.com/FInAT/FInAT>), a successor project to FIAT that produces abstract syntax for finite element evaluation rather than flat tables of numerical values.TSFC <cit.> already relies on FInAT to enable sum-factorization of tensor-product bases.If FInAT can provide rules for evaluating the matrix M in terms of local geometry on a per-finite element basis, then TSFC and other form compilers should be able to seamlessly (from the end-users' perspective) generate code for many new kinds of finite elements. plain | http://arxiv.org/abs/1706.09017v1 | {
"authors": [
"Robert C. Kirby"
],
"categories": [
"math.NA",
"65N30"
],
"primary_category": "math.NA",
"published": "20170627191215",
"title": "A general approach to transforming finite elements"
} |
[^∗] Based on archival observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. February 27, 2017 June 26, 2017 1Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA 2Max-Planck Institute for Astronomy, Koenigstuhl 17, D-69117 Heidelberg, Germany 3McMaster: Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada 4Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 5Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, v.co dell'Osservatorio 3, I-35122, Padova, Italy 6Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Padova, v.co dell'Osservatorio 5, I-35122, Padova, Italy 7Department of Astronomy, Indiana University, Bloomington, IN 47405, USA 1.0cm High-precision proper motions of the globular cluster 47 Tuc have allowed us to measure for the first time the cluster rotation in the plane of the sky and the velocity anisotropy profile from the cluster core out to about 13^'. These profiles are coupled with prior measurements along the line of sight and the surface-brightness profile, and fit all together with self-consistent models specifically constructed to describe quasi-relaxed stellar systems with realistic differential rotation, axisymmetry and pressure anisotropy. The best-fit model provides an inclination angle i between the rotation axis and the line-of-sight direction of 30^∘, and is able to simultaneously reproduce the full three-dimensional kinematics and structure of the cluster, while preserving a good agreement with the projected morphology. Literature models based solely on line-of-sight measurements imply a significantly different inclination angle (i=45^∘), demonstrating that proper motions play a key role in constraining the intrinsic structure of 47 Tuc. Our best-fit global dynamical model implies an internal rotation higher than previous studies have shown, and suggests a peak of the intrinsic V/σ ratio of ∼0.9 at around two half-light radii, with a non-monotonic intrinsic ellipticity profile reaching values up to 0.45. Our study unveils a new degree of dynamical complexity in 47 Tuc, which may be leveraged to provide new insights into the formation and evolution of globular clusters. Hubble Space Telescope Proper Motion (HSTPROMO) Catalogs of Galactic Globular Clusters. V.The rapid rotation of 47 Tuc traced and modeled in three dimensions^∗A. Bellini1, P. Bianchini2,3, A. L. Varri4, J. Anderson1, G. Piotto5,6, R. P. van der Marel1,E. Vesperini7, and L. L. Watkins1December 30, 2023 =====================================================================================================================================================================§ INTRODUCTIONGlobular-cluster (GC) formation, internal dynamical evolution and the effects of the external tidal field of the host galaxy are expected to leave a number of fingerprints on a cluster's structural,morphological and kinematical properties.Until recently, the dynamical characterization of most GCs has been limited to the surface-brightness or projected-star-count radial profiles. These profiles can shed light on some aspects of cluster dynamical evolution – such as the identification of the systems that have already evolved past the core collapse phase (see, e.g., ), and to calculate a number of fundamental global properties (see, e.g., ) – but they provide only a very partial view of a cluster's dynamical state. To construct a complete dynamical picture of clusters, we require an accurate understanding of both their internal kinematics and their detailed morphologies. Such information places key constraints on theoretical studies, and thus allows us to reconstruct cluster dynamical histories, and determine the role played by different dynamical processes <cit.>.After some early pioneering work on cluster kinematical properties (see, e.g.,and references therein), there has been a recent revival in observational studies of internal cluster kinematics based on ESO/VLT radial velocities (see, e.g., ) and Hubble Space Telescope (HST)-based, proper-motion (PM) measurements (see, e.g., , hereafter Paper I and Paper II, respectively).These kinematics have already been used to address a number of fundamental issues, including the existence of intermediate-mass black holes at the centers of GCs (see, e.g., ), the kinematical differences between multiple stellar populations (), differences between the velocity dispersion of stars with different masses and the degree of energy equipartition (see, e.g., , Paper IV), the presence and origin of anisotropy in the velocity distribution (Paper II), and the strength of cluster rotation and its possible link with the cluster morphology (see, e.g., ). Forthcoming data from the Gaia mission (see, e.g., ) will further enrich the observational landscape.On the theoretical side, renewed efforts are being made to expand the numerical and analytical tools necessary to interpret the results of these observational studies and gather a deeper understanding of their implications in the context of GC dynamical evolution.These efforts include studies proposing new distribution-function-based models of rotating and anisotropic models (see, e.g., ) extending the widely used (but limited to spherical symmetry and isotropic velocity distribution) models such as <cit.> or <cit.> models, and numerous numerical studies exploring, for example, the evolution of rotating and anisotropic models (see, e.g., the early studies of , and the more recent investigations by ), the evolution of the degree of energy equipartition (see, e.g., ) the role of potential escapers in the observed velocity dispersion profiles (see, e.g., ) and the kinematical implications of star loss from GCs (see, e.g., ).On the observational, proper-motion-based side, measurements of GC rotation in the plane of the sky had been completely lacking until <cit.> took advantage of a large number of photographic plates spanning half a century to measure a plane-of-the-sky differential rotation for the GC ω Centauri (see their Fig. 18). A few years later, <cit.> took advantage of the high-precision astrometric capabilities of HST to obtain a direct measurement of the plane-of-the-sky rotation of 47 Tuc (NGC 104) using two diametrically opposite fields ∼5^' from the cluster center.The authors exploited the unique feature of 47 Tuc to have a large number of relatively bright Small Magellanic Cloud (SMC) stars in the background, and SMC stars themselves could be used as reference objects. In contrast, much fainter galaxies or much rarer QSOs must be used as reference objects for the vast majority of the other GCs. A few other GCs analogous to 47 Tuc are: (1) NGC 362, which is also in front of SMC stars and will be the subject of a forthcoming paper in this series, (2) NGC 6652, which is in front of Sagittarius Dwarf Spheroidal (Sgr dSph) galaxy stars, and (3) NGC 6681, also in front of Sgr dSph stars. The plane-of-the-sky rotation of NGC 6681 has indeed been recently measured (with a null result) by <cit.>, but using background galaxies as a reference. The <cit.> work represents the third –and so far the last– observational, PM-based work on the subject.In this paper, we extend the pioneering work of <cit.> and measure, for the first time, the rotation curve of 47 Tuc in the plane of the sky from the center of the cluster out to about 13^' (about 4 half-light radii, ). We present state-of-the-art, high-precision, HST-based PM measurements for the cluster, which we combine with existing line-of-sight (LOS) velocities to determine the strength of the cluster present-day rotation and anisotropy in the velocity distribution. By combining the kinematical data with the known surface-brightness profile, we build a detailed self-consistent model of the cluster. This work represents the most complete dynamical characterization of any Galactic GC and clearly illustrates how only such a detailed study can reveal the intrinsic dynamical properties of a cluster and provides the needed constraints to explore the possible evolutionary paths leading to the present-day observed properties.The structure of this paper is the following: in Sect. <ref> we present the HST data set and the reduction techniques used to measure high-precision PMs. The results concerning the PM-based rotation of 47 Tuc in the plane of the sky and the velocity anisotropy are presented, respectively, in Sect. <ref> and <ref>. In Sect. <ref> we present the results of a detailed dynamical model fitting to the observational data using the distribution-function based models of <cit.>. Conclusions are summarized in Sect.<ref>.§ DATA SETS AND REDUCTIONWe measured PMs in four fields located at different radial distances from the center of 47 Tuc. These fields are roughly aligned along the same axis with respect to the cluster's center, and span over 1200 arcsecs (>20^') on the sky. In Fig. <ref> we show the footprints of these four fields superimposed on a Sloan Digital Sky Survey (SDSS) image of the cluster. The four fields are hereafter identified, from the East to the West, as: the inner field, the central field, the calibration field, and the outer field. We made use of exposures taken through three different HST cameras: the Wide-Field Planetary Camera 2 (WFPC2), both the High-Resolution Channel (HRC) and the Wide-Field Channel (WFC) of the Advanced Camera for Surveys (ACS), and the Ultraviolet-VISible (UVIS) channel of the Wide-Field Camera 3 (WFC3).The data reduction and analysis are based on -type exposures (or the equivalentformat for WFPC2), as they preserve the un-resampled pixel data for optimal stellar-profile fitting. All ACS/WFC and WFC3/UVIS exposures were corrected for charge-transfer-efficiency (CTE) defects <cit.>.Images were then reduced using the software family<cit.>, employing either single (HRC), spatially-varying (WFPC2), or spatially- and time-dependent empirical PSFs (WFC and UVIS, ).Stellar positions were corrected for geometric distortion using the state-of-the-art solutions provided by <cit.>.The photometric data of the central field comes directly from <cit.>. Photometry of the other fields was calibrated following the prescriptions given in <cit.>.Stellar positions in each exposure were transformed into a common, distortion-free, reference frame (the master frame). Only stars measured in at least four distinct exposures with a time baseline of at least six months were considered for the present analysis. PMs were computed using the central overlap method <cit.>, in which each exposure is considered as a stand-alone epoch. In a nutshell, we transformed stellar positionsas measured on the individual exposures on the master frame, by means of general, 6-parameter linear transformations. For each star, its master-frame transformed positions as a function of the epoch are fitted by a straight line, the slope of which gives us a direct measurement of the stellar motion. We applied a careful data-rejection procedure to remove outliers or mismatches (see Section 5.5 of Paper I for more details).Extensive simulations have demonstrated the reliability of both estimated PMs and PM errors.In the following, we describe the reduction procedures of each field. §.§ The central fieldThe PM catalog of the central field is that published in Paper I.We analyzed 433 exposures taken with the ACS (both HRC and WFC channels) and the WFC3/UVIS. The complete list of observations can be found in Table 7 of Paper I. The available time baseline used to compute the motion of each star goes from 0.52 to 10.32 years (median of 8.23), depending on the available overlap between different exposures.We have 103638 stars with measured PMs in the central field. We closely applied the prescriptions given in Section 7.5 of Paper I in order to select only high-quality PM measurements (53898 stars). §.§ The calibration fieldThis particular field was selected as one of the instrument-calibration fields for the ACS and WFC3 detectors, and has been repeatedly observed since 2002, when ACS was installed on-board HST. We chose a subsample of deep exposures taken from 2002 to 2014 with ACS/WFC and WFC3/UVIS (see Table 1).The final PM catalog contains 13466 objects, 12132 of which passed our high-quality selection criteria. §.§ Inner and outer fieldsThe inner and outer fields were observed with either ACS/WFC or WFC3/UVIS in only one epoch.All other available observations were taken with the WFPC2. The inner field consists of one single ACS pointing and all the available WFPC2 exposures that overlap it. The outer field consists of three marginally-overlapping UVIS pointings, roughly aligned along the Declination direction, and all the available WFPC2 exposures overlapping them. The complete list of observations for the inner and outer fields are reported in Tables 2 and 3, respectively.The PM-reduction tools presented in Paper I were not designed to be used on WFPC2 data, because the astrometric precision reachable with WFPC2 exposures is generally far less that what we can achieve with later HST optical imagers.The WFPC2 camera was composed of four 800×800 pixel^2 12-bit detectors, with a pixel scale of 99.6 mas pixel^-1 for the three WF chips, and 45.5 mas pixel^-1 for the PC chip.All four chips suffered from ∼16% vignetting.The incident light was split into four beams (one per chip) by a four-faced pyramid mirror, which provided a challenging registration of the observed relative chip positions. Finally, there is no CTE correction available for the WFPC2.Nevertheless, WFPC2 observations represent the only available first epochs for the inner and outer fields, and allowed us to compute PMs with a time baseline of 18.6 and 8 years, respectively. Our PM-reduction tools are scalable, and it is straightforward to include data coming from different instruments/cameras. We treated each WFPC2 chip independently, so to minimize inter-chip transformation errors. We modeled single-exposure expected astrometric errors as a function of the instrumental magnitude using all the available WFPC2 exposures in the two fields. The expected errors are used as a first-guess weight during the PM-fitting procedures (see Sect. 5.2 of Paper I for more details). The computed PMs for the inner and the outer fields are in agreement with with those computed in the central and calibration fields in terms of expected intrinsic values and associated errors.On the other hand, the available dataset does not allow us to adequately study and minimize the impact of systematic effects in the quoted PM errors, as we did for the ACS and UVIS detectors. As a result, we cannot as reliably study the internal kinematics of 47 Tuc stars (which relies the subtraction in quadrature of PM errors) using inner- and outer-field measurements as for the central and calibration field.The inner-field PM catalog contains 2187 objects, 2084 of which passed our high-quality selection criteria. The outer field consists of 648 total objects, of which 579 are identified as high-quality measurements.§ THE ROTATION OF 47 TUC IN THE PLANE OF THE SKYThe top panels of Figure <ref> show the field-of-view of each of the four fields around the center of the cluster. From left to right we have: (1) the inner field; (2) the central field; (3) the calibration field; and (4) the outer field. Concentric circles, in red, give an idea of the radial extension of the data.HST exposures all have different roll angles for different epochs, and axis rotation is one of the six parameters that are solved for when we transform stellar positions as measured on single exposures into the master frame. Because of this, any direct sign of cluster rotation (if present) is absorbed by the linear terms of the coordinate transformations. Since our PMs are relative to the bulk motion of the cluster, any other object in the field that is not a cluster member would have a systematic component in its PM measurement that is equal in size and with the opposite sign of the bulk rotation of the cluster.This systematic PM component is clearly visible in the PM diagrams shown in the middle panel of Fig. <ref>. Cluster stars are in black, while background SMC stars are marked as red crosses. As a reference, a black crosshair in each panel highlights the barycenter of SMC stars as measured in the central field where, because of symmetry, the systematic PM rotation component is minimized. The white crosses near the center of the crosshairs mark the median loci of SMC stars in each of the four fields. The mean PM of SMC stars in the inner field is shifted toward smaller μ_δ values, while that of the calibration and outer fields is shifted toward larger values of μ_δ.It is clear from the figure that SMC stars appear to rotate counter-clockwise with respect to the center of 47 Tuc, which directly translates into a clockwise rotation of 47 Tuc in the plane of the sky.The bottom panels of Fig. <ref> show the color-magnitude diagrams (CMDs) of 47 Tuc stars (black dots) and SMC stars (red crosses) in different magnitude/color combinations for the four fields. To select bona-fide SMC stars we took advantage of their location on both the PM diagrams and the CMDs. SMC stars on the PM diagram are clustered around a well-defined location, clearly distinct from that occupied by 47 Tuc stars. We preliminary selected SMC candidates as all the sources 4 sigma outside the distribution of 47 Tuc stars.This rough selection necessarily includes field stars and a few cluster members. We then iteratively computed 4σ-clipped median valuesandof the barycenter of SMC stars in the PM diagram, and kept only stars within 4 σ within this location. Finally, we excluded by-eye from the SMC selection all those stars that did not lie in the SMC region on the CMDs. We repeated one more time the process of computing the location of the barycenter of SMC stars in the PM diagram, and kept only SMC stars within 4 σ from their median location.The final sample of SMC stars is what we show in red in Fig. <ref>. We have, respectively: 96 SMC stars in the inner field, 164 in the central field, 1495 in the calibration field, and 235 in the outer field.In order to properly measure the apparent rotation of SMC stars, we need to define a reference point (the zero point) on the PM diagram with respect to which to compute their tangential component of the motion ().To do this, we selected all SMC stars within the largest circle that can be fully contained within the central field, and determined a 3σ-clipped estimate of their barycenter location on the PM diagram. This location has coordinates: =-4.797 , =1.244 , and it defines the center of the crosshairs in the middle panels of Fig. <ref>.We zero-pointed the PM of SMC stars to the reference point defined above, and computed their averageat different equally-populated radial intervals with respect to the cluster's center. A finer radial subdivision (about 28 stars per bin) is applied to the central field, where the fastest rotational variation is found (at the cost of larger measurement errors).We then derived a single rotation measurement for the inner field (96 stars), 10 measurements for the calibration field (about 150 stars each), and 2 measurements for the outer field (117 stars each). The computed quantities, as a function of the radial distance, are shown with errorbars in Fig. <ref>, and are listed in Table 4.We adopted the convention of using negativevalues in case of a clockwise rotation in the plane of the sky (North to West). The horizontal bars in Fig. <ref> indicate the size of the radial intervals within which eachvalue is computed. For completeness, we included (in red) the rotation value computed by <cit.>. All these quantities are listed in Table 4.The cluster has a solid-body-like rotation within our central field. A simple least-squares fit of the form =m r gives m=-1.65±0.10arcsec^-1. Then, the rotation profile slowly flatters at larger radii, to reach the highest value of -0.312±0.015at ∼ 390^'' (∼ 652) from the cluster center. The measured rotation slows down to about -0.21in the outermost regions probed by our data.The general shape of the rotation curve of 47 Tuc we have measured on the plane of the sky is qualitatively similar to that obtained by <cit.> using LOS measurements. There is, though, an important difference between the two rotation profiles: the PM-based profile has a rotation peak in the plane of the sky that is about twice as large as that measured with LOS velocities, which we model in Sect. <ref>.To give the reader a better sense of how the cluster is rotating on the plane of the sky, we show in Fig. <ref> the rotation map of the cluster derived with the datasets we analyzed. To obtain the map, we divided the datasets into either four quadrants (for the central field) or equally-populated regions (for the other three fields), and computed the zero-pointed median , components of the motion of SMC stars in each region. These values (with the opposite sign) are shown as vectors departing from the median location of SMC stars within each region. The length of the vectors, in , has been magnified by a factor 750, for clarity. For completeness, we show in red the rotation axis as measured from LOS data (136^∘ North to East, ). § VELOCITY-DISPERSION AND ANISOTROPY PROFILESThe careful reader might have noticed that the distribution of 47 Tuc stars in the PM diagram of the central field is rather circular, while this distribution is more flattened in the inner and calibration field, to become again somewhat more circular in the outer field (middle panels of Fig. <ref>). This is the effect of velocity-dispersion anisotropy.To properly measure the degree of velocity-dispersion anisotropy of the cluster as a function of radius, we proceeded as follows.First, we need to select stars with reliable PMs and similar masses in the four fields.As we mentioned earlier, PMs computed in the inner and outer fields are based on WFPC2 measurements, and might be affected by significant systematic effects. Our datasets do not allow us to adequately study and minimize the impact of systematic effects due to WFPC2 measurements. While these systematic errors are expected to only have second-order effects in the quoted PMs, they can significantly alter the estimated PM errors. Since PM errors are subtracted in quadrature when we want to compute velocity-dispersion profiles, under/overestimating PM errors could lead to incorrect profiles.In what follows, we will include measurements coming from the inner and the outer fields only, for completeness.To select stars of similar mass in the four fields, we limit our selections to the magnitude range 20<m_ F606W<22. The radial and tangential velocity-dispersion profiles were estimated using the same method as in <cit.>, which corrects the observed scatter for the individual PM uncertainties.The top panel of Fig. <ref> shows the radial velocity-dispersion profile σ_ RAD of 47 Tuc as a function of radius. In red we report the values computed in the inner and outer field, for completeness. The horizontal bars illustrate the radial extent over which each point is obtained. The central radial velocity dispersion is about 0.64 , or 13.65at a distance of 4.5 kpc (to be compared to 11.5for evolved stars, ).[We are assuming here that the LOS-based velocity-dispersion-profile value quoted in the Harris catalog, which is the average of available measurements in the literature, refers to the center of the cluster. This might actually not be true, as literature values are computed over different radial ranges. We will see later, in Section <ref>, that our σ-mass dependence modeling predicts a red-giant-branch- (RGB-) mass scaled central velocity-dispersion value closer to about 12.5rather than 11.5 .] The profile reaches a minimum value of about 0.31in the outermost point of the calibration-field (∼825, or about 495^'').The middle panel of the figure shows the tangential velocity-dispersion profile σ_ TAN computed in the same radial intervals as for σ_ RAD. The two vertical lines mark the location of the core radius r_ c=036 (216) and the half-light radius r_ h=317 (1902) (). Table 5 lists the values of the data points shown in Fig. <ref>.The deviation from isotropy (σ_ TAN/σ_ RAD-1) as a function of radius is shown in the bottom panel of Fig. <ref>. The horizontal line at 0 indicates an isotropic system. The center of the cluster is isotropic, with increasing radial anisotropy moving outward. It is worth noting that this trend agrees with what we saw in Paper II for the entire sample of 22 GCs, albeit over a smaller radial range.The degree of anisotropy is significant in the calibration field, i.e. at radial distances between 5^' (300^'') and 8^' (480^'').Both the radial and the tangential velocity-dispersion profiles of 47 Tuc seem to drop in the centermost radial bin. The first and the second points of the dispersion profiles are nonetheless consistent with each other to within less than one sigma. Changing the size of the first few radial bins produced similar trends, all consistent with being flat in the centermost regions within the errorbars. Also note that these profiles are based on relatively faint MS stars, which suffer the most from the highly-crowded conditions of the core of the cluster.The velocity-dispersion profile of 47 Tuc we published in Paper II, based on much brighter, higher S/N RGB stars, does not show any central drop. § DYNAMICAL MODELS The goal of this section is to provide a dynamical model to describe the comprehensive set of available observations, comprising the PM kinematics analyzed above in addition to the classic LOS data and photometry. We will use a family of physically-motivated distribution-function based models (), recently applied to a selected sample of Galactic GCs (, ).These self-consistent models have been specifically constructed to describe quasi-relaxed stellar systems with realistic differential rotation, axisymmetry and pressure anisotropy. The models are defined by four dimensionless parameters (concentration parameter Ψ, rotation strength parameter χ, and the parameters b and c determining the shape of the rotation profile). A full description of the distribution function and of the parameter space is given in <cit.>. Since these models allow only for single-mass component, they do not take into consideration the effects connected with mass segregation. The implications for this assumption are described in Sect. <ref>.We will use the PM-based profiles described in Sect. <ref> (tangential and radial velocity dispersion profiles and rotation profile in the plane of the sky), the LOS-based velocity profiles (velocity dispersion and rotation profiles) reported in <cit.>, and the photometric data from <cit.>. We do not consider in the fit procedure the data obtained from the inner and outer fields, as described in Sect. <ref>.§.§ Fitting procedureWe follow the same fitting procedure outlined in <cit.> and <cit.>, as summarized in the following. The comparison between the differentially-rotating axisymmetric models and the observations requires us to specify four dimensionless parameters and five additional quantities: three physical scales (i.e., the radial scale r_0, the central surface density Σ_0, and the velocity scale v_0), the inclination angle i between the rotation axis and the LOS direction, and the cluster distance.Since the adopted dynamical models are characterized by deviations from isotropy in configuration and velocity space, the choice of the inclination angle plays a fundamental role in the fitting procedure. To exploit such an additional degree of freedom, we initially assume the value i=30^∘, derived by applying Eqn. 8 of <cit.> (linking the average motion along the LOS to the one in the plane of the sky) to our data. We will later explore how the best-fit parameters change with different inclinations (i=25^∘–35^∘). Since PMs are measured in mas yr^-1 in the plane of the sky, a multiplying factor of 4.74 d is needed to convert these values to . We will assume the distance of 47 Tuc of d=4.5 kpc ().The fitting procedure is twofold. First, we determine the dimensionless parameters needed to reproduce the observed value of V/σ and the observed position of the rotation peak (for further details see Sect. 3.1 and 3.5 of ). Then we calculate the physical scales through χ^2 minimization for all the kinematic profiles (three dispersion profiles, two rotation curves) to obtain the radial scale r_0 and the velocity scale v_0, and then we fit the surface-density profile to obtain the central surface density Σ_0. This provides all the constrains needed to determine the best-fit dynamical model.We wish to emphasize that, during the fitting procedure, the projection of the self-consistent dynamical models is performed by sampling from the relevant distribution function a discrete set of N=2 048 000 particles and then by performing a rotation of such a discrete system to match the relevant inclination angle.The theoretical kinematic and photometric profiles are then calculated by following the same procedures applied for the construction of the observational profiles (i.e., by means of circular annuli in the projection plane). Any emerging constraint on the morphology and degree of anisotropy of the stellar system should be, therefore, considered as resulting properties of the best-fit model, which has been selected exclusively on the basis of the (spherically-averaged) kinematic and photometric information.As to the morphological characterization (see Sect. <ref>), the projected isodensity contours are calculated on the basis of the nonspherical projected number density distribution. The relevant ellipticity profile is then constructed by considering the ratio of the principal axes of approximately 60 isodensity contours, corresponding to selected values of the normalized projected number density in the range [0.9, 10^-3]; smooth profiles are then obtained by performing an average on subsets made of 10–20 individual ellipticity values. For completeness, we also explored the distance value of 4.15 kpc from <cit.> in our fitting procedures. The best-fit model based on the 4.15 kpc value provides comparable results to those based on the 4.5 kpc distance value. However, the Paper III-based model offers a better fit to the LOS velocity-dispersion profile, but a worse fit to the surface-brightness profile.§.§.§ Correction for energy-equipartition effectsSince PM measurements sample different kinematic tracers than LOS measurements –namely stars with lower mass than bright red giant stars– some caution is needed when applying a one-component dynamical model simultaneously to the three-dimensional kinematics. In particular, our PM-based velocity-dispersion profiles are constructed using MS stars within the magnitude range 20<m_ F606W<22. This implies a typical stellar mass between 0.62 and 0.47 M_⊙ (using aisochrone with [Fe/H]=-0.5, [α/Fe]=0.2, d=4.5 kpc and E(B-V)=0.04). The LOS measurements have instead a typical stellar mass of 0.83 M_⊙, since they sample only bright giant stars.Since GCs reach a state of partial energy equipartition (see e.g., , ), their kinematics are expected to show a mass dependence, with lower-mass stars having gradually higher velocity dispersions. <cit.> showed that the mass-dependence of kinematics depends on the relaxation condition of the cluster (see their Eqn. 6). Therefore, by knowing the relaxation state of a cluster it is possible to predict the shape of the velocity dispersion as a function of mass σ(m), and then rescale the PM-based dispersion profiles according to the calculated factor.Given n_ rel=T_ age/T_r_ c=169.6 (number of core relaxation times the cluster has experienced), the mass dependence of the velocity dispersion from Eqn. 3 of <cit.> isσ(m)=σ_0exp(-1/2m/m_ eq),with m_ eq=1.60 M_⊙ (from Eqn. 6 of ) and σ_0 a normalization factor. Given a mass of 0.83 M_⊙ for the LOS velocities and a median value of 0.54 M_⊙ for the stars used for the PM-based velocity-dispersion profiles (estimated via the adopted isochrone), the velocity dispersions of the two different mass-tracers are related by σ(0.83)/σ(0.54)=0.913.We rescale the tangential and radial PM dispersion profiles of the model by dividing them by this correction factor. We do not rescale the PM rotation profile, since we do not observe any signature of dependence of mass.Note that the central velocity-dispersion estimate based on the PM of evolved stars (Paper II) is 0.573±0.005 , which translates into 12.2±0.1at a distance of 4.5 kpc. The adopted rescaling factor of 0.913 implies a central velocity-dispersion for RGB stars of 12.46 , in full agreement with the value reported in Paper II. §.§ Results§.§.§ Projected propertiesThe results of the fit are reported in Figs. <ref>–<ref>. The azure, green and red lines in each plot correspond to the fit with inclination angles i=25^∘, 30^∘ and 35^∘, respectively. The structural parameters of the three different models do not differ significantly, however, the model with i=30^∘ gives a better fit. We will consider this model as our fiducial best-fit model and we report the best fit parameters and structural properties in Tables 6 and 7.Figure <ref> shows the rotation profile based in the LOS direction (right) and in the plane of the sky (left). The additional yellow lines represent the best-fit model derived by <cit.> assuming an inclination of 45^∘ and without accounting for the rotation in the plane of the sky. The figure clearly shows that our new best-fit model, assuming the new inclination value of i≃30^∘, is able to reproduce the three-dimensional rotational structure of 47 Tuc, whereas a higher inclination angle would fail in describing the rotation in the plane of the sky.Our model predicts an anisotropy profile that is in excellent agreement with the observations (Fig. <ref>), characterized by isotropy in the central cluster regions, and mild radial anisotropy in the intermediate regions. Moreover, we are able to simultaneously reproduce all the dispersion profiles (Fig. <ref>) and the surface-brightness profile (Fig. <ref>), employed for the fit.Since the adopted dynamical equilibria are axisymmetric, we can calculate the corresponding projected ellipticity profile, which we illustrate in Fig. <ref>, together with the ellipticity data currently available for 47 Tuc (from ). We recall that the ellipticity profile associated with the best-fit self-consistent model is a structural property completely determined by the values of the dimensionless parameters and physical scales identified during the model selection procedure. As already appreciated in our previous analysis (see ), we find a very good agreement with the available observational profile. Such a result allows us to confirm, with increased confidence, that the physical origin of the observed flattening of 47 Tuc is indeed the presence of internal rotation. We emphasize that such an agreement is nontrivial, especially in consideration of the reduced value of the inclination angle (in our previous study we adopted i=45^∘, while now we determined i=30^∘ to be more appropriate). The appreciable morphological consistency recovered, once again, in the current analysis should be interpreted as a manifestation of the more significant intrinsic rotation (as recovered from the additional constraints posed by new PM datasets), which compensates the effects of a less favorable LOS direction.For completeness, in Fig. <ref> we also show the projected isodensity contours of the discrete realization of the best-fit axisymmetric model as a function of the tridimensional radius in cylindrical coordinates. §.§.§ Intrinsic propertiesOur three-dimensional model allows us to explore the intrinsic kinematic and morphological structure of 47 Tuc.The derived intrinsic V/σ profile, measured in the equatorial plane perpendicular to the rotation axis, is characterized by a peak value of ∼0.9 reached at around 2 half-light radii (Fig. <ref>).This confirms that the internal rotation of 47 Tuc is higher than what has been reported in previous studies (e.g., the LOS-based value V/σ=0.25, ). Correspondingly, we can also characterize the three-dimensional structure of the cluster by means of the intrinsic ellipticity profile, as measured in the meridional plane of system (defined by the rotation axis and any of the principal axes on the equatorial plane) and expressed as a function of the semimajor axis (Fig. <ref>). The behavior of this profile in the central regions is approximately linear, as shaped by the solid-body like behavior of the rotation curve in that portion of the system. We emphasize that the radial location of the peak of the ellipticity profile does not correspond to the location of the peak of the intrinsic V/σ profile, as appropriate in the presence of a nontrivial coupling between the angular momentum and mass distribution within the system. We recall that the intrinsic morphological and kinematic structure of an axisymmetric (nonstratified) rotating equilibrium is more complex than a simple characterization expressed by means of unidimensional radial profiles, therefore, we include also a representation of the bidimensional maps depicting the structure of the density and angular momentum distribution, expressed in terms of the contours of the normalized isodensity and isovelocity surfaces evaluated on the meridional plane of the three-dimensional model. In the maps illustrated in Fig. <ref> the radial range on the equatorial plane is approximately equal to 2r_ h. From the density map (left panel) it may be appreciated that the maximum degree of flattening is reached in the intermediate regions of the system and that the shape of the isodensity contours becomes more complex as the density decreases (i.e., the outer contours are not monotonically increasing functions of the polar angle, hence the dimples on the rotation axis). From the kinematic map (right panel) it may be noted that the velocity field is not cylindrically-stratified, presenting an absolute maximum at about 300^''; the location of the peak of the intrinsic V/σ profile is determined by the interplay between the three-dimensional structure of such a velocity field and that of the trace of the velocity dispersion tensor. Equivalent maps may be calculated also for the meridional sections of the equipotential and isobaric surfaces.Lastly, we wish to emphasize that the current presence of such an appreciable degree of internal rotation in 47 Tuc carries important implications about the initial amount of angular momentum of the cluster. Indeed, many facets of the role played by internal rotation during the long-term evolution of collisional stellar systems must still be explored, but there is confirmed evidence that two-body relaxation and mass loss determine transport and loss of angular momentum in clusters (see, e.g., ), therefore any measurement of the presence of internal rotation at the present day should be considered as a lower limit of the initial angular momentum content. The emerging kinematic complexity of 47 Tuc (and a progressively increasing number of Galactic GCs) therefore offers an essential ingredient towards a more complete and fundamental understanding of the formation and early dynamical evolution of GCs.§.§.§ Some remarks on the modeling strategy We recognize that our modeling strategy has a number of limitations, especially regarding the description of the projected structural and kinematic profiles in the outer regions of the cluster. With particular reference to the behavior of the surface brightness profile and LOS velocity dispersion profile in the very outer parts (at radii > 1500^''), we interpret their discrepancies as due to the effects of the tidal field of the Milky Way on the structure and internal kinematics of the system.It is well known that, during the course of their dynamical evolution (as driven by two-body relaxation), collisional stellar systems tend to expand until they fill their Roche lobe, and then progressively start to lose mass. Such a process significantly affects the phase-space structure of their periphery, both by increasing the complexity of the kinematics of the stars that are energetically bound (e.g., ), and by determining the existence of a population of energetically unbound stars that are nonetheless spatially confined within the cluster (“potential escapers”, see ). In terms of projected observables, these evolutionary effects may manifest themselves in the form of surface-brightness profiles extending beyond the cut-off radius predicted by a simple spherical King model (i.e., the so-called “extra-tidal” structures identified in many globular clusters in Local Group galaxies, see e.g. ) and velocity dispersion profiles characterized by an untruncated, flattened behavior (e.g., see the studies of M15 and M92 byand NGC 5694 by ).In term of intrinsic properties, tidally-perturbed systems are also characterized by a flavor and degree of anisotropy that strongly depends on the strength of the tidal environment in which they have evolved (seeand additional remarks at the end of next paragraph).In this respect, we emphasize that simple, physically-based dynamical models of the kind adopted in our study, as defined by a quasi-Maxwellian distribution function, suitably modified near the tidal boundary and truncated above it, successfully capture the phase-space properties of the bulk of the cluster members by relying on a truncation prescription that heuristically mimic the effects of the tidal field (e.g.and, more recently, ).Of course, the choice of such a one-dimensional (energy) truncation prescription in the definition of the distribution function strongly affects the structural and kinematic properties of the resulting configurations (see ), but, by definition, they can not offer a realistic description of the tidal field (see ) or account for the existence of energetically unbound members (see ).We wish to emphasize that, despite their inherent simplicity, the self-consistent rotating models adopted here have nonetheless two advantageous properties. First, their phase-space truncation reduces, in the non-rotating limit, to the smooth <cit.> prescription, which is now often considered more successful in describing the outer regions of clusters than the traditional <cit.> model (see ), which demands only continuity in phase space. Second, their velocity dispersion tensor is characterized by isotropy in the central region, weak radial anisotropy in the intermediate regions, and tangential anisotropy in the outer parts. Such a behavior of the pressure tensor was not assigned a priori in the definition of the models, but it results from the requirement of self-consistency, positivity of the distribution function, and energy truncation in the presence of angular momentum as a second integral of the motion. Isotropy or mild tangentially-biased pressure anisotropy in the outer parts of a star cluster has been shown to be a natural result of the dynamical evolution of a collisional stellar system within an external tidal field, which induces a preferential loss of stars on radial orbits (e.g., see ).The other discrepancy, namely a model underestimate of the surface brightness in the very central region and in an intermediate region in the range (100^''–200^''), can instead be considered as a result of a χ^2-minimization employed by the fitting procedure that gives more weight to the three-dimensional kinematic data (see Sect. <ref>). As a result, the value of the best-fit half-light radius we obtain (r_h =240.7±10.9 arcsec) is larger than what is reported in previous literature works (e.g., r_h=190.2 arcsec,).We wish to emphasize that our study focuses on understanding the rotational structure of 47 Tuc. The kinematic profiles (the two rotation curves and the three velocity dispersion profiles) show a general agreement within the observational error bars. However our model underestimates the values of the rotation profiles in the intermediate region and overestimates the value of velocity dispersion along the LOS. These discrepancies could be interpreted in view of the intrinsic limitations of the choice of the distribution function defining the dynamical models under consideration. Indeed, alternative modeling strategies may offer additional degrees of freedom in the parametrization of the phase space, but we wish to note that differentially rotating equilibria are relatively rare, especially for globular cluster studies (seefor an application of the rotating models proposed by Wilson 1975 to the study of ω Cen). An alternative approach for the construction of phase-space equilibria is based on the use of actions instead of integrals of the motion; an example of this line of attack has been recently proposed by <cit.>.Finally, we stress that the velocity anisotropy profile and the ellipticity profile are not directly used in the fitting procedure and the respective figures show a comparison between the available data and the model prediction. This result may also be interpreted as a reassuring “a posteriori” validation of our choice of adopting axial symmetry and reduces the scope of exploring more general symmetries (e.g., triaxial configurations). A complementary path would consist in employing empirical models that are optimally designed to describe the data (e.g., Jeans models or orbit-based Schwarzschild models). Despite providing more freedom in the description of the data, the major drawback is they do offer a very limited connection to the underlying physical picture of the stellar system under consideration. However, for studies aimed at the understanding of specific physical ingredients (e.g., the possible presence of a intermediate-mass black hole in 47 Tuc, the exact mass-to-light ratio of the stellar population, or the amount of energy equipartition), we would certainly benefit from complementary and alternative modeling approaches from the one employed here. In addition, thanks to recent progress on the computational side, it has recently been proved that realistic N-body models, following the entire dynamical evolution of the system, are finally within reach, at least for selected Galactic globular clusters (e.g., see the model of M4 presented byor the DRAGON series by ), although it should be kept in mind that the performance cost of the exploration of a wide range of initial conditions is still far from negligible, especially for massive clusters such as 47 Tuc.In order to understand the limitations intrinsic to our modeling, we report in Table 8 a compilation of the total mass and the mass-to-light ratio for 47 Tuc from the literature obtained using a variety of dynamical modeling techniques. These include: i) distribution-function-based models (spherical and isotropic models, , spherical and anisotropic models, , and axisymmetric rotating models, ), ii) isotropic spherical Jeans models (), and iii) N-body and spherical Monte Carlo models (). Note that all these previous works, with the exception of <cit.>, do not take into consideration internal rotation, therefore they are intrinsically unable to reproduce our state-of-the-art three-dimensional kinematic measurements.From Table 8, it is evident that the estimates in the literature depend on the particular modeling techniques employed. The values of the total mass of 47 Tuc ranges between 5.5–10.7×10^5 M_⊙ with our model giving an intermediate value of 8.4×10^5 M_⊙, while the mass-to-light ratio ranges from 1.2–2 M_⊙/L_⊙, with our model giving 1.98 M_⊙/L_⊙, consistent with the upper limit. Given the agreement with these previous works, we are confident that our model is able to reproduce the global properties of 47 Tuc, despite some of the discrepancies reported above. § CONCLUSIONSWe have derived for the first time the plane-of-the-sky rotation of the GC 47 Tuc from the core out to 13^' (or 3–4 half-light radii, or ∼30% of the tidal radius), together with detailed radial and tangential velocity-dispersion profiles. The cluster's plane-of-the-sky kinematic information is coupled to literature LOS measurements of the same quantities and surface-brightness profiles, and simultaneously fit with state-of-the-art dynamical models.From the application of the rotating dynamical model we conclude that: * PMs are critically necessary if we hope to constrain the intrinsic structure of GCs. With the application of our dynamical model to both kinematics and photometry we have obtained a full three-dimensional description of 47 Tuc;* The higher rotation in the plane of sky with respect to the one along the LOS implies an inclination angle of the rotation axis of ≃30^∘;* Our best-fit dynamical model predicts an intrinsic V/σ profile that reaches values of ≃0.9 around two half-light radii from the cluster's center;* On the basis of our global dynamical analysis, we confirm that the observed flattening of the cluster is most likely due to its appreciable internal rotation, as the projected ellipticity profile determined by our model is in good agreement with the ellipticity measurements currently available for 47 Tuc. The three-dimensional morphological structure implied by our axisymmetric model is complex, and may be characterized by a non-monotonic intrinsic ellipticity profile, which reaches values of about 0.45 towards the intermediate regions of the cluster.This comprehensive dynamical investigation, based on new PM measurements of unprecedented accuracy, has allowed us to unveil a new degree of kinematic complexity in 47 Tuc. Such a superb characterization of the three-dimensional velocity space offers a novel and invaluable ground for the study of numerous aspect of collisional gravitational dynamics, from the long-forgotten role played by angular momentum to the tantalizing opportunity of finally exploring the phase space properties of its stellar populations, with the goal of providing a key contribution towards a more realistic dynamical paradigm for this class of stellar systemsAB acknowledges support from STScI grant AR-12845 (PI: Bellini). 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L. 2012, , 539, A65 [Zocchi et al.(2016)]2016MNRAS.462..696Z Zocchi, A., Gieles, M., Hénault-Brunet, V., & Varri, A. L. 2016, , 462, 696 | http://arxiv.org/abs/1706.08974v1 | {
"authors": [
"A. Bellini",
"P. Bianchini",
"A. L. Varri",
"J. Anderson",
"G. Piotto",
"R. P. van der Marel",
"E. Vesperini",
"L. L. Watkins"
],
"categories": [
"astro-ph.GA",
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"primary_category": "astro-ph.GA",
"published": "20170627180003",
"title": "Hubble Space Telescope Proper Motion (HSTPROMO) Catalogs of Galactic Globular Clusters. V. The rapid rotation of 47 Tuc traced and modeled in three dimensions"
} |
addressref=aff1,corref=aff1, [email protected]]APAlan Pryor Jr. addressref=aff2,[email protected] ]COColin Ophus addressref=aff1,[email protected] ]JMJianwei Miao [id=aff1] Department of Physics, University of California, Los Angeles,Knudsen Hall, 475 Portola Plaza,90095, Los Angeles, USA[id=aff2] National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory,1 Cyclotron Road,Berkeley, USA Alan Pryor [email protected] Department of Physics, University of California, Los Angeles, CA, USA Colin Ophus [email protected] NCEM, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA, USAJianwei Miao [email protected] Department of Physics, University of California, Los Angeles, CA, USA Simulation of atomic resolution image formation in scanning transmission electron microscopycan require significant computation times using traditional methods. A recently developed method, termed plane-wave reciprocal-space interpolated scattering matrix (PRISM), demonstrates potential for significant acceleration of such simulations with negligible loss of accuracy. Here we present a software package called Prismatic for parallelized simulation of image formation in scanning transmission electron microscopy (STEM) using both the PRISM and multislice methods. By distributing the workload between multiple CUDA-enabled GPUs and multicore processors, accelerations as high as 1000x for PRISM and 15x for multislice are achieved relative to traditional multislice implementations using a single 4-GPU machine. We demonstrate a potentially important application of Prismatic, using it to compute images for atomic electron tomography at sufficient speeds to include in the reconstruction pipeline. Prismatic is freely available both as an open-source CUDA/C++ package with a graphical user interface and as a Python package, PyPrismatic. Scanning Transmission Electron Microscopy PRISM Multislice GPU CUDA Electron Scattering Imaging Simulation High Performance Computing § INTRODUCTION Scanning transmission electron microscopy (STEM) has had a major impact on materials science <cit.>, especially for atomic-resolution imaging since the widespread adoption of hardware aberration correction <cit.>. Many large-scale STEM experimental techniques are routinely validated using imaging or diffraction simulations. Examples include electron ptychography <cit.>, 3D atomic reconstructions using dynamical scattering <cit.>, high precision surface atom position measurements on catalytic particles <cit.>, de-noising routines <cit.>, phase contrast imaging with phase plates <cit.>, new dynamical atomic contrast models <cit.>, atomic electron tomography (AET) <cit.>, and many others.The most commonly employed simulation algorithm for STEM simulation is the multislice algorithm introduced by Cowlie and Moodie <cit.>. This method consists of two main steps. The first is calculation of the projected potentials from all atoms into a series of 2D slices. Second, the electron wave is initialized and propagated through the sample. The multislice method is straightforward to implement and is quite efficient for plane-wave or single-probe diffraction simulations <cit.>.A large number of electron microscopy simulation codes are available, summarized in Table <ref>. Most of these codes use the multislice method, and many have implemented parallel processing algorithms for both central processing units (CPUs) and graphics processing units (GPUs).Recently some authors have begun using hybrid CPU+GPU codes for multislice simulation <cit.>. Multislice simulation relies heavily on the the fast Fourier transform (FFT) which can be computed using heavily optimized packages for both CPUs <cit.> and GPUs <cit.>. The other primary computational requirement of multislice calculations is large element-wise matrix arithmetic, which GPUs are very well-suited to perform <cit.>. Parallelization is important because STEM experiments may record full probe images or integrated values from thousands or even millions of probe positions <cit.>. Performing STEM simulations on the same scale as these experiments is very challenging, because in the conventional multislice algorithm the propagation of each STEM probe through the sample is computed separately. Furthermore, if additional simulation parameters are explored the number of required simulations can become even larger, requiring very large computation times even using a modern, paralellized implementation. To address this issue, we introduced a new algorithm called PRISM which offers a substantial speed increase for STEM image simulations <cit.>. In this manuscript, we introduce a highly-optimized multi-GPU simulation code that can perform both multislice and PRISM simulations of extremely large structures called Prismatic. We will briefly describe the multislice and PRISM algorithms, and describe the implementation details for our parallelized CPU and CPU+GPU codes. We perform timing benchmarks to compare both algorithms under a variety of conditions. Finally, we demonstrate the utility of our new code with typical use cases and compare with the popular package computem <cit.>. Prismatic includes a graphical user interface (GUI) and uses the cross-platform build system CMake <cit.>. All of the source code is freely available.Throughout this manuscript, we use the NVIDIA convention of referring to the CPU and GPU(s) as the host and device(s), respectively. § METHODS§.§ Description of Algorithms A flow chart of the steps performed in Prismatic are given in Fig. <ref>. Both multislice and PRISM share the same initial steps, where the sample is divided into slices which are used to compute the projected potential from the atomic scattering factors give in <cit.>. This step is shown schematically in Figs. <ref>a and b, and is implemented by using a precomputed lookup table for each atom type <cit.>. Figs.<ref>c-e show the steps in a multislice STEM simulation. First the complex electron wave Ψ representing the initial converged probe is defined, typically as an Airy disk function shown in Fig. <ref>c. This probe is positioned at the desired location on the sample surface in realspace, as in Fig. <ref>d. Next, this probe is propagated through the sample's potential slices defined in Fig. <ref>b. This propagation is achieved by alternating two steps.The first step is a transmission through a given potential slice V_p^2D over the realspace coordinates r⃗ψ_p+1(r⃗) =ψ_p(r⃗) exp[σ V_p^2D(r⃗)],where σ is the beam-sample interaction constant. Next, the electron wave is propagated over the distance t to the next sample potential slice, which is done in Fourier space over the Fourier coordinates q⃗Ψ_p+1(q⃗) =Ψ_p(q⃗) exp(- πλ |q⃗|^2 t),where λ is the electron wavelength. These steps are alternated until the electron probe has been propagated through the entire sample. Next, the simulated output is computed, which is typically a subset of the probe's intensity summed in Fourier space as shown in Fig. <ref>e. The steps given inFigs. <ref>c-e are repeated for the desired probe positions, typically a 2D grid. The simulation result can be a single virtual detector, an array of annular ring virtual detectors or the entire probe diffraction pattern for each probe location, giving a 2D, 3D or 4D output respectively. For more details on the multislice method we refer readers to Kirkland <cit.>.The PRISM simulation method for STEM images is outlined in Figs. <ref>f-k. This method exploits the fact that an electron scattering simulation can be decomposed into an orthogonal basis set, as in the Bloch wave method <cit.>. If we compute the electron scattering for a set of plane waves that forms a complete basis, these waves can each be multiplied by a complex scalar value and summed to give a desired electron probe. A detailed description of the PRISM algorithm is given in <cit.>.The first step of PRISM is to compute the sample potential slices as in Figs.<ref>a-b. Next, a maximum input probe semi-angle and an interpolation factor f is defined for the simulation. Fig. <ref>g shows how these two variables specify the plane wave calculations required for PRISM, where every f^th plane wave in both spatial dimensions inside the maximum scattering angle is required. Each of these plane waves must be propagated through the sample using the mutlislice method given above, shown in Fig. <ref>h. Once all of these plane waves have been propagated through the sample, together they form the desired basis set we refer to as the compact S-matrix. Next we define the location of all desired STEM probes. For each probe, a subset of all plane waves is cut out around the maximum value of the input STEM probe. The size length of the subset regions is d/f, where d is the simulation cell length. The probe coefficients for all plane waves are complex values that define the center position of the STEM probe, and coherent wave aberrations such as defocus or spherical aberration. Each STEM probe is computed by multiplying each plane wave subset by the appropriate coefficient and summing all wave subsets. This is equivalent to using Fourier interpolation to approximate the electron probe wavefunction. As long as the subset region is large enough to encompass the vast majority of the probe intensity, the error in this approximation will be negligible <cit.>. Finally, the output signal is computed for all probes as above, giving a 2D, 3D or 4D output array. As will be shown below, STEM simulations using the PRISM method can be significantly faster than using the multislice method.§ IMPLEMENTATION DETAILS§.§ Computational Model Wherever possible, parallelizable calculations in Prismatic are divided into individual tasks and performed using a pool of CPU and GPU worker threads that asynchronously consume the work on the host or the device, respectively. We refer to a GPU worker thread as a host thread that manages work dispatched to a single device context. Whenever one of these worker threads is available, it queries a mutex-synchronized dispatcher that returns a unique work ID or range of IDs. The corresponding work is then consumed, and the dispatcher requeried until no more work remains. This computational model, depicted visually in Fig. <ref>, provides maximal load balancing at essentially no cost, as workers are free to independently obtain work as often as they become available. Therefore, machines with faster CPUs may observe more work being performed on the host, and if multiple GPU models are installed in the same system their relative performance is irrelevant to the efficiency of work dispatch. The GPU workers complete most types of tasks used by Prismatic well over an order of magnitude faster than the CPU on modern hardware, and if a CPU worker is dispatched one of the last pieces of work then the entire program may be forced to unnecessarily wait on the slower worker to complete. Therefore, an adjustable early stopping mechanism is provided for the CPU workers.GPU calculations in Prismatic are performed using a fully asynchronous memory transfer and computational model driven by CUDA streams. By default, kernel launches and calls to the CUDA runtime API for transferring memory occur on what is known as the default stream and subsequently execute in order. This serialization does not fully utilize the hardware, as it is possible to simultaneously perform a number of operations such as memory transfer from the host to the device, memory transfer from the device to the host, and kernel execution concurrently. This level of concurrency can be achieved using CUDA streams. Each CUDA stream represents an independent queue of tasks using a single device that execute internally in exact order, but that can be scheduled to run concurrently irrespective of other streams if certain conditions are met. This streaming model combined with the multithreaded work dispatch approach described previously allow for concurrent two-way host/device memory transfers and simultaneous data processing. A snapshot of the output produced by the NVIDA Visual Profiler for a single device context during a streaming multislice simulation similar to those described later in this work verifies that Prismatic is indeed capable of such concurrency (Fig. <ref>). To achieve maximum overlap of work, each CUDA-enabled routine in Prismatic begins with an initialization phase where relevant data on the host-side is copied into page-locked (also called “pinned”) memory, which provides faster transfer times to the device and is necessary for asynchronous memory copying as the system can bypass internal staging steps that would be necessary for pageable memory <cit.>. CUDA streams and data buffers are then allocated on each device and copied to asynchronously. Read-only memory is allocated once per device, and read/write memory is allocated once per stream. It is important to perform all memory allocations initially, as any later calls to cudaMalloc will implicitly force synchronization of the streams. Once the initialization phase is over, a host thread is spawned for each unique CUDA stream and begins to consume work. §.§ Calculation of the Projected PotentialsBoth PRISM and multislice require dividing the atomic coordinates into thin slices and computing the projected potential for each. The calculation details are described by Kirkland and require evaluation of modified Bessel functions of the second kind, which are computationally expensive <cit.>. This barrier is overcome by precomputing the result for each unique atomic species and assembling a lookup table. Each projected potential is calculated on a supersampled grid, integrated, and cached. The sample volume is then divided into slices, and the projected potential for each slice is computed on separate CPU threads using the cached potentials. In principle this step could be GPU accelerated, but even for a large sample with several hundred thousand atoms the computation time is on the order of seconds and is considered negligible. §.§ PRISM Probe Simulations Following calculation of the projected potential, the next step of PRISM is to compute the compact S-matrix. Each plane wave component is repeatedly transmitted and propagated through each slice of the potential until it has passed through the entire sample, at which point the complex-valued output wave is stored in real space to form a single layer of the compact S-matrix. This step of PRISM is highly analogous to multislice except whereas multislice requires propagating/transmitting the entire probe simultaneously, in PRISM each initial Fourier component is propagated/transmitted individually. The advantage is that in PRISM this calculation must only be performed once per Fourier component for the entire calculation, while in multislice it must be repeated entirely at every probe position. Thus, in many sample geometries the PRISM algorithm can significantly out-perform multislice despite the overhead of the S-matrix calculation <cit.>.The propagation step requires a convolution operation which can be performed efficiently through use of the FFT. Our implementation uses the popular FFTW and cuFFT libraries for the CPU and GPU implementations, respectively <cit.>. Both of these libraries support batch FFTs, whereby multiple Fourier transforms of the same size can be computed simultaneously. This allows for reuse of intermediate twiddle factors, resulting in a faster overall computation than performing individual transforms one-by-one at the expense of requiring a larger block of memory to hold the multiple arrays. Prismatic uses this batch FFT method with both PRISM and multislice, and thus each worker thread will actually propagate a number of plane waves or probes simultaneously. This number, called the batch_size, may be tuned by the user to potentially enhance performance at the cost of using additional memory, but sensible defaults are provided.In the final step of PRISM, a 2D output is produced for each probe position by applying coefficients, one for each plane wave, to the elements of the compact S-matrix and summing along the dimension corresponding to the different plane waves. These coefficients correspond to Fourier phase shifts that scale and translate each plane wave to the relevant location on the sample in real space. The phase coefficients, which are different for each plane wave but constant for a given probe position, are precomputed and stored in global memory. Each threadblock on the device first reads the coefficients from global memory into shared memory, where they can be reused throughout the lifetime of the threadblock. Components of the compact S-matrix for a given output wave position are then read from global memory, multiplied by the relevant coefficient, and stored in fast shared memory, where the remaining summation is performed. This parallel sum-reduction is performed using a number of well-established optimization techniques including reading multiple global values per thread, loop unrolling through template specialization, and foregoing of synchronization primitives when the calculation has been reduced to the single-warp level. Once the realspace exit wave has been computed, the modulus squared of its FFT yields the calculation result at the detector plane. §.§ Multislice Probe Simulations The implementation of multislice is fairly straightforward. The initial probe is translated to the probe position of interest, and then is alternately transmitted and propagated through the sample. In practice this is accomplished by alternating forward and inverse Fourier transforms with an element-wise complex multiplication in between each with either the transmission or propagation functions. Upon propagation through the entire sample, the squared intensity of the Fourier transform of the exit wave provides the final result of the calculation at the detector plane for that probe position. For additional speed, the FFTs of many probes are computed simultaneously in batch mode. Thus in practice batch_size probes are transmitted, followed by a batch FFT, then propagated, followed by a batch inverse FFT, etc. §.§ Streaming Data for Very Large Simulations The preferred way to perform PRISM and multislice simulations is to transfer large data structures such as the projected potential array or the compact S-matrix to each GPU only once, where they can then be read from repeatedly over the course of the calculation. However, this requires that the arrays fit into limited GPU memory. For simulations that are too large, we have implemented an asynchronous streaming version of both PRISM and multislice. Instead of allocating and transferring a single read-only copy of large arrays, buffers are allocated to each stream large enough to hold only the relevant subset of the data for the current step in the calculation, and the job itself triggers asynchronous streaming of the data it requires for the next step. For example, in the streaming implementation of multislice, each stream possesses a buffer to hold a single slice of the potential array, and after transmission through that slice the transfer of the next slice is requested. The use of asynchronous memory copies and CUDA streams permits the partial hiding of memory transfer latencies behind computation (Fig. <ref>). Periodically, an individual stream must wait on data transfer before it can continue, but if another stream is ready to perform work the device is effectively kept busy. Doing so is critical for performance, as the amount of time needed to transfer data can become significant relative to the total calculation. By default, Prismatic uses an automatic setting to determine whether to use the single-transfer or streaming memory model whereby the input parameters are used to estimate how much memory will be consumed on the device, and if this estimate is too large compared with the available device memory then streaming mode is used. This estimation is conservative and is intended for convenience, but users can also forcibly set either memory mode.§.§ Launch Configuration All CUDA kernels are accompanied by a launch configuration that determines how the calculation will be carried out <cit.>. The launch configuration specifies the amount of shared memory needed, on which CUDA stream to execute the computation, and defines a 3D grid of threadblocks, each of which contains a 3D arrangement of CUDA threads. It is this arrangement of threads and threadblocks that must be managed in software to perform the overall calculation. The choice of launch configuration can have a significant impact on the overall performance of a CUDA application as certain GPU resources, such as shared memory, are limited. If too many resources are consumed by individual threadblocks, the total number of blocks that run concurrently can be negatively affected, reducing overall concurrency. This complexity of CUDA cannot be overlooked in a performance-critical application, and we found that the speed difference in a suboptimal and well-tuned launch configuration could be as much as 2-3x.In the reduction step of PRISM, there are several competing factors that must be considered when choosing a launch configuration. The first of these is the threadblock size. The compact S-matrix is arranged in memory such that the fastest changing dimension, considered to be the x-axis, lies along the direction of the different plane waves. Therefore to maximize memory coalescence, threadblocks are chosen to be as large as possible in the x-direction. Usually the result will be threadblocks that are effectively 1D, with BlockSize_y and BlockSize_z equal to one; however in cases where very few plane waves need to be computed the blocks may be extended in y and z to prevent underutilization of the device. To perform the reduction, two arrays of shared memory are used. The first is dynamically sized and contains as many elements as there are plane waves. This array is used to cache the phase shift coefficients to prevent unnecessary reads from global memory, which are slow. The second array has BlockSize_x*BlockSize_y*BlockSize_z elements and is where the actual reduction is performed. Each block of threads steps through the array of phase shifts once and reads them into shared memory. Then the block contiguously steps through the elements of the compact S-matrix for a different exit-wave position at each y and z index, reading values from global memory, multiplying them by the associated coefficient, and accumulating them in the second shared memory array. Once all of the plane waves have been accessed, the remaining reduction occurs quickly as all remaining operations occur in fast shared memory. Each block of threads will repeat this process for many exit-wave positions which allows efficient reuse of the phase coefficients from shared memory. The parallel reduction is performed by repeatedly splitting each array in half and adding one half to the other until only one value remains. Consequently, if the launch configuration specifies too many threads along the x-direction, then many of them will become idle as the reduction proceeds, which wastes work. Conversely, choosing BlockSize_x to be too small is problematic for shared memory usage, as the amount of shared memory per block for the phase coefficients is constant regardless of the block size. In this case, the amount of shared memory available will rapidly become the limiting factor to the achievable occupancy. A suitably balanced block size produces the best results.The second critical component of the launch configuration is the number of blocks to launch. Each block globally reads the phase coefficients once and then reuses them, which favors using fewer blocks and having each compute more exit-wave positions. However, if too few blocks are launched the device may not reach full occupancy. The theoretically optimal solution would be to launch the minimal amount of blocks needed to saturate the device and no more. Considering these many factors, Prismatic uses the following heuristic to choose a good launch configuration. At runtime, the properties of the available devices are queried, which includes the maximum number of threads per threadblock, the total amount of shared memory, and the total number of streaming multiprocessors. BlockSize_x is chosen to be either the largest power of two smaller than the number of plane waves or the maximum number of threads per block, whichever is smaller. The total number of threadblocks that can run concurrently on a single streaming multiprocessor is then estimated using BlockSize_x, the limiting number of threads per block, and the limiting number of threadblocks per streaming multiprocessor.The total number of threadblocks across the entire device is then estimated as this number times the total number of streaming multiprocessors, and then the grid dimensions of the launch configuration are set to create three times this many blocks, where the factor of three is a fudge factor that we found produces better results.§ BENCHMARKS§.§ Algorithm Comparison A total of four primary algorithms are implemented Prismatic, as there are optimized CPU and GPU implementations of both PRISM and multislice simulation. To visualize the performance of the different algorithms, we performed a number of benchmarking simulations spanning a range of sample thicknesses, sizes, and with varying degrees of sampling. Using the average density of amorphous carbon, an atomic model corresponding to a 100x100x100 Åcarbon cell was constructed and used for image simulation with various settings for slice thickness and pixel sampling. The results of this analysis are summarized in Fig. <ref>. These benchmarks are plotted as a function of the maximum scattering angleq_max, which varies inversely to the pixel size.The difference in computation time t shown in Fig. <ref> between traditional CPU multislice and GPU PRISM is stark, approximately four orders of magnitude for the “fast” setting where f=16, and still more than a factor of 500 for the more accurate case of f=4. For both PRISM and multislice, the addition of GPU acceleration increases speed by at least an order of magnitude. Note that as the thickness of the slices is decreased, the relative gap between PRISM and multislice grows, as probe calculation in PRISM does not require additional propagation through the sample.We have also fit trendline curves of the formt = A + Bq_max^n,where A and B are prefactors and n is the asymptotic power law for high scattering angles.We observed that most of the simulation types approximately approach n=2, which is unsurprising for both PRISM and multislice. The limiting operation in PRISM is matrix-scalar multiplication,which depends on the array size and varies as q_max^2. For multislice the computation is a combination of multiplication operations and FFTs, and the theoretical𝒪(nlogn) scaling of the latter is only slightly larger than 2, and thus the trendline is an approximate lower bound. The only cases that fall significantly outside the n=2 regime were the multislice GPU simulations with the largest slice separation (20 Å) and the “fast” PRISM GPU simulations where f=16. These calculations are sufficiently fast that the relatively small overhead required to compute the projected potential slices, allocate data, etc., is actually a significant portion of the calculation, resulting in scaling better than q_max^2. For the f=16 PRISM case, we observed approximately q_max^0.6 scaling, which translates into sub-millisecond calculation times per probe even with small pixel sizes and slice thicknesses.To avoid unnecessarily long computation times for the many simulations, particularly multislice, different numbers of probe positions were calculated for each algorithm, and thus we report the benchmark as time per probe. Provided enough probe positions are calculated to obviate overhead of computing the projected potential and setting up the remainder of the calculation, there is a linear relationship between the number of probe positions calculated and the calculation time for all of the algorithms, and computing more probes will not change the time per probe significantly. Here this overhead is only on the order of 10 seconds or fewer, and the reported results were obtained by computing 128x128 probes for PRISM CPU and multislice CPU, 512x512 for multislice GPU, and 2048x2048 for PRISM GPU. All of these calculations used the single-transfer memory implementations and were run on compute nodes with dual Intel Xeon E5-2650 processors, four Tesla K20 GPUs, and 64GB RAMfrom the VULCAN cluster within the Lawrence Berkeley National Laboratory Supercluster. §.§ Hardware Scaling Modern high performance computing is dominated by parallelization. At the time of this writing virtually all desktop CPUs contain at least four cores, and high end server CPUs can have as many as twenty or more. Even mobile phones have begun to routinely ship with multicore processors <cit.>. In addition to powerful CPUs, GPUs and other types of coprocessors such as the Xeon Phi <cit.> can be used to accelerate parallel algorithms. It therefore is becoming increasingly important to write parallel software that fully utilizes the available computing resources.To demonstrate how the algorithms implemented in Prismatic scale with hardware, we performed the following simulation. Simulated images of a 100x100x100 Å amorphous carbon cell were produced with both PRISM and multislice using 5 Å thick slices, pixel size 0.1 Å, and 80 keV electrons. This simulation was repeated using varying numbers of CPU threads and GPUs. As before, a varying number of probes was computed for each algorithm, specifically 2048x2048 for GPU PRISM, 512x512 for CPU PRISM and GPU multislice, and 64x64 for CPU multislice. This simulation utilized the same 4-GPU VULCAN nodes described previously. The results of this simulation are summarized in Fig. <ref>.The ideal behavior for the CPU-only codes would be to scale as 1/x with the number of CPU cores utilized such that doubling the number of cores also approximately doubles the calculation speed. Provided that the number of CPU threads spawned is not greater than the number of cores, the number of CPU threads can effectively be considered the number of CPU cores utilized, and this benchmark indicates that both CPU-only PRISM and multislice possess close to ideal scaling behavior with number of CPU cores available.The addition of a single GPU improves both algorithms by approximately a factor of 8 in this case, but in general the relative improvement varies depending on the quality and number of the CPUs vs GPUs. The addition of a second GPU improves the calculation speed by a further factor of 1.8-1.9 with 14 threads, and doubling the number of GPUs to four total improves again by a similar factor. The reason that this factor is less than two is because the CPU is doing a nontrivial amount of work alongside the GPU. This claim is supported by the observation that when only using two threads the relative performance increase is almost exactly a factor of two when doubling the number of GPUs. We conclude that our implementations of both algorithms scale very well with available hardware, and potential users should be confident that investing in additional hardware, particularly GPUs, will benefit them accordingly.§.§ Data Streaming/Single-Transfer BenchmarkFor both PRISM and multislice, Prismatic implements two different memory models, a single-transfer method where all data is copied to the GPU a single time before the main computation begins, and a streaming mode where asynchronous copying of the required data is triggered across multiple CUDA streams as it is needed throughout the computation. Streaming mode reduces the peak memory required on the device at the cost of redundant copies; however, the computational cost of this extra copying can be partially alleviated by hiding the transfer latency behind compute kernels and other copies (Fig. <ref>).To compare the implementations of these two memory models in Prismatic, a number of amorphous carbon cells of increasing sizes were used as input to simulations using 80 keV electrons, 20 mrad probe convergence semi-angle, 0.1 Å pixel size, 4 Å slice thickness, and 0.4 Å probe steps.Across a range of simulation cell sizes the computation time of the streaming vs. single-transfer versions of each code are extremely similar while the peak memory may be reduced by an order of magnitude or more (Fig. <ref>). For the streaming calculations, memory copy operations may become significant relative to the computational work (Fig. <ref>);however, this can be alleviated by achieving multi-stream concurrency.§.§ Comparison to existing methods All previous benchmarks in this work have measured the speed of the various algorithms included in Prismatic against each other; however, relative metrics are largely meaningless without an external reference both in terms of overall speed and resulting image quality. To this end, we also performed STEM simulations of significant size and compare the results produced by the algorithms in Prismatic and the popular package computem <cit.>. We have chosen a simulation cell typical of those used in structural atomic-resolution STEM studies, a complex Ruddlesden–Popper (RP) layered oxide.The RP structure we used contains 9 pseudo-cubic unit cells of perovskite strontium titanate structure, with two stacking defects every 4.5 1x1 cells that modify the composition and atomic coordinates. The atomic coordinates of this cell were refined using Density Functional Theory and were used for very-large-scale STEM image simulations <cit.>. This 9x1x1 unit cell was tiled 4x36x25 times resulting in final sample approximately 14 x 14 nm in-plane and 10 nm thick, containing roughly 1.4 million atoms. Simulations were performed with multislice as implemented in computem (specifically using the autostem module), multislice in Prismatic, and the PRISM method with f values of 4, 8 and 16 using 80 keV electrons, 1024 x 1024 pixel sampling, 20 mrad probe convergence semi-angle, and 5 Å thick potential slices. A total of 720x720 evenly spaced probes were computed for each simulation, and a total of 32 frozen phonon configurations were averaged to produce the final images, which are summarized in Fig. <ref>. The PRISM algorithms were run on the VULCAN GPU nodes while computem simulations utilized better VULCAN CPU nodes with dual Intel Xeon E5-2670v2 CPUs and 64GB RAM. The mean computation time per frozen phonon for the computem simulations was 709.8 minutes resulting in a total computation time of 15.8 days. The use of our GPU multislice code here provides an acceleration of about 15x, reducing the computation from more than two weeks to just over one day. The PRISM f=4 simulation is almost indistinguishable from the multislice results, and gives a 2.7x speed up over our GPU multislice simulation. For the f=8 PRISM simulation, some intensity differences are visible in the two bright field images, but the relative contrast of all atomic sites is still correct. This simulation required just over an hour, providing a speedup of 25X relative to our GPU multislice simulation. The f=16 PRISM result show substantial intensity deviations from the ideal result, but require just 43 seconds per frozen phonon configuration. The total difference in acceleration from CPU multislice to the fastest PRISM simulation shown in Fig. <ref> is just under three orders of magnitude. Ultimately, the user’s purpose dictates what balance of speed and accuracy is appropriate, but the important point is that calculations that previously required days or weeks on a computer cluster may now be performed on a single workstation in a fraction of the time. § APPLICATION TO ATOMIC ELECTRON TOMOGRAPHY One potentially important application of STEM image simulations is AET experiments. One of the ADF-STEM images from an atomic resolution tilt series of an FePt nanoparticle <cit.> is shown in Fig. <ref>a, with the corresponding linear projection from the 3D reconstruction shown in Fig. <ref>b. In this study and others, we have used multislice simulations to validate the tomographic reconstructions and estimate both the position and chemical identification errors <cit.>. One such multislice simulation is given in Fig. <ref>c. This simulation was performed at 300 kV using a 30 mrad STEM probe, with a simulation pixel size of 0.0619 Å and a spacing between adjacent probes of 0.3725 Å. The image results shown are for 16 frozen phonon configurations using a 41-251 mrad annular dark field detector. This experimental dataset includes some postprocessing and was obtained freely online <cit.>.The 3D reconstruction algorithm we have used, termed GENeralized Fourier Iterative REconstruction (GENFIRE), assumes that the projection images are linearly related to the potential of the reconstruction <cit.>. This assumption was sufficient for atomic resolution tomographic reconstruction, but the measured intensity has some non-linear dependence on the atomic potentials, due to effects such as exponential decrease of electrons in the unscattered STEM probe, channeling effects along atomic columns, coherent diffraction at low scattering angles and other related effects <cit.>. These effects can be seen in the differences between the images shown in Figs. <ref>b and c. The multislice simulation image shows sharper atomic columns, likely due to the channeling effect along atomic columns that are aligned close to the beam direction <cit.>. Additionally, there are mean intensity differences between the center part of the the particle (thickest region) and the regions closed to the surfaces in projection (thinnest regions). Including these dynamical scattering effects in the reconstruction algorithm would increase the accuracy of the reconstruction.However, Fig. <ref>h shows that the computation time for the multislice simulation is prohibitively high. Even using the Prismatic GPU code, each frozen phonon configuration for multislice require almost 7 hours. Using 16 configurations and simulating all 65 projection angles would require months of simulation time, or massively parallel simulation on a super cluster. An alternative is to use the PRISM algorithm for the image simulations, shown in Figs. <ref>d, e and f for interpolation factors of f=8, 16 and 32 respectively. Fig. <ref>g shows the relative errors of Figs. <ref>b-f, where the error is defined by the root-mean-square of the intensity difference with the experimental image in Fig. <ref>a, divided by the root-mean-square of the experimental image. Unsurprisingly, the linear projection shows the lowest error since it was calculated directly from the 3D reconstruction built using the experimental data. The multislice and PRISM f=8 and f=16 simulations show essentially the same errors within the noise level of the experiment. The PRISM f=32 has a higher error, and obvious image artifacts are visible in Figs. <ref>f. Thus, we conclude that using an interpolation factor f=16 produces an image of sufficient accuracy. This calculation required only 90 s per frozen phonon calculation, and therefore computing 16 configuration for all 65 tilt angles would require only 26 hours. One could therefore imagine integrating this simulation routine into the final few tomography reconstruction iterations to account for dynamical scattering effects and to improve the reconstruction quality.§ CONCLUSION We have presented Prismatic, an asynchronous, streaming multi-GPU implementation of the PRISM and multislice algorithms for image formation in scanning transmission electron microscopy. Both multislice and PRISM algorithms were described in detail as well as our approach to implementing them in a parallel framework. Our benchmarks demonstrate that this software may be used to simulate STEM images up to several orders of magnitude faster than using traditional methods, allowing users to simulate complex systems on a GPU workstation without the need for a computer cluster. Prismatic is freely available as an open-source C++/CUDA package with a graphical interface that contains convenience features such as allowing users to interactively view the projected potential slices, compute/compare individual probe positions with both PRISM and multislice, and dynamically adjust positions of virtual detectors. A command line interface and a Python package, PyPrismatic, are also available. We have demonstrated one potential application of the Prismatic code, using it to compute STEM images to improve the accuracy in atomic electron tomography. We hope that the speed of this code as well as the convenience of the user interface will have significant impact for users in the EM community. § COMPETING INTERESTSThe authors declare that they have no competing interests.§ AUTHOR'S CONTRIBUTIONSAP designed the software, implemented the CUDA/C++ versions of PRISM and multislice, programmed the graphical user interface and command line interface, and performed the simulations in this paper. CO conceived of the PRISM algorithm, wrote the original MATLAB implementations, and made the figures. AP and CO rote the manuscript. JM advised the project. All authors commented on the manuscript.§ ACKNOWLEDGEMENTS The computations were supported by a User Project at The Molecular Foundry using its compute cluster (VULCAN), managed by the High Performance Computing Services Group, at Lawrence Berkeley National Laboratory (LBNL), and supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231.§ DATA AVAILABILITYThe Prismatic source code, installers, and documentation with tutorials are freely available at www.prism-em.com bmc-mathphys | http://arxiv.org/abs/1706.08563v2 | {
"authors": [
"Alan Pryor Jr.",
"Colin Ophus",
"Jianwei Miao"
],
"categories": [
"physics.comp-ph"
],
"primary_category": "physics.comp-ph",
"published": "20170626185829",
"title": "A Streaming Multi-GPU Implementation of Image Simulation Algorithms for Scanning Transmission Electron Microscopy"
} |
The SU(4)-SU(2) crossover and spin filter properties of a double quantum dot nanosystem E. V. Anda December 30, 2023 =======================================================================================Most existing datasets for speaker identification contain samples obtained under quite constrained conditions, and are usually hand-annotated, hence limited in size. The goal of this paper is to generate a large scale text-independentspeaker identification dataset collected `in the wild'. We make two contributions. First, we propose a fully automated pipeline based on computer vision techniques to create the dataset from open-source media. Our pipeline involves obtaining videos from YouTube; performing active speaker verification using a two-stream synchronization Convolutional Neural Network (CNN), and confirming the identity of the speaker using CNN based facial recognition. We use this pipeline to curate VoxCeleb which contains hundreds of thousands of `real world' utterances for over 1,000 celebrities. Our second contribution is to apply and compare various state of the art speaker identification techniques on our dataset to establish baseline performance. We show that a CNN based architecture obtains the best performance for both identification and verification. Index Terms: speaker identification, speaker verification, large-scale, dataset, convolutional neural network§ INTRODUCTION ^†These authors contributed equally to this work.Speaker recognition under noisy and unconstrained conditions is an extremely challenging topic. Applications of speaker recognition are many and varied, ranging from authentication in high-security systems and forensic tests, to searching for persons in large corpora of speech data. All such tasks require high speaker recognition performance under `real world' conditions. This is an extremely difficult task due to both extrinsic and intrinsic variations; extrinsic variations include background chatter and music, laughter, reverberation, channel and microphone effects; while intrinsic variations are factors inherent to the speaker themself such as age, accent, emotion, intonation and manner of speaking, amongst others <cit.>.Deep Convolutional Neural Networks (CNNs) have given rise to substantial improvements in speech recognition, computer vision and related fields due to their ability to deal with real world, noisy datasets without the need for handcrafted features <cit.>. One of the most important ingredients for the success of such methods, however,is the availability of large training datasets. Unfortunately, large-scale public datasets in the field of speaker identification with unconstrained speech samples are lacking. While large-scale evaluations are held regularly by the National Institute of Standards in Technology (NIST), these datasets are not freely available to the research community. The only freely available dataset curated from multimedia is the Speakers in the Wild (SITW) dataset <cit.>, which contains speech samples of 299 speakers across unconstrained or `wild' conditions. This is a valuable dataset, but to create it the speech samples have been hand-annotated. Scaling it further, for example to thousands of speakers across tens of thousands of utterances, would require the use of a service such as Amazon Mechanical Turk (AMT). In the computer vision community AMT like services havebeen used to produce very large-scale datasets, such as ImageNet <cit.>.This paper has two goals. The first is to propose a fully automated and scalable pipeline for creating a large-scale `real world' speaker identification dataset. By using visual active speaker identification and face verification, our method circumvents the need for human annotation completely. We use this method to curate VoxCeleb, a large-scale dataset with hundreds of utterances for over a thousand speakers. The second goal is to investigate different architectures and techniques for training deep CNNs on spectrograms extracted directly from the raw audio files with very little pre-processing, and compare our results on this new dataset with more traditional state-of-the-art methods. VoxCeleb can be used for both speaker identification and verification. Speaker identification involves determining which speaker has produced a given utterance, if this is performed for a closed set of speakers then the task is similar to that of multi-class classification. Speaker verification on the other hand involves determining whether there is a match between a given utterance and a target model. We provide baselines for both tasks. The dataset can be downloaded from <http://www.robots.ox.ac.uk/ vgg/data/voxceleb>.§ RELATED WORKS For a long time, speaker identification was the domain of Gaussian Mixture Models (GMMs) trained on low dimensional feature vectors <cit.>. The state of the art in more recent times involves both the use of joint factor analysis (JFA) based methods which model speaker and channel subspaces separately <cit.>, and i-vectors which attempt to model both subspaces into a single compact, low-dimensional space <cit.>. Although state of the art in speaker recognition tasks, these methods all have one thing in common – they rely on a low dimensional representation of the audio input, such as Mel Frequency Cepstrum Coefficients (MFCCs). However, not only does the performance of MFCCs degrade rapidly in real world noise <cit.>, but by focusing only on the overall spectral envelope of short frames, MFCCs may be lacking in speaker-discriminating features (such as pitch information). This has led to a very recent shift from handcrafted features to the domain of deep CNNs which can be applied to higher dimensional inputs <cit.> and for speaker identification <cit.>. Essential to this task however, is a large dataset obtained under real world conditions.Many existing datasets are obtained under controlled conditions, for example: forensic data intercepted by police officials <cit.>, data from telephone calls <cit.>,speech recorded live in high quality environments such as acoustic laboratories <cit.>, or speech recorded from mobile devices <cit.>. <cit.> consists of more natural speech but has been manually processed to remove extraneous noises and crosstalk. All the above datasets are also obtained from single-speaker environments, and are free from audience noise and overlapping speech. Datasets obtained from multi-speaker environments include those from recorded meeting data <cit.>, or from audio broadcasts <cit.>. These datasets usually contain audio samples under less controlled conditions. Some datasets contain artificial degradation in an attempt to mimic real world noise, such as those developed using the TIMIT dataset <cit.>: NTIMIT, (transmitting TIMIT recordings through a telephone handset) and CTIMIT, (passing TIMIT files through cellular telephone circuits). Table <ref> summarises existing speaker identification datasets. Besides lacking real world conditions, to the best of our knowledge, most of these datasets have been collected with great manual effort, other than <cit.> which was obtained by mapping subtitles and transcripts to broadcast data. § DATASET DESCRIPTION VoxCeleb contains over 100,000 utterances for 1,251 celebrities, extracted from videos uploaded to YouTube. The dataset is gender balanced, with 55% of the speakers male. The speakers span a wide range of different ethnicities, accents, professions and ages. The nationality and gender of each speaker (obtained from Wikipedia) is also provided.Videos included in the dataset are shot in a large number of challenging multi-speaker acoustic environments. These include red carpet, outdoor stadium,quiet studio interviews, speeches given to large audiences, excerpts from professionally shot multimedia,and videos shot on hand-held devices. Crucially, all are degraded with real world noise, consisting of background chatter, laughter, overlapping speech, room acoustics, and there is a rangein the quality of recording equipment and channel noise. Unlike the SITW dataset, both audio and video for each speaker is released.Table <ref> gives the dataset statistics.§ DATASET COLLECTION PIPELINE This section describes our multi-stage approach for collecting a large speaker recognition dataset, starting from YouTube videos. Using this fully automated pipeline, we have obtained hundreds of utterances for over a thousand different Persons of Interest (POIs). The pipeline is summarised in Figure <ref> left, and key stages are discussed in the following paragraphs: Stage 1. Candidate list of POIs. The first stage is to obtain a list of POIs. We start from the list of people that appear in the VGG Face dataset <cit.> , which is based on an intersection of the most searched names in the Freebase knowledge graph, and the Internet Movie Data Base (IMDB). This list contains 2,622 identities, ranging from actors and sportspeople to entrepreneurs, of which approximately half are male and the other half female. Stage 2. Downloading videos from YouTube. The top 50 videos for each of the 2,622 POIs are automatically downloaded using YouTube search. The word `interview' is appended to the name of the POI in search queries to increase the likelihood that the videos contain an instance of the POI speaking, and to filter out sports or music videos. No other filtering is done at this stage.Stage 3. Face tracking. The HOG-based face detector <cit.> is used to detect the faces in every frame of the video. Facial landmark positions are detected for each face detection using the regression tree based method of <cit.>. The shot boundaries are detected by comparing colour histograms across consecutive frames. Within each detected shot, face detections are grouped together into face tracks using a position-based tracker. This stage is closely related to the tracking pipeline of <cit.>, but optimised to reduce run-time given the very large number of videos to process.Stage 4. Active speaker verification. The goal of this stage is to determine the audio-video synchronisation between mouth motion and speech in a video in order to determine which (if any) visible face is the speaker. This is done by using `SyncNet', a two-stream CNN described in <cit.> which estimates the correlation between the audio track and the mouth motion of the video. This method is able to reject the clips that contain dubbing or voice-over. Stage 5. Face verification.Active speaker face tracks are then classified into whether they are of the POI or not using the VGG Face CNN. This classification network is based on the VGG-16 CNN <cit.> trained on the VGG Face dataset (which is a filtered collection of Google Image Search results for the POI name). Verification is done by directly using this classification score with a high threshold. Discussion.In order to ensure that our system is extremely confident that a person is speaking (Stage 4), and that they have been correctly identified (Stage 5) without any manual interference, we set very conservative thresholds in order to minimise the number of false positives.Precision-recall curves for both tasks on their respective benchmark datasets <cit.> are shown in Figure <ref> right, and the values at the operating point are given in Table <ref>. Employing these thresholds ensures that although we discard a lot of the downloaded videos, we can be reasonably certain that the dataset has few labelling errors.This ensures a completely automatic pipeline that can be scaled up to any number of speakers and utterances (if available) as required.§ CNN DESIGN AND ARCHITECTUREOur aim is to move from techniques that require traditional hand-crafted features, to a CNN architecture that can choose the features required for the task of speaker recognition. This allows us to minimise the pre-processing of the audio data and hence avoid losing valuable information in the process.Input features.All audio is first converted to single-channel, 16-bit streams at a 16kHz sampling rate for consistency. Spectrograms are then generated in a sliding window fashion using a hamming window of width 25ms and step 10ms. This gives spectrograms of size 512 x 300 for 3 seconds of speech.Mean and variance normalisation is performed on every frequency bin of the spectrum. This normalisation is crucial,leading to an almost 10% increase in classification accuracy, as shown in Table <ref>. No other speech-specific preprocessing (e.g. silence removal, voice activity detection, or removal of unvoiced speech) is used. These short time magnitude spectrograms are then used as input to the CNN.Architecture.Since speaker identification under a closed set can be treated as a multiple-class classification problem, we base our architecture on the VGG-M <cit.> CNN, known for good classification performance on image data, with modifications to adapt to the spectrogram input.The fully connected fc6 layer of dimension 9 × 8 (support in both dimensions) is replaced by two layers – a fully connected layer of 9 × 1 (support in the frequency domain) and an average pool layer with support 1 × n, where n depends on the length of the input speech segment (for example for a 3 second segment, n=8). This makes the network invariant to temporal position but not frequency, and at the same time keeps the output dimensions the same as those of the original fully connected layer. This also reduces the number of parameters from 319M in VGG-M to 67M in our network, which helps avoid overfitting.The complete CNN architecture is specified in Table <ref>. Identification. Since identification is treated as a simple classification task, the output of the last layer is fed into a 1,251-way softmax in order to produce a distribution over the 1,251 different speakers. Verification. For verification, feature vectors can be obtained from the classification network using the 1024 dimension fc7 vectors, and a cosine distance can be used to comparevectors. However,it is better to learn an embedding by training a Siamese network with a contrastive loss <cit.>. Thisis better suited to the verification task as the network learns to optimize similarity directly,rather than indirectly via a classification loss. For the embedding network, the last fully connected layer (fc8) is modified so that the output size is 1024 instead of the number of classes. We compare both methods in the experiments.Testing. A traditional approach to handling variable length utterances at test time is tobreak them up into fixed length segments (e.g.3 seconds) and average the results on each segment to give a final class prediction.Average pooling, however allows the network to accommodate variablelength inputs at test time,as the entire test utterance can be evaluated at once by changing the size of the apool6 layer. Not only is this more elegant, it also leads to an increase in classification accuracy, as shown in Table <ref>.Implementation details and training. Our implementation is based on the deep learning toolbox MatConvNet <cit.> and trained on a NVIDIA TITAN X GPU.The network is trained using batch normalisation <cit.>and all hyper-parameters (e.g. weight decay, learning rates) use the default values provided with the toolbox. To reduce overfitting, we augment the data by taking random 3-second crops in the time domain during training. Using a fixed input length is also more efficient. For verification, the network is first trained for classification (excluding the test POIs for the verification task, see Section <ref>),and then all filter weights are frozen except for the modified last layer and the Siamese network trained with contrastive loss. Choosing good pairs for training is very important in metric learning. We randomly select half of the negative examples, and the other half using Hard Negative Mining, where we only sample from the hardest 10% of all negatives. § EXPERIMENTSThis section describes the experimental setup for both speaker identification and verification, and compares the performance of our devised CNN baseline to a number of traditional state of the art methods on VoxCeleb.§.§ Experimental setup Speaker identification. For identification, the training and the testing are performed on the same POIs. From each POI, we reserve the speech segments from one video for test. The test video contains at least 5 non-overlapping segments of speech.For identification, we report top-1 and top-5 accuracies. The statistics are given in Table <ref>.Speaker verification. For verification,all POIs whose name starts with an `E' are reserved for testing, since this gives a good balance of male and female speakers. These POIs are not used for training the network, and are only used at test time. The statistics are given in Table <ref>.Two key performance metrics are used to evaluatesystem performance for the verification task. The metrics are similar to those used by existing datasets and challenges, such as NIST SRE12 <cit.> and SITW <cit.>. The primary metric is based on the cost function C_det C_det = C_miss× P_miss× P_tar + C_fa× P_fa× (1-P_tar)where we assume a prior target probability P_tar of 0.01 and equal weights of 1.0 between misses C_miss and false alarms C_fa. The primary metric, C^min_det, is the minimum value of C_det for therange of thresholds.The alternative performance measure used here isthe Equal Error Rate (EER) which is the rate atwhich both acceptance and rejection errors are equal. This measure is commonly used for identityverification systems.§.§ Baselines GMM-UBM. The GMM-UBM system uses MFCCs of dimension 13 as input. Cepstral mean and variance normalisation (CMVN) is applied onthe features. Using the conventional GMM-UBM framework, a single speaker-independent universal background model (UBM) of 1024 mixture components is trained for 10 iterations from the training data. I-vectors/PLDA. Gender independent i-vector extractors <cit.> are trained on the VoxCeleb dataset to produce 400-dimensional i-vectors. Probabilistic LDA (PLDA) <cit.> is then used to reduce the dimension of the i-vectors to 200.Inference. For identification, a one-vs-rest binary SVM classifier is trained for each speaker m (m ∈ 1...K). All feature inputs to the SVM are L2 normalised and a held out validation set is used to determine the C parameter (determines trade off between maximising the margin and penalising training errors). Classification during test time is done by choosing the speaker corresponding to the highest SVM score.The PLDA scoring function <cit.> is used for verification. §.§ Results Results are given in Tables <ref>and <ref>.For both speaker recognition tasks, the CNN provides superior performance to the traditionalstate-of-the-art baselines. For identification we achieve an 80.5% top-1classification accuracy over 1,251 different classes, almost 20% higher than traditional state of the art baselines.The CNN architecture uses the average pooling layer for variable length test data.We also compare to two variants: `CNN-fc-3s', this architecture has a fully connected fc6 layer,and divides the test data into 3s segments and averages the scores. As is evident there is a considerable drop in performance compared to the average pooling original – partly due to the increased number of parameters that must be learnt; `CNN-fc-3s no var.norm.', this is the CNN-fc-3s architecture without the variance normalizationpre-processing of the input (the input is still mean normalized). The difference in performance between the two shows the importance of variance normalization for this data.For verification, the margin over the baselines is narrower, but still a significant improvement, with the embedding being the crucial step. § CONCLUSIONS We provide a fully automated and scalable pipeline for audio data collection and use it to create a large-scale speaker identification dataset called VoxCeleb, with 1,251 speakers and over 100,000 utterances. In order to establish benchmark performance, we develop a novel CNN architecture with the ability to deal with variable length audio inputs, which outperforms traditional state-of-the-art methods for both speaker identification and verification on this dataset. Acknowledgements. Funding for this research is provided by the EPSRCProgramme Grant Seebibyte EP/M013774/1 and IARPA grant JANUS.We would like to thank Andrew Senior for helpful comments. ieeetr | http://arxiv.org/abs/1706.08612v2 | {
"authors": [
"Arsha Nagrani",
"Joon Son Chung",
"Andrew Zisserman"
],
"categories": [
"cs.SD"
],
"primary_category": "cs.SD",
"published": "20170626214227",
"title": "VoxCeleb: a large-scale speaker identification dataset"
} |
[email protected] School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, ChinaStabilizing defects in liquid-crystal systems is crucial for many physical processes and applications ranging from functionalizing liquid-crystal textures to recently reported command of chaotic behaviors of active matters. In this work, we perform analytical calculations to study the curvature driven stability mechanism of defects based on the isotropic nematic disk model that is free of any topological constraint. We show that in a growing spherical disk covering a sphere the accumulation of curvature effect can prevent typical +1 and +1/2 defects from forming boojum textures where the defects are repelled to the boundary of the disk. Our calculations reveal that the movement of the equilibrium position of the +1 defect from the boundary to the center of the spherical disk occurs in a very narrow window of the disk area, exhibiting the first-order phase-transition-like behavior. For the pair of +1/2 defects by splitting a +1 defect, we find the curvature driven alternating repulsive and attractive interactions between the two defects. With the growth of the spherical disk these two defects tend to approach and finally recombine towards a +1 defect texture. The sensitive response of defects to curvature and the curvature driven stability mechanism demonstrated in this work in nematic disk systems may have implications towards versatile control and engineering of liquid crystal textures in various applications. Curvature-driven stability of defects in nematic textures over spherical disks Xiuqing Duan and Zhenwei Yao==============================================================================§ INTRODUCTION Functionalizing the rich variety of self-assembled liquid-crystal (LC) structures represents a trend in LC research <cit.>. Confining LCs in various geometries in the form ofdroplets <cit.>, shells <cit.> and fibers <cit.> using modern microfluidic technology and characterization methods opens the prospect of many application opportunities, and brings new scientific problems related to the creation and engineering of complex director arrangements <cit.>. LC textures can be strongly affected by the distribution and type of topological defects, which are singularities in the otherwise continuous LC director field <cit.>.The extraordinary responsiveness of LC makes the manipulation of defects a challenge in applications. Stabilizing defects in two-dimensional LC systems is directly related to arrangement of LC textures <cit.>, fabrication of controllable valency in colloid-LC-based artificial atoms <cit.>, modulation of coupled geometries where LC lives <cit.>, and relevant applications in active matter systems <cit.>.A prototype model to study the stability mechanism of defects in LC is the isotropic two-dimensional LC disk model with a single elastic constant <cit.>.In a flat freestanding LC disk, defects tend to move swiftly to the boundary to form a boojum texture, which is a two-dimensional version of its namesake in superfluid helium-3 <cit.>. A “virtual boojum" texture with a topological defect outside the sample has been predicted in planar circular LC domains by Langer and Sethna <cit.>, and it has been found to be a local energy extremal <cit.>.Sufficiently strong pinning boundary conditions can stabilize a defect within a circular LC domain <cit.>.Exploring other stability mechanisms of defects in LC samples in addition to imposing boundary conditions constitutes an underlying scientific problem towards versatile control and engineering of LC director arrangement. Confining LC over spherical surfaces can generate various regularly arranged stable defect patterns <cit.>. Vitelli and Nelson have studied two-dimensional nematic order coating frozen surfaces of spatially varying Gaussian curvature, and found the instability of a smooth ground-state texture to the generation of a single defect using free boundary conditions <cit.>. These results of LC order on closed spheres and topographies with varying curvature show that curvature suffices to provide a stability mechanism for defects even without imposing any pinning boundary condition. However, for LC order on a closed sphere, it is unknown to what extent the appearance of defects is energetically driven, while they must appear as a consequence of the spherical topology. To remove the topological constraint, we study nematic order, the simplest LC order, on a spherical disk. Here we emphasize that, due to the fundamentally distinct topologies of sphere and disk, the appearance of defects on spherical disks is not topologically required; the emergence of defects therein is purely geometrically driven. According to the continuum elasticity theory of topological defects in either LC or crystalline order, the stress caused by defects can be partially screened by Gaussian curvature <cit.>. Therefore, one expects the appearance of defects on a sufficiently curved spherical disk. It is of interest to identify the transition point for a defect to depart from the boundary of the disk, and illustrate the nature of the transition by clarifying questions such as: Will the defect move rapidly or gradually with the accumulation of curvature effect? Will the defect split as the nematic texture becomes more and more frustrated by the curvature? Once split, will the resulting defects become stable on the spherical disk?We perform analytical calculations based on the isotropic nematic disk model to address these fundamental problems. This theoretical model may be realized experimentally in Langmuir monolayers <cit.> and liquid-crystal films <cit.>deposited at the surface of water droplets whose curvature is controllable by tuning the droplet size <cit.>.Flat space experiments in these two-dimensional monolayer systems at air-water interface have revealed stable liquid-crystal phases <cit.>. In this work, we first discuss the two instability modes of a +1 defect over a flat disk, either sliding to the boundary or splitting to a pair of +1/2 defects. By depositing the nematic order over a spherical surface, we analytically show that bending deformation of a director field is inevitable everywhere, which implies the appearance of defects to release the curvature-driven stress.By comparing a flat and a spherical nematic disk of the same area, both containing a +1 defect at the center, we derive for the analytical expression for the difference of the Frank free energy, and show that the spherical disk always has higher energy.However, when the +1 defect deviates from the center of the disk, the free energy curves become qualitatively different for flat and spherical disks when the disk area exceeds some critical value. Specifically, the equilibrium position of the +1 defect rapidly moves from the boundary to the center of the spherical disk in a narrow window of the disk area, exhibiting the first-order phase-transition-like behavior. For the pair of +1/2 defects by splitting a +1 defect, we further show the curvature-driven alternating repulsive and attractive interactions between the two defects. When the spherical cap occupies more area over the sphere, the pair of +1/2 defects tend to approach until merging to a +1 defect texture. The recombination of the pair of +1/2 defects into a +1 defect is consistent with the result of the +1 defect case. These results demonstrate the fundamentally distinct scenario of defects in a spherical disk from that on a planar disk. We also briefly discuss the cases of nematic order on hyperbolic disks. In this work, the demonstrated distinct energy landscape of LC defects created by curvature is responsible for the stability of defects, and may have implications in the design of LC textures with the dimension of curvature. § MODEL AND METHODIn the continuum limit, the orientations of liquid-crystal molecules lying over a disk are characterized by a director field 𝐧(𝐱) that is defined at the associated tangent plane at 𝐱.The equilibrium nematic texture is governed by minimizing the Frank free energy <cit.>F= ∫_D f dA + λ (𝐧^2-1),where the integration is over the disk D. The Frank free energy densityf=1/2K_1(div 𝐧)^2+1/2K_3(𝐧×curl 𝐧)^2,where K_1 and K_3 are the splay and bending rigidities, respectively.The Lagrange multiplier λ is introduced to implement the constraint of 𝐧·𝐧=1. In general, λ is a function of coordinates.The twist term (𝐧·curl 𝐧)^2 vanishes in nematics confined on a sphere (see Appendix B). Equation (<ref>) has been widely used to analyze the deformation in nematic phases.For nematics on curved surfaces, the operators of divergence and curl in the Frank free energy are promoted to be defined on the curved manifold and carry the information of curvature. Note that the curl operator relies on the extrinsic geometry of the surface <cit.>. Note that the Frank free energy model in Eq. (<ref>) describes the distortion free energy of uniaxial nematics. A formalism based on the tensorial nematic order parameter has been proposed to characterize the distortion of both uniaxial and biaxial nematics and defects therein <cit.>. We work in the approximation of isotropic elasticity with K_1 = K_3. Under such an approximation, one can show that the free energy is invariant under the local rotation of the director field by any angle, whether the disk is planar or curved (see Appendix A). In other words, the energy degeneracy of the system becomes infinite when K_1 = K_3. Such configurational symmetry is broken when the ratio K_1/K_3 is deviated from unity. While the states selected by the differential in the values for K_1 and K_3 are of interest in other contexts such as in the ground states of spherical nematics <cit.>, here we work in the isotropic regime to highlight the curvature effect of substrates on the configuration of nematics.The general Euler-Lagrange equation of the Frank free energy on a curved surface 𝐱(u^1, u^2) is∂_j ∂ f/∂ (∂ n_i/∂ u^j)+∂_j √(g)/√(g)∂ f/∂ (∂ n_i/∂ u^j) - ∂ f/∂ n_i = -λ n_i,where i, j=1,2, and g is the determinant of the metric tensor. The second term in Eq.(<ref>) is due to the spatially varying g. The nematic textures studied in this work aresolutions to Eq.(<ref>).To characterize defects that are named disclinations in a two-dimensional director field, we perform integration of the orientation θ of the director 𝐧 with respect to any local reference frame along any closed loop Γ:∮_Γ dθ = kπ,where k is nonzero if Γ contains a defect.Unlike in a vector field where k can only be integers, two-dimensional nematics supports both integer and half-integer disclinations due to the apolarity of liquid-crystal molecules, i.e., 𝐧≡ -𝐧. § RESULTS AND DISCUSSION We first discuss the case of nematics on a planar disk.It is straightforward to identify the following solution to the Euler-Lagrange equation:𝐧= cos(φ+θ_0) 𝐞_1+sin(φ+θ_0)𝐞_2,where φ = arctan(y/x) is the polar angle, θ_0 is a constant, and 𝐞_i is the unit basis vector in Cartesian coordinates. The strength of the defect located at the origin of the coordinates is +1. The associated Lagrange multiplier is λ=K_1/(x^2+y^2). The contributions to the splay and bending terms in the free energy density are K_1 cos^2 θ_0/[2(x^2+y^2)] and K_3 sin^2 θ_0/[2(x^2+y^2)], respectively. When θ_0 increases from 0 to π/2, the +1 defect transforms from the radial (pure splay) to the azimuthal (pure bending) configurations. In this process, the sum of the splay and bending energies is an invariant under the isotropic elasticity approximation. The total free energy of the configuration in Eq.(<ref>) isF_+1, p = K_1/2∬_x^2+y^2 ≤ r_p^21/x^2+y^2 dxdy,where r_p is the radius of the planar disk. We show that the +1 defect at the center of the planar disk in Eq.(<ref>) is unstable and tends to slide to the boundary of the disk. For simplicity, we employ free boundary condition. Consider a +1 defect like in Eq.(<ref>) at (c, 0), where c ≤ r_p.Its free energy isF_+1,p(c)=K_1/2∬_x^2+y^2 ≤ r_p^21/(x-c)^2+y^2 dxdy.To avoid the singularity point at (c, 0) in the evaluation for F_+1,p(c), we take the derivative of F_+1,p(c) with respect to c. Physically, this procedure returns the force on the defect. While the free energy may diverge, a physical force must be finite. After some calculation, we haveF'_+1,p(c)=K_1 ∬_x^2+y^2 ≤ r_p^2x-c/[(x-c)^2+y^2]^2 dxdy = K_1 ∬_(x'+c)^2+y^2 ≤ r_p^2x'/(x'^2+y^2)^2 dx'dy,where variable substitution is applied in the last equality.The integral domain is shown in Fig. <ref>(a). The defect is located at x'=0 (i.e., x = c) and y=0. We see that the integration in the red region returns zero, since the integrand x'/(x'^2+y^2)^2 is an odd function of x'. In the rest region where x'<0, the integrand is negative. Therefore, F'_+1,p(c) is negative when the defect is deviated from the center of the disk. F'_+1,p(c=0)=0. In other words, once deviated from the center of the disk, the defect will slide to the boundary to reduce the free energy of the system. Figure <ref>(b) shows the numerical result on the dependence of F'_+1,p(c) on c/r_p.An alternative instability mode of the central +1 defect in the planar disk is to split into two +1/2 defects. Such a process may occur when the interaction energy of the two repulsive +1/2 defects dominates over the core energy of the defects. To analyze the energetics of the +1/2 defects, we construct the director field containing two +1/2 defects by cutting and moving apart an azimuthal configuration as shown in Fig.1(c), where the +1/2 defects are represented by red dots. The region between the two half azimuthal configurations is filled with a uniform director field. The Frank free energy of such a configuration isF_+1/2,p(c)=2K_1 ∫_0^r_p-cdx∫_0^√(r_p^2-(x+c)^2)dy 1/x^2+y^2,where the separation between the two defects is 2c. F'(0)=-2K_1/r_p<0. In Fig. <ref>(d), we plot F'_+1/2,p(c) versus c. The negative sign indicates the repulsive nature of the two +1/2 defects. The resulting +1/2 defects are ultimately pushed to the boundary of the disk under the repulsive interaction. In the preceding discussions, we employ the free boundary condition where directors at the boundary do not have preferred orientations. Another important class of boundary condition is to fix the orientation of the molecules at the boundary. Homeotropic and planar liquid-crystal samples are two typical cases, where the directors are perpendicular and parallel to the boundary, respectively. Imposing these pinning boundary conditions over the aster configuration can lead to spiral deformations <cit.>. Note that a recent study has demonstrated a dynamic consequence of the radial-to-spiral transition of a +1 defect pattern in the system of swimming bacteria in a liquid-crystal environment <cit.>.It is observed that the swimming mode of bacteria changes from bipolar to unipolar when the +1 defect pattern becomes spiral. For the general pinning boundary condition that the angle between 𝐧 and the tangent vector at the boundary is α (α∈ [0, π/2]), we obtain the solution to Eq.(<ref>): (n_r,n_φ)=( sinθ(α),cosθ(α) ),where n_r and n_φ are the components of 𝐧 in polar coordinates (r, φ), θ(α) = α ln(r/r_0)/ln(r_p/r_0), r_p and r_0 are the outer and inner radius of the planar disk as shown in Fig. <ref>. The magic spiral solution in Ref. <cit.> is a special case of α=π/2. The associated Lagrange multiplier is λ=(K_1/r^2) {1+α^2/[ln (r_p/r_0)]^2 }. The configuration of the solution in Eq.(<ref>) is plotted in Fig. <ref>. The originally straight radial lines deform to spiral curves to satisfy the boundary condition. The Frank free energy of the spiral configuration isF_spiral(α)=( 1+(α/lnr_p/r_0)^2 ) F_+1, p,where F_+1, p is the free energy of an aster configuration in a planar disk given in Eq.(<ref>).Eq.(<ref>) shows that the boundary effect does not enter the integral of F_+1, p. The energy cost associated with the spiral deformation conforms to a quadratic law with respect to the angle α. And its dependence on the size of disk is relatively weak in a logarithm relation.Now we discuss two-dimensional nematic texture confined on spherical disks. Consider a director field 𝐧 on a sphere 𝐧 = n_1(θ,φ)𝐞_θ + n_2(θ,φ)𝐞_φ, where𝐞_θ and 𝐞_φ are the unit tangent vectors in spherical coordinates. θ and φ are the polar and azimuthal angles, respectively.We first show that on spherical geometry a director field without any splay and bending deformations is impossible. Topology of the two-dimensional sphere dictates that a harmonic vector field on a sphere is impossible <cit.>. A vector field is called harmonic if it is divergence-free, irrotational, and tangent to the spherical surface. A director field is a vector field with the extra constraints of |𝐧|=1 and 𝐧≡ -𝐧. Therefore, it is a topological requirement that one cannot completely eliminate both bending and splay deformations in a director field living on a sphere.In addition to the above global analysis, we will further show that an irrotational director field is impossible at any point on a sphere. In other words, bending of a director field is inevitable everywhere on a spherical surface. We first present the general expressions for the divergence and curl of a director field over a smooth surface:div 𝐧 = 1/√(g)∂_i(√(g) n_i) andcurl n = (⋆ d n^♭)^♯, where ⋆ is the Hodge dual, ♭ and ♯ are the musical isomorphisms, d is exterior derivative (see Appendix B). Applying these expressions on a sphere, we havediv 𝐧 = cosθ/sinθn_1/R + 1/R∂ n_1/∂θ + 1/R sinθ∂ n_2/∂φ,andcurl n = 1/R sinθ (-∂ n_1/∂φ + n_2 cosθ + sinθ∂ n_2/∂θ) 𝐞_𝐫 - n_2/R𝐞_θ + n_1/R𝐞_φ ,where 𝐞_𝐫 is the unit normal vector. According to Eq.(<ref>), we clearly see that at least one of the last two terms must be nonzero. In contrast, Eq.(<ref>) shows that a divergence free director field with vanishing splay deformation without any bend deformation is possible. The simplest example is the direction field with only the azimuthal component: 𝐧 = 𝐞_φ. Such a director field is divergence free but with bending deformation. Note that in the calculation for the curl of the director field, we use the condition that the sphere is embedded in three-dimensional Euclidean space. The divergence of the director field does not depend on how the sphere is embedded in the Euclidean space. One can check that the twist term (𝐧·curl 𝐧)^2 = 0.The stability analysis of defects in nematic textures over spherical disks is based on the following expression for the Frank free energy density in spherical coordinates:f = K_1/2R^2(n_1 cosθ/sinθ+∂ n_1/∂θ+1/sinθ∂ n_2/∂φ)^2+K_3/2R^2 × [1+1/sin^2 θ (n_2 cosθ + sinθ∂ n_2/∂θ-∂ n_1/∂φ)^2 ],Note that the first term K_3/(2R^2) in the bending part represents the irremovable bending deformation of a director field over spherical substrates. This term vanishes in the limit of R→∞.One can check that for a divergence-free director field 𝐧 = 𝐞_φ, f=K_1/(2R^2sin^2θ).The singularities at θ=0 and θ=π correspond to the two +1 defects at the north and south poles. We first discuss if the +1 defect can be supported by spherical geometry. All the degenerate nematic configurations containing a +1 defect at the center of the spherical cap are characterized by the director field 𝐧=(c_1,c_2), where the constants c_1 and c_2 satisfy c_1^2+c_2^2=1. These degenerate states have the same Frank free energy:F_+1,s = K_1/2∬_θ, φ∈ D1/R^2 sin^2 θ (R^2 sinθ dθ dφ) = K_1/2∬_θ, φ∈ D1/sinθ dθ dφ,where the integration is over a spherical cap D with spherical radius R and geodesic radius r_s. And these states are solutions to the Euler-Lagrange equation (see Appendix C). In order to derive for F_+1,s - F_+1,p, the free energy difference of a +1 defect configuration on spherical and planar disks, we introduce the following coordinates transformation. For generality, the Cartesian coordinates of the center of the spherical cap are (c,0,√(R^2-c^2)) as shown in Fig. <ref>. The center of the spherical cap is located at the north pole for c=0. The region of the spherical cap is D = {(x,y,z)|x^2+y^2+z^2=R^2, (x-c)^2+y^2+(z-√(R^2-c^2))^2 ≤ r_s'^2}. r_s' is the Euclidean distance from the center to the boundary of the spherical cap. The area of such a spherical cap is S = π r_s'^2. Now we construct the stereographic projection from the spherical cap to the plane of equator. Specifically, we draw a line connecting the south pole of the sphere and any point at (x,y,z) or (θ, φ) on the spherical cap. The point on the spherical cap is thus projected to the intersection point (u, v) of this line and the equator plane. The projection is described by the formula(u, v) = (Rx/z+R,Ry/z+R),or, in terms of spherical coordinates,(u,v) = (R sinθcosφ/cosθ + 1,R sinθsinφ/cosθ +1 ).The stereographic projection has a convenient geometric property that any spherical cap not containing the point of projection (south pole) is projected to a circular disk on the equator plane:(u-u_0)^2+v^2 ≤ r_eq^2,whereu_0=2cR^2/-r_s'^2+2R(R+√(R^2-c^2)),andr_eq^2 =r_s'^2R^2(4R^2-r_s'^2)/[r_s'^2-2R(R+√(R^2-c^2))]^2.To guarantee that the spherical cap contains the north pole, it is required that r_s'^2 ≥ 2R(R-√(R^2-c^2)). Alternatively, c ≤ r_s' √(1-[r_s'/(2R)]^2) for given r_s'.On the other hand, the spherical cap occupies no more than half of a sphere, so r_s' ≤√(2)R.From the Jacobian of the coordinates transformation in Eq.(<ref>)∂(u,v)/∂ (θ,φ) = ( [R cosφ/1 + cosθ -R sinθsinφ/1 + cosθ;R sinφ/1 + cosθR sinθcosφ/1 + cosθ;]),anddudv =| ∂(u,v)/∂ (θ,φ)| dθ dφ = u^2+v^2/sinθ dθ dφ,we finally havedudv/u^2+v^2=dθ dφ/sinθ.We therefore obtain the desired expression for Eq.(<ref>) in the (u, v) coordinates:F_+1,s = K_1/2∬_D1/u^2+v^2 dudv,where the integral domain D={(u, v)| u^2+v^2 ≤ r_s'^2 R^2/(4R^2-r_s'^2) }. Note that now the integrands in Eq.(<ref>) and Eq.(<ref>) have the same functional form and can be conveniently compared. A subtle point worth mentioning is that the direct subtraction of Eq.(<ref>) from Eq.(<ref>) will lead to a wrong expression of Δ F = F_+1,s - F_+1,p = - (π/2) K_1 ln [4-(r_s'/R)^2]. One can check that Δ F fails to converge to the expected zero in the limit of R→∞. Here, the subtlety is from the fact that the integrands in Eq.(<ref>) and Eq.(<ref>) have singularity at the origin point. To eliminate this singularity, one has to cut off the small defect core. The integral domain of Eq.(<ref>) should be D={(u, v)| (a/2)^2 ≤ u^2+v^2 ≤ r_s'^2 R^2/(4R^2-r_s'^2) }, where a is the radius of the defect core. The prefactor of 1/2 is due to the shrink of the defect size in the previously introduced stereographic projection. The integral domain in Eq.(<ref>) also becomes a^2 ≤ x^2+y^2 ≤ r_p^2. To conclude, the change of the total free energy in the deformation of the planar to the spherical nematic disk in the constraint of fixed disk area A_d isΔ F=F_+1,s - F_+1,p=- π/2 K_1 ln( 1-(A_d/4π R^2) ).We check that Δ F approaches zero in the limit of R→∞, as expected. Equation (<ref>) shows that F_+1,s is always larger than F_+1,p.However, it will be shown that a +1 defect can be stabilized within a sufficiently curved spherical disk despite the higher energy in comparison with the planar disk case. We analyze the stability of the +1 defect from the derivative of the free energy with respect to its position in the disk.The expression for the free energy is rewritten in the new coordinates {x, y }, where x=u-u_0 and y=v:F_+1,s(c) = K_1/2∬_x^2+y^2 ≤ r_eq^21/(x+u_0)^2+y^2 dxdy= K_1 ∫ _-r_eq^r_eq dx 1/x+u_0arctan√(r_eq^2-x^2)/x+u_0,where u_0 and r_eq are given in Eq.(<ref>). From Eq.(<ref>), we haveF'_+1,s(c) = K_1 ∫_-r_eq^r_eq ( G_1 + G_2 + G_3 )dx,where G_1=-[u_0'(c)/(x+u_0)^2] arctan [√(r_eq^2-x^2)/(x+u_0)], G_2=- u_0'(c) √(r_eq^2-x^2)/[(x+u_0)(u_0^2-2u_0 x +r_eq^2)], and G_3=r_eqr_eq'(c)/[(u_0^2-2u_0 x+r_eq^2)√(r_eq^2-x^2)]. The G_3 term can be integrated out: K_1 ∫_-r_eq^r_eq G_3 dx= K_1 π r_eq r_eq'(c)/|u_0^2-r_eq^2|. Local analysis around the defect at x=-u_0 shows that both the G_1 and the G_2 terms are odd functions of x, and can be canceled in the integration of x near the defect. The singularity associated with the defect is therefore removed. Note that F'_+1,s(c) is negative in the large R limit, which is consistent with the planar disk case. Now we analyze zero points of F'_+1,s(c). The defect is stable at a zero point where the slope of the F'_+1,s(c) curve is positive.With the increase of c, numerical analysis shows that the G_1 term decreases and the G_3 term increases, both starting from zero at c=0. While the G_1 and the G_3 terms are comparable, the G_3 term is much smaller than either of them. The competition of the G_1 and the G_3 terms may lead to another zero point at the F'_+1,s(c) curve in addition to the unstable zero point at c=0.In Figs. <ref>(a)–<ref>(c), we plot F'_+1,s(c) versus c at typical values for r_s'. We see that the F'_+1,s(c) is negative and monotonously decreasing when the spherical cap is smaller than a critical value.With the increase of r_s', a second zero point appears at c=c^*, where a perturbed defect will be restored to the original equilibrium position. It indicates that the equilibrium position of the defect starts to depart from the boundary of the disk. We introduce the quantity r_d'/r_s' to characterize the equilibrium position of the defect over the spherical cap, where r_d' is the Euclidean distance between the center of the disk and the defect. The variation of the optimal position of the +1 defect with the size of the spherical cap is summarized in Fig. <ref>(d). A pronouncing feature of the r_d'/r_s' vs r_s'/R curve is the rapid decrease from unity to zero when r_s'/R varies by only about 0.1 %. It corresponds to the movement of the defect from the boundary to the center of the disk. Such a transition occurs in the narrow window of r_s' when the spherical cap occupies about half of the sphere. Note that the spherical cap becomes a half sphere when r_s'=√(2) R.Here, it is of interest to compare a +1 defect in nematics and a five-fold disclination in a two-dimensional hexagonal crystal on a sphere. Both nematic and crystalline order are frustrated on a sphere, leading to the proliferation of defects. The resulting defects in condensed matter orders are to screen the geometric charge of the substrate surface, which is defined to be the integral of Gaussian curvature. Over a spherical crystal, the topological charge of a five-fold disclination can be screened by a spherical cap of area A_0/12 (A_0 is the area of sphere), since 12 five-fold disclinations are required over a spherical crystal by topological constraint <cit.>. Topological analysis of a spherical nematics shows that a sphere can support two +1 defects, so the topological charge of a +1 defect can be screened by a spherical cap of area A_0/2. Our energetics calculation is consistent with such topological analysis; it is when the spherical cap becomes as large as a half sphere that a +1 defect will be energetically driven to move to the center of the disk.We proceed to discuss the split of a +1 defect into two +1/2 defects over a spherical cap. Like the case of the planar disk, we first construct the director field containing two +1/2 defects by cutting an azimuthal +1 defect configuration. As shown in Fig. <ref>(a), the resulting director field on the spherical cap is composed of three parts: the middle uniform region where 𝐧=(-z/√(x^2+z^2),0,x/√(x^2+z^2)), and the symmetric azimuthal configurations at the two sides. The origin of the Cartesian coordinates is at the center of the sphere, and the z-axis passes through the north pole.The two +1/2 defects are indicated by red dots in Fig.5. Their x-coordinates are x=± c. The center of the spherical cap is at the north pole. The Frank free energy density of the middle uniform configuration is f=K_1/[2(x^2+z^2)]. By putting them together and working in the Cartesian coordinates over the equator plane, we haveF_+ 1/2,s/2K_1= ∬_D_11/x^2+y^2dA+∬_D_21/R^2-y^2dA,where the surface element of the spherical cap dA=(R/√(R^2-x^2-y^2))dxdy, D_1 = {(x, y)| y∈[0, √(b^2-(x+c)^2)],x∈[0, b-c] }, and D_2 = {(x, y)| y∈[0, √(b^2-x^2)],x∈[0, c] }. b is the radius of the circular boundary of the spherical cap. b=r_s' √(1-(r_s'/2R)^2). From Eq.(<ref>), we haveF'_+1/2,s(c)/2K_1= ∫_0^b-c/RF_1(x)dx+ ∫_0^√(b^2-c^2)F_2(y)dy,where F_1(x)=R^2 (c+Rx)/{√(-b^2+c^2+R^2+2Rcx) √(b^2-(c+xR)^2) [-b^2+c(c+2Rx)]}, and F_2(y)= R/[(R^2-y^2)√(R^2-c^2-y^2)]. It is straightforward to show that F'_+1/2,s(0)=-√(R^2-b^2)/(Rb) < 0. It indicates the repulsive interaction between two infinitely close +1/2 defects. Numerical evaluation of Eq.(<ref>) shows that when the spherical disk is sufficiently large, the departing +1/2 defects can be stabilized within the disk. The plots of F'_+1/2,s(c) at typical values for b/R are shown in Figs. <ref>(b)–<ref>(d). We see that when b/R>0.93, the F'_+1/2,s(c) curve starts to hit the horizontal zero line, leading to the two zero points indicated by the blue and the green dots in Fig.<ref>(d). When the separation between the two defects is smaller than the value at the blue dot or larger than the value at the green dot, they repel with each other. In the regime between the two zero points, the defects attract with each other. The curvature-driven alternating repulsive and attractive regimes in the F'_+1/2,s(c) curve are indicated by the arrows in Fig.<ref>(d). The left zero point (blue dot) represents the equilibrium configuration of the +1/2 defects. In Fig. <ref>(e), we show the variation of the equilibrium position of the +1/2 defects with the size of the spherical disk. When the spherical cap occupies more area over the sphere, the distance between the two +1/2 defects in the equilibrium configuration shrinks. In the limit of a half sphere, the two +1/2 defects merge together, becoming a +1 defect. This result is consistent with our previous analysis of the +1 defect case, where the optimal position of the +1 defect over a half sphere is at the center of the disk. We proceed to discuss nematic order on Poincaré disk with constant negative Gaussian curvature <cit.>. The associated metric over a hyperbolic disk with Gaussian curvature K_G is characterized by ds^2 = 4(dx^2+dy^2)/(1+K_G r^2)^2, where r^2=x^2+y^2. The area element dA=4dxdy/(1+ K_G r^2)^2. For the director field n = n_1(x,y) 𝐞_1 + n_2(x,y)𝐞_2, where 𝐞_1 and 𝐞_2 are the orthogonal unit basis vectors, its divergence and curl are div n = (1/2)(1+ K_G r^2) (∂ n_1/∂ x + ∂ n_2/∂ y) - K_G (n_1x + n_2y), and curl n = (1/2)(1+ K_G r^2) (∂ n_2/∂ x - ∂ n_1/∂ y) - K_G (n_2x - n_1y), respectively (see Appendix B for the derivation of curl n).We first consider a defect-free uniform director field (n_1,n_2)=(cosθ_0, sinθ_0) whose associated Lagrange multiplier is λ=K_1 K_G (1+K_G r^2). θ_0 ∈ [0, π/2].The associated Frank free energy density is independent of θ_0: f = K_1 K_G^2r^2/2. We see that the uniform state in Poincaré disk has a non-zero energy density that increases with r in a power law. It is due to the special metric structure of the Poincaré disk.Now we consider a +1 defect configuration in the nematic texture on Poincaré disk. It can be characterized by n=[(c_2 x - c_1 y)𝐞_1+ (c_1 x + c_2 y) 𝐞_2] /√(x^2+y^2), where c_1 and c_2 are both constants satisfying c_1^2+c_2^2=1, such that the magnitude of 𝐧 is unity. Varying the value of c_1 from zero to unity, we obtain director fields from radial to azimuthal configurations. The associated Frank free energy density isf_+1,h = K_1/2(1- K_G r^2)^2/4r^2.Since f'_+1,h(r) = (K_1/2) (-1+K^2 r^4)/(2r^3) < 0, the Frank free energy density decreases with r. On the other hand, due to the homogeneity of the Poincaré disk, the optimal position of a +1 defect is always at the boundary of the disk.Finally, we discuss some effects that are not taken into consideration in our calculations.First, by introducing anisotropy in the elastic constants, the free energy varies with the local rotation of the director field. Despite the reduced energy degeneracy arising from the elasticity anisotropy, both radial and azimuthal configurations based on which our calculations are performed are still solutions to the Euler-Lagrange equation (see Appendix C). Therefore, introducing elasticity anisotropy does not change the major conclusions about the optimal positions of both +1 and +1/2 defects. Second, in addition to curvature, the thickness of liquid-crystal shells is an important parameter to control the number and orientation of defects <cit.>. It has been experimentally observed that thickness variation can produce a number of novel defect configurations over a spherical liquid-crystal shell <cit.>.It is of great interest to include the effect of thickness in a generalized Frank free energy model to account for these new experimental observations <cit.>.This is beyond the scope of this study.Third, spatial variations in nematic order parameter within defect cores contribute to the condensation free energy of topological defects <cit.>. Notably, nematic textures in defect core regions can exhibit featured patterns and energy profiles, such as highly biaxial nematic order in the cores of +1/2 defects <cit.> and local melting of the nematic ordering <cit.>. A recent study has demonstrated that the condensation energy associated with the defect core plays an important role in the formation of defects triggered by strong enough curvature <cit.>. In our study, we focus on the optimal locations of pre-existent defects. They are determined by the variation of the free energy with the positions of the defects, where the contribution from the defect core structures is canceled without considering the boundary effect of defects. § CONCLUSION In summary, we investigate the curvature-driven stability mechanism of LC defects based on the isotropic nematic disk model where the appearance of defects is not topologically required, and present analytical results on the distinct energy landscape of LC defects created by curvature. We show that with the accumulation of curvature effect both +1 and +1/2 defects can be stabilized within spherical disks. Specifically, the equilibrium position of the +1 defect will move abruptly from the boundary to the center of the spherical disk, exhibiting the first-order phase-transition-like behavior.We also find the alternating repulsive and attractive regimes in the energy curve of a pair of +1/2 defects, which leads to an equilibrium defect pair separation. The sensitive response of defects to curvature and the curvature-driven stability mechanism demonstrated in this work may have implications in the control of LC textures with the dimension of curvature.§ APPENDIX A: INFINITE DEGREE OF DEGENERACY IN THE ONE ELASTIC CONSTANT APPROXIMATION Let us consider a planar nematic disk.𝐧 = (cosθ(x,y),sinθ(x,y)) in Cartesian coordinates. In the one elastic constant approximation, the Frank free energy is F=1/2K_1 ∬_ x^2+y^2 ≤ r_p^2 |▿θ(x,y)|^2 dxdy. It is easily seen that rotating a director by a constant angle does not change the Frank free energy.For a nematic field on a sphere, by insertingn=(cosΨ(θ,φ),sinΨ(θ,φ)) in spherical coordinates into Eq.(<ref>), we obtain the expression for the Frank free energy densityf=K_1/2(1/R^2 sin^2 θ+ |∇Ψ|^2 +2cosθ/R^2 sin^2 θΨ_φ),where ∇Ψ=1/R∂Ψ/∂θe_θ+1/Rsinθ∂Ψ/∂φe_φ, and the notation Ψ_φ is an abbreviation for ∂Ψ / ∂φ. Obviously, the Frank free energy density is invariant under the transformation Ψ→Ψ+c.The conclusion that the nematic texture has infinite degree of degeneracy in the one elastic constant approximation can be generalized to any generally curved surface by writing the Frank free energy under the one constant approximation in the form ofF=1/2∫ dS g^ij(∂_i α - A_i) (∂_j α - A_j),where the integration is over an area element dS on the surface 𝐱(u^1, u^2), α(u^1,u^2) is the angle between 𝐧(u^1,u^2) and any local reference frame, and A_i is the spin connection <cit.>. The free energy is invariant under the rotation α(u^1,u^2)→α(u^1,u^2) +c. § APPENDIX B: CALCULATING CURL 𝐧 ON SPHERICAL GEOMETRYIn a coordinates independent expression, curl 𝐧=(⋆ d n^♭)^♯ <cit.>. The operators ⋆, ♭ and ♯ are to be explained below. ⋆ is an operator called Hodge dual. When applied on an antisymmetric tensor α=1/k !α_i_1,···,i_k e^i_1∧···∧ e^i_k, where e^i_1, ···, e^i_n are dual bases,⋆α = √(|g|)ε_i_1, ···, i_nα_j_1,···,j_kg^i_1j_1··· g^i_kj_k/k!(n-k)!e^i_k+1∧···∧ e^i_n.♭ and ♯ are the musical isomorphisms. X^♭=g_ijX^i dx^j, where X=X^i∂_i. ω^♯=g^ijω_i ∂_j, where ω=ω_i dx^i. Consider a vector field 𝐧 defined on a two-dimensional sphere. 𝐧 = n_1(θ,φ)𝐞_θ + n_2(θ,φ)𝐞_φ, where 𝐞_θ and 𝐞_φ are the unit tangent vectors in spherical coordinates. Applying the above formulas on such a vector field, we have n^♭ =n_1rdθ +n_2rsinθ dφ, dn^♭ =n_1drdθ +n_2sinθ drdφ+r(n_2cosθ +sinθ∂ n_2/∂θ- ∂ n_1/∂φ)dθ dφ,and⋆ dn^♭ =n_1sinθ dφ -n_2dθ+ 1/rsinθ(n_2cosθ +sinθ∂ n_2/∂θ-∂ n_1/∂φ)dr.We finally obtain Eq.(<ref>). It is of interest to note that the curl of a director field 𝐧 on a generally curved surface is curl 𝐧 = -τ_n𝐧 -c_n𝐭 + κ_nν, where {𝐧, 𝐭, ν} constitute the Darboux basis <cit.>. τ_n and c_n are the components of the extrinsic curvature tensor 𝐋. 𝐋_nn=c_n, and 𝐋_nt=𝐋_tn=-τ_n. In general, the extrinsic curvature influences the Frank free energy of nematics on a curved surface. It is only on a flat or spherical surfaceτ_n=0 and c_n is a constant. So the extrinsic curvature effect only contributes a constant term in the Frank free energy <cit.>. § APPENDIX C: EULER-LAGRANGE EQUATIONS IN CARTESIAN AND SPHERICAL COORDINATES In this appendix, we present the Euler-Lagrange equations in Cartesian and spherical coordinates derived from Eq.(<ref>), and show that both radial and azimuthal configurations are solutions to the Euler-Lagrange equations. We also show that the anisotropic elastic constants will not change the main result of curvature-driven alternating repulsive and attractive interactions between the two +1/2 defects due to the fact that the elastic modulus K_1 plays no role in the energy expression.In two-dimensional Cartesian coordinates, 𝐧=[n_1(x,y),n_2(x,y)]. The components of the director field in equilibrium nematic textures satisfy the following Euler-Lagrange equations:K_1 (∂^2 n_1/∂ x^2 + ∂^2 n_2/∂ x ∂ y) - K_3(∂^2 n_2/∂ x ∂ y - ∂^2 n_1/∂ y^2 )=-λ n_1,andK_1 (∂^2 n_2/∂ y^2 + ∂^2 n_1/∂ x ∂ y) + K_3(∂^2 n_2/∂ x^2 - ∂^2 n_1/∂ x ∂ y )=-λ n_2.It is found that both radial (n_1,n_2)= (x/√(x^2+y^2),y/√(x^2+y^2))and azimuthal (n_1,n_2) = (-y/√(x^2+y^2),x/√(x^2+y^2)) configurations satisfy the above Euler-Lagrange equations with λ =K_1/(x^2+y^2) and λ =K_3/(x^2+y^2), respectively. The spiral configuration 𝐧=c_1(x/√(x^2+y^2),y/√(x^2+y^2))+c_2(-y/√(x^2+y^2),x/√(x^2+y^2)) (c_1^2+c_2^2=1 and neither c_1 nor c_2 is 0) is the solution to the Euler-Lagrange equations only in the one elastic constant approximation.In spherical coordinates,𝐧=n_1𝐞_θ+n_2𝐞_φ. In equilibrium nematic textures, n_1 and n_2 satisfy the following Euler-Lagrange equations: K_1/R^2(-n_1/sin^2 θ+∂ n_1/∂θcosθ/sinθ+∂^2 n_1/∂θ^2-cosθ/sin^2 θ∂ n_2/∂φ +1/sinθ∂^2 n_2/∂θ∂φ)+K_3/r^2 sin^2 θ(∂^2 n_1/∂φ^2-∂ n_2/∂φcosθ -sinθ∂^2 n_2/∂θ∂φ) = -λ n_1, K_3/R^2(- n_2/sin^2 θ + cosθ/sin^2 θ∂ n_1/∂φ - 1/sinθ∂ ^2 n_1/∂θ∂φ + ∂^2 n_2/∂θ^2 + cosθ/sinθ∂ n_2/∂θ)+K_1/R^2 sinθ (cosθ/sinθ∂ n_1/∂φ+∂^2 n_1/∂θ∂φ +1/sinθ∂^2 n_2/∂φ^2) = -λ n_2.We remark that the equilibrium equations in spherical coordinates are invariant under uniform local rotation of the director field. Similarly, one can show that both radial and azimuthal configurations are solutions to the Euler-Lagrange equations with λ =K_1/(R^2 sin^2 θ) and λ =K_3/(R^2 sin^2 θ), respectively.The spiral configuration of n_1=c_1, n_2=c_2 (c_1 and c_2 are non-zero constants satisfying c_1^2+c_2^2=1) satisfies the equilibrium equation only when K_1 = K_3. For the two +1/2 defects configurations discussed in the main text, we show that introducing elasticity anisotropy does not change the curvature-driven alternating repulsive and attractive interactions between thedefects. For the two +1/2 defects configuration on a spherical disk where an azimuthal configuration is separated by a uniform configuration, the associated Frank free energy isF_+1/2,s/2K_3=∬ _D_11/x^2+y^2dA+∬ _D_21/R^2-y^2dA,where D_1 and D_2 are given below Eq.(<ref>). We see that since the entire defect configuration is divergence free, the parameter K_3 does not appear in the expression for the Frank free energy. 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"authors": [
"Xiuqing Duan",
"Zhenwei Yao"
],
"categories": [
"cond-mat.soft",
"cond-mat.mtrl-sci",
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.soft",
"published": "20170627185145",
"title": "Curvature-driven stability of defects in nematic textures over spherical disks"
} |
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